High-order Flux Reconstruction Method Research Articles

A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation

In this work, we present a positivity-preserving high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation augmented with a discrete velocity model that ensures the scheme is discretely conservative. Through modeling the internal degrees of freedom, the approach is further extended to polyatomic molecules and can encompass arbitrary constitutive laws. The approach is validated on a series of large-scale complex numerical experiments, ranging from shock-dominated flows computed on unstructured grids to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. The results show the ability of the scheme to directly resolve shock structures without any ad hoc numerical shock capturing method and correctly approximate turbulent flow phenomena in a consistent manner with the hydrodynamic equations.

Turbulence amplification in the shock wave/turbulent boundary layer interaction over compression ramp by the flux reconstruction method

Wall-resolved large eddy simulation on a supersonic turbulent boundary layer over a 24° compression ramp is performed under the framework of high order discontinuous methods for a free-stream Mach number M∞=2.9 and Reynolds number Re∞=5.966×106. The turbulent flow is resolved by the high order flux reconstruction method, and the shock is captured by a high-resolution, but stable weighted essentially non-oscillation limiter. The solver used in this paper is validated by the double Mach reflection case and the Taylor–Green vortex case. The results of shock wave/turbulent boundary layer interaction at the ramp corner are validated by the numerical simulations and the experimental data in the literature. The analysis of the physics behind the turbulence amplification at around the ramp corner is presented. The shear effects and the flow deceleration/acceleration are the main reasons of the turbulence amplification.

High-order flux reconstruction thermal lattice Boltzmann flux solver for simulation of incompressible thermal flows.

In this paper, a high-order solver combining the flux reconstruction (FR) method and the thermal lattice Boltzmann flux solver (FRTLBFS) is developed for accurately and efficiently simulating incompressible thermal flows. The conservative differential equations recovered from Chapman-Enskog analysis of the thermal lattice Boltzmann equation are solved by the high-order FR method. The thermal lattice Boltzmann method is only applied to reconstruct the local solution used for evaluating fluxes at the solution and flux points. Unlike the traditional Navier-Stokes-Boussinesq (NSB) solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current FRTLBFS are coupled and computed uniformly. In comparison with the recently developed high-order flux reconstruction thermal lattice Boltzmann method, the FRTLBFS holds advantages such as high-order accuracy, good stability, and compactness but is more efficient and low storage, since only macroscopic flow variables including density, velocity, and temperature are stored and evolved. In addition, the physical boundary conditions in FRTLBFS can be directly implemented by using the same method as in conventional NSB solvers. Numerical validations of the proposed method are implemented by simulating (a) the porous plate problem, (b) natural convection in a square cavity, (c) unsteady natural convection in a tall cavity, and (d) thermal lid-driven cavity flow. Numerical results demonstrate that the present solver is an attractive tool to simulate incompressible thermal flows due to its high-order accuracy, stability, and low memory cost.

Noise reduction mechanisms for insert-type serrations of the NACA-0012 airfoil

Trailing-edge serrations inspired by owls are capable of reducing broadband noise. In this study, the wall-resolved large-eddy simulations (LES) are carried out on the flow over NACA-0012 airfoil with additional serrated trailing edges. The computations are conducted with the high-order flux reconstruction method on unstructured meshes. Three kinds of serrations with different lengths are studied and compared with the straight trailing-edge case, and all three types of serration achieved a certain degree of noise reduction. Presently, the medium-length serration achieves the best noise reduction effect. The maximum decrease of overall sound pressure level is approximately 2.4 dB, implying that the length of serration has a substantial impact on the noise reduction effect. The serration has no significant effect on the upstream turbulence statistics, but it changes the flow structure near the serration, such as inducing side vortex pairs attached to the serration edges. Moreover, dynamic mode decomposition shows that the pressure structures vary with the serration length. For the most unstable hydrodynamic wave, the spanwise coherence of the mode structure of pressure in the upstream boundary layer is weakened. In addition, serrations can redistribute the dipole sources on the surfaces of airfoil and serrations. The destructive interference is enhanced to some extent, which is favourable for noise reduction. In contrast with LES simulations, the pure dipole analysis shows that the longest serration case seems to be the best. Furthermore, a recently developed noise theory is used to evaluate the influence of serrations on the flow noise sources qualitatively and quantitatively. It is found that the serrations can mitigate noise source intensity near the serration edges but increase the source intensity in the near wake. The combined effect of serration on the dipole source and flow noise source determines the overall noise reduction effect. To conclude, destructive interference plays a primary role in suppressing noise radiation by serrated trailing edges, and the dual effect of flow noise sources should be considered in future serration designs. As the influence of turbulence structure will make it more difficult to find the optimal serration parameters, the position of high-fidelity simulation will become increasingly important.

Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

R-adaptive algorithms for supersonic flows with high-order Flux Reconstruction methods

The present paper addresses the development and implementation of the first r-adaptive mesh refinement (r-AMR) algorithm for a high-order Flux Reconstruction solver. The r-refinement consists on nodal re-positioning while keeping the number of mesh nodes and their connectivity frozen. The developed algorithm is based on physics-driven spring-analogies, where the mesh can be seen as a network of fictitious springs. While AMR increases the local mesh density, the high-order Flux Reconstruction method potentially provides a more accurate detection of complex flow features over relatively coarser mesh, when compared to low-order methods. In this work, a concise overview of the Flux Reconstruction method and spring-based AMR techniques will be given, followed by some promising results of r-AMR applied to benchmark high-order steady-state supersonic flow simulations.

Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

We present a numerical analysis of linear multigrid operators for the high-order Flux Reconstruction method. The non-modal analysis is used to assess the short-term numerical dissipation in the context of 1D and 2D linear convection-diffusion. The effect of several parameters, namely the number of coarse-level iterations, the polynomial order and the combination of h- and p-multigrid is explored in an effort to identify the most efficient configurations. V-cycle p-multigrid is shown to have increased efficiency at higher polynomial orders, and the use of W-cycles and/or hp-multigrid appear to offer additional advantages. The effect of high Péclet numbers and high aspect-ratio cells is also explored in 2D, and both factors are shown to decrease the error dissipation. Finally, we relate the non-modal dissipation to the convergence rate of the multigrid in a series of manufactured solutions.

Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

Non-modal analysis of linear multigrid schemes for the high-order Flux Reconstruction method

A dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation

We present a dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation. The high-order explicit first stage, singly diagonal implicit Runge–Kutta (ESDIRK) method is employed to circumvent the restriction on the time step size. The pseudo transient continuation is coupled with the matrix-free restarted generalized minimal residual (GMRES) method to solve the nonlinear equations at each stage, except the first one, of ESDIRK. We use the spectral decay smoothness indicator as the refinement/coarsening indicator for p-adaptation. A dynamic load balancing technique is developed with the aid of the open-source library ParMETIS. The trivial cost, compared to implicit time stepping, of mesh repartitioning and data redistribution enables us to conduct p-adaptation and load balancing every time step. An isentropic vortex propagation case is employed to study the impact of element weights used in mesh repartitioning on parallel efficiency. We apply the p-adaptive solver for implicit large eddy simulation (ILES) of the transitional flows over a cylinder when Reynolds number (Re) is 3900 and the SD7003 wing when Re is 60000. Numerical experiments demonstrate that a significant reduction in the run time (up to 70%) and total number of solution points (up to 76%) can be achieved with p-adaptation.

A high-order flux reconstruction method for 3D mixed overset meshes

The use of overset meshes can significantly simplify grid generation for complex configurations, and is particularly desired for moving boundary problems as remeshing is often unnecessary. In the present study, we develop a high-order flux reconstruction (FR) solver for mixed overset meshes including the near-body, and background meshes. The main objective is to achieve uniformly high order accuracy in the entire computational domain on both the near body and the background grids. Two different approaches to handle the overset interfaces are evaluated for accuracy, efficiency and robustness. Waves passing across the overset interfaces are tested with both smooth and discontinuous waves. In the present study, we focus on non-moving boundary problems, and demonstrate the overall methodology for steady and unsteady flow problems.

High-order computational fluid dynamics simulations of a spinning golf ball

This paper presents the first high-order computational fluid dynamics (CFD) simulations of static and spinning golf balls at realistic flow conditions. The present results are shown to capture the complex fluid dynamics inside the dimples which lead to drag reduction versus a smooth sphere, and compare well to previous experimental and computational studies. The high-order flux reconstruction method has been paired with the artificial boundary overset method to enable simplified mesh generation and grid motion. The compressible Navier–Stokes equations are modeled using a scale-resolving large eddy simulation (LES) approach with no sub-grid models. The codes implementing these methods have been implemented for NVIDIA graphical processing units (GPUs), enabling large speedups over traditional computer hardware. The new method allows for the simulation of golf balls, and other objects at moderate Reynolds numbers, to be simulated in a matter of days on large computing clusters.

