I found a MATLAB routine, the code provided below, where numerical flux is calculated by using a method at the cell which share its face with a solid wall. And this numerical flux(H) is then used to solve the momentum equations(governing equations) in a finite volume method. Another different method is used to calculate the numerical fluxes of the other faces of the same cell that are share with another cells. But I can't figure out which method is applied in the former case. This code is written for a FVM for 2D inviscid supersonic flow. In this code, unL=contravariant velocity, qL=flow velocity. It will be convenient if you can explain me what is utL. I am totally unfamiliar with this term. It will be helpful for me as well if you can suggest me some methods for calculating flux at solid wall.
% The inputs are:
% UL: state vector in left cell (in the domain)
% n: normal pointing from boundary into the left cell
% gamma: Ratio of specific heats
%
% The outputs are:
% H: the flux into the left cell
% smax: the maximum propagation speed of disturbance
function [H, smax] = fluxWall(UL, n, gamma)
rL = UL(1); %density
uL = UL(2)/rL; % velocity in x-direction
vL = UL(3)/rL; % velocity in y-direction
unL = uL*n(1) + vL*n(2);
qL = sqrt(UL(2)^2 + UL(3)^2)/rL;
utL = sqrt(qL^2 - unL^2);
pL = (gamma-1)*(UL(4) - 0.5*rL*utL^2);
rHL = UL(4) + pL;
cL = sqrt(gamma*pL/rL);
smax = abs(unL) + cL;
H = zeros(size(UL));
H(2) = pL*n(1);
H(3) = pL*n(2);
If I understand your post, the flux at the wall is a BC and is not computed but is prescribed. The flux at the left should be for the face inside the fluid.
Dr. Filippo Maria Denaro,
Thanks for your reply. I am sorry for not mentioning that these fluxes are actually numerical fluxes(H) which are used to solve the momentum equations of governing equations.
This routine returns a numerical flux(H) of a cell that share at least one face with the wall. and n is the vector normal to that wall. Please take a look at the picture I have attached. Suppose, I want to calculate the numerical fluxes for the all three faces of the green colored cell. These numerical fluxes are then summed up and used to solve the governing equations for this cell. The way to approximate numerical flux for edge/wall will obviously be different than the other two edges of this green triangle. I just want to know how the researcher deal with the cell that shares a face with a solid wall while developing a finite volume method.
As I wrote, the face coincident with the wall requires a BC, that is a known condition for the flux not a computation. For example, the normal velocity is zero for not-permeable wall, the heat flux is known (zero or prescribed), etc.
The other two edges require the computation of the numerical fluxes in the interior of the fluid domain.
Dr. Filippo Maria Denaro,
Thanks again. This time i got it perfectly.
Now, I would like to know, in case of inviscid supersonic flow, what kind of CONDITIONS I can implement in a face that coincide with the wall? and I know nothing about that face except the normals and the state vector of the cell beside it.
On the wall you can set vn=0 so that the mass flux has zero normal component. It enters also into the normal component of the momentum and total energy. On the wall you must know the heat flux, If the wall is adiabatic it is just zero
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