Two Component Incompressible Flow

in the Headspace of a Filter DryerMIE 604 - Term Project

Aditya Dodda

April 30, 2015

1

Contents

Nomenclature 3

1 Problem statement 4

2 Governing equations 5

3 Numerical values for inputs 6

4 Meshing 6

5 Boundary conditions 7

6 Numerical methods 7

6.1 Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6.2 Vorticity and Stream Function . . . . . . . . . . . . . . . . . . . . . . . . . 10

6.3 Species mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Validation 11

7.1 Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

7.2 mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

8 Grid dependency and computation cost 14

9 Results 16

9.1 Problem statement as is . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

9.2 Problem statement with a minor modification . . . . . . . . . . . . . . . . 17

10 Conclusion 17

2

Nomenclature

Dimensionless

Dimensionless vorticity mass fraction of component 1 Dimensionless time Dimensionless stream function P Dimensionless pressure Pe Peclet number Re Reynolds number U Dimensionless velocity in x-direction V Dimensionless Velocity in y-direction X Dimensionless x Y Dimensionless y

Subscripts

i Index in y-direction j Index in x-direction

Variables

Viscosity Pa s Density kg/m3

D Diffusion coefficient m2/s

Lx Width of headspace m

Ly Height of headspace m

p Pressure Pa

t time s

u Velocity in x-direction m/s

v Velocity in y-direction m/s

v1 Inlet velocity of component 1 m/s

v2 Inlet velocity of component 2 m/s

v3 Outlet velocity m/s

x x m

y y m

3

1 Problem statement

u = 0v = v2 = 0

lpurge

u = 0v = v3

y= 0

loutlet

u = 0 , v = v1, = 1

Lx

Ly

At all other boundariesu = 0v = 0

z= 0

P

z= 0 at all boundaries

z is thedirection perpendicular

to the boundary

Figure 1: Domain showing the boundary conditions.

As seen in figure 1, in this problem there is component 1 that is entering at the bottom of

the domain, denoted by a thick line, at a velocity of v1. The mass fraction of component

1 is denoted by . Species 2 is entering from the dashed line section in the domain at

a velocity of v2. As there are only two components, the mass fraction of component 2 is

1 . The negative sign is used to denote the direction of velocity. A mixture of the twois exiting from the dotted line section at a velocity of v3. As the flow is incompressible,

v1Lx + v2lpurge = v3loutlet.

4

Initial conditions

u(t, x) = 0, v(t, x) = 1, (t, x) = 0

2 Governing equations

u

t+u2

x+uv

y= 1

p

x+

1

O2u (1)

v

t+uv

x+v2

y= 1

p

y+

1

O2v (2)

t+ u

x+ v

y= DO2 (3)

Scaled and dimensionless form of equations

U =u

v3, V =

uv

v3, P =

p

v23, =

tLxv3, s =

LxLy, Re =

Lxv3

, Pe =

v3LxD

U

+U2

X+ s

UV

Y= P

X+

1

Re

(2U

X2+ s2

2U

Y 2

)(4)

V

+UV

X+ s

V 2

Y= P

Y+

1

Re

(2V

X2+ s2

2V

Y 2

)(5)

+ U

X+ sV

Y=

1

Pe

(2

X2+ s2

2

Y 2

)(6)

Divergence in scaled form

U

X+ s

V

Y= 0

5

3 Numerical values for inputs

Most inputs are provided to replicate the values seen in experiments. For parameters that

are not known via experiments simple existing models are used.

From experiment,

v3 = 0.06 m/s, Lx = 0.08 m, Ly = 0.16m

From wolframalpha,

= 0.05 kg/m3, = 2 105 Pa s

Diffusion coefficient D = 1 105 m2/s using elementary theory of diffusion coefficient ingases.

From these,

Re = 12, P e = 480, s = 0.5

4 Meshing

The meshing is over a square domain of sides with length unit 1. This is possible as s takes

care of the stretching in the y direction. lpurge and loutlet are both 0.1 units.

A uniform mesh is used in this problem. To capture the boundary values correctly, the

minimum grid spacing dX = dY = 0.1. Consequently, halving the value of dX also captures

the boundary values accurately. This process can be repeated till the desired fineness is

achieved.

All the values are at cell centers and the cell is the same for U, V, P & .

Index i & j represent the y & x directions, respectively.

6

5 Boundary conditions

The boundary conditions are extracted from an excel file. The can handle Neumann and

Dirichlet boundary conditions.

Dirichlet

Suppose the boundary condition is u(t, x = 0, y) = 1.2. Then the excel file would have 0

for the type of boundary condition and 1.2 for the value.

When solving the equations, x = 0 corresponds to j = 1/2. When face values are calculated

for advection, the Dirichlet condition is directly used. However, for diffusion calculations,

value at i, j = 0 are necessary. This a ghost point.

Therefore, ui,0 = 2ui,1/2 ui,1 where, ui,1/2 comes directly from the boundary condition.

Neumann

Suppose the boundary condition isu

x(t, x = 0, y) = 2.3. Then the excel file would have 1

for the type of boundary condition and 2.3 for the value.

Unlike the case of Dirichlet condition, ui,0 is easily obtained as ui,1 2.3 dx. Then ui,1/2is calculated as

ui,0 + ui,12

.

6 Numerical methods

6.1 Navier-Stokes

The Navier-Stokes equations (4 and 5) are solved in a similar manner as they are of similar

form.

7

Advection

The advection terms are solved using TVD with superbee as the flux limiter. The can take

in other forms of flux limiters as well. Two flux limiters tested for advection were minmod

and superbee. Of these, superbee performed a little better.

