Upwind scheme (in a cell with zero velocity)
 

I am somewhat new to CFD and am wondering how the first order upwind scheme would be applied to a cell with a zero velocity. It is my understanding that, in the upwind scheme, the sign of the cell velocity is used to determine which cell velocities are applied at the cell faces. But what happens when the cell velocity is zero (how is the upwind direction determined)? This is probably not important in a large region of zero velocity cells, but I'm thinking about a zero velocity cell that is adjacent to a non-zero velocity cell.

where you have zero velocity there is no convective flux contributing to the balance....

But wouldn't it matter in a case like this?

|------------------|-------------------|-----------------|
|-------i-1---------|---------i---------|------- i+1-------|
|-----u(i-1)=1----- |-------u(i)=0------|----- u(i+1)=0---|
|------------------ |------------------|------------------|

---------------u(i-1/2)=???---------u(i+1/2)=0

So, for cell (i), shouldn't the first order upwind scheme assign a value of
u(i-1/2)=1? But how would the upwind scheme determine this, because the velocity at cell (i) is zero?

- In FD scheme, you evaluate u*du/dx which is zero at node i as u(i)=0.

- In FV scheme you need to evaluate the flux difference f(i+1/2) - f(i-1/2). Now you can evaluate the velocity at the interface by means of linear interpolation:

u(i-1/2) = 0.5*(u(i)+u(i-1)); u(i+1/2) = 0.5*(u(i)+u(i+1))

Then the sign will be the direction for evaluating the upwinded flux.

Note that your example is a discontinuity travelling

Yes. Thank you very much, I think I understand now. For computing the upwind flux direction, the velocity at the cell interface is computed by averaging the velocities at the cell centers that bound the interface. But, for the first-order upwind scheme (once the upwind direction has been determined), the interface fluxes are evaluated using the appropriate cell center velocity.