routes to transition in shear flowstransition in shear flows is a phenomenon still not fully...
TRANSCRIPT
![Page 1: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/1.jpg)
ROUTES TO TRANSITION IN SHEAR FLOWS
Alessandro Bottaro
with contributions from:S. Zuccher, I. Gavarini, P. Luchini and F.T.M. Nieuwstadt
![Page 2: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/2.jpg)
1. TRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between predictions from the classical linear stability theory (Recrit) and experimentals results (Retrans)
Poiseuille Couette Hagen-Poiseuille BlasiusRecrit 5772 ∞ ∞ ~ 500
Retrans ~ 1000 ~400 ~2000 ~400
![Page 3: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/3.jpg)
2. TRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD.
![Page 4: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/4.jpg)
THE TRANSIENT GROWTH
THE MECHANISM: a stationary algebraic instability exists in the inviscid system (“lift-up” effect). In the viscous case the growth of the streaks 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%
![Page 5: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/5.jpg)
Proposition:Optimal and robust control of streaks during their initial development phase, in pipes and boundary layers by
acting at the level of the disturbances (“cancellation control”) acting at the level of the mean flow (“laminar flow control”)
![Page 6: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/6.jpg)
Optimal cancellation control of streaks
Cancellation Control
Known base flow U(r); disturbance O(ε) control O(ε)
Disturbance field(system’s state)Control O(ε)
![Page 7: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/7.jpg)
Optimal laminar flow control of streaks
Laminar Flow Control
Compute base flow O(1) together with the disturbance field O(ε) control O(1)
Whole flow field(system’s state)Control O(1)
![Page 8: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/8.jpg)
OPTIMAL CONTROL: A PARABOLIC MODEL PROBLEM
ut + uux = uxx + S u(t,x): state of the systemu(0,x)=u0(x) u0(x): initial conditionu(t,0)=uw(t) uw(t): boundary controlu(t,∞)=0 S(t,x): volume control
Suppose u0(x) is known; we wish to find thecontrols, uw and S, that minimize the functional:
I(u, uw, S) = u2 dt dx + α2 uw2 dt + β2 S2 dt dx
disturbance norm energy needed to control
![Page 9: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/9.jpg)
I(u,uw,S) = u2 dt dx + α2 uw2 dt + β2 S2 dt dx
α=β=0 no limitation on the cost of the control
α and/or β small cost of employing uw and/or S is not important
α and/or β large cost of employing uw and/or S is important
![Page 10: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/10.jpg)
Optimal control
For the purpose of minimizing I, let us introduce an augmented functional L = L(u,uw,S,a,b), with a(t,x) and b(t)Lagrange multipliers, and let us minimize the new objective functional
L(u, uw, S, a, b) = I + a (ut + uux - uxx - S) dt dx +
+ b(t) [u(t,0)-uw(t)] dt
Constrained minimization of I
Unconstrained minimization of L
![Page 11: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/11.jpg)
Optimal control
Each directional derivative must independentlyvanish for a relative minimum of L to exist.
For example it must be:
termsinitialandboundarydt)0,t(ub
dtdxuauuauauu2dt)0,t(ub
dtdxuauuauuauaududIδu
dudL
xxxt
xxxxt
+δ+
+δ−δ−δ−δ=δ+
+δ−δ+δ+δ+δ=
∫
∫∫∫
∫∫
![Page 12: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/12.jpg)
Optimal control
Since δu is an arbitrary variation, the double integralvanish if and only if the linear adjoint equation
at + uax = - axx + 2u
is satisfied, together with:a(T,x)=0 terminal conditiona(t,0)=a(t,∞)=0 boundary conditions
The vanishing of the other directional derivatives is accomplished by letting:
)x,t(a)x,t(S2iff0δSdSdL
)0,t(a)t(b)t(u2iff0δududL
2
xw2
ww
=β=
==α=
![Page 13: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/13.jpg)
Optimal control
Resolution algorithm:
adjoint equationsfor dual state a
direct equationsfor state u
(using uw and S)
optimalityconditions:
uw(t)=ax(t,0)/2α2
S(t,x)=a(t,x)/2β2
convergencetest
Optimality conditions are enforced by employing a simple gradient or a conjugate gradient method
![Page 14: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/14.jpg)
ROBUST CONTROL: A PARABOLIC MODEL PROBLEM
ut + uux = uxx + S u(t,x): state of the systemu(0,x)=u0(x) u0(x): initial conditionu(t,0)=uw(t) uw(t): boundary controlu(t,∞)=0 S(t,x): volume control
Now u0(x) is not known; we wish to find the controls, uw and S, and the initial condition u0(x) that minimize the functional:
I1(u, u0, uw, S) = u2 dt dx + α2 uw2 dt + β2 S2 dt dx -
- γ2 u02 dx
i.e. we want to maximize over all u0.
![Page 15: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/15.jpg)
Robust control
This non-cooperative strategy consists in finding the worst initial condition in the presence of the bestpossible control.
