chapter 12 finite-volume (control-volume) method-introduction

CHAPTER 12

Finite-Volume (control-Volume) Method-Introduction

12-1 Introduction (1)

In developing what has become known as the finite-volume method, the conservation principles are applied to a fixed region in space known as a control volume, are somewhat interchangeably in the literature.

12-1 Introduction(2) In the finite-volume approach, a point of view is taken that

is distinctly different from finite-difference method(or Taylar-series method ). In the Taylar-series method, we accepted the PDE as the correct and appropriate from of the conservation principle(physical law) governing our problem and merely turned to mathematical tools to develop algebraic approximations to derivatives. We never again considered the physical law represented by the PDE. In the finite-volume method, the conservation statement is applied in a form applicable to a region in space (control volume).

12-1 Introduction(3)

This integral form of the conservation statement is usually well known from the first principles, or it can in most cases, be developed from the PDE form of the conservation from.


The feature of the FV method is shared in common with the finite-element methods. The FV procedure can, in fact, be considered as a variant of the finite-element method, although it is, from another point of view, just a particular type of finite-difference method.


As an example, consider unsteady 2-D heat conduction in a rectangular-shaped solid. The problem domain is divided up into control volume with associated points. We can establish the control volumes first and place grid points in the centers of the volumes (cell-centered method) or establish the grid first and then fix the boundaries of the control volumes (cell-vertex method) by, for example placing the boundaries halfway between grid points.

12-1 Introduction(6) The General Differential Equation

The differential equation obeying the generalized conservation principle can be written by the general differential equation as

)1(

sv

t

:dependent variable, such as velocity components (u,v,w), h or T, k, ε concentration, etc.

12-1 Introduction(7) : diffusion coefficientsS: source term

The four terms of eq.(1) are the unsteady term, the convection term, the diffusion term and the source term.

*Note: The “conservation form” of the PDE is also referred to as “conservation law form” or “divergence form”, i.e., all spatial derivatives appear purely as divergences.


Conservation form of the governing equations of fluid flow

C

T

M

SCDcvt

cSpecies

STkhvt

hEnergy

Svpvvt

vMometum

vt

Mass

:

:

:

0:


One-way and two-way coordinates :1. Definitions: a two-way coordinate is such that

the conditions at a given location in that coordinate are influenced by changes in conditions on either side of that location. A one-way coordinate is such that the conditions at a given location in that coordinate are influenced by changes in the conditions on only one side of that location.


2. Examples: one-dimensional steady heat conduction in a rod provides one example of a two-way coordinate. The temperature of any given point in the rod can be influenced by changing the temperature of either end. Normally, space coordinates are two-way coordinates. Time, on the other hand, is always a one-way coordinate. During the unsteady cooling of a solid, the temperature at a given instant can be influenced by changing only these conditions that prevailed before that instant.


3. Space as a one-way coordinate:

If there is a strong unidirectional flow in the coordinate direction, then significant influences travel only from upstream. The conditions at a given point are then affected largely by the upstream conditions, and very little by the downstream ones. It is true that convection is a one-way process, but diffusion (which is always present) has two-way influences. However., then the flow rate is large, condition overpowers diffusion and thus make the space coordinate nearly one-way.

12-1 Introduction(12) 4. Parabolic, elliptic, hyperbolic:

a) The term parabolic indicates a one way behavior, while elliptic signifies the two-way concept.

b) It would be more meaningful if situations were described as being parabolic or elliptic in a given coordinate. Thus, the unsteady heat condition problem, which is normally called parabolic, is actually parabolic in time and elliptic in all coordinate. A two-dimension boundary layer is parabolic in the stream wise coordinate and elliptic in the cross-stream coordinate

12-1 Introduction(13) c) A hyperbolic problem has a kind of one-way beha

vior, which is, however, not along coordinate directions but along special-lines called characteristics.

d) A situation is parabolic if there exists at least one one-way coordinate: otherwise, it is elliptic.

e) A flow with one one-way space coordinate is sometimes called a boundary-layer-type flow, while a flow with all two-way coordinate is referred to as a recirculating flow.


5. Computational implications: The motivation for the foregoing discussion about

one-way and two-way coordinates is that, it a one-way coordinate can be identified in a given situation, substantial economy of computer storage and computer time is possible.

12-2 An Illustrative Example(1)The FV method used the integral form of

the conservation equation(eq.1) as the starting point:

)2( CVACVCV

dVsdAndVsVd

• Let us consider steady one-dimensional heat conduction governed by

)3(0

sdx

dTk

dx

d

12-2 An Illustrative Example(2)

1. Preparation: To derive the discrerization equation, we shall employ the grid-point cluster shown in Fig.1. We focus attention on the grid point P, which has the grid points E and W as its neighbors.(E denotes the east side, while W stands for the west side). The dashed lines show the faces of the control volume. The letters e and w denote these faces.


For one-dimensional problem under consideration, we shall assume a unit thickness in the y and z directions. Thus, the volume of the cv shown is △x × 1 ×1. If we integrate eq(3) over the cv, we get

W P E

(δ x)w (δ x)e

△ x

)4(0

e

wwe

sdxx

Tk

x

Tk

Fig. 1w e

12-2 An Illustrative Example(4)2. Profile assumption: To make further further progress,

we need a profile assumption or an interpolation formula. Here, linear interpolation functions are used between the grid points, as shown in Fig 2.

