Introduction to Hydrodynamic Instabilities Fran¸ cois Charru Institut de M´ ecanique des Fluides de Toulouse CNRS – Univ. Toulouse ´ Ecole d’´ et´ e sur les Instabilit´ es et Bifurcations en M´ ecanique Quiberon 2015 19th September 2015 1 / 95
Contents Instabilities of fluids at rest 1 Stability of open flows: basic ideas 2 Inviscid instability of parallel flows 3 Viscous instability of parallel flows 4 Nonlinear evolution of systems with few degrees of freedom 5 Nonlinear dispersive waves 6 Nonlinear dynamics of dissipative systems 7 19th September 2015 2 / 95
References This presentation is based on the book Hydrodynamic Instabilities (F. Charru 2011, Cambridge Univ. Press) where the references of all the pictures can be found. Other references: Textbooks on Fluid Mechanics Guyon E., Hulin J.P. & Petit L. 2012 Hydrodynamique Physique , CNRS Eds. Landau L. & Lifschitz E. 1987 Fluid Mechanics , Butterworth-Heinemann. Tritton D.J. 1988 Physical Fluid Dynamics , Clarendon Press. Specialized Textbooks Drazin P.G. 2002 Introduction to Hydrodynamic Stability , Cambridge UP. Huerre P. & Rossi M. 1998 Hydrodynamic Instabilities in Open Flows . Eds Godr` eche C. & Manneville P., Cambridge UP. Manneville P. 1990 Dissipative Structures and Weak Turbulence , Academic Press. Schmid P.J. & Henningson D.S. Stability and Transition in Shear Flows , Springer-Verlag. 19th September 2015 3 / 95
Instabilities of fluids at rest 1. Instabilities of fluids at rest 19th September 2015 4 / 95
Instabilities of fluids at rest Gravity-driven Rayleigh-Taylor instability (1) Pending drops under a suspended liquid film Descending fingers of salt water into fresh water 19th September 2015 5 / 95
Instabilities of fluids at rest Gravity-driven Rayleigh-Taylor instability (2) Analysis with viscosity and bounding walls neglected. Base state: fluids at rest with horizontal interface, hydrostatic pressure distribution. Perturbed flow: 19th September 2015 6 / 95
Instabilities of fluids at rest Gravity-driven Rayleigh-Taylor instability (3) Linearized perturbation equations and perturbations ∝ e i ( kx − ω t ) ω 2 = ( ρ 1 − ρ 2 ) gk + k 3 γ → Dispersion relation ρ 1 + ρ 2 → Instability (complex ω ) when ρ 1 < ρ 2 , with growth rate: ( l c capillary length, τ ref capillary time) 1 0.5 ! i " ref 0 ! 0.5 ! 1 0 0.5 1 1.5 kl c → Long-wave instability 19th September 2015 7 / 95
Instabilities of fluids at rest Instabilities related to Rayleigh-Taylor Inertial instability of accelerated flows (Taylor 1950) Gravitational instability in astrophysics (Jeans 1902) ω 2 = c 2 s k 2 − 4 π G ρ 0 . 19th September 2015 8 / 95
Instabilities of fluids at rest Capillary Rayleigh-Plateau instability jet of water drops on a spider web r − , p + − → r + , p − − → ω i ka 19th September 2015 9 / 95
Instabilities of fluids at rest Buoyancy-driven Rayleigh-B´ enard instability Linear stability analysis → dispersion curve Bifurcation parameter: Rayleigh number Ra = α p g ( T 1 − T 2 ) d 3 νκ 19th September 2015 10 / 95
Instabilities of fluids at rest A toy-model: convection in an annulus (1) (Welander 1967) Base state: fluid at rest with temperature z T = T 0 − T 1 a = T 0 + T 1 cos φ. Momentum conservation: ∂ U ∂ t = − 1 ∂ p ∂φ + α g ( T − T ) sin φ − γ U . ρ a Energy conservation: ∂ T ∂ T ∂ t + U ∂φ = k ( T − T ) . a Temperature sought for as T ( t , φ ) = T + T A ( t ) sin φ − T B ( t ) cos φ, 19th September 2015 11 / 95
Instabilities of fluids at rest A toy-model: convection in an annulus (2) The change of scales X ∝ U , Y ∝ T A , Z ∝ T B , τ ∝ t , then provides the Lorenz system (1963) ∂ τ X = − PX + PY ∂ τ Y = − Y − XZ + RX ∂ τ Z = − Z + XY where P = k /γ, R = α gT 1 / 2 γ ka . Stability analysis of the fixed point (0 , 0 , 0) (fluid at rest) → Supercritical pitchfork bifurcation at R c 1 = 1 (convection) Chaotic behavior beyond R c 2 ( P ) > R c 1 via a subcritical Hopf bifurcation (Lorenz strange attractor). 19th September 2015 12 / 95
Instabilities of fluids at rest Thermocapillary B´ enard-Marangoni instability 19th September 2015 13 / 95
Instabilities of fluids at rest Saffman-Taylor instability of fronts between viscous fluids 19th September 2015 14 / 95
Stability of open flows: basic ideas 2. Stability of open flows: basic ideas 19th September 2015 15 / 95
