optimization and control of a separated boundary-layer flowsicogif/publications/aiaa... ·...

Optimization and control of a separated boundary-layer

flow

Journal: 2011 Hawaii Summer Conferences

Manuscript ID: Draft

luMeetingID: 2225

Date Submitted by the Author:

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Contact Author: PASSAGGIA, Pierre-Yves

http://mc.manuscriptcentral.com/aiaa-mfd11

2011 Hawaii Summer Conferences

Optimization and control of a separated

boundary-layer flow

P.-Y. Passaggia∗ and U. Ehrenstein †

Institut de Recherche sur les Phenomenes Hors Equilibre (IRPHE),

CNRS / Aix-Marseille Universite, F-13384 Marseille, France

In order to investigate the possibilities of controlling wall-bounded flows, optimal controlof the nonlinear dynamics of a separated boundary-layer flow is studied using a localizedblowing-suction at the wall. Control laws are computed using a formulation based on theaugmented Lagrangian approach using the full knowledge of the flow dynamics. The casestudy consists of a laminar separated boundary-layer flow over a bump.

I. Introduction

Flow separation is ubiquitous to wall bounded flows when subject to an adverse pressure gradient orgeometrical devices. The resulting recirculation bubble are source of instability phenomena leading to a lossof aerodynamic performances. It is essential to propose control solutions capable of minimizing there impacton the entire flow. In this paper, an adjoint-based optimization technique is presented in order to control theflow behind an obstacle. The feedback control of such dynamics has been recently investigated with successin the case of a flow over a shallow cavity2 using model reduction based on the linearized perturbationdynamics.6 However, model reduction based on the linear dynamics is likely to be not reliable as soon asabsolute/convective instabilities coexist in the nonlinear regime. In order to assess the possibility of feedbackcontrol of such flow dynamics, the augmented Lagrangian approach has been implemented. This formulationtakes into account the entire flow dynamics and was for instance shown to relaminarize a turbulent channelflow.4 This formulation can also be adapted to the computation of optimal perturbations which are knownto play an important role in the transient dynamics of open flows. In the present investigation, the flowis a laminar boundary layer where flow separation is triggered by a bump. This configuration produces along recirculation bubble which becomes unstable beyond a critical Reynolds number.5 This flow has beenstudied numerically by Marquillie & Ehrenstein7 and their algorithm has been adapted to the present study.For the optimization problem with constraints, the gradient method based on a formulation of the adjointNavier-Stokes system has been implemented. The control problem of the fully developed nonlinear dynamics,detailed in section 2, is formulated using a cost function based on either the integral or the terminal timekinetic energy of the perturbation. The effect of nonlinearities on the optimal initial disturbance and thecontrol of the fully developed dynamics is presented in section 3. Finally the results are discussed in section4.

II. Formulation and numerical procedure

The flow is decomposed into a baseflow and a perturbation and the Navier-Stokes system is written

∂tu + (U · ∇)u + (u · ∇)U + (u · ∇)u +∇p− (1/Re)∆u = 0, ∇ · u = 0 (1)

where U is the velocity of the baseflow and u, p are the velocity and pressure pertubation, respectively.The Reynolds number

∗PhD Student, IRPHE, 49 rue F. Joliot Curie, B.P. 146, F-13384 Marseille Cedex 13, France†Professor, IRPHE, Aix-Marseille Universite

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American Institute of Aeronautics and Astronautics

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Re =δ∗U∞ν

(2)

is based on the displacement thickness δ∗ of the Blasius profile imposed at the entrance of the domain. Here,the numerical procedure of Marquillie & Ehrenstein7 has been adapted for the optimization formulation.The bump is taken into account by use of a mapping which transforms the physical domain into a cartesianone. The streamwise direction x is discretized using finite differences whereas y, the wall-normal direction,is discretized using Chebyshev-collocation. The time integration is performed using a second order semi-implicit scheme. The difference between the present formulation and the one described in Marquillie &Ehrenstein comes from the algorithm used to solve the pressure. In the optimization formulation, the choiceof outflow boundary condition is essential. In order to ensure the compatibility between the Navier-Stokessystem and its adjoint, stress-free outflow boundary condition are to be implemented which is achieved usingthe influence matrix technique which couples pressure and velocity8