T

x

Fig. 2

W Ew e

δxw δxe

Δp


3. The discrerization equation: If we evaluate the derivatives dT/dx in eq.(4) from the piecewise-linear profile, the resulting equation will be

)5(0

xs

x

TTk

x

TTk

w

wpw

e

pEe


xsbaaax

ka

x

kaWhere

bTaTaTa

s

WEPw

wW

e

eE

WWEEpp

,,,

)6(

:form following theinto

eq(5)tion discretiza cast the touseful isIt

cv. over the s of valueaverage theis where

12-2 An Illustrative Example(7)4. Comments:

a) In general, it is convenient to extend eq.(6) into multidimensional form as

where nb denotes a neighbor, and the summation is to be taken over all the neighbors.

b) In deriving eq(6), we have used the simplest profile assumption that enabled us to evaluate dT/dx. Of course, many other interpolation functions would have been possible.

)7( bTaTa nbnbPP

12-2 An Illustrative Example(8)c) Further, it is important to understand that we need

not use the same profile for all quantities.

d) Even for given variable, the same profile assumption need not be used for all terms in the equation.


5. Treatment of source term:

The discretization equations will be solved by the techniques for linear algebraic equations. The procedure for “linearizing” a given S~T relationship is necessary. Here, it is sufficient to express the overage value S as

PPC TSSS


Where Sc stands for the constant part of S, while Sp is the coefficient of Tp. With the linearized source expression, the discretization equation will become

xSb

xsaaa

x

ka

x

kawhere

bTaTaTa

C

pWEP

w

wW

e

eE

WWEEPP

12-3 The Four Basic Rules(1)Rule 1:Consistency at a control-volume face

-When a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization equations for the two control volumes

Rule 2:Positive coefficients-All coefficients (ap and neighbor coefficients anb) mu

st always be positive.

12-3 The Four Basic Rules(2)

Rule 3:Negative-slope linearization of the source term

-When the source term is linearized as S=SC+SPTP, the coefficient SP must always be less than or equal to zero.

Rule 4:Sum of the neighbor coefficients-We require nbP aa

CHAPTER 13

The Finite Volume Method for Diffusion Problems

13-1 Steady One-dimensional Condition(1)

The Basic Equation

The Discretization Equation)1(0

Sdx

dTk

dx

d

xSb

xSaaa

x

ka

x

kawhere

bTaTaTa

C

PWEp

w

wW

e

eE

WWEEPP

)2(


The Grid Spacing1. For the grid points shown in 8.4, it it not necessary

that the distances (δx)e and (δx)w be equal. Indeed, the use of non-uniform grid spacing is often desirable, for it enables us to deploy computing power effectively. In general, we shall obtain an accurate solution only when the grid is sufficiently fine, but there is no need to employ a fine grid in regions where the dependent variable T changes rather slowly with x. On the other hand, a fine grid is required where the T~x variation is steep.


2. A misconception seems prevail that non-uniform grid lead to less accuracy than do uniform grids. There is no sound basis for such an assertion. Also there are no universal rules about what maximum (or minimum) ratio the adjacent grid intervals should maintain.


3. Since the T~x distribution is not known before the problem is solved, how can we design an appropriate non-uniform grid?

First: One normally has some qualitative expectations about the solution, from which some guidance can be obtained.

second: preliminary coarse-grid solutions can be used to find the pattern of

the T~x variation; then a suitable non- uniform grid can be constructed.


The Interface Conductivity1. The most straightforward procedure for

obtaining the interface conductivity ke is to assume a linear variation of k between points P and E

x

EP e

(δx)e

(δx)e+(δx)e-


If the interface e were midway between grid points, fe would be 0.5, and ke would be he arithmetic mean of kp and kE.

)4(

)3()1(,

e

ee

Eepee

x

xfwhere

kfkfkThen


2.We shall shortly show that this simple-minded approach leads to rather incorrect implications in some cases and cannot accurately handle the abrupt changes of conductivity that may occur in composite materials. Fortunately, a much better alternative is available.

3.Our main objective is to obtain a good representation for the heat flux qe at the interface via


For the composite slab between points P and E, a steady one-dimensional analysis (without sources) lead to

)5(

e

Epee x

TTkq

)6(

E

e

P

e

EPe

kx

kx

TTq

3.Our main objective is to obtain a good

representation for the heat flux qe at the

interface via


Combination of Eqs.(4) —— (6) yields

)7(1

1

E

e

p

ee k

f

k

fk

When the interface e is placed midway between p and E, we have fe=0.5; then

)9(2

)8(5.0 111

Ep

Epe

Epe

kk

kkkor

kkk

Eq. (9) show that ke is the harmonic mean of kp and kE, rather than the arithmetic mean.


4.

A similar expression can be written for aW.

1

E

e

p

eE k

x

k

xa


5. The recommended interface conductivity formula (7) is based on the steady, no-source, one-dimensional situation in which the conductivity varies in a stepwise fashion from one control volume to the next. Even in situations with nonzero sources or with continuous variation of conductivity, it performs much better then the arithmetic-mean formula.