U

= U

2

X sUV

Y(7)

U2

X=

U2i,j+1/2 U2i,j1/2dX

(8)

U2i,j+1/2 = Ui,j+1/2[UUWi,j+1/2

(ri,j+1/2

) (UUWi,j+1/2 Ui,j+1/2

)](9)

UUWi,j+1/2 =

Ui,j, if Ui,j+1/2 0

Ui,j+1, if Ui,j+1/2 < 0

(10)

ri,j+1/2 =Ui,j Ui,j1Ui,j+1 Ui,j (11)

(r) = max(0,min (1, 2r) ,min (2, r)) (12)

Similar procedure is used for UVY

. All these calculations are done for interior faces in the

domain. The flux at faces on the boundary are evaluated from boundary conditions.

Analogous algorithm is used for the advection terms in eq 5.

The code for this is in advection navier stokes tvd.m.

Diffusion

The output of the advection solver is fed to the diffusion solver (diffusion solver.m).

The diffusion terms were calculated using the ADI algorithm for Crank-Nicolson method.

Tridiagonal matrix algorithm was used in the process of solving matrices. (This implementation

is identical to that used in the homework assignments.)

8

U Ud

=1

2Re

(2U

X2+2U

Y 2

)

U Ud

=1

2Re

(2U

X2+2U

Y 2

)Analogous steps for V

Pressure solver (2P

X2+ s2

2P

Y 2

)=

1

d

(U

X+ s

V

Y

)

U

X=Ui,j+1/2 Ui,j1/2

dX

V

Y=V i+1/2,j V i1/2,j

dY

Conjugate gradient method is used to solve the pressure equation. The code corresponding

to this section is in pressure solver.m.

Pressure correction

Un+1 = U dt PX

V n+1 = V s dtPY

P

X=Pi,j+1/2 Pi,j1/2

dX

P

Y=Pi+1/2,j Pi1/2,j

dY

9

6.2 Vorticity and Stream Function

Vorticity and the stream function are solved for using Un+1. The code for this is in

vorticity.m and stream function.m.

=V

X sU

Y(2

X2+ s2

2

Y 2

)=

6.3 Species mass balance

Advection

= U

X sV

Y(13)

i,j+1/2 =[UWi,j+1/2

(ri,j+1/2

) (UWi,j+1/2 i,j+1/2

)](14)

UWi,j+1/2 =

i,j, if U

n+1i,j+1/2 0

i,j+1, if Un+1i,j+1/2 < 0

(15)

ri,j+1/2 =i,j i,j1i,j+1 i,j (16)

(r) = max(0,min (1, 2r) ,min (2, r)) (17)

U

X= Un+1i,j

i,j+1/2 i,j1/2dX

(18)

Diffusion

=

1

Pe

(2

X2+ s2

2

Y 2

)This is identical to the diffusion part of the Navier-Stokes equation. The same function

(diffusion solver.m) is used to solve the diffusion portion by passing the appropriate coefficients

to the solver.

10

7 Validation

7.1 Navier-Stokes

The code was validated for a a lid-driven cavity problem in an unit square for Re = 20.

The output was matched to the output of the code mentioned in http://math.mit.edu/

~stoopn/18.086/code/mit18086_navierstokes.pdf.

X0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

U

-0.25

-0.2

-0.15

-0.1

-0.05

0 Re = 20, N = 40, CFL = 0.8

SourceMy code

Y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V

-0.01

0

0.01

0.02

0.03SourceMy code

Figure 2: Validation of Navier-Stokes solver by comparing to output from code providedin the link mentioned. Re = 20 with 40 x 40 grid. U and V ar plotted at Y = 0.5 and X= 0.5, respectively.

11

7.2 mass balance

Advection

To validate the advection of in x and y directions, the boundary conditions were set

to mimic a case of step wave in each of the directions respectively. As seen in figure 3,

the numerical solution matches the analytical solution very well (RMS error = 0.08). The

error is reduced with further discretization. Also, in case of the problem being dealt with,

the presence of diffusion will reduce the need to have a very high level of discretization to

capture a wave front accurately.

The governing equation for the case of x-direction,

= U

X

V (,X, Y ) = 0, U(,X, Y ) = 1

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t = 0.31667, rms = 0.079057

AnalyticalNumerical

(a) Validation of advection of inx-direction

y0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t = 0.31667, rms = 0.079057

AnalyticalNumerical

(b) Validation of advection of iny-direction

Figure 3: Validation of advection of

12

Diffusion

For validating diffusion of , the following problem was setup. Only the validation in

x-direction is discussed here. The validation is y-direction is identical. s = 1 in both cases.

( = 0, X, Y ) = sin(X), (,X = 0, Y ) = 0, (,X = 1, Y ) = 0,

Y(,X, Y = 0) = 0,

Y(,X, Y = 1) = 0

V (,X, Y ) = 0, U(,X, Y ) = 0

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t = 0.48, rms = 8.3849e-05

AnalyticalNumerical

(a) Validation of diffusion of in thex-direction

y0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t = 0.48, rms = 8.3849e-05

AnalyticalNumerical

(b) Validation of diffusion of in they-direction

Figure 4: Validation of diffusion of

The governing equation is,

= 4

The solution to this problem is,

(,X, Y ) = exp(pi2) sin(piX)

13

As seen in figure 4, there is a very good (RMS error = 8.4e-5) agreement between the

analytical and numerical solution.

8 Grid dependency and computation cost

Y0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(X

=0.5,

Y)

0

0.2

0.4

0.6

0.8

1

0.10.050.0250.0125

X0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(X

,Y=0

.5)

0

0.2

0.4

0.6

0.8

1

0.10.050.0250.0125

Figure 5: Grid dependency of solution. The plot legends show the grid spacing in eachcase. CFL = 1.5 was used without stability issues.

