assessingturbulencemodelsforlarge-eddysimulation using...

Assessing turbulence models for large-eddy simulationusing exact solutions to the Navier-Stokes equations

Bachelor Project Mathematics

July 5 2016

Student Daniel Ward

Daily Supervisor Maurits H Silvis MSc

First Supervisor Prof dr ir RWCP Verstappen

Second Supervisor Dr AE Sterk

Abstract

We study the behaviour of turbulent fluid flows behaviour that is governed by the Navier-Stokes equations Due to the extensive detail entailed in a turbulent flow it is difficult tosolve the Navier-Stokes equations numerically A large-eddy simulation (LES) using a filteringoperation solves for the large-scale motions in a flow and smooths over small-scale motionhence requiring a turbulence model for these small-scale motions Using conditions derivedfrom and exact solutions to the Navier-Stokes equations we investigate a number of turbulencemodels for LES both with and without explicit filtering We found that the Vortex-Stretching-based eddy-viscosity model was most endorsed in the case without explicit filtering and thatthe Vreman QR Gradient and Vortex-Stretching-based models were equally endorsed by thecase with explicit filtering with the Smagorinsky model being the least endorsed To reducethe bias that may arise when exact solutions are similar we then considered classes of flowsto further determine which models are the most endorsed We considered classes based onVremanrsquos flow classes and on the principal and combined invariants of flows Once again theVortex-Stretching-based eddy-viscosity model was usually the most endorsed As more exactsolutions to the Navier-Stokes equations are found they can be easily added to the report tofurther endorse or oppose the models We also considered Vremanrsquos paper [16] and did notfind anything to contradict his results We did extend his results slightly by finding a solutionoutside of his zero-subfilter-dissipation classes with zero subfilter dissipation

2

Contents

1 Introduction 4

2 The Navier-Stokes Equations 521 Introduction 522 Exact Solutions of the Navier-Stokes Equations 623 Characterisation of Flows 20

231 Characterising Solutions using Invariants 20232 Vremanrsquos Characterisation of Flows 21

3 Conditions for Large-Eddy Simulation Models 2231 Large-Eddy Simulation 2232 Subfilter-Scale Models 2233 Conditions for Large-Eddy Simulation without Explicit Filtering 2334 Conditions for Large-Eddy Simulation with Explicit Filtering 24

4 Analysis of Large-Eddy Simulation Models 2541 Large-Eddy Simulation Without Explicit Filtering 2542 Large-Eddy Simulation With Explicit Filtering 26

421 Example of a Test for a Model 27422 Testing the Models 28

43 Vremanrsquos True Subfilter Dissipation Claim 2944 Classes of Flows 29

441 Vremanrsquos Velocity Gradient Flow Classes 30442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 31443 Classes using the Third Principal Invariant 33444 Two-component Flow Classes 35

5 Conclusions 37

Appendices 39

A Invariants of Flows 39

B Vremanrsquos Flow Classes 46

C Extended Tables for Flow Classes 47C1 Vremanrsquos Velocity Gradient Flow Classes 47C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 48C3 Classes using the Third Principal Invariant 49C4 Two-Component Flow Classes 50

3

1 Introduction

In this project we will study and test models for turbulent fluid flows There is no all-encompassingdefinition of a turbulent flow there are however a number of characterising properties A turbulentflow is said to be irregular in that it appears to be random Turbulent flows are chaotic and henceusually described statistically due to the difficulty in a deterministic approach Turbulent flowsare diffusive in that there is a high rate of momentum heat and mass transfer They require apersistent source of input energy in order to sustain the turbulent nature of the flow as kineticenergy is quickly converted into internal energy A large Reynolds number is characteristic of aturbulent flow [5]

The Navier-Stokes equations are a set of fundamental governing equations for any Newtonian fluidAn exact solution to the Navier-Stokes equations is in the form of a flow velocity and a pressureterm and hence a fluid flow can be completely described by an exact solution to the Navier-Stokesequations It is still a non-trivial task to find such solutions although there are a number in literature[17]

The Navier-Stokes equations have no closed form solution resulting in difficulty when analysingcomplicated flows in complex geometries This results in difficulty when solving the Navier-Stokesequations explicitly So a mathematical model known as Large Eddy Simulation (LES) can insteadbe used to model turbulent flows LES uses what is known as a filtering operation to reduce thecomputational cost of a simulation by ignoring small-scale motion What this means however isthat information about these small-scale motions is lost This information is often non-trivial sosubfilter-scale models (or turbulence models) are used in order to approximate the effects of small-scale motions on the large-scale fluid flow

In this project we will use the behaviour of exact solutions to the Navier-Stokes equations to testsubfilter-scale models Applying a subfilter-scale model to a flow has two effects One is a bodyforce on the velocity field called a subfilter force and the other is a transfer of energy from largeto small scales of motion called a subfilter dissipation This results in respectively an extra termbeing added to the Navier-Stokes equations and to the equation of the kinetic energy of a flow

We will then consider conditions for subfilter-scale models in two different cases In LES withoutexplicit filtering we will consider how a subfilter-scale model behaves for an exact solution to theNavier-Stokes equations by checking if the subfilter force and subfilter dissipation are zero for anexact solution If so this can be interpreted as the model being inactive and hence this exactsolution can be said to lsquoendorsersquo the model In LES with explicit filtering we compare a model termwith a term in the filtered Navier-Stokes equations (called the true turbulent stress of a flow) forexact solutions If the two terms are both zero we will conclude that the exact solution lsquostronglyrsquoendorses the model and if the two terms are both non-zero we will conclude that the exact solutionlsquoweaklyrsquo endorses the model

