加权型紧致格式与加权本质无波动格式的比较

线性紧致格式和加权本质无波动格式是两种典型的高阶精度数值格式,它们各有优缺点.线性紧致格式在具有高阶精度的同时,格式的分辨率也比较高,耗散低,是计算多尺度流场结构的较好格式,但是不能计算具有强激波的流场.加权本质无波动格式是一种高阶精度捕捉激波格式,鲁棒性好,但耗散比较高,分辨率也不理想.近年来,在莱勒的线性紧致格式基础上,采用加权本质无波动格式捕捉激波思想,发展了一系列加权型紧致格式.本文较全面地比较了加权型紧致格式和加权本质无波动格式,包括构造方法、鲁棒性、分辨率、耗散特性、收敛特性以及并行计算效率.结果表明,现有的加权型紧致格式基本保持了加权本质无波动格式的性质,对于气动力等宏观量的计算,比加权本质无波动格式没有明显的优势.      

应用于计算气动声学的优化有限紧致格式

对于含间断的计算气动声学问题,数值计算的格式不仅要求低耗散低色散的设计,对短波具有较高的分辨率,还要求能捕捉激波.中心紧致格式具有高精度,具有无耗散和低色散特征,但不能捕捉间断和激波;WENO格式处理间断较为成功,而耗散和色散误差相对较大.有限紧致格式可以将紧致格式与WENO格式相结合构造成混合格式,利用光滑因子之间的关系对激波区域进行自动判断,将传统的全域求解的紧致格式划分为有限的局部紧致求解,间断点上的激波捕捉铜梁自动作为局部紧致求解的边界通量,在在光滑区域具有紧致格式的高精度低耗散性质,在激波附近不产生非物理振荡.本文利用有限紧致格式思想,构造了新的适合于气动声学问题的优化有限紧致格式,将其应用于计算气动声学一维标准测试问题,对相关格式的模拟性能进行了评估,显示该格式在宽频声波传播和含有间断的声波传播模拟方面具有优势.     

尺度无关高阶WCNS格式

高速流场的数值模拟中, 既要保证对小尺度结构的高保真分辨, 又要实现对激波稳定、无振荡地捕捉.当前工程中广泛应用的高精度数值格式虽然都能一定程度地满足上述两种要求, 但仍与理想目标存在较大差距.例如, 模拟雷诺应力模型等小尺度问题时, 高精度格式在间断解附近易产生数值振荡.基于高精度格式所存在的上述问题, 本文引入去尺度函数, 探索了一种更加简单稳定的非线性权重构造方法, 并将其应用于7阶精度加权紧致非线性格式WCNS, 提出了一种尺度无关的7阶WCNS格式.该格式的性能与灵敏度参数和尺度因子的选择无关, 并且在小尺度下仍可以有效捕捉流场激波.同时, 该格式在间断处具有基本无振荡性质, 且在任意尺度函数下保持尺度无关, 并且在极值点处也能保持最优精度.本文还推导了7阶D权函数的形式.最后, 在一维线性对流方程中验证了新格式在流场光滑区能够达到设计精度, 并通过一系列数值实验证明了尺度无关的7阶WCNS格式在激波捕捉能力上具有良好表现, 为WCNS格式改进和解决可压缩湍流等非线性问题提供了一种新途径.      

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。     

一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用

由于映射操作会带来额外的计算时间消耗,传统加映射的WENO格式存在计算效率低的缺陷.为了提高传统加映射WENO格式的计算效率,通过利用标准符号函数的一种近似逼近函数构造出一族近似常值映射函数,本文提出了一种新的加映射WENO格式,称为WENO-ACM.新映射函数满足文献中已有WENOPM6格式映射函数的全部设计要求,其中WENO-PM6是一种为了克服经典WENO-M格式潜在的精度丢失缺陷而提出的格式.新格式保留了WENO-PM6在低耗散和高分辨率方面的优势,同时,显著的减少了每个时间步映射过程中的数学运算操作数,进而在计算效率方面获得了明显的提升.理论分析表明,新格式在即使包含临界点的光滑区域也能够获得最佳收敛精度.对近似色散关系的研究表明,新格式的频谱特性也得到了显著的提升.对大量标准测试算例进行了模拟计算,包括精度测试、激波管问题、激波-熵波相互作用、爆炸波相互碰撞、二维黎曼问题、双马赫反射、前台阶流动、瑞利-泰勒不稳定性和开尔文-亥姆霍兹不稳定性问题等.与广泛认可的WENO-JS, WENO-M, WENO-PM6格式综合比较发现,新提出的WENO-ACM格式在高效率、低数值耗散...     

