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。     

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。     

一种改进混合迎风格式在翼型流场激波捕捉中的应用

波阻是飞行器超音速飞行的关键设计因素,精确捕捉激波在流场中的位置,是数值模拟含激波流场和精确计算波阻的一个重要研究内容.本文基于网格节点有限体积空间离散方法,采用AUSM格式与FVS格式的混合格式(MAUSM方法)计算对流通量,从而抑制在数值模拟流场出现的激波处振荡和过冲现象,确保AUSM准确捕获接触间断的特性和FVS格式捕捉激波的能力.本文使用MAUSM方法分别计算了在跨声速和超声速条件下的NACA0012翼型流场,并与中心差分格式的计算结果进行比较.结果表明,对于存在激波的翼型流场,MAUSM方法是有效的.     

超声速平面混合层小激波的形成与演变

为了揭示超声速混合层中小激波形成机理及其与涡相互作用的演变过程,本文基于大涡模拟（LES）方法,结合五阶精度混合TCD／WENO格式,对超声速平面混合层在对流马赫数为Mc=0．65条件下的流场结构进行了数值模拟,数值结果详细描述了超声速混合层中小激波的形成过程。研究了小激波形成后,随涡运动而产生的变形、脱落及发展过程。同时,对混合层双涡合并过程中,小激波与相邻涡相互作用所产生的变形与演变过程进行了讨论。     

基于非结构/混合网格的高阶精度格式研究进展

尽管以二阶精度格式为基础的计算流体力学(CFD) 方法和软件已经在航空航天飞行器设计中发挥了重要的作用, 但是由于二阶精度格式的耗散和色散较大, 对于湍流、分离等多尺度流动现象的模拟, 现有成熟的CFD 软件仍难以给出满意的结果, 为此CFD 工作者发展了众多的高阶精度计算格式. 如果以适应的计算网格来分类, 一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法. 由于非结构/混合网格具有良好的几何适应性, 基于非结构/混合网格的高阶精度格式近年来备受关注. 本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展, 重点介绍了空间离散方法, 主要包括k-Exact 和ENO/WENO 等有限体积方法, 间断伽辽金(DG) 有限元方法, 有限谱体积(SV) 和有限谱差分(SD) 方法, 以及近来发展的各种DG/FV 混合算法和将各种方法统一在一个框架内的CPR (correctionprocedure via reconstruction) 方法等. 随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题, 包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等. 在综述过程中, 介绍了各种方法的优势与不足, 其间介绍了作者发展的基于"静动态混合重构" 的DG/FV 混合算法. 最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.     

环形激波绕射、反射和聚焦流场的间断有限元模拟

采用间断有限元方法对环形激波在圆柱形激波管内绕射、反射和聚焦流场进行了数值模拟。将二维守恒方程的间断有限元方法发展到轴对称Euler方程,并对环形激波绕后台阶流动进行了数值计算。计算结果表明,采用间断有限元方法能够有效地捕捉运动激波在圆柱形激波管内传播的复杂流场结构;在聚焦点附近,数值解具有较大的梯度变化,表明该方法对间断解具有较强的捕捉能力,在聚焦点附近不会产生振荡或抹平间断现象。     

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。     

求解一维理想磁流体方程的移动网格熵稳定格式

磁流体方程的数值求解在等离子体物理学、天体物理研究以及流动控制等领域具有重要意义，本文构造了用于求解理想磁流体动力学方程的基于移动网格的熵稳定格式，此方法将Roe型熵稳定格式与自适应移动网格算法结合，空间方向采用熵稳定格式对磁流体动力学方程进行离散，利用变分法构造网格演化方程并通过Gauss-Seidel迭代法对其迭代求解实现网格的自适应分布，在此基础上采用守恒型插值公式实现新旧节点上的量值传递，利用三阶强稳定Runge-Kutta方法将数值解推进到下一时间层。数值实验表明，该算法能有效捕捉解的结构(特别是激波和稀疏波),分辨率高，通用性好，具有强鲁棒性。     

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。     

一种改进的中心-WENO混合格式

基于中心差分与WENO格式混合可以改善WENO格式耗散特性的思想,在理论推导的基础上,给出了一种用于激波捕捉计算的守恒型中心-WENO混合格式,该混合格式可视为三阶WENO格式和二阶中心差分格式的加权平均。在数值研究现有加权函数的基础上,给出了适用于该混合格式的加权函数,使其能够自适应地调整数值耗散以捕捉激波间断。数值结果表明:与三阶WENO格式相比,混合格式HY3_4能够降低数值耗散,更陡峭地捕捉间断,对复杂流场结构具有较高的分辨率;混合格式HY3_5对于包含高压比激波间断流场结构,能给出无振荡、低耗散的结果。     

Analysis of High-order Explicit LES Dynamic Modeling Applied to Airfoil Flows

A high-order low dissipative numerical framework is discussed to tackle simultaneously the modeling of unresolved sub-grid scale flow turbulence and the capturing of shock waves. The flows around two different airfoil profiles are simulated using a Spectral Difference discretisation scheme. First, a transitional, almost incompressible, low Reynolds number flow over a Selig-Donovan 7003 airfoil. Second, a high Reynolds number flow over a RAE2822 airfoil under transonic conditions. These flows feature both laminar and turbulent flow physics and are thus particularly challenging for turbulence sub-grid scale modeling. The accuracy of the recently developed Spectral Element Dynamic Model, specifically capable of detecting spatial under-resolution in high-order flow simulations, is evaluated. Concerning the test in transonic conditions, the additional complexity due to the presence of shock waves has been handled using an artificial viscosity shock-capturing technique based on bulk viscosity. To mitigate the impact of the shock-capturing on turbulence dissipation, it was necessary to combine the high-order modal-type shock detection with a usual sensor measuring the local flow compressibility.

