Van Leer flux for 3D
 

Hello everyone,

I'm trying to implement Van Leer flux for 3D, and Toro's book on Riemann solvers only mention the x-split equations for the flux.

I think it's fairly understandable that they expected the reader to derive the y and z split equations.I think it's straight forward, but wanted to verify if I was correct or wrong before committing to building the solver.

I tried to form the positive flux equation for the y and z direction fluxes.

Are the equations correct?

Well, if you strictly rewrite the 1D counterparts in the generic (i,j,k) directions, you apply an index permutation. Formally, that should imply also a directional Mach number Mi=ui/a. Differently you have to consider the velocity magnitude.
Anyway, a real multi-dimensional implementation is somehow more complicate.

Yes you're correct. The Mach number will be different for each direction and of form Mi = ui/a.

So, the formulas I derived previously, were wrong, and possibly unnecessary.

Because, in every reference paper/book, they're not giving the full flux equations for the x,y,z directions in 3D.

Even in this paper, only the Mach number difference is shown: https://www.abcm.org.br/anais/encit/...gos/1_4669.pdf

At section 4.2 it says similar expressions are obtained by using the Mach numbers My = v/a and Mz = w/a.

I'm starting to believe that we only need to calculate the Mach number for each direction (Mx, My, Mz), and substitute those in the equation, and that'll actually be the flux contribution in the x,y,z directions.

That is, Fx = Fx(rho,a,Mx) is given to us, so Fy = Fy(rho,a,My), and Fz = Fz(rho,a,Mz). (This kind of makes sense to me, as neither rho, or a, changes as in the different directions. They remain constant in the control volume, but the Mach number can change in different directions, as we're splinting the velocity it into parts, and using that to calculate the Mach number)

That is, we use the same equation, and we just substitute different values of Mach number to get the fluxes in x,y,z direction.

Am I missing anything else?

That's one reason I love unstructured methods, they really take 1/3 of the effort in 3D [emoji6]

I found a paper which does multidimensional van leer flux splitting for cartesian and curvilinear coordinates: https://arc.aiaa.org/doi/10.2514/3.9858

The paper, and even Toro's book, once mentioned that we can use a rotation matrix that will allow us to split the three dimensional flux vector, by treating it similar to a one dimensional case..

Seems possible, but my trust is low.

I might just use Roe flux for tetrahedral elements in the end, if this doesn't work.