Simple MATLAB Code for solving Navier-Stokes Equation (Finite Difference Method, Explicit Scheme)

(1)

Email: [email protected]

The Navier – Stokes Equation

In physics, the Navier – Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes,

describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid

motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term

(proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.

The general form of Navier – Stokes equation in vector expression is

.

v

v v

p

T

f

t







_



      





_









 

 

Here,

T

is the deviatoric stress tensor,



f



is the body force. Replacing

T

with



2

v







, the equation yields

2

.

v

v v

p

v

f

t







_



      





_











 

 

Continuity Equation

0 u

u

x

y



_



_



Momentum Equation (x - dir)

2 2 2 2

1 u

u

p

u

v

t

x

y



x



x

y







_



_



_{ }



_



_







_



_



;



= kinematic viscosity

Momentum Equation (y - dir)

2 2 2 2

1 v

v

p

v

u

v

t

x

y



y



x

y







_



_



_{ }



_



_









_



_

The Pressure – Poisson equation (Poisson equation for pressure) is a derived equation to relate the pressure

with momentum equation. It has been derived using the continuity equation as constrain for momentum

equation. Adding the partial derivative of x – momentum w.r.t. x and the partial derivative of y –

momentum w.r.t. y and then applying the continuity equation yields the Pressure – Poisson equation. Hence,

Pressure Equation (Pressure – Poisson Equation)

2 2 2 2 2 2

2 p

p

u

v

u

u v

v

x

y



t

x

y

x

y x

y



_

_

_





_



_{ }

_

 

_



_







_

 

_



_















_

 













 







_







Derivation of Pressure Equation

Let’s take the partial derivative of x – momentum w.r.t. x

2 2 2 2 2 2

1 u

u u

u

v u

u

p

u

v

t

x

x x

x

x y



x



x

y





 



_{ }

 

_



_

 

_



_{ }



_

 

_











 





 



 









2 2 2

Source: Computational Fluid Dynamics, ME 702 (Online Material) - Dr. Lorena A Barba, Professor, Department of Mechanical Engineering, Boston University Acknowledgement: Boston University – for sharing the course content at BU You Tube Channel (http://www.youtube.com/user/bu)

(2)

Now take the partial derivative of y – momentum w.r.t. y

2 2 2 2 2 2 2

1 v

u v

v

v v

v

p

v

u

v

t

y

y x

x y

y y

y



y



y

x

y









 



 



 





 



_



_





 

_

_

 



_



_

2 2

Let’s add last two equations. Now, the L.H.S. of the added equation,

2 2 2 2 2 ₂ 2 ₂

u

v

u

x

x y

u

u v

v u

v

t

x

v

x y

y

v

x

y

x

x y

y















_







_

 

_

 

_



_















 





 















 

_



2 2

u

u v

v u

v

t

x

y x

x

u

v

u

x

u

v

x

y

u

v

y

x

y





 



















 





_







_



_

_









 

_





























 









 















Using the continuity equation and keeping the continuity part associated with time derivative,

L.H.S

2 2

2 u

v

u

u v

v

t

x

y

x

y x

y







 









 





_



_



_

_



_

 













 











Now, for the R.H.S

2 2 2 2 2 2 2 2 2 2 2 2

1 p

p

u

v

x

y



x

y

x

y













 







 







_



_



_

_



_



_



_

















v 





2 2 3 3 3 3 2 2 3 2 2 3

1 p

p

u

v

x

y



x

x y













 

_



_



_





 









y







3 2 2 2 2 3 3 2 3 3 3 2

1 p

p

x

y

u

v

x

y

u

v

x

y







_











 

_



_



_





 





















2 2 2 2 2 2 2 2

1 u

v

u

v

y

p

x

y

x

y

x





















 

_



_



_

_









_









_

_

_











_



_







Again, using the continuity equation,

R.H.S

2 2 2 2

1 p

p

x

y









 

_



_







Now, combining both hand sides,

2 2 2 2 2 2

1

2 p

p

u

v

u

u v

v

x

y

t

x

y

x

y x

y









 













 







_



_



_



_



_

_



_



 













 















2 2 2 2 2 2

2 p

p

u

v

u

u v

v

x

y



t

x

y

x

y x

y



_

_

_





 









 







 



_



_



_

_



_







_

 













 







_







This is similar to the Poisson equation



2

p

  .

f

Email: [email protected] Source: Computational Fluid Dynamics, ME 702 (Online Material) - Dr. Lorena A Barba, Professor, Department of Mechanical Engineering, Boston University Acknowledgement: Boston University – for sharing the course content at BU You Tube Channel (http://www.youtube.com/user/bu)

(3)

inite Difference Discretization of Navier – Stokes Equation

F

st

Let’s discretize the Navier – Stokes equation taking 1 order forward difference for time discretization, 1

st

order backward difference for 1

st

order space derivative ( v





), 2

nd

order central difference for 2

nd

order

space derivative (



2

v



) and 1

st

order central difference for pressure discretization,

Discretized Navier – Stokes equation in x – direction

1 1, , 1

1

n n n n n n n ij ij n ij i j n ij i j i

u





u



u

_

u



u

_

p

_1, _1, _1, _1, _, ₁ _, ₁ 2 2

2

n n n n n n n j i j i j ij i j i j ij i j ij ij

p

u

v

t

x

y



x



x

y













 











 





_





_



_



_

iscretized Navier – Stokes equation in y – direction

D

1 1, , 1

1

n n n n n n n ij ij n ij i j n ij i j i

v





v



v

_

v



v

_

p

__1, _1, _1, _1, _, ₁ _, ₁ 2 2

2

n n n n n n n j i j i j ij i j i j ij i j ij ij

p

v

u

v

u

v

t

x

y



y



x

y





   











 





_





_



_



_

or discretized pressure equation, let’s take 2

nd

order central difference for pressure and 1

st

order central

F

difference in space. Hence, the discretized pressure equation is –

2 1, 1, , 1 , 1 1, 1, 1, 1, , 1 , 1 2 2 , 1 , 1 1, 1, , 1 ,

1 

u

in j







2

n n n n n i j i j i j i j i j n n n n n n i j ij i j i j ij i j n n n n n i j i j i j i j i j i

u

v

u

t

x

y

x

p

x

y

_u

_v

y

x



            

 









 





_

_

_

_{ }

_

_













_

_{ }

_



 



_

_

_

_

_

_





_



_



_

























₂



1

2

n j

y















_



_

















y transposing,

B

















1 1, , 1 1, 1, 1, 1, 2 , 1

1

2

n n n n n n n n n n ij ij ij ij i j ij ij i j i j i j n n n n i j ij i j i j

t

u

v

u

p

x

y

x

t

u

x





       



















_

_



_

_



_

_



























_

_













, 1



2 n n ij i j

t

u

y



















_





























1 1, , 1 , 1 , 1 1, 1, 2 , 1

1

2

n n n n n n n n n n ij ij ij ij i j ij ij i j i j i j n n n n i j ij i j i j

t

v

u

v

p

x

y

t

v

x





       



















_

_



_

_



_

_



























_

_













, 1



2 n n ij i j

t

v

y



















_















  

 













2 2 ₂ ₂ 1, 1, , 1 , 1 1, 1, , 1 , 1 2 2 2 2 2 2 2 1, 1, 2 2

2

n n n n n n n n i j i j i j i j i j i j i j i j n ij n n i j i j

p

y

p

x

_x

_y

u

v

p

x

y

x

y

x

y

t

u

x

y

x

y



         











 

_

 



_















_

_



_



_



  







_

_



 





_{ }

_{ }

_

_







_

_



_



_



  

_

_







2 , 1 , 1 1, 1, , 1 , 1

2

n n n n n n i j i j i j i j i j i j

u

v

y

x

y

     



_

_

_

_

_{ }

_









_



_

 

_{ }









_

_

_{ }













(4)

(0, 0) y (2 ux,2 = 1 x , 2) vx,2 = 0 u0,y = 0 v0,y 0 ux,0 = 0 = vx,0 = 0 u2,y = 0 v2,y 0 = px,2 = 0 0 p y    0 p x    0 p x    (2, 0) (0, 2)

ATLAB Code for Solution – Explicit Scheme

M

Case 1: Cavity Flow

Let’s consider a square domain for the cavity flow problem with arm length of 2 units. For all the walls,

itial Condition

except the top wall, no slip condition has been considered. The top wall has been considered as free stream.

