turbulence intro

8/13/2019 Turbulence Intro

1/9

Computational Fluid Dynamics

ClassicalTurbulence

Modeling

Grtar TryggvasonFall 2011


A jet in a cross flow

cross section of a jet

Most engineering problems involveturbulent flows. Such flows involveare highly unsteady and contain a

large range of scales. However, in

most cases the mean or averagemotion is well defined.


Flow over a sphere

The drag depends on the separation point


A modest Reynolds number the separated boundary layer remains initiallylaminar (left), before becoming turbulent. If the boundary layer is tripped (right)

it becomes turbulent, so that it separates farther rearward. Theoverall drag is thereby dramatically reduced, in a way that occurs

naturally on a smooth sphere only at a Reynolds numbers ten timesas great. ONERA photograph, Werle 1980.From "An Album of Fluid Motion," by Van Dyke, Parabolic Press.

Instantaneous flow past asphere at R = 15,000.

Instantaneous flow past a sphereat R = 30,000 with a trip wire


Examples of Reynolds numbers:

Flow around a 3 m long car at

100 km/hr:

Flow around a 100 m long submarineat 10 km/hr:

Re =LU

v=

3!27.78

1.5!10"5=5.5 !10

6

Kinematic viscosity(~20 C)

Water "= 10-6m2/s

Air "= 1.5 !10-5m2/s

1km/hr = 0.27778 m/s

Re =LU

v=

100!2.78

10"6

=2.78!108

Water flowing though a 0.01 m diameter pipe with a velocityof 1 m/s

Re =LU

v=

0.01!1

10"6

=104


Reynolds Averaged Navier-Stokes (RANS): Onlythe averaged motion is computed. The effect of

fluctuations is modeled

Large Eddy Simulations (LES): Large scalemotion is fully resolved but small scale motion ismodeled

Direct Numerical Simulations (DNS): Every lengthand time scale is fully resolved

2/9


ReynoldsAveraged

Navier-StokesEquations


To solve for the mean motion, we derive equations for the

mean motion by averaging the Navier-Stokes equations.The velocities and other quantities are decomposed into the

average and the fluctuation part

a = A+ a'

< a > = A

< a' > = 0

< a + b > = A + B

= cA

=!A

Defining an averagingprocedure that satisfies

the following rules:

This will hold forspatial averaging,

temporal averaging,and ensamble

averaging


There are several ways to define the proper averages

For homogeneous turbulence we can use the space average

For steady turbulence flow we can use the time average

For the general case we use the ensemble average

< a > =

1

Ladx

0

!

< a > =1

Tadt

0

T

!

< a > = ar(x,t)ensambles! a = A+ a'


!

!tu +" # uu = $

1

%"p+&"2u

u = U+ u'

p = P + p'

< a > = A

< a' > = 0

=cA

=!A

Start with the Navier-Stokes equations

Decompose the pressure and velocityinto mean and fluctuations:

a = A+ a'

Or, in general, for anydependant variable:


!

!t

U+ " #UU= $1

%

"P+&"2U$ "# < u'u' >

Applying the averaging to the Navier-Stokesequations results in:

< u'u'>=

 

 < v'v'> < v'w'>

 < v'w'> < w 'w'>

"

### %

&&&

Reynolds stress tensor


Physical interpretation

< uv >

Fast moving fluid particle

Slow moving fluid particle

Net momentum transferdue to velocity fluctuations


3/9


Closure:

Since we only have an equation for the mean flow,the Reynolds stresses must be related to the mean

flow.

No rigorous process exists for doing this!

THE TURBULENCE PROBLEM


Zero and Oneequation models


Introduce the turbulent eddy viscosity

!T=

l0

2

t0

ij=!"

T

#Ui

#xj

+

#Uj

#xi

$

%&

'

()

where


Zero equation models

!T =l02 dU

dy

Prandtlmixing length

l0=!y

Smagorinsky model

Baldvin-Lomaz model

!T= l

0

22SijSij( )

1/ 2

!T =l02

"i"i( )1/ 2

Sij=1

2

!Ui

!xj+

!Uj

!xi

#$$

&''

!i=

"Ui

"x j #

"Uj

"xi

%&

& ()

)


One equation models

!T= k

1/ 2t0

Where kis obtained by an equation describing its

temporal-spatial evolution

However, the problem with zero and one equationmodels is that t

0and l

0are not universal. Generally, it is

found that a two equation model is the minimum needed

for a proper description


Two equation

models

4/9


To characterize the turbulence it seems reasonable tostart with a measure of the magnitude of the velocityfluctuations. If the turbulence is isotropic, the turbulent

kinetic energy can be used:

k=1

2< u'u'> + < v 'v'> + < w 'w'>( )

The turbulent kinetic energy does, however, notdistinguish between large and small eddies.