We will also consider the classification of fluid flows in order to further inspect the models andreduce the bias that may occur when solutions have similar properties We will use properties of theinvariants of a flow to characterise flows in a coordinate-independent way Using Vremanrsquos paperldquoAn eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory and applicationsrdquo[16] we will place the fluid flows into Vremanrsquos flow classes We will also determine whether ourresults say anything beyond Vremanrsquos work

4

2 The Navier-Stokes Equations

21 Introduction

The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by

partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3 (1)

These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid

The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term

ujpartuipartxjequiv

3sumj=1

ujpartuipartxj

= u1partuipartx1

+ u2partuipartx2

+ u3partuipartx3

=

(u1

part

partx1+ u2

part

partx2+ u3

part

partx3

)ui

= (u middot nabla)ui

(2)

Similarly for the diffusion term

νpart2uipartxjpartxj

equiv ν3sumj=1

part2uipartxjpartxj

= νpart2uipartx1partx1

+part2uipartx2partx2


= ν

(part2

partx1partx1+

part2

partx2partx2+

part2

partx3partx3

)ui

= νnabla2ui

(3)

It therefore holds that in vector notation the Navier-Stokes equations are given by

partu

partt+ (u middot nabla)u = minus1

ρnablap+ νnabla2u (4)

It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by

partuipartxi

= 0 (5)

in index notation and

nabla middot u = 0 (6)

5

in vector notation

There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables

Let

ulowasti =uiv xlowasti =

xi` tlowast =

t

`v plowast =

p

ρv2 (7)

where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are

partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)

where Re is the Reynolds number and is defined by the dimensionless Re =v`

ν Similarly in vector

notation the scaled incompressible Navier-Stokes equations are

partu

partt+ (u middot nabla)u = minusnablap+

1

Renabla2u

nabla middot u = 0(9)

By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found

part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)

The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models

22 Exact Solutions of the Navier-Stokes Equations

By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation

Solution 1 Linear Velocity

If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie

part2uipartxjpartxj

= 0 (11)

6

then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation

ui = aijxj + βi (12)

If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution

u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)

where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)

Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1

Solution 2 Planar Flow

Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found

u1(x y z t) = Az2 +Bz + C

u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7

where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form

ui(xi xj xk t) = Ax2k +Bxk + C

uj(xi xj xk t) = Dx2k + Exk + F

uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i

Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1

Solution 3 Velocity as a function of time t

If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows

u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)

p(x y z t) = minusxpartf(t)

parttminus y partg(t)

parttminus z parth(t)

partt

(16)

where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified

8

Solution 4 Axisymmetric Flow

Using the text by Irmay [7] a solution is defined as follows

u1(x y z t) = kxz

u2(x y z t) = kyz

u3(x y z t) = minuskz2

p(x y z t) = minus2kz

Reminus k2z4

2

(17)

where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3

Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1

Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1

Solution 5 2D Taylor Solutions

Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing

u1 = minuspartψparty

u2 =partψ

partx u3 = 0 (18)

In the text the stream function is not given however it was worked out that

ψ = minuseminusπ2t

Re (cos(πx) + cos(πy))

π (19)

The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below

9

was instead found)

u1(x y z t) = minus sin(πy) eminus(π2t)Re

u2(x y z t) = sin(πx) eminus(π2t)Re

u3(x y z t) = 0

p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re

(20)

Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0

Solution 6 Generalized Beltrami Flow

Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely

u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re

u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re

u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re

p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)

+AC cos(πy) sin(πz)] eminus(2π2t)Re

(21)

where ABC isin R

10

Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0

Solution 7 Stagnation Flow

In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution

u1(x y z t) =a

kx

u2(x y z t) = minusaky +

k

Re

partψ(y z)

partz

u3(x y z t) = minus k

Re

partψ(y z)

party

(22)

Here we choose the stream function to be ψ(y z) = yz Then a solution is given by

u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y

u3(x y z t) = minus kzRe

p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)

where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however

11

Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2

Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2

Solution 8 Vortex in Stagnation Flow

As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes

u1(x y z t) =1

Rex

u2(x y z t) = minus 1

Rey minus k sin z

u3(x y z t) = 0

p(x y z t) = minus 1

2Re2(x2 + y2)

(24)

where a and k are as in Sol 7 (Stagnation Flow)

12

Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10

Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10

Solution 9 Burgers Vortex

The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows

ur(r ϕ z t) = minus 2r

c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re

p(r ϕ z t) = minusc2Ω2F

(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)

This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates

u1(x y z t) = minus 2x

c2Reminus Ω

h(x y)(1minus exp (minush(x y))) y

u2(x y z t) = minus 2y

c2Re+

Ω

h(x y)(1minus exp (minush(x y)))x

u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)

and Ω and c are constants

Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown

Solution 10 Another Vortex Solution

There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates


c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re

p(r ϕ z t) = minus a2

2r2minus 2r2 + 8z2

c2Re2

(29)

which can also be converted into Cartesian coordinates

14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)

where a and c are constants

Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown

Solution 11 Pseudo-Motion Shots of the Second Kind

In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by

ur(r ϕ z t) =2

Re

1

r

uϕ(r ϕ z t) = A1r3 +B1r +

C1

r

uz(r ϕ z t) = A2r2 log(r) +B2r

2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)

where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is

15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)

Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10

Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10

Solution 12 Flows With Constant Whirl

In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by

ur(r ϕ z t) = minus 1

Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)

where we define g(t) =r2Re

2t When converted into Cartesian coordinates we find

16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)

Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z

17

Solution 13 Rectilinear and Uniform Translation of a Sphere

In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by

u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)

u3(x y z t) = 1minus 3

4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)

p(x y z t) =3a(aminusR)(a+R)

32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2

) (a3 + 3aR2 minus 4R3

)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)

where a and R are constants

Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3

Solution 14 Radially Dependent Flow

A solution with velocity in the z-direction dependent only on the distance from the z-axis is as

18

follows in cylindrical coordinates

ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0

uz(r ϕ z t) = A(r2 minus a2

)

p(r ϕ z t) =4Az

Re

(36)

where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)

Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10


Solution 15 Rankine Vortex

A solution discovered by Rankine [1] is a vortex flow of the following form

ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19

where a and R are constants In Cartesian coordinates the flow has the form

u1(x y z t) = minus ay

2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)

This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis

Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1

radic2π

23 Characterisation of Flows

It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised

231 Characterising Solutions using Invariants

A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by

Gij =

[partuipartxj

] (39)

20

for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby

PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)

Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by

Sij =1

2

(partuipartxj

+partujpartxi

) (41)

and the rate-of-rotation tensor Ω by

Ωij =1

2

(partuipartxjminus partujpartxi

) (42)

Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω

Rather than considering the principal invariants we can also consider the combined invariants

I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)

We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants

PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)

In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A

Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants

232 Vremanrsquos Characterisation of Flows

Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example

[partuipartxj

]=

0 0 0lowast 0 0lowast 0 0

(45)

where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity

21

gradient Flows of the form in Eq (45) have a velocity field of the form

u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)

where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441

3 Conditions for Large-Eddy Simulation Models

31 Large-Eddy Simulation

Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation

partuipartxj

=partuipartxj

(47)

The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by

partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where

τij = uiuj minus uiuj (49)

is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod

ij in practical applications

32 Subfilter-Scale Models

We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo

true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34

All the models barring one are of the same form That is each of the eddy-viscosity models arewritten

τmodij minus 1

3τmodkk δij = minus2νeSij (50)

with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each

22

of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows

Smagorinsky [12] νSe = (CSδ)2radic

2I1 (51)

Vreman [16] νVe = (CV δ)2

radicQGGT

PGGT (52)

QR [13 14 15] νQRe = (CQRδ)2 max0minusI3

I1 (53)

Vortex-Stretching [11] νV Se = (CV Sδ)2radic

2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities

PGGT = I1 minus I2 QGGT =1

4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)

There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by

Gradient [4 9] τmodG = CGδ

(S2 minus Ω2 minus (SΩminus ΩS)

) (56)

The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined

33 Conditions for Large-Eddy Simulation without Explicit Filtering

The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod

ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by

partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part

partxjτmodij (v) (57)

It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition

part

partxjτmodij (v) = 0 (58)

is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find

part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23

Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether

vipart


We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy

d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV

[minus2νSijSij minus

(minusSijτmod

ij (v))]dV

(61)

where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is

Sijτmodij (v) = 0 (62)

We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the

subgrid dissipation

34 Conditions for Large-Eddy Simulation with Explicit Filtering

As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that

partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)

In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod

ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod

ij (v) respectively LES with explicit filtering is thengiven by

partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part


Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms

part

partxjτij (65)

and

part


24

are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation

Sijτij (67)

(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation

Sij(v)τmodij (v) (68)

are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)

4 Analysis of Large-Eddy Simulation Models

In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering

41 Large-Eddy Simulation Without Explicit Filtering

We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table

Velocity Field amp Kinetic Energy Conditions [Dopart

partxjτmodij (v) and vi

part

partxjτmodij (v) equal zero]

Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 ndash ndash 0 0

Vortex-Stretching 0 0 0 0 0 0 0 0 0

Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj

τmodij (v) = 0 for respective

models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj

τmodij (v) 6= 0 For the Velocity

Field condition a lsquo0rsquo result means that partpartxj

τmodij (v) = 0 for respective models and exact solutions

whereas a lsquolowastrsquo result means that partpartxj

τmodij (v) 6= 0 The Gradient model can take both positive and

negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical

We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22

25

Dissipation Condition [Does Sijτmodij (v) = 0]

Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0


Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas

a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative

values so when the sign of the model is known it is included in the table instead of a lsquorsquo

We then sum all the results to find the total number of endorsements for each model described inSection 32

Smagorinsky Vreman QR Gradient Vortex-Stretching

lsquoVelocityKineticEnergyrsquo Endorsements

4 6 8 9 9

lsquoDissipationrsquoEndorsements

2 5 7 7 9

Total Endorsements 6 11 15 16 18

Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least

Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]

42 Large-Eddy Simulation With Explicit Filtering

In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod

ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each

exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)

26

421 Example of a Test for a Model

The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)

The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as

u(x y z t) =

0ax+ bcx+ d

(69)

Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence

[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)

Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that

[τij(u)] =[uiuj minus uiuj

]=

lowast 0 middot ax+ b 0 middot cx+ d

ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast

=

lowast 0 00 lowast lowast0 lowast lowast

(71)

where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation

Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0

0 lowast lowast0 lowast lowast

= 0 (72)

Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =

Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that

τmodij = minus2(CQRδ)

2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)

it holds that τmodij (u) = 0 so Sij(u)τmod

ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model

27

422 Testing the Models

Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4

Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]

Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0


Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod

ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate

Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod

ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod

ij (u) 6= 0 For Sol 4 to

13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse

the Smagorinsky model


lsquoStrongrsquo Endorsements 2 5 5 5 5

lsquoWeakrsquo Endorsements 10 10 9 6

Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero

A lsquoweakrsquo endorsement means that both were found to be non-zero

Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement

The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this

28

completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated

43 Vremanrsquos True Subfilter Dissipation Claim

Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows

The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows

GLV =

0 0 0a 0 0c 0 0

(75)

and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results

We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij

We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]

