A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.     

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 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.     

High‐order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations

In this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under‐resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high‐order spectral methods by using high‐order ansatz functions up to 12th order. We employ polynomial de‐aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high‐order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at Re_D=3900, a confined periodic hill flow at Re_h=2800 and the transitional flow over a SD7003 airfoil at Re_c=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.     

Wall modeling via function enrichment within a high‐order DG method for RANS simulations of incompressible flow

We present a novel approach to wall modeling for the Reynolds‐averaged Navier‐Stokes equations 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_τ=180 and up to Re_τ=100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re_H=10595 and Re_H=19000.     

Assessment of the finite volume method applied to the v2 − f model

The objective of this paper is to assess the accuracy of low‐order finite volume (FV) methods applied to the v² ? f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3 :1–13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers ‐ the topic at hand is simply whether the mathematical properties of the v² ? f model can be captured by the given, widespread, numerical method. The v² ? f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The v² ? f model is studied in a statistically one‐dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher‐order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second‐order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three‐point Gauss–Lobatto quadrature), as well as a higher‐order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y⁺=1. Copyright © 2009 John Wiley & Sons, Ltd.     

An algebraic factorisation scheme for spectral element solution of incompressible flow and scalar transport

A spectral element algorithm for solution of the unsteady incompressible Navier–Stokes and scalar (species/heat) transport equations is developed using the algebraic factorisation scheme. The new algorithm utilises Nth order Gauss–Lobatto–Legendre points for velocity and the scalar, while (N-2)th order Gauss–Legendre points are used for pressure. As a result, the algorithm does not require inter-element continuity for pressure and pressure boundary conditions on solid surfaces. Implementations of the algorithm are performed for conforming and non-conforming grids. The latter is accomplished using both the point-wise matching and integral projection methods, and applied for grids with both polynomial and geometric non-conformities. Code validation cases include the unsteady scalar convection equation, and Kovasznay flow in two- and three-dimensional domains. Using cases with analytical solutions, the algorithm is shown to achieve spectral accuracy in space and second-order accuracy in time. The results for the Boussinesq approximation for buoyancy-driven flows, and the species mixing in a continuous flow micro-mixer are also included as examples of applications that require long-time integration of the scalar transport equations.     

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.     

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number

This paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor–Green vortex at Re = 1600 is considered. The results are compared with those obtained using a pseudo‐spectral solver, converged on a 512³ grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 288³, the fourth‐order accurate discontinuous Galerkin method (DGM) solution (p = 3) is already very close to the pseudo‐spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 384³, the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under‐resolved grid (resolution of 192³). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth‐order DGM is also compared with an energy conserving fourth‐order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 768³ DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright © 2013 John Wiley & Sons, Ltd.     

基于特征分裂有限元准隐格式的共轭传热整体耦合数值模拟方法

共轭传热现象在科学和工程领域中大量存在. 随着计算能力的发展, 对共轭传热现象进行准确有效的数值模拟, 成为科学研究和工程设计上的重要挑战.共轭传热数值模拟的方法可以分为两大类: 分区耦合和整体耦合.本文采用有限元法对共轭传热问题进行整体耦合模拟. 固体传热求解采用标准的伽辽金有限元方法.流动求解采用基于特征分裂的有限元方法. 该方法是一种重要的求解流动问题的有限元方法, 可以使用等阶有限元. 该方法的准隐格式与其他格式相比, 具有时间步长大的特点. 将稳定项中的时间步长与全局时间步长分开, 改进了准隐格式的稳定性. 基于改进的特征分裂有限元方法的准隐格式, 发展了一种层流共轭传热数值模拟的整体耦合方法. 采用这种方法可以将流体计算域和固体计算域作为一个整体划分有限元网格, 并且所有变量都可以采用相同的插值函数, 从而有利于程序的实现. 通过对典型问题的模拟, 验证了这种方法的准确性. 本工作还研究了固体区域时间步长对定常共轭传热问题数值模拟收敛性的影响.      

H 1 space-time discontinuous finite element method for convection-diffusion equations

An H~1 space-time discontinuous Galerkin (STDG) scheme for convectiondiffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H~1 Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L~∞ (H~1 ) norm is derived. The numerical exper- iments are presented to verify the theoretical results.     

