Boundary-Layer Linear Stability Theory

Total Page:16

File Type:pdf, Size:1020Kb

Boundary-Layer Linear Stability Theory 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. ..
Recommended publications
  • Blasius Boundary Layer
    3 Fluid flow at a large Reynolds number. In our first application of matched asymptotic expansions to problems of fluid dynam- ics, let us consider a two-dimensional flow past a solid body, B, at a large Reynolds number. In non-dimensional variables we have the Navier-Stokes equations derived in Section 1 (see equations (1.71)-(1.74)) which, omitting over bars, can be written as ∂u ∂u ∂u ∂p ∂2u ∂2u! + u + v = − + Re−1 + , (3.1) ∂t ∂x ∂y ∂x ∂x2 ∂y2 ∂v ∂v ∂v ∂p ∂2v ∂2v ! + u + v = − + Re−1 + , (3.2) ∂t ∂x ∂y ∂y ∂x2 ∂y2 ∂u ∂v + = 0. (3.3) ∂x ∂y As boundary conditions we take a uniform stream far from the body, u → 1, v → 0, p → 0 as x2 + y2 → ∞, (3.4) and the no-slip conditions on the surface of the body, u = v = 0 at r = ∂B, (3.5) r being the position vector in the (x, y)-plane. One needs to be a little careful with formulating far-field conditions, especially when the solid surface extends to infinity downstream, however in every particular case this should not be an issue. 3.1 Outer limit - inviscid flow. Let Re → ∞. We attempt solution for the flow as an expansion, u = u0(x, y, t) + ..., v = v0(x, y, t) + ..., p = p0(x, y, t) + ..., (3.6) where the neglected terms are small but at this stage we do not know how small. The Navier-Stokes equations suggest correction terms of order Re−1but in fact the corrections prove to be O(Re−1/2).
    [Show full text]
  • Solution Bounds, Stability and Attractors' Estimates of Some Nonlinear Time-Varying Systems
    1 Solution Bounds, Stability and Attractors’ Estimates of Some Nonlinear Time-Varying Systems Mark A. Pinsky and Steve Koblik Abstract. Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for a broad class of the corresponding systems. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and estimates the stability basins and trapping regions for the corresponding systems. Finally, we conform the theoretical results in representative simulations. Key Words. Comparison principle, Lipschitz condition, stability criteria, trapping/stability regions, solution bounds AMS subject classification. 34A34, 34C11, 34C29, 34C41, 34D08, 34D10, 34D20, 34D45 1. INTRODUCTION. We are going to study a system defined by the equation n (1) xAtxftxFttt , , [0 , ), x , ft ,0 0 where matrix A nn and functions ft:[ , ) nn and F : n are continuous and bounded, and At is also continuously differentiable. It is assumed that the solution, x t, x0 , to the initial value problem, x t0, x 0 x 0 , is uniquely defined for tt0 . Note that the pertained conditions can be found, e.g., in [1] and [2].
    [Show full text]
  • Strange Attractors and Classical Stability Theory
    Nonlinear Dynamics and Systems Theory, 8 (1) (2008) 49–96 Strange Attractors and Classical Stability Theory G. Leonov ∗ St. Petersburg State University Universitetskaya av., 28, Petrodvorets, 198504 St.Petersburg, Russia Received: November 16, 2006; Revised: December 15, 2007 Abstract: Definitions of global attractor, B-attractor and weak attractor are introduced. Relationships between Lyapunov stability, Poincare stability and Zhukovsky stability are considered. Perron effects are discussed. Stability and instability criteria by the first approximation are obtained. Lyapunov direct method in dimension theory is introduced. For the Lorenz system necessary and sufficient conditions of existence of homoclinic trajectories are obtained. Keywords: Attractor, instability, Lyapunov exponent, stability, Poincar´esection, Hausdorff dimension, Lorenz system, homoclinic bifurcation. Mathematics Subject Classification (2000): 34C28, 34D45, 34D20. 1 Introduction In almost any solid survey or book on chaotic dynamics, one encounters notions from classical stability theory such as Lyapunov exponent and characteristic exponent. But the effect of sign inversion in the characteristic exponent during linearization is seldom mentioned. This effect was discovered by Oscar Perron [1], an outstanding German math- ematician. The present survey sets forth Perron’s results and their further development, see [2]–[4]. It is shown that Perron effects may occur on the boundaries of a flow of solutions that is stable by the first approximation. Inside the flow, stability is completely determined by the negativeness of the characteristic exponents of linearized systems. It is often said that the defining property of strange attractors is the sensitivity of their trajectories with respect to the initial data. But how is this property connected with the classical notions of instability? For continuous systems, it was necessary to remember the almost forgotten notion of Zhukovsky instability.
