Linear and non-linear stability analysis of incompressible boundary layer over a two-dimensional hump

Abstract Linear and non-linear stability analysis of boundary layer over a two-dimensional smooth hump is carried out by using parabolized stability equations (PSEs). Overall destabilization influence of the hump is confirmed for both linear and non-linear evolution of instability waves. For linear stability, effects of several parameters including frequency, hump height, and wave obliqueness are examined. Comparisons of the linear PSE results with those from linear stability theory (LST) are also given. The discrepancy between the LST and linear PSE results mainly observed in the vicinity of the hump is found to be caused by both the flow non-parallelism and the transient behavior of PSE approach. The transient behavior sometimes accompanying wiggling in the growth rate curve is found to vary depending on the flow configurations such as frequency and hump height. Non-linear PSE analysis is carried out to examine non-linear evolution of Tollmien–Schlichting wave and subharmonic breakdown. Influence of frequency, hump height, and initial amplitude is investigated. The results demonstrate that disturbances can be amplified very much by the non-linear interaction due to the presence of hump in spite of very low level of disturbance amplitude, which suggests that the hump can bring forth earlier breakdown even for the case of rather small initial amplitude disturbances. Through analyses on subharmonic breakdown, it is found that the amplitude of primary wave affects most the growth of the secondary wave. The subharmonic mode appears to be unstable over a wide range of spanwise wave number in agreement with the findings from previous studies on the secondary instability.

[1]  A. Nayfeh,et al.  Effect of a bulge on the subharmonic instability of subsonic boundary layers , 1992 .

[2]  T. Herbert.  PARABOLIZED STABILITY EQUATIONS , 1994 .

[3]  U. Rist,et al.  Accuracy of Local and Nonlocal Linear Stability Theory in Swept Separation Bubbles , 2009 .

[4]  Fei Li,et al.  On the nature of PSE approximation , 1996 .

[5]  M. Malik.  Numerical methods for hypersonic boundary layer stability , 1990 .

[6]  A. Nayfeh,et al.  Effect of suction on the stability of subsonic flows over smooth backward-facing steps , 1990 .

[7]  Chau-Lyan Chang.  Langley Stability and Transition Analysis Code (LASTRAC) Version 1.2 User Manual , 2004 .

[8]  A. Nayfeh,et al.  Effect of a bulge on the subharmonic instability of boundary layers , 1990 .

[9]  M. Y. Hussaini,et al.  Linear and nonlinear PSE for compressible boundary layers , 1993 .

[10]  Meelan Choudhari,et al.  Secondary instability of crossflow vortices and swept-wing boundary-layer transition , 1999, Journal of Fluid Mechanics.

[11]  Chau-Lyan Chang,et al.  The Langley Stability and Transition Analysis Code (LASTRAC) : LST, Linear and Nonlinear PSE for 2-D, Axisymmetric, and Infinite Swept Wing Boundary Layers , 2003 .

[12]  Meelan Choudhari,et al.  Hypersonic viscous flow over large roughness elements , 2009 .

[13]  A. Nayfeh,et al.  Effect of wall cooling on the stability of compressible subsonic flows over smooth humps and backward-facing steps , 1989 .

[14]  P S Klebanoff,et al.  MECHANISM BY WHICH A TWO-DIMENSIONAL ROUGHNESS ELEMENT INDUCES BOUNDARY-LAYER TRANSITION: ROUGHNESS INDUCED TRANSITION , 1972 .

[15]  A. Dovgal,et al.  Hydrodynamic Instability and Receptivity of Small Scale Separation Regions , 1990 .

[16]  V. Ya. Levchenko,et al.  The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer , 1984, Journal of Fluid Mechanics.

[17]  Seung-O Park,et al.  Stability analysis of a boundary layer over a hump using parabolized stability equations , 2011 .

[18]  A. H. Nayfeh,et al.  Stability of compressible boundary layers over a smooth backward-facing step , 1989 .

[19]  Tuncer Cebeci,et al.  Numerical and Physical Aspects of Aerodynamic Flows , 1986 .

[20]  R. Joslin,et al.  Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory , 1993 .

[21]  Paul Fischer,et al.  Roughness-Induced Transient Growth , 2005 .

[22]  A. Nayfeh,et al.  Effect of a hump on the stability of subsonic boundary layers over an airfoil , 1996 .

[23]  J. Edwards,et al.  On the Effects of Surface Roughness on Boundary Layer Transition , 2009 .

[24]  Venkit Iyer,et al.  Transition prediction and control in subsonic flow over a hump , 1993 .

[25]  Tuncer Cebeci,et al.  Prediction of Transition Due to Isolated Roughness , 1989 .

[26]  D. Henningson,et al.  On a Stabilization Procedure for the Parabolic Stability Equations , 1998 .

[27]  Hermann F. Fasel,et al.  Numerical investigation of the three-dimensional development in boundary-layer transition , 1987 .

[28]  Yong-Sun Wie,et al.  Effect of surface waviness on boundary-layer transition in two-dimensional flow , 1998 .

[29]  Chau-Lyan Chang,et al.  Oblique-mode breakdown and secondary instability in supersonic boundary layers , 1994, Journal of Fluid Mechanics.

[30]  T. Herbert.  Secondary Instability of Boundary Layers , 1988 .

[31]  M. Malik,et al.  Link between flow separation and transition onset , 1995 .

[32]  Ali H. Nayfeh,et al.  Effect of bulges on the stability of boundary layers , 1988 .

[33]  Ronald D. Joslin,et al.  Validation of three-dimensional incompressible spatial direct numerical simulation code: A comparison with linear stability and parabolic stability equation theories for boundary-layer transition on a flat plate , 1992 .

[34]  Siegfried Wagner,et al.  Humps/Steps Influence on Stability Characteristics of Two-Dimensional Laminar Boundary Layer , 2003 .