ROUTES TO TRANSITION IN SHEAR FLOWS

Transcription

1 ROUTES TO TRANSITION IN SHEAR FLOWS Alessandro Bottaro with contributions from: D. Biau, B. Galletti, I. Gavarini and F.T.M. Nieuwstadt ROUTES TO TRANSITION IN SHEAR FLOWS

2 Osborne Reynolds, An experimental investigation of the circumstances which determine whether motion of water shall be direct or sinuous and of the law of resistance in parallel channels, Royal Society, Phil. Trans. 1883

3 `the colour band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of coloured water... On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls, showing eddies.' Re trans 13000

4 Hydrodynamic stability theory followed a few years later: W. M F.Orr, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Proc. Roy. Irish Academy, 1907 A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklaerung der turbulenten Fluessigkeitsbewegungen, Proc. 4th International Congress of Mathematicians, Rome, 1908 Hints on the solution of the stability equations for the flow in a pipe arrived only much later (C.L. Pekeris, 1948), just to show that Re crit (!!)

5 STILL TODAY, TRANSITION IN SHEAR FLOWS IS STILL NOT FULLY UNDERSTOOD. For the simplest parallel flows there is poor agreement between predictions from the classical linear stability theory (Re crit ) and experimentals results (Re trans ) Poiseuille Couette Hagen-Poiseuille Square duct Re crit 5772 Re trans ~2000 ~400 ~2000 ~2000

6 THE TRANSITION PROCESS Receptivity phase: the flow filters environmental disturbances Initial phase ROUTE 1: TRANSIENT GROWTH ROUTE 2: EXPONENTIAL GROWTH Late, nonlinear stages of transition

7 SMALL AMPLITUDE DISTURBANCES Linearized Navier-Stokes equations [to make things simple: cartesian coordinates, U=U(y)] u u v x t t v y Uu Uv + x x w + = vu ' = 1 w t + Uw x = p z + ν w ρ with homogeneous boundary conditions for (u, v, w) at y=±h Goal: SPATIAL EVOLUTION OF DISTURBANCES z = 0 1 p ρ y 1 p ρ x + ν v + ν u

8 ROUTE 1: TRANSIENT GROWTH ROUTE 1: TRANSIENT GROWTH THE MECHANISM: a stationary algebraic instability exists in the inviscid system ( lift-up effect). In the viscous case the growth of the disturbance energy is hampered by diffusion transient growth P.H. Alfredsson and M. Matsubara (1996); streaky structures in a boundary layer. Free-stream speed: 2 [m/s], free-stream turbulence level: 6%

9 ROUTE 1: TRANSIENT GROWTH INVISCID ANALYSIS: the physics suggests to employ different length and velocity scales along different directions (introducing a parameter ε << 1) x h/ε y, z h U, u U max v, w εu max p ρ(ε U max ) 2 so that the steady inviscid equations to leading order are: u x Uu + v x y + w z + vu' = 0 = 0 Uv Uw x x = p = p y z

10 ROUTE 1: TRANSIENT GROWTH Look for similarity solutions of the form: u(x, y,z) = x λ û(y) e iβz + c.c. v(x, y,z) = x λ 1 vˆ(y) e iβz + c.c. w(x, y,z) = x λ 1 ŵ(y) e iβz + c.c. p(x, y,z) = x λ 2 pˆ(y) e iβz + c.c. By substituting into the equations it is easy to find that λ = 1 is a solution, so that: U' iβz u(x, y, z) = x_ vˆ 0 (y) e ALGEBRAIC GROWTH U v(x, y, z) = vˆ 0 i w (x, y, z) = β (y) vˆ e 0 y iβz vˆ 0 U' U e iβz

11 ROUTE 1: TRANSIENT GROWTH VISCOUS ANALYSIS h 2 Classical approach: use ( U ) max, h,, ρumax Umax as scales and find the Orr-Sommerfeld/Squire max system, with Re = and U Two-scale approach: use ( ) as time scale ε h U max and find a reduced Os/Squire system ν h [( 1 ) ] t + U x Re U'' x ( 1 + U Re ) η = U' v t x η = z u x w [( t + U x 2 ) 2 U'' x ] ( + U ) η = U' v t x 2 z z v = 0 v = 0

