三维非平行流边界层稳定性研究
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摘要
本文从Navier-Stokes(N-S)方程出发,导出了不同坐标系下的抛物化稳定性方程(PSE),研究了三维物体的边界层稳定性问题,以及三维扰动的C型失稳的空间演化过程。
对于曲线坐标系下的抛物化稳定性方程,文中发展了求解的高效数值方法:引进法向变换,使得在临界层与壁面之间的扰动量变化最快的区域有更多的法向网格点;导出包含边界邻域在内的完全四阶精度的法向有限差分格式,这对方程精确离散至关重要;采用全局法和局部法相结合的方法及其新的迭代公式,大大加速收敛并得到更精确的特征值。算例给出了扰动幅值增长和形状函数等演化曲线,并对机翼、机身的边界层稳定性进行了分析和研究。
借助Fourier级数的方法将扰动波分解为基本模态和高频模态,研究了三维扰动的非线性边界层稳定性问题。采用高精度的高阶谱方法及其对无限区域问题的有效代数变换,提高了计算精度和收敛速度。运用Landau展开给出了二维谐波的初始解、通过求解线性PSE而得到TS波和C型亚谐扰动的初始站值、以及近似地模拟初始站的均流变形值,以此为初始条件,并结合迭代-预估校正方法来实现非线性PSE的求解和推进,详细地研究了非平行边界层三维扰动C型失稳过程。
文中通过对三维物体的非平行边界层稳定性的研究,确定最不稳定波的扰动参数和最易失稳的流向、展向位置;对三维扰动的非平行非线性边界层稳定性的研究,得到更精确的扰动放大因子值。这些都为边界层的转捩位置的确定和机理研究提供了可靠的依据,具有重要的理论意义和实用价值。
本文的研究工作是在教育部博士点基金(20030287003)的资助下完成的。
In this thesis, the parabolic stability equations (PSE) in the different coordinate systems are driven from Navier-Stokes (N-S) equations. The stability problems of the boundary layers for three-dimensional body,and the spatial evolution process of the C-type instability are studied.
The PSE in the curve coordinate systems are solved by the numerical techniques developed efficiently in this paper. A normal transform is introduced, and there are enough much grids in the region between the critical layer and the wall, where the variation of the disturbance is the quickest. The finite-difference of governing equations with fourth-order accuracy in the normal direction is utilized in full regions including points close the wall, and is very important for accurately discreting equations. The combination of global and local methods is implemented, and a new iterative formula is derived. Thus, the more precision eigenvalues can be obtained, and the convergence is made rapidly. In the examples we get some evolution curves, such as the growth rates and shape functions, which are used to analyze the stability problems for the boundary layers of the fuselage and the wing.
With the Fourier series technique, the disturbance is discomposed into predominant mode and high frequency harmonics. The nonlinear stability of three-dimensional disturbances for the Blasius boundary layers is studied. We adopted here the high accuracy method of using the expansions in orthogonal functions along the normal direction and the effective algebraic mapping to deal with the problem in the infinite region. The initial solutions of two-dimensional harmonic waves are given by Landau expansion, and the mean-flow-distortion is calculated by the approximation equation. We can get the initial conditions of TS wave and sub-harmonic wave in the C-type instability by linear PSE method. Furthermore, we employ iteration and Predictor-Corrector-Approach to solve the nonlinear equations in order to implement the marching procedure. We investigate detailedly the process of the C-type instability for nonparallel boundary layers with three-dimensional disturbances.