Iteration

1. Start with a guess or estimate for the values of T at all grid points.

2. From these guessed T´s, calculate tentative values of the coefficients in the discretization equation.

3. Solve the nominally set of algebraic equations to get new values of T.

4. With these T´s as better guesses, return to step 2 and repeat the process until further repetitions cease to produce significant changes in the values of T.

13-1 Steady One-dimensional Condition(13) Source-Term Linearization

Tp*: the guess value or the previous-iteration value of Tp

Example 1: Given S=5-4T

-Sc=5, Sp=-4—recommended

-Sc=5-4Tp *, Sp=0 —not impractical

-Sc=5+7Tp *,Sp=-11 —a steeper S~T relationship, will slow down the convergence


Example 2: Given S=3+7T

1. Sc=3,Sp=7 —this is not acceptable, as it makes Sp positive. The presence of a positive Sp many cause divergence.

2. Sc=3+7Tp*, Sp=0 —this is the practice one should follow.

3. Sc=3+9Tp*, Sp=-2 —this is an artificial creative Sp. It will, in general, slow down the convergence.


Example 3: Given S=4-5T3

1. Sc=4-5Tp*3, Sp=0—this is the lazy-person approach.

2. Sc=4, Sp=-5Tp*2—this given S~T curve is steeper than this implies.


This linearization represents the tangent to the S~T curve at Tp*

2*3*

2*3*

*2*3*

**

*

15,104,

15104

1554

pppc

ppp

pppp

pp

TSTSThus

TTT

TTTT

TTdT

dSSS

3. Recommended method:


4. Sc=4+20Tp*, Sp=-25Tp*2—This givens a steeper S~T curve, which would slow down convergence

S

T

(1)

(2)

(3)(4)


Boundary Conditions:1. Typically, three kinds of boundary conditions

are encountered in heat condition. These are -Given boundary temperature.

-Given boundary heat flux

-Boundary heat flux specified via a heat transfer

coefficient and the temperature of the surrounding

fluid.


2. If the boundary temperature is given, no particular difficulty arises, and no additional equations are required. When the boundary temperature is not given, we need to construct an additional equation for TB. This is done by integrating the differential equation over the “half” control volume shown adjacent to the boundary in the following Figure.


B I W P E

Typical C.V.

“Half” C.V.

B I

(δx)i

iΔx

qB

Fig 1

Fig 2


3. Apply the principles of energy conservation

xsdx

dTk

dx

dTkdxs

dx

dTk

dx

d

Bi

i

B

Applying the principles of energy conservation over C.V. of Fig.2 and noting that the heat flux q stands for -k(dT/dx), we get

xSaaqxSbx

kawhere

bTaTa

xTSSdx

TTkq

xTSSqq

PIBBci

iI

IIBB

Bpci

IBiB

BpciB

,,

0

0)(


4. If qB is specified in terms of a heat transfer coefficient h and a surrounding-fluid temperature Tf such that

qB =h(Tf-TB)

Then, the equation for TB becomes

hxSaa

hTxSbx

kawhere

bTaTa

PIB

fCi

iI

IIBB

,,


Solution of the Linear Algebra Equation (TDMA)

1. The discretization equations can be written as

)1(11 iiiiiii dTcTbTa

13-2 Unsteady One-Dimensional Condition(1)

The general Discretization Equation1. Unsteady one-dimensional heat-conduction e

quation

)2(

becomes eq.(1) constant, are c, if

)1(

x

Tk

xt

Tc

x

Tk

xt

cT


2. The discretization equation

W P Ew e

0

)3(

pP

e

w

tt

t

e

w

tt

t

e

w

tt

t

TTxcdtdxt

Tc

dxdtx

Tk

xdtdx

t

Tc


where f is a weighting factor between 0 and 1 from Eqs.(4) and (5), we can get

tt

t ppp

tt

tw

wpw

e

pEepP

tTfTfdtT

dtx

TTk

x

TTkTTxc

)5()1(

)4(

0

0

)6(1

0000

000

w

wpw

e

pEe

w

wpw

e

pEepp

x

TTk

x

TTkf

x

TTk

x

TTkfTT

t

xc


00

00

00

)6(11

11

pwEpp

w

ww

e

eE

PWEP

WWWEEEpp

afafaat

xca

x

ka

x

ka

where

Tafafa

TffTaTffTaTa


1. Example: Crank-Nicolson, and Fully Implicit Schemes

stablenallyunconditioschemeimplicitFullyf

schemeNicolsonCrankfk

xctcriteriastabilityschemelicitf

:1

:5.02

:exp:02


2. Variation of temperature with time for three different schemes

Explicit

Fully implicit

Crank-Nicolson

t tt+Δt

0pT

pT

pT


3. Why would we prefer the fully implicit scheme?

a) For f=0 (explicit) scheme

→ apTp= aETE0 + awTw

0 +( ap0-aE-aW ) TP

0

This means that TP is not related to other unknows such as TE or TW, but is explicitly obtainable in terms of known temperature TP

0, TE0, TW

0.


b) For f=0.5, the coefficient of TP0 in eq(6) becomes

aP0-(aE + aW )/2. For uniform conductivity and unif

orm grid spacing, this coefficient can be seen to be ρc(∆x/∆t)-k/ ∆x. Whenever the time step is not sufficiently small, this coefficient could become negative, with its potential for physically unrealistic results.

c) For f=1, the coefficient of TP0 in eq(6) must never

become negative. It is for this reason that we shall adopt the fully implicit scheme in this book.


