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O N Viscous, Inviscid and Centrifugal Instability M On viscous, inviscid and centrifugal instability mechanisms in compressible boundary layers, including non-linear vortex/wave interaction and the effects of large Mach number on transition. Submitted to the University of London as a thesis for the degree of Doctor of Philosophy. Nicholas Derick Blackaby, University College. March 1991. 1 ProQuest Number: 10609861 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10609861 Published by ProQuest LLC(2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 A b stract The stability and transition of a compressible boundary layer, on a flat or curved surface, is considered using rational asymptotic theories based on the large size of the Reynolds numbers of concern. The Mach number is also treated as a large parameter with regard to hypersonic flow. The resulting equations are simpler than, but consistent with, the full Navier Stokes equations, but numer­ ical computations are still required. This approach also has the advantage that particular possible mechanisms for instability and/or transition can be studied, in isolation or in combination, allowing understanding of the underlying physics responsible for the breakdown of a laminar boundary layer. The nonlinear interaction of Tollmien-Schlichting waves and longitudinal vor­ tices is considered for the entire range of the Mach number; it is found that com­ pressibility has significant effects on the solution properties. The arguments break­ down when the Mach number reaches a certain, large, size due to the ‘collapse’ of the multi-layered boundary layer present and thus we are naturally led on to inves­ tigate this new regime where ‘non-parallelism’ must be incorporated in the theory. Also, the effects of compressibility are then more significant, analytically, causing the governing equations to be more complicated, and further analytic progress relies on shorter scales being employed for any perturbations to the basic flow. The numerical solution is discussed along with a non-linear asymptotic solution capturing a ‘finite-time break-up’ of the interactive boundary layer. This work suggests that for larger Mach numbers the crucial non-linear in­ teraction is between inviscid modes and Gortler vortices and these are discussed in the remaining chapters. The inviscid modes are studied initially with no shock present, before the theory is modified for the inclusion of shock-wave/boundary layer interaction. In the last chapter the Gortler-vortex mechanism for large Mach numbers is considered. 2 C ontents page Title p a g e ...................................................................................................................................... 1 A b stra c t ...........................................................................................................................................2 C o n ten ts .......................................................................................................................................... 3 Acknowledgements .................................................................................................................. 10 Chapter 1 General Introduction. 1.1 Boundary layer stability theory ...................................................................11 1.1.1 Historical background ......................................................................... 11 1.1.2 The inviscid, viscous and centrifugal instabilities ....................... 18 1.2 The present thesis ............................................................................................ 21 Chapter 2 Compressible boundary layer theory. 2.1 Introduction .......................................................................................................24 2.1.1 The parameters of dynamical similarity ......................................... 24 2.1.2 Formulation .............................................................................................25 2.1.3 Real and ideal gases ........................................................................... 26 2.1.4 The ratio of specific heats and the Prandtl number. 27 2.1.5 The constitutive relationship between viscosity and temperature ............................................................................................28 2.1.6 The free-stream temperature: what value should be chosen? 29 2.2 Non-interactive steady flows ........................................................................ 33 2.2.1 The similarity solution .........................................................................33 2.2.2 Wall shear of base flow: the model Chapman-fluid. 34 2.2.3 Wall shear of base flow: Sutherland-fluid ..................................... 35 2.3 Compressible triple deck theory ...................................................................37 2.3.1 Significant advances in the theory .................................................... 37 2.3.2 The Sutherland-fluid triple-deck scales ...........................................39 2.3.3 Linear stability: the eigenrelation for Tollmien- Schlichting waves .................................................................................. 44 2.3.4 The large Mach number limit ............................................................46 3 Chapter 3 Non-linear Tollmien-Schlichting/vortex interaction in compressible boundary-layer flows. 3.1 Introduction .......................................................................................................59 3.2 Formulation ....................................................................................................... 61 3.2.1 Discussion ................................................................................................61 3.2.2 The 3-D compressible triple-deck equations ..................................63 3.2.3 The interaction scales...........................................................................68 3.3 The derivation of the interaction equations ..............................................71 3.3.1 The lower deck .......................................................................................71 3.3.2 The upper-deck and pressure-displacement law .................... 74 3.3.3 The compatibility relations and the interaction coefficients. 77 3.3.4 The buffer layer ..................................................................................... 78 3.4 An alternative derivation of the interaction coefficients; general TS-eigenr elation ............................................................................................... 80 3.4.1 The ratio b/a ..........................................................................................82 3.4.2 The ratio c/a ..........................................................................................83 3.5 The interaction equations ..............................................................................86 3.5.1 Solution of interaction equations by Fourier transforms. 87 3.5.2 Possible limiting forms for large-X ................................................. 89 3.5.3 The transonic and hypersonic limits ................................................91 3.6 Results, discussion and conclusions ............................................................ 96 3.6.1 The interaction equations renormalised ......................................... 97 3.6.2 The incompressible case (M 00 = 0)..................................................99 3.6.3 The subsonic, supersonic and hypersonic cases ......................... 103 3.6.4 Further discussion and closing remarks ........................................108 Chapter 4 The two-tiered interactive structure governing the viscous stability of supersonic flow in the hypersonic limit. 4.1 Introduction ...................................................................................................109 4.1.1 Introductory discussion ..................................................................... 109 4.1.2 An alternative, physical argument .................................................110 4.1.3 The multi-layered upper-branch structure at large Mach number .......................................................................................116 4 4.2 Formulation ..................................................................................................... 125 4.2.1 Introduction ..........................................................................................125 4.2.2 The lower-tier (boundary-layer) ..................................................... 127 4.2.3 The upper-tier (upper-deck) ............................................................ 131 4.2.4 The pressure-displacement law ....................................................... 133 4.3 The two-tier structure: properties and resulting difficulties. 136 4.3.1 Introduction and review ....................................................................136 4.3.2 The linearised problem ......................................................................138 4.3.3 Comments on the numerical solution of the
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