ME 5302 Computational fluid mechanics

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1 ME 5302 Computational fluid mechanics Assignment Part II Thursday, 23 April 2015 National University of Singapore ARAVIND BASKAR A

2 Question 1 Solution: Problem Statement:

3 Analysis: The conservative form of the navier stokes equation has been employed for the analysis. Pressure correction is used for updating the values of velocity and temperature. The algorithm for solving the problem is as shown below: Input parameters Boundary Conditions & Stability Criteria Temporary Velocity using NS equation - Conservative form Pressure Correction and error check Update Velocity Energy equation solution Plot Results Conservative form of Navier Stokes Equation is as follows: u t + u2 x + uv x = Pr u ( 2 x u P y2) x v t + v2 y + uv y = Pr u ( 2 x u P y2) + Pr. Ra. T x T t + ut x + vt y = Pr T ( 2 x T y 2)

4 The plots at various resolutions are as follows: Fig (a) Grid Size:30x30, Velocity profiles at Ra=10 3 Fig (b) Grid Size: 30x30, Velocity contour at Ra=10 3

5 X 10 3 X Fig (c) Grid Size: 30x30, Velocity contour at Ra=10 5

6 Fig (c) Grid Size: 30x30, Velocity contour at Ra=10 5 X 10-1

7 Fig (c) Grid Size: 30x30, Velocity contour at Ra=10 3 Fig (c) Grid Size: 30x30, Velocity contour at Ra=10 3

8 Fig (c) Grid Size: 20x20, Velocity contour at Ra=10 3 X 10-1

9 Fig (d) Grid Size: 20x20, Velocity Contour

10 Matlab Code % ME Computational Fluid Dynamics % Assignment - Cavity Flow with natural convection % Date - 23/4/2015 % Aravind Baskar - A J % Input parameters nx=10;ny=10; % Grid Points in x & y hx=1/nx;hy=1/ny; % Grid Size in x & y T_left=1; % Left Wall Temperature u_top=0; % Initial Velocity Ra=1e5; % Rayleigh Number Pr=0.7; % Prandtl Number tf=0.15; % Total time Beta=1.5; % Correction factor MaxIt=100; % Iterations MaxErr=0.001; % Error dt=1e-4; % Time step size Maxstep=tf/dt; % Time steps % Initialize vectors u=zeros(nx+1,ny+2); u_temp=zeros(nx+1,ny+2); v=zeros(nx+2,ny+1); v_temp=zeros(nx+2,ny+1); P=zeros(nx+2,ny+2); T=ones(nx+2,ny+2); t=0; %% Time Loop % Boundary conditions for n=1:maxstep %% Left Wall u(1,2:ny+1)=0; v(1,1:ny+1)=2*0-v(2,1:ny+1); T(1,2:ny+1)=T_left; %% Right Wall u(nx+1,2:ny+1)=0; v(nx+2,1:ny+1)=2*0-v(nx+1,1:ny+1); T(nx+1,2:ny+1)=T(nx+1,2:ny+1); %% Top Wall v(2:nx+1,1)=0;

11 u(1:nx+1,ny+2)=2*u_top-u(1:nx+1,ny+1); T(2:nx+1,ny+2)=T(2:nx+1,ny+1); %% Bottom Wall v(2:nx+1,ny+1)=0; u(1:nx+1,1)=2*0-u(1:nx+1,2); T(2:nx+1,1)=T(2:nx+1,2); % Temporary velocity and temperature for i=2:nx for j=2:ny+1 usqr_1=(0.5*(u(i+1,j)+u(i,j)))^2; usqr_2=(0.5*(u(i-1,j)+u(i,j)))^2; uv_x1=(0.5*(u(i,j)+u(i,j+1)))*(0.5*(v(i,j)+v(i+1,j))); uv_x2=(0.5*(u(i,j)+u(i,j-1)))*(0.5*(v(i,j-1)+v(i+1,j-1))); A=((usqr_1-usqr_2)/hx)+((uv_x1-uv_x2)/hy); D2_u=((u(i+1,j)-(2*u(i,j))+u(i-1,j))/hx^2)+((u(i,j+1)-(2*u(i,j))+u(i,j- 1))/hy^2); u_temp(i,j)=u(i,j)+(dt*((pr*d2_u)-a)); for i=2:nx+1 for j=2:ny vsqr_1=(0.5*(v(i,j+1)+v(i,j)))^2; vsqr_2=(0.5*(v(i,j)+v(i,j-1)))^2; uv_y1=(0.5*(u(i,j)+u(i,j+1)))*(0.5*(v(i,j)+v(i+1,j))); uv_y2=(0.5*(u(i-1,j+1)+u(i-1,j)))*(0.5*(v(i,j)+v(i-1,j))); B=((vsqr_1-vsqr_2)/hy)+((uv_y1-uv_y2)/hx); D2_v=((v(i+1,j)-(2*(v(i,j)))+v(i-1,j))/hx^2)+((v(i,j+1)-(2*(v(i,j)))+v(i,j- 1))/hy^2); v_temp(i,j)=v(i,j)+(dt*((pr*d2_v)+(pr*ra*t_temp)-b)); % Pressure Correction for it=1:maxit P_chk=P; for i=2:nx+1 for j=2:ny+1 if(i==2&&j==2) (i==nx+1&&j==2) (i==2&&j==ny+1) (i==nx+1&&j==ny+1) C=1; elseif(i>=3&&i<=nx&&j==2) (i==2&&j>=3&&j<=ny) (i==nx+1&&j>=3&&j<=ny ) (i>=3&&i<=nx&&j==ny+1) C=2/3;

