13th Step of the 12 steps to Navier-Stokes 😑

     Indeed, the 13th step now exists! This case is called the case of the backward facing step (BFS)! ⬜ This is an unofficial continuation to this. If I get it approved by Dr. Barba, then it will be official. The original series is in Python but I coded this in MATLAB without using many MATLAB specific functions so the code can be translated to other programing languages 🖧 quite easily.

The Code

%% clear and close

close all

clear

clc

%% define spatial and temporal grids

l_square = 1; % length of square

h = l_square/50; % grid spacing

dt = 0.1; % time step

L = 21; % cavity length

D = 2; % cavity depth

Nx = round((L/h)+1); % grid points in x-axis

Ny = round((D/h)+1); % grid points in y-axis

nu = 0.000015111; % kinematic viscosity

Uinf = 0.0060444; % free stream velocity / inlet velocity / lid velocity

cfl = dt*Uinf/h; % % cfl number

travel = 4; % times the disturbance travels entire length of computational domain

TT = travel*L/Uinf; % total time

ns = TT/dt; % number of time steps

Re = l_square*Uinf/nu; % Reynolds number

rho = 1.2047; % fluid density

%% initialize flowfield

u = zeros(Nx,Ny); % x-velocity

v = zeros(Nx,Ny); % y-velocity

p = zeros(Nx,Ny); % pressure

i = 2:Nx-1; % spatial interior nodes in x-axis

j = 2:Ny-1; % spatial interior nodes in y-axis

[X, Y] = meshgrid(0:h:L, 0:h:D); % spatial grid

maxNumCompThreads('automatic'); % select CPU cores

%% solve 2D Navier-Stokes equations

for nt = 1:ns

pn = p;

p(i, j) = (pn(i+1, j)+pn(i-1, j)+pn(i, j+1)+pn(i, j-1))/4 ...

-h*rho/(8*dt)*(u(i+1, j)-u(i-1, j)+v(i, j+1)-v(i, j-1)); % pressure poisson

p(1, :) = p(2, :); % dp/dx = 0 at x = 0

p(Nx, :) = 0; % p = 0 at x = L

p(:, 1) = p(:, 2); % dp/dy = 0 at y = 0

p(:, Ny) = p(:, Ny-1); % dp/dy = 0 at y = D

p(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

p(round(1*Nx/L), round(1:1*Ny/D)) = p(round(1*Nx/L)+1, round(1:1*Ny/D)); % dp/dx = 0 at x = 1 and y = 0 to 1

p(1:round(1*Nx/L), round(1*Ny/D)) = p(1:round(1*Nx/L), round(1*Ny/D)+1); % dp/dy = 0 at x = 0 to 1 and y = 1

p(1:round(1*Nx/L), 1) = p(1:round(1*Nx/L), 2); % dp/dy = 0 at x = 0 to 1 and y = 1

un = u;

vn = v;

u(i, j) = un(i, j)-dt/(2 * h)*(un(i, j).*(un(i+1, j)-un(i-1, j))+vn(i, j).*(un(i, j+1)-un(i, j-1))) ...

-dt/(2*rho*h)*(p(i+1, j)-p(i-1, j)) ...

+nu*dt/h^2*(un(i+1, j)+un(i-1, j)+un(i, j+1)+un(i, j-1)-4*un(i, j)); % x-momentum

u(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

u(1, round(1:1*Ny/D)) = 0; % u = 0 at x = 0 and y = 0 to 1

u(1, round(1*Ny/D:2*Ny/D)) = Uinf; % u = Uinf at x = 0 and y = 1 to 2

u(round(1*Nx/L), round(1:1*Ny/D)) = 0; % u = 0 at x = 1 and y = 0 to 1

u(1:round(1*Nx/L), round(1*Ny/D)) = 0; % u = 0 at x = 0 to 1 and y = 1

u(1:round(1*Nx/L), 1) = 0; % u = 0 at x = 0 to 1 and y = 1

u(Nx, :) = u(Nx-1, :); % du/dx = 0 at x = L

u(:, 1) = 0; % u = 0 at y = 0

u(:, Ny) = 0; % u = 0 at y = D

v(i, j) = vn(i, j)-dt/(2*h)*(un(i, j).*(vn(i+1, j)-vn(i-1, j))+vn(i, j).*(vn(i, j+1)-vn(i, j-1))) ...

