MINIMAL DEFECTS
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MINIMAL DEFECTS
Damien Biau & Alessandro Bottaro
DICAT, University of Genova, Italy
OPTIMAL PATHS TO OPTIMAL PATHS TO TRANSITION IN A DUCTTRANSITION IN A DUCT
Relevant references: Galletti & Bottaro, JFM 2004; Bottaro et al. AIAA J. 2006; Biau et al. JFM 2008; Wedin et al. PoF 2008; Biau & Bottaro, Phil. Trans. 2008, Special Issue on the
125th Anniversary …
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HYDRODYNAMIC STABILITY THEORY
The flow in a duct is asymptotically stable to small perturbations
Is transient growth the solution?Butler & Farrell (1993)Trefethen, Trefethen Reddy & Driscoll (1993)Schmid & Henningson (2001)
Transient growth is related to unstructured operator’sperturbations, i.e. to the pseudospectrum of the linearstability operator: [L(U,Re) + ] v = 0
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The case studied
dP/dx = const.characteristic length = h [channel height]characteristic speed = u
[u2= -(h/4dp/dx]
Re = 150 (MARGINAL VALUE)
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Governing equations:
Linearized disturbance equations: Linearized disturbance equations:
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OPTIMAL PERTURBATIONS: Traditional functional optimization with adjoints
The adjoint problem reads:
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G = 873,1 at t+ = 1.31 G = 869.03 at t+ = 2.09
uopt = 0 at t+ = 0
uopt ≠ 0 at t+ = 0
t+ t+
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Nonlinear evolution of the optimal disturbances ( + random noise)(streamwise periodic duct of length 4)
F = skin friction factor (F = 0.0415 from DNS at Re = 150 (Reb = 2084))
At t = 0: = 0 E0 = 10-1 !! = 1 E0 = 7.8 x 10-3
= 2 E0 = 4.4 x 10-3
fully developed turbulence
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Excellent agreement with Gavrilakis, JFM 1992.
Streamwise/time averaged turbulent flow
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The = 1 mode
The threshold value is E0 = 7.8 x 10-3. In fact the optimal perturbation for = 1 is not so important. What matters is the distorted field which emerges at t ~ 0.8.Such a field is subject to a strong amplification.
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Partial conclusions 1
• Global optimal disturbances are interesting concepts, with probably little connection to transition
• The key to transition is to set up a distorted base flow with certain features
• The distorted base flow can be set up efficiently by sub-optimal disturbances in the form of streamwise travelling waves. In fact, any initial condition in the form of a travelling wave of sufficiently large amplitude is capable to do it!
• Is there any scope for studying optimal perturbations?
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What happens past t ~ 0.8?
Streamwise velocity showing a non-staggered array of vortices (similar to H-type transition)
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What happens past t ~ 0.8?
The secondary flow does not oscillate around the 8-vortex state,but around a 4-vortex state, with active walls which alternate ...… eventually relaminarisation ensues (and the lifetime of the turbulence depends non-monotonically on the streamwise dimension of the computational box)
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THE PHASE SPACE PICTURE
EU = mean flow energyReb = Reynolds number based on bulk velocity
lower branch solution upper branch solution (qualitative)
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Partial conclusions 2
• A saddle point appears in phase space to structure the turbulence; after leaving the unstable manifold of the saddle, the trajectory loops in phase space a few times, with the flow spending most of the time in the vicinity of the saddle, until the flow eventually relaminarizes. • The lifetime depends on the box size.
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HYDRODYNAMIC STABILITY THEORY
Distorted base flow: minimal defectsBottaro, Corbett & Luchini (2003)Biau & Bottaro (2004)Ben-Dov & Cohen (2007)
The growth of instabilities on top of a distorted base flow(related to dynamical uncertainties and/or poorly modeledterms) is related to structured operator’s perturbations, i.e.to the structured pseudospectrum of the linear stabilityoperator.
