Numerical Modeling of Natural Convection in Square Cavity

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% Plot Streamlines & Isotherms for Natural Convection across Square Cavity

% Defining constant properties and physical parameters

w=1; % Breadth

h=1; % Length

A=h/w; % Aspect Ratio of Enclosure

Pr=0.7; % Prandtl Number as given in the Reading Material

Ra=1000; % Rayleigh Number

% To include the Conduction between the sides without Natural Convection

% Dividing square sides into Nodal points/Grid

dx=0.01; % Grid length x-direction

dy=0.01; % Grid Length y-direction

X=(0:dx:1);

Thetafn=zeros(101,101); % Dimensionless Temperature

for i=1:101

  for j=1:101

   Thetafn(i,j)=1-X(j);

  end

end

% Iterative Procedure till Convergence is achieved

% Defining the 2-D Array of desired output parameters

Stmfn=ones(101,101); % Stream Function

Vortfn=ones(101,101); % Vorticity

% Assigning Relaxation Factors < 1

r=0.9;

rb=0.95; % rb>r

for n=1:500

% Assigning Output parameters with previous iteration values

for j=1:101

  for i=1:101

    Vortfn_prev(i,j)=Vortfn(i,j);

    Stmfn_prev(i,j)=Stmfn(i,j);

    Thetafn_prev(i,j)=Thetafn(i,j);

  end

end

% Computing Vorticity at nodal points within Square cavity

% Internal points

for j=2:100

  for i=2:100

    Vortfn(i,j)=((((-1)/(4*dx*dy*Pr))*(((Stmfn(i-1,j)-Stmfn(i+1,j))...

    *(Vortfn(i,j+1)-Vortfn(i,j-1)))-((Stmfn(i,j+1)-Stmfn(i,j-1))...

    *(Vortfn(i-1,j)-Vortfn(i+1,j)))))+((Vortfn(i,j+1)+Vortfn(i,j-...  1))/(dx^2))...     

     +((Vortfn(i-1,j)+Vortfn(i+1,j))/(dy^2))...

     -(Ra*((Thetafn(i,j+1)-Thetafn(i,j-1))/(2*dx))))/((2/(dx^2))+(2/(dy^2)));

    Vortfn(i,j)=Vortfn_prev(i,j)+(r*(Vortfn(i,j)-Vortfn_prev(i,j)));

  end

end

% Vorticity in Edge/Side Nodal points