In

:

; for all (x,y)

oundary Condition

0;

v

0;

p

0 



u

B

:

Wall

,

Bottom

:

u

_x_,0

 ,

0 v

_x_,0



0

y

p

y

_



_



Left Wall

:

u

_0,_y

 ,

0 v

_0,_y

 ,

0

x

p

x

_



_



Right Wall

:

u

_0,2

 ,

0 v

_0,2



0 ,

2

0

x

p

x

_



_



Top Wall

,

ase 1: MATLAB Code for Cavity Flow

:

u

_x_,2

 ,

1 v

_x_,2



0 p

_x_,2



0 C

%%%%%%%%%%%%%%%%%%%%%% Start of the code %%%%%%%%%%%%%%%%%%%%%%

all

= 21; % number of nodes in x-direction

... -direction c cl clear close all %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Initial Condition % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nx

ny = 21; % number of nodes in y-direction

nt = 10; % number of time steps

ime steps

nit = 10; % number of artificial t

% ... for pressure vis = 0.1; % Viscosity rho = 1.0; % Density x Lx = 2; % Length in Ly = 2; % Length in y-direction

(5)

dx = Lx/(nx-1); % grid spacing in x-direction

omponent

ction

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(:,:,1) = u(:,:);

:nt+1 % loop over time

for calculating the ...

j,i) = rho*(((((u(j,i+1)-u(j,i-1))/(2*dx))+... 1)... r -1 (j,i+1)+p(j,i-1))*(dy^2)... +dy^2))... p(:,2); p(:,nx) = p(:,nx-1);

dy = Ly/(ny-1); % grid spacing in y-direction

dt = 0.01; % time step size

x = 0:dx:Lx; % node x-ordinates

y = 0:dy:Ly; % node y-ordinates

u = zeros(ny,nx); % nodal velocity x-c

v = zeros(ny,nx); % nodal velocity y-component

p = zeros(ny,nx); % nodal pressure

ocity x-dire

un = zeros(ny,nx); % time marched vel

vn = zeros(ny,nx); % time marched velocity x-direction

pn = zeros(ny,nx); % temporary pressure for calculation

= zeros(ny,nx); % nodal source term value from pressure

b % % % Boundary Condition % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,x] = meshgrid(y,x); [y u(ny,:) = 1; %% %%% % % % Calculation % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uT vT(:,:,1) = v(:,:); pT(:,:,1) = p(:,:); for it = 1

for i = 2:nx-1 % this i,j loop is

for j = 2:ny-1 % ... body force from pressure equation

b( ((v(j+1,i)-v(j-1,i))/(2*dy)))*(1/dt))+((u(j,i+ - u(j,i-1))/(2*dx)).^2+(2*((u(j,i+1)... 2*dx)))... -u(j,i-1))/(2*dy))*((v(j+1,i)-v(j-1,i))/( +((v(j+1,i)-v(j-1,i))/(2*dy)).^2); end end iit = 1:nit+1 fo pn = p; :nx-1 for i = 2 for j = 2:ny p(j,i) = ((p +(p(j+1,i)+p(j-1,i))*(dx^2))/(2*(dx^2 +(b(j,i)*dx^2*dy^2)/(2*(dx^2+dy^2)); end end ) = p(:,1 p(1,:) = p(2,:); p(ny,:) = p(ny-1,:); end

(6)

= u; vn = v; % assigning velocity values to nth-time values

r i = 2:nx-1 % this i,j loop is for calculating velocity

.. 2)); j,i) = vn(j,i)-un(j,i)*(vn(j,i)-vn(j,i-1))*(dt/dx)... *... 2)); %%%%%%%%%%%%%%%%% Surface Plotting %%%%%%%%%%%%%%%%%%%%%% gure(1) )

Plot: Velocity Component, u');

t, u'); gure(2) ) rt(u.^2+v.^2)') y Field, U'); t, v'); gure(4) ) Plot: Pressure, P'); y,u',10) ity Component, u'); un

