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Review Articles

# Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

[+] Author and Article Information
Denis Sipp

ONERA-DAFE, 8 rue des Vertugadins, F-92190 Meudon, Francedenis.sipp@onera.fr

Olivier Marquet

ONERA-DAFE, 8 rue des Vertugadins, F-92190 Meudon, Franceolivierket@yahoo.fr

Philippe Meliga1

ONERA-DAFE, 8 rue des Vertugadins, F-92190 Meudon, Francephilippe.meliga@epfl.ch

Alexandre Barbagallo

Yet, for a given base-flow, poor results are expected from such a comparison, since the weakly-non-linear analysis presented in Sec. 3 blows up in the case of weakly-non-parallel flows (44,60), while the validity domain of the nonlinear local criterion by Pier and Huerre (46) is precisely restricted to weakly-non-parallel flows. Still, both approaches are complementary and concern different base-flows (weakly-non-parallel base-flows for the local approach and strongly-non-parallel ones for the global approach).

Note that $‖⋅‖$ acts on a vector and not on a vector field. On the other hand, dependent on the specific context, $⟨⋅,⋅⟩$ represents a scalar product acting on scalar fields or vector fields so that $⟨‖u‖,‖u‖⟩=⟨u,u⟩$ yields the energy of the flow-field $u$.

In fact, $û21=duB/dε$ since the base-flow $uB(ε)$ depends on the Reynolds number $ε$.

Chomaz (44) argued that the more the flow is parallel, the smaller $|μr+νr|$. This stems from the fact that, the more the flow is parallel, the further apart are the spatial supports of the direct and adjoint global modes. Hence, the mean flow and second-order harmonics have less and less impact on the dynamics since their support is more and more outside the wavemaker region (see Sec. 42 for definition). In this case, one has to resort to a strongly-non-linear approach, as presented by Pier and Huerre (46).

The cylinder bifurcation corresponds to a supercritical instability; i.e., the flow is unstable solely for supercritical parameters $ε>0$. If $μr+νr<0$, then the bifurcation would be subcritical and an instability of an open-flow may arise for subcritical parameters $ε<0$ but only for finite-amplitude perturbations (130-131).

If $⟨ ⟩T$ denotes the process of averaging over time, we thus obtain $uM=⟨u(t)⟩T$. Letting $u=uM+u′$ with $⟨u′⟩T=0$ and averaging Eq. 1, the following equation governing the mean flow is obtained: $R(uM)=−⟨R(u′)⟩T$. It is noted that the mean flow $uM$ is not a base-flow, i.e., a solution of Eq. 2. For our case, we get $u′=ε[Aeiω0tû1A(x,y)+c.c.]$ at the dominant order.

It should be noted that this approach is only rigorously justified in the case of a marginal global mode forced in the vicinity of its natural frequency. In fact, it is the entire sum in Eq. 20 that should be considered as the functional objective and not just the response in a particular component. The relevant concept here should be the singular value decomposition of the resolvent that seeks the maximum response associated with a given forcing energy. This will be further discussed in the section dealing with noise-amplifiers in Sec. 6.

Meliga (136) analyzed this gradient in the case of compressible Navier–Stokes equations. He showed for an axisymmetric bluff body how the sensitivity fields may be used to study the effect of compressibility on the instability.

The base-flow correction $duB/dε$ is defined by $R(uB+duB,ε+dε)=0$. Linearizing this equation and noting that $∂R/∂ε=−ΔuB$, we obtain $AduB−ΔuBdε=0$, which yields $duB/dε=A−1(ΔuB)$. Note that $Δ$ refers here to the matrix related to the Laplace operator.

The variation of the eigenvalue $dλ$ with respect to an increase in the Reynolds number $dε$—with the base-flow $uB$ frozen—may be obtained from Eq. 25, using the following perturbation matrix: $δA=−Δ$, i.e., the negative of the matrix standing for the Laplace operator.

This is incorrect if the small control cylinder is located in a shear flow. In this case, a lift force must also be taken into account.

For this, a control cylinder of a sufficiently small diameter is chosen such that the Reynolds number based on the local velocity of the base-flow and the diameter of the small control cylinder is lower than $Rec=47$.

It may be shown that this function is also equal to $Ĥ(ω)=M(iωI−A)−1PsC$.

1

Present address: EPFL-LFMI, CH-1015 Lausanne, Switzerland.

Appl. Mech. Rev 63(3), 030801 (Apr 27, 2010) (26 pages) doi:10.1115/1.4001478 History: Received November 19, 2009; Revised March 15, 2010; Published April 27, 2010; Online April 27, 2010

## Abstract

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier–Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows.

