CHEMICAL ENGINEERING DEPARTMENT

INDIAN INSTITUTE OF TECHNOLOGY MADRAS

CFDT PROJECT Steady Flow of an Incompressible Fluid in a

Lid Driven Cavity

Jason Ryan Picardo CH11D026

4/5/2012

Course: Computational Fluid Dynamics Tech

Course No: CH 6020

2

Table of Contents

Declaration

1 Introduction ........................................................................................................................ 4

2 The SIMPLE Method ......................................................................................................... 5

3 Discretization of the Governing Equations ........................................................................ 6

3.1 Momentum Equations ................................................................................................. 7

3.2 Pressure Correction Equation ...................................................................................... 9

3.3 Boundary Conditions................................................................................................. 11

3.4 Under-Relaxation of Pressure and Velocity Fields ................................................... 12

4 Results and Discussion ..................................................................................................... 13

5 Comparison of Results with Published Work................................................................... 16

6 Conclusions ...................................................................................................................... 17

Appendix .......................................................................................................................... 18

References ........................................................................................................................ 24

3

Declaration

The work presented in this report is the original work done by Jason Ryan Picardo, Roll

No. CH11D026 of the Chemical Engineering Department, IIT Madras towards partial

requirement for the completion of the course CH6020- Computational Fluid Dynamics Tech

in the Jan-May semester of 2012.

Jason Ryan Picardo

4

1 Introduction

In this report a Computational Fluid Dynamics (CFD) solution to the 'Lid Driven Cavity'

problem is presented. The steady flow of an incompressible fluid is considered. The SIMPLE

method (Semi-Implicit Method for Pressure-Linked Equations), originally devised by

Patankar and Spalding,

1 to deal with incompressible steady flows, is adopted.

The 'Lid Driven Cavity' problem is a standard problem that is often used to test new CFD

codes and solution methods. The problem domain has a simple geometry with easily

implementable boundary conditions. These characteristics have made it a popular benchmark

problem. A standard two-dimensional case consists of a rectangular cavity with its top lid

moving at a constant velocity (Vw). A schematic of the problem is depicted in Figure 1. The

cavity is assumed to be infinite in the z-direction. The motion of the top lid creates a 2-D

circulatory flow pattern inside the cavity. While the flow remains steady and laminar at

moderate Reynolds numbers (Re), it does exhibit vortices, which grow with Re, at the lower

sharp corners.

Figure 1. Schematic of the Lid Driven Cavity problem. Vw is the constant velocity at which

the wall moves. The cavity has a height H in the y direction and a width W in the x-direction.

The special case of H = W corresponds to a square cavity.

The SIMPLE method is briefly described in Section 2. The discretized equations and the

pressure correction equation is presented in Section 3. Results for various Re are presented

and discussed in Section 4. In order to validate the present calculations, some results are

compared with the work of Ghia and co-workers 2

in Section 0. Several obstacles were faced

during the development of the code and the implementation of the SIMPLE method. This

lead to new learning and insights which are discussed in the Conclusions.

5

2 The SIMPLE Method

The SIMPLE method, first formulated by Patankar and Spalding,

1 has proved to be an

effective method for the solution of CFD problems involving incompressible fluids,

especially when the steady solution is of interest. The method is described in detail in a text

by Patankar. 3

It is essentially a pressure correction type method. For any given 'guessed'

pressure field, the velocity fields are readily computed using the Navier-Stokes (N-S)

Momentum Equations (density is a constant for incompressible fluids). However, the

computed velocity fields will not, in general, satisfy continuity unless the guessed pressure

field is 'correct'. In a pressure correction method, a pressure correction equation is developed

which applies a correction to the initial guessed pressure such that the corresponding velocity

fields satisfy the Continuity Equation. In the SIMPLE method, the pressure correction

equation is only approximate, as shall be seen in Section 3.2. Moreover, the N-S Momentum

equations are nonlinear and highly coupled. Thus an iterative method is necessary to arrive at

the final solution. A flow chart of the major steps in the SIMPLE method is presented in

Figure 2.