-dt/(2*rho*h)*(p(i, j+1)-p(i, j-1)) ...

+ nu*dt/h^2*(vn(i+1, j)+vn(i-1, j)+vn(i, j+1)+vn(i, j-1)-4*vn(i, j)); % y-momentum

v(round(1:1*Nx/L), round(1:1*Ny/D)) = 0; % step geometry

v(1, round(1:1*Ny/D)) = 0; % v = 0 at x = 0 and y = 0 to 1

v(1, round(1*Ny/D:2*Ny/D)) = 0; % v = Uinf at x = 0 and y = 1 to 2

v(round(1*Nx/L), round(1:1*Ny/D)) = 0; % v = 0 at x = 1 and y = 0 to 1

v(1:round(1*Nx/L), round(1*Ny/D)) = 0; % v = 0 at x = 0 to 1 and y = 1

v(1:round(1*Nx/L), 1) = 0; % v = 0 at x = 0 to 1 and y = 1

v(Nx, :) = v(Nx-1, :); % dv/dx = 0 at x = L

v(:, 1) = 0; % u = 0 at y = 0

v(:, Ny) = 0; % u = 0 at y = D

end

%% post-processing

velocity_magnitude = sqrt(u.^2 + v.^2); % velocity magnitude

u1 = u; % u-velocity for plotting with box

v1 = v; % v-velocity for plotting with box

p1 = p; % p-velocity for plotting with box

u1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

v1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

p1(1:round(1*Nx/L) , 1:round(1*Ny/D)) = NaN; % step geometry

velocity_magnitude1 = sqrt(u1.^2 + v1.^2); % velocity magnitude with box

%% Visualize velocity vectors and pressure contours

hold on

contourf(X, Y, u1', 64, 'LineColor', 'none'); % contour plot

set(gca, 'FontSize', 20)

% skip = 20;

% quiver(X(1:skip:end, 1:skip:end), Y(1:skip:end, 1:skip:end),...

% u1(1:skip:end, 1:skip:end)', v1(1:skip:end, 1:skip:end)', 1, 'k','LineWidth', 0.1); % Velocity vectors

% hh = streamslice(X, Y, u1', v1',2); % Streamlines

% set(hh, 'Color', 'k','LineWidth', 01);

colorbar; % Add color bar

colormap parula % Set color map

axis equal % Set true scale

xlim([0 L]); % Set axis limits

ylim([0 D]);

xticks([0 9 L]) % Set ticks

yticks([0 D]) % Set ticks

xt = [0 21]; % draw top wall

yt = [2 2];

xb = [1 21]; % draw bottom wall

yb = [0 0];

xbox = [0 1 1 0 0]; % draw box

ybox = [0 0 1 1 0];

plot(xbox, ybox, 'k', 'LineWidth', 2)

plot(xt, yt, 'k', 'LineWidth', 2)

plot(xb, yb, 'k', 'LineWidth', 2)

% clim([0 max(velocity_magnitude(:))]) % legend limits

% title('Velocity [m/s]');

xlabel('x [m]');

ylabel('y [m]');

     The results from this code at Re 400 are presented in Fig. 1. The re-attachment length is ~8 m from the trailing-edge of the box, which is same as previously published results, from example in [1].

Fig. 1, post-processes results

     Thank you for reading! If you want to hire me as your next PhD student, please do reach out!

References

 [1] Irisarri, D., Hauke, G. Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations. Calcolo 56, 38 (2019). https://doi.org/10.1007/s10092-019-0332-5

Saithe Fish UDF (ANSYS Fluent)

     This post is about Fish Simulation in ANSYS Fluent using a User Defined Function (UDF). The UDF is mentioned below. The flow conditions are taken from [1]. This goes with the videos shown in Fig. 1-2. The CAD files for t=0 are available here (for UDF 01).

Fig. 1, Animation of motion achieved through the UDF 01.

 

 Fig. 2, Animation of motion achieved through the UDF 02 (Validated).

     The results of present simulations are compared with [1]. The results are in excellent agreement as the C_{l, max} from [1] is at 1.57 while the maximum C_{l, max} from present simulation is at 1.6. The drag coefficient [1], C_{d, max} in [1] is at 0.164; while from the present simulations I got, 0.151. The C_{d, avg} comes out to be 0.072 [1] form the present simulations. I got a value of 0.064 from he present simulation. These would gradually become more accurate with mesh refinement, which I will certainly do if I send some ideas I have for peer review.

