A robust low‐dissipation AUSM‐family scheme for numerical shock stability on unstructured grids

The simple low‐dissipation advection upwind splitting method (SLAU) scheme is a parameter‐free, low‐dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional‐shock‐detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

The Harten–Lax–van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High‐order accuracy is achieved using the weighted average flux (WAF) and MUSCL‐Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten–Hyman entropy fix method using Harten–Lax–van Leer‐type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method. Copyright © 2009 John Wiley & Sons, Ltd.     

一种健壮的混合Roe黎曼求解器

使用Roe格式计算多维流动问题时,在强激波附近会出现数值激波不稳定现象。带有剪切粘性的HLLEC格式不仅可以捕捉接触间断,而且表现出很好的稳定性。混合Roe格式和HLLEC格式来消除数值激波不稳定性。在强激波附近,通过激波面法向和网格界面法向的夹角来定义开关函数,使得数值通量在激波面横向切换成HLLEC格式。在其余地方,数值通量依然使用Roe格式来计算。数值试验表明,混合格式不仅消除了Roe格式的数值激波不稳定性,还最大程度地减少了HLLEC格式所带来的剪切耗散,保留了Roe格式高分辨率的优点。     

管系内一维可压缩流动数值建模与特性分析

针对复杂管系内可压缩流体,基于有限体积法,采用HLLC(Harten-Lax-vanLeerContact)格式和黎曼求解器构建了有限控制体数值离散方法,引入虚拟节点用于连接有限控制体,借助虚拟节点给出控制体之间数值通量的计算格式,发展了一种管道内一维流动数值建模方法。针对含有分支管路的管系,在管道连接部位构建了分支管路拟一维流动数值计算模型。基于所发展的一维流动数值方法,建立了变径管道和含60°分支管道内流动计算模型,验证了该方法的收敛性和有效性;基于虚拟节点的数值格式处理变径管激波问题具有一定精度优势。研究了变径管和分支管模型中可压缩流体激波、稀疏波等的传播机理,分析了管径对相邻支管压力的影响,为工程管路设计提供了参考。     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

Numerical solutions of Euler equations by using a new flux vector splitting scheme

A new flux vector splitting scheme has been suggested in this paper. This scheme uses the velocity component normal to the volume interface as the characteristic speed and yields the vanishing individual mass flux at the stagnation. The numerical dissipation for the mass and momentum equations also vanishes with the Mach number approaching zero. One of the diffusive terms of the energy equation does not vanish. But the low numerical diffusion for viscous flows may be ensured by using higher-order differencing. The scheme is very simple and easy to be implemented. The scheme has been applied to solve the one dimensional (1D) and multidimensional Euler equations. The solutions are monotone and the normal shock wave profiles are crisp. For a 1D shock tube problem with the shock and the contact discontinuities, the present scheme and Roe scheme give very similar results, which are the best compared with those from Van Leer scheme and Liou–Steffen's advection upstream splitting method (AUSM) scheme. For the multidimensional transonic flows, the sharp monotone normal shock wave profiles with mostly one transition zone are obtained. The results are compared with those from Van Leer scheme, AUSM and also with the experiment.     

AUSM-like expression of HLLC and its all-speed extension

Advection upstream splitting method (AUSM) and Harten-Lax-van Leer with contact (HLLC) are two popular families of flux functions. The AUSM is simple and requires no eigenstructure, which facilitates its extensions to general equations of state. Furthermore, one of its variants, simple low-dissipation AUSM (SLAU), is applicable to all speeds and features removal of parameter setting by the user. HLLC, on the other hand, clearly defines three distinct waves in Riemann problem, namely, left-running and right-running acoustic waves, and entropy wave. This paper demonstrates that HLLC can be written in a very similar form with the AUSM family and that the similar manner in extending AUSM family to all speeds is easily incorporated into HLLC in this AUSM-like form. Then, we combine the strengths of the both flux functions and offer a new inviscid numerical flux function within the framework of monotone upwind scheme for conservation laws (MUSCL) in computational fluid dynamics (CFD) for Euler and Navier-Stokes equations. The resultant HLLC with low dissipation (HLLCL) numerical flux can compute low Mach number flows and sound propagations at the same time with high accuracy, as demonstrated by one-dimensional and two-dimensional numerical examples. Furthermore, the results indicate that its extensions to general fluids such as supercritical fluids are encouraging.     