Stability of open flows: basic ideas Forced flow: canonic forcings Consider the (1D) linearized evolution equation for u ( x , t ) L u ( x , t ) = S ( x , t ) L : differential linear operator involving x - and t -derivatives S ( x , t ): forcing. Three types of elementary forcing functions of special importance: S ( x , t ) = F ( x ) δ ( t ) ( initial value problem ) S ( x , t ) = δ ( x ) δ ( t ) ( impulse response problem ) S ( x , t ) = δ ( x ) H ( t ) e − i ω t ( periodic forcing problem ) where δ and H are the Dirac and Heaviside functions. 19th September 2015 16 / 95
Stability of open flows: basic ideas Impulse response – Definitions Spatiotemporal evolution of a disturbance localized at x = 0 at t = 0 (a) (a): Linearly stable flow (b): Linearly unstable flow, convective instability t (c): Linearly unstable flow, absolute instability 0 0 20 40 x (b) (c) t t 0 0 0 20 40 0 20 40 x x 19th September 2015 17 / 95
Stability of open flows: basic ideas Illustration: waves on a falling film (1) Natural waves Forced waves, 5.5 Hz 19th September 2015 18 / 95
Stability of open flows: basic ideas Illustration: waves on a falling film (2) A perturbation generated at x = t = 0 amplifies while it is convected downstream: x = 44 cm x = 97 cm 19th September 2015 19 / 95
Stability of open flows: basic ideas Stability criteria It can be shown that: A necessary and sufficient condition for stability is that the growth rates of all the modes with real wavenumber k are negative (temporal stability) The criterion for absolute instability is that there exists some wavenumber k 0 with zero group velocity and positive growth rate. A convective instability amplifies any unstable perturbation, and advects it downstream (“noise amplifier”) An absolutely unstable flow responds selectively to the perturbation with zero group velocity: it behaves like an oscillator with its own natural frequency. 19th September 2015 20 / 95
Inviscid instability of parallel flows 3. Inviscid instability of parallel flows Large Reynolds number flows (negligible viscous effects) Far from solid boundaries 19th September 2015 21 / 95
Inviscid instability of parallel flows Illustration 1: tilted channel (Reynolds 1883, Thorpe 1969) t = 0 t = 0 + 19th September 2015 22 / 95
Inviscid instability of parallel flows Illustration 2: wind in a stratified atmosphere Flowing layer Air layer at rest 19th September 2015 23 / 95
Inviscid instability of parallel flows Illustration 3: rising mixing layer Water Water + Air 19th September 2015 24 / 95
Inviscid instability of parallel flows Illustration 4: jet Jet of carbone dioxyde 6 mm in diameter issuing into air at a speed of 40 m s − 1 (Re = 30 000). 19th September 2015 25 / 95
Inviscid instability of parallel flows Illustration 5: wake inflection Wake of a cylinder in water flowing at 1.4 cm s − 1 (Re = 140). 19th September 2015 26 / 95
Inviscid instability of parallel flows General results – Base flow Ignoring viscous effects, and with unit scales L , V and ρ V 2 , the governing equations are the Euler equations div U = 0 , ∂ t U + ( U · grad ) U = − grad P . These equations have the family of base solutions U ( x , t ) = U ( y ) e x , P ( x , t ) = P , corresponding to parallel flow. 19th September 2015 27 / 95
Inviscid instability of parallel flows General results – Linearized stability problem Linearized equations for the perturbed base flow U + u , P + p div u = 0 , ( ∂ t + U ∂ x ) u + v ∂ y U e x = − grad p . Thanks to the translational invariance in t , x and z , the solution can be sought in the form of normal modes such as u ( y ) e i ( k x x + k z z − ω t ) + c . c ., u ( x , t ) = ˆ whose amplitudes ˆ u ( y ) , ... satisfy the homogeneous system: i k x ˆ u + ∂ y ˆ v + i k z ˆ w = 0 , i ( k x U − ω )ˆ u + ∂ y U ˆ v = − i k x ˆ p , i ( k x U − ω )ˆ v = − ∂ y ˆ p , i ( k x U − ω ) ˆ w = − i k z ˆ p . with the conditions that the perturbations fall off for y → ±∞ or that v ( y 1 ) = ˆ ˆ v ( y 2 ) = 0 at impermeable walls. 19th September 2015 28 / 95
Inviscid instability of parallel flows General results – Dispersion relation The above system can formally be written as the generalized eigenvalue problem L φ = ω M φ, where φ = (ˆ u , ˆ v , ˆ w , ˆ p ) and L , M linear differential operators. This problem has a nonzero solution φ only if the operator L − ω M is noninvertible, i.e. , if for a given wave number the frequency ω is an eigenvalue. This condition can be written formally as D ( k , ω ) = 0 , which is the dispersion relation of perturbations of infinitesimal amplitude. 19th September 2015 29 / 95
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