基于色散关系分析的高精度气动声学计算方法研究
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摘要
气动声学问题与流体力学问题有着本质的区别,直接数值模拟气动声的产生与传播问题时,对数值算法的要求远远高于计算流体力学。为了正确捕捉波的色散性、耗散性、相位速度与群速度等传播特征,数值离散格式的色散和耗散误差必须尽可能的小,同时,在数值处理开放边界或者半开放边界问题时无明显反射。因此,发展高分辨率的数值离散格式与高性能的无反射边界条件是计算气动声学(Computational Aeroacoustics,简称CAA) CAA的两个关键问题。
本文根据CAA的高精度技术需求,针对这两个关键问题,从数值格式与控制方程的色散关系(频率与波数关系)出发,研究了高精度离散格式的误差行为、高精度空间数值离散格式与高精度无反射边界条件。主要包括：高精度有限差分格式的误差分析,显式有限差分格式的保色散优化,非均匀网格的高精度紧致差分格式,适用斜流计算的线性与非线性完全耦合层(Perfectly Matched Layer,简称PML)吸收边界条件。采用国际标准CAA算例验证、测试与分析各格式及吸收边界条件的求解性能。本论文的主要研究内容如下：
(1)提出了一种构造高精度色散保持显式有限差分格式的优化方法。为了获得高精度的数值离散格式,提高离散格式数值模拟长距离、长时间传播波的能力。首先,通过傅立叶与拉普拉斯变换法则将有限差分格式变换至频率与波数空间；然后,采用修正误差传播分析方法和Neumann误差分析方法,从理论上详细推导了高精度有限差分格式误差构成、误差传播以及误差累积的数学公式,为发展低耗散低色散高精度数值格式提供理论依据；在已有DRP优化思想的基础上,重点考虑群速度对格式色散性能的影响,提出了一种新的DRP优化方法。该优化方法以绝对误差最小积分值为目标函数，格式系数与积分上限为优化变量,色散、耗散及群速度误差为约束条件,采用序列二次规划方法,通过迭代计算,寻找最优解,获得高精度有限差分格式系数及积分上限值。基于该优化方法,优化了7点、9点及11点有限差分格式,并分析其色散、耗散与群速度性能,最后通过一些典型算例验证误差分析及优化方法所得的结论。
(2).提出了一种构造非均匀网格高精度紧致差分格式的方法。针对非均匀网格容易导致紧致差分格式产生较大误差、引起虚假振荡等问题,首先,基于光滑的思想,通过泰勒展开分析方法,详细给出了内点与边界点紧致离散格式系数的通用求解表达式；然后,根据该理论公式构造了一个3对角6阶精度的紧致差分格式。通过傅立叶与拉普拉斯变换法则将该紧致差分格式变换至频率与波数空间,推导出该格式数值耗散与色散的理论计算公式,并以均匀网格、扰动网格、拉伸网格及突变网格4种典型网格为例,详细研究该格式对不同网格的耗散与色散特性,重点研究了扰动因子对其的影响；采用特征值方法,推导了半离散格式满足渐近稳定的条件,并研究了不同网格下该紧致离散格式的渐近稳定性。最后,通过一些典型算例测试了该格式的求解性能,数值计算结果表明所得数值解与精确解吻合,本文所提出的高精度紧致差分格式,在进行非均匀网格气动声学数值模拟时,能够抑制因网格不均匀而引起的非物理振荡,计算准确。该格式构造方便,精度高,显示了其在非均匀网格模拟中的优越性。
(3)提出了一种构造二维对流线性PML吸收边界条件的七步通用方法。首先,通过傅立叶与拉普拉斯变换法则,将对流线性控制方程转换至频率与波数空间,并建立控制方程的色散关系；然后,根据该色散关系轨迹分析流场内各物理波的群速度与相位速度方向,提出了保频型与变频型两种时间与空间坐标变换关系式：基于这两类坐标变化关系式,发展了两类二维斜流线性PML吸收边界条件,并对各层吸收边界条件的控制方程进行了理论推导；采用特征值方法,分别研究了Hu的分裂型、本文建立的保频型与变频型斜流线性PML吸收边界条件的动态稳定性,重点研究了来流马赫数、波数、吸收系数及额外增加的吸收项对其稳定性的影响；最后,通过一些典型算例测试了这些斜流线性PML吸收边界条件的数值求解精度与可靠性,数值计算结果表明：PML域内各物理波具有一致的群速度与相位速度方向仅为PML稳定的必要条件而非充分条件；Hu的分裂型斜流线性PML吸收边界条件不稳定,保频型斜流线性PML吸收边界条件条件稳定,变频型斜流线性PML吸收边界条件稳定；变频型斜流线性PML吸收边界条件对声波、涡波与熵波具有良好的吸收特性,非常适合用来计算斜流CAA问题。
