a b s t r a c tIn this work we investigate the use of linearly implicit Rosenbrock-type Runge-Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier-Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier-Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil.

SUMMARYDiscontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very highorder accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p-multigrid solution strategy has been developed, which is based on a semi-implicit Runge-Kutta smoother for high-order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p-multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases.

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In this work the capabilities of a recently developed high-order accurate Discontinuous Galerkin (DG) solver, named MIGALE, are assessed for the computation of turbomachinery flows. The Reynolds averaged Navier-Stokes equations coupled with the two equations k-ω turbulence model are solved to predict the flow features, either in a fixed or rotating reference frame, to simulate the fluid flow around bodies that operate under an imposed steady rotation. To ensure, by design, the positivity of all thermodynamic variables at a discrete level, a set of primitive variables based on pressure and temperature logarithms is used. The flow fields through the MTU T106A low-pressure turbine cascade and the NASA Rotor 37 axial compressor have been computed up to fourth-order of accuracy and compared to the experimental and numerical data available in the literature.

Abstract. This chapter presents high-order DG solutions of the RANS and k-ω turbulence model equations for transonic flows around aeronautical configurations. A directional shock-capturing term, proportional to the inviscid residual, is employed to control oscillations around shocks. Implicit time integration is applied to the fully coupled RANS and k-ω equations. Several high-order DG results of 2D and 3D transonic turbulent test cases proposed within the ADIGMA project demonstrate the capability of the method.