14

Table 1: Details of grid dependency test results

dXdA Simulation time (s) 1

N

(X = 0.5, Y ) 1

N

(X, Y = 0.5)

0.1 0.7152 0.8635 0.8301 0.80310.05 0.7230 3.0642 0.8335 0.8054

0.025 0.7206 15.5306 0.8293 0.80280.0125 0.7178 221.3552 0.8265 0.7999

To look into grid dependency, the problem statement was solved at dX = 0.1, 0.05, 0.025

and 0.0125. This translates to 10x10, 20x20, 40x40 and 80x80 grid point mesh. After

solving, (X = 0.5, Y ) & (X, Y = 0.5) were plotted. These plots are shown in figure 5.

4 5 6 7 8 9-2

0

2

4

6

8

10

12

14

16

18log(t) vs. log(N2)Reference line: Slope = 2

Figure 6: Log plot of simulation time vs number of grid points N2

In table 1, the first column is the grid spacing. The second column, gives the sum of in

the domain. The third column is the time taken for simulating 20 s of real time. Columns

4 and 5 correspond to plots in figure 5. They give the average represented by the lines.

15

From figure 6, it is clear that the log of simulation time increase linearly with number of

grid points initially and then moves to a slope = 2. Most the computation time is taken up

in solving the Laplace equation for pressure using conjugate gradient method. The second

highest time consuming routine is the tridiagonal matrix algorithm. The reason for this is

due to shear number of calls made to the function as the diffusion part needs to be solved

thrice, once each for U, V & .

From the mean value of (X = 0.5, Y ) & (X, Y = 0.5) shown in table 1, it is easy to

conclude that the benefit of increasing fineness of the grid is lost after dX = 0.025. For

this reason, the results in the next section will correspond to dX = 0.025.

9 Results

9.1 Problem statement as is

As the Re is much smaller than Pe, the velocity field reaches steady state much faster than

.

From figures 7, 8, 9, 10 and 11 (look in appendix) component 1 almost completely fills

up the headspace in 60 seconds. This is very much of interest as drying is a very slow

process. The fact that the headspace fills up so fast would mean that the data from the

mass spectrometer is reasonably reliable. From, experiments the mass spectrometer takes

about 30 seconds to start reading component 1 after the boiling is visible. This about what

is seen from the simulations. The t in the titles of the images is real time.

16

9.2 Problem statement with a minor modification

The only difference in this section is that at t = 30 s, at Y = 0 will cease to enter the

domain. This means (,X, Y = 0) = 0. In addition, v1 = 0 and v3 = v2.

From figures 12, 13, 14, 15 and 16 (look in appendix) after t = 30 s, it takes a long

time to purge the headspace of component 1. This is in agreement with what is observed

experimentally.

10 Conclusion

The code is robust and simulate the experiment procedure. There is definite qualitative

match between the model results and experiments. The implementation is robust enough

to capture changes in boundary conditions.

17

Appendix

Figures for problem statement

U, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 0.1

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

5

10

15

20

25

30

35

0

0

0.10.2

, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 0.1

0 0.5 10

0.5

1

1.5

2

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Figure 7: Solution at t = 0.1

18

U, t: 15

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 15

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 15

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.10.20.30.40.50.60.70.80.90.950.99

, t: 15

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 15

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 8: Solution at t = 15

19

U, t: 30

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 30

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 30

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.20.30.40.50.60.7

0.80.8

0.90.9

0.950.99

, t: 30

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 30

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 9: Solution at t = 30

20

U, t: 45

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 45

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 45

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.50.60.70.80.90.9

50.99

, t: 45

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 45

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 10: Solution at t = 45

21

U, t: 60

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 60

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 60

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.70.80.90.950.9

9

, t: 60

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 60

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 11: Solution at t = 60

22

Figures for modified problem statement

U, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 0.1

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

5

10

15

20

25

30

35

0

0

0.10.2

, t: 0.1

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 0.1

0 0.5 10

0.5

1

1.5

2

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Figure 12: Solution at t = 0.1

23

U, t: 15

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 15

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 15

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.10.20.30.40.50.60.70.80.90.950.99

, t: 15

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 15

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 13: Solution at t = 15

24

U, t: 30

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 30

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 30

0 0.5 10

0.5

1

1.5

2

-1

-0.8

-0.6

-0.4

-0.2

00.20.30.40.50.60.7

0.80.8

0.90.9

0.950.99

, t: 30

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 30

0 0.5 10

0.5

1

1.5

210-3

-10

-5

0

5

Figure 14: Solution at t = 30

25

U, t: 45

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 45

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 45

0 0.5 10

0.5

1

1.5

210-4

-6

-4

-2

0

2

4

6

0.40.50.60.70.8

0.90.95

0.99

, t: 45

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 45

0 0.5 10

0.5

1

1.5

210-6

-20

-15

-10

-5

0

5

Figure 15: Solution at t = 45

26

U, t: 60

0 0.5 10

0.5

1

1.5

2

0

0.5

1

1.5V, t: 60

0 0.5 10

0.5

1

1.5

2

-0.2

0

0.2

0.4

0.6

0.8

1

P, t: 60

0 0.5 10

0.5

1

1.5

210-4

-6

-4

-2

0

2

4

6

0.40.50.6 0.70.8

0.9

0.95

0.99

, t: 60

0 0.5 10

0.5

1

1.5

2

0

0.2

0.4

0.6

0.8

1

, t: 60

0 0.5 10

0.5

1

1.5

210-6

-20

-15

-10

-5

0

5

Figure 16: Solution at t = 60

27

main.m

%clear; %close all; clc;

%% Collecting boundariesfilename = 'boundaries actual.xlsx';% filename = 'boundaries.xlsx';sheets = {'u','v','phi'};

sides = {'north', 'east','south','west'};

Bound = boundaries(filename, sheets, sides);for gridSep = 0.025%[0.1 0.05 0.025 0.0125]

%% Gridx edge = (0:gridSep:1)';y edge = x edge;dx = diff(x edge);dy = diff(y edge);

x = (x edge(2:end) + x edge(1:end1))/2;y = (y edge(2:end) + y edge(1:end1))/2;

