A divergence-conforming hybridized discontinuous Galerkin method for the incompressible Reynolds-averaged Navier-Stokes equations

We present a hybridized discontinuous Galerkin (HDG) method for the incompressible Reynolds-averaged Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The method extends upon an HDG method recently introduced by Rhebergen and Wells for the incompressible Navier-Stokes equations. With a special choice of velocity and pressure spaces for both element and trace degrees of freedom (DOFs), the method returns pointwise divergence-free mean velocity fields and properly balances momentum and energy. We further examine the use of different polynomial degrees and meshes to see how the order of the scalar eddy viscosity affects the convergence of the mean velocity and pressure fields, specifically for the method of manufactured solutions. As is standard with HDG methods, static condensation can be employed to remove the element DOFs and thus dramatically reduce the global number of DOFs. Numerical results illustrate the effectiveness of the proposed methodology.     

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

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 the literature. This topic is carefully investigated in the present work by the example of the three‐dimensional 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 and preconditioners as well as matrix‐free implementations are necessary ingredients in achieving 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 3·10⁸ up to 1·10⁹ DoFs/s (degrees of freedom per second) on one Intel Haswell node with 28 cores. Compared to performance results published within the last five years for high‐order discontinuous Galerkin discretizations of the compressible Navier‐Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.     

A high-order discontinuous Galerkin method for the incompressible Navier-Stokes equations on arbitrary grids

A high-order discontinuous Galerkin (DG) method is proposed in this work for solving the two-dimensional steady and unsteady incompressible Navier-Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax-Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.     

A discontinuous Galerkin scheme for the numerical solution of flow problems with discontinuities

In this paper, a new discontinuous Galerkin finite element method for the numerical solution of flow problems with discontinuities is presented. The method is based on the limitation in every cell of the difference between the extrema values and the mean value of the numerical solution. The algorithm and technical details for the implementation of the method are presented in one‐and two‐dimensional problems. Numerical experiments for classical test problems are solved on unstructured triangulations to demonstrate the performance of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

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

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

In this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two‐phase flow

A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 John Wiley & Sons, Ltd.     

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size. To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L²‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order continuous and discontinuous Galerkin methods for wave problems

Three Galerkin methods—continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin—are compared in terms of performance and computational efficiency in 2‐D scattering problems for low and high‐order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high‐order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high‐order elements are adopted, both of them clearly outperforming compact discontinuous Galerkin. Copyright © 2013 John Wiley & Sons, Ltd.     

Higher‐order and adaptive discontinuous Galerkin methods with shock‐capturing applied to transonic turbulent delta wing flow

Discontinuous Galerkin (DG) methods allow high‐order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight‐sided elements. DG discretizations with higher polynomial degrees must, however, be stabilized in the vicinity of discontinuities of flow solutions such as shocks. In this article, we device a consistent shock‐capturing method for the Reynolds‐averaged Navier–Stokes and k‐ ω turbulence model equations based on an artificial viscosity term that depends on element residual terms. Furthermore, the DG method is combined with a residual‐based adaptation algorithm that targets at resolving all flow features. The higher‐order and adaptive DG method is applied to a fully turbulent transonic flow around the second Vortex Flow Experiment (VFE‐2) configuration with a good resolution of the vortex system.Copyright © 2013 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

Large eddy simulation of turbulent channel flow using an algebraic model

In this paper an algebraic model from the constitutive equations of the subgrid stresses has been developed. This model has an additional term in comparison with the mixed model, which represents the backscatter of energy explicitly. The proposed model thus provides independent modelling of the different energy transfer mechanisms, thereby capturing the effect of subgrid scales more accurately. The model is also found to depict the flow anisotropy better than the linear and mixed models. The energy transfer capability of the model is analysed for the isotropic decay and the forced isotropic turbulence. The turbulent plane channel flow simulation is performed over three Reynolds numbers, Re_τ=180, 395 and 590, and the results are compared with that of the dynamic model, Smagorinsky model, and the DNS data. Both the algebraic and dynamic models are in good agreement with the DNS data for the mean flow quantities. However, the algebraic model is found to be more accurate for the turbulence intensities and the higher‐order statistics. The capability of the algebraic model to represent backscatter is also demonstrated. Copyright © 2005 John Wiley & Sons, Ltd.     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

Revisit of dilation-based shock capturing for discontinuous Galerkin methods

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement.     

Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

In this paper, a high‐order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second‐order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG‐MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical comparison of hybridized discontinuous Galerkin and finite volume methods for incompressible flow

In this study, a hybridizable discontinuous Galerkin method is presented for solving the incompressible Navier–Stokes equation. In our formulation, the convective part is linearized using a Picard iteration, for which there exists a necessary criterion for convergence. We show that our novel hybridized implementation can be used as an alternative method for solving a range of problems in the field of incompressible fluid dynamics. We demonstrate this by comparing the performance of our method with standard finite volume solvers, specifically the well‐established finite volume method of second order in space, such as the icoFoam and simpleFoam of the OpenFOAM package for three typical fluid problems. These are the Taylor–Green vortex, the 180‐degree fence case and the DFG benchmark. Our careful comparison yields convincing evidence for the use of hybridizable discontinuous Galerkin method as a competitive alternative because of their high accuracy and better stability properties. Copyright © 2014 John Wiley & Sons, Ltd.     

An internal penalty discontinuous Galerkin method for simulating conjugate heat transfer in a closed cavity

Using the discontinuous Galerkin (DG) method for conjugate heat transfer problems can provide improved accuracy close to the fluid‐solid interface, localizing the data exchange process, which may further enhance the convergence and stability of the entire computation. This paper presents a framework for the simulation of conjugate heat transfer problems using DG methods on unstructured grids. Based on an existing DG solver for the incompressible Navier‐Stokes equation, the fluid advection‐diffusion equation, Boussinesq term, and solid heat equation are introduced using an explicit DG formulation. A Dirichlet‐Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux of interface quadrature points in the fluid‐solid interface. Formal h and p convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved. The algorithm is then further validated against 3 existing benchmark cases, including a thermally driven cavity, conjugate thermally driven cavity, and a thermally driven cavity with conducting solid, at Rayleigh numbers from 1000 to 100 000. The computational effort is documented in detail demonstrating clearly that, for all cases, the highest‐order accurate algorithm has several magnitudes lower error than first‐ or second‐order schemes for a given computational effort.     

One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

Time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process for incompressible steady flow

The time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process (EFTG‐NSD) is proposed for the simulation of steady flows. The goal of the present paper is twofold. One is to raise the efficiency of the time‐related methods for solving steady flow problems, and the other is to obtain a good stability. The EFTG‐NSD method, which uses the time‐related Navier–Stokes equations to describe steady flows, does not care about the intermediate process and obtains solution of steady flows through time marching. Different from the classical time‐related fractional step methods, the EFTG‐NSD method decouples the Navier–Stokes equations without any operator‐splitting and correction. Because the elimination of correction at each iteration step reduces the computation cost, the EFTG‐NSD method possesses higher computation efficiency. In addition, the EFTG‐NSD method has a good stability due to the use of the Taylor–Galerkin formula in time and space discretization. Furthermore, the method combining element‐free Galerkin method with Taylor–Galerkin method is an important supplement of the element‐free Galerkin method for solving flow problems. Copyright © 2011 John Wiley & Sons, Ltd.