% to calculate (n+1)th-time values

fo for j = 2:ny-1 u(j,i) = un(j,i)-un(j,i)*(un(j,i)-un(j,i-1))*(dt/dx)... -vn(j,i)*(un(j,i)-un(j-1,i))*(dt/dy)... -(1/rho)*(p(j,i+1)-p(j,i-1))*(dt/(2*dx)). +(vis/rho)*((un(j,i+1)-(2*un(j,i))+un(j,i-1))*... (dt/dx^2)+(un(j+1,i)-(2*un(j,i))+un(j-1,i))*(dt/dy^ v( -vn(j,i)*(vn(j,i)-vn(j-1,i))*(dt/dy)... -(1/rho)*(p(j+1,i)-p(j-1,i))*(dt/(2*dy))... +(vis/rho)*((vn(j,i+1)-(2*vn(j,i))+vn(j,i-1)) (dt/dx^2)+(vn(j+1,i)-(2*vn(j,i))+vn(j-1,i))*(dt/dy^ end end %%% %% fi surf(x,y,u' title('Surface xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Velocity componen

fi

surf(x,y,v'

title('Surface Plot: Velocity Component, v'); xlabel('x - ordinate');

ylabel('y - ordinate');

zlabel('Velocity component, v');

figure(3) surf(x,y,sq

title('Surface Plot: Velocit

xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Velocity componen

fi surf(x,y,p' title('Surface xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Pressure, p'); %%%%%%%%%%%%%%%%%%%%%% Contour Plotting %%%%%%%%%%%%%%%%%%%%%% figure(5) contourf(x,

title('Contour: Veloc

xlabel('x - ordinate'); ylabel('y - ordinate');

(7)

gure(6) y,v',10) ity Component, v'); gure(7) y,sqrt(u.^2+v.^2)') y,p',10) ure, P'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

:,1) = 0; u(:,nx) = 0; u(1,:) = 0; u(ny,:) = 1;

%%%%%%%%%%%%%%%%%%%%%%% End of the code %%%%%%%%%%%%%%%%%%%%%%%

fi

contourf(x,

fi

contourf(x,

title('Surface Plot: Velocity Field, U'); xlabel('x - ordinate');

ylabel('y - ordinate');

zlabel('Velocity component, v');

figure(8) contourf(x,

title('Contour: Press

xlabel('x - ordinate'); ylabel('y - ordinate'); u( v(:,1) = 0; v(:,nx) = 0; v(1,:) = 0; v(ny,:) = 0; end

(8)

Fig.3

: Surface plot for velocity y – component

Fig.4

: Contour plot for velocity y – component

Fig.5

: Surface plot for velocity field, U

Fig.6

: Contour plot for velocity field, U

Fig.7

: Surface plot for pressure, p

Fig.8

: Contour plot for pressure, p

(9)

ase 2: Channel Flow

C

Let’s consider a square domain for the channel flow problem with arm length of 2 units. For the top and

itial Condition

bottom wall, no slip condition has been considered. The side walls have been considered as periodic.

In

:

; for all (x,y)

oundary Condition

0;

v

0;

p

0 



u

B

:

Wall

,

Bottom

:

u

_x_,0

 ,

0 v

_x_,0



0

y

p

y

_



_



Left Wall

:

u

0,y

 ,

0 v

0,y

 ,

0

x

p

x

_







Right Wall

:

u

_0,2

 ,

0 v

_0,2



0 ,

2

0

x

p

x

_



_



Top Wall

:

u

_x_,2

 ,

0 v

_x_,2



0 ,

2

0

y

p

y

_



_



ase 2: Channel Flow

C

%%%%%%%%%%%%%%%%%%%%%% Start of the code %%%%%%%%%%%%%%%%%%%%%%

all %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % .... -direction tion x (2, 2) (0, 0) y ux,2 = 0 c cl clear close all % %

% Initial & Boundary Condition

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

= 21; % number of nodes in x-direction

nx

ny = 21; % number of nodes in y-direction

nt = 10; % number of time steps

ime steps .

nit = 10; % number of artificial t

% ... for pressure vis = 0.1; % Viscosity rho = 1.0; % Density x Lx = 2; % Length in Ly = 2; % Length in y-direction

x = Lx/(nx-1); % grid spacing in x-direc

d

dy = Ly/(ny-1); % grid spacing in y-direction

vx,2 = 0 u0,y = 0 v0,y 0 ux,0 = 0 = vx,0 = 0 u2,y = 0 v2,y 0 = 0 p y    0 p x    0 p x    (2, 0) (0, 2) 0 p y   

Periodic

(10)

= 0:dx:Lx; % node x-ordinates

= zeros(ny,nx); % nodal body force term from pressure

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

= 1:nt+1 % loop over time

for calculating the ...

1)...

:nx-(j,i+1)+p(j,i-1))*(dy^2)...

+dy^2))...