[X,Y] = meshgrid(x,y);[dX,dY] = meshgrid(dx,dy);

Ny = size(X,1);Nx = size(X,2);

s = 0.5;% s = Lx/Ly

%% Boundary values for solvingBoundVals = boundary values(Bound,sheets,sides,x,y,Nx,Ny);

%%CFL = 1.5;

COMPUTE STREAM FUNCTION = 1;VISUALIZE = 1;TYPE = 'superbee';COMPARE DISCRETIZATION = 0;

[maxV, minV] = maximum velocity(BoundVals);

D = 1e5;Lx = 0.08;rho = 0.05;mu = 2e5;

28

vOut = 6e2;tfReal = 60;

% InitializePhi = 0.0*ones(size(X));U = 0.0*ones(size(X));V = 1.0*ones(size(X));Ap = matrix poisson eqn(dX(1,1),dY(1,1),Nx,Ny,s);A sparse = sparse(Ap);

if abs(maxV) > 2*eps && D > 0

Pe = vOut*Lx/D;Re = (Lx*vOut*rho)/mu;tau = Lx/vOut;

dt = CFL*dx(1)/abs(maxV);

g = (1/Pe)*(dt/2);gu = (1/Re)*(dt/2);gv = (1/Re)*(dt/2);

tf = tfReal/tau;

NS COMPUTE = 1;Phi COMPUTE = 1;Phi ADVECTION = 1;ADVECTION = 1;

elseif D > 0

tau = Lx2/D;

tf = tfReal/tau;dt = tf/100;

g = 1*(dt/2);

NS COMPUTE = 0;Phi COMPUTE = 1;Phi ADVECTION = 0;ADVECTION = 0;

elseif D

g = 0;

NS COMPUTE = 0;Phi COMPUTE = 1;Phi ADVECTION = 1;ADVECTION = 1;

end

Nt = floor(tf/dt)+1;Ntsim = Nt;

plotSteps = 20;tmat = 0:tf/plotSteps:tf;timePoints = interp1(dt*(0:Ntsim),1:Ntsim+1,tmat,'nearest');timePoints = unique(timePoints);timePoints = timePoints(isnan(timePoints));

%% Grid point coefficients for diffusion dsolver[dxIn,dxMid,dxOut,dyIn,dyMid,dyOut] = ...

grid spacing matrixes(dX,dY,Nx,Ny);

%% Solving

Ustore = cell(Ntsim+1,1);Vstore = cell(Ntsim+1,1);Pstore = cell(Ntsim+1,1);Phistore = cell(Ntsim+1,1);

U0 = U;V0 = V;Phi0 = Phi;

Ustore{1} = U0;Vstore{1} = V0;Phistore{1} = Phi;Pstore{1} = zeros(size(X));

endPoint = Ntsim;

normCheckUV = 0;normCheckPhi = 0;changeBC = 0;startTime = tic;for ii = 1:Ntsim

storeIndx = ii+1;% Solution for NavierStokesif NS COMPUTE == 1

% TVD Advectionif ADVECTION > 0

if ii == 1

30

U n = U;V n = V;

elseif ii == 2U n = Ustore{storeIndx1};V n = Vstore{storeIndx1};

elseU n = Ustore{storeIndx1};V n = Vstore{storeIndx1};

end[Us, Vs] = advection navier stokes tvd(U n,V n, ...

dX,dY,dt,BoundVals,s,TYPE);else

%If no advectionUs = U0;Vs = V0;

end

% Implicit DiffusionUss = diffusion solver(Us,gu,gu*s2,dy,dx, ...

BoundVals.u.Xb,BoundVals.u.Yb, ...dxIn,dxMid,dxOut,dyIn,dyMid,dyOut);

Vss = diffusion solver(Vs,gv,gv*s2,dy,dx,...BoundVals.v.Xb,BoundVals.v.Yb, ...dxIn,dxMid,dxOut,dyIn,dyMid,dyOut);

% Poisson equation[P, B] = pressure solver(A sparse,Uss,Vss,dt,Nx,Ny, ...

dX,dY,BoundVals,s);

if mod(ii,20)clcdisp(['Time: ',num2str(ii*dt*tau),' s'])disp(['Simulation time: ',num2str(toc(startTime)),' s']);disp(['Divergence check: ',num2str(sum(B(:))/(Nx*Ny))])

end

% Pressure correction[Usss, Vsss] = pressure correction(P,Uss,Vss,dX,dY,dt,Nx,Ny,s);

Unorm = norm(Usss U0)/(Nx*Ny);Vnorm = norm(Vsss V0)/(Nx*Ny);

Ustore{storeIndx} = Usss;Vstore{storeIndx} = Vsss;Pstore{storeIndx} = P;

U0 = Ustore{storeIndx};V0 = Vstore{storeIndx};

if sum(abs(Unorm)/maxV) < 1e6 && sum(abs(Vnorm)/maxV) < 1e6normCheckUV = 1;

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elsenormCheckUV = 0;

endend

% Solution for phiif Phi COMPUTE == 1

if Phi ADVECTION == 1Phis = advection phi(Phi0,dX,dY,dt,BoundVals,U0,V0,s,TYPE);

elsePhis = Phi0;

end

[Phiss, Phiint] = ...diffusion solver(Phis,g,g*s2,dy,dx, ...BoundVals.phi.Xb,BoundVals.phi.Yb, ...dxIn,dxMid,dxOut,dyIn,dyMid,dyOut);

Phistore{ii} = Phiss;

phinorm = norm(Phiss Phi0)/(Nx*Ny);

Phi0 = Phistore{ii};

if sum(abs(phinorm)) < 1e6normCheckPhi = 1;

elsenormcheckPhi = 0;

end

if changeBC == 0 && ii*dt*tau > 30BoundVals.phi.Yb(:,4) = 0;BoundVals.phi.Yb(:,3) = 1;

