A method for optimal control in boundary-layer flow

Alexandru Dumitrache1,∗

1 Institute of Mathematical Statistics and Applied Mathematics, P.O. Box 1-24, 010145 Bucharest, Romania.

A methodology for active flow control which couples unsteady flow fields and controls is described. Active-control methodsare used to maintain laminar flow in a region in which the natural instabilities lead to turbulent flow. The simplest formof control which might achieve this objective is the wave-cancellation approach. The case of boundary layer instabilitysuppression is considered as the initial validation and test case. Control is effected through the injection or suction of fluidthrough a single orifice on the boundary. The optimal control theory provides an approach which does not require a prioriknowledge of the flow.

1 Introduction

Controlling the unsteady flow in a wide variety of configurations (engine inlets and nozzles, combustors, aircraft, etc.) can leadto improve efficiency and performance [1]. One goal is to obtain a good control system, at least to inhibit all instabilities whichdeviate the flow state from laminar to turbulent. The wave cancellation approach [2]-[4] assumes that a wave-like disturbancecan be linearly cancelled by introducing another wave with similar amplitude but opposite in phase.The wave cancellationproblem is a test case for a new flow-control methodology. The optimal control theory provides an approach which does notrequire a priori knowledge of the flow characteristics.

For boundary layer case, an objective is defined as a ”distance” between the measured flow stresses, and their correspondingdesired values along the wall and over a specified length of time. To control the flow, unsteady injection and suction are usedalong a small orifice. The time-dependence of the control is the natural results of the minimization of the objective functional.There is a sensor (an objective functional) that transmit information to a controller that in turn transmit information to theactuator.

2 The optimization mathematical model

Let Ω denote the flow domain which is boundary layer [x ≥ 0, 0 ≤ y ≤ h], where h is the freestream distance for theboundary layer. Let b be its boundary and let (0, T ) be the considered time interval. [x = 0, 0 ≤ y ≤ h] is the inflow partof the boundary and [x ≥ 0, y = 0] is the part of the boundary on which control is applied. Solid walls are denoted by w.[x ≥ 0, y = h] is the upper boundary. The control is activated over a given time interval t ∈ (T0, T1 ⊂ (0, T ). The flow field,described by the velocity vector (u, v) and the scalar pressure p, is obtained by solving the following momentum and massconservation equations

ut + uux + vuy + px −−v(2ux)x − v(uy + vx)y = 0 (1)

ut + uvx + vvy + py − v(uy + vx)x − v(2vx)y = 0 (2)

uy + vx = 0 (3)

all in (0, T )×Ω and subjected to initial and boundary conditions: (u, v)|t=0 = (u0, v0) in Ω; (u, v)|Γa= (g1, g2) in (T0, T1)

and (u, v)|Γa= (0, 0) in (0, T ) \ [T0, T1]; (u, v)|Γi

= (ui, vi) and (u, v)|Γa= (0, 0) in (0, T ); (u, v, p) → base flow,

ux, vx → 0 as x → ∞; u|Γe= U∞ and p − 2νvy|Γe

in (0, T ), where U∞ and P∞ denote the free-stream flow speed andpressure, respectively. The initial velocity vector (u0, v0) and the inflow velocity (ui, vi) are assumed given and the baseflow is assumed to be Blasius flow. As part of the optimization process, when the fluid is injected or sucked through controlboundary, the control functions q1(t, x) and q2(t, x) are to be determined.

The objective functional. Let us consider a finite part of the lower boundary [x ≥, y = 0], disjointed from it and the timeinterval (Ta, Tb) ⊂ (0, T ). Let consider the functional

F(u, v, p, q1, q2) =α1

2

∫ Tb

Ta

∫Γl

|τ1 − τa|2dbdt +

α2

2

∫ Tb

Ta

∫Γl

|τ2 − τb|2dbdt +

β1

2

∫ T1

T0

∫Γa

(|(q1)t|2 + |q1|

2)dbdt

+β2

2

∫ T1

T0

∫Γa

(|(q2)t|2 + |q2|

2)dbdt, (4)

∗ Corresponding author E-mail: alex dumitrache@yahoo.com, Phone: +00 40 21 3182433, Fax: +00 40 21 3182439

PAMM · Proc. Appl. Math. Mech. 7, 1061103–1061104 (2007) / DOI 10.1002/pamm.200700117

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fig. 1 Actuator response for control (C2) and wavecancellation (WC) with discrete time.