Figure 2. Flow Chart of the SIMPLE Method

Before the SIMPLE method can be implemented, the N-S Momentum Equations must be

suitably discretized. The exact form of the pressure correction equation will depend on the

manner in which the Momentum equations are discretized. Suitable boundary conditions

must also be specified with a means to implement them in the CFD code. This is dealt with in

the next section.

6

3 Discretization of the Governing Equations

In this work a staggered grid is used, following Patankar.

3 This prevents the occurrence of

checkerboard oscillations. The pressure is calculated at the (i , j) nodes while the x and y

velocity components are calculated at (i+1/2 , j) and (i, j+1/2) respectively. Thus the grid for

the x- velocity is staggered to the right (in the positive x direction) and the grid for the y-

velocity is staggered to the top ( in the positive y direction) with respect to the pressure grid.

A typical staggered grid is depicted in Figure 3.

Figure 3. A typical staggered grid. The pressure is calculated at the (i , j) nodes while the x

and y velocity components are calculated at (i+1/2 , j) and (i, j+1/2) respectively.

The non-dimensional continuity equation for an incompressible fluid and the N-S

Momentum Equations in conservative form are presented below.

0

y

v

H

W

x

u

(1)

2

22

2

21

y

u

H

W

x

u

Rex

P

y

vu

H

W

x

uu

t

u

(2)

2

22

2

21

y

v

H

W

x

v

Rey

P

y

vv

H

W

x

uv

t

v

(3)

7

Here, the following characteristic scales are used to non-dimensionalize the equations.

WuReuP

u

WtHyWxVvu

ccc

c

cccwcc

2

(4)

The SIMPLE method elucidated by Patankar

3 is based on a Finite Volume Method (FVM)

discretization of the equations. Here we adopt a Finite Difference (FDM) discretization

scheme involving a fully implicit forward difference in time and central differences in space

for both the convective and diffusive terms. A similar scheme is presented in the discussion

of the SIMPLE method by Anderson.

4 The use of the conservative form of the N-S equations

leads to a final set of discretized equations which are similar to those obtained using the

FVM.

4

In this problem we are interested in obtaining the steady flow solution. In spite of this we

use the unsteady N-S Momentum Equations. It is much easier to converge to the solution

from an arbitrary initial guess of the pressure and velocity fields if the unsteady equations are

used. It must be noted that the SIMPLE method does not result in time accurate solutions.

Here, the discrete time steps should be interpreted as sequential iteration steps which

ultimately converge to the steady flow solution. The time step value (t) is simply a

parameter which affects the speed of convergence and the stability of the iterations. High

values can cause the iterations to become unstable. Anderson uses a fully explicit time

differencing scheme which makes the calculations of the velocity fields simple. However,

this makes the method susceptible to instability at relatively large values of the time step. In

this work, a fully implicit scheme is adopted. While this requires the simultaneous solution of

the equations at all grid points, it provides greater stability to the method.

3.1 Momentum Equations

Based on the previous discussion, a finite difference (FDM) based discretization of the X-

Momentum equation yields

2

1

121

1

21

1

121

2

2

1

21

1

21

1

23

1121

1

121

1

21

1

23

1

21

1

21

221

2

2

y

uuu

H

W

x

uuu

Re

x

PP

y

uvsuvn

H

W

x

uuuu

t

uu

n

j,i

n

j,i

n

j,i

n

j,i

n

j,i

n

j,i

n,

j,i

n,

j,ij,i

nn,

j,i

nn,

j,i

nn,

j,i

nn,n,

j,i

n

j,i

(5)

8

After collecting terms Eq. (5) results in a linear equation for the velocity at (i+1/2 , j) in

terms of the neighbouring nodes.

2

2

1

2

1

2

1

2

1

2

221

21211

21211

211

2

2

1

121

2

2

1

1212

211

21

2

231

232

2

2

1

21

n,

j,i

n,

j,in,

n,

j,i

n,

j,in,

n,

j,i

n,

j,i

n,

j,i

n,

n

j,i

n,

n

j,i

n,

j,in

j,i

n,

j,in

j,i

n

j,i

vvvs

vvvn

with

t

u

x

PP

H

W

yReH

W

y

vsu

H

W

yReH

W

y

vnu

xRex

uu

xRex

uu

yReH

W

xRetu

(6)

Similarly, the Y- Momentum equation may be discretized to give the following equation

for the node (i, j+1/2)

2

2

1

2

1

2

1

2

1

2

221

21121

21121

211

2

1

211

2

1

211

2

2

211

21

2

2

231

232

2

2

1

21

n,

j,i

n,

j,in,

n,

j,i

n,

j,in,

n,

j,i

n,

j,i

n,

j,i

n,

n

j,i

n,

n

j,i

n,

j,in

j,i

n,

j,in

j,i

n

j,i

uuub

uuuf

with

t

v

y

PP

xRex

ubv

xRex

ufv

H

W

yReH

W

y

vv

H

W

yReH

W

y

vv

yReH

W

xRetv

(7)

Clearly these equations must be solved simultaneously for all nodes. This may be done

using a Gauss-Siedel iterative solution method with under-relaxation. In the calculations an

under-relaxation factor of 0.7 is used. Thus given an initial pressure and velocity field

(denoted by superscripted asterisk) new velocity fields may be calculated which satisfy the

N-S momentum equations. This takes care of the first step in the iterative loop in Figure 2.

9

However, the computed velocity field will not, in general, satisfy the continuity equation

since the pressure field was an initial guess value. The pressure correction equation is

necessary to rectify this by incorporating suitable corrections to the pressure and velocity

fields.

3.2 Pressure Correction Equation

In this section the discretized pressure correction equation is developed which enforces

conservation of mass on the calculated velocity fields. The Equation of Continuity (Mass

Conservation Equation) for an incompressible fluid is given in Eq. (1). The discretized

equation for the node (i , j) is presented below. Central differences are used for both

derivatives.

021212121

y

v~v~

H

W

x

u~u~ j,ij,ij,ij,i

(8)

u~ and v~ are velocity fields which satisfy continuity. The velocity fields computed from the

momentum equations, in the previous step, will not satisfy the continuity equation. A velocity

correction ( u and v ) is defined such that the corrected velocity field satisfies the continuity

Eq. (8).

vvv~uuu~ (9)

The velocity correction ( u and v ) may be calculated from a corresponding correction to

the pressure field ( P ).

PPP~

(10)

If these two corrections are made to satisfy the N-S Momentum Equations (Eqs. (6(7) then

the continuity satisfying velocity field ( u~ and v~ ) and the corrected pressure field ( P~

) will

satisfy the governing equations of fluid motion for incompressible flow. This would then be

the solution to the problem. However, to simplify calculations, only the direct effect of the

pressure correction on the velocity at any node is accounted for i.e. the terms involving

velocities at neighbouring nodes are dropped.

10

2

2

2

1

21

1

21

221

yReH

W

xRetA

with

yA

PPv

xA

PPu

j,ij,i

j,i

j,ij,i

j,i

(11)

This simplification will lead to a velocity and pressure field which satisfy mass

conservation but not the N-S momentum equations. This is step two in the SIMPLE

procedure (Figure 2). These fields are then taken as the new initial guesses and the process is

repeated. Finally, the iterations will converge to a solution that satisfies the governing

equations. The simplification in the pressure correction equation does not affect the final

steady flow solution.

The velocity correction is substituted into Eq. (8) along with the velocity field calculated in

Step 1 of Figure 2 from Eqs. (6)) and ((7) to yield a discretized equation for the pressure

correction.

2

2

2

21212121

2

21

2

21

2121

2

22

221

11

1122

yReH

W

xRetA

y

vv

H

W

x

uu)j,i(q

with

)j,i(qH

W

yAP

H

W

yAP

xAP

xAP

H

W

yAxAP

j,ij,ij,ij,i

j,ij,i

j,ij,ij,i

(12)

This is a pressure correction formula which has the form of a Poisson equation for the

corrected pressure. q(i , j) is a source term which is in fact the residuals from the continuity

equation of the incorrect velocity field at every node. When q(i , j) is identically zero, it

implies that the field computed at Step 1 of Figure 2 satisfies continuity and is in fact the

solution. This is reflected by the fact that the pressure corrections calculated from Eq. (12)

will be zero. Hence the values of q(i , j) serve as a useful criteria to check for convergence.

