Boundary-Layer Linear Stability Theory Leslie M. Mack Jet Propulsion Laboratory California Institute of Technology Pasadena, California 91109 U.S.A. AGARD Report No. 709, Part 3 1984 Contents Preface 9 1 Introduction 10 1.1 Historical background . 10 1.2 Elements of stability theory . 11 I Incompressible Stability Theory 14 2 Formulation of Incompressible Stability Theory 15 2.1 Derivation of parallel-flow stability equations . 15 2.2 Non-parallel stability theory . 17 2.3 Temporal and spatial theories . 18 2.3.1 Temporal amplification theory . 18 2.3.2 Spatial amplification theory . 19 2.3.3 Relation between temporal and spatial theories . 20 2.4 Reduction to fourth-order system . 21 2.4.1 Transformation to 2D equations - temporal theory . 21 2.4.2 Transformation to 2D equations - spatial theory . 22 2.5 Special forms of the stability equations . 23 2.5.1 Orr-Sommerfeld equation . 23 2.5.2 System of first-order equations . 23 2.5.3 Uniform mean flow . 24 2.6 Wave propagation in a growing boundary layer . 25 2.6.1 Spanwise wavenumber . 26 2.6.2 Some useful formulae . 27 2.6.3 Wave amplitude . 28 3 Incompressible Inviscid Theory 29 3.1 Inflectional instability . 30 3.1.1 Some mathematical results . 30 3.1.2 Physical interpretations . 31 3.2 Numerical integration . 31 3.3 Amplified and damped inviscid waves . 32 3.3.1 Amplified and damped solutions as complex conjugates . 32 3.3.2 Amplified and damped solutions as R ! 1 limit of viscous solutions . 33 4 Numerical Techniques 35 4.1 Types of methods . 35 4.2 Shooting methods . 35 4.3 Gram-Schmidt orthonormalization . 36 4.4 Newton-Raphson search procedure . 37 1 5 Viscous Instability 38 5.1 Kinetic-energy equation . 38 5.2 Reynolds stress in the viscous wall region . 39 6 Numerical Results - 2D Boundary Layers 41 6.1 Blasius boundary layer . 41 6.2 Falkner-Skan boundary layers . 50 6.3 Non-similar boundary layers . 52 6.4 Boundary layers with mass transfer . 52 6.5 Boundary layers with heating and cooling . 53 6.6 Eigenvalue spectrum . 53 7 Harmonic Point Sources of Instability Waves 55 7.1 General remarks . 55 7.2 Numerical integration . 57 7.3 Method of steepest descent . 58 7.4 Superposition of point sources . 61 7.5 Numerical and experimental results . 62 II Compressible Stability Theory 65 8 Formulation of Compressible Stability Theory 66 8.1 Introductory remarks . 66 8.2 Linearized parallel-flow stability equations . 67 8.3 Normal-mode equations . 69 8.4 First-order equations . 71 8.4.1 Eighth-order system . 71 8.4.2 Sixth-order system . 71 8.5 Uniform mean flow . 71 9 Compressible Inviscid Theory 74 9.1 Inviscid equations . 74 9.2 Uniform mean flow . 75 9.3 Some mathematical results . 76 9.4 Methods of solution . 77 9.5 Higher modes . 78 9.5.1 Inflectional neutral waves . 78 9.5.2 Noninflectional neutral waves . 80 9.6 Unstable 2D waves . 83 9.7 Three-dimensional waves . 83 9.8 Effect of wall cooling . 86 10 Compressible Viscous Theory 89 10.1 Effect of Mach number on viscous instability . 89 10.2 Second mode . 93 10.3 Effect of wall cooling and heating . 93 10.4 Use of sixth-order system for 3D waves . 95 10.5 Spatial theory . 97 11 Forcing Theory 99 11.1 Formulation and numerical results . 99 11.2 Receptivity in high-speed wind tunnels . 100 11.3 Reflection of sound waves from a laminar boundary layer . 101 11.4 Table of boundary-layer thicknesses . 103 2 III Three-Dimensional Boundary Layers 105 12 Rotating Disk - A Prototype 3D Boundary Layer 106 12.1 Mean boundary layer . 106 12.2 Crossflow instability . 107 12.3 Instability characteristics of normal modes . 108 12.4 Wave pattern from a steady point source . 110 13 Falker-Skan-Cooke Boundary Layers 114 13.1 Mean boundary layer . 114 13.2 Boundary layers with small crossflow . 117 13.3 Boundary layers with crossflow instability only . 119 13.4 Boundary layers with both crossflow and streamwise instability . 122 14 Transonic Infinite-Span Swept-Wing Boundary Layer 125 14.1 Mean boundary layer . 125 14.2 Crossflow instability . 129 14.3 Streamwise instability . 131 14.4 Wave amplitude . 134 A Coefficient Matrix of Compressible Stability Equations 137 B Freestream Solutions of Compressible Stability Equations 140 3 List of Figures 1.1 Typical neutral-stability curves. 12 3.1 Alternative indented contours for numerical integration of inviscid equations. 33 3.2 Inviscid temporal damping rate vs. wavenumber for Blasius boundary layer. 34 6.1 Neutral-stability curves for Blasius boundary layer: (a) F vs. R; (b) αr vs. R; (c) c vs. R; − · −, σmax; − − −,(A=A0)max; both maxima are with respect to frequency at constant R. 42 6.2 Distribution of 2D spatial amplification rate with frequency in Blasius boundary layer at R = 600 and 1200. ..