     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

Dilation‐based shock capturing for high‐order methods

In this paper, we introduce a shock‐capturing artificial viscosity technique for high‐order unstructured mesh methods. This artificial viscosity model is based on a non‐dimensional form of the divergence of the velocity. The technique is an extension and improvement of the dilation‐based artificial viscosity methods introduced in Premasuthan et al., 15 and further extended in Nguyen and Peraire 27 . The approach presented has a number attractive properties including non‐dimensional analytical form, sub‐cell resolution, and robustness for complex shock flows on anisotropic meshes. We present extensive numerical results to demonstrate the performance of the proposed approach. Copyright © 2016 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.     

Calculation of supersonic steady-state flow in axisymmetric nozzles with an upwind difference scheme

This paper is on the application of the upwind difference scheme proposed by the author^[1] to the calculation of supersonic steady-state flow in axisymmetric nozzles. The upwind scheme is conservative (or weakly conservative), it yields results approximating those from the characteristic relations, and it has corresponding boundary difference schemes. The entropy phenomenon in the calculation of shock reflection on boundaries with the shock-capturing method will be discussed and a correction of this phenomenon will be proposed. From numerical experiments on an arbitrary nozzle, it is seen that the upwind difference scheme, its corresponding boundary scheme, and the improved treatment of shock reflection work well for the calculation of supersonic steady-state flow in axisymmetric nozzles.     

A discontinuous Galerkin immersed boundary solver for compressible flows: Adaptive local time stepping for artificial viscosity–based shock-capturing on cut cells

We present a higher-order cut cell immersed boundary method (IBM) for the simulation of high Mach number flows. As a novelty on a cut cell grid, we evaluate an adaptive local time stepping (LTS) scheme in combination with an artificial viscosity–based shock-capturing approach. The cut cell grid is optimized by a nonintrusive cell agglomeration strategy in order to avoid problems with small or ill-shaped cut cells. Our approach is based on a discontinuous Galerkin discretization of the compressible Euler equations, where the immersed boundary is implicitly defined by the zero isocontour of a level set function. In flow configurations with high Mach numbers, a numerical shock-capturing mechanism is crucial in order to prevent unphysical oscillations of the polynomial approximation in the vicinity of shocks. We achieve this by means of a viscous smoothing where the artificial viscosity follows from a modal decay sensor that has been adapted to the IBM. The problem of the severe time step restriction caused by the additional second-order diffusive term and small nonagglomerated cut cells is addressed by using an adaptive LTS algorithm. The robustness, stability, and accuracy of our approach are verified for several common test cases. Moreover, the results show that our approach lowers the computational costs drastically, especially for unsteady IBM problems with complex geometries.     

Unstructured adaptive remeshing finite element method for dusty shock flow

The passage of planar shocks in a dusty gas was investigated to note effects due to particle loading and initial shock Mach number. Two-phase flow equations have been added to a conservative, monotonic flow solver to allow study of compressible particle and droplet flows, which are of importance for shock propagation in two-phase flows and spray propulsion systems. The formulation developed herein employed a conservative Eulerian treatment for the gas and particle phases. The computations were performed using the finite element method-flux corrected transport (FEM-FCT) scheme, which has shown excellent predictive capability of various compressible flows which include both strong and weak shocks. The flux limiting technique was modified to provide monotonic particle velocity fields to increase the scheme's computational stability. Adaptive unstructured methodology based on adapting to high gradients of both the fluid and particle densities was used in conjunction with the conservative shock-capturing scheme to adequately resolve strong flowfield gradients. The shock attenuation of this scheme was then compared with previous experimental and numerical results and was found to yield robust predictions. Various interphase coupling terms were also considered to note their effect on the shock attenuation.This article was processed using Springer-Verlag t_ex Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

A vertex-centred finite volume method with shock detection

This paper introduces a vertex-centred finite volume method for compressible viscous flow incorporating a new shock detection procedure. The discretization is designed to be robust and accurate on the highly stretched and curved meshes necessary for resolving turbulent boundary layers around the leading edge of an aerofoil. Details of the method are described for two-dimensional problems and the natural extension of three-dimensional multiblock meshes is discussed. The shock detection procedure is used to limit the range of the shock-capturing dissipation specifically to regions containing shocks. For transonic turbulent flow this is shown to improve the boundary layer representation significantly.     

TVD格式在超音速喷管三维粘性流动求解中的应用

详细给出了任意三维曲线坐标系中Ｎｏｖｉｅｒ－Ｓｔｏｋｅｓ方程的对流项ＴＶＤ格式的构造过程，建立了数值求解三维粘性流动的计算方法，应用该方法对三维超音速喷管中有激波及无激波情况下的两种工况的层流流场进行了数值求解，并与实验做了对比。结果表明本文建立的计算方法具有较高的精度，同时也证明ＴＶＤ格式具有分辩率高，稳定收敛等优点，为进一步开展叶栅流场及紊流的研究打下了基础。