12 else C=1/2; P(i,j)=((C*Beta/(hx^2+hy^2))*((hy^2*(P(i+1,j)+P(i-1,j)))+... (hx^2*(p(i,j+1)+p(i,j-1)))-(((hx*hy^2)/dt)*(u_temp(i,j)-... u_temp(i-1,j)))-(((hx^2*hy)/dt)*(v_temp(i,j)-v_temp(i,j-1)))))+ ((1-Beta)*(P(i,j))); Err=0; for i=2:nx+1 for j=2:ny+1 Err=Err+(abs(P_chk(i,j)-P(i,j-1))); ; ; if Err<=MaxErr break % Velocity Updation for i=2:nx for j=2:ny+1 u(i,j)=u_temp(i,j)-((dt/hx)*(p(i+1,j)-p(i,j))); for i=2:nx+1 for j=2:ny v(i,j)=v_temp(i,j)-((dt/hy)*(p(i,j+1)-p(i,j))); for i=2:nx+1 for j=2:ny+1 A_T=(0.5*(u(i-1,j)+u(i,j)))*((T(i+1,j)-T(i-1,j))/(2*hx))+((0.5*(v(i,j- 1)+v(i,j)))*((T(i,j+1)-T(i,j-1))/(2*hy))); D2_T=((T(i+1,j)-(2*T(i,j))+T(i-1,j))/hx^2)+(T(i,j+1)-(2*T(i,j))+T(i,j- 1)/hy^2); T(i,j)=T(i,j)+(dt*(D2_T-A_T)); t=t+dt;

13 % Plotting results P(1:nx+2,1)=0;P(1:nx+2,ny+2)=0; P(1,1:ny+2)=0; P(nx+2,1:ny+2)=0; u_cnt(1:nx+1,1:ny+1)=0.5*(u(1:nx+1,1:ny+1)+u(1:nx+1,2:ny+2)); v_cnt(1:nx+1,1:ny+1)=0.5*(v(1:nx+1,1:ny+1)+v(2:nx+2,1:ny+1)); P_cnt(1:nx+1,1:ny+1)=0.25*(P(1:nx+1,1:ny+1)+P(2:nx+2,1:ny+1)+P(1:nx+1,2:ny+2)+P(2:nx+2, 2:ny+2)); wt(1:nx+1,1:ny+1)=((v(2:nx+2,1:ny+1)-v(1:nx+1,1:ny+1))/hx)-((u(1:nx+1,2:ny+2)- u(1:nx+1,1:ny+1))/hy); T_cnt(1:nx+1,1:ny+1)=0.25*(T(1:nx+1,1:ny+1)+T(2:nx+2,1:ny+1)+T(1:nx+1,2:ny+2)+T(2:nx+2, 2:ny+2)); x(1:nx+1)=(0:nx); y(1:ny+1)=(0:ny); figure(1), quiver(x,y,(rot90(fliplr(u_cnt))),(rot90(fliplr(v_cnt)))),... xlabel('nx'),ylabel('ny'),title('velocity Vector Plot');axis('square','tight'); figure(2), subplot(211),contourf(x,y,rot90(fliplr(u_cnt)),30),... xlabel('nx'),ylabel('ny'),title('u-velocity');axis('square','tight');colorbar; subplot(212),contourf(x,y,rot90(fliplr(v_cnt)),30),... xlabel('nx'),ylabel('ny'),title('v-velocity');axis('square','tight');colorbar; figure(3),contourf(x,y,rot90(fliplr(p_cnt)),30),... xlabel('nx'),ylabel('ny'),title('pressure');axis('square','tight');colorbar; figure(4),contourf(x,y,rot90(fliplr(wt)),30),... xlabel('nx'),ylabel('ny'),title('vorticity');axis('square','tight');colorbar; figure(5),contourf(x,y,rot90(fliplr(t_cnt)),30),... xlabel('nx'),ylabel('ny'),title('temperature');axis('square','tight');colorbar; Inferences: The velocity of flow is increased as the Rayleigh number is increased from 10 3 to Convergence time reduces as the Rayleigh number is increased. Velocity contours are almost zero at the centre and the gradient increases as it nears the walls. Temperature can undergo an inversion at the centre of the cavity Temperature gradient is increased as it nears the walls as Ra increases Ra Numin Numax Nuavg Umax Vmax Numin Numax Nuavg Umax Vmax 30X30 20X

14 References: 1. Numerical simulation of natural convection in a square cavity by simple generalized differential quadrature method by C.Shu,K.H.A Wee, Computer & Fluids Journal (Pg ). 2. D.C. Wan, B.S.V. Patnaik and G.W. Wei, A new benchmark quality solution for the buoyancydriven cavity by discrete singular convolution. Taylor & Francis, Numerical Heat Transfer, Part B, Vol. 40, pages , D.A. Mayne, A.S. Usmani and M. Crapper, h-adaptive finite element solution of high Rayleigh number thermally driven cavity problem. International Journal of Numerical Methods for Heat & Fluid Flow,Vol. 10, No. 6, pages , 2000.

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