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。     

无粘可压缩流动的改进高精度方法

为更准确捕捉复杂流场的流动细节,通过对WENO格式的光滑因子进行改进,发展了一种新的五阶WENO格式。对三阶ENO格式进行加权可以得到五阶WENO格式,但是不同的加权处理,WENO格式在极值处保持加权基本无振荡的效果不同,本文构造了二阶精度的局部光滑因子,及不含一阶二阶导数的高阶全局光滑因子,从而实现WENO格式在极值处有五阶精度。基于改进五阶WENO格式,对一维对流方程、一维和二维可压缩无粘问题进行算例验证,并与传统WENO-JS格式和WENO-Z格式进行比较。计算结果表明,改进五阶WENO格式有较高的精度和收敛速度,有较低的数值耗散,能有效捕捉间断、激波和涡等复杂流动。     

高精度格式WCNS-E-5的Fourier分析与应用

对高精度加权紧致非线性格式WCNS-E-5进行了理论分析和应用研究。首先采用Fourier方法分析了WCNS-E-5与WCNS-5及其它高阶格式(迎风偏置格式EUW-5与标准格式)的差分误差特性,研究了它们在分辨效率方面的特性和相位误差在多维方向上的各向异性特性。分析结果显示WCNS-E-5与WCNS-5在色散与耗散特性方面优于EUW-5,分辨效率也普遍高于EUW-5和格式,而且它们的相位速度在多维方向上传播所表现的各向异性特性与其它高阶格式一致。WCNS-E-5的高精度特性与WCNS-5的一致,但在计算过程中少了三对角矩阵求解因而它的效率更高,于是采用WCNS-E-5数值模拟了二维/三维复杂流场,算例包括单涡斜向运动、二维Rie-mann问题以及存在分离的大攻角钝锥流动问题。计算结果体现了WCNS-E-5具有很低的数值耗散误差,它对激波、膨胀波和接触面等间断具有很好的捕捉能力,得到的图像清晰光滑,准确再现了真实流动现象。     

满足"抑制波动原则"的广义紧致格式的特性及应用

详细分析了满足“抑制波动原则”和“稳定性原则”的五阶精度三点广义紧致格式的数值特性。该格式可以无波动地捕捉激波，并五具有类似于谱方法的高分辨率，以及很好的各向同性性质。另外还采用了时间相关的边界处理方法，以保证边界点上也满足“抑制波动原则”和“稳定性原则”。典型数值实验表明，本文格式及其边界处理方法能够有效求解Euler方程与N—S方程。     

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.     

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high‐order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high‐order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock‐capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high‐order weighted compact nonlinear scheme is not only robust for shock‐capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.     

A parameter-free ε-adaptive algorithm for improving weighted compact nonlinear schemes

In this paper, we propose a parameter-free algorithm to calculate ε, a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth-order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high-order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high-order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high-order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth-order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders.     

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.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Implicit large eddy simulation of a supersonic flat-plate boundary layer flow by weighted compact nonlinear scheme

The performance of implicit large eddy simulation (ILES) of a supersonic flat-plate turbulent boundary layer flow by weighted compact nonlinear scheme (WCNS) has been investigated. In view of features of WCNS and ILES, it was expected that ILES by WCNS could be an efficient approach to perform LES of supersonic turbulent flows. The flowfield calculated by WCNS was of lower turbulent intensity compared with an explicit LES data obtained by a numerical scheme of the same order of accuracy on a computational grid of similar resolution. It was concluded that the numerical dissipation inherent in WCNS is so large that applying WCNS to ILES of this flowfield is inefficient compared with explicit LES.     

Improvement of the weighted essentially nonoscillatory scheme based on the interaction of smoothness indicators

The weighted essentially nonoscillatory scheme is improved by introducing new smoothness indicators that evaluate the interactions among the classical smoothness indicators suggested by Jiang and Shu. The effect of the key parameters in the new smoothness indicators on the scheme is systematically investigated. The improved scheme has smaller dissipation with larger weight assignment to the discontinuous stencils and higher numerical accuracy with weights closer to the ideal weights. To verify the theory, benchmark problems governed by the linear transport equation, the 1‐dimensional nonlinear Burgers equation, and the Euler equations are conducted and analyzed, respectively. Better computational performances both on numerical resolution and accuracy are shown in the comparisons with other classical weighted essentially nonoscillatory schemes.     

基于模态分解的轴对称超声速射流啸声产生位置数值分析

不完全膨胀超声速射流的势核中会产生准周期的激波栅格结构, 其与剪切层内拟序结构的相互作用会产生激波噪声. 啸声是主要向上游方向传播的、具有离散频率的高强度激波噪声, 其产生是受一种非线性的声反馈环机制驱动. 精确定位啸声的声源位置是定量理解啸声反馈环机制和发展准确的啸声预测模型的一个关键所在. 为了分析近场啸声, 本文采用高精度数值方法直接求解轴对称可压缩Navier-Stokes方程, 数值模拟了完全膨胀射流马赫数为1.10和1.15的圆形声速喷管欠膨胀超声速冷射流, 得到了A₁和A₂两种轴对称模态啸声. 通过傅里叶模态分解、本征模态分解和动态模态分解, 分析了射流时序压力场和速度场, 研究了啸声关联拟序流动结构的空间演化, 精确定位了轴对称模态啸声的声源位置. 研究表明: 啸声关联拟序流动结构存在饱和态区域, 啸声声波是在其饱和态区域产生并向外传播; 在本文所涉及的射流马赫数范围内, A₁和A₂两种轴对称模态啸声的有效声源位置分别是在第4和第3个激波栅格结构的尾缘.      

随机振动的一种加权等价线性化方法

加权等价线性化方法是研究非线性随机振动的一种有效近似方法。关健在于找到一个合适的权函数使之对多数非线性问题都有比较满意的结果。本文提出一种类似峰值概率密度函数的权函数,由此构成一种加权等价线性化方法,借几个各具特点的非线性振动系统进行了可行性验证,表明与一般的等价线性化方法相比,本法所得的均方响应精度有相当程度的改善。