Stabilization
When solving the linearized Euler (LE) and linearized Navier–Stokes (LNS) equations using the finite element method (the standard Galerkin formulation), it can be shown that the method looses its good approximation characteristics when convective terms are present and when these terms dominate locally (Ref. 19). This can lead to spurious numerical oscillations. To remedy these oscillations, stabilization is used.
It can be shown that if the cell Péclet number has a value larger than one, Pec > 1, the solution will oscillate. In the linearized Navier–Stokes interface, the variables lnsf.CellPe_th and lnsf.CellPe_v can be plotted to assess the characteristic number when comparing convection to viscous and thermal diffusion, respectively.
The Stabilization Methods
In the linearized Navier–Stokes (LNS) and linearized Euler (LE) interfaces, two stabilization methods are implemented. The Galerkin least squares (GLS) and the streamline upwind Petrov–Galerkin (SUPG) stabilization. It is in general recommended to use the GLS method. The SUPG method is implemented for completeness and can be used by experienced users. There is also the option to not use stabilization or to use the legacy method from version 5.2a and older. The legacy method for the LE interface is based on the implementation discussed in Ref. 10.
The implementation of the GLS stabilization methods follows the one discussed in Ref. 14, 19, 21, and 23. A general overview about stabilization methods can be found in Ref. 22. The GLS method combines stability and accuracy and the method order of accuracy is O(h2p+1) where p is the shape function order and h is the local mesh size. The SUPG method has an accuracy order of O(hp+1/2) for convection dominated problems and O(hp+1) for diffusion dominated problems.
The default discretization for the LNS and LE interfaces is to use linear elements for all the dependent variables (P1-P1-P1). This effectively removes the stabilization on the diffusive parts of the equation. The GLS method is still superior to the SUPG method as it also stabilizes the reactive terms. These are the terms where gradients of the background mean flow fields enter. If no stabilization is used, then set a P1-P2-P2 discretization for the dependent variables as it ensures a more stable numerical scheme.
The equation residuals can be visualized in the LE and the LNS interfaces by plotting, for example, lnsf.res_e (equation residual for the energy equation), lnsf.res_mx (equation residual for the momentum equation x-component), lnsf.res_my, lnsf.res_mz, and lnsf.res_e (equation residual for the energy equation).