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## Figures

Figure 1

Flow around a cylinder for Re=47. Base-flow uB visualized by isocontours of streamwise velocity. Adapted from Ref. 38.

Figure 2

Flow around a cylinder for Re=47. Marginal global mode characterized by the frequency ω=0.74. The structure is visualized by isocontours of the real part of the cross-stream velocity (R(v̂)). Adapted from Ref. 38.

Figure 3

Flow around a cylinder. Strouhal number versus Reynolds number. The thick solid line refers to experimental results (52), the thin solid line to a global linear stability analysis on the base-flow, and the symbols to a global linear stability analysis on the mean flow. Adapted from Ref. 49.

Figure 4

Flow around a cylinder. Flow stabilization regions obtained experimentally for various Reynolds numbers. Adapted from Ref. 67.

Figure 5

Flow over an open cavity. Configuration and location of the actuator and sensor. Adapted from Ref. 141.

Figure 6

Flow around a cylinder for Re=47. Marginal adjoint global mode. The structure is visualized by isocontours of the real part of the cross-stream velocity (R(v̂)). Adapted from Ref. 38.

Figure 7

Flow around a cylinder. (a) Bifurcation diagram. (b) The least damped eigenvalues in the (σ,ω)-plane for subcritical, critical, and supercritical Reynolds numbers.

Figure 8

Flow around a cylinder for Re=47. Streamwise velocity on the symmetry axis for the base-flow u0 (dotted line), for the correction of the base-flow û21 (continuous line) and for the correction of the mean flow û2M (dashed line). Adapted from Ref. 38.

Figure 9

Open-loop control by action on the base-flow by an external forcing. Diagram displaying the law σ(f).

Figure 10

Flow around a cylinder at Re=47 and sensitivities associated with a modification of the base-flow. (a) Sensitivity of the amplification rate. (b) Sensitivity of the frequency. Adapted from Ref. 135.

Figure 11

Flow around a cylinder. (a) Wavemaker region for Re=50 according to Giannetti and Luchini (59). (b) Wavemaker region for Re=47 identified by the field W in the vicinity of the bifurcation threshold.

Figure 12

Flow around a cylinder at Re=47 and sensitivities associated with a steady forcing of the base-flow. (a) Sensitivity of the amplification rate. (b) Sensitivity of the frequency. Adapted from Ref. 135.

Figure 13

Flow around a cylinder at Re=47. (a) Variation of the amplification rate with respect to the placement of a control cylinder of infinitesimal size located at the current point. (b) Associated variation of the frequency. Adapted from Ref. 135.

Figure 14

Flow around a cylinder. Stabilization zones for the unsteadinesses as obtained by the sensitivity approach for different Reynolds numbers. The results should be compared with the experimental results displayed in Fig. 4. Adapted from Ref. 135.

Figure 15

Flow over an open cavity for Re=7500. (a) Spectrum of the matrix A, (b) real part of the streamwise velocity of the most unstable global mode, (c) same for the unstable global mode with the lowest frequency, (d) likewise for the most unstable adjoint global mode, and (e) likewise for the unstable adjoint global mode with the lowest frequency. Adapted from Ref. 141.

Figure 16

Flow over an open cavity for Re=7500 visualized by streamwise velocity contours and velocity vectors. (a) Base-flow. (b) Control matrix C. Adapted from Ref. 141.

Figure 17

Flow over an open cavity for Re=7500. Spectrum of the flow with the eigenvalues colored according to the criterion Γj. Adapted from Ref. 141.

Figure 18

Flow over an open cavity for Re=7500. Transfer function |Ĥ(ω)| representative of the input-output dynamics of the stable subspace. Adapted from Ref. 141.

Figure 19

Flow over an open cavity for Re=7500. (a) Singular values of the Hankel matrix. ((b)–(e)) Streamwise velocity of the 1st, 2nd, 9th, and 13th balanced modes. Adapted from Ref. 141.

Figure 20

Flow over an open cavity for Re=7500. Approximation error of reduced-order-models versus their dimension. (a) balanced modes, (b) global modes, and (c) POD-modes. In (a), the continuous curves represent the upper and lower bounds of the error 52,53. Adapted from Ref. 141.

Figure 21

Flow over an open cavity for Re=7500. Linearized direct numerical simulations with a controller and an estimator obtained by the LQG approach. (a) Reduced-order-model consisting of eight unstable global modes and p balanced modes. (b) Likewise, but p POD-modes. Adapted from Ref. 141.

Figure 22

Boundary-layer flow over a flat plate for Re=200,000. (a) Frequency response of the flow μ12(ω), (b) real part of streamwise momentum forcing for f̂1 at ων/U∞2=0.00018, and (c) associated optimal response û1 (real part of streamwise velocity).

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