Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

T. Tachim Medjo; R. Temam; M. Ziane

doi:10.1115/1.2830523

Applied Mechanics Reviews | Volume 61 | Issue 1 | Review Article

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Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

T. Tachim Medjo, R. Temam and M. Ziane

[+] Author and Article Information

T. Tachim Medjo

Department of Mathematics, Florida International University, DM413B University Park, Miami, FL 33199

R. Temam

The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405

M. Ziane

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

In the space periodic case over $Ω = {(0, 2 π)}^{3}$ , $H^{- 1} (Ω)$ , $s > 0$ is the space of function $u = \sum {\hat{u}}_{k} e^{i k \cdot x}$ , and $k = (k_{1}, k_{2}, k_{3})$ , and $x = (x_{1}, x_{2}, x_{3})$ , such that the norm $\sum_{k} {∣ k ∣}^{- 2 s} {∣ {\hat{u}}_{k} ∣}^{2} < 0$ , thus giving less predominance to the high modes. On the contrary the norm of $H^{s} (Ω)$ , $s > 0$ , $\sum_{k} {∣ k ∣}^{2 s} {∣ {\hat{u}}_{k} ∣}^{2}$ gives enhanced weight to the height modes.

Otherwise, the magnetohydrodynamics (MHD) or radiation equations must be included and the control problem must be fully redefined (objectives, cost functional, etc.).

Robust control is addressed in Sec. 4.

Appl. Mech. Rev 61(1), 010802 (Feb 20, 2008) (23 pages) doi:10.1115/1.2830523 History: Published February 20, 2008

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In this article, we review from the mathematical and numerical points of view some of the recent progresses in the area of flow control.

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Figures

Classification of flow control strategies

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Figure 1

Classification of flow control strategies

Different control loops for active flow control. (a) Predetermined, open-looped control; (b) reactive, feedforward, open-loop control; (c) reactive, feedback, closed-loop control.

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Figure 2

Different control loops for active flow control. (a) Predetermined, open-looped control; (b) reactive, feedforward, open-loop control; (c) reactive, feedback, closed-loop control.

Performance of optimized controls for formulations based on Eq. 213 as a function of the optimization horizon t1 as computed in direct numerical simulations at Reτ=100: history of turbulent kinetic energy (see Ref. 32)

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Figure 3

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Figure 4

Performance of optimized controls for formulations based on Eq. 213 as a function of the optimization horizon t1 as computed in direct numerical simulations at Reτ=100: history of drag. For small optimization horizons (t1=O(1)), approximately 20% drag reduction is obtained, a result which can be obtained with a variety of other approaches. For sufficiently large optimization horizons (t1⩾20), the flow is returned to the region of stability of the laminar flow and the flow relaminarizes, resulting in a 57.2% drag reduction with no further effort required (see Ref. 32).

Side view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Large eddies are visualized using a sheet of laser fluorescent dye (see Ref. 4).

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Figure 5

Side view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Large eddies are visualized using a sheet of laser fluorescent dye (see Ref. 4).

Top view of a low Reynolds number turbulent boundary layer, Reθ=742. Wind tunnel experiment. Pockets, believed to be the fingerprints of typical eddies, are visualized using dense smoke illuminated with a sheet of laser (see Ref. 4).

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Figure 6

Top view of a low Reynolds number turbulent boundary layer, Reθ=742. Wind tunnel experiment. Pockets, believed to be the fingerprints of typical eddies, are visualized using dense smoke illuminated with a sheet of laser (see Ref. 4).

Top view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Low-speed streak are visualized using a sheet of laser fluorescent dye (see Ref. 4).

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Figure 7

Top view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Low-speed streak are visualized using a sheet of laser fluorescent dye (see Ref. 4).

Top view of an artificially generated spot in a laminar boundary layer. The displacement thickness Reynolds number at the spot’s initiation is Reθ=625 (see Ref. 4).

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Figure 8

Top view of an artificially generated spot in a laminar boundary layer. The displacement thickness Reynolds number at the spot’s initiation is Reθ=625 (see Ref. 4).

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Medjo T, Temam RR, Ziane MM. Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects. ASME. Appl. Mech. Rev. 2008;61(1):010802-010802-23. doi:10.1115/1.2830523.

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Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

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