Primary, secondary instabilities and control of the rotating-disk boundary layer

Transcription

1 Primary, secondary instabilities and control of the rotating-disk boundary layer Benoît PIER Laboratoire de mécanique des fluides et d acoustique CNRS Université de Lyon École centrale de Lyon, France LMS/Leicester Workshop on the Stability and Transition of Rotating Boundary-Layer Flows 8 May 2009 Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 1 Introduction 2 Natural flow Base flow Linear stability analysis Nonlinear analysis Secondary stability analysis 3 Open-loop control General strategy Application to the rotating-disk boundary layer 4 Conclusions and outlook Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

2 Some flow visualisations Rotating bodies Swept wings (Kohama 2000) (Bippes 1999) Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 The rotating-disk flow Introduction no intrinsic geometrical length scale self-similar basic flow boundary layer thickness δ = ν ω disk ru(z) U(r,z) = rv (z) W (z) turbulent régime laminar flow (Kohama 1984) transition at r 500δ Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

3 Typical 3D boundary layers rotating disk swept wing Common features: crossflow component near the wall inflection point strong inviscid instability secondary instabilities growth and saturation of crossflow vortices transition Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Introduction Known features and questions Laminar turbulent transition, near R 500 R < R sc 280 stability: all perturbations are damped R sc < R < R ca 500 convective instability: spatial exponential amplification of external perturbations R > R ca 500 absolute instability: perturbations grow at fixed r self-sustained finite-amplitude fluctuations nonlinear wavetrains secondary instabilities Precise dynamics in transition region? Spatial distribution of weakly and fully nonlinear fluctuations? Characteristic frequencies? Sensitivity to external noise, to disk roughness? Controllability of natural dynamics? Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

4 Self-similar laminar basic flow no characteristic length scale U(r,z) = ru(z) rv (z) W (z) constant boundary layer thickness δ = ν Ω Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Quasi-parallel flow assumption Base flow Weak radial development of basic flow 1 R = boundary layer thickness typical radii of interest 1 Transition occurs at R 500 Local properties at given radial location R are then derived from RU(z) U(z;R) = RV (z) W (z) 3D flow, homogenous in θ and r R effective local Reynolds number Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

5 Local linear dispersion relation Separate total flow fields into basic and perturbation quantities as U(z;R) + u(r,θ,z,t). Linearize governing equations and write small-amplitude perturbations in normal-mode form as u(r,θ,z,t) = u l (z)exp i(αr + βθ ωt). Solution of eigenvalue problem leads to ω = Ω l (α,β;r) with ω, α complex, β integer, R real. Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Temporal growth rate α real Natural flow Linear stability analysis Isolevels Ω l i = 0, 0.5,..., 3.5 ½ ¼ Ê ¼¼ ½¼¼ ¼ Ê ¼¼ ¼ ¼ ¼º¾ ¼º Ê ¼¼ Ê ¼¼ Instability whenever Ω l i > 0 ¼º «Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

6 Local absolute instability analysis (Lingwood 1995) Absolute frequency ω 0 is defined as the frequency observed at a fixed spatial location in the long-time linear response to an initial impulse. Vanishing radial group velocity condition: ω 0 (β;r) = Ω l (α 0,β;R) with Ω l α = 0 ω 0,i < 0 convective instability perturbations propagate radially outwards ω 0,i > 0 absolute instability perturbations grow at fixed radial location Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Absolute growth rate ω 0,i (β; R) Linear stability analysis β R Onset of absolute instability at R ca = and β ca = 68. Critical absolute frequency ω ca = Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

7 Response to localized impulse transient response final state (Lingwood 1996) amplified wavepacket is advected outwards in a spiralling trajectory inner boundary of wavepacket asymptotically reaches a critical radius turbulent régime prevails beyond this radius without ext. perturbation Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 However... Natural flow Linear stability analysis R ca closely corresponds to experimentally observed transition radius However: Fully linearized dynamics does not display global instability DNS, Davies & Carpenter (2002) Analytic derivation, Peake & Garrett (2002) Nonlinear régime must be accounted for to understand self-sustained behaviour The rotating disk configuration displays all the desirable features required by the theory of elephant nonlinear global modes (Pier, Huerre & Chomaz, 2001) Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

8 Elephant nonlinear global mode basic advection CU R ca AU Stationary front at transition station CU/AU generates downstream propagating nonlinear wavetrain tunes entire system to global frequency ω ca 0 = ω 0 (R ca ) with ω 0,i (R ca ) = 0 AU region is a sufficient condition for nonlinear global instability Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Nonlinear analysis Predicted behaviour of the rotating disk flow in central CU region R < R ca unperturbed basic flow at marginal AU station R = R ca stationary front of frequency ω0 ca 50.5 in outer AU region R > R ca finite-amplitude saturated spiral vortices Questions: Do saturated waves exist in this configuration? Why is a turbulent state observed instead? What about secondary instabilities of the saturated vortices? R ca Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

9 Local saturated crossflow vortices Whenever Ω l i (α,β;r) > 0 linear exponential temporal growth of spatially periodic perturbations nonlinear quadratic interactions produce higher harmonics amplitude saturation, finite-amplitude periodic wavetrain of the form u(r,θ,z,t) = n u n (z)expni(αr + βθ ωt) real nonlinear frequency ω Nonlinear dispersion relation ω = Ω nl (α,β;r) Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Nonlinear waves at β = 68 Nonlinear analysis α Nonlinear frequency Ω nl (α,β;r) R Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