To distinguish between large and small eddies we need tointroduce a new quantity that describe

!" # $u'i$u'i

$x j$x j

Usually, the turbulent dissipation rate is used

Smaller eddiesdissipate faster


!T=C

k2

"

!

!tU+" #UU= $

1

%"P + &+&

T( )"2U

Solve for the average velocity

Where the turbulent kinematic eddy viscosity isgiven by


!k

!t+Uj

!k

!x j="ij

!Ui!x j

#$+!

!x j%!k

!x j#1

2ui

'ui

'uj

' #1

&p'uj

'

()

+,

The exact k-equation is:

where !ij = " ui'uj

'

The exact epsilon-equation is considerably more complexand we will not write it down here.

Both equations contain transport, dissipation and

production terms that must be modeled


!k

!t+U"#k=# " D

k#k+ production $dissipation

!"

!t+U#$"=$ # D

"$"+ production %dissipation

The general for for the equations for k and epsilon is:

These terms must be modeled

Closure involves proposing a form for the missing termsand optimizing free coefficients to fit experimental data


Here

The k-epsilon model

!T=C

k2

"

!ij ==

2

3k"ij#$T

%Ui

%x j+

%Uj

%xi

'((

*++and

C1 =0.09; C2 =1.0; C3 =0.769; C4 =1.44; C5 =1.92

Dk

Dt= +!" (#+C

2#T)!k- $ij

%Ui

%x j&'

D!

Dt

=" # ($+C3$T)"!+C4!

k

%ij&Ui

&x j'C5

!2

kProduction Dissipation

Turbulenttransport

5/9


Other two equation turbulence models:RNG k-epsilon

Nonlinear k-epsilonk-enstrophyk-lok-reciprocal timeetc


Turbulent transport of energy and species concentrations is

modeled in similar ways.

For temperature we have:

!T

!t+" # uT =$"

2T

u = U+ u'

T = +T'

!< T>

!t+" # U = $"

2< T> %"# < UT>

Gradient Transport Hypothesis:

!"T# < T>


ModelPredictions


Spreading rates:

exp k-e CmottPlane jet 0.10 - 0.11 0.108 0.102

Round jet 0.085-0.095 0.116 0.095Mixing layer 0.13 - 0.17 0.152 0.154


From: C.G. Speziale: Analytical Methods for theDevelopment of Reynolds-stress closure in Turbulence.

Ann Rev. Fluid Mech. 1991. 23: 107-157


6/9


Results




Wall boundedturbulence


Wall bounded turbulence

Fundamental assumption: determined by localvariables only

Mean flow

Only the meanshear rate andthe properties of

the fluid are

important

!w =dU

dy, ", #


Define a shear velocity:

v*=

!w

"

!w = du

dy, ", #

kg /ms2[ ], kg /m3[ ], m2 /s[ ]

Normalize the length andvelocity near the wall

u+

=u

v*

y+

=y v

*

v

Called wall variables


Thus, the velocity near the wall is

Velocity versusdistance from wall

u+

y+

u+

= y+

u+

=1

!

lny+

+C

10

!=0.4

C=5.5

v* =!w

"

!w = du

dy

u

+

=u

v*

y+ =y v*

v

Bufferlayer

Outerlayer

Viscoussub-layer


For a practicalengineering problem

L = 1m; U = 1m/s; "= 10-6(water)

The Reynolds number is therefore:

For a flat plate, the average drag coefficient is

Re =LU

v=10

6

CD

=

FD

1

2!U

2LW

CD

=0.592Re!1/ 5

where

CD

=0.0037

!w

=

FD

LW= C

D

1

2"U

2=3.74

Thusand

And we findv

*=0.06


7/9


y =10!

!*=

10"10#6

0.06=1.667"10

#4m =0.1667 mm

Thickness of the viscous sub-layer

Find the thickness of the boundary layer

!

L=0.37Re

"1/ 5

!