UDF 01:

#include "udf.h"

#include "unsteady.h"

#include "dynamesh_tools.h"

#include "math.h"

DEFINE_GRID_MOTION(dynamic,domain,dt,time,dtime)

{

 Thread *tf = DT_THREAD(dt);

 face_t f;

 Node *v;

 int n;

 double x, y, y_ref_previous, y_ref;

 SET_DEFORMING_THREAD_FLAG(THREAD_T0(tf));  

 begin_f_loop(f,tf) {

  f_node_loop(f,tf,n) {

   v = F_NODE(f,tf,n);

   if (NODE_POS_NEED_UPDATE(v)) {

    NODE_POS_UPDATED(v);

    x = NODE_X(v);

    real amplitude = 0.02 + 0.01*x + 0.1*x*x;

    y_ref_previous = amplitude * cos(2*M_PI*x + 2*M_PI*0.8*(PREVIOUS_TIME));

    y_ref = amplitude * cos(2*M_PI*x + 2*M_PI*0.8*(CURRENT_TIME));

     

if (NODE_Y(v) > y_ref_previous){

     NODE_Y(v) = y_ref+fabs(NODE_Y(v)-y_ref_previous);

    }

    else 

     if (NODE_Y(v) < y_ref_previous){

      NODE_Y(v) = y_ref-fabs(NODE_Y(v)-y_ref_previous);

     }

     else {

      NODE_Y(v) = y_ref;

     }

    }

   }

  }

 }

 end_f_loop(f,tf);

UDF 02 (Validated):

#include "udf.h"

DEFINE_GRID_MOTION(dynamic,domain,dt,time,dtime)

{

 Thread *tf = DT_THREAD(dt);

 face_t f;

 Node *v;

 int n;

 double x, y_ref_previous, y_ref, amplitude, fr;

 SET_DEFORMING_THREAD_FLAG(THREAD_T0(tf));  

 begin_f_loop(f,tf) {

  f_node_loop(f,tf,n) {

   v = F_NODE(f,tf,n);

   if (NODE_POS_NEED_UPDATE(v)) {

    NODE_POS_UPDATED(v);

    x = fabs(NODE_X(v));

    amplitude = 0.02 - 0.0825 * x + 0.1625 * x * x;

    fr = 2;

    y_ref_previous = amplitude * cos(2 * M_PI * x - 2 * M_PI * fr * (PREVIOUS_TIME));

    y_ref = amplitude * cos(2 * M_PI * x - 2 * M_PI * fr * (CURRENT_TIME));

     

if (NODE_Y(v) > y_ref_previous){

     NODE_Y(v) = y_ref + fabs(NODE_Y(v) - y_ref_previous);

    }

    else 

     if (NODE_Y(v) < y_ref_previous){

      NODE_Y(v) = y_ref - fabs(NODE_Y(v) - y_ref_previous);

     }

     else {

      NODE_Y(v) = y_ref;

     }

    }

   }

  }

 }

 end_f_loop(f,tf);

If you want to hire me as you next shining PhD/Master student or collaborate in research, please reach out! Thank you for reading!

References

[1] Gen-Jin Dong, Xi-Yun Lu; Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Physics of Fluids 1 May 2007; 19 (5): 057107. https://doi.org/10.1063/1.2736083

Lid-Driven Cavity MATLAB Code

     MATLAB code for 2D Lid-Driven Cavity. Includes labeled commands, plotting and is less than 100 lines of code. Resume possible, to resume comment close all, clear, clc, u, v, p and then run the code. Results are available here:

%% clear and close

close all

clear

clc

%% define spatial and temporal grids

h = 1/10; % grid spacing

cfl = h; % cfl number

L = 1; % cavity length

D = 1; % cavity depth

Nx = round((L/h)+1); % grid points in x-axis

Ny = round((D/h)+1); % grid points in y-axis

nu = 0.000015111; % kinematic viscosity

Uinf = 0.0015111; % free stream velocity / lid velocity

dt = h * cfl / Uinf; % time step

travel = 2; % times the disturbance travels entire length of computational domain