A sensitivity analysis on the parameter of the GLS method for a second‐gradient theory of incompressible flow

Using a non‐conforming C⁰‐interior penalty method and the Galerkin least‐square approach, we develop a continuous–discontinuous Galerkin finite element method for discretizing fourth‐order incompressible flow problems. The formulation is weakly coercive for spaces that fail to satisfy the inf‐sup condition and consider discontinuous basis functions for the pressure field. We consider the results of a stability analysis through a lemma which indicates that there exists an optimal or quasi‐optimal least‐square stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields, and on the geometry of the finite element in the mesh. We provide several numerical experiments illustrating such dependence, as well as the robustness of the method to deal with arbitrary basis functions for velocity and pressure, and the ability to stabilize large pressure gradients. We believe the results provided in this paper contribute for establishing a paradigm for future studies of the parameter of the Galerkin least square method for second‐gradient theory of incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order element‐based Galerkin method for the barotropic vorticity equation

A high‐order element‐based Galerkin method is developed to solve the non‐divergent barotropic vorticity equation (BVE). The solution process involves solving a conservative transport equation for the vorticity fields and a Poisson equation for the stream function fields. The discontinuous Galerkin method is employed for solving the transport equation and a spectral element method (continuous Galerkin) is used for the Poisson equation. A third‐order strong stability preserving explicit Runge–Kutta scheme is used for time integration. A series of tests have been performed to validate the model, which include the evolution of an idealized tropical cyclone and interaction of dual vortices in close proximity. The numerical convergence study is performed by solving the BVE on the sphere where the analytic solution is known. The test results are consistent with physical observations, and the model exhibits exponential convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Streamline upwind finite element method for conjugate heat transfer problems

This paper presents a combined finite element method for solving conjugate heat transfer problems where heat conduction in a solid is coupled with heat convection in viscous fluid flow. The streamline upwind finite element method is used for the analysis of thermal viscous flow in the fluid region, whereas the analysis of heat conduction in solid region is performed by the Galerkin method. The method uses the three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the proposed method is to consistently couple heat transfer along the fluid-solid interface. Three test cases, i.e. conjugate Couette flow problem in parallel plate channel, counter-flow in heat exchanger, and conjugate natural convection in a square cavity with a conducting wall, are selected to evaluate the efficiency of the present method. The English text was polished byYunming Chen.     

Implicit LES of free and wall‐bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method

This paper presents the second validation step of a compressible discontinuous Galerkin solver with symmetric interior penalty (DGM/SIP) for the direct numerical simulation (DNS) and the large eddy simulation (LES) of complex flows. The method has already been successfully validated for DNS of an academic flow and has been applied to flows around complex geometries (e.g. airfoils and turbomachinery blades). During these studies, the advantages of the dissipation properties of the method have been highlighted, showing a natural tendency to dissipate only the under‐resolved scales (i.e the smallest scales present on the mesh), leaving the larger scales unaffected. This phenomenon is further enhanced as the polynomial order is increased. Indeed, the order increases the dissipation at the largest wave numbers, while its range of impact is reduced. These properties are spectrally compatible with a subgrid‐scale model, and hence DGM may be well suited to be used for an implicit LES (ILES) approach. A validation of this DGM/ILES approach is here investigated on canonical flows, allowing to study the impact of the discretisation on the turbulence for under‐resolved computations. The first test case is the LES of decaying homogeneous isotropic turbulence (HIT) at very high Reynolds number. This benchmark allows to assess the spectral behaviour of the method for implicit LES. The results are in agreement with theory and are even slightly more accurate than other numerical results from literature, obtained using a pseudo‐spectral (PS) method with a state‐of‐the‐art subgrid‐scale model. The second benchmark is the LES of the channel flow. Three Reynolds numbers are considered: Re_τ=395, 590 and 950. The results are compared with DNS of Moser et al. and Hoyas et al., also using PS methods. Both averaged velocity and fluctuations are globally in good agreement with the reference, showing the ability of the method to predict equilibrium wall‐bounded flow turbulence. To show that the method is able to perform accurate DNS, a DNS of HIT at Re_λ=64 and a DNS of the channel flow at Re_τ=180 are also performed. The effects of the grid refinement are investigated on the channel flow at Re_τ=395, highlighting the improvement of the results when refining the mesh in the spanwise direction. Finally, the modification of the ILES parameters, that is the Riemann solver and of the SIP coefficient, is studied on both cases, showing a significant influence on the choice of the Riemann solver. Copyright © 2015 John Wiley & Sons, Ltd.     

A stabilization for three‐dimensional discontinuous Galerkin discretizations applied to nonhydrostatic atmospheric simulations

A discontinuous Galerkin nonhydrostatic atmospheric model is used for two‐dimensional and three‐dimensional simulations. There is a wide range of timescales to be dealt with. To do so, two different implicit/explicit time discretizations are implemented. A stabilization, based upon a reduced‐order discretization of the gravity term, is introduced to ensure the balance between pressure and gravity effects. While not affecting significantly the convergence properties of the scheme, this approach allows the simulation of anisotropic flows without generating spurious oscillations, as it happens for a classical discontinuous Galerkin discretization. This approach is shown to be less diffusive than usual spatial filters. A stability analysis demonstrates that the use of this modified scheme discards the instability associated with the usual discretization. Validation against analytical solutions is performed, confirming the good convergence and stability properties of the scheme. Numerical results demonstrate the attractivity of the discontinuous Galerkin method with implicit/explicit time integration for large‐scale atmospheric flows. Copyright © 2015 John Wiley & Sons, Ltd.