By investigating the nonparallel boundary layers of three-dimensional body, we
对于曲线坐标系下的抛物化稳定性方程,文中发展了求解的高效数值方法:引进法向变换,使得在临界层与壁面之间的扰动量变化最快的区域有更多的法向网格点;导出包含边界邻域在内的完全四阶精度的法向有限差分格式,这对方程精确离散至关重要;采用全局法和局部法相结合的方法及其新的迭代公式,大大加速收敛并得到更精确的特征值。算例给出了扰动幅值增长和形状函数等演化曲线,并对机翼、机身的边界层稳定性进行了分析和研究。
借助Fourier级数的方法将扰动波分解为基本模态和高频模态,研究了三维扰动的非线性边界层稳定性问题。采用高精度的高阶谱方法及其对无限区域问题的有效代数变换,提高了计算精度和收敛速度。运用Landau展开给出了二维谐波的初始解、通过求解线性PSE而得到TS波和C型亚谐扰动的初始站值、以及近似地模拟初始站的均流变形值,以此为初始条件,并结合迭代-预估校正方法来实现非线性PSE的求解和推进,详细地研究了非平行边界层三维扰动C型失稳过程。
文中通过对三维物体的非平行边界层稳定性的研究,确定最不稳定波的扰动参数和最易失稳的流向、展向位置;对三维扰动的非平行非线性边界层稳定性的研究,得到更精确的扰动放大因子值。这些都为边界层的转捩位置的确定和机理研究提供了可靠的依据,具有重要的理论意义和实用价值。
本文的研究工作是在教育部博士点基金(20030287003)的资助下完成的。
In this thesis, the parabolic stability equations (PSE) in the different coordinate systems are driven from Navier-Stokes (N-S) equations. The stability problems of the boundary layers for three-dimensional body,and the spatial evolution process of the C-type instability are studied.
The PSE in the curve coordinate systems are solved by the numerical techniques developed efficiently in this paper. A normal transform is introduced, and there are enough much grids in the region between the critical layer and the wall, where the variation of the disturbance is the quickest. The finite-difference of governing equations with fourth-order accuracy in the normal direction is utilized in full regions including points close the wall, and is very important for accurately discreting equations. The combination of global and local methods is implemented, and a new iterative formula is derived. Thus, the more precision eigenvalues can be obtained, and the convergence is made rapidly. In the examples we get some evolution curves, such as the growth rates and shape functions, which are used to analyze the stability problems for the boundary layers of the fuselage and the wing.
With the Fourier series technique, the disturbance is discomposed into predominant mode and high frequency harmonics. The nonlinear stability of three-dimensional disturbances for the Blasius boundary layers is studied. We adopted here the high accuracy method of using the expansions in orthogonal functions along the normal direction and the effective algebraic mapping to deal with the problem in the infinite region. The initial solutions of two-dimensional harmonic waves are given by Landau expansion, and the mean-flow-distortion is calculated by the approximation equation. We can get the initial conditions of TS wave and sub-harmonic wave in the C-type instability by linear PSE method. Furthermore, we employ iteration and Predictor-Corrector-Approach to solve the nonlinear equations in order to implement the marching procedure. We investigate detailedly the process of the C-type instability for nonparallel boundary layers with three-dimensional disturbances.
By investigating the nonparallel boundary layers of three-dimensional body, we
引文
[1] 周恒,赵耕夫.流动稳定性.国防工业出版社,2003.
[2] Nayfeh A H.Stability of Three-dimensional boundary layer[J]. AIAA J, 1980,18(4):406-416.
[3] Craik A D D. Non-linear resonant instability in boundary layers. J.F.M, 1971(50):393～413.
[4] Herbert T. Secondary instability of plane channel flow to subharmonic three-dimensional disturbance. Physics of Fluids, 1983(26):871～874.
[5] Herbert T. Boundary-layer transition analysis and prediction revisited [R]. AIAA-91-0737,1991.
[6] Haj-Hariri H. Characteristics analysis of the parabolized stability equations. Studies Applied Mathematics. 1994,92(1):41～53
[7] 郭乃龙. 三维不可压边界层抛物化稳定性方程的椭圆特性研究. 南京航空航天大学学报.1999, 20(2):104-106.
[8] 朱国祥.三维可压缩边界层计算及其稳定性研究[硕士学位论文].南京航空航天大学.2003.
[9] Joseph. Simulation of Diffuser Duct Flowfields Using a Three Dimensional Euler/Navier-Stockes Algorithm. AIAA-86-0310, January 6-9,1986.
[10] Bertolotti F P, Herbert T,Spalart P R.Linear and Nonlinear stability of the Blasius Boundary Layer.J Fluid Mech, 1992,242:441～474.
