Hydrodynamic stability Jan Pralits Department of Chemical, Civil and Environmental Engineering University of Genoa, Italy jan.pralits@unige.it July 8-10, 2013 Corso di dottorato in Scienze e Tecnologie per l’Ingegneria (STI): Fluidodinamica e Processi dell’Ingegneria Ambientale (FPIA) Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 1 / 77
Outline Outline stability analysis Topic : Hydrodynamic stability Hours : 10h Content : 1 Introduction 2 Definitions 3 Modal analysis (2h) 4 Nonmodal analysis (2.5h) 5 Optimal perturbations (Constrained optimization) (2.5h) Exercises : (3h) 6 Aim : Overview of main concepts; Provide you with tools and let you test them Book : Schmid P. J. & Henningson D. S., Stability and Transition in Shear Flows , Springer Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 2 / 77
Examples Poiseuille flow The evolution of the linearized equations give us the dynamics of infinitesimal perturbations, potentially leading to transition. Q1 : What is the behaviour for t → ∞ ? A1 : Modal analysis will give the answer. Q2 : How large can the amplification be for finite t ? A2 : Nonmodal analysis will give the answer. Poiseuille: Re=10000 Poiseuille: Re=10000 Poiseuille: Re=10000 2 10 1 0.6 |v(y,0)| d/dt(ln E) |v(y,T)| λ 0.5 0.5 0.4 d/dt(ln(E)) 0.3 1 0 E 10 y 0.2 0.1 −0.5 0 0 10 −0.1 −1 0 20 40 60 80 100 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 t |v(y,t)| t Perturbations Energy Growth rate Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 3 / 77
Examples Aeroelasticity Q1 : What is the behaviour for finite and infinite t ? L A1 : Answer from nonmodal and modal stability analysis. U ∞ K Θ Θ M a.c. Q2 : Can we determine an optimal way to control c.g. instabilities ? e K w w A2 : Constrained optimization is a useful tool. δ Optimal perturbations ↔ Nonmodal growth 2 10 1 10 ||w|| 0 10 −1 10 0 2 4 6 8 10 12 14 16 18 20 t Movie 2 Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 4 / 77
Introduction Hydrodynamic stability Hydrodynamic stability theory is concerned with the respons of laminar flow to a disturbance of small or moderate amplitude. The flow is generally defined as Stable : If the flow returns to its original laminar state. Unstable : If the disturbance grows and causes the laminar flow to change into a different state. Stability theory deals with the mathematical analysis of the evolution of disturbances superposed to a laminar base flow. In many cases one assumes the disturbances to be small so that further simplifications can be justified. In particular, a linear equation governing the evolution of disturbances is desirable. As the disturbance velocities grow above a few % of the base flow, nonlinear effects become important and linear equations no longer accurately predict the disturbance evolution. Although the linear equations have a limited region of validity they are important in detecting physical growth mechanisms and identifying dominant disturbance types. Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 5 / 77
Introduction Reynolds pipe flow experiment (1883) Original 1883 appartus Dye into center of pipe Critical Re = 13 . 000 Lower today due to traffic Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 6 / 77
Introduction History of shear flow stability and transition Reynolds pipe flow experiment (1883) Rayleigh’s inflection point criterion (1887) Orr (1907) Sommerfeld (1908) viscous eq. Heisenberg (1924) viscous channel solution Tollmien (1931) Schlichting (1933) viscous Boundary Layer solution Schubauer & Skramstad (1947) experimental TS-wave verification Klebanoff, Tidstr¨ om & Sargent (1962) 3D breakdown Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 7 / 77
Introduction Routes to transition : highly dependent on Tu ¡ Tu ∼ 0 . 1% ¡ ¡ ¡ ¡ Tu ∼ 10% ¡ ¡ Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 8 / 77
