基于PSE的可压缩流边界层稳定性研究
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摘要
本文主要采用PSE(Parabolized Stability Equations,抛物化稳定性方程)的新理论和方法研究可压缩流边界层稳定性问题。PSE理论和方法的基本思想是将行进的T-S波在空间和时间上的正弦变量从流场中分离出来,留下的扰动流场由速度剖面、特征波长和在流向作缓慢变化的增长率组成,利用流向慢变特性,其稳定性方程通过忽略慢变函数的流向二阶及其高阶导数而抛物化,得到了抛物化稳定性方程。由于既没有引进平行近似也没有对初始振幅的限制,因而可用于同时考虑非平行、非线性的边界层稳定性研究,适用于一般对流稳定性问题。文中PSE方法的计算结果与相关文献数据吻合甚好。
首先研究了可压缩流非平行边界层稳定性问题。从Navier-Stokes方程出发推导出可压缩流的抛物化稳定性方程;发展了精确的数值方法,采用高精度的数值离散和高效的代数变换,大大提高了计算精度和收敛速度;依据抛物化稳定性方程的特征,采用预估校正迭代法和空间推进求解,并使至关重要的限制可压缩流扰动质量变化的正规化条件得到满足,保证了数值计算的稳定;分析了包括扰动频率、展向波数、壁面温度以及来流马赫数等各种参数对流动稳定性的影响。
文中着重研究了近年来特别受到关注的超音速和高超音速边界层稳定性的热门问题。针对高马赫数边界层稳定性方程特性,采用复平面上的锯齿型积分路径以及泰勒级数扩展边界层流动到复平面的方法,有效地处理数值积分路径中的奇点问题,得到了高马赫数边界层的多重不稳定模态。采用PSE方法对超音速和高超音速的非平行边界层稳定性的第一模态和第二模态进行了空间推进求解。
对于可压缩流边界层的非线性稳定性问题,则是通过快速Fourier变换(FFT),将扰动波分解为基本模态和高频模态,导出非线性抛物化稳定性方程(NPSE),研究超音速和高超音速边界层中T-S波与三维亚谐波及其高阶谐波之间的非线性稳定性问题。分析了谐波之间的非线性相互作用以及不同振幅背景的T-S波和亚谐波对流动失稳的影响,结果清楚地展示了所生成的流向涡和展向涡等经历的非线性演化过程和物理特征。该方法所用计算时间比DNS方法成数量级地减少,表明NPSE是研究非线性边界层稳定性的一个强有力的工具。
文中还对极为复杂的后掠机翼边界层稳定性进行了研究。先导出曲线坐标系的边界层方程,并对其精确求解,得到稳定性计算所需的基本流参数;然后推导出非正交曲线坐标系的稳定性方程,采用高效数值方法精确求解。研究了包括横流稳定性在内的后掠机翼的边界层稳定性问题,详细分析了扰动频率、波角、流向和展向位置等各种参数对流动稳定性的影响,并对机翼表面不同区域的稳定性的特征进行了分析和比较,所有这些将为机翼表面的流动转捩和边界层控制等工程应用提供可靠依据。
概言之,本文对可压缩流边界层稳定性问题进行了较为系统的数值计算和分析研究,研究的速度范围大,包括亚音速、超音速和高超音速流动,涉及到稳定性的多重模态,特别是第二模态对高马赫数流动稳定性的研究至关重要;研究了稳定性问题的难点,非平行稳定性和非线性稳定性,特别是采用NPSE研究了扰动波和复杂涡系的非线性演化过程,有着重要的理论意义;而对于像后掠机翼这样的三维物体的复杂边界层稳定性的研究,对于飞机设计等工程应用,则会有重要实用价值。
In this article, the new theory and methods of PSE(Parabolized Stability Equations) are used to research the stability of the compressible boundary layer flows. The principal idea of PSE is that the disturbance flow field can be resolved into the sinusoidal variation in space and time of the traveling Tollmien-Schlichting (T-S) wave component, and the residual field with a certain velocity profile and a characteristic wavelength and growth rate for cases of general convective instability. For a weakly nonparallel basic flow, the growth rate of the remaining field in the disturbance varies slowly in the streamwise direction. The stability equations can be parabolized by neglecting the smaller second and higher derivatives of the slowly varying functions with respect to the streamwise direction, resulting in the parabolized stability equations. Because the parallel approximation and the limitation of the initial amplitude are not introduced, PSE approach can be utilized to analyze simultaneously the non-parallel and nonlinear stability problem. The computed results of PSE in this article agree well with the relevant data in references.
