An energy-preserving unconditionally stable method for DNS and LES simulations on unstructured grids

  • Trias, Francesc Xavier (Technical University of Catalonia)
  • Santos, Daniel (Technical University of Catalonia)
  • Hopman, Johannes (Technical University of Catalonia)
  • Gorobets, Andrey (Keldysh Institute of Applied Mathematics)
  • Oliva, Assensi (Technical University of Catalonia)

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The essence of turbulence are the smallest scales of motion. They result from a subtle balance between convective transport and diffusive dissipation. Mathematically, these terms are governed by two differential operators differing in symmetry: the convective operator is skew-symmetric whereas the diffusive is symmetric and positive-definite. On the other hand, accuracy and stability need to be reconciled for simulations of turbulent flows in complex geometries. With this in mind, an energy-preserving discretization for unstructured grids was proposed in Ref.[1]: it exactly preserves the symmetries of the underlying differential operators on collocated grids. Hence, unlike other formulations, the discrete convective operator transports energy from a resolved scale of motion to other resolved scales without dissipating energy, as it should do from a physical point-of-view. Therefore, we think that apart from being a right approach for large-scale DNSs of turbulence, it also forms a solid basis for testing subgrid scale LES models. The discretization is based on only five operators (i.e. matrices): the cell-centered and staggered control volumes (diagonal matrices), the face normal vectors, the cell-to-face interpolation, and the cell-to-face divergence operator. Therefore, it constitutes a robust approach that can be easily implemented in already existing codes such as OpenFOAM [2]. Moreover, for the sake of cross-platform portability and optimization, CFD algorithms must rely on a very reduced set of (algebraic) kernels (e.g. sparse-matrix vector product, SpMV; dot product; linear combination of vectors). This imposes restrictions and challenges that need to be addressed such as the inherent low arithmetic intensity of the SpMV, the reformulation of flux limiters [3] or the efficient computation of eigenbounds to determine the time-step. Results showing the benefits of symmetry-preserving discretizations will be presented together with novel methods aiming to keep a good balance between code portability, numerical robustness and performance.