Lagrangian Coherent Structures in TwoDimensional Turbulence

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Fluid Flow Separation by the Method of
Averaging George Haller
Mechanical Engineering MIT
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Collaborators Mohammed-Reza Alam (MIT), Raul
Coral (MIT), Olivier Grunberg (MIT), Guus Jacobs
(Brown/MIT), Mustafa Sabri Kilic (MIT), Anatolij
Neishtadt (RSI), Thomas Peacock (MIT), Hayder
Salman (UNC), Amit Surana (MIT)
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• Flow separation
• Detachment of fluid from a boundary
• Causes loss of aerodynamic performance
• Creates mixing

Achayara et al. 1993 Re30,000
Diffusers
Airfoils
Islands
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Steady 2D Separation
Nonhyperbolic unstable manifold

• L. Prandtl 1904
• Steady separation takes place at
• vanishing skin friction
• negative skin-friction gradient

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Unsteady 2D Separation
Example Unsteady separation bubble flow
Ghosh et al., JFM, 1998
Common belief separation is again at
Sears Telionis SIAM J. Appl.
Math,1975 Vanishing wall-shear and
accompanying flow reversal do not, in general,
denote separation in any meaningful sense in
unsteady flow.
Where do unsteady flows separate?
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Exact separation theory for general 2D Flows
Definition Fixed unsteady separation occurs
along time-dependent nonhyperbolic unstable
manifolds
But where are these manifolds?
Theorem 1 Sufficient condition For a locally
mass-conserving flow
1.
(generalizes Prandtls 1st criterion)
(generalizes Prandtls 2nd criterion Somewhat
restrictive!
2.
Haller, J. Fluid Mech. 2004
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Idea for improvement try averaging
Only works if (u,v) is Hamiltonian
(incompressible) and has steady mean
Next idea make flow Hamiltonian at least on the
wall
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Averaged system if the velocity field has
steady asymptotic mean,

• Apply Prandtls steady result
• Use topological techniques to show
• separation manifold persist in full system

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Theorem 2 Kilic, Haller, and Neishtadt, Phys.
Fluids. 2005 If v(x,y) has steady asymptotic
mean and bounded variation, then the sufficient
separation conditions can be strengthened to
1.
2.
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Example 1 Periodically forced incompressible
separation bubble
Separation shape prediction
from
higher-order averaging
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Example 2 Oscillating rotor experiment
(with R. Coral and T. Peacock, MIT)
Experimental set-up
Sample result Randomly oscillating rotor
Horizontally moving vertical rotor creates
separation on vertical wall of tank
Separation point correctly predicted by our theory
Instantaneous zero of skin friction moves randomly
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Moving separation (with A. Surana)
Cannot be an unstable manifold
Assume
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Averaged equations
Unperturbed system (d0)
Perturbed system (dgt0)
?
f
f
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Theorem 3 Existence of a ghost manifold
Assume that Then for small dgt0, exists an
nearby locally invariant manifold
d0.05
f
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Ongoing work (with Amit Surana (MIT), Guus Jacobs
(Brown/MIT) )
3D steady and unsteady separation
Backward-facing step
Lid-driven cavity