An interior penalty stabilised incompressible discontinuous Galerkin-Fourier solver for implicit large eddy simulations

We present an implicit Large Eddy Simulation (iLES) h/p high order (≥2) unstructured Discontinuous Galerkin–Fourier solver with sliding meshes. The solver extends the laminar version of Ferrer and Willden, 2012 [34], to enable the simulation of turbulent flows at moderately high Reynolds numbers in the incompressible regime. This solver allows accurate flow solutions of the laminar and turbulent 3D incompressible Navier–Stokes equations on moving and static regions coupled through a high order sliding interface. The spatial discretisation is provided by the Symmetric Interior Penalty Discontinuous Galerkin (IP-DG) method in the x–y plane coupled with a purely spectral method that uses Fourier series and allows efficient computation of spanwise periodic three-dimensional flows.

Since high order methods (e.g. discontinuous Galerkin and Fourier) are unable to provide enough numerical dissipation to enable under-resolved high Reynolds computations (i.e. as necessary in the iLES approach), we adapt the laminar version of the solver to increase (controllably) the dissipation and enhance the stability in under-resolved simulations. The novel stabilisation relies on increasing the penalty parameter included in the DG interior penalty (IP) formulation. The latter penalty term is included when discretising the linear viscous terms in the incompressible Navier–Stokes equations. These viscous penalty fluxes substitute the stabilising effect of non-linear fluxes, which has been the main trend in implicit LES discontinuous Galerkin approaches.

The IP-DG penalty term provides energy dissipation, which is controlled by the numerical jumps at element interfaces (e.g. large in under-resolved regions) such as to stabilise under-resolved high Reynolds number flows. This dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps, thus providing an efficient stabilization mechanism for iLES.

The IP-DG stabilisation is complemented with a Spectral Vanishing Viscosity (SVV) method, in the z-direction, to enhance stability in the continuous Fourier space. The coupling between the numerical viscosity in the DG plane and the SVV damping, provides an efficient approach to stabilise high order methods at moderately high Reynolds numbers.

We validate the formulation for three turbulent flow cases: a circular cylinder at Re=3900, a static and pitch oscillating NACA 0012 airfoil at Re=10000 and finally a rotating vertical-axis turbine at Re=40000, with Reynolds based on the circular diameter, airfoil chord and turbine diameter, respectively. All our results compare favourably with published direct numerical simulations, large eddy simulations or experimental data.

We conclude that the DG-Fourier high order solver, with IP-SVV stabilisation, proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery.