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 the pressure in the 푣-equation
휕 휕 휕 1 + 푈 훻² − 푈′′ − 훻4 푣 = 0 휕푡 푏 휕푥 푏 휕푥 푅푒
• This is the Orr-Sommerfeld equation. It governs the dynamics of the wall-normal velocity component of the perturbation.
GENERAL INTRODUCTION TO 37 HYDRODYNAMIC INSTABILITIES Squire equation
• The normal vorticity is given by
휕푢 휕푤 휂 = − 휕푧 휕푥
• Its governing equation is
휕 휕 1 휕푣 + 푈 − 훻² 휂 = −푈′ 휕푡 푏 휕푥 푅푒 푏 휕푧
• This is the Squire equation. It governs the dynamics of the horizontal flow (푢, 푤).
GENERAL INTRODUCTION TO 38 HYDRODYNAMIC INSTABILITIES Orr-Sommerfeld-Squire equations
• The Orr-Sommerfeld-Squire (OSS) equations read
휕푣 휕 휕 1 = −푈 훻² + 푈′′ + 훻4 푣 휕푡 푏 휕푥 푏 휕푥 푅푒
휕휂 휕 1 휕푣 = −푈 + 훻2 휂 − 푈′ 휕푡 푏 휕푥 푅푒 푏 휕푧
• In matrix form
휕 푣 퓛 0 푣 = 푂푆 휕푡 휂 퐂 퓛푆 휂
GENERAL INTRODUCTION TO 39 HYDRODYNAMIC INSTABILITIES Orr-Sommerfeld-Squire equations
• The dynamics of the cross-stream velocity 푣 are decoupled from the dynamics of the normal vorticity 휂.
• The linear stability of the Squire equation is dictated by the linear stability of the Orr-Sommerfeld one.
• As a consequence, to determine the asymptotic time- evolution (푡 → ∞) of an infinitesimal perturbation, it is sufficient to consider the Orr-Sommerfeld equation only.
GENERAL INTRODUCTION TO 40 HYDRODYNAMIC INSTABILITIES Linear stability of the OS equation
• The OS equation is autonomous in time 푡, and in the space coordinates 푥 and 푧.
• Its solutions can be sought in the form of normal modes
푣 푥, 푦, 푧, 푡 = 푣 푦 푒푖(훼푥+훽푧−휔푡) + 푐. 푐 = ℜ 푣 푦 푒푖 훼푥+훽푧−훼 푐푟+푖푐푖 푡 훼푐푖푡 = 푣 (푦) cos 훼 푥 − 푐푟 푡 + 훽푧 푒 with 훼 the streamwise wavenumber of the perturbation, 훽 its spanwise wavenumber, 휔 the complex angular frequency, 푐푟 the phase speed, 훼푐푖 the temporal growth rate.
GENERAL INTRODUCTION TO 41 HYDRODYNAMIC INSTABILITIES Linear stability of the OS equation
Introducing the normal mode ansatz into the OS equation yields
1 푈 − 푐 퐷2 − 푘2 − 푈′′ − (퐷2 − 푘2)² 푣 = 0 푖훼푅푒 with 푘² = 훼² + 훽² and 휕² 퐷² = 휕푦²
GENERAL INTRODUCTION TO 42 HYDRODYNAMIC INSTABILITIES Squire transformation
• In 1933, Squire proposed a change of variables to reduce the 3D problem to an equivalent 2D one.
• Assuming that 훼 훼 = 훼² + 훽² , 휔 = 휔 ,훼 푅푒 = 훼푅푒 and 푣 = 푣 , 훼 2퐷
the OS equation reduces to
1 푈 − 푐 퐷2 − 훼 2 − 푈′′ − (퐷2 − 훼 2)² 푣 = 0 푖훼 푅푒2퐷
with 휔 > 휔 and 푅푒2퐷 < 푅푒.
GENERAL INTRODUCTION TO 43 HYDRODYNAMIC INSTABILITIES Squire theorem (1933)
Theorem: For any three-dimensional unstable mode (훼, 훽, 휔) of temporal growth rate 휔푖 there is an associated two- dimensional mode (훼 , 휔 ) of temporal growth rate
휔 휔 = 훼² + 훽² 푖 푖 훼
which is more unstable since 휔 푖 > 휔푖. Therefore, when the problem is to determine an instability condition, it is sufficient to consider only two-dimensional perturbations.
GENERAL INTRODUCTION TO 44 HYDRODYNAMIC INSTABILITIES Summary
GENERAL INTRODUCTION TO 45 HYDRODYNAMIC INSTABILITIES Summary
• Hydrodynamic instabilities are ubiquitous in nature.
• Though the Navier-Stokes equations are nonlinear PDE’s, the kinetic energy transfer from the base flow to the perturbation is governed by a linear equation (Reynolds- Orr equation).
• Hence, as a first step toward our understanding of transition to turbulence, investigating the dynamics of infinitesimally small perturbations governed by the linearized Navier-Stokes equations can prove useful.
GENERAL INTRODUCTION TO 46 HYDRODYNAMIC INSTABILITIES Summary
• Investigating the linear stability of a given system is a four- step procedure:
1. Compute an equilibrium solution 퐐b of the original nonlinear system.
2. Linearize the equations in the vicinity of 퐐푏. 3. Use the normal mode ansatz to formulate the problem as an eigenvalue problem. 4. Solve the eigenvalue problem.
• The linearly stable or unstable nature of 퐐푏 is governed by the eigenspectrum of the Jacobian matrix.
GENERAL INTRODUCTION TO 47 HYDRODYNAMIC INSTABILITIES Summary
• For the Navier-Stokes equations, using the parallel flow assumption greatly reduces the complexity of the perturbation’s governing equations.
