A new,high-resolution shock-capturing hybrid scheme of flux vector splitting-Harten's TVD

The paper presents a new high-resolution hybrid scheme combining implicit flux vector splitting with Harten's TVD, which is proved suitable for shock-capturing calculation in gasdynamics. Fluxsplitting procedures are applied to discretize the implicit part of the Euler equations whereas Harten's numerical fluxes are used to calculate the residual of steady-state solutions. It ensures good shock-capturing properties and produces sharp numerical discontinuities without oscillations. It excludes expansion shocks and leads only to physically relevant solutions. The block-line-Gauss-Seidel relaxation procedure (block-LGS) is used to solve the resulting difference equations. The time step and the CFL number are much larger than those in the linearized block-alternating-direction-implicit approximate factorization method (block-ADI). Numerical experiments suggest that the hybrid scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the steady-state solution. Hence scheme seems to lead to an effective nonoscillatory shock capturing method for steady transonic flow. Project Supported by National Natural Science Foundation of China     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

A hybrid scheme for the numerical simulation of shock/discontinuity problems

To address accuracy issues for direct numerical simulation, a hybrid scheme based on the weighted compact scheme (WCS) and weighted essentially non-oscillatory (WENO) scheme is developed. The new hybrid method incorporates the advantages of both schemes. Time integration is performed using the fourth-order total variation diminishing Runge–Kutta method with a characteristic filter. The accuracy of the scheme is assessed using several benchmark problems. Results show that the proposed scheme produces a more accurate solution for problems involving shocks and discontinuities in comparison with the traditional shock-capturing methods.     

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

基于非结构网格的TTGC有限元格式的实现及在超声速流动中的应用

基于非结构网格系统,实现了时空三阶精度的TTGC有限元格式,并在三阶TTGC格式上发展了基于人工粘性的激波捕捉技术。在非结构网格下,采用这种方法对若干典型的超声速流动问题(SOD激波管、马赫数为3的前台阶流动以及马赫数为8的高超声速圆柱流动)进行了验证计算。结果表明,TTGC格式分辨率高,在粗糙网格下能够准确的模拟超声速流场中的激波、接触间断等复杂流动现象,并且能有效的控制间断附近的数值色散现象。与传统的有限体积方法相比,本文实现的TTGC有限元格式在模拟超声速流动问题方面具有格式精度高、数值耗散小等优点。     

A high‐resolution hybrid scheme for hyperbolic conservation laws

A new hybrid scheme is proposed, which combines the improved third‐order weighted essentially non‐oscillatory (WENO) scheme presented in this paper with a fourth‐order central scheme by a novel switch. Two major steps have been gone through for the construction of a high‐performance and stable hybrid scheme. Firstly, to enhance the WENO part of the hybrid scheme, a new reference smoothness indicator has been devised, which, combined with the nonlinear weighting procedure of WENO‐Z, can drive the third‐order WENO toward the optimal linear scheme faster. Secondly, to improve the hybridization with the central scheme, a hyperbolic tangent hybridization switch and its efficient polynomial counterpart are devised, with which we are able to fix the threshold value introduced by the hybridization. The new hybrid scheme is thus formulated, and a set of benchmark problems have been tested to verify the performance enhancement. Numerical results demonstrate that the new hybrid scheme achieves excellent performance in resolving complex flow features, even compared with the fifth‐order classical WENO scheme and WENO‐Z scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Uniform algorithm for all-speed shock-capturing schemes

Many ideas exist for the development of shock-capturing schemes, such as Roe, Harten–Lax–van Leer (HLL) and advection upstream splitting method (AUSM) families, and their extension for all-speed flow. A uniform algorithm that expresses the three families in the same framework is proposed in this study. The algorithm has an explicit physical meaning, provides new understanding and comparison of the mechanism of schemes, and may play a significant role in further research. As an example of applying the uniform algorithm, the low Mach number behaviour of the schemes is analysed. A clear and simple explanation is provided based on the wall boundary, and a concise rule is proposed to determine whether a scheme has satisfied low Mach number behaviour.     

The three-point fifth-order accurate generalized compact scheme and its applications

A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics. The project supported by the National Natural Science Foundation of China (19972038) and Foundation of the National CFD Laboratory of China     

群桩作用机理的数值试验研究

 在分析桩土支撑体系及其相互作用关系的基础上，利用有限元分析软件ANSYS建立了群桩 体系计算模型. 通过对均质土体例题计算结果分析，可以看出利用该建模方法与传统解答有 很好的一致性. 根据辽宁工程技术大学实验馆场地土实测资料，利用该建模方法，求解出了承台与桩的 荷载分担比，对安全、经济地进行群桩基础设计具有重要指导作用和应用价值.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