Procedure: like previously we introduce a Lagrangianfunctional, including a statement on u0
L1(u, u0, uw, S, a, b, c) = I1 + a (ut + uux - uxx - S) dt dx +
b(t) [u(t,0)-uw(t)] dt + c(x) [u(0,x)-u0(x)] dx
![Page 16: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/16.jpg)
Robust control
Vanishing of yields: 2γ2u0(x)=a(0,x)
Robust control algorithm requires
alternating ascent iterations to find u0 and descent
iterations to find uw and/or S.
Convergence to a saddle point in the space of variables.
00δu
dudL
2γ2u0(x)=a(0,x)
![Page 17: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/17.jpg)
Optimal cancellation control of streaks
Optimal control of streaks in pipe flow
Spatially parabolic model for the streaks: structures elongated in the streamwise direction x
Long scale for x: R ReShort scale for r: R
Fast velocity scale for u: UmaxSlow velocity scale for v, w: Umax/Re
Long time: R Re/Umax
![Page 18: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/18.jpg)
Optimal cancellation control of streaks
Optimal control of streaks in pipe flow
with
![Page 19: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/19.jpg)
withwith
Optimal cancellation control of streaks
![Page 20: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/20.jpg)
Optimal disturbance at x=0
Initial condition for the state: optimal perturbation, i.e. the disturbance which maximizes the gain G in the absence of control.
![Page 21: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/21.jpg)
Optimal disturbance at x=0
![Page 22: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/22.jpg)
Optimal cancellation control of streaks
Optimal control
with
where is the control statement=
![Page 23: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/23.jpg)
Optimal cancellation control of streaks
Order of magnitude analysis:
increases Small
increases Flat
Equilibrium
increases Cheap
![Page 24: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/24.jpg)
Optimal cancellation control of streaks
As usual we introduce a Lagrangian functional, weimpose stationarity with respect to all independentvariables and recover a system of direct and adjointequations, coupled by transfer and optimality conditions. The system is solved iteratively; at convergence we have the optimal control.
![Page 25: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/25.jpg)
Optimal cancellation control of streaks
![Page 26: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/26.jpg)
Optimal cancellation control of streaks
![Page 27: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/27.jpg)
Optimal cancellation control of streaks
![Page 28: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/28.jpg)
Optimal cancellation control of streaks
![Page 29: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/29.jpg)
Robust cancellation control of streaks
![Page 30: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/30.jpg)
Robust cancellation control of streaks
![Page 31: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/31.jpg)
Optimal and robust cancellation control of streaks
• In theory it is possible to optimally counteract disturbancespropagating downstream of an initial point(trivial to counteract a mode)
• Physics: role of buffer streaks
• Robust control laws are available
• Next:• Feedback control, using the framework recently
proposed by Cathalifaud & Bewley (2004)
• Is it technically feasible?
![Page 32: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/32.jpg)
Optimal laminar flow control of streaks
Optimal control of streaks in boundary layer flow
Spatially parabolic model for the streaks: steady structures elongated in the streamwise direction x
Long scale for x: LShort scale for y and z: δ = L/Re
Fast velocity scale for u: U∞Slow velocity scale for v, w: U∞ /Re
![Page 33: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/33.jpg)
Optimal laminar flow control of streaks
Optimal control of streaks in boundary layer flow
with
![Page 34: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/34.jpg)
Optimal laminar flow control of streaks
::
with the gain given by:
![Page 35: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/35.jpg)
Optimal laminar flow control of streaks
Lagrangian functional:
Stationarity of Optimality system
![Page 36: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/36.jpg)
Optimal laminar flow control of streaks
__________
![Page 37: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/37.jpg)
Optimal laminar flow control of streaks
![Page 38: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/38.jpg)
Optimal laminar flow control of streaks
![Page 39: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/39.jpg)
Laminar flow control creates a thin boundary layer …
… what about the skin friction then?
Prandtl low-Re turbulent correlation:
![Page 40: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/40.jpg)
Robust laminar flow control of streaks
![Page 41: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/41.jpg)
Robust laminar flow control of streaks
![Page 42: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/42.jpg)
Optimal and robust laminar flow control of streaks
• Mean flow suction can be found to optimally damp thegrowth of streaks in the linear and non-linear regimes
• Both in the optimal and robust control case the controllaws are remarkably self-similar Bonus for applications
• No need for feedback
• Technically feasible (cf. ALTTA EU project)
![Page 43: Routes to transition in shear flowsTRANSITION IN SHEAR FLOWS IS A PHENOMENON STILL NOT FULLY UNDERSTOOD. For the simplest parallel or quasi-parallel flows there is poor agreement between](https://reader033.vdocuments.mx/reader033/viewer/2022050504/5f966dfcccd6d84d3722436d/html5/thumbnails/43.jpg)
• Optimal control theory is a powerful tool
• Optimal feedback strategies underway
• Optimal control via tailored magnetic fields should be possible