一种健壮的TV通量分裂格式

随着计算流体力学的快速发展，设计精确、高效并且健壮的数值格式变得尤为重要。Toro等^[8]提出的TV通量分裂格式表现出简单、高效和精确分辨接触间断等优点，但是在计算一些多维算例时会出现数值激波不稳定现象。两波近似的HLL格式在计算中非常高效和健壮，但是不能分辨接触间断大大地限制了其应用。本文对TV通量分裂格式进行稳定性分析，据此提出一种混合格式来消除TV格式的数值激波不稳定性。数值试验表明，本文构造的混合格式不仅保留了原始TV格式的优点，而且具有更好的健壮性，在计算二维问题时不会出现数值激波不稳定现象。     

一种鲁棒的HLLC格式及其稳定性分析

精确捕捉接触波和剪切波的Godunov型数值方法，如流行的HLLC格式，在模拟高超声速流动问题时会出现激波异常现象。对HLLC格式进行稳定性分析发现，流体主流方向的扰动都能有效衰减，但是横向的密度与剪切速度的扰动不会衰减。具有特殊对称性的二维Sedov爆轰波问题证明了横向通量和不稳定现象之间的密切联系。利用压力比和马赫数来探测数值激波层亚声速区的横向网格界面，并且在该界面的数值通量上增加熵波粘性和剪切波粘性来构造一种激波稳定的HLLC格式。分析表明，在熵波粘性和剪切波粘性的作用下，横向的所有扰动都会衰减。一系列数值测试证明了新格式不仅可以成功地抑制各类激波异常现象，还保留了原HLLC格式低耗散性的优点。     

Harten-Lax-van Leer-contact (HLLC) approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver is built with elastic waves (HLLCE) for one-dimensional elastic-plastic flows with a hypoelastic constitutive model and the von Mises’ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the presented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves (TRRSE).     

一种激波稳定的对流-压力通量分裂格式

针对欧拉方程三种流行的对流-压力通量分裂方法(Liou-Steffen,Zha-Bilgen和Toro-Vázquez)进行特征分析,进而提出一种新的对流-压力通量分裂格式。采用Zha-Bilgen分裂方法将欧拉方程的通量分裂成对流项和压力项两部分,使用TV格式来计算这两部分的数值通量。利用压力比构造激波探测函数,并且在强激波附近的亚声速区域增加TV格式的剪切粘性来克服数值模拟中的激波不稳定性。数值算例的计算结果表明,新的对流-压力通量分裂格式不仅保留了原始TV格式精确分辨接触间断的优点,而且具有更好的鲁棒性,在数值模拟多维强激波问题时不会出现不稳定现象。因此,该格式是一种精确并且具有强鲁棒性的数值方法,可以广泛地应用于可压缩流体的数值计算中。     

Strategies to cure numerical shock instability in the HLLEM Riemann solver

The HLLEM scheme is a popular contact and shear preserving approximate Riemann solver that is known to be plagued by various forms of numerical shock instability. In this paper, we clarify that the shock instability exhibited by this scheme is primarily triggered by the spurious activation of the antidiffusive terms present in the first and third Riemann flux components on the transverse interfaces adjoining the shock front due to numerical perturbations. These erroneously activated terms are shown to counteract the favorable damping mechanism provided by its inherent HLL-type diffusive terms, causing an unphysical variation of the conserved quantity ρu both along and across the numerical shock. To prevent this, two distinct strategies are proposed termed as S elective W ave M odification and A nti D iffusion C ontrol. The former focuses on enhancing the quantity of the favorable HLL-type dissipation available on these critical flux components by carefully increasing the magnitudes of certain nonlinear wave speed estimates, while the latter focuses on directly controlling the magnitude of these critical antidiffusive terms. A linear perturbation analysis is performed to gauge the effectiveness of these cures and to estimate a von Neumann–type stability bounds on the CFL number associated with their use. Results from a variety of classic shock instability test cases show that the proposed strategies are able to provide excellent shock stable solutions even on grids that are highly elongated across the shock front without compromising the accuracy on inviscid contact or shear dominated viscous flows.     