The Fully Implicit Discretitation Equation1. aP TP

=aE TE

+aW TW

+b

Where

xSaaaa

TaxSb

t

xca

x

ka

x

ka

PPWEP

ppC

PW

WW

e

eE

0

00

0 ,,,

2. It can be seen that, as ∆t →∞ , this equation reduces to our steady-state discretization equation.

13-3 Two-And Three-Dimensional Situations(1)

Discretization Equation for Two Dimensions

s

y

Tk

yx

Tk

xt

Tc

sy

Tk

yx

Tk

xt

cT

then constant,c , if


y

x

N

EW

S

∆x

∆y

s

ew

n

p


→aP TP =aE

TE +aW

TW +aN TN

+b

where

bTaTaTaTaTaTaTa

yxSaaaaaa

TayxSb

t

yxca

y

xka

y

xka

x

yka

x

yka

BBTTSSNNWWEEPP

PPNSWEP

ppC

P

S

sW

n

nN

W

WW

e

eE

Dimensions Threefor Equation tion Discretiza

,

,,,,

0

00

0

13-4 Overrelaxation and Underrelaxation(1)Overrelaxation is often used in conjunction with th

e Gauss-Seidel method, the resulting scheme being known as successive Over-Relaxation (SOR). With the line-by-line method, the use of overrelaxation is less common. Underrelaxation is a very useful device for nonlinear problem. It is often employed to avoid divergence in the iterative solution of strongly nonlinear equations.

13-4 Overrelation and Underrelaxation(2)The general discretization equation of the form is

)4(1or

)4(

iteration. previous thefrom of value theis

)3(

)2(

)1(

*

**

*

**

bTa

bTaTa

aTa

bTaTT

TT

Ta

bTaTT

a

bTaT

bTaTa

PP

nbnbPP

PP

nbnbPP

PP

PP

nbnbPP

P

nbnbP

nbnbPP

13-4 Overrelation and Underrelaxation(3)At first, it should be noted that, when the iterations,

that is, TP becomes equal to TP*. Eq. (4a) implies t

hat the converged values of T do satisfy the original Eq.(1). Any relaxation scheme, of course must possess this property; the final converged solution, although obtained through the use of arbitrary relaxation factors or similar devices, must still satisfy the original discretization equation.

13-4 Overrelation and Underrelaxation(4)

There are no general rules for choosing the best value of α. The optimum value depends upon a number of factors, such as the nature of the problem, the number of grid points, the grid spacing, and the iterative procedure used. Usually, a suitable value of α can be found by experience.

CHAPTER 14

Convection and Diffusion

14-1 Convection-Diffusion Term

The convection term has an inseparable connection with the diffusion term, and therefore, the two terms need to be handled as one unit.

Governing Equations

Sxx

uxt

uxt

jjj

j

jj

0

14-2 Steady One-Dimensional Convection and Diffusion(1)

Governing Equations

)1(

tan0

xdx

du

dx

d

tconsuorudx

d


A Preliminary Derivation1. Integration of Eq (1) over the C.V. shown in

Fig.1 gives

)2(

wewe dx

d

dx

duu

W P Ew e

(δx)w (δx)e

C.V.


2. Diffusion term: The same way of chapter 9.

3. Convection term:

WPwPEe and 2

1

2

1

The factor ½ arises from the assumption of the interfaces being midway. Now, Eq (1) can be written as

w

WPw

e

PEe

WPwPEe

xx

uu

2

1

2

1


4. a)

b) Both have the same dimensions; F indicates the strength of the convection (or the flow), while D is the diffusion conductance.

c) D always remains positive, F can take either positive or negative values depending on the direction of the fluid flow.

xDuF

,


5. Discretization Equation:

)3(22

)3(2

)3(2

)3(

cFFaa

FD

FDa

bF

Da

aF

DaWhere

aaa

weWE

ww

eeP

wwW

eeE

WWEEPP


6. Discussion:a) Since by continuity Fe=Fw, we do get the property

aP= aE + aW

b) Eq (3) is also known as the central-difference scheme and is the natural outcome of a Taylor-series formulation.


c) Example: De=Dw=1 and Fe=Fw=4

Consider two sets of values:

i. If ΦE=200 and ΦW=100 , then ΦP=50!

ii. If ΦE=100 and ΦW=200 , then ΦP=250!

Since , in reality, cannot fall outside the range of 100~200 established by its neighbors, these results are clearly unrealistic.


d) Eqs (3a)-(3c) indicate that the coefficients could, at times, become negative. When |F| exceeds 2D, then, depending on whether F is positive or negative, where is a possibility of aE or aW becoming negative. This will be a violation of one of the basic rules.


The Upwind Scheme1. The upwind scheme recognizes that the weak

point in the preliminary formulation is the assumption that the convected property Φe at the interface is the average of ΦE and ΦP , and it proposes a better prescription. The formulation of the diffusion term is left unchanged, but the convection term is calculated from the following assumption:


The value of Φ at an interface is equal to the value of Φ at the grid point on the upwind side of the face.

thus, Φe = Φp if Fe>0—(4a)

and, Φe = ΦE if Fe<0—(4b)

The value of Φw can be defined similarly.

