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The present paper addresses the numerical solution of turbulent flows with high-order discontinuous Galerkin methods for discretizing the incompressible Navier-Stokes equations. The efficiency of high-order methods when applied to under-resolved problems is an open issue in literature. This topic is carefully investigated in the present work by the example of the 3D Taylor-Green vortex problem. Our implementation is based on a generic high-performance framework for matrix-free evaluation of finite element operators with one of the best realizations currently known. We present a methodology to systematically analyze the efficiency of the incompressible Navier-Stokes solver for high polynomial degrees. Due to the absence of optimal rates of convergence in the under-resolved regime, our results reveal that demonstrating improved efficiency of high-order methods is a challenging task and that optimal computational complexity of solvers, preconditioners, and matrix-free implementations are necessary ingredients to achieve the goal of better solution quality at the same computational costs already for a geometrically simple problem such as the Taylor-Green vortex. Although the analysis is performed for a Cartesian geometry, our approach is generic and can be applied to arbitrary geometries. We present excellent performance numbers on modern, cache-based computer architectures achieving a throughput for operator evaluation of 3e8 up to 1e9 DoFs/sec on one Intel Haswell node with 28 cores. Compared to performance results published within the last 5 years for high-order DG discretizations of the compressible Navier-Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.

We present a robust and accurate discretization approach for incompressible turbulent flows based on high-order 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-of-the-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.

We extend the approach of wall modeling via function enrichment to detached-eddy simulation. The wall model aims at using coarse cells in the near-wall region by modeling the velocity profile in the viscous sublayer and log-layer. However, unlike other wall models, the full Navier-Stokes equations are still discretely fulfilled, including the pressure gradient and convective term. This is achieved by enriching the elements of the high-order discontinuous Galerkin method with the law-of-the-wall. As a result, the Galerkin method can "choose" the optimal solution among the polynomial and enrichment shape functions. The detached-eddy simulation methodology provides a suitable turbulence model for the coarse near-wall cells. The approach is applied to wall-modeled LES of turbulent channel flow in a wide range of Reynolds numbers. Flow over periodic hills shows the superiority compared to an equilibrium wall model under separated flow conditions.

We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators based on sum factorization on quadrilateral and hexahedral meshes. We identify a set of kernels for fast quadrature on cells and faces targeting a wide class of weak forms originating from linear and nonlinear partial differential equations. Different algorithms and data structures for the implementation of operator evaluation are compared in an in-depth performance analysis. The sum factorization kernels are optimized by vectorization over several cells and faces and an even-odd decomposition of the one-dimensional compute kernels. In isolation our implementation then reaches up to 60\% of arithmetic peak on Intel Haswell and Broadwell processors and up to 50\% of arithmetic peak on Intel Knights Landing. The full operator evaluation reaches only about half that throughput due to memory bandwidth limitations from loading the input and output vectors, MPI ghost exchange, as well as handling variable coefficients and the geometry. Our performance analysis shows that the results are often within 10\% of the available memory bandwidth for the proposed implementation, with the exception of the Cartesian mesh case where the cost of gather operations and MPI communication are more substantial.

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for coarse spatial resolutions and small time step sizes. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly., Comment: 31 pages

We present a novel approach to hybrid RANS/LES wall modeling based on function enrichment, which overcomes the common problem of the RANS-LES transition and enables coarse meshes near the boundary. While the concept of function enrichment as an efficient discretization technique for turbulent boundary layers has been proposed in an earlier article by Krank & Wall (J. Comput. Phys. 316 (2016) 94-116), the contribution of this work is a rigorous derivation of a new multiscale turbulence modeling approach and a corresponding discontinuous Galerkin discretization scheme. In the near-wall area, the Navier-Stokes equations are explicitly solved for an LES and a RANS component in one single equation. This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard high-order polynomial basis resolves turbulent eddies where the mesh is sufficiently fine and the enrichment automatically computes the ensemble-averaged flow if the LES mesh is too coarse. As a result of the derivation, the RANS model is consistently applied solely to the RANS degrees of freedom, which effectively prevents the typical issue of a log-layer mismatch in attached boundary layers. As the full Navier-Stokes equations are solved in the boundary layer, spatial refinement gradually yields wall-resolved LES with exact boundary conditions. Numerical tests show the outstanding characteristics of the wall model regarding grid independence, superiority compared to equilibrium wall models in separated flows, and achieve a speed-up by two orders of magnitude compared to wall-resolved LES.

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the hybridized discontinuous Galerkin method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. Despite similar solver complexity of the matrix-based HDG solver and matrix-free geometric multigrid schemes with continuous and discontinuous Galerkin finite elements, the latter offer up to order of magnitude faster time to solution, even after including the superconvergence effects. This difference is because of vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A roofline performance model confirms the advantage of the matrix-free implementation.

We present a novel approach to wall modeling for RANS within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high-order discontinuous Galerkin solver for incompressible flow complemented by the Spalart-Allmaras closure model. As benchmark examples we present turbulent channel flow starting from $Re_{\tau}=180$ and up to $Re_{\tau}=100{,}000$ as well as flow past periodic hills at Reynolds numbers based on the hill height of $Re_H=10{,}595$ and $Re_{H}=19{,}000$.

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 $Re_{\tau}=180$ as well as $590$., Comment: 28 pages, in preparation for submission to Journal of Computational Physics