44 Classes of Flows

It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A

29

441 Vremanrsquos Velocity Gradient Flow Classes

We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows

Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)

Q7minus9 = Q7 cupQ8 cupQ9 (77)

where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1

Endorsements for models using solutions in Q7minus9 for LES without explicit filtering



3 5 6 6 6


2 5 6 6 6

Percentage Endorsed 417 833 100 100 100

Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary




1 1 2 3 3


0 0 1 0 3



All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very

30

low

Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1

Endorsements for models using solutions in Q7minus9 for LES with explicit filtering



lsquoWeakrsquo Endorsements 1 1 0 0 0









We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))

442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes

We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining

31

classes Then

V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)

Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)

Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)

Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)

Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)

Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)

Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)

(79)

We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2

Endorsements for models using solutions in V0 for LES without explicit filtering



2 4 4 4 4


1 4 4 4 4


Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary

Endorsements for models using solutions in Vlowast for LES without explicit filtering



2 2 4 5 5


1 1 3 2 5


Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary

All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the

32

Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow

Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2

Endorsements for models using solutions in V0 for LES with explicit filtering






Endorsements for models using solutions in Vlowast for LES with explicit filtering






We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements

443 Classes using the Third Principal Invariant

Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)

We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That

33

is

R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)

Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)


Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)

(80)

Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)

Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)

Sol 13 (Sphere Translation)(81)

We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3

Endorsements for models using solutions in R0 for LES without explicit filtering



3 5 7 7 7


2 5 7 6 7


Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary

Endorsements for models using solutions in Rlowast for LES without explicit filtering



1 1 1 2 2


0 0 0 0 2


Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary

The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small

34

Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3

Endorsements for models using solutions in R0 for LES with explicit filtering






Endorsements for models using solutions in Rlowast for LES with explicit filtering






As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes

444 Two-component Flow Classes

Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy

I3 = I4 = I5 minus1

2I1I2 = 0 (82)

We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that

I3 minus I4 = 0 (83)

What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are

C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)

Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)

Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)

35

and we call the remaining flows three-component flows They are

C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)


Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)

Sol 13 (Sphere Translation)

(85)

Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4

Endorsements for models using solutions in C2 for LES without explicit filtering



3 5 7 6 7


2 5 7 6 7


Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary




1 1 1 3 2


0 0 0 0 2



The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model

36


Endorsements for models using solutions in C2 for LES with explicit filtering












As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2

5 Conclusions

In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust

37

We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses

In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes

This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold

38

Appendices

A Invariants of Flows

Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows

PG = 0 QG = 0 RG = 0

and the combined invariants are

I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2

Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1

PG = 0 QG = 0 RG = 0

and the combined invariants are similar in that

I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2

Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows

PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0

Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows

PG = 0 QG = minus3a2z2 RG = minus2a2z2

39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)

Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows

PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0


I1 =π2

2eminus

2π2tR (cos(πx)minus cos(πy))2

I2 = minusπ2

2eminus

2π2tR (cos(πx) + cos(πy))2

I3 = 0

I4 = 0

I5 = minusπ4

32eminus

4π2tR (cos(2πx)minus cos(2πy))2

Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows

PG = 0

QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))

RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))

40

and the terms for the combined invariants are

I1 =π2

2eminus

2π2tR

(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)

)

I2 = minusπ2

2eminus

2π2tR

(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)

)

I3 =3π3

4eminus

3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))

I4 =π3

4eminus

3π2tR

(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+

C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))

I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2

)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)

minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+

A(B minus C)(B + C) cos(2πz)) +B4 + C4)

Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows

PG = 0 QG = minusa2

k2+ anminus k2n2 RG =

an(aminus k2n

)k


I1 =2a2

k2minus 2an+ 2k2n2 I2 = 0 I3 =

3an(aminus k2n

)k

I4 = 0 I5 = 0

Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows

PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)

I3 = minus3k2 cos2(z)

4Re

I4 =k2 cos2(z)

4Re

I5 = minusk2 cos2(z)

(k2Re2 cos2(z) + 2

)8Re2

41

Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows

PG = 0

QG =1

2

(minus 24

c4Re2minus

8eminus2h(xy)(eh(xy) minus 1

)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2

h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2

(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))

where h(x y) is as defined in Eq (28) The combined invariants are

I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2

)2)minus 2c6Re2Ω2eh(xy)

(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)

I2 = minus2Ω2eminus2h(xy)

I3 =1

c6Re3 (x2 + y2)2

I4 =4Ω2eminus2h(xy)

c2Re

I5 = minus 2Ω2eminus4h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)

Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows

PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42

Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows

PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64

)minus C1Re2(A2 + 2B2)2

(x2 + y2

) ))

43

Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows

PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2

I2 = minusRe2eminusRe(x2+y2)

2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2

)2)minus 8Re3t(x2 + y2

)minus 16Re2t2 minus 16t3e

Re(x2+y2)2t

)128t6 (x2 + y2)

2

Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows

PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10

I2 = minus9(a3 minus aR2

)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20

Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows

PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2

Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows

PG = 0 QG = 0 RG = 0


I1 = 0 I2 = minus a2

2π2R4 I3 = 0 I4 = 0 I5 = 0

45

B Vremanrsquos Flow Classes

In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0

0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0

0 0 0lowast 0 lowast0 0 0

0 0 0

0 0 0lowast lowast 0

sube Q7

46

C Extended Tables for Flow Classes

C1 Vremanrsquos Velocity Gradient Flow Classes



part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0


Table 22 Table 1 rearranged into the flow classes described in Section 441


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47

C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes



part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48

C3 Classes using the Third Principal Invariant



part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49

C4 Two-Component Flow Classes



part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References

[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0

[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007

[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963

[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017

S002211207900001X

[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd

chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016

[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006

[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L

[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005

[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973

[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]