    [Show full text]
  • Chapter 8 Stability Theory
    Chapter 8 Stability theory We discuss properties of solutions of a first order two dimensional system, and stability theory for a special class of linear systems. We denote the independent variable by ‘t’ in place of ‘x’, and x,y denote dependent variables. Let I ⊆ R be an interval, and Ω ⊆ R2 be a domain. Let us consider the system dx = F (t, x, y), dt (8.1) dy = G(t, x, y), dt where the functions are defined on I × Ω, and are locally Lipschitz w.r.t. variable (x, y) ∈ Ω. Definition 8.1 (Autonomous system) A system of ODE having the form (8.1) is called an autonomous system if the functions F (t, x, y) and G(t, x, y) are constant w.r.t. variable t. That is, dx = F (x, y), dt (8.2) dy = G(x, y), dt Definition 8.2 A point (x0, y0) ∈ Ω is said to be a critical point of the autonomous system (8.2) if F (x0, y0) = G(x0, y0) = 0. (8.3) A critical point is also called an equilibrium point, a rest point. Definition 8.3 Let (x(t), y(t)) be a solution of a two-dimensional (planar) autonomous system (8.2). The trace of (x(t), y(t)) as t varies is a curve in the plane. This curve is called trajectory. Remark 8.4 (On solutions of autonomous systems) (i) Two different solutions may represent the same trajectory. For, (1) If (x1(t), y1(t)) defined on an interval J is a solution of the autonomous system (8.2), then the pair of functions (x2(t), y2(t)) defined by (x2(t), y2(t)) := (x1(t − s), y1(t − s)), for t ∈ s + J (8.4) is a solution on interval s + J, for every arbitrary but fixed s ∈ R.
    [Show full text]
  • Turbulence, Entropy and Dynamics
    TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions
    [Show full text]
  • Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory
    MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 – p.1/17 Stability in the sense of Lyapunov A dynamic system x˙ = f(x) is Lyapunov stable or internally stable about an equilibrium point xeq if state trajectories are confined to a bounded region whenever the initial condition x0 is chosen sufficiently close to xeq. Mathematically, given R> 0 there always exists r > 0 so that if ||x0 − xeq|| <r, then ||x(t) − xeq|| <R for all t> 0. As seen in the figure R defines a desired confinement region, while r defines the neighborhood of xeq where x0 must belong so that x(t) does not exit the confinement region. R r xeq x0 x(t) MCE503 – p.2/17 Stability in the sense of Lyapunov... Note: Lyapunov stability does not require ||x(t)|| to converge to ||xeq||. The stronger definition of asymptotic stability requires that ||x(t)|| → ||xeq|| as t →∞. Input-Output stability (BIBO) does not imply Lyapunov stability. The system can be BIBO stable but have unbounded states that do not cause the output to be unbounded (for example take x1(t) →∞, with y = Cx = [01]x). The definition is difficult to use to test the stability of a given system. Instead, we use Lyapunov’s stability theorem, also called Lyapunov’s direct method. This theorem is only a sufficient condition, however. When the test fails, the results are inconclusive. It’s still the best tool available to evaluate and ensure the stability of nonlinear systems.