• Making use of the Orr-Sommerfeld-Squire equations decreases the dimension of the problem from ℝ4푛 to ℝ2푛.
• The linear (in)stability of the flow is solely governed by the Orr-Sommerfeld equation, thus further reducing the dimension of the problem down to ℝ푛.
• Thanks to the Squire theorem, it is sufficient to investigate the linear stability of two-dimensional perturbations to determine the instability condition.
GENERAL INTRODUCTION TO 48 HYDRODYNAMIC INSTABILITIES Inviscid instability of parallel flows Rayleigh equation, Rayleigh, Fjørtotf and Howard theorems and the vortex sheet instability
49 GENERAL INTRODUCTION TO HYDRODYNAMIC INSTABILITIES Rayleigh equation
• The linear (in)stability of a viscous parallel flow is governed by the Orr-Sommerfeld equation
1 푈 − 푐 퐷2 − 훼2 − 푈′′ − (퐷2 − 훼2)² 푣 = 0 푖훼푅푒
• In the inviscid limit (푅푒 → ∞), it reduces to the Rayleigh equation
푈 − 푐 퐷2 − 훼2 − 푈′′ 푣 = 0
GENERAL INTRODUCTION TO HYDRODYNAMIC INSTABILITIES Rayleigh equation
• It can be useful to introduce the stream function 휓
휕휓 휕휓 푢 = , 푣 = − 휕푦 휕푥
• The Rayleigh equation for the normal mode 휓 then reads
푈 − 푐 퐷2 − 훼2 − 푈′′ 휓 = 0
with appropriate boundary conditions.
GENERAL INTRODUCTION TO 51 HYDRODYNAMIC INSTABILITIES Rayleigh inflection point theorem
Theorem: The existence of an inflection point in the velocity profile of a parallel flow is a necessary (but not sufficient) condition for linear instability.
GENERAL INTRODUCTION TO 52 HYDRODYNAMIC INSTABILITIES Rayleigh inflection point theorem
Demonstration: Let us assume the flow is unstable so that
푐푖 ≠ 0 and 푈 − 푐 ≠ 0
Dividing the Rayleigh equation by 푈 − 푐, multiplying by 휓 ∗ and integrating from 푦 = −1 to 푦 = 1 gives
1 1 ′′ 2 2 푈 2 퐷휓 + 훼2 휓 푑푦 + 휓 푑푦 = 0 −1 −1 푈 − 푐
GENERAL INTRODUCTION TO 53 HYDRODYNAMIC INSTABILITIES Rayleigh inflection point theorem
Let us consider only the imaginary part of this integral
1 푈′′ 1 푐 푈′′ 2 푖 2 ℑ 휓 푑푦 = 2 휓 푑푦 = 0 −1 푈 − 푐 −1 푈 − 푐
By assumption, we have 푐푖 ≠ 0 and this integral must vanish. As a consquence 푈′′(푦) must change sign, i.e., the velocity profile must have an inflection point.
GENERAL INTRODUCTION TO 54 HYDRODYNAMIC INSTABILITIES Rayleigh inflection point theorem
GENERAL INTRODUCTION TO 55 HYDRODYNAMIC INSTABILITIES Rayleigh inflection point theorem
• According to the Rayleigh theorem : o Velocity profile (a) is stable (in the inviscid limit). o Vecolity profiles (b) and (c) can potentially be unstable.
• One important conclusion of this theorem is that, if the effect of viscosity on the pertubation is neglected, both the Poiseuille flow and the Blasius boundary layer flow are stable (see profile (a) )..
GENERAL INTRODUCTION TO 56 HYDRODYNAMIC INSTABILITIES Fjørtoft theorem
Theorem: For a monotonic velocity profile, a necessary (but still not sufficient) condition for instability is that the inflection point corresponds to a vorticity maximum.
GENERAL INTRODUCTION TO 57 HYDRODYNAMIC INSTABILITIES Fjørtoft theorem
• In the inviscid limit, according to the Fjørtoft theorem : o Velocity profiles (a) and (b) are stable. o Vecolity profile (c) can potentially be unstable.
GENERAL INTRODUCTION TO 58 HYDRODYNAMIC INSTABILITIES Application to KH-instability
‘’Bernoulli effect’’-like explanation of the Kelvin-Helmholtz instability.
GENERAL INTRODUCTION TO 59 HYDRODYNAMIC INSTABILITIES Application to KH-instability (Exercises)
GENERAL INTRODUCTION TO 60 HYDRODYNAMIC INSTABILITIES Summary
GENERAL INTRODUCTION TO 61 HYDRODYNAMIC INSTABILITIES Summary
• If we assume that the inertial effects are much larger than the viscous ones (푅푒 → ∞), the Orr-Sommerfeld equation reduces to the Rayleigh equation.
• Rayleigh theorem states that an inflection point in the velocity profile is a necessary (but not sufficient) condition for inviscid instability.
• Fjørtoft theorem states that this inflection point needs to correspond to a maximum in the vorticity distribution. This is however still just a necessary (but not sufficient) condition for inviscid instability.
GENERAL INTRODUCTION TO 62 HYDRODYNAMIC INSTABILITIES Summary
• Ignoring all effects of viscous diffusion leads to an unbounded growth rate at large wave numbers (small wavelength).
• Despite this limitation, the vortex sheet problem enable a relatively good understanding of the Kelvin-Helmholtz instability process.
• More realistic models, as the broken line velocity profile, avoids the divergence of the growth rate at large wave numbers while retaining the invisicid approximation.
GENERAL INTRODUCTION TO 63 HYDRODYNAMIC INSTABILITIES