(4)提出了三种适用斜流计算的二维非线性PML吸收边界条件。首先,通过伪时均流假设,建立了扰动量的守恒型与非守恒型控制方程；然后,采用变频时间与空间坐标变换关系,将这些扰动量控制方程变换至新的时间与空间坐标系：采用复数变换方法,分别构建了非守恒型、守恒非分裂型与守恒分裂型斜流非线性PML吸收边界条件,并推导了各层PML吸收边界条件的理论公式；通过线性假设,将非线性问题转化为线性问题,采用特征值分析方法,对建立的两种守恒型非线性PML吸收边界条件进行稳定性分析,详细研究了来流马赫数、波数、吸收系数及增加的吸收项对其稳定性的影响；最后,通过一些典型算例测试和分析了这些斜流非线性PML吸收边界条件的数值求解性能,数值计算结果表明：当角层非线性PML吸收边界条件中未额外增加吸收项时,系统特征值的最大虚部均为正数,吸收边界条件不稳定,当角层非线性PML吸收边界条件中额外增加吸收项时,系统特征值的最大虚部均为负数,吸收边界条件稳定；非守恒型斜流非线性PML吸收边界条件的吸收效果最好,分裂型PML次之,非分裂型PML反射量最大。本文建立的非守恒型、非分裂型和分裂型PML吸收边界条件均稳定,能够有效抑制边界反射,提出的构建斜流非线性PML吸收边界条件的方法切实可行。
(5)提出了适用斜流计算的三维线性与非线性PML吸收边界条件。在二维线性与非线性斜流PML吸收边界条件的基础上,将其构造方法拓展至三维斜流PML吸收边界条件的理论推导过程中。将三维斜流PML计算域分为26块,7大类。根据流场内各物理波的色散关系轨迹,提出了一组适用三维斜流PML构造的变频型时间与空间坐标变换关系式。基于这组坐标变换关系,通过复数变换法,构建了三维斜流线性、三维斜流非守恒非线性、三维斜流守恒非分裂型与分裂型非线性PML吸收边界条件,详细交代了各角层PML吸收边界条件中吸收项的施加原则。此外,还重点研究了来流马赫数、波数、吸收系数及增加的吸收项对各角层PML吸收边界条件稳定性的影响。最后,通过一些典型算例测试和分析了三维斜流线性与三维斜流守恒分裂型非线性PML吸收边界条件的吸收性能,数值计算结果表明：本文建立的三维斜流线性与非线性PML吸收边界条件对声波与涡环均具有良好的吸收性能,无明显反射现象产生,计算稳定。
Aeroacoustic problems are very different from standard aerodynamic and fluid mechanic problems naturally. It is important to recognize that the numerical algorithms for computational aeroacoustics (CAA) are more critical than for computational fluid dynamical (CFD). In order to accurately capture the dispersion, dissipation, phase velocity and group velocity characteristics of aeroacoustic wave, the numerical scheme must exhibit a minimum of dispersion and dissipation error. Furthermore, non-reflecting boundary condition is also a critical component for open and half open computational boundary problems. Therefore, high resolution numerical schemes and high powered non-reflecting boundary conditions are two key problems of CAA.