TT = travel * L / Uinf; % total time

ns = TT / dt; % number of time steps

l_square = 1; % length of square

Re = l_square * Uinf / nu; % Reynolds number

rho = 1.2047; % fluid density

%% initialize flowfield

u = zeros(Nx,Ny); % x-velocity

v = zeros(Nx,Ny); % y-velocity

p = zeros(Nx,Ny); % pressure

i = 2:Nx-1; % spatial interior nodes in x-axis

j = 2:Ny-1; % spatial interior nodes in y-axis

[X, Y] = meshgrid(0:h:L, 0:h:D); % spatial grid

maxNumCompThreads('automatic'); % select CPU cores

%% solve 2D Navier-Stokes equations

for nt = 1:ns

    pn = p;

    p(i, j) = ((pn(i+1, j) + pn(i-1, j)) * h^2 + (pn(i, j+1) + pn(i, j-1)) * h^2) ./ (2 * (h^2 + h^2)) ...

        - h^2 * h^2 / (2 * (h^2 + h^2)) * (rho * (1 / dt * ((u(i+1, j) - u(i-1, j)) / (2 * h) + (v(i, j+1) - v(i, j-1)) / (2 * h)))); % pressure poisson

    p(1, :) = p(2, :); % dp/dx = 0 at x = 0

    p(Nx, :) = p(Nx-1, :); % dp/dx = 0 at x = L

    p(:, 1) = p(:, 2); % dp/dy = 0 at y = 0

    p(:, Ny) = 0; % p = 0 at y = D 

    un = u;

    vn = v;

    u(i, j) = un(i, j) - un(i, j) * dt / (2 * h) .* (un(i+1, j) - un(i-1, j)) ...

        - vn(i, j) * dt / (2 * h) .* (un(i, j+1) - un(i, j-1)) - dt / (2 * rho * h) * (p(i+1, j) - p(i-1, j)) ...

        + nu * (dt / h^2 * (un(i+1, j) - 2 * un(i, j) + un(i-1, j)) + dt / h^2 * (un(i, j+1) - 2 * un(i, j) + un(i, j-1))); % x-momentum

    u(1, :) = 0; % u = 0 at x = 0

    u(Nx, :) = 0; % u = 0 at x = L

    u(:, 1) = 0; % u = 0 at y = 0

    u(:, Ny) = Uinf; % u = Uinf at y = D

    v(i, j) = vn(i, j) - un(i, j) * dt / (2 * h) .* (vn(i+1, j) - vn(i-1, j)) ...

        - vn(i, j) * dt / (2 * h) .* (vn(i, j+1) - vn(i, j-1)) - dt / (2  * rho * h) * (p(i, j+1) - p(i, j-1)) ...

        + nu * (dt / h^2 * (vn(i+1, j) - 2 * vn(i, j) + vn(i-1, j)) + dt / h^2 * (vn(i, j+1) - 2 * vn(i, j) + vn(i, j-1))); % y-momentum

    v(1, :) = 0; % v = 0 at x = 0

    v(Nx, :) = 0; % v = 0 at x = L

    v(:, 1) = 0; % v = 0 at y = 0

    v(:, Ny) = 0; % v = 0 at y = D

end

%% post-processing

velocity_magnitude = sqrt(u.^2 + v.^2); % velocity magnitude

% Visualize velocity vectors and pressure contours

hold on

contourf(X, Y, velocity_magnitude', 64, 'LineColor', 'none'); % contour plot

set(gca,'FontSize',40)

% skip = 20;

% quiver(X(1:skip:end, 1:skip:end), Y(1:skip:end, 1:skip:end),... % Velocity vectors

%     u1(1:skip:end, 1:skip:end)', v1(1:skip:end, 1:skip:end)', 1, 'k','LineWidth', 0.1);

hh = streamslice(X, Y, u', v', 5); % Streamlines

set(hh, 'Color', 'k','LineWidth', 0.1);

colorbar; % Add color bar

colormap hsv % Set color map

axis equal % Set true scale

xlim([0 L]); % Set axis limits

ylim([0 D]);

xticks([0 L]) % Set ticks

yticks([0 D]) % Set ticks

clim([0 0.95*max(velocity_magnitude(:))]) % Legend limits

title('Velocity [m/s]');

xlabel('x [m]');

ylabel('y [m]');

Cite as: Fahad Butt (2023). Lid-Driven Cavity (https://fluiddynamicscomputer.blogspot.com/2023/09/lid-driven-cavity-matlab-code.html), Blogger. Retrieved Month Date, Year