W P Ew e

wewe FFuu


2. We shall define 【 A,B 】 to denote the greater of A and B . Then, the upwind scheme implies

)5(0,0, eEepee FFF


3. The discretization equation becomes

)6(

0,10,

)6(0,

)6(0,

)6(

cFFaa

FDFDa

bFDa

aFDawhere

aaa

weWE

wweeP

wwW

eeE

WWEEpp


4. Discussion:

It is evident from Eqs.(6) that no negative coefficients would arise, thus, the solutions will always be physically realistic.


The Exact Solution

)1(

xxu

x

can be solved exactly if Γ is taken to be constant.(ρu is constant)


If a domain 0≦x≦L is used, with the boundary conditions

diffusion. and convection

of strengths theof ratio theis seen that becan It

by definednumber Peclet a is P where

)7(1exp

1exp

is Eq(1) ofsolution The

Lx

0x

0

0

0

P

uLp

PL

xP

At

At

L

L


2. The nature of the exact solution:

фL

ф0

0 L

ф

x

P>>1

P=1P=0

P= -1

-P>>1

Fig.2 Exact solution for the one-dimensional convection-diffusion problem


a) In the limit of zero Peclet number, we get the pure-diffusion (or conduction) problem, and the ф~x variation is linear.

b) When the flow is in the positive x direction (i.e., for positive values of P), the values of ф in the domain seem to be more influenced by the upstream value ф0.

c) For a large positive value of P, the value of ф remains very close to the upstream value ф0 over much of the domain.


d) When the fluid flows in the negative x direction, Φ L becomes the upstream value, which dominates the values of Φ in the domain.

e) When a large negative P, the value Φ of over most of the region is very nearly equal to Φ L .


3. Implications:a) It is easy to see why our preliminary derivation

failed to give a satisfactory formulation. The Φ ~x profile is far from being linear except for small values of |P|.

b) Where |P| is large, the value of Φ at x=L/2 (the interface is nearly equal to the value of Φ at the upwind boundary. This is precisely the assumption made in the upwind scheme; but there it is used for all values of |P|, not just for large value.


c) Where |P| is large, dΦ/dx is nearly zero at x=L/2. Thus the diffusion is almost absent. The upwind scheme always calculates the diffusion term from a linear Φ~x profile and thus overestimates diffusion at large value of |P|.


The exponential scheme1. It is useful to consider a total flux J that is ma

de of the convection flux ρu and the diffusion flux -Γd /dx. Thus,

)8(dx

duJ

with this definition eq (1) becomes

)9(0 dx

dJ

which, when integrated over the C.V. shown in fig.1 gives )10(0 we JJ


2. The substitution of eq. (9) into eq. (8) would give the expression for Je:

e

e

e

eee

e

Eppee

D

Fxupwhere

pFJ

)10(1exp


3. Finally, substitution of eq. (11) and a similar expression for Jw into eq. (10) leads to

)14(

)14(1/exp

/exp

)14(1/exp

)13(

)12(01exp1exp

cFFaaa

bDF

DFFa

aDF

Fawhere

aaa

pF

pF

weWEp

ww

wwww

ee

eE

WWEEPP

w

pWWw

e

Eppe


4. Discussions:a) These coefficients expressions define the exponent

ial scheme. When used for the steady one-dimensional problem, this scheme is guaranteed to produce the exact solution for any value of the Peclet number and for any number of grid points.


b) Despite its highly desirable behavior, it is not widely used becausei. Exponentials are expensive to compute.

ii. Since the scheme is not for two- or three-dimensional situations, nonzero sources, etc., the extra expense of computing the exponentials does not seem to be justified.


The Hybrid scheme1. To appreciate the connection between the

exponential scheme and the hybrid scheme, we shall plot aE/De vs. pe as follows:


5

4

3

2

e

E

D

a

ee

E pD

a

21 e

e

E p

D

a

0e

E

D

aExact

1exp

thatdeduce weeq(14), From

e

e

e

E

p

p

D

a

Fig. 3 1 2 3 4 5pe


2. From Fig 3, we can get

2

p1 is tangent the,0p )

p p )

0 p )

ee

ee

e

e

E

e

E

e

E

D

aAtc

D

aForb

D

aFora


3. The hybrid scheme is made up of these three straight lines of Fig.3, so that

0 2p )

2

p1 2p2- )

p 2p )

e

ee

ee

e

E

e

E

e

E

D

aForc

D

aForb

D

aFora


4. These expressions can be combined into a compact form as follows:

0 ,2

,

0 ,2

1 ,

ee

eE

eeeE

FDFaor

ppDa


5. The significance of the hybrid scheme can be understood by observing that

a) It is identical with the central-difference scheme for –2 < pe < 2

b) Outside this range it reduces to the upwind scheme in which the diffusion has been set equal to zero.

c) The name hybrid is indicative of a combination of the central-difference and upwind scheme, but it is best to consider the hybrid scheme as the three-line approximation to the exact curve, shown in Fig.3


6. The convection-diffusion discretization for the hybrid scheme can now be written as

weWEP

wwww

eeE

WWEEPP

FFaaa

FDFa

FDFa

where

aaa

0 ,2

,

0 ,2

, e


The Power-Law scheme1. It can be seen from Fig.3, that the departure

of the hybrid scheme from the exact curve is rather large. A better approximation to the exact curve is given by the Power-law scheme.


