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General Introduction to Hydrodynamic Instabilities L KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Jean-Christophe Loiseau Professor at KTH Mechanics Postdoc at KTH Mechanics Email: [email protected] Email: [email protected] Outline • General introduction to Hydrodynamic Instabilities • Linear instability of parallel flows o Inviscid vs. viscous, temporal vs. spatial, absolute vs. convective instabilities • Non-modal instabilities o Transient growth, resolvent, receptivity, sensitivity, adjoint equations • Extension to complex flows situations and non-linear instabilities GENERAL INTRODUCTION TO 3 HYDRODYNAMIC INSTABILITIES References • Books: o Charru, Hydrodynamic Instabilities, Cambridge Univ. Press o Drazin, Introduction to Hydrodynamic Stability, Cambridge Univ. Press o Huerre & Rossi, Hydrodynamic Instabilities in Open Flows, Cambridge Univ. Press o Schmid & Henningson, Stability and Transition in Shear Flows, Springer-Verlag • Articles: o P. Schmid & L. Brandt, Analysis of fluid systems: stability, receptivity, sensitivity. Appl. Mech. Rev 66(2), 024803, 2014. GENERAL INTRODUCTION TO 4 HYDRODYNAMIC INSTABILITIES Introduction Illustrations, simple examples and local hydrodynamic stability equations. 5 GENERAL INTRODUCTION TO HYDRODYNAMIC INSTABILITIES Illustrations What are hydrodynamic instabilities? GENERAL INTRODUCTION TO 6 HYDRODYNAMIC INSTABILITIES Transition in pipe flow Pioneer experiment by Osborn Reynolds in 1883. GENERAL INTRODUCTION TO 7 HYDRODYNAMIC INSTABILITIES Smoke from a cigarette. Three different flow regimes can easily be identified : laminar, transition and turbulence. GENERAL INTRODUCTION TO 8 HYDRODYNAMIC INSTABILITIES Kelvin-Hemholtz billows Named after Lord Kelvin and Hermann von Helmholtz. One of the most common hydrodynamic instabilities. GENERAL INTRODUCTION TO 9 HYDRODYNAMIC INSTABILITIES Kelvin-Hemholtz billows Named after Lord Kelvin and Hermann von Helmholtz. One of the most common hydrodynamic instabilities. GENERAL INTRODUCTION TO 10 HYDRODYNAMIC INSTABILITIES von-Karman vortex street Named after the engineer and fluid dynamicist Theodore von Karman. GENERAL INTRODUCTION TO 11 HYDRODYNAMIC INSTABILITIES Rayleigh-Taylor instability Instability of an interface between two fluids of different densities. GENERAL INTRODUCTION TO 12 HYDRODYNAMIC INSTABILITIES Some definitions, mathematical formulation and a simple example. How do we study them? GENERAL INTRODUCTION TO 13 HYDRODYNAMIC INSTABILITIES Stability? Stable Unstable Neutral Conditionaly stable We will mostly discuss about linear (in)stability in this course. GENERAL INTRODUCTION TO 14 HYDRODYNAMIC INSTABILITIES Some definitions Let us consider a nonlinear dynamical system 푑퐐 = 푓(퐐, 푅푒) 푑푡 For a given value of the control parameter 푅푒, equilibrium solutions of the system are given by 푓 퐐푏, 푅푒 = 0 GENERAL INTRODUCTION TO 15 HYDRODYNAMIC INSTABILITIES Some definitions Unconditional stability Conditional stability Unconditional instability 푅푒푔 푅푒푐 Three situations can be encountered depending on the value of 푅푒: 1. 푅푒 < 푅푒푔 : The equilibrium is unconditionally stable. Whatever the shape and amplitude of the perturbation, it decays and the system return to its equilibrium position. 