The stabilities of unparallel compressible boundary layer flows are studied. Parabolized stability equations are derived from fully compressible Navier-Stokes equations. Developed accuracy methods, including high order differential methods and effective algebraic transformation, have improved the accuracy of the numerical solution and the velocity of convergence. According the feature of PSE, as predict-correction and spatial marching procedures are implemented, and the critical normalization condition of limiting the variation of disturbance mass is satisfied, the stability of numerical computa-tion can be guaranteed. The influences of various parameters, such as frequencies, spanwise wave numbers of disturbances, wall temperature and Mack number of free flows, on the stability are ana-lyzed.
Recently, the supersonic and hypersonic boundary layer stabilities are the important topic of dis-cussion, and are emphasized in this article. According to the characteristic of the stability equations for high Mach number boundary layer flows, the saw tooth integral path on the complex plane and the expansion of the basic flow onto the complex plane by using Taylor series are adopted to deal with the singularity in the numerical integral process. Multiple unstable modes are obtained. Moreover, the spatial marching of the first modes and second modes are carried out by using PSE method.
Next, we study nonlinear stability of the compressible boundary layer flows. Nonlinear parabo-lized stability equations(NPSE) are derived by the decomposition of the disturbance waves into basic modes and high order modes using fast Fourier transformation (FFT). The nonlinear stability among the T-S wave, the three-dimensional subharmonic wave and the produced high order waves in super- sonic boundary layer flows are researched. The nonlinear interaction of harmonic waves and the effect of different initial amplitudes of T-S wave and subharmonic wave on the stability are analyzed in de-tail. The nonlinear evolution and physic properties of the streamwise and spanwise vortexes and so on are clearly exhibited. The computational costs of NPSE method are much smaller than that of DNS method, which indicates that NPSE method is a powerful tool for the studies of the nonlinear stability.
The boundary layer stabilities of the swept wing, which are extremely complicated, are also re-searched. Firstly, boundary layer equations in the curvilinear coordinates are derived and solved ex-actly, and the basic flow parameters are attained. Secondly, the stability equations in the non-orthogonal curvilinear coordinates are derived, and solved by using accurate numerical methods. The stability problems including crossflow instability are investigated. The influences of various pa-rameters, such as frequencies, wave angles, and the streamwise and spanwise positions, on the stabil-ity are further analyzed. The boundary layer stabilities on different wing surface regions are compared, which provide the reliable basis for engineering application of the transition prediction and the boundary layer controls.
In conclusion, a systematic study of the compressible boundary layer flow stabilities is con-ducted. The stabilities of extensive velocity ranges, including subsonic, supersonic and hypersonic boundary layer flows, are researched. The multiple stability modes, specially the second modes which are most important for high Mach number flows, are used in the computation. The nonparallel and nonlinear stability problems, especially the nonlinear evolution of the disturbance waves and the complex vortices, are studied by using NPSE, which have important theoretical sense. The stability researches of the boundary layer for three dimensional objects, such as swept wing, are very valuable for engineering application like aircraft design.
首先研究了可压缩流非平行边界层稳定性问题。从Navier-Stokes方程出发推导出可压缩流的抛物化稳定性方程;发展了精确的数值方法,采用高精度的数值离散和高效的代数变换,大大提高了计算精度和收敛速度;依据抛物化稳定性方程的特征,采用预估校正迭代法和空间推进求解,并使至关重要的限制可压缩流扰动质量变化的正规化条件得到满足,保证了数值计算的稳定;分析了包括扰动频率、展向波数、壁面温度以及来流马赫数等各种参数对流动稳定性的影响。
文中着重研究了近年来特别受到关注的超音速和高超音速边界层稳定性的热门问题。针对高马赫数边界层稳定性方程特性,采用复平面上的锯齿型积分路径以及泰勒级数扩展边界层流动到复平面的方法,有效地处理数值积分路径中的奇点问题,得到了高马赫数边界层的多重不稳定模态。采用PSE方法对超音速和高超音速的非平行边界层稳定性的第一模态和第二模态进行了空间推进求解。
对于可压缩流边界层的非线性稳定性问题,则是通过快速Fourier变换(FFT),将扰动波分解为基本模态和高频模态,导出非线性抛物化稳定性方程(NPSE),研究超音速和高超音速边界层中T-S波与三维亚谐波及其高阶谐波之间的非线性稳定性问题。分析了谐波之间的非线性相互作用以及不同振幅背景的T-S波和亚谐波对流动失稳的影响,结果清楚地展示了所生成的流向涡和展向涡等经历的非线性演化过程和物理特征。该方法所用计算时间比DNS方法成数量级地减少,表明NPSE是研究非线性边界层稳定性的一个强有力的工具。
文中还对极为复杂的后掠机翼边界层稳定性进行了研究。先导出曲线坐标系的边界层方程,并对其精确求解,得到稳定性计算所需的基本流参数;然后推导出非正交曲线坐标系的稳定性方程,采用高效数值方法精确求解。研究了包括横流稳定性在内的后掠机翼的边界层稳定性问题,详细分析了扰动频率、波角、流向和展向位置等各种参数对流动稳定性的影响,并对机翼表面不同区域的稳定性的特征进行了分析和比较,所有这些将为机翼表面的流动转捩和边界层控制等工程应用提供可靠依据。
概言之,本文对可压缩流边界层稳定性问题进行了较为系统的数值计算和分析研究,研究的速度范围大,包括亚音速、超音速和高超音速流动,涉及到稳定性的多重模态,特别是第二模态对高马赫数流动稳定性的研究至关重要;研究了稳定性问题的难点,非平行稳定性和非线性稳定性,特别是采用NPSE研究了扰动波和复杂涡系的非线性演化过程,有着重要的理论意义;而对于像后掠机翼这样的三维物体的复杂边界层稳定性的研究,对于飞机设计等工程应用,则会有重要实用价值。