In this work the root mean square value ( ) of q(i , j) across all nodes is used as the

convergence criterion (Eq. (13)).

11

y

vv

H

W

x

uu)j,i(q

with

)j,i(q

j,ij,ij,ij,i

j,i

21212121

2

(13)

As discussed in Section 3, the solution at various 'time steps' does not correspond to any

physically relevant transient solution due to the approximation in the pressure correction

equation. They should be interpreted solely as iteration steps which converge to the steady

solution.

4 Nevertheless, if the velocity correction is not relaxed, then the SIMPLE procedure

converges to the final solution via a series of intermediate velocity fields which satisfy

continuity. As Patankar

3 points out, this is advantageous in terms of numerical stability and

for the calculation of scalar transport variables (heat, concentration etc.) as they can be

calculated based on velocity fields which satisfy mass conservation.

3.3 Boundary Conditions

Boundary conditions must be specified for the momentum equations (Eqs. (6) and (7)) and

the pressure correction equation (Eq. (12)). The velocity components are zero everywhere

along the stationary walls from the no-slip condition and the kinematic condition. Along the

top moving wall the y-velocity is zero while the x-velocity is taken to be that of the wall (1 in

dimensionless variable form- see Eq.(4) and Figure 1).

A homogeneous Neumann condition is applied for the derivative of the pressure correction

field. This is justifiable since the normal components of velocity are zero at all the walls

according to the kinematic condition.

3,

4

100

100

,yaty

P

,xatx

P

(14)

12

3.4 Under-Relaxation of Pressure and Velocity Fields

The iterations may become divergent for many problems. To ensure convergence an under-

relaxation factor may be used while updating the pressure and velocity fields.

vvv

uuu

PPP

v

nn,

u

nn,

P

n,n,

11

11

1

(15)

The value must be between zero and one for under-relaxation. A value of zero corresponds

to no correction. Patankar recommends values of P = 0.8 and u = v = 0.5. Anderson uses a

more conservative estimate of P = 0.1 for solution of the Couette flow problem. In this

work, a value of P = 0.1 , 0.05 is used depending on Re and the number of grid points.

Smaller values of relaxation are required at higher Re and a larger number of grid points. u

and v are both set to one. Thus the intermediate velocity field at various iterations satisfies

mass conservation.

The t value also acts as an additional relaxation factor and can be adjusted to affect

convergence rate and stability. For the Couette problem Anderson uses a small value of t =

0.001 since an explicit time differencing scheme is adopted in that work. Here, a larger value

of t = 0.01 is used as a fully implicit scheme of time discretization has been adopted

(Section 3.1).

13

4 Results and Discussion

A code is developed in MATLAB based on the previously outlined solution method. A

sample code for a particular case of Re = 100 and a 80 X 80 grid is given in the Appendix.

Two simulations were carried out for the case of a square cavity. The first at a Re = 100 and

the second at Re = 400. An 80 x 80 grid was used for the first case and a 81 X 81 grid was

used for the second. These computations took about 16 and 24 hrs respectively. The

convergence history of the computations are presented in Figure 4. Here, is the

convergence criterion defined in Eq. (13).

Figure 4. Convergence history for the calculations for the cases of (a) Re = 100 on 80 X 80

grid and (b) Re = 400 on 81 X 81 grid.

Streamline plots of the velocity field for both cases are presented in Figure 5. The

circulatory flow pattern is clearly visible. The vortices formed at the lower edges of the cavity

due to boundary layer separation grow as the Reynolds number is increased. This is due to

the dominance of inertial forces over the viscous forces. In both cases, the top lid has a

positive scaled velocity of 1 (in the positive x direction).

In Figure 6, contours of the x and y velocity components are displayed for the case of Re =

400 on 81 X 81 grid. The velocity components switch direction across the cavity, as is

consistent with a circulatory flow. Contours of pressure are displayed for the same case in

Figure 7. The stagnation point is clearly visible at the top right corner of the cavity where the

pressure is highest. The vortex regions at the two lower corners of the cavity are at a

relatively higher pressure. This is characteristic of boundary layer separation situations.