L=0.0233m =23.3mm

To resolve the viscous sublayer at the same time as theturbulent boundary layer would require a large number of

grid points

The average thickness of the viscous sub-layer is 10 in units of y+:


To deal with this problem it is common to use wallfunctionswhere the mean velocity is matched with

an analytical approximation to the viscosus sublayer.

For a reference, see: Patel, Rodi, and Scheuerer,

Turbulence Models for Near-Wall and Low ReynoldsNumber Flows: A Review. AIAA Journal, 23 (1985),1308-1319


Second order closure


The k-epsilon and other two equation modelshave several serious limitations, including the

inability to predict anisotropic Reynolds stress

tensors, relaxation effects, and nonlocaleffects due to turbulent diffusion.

For these problems it is necessary to modelthe evolution of the full Reynolds stress

tensor


Derive equations for the Reynolds stresses:

!ui

!t+"u

iu

j = #

1

$"p +%"2u

i

The Navier-Stokes equations in component form:

ui

!ui!t

+"uiuj = - 1#"p+$"2u

i

&'

)*

Multiply the equation by the velocity

and averaging leds to equations for

!

!tu

iu

j


The new equations contain terms like

which are not known. These terms are therefore modeled

uiu

iu

j

The Reynolds stress model introduces 6 new equations(instead of 2 for the k-e model. Although the models

have considerably more physics build in and allow, for

example, anisotrophy in the Reynolds stress tensor,these model have yet to be optimized to the point that

they consistently give superior results.

For practical problems, the k-e model or more recentimprovements such as RNG are therefore most commonly used!


8/9


DirectNumerical

Simulations


In direct numerical simulations the full unsteadyNavier-Stokes equations are solved on asufficiently fine grid so that all length and time

scales are fully resolved. The sizeof the

problem is therefore very limited. The goal ofsuch simulations is to provide both insight andquantitative data for turbulence modeling


Channel Flow

Streamwise velocity

Flow direction

Periodicstreamwise

andspanwise

boundaries

Wall


Streamwise vorticity


Streamwise vorticity

Turbulentshear stress

Turbulent eddies generate anearly uniform velocity profile

Channel Flow


Turbulence are intrinsically linked to vorticity, yetlaminar flows can also be vortical so looking at the

vorticity is not sufficient to understand what is

going on in a turbulent flows. Several attemptshave been made to define properties of the

turbulent flows that identifies vortices (as opposedto simply vortical flows.

One of the most successful method is the lambda-2method of Hussain.


9/9


Visualizing turbulence

!u =

"u

"x

"u

"y

"u

"z

"v

"x

"v

"y

"v

"z

"w

"x

"w

"y

"w

"z

$

%%%%%%

'

((((((

S =1

2!u + !Tu( ) =

1

2

2"u

"x

"u

"y+

"v

"x

"u

"z+

"w

"x

"v

"x+

"u

"y2"v

"y

"v

"z+

"w

"y

"w

"x+

"u

"z

"w

"y+

"v

"z2"w

"z

$

%%%%%%

'

((((((

!=1

2"u - "Tu( ) =

1

2

0 #u

#y$#v

#x

#u

#z$#w

#x

#v

#x$#u

#y0

#v

#z$#w

#y

#w

#x$#u

#z

#w

#y$#v

#z0

&

''''''

)

******


It can be shown that the second eigenvalue of

S2

+!2

define vortex structures

Referece: J. Jeong and F. Hussain, "On the identification ofa vortex," Journal of Fluid Mechanics, Vol. 285, 69-94,1995.

Other quantities have also been used, such as thesecond invariant of the velocity gradient:

Q =!ui

!x j

!uj

!x i

!2


!2= "0.3

!2= "0.2


Turbulence models are used to allow us tosimulate only the averaged motion, not theunsteady small scale motion.

Turbulence modeling rest on the assumptionthat the small scale motion is universaland

can be described in terms of the large scalemotion.

Although considerable progress has beenmade, much is still not known and results from

calculations using such models have to beinterpreted by care!


For more information:

D. C. Wilcox, Turbulence Modeling for

CFD (2nded. 1998; 3rded. 2006).

The author is one of the inventors ofthe k-#model and the book promotes ituse. The discussion is, however,

general and very accessible, as well as

focused on the use of turbulence

modeling for practical applications inCFD

turbulence intro

Documents