Introduction More examples of instabilities I Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 9 / 77
Introduction More examples of instabilities II Movie 2 Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 10 / 77
Definitions Disturbance equations I ∂ u i ∂ u i − ∂ p + 1 Re ∇ 2 u i = − u j ∂ t ∂ x j ∂ x i ∂ u i = 0 ∂ x i u 0 u i ( x i , 0) = i ( x i ) ¡ u i ( x i , t ) = 0 on solid boundaries U ∗ ∞ δ ∗ /ν ∗ Re = U i + u ′ u i = decomposition i P + p ′ p = Introduce decomposition, drop primes, subtract eq’s for { U i , P } ∂ u i ∂ u i ∂ U i − ∂ p + 1 ∂ u i Re ∇ 2 u i − u j = − U j − u j ∂ t ∂ x j ∂ x j ∂ x i ∂ x j ∂ u i = 0 ∂ x i Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 11 / 77
Definitions Disturbance equations II ∂ u i ∂ u i − ∂ p + 1 Re ∇ 2 u i = − u j ∂ t ∂ x j ∂ x i ∂ u i = 0 ∂ x i u 0 u i ( x i , 0) = i ( x i ) ¡ u i ( x i , t ) = 0 on solid boundaries U ∗ ∞ δ ∗ /ν ∗ Re = U i + u ′ u i = decomposition i P + p ′ p = Introduce decomposition, drop primes , linearize ∂ u i ∂ u i ∂ U i − ∂ p + 1 ∂ u i Re ∇ 2 u i − u j = − U j − u j ∂ t ∂ x j ∂ x j ∂ x i ∂ x j ∂ u i = 0 ∂ x i Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 12 / 77
Definitions Disturbance equations III ∂ u i ∂ u i − ∂ p + 1 Re ∇ 2 u i = − u j ∂ t ∂ x j ∂ x i ∂ u i = 0 ∂ x i u 0 u i ( x i , 0) = i ( x i ) ¡ u i ( x i , t ) = 0 on solid boundaries U ∗ ∞ δ ∗ /ν ∗ Re = U i + u ′ u i = decomposition i P + p ′ p = Linearised Navier-Stokes equations , ∂ u i ∂ u i ∂ U i − ∂ p + 1 Re ∇ 2 u i = − U j − u j ∂ t ∂ x j ∂ x j ∂ x i ∂ u i = 0 ∂ x i Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 13 / 77
Definitions Stability definitions I E ( t ) = 1 � u i ( t ) u i ( t ) d Ω 2 Ω E ( t ) Stable : lim E (0) → 0 t →∞ Conditionally stable : ∃ δ > 0 : E (0) < δ ⇒ stable Globally stable : Conditionally stable with δ → ∞ Globally stable and dE Monotonically stable : dt ≤ 0 ∀ t > 0 Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 14 / 77
Definitions Critical Reynolds numbers Re E : Re < Re E flow monotonically stable Re G : Re < Re G flow globally stable Re L : Re < Re L flow linearly stable ( δ → 0) ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Initial energy E vs the Reynolds number Re Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 15 / 77
Definitions Critical Reynolds numbers Flow Re E Re G Re tr Re L Hagen-Poiseuille 81.5 − 2000 ∞ Plane Poiseulle 49.6 − 1000 5772 Plane Couette 20.7 125 360 ∞ Critcial Reynolds numbers for a number of wall-bounded shear flows compiled from the literature. Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 16 / 77
Definitions Reynolds-Orr equation Scalar multiplication of linearised Navier-Stokes equations with u i ∂ u i ∂ U i − 1 ∂ u i ∂ u i u i = − u i u j ∂ t ∂ x j Re ∂ x j ∂ x j ∂ � − 1 2 u i u i U j − 1 2 u i u i u j − u i p δ ij + 1 ∂ u i � + Re u i ∂ x j ∂ x j integrate in space (Ω) , vanishing perturbation at the boundaries ⇒ dE ∂ U i d Ω − 1 ∂ u i ∂ u i � � = − u i u j d Ω dt ∂ x j Re ∂ x j ∂ x j Ω Ω Nonlinear terms have dropped out RHS : exchange of energy with the base flow and energy dissipation due to viscosity Theorem : Linear mechanisms required for energy growth Proof : 1 dE dt independent of disturbance amplitude E Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 17 / 77
Linear Inviscid Analysis Inviscid Analysis Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 18 / 77
Linear Inviscid Analysis Parallel shear flows : U i = U ( y ) δ 1 i I ∂ u ∂ t + U ∂ u − ∂ p ∂ x + vU ′ = ∂ x ∂ v ∂ t + U ∂ v − ∂ p ∂ x + = ∂ y ∂ w ∂ t + U ∂ w − ∂ p ∂ x + = ∂ z ∂ x + ∂ v ∂ u ∂ y + ∂ w = 0 ∂ z Initial conditions : { u , v , w } ( x , y , z , t = 0) = { u 0 , v 0 , w 0 } ( x , y , z ) Boundary conditions : v ( x , y = y 1 , z , t ) · n = 0 solid boundary 1 v ( x , y = y 2 , z , t ) · n = 0 solid boundary 2 Jan Pralits (University of Genoa) Hydrodynamic stability July 8-10, 2013 19 / 77
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