II.A. Unstable baseflow

At a supercritical Reynolds number of 650 used in the presnet investigation, a large domain with lengthL = 250 and a height H = 100 has been considered to minimize finite boundary effects. The first step ofthe analysis consists in generating a baseflow and the method of Akervik et al.1 ( called selective frequencydamping) has been implemented in order to converge towards a stationary solution of the Navier-Stokessystem. The method consists in solving the coupled system

q = f(q, Re)− χ(q− q)˙q = (q−q)

Λ

(3)

where f represents the Navier-Stokes system, q contains the velocity U and the pressure P of the baseflow.Here Λ is a cutoff frequency and χ is a damping coefficient. In the present case, the value of Λ correspondsto the dominant frequency measured using a probe localized inside the shear layer. A good estimation ofthe parameter χ is given by the growth rate of the linear instability, provided by the global stability analysisof the flow.5 For Re = 650, the parameters used are ∆ = 6.37, χ=0.03 and figure 1 shows the resultingbaseflow.

Figure 1. Baseflow for Re = 650

II.B. Augmented Lagrangian and gradient

In the case of the control, the aim is to minimize the kinetic energy in the entire domain and the cost functionis either

J1,int(φ,u) =1

2

∫ T1

T0

∫Ωo

u · u dx dt+ γ

∫ T1

T0

∫Γc

gφ · gφ ds dt (4)

when energy integral is considered or

J1,term(φ,u) =1

2

∫Ωo

u(T1) · u(T1) dx dt+ γ

∫ T1

T0

∫Γc

gφ · gφ ds dt (5)

for energy optimization at time T1. The portion of the boundary where the control is applied is denoted byΓc, whereas g(x) defines the spatial distribution (taken as uniform in the present case) of the blowing-suction.The control has been applied at three different locations (each one of length 5) in the vicinity of the bump(cf. figure 2). The function φ(t) is the control law and γ refers to the cost of the control. The optimization

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time interval is [T0, T1]. The aim is to minimize the functional J1, u being the solution of the Navier-Stokessystem written as

f(U,u, p) = 0, g(u, φ) = 0 with g(u, φ) = u|Γc− gφ. (6)

In the case of the optimal perturbation analysis, the objective function becomes

J2(φ,u) =1

2

∫Ω

u(T1) · u(T1) dx (7)

the unknown being now the initial condition u0, that is u is now solution of

f(U,u, p) = 0, g(u,u0) = u(x, 0)− u0 = 0, h(u0, ε) =

∫Ω

u0 · u0 dx− ε = 0 (8)

where ε is the amplitude of the initial condition u0. Lagrange multipliers u+, p+, φ+ are introduced and in

Figure 2. Sketch of the domain and of the control setup

order to solve the optimization problem, the Lagrangian L

L(u,u+, p, p+, φ, φ+, ε, ε+) = J1,int(φ,u)− < f(U,u, p), (u+, p+) > − < g(u, φ), φ+ > (9)

is to be rendered stationary. Cancelling the Frechet derivatives with respect to the Lagrange multipliersconsist in imposing the Navier-Stokes constraints. The derivatives with respect to the state variables (u, p)writes

Du,pL · (u, p) =

∫ T1

T0

∫Ω

u · u dxdt−Du,p < f(u, p), (u+, p+) > ·(u, p)−∫ T1

T0

∫Γc

u · gφ+ ds dt (10)

The expression of Du,p < f(u, p), (u+, p+) > ·(u, p) is detailed for example in Bewley et al.3 and cancelingthe above expression is equivalent to solve the adjoint system

−∂tu+ − (U · ∇)u+ + (∇U)tu+ − (u · ∇)u+ + (∇u)tu+ +∇p+ − (1/Re)∆u+ = u, ∇ · u+ = 0. (11)

The right hand side of the adjoint momentum equation is zero in the case of a terminal time optimization.The adjoint system is integrated backward in time from T1 to T0, starting with u+(T1) = 0 in the intgraltime formulation whereas for terminal time u+(T1) = u(T1). The integrations by part necessary to obtainthe adjoint system introduce the integral on the boundary ∂Ω∫ T1

T0

∫∂Ω

[1

Reu+(∇u) · n− 1

Re(∇u+ · n)u− (U · n)u+ · u− (u · n)u+ · u + (p+ · n)u− u+(p · n)] ds dt. (12)

The stress-free conditions 1Re∇u · n− pn = 0 being imposed at the outflow of the domain ∂ΩS , the outflow

condition for the adjoint system (implemented using the influence matrix)

− 1

Re(∇u+)n|∂ΩS

− (U · n)u+|∂ΩS− (u · n)u+|∂ΩS

+ p+n|∂ΩS= 0, u+|∂Ω−∂ΩS

= 0 (13)

cancels this integral on ∂Ω− Γc. Finally, the contribution on Γc is cancelled taking

gφ+ = − 1

Re∇u+ · n + p+ n (14)

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in (10). The derivative of the Lagrangian with respect to the control variable φ becomes

DφL · φ = γ

∫ T1

T0

∫Γc

gφ · gφ ds dt−∫ T1

T0

∫Γc

(−gφ) · gφ+ds dt (15)

and taking into account (14), the gradient for the control is

∇φJ1(φ) = γφ

∫Γc

g · g ds +

∫Γc

(− 1

Re∇u+n + p+n) · g ds. (16)

The same procedure is used for the optimal perturbation, the expression of the gradient now being

∇u0J2(u0) = −u+

0 + 2ε+u0 with

u+

0 = u+(x, T0)

ε+ =√∫

Ωu+(T0) · u+(T0) dx/4ε

(17)

An iterative gradient algorithm

φk+1 = φk − αk∇φJ (φk,u+,k).

is used to converge towards the optimal solution. A main difficulty is to appropriately choose the step αk andthis parameter is optimized using a classical line search procedure. Once the optimal control is computedover the first time interval, the time marching is insured using a ”recieding-predictive control” algorithmdescribed in4 with a time shift Ta = ∆T/2 equal to half the time of optimization in our case (see figure 3).

Figure 3. Schematic of the recieding-predictive algorithm

III. Optimal growth and control

The linear optimal perturbation analysis5 showed that perturbations localized near the separation pointare responsible for the largest possible transient growth of the system. In order to examine the effect of

Figure 4. a) Evolution of the structure of the optimal perturbation as a function of the initial amplitude, b) transientgrowth of optimal perturbations as a function of initial amplitude for T1 = 150.

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nonlinearities on these optimal perturbations, the initial amplitude of the disturbance is first set to smallvalues in order to maintain the flow close to the linear regime. Then the amplitude of the initial disturbanceis increased from 10−8 to 10−4 and successive optimization times were considered between T1 = 50 andT1 = 150. Figure 4a shows the shape of the optimal initial disturbance for three amplitudes consideredand its structure is seen to spread in the shear layer, as well as upstream the bump when the ampliudeis increased. This clearly shows that the nonlinear flow is not only sensitive to actuators located on theseparation point, but also to regions located inside the recirculation bubble as well as upstream the bump.Figure 4b shows the energy evolution for the time horizon T1 = 150. The flow perturbation starts to saturateat T = 120 when the highest initial perturbation amplitude is considered. The possibilities of controlling thefully developed dynamics is then investigated. This unstable dynamics is characterized by a low frequencyoscillations of the recirculation bubble5 . The uncontrolled flow energy is shown as the solid line in figure 5and the ”flapping” frequency of Λ ≈ 200 is clearly visible.

Figure 5. a) Energy of the perturbation for the uncontrolled case ( ), control using the integral time formulationwith a single actuator (.....), control with 3 actuators ( . . ) , control with 3 actuators (− − −) using the terminaltime formulation and b) associated control laws φ for the integral time formulation.

Ideally one would like to take arbitrarily long time horizons for optimization. However, the direct solution intime u is input of the adjoint system and memory constraints put a bound on the maximum time T1 whichcan be considered. Here we have chosen T1 = 300 which is larger than the flapping period. The optimizationprocedure implies 5 to 7 computations of the gradient whereas 4 to 7 evaluations of the objective function arenecessary to evaluate the optimal step α. A first actuator is introduced at the separation point shown as 1 infigure 2, where the nonlinear optimal perturbation dominates. The problem is then solved in time iterating

Figure 6. Time-averaged streamlines for the uncontrolled flow a), and for the controlled flow b) using the integralformulation and 3 actuators.

back and forth the direct and the adjoint system to achieve convergence for each time interval. Using thissingle actuator, the control is capable of diminishing the energy to half of the uncontrolled value withouthowever achieving definite control, as seen in figure 5. In the attempt to improve the control, two additionalactuators shown as 2 and 3 in figure 2 have been added. The corresponding energy evolutions are depictedin figure 5: although the energy is decreased further for T > 500 the perturbation energy cannot be reducedto zero. As shown in Marquillie & Ehrenstein,7 in the perturbed nonlinear regime, the mean recirculation

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Figure 7. Instantenous snapshot of the vorticity at time T = 600 for the uncontrolled flow a), for the controlled flow b)using the integral formulation and 3 actuators.

length diminishes when compared with that of the baseflow. Therefore the recirculation length is likely toprovide informations about the quality of the control. Figure 6 shows an increase from approximately 48.5for the uncontrolled case to 63.4 when the control is applied. Note that the mean values have been computedfor the interval 450 ≤ t ≤ 900. A snapshot of the vorticity at time t = 600 is depicted in figure 7. Theuse of terminal time formulation in the objective function has been advocated by ref 3, which in the case ofthe turbulent channel flow was capable to achieve a relaminarization. Here we imposed finite time energyoptimization starting at time 350, however for the present flow dynamics the results are comparable withthe integral time formulation, as can be seen in figure5 .

IV. Discussion

The optimal growth and control of a separated laminar boundary layer flow over a bump is investigatednumerically using an adjoint-based augmented Lagrangian approach. In this configuration, stress-free outflowboundary conditions compatible between the Navier-Stokes system and its adjoint were implemented, usingthe influence-matrix approach. The problem of optimal perturbations in the nonlinear regime has beenstudied, highlighting the regions of the flow which are the most sensitive to actuator devices. It is shown thatin the nonlinear regime the flow dynamics becomes sensitive to perturbations aligned along the whole shear-layer, in contrast to the linear regime which is only sensitive to upstream perturbations. The control problemusing blowing-suction at the wall has been addressed as well, using a ”recieding-predictive” algorithm for timemarching. The unstable dynamics related to the flapping of the recirculation bubble is clearly attenuated buta full control of the nonlinear dynamics could not be achieved and this point is still under investigations. Forinstance to take larger time horizons and/or multiple actuators is likely to improve the control capabilities.

Acknowledgements

The authors gratefully acknowledge the financial support of the Agence Nationale pour la Recherche (ref.ANR-09-SYSC-011).

References

1E. Akervik, L. Brandt, D. Henningson, J. Hœpffner, O. Marxen, and P. Schlatter. Steady solutions of the navier-stokesequations by selective frequency damping. Phys. of Fluids, 18:068102, 2006.

2E. Akervik, J. Hœpffner, U. Ehrenstein, and D. Henningson. Optimal growth, model reduction and control in a separatedboundary-layer flow using global eigenmodes. J. Fluid Mech., 579:305 – 314, 2007.

3T.R. Bewley. Flow control: new challenges for a new renaissance. Prog. in Aero. Sci., 37:21–58, 2001.4T.R. Bewley, P. Moin, and R. Temam. Dns-based predictive control of turbulence: an optimal benchmark of feedback

algortihms. J. Fluid Mech., 447:179–225, 2001.5U. Ehrenstein and F. Gallaire. Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J.

Fluid Mech., 614:315–327, 2008.6J. Kim and T. R. Bewley. A linear systems approach fo flow control. Annu. Rev. Fluid Mech., 39:383–417, 2007.7M. Marquillie and U. Ehrenstein. On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid

Mech., 490:169– 188, 2003.8R. Peyret. Spectral Methods for Incompressible Flows. Springer, 2002.

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