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ROUTES TO TRANSITION IN SHEAR FLOWS
ROUTES TO TRANSITION IN SHEAR FLOWS
Alessandro Bottaro
with contributions from:D. Biau, B. Galletti, I. Gavarini and F.T.M. Nieuwstadt
Osborne Reynolds, 1842-1912
“An experimental investigation of the circumstances which determinewhether motion of water shall be direct or sinuous and of the law ofresistance in parallel channels”, Royal Society, Phil. Trans. 1883
`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.'
Retrans ≈ 13000
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 Erklaerungder 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
(!!)Recrit → ∞
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 (Recrit) and experimentals results (Retrans)
Poiseuille Couette Hagen-Poiseuille Square ductRecrit 5772 ∞ ∞ ∞
Retrans ~2000 ~400 ~2000 ~2000
THE TRANSITION PROCESS
Receptivity phase: the flow filters environmental disturbances
Initial phaseROUTE 1: TRANSIENT GROWTHROUTE 2: EXPONENTIAL GROWTH
Late, nonlinear stages of transition
SMALL AMPLITUDE DISTURBANCESLinearized Navier-Stokes equations[to make things simple: cartesian coordinates, U=U(y)]
with homogeneous boundary conditions for (u, v, w) at y=±h
Goal: SPATIAL EVOLUTION OF DISTURBANCES
wp1Uww
vp1Uvv
up1'vUUuu
0wvu
zxt
yxt
xxt
zyx
∆ν+ρ
−=+
∆ν+ρ
−=+
∆ν+ρ
−=++
=++
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 boundarylayer. Free-stream speed: 2 [m/s], free-stream turbulence level: 6%
INVISCID ANALYSIS: the physics suggests to employ different length and velocity scales along different directions (introducing a parameter ε << 1)
so that the steady inviscid equations to leading order are:
x → h/ε y, z → h
U, u → Umax v, w → ε Umax
p → ρ (ε Umax)2
ROUTE 1: TRANSIENT GROWTH
zx
yx
x
zyx
pUw
pUv
0'vUUu
0wvu
−=
−=
=+
=++
Look for similarity solutions of the form:
By substituting into the equations it is easy to find thatλ = 1 is a solution, so that:
_
.c.ce)y(px)z,y,x(p
.c.ce)y(wx)z,y,x(w
.c.ce)y(vx)z,y,x(v
.c.ce)y(ux)z,y,x(u
zi2
zi1
zi1
zi
+=
+=
+=
+=
β−λ
β−λ
β−λ
βλ
zi0y0
zi0
zi0
eU
'Uvvi)z,y,x(w
e)y(v)z,y,x(v
e)y(vU
'Ux)z,y,x(u
β
β
β
−
β=
=
−= ALGEBRAIC GROWTH
ROUTE 1: TRANSIENT GROWTH
ROUTE 1: TRANSIENT GROWTH
VISCOUS ANALYSIS
maxUh
ε
• Classical approach: use ( )
as scales and find the Orr-Sommerfeld/Squire
system, with and
• Two-scale approach: use ( ) as time scale
and find a reduced Os/Squire system
2max
maxmax U,
Uh,h,U ρ
ν=
hURe max
( )[ ]( ) v'UReU
0v''UReU
z1
xt
x1
xt
∂−=η∆−∂+∂
=∂−∆∆−∂+∂−
−
wu xz ∂−∂=η
( )[ ]( ) v'UU
0v''UU
z2xt
x22xt
∂−=η∆−∂+∂=∂−∆∆−∂+∂
ROUTE 1: TRANSIENT GROWTH
[ ] [ ] .c.ce)y(ˆ,v,v )tzx(i )p()p( +η=η ω−β+α β,Normal modes: ℜ∈ω,
Poiseuille flow: eigenmodes full system vs parabolic model
Re = 2000, ω = 0, β = 1.91 Re = 2000, ω = 0.3, β = 1.91
Parabolic: αp = α Re, ωp = ω Re Full: α, ω
ROUTE 1: TRANSIENT GROWTH
“Optimal” disturbances: most dangerous initial conditions at x = 0, i.e. those that maximize the output disturbance energy
dzdy)wwvvuu(
dzdywwvvuu
)0(E)x(E)x(G
000S 000
xS
++
++==
∫∫∫∫
In boundary layer scalings it is obvious that, sinceu = ❡ (Umax) and (v,w) = ❡ (Umax/Re), G is rendered maximum when u0 = 0. Then:
∫∫∫∫
+≈
S 0000
S x2
dzdy)wwvv(dzdyuu
ReG
ROUTE 1: TRANSIENT GROWTH
Decompose the generic disturbance vector q as a sum of normal modes:
Then:
)tzx(inn
nnne)y(ˆ)t,z,y,x( ω−β+α= qq κ
mnmn
mnmn21
1mmnmnn
1
1mm
xixinn
2
B)x(A
Redy]wwvv[
dyueuRe)x(G
mn
κκκκ
κκ
κκ=
+≈
∫
∫
−
−
α+α−
Rayleigh quotient. The largest gain at each x is the
largest eigenvalue solution of2Reλ
mnm2mnm BReλ)x(A κκ =
ROUTE 1: TRANSIENT GROWTH
Optimal growth, Re = 1000, ω = 0, β = 1.91 (Poiseuille flow)Re = 500, ω = 0, β = 1.58 (Couette flow)
ROUTE 1: TRANSIENT GROWTH
Iso-G, Poiseuille flow, Re = 2000
ROUTE 1: TRANSIENT GROWTH
Optimal growth, Poiseuille flow, Re = 2000 & 5000, ω = 0, β = 1.91
Viscous
Inviscid
Viscous
Analytical
ROUTE 1: TRANSIENT GROWTH
Optimal symmetric disturbance and optimal streak at xopt,Poiseuille flow, Re = 2000, ω = 0, β = 1.91
ROUTE 1: TRANSIENT GROWTH
Optimal antisymmetric disturbance and optimal streak at xopt,Poiseuille flow, Re = 2000
ROUTE 1: TRANSIENT GROWTH
The accuracy of the parabolic system degrades with ω
Percentage error, , Poiseuille flow, Re = 2000elliptparab GG −
elliptG
ROUTE 1: TRANSIENT GROWTH
Optimal disturbances, spatial results ...