14

Figure 5. Streamline plots for (a) Re = 100 on 81 X 81 grid and (b) Re = 400 on 81 X 81

grid.

Figure 6. Contours of the x and y components of velocity for the case of Re = 400 on 81 X

81 grid.

Figure 7. Contours of the pressure for the case of Re = 400 on 81 X 81 grid.

15

First, an interesting variant of the present problem is briefly considered. That is the case

when both the top and bottom plates are moved at constant velocities, in the same direction.

The resultant flow field will have some relation to the flow field inside slugs/drops moving

within a second, immiscible, continuous fluid. Both fluids flow in a channel of diameter

comparable to the slug/drop diameter. Such a scenario arises in two phase flow in

microchannels and has several technological applications. In particular, mass transfer in

liquid-liquid extraction in micro channels is enhanced due to the circulatory flow inside the

slugs/drops.

5 Figure 8 displays the streamlines for a first computation at Re = 5 and a grid of

40 X 40. This low Re is typical for the case of a microchannel flow.

Figure 8. Streamlines for the two moving wall problem- reminiscent of circulation in

drops/slugs. Re = 5 with a grid of 40 X 40.

The flow field is symmetric as expected, with two counter rotating central vortices. No

secondary vortices are formed since Re is very small. The velocity contours are presented in

Figure 9 below. Here, both the plates are moved in the negative x direction uniformly.

Figure 9. Contours of the x and y components of velocity for the the two moving wall

problem. Re = 5 with a grid of 40 X 40.

16

5 Comparison of Results with Published Work

In this section, the calculations done using the SIMPLE scheme are compared with the

results of Ghia and co-workers.

2 The authors use a vorticity-stream function formulation of

the 2D N-S Momentum Equations with a coupled strongly implicit multigrid method. Tabular

results of velocity are presented which allows for convenient comparison. Figure 10 displays

the comparison of results for Re = 100 and Re = 400. Ghia and co-workers

2 use a

comparatively fine mesh of 129 X 129 grids. The results of the present work compare well

for the case of Re = 100 , considering the coarse 81 X 81 grid used. However, at the higher

Re = 400 a discrepancy is observed in the results. Clearly, a finer mesh must be used. This

will require a longer computation time.

Figure 10. Comparison of results of the present work with the results of Ghia and co-

workers.

2 Comparisons are depicted for Re = 100 on 80 X 80 grid and Re = 400 on 81 X 81

grid with the corresponding results of Ghia et. al. on a 129 X 129 grid.

17

6 Conclusions

The SIMPLE method is successfully applied to solve the 'Lid Driven Cavity' problem. A

finite difference scheme with a fully implicit time discretization is used. The method is

implemented as a code in MATLAB. The calculations were found to converge

monotonically. The results calculated for Re = 100, 400 on relatively coarse grids are

compared with the published work of Ghia and co-workers

2. The comparison reveals a

satisfactory agreement for lower Reynolds numbers (Re = 100). While a finer mesh must be

used for better results, particularly at high Re, the SIMPLE approach is found to be adequate

to solve this steady, incompressible fluid flow problem.

An important point is the use of the unsteady N-S Momentum Equations as opposed to the

steady state equations. In prior attempts at developing a SIMPLE based CFD solution, the

steady state equations were also used without much success by the author. It is found that the

SIMPLE method with the steady N-S Momentum equations does not converge for problems

with Re > 50 and/or with a grid larger than 20 X 20. Even when the iterations converge, a

very small relaxation factor of P = 0.001. This is to be contrasted with the value of 0.1/0.05

which is successfully used in the calculations with the unsteady equations. It seems that the

SIMPLE method based on the unsteady equations is much more suited to the calculation of

the final steady flow field via a series of iterations beginning with an arbitrary initial guess.

The values of the variables at successive iterations do not correspond to a transient solution

since the SIMPLE pressure correction equation is only approximate. Hence the time step, t

is a factor similar to a relaxation factor which controls the rate of convergence and the

stability of the iterations. It must be noted that large values of t result in a set of discretized

equations which tend to the steady state discretized equations. Thus, based on the previous

discussion, small values of t should be used to ensure convergence.