Based on the technical requirements of CAA, in view of the dispersion relation between numerical schemes and controlled equations, the error behavior of high order accuracy finite difference schemes, high order accuracy finite difference schemes and non-reflecting boundary conditions are studied in this paper. Mainly includes:the error analysis of high order accuracy finite difference scheme, the optimization of low-dispersion explicit finite difference scheme, the construction of high order accuracy compact finite difference scheme on non-uniform meshes, the construction of linear and nonlinear perfectly matched layer (PML) absorbing boundary condition for oblique flow calculation. The accuracy of proposed schemes and non-refection PML absorbing boundary conditions is demonstrated by CAA benchmark problems. The detailed research works and results of this dissertation are listed as follows:
(1) An optimization method for constructing high order accuracy explicit finite difference scheme is proposed. In order to obtain high order accuracy numerical schemes and improve the accuracy for computing the propagation of aeroacoustic wave in long time or long distance. First, the finite difference scheme is transformed into frequency and wavenumber space by applying Fourier and Laplace transformation method. Then the mathematical formula of error construction, error propagation and error accumulation of high order finite difference schemes are derived in detail using Neumann error analysis method and correction error propagation analysis method. This can provide a theoretical basis for developing low dissipation and low dispersion higher order accuracy numerical schemes. Based on the idea of dispersion relation preserving optimization, a new strategy to optimize finite difference schemes in spectral domain is proposed considering the effects of group velocity on dispersion characteristics. The objective of optimization is minimizing the integrated absolute error with a strict tolerance for dispersion, dissipation and group velocity errors. The scheme coefficients and optimized wavenumber domain are determined using iterative calculations and sequential quadratic programming method. The7-point,9-point and11-point finite difference schemes are optimized using the new strategy, and the dispersion, dissipation and group characteristics of these optimized schemes are analyzed. At last, the results of error analysis and optimization method are validated by CAA benchmark problems.
(2) A new method for constructing high order accuracy compact finite difference scheme on non-uniform meshes is proposed. When the classic compact finite difference scheme was applied to practical problems using non-uniform meshes, the spurious numerical oscillations would be excited. Based on the smooth techniques, the general mathematical expressions of inner and boundary point scheme's coefficients are derived through Taylor expansion method. Then a sixth order tridiagonal compact scheme is presented based on the theory expressions. The numerical dispersion and dissipation on different mesh types are analyzed by applying Fourier and Laplace transformation to the sixth order tridiagonal compact scheme. This paper mainly studies the influence of uniform mesh, perturbed mesh, stretched mesh and sudden mesh on the numerical dispersion and dissipation, especially for the perturbed factor. Furthermore, the asymptotic stability of semi discrete scheme on different mesh types is researched using eigenvalue method. At last, the resolution performance of the scheme is demonstrated using CAA benchmark problems. Numerical results show that the numerical solutions for CAA benchmark problems agree well with the theoretical solutions. The compact difference scheme proposed in this paper is stability, accurate and simple for CAA problems on non-uniform meshes, which shows advantages in the simulation of CAA problems on non-uniform meshes.
(3) A seven steps general method for constructing PML absorbing boundary conditions in case of convection flow is proposed. First, the dispersion relation of linear controlled equations is established by employing Fourier and Laplace transform method. Then the propagation directions of the phase and group velocities of the physical waves are analyzed according to the dispersion relation. Two kinds of proper space-time transformations are determined under the hypothesis of unchanged frequency (UCF) and changed frequency (CF), respectively. UCFPML and CFPML absorbing boundary conditions for linear Euler equations in two dimensions are developed when mean velocities strike the boundary at an arbitrary angle, and the UCFPML and CFPML equations for the side layers and corner layers of a rectangular domain will be derived independently. Furthermore, the stability of linear wave in Hu's split PML, UCFPML and CFPML layers are analyzed using eigenvalve method. The influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail. Finally, these PML absorbing boundary conditions for linear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problem. The phase velocity should be consistent with the direction of group velocities of the physical waves, which is a necessary but not sufficient condition for stable PML absorbing boundary conditions. For the three PML absorbing boundary conditions, Hu's split PML equations are instability, UCFPML equations are stable with some conditions, and CFPML equations are very stable. Moreover, the acoustic, vorticity and entropy wave will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the CFPML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow.