12 ROUTE 1: TRANSIENT GROWTH Normal modes: i ( α(p) x+βz ω(p) t) [ v, η] = [ vˆ, ηˆ ](y)e + c.c., β,ω R Poiseuille flow: eigenmodes full system vs parabolic model Re = 2000, ω = 0, β = 1.91 Re = 2000, ω = 0.3, β = 1.91 Parabolic: α p = α Re, ω p = ω Re Full: α, ω

13 ROUTE 1: TRANSIENT GROWTH Optimal disturbances: most dangerous initial conditions at x = 0, i.e. those that maximize the output disturbance energy G(x) = E(x) E(0) = (u S uu + vv + ww u + v S v 0 + w x 0 w dy 0 ) dz dy dz In boundary layer scalings it is obvious that, since u = (U max ) and (v,w) = (U max /Re), G is rendered maximum when u 0 = 0. Then: G Re 2 (v v S uu + x w dy dz w S ) dy dz

14 ROUTE 1: TRANSIENT GROWTH Decompose the generic disturbance vector q as a sum of normal modes: q(x, y,z, t) = κ qˆ n n (y) e i( α n x+β n z ω n t) Then: G(x) Re κ κ n n û [vˆ n n e iα vˆ m n x+ iα + ŵ Rayleigh quotient. The largest gain at each x is the λ largest eigenvalue 2 solution of Re m n x ŵ û m m κ ] κ m m dy dy = Re 2 κ n A κ n nm B (x) κ nm κ m m A nm (x) κ = m λ Re 2 B nm κ m

15 ROUTE 1: TRANSIENT GROWTH Optimal growth, Re = 1000, ω = 0, β = 1.91 (Poiseuille flow) Re = 500, ω = 0, β = 1.58 (Couette flow)

16 ROUTE 1: TRANSIENT GROWTH Iso-G, Poiseuille flow, Re = 2000

17 ROUTE 1: TRANSIENT GROWTH Optimal growth, Poiseuille flow, Re = 2000 & 5000, ω = 0, β = 1.91 Inviscid Viscous Viscous Analytical

18 ROUTE 1: TRANSIENT GROWTH Optimal symmetric disturbance and optimal streak at x opt, Poiseuille flow, Re = 2000, ω = 0, β = 1.91

19 ROUTE 1: TRANSIENT GROWTH Optimal antisymmetric disturbance and optimal streak at x opt, Poiseuille flow, Re = 2000

20 ROUTE 1: TRANSIENT GROWTH The accuracy of the parabolic system degrades with ω G Percentage error, parab G ellipt, Poiseuille flow, Re = 2000 G ellipt

21 ROUTE 1: TRANSIENT GROWTH Optimal disturbances, spatial results... vscorresponding temporal results t

22 ROUTE 1: TRANSIENT GROWTH Square duct, exit result, no optimization

23 ROUTE 1: TRANSIENT GROWTH Square duct Optimal inlet flow Outlet streaks

24 ROUTE 1: TRANSIENT GROWTH Square duct E vw E x 2 E u x/re

25 ROUTE 1: TRANSIENT GROWTH Optimal perturbations G max /Re 2 x opt /Re Poiseuille 2.41* β opt = 1.91 Couette 3.39* β opt = 1.58 Pipe 1.03* m opt = 1 Square duct 1.12*

26 ROUTE 2: EXPONENTIAL GROWTH ROUTE 2: EXPONENTIAL GROWTH Preliminary observation: eigenvalues of the OS/Squire system are very sensitive to operator perturbations E Λ ε (L) = { α C : α Λ(L + E), with E such that E ε}

27 ROUTE 2: EXPONENTIAL GROWTH Consider a very particular operator perturbation, a distortion of the mean flow U(y) (induced by whatever environmental forcing) { α C : α Λ[L(U + δu)], with δu ε} Λ δ U ( L) = ref OS equation: L (U, α; ω, β, Re) v = 0 With a base flow variation δu(y): δlv + Lδv= 0 L L δu v + δα v + L δv = U α 0