… vs corresponding temporal results
t
ROUTE 1: TRANSIENT GROWTH
Square duct, exit result, no optimization
ROUTE 1: TRANSIENT GROWTH
Square duct
Optimal inlet flow Outlet streaks
ROUTE 1: TRANSIENT GROWTH
Square duct
E
x/Re
x2
Eu
Evw
ROUTE 1: TRANSIENT GROWTH
Optimal perturbations
Gmax/Re2 xopt/Re
Poiseuille 2.41*10-4 0.057 βopt = 1.91
Couette 3.39*10-4 0.073 βopt = 1.58
Pipe 1.03*10-4 0.033 mopt = 1
Square duct 1.12*10-4 0.039
ROUTE 2: EXPONENTIAL GROWTH
ROUTE 2: EXPONENTIAL GROWTH
Preliminary observation: eigenvalues of the OS/Squire system are very sensitive to operator perturbations E
{ }ε≤+Λ∈α∈α=Λε EthatsuchEwith),EL(:C)L(
ROUTE 2: EXPONENTIAL GROWTH
Consider a very particular operator perturbation, a distortion of the mean flow U(y) (induced by whatever environmental forcing) →
OS equation: L (U, α; ω, β, Re) v = 0
With a base flow variation δU(y):
δL v + L δv = 0
0vLvLvULU =δ+
α∂∂
δα+∂∂
δ
{ }ε≤δδ+Λ∈α∈α=Λδ Uwith)],UU(L[:C)L( refU
ROUTE 2: EXPONENTIAL GROWTH
Projecting on a, eigenfunction of the adjoint system (L*a=0) we find
and hence,
In practice, for each eigenvalue αn we can tie the base flow variation δU to the ensuing variation δα via a sensitivity function GU
0vLavLavULUa =δ⋅+
α∂∂
⋅δα+∂∂
δ⋅
∫−
δ==
α∂∂⋅
∂∂δ⋅
−=δα1
1U dyUG...
vLa
vULUa
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
ROUTE 2: EXPONENTIAL GROWTH
drUGr U
1
0δ=δα ∫
Corresponding ∞ norm of rGu. Modes arranged in order of increasing |αi|
ROUTE 2: EXPONENTIAL GROWTH
SENSITIVITY FUNCTIONS
Mode 22 (solid); mode 24 (dashed);103*mode 1 (dash-dotted)
ROUTE 2: EXPONENTIAL GROWTH
“Optimization”
Look for optimal base flow distortion of given norm ε, so that the growth rate of the instability (-αi) is maximized:
Necessary condition is that:
ε−−γ+α=α ∫
−
1
1
2refii dy)UU(Min)(Min
0dyU)UU(21
1refi =
δ−γ+δα ∫
−
ROUTE 2: EXPONENTIAL GROWTH
Employing the previous result:
A simple gradient algorithm can be used to find the new base flow that maximizes the growth rate, for any αn and for any given base flow distortion norm ε:
with
[ ] 0dyU)UU(2)GIm(1
1refU =δ−γ+∫
−
[ ] 21
1
1
2)i(U
)i(
4
)GIm(
ε=γ
∫−µ
[ ])UU(2)GIm(UU ref)i()i()i(
U)i()1i( −γ+ϑ+=+
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 αi of mode 22.
ROUTE 2: EXPONENTIAL GROWTH
U
Optimally distorted base flow vs Hagen-Poiseuille flow.The curve of (U’/r)’/20 indicates an inflectional instability
ROUTE 2: EXPONENTIAL GROWTH
Re=3000
Re=2300Re=1800
Re=1760
Growth rate as function of ω for m = 1 and ε = 10-5
ROUTE 2: EXPONENTIAL GROWTH
ε=5*10-5
ε=10-5ε=5*10-6
ε=2*10-6
Neutral curves for m = 1 and ε = 10-5. Symbols give Recrit
FULL NONLINEAR SIMULATIONS
dtrdrdeˆT21)x(E
2T 2
0
1
0
)tkj(i
mj nk
)n,m(dis ϑ= ∫ ∫ ∫∑ ∑
+τ
τ
πω−ϑ
±= ±=u
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 0.002. Re = 3000, m = 1, n = 1, ε = 2.5*10-5.
CONCLUSIONS
Transient growth in space can be described by a parabolic systemTransient 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 two-way (possibly unphysical?) coupling between OS and Squire equations is not allowed)
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
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
x
R1θ
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