    [Show full text]
  • Boundary Layers
    1 I-campus project School-wide Program on Fluid Mechanics Modules on High Reynolds Number Flows K. P. Burr, T. R. Akylas & C. C. Mei CHAPTER TWO TWO-DIMENSIONAL LAMINAR BOUNDARY LAYERS 1 Introduction. When a viscous fluid flows along a fixed impermeable wall, or past the rigid surface of an immersed body, an essential condition is that the velocity at any point on the wall or other fixed surface is zero. The extent to which this condition modifies the general character of the flow depends upon the value of the viscosity. If the body is of streamlined shape and if the viscosity is small without being negligible, the modifying effect appears to be confined within narrow regions adjacent to the solid surfaces; these are called boundary layers. Within such layers the fluid velocity changes rapidly from zero to its main-stream value, and this may imply a steep gradient of shearing stress; as a consequence, not all the viscous terms in the equation of motion will be negligible, even though the viscosity, which they contain as a factor, is itself very small. A more precise criterion for the existence of a well-defined laminar boundary layer is that the Reynolds number should be large, though not so large as to imply a breakdown of the laminar flow. 2 Boundary Layer Governing Equations. In developing a mathematical theory of boundary layers, the first step is to show the existence, as the Reynolds number R tends to infinity, or the kinematic viscosity ν tends to zero, of a limiting form of the equations of motion, different from that obtained by putting ν = 0 in the first place.
    [Show full text]
  • Engineering Fluid Dynamics CTW-TS Bachelor Thesis Remco Olimulder
    Faculty of Advanced Technology Engineering Fluid Dynamics The boundary layer of a fluid CTW-TS stream over a flat Bachelor thesis plate Date Januari 2013 Remco Olimulder TS- 136 ii Abstract In the flow over a solid surface a boundary layer is formed. In this study this boundary layer is investi- gated theoretically, numerically and experimentally. The equations governing the incompressible laminar flow in thin boundary layers have been derived following Prandtl’s theory. For the case of two-dimensional flow over an infinite flat plate at zero incidence, the similarity solution of Blasius is formulated. In addition the similarity solution for the flow stagnating on an infinite flat plate has been derived. Subsequently the Integral Momentum Equation method of von Kármán is derived. This method is then used to numerically solve for the flow in the boundary layer along the flat plate and for the stagnation-point flow. For the experimental part of the investigation a wind-tunnel model of a flat plate, of finite thickness, was placed inside the silent wind tunnel. The leading-edge of the wind-tunnel model has a specific smooth shape, designed such that the boundary layer develops like the boundary layer along a plate of infinitesimal thickness. The pressure distribution along the wind-tunnel model, obtained from a CFD calculation has been used as input for the Integral Momentum Equation method to predict the boundary layer development along the model and compare it to Blasius’ solution. The velocity distribution in the boundary layer has been measured using hot-wire anemometry (HWA) at three distances from the leading edge.
    [Show full text]
  • Analysis of ODE Models
    Analysis of ODE Models P. Howard Fall 2009 Contents 1 Overview 1 2 Stability 2 2.1 ThePhasePlane ................................. 5 2.2 Linearization ................................... 10 2.3 LinearizationandtheJacobianMatrix . ...... 15 2.4 BriefReviewofLinearAlgebra . ... 20 2.5 Solving Linear First-Order Systems . ...... 23 2.6 StabilityandEigenvalues . ... 28 2.7 LinearStabilitywithMATLAB . .. 35 2.8 Application to Maximum Sustainable Yield . ....... 37 2.9 Nonlinear Stability Theorems . .... 38 2.10 Solving ODE Systems with matrix exponentiation . ......... 40 2.11 Taylor’s Formula for Functions of Multiple Variables . ............ 44 2.12 ProofofPart1ofPoincare–Perron . ..... 45 3 Uniqueness Theory 47 4 Existence Theory 50 1 Overview In these notes we consider the three main aspects of ODE analysis: (1) existence theory, (2) uniqueness theory, and (3) stability theory. In our study of existence theory we ask the most basic question: are the ODE we write down guaranteed to have solutions? Since in practice we solve most ODE numerically, it’s possible that if there is no theoretical solution the numerical values given will have no genuine relation with the physical system we want to analyze. In our study of uniqueness theory we assume a solution exists and ask if it is the only possible solution. If we are solving an ODE numerically we will only get one solution, and if solutions are not unique it may not be the physically interesting solution. While it’s standard in advanced ODE courses to study existence and uniqueness first and stability 1 later, we will start with stability in these note, because it’s the one most directly linked with the modeling process.
