Towards industrial large eddy simulation using the FR/CPR method

Journal of Computational Physics

2018

Self Cite

Large eddy simulation (LES) has been increasingly used to tackle vortex-dominated turbulent flows. In LES, the quality of the simulation results hinges upon the quality of the numerical discretizations in both space and time. It is in this context we perform a Fourier analysis of several popular methods in LES including the discontinuous Galerkin (DG), finite difference (FD), and compact difference (CD) methods. We begin by reviewing the semi-discrete versions of all methods under-consideration, followed by a fully-discrete analysis with explicit Runge-Kutta (RK) time integration schemes. In this regard, we are able to unravel the true dispersion/dissipation behavior of DG and Runge-Kutta DG (RKDG) schemes for the entire wavenumber range. The physical-mode is verified to be a good approximation for the asymptotic behavior of these DG schemes in the low wavenumber range. After that, we proceed to compare the DG, FD, and CD methods in dispersion and dissipation properties. Numerical tests are conducted using the linear advection equation to verify the analysis. In comparing different methods, it is found that the overall numerical dissipation strongly depends on the time step. Compact difference (CD) and central finite difference (FD) schemes, in some particular settings, can have more numerical dissipation than the DG scheme with an upwind flux. This claim is then verified through a numerical test using the Burgers' equation. 30 22, 24] defined as the one that approximates the exact dispersion relation for a range of wavenumbers while regarding other modes as spurious. Recently, Moura et al. [27] provided new interpretations on the role of spurious or secondary modes. In their work, these modes are replicates of the physical-mode along the wavenumber axis and they improve the accuracy of the scheme. Vanharen et al. [29] concluded that after a large number of iterations, high-order schemes behave in dispersion and dissipation according to the 35 physical-mode asymptotically, for wavenumbers less than π. Nevertheless, the complete behavior of DG-type high-order schemes in dispersion and dissipation based on all eigenmodes has not been studied before. In this paper, we provide a first attempt to achieve this goal.Whilst there exists abundant work on the analysis of both high-order and low-order schemes or classical finite difference/finite-volume schemes, little attention was given to comparing the DG, FD, and CD schemes 40 of the same order of accuracy. The DG method, originally introduced by Reed and Hill [30] to solve the neutron transport equation, is chosen in this study as a representative of the high-order polynomialbased methods capable of handling unstructured grids including the spectral difference (SD) [31], and the flux reconstruction (FR) or correction procedure via reconstruction (CPR) methods [32]. LaSaint and Raviart [33] performed an error analysis for the DG method. It was then further developed for convection-45 dominated problems and fluid dynamics by many researches, see for example ([34, 35, 36,...

Section: Introductionmentioning

confidence: 99%

Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Alhawwary

Journal of Computational Physics

2018

Self Cite

“…With the recent development in high-order numerical methods and computer hardware, especially GPU architectures, LES has been increasingly used in industrial applications, especially in turbomachinery [12,13], for mission-critical problems. Flow problems with Reynolds numbers of several million have been tackled successfully.…”

Section: Introductionmentioning

confidence: 99%

Benchmark for scale-resolving simulation with curved walls: the Taylor Couette flow

Jourdan

2021

Adv. Aerodyn.

Flow between rotating concentric cylinders, or the Taylor Couette flow, has been studied extensively because of its rich physics, ranging from axisymmetric steady laminar flow, to fully developed turbulent flow. In the present study, we advocate the use of this problem as a benchmark case for scale-resolving simulation, such as large eddy simulation (LES) and direct numerical simulation (DNS). The problem is attractive because of its simple geometry, simple boundary conditions, and complex physics involving wall-shear induced and centrifugal instability. Unlike the well-known fully developed channel flow, this problem has a curved wall boundary, and it is unnecessary to add a source term to the governing equations to sustain the fully developed turbulent flow. A p-refinement study for Re = 4000 is performed first to establish DNS data, including the time history of enstrophy, which can be used as an accuracy and resolution indicator to evaluate numerical methods, and is orders of magnitude faster than using the mean flow quantities and Reynolds stresses to evaluate solution quality. Finally, an hp-refinement study is performed to establish the relative accuracy and efficiency of high-order schemes of various accuracy.

“…High-order computational fluid dynamics (CFD) methods have been attracting much research attention in past decades due to their high-resolution and low-dissipation properties that enable high-fidelity simulation of intricate flows. High-order spatial discretization methods, such as discontinuous Galerkin (DG) methods [1,2,3] and FR/CPR methods [4,5,6,7], have shown their capabilities of dealing with turbulent flows [8,9,10,11,12,13,14,15]. Usually, the high-order explicit strong stability preserving Runge-Kutta (SSPRK) methods [16] are used to integrate the semidiscretized governing equations for unsteady flow simulation.…”

Section: Introductionmentioning

confidence: 99%

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

2020

J Sci Comput

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) solution of the unsteady Navier-Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method (GMRES) solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.