Legendre Series Solution to Rayleigh Stability Equation of Mixing Layer

GUO Xin; WANG Xian; XU Ding; XIE Gong-nan

doi:10.3879/j.issn.1000-0887.2013.08.002

Volume 34 Issue 8

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2013 > 34(8): 782-794

GUO Xin, WANG Xian, XU Ding, XIE Gong-nan. Legendre Series Solution to Rayleigh Stability Equation of Mixing Layer[J]. Applied Mathematics and Mechanics, 2013, 34(8): 782-794. doi: 10.3879/j.issn.1000-0887.2013.08.002

Citation:

PDF( 865 KB)

Legendre Series Solution to Rayleigh Stability Equation of Mixing Layer

doi: 10.3879/j.issn.1000-0887.2013.08.002

¹Xi’an Research Institute of China Coal Technology & Engineering Group Corp, Xi’an 710054, P.R.China;²State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, P.R.China;³Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, Xi’an 710049, P.R.China

Received Date: 2013-05-30
Rev Recd Date: 2013-06-15
Publish Date: 2013-08-15

Abstract

Abstract

Based on the fixed point concept in functional analysis, the fixed point method (FPM) was used to analyze the invisicd stability equation of the mixing layer, and an explicit semi-analytical solution in Legendre series form was obtained. It is different from other existing analytical methods, such as the well-known perturbation technique, because FPM can obtain a uniformly convergent solution in the full infinite flow domain. Meanwhile, the present Legendre series solution is valid to all wave numbers. What’s more, in the framework of FPM, the eigenvalue can be determined by the solvability condition in a straightforward manner. Finally, the comparison between FPM and other numerical methods shows that FPM is of high accuracy and efficiency.
- fixed point method,
- solvability condition,
- Legendre series solution,
- Rayleigh stability equation,
- mixing layer

FullText(HTML)

References(25)

References

[1]	罗纪生, 吕祥翠. 超音速混合层稳定性分析及增强混合的研究[J].力学学报, 2004, 36 (2): 202-207. (LUO Ji-sheng, Lü Xiang-cui. Investigation on stability of 3-D supersonic mixing layer and method of enhancing the mixing[J].Acta Mechanica Sinica,2004, 36 (2): 202-207.(in Chinese))
[2]	倪慧, 罗纪生, 何立忠. 三维可压缩混合层中扰动演化的研究[J].空气动力学学报, 2004, 22-(4): 416421.(NI Hui, LUO Ji-sheng, HE Li-zhong. An investigation for the evolvement of disturbances in 3-D compressible mixing layer[J].Acta Aerodynamica Sinica,2004, 22 (4): 416-421.(in Chinese))
[3]	潘宏禄, 马汉东, 王强. 高对流Mach数三维混合层转捩特性分析及小激波结构模拟[J].空气动力学学报, 2008, 26 (3): 275-281.(PAN Hong-lu, MA Han-dong, WANG Qiang. Transition coherent structures and shocklets in 3-D spatial developing mixing layers at high convective Mach numbers[J].Acta Aerodynamica Sinica,2008, 26 (3): 275-281.(in Chinese))
[4]	杨武兵, 庄逢甘, 沈清, 易仕和, 何霖, 赵玉新. 超声速混合层中扰动增强混合实验[J].力学学报, 2010, 42 (3): 373-382. (YANG Wu-bing, ZHUANG Feng-gan, SHEN Qing, YI Shi-he, HE Lin, ZHAO Yu-xin. Experimental study on perturbation mixing enhancement in supersonic mixing layers[J].Acta Mechanica Sinica,2010, 42 (3): 373-382.(in Chinese))
[5]	周强, 何枫, 沈孟育. 可压缩混合层中的涡结构和激波[J].空气动力学学报, 2010, 28 (3): 245-249. (ZHOU Qiang, HE Feng, SHEN Meng-yu. Vortex structures and shocks in the compressible mixing layer[J].Acta Aerodynamica Sinica,2010, 28 (3): 245-249.(in Chinese))
[6]	Criminale W, Jackson T, Joslin R.Theory and Computation of Hydrodynamic Stability [M]. United Kingdom:Cambridge University Press, 2003.
[7]	Michalke A. The instability of free shear layers[J]. Progress in Aerospace Sciences,1972, 12 (1): 213-216.
[8]	Ho C M, Huerre P. Perturbed free shear layers[J]. Annual Review of Fluid Mechanics,1984, 16 (1): 365-422.
[9]	Betchov R, Szewczyk A. Stability of a shear layer between parallel streams[J]. Physics of Fluids,1963, 6 (10): 1391-1396.
[10]	Drazin P G, Howard L N. Hydrodynamic stability of parallel flow of inviscid fluid[J].Advances in Applied Mechanics,1966, 9 (1): 1-89.
[11]	Drazin P G, Howard L N. The instability to long waves of unbounded parallel inviscid flow[J].J Fluid Mech,1962, 14 (2): 257-283.
[12]	Michalke A. On the inviscid instability of the hyperbolic tangent velocity profile[J]. J Fluid Mech,1964, 19 (4): 543-556.
[13]	Drazin P G, Reid W H.Hydrodynamic Stability [M]. United Kingdom:Cambridge University Press, 2004.
[14]	Heisenberg W. ber stabilitat und turbulenz von flüssigkeitsstromen[J]. Ann Phys,1924, 74 (4): 577-627.
[15]	Tollmien W.The Production of Turbulence[M]. Washington:NACATM609, 1931.
[16]	Tatsumi T, Gotoh K, Ayukawa K. The stability of a free boundary layer at large Reynolds numbers[J].Journal of the Physical Society of Japan,1964, 19 (10): 1966.
[17]	Peyret R.Spectral Methods for Incompressible Viscous Flow [M]. New York:Springer, 2002.
[18]	Hussaini M Y, Zang T A. Spectral methods in fluid dynamics[J]. Annual Review of Fluid Mechanics,1987, 19 (1): 339-367.
[19]	Canuto C, Hussaini M Y, Quarteroni A, Zang T A.Spectral Methods:Fundamentals in Single Domains [M]. Berlin: Springer, 2006.
[20]	Xu D, Guo X. Fixed point analytical method for nonlinear differential equations[J]. J Comput Nonlinear Dyn,2013, 8 (1): 011005.
[21]	Zeidler E.Nonlinear Functional Analysis and Its ApplicationsⅠ: FixedPoint Theorems [M]. New York: Springer,1986.
[22]	Orszag S A. Accurate solution of the orr-sommerfeld stability equation[J]. J Fluid Mech,1971, 50 (4): 689-703.
[23]	Burden R L, Faires J D.Numerical Analysis [M]. Boston:Brooks Cole, 2010.
[24]	Garcia R V. Barotropic waves in straight parallel flow with curved velocity profile[J]. Tellus,1956, 8 (1): 82-93.
[25]	Boguslawski A. Inviscid instability of the hyperbolic-tangent velocity profilespectral“tau”solution[J]. Task Quarterly,2001, 5 (2): 155-164.