[L(U, , Re) + ] v = 0 Unstructured perturbations
[L(U + U, , Re)] v = 0 Structured perturbations
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Input-output framework
Instead of optimising the input (the vortex), we optimise the output (the streak)!
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SENSITIVITY ANALYSIS
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SENSITIVITY ANALYSIS
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MINIMAL DEFECT
constraintconstraint
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We do not look directly at the spectrum oftemporal eigenvalues of the direct and adjoint equations, instead we iterate the PDE’s for long time, until the leading eigenvalue emerges.
Advantage: if we decided to iterate for a fixed (short) time, we could find the optimal defect for transient growth …
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Minimal defects for = 1, Re = 150
( = 1 corresponds to 0.16 of the laminar flow energy)
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Minimum defect for = 1: Unstable eigenmode:modulus of the streamwise streamwise vel. perturbationvelocity defect
(the white line corresponds to u = Ub)
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Partial conclusions 3
• Minimal defects represent the optimisation of the output (rather than the input, as customarily done in optimal perturbation analysis)
• They can lead to transition as efficiently as suboptimal perturbations
• Both minimal defects and suboptimals are more efficient for initial ‘s larger than zero (i.e. when the disturbances have small longitudinal dimensions)
• We have not closed the cycle yet ... we need to include the feedback !
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Self sustained process
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A NEW OPTIMISATION APPROACH:i.e. maximising the feedback
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The initial process of transition:
1. Algebraic growth of a O() travelling wave2. Generation of weak streamwise vortices O()
by quadratic interactions3. Production of a strong streak O(2Re) by lift-up4. Wavy instability of the streak to close the loop
U0(y,z) U(y,z,t) u(x,y,z,t)0 V(y,z,t) v(x,y,z,t)0 W(y,z,t) w(x,y,z,t)P0(x) P(y,z,t) p(x,y,z,t)
+ +
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Inserting into the Navier-Stokes equations we obtain equations for
STREAMWISE-AVERAGED FLOW
plus FLUCTUATIONS
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Goal: maximise the feedback of the wave onto the rolls, i.e. maximise Reynolds stress terms or, more simply, the energyGAIN of the wave over a fixed time interval:
G = e(T)/e(0)
with
Tool: the usual Lagrangian optimisation, with adjoints
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ADJOINT EQUATIONS:
with SENSITIVITY FUNCTIONS
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Ansatz for the fluctuations:
acceptable as long as in the first stage of the process the large
scale coherent wave is but mildly affected by the behaviour of
features occurring at smaller space-time scales
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Optimisation result for Re = 150, = 1, T = 1, e0 = 3x10-3
Energy of the defect
Energy of the wave
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Parametric study
Wave Defect
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Two comments:
small defect is not created (classical optimal perturbations are of little use)
large cut off length “minimal channel”
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Comparison between the minimal defect (left)
and the optimal streak at t = 0.6 (centre)
DNS at t = 0.6
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Evolution of the skin friction in time
“edge state”
turbulent mean value
laminar value
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edge
laminar fixed point
chaos
THE EDGE MEDIATES LAMINAR-TURBULENT TRANSITION,it is the stable manifold of the hyperbolic fixed point
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The edge state solution
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Simplified phase diagram
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Global optimal perturbations saddle node state “edge states” (Wedin et al. 2007)
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Partial conclusions 4
• A model has been developed which closes the loop and combines transient growth, exponential amplification, defects, the “exact coherent states” and the SSP
• In the very initial stages of transition we need growth of a wave andfeedback onto the mean flow, to create a vortex. The lift-up then
generates the streaks, which break down and re-generate the wave.
• The initial conditions in the form of elongated streaks (the optimal disturbance) is inefficient at yielding transition because it fails at modifying the mean flow
• Our new simplified model yields a flow which sits initially on a trajectory directed towards the saddle point on the edge surface; direct
simulations confirm the suitability of the proposed model
• The basic flow structure of the edge state is two vortex pairs, overlapping one another, with the stronger pair sitting on a wall
• LINEAR equations can go a long way !!!