2. 푅푒푐 < 푅푒: The equilibrium is unconditionally (linearly) unstable. At least one infinitesimal perturbation will always depart away from it. 3. 푅푒푔 ≤ 푅푒 ≤ 푅푒푐 : The stability of the equilibrium depends on the shape and finite-amplitude of the perturbation. Determining the shape and amplitude of such perturbation usually requires solving a complex nonlinear problem. GENERAL INTRODUCTION TO 16 HYDRODYNAMIC INSTABILITIES Mathematical formulation of the problem • The first part of this course concerns linear stability analysis, that is the determination of the unconditional linear instability threshold 푅푒푐. • The dynamics of an infinitesimal perturbation 퐪 can be studied by linearizing the system in the vicinity of the equilibrium 퐐푏 푑퐪 = 퐉퐪 푑푡 with 퐉 the Jacobian matrix of the system evaluated at 퐐푏. GENERAL INTRODUCTION TO 17 HYDRODYNAMIC INSTABILITIES Mathematical formulation of the problem The 푖푗-th entry of the Jacobian matrix evaluated in the vicinity of 퐐푏 is given by 휕푓푖 퐽푖푗 = 휕푄푗 퐐푏 GENERAL INTRODUCTION TO 18 HYDRODYNAMIC INSTABILITIES Mathematical formulation of the problem • This linear dynamical system is autonomous in time. Its solutions can be sought in the form of normal modes 퐪 푡 = 퐪 푒 휆푡 + 푐. 푐 with 휆 = 휎 + 푖휔. • Injecting this form for 퐪 푡 into our linear system yields the following eigenvalue problem 휆퐪 = 퐉퐪 GENERAL INTRODUCTION TO 19 HYDRODYNAMIC INSTABILITIES Mathematical formulation of the problem The linear (in)stability of the equilibrium 퐐푏 then depends on the value of the growth rate 휎 = ℜ(휆) of the leading eigenvalue : 1. If 휎 < 0, the system is linearly stable (푅푒 < 푅푒푐). 2. If 휎 > 0, the system is linearly unstable (푅푒 > 푅푒푐). 3. If 휎 = 0, the system is neutrally stable (푅푒 = 푅푒푐). GENERAL INTRODUCTION TO 20 HYDRODYNAMIC INSTABILITIES Mathematical formulation of the problem The value of 휔 = ℑ(휆) characterizes the oscillatory nature of the perturbation : 1. If 휔 ≠ 0, the perturbation oscillates in time. 2. If 휔 = 0, the perturbation has a monotonic behavior. GENERAL INTRODUCTION TO 21 HYDRODYNAMIC INSTABILITIES Exercise • Consider the equation of motion of a damped pendulum. 2 휃 = −푘휃 − 휔0 sin(휃) • Introducing 푥 = 휃 and 푦 = 휃 , this equation can be rewritten as a 2 × 2 system of first order ODE’s 푥 = 푦 2 푦 = −푘푦 − 휔0 sin(푥) GENERAL INTRODUCTION TO 22 HYDRODYNAMIC INSTABILITIES Exercise 1. Compute the two equilibrium solutions of this system. 2. Derive the linear equations governing the dynamics of an infinitesimal perturbation. 3. Study the linear stability of a) the first equilibrium. Is it linearly stable or unstable? b) the second one. Is it linearly stable or unstable? GENERAL INTRODUCTION TO 23 HYDRODYNAMIC INSTABILITIES Navier-Stokes, Reynolds-Orr and Orr-Sommerfeld-Squire equations Local hydrodynamic stability analysis GENERAL INTRODUCTION TO 24 HYDRODYNAMIC INSTABILITIES Navier-Stokes equations • The dynamics of an incompressible flow of Newtonian fluid are governed by 휕퐔 1 = − 퐔 ∙ 훻 퐔 − 훻푃 + 훻2퐔 휕푡 푅푒 훻 ∙ 퐔 = 0 GENERAL INTRODUCTION TO 25 HYDRODYNAMIC INSTABILITIES Navier-Stokes equations • It is a system of nonlinear partial differential equations (PDE’s). The variables depend