28 ROUTE 2: EXPONENTIAL GROWTH Projecting on a, eigenfunction of the adjoint system (L*a=0) we find L L a δu v+δα a v+ a Lδv = 0 U α and hence, L a δu v 1 δα = U =... = G U δu dy L a v 1 α In practice, for each eigenvalue α n we can tie the base flow variation δu to the ensuing variation δα via a sensitivity function G U

29 ROUTE 2: EXPONENTIAL GROWTH HAGEN-POISEUILLE FLOW Spectrum of eigenvalues at Re = 3000, m = 1, ω = 0.5. The circle includes the two most receptive eigenvalues

30 ROUTE 2: EXPONENTIAL GROWTH δα = 1 0 rg U δu dr Corresponding norm of rg u. Modes arranged in order of increasing α i

31 ROUTE 2: EXPONENTIAL GROWTH SENSITIVITY FUNCTIONS Mode 22 (solid); mode 24 (dashed); 10 3 *mode 1 (dash-dotted)

32 ROUTE 2: EXPONENTIAL GROWTH Optimization Look for optimal base flow distortion of given norm ε, so that the growth rate of the instability (-α i ) is maximized: Min( α i ) = Min α i + γ 1 1 (U U ref ) 2 dy ε Necessary condition is that: 1 δαi + γ 2(U Uref ) δu dy = 1 0

33 ROUTE 2: EXPONENTIAL GROWTH Employing the previous result: 1 [ Im(G U ) + 2γ(U Uref )] δu dy = 0 1 A simple gradient algorithm can be used to find the new base flow that maximizes the growth rate, for any α n and for any given base flow distortion norm ε: U (i+ 1) = U (i) + ϑ [ (i) Im(G ) 2 (U U )] (i) (i) + γ U ref with γ (i) = µ [ (i) ] 2 Im(G ) 2 4ε U

34 ROUTE 2: EXPONENTIAL GROWTH Re = 3000, m = 1, ω = 0.5. HP flow (circles), OD flow (triangles) with ε = 2.5*10-5 which minimizes α i of mode 22.

35 ROUTE 2: EXPONENTIAL GROWTH U Optimally distorted base flow vs Hagen-Poiseuille flow. The curve of (U /r) /20 indicates an inflectional instability

36 ROUTE 2: EXPONENTIAL GROWTH Re=3000 Re=1800 Re=2300 Re=1760 Growth rate as function of ω for m = 1 and ε = 10-5

37 ROUTE 2: EXPONENTIAL GROWTH ε=10-5 ε=5*10-6 ε=5*10-5 ε=2*10-6 Neutral curves for m = 1 and ε = Symbols give Re crit

38 FULL NONLINEAR SIMULATIONS

39 E (m,n) dis (x) j= ± m k= ± n 1 2T τ+ T 2π = τ uˆ e i( jϑ kωt) 2 rdrdϑdt Spatial evolution of the disturbance energy for the Fourier modes (m, n), with ω = 0.5. Initial amplitude of the (1,1) eigenmode (shown with thick blue line) is Re = 3000, m = 1, n = 1, ε = 2.5*10-5.

40 CONCLUSIONS Transient growth in space can be described by a parabolic system Transient growth is related to the non-normality of OS-Squire eigenmodes (L. N. Trefethen et al., 1993) OS eigenmodes are very sensitive to base flow variations (δu-pseudospectrum, the growth is less than for the ε-pseudospectrum since twoway (possibly unphysical?) coupling between OS and Squire equations is not allowed)

41 CONCLUSIONS Exponential growth can take place even in nominally subcritical conditions for mild distortions of the base flow Poiseuille flow, Re = 3000, ω = 0.5 Optimal distortion

42 CONCLUSIONS Transition is likely to be provoked by the combined effect of algebraic and exponentially growing disturbances, the ultimate fate of the flow being decided by the prevailing receptivity conditions of the flow

43 R 1 θ x