(4) Three kinds of nonlinear PML absorbing boundary conditions in two dimensional are proposed in case of oblique flow. First, the pseudo mean flow is introduced for formulating the time-dependent fluctuation nonlinear equations in conservation and primitive form. Then these time-dependent fluctuation nonlinear equations are transformed into new space-time coordinates using CF space-time transformations proposed in section4. For nonlinear Euler equations in oblique flow in two dimensions, the primitive PML (PPML), unsplit conservation PML (USCPML) and split conservation PML (SCPML) absorbing boundary conditions are established by applying the PML complex change of time-dependent fluctuation nonlinear equations in new space-time coordinates. These PML equations for the side layers and corner layers of a rectangular domain are also formulated independently. The nonlinear stability analysis of nonlinear PML equations is translated into linear stability analysis using linearity hypothesis, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail, and the importance of added absorption term is emphasized. Finally, these PML absorbing boundary conditions for nonlinear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problems. Numerical results show that the imaginary part of eigenvalue is nonpositive for every eigenvalue when the absorption terms are added to corner PML equations. Therefore, these PML equations will be dynamically stable. Otherwise, these PML equations will be not stable if absorption terms are not added to corner PML equations. Among the three nonlinear PML absorbing boundary conditions, PPML is the best, SCPML is better, USCPML is the worst one. The three nonlinear PML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow due to no boundary reflections in the PML domain, so the method for developing nonlinear PML absorbing boundary conditions in case of oblique flow is effective and feasible.
(5) The linear and nonlinear PML absorbing boundary conditions in three dimensions for the more general case of an oblique mean flow are proposed. The goal of this work is to further extend the two dimensional linear and nonlinear PML methodology in three dimensions. The three dimensional PML computational domain is divided into twenty six subdomains and seven categories. According to the path of dispersion relationship of physical waves, a series of appropriate space-time transformations for correcting the inconsistencies in phase and group velocity is presented with the hypothesis of unchanged frequency. Based on the space-time transformations, the derivations of linear PML, primitive nonlinear PML (PPML), unsplit conservation nonlinear PML (USCPML) and split conservation nonlinear PML (SCPML) equations are illustrated in details using PML complex transformation method, and the adding principle of absorption term for corner PML equations is emphasized. In addition, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability of corner PML equations is studied. Finally, the validity and efficiency of proposed equations as linear and nonlinear PML absorbing boundary conditions are demonstrated by numerical examples. The numerical results show that the acoustic wave and vortex ring will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the proposed PML absorbing boundary conditions are effective and stable for computing CAA problems when mean velocities strike the boundary at an arbitrary angle.
