Finite Volume Method

Chapter 6 Finite Volume Method

6.1 Transformation of the Navier-Stokes Equa- tions in the Finite Volume Method

The Navier Stokes equation @u i

is fulfilled within each mesh element (finite volume U) in the integral sense. For that it is integrated over the volume U :

Application of the Gauss theorem results in Z

pE e i E grad u i ndS E (6.3) @t

u i dUC u i uE E ndS D F i dU

US

The same procedure applied to the continuity equation gives

uE E ndS D 0

Figure 6.1: Staggered arrangement of finite volumes.

6.2 Sample

Let us consider the two dimensional transport equation without the diffusion term

In the integral form this equation reads

pE e i ndS E (6.6) @t

u i dUC u i uE E ndS D

We use the staggered grid (Fig. 13.2). The pressure is stored at the volume centers. The u x velocity is stored at the centers of vertical faces, whereas the velocity u y component at centers of horizontal faces. The x- equation is satisfied for volumes displaced in x-direction, whereas the y-equation for these displaced in y-direction. Below we consider approximations of different terms:

6.2.1 Pressure and unsteady terms

Source (pressure) term for x-equation: p Z

Q 1 D p Ee 1 p e S e p w S w D p i C1j p ij

Unsteady term for x-equation:

Pressure term for y-equation: p Z

Q 2 D p Ee 2 p n S n p s S s D p ij C1 p ij

Unsteady term for y-equation:

2 u yij

nC1

u yij

u y dU

@t

6.2.2 Convection term o f the x-equation

The integrand in convection term u i EuEn is represented in the table 6.1. Table 6.1: EnEu and u i at different sides. x-equation

The necessary velocities are approximated as shown in the table 6.2. Herewith the convection term has the form presented in the table 6.3.

6.2.3 Convection term o f the y-equation

The integrand in convection term u i EuEn is represented in the table 6.4. The necessary velocities are approximated as shown in the table 6.5. Herewith the convection term has the form presented in the table 6.6.

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Table 6.2: Velocities at different sides. x-equation velocity

Table 6.3: Convection flux. x-equation side

flux

C east 2 4 .u xij u xi C1j /

Table 6.4: E nE u and u i at different sides. y-equation

Table 6.5: Velocities at different sides. y-equation velocity

6.2.4 X-equation approximation

2 u xij

nC1

u xij

Table 6.6: Convection flux. y-equation side

6.2.5 Y-equation approximation

6.3 Explicit scheme

The next task is to specify the upper index in X and Y equations. If the index is n we get fully explicit scheme which is similar to that derived above for finite difference method

2 u xij

nC1

u xij

.u 2

xij Cu xi C1j /

.u xij Cu xi 1j / C

.u n xij Cu xij C1 /.u yij Cu yi C1j / .u xij Cu xij 1 /.u yij 1 Cu yi C1j 1 /

nn

p ij

i C1j

2 u yij

nC1

u yij

.u n yij Cu yi C1j /.u xij Cu xij C1 / .u yij Cu yi 1j /.u xi 1j Cu xi 1j C1 /C

nn

(6.12) The Poisson equation for pressure is derived in the same manner as above

p ij

ij C1

for finite difference method. For that the equations (6.11) is differentiated on x, whereas the equation (6.12) is differentiated on y. Then both results

are summed under assumptions that both u n

nC1

and u ij are divergence free:

This equation is coupled with equations (6.11) and (6.12). The explicit scheme has advantage that the solution at the time instant n C 1 is ex- plicitly expressed through the solution at time instant n. The solution of linear algebraic equations which is the most laborious numerical procedure is necessary only for the solution of the Poisson equation. The momentum

and u yij are computed then from simple algebraic formula (6.11) and (6.12). The big disadvantage of the explicit method is the limitation forced by the Courant

equations (6.11) and (6.12) are solved explicitly. Velocities u nC1

nC1

xij

numerical stability. This disadvantage can be overcome within the implicit schemes.

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6.4 Implicit scheme

If the index is n C 1 we get implicit scheme

(6.14) The Poisson equation for pressure is derived in the same manner as above

for finite difference method. For that the equations (6.13) is differentiated on x, whereas the equation (6.14) is differentiated on y. Then both results

are summed under assumptions that both u nC1

and u ij are divergence free:

The resulting Poisson equation can not be solved because both the r.h.s. (velocities) and the l.h.s (pressure) depend on n C 1. The term on r.h.s. cannot be computed until the computation of velocity field at time n C 1 is completed and vice versa. Other problem is that the equations (6.13) and (6.14) are non linear.

6.5 Iterative procedure for implicit scheme

To solve the nonlinear system and the whole system of equations we use the iterative procedure. Let m be an iteration number. The nonlinear term is represented in form:

.m/ @u .m 1/

@u

@x j

@x j

The velocity u j is taken from the previous iteration .m 1/. The system (6.11) and (6.12) is rewritten in the form

q .m/ xi 1j u xi 1j Cq xij u xij Cq xi C1j u xi C1j Cq xij 1 u xij 1 Cq xij C1 u xij C1 C (6.16)

q .m/ yi 1j u yi 1j Cq yij u yij Cq yi C1j u yi C1j Cq yij 1 u yij 1 Cq yij C1 u yij C1 C (6.17)

.m/

.m/

.m/

.m/

C .m/ p

.m/

p ij

ij C1

yij

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Dividing the equations (6.16) by q xij and (6.17) by q yij we obtain

a .m/ xi 1j u xi 1j C u xij C a xi C1j u xi C1j C a xij 1 u xij 1 C a xij C1 u xij C1 C (6.18)

a .m/ yi 1j u yi 1j C u yij C a yi C1j u yi C1j C a yij 1 u yij 1 C a yij C1 u yij C1 C (6.19)

where a x;ykl Dq x;ykl =q x;yij and R x;ykl Dr x;ykl =q x;yij . In what follows we use the operator form of equations (6.18) and (6.19):

u D Au C Bp C C

6.6 Pressure correction method

The velocity field satisfying the equation (6.20) is the solution of the lin- earized Navier Stokes equation. It doesn’t fulfill the continuity equation. The iterative solution satisfying the whole system of equations is computed using the pressure correction method.

The iterative scheme consists of following steps. First, the intermediate so- lution is calculated with pressure taken from the previous iteration:

u .m 1/ D Au C Bp CC (6.21) The velocity and pressure corrections

(6.22) are computed within next steps. Substitution of (6.22) into (6.20) gives

.u 0 Cu / D A.u Cu / C B.p Cp /CC (6.23) Since u satisfies the equation (6.21) the equation for the velocity correction

0 0 .m 1/

reads

(6.24) The velocity at the iteration .m/ is

0 0 u 0 D Au C Bp

0 u 0 Du C Au C Bp

.m/

It should satisfy the continuity equation ru .mC1/ D0 (6.26)

what results in

0 ru 0 D rBp rAu

6.7 SIMPLE method

A very popular pressure correction method is the SIMPLE method. The main assumption of this method is neglect of the term rAu 0 in (6.27) and (6.24):

Figure 6.2: SIMPLE algorithm.

The equation (6.28) is the Poisson equation for the pressure correction p 0 . The solution algorithm is summarized in Fig. 6.2. Let the solution is known

at time slice n, the solution at the next time instant n C 1 is seeking. In the first iteration all quantities are taken from the previous time instant

u n .mD1/ .mD1/

Du x;yij ;p ij

Dp ij

x;yij

At each time instant the inner loop iterations are performed until residuals are getting smaller than some threshold

x;yij

u x;yij j<" u ;

As soon as the inner loop iterations are converged the solution at time in- stant n C 1 is equaling to the solution from the last iteration and the next time instant is computed. The structure of the inner loop is shown in Fig. 6.2.

6.7.1 Pressure correction equation

Let us consider the pressure correction equation (6.28) in details.This equa- tion is solved for the control volume shown in Fig. 6.3. The divergency

@f x C @f operator rf D y

@y is represented within Finite Volume Method as follows Z

Therefore, the right hand side of the equation (6.28) takes the form Z

ru dUD u n dS D .u xij u xi 1j C u yij u yij 1 (6.30)

As follows from (6.18) and (6.19) the operator Bp 0 has the following values at faces of the control volume

0 0 0 0 0 .Bp 0 / yij D p ij C1 p ij yij ; .Bp / yij 1 D p ij p ij 1 yij 1 : (6.31)

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Substitution of (6.30) and (6.31) into (6.28) results in

i C1j p i C1j C 0 0 ˇ 0 ij p ij C ˇ i 1j p i 1j C ˇ ij C1 p ij C1 C ˇ ij 1 p ij 1 D c ij (6.33) where

6.7.2 Summary of the SIMPLE algorithm

We introduce one dimensional numbering instead of two dimensional one according to the rule

˛ D .i 1/N y C j

Let the solution at the time instant n be known. The task is to find the solution at the time n C 1. At each time instant the guess solution is taken from the previous time instant:

u .1/ Du n ;p .1/ x;y˛ x;y˛ ˛ Dp n ˛

The solution is found within the next substeps:

i) Calculation of the auxiliary velocity u x;y˛ from two independent sys- tems of linear algebraic equations:

a x˛ N y u x˛ N y C u x˛ C a x˛CN y u x˛CN y C a x˛ 1 u x˛ 1 C a x˛C1 u x˛C1 C

p .m 1/

C .m 1/

x˛ DR x˛

˛CN y

ii) Calculation of the pressure correction p 0 ˛ from the system of linear algebraic equations:

˛CN y p ˛CN y C ˇ ˛ p

˛ C ˇ ˛N y p 0 ˛N y C ˇ ˛C1 p ˛C1 C ˇ ˛1 p ˛1 Dc ˛

iii) Calculation of the velocity correction u 0 x;y˛ :

iv) Correction of the velocity and pressure: u .m/

x;y˛ Du x;y˛ C u x;y˛ ; p ˛ Dp ˛

0 .m/

.m 1/

Figure 6.3: Control volume used for the pressure correction equation.

max .m 1/ ju

.m/

u x;y˛ j<" u ;

If these conditions are not fulfilled then

x;y˛ ;p ˛

and go to the step i). Otherwise the calculation at the time moment n C1 is completed

u .m/

nC1

x;y˛ Du x;y˛ ;p ˛ Dp ˛

.m/

nC1

and one proceeds to the next time instant n C 2.

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