    [Show full text]
  • Apollo Afterbody Heat Transfer and Pressure with and Without Ablation at M, of 5.8 to 8.3
    https://ntrs.nasa.gov/search.jsp?R=19660027030 2020-03-24T01:55:18+00:00Z View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NASA Technical Reports Server NASA TECHNICAL NOTE NASA TN D-3620 0 N 40 @? P z c e c/I e z APOLLO AFTERBODY HEAT TRANSFER AND PRESSURE WITH AND WITHOUT ABLATION AT M, OF 5.8 TO 8.3 by George Lee and Robert E. Sandell Ames Research Center Moffett Field, Cali$ NATIONAL AERONAUTICS AND SPACE ADMINISTRATION b WASHINGTON, D. C. SEPTEMBER 1966 1 3” TECH LIBRARY KAFB, NM APQLLO AFTERBODY HEAT TRANSFER AND PRESSURE WITH AND WITHOUT ABLATION AT Moo OF 5.8 TO 8.3 By George Lee and Robert E. Sundell Ames Research Center Moffett Field, Calif. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price 62.00 APOUO AFTEIiBODY HEAT TRANSFER AND PRESSURE WITH AND WITHOUT ABLATION AT I!&,OF 3.8 TO 8.3 '' By George Lee and Robert E. Sundell Ames Research Center SUMMARY Apollo afterbody pressures and heat transfer with and without an ablating nose cap have been measured at Mach numbers 5.8 to 8.3, stream enthalpies of 1.628~106to 9.296~106J/kg, and Reynolds numbers based on diameter of 200 to 22,000. Correlations of the data and comparisons with theories have been made. INTRODUCTION Ablation type heat shields, commonly used for the thermal protection of the stagnation region of spacecraft, inject gases into the stream.
    [Show full text]
  • Combinatorial and Analytical Problems for Fractals and Their Graph Approximations
    UPPSALA DISSERTATIONS IN MATHEMATICS 112 Combinatorial and analytical problems for fractals and their graph approximations Konstantinos Tsougkas Department of Mathematics Uppsala University UPPSALA 2019 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 15 February 2019 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Ben Hambly (Mathematical Institute, University of Oxford). Abstract Tsougkas, K. 2019. Combinatorial and analytical problems for fractals and their graph approximations. Uppsala Dissertations in Mathematics 112. 37 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2739-8. The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs. The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.
    [Show full text]
  • Stability Theory for Ordinary Differential Equations 1. Introduction and Preliminaries
    JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 1, June 1988 Stability Theory for Ordinary Differential Equations Sung Kyu Choi, Keon-Hee Lee and Hi Jun Oh ABSTRACT. For a given autonomous system xf = /(a?), we obtain some properties on the location of positive limit sets and investigate some stability concepts which are based upon the existence of Liapunov functions. 1. Introduction and Preliminaries The classical theorem of Liapunov on stability of the origin 鉛 = 0 for a given autonomous system xf = makes use of an auxiliary function V(.x) which has to be positive definite. Also, the time deriva­ tive V'(⑦) of this function, as computed along the solution, has to be negative definite. It is our aim to investigate some stability concepts of solutions for a given autonomous system which are based upon the existence of suitable Liapunov functions. Let U C Rn be an open subset containing 0 G Rn. Consider the autonomous system x' = /(⑦), where xf is the time derivative of the function :r(t), defined by the continuous function f : U — Rn. When / is C1, to every point t()in R and Xq in 17, there corresponds the unique solution xq) of xf = f(x) such that ^(to) = ⑦⑴ For such a solution, let us denote I(xq) = (aq) the interval where it is defined (a, ⑵ may be infinity). Let f : (a, &) -나 R be a continuous function. Then f is decreasing on (a, b) if and only if the upper right-sided Dini derivative D+f(t) = lim sup f〔t + h(-f(t) < o 九 一>o+ h Received by the editors on March 25, 1988.
    [Show full text]