2. The Power-law expressions for aE can be written as

)15(p1.01 10p0 )

)15(p1.01 0p10- )

)15(p 10p )

5ee

5ee

ee

cD

aForc

bpD

aForb

aD

aFora

e

E

ee

E

e

E


A Generalized Formulation1. The general convection-diffusion formulation

can be written as

wewEp

www

eeE

wwEEpp

FFaaa

FpADa

FpADawhere

aaa

0,

0,


2. The function A(|p|) for different schemes

1exp)(

1.01,0

5.01,0

1

5.01

5

p

pexactlExponentia

plawPower

pHybrid

Upwind

pdifferenceCentral

14-3 Discretization Equation for Two Dimensions(1)

Discretization Equation of 2-D

)1(

sy

J

x

J

tyx

where Jx and Jy are the total (convection plus diffusion) fluxes defined by

)1(

)1(

by

vJ

ax

uJ

y

x


The integration of eq(1) over the c.v. shown in Fig.1 would give

y

x

N

EW

S

∆x

∆y

ew

n

p

Jn

Je

Js

Jw

Fig.2

s


)2(

00

yxSS

JJJJt

yx

ppc

snwepppp


In a similar manner, we can integrate the continuity eq

0y

F

x

F

tyx

over the c.v. and obtain

uFwhere

FFFFt

yxsnwe

pp

)3(0

0


)4(

)3()2(0

0

yxSSFJFJ

FJFJt

yx

ppcpsspnn

pwwpeep

pp

p


The final discretization equation

baaaaa

FpADa

FpADawhere

aFJ

aFJ

SSNNWEEpp

wwWW

eeEE

pwwpww

EpEpee

as written be

nowcan equation tion discretiza D-2 The

0,

0,


where

yxSaaaaaa

payxSb

t

yxa

FpADa

FpADa

FpADa

FpADa

ppSNwEp

ppC

pp

sssS

nnnN

wwwW

eeeE

0

00

00

0,

0,

0,

0,

14-4 One-Way Space Coordinate(1)Time is a one-way coordinate. The convection-diff

usion formulation reveals that a space coordinate can also become a one-way.

What makes a space coordinate One-Way?

When the Peclet number exceeds10, the Power-law scheme will set the downstream-neighbor coefficient equal to zero. (the hybrid scheme does this for a Peclet number greater than 2.)

14-4 One-Way Space Coordinate(2)

Suppose that, in the 2-D situation shown in Fig.2 , there is a high flow rate in the positive x direction. Then, for all the grid points along a y-direction line, the coefficient aE will be zero. In other words, p will be dependent on W , N , and S , but not on aE. Thus the x coordinate will become a one-way coordinate since the value at any point will be uninfluenced by any of the downstream values.


w

N

p

S

E

Fig.2

14-4 One-Way Space Coordinate(4)The outflow Boundary Conditions

At the outflow boundary shown in Fig.3, for example, one may not know the temperature or the heat flux. How can we then solve the problem? The answer is surprisingly simple: No boundary-condition information is needed at an outflow boundary. For all grid points p next to the outflow boundary, the coefficient a

E will be zero, and hence no boundary values will be needed. In other words, the region near the outflow boundary exhibits, for large Peclet number, local one-way behavior.


Inflow boundary

outflow boundary

Fig 3

N

P

SEW


A particularly bad choice of an outflow-boundary location is the one in which there is an “inflow” over a part of it. An example of this shown in Fig.4. For such a bad choice of the boundary, no meaningful solution can be obtained.


Bad Good

Fig. 4

CHAPTER 15

Calculation of The Flow Field

15-1 Need for a Special Procedure

The real difficulty in the calculation of the velocity field lies in the unknown pressure field. The pressure gradient forms a part of source term for a momentum equation.

15-2 Some Related Difficulties(1)

Representation of the Pressure-Gradient Term

1. To integrate –dp/dx over the control volume shown in Fig.1, we can get pw-pe.

W P Ew e

c.v.Fig. 1

x


2. To express pw-pe in terms of the grid-point pressure, we may assume a piecewise-linear profile for pressure. Therefore, we can get

222EWEppW

ew

pppppppp

This means that the momentum equation will contain the pressure difference between two alternate grid points, and not between adjacent ones.


3. There is another implication that is far more serious. It can be seen from Fig.2, where a pressure field is proposed in terms of the grid-point values of pressure.

p=100 500 100 500 100 500

Fig. 2such a zig-zag field cannot be regarded as realistic; but for any grid point p, the corresponding pW-pE can be seen to zero, since the alternate pressure values are everywhere equal.


Representation of the continuity Equation

If we integrate the continuity equation over the c.v. shown in Fig1, we have

ue-uw=0


Once again, the use of a piecewise-linear profile for u and of the midway locations of the control-volume faces leads to

0022

WEpWEp uuor

uuuu

Thus, the discretized continuity equation demands the quality of velocities at alternate grid points and not at adjacent ones.