本文根据CAA的高精度技术需求,针对这两个关键问题,从数值格式与控制方程的色散关系(频率与波数关系)出发,研究了高精度离散格式的误差行为、高精度空间数值离散格式与高精度无反射边界条件。主要包括：高精度有限差分格式的误差分析,显式有限差分格式的保色散优化,非均匀网格的高精度紧致差分格式,适用斜流计算的线性与非线性完全耦合层(Perfectly Matched Layer,简称PML)吸收边界条件。采用国际标准CAA算例验证、测试与分析各格式及吸收边界条件的求解性能。本论文的主要研究内容如下：
(1)提出了一种构造高精度色散保持显式有限差分格式的优化方法。为了获得高精度的数值离散格式,提高离散格式数值模拟长距离、长时间传播波的能力。首先,通过傅立叶与拉普拉斯变换法则将有限差分格式变换至频率与波数空间；然后,采用修正误差传播分析方法和Neumann误差分析方法,从理论上详细推导了高精度有限差分格式误差构成、误差传播以及误差累积的数学公式,为发展低耗散低色散高精度数值格式提供理论依据；在已有DRP优化思想的基础上,重点考虑群速度对格式色散性能的影响,提出了一种新的DRP优化方法。该优化方法以绝对误差最小积分值为目标函数，格式系数与积分上限为优化变量,色散、耗散及群速度误差为约束条件,采用序列二次规划方法,通过迭代计算,寻找最优解,获得高精度有限差分格式系数及积分上限值。基于该优化方法,优化了7点、9点及11点有限差分格式,并分析其色散、耗散与群速度性能,最后通过一些典型算例验证误差分析及优化方法所得的结论。
(2).提出了一种构造非均匀网格高精度紧致差分格式的方法。针对非均匀网格容易导致紧致差分格式产生较大误差、引起虚假振荡等问题,首先,基于光滑的思想,通过泰勒展开分析方法,详细给出了内点与边界点紧致离散格式系数的通用求解表达式；然后,根据该理论公式构造了一个3对角6阶精度的紧致差分格式。通过傅立叶与拉普拉斯变换法则将该紧致差分格式变换至频率与波数空间,推导出该格式数值耗散与色散的理论计算公式,并以均匀网格、扰动网格、拉伸网格及突变网格4种典型网格为例,详细研究该格式对不同网格的耗散与色散特性,重点研究了扰动因子对其的影响；采用特征值方法,推导了半离散格式满足渐近稳定的条件,并研究了不同网格下该紧致离散格式的渐近稳定性。最后,通过一些典型算例测试了该格式的求解性能,数值计算结果表明所得数值解与精确解吻合,本文所提出的高精度紧致差分格式,在进行非均匀网格气动声学数值模拟时,能够抑制因网格不均匀而引起的非物理振荡,计算准确。该格式构造方便,精度高,显示了其在非均匀网格模拟中的优越性。
(3)提出了一种构造二维对流线性PML吸收边界条件的七步通用方法。首先,通过傅立叶与拉普拉斯变换法则,将对流线性控制方程转换至频率与波数空间,并建立控制方程的色散关系；然后,根据该色散关系轨迹分析流场内各物理波的群速度与相位速度方向,提出了保频型与变频型两种时间与空间坐标变换关系式：基于这两类坐标变化关系式,发展了两类二维斜流线性PML吸收边界条件,并对各层吸收边界条件的控制方程进行了理论推导；采用特征值方法,分别研究了Hu的分裂型、本文建立的保频型与变频型斜流线性PML吸收边界条件的动态稳定性,重点研究了来流马赫数、波数、吸收系数及额外增加的吸收项对其稳定性的影响；最后,通过一些典型算例测试了这些斜流线性PML吸收边界条件的数值求解精度与可靠性,数值计算结果表明：PML域内各物理波具有一致的群速度与相位速度方向仅为PML稳定的必要条件而非充分条件；Hu的分裂型斜流线性PML吸收边界条件不稳定,保频型斜流线性PML吸收边界条件条件稳定,变频型斜流线性PML吸收边界条件稳定；变频型斜流线性PML吸收边界条件对声波、涡波与熵波具有良好的吸收特性,非常适合用来计算斜流CAA问题。
(4)提出了三种适用斜流计算的二维非线性PML吸收边界条件。首先,通过伪时均流假设,建立了扰动量的守恒型与非守恒型控制方程；然后,采用变频时间与空间坐标变换关系,将这些扰动量控制方程变换至新的时间与空间坐标系：采用复数变换方法,分别构建了非守恒型、守恒非分裂型与守恒分裂型斜流非线性PML吸收边界条件,并推导了各层PML吸收边界条件的理论公式；通过线性假设,将非线性问题转化为线性问题,采用特征值分析方法,对建立的两种守恒型非线性PML吸收边界条件进行稳定性分析,详细研究了来流马赫数、波数、吸收系数及增加的吸收项对其稳定性的影响；最后,通过一些典型算例测试和分析了这些斜流非线性PML吸收边界条件的数值求解性能,数值计算结果表明：当角层非线性PML吸收边界条件中未额外增加吸收项时,系统特征值的最大虚部均为正数,吸收边界条件不稳定,当角层非线性PML吸收边界条件中额外增加吸收项时,系统特征值的最大虚部均为负数,吸收边界条件稳定；非守恒型斜流非线性PML吸收边界条件的吸收效果最好,分裂型PML次之,非分裂型PML反射量最大。本文建立的非守恒型、非分裂型和分裂型PML吸收边界条件均稳定,能够有效抑制边界反射,提出的构建斜流非线性PML吸收边界条件的方法切实可行。
(5)提出了适用斜流计算的三维线性与非线性PML吸收边界条件。在二维线性与非线性斜流PML吸收边界条件的基础上,将其构造方法拓展至三维斜流PML吸收边界条件的理论推导过程中。将三维斜流PML计算域分为26块,7大类。根据流场内各物理波的色散关系轨迹,提出了一组适用三维斜流PML构造的变频型时间与空间坐标变换关系式。基于这组坐标变换关系,通过复数变换法,构建了三维斜流线性、三维斜流非守恒非线性、三维斜流守恒非分裂型与分裂型非线性PML吸收边界条件,详细交代了各角层PML吸收边界条件中吸收项的施加原则。此外,还重点研究了来流马赫数、波数、吸收系数及增加的吸收项对各角层PML吸收边界条件稳定性的影响。最后,通过一些典型算例测试和分析了三维斜流线性与三维斜流守恒分裂型非线性PML吸收边界条件的吸收性能,数值计算结果表明：本文建立的三维斜流线性与非线性PML吸收边界条件对声波与涡环均具有良好的吸收性能,无明显反射现象产生,计算稳定。
Aeroacoustic problems are very different from standard aerodynamic and fluid mechanic problems naturally. It is important to recognize that the numerical algorithms for computational aeroacoustics (CAA) are more critical than for computational fluid dynamical (CFD). In order to accurately capture the dispersion, dissipation, phase velocity and group velocity characteristics of aeroacoustic wave, the numerical scheme must exhibit a minimum of dispersion and dissipation error. Furthermore, non-reflecting boundary condition is also a critical component for open and half open computational boundary problems. Therefore, high resolution numerical schemes and high powered non-reflecting boundary conditions are two key problems of CAA.