[11] Wang M J.Stability Analysis of Three-Dimensional Boundary Layers With parabolized Stability Equations[D]. Ohio State University,1994.
[12] Bertolotti F P. Linear and nonlinear stability of boundary layers with streamwise varying properties [D]. The Ohio State University, 1991.
[13] Malik M R. Numerical Methods for Hypersonic Boundary layer Stability. Journal of Computational Physics,1990,86:376～413.
[14] Sanjiva K L. Compact Finite Difference Schemes With Spectral-Like Resolution. Journal of Computational Physics,1992,103:16～42.
[15] Orszag S A. Accurate Solution of the Orr-Sommerfeld Equation. J Fluid Mech,1971,50:689～703.
[16] Hu S H,Xiaolin Zhong. Nonparallel Stability Analysis of Compressible Boundary Layer Using 3-D PSE. AIAA 99-0813, January 11-14,1999.
[17] Saric W S,Reed H L,Kerschen E J. Boundary layer Receptivity to freestream disturbance. Annual Review of Fluid Mechanics, 2002,34:291～319.
[18] Tadjfar M, Bodonyi R J. Receptivity of a laminar boundary layer to the interaction of a three-dimensional roughness element with time-harmonic free-stream disturbances. J Fluid Mech. 1992,242:701～720.
[19] Herbert T,Bertolotti F P. Stability analysis of nonparallel boundary layers. J.Bull.Am.Phys.soc,1992,32:2079.
[20] Malik M R, Li F. Transition Studies for Swept Wing Flows Using PSE. AIAA-93-0077, 1993.
[21] Dr. Hermann Schlichting. Boundary Layer Theory. McGraw-Hill Book Company, ISBN 0-07-055334-3, 1979:385～387,464～465,
[22] 是勋刚. 湍流. 天津大学出版社,ISBN 7-5617-0566-7,1994.
[23] Haj-Hariri H. Transformations reducing the order of the parameter in differential eigenvalue problems. J.comp.Phys. 1988,77.
[24] 王强,傅德薰,马延文. 不可压扰动方程高精度对称精致差分数值解法与应用. 空气动力学学报. 1999,17(3):292～298.
[25] Balakumar P . Evolution of Disturbances in Three-Dimensional Boundary Layers.AIAA 2000-0145, 38th Aerospace Sciences Meeting & Exhibit, January 11-14, 2000.
[26] Lehoucq R B, Sorensen D C. Deflation Techniques for an Implicitly Re-Started Arnoldi Iteration. SIAM J. Matrix Analysis and Applications, 1996, 17:789～821.
[27] Lehoucq R B, Sorensen D C, Yang C. ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia, 1998.
[28] Sorensen D C, Implicit Application of Polynomial Filters in a k-Step Arnoldi Method, SIAM J. Matrix Analysis and Applications, 1992, 13:357～385.
[29] 威尔金森 J H.代数特征问题.科学出版社,1987:178～181,333～334,654～661.
[30] Cebeci T, Bradshow P. Momentum Transfer in Boundary Layer. Hemispere Publishing Corporation, ISBN 0-07-010300-3, 1977:281～300.
[31] Malik M R, Orszag S A . Efficient Computation of the Stability of Three-Dimesional Compressible Boundary Layers.AIAA-81-1277,1981.
[32] 唐登斌.后掠翼可压缩三维边界层稳定性计算.航空学报, 1992,13(1):A1～A7.
[33] 袁慰平,张令敏. 等数值分析. 东南大学出版社,ISBN 7-81023-657-1/O?60. 1998: 419～433,456～459
[34] Klebanoff P S, Tidstorm K D, Sargent L M. The Three-Dimensional Nature of Boundary Layer Instability. J.F.M. 1962(12):1～34.
[35] Saric W S, Kozlor V V, Levehenko V Y. Forced and Unforced Subharmonic Resonance in Boundary Layer Transition. AIAA, 22nd Aerospace Sciences Meeting January 9-12. 1984/Reno,Nevada.
[36] Herbert T. Secondary Instability of Boundary Layers. Ann. Rev. Fluid Mech. 1998(20):487～526.