二维激波与剪切层相互作用的直接数值模拟研究

采用五阶weighed esseritially non-oscillatory (WENO) 格式和三阶total variation diminishing (TVD) Runge-Kutta 格式, 通过求解二维非定常Navier-Stokes 方程, 直接数值模拟了激波与剪切层相互作用, 目的在于揭示激波与剪切层相互作用过程中噪声产生的机理. 研究发现:(1) 当入射激波穿过剪切层时, 剪切层中心位置向下层区域偏移;(2) 入射激波穿过剪切层产生小激波, 在小激波与剪切层接触点处产生声波并向外辐射;(3) 反射激波穿过剪切层后形成了分段弧状激波;(4) 当反射激波穿过剪切层时, 激波在鞍点处泄漏并向外辐射声波, 这是一种激波泄漏机制.     

Damping numerical oscillations in hybrid solvers through detection of Gibbs phenomenon

A Gibbs phenomenon detector that is useful in damping numerical oscillations in hybrid solvers for compressible turbulence is proposed and tested. It is designed to function in regions away from discontinuities where commonly used discontinuity sensors are ineffective. Using this Gibbs phenomenon detector in addition to a discontinuity sensor for combining central and shock capturing schemes provides an integrated way of dealing with numerical oscillations generated by shock waves and contact lines that are normal to the flow. When complete suppression of numerical oscillations is not possible, they are sufficiently localized. Canonical tests and large eddy simulations show that inclusion of the proposed detector does not cause additional damping of ‘well‐resolved’ physical oscillations. Copyright © 2017 John Wiley & Sons, Ltd.     

High‐order shock capturing schemes for turbulence calculations

This work investigates high‐order central compact methods for simulating turbulent supersonic flows that include shock waves. Several different types of previously proposed characteristic filters, including total variation diminishing, monotone upstream‐centered scheme for conservation laws, and weighted essentially non‐oscillatory filters, are investigated in this study. Similar to the traditional shock capturing schemes, these filters can eliminate the numerical instability caused by large gradients in flow fields, but they also improve efficiency compared with classical shock‐capturing schemes. Adding the nonlinear dissipation part of a classical shock‐capturing scheme to a central scheme makes the method suitable for incorporation into any existing central‐based high‐order subsonic code. The amount of numerical dissipation to add is sensed by means of the artificial compression method switch. In order to improve the performance of the characteristic filters, we propose a hybrid approach to minimize the dissipation added by the characteristic filter. Through several numerical experiments (including a shock/density wave interaction, a shock/vortex interaction, and a shock/mixing layer interaction) we show that our hybrid approach works better than the original method, and can be used for future turbulent flow simulations that include shocks. Copyright © 2009 John Wiley & Sons, Ltd.     

Study on the numerical schemes for hypersonic flow simulation

Hypersonic flow is full of complex physical and chemical processes, hence its investigation needs careful analysis of existing schemes and choosing a suitable scheme or designing a brand new scheme. The present study deals with two numerical schemes Harten, Lax, and van Leer with Contact (HLLC) and advection upstream splitting method (AUSM) to effectively simulate hypersonic flow fields, and accurately predict shock waves with minimal diffusion. In present computations, hypersonic flows have been modeled as a system of hyperbolic equations with one additional equation for non-equilibrium energy and relaxing source terms. Real gas effects, which appear typically in hypersonic flows, have been simulated through energy relaxation method. HLLC and AUSM methods are modified to incorporate the conservation laws for non-equilibrium energy. Numerical implementation have shown that non-equilibrium energy convect with mass, and hence has no bearing on the basic numerical scheme. The numerical simulation carried out shows good comparison with experimental data available in literature. Both numerical schemes have shown identical results at equilibrium. Present study has demonstrated that real gas effects in hypersonic flows can be modeled through energy relaxation method along with either AUSM or HLLC numerical scheme.     