Based on the technical requirements of CAA, in view of the dispersion relation between numerical schemes and controlled equations, the error behavior of high order accuracy finite difference schemes, high order accuracy finite difference schemes and non-reflecting boundary conditions are studied in this paper. Mainly includes:the error analysis of high order accuracy finite difference scheme, the optimization of low-dispersion explicit finite difference scheme, the construction of high order accuracy compact finite difference scheme on non-uniform meshes, the construction of linear and nonlinear perfectly matched layer (PML) absorbing boundary condition for oblique flow calculation. The accuracy of proposed schemes and non-refection PML absorbing boundary conditions is demonstrated by CAA benchmark problems. The detailed research works and results of this dissertation are listed as follows:
(1) An optimization method for constructing high order accuracy explicit finite difference scheme is proposed. In order to obtain high order accuracy numerical schemes and improve the accuracy for computing the propagation of aeroacoustic wave in long time or long distance. First, the finite difference scheme is transformed into frequency and wavenumber space by applying Fourier and Laplace transformation method. Then the mathematical formula of error construction, error propagation and error accumulation of high order finite difference schemes are derived in detail using Neumann error analysis method and correction error propagation analysis method. This can provide a theoretical basis for developing low dissipation and low dispersion higher order accuracy numerical schemes. Based on the idea of dispersion relation preserving optimization, a new strategy to optimize finite difference schemes in spectral domain is proposed considering the effects of group velocity on dispersion characteristics. The objective of optimization is minimizing the integrated absolute error with a strict tolerance for dispersion, dissipation and group velocity errors. The scheme coefficients and optimized wavenumber domain are determined using iterative calculations and sequential quadratic programming method. The7-point,9-point and11-point finite difference schemes are optimized using the new strategy, and the dispersion, dissipation and group characteristics of these optimized schemes are analyzed. At last, the results of error analysis and optimization method are validated by CAA benchmark problems.
(2) A new method for constructing high order accuracy compact finite difference scheme on non-uniform meshes is proposed. When the classic compact finite difference scheme was applied to practical problems using non-uniform meshes, the spurious numerical oscillations would be excited. Based on the smooth techniques, the general mathematical expressions of inner and boundary point scheme's coefficients are derived through Taylor expansion method. Then a sixth order tridiagonal compact scheme is presented based on the theory expressions. The numerical dispersion and dissipation on different mesh types are analyzed by applying Fourier and Laplace transformation to the sixth order tridiagonal compact scheme. This paper mainly studies the influence of uniform mesh, perturbed mesh, stretched mesh and sudden mesh on the numerical dispersion and dissipation, especially for the perturbed factor. Furthermore, the asymptotic stability of semi discrete scheme on different mesh types is researched using eigenvalue method. At last, the resolution performance of the scheme is demonstrated using CAA benchmark problems. Numerical results show that the numerical solutions for CAA benchmark problems agree well with the theoretical solutions. The compact difference scheme proposed in this paper is stability, accurate and simple for CAA problems on non-uniform meshes, which shows advantages in the simulation of CAA problems on non-uniform meshes.