15-3 A Remedy : The staggered Grid(1)

The difficulties described so far can be resolved by recognizing that we do not have to calculate all the variables for the same grid points. We can, if we wish, employ a different grid for each dependent variable.


Staggered grid:1. The velocity components are calculated for

the points that lie on the face of the control volume, as shown in Fig.3.

2. Other variables are calculated for the grid points(small circles).

Fig.3 →=ui,, ↑=vi ,

=other variables


3. The important advantages are twofold:a) For a typical c.v. shown in Fig.3, it is easy to see t

hat the discretized continuity equation would contain the difference of adjacent velocity components, and this would prevent a wavy velocity field.

b) The second important advantage of the staggered grid is that the pressure difference between two adjacent grid points now becomes the natural driving force for the velocity component located between these grid points.

15-4 The Momentum Equations(1)

The discretization equation (2-D) can be written as

)1(

)1(

)1(

cAppbwawasimilarly

bAppbvava

aAppbuaua

ttPnbnbtt

nNPnbnbnn

eEPnbnbee

15-4 The Momentum Equations(2)

The pressure gradient is not included in the

source-term SC and SP.

e

Fig.4a c.v. for u

n

Fig.4b c.v. for v

15-4 The Momentum Equations(3)The momentum equation can be solved when the

pressure field is given or is somewhat estimated. Unless the correct pressure field is employed, the resulting velocity field will not satisfy the continuity equation. Such an imperfect field based on a guessed pressure field p* will be denoted by u*,v*,w*. This “starred” velocity field will result from the solution of the following equations:

)2(

)2(

)2(

****

****

****

cAppbwawa

bAppbvava

aAppbuaua

ttPnbnbtt

nNPnbnbnn

eEPnbnbee

15-5 The Pressure and Velocity Corrections(1)One aim is to find a way of improving the

guessed pressure p* such that the resulting starred velocity field will progressively get closer to satisfying the continuity equation. Let us propose that the correct pressure p is obtained from

)3(* pppwhere p’ will be called the pressure correction.

15-5 The Pressure and Velocity Corrections(2)

Similarly, we can get

scorrectionvelocitywvu

cwww

bvvv

auuu

,,

),4(

),4(

),4(

*

*

*


If (1a)-(2a), we have

)5( eEpnbnbee Appuaua

At this point, we shall boldly decide to drop the term ∑anbu' nb from eq.(5) and the result is

e

ee

EpeeEpee

a

Adwhere

ppduorppua

)6(


Eq.(6) will be called the velocity-correction formula, which can also be written as

)7(

)7(,

)7(

*

*

*

cppdww

bppdvvSimilarly

appduu

tpttt

npnnn

Epeee

15-6 The Pressure-Correction Equation(1)

The discretization eq. of continuity eq:

)8(0

0

0

yxwwxzvv

zyuut

zyx

z

w

y

v

x

u

t

btsn

wepp


Substituting eqs(7) into eq(8), we can obtain

)10(

,,

,,

,,

)9(

****

**0

yxwwxzvv

zyuut

zyxb

aaaaaaa

zydazyda

zydazyda

zydazydawhere

bpapapapapapapa

tbns

ewpp

BTSNWEp

bbBttT

ssSnnN

wwWeeE

BBTTSSNNWWEEpp


If b is zero, it means that the stared velocity, in conjunction with the available value of (ρp

0- ρp), do satisfy the continuity equation and no pressure correction is needed. The term b this represents a “ mass source”, which the pressure corrections must be annihilate.

15-7 The SIMPLE Algorithm(1)

SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations.

Sequence of operations:1. Guess the pressure p*.2. Solve the momentum eqs, such as (2a)~(2c), to obtain u*,v*,

w*.3. Solve p' eq.4. Calculate p=p*+p '5. Calculate u,v,w from (7a)~(7c)6. Solve other Φ‘s.7. Treat the corrected pressure p as a new guessed pressure p*,

return to step (2) and repeat the whole procedure until a converged solution is obtain.


Discussion of the Pressure-Correction Equation

1. If expressions such as anbu'nb were retained, they would have to be expressed in terms of the pressure corrections and the velocity corrections at the neighbors of unb. The omission of the ∑anbu'nb term enables us to cast the p equation in the same form as the general Φ equation, and to adopt a sequential, one-variable-at-a-time, solution procedure.


2. The words semi-implicit in the name SMPLE have been used to acknowledge the omission of the term ∑anbu'nb . This term represents on indirect or implicit influence of the pressure correction on velocity. Pressure corrections at nearby locations can alter the neighboring velocities and thus a velocity correction at the point under consideration. We do not include this influence and thus work with a scheme that is only partially, and not totally, implicit.


3. The omission of any term, would of course, be unacceptable if it meant that the ultimate solution would not be true solution of the discretized forms of the momentum and continuity equation. It also happens that the converged solution given by SIMPLE does not contain any error resulting from the omission of ∑anbu'nb


4. The mass source b serves as a useful indicator of the convergence of the fluid-flow solution. The iterations should be continued until the value of b everywhere becomes sufficiently small.

5. The pressure-correction can be seen to be merely an intermediate algorithm that leads us to the correct pressure field, but has no direct effect on the final solution.