HLLE++格式在高马赫数空腔流动模拟中的应用

空腔流动存在剪切层运动、涡脱落与破裂,以及激波与激波、激波与剪切层、激波与膨胀波和激波/涡/剪切层相互干扰等现象,流动非常复杂,特别是高马赫数(M>2)时,剪切层和激波更强,激波与激波干扰更严重,对数值格式的要求更高,既需要格式耗散小,对分离涡等有很高的模拟精度,又需要格式在激波附近具有较大的耗散,可以很好地捕捉激波,防止非物理解的出现。Roe和HLLC等近似Riemann解格式在高马赫数强激波处可能会出现红玉现象,而HLLE++格式大大改善了这种缺陷,在捕捉高超声速激波时避免了红玉现象的发生,同时还保持在光滑区域的低数值耗散特性。本文在结构网格下HLLE++格式的基础上,通过改进激波探测的求解,建立了基于非结构混合网格的HLLE++计算方法,通过无粘斜坡算例,验证了HLLE++格式模拟高马赫数流动的能力,并应用于高马赫数空腔流动的数值模拟,开展了网格和湍流模型影响研究,验证了方法模拟高马赫数空腔流动的可靠性和有效性。     

A simple hybrid finite volume solver for compressible turbulence

A simple, explicit, hybrid finite volume method for simulating compressible turbulence is developed by combining a fourth‐order central scheme and a shock‐capturing simple low‐dissipation advection upstream splitting method. The total flux on each of the cell faces is computed as a weighted average of central/nondissipative and upwind/dissipative fluxes. The weights are determined using an unphysical oscillation sensor in addition to a more traditional discontinuity sensor used in earlier studies. Shocks are well captured, but overshoots in density are predicted around contact discontinuities that are normal to the flow. The use of the latter sensor effectively prevents these overshoots from generating spurious oscillations that travel away from the contact lines. The efficacy of the proposed method for direct or large‐eddy simulations of supersonic turbulence is established using several canonical test problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Solving the Savage–Hutter equations for granular avalanche flows with a second‐order Godunov type method on GPU

In this article, we apply Davis's second‐order predictor‐corrector Godunov type method to numerical solution of the Savage–Hutter equations for modeling granular avalanche flows. The method uses monotone upstream‐centered schemes for conservation laws (MUSCL) reconstruction for conservative variables and Harten–Lax–van Leer contact (HLLC) scheme for numerical fluxes. Static resistance conditions and stopping criteria are incorporated into the algorithm. The computation is implemented on graphics processing unit (GPU) by using compute unified device architecture programming model. A practice of allocating memory for two‐dimensional array in GPU is given and computational efficiency of two‐dimensional memory allocation is compared with one‐dimensional memory allocation. The effectiveness of the present simulation model is verified through several typical numerical examples. Numerical tests show that significant speedups of the GPU program over the CPU serial version can be obtained, and Davis's method in conjunction with MUSCL and HLLC schemes is accurate and robust for simulating granular avalanche flows with shock waves. As an application example, a case with a teardrop‐shaped hydraulic jump in Johnson and Gray's granular jet experiment is reproduced by using specific friction coefficients given in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

群速度控制格式及二维Riemann解

强激波和强接触间断的数值模拟一直是计算流体力学里一个富有挑战性的课题,它们是很多实际流动的基础。三阶迎风紧致格式是一种具有较高分辨率的高精度方法,但是在计算激波时仍有数值振荡产生。本文根据数值解的群速度特性,在三阶迎风紧致格式的基础上提出了一种群速度控制格式,使得能够正确模拟含有强激波和强接触间断的复杂流动。在此基础上构造了求解包含大压力比和密度比的二维界面问题的数值方法。计算结果表明,方法对激波和接触间断的分辨效果是令人满意的。