改进的五阶WENO-Z+格式

WENO-ZWENO-Z$+\!$格式的性能提升依赖于新增项的作用,该项的作用是在WENO-Z格式的基础上进一步增大欠光滑子模板上的权重. 系数$\lambda$被设置用来调控该项的作用, 以避免负耗散. 本文指出了WENO-Z$+\!$格式的缺陷,其所采用$\lambda $的取值方式既不能充分发挥格式的潜力, 也未完全消除负耗散;提出$\lambda $的值应随当地流场数据变化,方能充分发挥新增项在降低数值耗散、提高分辨率上的潜力. 基于此,本文重新设计了$\lambda $的计算公式,该公式能自适应地调控新增项的作用: 只在欠光滑子模板上的权重容易过度增大的地方削弱该项的作用,以避免负耗散; 在其他地方则充分发挥新增项的作用,最大限度增大欠光滑子模板上的权重, 提高格式的分辨率.将使用该系数公式的新格式命名为WENO-Z++, 并对其数值性能进行了系统的研究.理论分析表明, 新格式在间断处具有基本无振荡(essentially non-oscillatory,ENO)特性和更低的数值耗散. 对近似色散关系(approximate dispersion relation,ADR)的研究表明,新格式有效地避免了因过度增大欠光滑子模板上的权重而带来的负耗散,其频谱特性也得到了显著提升.本文还推导了使新格式在极值点处也能保持最优阶的精度的参数设置.一系列求解Euler方程的数值试验表明,新格式的激波捕捉能力和对复杂流场结构的分辨率都显著优于原WENO-Z$+\!$格式.}     

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 third-order compact nonlinear scheme for compressible flow simulations

Reynolds-averaged Navier-Stokes simulations based on second-order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin-friction drag and aerodynamic heating, as well as in simulations of complex flows such as large-scale separation and transition. A remedy for this is the development of high-order schemes, by which numerically induced dissipation and dispersion errors of low-order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high-resolution nonlinear shock-capturing methods. A stencil-selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third-order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh-Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

This paper establishes the entropy convergence of a new two-value high resolution finite volume scheme with slope relaxation for conservation laws. This scheme, motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities, unifies and evolves slopes directly with a slope relaxation equation that governs the evolution of slopes in both smooth and discontinuous regions. Proper choices of slopes are realized adaptively via a relaxation parameter. The scheme is shown to be total-variation-bounded (TVB) stable and satisfies cell-entropy inequalities.     

对称性与激波捕捉中的非物理波动问题

众所周知传统的一些激波捕捉格式（如Godunov，Roe格式）仅靠格式本身的耗散无法完全抑制激波下游的振荡。本文从微分方程拥有的内在对称性角度对此问题进行探讨。指出了Lax—wendroff格式数值求解无粘Burgers方程的问题出现空间振荡与对称破缺之间的内在联系。本文的研究表明：对称破缺可能是在激波附近导致数值解出现三种异常现象的内在机制。     

一类具有自适应性质的组合TVD格式的构造

本文分析比较了些典型的ＴＶＤ格式，发现各种ＴＶＤ格式对解的分辨能力有差别，格式中存在耗散性和压缩性的矛盾。在分析了格式的构造以及产生耗散、压缩作用的原因后，本文提出了一种具有自适应能力的ＴＶＤ格式的组合形式，该格式在解的剧烈变化区域，有较高的分辨能力；在解的光滑区域避免了由于压缩性过强造成的失真，数值试验表明该格式具有良好的性质．     

On the accuracy and efficiency of CFD methods in real gas hypersonics

A study of viscous and inviscid hypersonic flows using generalized upwind methods is presented. A new family of hybrid flux-splitting methods is examined for hypersonic flows. The hybrid method is constructed by the superposition of the flux-vector-splitting (FVS) method and second-order artificial dissipation in the regions of strong shock waves. The conservative variables on the cell faces are calculated by an upwind extrapolation scheme to third-order accuracy. A second-order-accurate scheme is used for the discretization of the viscous terms. The solution of the system of equations is achieved by an implicit unfactored method. In order to reduce the computational time, a local adaptive mesh solution (LAMS) method is proposed. The LAMS method combines the mesh-sequencing technique and local solution of the equations. The local solution of either the Euler or the NAVIER-STOKES equations is applied for the region of the flow field where numerical disturbances die out slowly. Validation of the Euler and NAVIER-STOKES codes is obtained for hypersonic flows around blunt bodies. Real gas effects are introduced via a generalized equation of state.     

有限谱ENO格式及其应用

首先对王健平提出的有限谱法^[1-3]做了进一步的理论研究，发现了一些新的有限谱法的插值基函数组，并互将有限谱法应用在ENO格式中，构造了有限谱ENO计算格式，然后通过对一维Euler方程的几个经典的模型问题和二维湍流与弱激波相互作用问题的数值计算，并且与理论解或准精确解进行比较分析，从而表明了此格式对于激波和其他间断具有较高的分辨率，在激波附近基本上没有明显的数值振荡，而且对于流场中的细致结构也具有相当高的精度。