An interesting variant of the problem occurs when both the top and bottom plates are

moved at constant velocities, in the same direction. The resultant flow field is reminiscent of

the flow field inside slugs/drops moving within a second, immiscible, continuous fluid. This

flow field arises in microchannel flows and has much technological relevance.

Since the computation time is quite large, this report presents only the first results of the

application of the SIMPLE method to the Lid Driven Cavity problem. However, there are

several aspects which entail further calculations. Firstly, simulations should be run for higher

grid sizes in an attempt to establish grid independency. These results should then be

compared with published work.

18

7 Appendix

A sample MATLAB code for a particular case of Re = 100 and a 80 X 80 grid.

% This code solves the moving lid problem. The steady state solution is % computed for the case of incompressible viscous flow via the SIMPLE % algorithm of Patankar and Spalding. % Author Details % Jason Ryan Picardo- CH11D026 % Chemical Engg Dept- IIT Madras %TIME IMPLICIT clear all

%Problem Parameters Uw=1; %velocity of moving lid (m/s) W=1; %Width of cavity (m) H=1; %Height of cavity (m) Re=100; %Reynolds No %Convergence array convergence=zeros(1,2000); k=0;

%Grid Properties np=80; nu=np-1; nv=np-1;

% x is alligned along the width (index i) origin at bottom left dx=1/(np-1); % y is alligned along the height (index i) origin at bottom left dy=1/(np-1); %artificial time dt=0.01; %Initialization Pstr=zeros(np,np); Ustr=zeros(np,nu); Vstr=zeros(nv,np); %A(j,i) Ustr(np,:)=Uw; qsum=1;

%--------------------------------------------------------------------------

---------- %SIMPLE LOOP

while sqrt(qsum)>10^-3

k=k+1; U=zeros(np,nu); U(np,:)=Uw; V=zeros(nv,np); relax=0.7; %Calculation of U field from Pstr and Ustr and Vstr U(np,:)=Uw; % top boundary moving, all other boundaries are at 0

residual1=1; residual2=1;

19

while residual1>10^(-8) if(residual1>residual2&&residual1>100) display('U cal failed') display(residual1) break; end residual2=residual1;

for i=2:nu-1 for j=2:np-1

Vnstr=(Vstr(j,i+1)+Vstr(j,i))/2; Vsstr=(Vstr(j-1,i+1)+Vstr(j-1,i))/2; U(j,i)=(1-

relax)*U(j,i)+relax*((1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*(Ustr(j,i

)/dt+U(j,i+1)*(-Ustr(j,i+1)/(2*dx)+(1/Re)/dx^2)+... U(j,i-1)*(Ustr(j,i-1)/(2*dx)+(1/Re)/dx^2)+U(j+1,i)*(-

Vnstr/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)+U(j-

1,i)*(Vsstr/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)... -1/(dx)*(Pstr(j,i+1)-Pstr(j,i)))); end end

delF=ones(1,(nu-2)*(np-2)); c=1; for i=2:nu-1 for j=2:np-1 Vnstr=(Vstr(j,i+1)+Vstr(j,i))/2; Vsstr=(Vstr(j-1,i+1)+Vstr(j-1,i))/2; delF(c)= U(j,i)-

(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*(Ustr(j,i)/dt+U(j,i+1)*(-

Ustr(j,i+1)/(2*dx)+(1/Re)/dx^2)+... U(j,i-1)*(Ustr(j,i-1)/(2*dx)+(1/Re)/dx^2)+U(j+1,i)*(-

Vnstr/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)+U(j-

1,i)*(Vsstr/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)... -1/(dx)*(Pstr(j,i+1)-Pstr(j,i))); c=c+1; end end

sum=0; for c=1:length(delF) sum=sum+delF(c)^2; end

residual1=sqrt(sum/length(delF)); end %display('U calculation') %display(residual);

%Calculation of V field from Pstr and Ustr and Vstr residual1=1; residual2=1; relax=0.7;

while residual1>10^(-8)

if(residual1>residual2&&residual1>100)