[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]

[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2

[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4

[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010

[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014

[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131

[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111

51

Introduction

The Navier-Stokes Equations

Introduction

Exact Solutions of the Navier-Stokes Equations

Characterisation of Flows

Characterising Solutions using Invariants

Vremans Characterisation of Flows

Conditions for Large-Eddy Simulation Models

Large-Eddy Simulation

Subfilter-Scale Models

Conditions for Large-Eddy Simulation without Explicit Filtering

Conditions for Large-Eddy Simulation with Explicit Filtering

Analysis of Large-Eddy Simulation Models

Large-Eddy Simulation Without Explicit Filtering

Large-Eddy Simulation With Explicit Filtering

Example of a Test for a Model

Testing the Models

Vremans True Subfilter Dissipation Claim

Classes of Flows

Vremans Velocity Gradient Flow Classes

Vremans Zero-True-Subfilter-Dissipation Flow Classes

Classes using the Third Principal Invariant

Two-component Flow Classes

Conclusions

Appendices

Invariants of Flows

Vremans Flow Classes

Extended Tables for Flow Classes




Two-Component Flow Classes

Abstract

We study the behaviour of turbulent fluid flows behaviour that is governed by the Navier-Stokes equations Due to the extensive detail entailed in a turbulent flow it is difficult tosolve the Navier-Stokes equations numerically A large-eddy simulation (LES) using a filteringoperation solves for the large-scale motions in a flow and smooths over small-scale motionhence requiring a turbulence model for these small-scale motions Using conditions derivedfrom and exact solutions to the Navier-Stokes equations we investigate a number of turbulencemodels for LES both with and without explicit filtering We found that the Vortex-Stretching-based eddy-viscosity model was most endorsed in the case without explicit filtering and thatthe Vreman QR Gradient and Vortex-Stretching-based models were equally endorsed by thecase with explicit filtering with the Smagorinsky model being the least endorsed To reducethe bias that may arise when exact solutions are similar we then considered classes of flowsto further determine which models are the most endorsed We considered classes based onVremanrsquos flow classes and on the principal and combined invariants of flows Once again theVortex-Stretching-based eddy-viscosity model was usually the most endorsed As more exactsolutions to the Navier-Stokes equations are found they can be easily added to the report tofurther endorse or oppose the models We also considered Vremanrsquos paper [16] and did notfind anything to contradict his results We did extend his results slightly by finding a solutionoutside of his zero-subfilter-dissipation classes with zero subfilter dissipation

2

Contents

1 Introduction 4








5 Conclusions 37

Appendices 39




3

1 Introduction







4


21 Introduction


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3 (1)



ujpartuipartxjequiv

3sumj=1

ujpartuipartxj

= u1partuipartx1

+ u2partuipartx2

+ u3partuipartx3

=

(u1

part

partx1+ u2

part

partx2+ u3

part

partx3

)ui


(2)



equiv ν3sumj=1

part2uipartxjpartxj




= ν

(part2

partx1partx1+

part2

partx2partx2+

part2

partx3partx3

)ui

= νnabla2ui

(3)


partu




partuipartxi

= 0 (5)



5

in vector notation


Let


xi` tlowast =

t

`v plowast =

p

ρv2 (7)


partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)




partu


1

Renabla2u



part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)






part2uipartxjpartxj

= 0 (11)

6




u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







Contents

1 Introduction 4








5 Conclusions 37

Appendices 39




3

1 Introduction







4


21 Introduction


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3 (1)



ujpartuipartxjequiv

3sumj=1

ujpartuipartxj

= u1partuipartx1

+ u2partuipartx2

+ u3partuipartx3

=

(u1

part

partx1+ u2

part

partx2+ u3

part

partx3

)ui


(2)



equiv ν3sumj=1

part2uipartxjpartxj




= ν

(part2

partx1partx1+

part2

partx2partx2+

part2

partx3partx3

)ui

= νnabla2ui

(3)


partu




partuipartxi

= 0 (5)



5

in vector notation


Let


xi` tlowast =

t

`v plowast =

p

ρv2 (7)


partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)




partu


1

Renabla2u



part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)






part2uipartxjpartxj

= 0 (11)

6




u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







1 Introduction







4


21 Introduction


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3 (1)



ujpartuipartxjequiv

3sumj=1

ujpartuipartxj

= u1partuipartx1

+ u2partuipartx2

+ u3partuipartx3

=

(u1

part

partx1+ u2

part

partx2+ u3

part

partx3

)ui


(2)



equiv ν3sumj=1

part2uipartxjpartxj




= ν

(part2

partx1partx1+

part2

partx2partx2+

part2

partx3partx3

)ui

= νnabla2ui

(3)


partu




partuipartxi

= 0 (5)



5

in vector notation


Let


xi` tlowast =

t

`v plowast =

p

ρv2 (7)


partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)




partu


1

Renabla2u



part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)






part2uipartxjpartxj

= 0 (11)

6




u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








21 Introduction


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3 (1)



ujpartuipartxjequiv

3sumj=1

ujpartuipartxj

= u1partuipartx1

+ u2partuipartx2

+ u3partuipartx3

=

(u1

part

partx1+ u2

part

partx2+ u3

part

partx3

)ui


(2)



equiv ν3sumj=1

part2uipartxjpartxj




= ν

(part2

partx1partx1+

part2

partx2partx2+

part2

partx3partx3

)ui

= νnabla2ui

(3)


partu




partuipartxi

= 0 (5)