= u; vn = v;

r i = 2:nx-1 % this i,j loop is for calculating velocity

..

dt = 0.01; % time step size

x

y = 0:dy:Ly; % node y-ordinates

u = zeros(ny,nx); % nodal velocity x-component

v = zeros(ny,nx); % nodal velocity y-component

p = zeros(ny,nx); % nodal pressure

un = zeros(ny,nx); % time marched velocity x-direction

vn = zeros(ny,nx); % time marched velocity x-direction

pn = zeros(ny,nx); % temporary pressure for calculation

b

f = ones(ny,nx); % nodal source term

[y,x] = meshgrid(y,x); % % % Calculation % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for it

for i = 2:nx-1 % this i,j loop is

for j = 2:ny-1 % ... body force from pressure equation

b(j,i) = rho*(((((u(j,i+1)-u(j,i-1))/(2*dx))+... ((v(j+1,i)-v(j-1,i))/(2*dy)))*(1/dt))+((u(j,i+ - u(j,i-1))/(2*dx)).^2+(2*((u(j,i+1)... -u(j,i-1))/(2*dy))*((v(j+1,i)-v(j-1,i))/(2*dx)))... +((v(j+1,i)-v(j-1,i))/(2*dy)).^2); end end

for iit = 1:nit+1 pn = p; 1 for i = 2 for j = 2:ny-1 p(j,i) = ((p +(p(j+1,i)+p(j-1,i))*(dx^2))/(2*(dx^2 +(b(j,i)*dx^2*dy^2)/(2*(dx^2+dy^2)); end end p(:,1) = p(:,2); p(:,nx) = p(:,nx-1); p(1,:) = p(2,:); p(ny,:) = p(ny-1,:); end un fo for j = 2:ny-1 u(j,i) = un(j,i)-un(j,i)*(un(j,i)-un(j,i-1))*(dt/dx)... -vn(j,i)*(un(j,i)-un(j-1,i))*(dt/dy)... -(1/rho)*(p(j,i+1)-p(j,i-1))*(dt/(2*dx)). +(vis/rho)*((un(j,i+1)-(2*un(j,i))+un(j,i-1))*... (dt/dx^2)+(un(j+1,i)-(2*un(j,i))+un(j-1,i))... *(dt/dy^2))+(f(j,i)*dt);

(11)

j,i) = vn(j,i)-un(j,i)*(vn(j,i)-vn(j,i-1))*(dt/dx)... *... 2)); % %%%%%%%%%%%%%%%%% Surface Plotting %%%%%%%%%%%%%%%%%%%%%% figure(1) )

Plot: Velocity Component, u');

t, u');

)

Plot: Velocity Component, v');

t, v'); gure(3) rt(u.^2+v.^2)') y Field, U'); t, v'); gure(4) ) Plot: Pressure, P'); Contour Plotting %%%%%%%%%%%%%%%%%%%%%% y,u',10) ity Component, u'); gure(6) y,v',10) ity Component, v'); gure(7) y,sqrt(u.^2+v.^2)') eld, U'); v( -vn(j,i)*(vn(j,i)-vn(j-1,i))*(dt/dy)... .. -(1/rho)*(p(j+1,i)-p(j-1,i))*(dt/(2*dy)). +(vis/rho)*((vn(j,i+1)-(2*vn(j,i))+vn(j,i-1)) (dt/dx^2)+(vn(j+1,i)-(2*vn(j,i))+vn(j-1,i))*(dt/dy^ end end %% %% surf(x,y,u' title('Surface xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Velocity componen

gure(2) fi surf(x,y,v' title('Surface xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Velocity componen

fi

surf(x,y,sq

title('Surface Plot: Velocit

xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Velocity componen

fi surf(x,y,p' title('Surface xlabel('x - ordinate'); ylabel('y - ordinate'); zlabel('Pressure, p'); %%%%%%%%%%%%%%%%%%%% %% figure(5) contourf(x,

fi

contourf(x,

fi

contourf(x,

title('Surface Plot: Velocity Fi

(12)

t, v');

gure(8)

y,p',10)

:,1) = 0; u(:,nx) = 0; u(1,:) = 0; u(ny,:) = 0; zlabel('Velocity componen

fi

contourf(x,

title('Contour: Pressure, P'); xlabel('x - ordinate'); ylabel('y - ordinate'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u( v(:,1) = 0; v(:,nx) = 0; v(1,:) = 0; v(ny,:) = 0; end