BoundVals.v.Yb(:,4) = 0;BoundVals.v.Yb(BoundVals.v.Yb(:,2) > 0,2) = abs(min(BoundVals.v.Yb(:,2)));

changeBC = 1;end

end

if normCheckUV ==1 && normCheckPhi == 1%endPoint = ii;disp(['Simulation stopped at ', ...

num2str(dt*ii*tau),' seconds as tolerance was met.'])break;

end

32

endcomputetime = toc(startTime);disp(['Time to solve: ',num2str(computetime)])

if NS COMPUTE == 1disp(['Pe: ',num2str(Pe)])disp(['Re: ',num2str(Re)])

end%%

if COMPUTE STREAM FUNCTION == 1 && NS COMPUTE == 1Asf = matrix stream function(dX(1,1),dY(1,1),Nx,Ny,s);Asf sparse = sparse(Asf);vorticityStore = cell(Ntsim,1);streamFunction = cell(Ntsim,1);divergence = cell(Ntsim,1);streamStarttime = tic;for ii = 1:endPoint

[vortTmp, Ux, Uy, Vx, Vy] = ...vorticity(Ustore{ii},Vstore{ii},dX,dY,BoundVals,s);

SFtmp = stream function(Asf sparse,vortTmp,Nx,Ny);vorticityStore{ii} = vortTmp;streamFunction{ii} = SFtmp;divergence{ii} = Ux+Vy;

endstreamStoptime = toc(streamStarttime);disp(['Time to compute stream function : ', ...

num2str(streamStoptime)])end

%% Visualization

if VISUALIZE == 1h = figure(1);set(h,'Position',[400 100 1400 800])for ii = [2 300 600 900 1200]%[timePoints(timePoints < endPoint) endPoint]

colormap grayif NS COMPUTE == 1

subplot(2,3,1)contour(X,Y/s,Ustore{ii}, ...

maxV:2*maxV/10:maxV)caxis([0.3 1.5])title(['U, t: ',num2str(dt*ii*tau)])axis equalxlim([0 1])ylim([0 1]/s)colorbardrawnow

33

subplot(2,3,2)contour(X,Y/s,Vstore{ii}, ...

maxV:2*maxV/10:maxV)caxis([0.3 1.1])title(['V, t: ',num2str(dt*ii*tau)])axis equalxlim([0 1])ylim([0 1]/s)colorbardrawnow

subplot(2,3,4)contour(X,Y/s,Pstore{ii})title(['P, t: ',num2str(dt*ii*tau)])axis equalxlim([0 1])ylim([0 1]/s)colorbardrawnow

end

%if Phi COMPUTE ==1

subplot(2,3,5)contour(X,Y/s,Phistore{ii},[0.1:0.1:0.9 0.95 0.99 1], ...

'Showtext','on')zlim([0.1 1.1])caxis([0.01 1.01])title(['\phi, t: ',num2str(dt*ii*tau)])axis equalxlim([0 1])ylim([0 1]/s)colorbardrawnow

end

if COMPUTE STREAM FUNCTION == 1 && NS COMPUTE == 1subplot(2,3,3)contour(X,Y/s,streamFunction{ii},20)title(['\xi, t: ',num2str(dt*ii*tau)])axis equalxlim([0 1])ylim([0 1]/s)colorbardrawnow

end

saveas(h, ...['solution mod ',strrep(num2str(dt*ii*tau),'.',' ')],'fig')

saveas(h, ...['solution mod ',strrep(num2str(dt*ii*tau),'.',' ')],'epsc')

34

end

% close(h)end

%%

if COMPARE DISCRETIZATION == 1figure(100)

subplot(2,1,1)hold onplot(y,mean(Phistore{endPoint}(:,[Nx Nx+2]/2),2))ylim([0 1])

subplot(2,1,2)hold onplot(x,mean(Phistore{endPoint}([Ny Ny+2]/2,:)))ylim([0 1])

end

%%fileSave = ['GridSep',strrep(num2str(gridSep),'.',' '), ...

' ',strrep(char(datetime),':',' ')];save(fileSave)

end%% End

diffusion solver.m

function [Unew, Uint] = diffusion solver(U,gx,gy,dy,dx,Xb,Yb, ...dxIn,dxMid,dxOut,dyIn,dyMid,dyOut)

ny = size(U,1);nx = size(U,2);

Us = U;

U i j = U;U endp1 j = ((2*Yb(:,1)1)'.*U(end,:))+ ...

(Yb(:,1)'*(dy(end) 2) + 2).*Yb(:,2)';U ip1 j = [U(2:end, :); U endp1 j];U 0 j = ((2*Yb(:,3)1)'.*U(1,:))+ ...

(Yb(:,3)'*(dy(1) 2) + 2).*Yb(:,4)';U im1 j = [U 0 j; U(1:end1,:)];

RHSmat explicit y = gy*dyOut.*U ip1 j + ...(1gy*dyMid).*U i j + gy*dyIn.*U im1 j;

35

for ii = 1:ny % solving for Us corresponding to each block using TDMAa = gx*dxOut(ii,2:end)';b = (1+gx*dxMid(ii,1:end))';b(1) = b(1) gx*dxIn(ii,1)*(2*Xb(ii,3) 1);b(end) = b(end) gx*dxOut(ii,end)*(2*Xb(ii,1) 1);c = gx*dxIn(ii,1:end1)';d = RHSmat explicit y(ii,1:nx)';d(1) = d(1) + gx*dxIn(ii,1)*((dx(1) 2)*Xb(ii,3) + 2)*Xb(ii,4);d(end) = d(end) + gx*dxOut(ii,end)*((dx(end) 2)*Xb(ii,1) + 2)*Xb(ii,2);Us(ii,1:nx) = tdma(a,b,c,d);

end

Uint = Us;U = Us;

U i j = U;U i endp1 = ((2*Xb(:,1)1).*U(:,end)) + ...

(Xb(:,1)*(dx(end) 2) + 2).*Xb(:,2);U i jp1 = [U(:,2:end) U i endp1];U i 0 = ((2*Xb(:,3)1).*U(:,1)) + ...

(Xb(:,3)*(dx(1) 2) + 2).*Xb(:,4);U i jm1 = [U i 0 U(:,1:end1)];

RHSmat explicit x = gx*dxOut.*U i jp1 + ...(1gx*dxMid).*U i j + gx*dxIn.*U i jm1;

for jj = 1:nxa = gy*dyOut(2:end,jj);b = (1+gy*dyMid(1:end,jj));b(1) = b(1) gy*dyIn(1,jj)*(2*Yb(jj,3) 1);b(end) = b(end) gy*dyOut(end,jj)*(2*Yb(jj,1) 1);c = gy*dyIn(1:end1,jj);d = RHSmat explicit x(1:ny,jj);d(1) = d(1) + gy*dyIn(1,jj)*((dy(1) 2)*Yb(jj,3) + 2)*Yb(jj,4);d(end) = d(end) + gy*dyOut(end,jj)*((dy(end) 2)*Yb(jj,1) + 2)*Yb(jj,2);Us(1:ny,jj) = tdma(a,b,c,d);

end

Unew = Us;

end

advection navier stokes tvd.m

function [Us, Vs] = advection navier stokes tvd(U,V, ...dX,dY,dt,BoundVals,s,type)

u = BoundVals.u;v = BoundVals.v;

36

[,,Ubottom,Uleft] = Uedges(u,U,dX,dY);Umodx = [Uleft U];Umody = [Ubottom; U];

[,,Vbottom,Vleft] = Uedges(v,V,dX,dY);Vmodx = [Vleft V];Vmody = [Vbottom; V];

[, , Umidx, Umidy] = gradient vel(U,dX,dY,u);[, , Vmidx, Vmidy] = gradient vel(V,dX,dY,v);

% U

bUx = sign(Umidx(:,2:end1)) >= 0;UxUpwind = (bUx.*U(:,1:end1) + (1bUx).*U(:,2:end));

rUx = (Umodx(:,2:end1) Umodx(:,1:end2))./ ...(Umodx(:,3:end) Umodx(:,2:end1));

psiUx = fluxLimiter(rUx,type);

bUy = sign(Vmidy(2:end1,:)) >= 0;UyUpwind = (bUy.*U(1:end1,:) + (1bUy).*U(2:end,:));

rUy = (Umody(2:end1,:) Umody(1:end2,:))./ ...(Umody(3:end,:) Umody(2:end1,:));

psiUy = fluxLimiter(rUy,type);

FUx = [Umidx(:,1).2 Umidx(:,2:end1).*(UxUpwind ...psiUx.*(UxUpwind Umidx(:,2:end1))) Umidx(:,end).2];

FUy = [Vmidy(1,:).*Umidy(1,:); Vmidy(2:end1,:).*(UyUpwind ...psiUy.*(UyUpwind Umidy(2:end1,:))); Vmidy(end,:).*Umidy(end,:)];

Us = U dt*(diff(FUx,[],2)./dX + s*diff(FUy)./dY);

% V

bVx = sign(Umidx(:,2:end1)) >= 0;VxUpwind = (bVx.*V(:,1:end1) + (1bVx).*V(:,2:end));

rVx = (Vmodx(:,2:end1) Vmodx(:,1:end2))./ ...(Vmodx(:,3:end) Vmodx(:,2:end1));

psiVx = fluxLimiter(rVx,type);

bVy = sign(Vmidy(2:end1,:)) >= 0;VyUpwind = (bVy.*V(1:end1,:) + (1bVy).*V(2:end,:));

rVy = (Vmody(2:end1,:) Vmody(1:end2,:))./ ...(Vmody(3:end,:) Vmody(2:end1,:));

37

psiVy = fluxLimiter(rVy,type);

FVx = [Umidx(:,1).*Vmidx(:,1) Umidx(:,2:end1).*(VxUpwind ...psiVx.*(VxUpwind Vmidx(:,2:end1))) Umidx(:,end).*Vmidx(:,end)];

FVy = [Vmidy(1,:).2; Vmidy(2:end1,:).*(VyUpwind ...psiVy.*(VyUpwind Vmidy(2:end1,:))); Vmidy(end,:).2];

Vs = V dt*(diff(FVx,[],2)./dX + s*diff(FVy)./dY);

end

pressure solver.m

function [P, B] = pressure solver(A, U,V,dt,nx,ny,dX,dY,BoundVals,s)

[Ux, , , ] = gradient vel(U,dX,dY,BoundVals.u);[, Vy, , ] = gradient vel(V,dX,dY,BoundVals.v);

B = (1/dt)*(Ux + s*Vy);

b = reshape(B,[],1);

%% Conjugate gradient method

% Tolerance for conjugate gradient methodTol = 1e8;

% Maximum number of iterationsnIter = 2*nx*ny;

% Solving for pressure[Pvec, ] = conjugateGradient(A,b,nIter,Tol);

% Reshaping vector to matrixP = reshape(Pvec,ny,nx);end

pressure correction.m

function [Usss, Vsss] = pressure correction(P,Uss,Vss,dX,dY,dt,nx,ny,s)Px = zeros(ny,nx+1);Py = zeros(ny+1,nx);

Px(:,2:end) = P;Px(:,1:end1) = Px(:,1:end1) + P;Px(:,2:end1) = Px(:,2:end1)/2;

