[matthias]

Hi,

I have two additional terms for convection in NS equation.

div((U*U)+(U*S) + (S*U))

where U is velocity vector and S another vector. In OF notation the equation can be written as

fvm::div(phi, U) + fvc::div(U*S) + fvc::div(S*U)

I would like to discretize all terms implicitly. So one can rewrite the equation in OF notation as

fvm::div(phi ,U) + fvm::div(phiS, U) + fvc::div(U*S)

with ssf phiS = fvc::interpolate(S) & mesh.Sf(). But i have no idea, how to discretize the last explicit term implicitly. Is it possible to rewrite it in an implicit way? Maybe someone has an advice for me.


Best regards

Matthias

[ngj]

Hallo Matthias,

It is a bit unclear, how your vector manipulations are defined, but why does (S*U) differ from (U*S)?

Is the one a scalar and the other a tensor, i.e. differs in terms of either inner or outer products?

Kind regards,

Niels

[matthias]

Hi Niels,

sorry for that. The vector operation (*) between vector U and S is the outer product. The final tensors (U*S) and (S*U) are different and therefore the divergence operation too. In the matrix Ueqn you solve for U applying an explicit or implicit discretization for convection. The term div(U*U) can be easily discretized using face flux phi and U as fvm::div(phi, U). That's the standard way in OF.
Here we have two additional terms (U*S) and (S*U), which have to be added to the stresses (U*U) + (U*S) + (S*U). Applying the divergence (or convection) you can write (in OF notation) fvm::div(phi, U)+fvc::div(U*S)+fvm::div(phiS,U) where phiS is the flux of S interpolated on the faces. The first term and the third term is now on the lhs of the matrix Ueqn whereas the second term remains on the rhs as it is explicit. Since we are solving for U in the matrix Ueqn we cannot write fvm::div(phi, S).
I'm looking for a way to make the term fvc::div(U*S) also implicit to improve the stability of the underlying turbulence model (or more precisely the numerical solution procedure of the model). In the original modeling all terms have been solved explicitely.
Best regards

Matthias

[ngj]

Hi Matthias,

Sorry, of course the are different. By applying the continuity equation (I assume incompressible flow) to the last term, it becomes the inner product of the velocity vector and the gradient of the vector S. Therefore, to have a completely implicit implementation of this you will need the (a) block coupled approach for the momentum equation. Otherwise, you will not be able to handle all of the velocity components implicitly.

However, depending on the sign of grad(S) you want to treat this term either as an implicit or explicit contribution due to stability reasons.

Kind regards,

Niels