Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows

Abstract

We present a robust and accurate discretization approach for incompressible turbulent flows based on highorder discontinuous Galerkin methods. The DG discretization of the incompressible Navier-Stokes equations uses the local Lax-Friedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocity-pressure coupling terms. Stability of the discretization approach for under-resolved, turbulent flow problems is realized by a purely numerical stabilization approach. Consistent penalty terms that enforce the incompressibility constraint as well as inter-element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under-resolved regime. The penalty parameters are derived by means of dimensional analysis using penalty factors of order 1. Applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows. The proposed approach is applicable independently of the solution strategy used to solve the incompressible Navier-Stokes equations, i.e., it can be used for both projection-type solution methods as well as monolithic solution approaches. Since our approach is based on consistent penalty terms, it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems. Robustness and accuracy are verified for the Orr-Sommerfeld stability problem, the Taylor-Green vortex problem, and turbulent channel flow. Moreover, the accuracy of high-order discretizations as compared to low-order discretizations is investigated for these flow problems. A comparison to state-ofthe-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.

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

“…The factor highlights that the time‐step size has to be reduced for increasing polynomial degrees, where the exponent r of the polynomial degree k u of the discrete velocity solution is discussed in more detail below. We will demonstrate numerically that an exponent of r =1.5 accurately models the dependency on the polynomial degree, which has, for example, been used in the work of Fehn et al, as opposed to exponents of r =1 or r =2 used in other literature works (see also the work of Hesthaven and Warburton). For high‐Reynolds‐number flow problems with high‐spatial‐resolution requirements as considered in this work, one typically observes that the CFL condition is restrictive in the sense that the temporal discretization error is negligible for Courant numbers close to the critical value.…”

“…In previous works, the efficiency has been evaluated from different perspectives by concentrating on one of the three influence factors. For example, the accuracy of high‐order DG discretizations has been analyzed in detail in the work of Fehn et al for laminar flow problems and in another work of Fehn et al for turbulent flow problems. In the context of DG discretizations of the compressible Navier‐Stokes equations, the accuracy of high‐order methods has been investigated, eg, in the works of Gassner and Beck and Carton de Wiart et al for the Taylor‐Green vortex problem.…”

“…The viscous term as well as the negative Laplace operator in the pressure Poisson equation are discretized using the symmetric interior penalty method . Accordingly, the weak DG formulation of the above operators can be summarized as follows (see the works of Fehn et al and the references therein for more detailed derivations): …”

“…Similar terms have first been proposed in the works of Steinmoeller et al and Joshi et al as a means to stabilize the spatially discretized pressure projection operator and have been used in the work of Krank et al in the context of LES in combination with the dual splitting scheme. Recently, the divergence and continuity penalty terms have been analyzed rigorously in the work of Fehn et al, where it has been demonstrated that these terms lead to a robust and accurate discretization scheme for under‐resolved flows independently of the solution strategy used to solve the incompressible Navier‐Stokes equations. These penalty terms can be motivated from the point of view of mass conservation as well as energy stability and might be interpreted as a realization of divergence‐free H (div) elements in a weak sense.…”

“…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 .…”