DNS and LES on unstructured grids: playing with matrices to preserve symmetries using a minimal set of algebraic kernels

Summary

The essence of turbulence are the smallest scales of motion. They result from a subtle balance between two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and negative-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows in complex configurations. With this in mind, a fully-conservative discretization method for collocated unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: it preserves the symmetries of the differential operators and it has shown to be a very suitable approach for DNS and LES. On the other hand, an efficient cross-platform portability is nowadays one of the greatest challenges for CFD codes. In this regard, our leitmotiv reads: relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convection and diffusion matrices which are needed to determine the time-step `a la CFL. A new inexpensive method that allows this, without explicitly constructing these time-dependent matrices is proposed and tested. It only requires a sparse-matrix vector product where only the vector changes on time. Hence, apart from being significantly more efficient than the standard CFL condition, cross-platform portability is straightforward.

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