3D Navier-Stokes + Heat Transfer

variational formulation
9

I added LSIC and to my surprise it made the solver converge at higher velocites (2x5.67 m/s)

The LSIC constant, often denoted \tau_\text{C}, is typically selected as

\tau_{\text{C}} = h^2/\tau_{\text{M}}\text{ ,}

which is motivated by the original error analysis for a linearized model problem.

In order to make SUPG/PSPG converge I had to use this formula with Reynolds number for Cinv

There should be some velocity-independent value for C_\text{inv}. (In principle, you can compute it exactly by solving a local eigenvalue problem in each element, as discussed in this paper, but, in practice, everyone just uses some heuristic formula in terms of the element degree). For purposes of nonlinear convergence, it can help to use “smoothed” versions of the \text{min} formula that comes from the analysis, e.g.,

\text{min}\,\{a,b\}~~~\to~~~\left(a^{-2} + b^{-2}\right)^{-1/2}\text{ ,}

and generalizations of “h” based on the full Jacobian of the mapping from the parent element to physical space (as in the FSI example I linked above), but this is, again, a separate issue from accuracy of the resulting converged algebraic solution.

I will mention, though, that when I’ve previously tried using an SUPG/PSPG(-ish) formulation with Taylor–Hood elements, I needed larger values of C_{\text{inv}} than I was used to using. (This type of stabilization is mostly used with equal-order interpolation of velocity and pressure, because PSPG stabilization means you no longer need to worry about inf-sup stability of the velocity–pressure pair.)

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