6. The pressure-correction equation is prone to divergence unless some under-relaxation is used. A generally successful practice can be described as follows: we under-relax u*,v*,w* while solving the momentum equations (with a relaxation factor α=0.5). Also, we employ

8.0,* pp ppp It is not implied that these values are the optimum ones or will even produce divergence for all problems. The optimum relaxation factor values are usually problem-dependent.


Boundary Conditions for the Pressure-Correction Equation

1. There are two kinds of conditions at a boundary. Either the pressure at the boundary is given (and the velocity is unknown) or the velocity component normal to the boundary is specified.

2. Given pressure at the boundary:

If the guessed pressure field p* is arranged such that at a boundary p*=pgiven, then the value of p' at the boundary will be zero.


3. Given normal velocity at the boundary:

As shown in Fig.1, the velocity ue is given. It the derivation of the p' equation for the c.v. shown, the flow rate across the boundary face should not expressed in terms of ue

* and a correction, but in terms of ue itself

ue((given)p

N

S

E

Fig.1

bpapapapapa

uuu

NNSSWWEEpp

e

,*

Then, p'E will not appear or aE will be zero in the p' equation. Thus no information about p'E will be needed.

*Note:


The Relative Nature of Pressure1. Since no boundary pressure is specified and al

l the boundary coefficients such as aE will be zero, the p' equation is left without any means of establishing the absolute value of p'. The coefficients of the p' equation are such that aP=∑a

nb ; this means that p' and p' +c (c is an arbitrary constant) would both satisfy the p' equation.


2. Only difference in the pressure are meaningful (p=p*+p'), and these are not altered by an arbitrary constant to the p' field. Pressure is then a relative variable, not a absolute one.


3. In many problems, the value of the absolute pressure is much larger than the local differences in pressure that are encountered. If the absolute values of pressure were for p, round-off errors would arise in calculating differences like pp-pE. It is, therefore, best to set p=0 as a reference value at a suitable point and to calculate all other values of p as pressures relation to start from p'=0 as guess for all point, so that the solution for p' does not acquire a large absolute value.

15-8 A Revised Algorithm: SIMPLER(1)

Motivation1. SIMPLER stands for SIMPLE Revised.

2. In most cases, it is reasonable to suppose that the pressure-correction equation does a fairly good job of correcting the velocities, but a rather poor job of correcting the pressure.


The Pressure Equation1. The momentum equation is first written as

)4(

getcan we, eq. continuity into (3c)~(3a) ngsubstituti

)3(ˆ

)3(ˆcan write wesimilarly,

)3(ˆ

)2(ˆ

asten first writ ucity pseudovelo a define weNow,

)1(

e

bpapapapapapapa

cppdww

bppdvv

appduu

a

buau

ppda

buau

BBTTSSNNWWEEpp

Tpttt

Npnnn

Epeee

e

nbnbe

Epee

nbnbe


2. Although the pressure eq and p' eq are almost identical, there is one major difference:

No approximations have been introduced in the derivation of the pressure equation. Thus, if a correct velocity field were used to calculate the pseudo-velocities, the pressure equation would at once give the correct pressure.


The SIMPLER Algorithm

The revised algorithm consists of solving the pressure equation to obtain the pressure field and solving the pressure-correction equation only to correct the velocities.


The sequence of operations can be stated as:

1. Start with a guessed velocity field.

2. Calculate pseudo-velocity û,v,ŵ from eqs (1),(2).

3. Solve pressure equation by eq.(4) to obtain the pressure field.

4. Treat this pressure field as p*, solve momentum eq to obtain u*,v*,w* solve p' eq.

5. Solve p' eq

6. Correct the velocity field (ue=u*+de(p'p-p'e),etc), but do not correct the pressure.

7. Solve other Φ‘s if necessary.

8. Return to step 2 and repeat until convergence.


Discussion1. In general, since the pressure-correction

equation produces reasonable velocity fields, and the pressure equation works out the direct consequence of a given velocity field, convergence to the final solution should be much faster.


2. In SIMPLE, a guessed pressure field plays an important role. On the other hand, SIMPLER does not use guessed pressures, but extracts a pressure field from a given velocity field.

3. Because of the close similarity between the pressure equation and the pressure-correction equation, the discussion in previous section about boundary conditions for p eq. is also relevant to the pressure equation.


4. Although SIMPLER has been found to give faster convergence than SIMPLE, it should be recognized that one iterations of SIMPLER involves more computational effort. Since SIMPLER requires fewer iterations for convergence, the additional effort per iteration is more than compensated by the overall saving of effort

15-9 The SIMPLEC Algorithm

SIMPLEC stands for SIMPLE-Consistent. It follows the same steps as the SIMPLE

algorithm. The u-velocity correction equation of

SIMPLEC is given by

eEpnbnbee

nbe

eeeeee

Appuauanote

aa

Adppdu

:

,

15-10 Convergence Criterion

apΦp=∑anbΦnb+b

Residual R=∑anbΦnb+b- apΦp

Obviously, when the discretization eq is satisfied, R will be zero. A suitable convergence criterion is to require that the largest value of |R| be less than a certain small number.

chapter 12 finite-volume (control-volume) method-introduction

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