Discretization of convective term in coupled approach
 

Hello all,

I would like to ask for your opinion on the spatial discretization of the convective term in the momentum equation for the case of a non-linear coupled approach.

Which of the three options (if any) do you think makes the most sense? Picture attached for the case of the west side CV face.

I understand you are talking about density based solver when you say coupled approach.

In the coupled approach, taking an example of 1D solver, you will be typically solving for
density, momentum and energy.

momentum is ( rho. velocity).

The momentum is the quantity that would be convected with velocity v or u whatever you want to call.

What does it mean, is that you have a flux variable that is velocity dot area. Then you have transport variable ie momentum, interpolated by some scheme (upwind, second order upwind , MUSCL etc etc). Then you set up a Reimann problem and solve it.

The flux will also contain some sort of dissipation , see ROE approach or the flux could be separated into two fluxes based on characteristics. (AUSM type).

Perhaps I misused the term coupled, so I will expand the original question a bit. What I want to do is solve the 3 transport equations plus the equation of state for a 1D compressible flow in coupled manner (so all 4 equations at once by using the Newton-Raphson method since there are non-linear terms present).

I would like to avoid a linear solver (which I believe discretizes the convective term in the same way as the segregated approach - so (rho*u) of the previous iteration multiplied with the new u) so that means that with a non-linear solver I am stuck with the rho*u^2 term which I have to discretize in space.

This is what I am unsure how to do. The 3 ideas that popped into my mind are in the attached picture above, so I wanted to check if any of them make sense.

By the way, when you mentioned that you interpolate the momentum based on upwind or other scheme. Don't you actually do this for the velocity term? So if I write (rho*u')*u'' wouldn't you upwind u''? Or did I misunderstand your answer?

I would always consider the discretization of the total flux for fulfilling the conservative property

Well in any case, you need to evaluate the function and its partial derivatives to construct jacobian for it.

One solver that does it is Polyflow.


Well there are two ways you can construct these variables at face.

First one is that you interpolate everything by CDS that is rho.velocity.velocity is say average from both side. (This is very likely to be unstable for you, because of nature of CDS).
Second way is that you construct transported variable by schemes like upwind etc.
This transported variable is transported by velocity.

Now the main issue for you to understand is for rho.velocity.velocity whether to transport velocity with flux rho.velocity or whether to transport momentum with flux of velocity (not rho . velocity).

The answer could be guessed from list of transported variables. You will observe that the continuity is transport equation for density. That means that density is transported as well. It means you need density from scheme too.

Once we know density is transported, you see that if velocity is also transported then momentum has to be transported with flux as velocity dot area.

That means you transport density, transport momentum with velocity.

In conclusion transport variables with velocity constructed by CDS scheme.

Thank you both for your explanations!