(3) A seven steps general method for constructing PML absorbing boundary conditions in case of convection flow is proposed. First, the dispersion relation of linear controlled equations is established by employing Fourier and Laplace transform method. Then the propagation directions of the phase and group velocities of the physical waves are analyzed according to the dispersion relation. Two kinds of proper space-time transformations are determined under the hypothesis of unchanged frequency (UCF) and changed frequency (CF), respectively. UCFPML and CFPML absorbing boundary conditions for linear Euler equations in two dimensions are developed when mean velocities strike the boundary at an arbitrary angle, and the UCFPML and CFPML equations for the side layers and corner layers of a rectangular domain will be derived independently. Furthermore, the stability of linear wave in Hu's split PML, UCFPML and CFPML layers are analyzed using eigenvalve method. The influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail. Finally, these PML absorbing boundary conditions for linear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problem. The phase velocity should be consistent with the direction of group velocities of the physical waves, which is a necessary but not sufficient condition for stable PML absorbing boundary conditions. For the three PML absorbing boundary conditions, Hu's split PML equations are instability, UCFPML equations are stable with some conditions, and CFPML equations are very stable. Moreover, the acoustic, vorticity and entropy wave will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the CFPML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow.
(4) Three kinds of nonlinear PML absorbing boundary conditions in two dimensional are proposed in case of oblique flow. First, the pseudo mean flow is introduced for formulating the time-dependent fluctuation nonlinear equations in conservation and primitive form. Then these time-dependent fluctuation nonlinear equations are transformed into new space-time coordinates using CF space-time transformations proposed in section4. For nonlinear Euler equations in oblique flow in two dimensions, the primitive PML (PPML), unsplit conservation PML (USCPML) and split conservation PML (SCPML) absorbing boundary conditions are established by applying the PML complex change of time-dependent fluctuation nonlinear equations in new space-time coordinates. These PML equations for the side layers and corner layers of a rectangular domain are also formulated independently. The nonlinear stability analysis of nonlinear PML equations is translated into linear stability analysis using linearity hypothesis, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability is studied in detail, and the importance of added absorption term is emphasized. Finally, these PML absorbing boundary conditions for nonlinear Euler equations in the case of oblique mean flow are validated by computing the CAA benchmark problems. Numerical results show that the imaginary part of eigenvalue is nonpositive for every eigenvalue when the absorption terms are added to corner PML equations. Therefore, these PML equations will be dynamically stable. Otherwise, these PML equations will be not stable if absorption terms are not added to corner PML equations. Among the three nonlinear PML absorbing boundary conditions, PPML is the best, SCPML is better, USCPML is the worst one. The three nonlinear PML absorbing boundary conditions are effective for computing CAA problems in case of oblique flow due to no boundary reflections in the PML domain, so the method for developing nonlinear PML absorbing boundary conditions in case of oblique flow is effective and feasible.
(5) The linear and nonlinear PML absorbing boundary conditions in three dimensions for the more general case of an oblique mean flow are proposed. The goal of this work is to further extend the two dimensional linear and nonlinear PML methodology in three dimensions. The three dimensional PML computational domain is divided into twenty six subdomains and seven categories. According to the path of dispersion relationship of physical waves, a series of appropriate space-time transformations for correcting the inconsistencies in phase and group velocity is presented with the hypothesis of unchanged frequency. Based on the space-time transformations, the derivations of linear PML, primitive nonlinear PML (PPML), unsplit conservation nonlinear PML (USCPML) and split conservation nonlinear PML (SCPML) equations are illustrated in details using PML complex transformation method, and the adding principle of absorption term for corner PML equations is emphasized. In addition, the influence of mean flow velocities, wavenumbers, absorption parameters and added absorption term on stability of corner PML equations is studied. Finally, the validity and efficiency of proposed equations as linear and nonlinear PML absorbing boundary conditions are demonstrated by numerical examples. The numerical results show that the acoustic wave and vortex ring will be exponentially decreasing with minimizing boundary reflections in the PML domain. Therefore, the proposed PML absorbing boundary conditions are effective and stable for computing CAA problems when mean velocities strike the boundary at an arbitrary angle.
引文
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