[37] Herbert T. Analysis of the Subharmonic Three Dimensional Disturbance in Unstable Sheer Flows. AIAA-83-1769,1983.
[38] 赵耕夫,陆昌根. 平面泊肖流三维亚谐扰动的稳定性研究. 天津大学学报,特刊,1991:64～71.
[39] Wang D Y, Zhao G F. On Secondary Instability With Respect to Three Dimensional Subharmonic Disturbance in Boundary Layer. ACTA MECH. SINICA. 1992,8(3):231～236.
[40] 刘志. 平板边界层 C-型失稳后的三波演化问题[硕士论文]. 天津大学.1987.
[41] 白玉川. 层流边界层 C 型失稳研究. 空气动力学学报. 1995,13(1):92～98.
[42] 陆昌根,赵耕夫. Blasius 边界层 C-型失稳理论分析. 空气动力学学报. 1995,13(3):329～333.
[43] 唐登斌. 粘性流体动力学. 南京航空航天大学,1997.
[44] Wang W Z, Tang D B. Studies On Nonlinear Stability of Three-Dimensional H-Type Disturbance. Acta Mechanica Sinica. 2003,19(6):517～526.
[45] Herbert T. On the Stability of 3D Boundary Layers. AIAA-97-1961,1997.
[46] Grosch C E, Orszag S A. Numerical Solution of Problem in Unbounded Regions: Coordinate Transforms. Journal of Computational Physics 25, 1997:273～296.
[2] Nayfeh A H.Stability of Three-dimensional boundary layer[J]. AIAA J, 1980,18(4):406-416.
[3] Craik A D D. Non-linear resonant instability in boundary layers. J.F.M, 1971(50):393～413.
[4] Herbert T. Secondary instability of plane channel flow to subharmonic three-dimensional disturbance. Physics of Fluids, 1983(26):871～874.
[5] Herbert T. Boundary-layer transition analysis and prediction revisited [R]. AIAA-91-0737,1991.
[6] Haj-Hariri H. Characteristics analysis of the parabolized stability equations. Studies Applied Mathematics. 1994,92(1):41～53
[7] 郭乃龙. 三维不可压边界层抛物化稳定性方程的椭圆特性研究. 南京航空航天大学学报.1999, 20(2):104-106.
[8] 朱国祥.三维可压缩边界层计算及其稳定性研究[硕士学位论文].南京航空航天大学.2003.
[9] Joseph. Simulation of Diffuser Duct Flowfields Using a Three Dimensional Euler/Navier-Stockes Algorithm. AIAA-86-0310, January 6-9,1986.
[10] Bertolotti F P, Herbert T,Spalart P R.Linear and Nonlinear stability of the Blasius Boundary Layer.J Fluid Mech, 1992,242:441～474.
[11] Wang M J.Stability Analysis of Three-Dimensional Boundary Layers With parabolized Stability Equations[D]. Ohio State University,1994.
[12] Bertolotti F P. Linear and nonlinear stability of boundary layers with streamwise varying properties [D]. The Ohio State University, 1991.
[13] Malik M R. Numerical Methods for Hypersonic Boundary layer Stability. Journal of Computational Physics,1990,86:376～413.
[14] Sanjiva K L. Compact Finite Difference Schemes With Spectral-Like Resolution. Journal of Computational Physics,1992,103:16～42.
[15] Orszag S A. Accurate Solution of the Orr-Sommerfeld Equation. J Fluid Mech,1971,50:689～703.
[16] Hu S H,Xiaolin Zhong. Nonparallel Stability Analysis of Compressible Boundary Layer Using 3-D PSE. AIAA 99-0813, January 11-14,1999.
[17] Saric W S,Reed H L,Kerschen E J. Boundary layer Receptivity to freestream disturbance. Annual Review of Fluid Mechanics, 2002,34:291～319.
[18] Tadjfar M, Bodonyi R J. Receptivity of a laminar boundary layer to the interaction of a three-dimensional roughness element with time-harmonic free-stream disturbances. J Fluid Mech. 1992,242:701～720.
[19] Herbert T,Bertolotti F P. Stability analysis of nonparallel boundary layers. J.Bull.Am.Phys.soc,1992,32:2079.