Open - Source coded 3D Navier–Stokes equations in C++

     Here are 3D Navier–Stokes equations configured for lid-driven cavity flow. The syntax is C++. The results from this code are shown in Fig. 1. From the pressure-Poisson equation, I removed mixed derivative terms to improve solution stability. 😆

p[i][j][k] = ((p[i+1][j][k] + p[i-1][j][k]) * h * h + (p[i][j+1][k] + p[i][j-1][k]) * h * h + (p[i][j][k+1] + p[i][j][k-1]) * h * h) / (2 * (h * h + h * h + h * h)) - h * h * h * h * h * h / (2 * (h * h + h * h + h * h)) * (rho * (1 / dt * ((u(i+1, j, k) - u(i-1, j, k)) / (2 * h) + (v(i, j+1, k) - v(i, j-1, k)) / (2 * h) + (w(i, j, k+1) - w(i, j, k-1)) / (2 * h))));

p[0][j][k] = p[1][j][k];

p[num_i - 1][j][k] = p[num_i - 2][j][k];

p[i][0][k] = p[i][1][k];

p[i][num_j - 1][k] = p[i][num_j - 2][k];

p[i][j][0] = p[i][j][1];

p[i][j][num_k - 1] = 0.0;

u[i][j][k] = u[i][j][k] - u[i][j][k] * dt / (2 * h) * (u[i+1][j][k] - u[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (u[i][j+1][k] - u[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (u[i][j][k+1] - u[i][j][k-1]) - dt / (2 * rho * h) * (p[i+1][j][k] - p[i-1][j][k]) + nu * (dt / (h * h) * (u[i+1][j][k] - 2 * u[i][j][k] + u[i-1][j][k]) + dt / (h * h) * (u[i][j+1] [k] - 2 * u[i][j][k] + u[i][j-1][k]) + dt / (h * h) * (u[i][j][k+1] - 2 * u[i][j][k] + u[i][j][k-1]));

u[0][j][k] = 0.0;

u[num_i - 1][j][k] = 0.0;

u[i][0][k] = 0.0;

u[i][num_j - 1][k] = 0.0;

u[i][j][0] = 0.0;

u[i][j][num_k - 1] = -Uinf;

v[i][j][k] = v[i][j][k] - u[i][j][k] * dt / (2 * h) * (v[i+1][j][k] - v[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (v[i][j+1][k] - v[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (v[i][j][k+1] - v[i][j][k-1]) - dt / (2 * rho * h) * (p[i][j+1][k] - p[i][j-1][k]) + nu * (dt / (h * h) * (v[i+1][j][k] - 2 * v[i][j][k] + v[i-1][j][k]) + dt / (h * h) * (v[i][j+1][k] - 2 * v[i][j][k] + v[i][j-1][k]) + dt / (h * h) * (v[i][j][k+1] - 2 * v[i][j][k] + v[i][j][k-1]));

v[0][j][k] = 0.0;

v[num_i - 1][j][k] = 0.0;

v[i][0][k] = 0.0;

v[i][num_j - 1][k] = 0.0;

v[i][j][0] = 0.0;

v[i][j][num_k - 1] = 0.0;

w[i][j][k] = w[i][j][k] - u[i][j][k] * dt / (2 * h) * (w[i+1][j][k] - w[i-1][j][k]) - v[i][j][k] * dt / (2 * h) * (w[i][j+1][k] - w[i][j-1][k]) - w[i][j][k] * dt / (2 * h) * (w[i][j][k+1] - w[i][j][k-1]) - dt / (2 * rho * h) * (p[i][j][k+1] - p[i][j][k-1]) + nu * (dt / (h * h) * (w[i+1][j][k] - 2 * w[i][j][k] + w[i-1][j][k]) + dt / (h * h) * (w[i][j+1][k] - 2 * w[i][j][k] + w[i][j-1][k]) + dt / (h * h) * (w[i][j][k+1] - 2 * w[i][j][k] + w[i][j][k-1]));

w[0][j][k] = 0.0;

w[num_i - 1][j][k] = 0.0;

w[i][0][k] = 0.0;

w[i][num_j - 1][k] = 0.0;

w[i][j][0] = 0.0;

w[i][j][num_k - 1] = 0.0;

Fig. 1, Velocity and pressure iso-surfaces

     Of course constants need to be defined, such as grid spacing in space and time, density, kinematic viscosity. These equations have been validated, as you might have read and here already! Happy coding!

     If you want to hire me as your PhD student in the research projects related to turbo-machinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.