10 Structure of finite-amplitude waves At R = 500, β = 68, α = 0.35 ω = 50.5 z 3 2 azimuthal velocity field over 2 wavelengths new inflection points further instabilities v 1 2 Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Secondary stability analysis Secondary stability analysis Primary finite-amplitude equilibrium solution u(r,θ,z,t) = n u n (z)expin(αr + βθ ωt) is 2π-periodic in φ αr + βθ ωt. Secondary (in)stability by Floquet theory perturbation in normal-mode form ( ) û(r,θ,z,t) = û n (z)expin(αr + βθ ωt) exp i(ˆαr + ˆβθ ˆωt) n ˆα secondary radial wavenumber (complex) ˆβ secondary azimuthal modenumber (integer) Secondary dispersion relation: ˆω = ˆΩ l (ˆα, ˆβ;α,β;R) Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

11 Secondary absolute instability Primary nonlinear structure affected by secondary disturbances only for secondary absolute instability. Secondary absolute frequency ˆω 0 and wave number ˆα 0 are obtained by a pinching condition in the complex ˆα-plane (Brevdo & Bridges 1996): ˆω 0 (ˆβ;α,β;R) ˆΩ l (ˆα 0, ˆβ;α,β;R) with ˆα 0 defined by ˆΩ l ˆα (ˆα 0, ˆβ;α,β;R) = 0 Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Natural flow Secondary absolute growth rate Secondary stability analysis ˆω 0,i Im ˆω 0 (ˆβ;α,β;R) At onset of primary nonlinearity: R = 508, α = 0.35, β = 68: ˆω 0,i ˆβ Strong secondary absolute instability! Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

12 Revised self-sustained behaviour in central CU region R < R ca unperturbed basic flow at marginal AU station R ca 507 stationary front in outer AU region R > R ca front generates nonlinear spiral vortices that are already absolutely unstable Transition to turbulent state immediately occurs at R = R ca R ca Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Goal and method Open-loop control General strategy Delay transition to larger radius reduce energy losses, noise... Low-tech method usable in practical situations: No real-time computations No real-time measurements No closed-loop control methods Apply localized perturbations in the BL upstream of transition Modify the naturally selected flow behaviour Take advantage of primary BL instabilities to control secondary instabilities Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

13 Spatial response to localized harmonic forcing One-dimensional spatially developing system (eg CGL equation) ω f α nl+ α l α l+ α l+ X f X nl X n X < X f upstream exponential decay X = X f localized forcing of frequency ω f X f < X < X nl downstream exponential growth X = X nl response reaches finite amplitude X nl < X < X n finite-amplitude response Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Open-loop control General strategy Forced response vs. self-sustained oscillations X f X nl α nl+ α l α l+ α l+ α nl+ source of self-sustained oscillations located at X ca front tunes entire system α l α l+ X ca Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

14 Control of natural oscillations Numerical simulation of 1D complex Ginzburg Landau equation with spatially varying coefficients [B. Pier, Proc. R. Soc. Lond. A, 2003] Natural oscillations at ω0 ca = 0.4 Forcing at ω f = 1 switched on at t = 0 t X f X Effective control: When forced response reaches finite amplitude upstream of the front, this front is overwhelmed and the intrinsic oscillations are replaced by the forced reponse. X ca Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Open-loop control Application to the rotating-disk boundary layer Control strategy applied to the rotating-disk flow Modify primary nonlinear state to avoid secondary instability. The transition station R ca = 507 acts as a source for the naturally selected spiral vortices. Perturb this source to replace the natural state by spiral vortices of our own choice. Apply localized periodic forcing at R f in the convect. unstable region: linear spatial response for R f < R < R nl, nonlinear spatial response for R > R nl, control dominates whenever R nl < R ca. Available control parameters: frequency ω f azim. modenumber β f. Careful choice of ω f and β f delays secondary instability. Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

15 Secondary absolute instability features Secondary absolute growth rate at R = 500 by Floquet analysis of primary NL spiral vortice [α: radial wavenumber β: azimuthal modenumber] β naturally selected vortices: β = 68, α = 0.35, ω = 50.5 ˆω 0,i = ˆω 0,i < 0 40 ˆω 0,i > 0 convective absolute α control at: β = 34, α = 0.57, ω = 46 ˆω 0,i = 1.06 Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Transition delay Open-loop control Application to the rotating-disk boundary layer Primary finite-amplitude spiral waves follow the nonlinear spatial branch α nl (R;ω f,β f ), determined by forcing frequency ω f and modenumber β f. Evolution of secondary absolute growth rate ˆω max 0,i along these branches ˆω 0,i max 1.5 ω f = ω0 ca,β f = β0 ca 1.0 ω f = 50, β f = ω f = 40, β f = 30 ω 0.0 f = 45, β f = R Onset of sec. absolute instability may be postponed to nearly R = 600. [B. Pier, J. Eng. Math. 2007] Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38

16 Transition delay Apply harm. forcing at R f to replace intrinsic behaviour by spatial response. R ca R ca R f R nl R Choose forcing characteristics to delay sec. inst. (and transition) to R. Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38 Conclusions Conclusions and outlook Complete map obtained of primary and secondary stability properties Control strategy: Apply localized harmonic forcing upstream of transition natural behaviour suppressed, entire system tuned to ext. forcing secondary instability delayed by about 100 boundary layer units Low-resources control: control parameters determined only once, no real-time measurements or computations Low-cost control: by exponential growth of spatial response in central region Benoît PIER (LMFA, Lyon) Rotating-disk instabilities and control Leicester, 8 May / 38