Fig. 2 Sensor-measured shear stress for control (C2)and wave cancellation (WC) with discrete time.

where τa(t, x) and τb(t, x) are given shear and normal stresses, respectively, defined on (Ta, Tb) on this boundary, τ1 = νuy

and τ2 = −p + 2νvy are the current shear and normal stresses, respectively, exerted by the fluid on the bounding wall. Thisboundary segment can be considered as a sensor which measures the stresses on the wall and, in (4) is the part of the boundary,along which one matches the shear and normal stresses to the given functions τa and τb over a given time interval, (Ta, Tb).The third and fourth terms in (4) limit the size of the control and the constants αi, βi, i = 1, 2 are used to adjust the relativeimportance of some terms.

Thus the optimization problem consists in finding u, v, p, q1 and q2 such that the functional F given in (4) is minimizedsubject to the requirements (1)-(3), subjected to initial and boundary conditions, and (4) are satisfied.

The Optimality System for the Boundary-Layer Flow. The method of Lagrange multipliers is used to enforce the con-straints (1)-(3) and the second boundary conditions. Then the Lagrangian functional is introduced. Through the introductionof Lagrange multipliers, the constrained optimization problem is changed into the unconstrained problem. Each arguments ofthe Lagrange functional may be varied independently. The first order necessary condition that stationary points must satisfyis that the first variation with respect to each of its arguments vanishes at those points. Now set the first variations of theLagrangians with respect to the state variables u, v, and p equal to zero and one obtains the adjoint or co-state equations. Thefirst-order necessary conditions δL/δq1 = 0 and δL/δq2 = 0 are usually called the optimality conditions.

We now have the full optimality system for boundary-layer flow on flat plate whose solutions determine the optimal states,control, and adjoint states: state equations, co-state equations and optimality equations.

Numerical results. The optimal control methodology is applied to a boundary-layer flow having a single instability wavethat can be characterized by a discrete frequency within the spectrum. These discrete small-amplitude instabilities can besuppressed through wave cancellation using known information about the wave. Hence, the optimal control is ”known” forvalidation of the present DNS/optimal control theory numerical approach in which the instability is to be suppressed withoutany a priori knowledge of said instability.

First the N-S equations are solved for the state variables. Then co-state equations are solved for the adjoint variables.Then, using these adjoint variables, the controls q1 and q2 are then found by solving the optimality equations. The procedure

is repeated until satisfactory convergence is achieved. In the present study only normal injection or suction control is allowed.Also we only match the normal stress along the wall so that we choose α1 = 0 in the functional (4) (Figure 3 and Figure 4).

3 Conclusions

The coupled Navier-Stokes equations, adjoint Navier-Stokes, and optimality condition equations were solved and validatedfor the flow-control problem of instability wave suppression in a at plate boundary layer. By solving the above system,optimal controls were determined that met the objective of minimizing the perturbation normal stress along a portion of thebounding wall. As a result, the optimal control was found to be an effective means for suppressing two-dimensional, unstableTollmien-Schlichting travelling waves.

References

[1] M. Gad-el-Hak and D. M. Bushnell, Separation Control: Review, J. Fluids Engineering, 113, 5–30 (1991).[2] S. Biringen, Active Control of Transition by Periodic Suction-Blowing, Physics of Fluids, 27, 1345–1347 (1984).[3] D. M. Ladd and E. W. Hendricks, Active Control of 2-D Instability Waves on an Axisymmetric Body, Experiments in Fluids 6, 69–70

(1988).[4] R. D. Joslin, G. Erlebacher and M. Y.Hussaini, Active Control of Instabilities in Laminar Boundary-Layer Flow. An Overview, NASA

CR 195016, ICASE Report No. 94-97 (1994).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Minisymposia – 06 Optimization 1061104