5

in vector notation


Let


xi` tlowast =

t

`v plowast =

p

ρv2 (7)


partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)




partu


1

Renabla2u



part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)






part2uipartxjpartxj

= 0 (11)

6




u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







in vector notation


Let


xi` tlowast =

t

`v plowast =

p

ρv2 (7)


partuipartt

+ ujpartuipartxj

= minus partp

partxi+

1

Re

part2uipartxjpartxj

i = 1 2 3 j = 1 2 3

partuipartxi

= 0

(8)




partu


1

Renabla2u



part

partt

(1

2uiui

)+ uj

part

partxj

(1

2uiui

)= minusui

partp

partxi+

2

Reui

part2uipartxjpartxj

(10)






part2uipartxjpartxj

= 0 (11)

6




u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










u1(x y z t) = 0

u2(x y z t) = ax+ b

u3(x y z t) = cx+ d

p(x y z t) = f(t)

(13)






u2(x y z t) = Dz2 + Ez + F

u3(x y z t) = 0

p(x y z t) =2A

Rex+

2D

Rey + f(t)

(14)

7




uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










uk(xi xj xk t) = 0

(15)

for i 6= j 6= k 6= i




u1(x y z t) = f(t)

u2(x y z t) = g(t)

u3(x y z t) = h(t)




partt

(16)


8



u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows









u1(x y z t) = kxz

u2(x y z t) = kyz



Reminus k2z4

2

(17)







u2 =partψ

partx u3 = 0 (18)




π (19)


9

was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







was instead found)



u3(x y z t) = 0


(20)









(21)

where ABC isin R

10




u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










u1(x y z t) =a

kx


k

Re

partψ(y z)

partz


Re

partψ(y z)

party

(22)


u1(x y z t) =a

kx

u2(x y z t) =

(minusak

+k

Re

)y


p(x y z t) = minus1

2

((axk

)2minus((

k

Reminus a

k

)y

)2

minus(kz

Re

)2)

(23)


11





u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows











u1(x y z t) =1

Rex


Rey minus k sin z

u3(x y z t) = 0


2Re2(x2 + y2)

(24)


12






c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows












c2Re

uϕ(r ϕ z t) =Ωc2

r

(1minus eminusr

2c2)

uz(r ϕ z t) =4z

c2Re


(r2c2

)2

minus 2

Re2r2 + 4z2

c4

(25)

where

F (ξ) =

infinintξ

(1minus eminusx)2

x2dx (26)



c2Reminus Ω



c2Re+

Ω


u3(x y z t) =4z

c2Re

(27)

13

where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







where

h(x y) =x2 + y2

c2 (28)






c2Re

uϕ(r ϕ z t) =a

r

uz(r ϕ z t) =4z

c2Re


2r2minus 2r2 + 8z2

c2Re2

(29)


14


c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








c2Reminus ay

x2 + y2


c2Re+

ax

x2 + y2

u3(x y z t) =4z

c2Re

(30)





ur(r ϕ z t) =2

Re

1

r


C1

r


2 + C2

p(r ϕ z t) =A2

1r6

6+A1B1r

4

2+A1C1r

2 +B2

1r2

2minus C2

1

2r2minus 2

r2Re2+ 2B1C1 log(r)

+2A2z

Reminus 4B1ϕ

Re

(31)


15

u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







u1(x y z t) =1

Re

2xminus C1Rey

x2 + y2minusA1y

(x2 + y2

)minusB1y

u2(x y z t) =1

Re

2y + C1Rex

x2 + y2+A1x

(x2 + y2

)+B1x

u3(x y z t) =1

2A2

(x2 + y2

)log(x2 + y2

)+B2

(x2 + y2

)+ C2

(32)






Re

1

r

uϕ(r ϕ z t) =1

rradicte(minus

12 g(t))

uz(r ϕ z t) = 0

p(r ϕ z t) =1

4

2(minus e

minusg(t)

t minus 1Re2

)r2

minus 1

t2Re

infinintminusg(t)

eminust

tdt

(33)



16


x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








x2 + y2

(yradict

exp

(minus

Re(x2 + y2

)4t

)+

x

Re

)

u2(x y z t) =1

x2 + y2

(xradict

exp

(minus

Re(x2 + y2

)4t

)minus y

Re

)

u3(x y z t) = 0

(34)


17



u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows









u1(x y z t) =3

4

axz

R3

(a2

R2minus 1

)

u2(x y z t) =3

4

ayz

R3

(a2

R2minus 1

)


4

a

Rminus 1

4

a3

R3minus 3

4

az2

R3

(a2

R2minus 1

)


32R10Re

((R2Re

(x2 + y2

)minus 2R2Rez2


)+ 3aRez4

(R2 minus a2

)minus 16R5z

)

(35)





18


ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








ur(r ϕ z t) = 0

uϕ(r ϕ z t) = 0


)

p(r ϕ z t) =4Az

Re

(36)






ur(r ϕ z t) = 0

uϕ(r ϕ z t) =ar

2πR2

uz(r ϕ z t) = 0

p(r ϕ z t) =a2r2

8π2R4

(37)

19



2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows









2πR2

u2(x y z t) =ax

2πR2

u3(x y z t) = 0

p(x y z t) =a2(x2 + y2

)8π2R4

(38)



radic2π





Gij =

[partuipartxj

] (39)

20


PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








PG = tr(G)

QG =1

2

((tr(G))2 minus tr

(G2))

RG = det(G)

(40)


Sij =1

2

(partuipartxj

+partujpartxi

) (41)


Ωij =1

2


) (42)





PG = 0

QG = minus1

2(I1 + I2)

RG =1

3(I3 + 3I4)

(44)





[partuipartxj

]=


(45)


