On the entropy-viscosity method for flux reconstruction

  • Font, Bernat (Barcelona Supercomputing Center)
  • Miró, Arnau (Barcelona Supercomputing Center)
  • Lehmkuhl, Oriol (Barcelona Supercomputing Center)

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Recent advances in modern computer architectures for high-performance computing are paving the path towards a wider adoption of high-order (HO) methods within the computational fluid dynamics (CFD) community. Compared to traditional low-order methods, HO methods promise to achieve an arbitrary level of accuracy at a reduced computational cost. During the last decade, research on HO methods for CFD has been focused on achieving a similar level of maturity as traditional low-order methods. To do so, classical turbulence models, shock-capturing schemes, and convection schemes, among other key features, need to be revisited for the HO approach. With this purpose, we investigate the entropy-viscosity shock-capturing scheme by Guermond (2011) for the flux reconstruction (FR) method, originally proposed by Huynh (2007). In the recent years, FR has been gaining attention due to its simple formulation and unifying framework, being able to recover other high-order methods such as nodal discontinuous Galerkin and spectral difference methods. The entropy viscosity scheme has been implemented in a 1-D flux reconstruction solver written in Julia. In contrast to other works, the implementation of the entropy-viscosity scheme is performed element-wise, ie. a single elemental viscosity is used taken as the maximum absolute norm of the values computed at the element solution points. This has proven to be more stable than the point-wise counterpart. Encouraging preliminary results have been obtained for the 1D Burgers equations. Two initial conditions have been explored, a sine and a square wave. The solution is obtained using polynomials from third to fifth degree, and Gauss-Legendre and Gauss-Lobatto (GLL) quadrature points. The numerical experiments show that the entropy viscosity is needed to stabilize the shock when using GLL quadrature nodes and a Roe solver. For both quadrature points, oscillations are reduced on the shock interface without completely eliminating them, and the shock is better captured with increasing polynomial degree. Moreover, the addition of the entropy viscosity allows to correctly capture the square wave shock, although some dissipation is present since the shape of the wave is not completely recovered. The next step is to test this method in the well-known Sod shock problem for the 1D Euler equations. Moreover, split formulations of the governing equations will be explored as a de-aliasing mechanism.