20

display('V cal failed') display(residual1) break; end

residual2=residual1;

for i=2:np-1 for j=2:nv-1

Ufstr=(Ustr(j+1,i)+Ustr(j,i))/2; Ubstr=(Ustr(j+1,i-1)+Ustr(j,i-1))/2; V(j,i)=(1-

relax)*V(j,i)+relax*((1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*(Vstr(j,i

)/dt+V(j,i+1)*(-Ufstr/(2*dx)+(1/Re)/dx^2)+... V(j,i-1)*(Ubstr/(2*dx)+(1/Re)/dx^2)+V(j+1,i)*(-

Vstr(j+1,i)/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)+V(j-1,i)*(Vstr(j-

1,i)/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)... -1/((dy*H/W))*(Pstr(j+1,i)-Pstr(j,i)))); end end

delF=ones(1,(nv-2)*(np-2)); c=1; for i=2:np-1 for j=2:nv-1 Ufstr=(Ustr(j+1,i)+Ustr(j,i))/2; Ubstr=(Ustr(j+1,i-1)+Ustr(j,i-1))/2; delF(c)= V(j,i)-

(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*(Vstr(j,i)/dt+V(j,i+1)*(-

Ufstr/(2*dx)+(1/Re)/dx^2)+... V(j,i-1)*(Ubstr/(2*dx)+(1/Re)/dx^2)+V(j+1,i)*(-

Vstr(j+1,i)/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)+V(j-1,i)*(Vstr(j-

1,i)/(2*(dy*H/W))+(1/Re)/(dy*H/W)^2)... -1/((dy*H/W))*(Pstr(j+1,i)-Pstr(j,i))); c=c+1; end end

sum=0; for c=1:length(delF) sum=sum+delF(c)^2; end

residual1=sqrt(sum/length(delF)); end %display('V calculation') %display(residual)

%Updating velocity fields Ustr=U; Vstr=V;

%calculation of error in continuity from Ustr and Vstr Q=zeros(np,np); qsum=0; for i=2:np-1 for j=2:np-1 Q(j,i)=(Ustr(j,i)-Ustr(j,i-1))/dx+(Vstr(j,i)-Vstr(j-1,i))/(dy*H/W); qsum=qsum+Q(j,i)^2;

21

end end display(k) sqrt(qsum) convergence(k)=sqrt(qsum);

if(k>2&&convergence(k)>1.5*convergence(k-1)) display('Iterations are diverging') break; end

if(sqrt(qsum)<10^-3) display('convergence achieved to 10^-3 in continuity') break end save('Re_V4_100_np_80_con

v_o3','Ustr','Vstr','Pstr','convergence','k','Re','np','dx','dy') %Calculation of P correction from Pstr and Ustr and Vstr Pcor=zeros(np,np); relax=0.7; residual1=1; residual2=1;

while residual1>10^(-8)

if(residual1>residual2&&residual1>100) display('Pcor cal failed') display(residual1) break; end

residual2=residual1;

A=-(1/(dx))*(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2)); B=-(1/((dy*H/W)))*(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2));

for i=2:np-1 for j=2:np-1

if(i==2&&j==2) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+B/(dy*H/W)*

Pcor(i,j+1)+Q(i,j))); elseif(i==2&&j==np-1) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+B/(dy*H/W)*

Pcor(i,j-1)+Q(i,j))); elseif(i==np-1&&j==2) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i-

1,j)+B/(dy*H/W)*Pcor(i,j+1)+Q(i,j))); elseif(i==np-1&&j==np-1) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+B/(dy*H/W))*(+A/dx*Pcor(i-

1,j)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j))); elseif(i==2) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i+1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j))); elseif(i==np-1)

22

Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i-1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j))); elseif(j==2) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(2*A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pcor

(i-1,j)... +B/(dy*H/W)*Pcor(i,j+1)+Q(i,j))); elseif(j==np-1) Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(2*A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pcor