21


u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








u1 = c1 (constant)

u2 = f2(x1)

u3 = f3(x1)

(46)





partuipartxj

=partuipartxj

(47)


partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (48)

where








τmodij minus 1



22



2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows









2I1 (51)


radicQGGT

PGGT (52)


I1 (53)


2I1

(I5 minus 1

2I1I2

minusI1I2

)32

(54)

with the quantities


4(I1 + I2)2 + 4

(I5 minus

1

2I1I2

) (55)




) (56)





partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part



part

partt

(1

2vivi

)+ vj

part

partxj

(1

2vivi

)= minusvi

1

ρ

partq

partxi+ νvi

part2vipartxjpartxj

minus vipart


23


vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








vipart



d

dt

intV

1

2vividV =

intS

[minusvj

(1

2vivi

)minus vj

q

ρ+ 2νviSij + viτ

modij (v)

]njdS

+

intV


(minusSijτmod

ij (v))]dV

(61)




subgrid dissipation



partuipartt

+ ujpartuipartxj

= minus1

ρ

partp

partxi+ ν

part2uipartxjpartxj

minus part

partxjτij (63)




partvipartt

+ vjpartvipartxj

= minus1

ρ

partq

partxi+ ν

part2vipartxjpartxj

minus part



part

partxjτij (65)

and

part


24


Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








Sijτij (67)










part


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0













25


Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

Gradient 0 0 0 0 + 0 0 0








4 6 8 9 9


2 5 7 7 9








26




u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










u(x y z t) =

0ax+ bcx+ d

(69)


[Sij(u)

]=

[1

2

(partuipartxj

+partujpartxi

)]=

0 a

2c2

a2 0 0

c2 0 0

=

0 a

2c2

a2 0 0

c2 0 0

(70)



]=



=


(71)


Sijτij = tr

0 a2

c2

a2 0 0c2 0 0

middotlowast 0 0


= 0 (72)




2 max0minusI3I1

Sij (73)

As

I1 = tr(S2) =

1

2(a2 + c2) and I3 = tr(S

3) = 0 (74)



27




Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0

















28





GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows











GLV =

0 0 0a 0 0c 0 0

(75)




44 Classes of Flows


29









3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows















3 5 6 6 6


2 5 6 6 6






1 1 2 3 3


0 0 1 0 3




30

low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







low

















31

classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







classes Then








(79)





2 4 4 4 4


1 4 4 4 4






2 2 4 5 5


1 1 3 2 5




32



















33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows

























33

is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







is





(80)








3 5 7 7 7


2 5 7 6 7






1 1 1 2 2


0 0 0 0 2




34

















I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows























I3 = I4 = I5 minus1

2I1I2 = 0 (82)


I3 minus I4 = 0 (83)





35






(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows












(85)





3 5 7 6 7


2 5 7 6 7






1 1 1 3 2


0 0 0 0 2




36















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows





















5 Conclusions


37




38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










38

Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







Appendices



PG = 0 QG = 0 RG = 0


I1 =1

2

(a2 + c2

) I2 = minus1

2

(a2 + c2

) I3 = 0 I4 = 0 I5 = minus1

8

(a2 + c2

)2


PG = 0 QG = 0 RG = 0


I1 =1

2

((2dz + e)

2+ (2gz + h)

2)

I2 = minus1

2

((2dz + e)

2+ (2gz + h)

2)

I3 = 0

I4 = 0

I5 = minus1

8

((2dz + e)

2+ (2gz + h)

2)2


PG = 0 QG = 0 RG = 0


I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0



39


I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








I1 =a2

2

(x2 + y2 + 12z2

)

I2 = minusa2

2(x2 + y2)

I3 = minus3a3z

4

(x2 + y2 + 8z2

)

I4 =a3z

4

(x2 + y2

)

I5 = minusa4

8

(x2 + y2

) (x2 + y2 + 10z2

)




I1 =π2

2eminus


I2 = minusπ2

2eminus


I3 = 0

I4 = 0

I5 = minusπ4

32eminus



PG = 0



40


I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








I1 =π2

2eminus

2π2tR


)

I2 = minusπ2

2eminus

2π2tR


)

I3 =3π3

4eminus


I4 =π3

4eminus

3π2tR



I5 = minusπ4

8eminus

4π2tR

(A4 + C2

(B2 minusA2





PG = 0 QG = minusa2


an(aminus k2n

)k


I1 =2a2


3an(aminus k2n

)k

I4 = 0 I5 = 0


PG = 0 QG = minus 1

Re2 RG = 0


I1 =k2

2cos2(z) +

2

Re2

I2 = minusk2

2cos2(z)


4Re

I4 =k2 cos2(z)

4Re


(k2Re2 cos2(z) + 2

)8Re2

41


PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








PG = 0

QG =1

2

(minus 24

c4Re2minus


)2Ω2x2y2

h(x y)2

+16eminus2h(xy)

(eh(xy) minus 1

)Ω2x2y2


(x2 + y2)2

)

RG =4eminus2h(xy)

c6Re3 (x2 + y2)2

(e2h(xy)

(4(x2 + y2

)2 minus c8Re2Ω2)

+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2

)minus c6Re2Ω2

(c2 + 2

(x2 + y2

)))


I1 =2eminus2h(xy)

c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 12

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


I3 =1

c6Re3 (x2 + y2)2


c2Re


c4Re2 (x2 + y2)2

(e2h(xy)

(c8Re2Ω2 + 4

(x2 + y2


(c2 + x2 + y2

)+ c4Re2Ω2

(c2 + x2 + y2

)2)


