A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

Abstract

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at Re τ = 180 as well as 590.

Efficiency of high‐performance discontinuous Galerkin spectral element methods for under‐resolved turbulent incompressible flows

“…The diagonal of the matrix required by the Chebyshev smoother is precomputed in the setup phase. Accordingly, all components of the multigrid method including level transfer operators are implemented in a matrix‐free way, and we refer to the work of Krank et al for more detailed information. For the projection equation as well as the Helmholtz‐like equation, the inverse mass matrix is an effective preconditioner due to the relatively small time‐step sizes related to the CFL condition (see also the work of Shahbazi et al).…”

“…Despite the full flexibility of DG methods with respect to geometry representation and nonlinear operators, our implementation is close to being memory bandwidth bound and, thus, almost reaching the throughput of simple finite difference stencils. Finally, all solver components have been completely parallelized with the message passing interface (MPI) and tuned to scale to processors and tens of billions of unknowns as shown in the work of Krank et al, particularly the iterative linear solvers, even though the computations performed in this work with up to 4·10 9 unknowns run on a smaller scale. In the field of high‐order continuous finite element methods, strong scaling experiments up to 5·10 5 cores and approximately 2 billion grid points have been shown in the work of Offermans et al for the spectral element solver Nek5000.…”

“…It is based on the high‐order dual splitting scheme for discretization in time, which has first been used in the work of Hesthaven and Warburton in combination with DG methods for discretization in space. Instabilities of this discretization approach occurring for small time‐step sizes have been discussed and solved in the works of Krank et al and Fehn et al by a proper DG discretization of the velocity‐pressure coupling terms. A stabilization of the DG discretization for under‐resolved flows based on a consistent divergence penalty term and a consistent continuity penalty term has been developed in the works of Krank et al and Fehn et al, which renders this approach a highly attractive candidate for implicit LES.…”

“…Instabilities of this discretization approach occurring for small time‐step sizes have been discussed and solved in the works of Krank et al and Fehn et al by a proper DG discretization of the velocity‐pressure coupling terms. A stabilization of the DG discretization for under‐resolved flows based on a consistent divergence penalty term and a consistent continuity penalty term has been developed in the works of Krank et al and Fehn et al, which renders this approach a highly attractive candidate for implicit LES. For example, this approach has been applied to large‐scale computations of turbulent flows such as the direct numerical simulation of periodic hill flow in the work of Krank et al Our implementation is based on high‐performance matrix‐free methods for tensor product finite elements developed recently in the works of Kronbichler and Kormann for both continuous and discontinuous Galerkin methods, exhibiting excellent performance characteristics .…”

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

“…While the focus is on the node‐level performance of the matrix‐free implementation in the present work, we mention that our approach is also well suited for massively parallel computations on modern high‐performance computing clusters. For parallel runs on large‐scale supercomputers, the implementation uses an efficient Message Passing Interface (MPI) parallelization where parallel scalability of up to cores has been shown in the works of Krank et al and Kronbichler and Wall …”