(i-1,j)... +B/(dy*H/W)*Pcor(i,j-1)+Q(i,j))); else Pcor(i,j)=(1-

relax)*Pcor(i,j)+(relax)*(1/(2*A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pc

or(i-1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j))); end end end

delF=ones(1,(np-2)*(np-2)); c=1; for i=2:np-1 for j=2:np-1

if(i==2&&j==2) delF(c)=Pcor(i,j)-

1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+B/(dy*H/W)*Pcor(i,j+1)+Q(i,j)); elseif(i==2&&j==np-1) delF(c)=Pcor(i,j)-

1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); elseif(i==np-1&&j==2) delF(c)=Pcor(i,j)-1/(A/dx+B/(dy*H/W))*(A/dx*Pcor(i-

1,j)+B/(dy*H/W)*Pcor(i,j+1)+Q(i,j)); elseif(i==np-1&&j==np-1) delF(c)=Pcor(i,j)-1/(A/dx+B/(dy*H/W))*(+A/dx*Pcor(i-

1,j)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); elseif(i==2) delF(c)=Pcor(i,j)-

1/(A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i+1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); elseif(i==np-1) delF(c)=Pcor(i,j)-1/(A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i-

1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); elseif(j==2) delF(c)=Pcor(i,j)-

1/(2*A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pcor(i-1,j)... +B/(dy*H/W)*Pcor(i,j+1)+Q(i,j)); elseif(j==np-1) delF(c)=Pcor(i,j)-

1/(2*A/dx+B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pcor(i-1,j)... +B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); else delF(c)=Pcor(i,j)-

1/(2*A/dx+2*B/(dy*H/W))*(A/dx*Pcor(i+1,j)+A/dx*Pcor(i-1,j)... +B/(dy*H/W)*Pcor(i,j+1)+B/(dy*H/W)*Pcor(i,j-1)+Q(i,j)); end c=c+1;

23

end end

sum=0; for c=1:length(delF) sum=sum+delF(c)^2; end

residual1=sqrt(sum/length(delF));

end

Pcor(1,:)=Pcor(2,:); Pcor(np,:)=Pcor(np-1,:); Pcor(:,np)=Pcor(:,np-1); Pcor(:,1)=Pcor(:,2); %display('Pcor calculation') %display(residual); Pcor;

Ucor=zeros(np,nu); Vcor=zeros(nv,np);

for i=2:nu-1 for j=2:np-1

Ucor(j,i)=(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*... (-1/(dx)*(Pcor(j,i+1)-Pcor(j,i))); end end

for i=2:np-1 for j=2:nv-1

Vcor(j,i)=(1/(1/dt+2*(1/Re)/dx^2+2*(1/Re)/(dy*H/W)^2))*... (-1/((dy*H/W))*(Pcor(j+1,i)-Pcor(j,i))); end end alp=0.1; al=1; Ustr=Ustr+al*Ucor; Vstr=Vstr+al*Vcor; Pstr=Pstr+alp*Pcor; %calculation of error in continuity from Ustr and Vstr-test % Q=zeros(np,np); % qsum=0; % for i=2:np-1 % for j=2:np-1 % Q(j,i)=(Ustr(j,i)-Ustr(j,i-1))/dx+(Vstr(j,i)-Vstr(j-1,i))/dy; % qsum=qsum+Q(j,i)^2; % end % end % qsum; end

save('Re_V4_100_np_80_conv_o3','Ustr','Vstr','Pstr','convergence','k','Re',

'np','dx','dy')

24

8 References

1. Patankar, S. V.; Spalding, D. B. A calculation procedure for Heat, Mass and Momentum

Transfer in Three-Dimensional Parabolic Flows. Int. J. Heat Mass Transfer 1972, 15,

1787-1806.

2. Ghia, U.; Ghia, K. N.; Shin, C. T. High-Re Solutions for Incompressible Flow using the

Navier-Stokes Equations and a Multigrid Method. J. Comput. Phys. 1982, 48, 387-411.

3. Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere: New York, 1980.

4. Anderson, J. D. J. Computational Fluid Dynamics. The basics with applications.;

McGraw-Hill: New York, 1995.

5. Fries, D. M.; Voitl, T.; Rohr, P. R. Liquid Extraction of Vanillin in Rectangular

Microreactors. Chem. Eng. Technol. 2008, 31 (8), 1182-1187.