PG = 0 QG =12

c4Re2minus d2

(x2 + y2)2 RG =

16

c6Re3minus 4d2

c2Re (x2 + y2)2


I1 =24

c4Re2+

2d2

(x2 + y2)2 I2 = 0 I3 =

48

c6Re3minus 12d2

c2Re (x2 + y2)2 I4 = 0 I5 = 0

42


PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








PG = 0

QG =(A1

(x2 + y2

)+B1

) (3A1

(x2 + y2

)+B1

)+ 2A1C1 minus

C21Re2 + 4

Re2 (x2 + y2)2

RG = 0


I1 =1

2

(Re2

((x2 + y2

)3 (4A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 8A1C1

(x2 + y2

)2+ 4C2

1

)+ 16

Re2 (x2 + y2)2

+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

))

I2 = minus2(2A1

(x2 + y2

)+B1

)2 minus 1

2x2(A2 log

(x2 + y2

)+A2 + 2B2

)2 minus 1

2y2(A2 log

(x2 + y2

)+A2 + 2B2

)2

I3 = minus3(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I4 =

(A2 log

(x2 + y2

)+A2 + 2B2

)22Re

I5 = minus 1

8Re2 (x2 + y2)

(2A2

2

(Re2

(3(x2 + y2

)3 (3A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8A2B2

(Re2

((x2 + y2

)3 (9A2

1

(x2 + y2

)+ 4B2

2

)minus 6A1C1

(x2 + y2

)2+ C2

1

)+ 4)

+ 8

(8A2

1

(x2 + y2

)(Re2

(C1 minusA1

(x2 + y2

)2)2+ 4

)+B2

2

(Re2

(C1 minus 3A1

(x2 + y2

)2)2+ 4

)+ 2B4

2Re2(x2 + y2

)3)+A4

2Re2(x2 + y2

)3+ 8A3

2B2Re2(x2 + y2

)3)minus 1

8Re2 (x2 + y2)2

(2B2

1

(Re2

((x2 + y2

)3 (8A2

1

(x2 + y2

)+ (A2 + 2B2)2

)minus 16A1C1

(x2 + y2

)2+ 8C2

1

)+ 32

)+A2

(x2 + y2

)log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) (2(Re2

( (x2 + y2

)2 ( (x2 + y2

) (9A2

1

(x2 + y2

)+ (A2 + 2B2)2

)+ 6A1B1

(x2 + y2

)+B2

1

)minus 2C1

(x2 + y2

) (3A1

(x2 + y2

)+B1

)+ C2

1

)+ 4)

+A2Re2(x2 + y2

)3log(x2 + y2

) (A2 log

(x2 + y2

)+ 2A2 + 4B2

) )+ 4B1

(x2 + y2

) (16A3

1Re2(x2 + y2

)4 minus 32A21C1Re2

(x2 + y2

)2+A1

(Re2

(3(A2 + 2B2)2

(x2 + y2

)3+ 16C2

1

)+ 64


(x2 + y2

) ))

43


PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








PG = 0 QG = minus2

Re2+ 1

t2 eminus

Re(x2+y2)2t

(Re(x2 + y2

)+ 2t

)2 (x2 + y2)

2 RG = 0


I1 =

eminusRe(x2+y2)

2t

(Re4

(x2 + y2

)2+ 8Re3t

(x2 + y2

)+ 16Re2t2 + 16t3e

Re(x2+y2)2t

)8Re2t3 (x2 + y2)

2


2t

8t3

I3 = 0

I4 = 0

I5 =

eminusRe(x2+y2)

t

(Re4

(minus(x2 + y2



Re(x2+y2)2t

)128t6 (x2 + y2)

2


PG = 0

QG = minus27a2z2

(a2 minusR2

)216R10

RG = minus27a3z3

(a2 minusR2

)332R15


I1 =9(a3 minus aR2

)2 (x2 + y2 + 12z2

)32R10


)2 (x2 + y2

)32R10

I3 =81z

(aR2 minus a3

)3 (x2 + y2 + 8z2

)256R15

I4 =(27(a3 minus aR2)3(x2 + y2)z)

(256R15)


)4 (x2 + y2

) (x2 + y2 + 10z2

)2048R20


PG = 0 QG = 0 RG = 0

44


I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows








I1 = 2(x2 + y2

) I2 = minus2

(x2 + y2

) I3 = 0 I4 = 0 I5 = minus2

(x2 + y2

)2


PG = 0 QG = 0 RG = 0



2π2R4 I3 = 0 I4 = 0 I5 = 0

45



0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows









0 0 00 0 0

sube Q9

0 0 0lowast 0 00 0 0

0 lowast 0

0 0 00 0 0

0 0 0

0 0 0lowast 0 0

0 0 lowast

0 0 00 0 0

0 0 0

0 0 00 lowast 0

0 0 0

0 0 lowast0 0 0

sube Q8


0 lowast 0

0 0 00 lowast 0

0 0 lowast

0 0 lowast0 0 0

0 lowast lowast

0 0 00 0 0


0 0 0


sube Q7

46





part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows











part


Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





Q7minus9 Q0minus6

Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




47




part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










part


V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





V0 Vlowast

Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




48




part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










part


R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





R0 Rlowast

Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




49




part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows










part


C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0 0 0

Vreman 0 0 0 0 0 0

QR 0 0 0 0 0 0 0 0

Gradient 0 0 0 0 0 0 0 0 0




C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0





C2 C3

Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13

Exact 0 0 0 0 0

Smagorinsky 0 0

Vreman 0 0 0 0 0

QR 0 0 0 0 0 0 0




50

References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







References





S002211207900001X















51

Introduction


Introduction














Testing the Models


Classes of Flows





Conclusions

Appendices

Invariants of Flows







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