38

Py(2:end,:) = P;Py(1:end1,:) = Py(1:end1,:) + P;Py(2:end1,:) = Py(2:end1,:)/2;

Usss = Uss ...dt*(Px(:,2:end) Px(:,1:end1))./dX;

Vsss = Vss ...dt*s*(Py(2:end,:) Py(1:end1,:))./dY;

end

advection phi.m

function Phis = advection phi(Phi,dX,dY,dt,BoundVals,U,V,s,type)

phi = BoundVals.phi;u = BoundVals.u;v = BoundVals.v;

[,,Phibottom,Phileft] = Uedges(phi,Phi,dX,dY);Phimodx = [Phileft Phi];Phimody = [Phibottom; Phi];

[, , Umidx, ] = gradient vel(U,dX,dY,u);[, , , Vmidy] = gradient vel(V,dX,dY,v);[, , Phimidx, Phimidy] = gradient vel(Phi,dX,dY,phi);

bPhix = sign(Umidx(:,2:end1)) >= 0;PhixUpwind = (bPhix.*Phi(:,1:end1) + (1bPhix).*Phi(:,2:end));

rx = (Phimodx(:,2:end1) Phimodx(:,1:end2))./ ...(Phimodx(:,3:end) Phimodx(:,2:end1));

psix = fluxLimiter(rx,type);

bPhiy = sign(Vmidy(2:end1,:)) >= 0;PhiyUpwind = (bPhiy.*Phi(1:end1,:) + (1bPhiy).*Phi(2:end,:));

ry = (Phimody(2:end1,:) Phimody(1:end2,:))./ ...(Phimody(3:end,:) Phimody(2:end1,:));

psiy = fluxLimiter(ry,type);

Fx = [Phimidx(:,1) (PhixUpwind ...psix.*(PhixUpwind Phimidx(:,2:end1))) Phimidx(:,end)];

Fy = [Phimidy(1,:); (PhiyUpwind ...psiy.*(PhiyUpwind Phimidy(2:end1,:))); Phimidy(end,:)];

39

Phis = Phi dt*(U.*diff(Fx,[],2)./dX + s*V.*diff(Fy)./dY);

end

conjugateGradient.m

function [u, U] = conjugateGradient(A,b,nIter,Tol)

if nargin == 3Tol = 1e8;

end

U = cell(nIter+1,1);x = zeros(size(b));r = b A*x;p = r;cnt = 1;U{cnt} = x;rmserr = 10*Tol;while rmserr > Tol

z = A*p;rpr = r'*r;a = rpr/(p'*z);x = x + a*p;r1 = r a*z;

rmserr = norm(r)/sqrt(length(r));

b = (r1'*r1)/rpr;p = r1 + b*p;

r = r1;

cnt = cnt+1;

U{cnt} = x;end

if cnt == nIter && nIter = 1disp('Max iter reached')

end

U = U(1:cnt);u = x;end

vorticity.m

40

function varargout = vorticity(U,V,dX,dY,BoundVals,s)

[Ux, Uy, , ] = gradient vel(U,dX,dY,BoundVals.u);[Vx, Vy, , ] = gradient vel(V,dX,dY,BoundVals.v);

varargout{1} = Vx s*Uy;varargout{2} = Ux;varargout{3} = Uy;varargout{4} = Vx;varargout{5} = Vy;

end

stream function.m

function SF = stream function(A sparse,vort,Nx,Ny,s)

% Tolerance for conjugate gradient methodTol = 1e8;

% Maximum number of iterationsnIter = 2*Nx*Ny;

b = reshape(vort,[],1);

% startTime = tic;[SFvec, ] = conjugateGradient(A sparse,b,nIter,Tol);% stopTime = toc(startTime);

SF = reshape(SFvec,Nx,Ny);

end

matrix poisson eqn.m

function A = matrix poisson eqn(dx,dy,Nx,Ny,s)

d2x = 1/dx2;d2y = (s2)/dy2;

A1 = diag(ones(Ny,1)*2*(d2y+d2x),0) + diag(ones(Ny1,1)*d2y,1) + ...diag(ones(Ny1,1)*d2y,1);

A1(1,1) = A1(1,1) + d2y;A1(Ny,Ny) = A1(Ny,Ny) + d2y;

A3 = A1 + diag(ones(Ny,1)*d2x,0);

A2 = diag(ones(Ny,1)*d2x,0);

41

A = zeros(Ny*Nx,Ny*Nx);

for ii = 2:Nx1

A((ii1)*Ny + (1:Ny),(ii1)*Ny + (1:Ny)) = A1;A((ii1)*Ny + (1:Ny),(ii)*Ny + (1:Ny)) = A2;A((ii1)*Ny + (1:Ny),(ii2)*Ny + (1:Ny)) = A2;

end

A(1:Ny,1:Ny) = A3;A(1:Ny,Ny+(1:Ny)) = A2;

A((Nx1)*Ny+(1:Ny),(Nx1)*Ny+(1:Ny)) = A3;A((Nx1)*Ny+(1:Ny),(Nx2)*Ny+(1:Ny)) = A2;

% Making P(X(1,1),Y(1,1)) = 0A(1,1) = 2*A(1,1);

end

matrix stream function.m

function A = matrix stream function(dx,dy,Nx,Ny,s)

d2x = 1/dx2;d2y = s2/dy2;

A1 = diag(ones(Ny,1)*2*(d2y+d2x),0) + diag(ones(Ny1,1)*d2y,1) + ...diag(ones(Ny1,1)*d2y,1);

A1(1,1) = A1(1,1) d2y;A1(Ny,Ny) = A1(Ny,Ny) d2y;

A3 = A1 diag(ones(Ny,1)*d2x,0);

A2 = diag(ones(Ny,1)*d2x,0);

