Open-loop

control of

a separated

boundary

layer

3rd GDR Symposium "Flow Separation Control"7th-8th Nov. 2013, Ecole

Centrale Lille

U. EhrensteinIRPHE, Aix Marseille Univ.

E. Boujo, F. GallaireLFMI, EPFL

MotivationSeparated flows are everywhere:

Bluff bodies

Convex walls

Adverse pressure gradients

MotivationControl design by trial and

error: successful

but tedious

[Strykowski

& Sreenivasan, JFM 1990]

Vortex-shedding

suppression with

a small

control cylinder:

Motivation

Sensitivity

fields

give

the

effect

of:– flow

modification:

– volume control:– wall

control:

[Marquet, Sipp

& Jacquin, JFM 2008]Sensitivity

of

leading

eigenmode's

growthrate

to a small

control cylinder

Alternative: sensitivity

analysis. -

Adjoint-based, one-shot

method.

-

Well-established

for eigenvalues.

[Hill 1992; Giannetti

& Luchini

2003; Marquet et al. 2008; Meliga

et al. 2010]

destabilising

effect

stabilising

effect

Open-loop

control of

a separated

boundary

layer

•

Noise amplification in the bump flow–

Optimal gain

–

DNS

•

Sensitivity analysis and control–

Optimal gain

–

Recirculation length

Open-loop

control of

a separated

boundary

layer

•

Noise amplification in the bump flow–

Optimal gain

–

DNS

•

Sensitivity analysis and control–

Optimal gain

–

Recirculation length

•

2D bump

on a flat plate, in a developing

boundary

layer

•

Long recirculation region

•

Bifurcation from

stationary

to unsteady

at ≈600

[Ehrenstein & Gallaire 2008]

Bump flow

displacement thickness

bump height [Bernard et al. 2003]

Bump flow: a noise-amplifier flow

•

Large transient growth [Ehrenstein & Gallaire 2008]

•

Large optimal harmonic response

(similar to pressure-induced laminar separation bubbles

[Alizard et al. 2009], and backward-facing step

[Dergham

et al. 2013]).

optimal gain

frequency

Optimal gain•

Linearize the perturbations equations around the steady-state base flow:

•

Optimal gain:

solution of the eigenvalue problem

•

[Forcing→velocity] relationship given by the resolvent :

•

Harmonic forcing: Steady-state response:

[Åkervik et al. 2008, Alizard et al. 2009, Garnaud et al. 2013, Dergham et al. 2013, Sipp & Marquet 2013]

Optimal forcing and response

Open-loop control of a separated boundary layer

•

Noise amplification in the bump flow–

Optimal gain

–

DNS

•

Sensitivity analysis and control–

Optimal gain

–

Recirculation length

DNS: harmonic forcing

• Choose a particular forcing, localized upstream:

Energy of the perturbations: Steady-state mean energy:

Power spectrum:

• Force the flow harmonically, e.g. at :

transition

DNS: stochastic

forcingForce the

flow

with

white

noise

(zero-mean, unit variance)

DNS power

spectra

at

different

locations (and

global, linear

optimal gain)

Steady-state

mean

energy

(Optimal response

at

) )

Open-loop control of a separated boundary layer

•

Noise amplification in the bump flow–

Optimal gain

–

DNS

•

Sensitivity analysis and control–

Optimal gain

–

Recirculation length

Sensitivity of optimal gain

• Sensitivity to flow modification:

• Optimal gain:

[Brandt et al., 2011]

• Sensitivity to control: solution of a linear system forced by :

Sensitivity of optimal gainSensitivity to volume control:

Focus on most amplified frequencies. Choose location where control cylinder has a reducing effect.

Difficult to find a location where optimal gain can be reduced at all frequencies..

Sensitivity of optimal gainVolume control:

• Small reduction.• Non-linear effects limitation.

•

most sensitive,

•

sign does not change with ω.

Sensitivity to wall control:

At the bump summit:

Sensitivity of optimal gain

Sensitivity of optimal gainWall control at the bump summit

•

Efficient at all frequencies

•

Large authority

(log. scale)

(lin. scale)

Sensitivity analysis

Non-linear

Harmonic forcing,

DNS: forcing + wall suction

Stochastic forcing

DNS: forcing + wall suctionRestabilize the flow from the bifurcated state.

Control turned ON at t=1000

Perturbation energy

Streamwise velocity at (x,y)=(80,1)

Total vorticity

Open-loop control of a separated boundary layer

•

Noise amplification in the bump flow–

Optimal gain

–

DNS

•

Sensitivity analysis and control–

Optimal gain

–

Recirculation length

Recirculation lengthRecirculation length increases with Re:

Cylinder

Backward-facing step

[Giannetti et Luchini 2007]

[Sinha et al. 1981]

Longer recirculation means:-

stronger backflow, more shear,

-

more length for perturbations to grow.[Brown & Roshko 1974]

is a macroscopic key parameter in separated flows, which is interesting to control.

Sensitivity of recirculation length• Reattachment at the wall: zero wall shear stress

• Sensitivity to flow modification:

•

Sensitivity to control: similar to optimal gain (solution of linear system forced by )

Sensitivity of recirculation length

Largest effect of wall control at the bump summit. Identify the same region as for optimal gain.

Streamwise velocity ux

Sensitivity analysis

Non-linear

Flow rate

uncontrolled

controlled, W=-0.035

Control of recirculation lengthWall suction at the bump summit

Recirculation length

Conclusion

•

Sensitivity analysis useful to identify regions of largest effect for steady open-loop control in separated flows.

•

Applied sensitivity analysis to optimal gain and recirculation length and obtained similar conclusions.

•

Designed a wall control configuration which successfully reduces recirculation length and energy amplification, and delays noise-

induced transition.