In this article, the new theory and methods of PSE(Parabolized Stability Equations) are used to research the stability of the compressible boundary layer flows. The principal idea of PSE is that the disturbance flow field can be resolved into the sinusoidal variation in space and time of the traveling Tollmien-Schlichting (T-S) wave component, and the residual field with a certain velocity profile and a characteristic wavelength and growth rate for cases of general convective instability. For a weakly nonparallel basic flow, the growth rate of the remaining field in the disturbance varies slowly in the streamwise direction. The stability equations can be parabolized by neglecting the smaller second and higher derivatives of the slowly varying functions with respect to the streamwise direction, resulting in the parabolized stability equations. Because the parallel approximation and the limitation of the initial amplitude are not introduced, PSE approach can be utilized to analyze simultaneously the non-parallel and nonlinear stability problem. The computed results of PSE in this article agree well with the relevant data in references.
The stabilities of unparallel compressible boundary layer flows are studied. Parabolized stability equations are derived from fully compressible Navier-Stokes equations. Developed accuracy methods, including high order differential methods and effective algebraic transformation, have improved the accuracy of the numerical solution and the velocity of convergence. According the feature of PSE, as predict-correction and spatial marching procedures are implemented, and the critical normalization condition of limiting the variation of disturbance mass is satisfied, the stability of numerical computa-tion can be guaranteed. The influences of various parameters, such as frequencies, spanwise wave numbers of disturbances, wall temperature and Mack number of free flows, on the stability are ana-lyzed.
Recently, the supersonic and hypersonic boundary layer stabilities are the important topic of dis-cussion, and are emphasized in this article. According to the characteristic of the stability equations for high Mach number boundary layer flows, the saw tooth integral path on the complex plane and the expansion of the basic flow onto the complex plane by using Taylor series are adopted to deal with the singularity in the numerical integral process. Multiple unstable modes are obtained. Moreover, the spatial marching of the first modes and second modes are carried out by using PSE method.
Next, we study nonlinear stability of the compressible boundary layer flows. Nonlinear parabo-lized stability equations(NPSE) are derived by the decomposition of the disturbance waves into basic modes and high order modes using fast Fourier transformation (FFT). The nonlinear stability among the T-S wave, the three-dimensional subharmonic wave and the produced high order waves in super- sonic boundary layer flows are researched. The nonlinear interaction of harmonic waves and the effect of different initial amplitudes of T-S wave and subharmonic wave on the stability are analyzed in de-tail. The nonlinear evolution and physic properties of the streamwise and spanwise vortexes and so on are clearly exhibited. The computational costs of NPSE method are much smaller than that of DNS method, which indicates that NPSE method is a powerful tool for the studies of the nonlinear stability.
The boundary layer stabilities of the swept wing, which are extremely complicated, are also re-searched. Firstly, boundary layer equations in the curvilinear coordinates are derived and solved ex-actly, and the basic flow parameters are attained. Secondly, the stability equations in the non-orthogonal curvilinear coordinates are derived, and solved by using accurate numerical methods. The stability problems including crossflow instability are investigated. The influences of various pa-rameters, such as frequencies, wave angles, and the streamwise and spanwise positions, on the stabil-ity are further analyzed. The boundary layer stabilities on different wing surface regions are compared, which provide the reliable basis for engineering application of the transition prediction and the boundary layer controls.
In conclusion, a systematic study of the compressible boundary layer flow stabilities is con-ducted. The stabilities of extensive velocity ranges, including subsonic, supersonic and hypersonic boundary layer flows, are researched. The multiple stability modes, specially the second modes which are most important for high Mach number flows, are used in the computation. The nonparallel and nonlinear stability problems, especially the nonlinear evolution of the disturbance waves and the complex vortices, are studied by using NPSE, which have important theoretical sense. The stability researches of the boundary layer for three dimensional objects, such as swept wing, are very valuable for engineering application like aircraft design.
引文
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