Implicit Discontinuous Galerkin methods with efficient time integration for incompressible variable density and compressible turbulent flows

With this research work we want to investigate the use of the state of art implicit high-order time integration schemes to integrate in time high-order Discontinuous Galerkin (DG) space discretization of compressible and incompressible flow model equations. The final goal is to demonstrate the capabilities of an high-order accurate, both in space and time, simulation of turbulent flows. DG methods proved to be very well suited for the Direct Numerical Simulation (DNS) and the Large Eddy Simulation (LES) of turbulent flows thanks to good dissipation and dispersion properties. However, accurate turbulent flow simulations imply long term and stiffly stable time integrations. Explicit Singly Diagonally Implicit Runge-Kutta (ESDIRK) schemes, linearly implicit Rosenbrock-type Runge-Kutta (Rosenbrock) schemes and linearly implicit two-step peer (Peer) methods are implicit time integration schemes that provide very high accuracy combined with excellent stability properties. Nevertheless, since they are implicit schemes they entail the solution of several systems on linear or non-linear equations by means of iterative methods and thus can require an high computational cost. In order to reduce this cost we developed, implemented and validated the automatic step-size control, the initial guess approach and the stopping criterion for iterative methods. Furthermore, we derived a new starting procedure able to preserve the accuracy order of non self-starting multi-step Peer schemes. The potential of the proposed high-order coupling between DG method and implicit temporal schemes has been exhaustively examined on compressible and incompressible benchmark test cases and demonstrated by computing the implicit Large Eddy simulation of massively separated compressible flows over periodic hills at Re = 10595, a challenging and deeply analysed turbulent test case that is part of the test case repository defined inside the EU project TILDA (Towards Industrial LES/DNS in Aeronautics - Paving the Way for Future Accurate CFD) at which the Computational Fluid Dynamics group of Università degli Studi di Bergamo is engaged. In this work we present, in addition, a high-order Discontinuous Galerkin approach for the simulation of variable density incompressible (VDI) flows developed inside the International Research Training Group project DROPIT (Droplet Interaction Technologies). The purpose of this project is the development of an high accurate and efficient method for the thorough investigation of interface problems for incompressible flows. The method is fully implicit and applies to the VDI Navier–Stokes equations. More in particular, the density is treated as a purely advected property tracking possibly multiple (more than two) fluids. Furthermore, the fluids interface is captured in a diffuse fashion by the high-degree polynomial solution thus not requiring a geometrical reconstruction and preserving the mass conservation. Density over/undershoots, spurious oscillations at flows interfaces and Godunov numerical fluxes at inter-element boundaries are numerical issues investigated during the development of the present approach. Promising results on numerical experiments involving high-density ratios (water-air) and the possible interaction of more than two fluids have been obtained using a very high-order polynomial representation of the solution on relatively coarse grids.