Discontinuous Galerkin methods for the Reynolds averaged Navier-Stokes equations: turbulence modeling aspects and applications in aeronautics (part II)

In these notes we review the high-order Discontinuous Galerkin (DG) discretization of the Reynolds-averaged Navier-Stokes (RANS) equations closed with the Spalart-Allmaras (SA) one-equation and the Wilcox two-equation k-w (based on !e = log(w) in place of w) turbulence models. The present form of our implementations was introduced in Bassi et al. (2005) for compressible flows and in Crivellini et al. (2013) for incompressible flows. Moreover, we present the approach we took to implement the Explicit Algebraic Reynolds Stress Model (EARSM) of Wallin and Johansson (2000) into our high-order DG code MIGALE. The main objective of this work, performed within the IDIHOM project, was to investigate if the improved modelling of Reynolds stresses provided by the EARSM model, coupled with a high-order accurate numerical discretization, could result in better prediction of aerodynamic quantities of engineering interest at an acceptable additional computational cost. Papers about the high-order DG space discretization of the RANS equations are not very numerous and only deal with compressible flows. Persson et al. (2007) solved the RANS and the SA turbulence model equations using an artificial viscosity term in order to stabilize the discretization of the turbulence model equation. Oliver and Darmofal (2007) also developed a DG solver for the RANS equations and the SA model. They added an artificial dissipation, governed by an additional PDE, to avoid numerical oscillations in under-resolved regions. Landman et al. (2008) solved the RANS equations coupled with both the SA and k-! turbulence models. For the k-! model they used the same approach adopted by Bassi et al. (2005), while for the SA model a strong instability for negative values of eddy viscosity was observed. Therefore they implemented a limiting technique whereby negative values of the computed turbulence variable were reset to zero after each Newton iteration. Finally, Hartmann et al. (2010a) developed an adaptive high-order DG code for the RANS and k-! equations based on the same formulation of the k-!e equations proposed by Bassi et al. (2005). The Boussinesq hypothesis, at the basis of both the SA and k-! models, assumes that the principal axes of the Reynolds stress tensor are coincident with those of the mean strain-rate tensor. While for many flows of engineering interest models based on this as- sumption provide excellent predictions, nevertheless, in several flow situations, predicted flow properties largely di↵er from measurements. In order to improve the prediction ca- pabilities of turbulence models several authors replaced the Boussinesq linear constitutive law with a nonlinear relation that could embed the modelling level proper of the Reynolds stress models, see, e.g., Pope (1975), Rodi (1976), Taulbee (1992), Gatski and Speziale (1993). In this context Johansson and Wallin (1996) and Girimaji (1996) independently found, for 2D mean flows, the first “self-consistent”, closed and fully explicit solution of the nonlinear implicit algebraic equation, involving the mean flow strain-rate and rate-of- rotation tensors, for the Reynolds stress anisotropy tensor. Moreover, it was shown in Wallin and Johansson (2000) that good approximations are easily found for general 3D mean flows. The good properties of EARSM, in terms of compatibility with the Navier- Stokes equations, were demonstrated in the a priori analysis reported in Naji et al. (2004), comparing the model results with those obtained from DNS. An interesting feature of the Wallin and Johansson model is that it can easily fit in already existing turbulence models implementations. Improved modelling of turbulence effects coupled with an high-order accurate numerical discretization can provide a reasonably cheap compromise between standard RANS plus two-equation turbulence models simulations and more resolved but computational intensive simulations of turbulent flows.