On the spatio-temporal stability of primary and secondary crossflow vortices in a three-dimensional boundary layer

To examine possible links between a global instability and laminar–turbulent breakdown in a three-dimensional boundary layer, the spatio-temporal stability of primary and secondary crossflow vortices has been investigated for the DLR swept-plate experiment. In the absence of any available procedure for the direct verification of pinching for three-dimensional wave packets the alternative saddle-point continuation method has been applied. This procedure is known to give reliable results only in a certain vicinity of the most unstable ray. Therefore, finding no absolute instability by this method does not prove that the flow is absolutely stable. Accordingly, our results obtained this way need to be confirmed experimentally or by numerical simulations. A geometric interpretation of the time-asymptotic saddle-point result explains certain convergence and continuation problems encountered in the numerical wave packet analysis. Similar to previous results, all our three-dimensional wave packets for primary crossflow vortices were found to be convectively unstable. Due to prohibitive CPU time requirements the existing procedure for the verification of pinching for two-dimensional wave packets of secondary high-frequency instabilities could not be implemented. Again saddle-point continuation was used. Surprisingly, all two-dimensional wave packets of high-frequency secondary instabilities investigated were also found to be convectively unstable. This finding was corroborated by recent spatial direct numerical simulations of Wassermann & Kloker (2001) for a similar problem. This suggests that laminar–turbulent breakdown occurs after the high-frequency secondary instabilities enter the nonlinear stage, and spatial marching techniques, such as the parabolized stability equation method, should be applicable for the computation of these nonlinear states.

[1]  P. Jeschke,et al.  Longitudinal vortices in a laminar natural convection boundary layer flow on an inclined flat plate and their influence on heat transfer , 2001, Journal of Fluid Mechanics.

[2]  Thomas J. Bridges,et al.  Absolute and convective instabilities of spatially periodic flows , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Michael Gaster,et al.  Propagation of Linear Wave Packets in Laminar Boundary Layers , 1981 .

[4]  R. Lingwood.  On the Application of the Briggs' and Steepest‐Descent Methods to a Boundary‐Layer Flow , 1997 .

[5]  Markus J. Kloker,et al.  Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer , 2002, Journal of Fluid Mechanics.

[6]  T. Brooke Benjamin,et al.  Theory of the vortex breakdown phenomenon , 1962, Journal of Fluid Mechanics.

[7]  R. Seydel.  Practical Bifurcation and Stability Analysis , 1994 .

[8]  Jean-Marc Chomaz,et al.  The effect of swirl on jets and wakes: Linear instability of the Rankine vortex with axial flow , 1998 .

[9]  N. So.  On the breakdown of boundary layer streaks , 2022 .

[10]  M. Gaster.  The development of three-dimensional wave packets in a boundary layer , 1968, Journal of Fluid Mechanics.

[11]  L. Brevdo.  Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows , 1991 .

[12]  P. Monkewitz,et al.  LOCAL AND GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS , 1990 .

[13]  N. Peake,et al.  The long-time behaviour of incompressible swept-wing boundary layers subject to impulsive forcing , 1998, Journal of Fluid Mechanics.

[14]  Erik Janke,et al.  On the Secondary Instability of Three-Dimensional Boundary Layers , 2000 .

[15]  Bohr,et al.  Growth shapes and turbulent spots in unstable systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  T. Bridges,et al.  Linear pulse structure and signalling in a film flow on an inclined plane , 1999, Journal of Fluid Mechanics.

[17]  Stefan Hein,et al.  Nonlinear equilibrium solutions in a three-dimensional boundary layer and their secondary instability , 2000, Journal of Fluid Mechanics.

[18]  T. Benjamin.  The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane , 1961, Journal of Fluid Mechanics.

[19]  Jean-Marc Chomaz,et al.  Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response , 1998, Journal of Fluid Mechanics.

[20]  J. Chomaz,et al.  Absolute and convective secondary instabilities in spatially periodic shear flows , 1997 .

[21]  H. Bippes,et al.  Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability , 1999 .

[22]  Alessandro Bottaro,et al.  On the linear breakdown of Görtler vortices , 1996 .

[23]  DNS of Laminar-Turbulent Transition in a 3D Aerodynamics Boundary-Layer Flow , 2001 .

[24]  A. Stolte.  Investigation of Transition Scenarios in Boundary-Layer Flows , 1999 .