On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method

The high-order flux reconstruction (FR) method on unstructured hexahedral grids is coupled with the localized artificial diffusivity (LAD) scheme for aiming at accurate simulation of shock-turbulence interaction. In order to overcome known robustness issues particularly with the high-order (r>0) derivative formulation, important properties of the artificial bulk viscosity (ABV) profile affecting the solution are investigated first. For the purpose of comparison, an approximated Gaussian filter and modified restriction-prolongation (RP) filters are developed and tested for the present FR-LAD approach. Then, we propose a multidimensional RP filter on unstructured quadrilateral and hexahedral grids to address the issues on non-Cartesian grids. It is shown that a simple extension of a one-dimensional RP filter for two-dimensional grids may result in insufficient smoothing of ABV with r=2, and that the proposed multidimensional RP filter can provide smooth ABV with both r=0 and r=2. The proposed FR-LAD scheme is tested for typical shock-related problems, including the 1D shock tube, 1D shock-entropy wave interaction, 2D steady shock flows, and 2D shock-vortex interaction. The FR-LAD scheme has favorable properties of subcell shock capturing with the length scale of O(h/p) and superior preservation of high-order accuracy for smooth flows. Finally, LES of an overexpanded supersonic jet is performed to demonstrate its capability for practical applications. The proposed FR-LAD scheme can be an attractive candidate for LES or DNS of compressible flows involving shocks, contact discontinuities, turbulence, and their interactions on unstructured meshes.

A study on the behaviour of high-order flux reconstruction method with different low-dissipation numerical fluxes for large eddy simulation

ABSTRACTThis work focuses on the numerical dissipation features of high-order flux reconstruction (FR) method combined with different numerical fluxes in turbulence flows. The famous Roe and AUSM+ numerical fluxes together with their corresponding low-dissipation enhanced versions (LMRoe, SLAU2) and higher resolution variants (HR-LMRoe, HR-SLAU2) are incorporated into FR framework, and the dissipation interplay of these combinations is investigated in implicit large eddy simulation. The numerical dissipation stemming from these convective numerical fluxes is quantified by simulating the inviscid Gresho vortex, the transitional Taylor–Green vortex and the homogenous decaying isotropic turbulence. The results suggest that low-dissipation enhanced versions are preferential both in high-order and low-order cases to their original forms, while the use of HR-SLAU2 has marginal improvements and the HR-LMRoe leads to degenerated solution with high-order. In high-order the effects of numerical fluxes are reduced, and their viscosity may not be dissipative enough to provide physically consistent turbulence when under-resolved.

A high-order flux reconstruction method with adaptive mesh refinement and artificial diffusivity on unstructured moving/deforming mesh for shock capturing

This article presents new progress of shock capturing on unstructured dynamic grids by using adaptive mesh refinement (AMR) method in conjunction with artificial diffusivity (AD) under the framework of flux reconstruction (FR)/correction procedure via reconstruction (CPR) method. The proposed AMR algorithm is applicable even when the grid undergoes dynamic motion. It features an innovative data structure that is capable of direct addressing in managing the system of cells on multiple levels of refinement. A conservative mortar method is used to handle non-conforming interfaces resulted from cell subdivision. The AD is added in regions of strong dilatation to counter the Gibbs oscillation that is disruptive to high-order solutions when shock discontinuities exist. Several test cases are used to verify that this FR-AMR-AD framework can achieve high-order accuracy for smooth flows, and obtain stable solution with the presence of shock discontinuity. By further limiting the shock within subdivided adjacent cells, a sharp shock solution can be obtained with reduced amount of AD addition, hence errors from adding dissipation, on dynamically refined grids.

A high-order solver for unsteady incompressible Navier–Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping

We report development of a high-order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. An implicit non-linear lower–upper symmetric Gauss–Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. This compact high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids.

Study on Two-dimensional Wing Flutter Analysis by a High-order Flux Reconstruction Method

Wing flutter analyses are carried out by a newly developed flow solution method using high-order flux reconstruction approach. The significance steps to be achieved are the moving boundary/moving grid technique in high-Reynolds number flows. The k-ω turbulence model is implemented for the objectives. The method is first validated by a forced oscillation and then fluid-structure coupled flutter simulations are carried out in subsonic flows. The predicted flutter boundary well agreed with previous numerical studies. It is found that 3rd - and 4th -order spatial accuracy produces almost the identical results in the presented cases, while the viscous effects the flutter boundary. Preliminary results in transonic flow with shock waves are also included.