[20] Malik M R, Li F. Transition Studies for Swept Wing Flows Using PSE. AIAA-93-0077, 1993.
[21] Dr. Hermann Schlichting. Boundary Layer Theory. McGraw-Hill Book Company, ISBN 0-07-055334-3, 1979:385～387,464～465,
[22] 是勋刚. 湍流. 天津大学出版社,ISBN 7-5617-0566-7,1994.
[23] Haj-Hariri H. Transformations reducing the order of the parameter in differential eigenvalue problems. J.comp.Phys. 1988,77.
[24] 王强,傅德薰,马延文. 不可压扰动方程高精度对称精致差分数值解法与应用. 空气动力学学报. 1999,17(3):292～298.
[25] Balakumar P . Evolution of Disturbances in Three-Dimensional Boundary Layers.AIAA 2000-0145, 38th Aerospace Sciences Meeting & Exhibit, January 11-14, 2000.
[26] Lehoucq R B, Sorensen D C. Deflation Techniques for an Implicitly Re-Started Arnoldi Iteration. SIAM J. Matrix Analysis and Applications, 1996, 17:789～821.
[27] Lehoucq R B, Sorensen D C, Yang C. ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia, 1998.
[28] Sorensen D C, Implicit Application of Polynomial Filters in a k-Step Arnoldi Method, SIAM J. Matrix Analysis and Applications, 1992, 13:357～385.
[29] 威尔金森 J H.代数特征问题.科学出版社,1987:178～181,333～334,654～661.
[30] Cebeci T, Bradshow P. Momentum Transfer in Boundary Layer. Hemispere Publishing Corporation, ISBN 0-07-010300-3, 1977:281～300.
[31] Malik M R, Orszag S A . Efficient Computation of the Stability of Three-Dimesional Compressible Boundary Layers.AIAA-81-1277,1981.
[32] 唐登斌.后掠翼可压缩三维边界层稳定性计算.航空学报, 1992,13(1):A1～A7.
[33] 袁慰平,张令敏. 等数值分析. 东南大学出版社,ISBN 7-81023-657-1/O?60. 1998: 419～433,456～459
[34] Klebanoff P S, Tidstorm K D, Sargent L M. The Three-Dimensional Nature of Boundary Layer Instability. J.F.M. 1962(12):1～34.
[35] Saric W S, Kozlor V V, Levehenko V Y. Forced and Unforced Subharmonic Resonance in Boundary Layer Transition. AIAA, 22nd Aerospace Sciences Meeting January 9-12. 1984/Reno,Nevada.
[36] Herbert T. Secondary Instability of Boundary Layers. Ann. Rev. Fluid Mech. 1998(20):487～526.
[37] Herbert T. Analysis of the Subharmonic Three Dimensional Disturbance in Unstable Sheer Flows. AIAA-83-1769,1983.
[38] 赵耕夫,陆昌根. 平面泊肖流三维亚谐扰动的稳定性研究. 天津大学学报,特刊,1991:64～71.
[39] Wang D Y, Zhao G F. On Secondary Instability With Respect to Three Dimensional Subharmonic Disturbance in Boundary Layer. ACTA MECH. SINICA. 1992,8(3):231～236.
[40] 刘志. 平板边界层 C-型失稳后的三波演化问题[硕士论文]. 天津大学.1987.
[41] 白玉川. 层流边界层 C 型失稳研究. 空气动力学学报. 1995,13(1):92～98.
[42] 陆昌根,赵耕夫. Blasius 边界层 C-型失稳理论分析. 空气动力学学报. 1995,13(3):329～333.
[43] 唐登斌. 粘性流体动力学. 南京航空航天大学,1997.
[44] Wang W Z, Tang D B. Studies On Nonlinear Stability of Three-Dimensional H-Type Disturbance. Acta Mechanica Sinica. 2003,19(6):517～526.
[45] Herbert T. On the Stability of 3D Boundary Layers. AIAA-97-1961,1997.
[46] Grosch C E, Orszag S A. Numerical Solution of Problem in Unbounded Regions: Coordinate Transforms. Journal of Computational Physics 25, 1997:273～296.