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

[26]  P. Manneville,et al.  Local Versus Global Concepts in Hydrodynamic Stability Theory , 1997 .

[27]  R. Seydel.  Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .

[28]  J. Chomaz,et al.  Absolute and convective nature of the Eckhaus and zigzag instability with throughflow , 1999 .

[29]  William S. Saric,et al.  A high-frequency, secondary instability of crossflow vortices that leads to transition , 1991 .

[30]  Valery Chernoray,et al.  Experimental study of K-regime of breakdown in straight and swept wing boundary layers , 2000 .

[31]  Mujeeb R. Malik,et al.  Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction and secondary instability , 1994, Journal of Fluid Mechanics.

[32]  Michael F. Schatz,et al.  Instability in a spatially periodic open flow , 1995 .

[33]  I. Grant,et al.  An experimental investigation of the formation and development of a wave packet in a laminar boundary layer , 1975, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[34]  W. Koch,et al.  Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow☆ , 1977 .

[35]  H. Deyhle,et al.  Disturbance growth in an unstable three-dimensional boundary layer and its dependence on environmental conditions , 1996, Journal of Fluid Mechanics.

[36]  F. Bertolotti.  On The Birth and Evolution of Disturbances in Three-Dimensional Boundary Layers , 1996 .

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

[38]  Antoine Sellier,et al.  Inviscid instability of the Batchelor vortex: Absolute-convective transition and spatial branches , 1999 .

[39]  R. Lingwood,et al.  Absolute instability of the boundary layer on a rotating disk , 1995, Journal of Fluid Mechanics.

[40]  K. Afanasiev,et al.  Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis , 1999 .

[41]  R. Lingwood.  On the impulse response for swept boundary-layer flows , 1997, Journal of Fluid Mechanics.

[42]  O. S. Ryzhov,et al.  Streamwise absolute instability of a three-dimensional boundary layer at high Reynolds numbers , 1998, Journal of Fluid Mechanics.

[43]  M. Gaster.  A theoretical model of a wave packet in the boundary layer on a flat plate , 1975, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[44]  F. Waleffe.  On a self-sustaining process in shear flows , 1997 .

[45]  Masaharu Matsubara,et al.  Free-stream turbulence, streaky structures and transition in boundary layer flows , 2000 .

[46]  R. Lingwood,et al.  An experimental study of absolute instability of the rotating-disk boundary-layer flow , 1996, Journal of Fluid Mechanics.

[47]  P. Alfredsson,et al.  Secondary instability in rotating channel flow , 1998, Journal of Fluid Mechanics.

[48]  Gordon Erlebacher,et al.  Numerical experiments in supersonic boundary‐layer stability , 1990 .

[49]  S. Widnall,et al.  Examination of group-velocity criterion for breakdown of vortex flow in a divergent duct , 1980 .

[50]  Edward B. White,et al.  Stages of Swept-Wing Transition , 2001 .

[51]  W. O. Criminale,et al.  The growth of localized disturbances in a laminar boundary layer , 1962, Journal of Fluid Mechanics.

[52]  Fei Li,et al.  Fundamental and subharmonic secondary instabilities of Görtler vortices , 1995, Journal of Fluid Mechanics.

[53]  M. Malik,et al.  Nonlinear Crossflow Disturbances and Secondary Instabilities in Swept-Wing Boundary Layers , 1996 .

[54]  Mingjun Wei,et al.  Absolute and convective instability character of slender viscous vortices , 2000 .

[55]  D. I. A. Poll,et al.  Some observations of the transition process on the windward face of a long yawed cylinder , 1985, Journal of Fluid Mechanics.

[56]  R. Deissler.  The convective nature of instability in plane Poiseuille flow , 1987 .

[57]  P. Huerre.  On the Absolute/Convective Nature of Primary and Secondary Instabilities , 1988 .

[58]  Brian J. Tester,et al.  The optimization of modal sound attenuation in ducts, in the absence of mean flow , 1973 .

[59]  L. Brevdo.  Convectively Unstable Wave Packets in the Blasius Boundary Layer , 1995 .

[60]  N. Marcuvitz,et al.  Electron-stream interaction with plasmas , 1965 .

[61]  Yasuaki Kohama,et al.  Stability characteristics of stationary crossflow vortices in three-dimensional boundary layer , 1999 .

[62]  H. Oertel,et al.  Mathematische Analyse der Bereiche reibungsbehafteter Strömungen , 1995 .