Modelling of boundary layer stability

by Andersson, Paul

Abstract (Summary)
A scenario for bypass transition to turbulence likely to occur in natural transition in a at plate boundary layer ow has been studied. The exponential growth or decay oftwo-dimensional wave disturbances, known as Tollmien{Schlichting waves, has long been the classical starting point for theoretical investigations of transition from laminar to turbulent ow. Its failure to explain experimentally observed transition for many ows has attracted intense interest to the recently revealed existence of non-modal growth mechanisms. This thesis focus mainly on transition emanating from the non-modal transient growth of streamwise streaks. Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to high levels of free-stream turbulence. The upstream disturbances experiencing maximum spatial energy growth have been calculated numerically using techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise aligned vortices developing downstream into streamwise streaks which are in good agreement with experimental measurements. The maximum spatial energy growth was found to scale linearly with the distance from the leading edge. Based on these results, a simple model for prediction of transition location is proposed. However, the non-modal growth of streamwise streaks only represent the initial phase of transition. If the disturbance energy of the streaks becomes su ciently large, secondary instabilities can take place and provoke early breakdown and transition, overruling the theoretically predicted modal decay. Using linear Floquet theory the temporal, inviscid secondary instability of these streaks were studied to determine the characteristic features of their breakdown. The critical streak amplitude beyond which streamwise travelling waves are excited is typically of order 26% of the free-stream velocity. The sinuous secondary instability mode was found to represent the most dangerous symmetry for travelling disturbances. Also the numerical-stability consequences of the remaining ellipticity in the Parabolic Stability Equations (PSE) are studied. The equations are found to constitute an ill-posed Cauchy problem. Suggestions of how to make the equations well-posed and to remove the methods otherwise intrinsic step-size restriction are given. Descriptors: laminar-turbulent transition, boundary layer ow, non-parallel e ects, adjoint equations, transient growth, optimal disturbances, streamwise streaks, streak instability, secondary instability, transition modelling, free-stream turbulence, parabolic stability equations, ill-posed equations. Preface This thesis considers the stability of boundary layer ows and modelling aspects. The thesis is based on and contains the following papers. Paper 1. Andersson, P., Henningson, D. S. & Hanifi, A. 1998 On a stabilization procedure for the parabolic stability equations. J. Eng. Math. 33, 311{332. Paper 2. Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134{150. Paper 3. Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 1999 On the breakdown of boundary layer streaks. submitted to J. Fluid Mech. Paper 4. Andersson, P., Bottaro, A., Henningson, D. S. & Luchini, P. 1999 Secondary instability of boundary layer streaks based on the shape assumption. TRITA-MEK, Technical Report 1999:13, Dept. of Mechanics, KTH, Stockholm, Sweden. Paper 5. Andersson, P. 1999 On the modelling of streamwise streaks in the Blasius boundary layer. TRITA-MEK, Technical Report 1999:14, Dept. of Mechanics, KTH, Stockholm, Sweden. The papers are here re-set in the present thesis format, and some minor corrections have been made as compared to published versions. The rst part of the thesis is both a short introduction to the eld and a summary of the most important results presented in the papers given above. iv
Bibliographical Information:


School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation



Date of Publication:01/01/1999

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