A = zeros(Ny*Nx,Ny*Nx);

for ii = 2:Nx1

A((ii1)*Ny + (1:Ny),(ii1)*Ny + (1:Ny)) = A1;A((ii1)*Ny + (1:Ny),(ii)*Ny + (1:Ny)) = A2;A((ii1)*Ny + (1:Ny),(ii2)*Ny + (1:Ny)) = A2;

end

A(1:Ny,1:Ny) = A3;

42

A(1:Ny,Ny+(1:Ny)) = A2;

A((Nx1)*Ny+(1:Ny),(Nx1)*Ny+(1:Ny)) = A3;A((Nx1)*Ny+(1:Ny),(Nx2)*Ny+(1:Ny)) = A2;

end

gradient vel.m

function [Ux, Uy, Umidx, Umidy] = gradient vel(U,dX,dY,u,V,v)

if nargin == 4

[U endp1 j,U i endp1,U 0 j,U i 0] = Uedges(u,U,dX,dY);

Umidx = [(U i 0 + U(:,1))/2 ...(U(:,2:end) + U(:,1:end1))/2 ...(U(:,end) + U i endp1)/2];

Umidy = [(U 0 j + U(1,:))/2; ...(U(2:end,:) + U(1:end1,:))/2; ...(U(end,:) + U endp1 j)/2];

elseif nargin == 6

[U endp1 j,U i endp1,U 0 j,U i 0] = Uedges(u,U,dX,dY);[V endp1 j,V i endp1,V 0 j,V i 0] = Uedges(v,V,dX,dY);

Z = U.*V;Z i 0 = U i 0.*V i 0;Z i endp1 = U i endp1.*V i endp1;Z 0 j = U 0 j.*V 0 j;Z endp1 j = U endp1 j.*V endp1 j;

Umidx = [(Z i 0 + Z(:,1))/2 ...(Z(:,2:end) + Z(:,1:end1))/2 ...(Z(:,end) + Z i endp1)/2];

Umidy = [(Z 0 j + Z(1,:))/2; ...(Z(2:end,:) + Z(1:end1,:))/2; ...(Z(end,:) + Z endp1 j)/2];

elsedisp('Wrong number of inputs')

end

Ux = (Umidx(:,2:end) Umidx(:,1:end1))./dX;Uy = (Umidy(2:end,:) Umidy(1:end1,:))./dY;

end

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Uedges.m

function [Utop,Uright,Ubottom,Uleft] = Uedges(p,P,dX,dY)Xb = p.Xb;Yb = p.Yb;Uleft = ((2*Xb(:,3)1).*P(:,1)) + ...

(Xb(:,3).*(dX(:,1) 2) + 2).*Xb(:,4);Uright = ((2*Xb(:,1)1).*P(:,end)) + ...

(Xb(:,1).*(dX(:,end) 2) + 2).*Xb(:,2);Ubottom = ((2*Yb(:,3)1)'.*P(1,:))+ ...

(Yb(:,3)'.*(dY(1,:) 2) + 2).*Yb(:,4)';Utop = ((2*Yb(:,1)1)'.*P(end,:))+ ...

(Yb(:,1)'.*(dY(end,:) 2) + 2).*Yb(:,2)';end

fluxLimiter.m

function P = fluxLimiter(r,type)

switch typecase 'superbee'

P = max(0,max(min(1,2*r),min(2,r)));case 'minmod'

P = max(0,min(1,r));case 'Koren'

P = max(0,min(min(2*r,(2+r)/3),2));case 'van leer'

P = 2*r./(1 + r);P(isinf(P)) = 0;P(isnan(P)) = 0;

case 'UMIST'P = max(0, ...

min( ...min(...min(2*r,(0.25+0.75*r)), ...(0.75+0.25*r)),...2)....);

otherwisedisp('superbee')P = max(0,min(1,r));

end

end

boundary values.m

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function BoundVals = boundary values(Bound,sheets,sides,x,y,Nx,Ny)

for kk = 1:length(sheets)BoundVals.(sheets{kk}).Xb = zeros(Ny,4);BoundVals.(sheets{kk}).Yb = zeros(Nx,4);cnt = 1;for ii = [2 4]

data = Bound.(sheets{kk}).(sides{ii});for jj = 1:size(data,1)

BoundVals.(sheets{kk}).Xb(y >= data(jj,1) & ...y = data(jj,1) & ...y = data(jj,1) & ...x = data(jj,1) & ...x

tdma.m

function x = tdma(a,b,c,d)

n = length(d);

a = [0; a];c = [c; 0];

c new = zeros(size(c));d new = zeros(size(d));

c new(1) = c(1)/b(1);for ll = 2:n1;

c new(ll) = c(ll)/(b(ll) a(ll)*c new(ll1));end

d new(1) = d(1)/b(1);for ll = 2:n;

d new(ll) = (d(ll) a(ll)*d new(ll1))/ ...(b(ll) a(ll)*c new(ll1));

end

x = zeros(size(d));

x(n) = d new(n);for kk = n1:1:1

x(kk) = d new(kk) c new(kk).*x(kk+1);end

end

maximum velocity.m

function [maxV, minV] = maximum velocity(b)

a1 = b.u.Xb(:,[2 4]);a2 = b.u.Yb(:,[2 4]);a3 = b.v.Xb(:,[2 4]);a4 = b.v.Yb(:,[2 4]);

A = [a1(:); a2(:); a3(:); a4(:)];

maxV = max(abs(A(:)));minV = min(abs(A(:)));

if sum(A(:) < 0)/length(A(:)) == 1

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tmpV = maxV;maxV = minV;minV = tmpV;

end

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NomenclatureProblem statementGoverning equationsNumerical values for inputsMeshingBoundary conditionsNumerical methodsNavier-StokesVorticity and Stream FunctionSpecies mass balance

ValidationNavier-Stokes mass balance

Grid dependency and computation costResultsProblem statement as isProblem statement with a minor modification

Conclusion