on both time and space. • In the rest of this course, we will assume that the 푇 equilibrium solution (or base flow) 퐐푏 = (퐔푏,푃푏) is given. GENERAL INTRODUCTION TO 26 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • Assume a velocity field of the form 퐔 = 퐔 + 퐮 GENERAL INTRODUCTION TO 27 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • Plugging it into the Navier-Stokes equations, the governing equations for the fluctuation 퐮 read 훻 ∙ 퐮 = 0 휕퐮 1 = − 퐮 ∙ 훻 퐔 − 퐔 ∙ 훻 퐮 − 훻푝 + 훻2퐮 − 퐮 ∙ 훻 퐮 휕푡 푅푒 Linear terms Nonlinear term GENERAL INTRODUCTION TO 28 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • Multiplying from the left by 퐮 gives an evolution equation for the local kinetic energy 1 휕 1 퐮 ∙ 퐮 = −퐮 ∙ 퐮 ∙ 훻 퐔 − 퐮 ∙ 퐔 ∙ 훻 퐮 + 퐮 ∙ 훻2퐮 2 휕푡 푅푒 −퐮 ∙ 훻푝 − 퐮 ∙ 퐮 ∙ 훻 퐮 GENERAL INTRODUCTION TO 29 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • After integrating over the whole volume, the Reynolds-Orr equation governing the evolution of the total kinetic energy of the perturbation finally reads 푑퐸 1 = − 퐮 ∙ 퐮 ∙ 훻 퐔 푑푉 − 훻퐮: 훻퐮 푑푉 푑푡 푉 푅푒 푉 Production Dissipation • The evolution of the perturbation’s kinetic energy results from a competition between production and dissipation. GENERAL INTRODUCTION TO 30 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • These two terms only involve linear mechanisms whether or not we initially considered the nonlinear term in the momentum equation. • The non-linear term is energy-conserving. It only scatters the energy along the different velocity components and length scales. GENERAL INTRODUCTION TO 31 HYDRODYNAMIC INSTABILITIES Reynolds-Orr equation • In the case of the Navier-Stokes equations, investigating the dynamics of infinitesimal perturbation allows one to: 1. Identify the critical Reynolds number beyond which the steady equilibrium flow is unconditionally unstable. (Linear stability analysis) 2. Highlight the underlying physical mechanisms through which any kind of perturbation (linear or nonlinear) relies to grow over time. (Reynolds-Orr equation) GENERAL INTRODUCTION TO 32 HYDRODYNAMIC INSTABILITIES Parallel flow assumption • For the sake of simplicity, in the rest of the course we will assume a base flow of the form 퐔푏 = 푈 푦 , 0,0 • The base flow only depends on the cross-stream coordinate. We neglect the streamwise evolution of the flow. GENERAL INTRODUCTION TO 33 HYDRODYNAMIC INSTABILITIES Examples of parallel flows GENERAL INTRODUCTION TO 34 HYDRODYNAMIC INSTABILITIES Linearized Navier-Stokes equations • In the most general case (3D base flow and perturbation), the linearized Navier-Stokes equations read 휕퐮 1 2 = − 퐔푏 ∙ 훻 퐮 − (퐮 ∙ 훻)퐔푏 − 훻푝 + 훻 퐮 휕푡 푅푒 훻 ∙ 퐮 = 0 GENERAL INTRODUCTION TO 35 HYDRODYNAMIC INSTABILITIES Linearized Navier-Stokes equations • Based on the parallel flow assumption used for 퐔푏, these equations simplify to 휕푢 휕푢 휕푝 1 = −푈 − 푈′ 푣 − + 훻2푢 휕푡 푏 휕푥 푏 휕푥 푅푒 휕푣 휕푣 휕푝 1 = −푈 − + 훻2푣 휕푡 푏 휕푥 휕푦 푅푒 휕푤 휕푤 휕푝 1 = −푈 − + 훻2푤 휕푡 푏 휕푥 휕푧 푅푒 휕푢 휕푣 휕푤 + + = 0 휕푥 휕푦 휕푧 GENERAL INTRODUCTION TO 36 HYDRODYNAMIC INSTABILITIES Orr-Sommerfeld equation • Taking the divergence of the momentum equations gives 휕푣 훻²푝 = −2푈′ 푏 휕푥 • One can now eliminate
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