Second-order sensitivity in the cylinder wake: Optimal spanwise-periodic wall actuation and wall deformation

Abstract

Two-dimensional (2D) flows can be controlled efficiently using spanwise "waviness", i.e. a control (e.g. wall blowing/suction or wall deformation) that is periodic in the spanwise direction. This study tackles the global linear stability of 2D flows subject to small-amplitude 3D spanwise-periodic control. Building on previous work for parallel flows [1], an adjoint method is proposed for computing the second-order sensitivity of eigenvalues. Since such control has indeed a zero net first-order (linear) effect, the second-order (quadratic) effect prevails. The sensitivity operator allows one (i) to predict the effect of any control without actually computing the controlled flow, and (ii) to compute the optimal control (and an orthogonal set of sub-optimal controls) for stabilization/destabilization or frequency modification. The proposed method takes advantage of the very spanwise-periodic nature of the control to reduce computational complexity (from a fully 3D problem to a 2D problem). The approach is applied to the leading eigenvalue of the laminar flow around a circular cylinder, and two kinds of spanwise-harmonic control are explored: wall actuation via blowing/suction, and wall deformation. Decomposing the eigenvalue variation, it is found that the 3D contribution (from the spanwise-periodic first-order flow modification) is generally larger than the 2D contribution from the mean flow correction (spanwise-invariant second-order flow modification). Over a wide range of control spanwise wavenumber, the optimal control for flow stabilization is top-down symmetric, leading to varicose streaks in the cylinder wake. Analyzing the competition between amplification and stabilization shows that optimal varicose streaks are not significantly more amplified than sinuous streaks but have a stronger stabilizing effect. The optimal wall deformation induces a flow modification very similar to that induced by the optimal wall actuation. In general, spanwise and tangential actuation have a small contribution to the optimal control, so normal-only actuation is a good trade-off between simplicity and effectiveness. Our method opens the way to the systematic design of optimal spanwise-periodic control for a variety of control objectives other than linear stability properties.

The effect of streaks on the instability of jets

“…The same observation had been reported before from studies of instability control in plane two-dimensional flows via spanwise-periodic base flow modifications (Hwang & Choi 2006; Tammisola et al. 2014; Boujo, Fani & Gallaire 2019). If, however, in the present configuration, rolls and streaks in jets cause first-order variations in the instability eigenvalues, then a sensitivity analysis will allow us to identify roll shapes that optimally destabilise linear eigenmodes.…”

“…In the context of streaks in plane shear layers, Marant & Cossu (2018) demonstrated that this analysis needs to be expanded to second order, if one wishes to correctly retrieve the quadratic dependency of eigenmodes on streak amplitude. The same observation had been reported before from studies of instability control in plane two-dimensional flows via spanwise-periodic base flow modifications (Hwang & Choi 2006;Tammisola et al 2014;Boujo, Fani & Gallaire 2019). If, however, in the present configuration, rolls and streaks in jets cause first-order variations in the instability eigenvalues, then a sensitivity analysis will allow us to identify roll shapes that optimally destabilise linear eigenmodes.…”

“…Optimisation could be performed on the second-order sensitivity operator to identify azimuthally non-uniform base flow modifications for maximum change of an eigenmode (Boujo, Fani & Gallaire 2015; Boujo et al. 2019).

Figure 5.First-order sensitivity of the growth rate with respect to base flow modifications: ( a ) an instability eigenmode () of the axisymmetric base flow (axial velocity); associated sensitivity with respect to ( b ) radial velocity, ( c ) azimuthal velocity and ( d ) streamwise velocity changes in the base flow.

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Non-uniform suction control of flow around a circular cylinder

“…The separation angle of the flow was not investigated. Very recently, the optimum spanwise-varying suction/blowing control of a 3D circular cylinder in 2D flow was determined using eigenmode analysis by Boujo et al 21 Due to the nature of this method, though, it can only be used to optimise for stabilisation or frequency modification which does not necessarily coincide with minimised drag or the elimination of separation. All of these studies were performed by numerical methods; non-uniform suction has not been explored substantially by physical experimentation.…”

Sensitivity gradients of surface geometry modifications based on stability analysis of compressible flows

“…On the stabilization of three-dimensional flows, Tammissola [20] and Tammissola et al [21] used a second-order perturbation analysis on the optimization of spanwise-periodic shaping and actuation of a cylindrical body to stabilize the vortex shedding. Boujo et al [22,23] developed an adjoint method to extract the second-order sensitivity of the leading eigenvalues, predicting optimal wall actuation and wall deformation on spanwise-periodic flows, and reducing the complexity of a three-dimensional problem to a two-dimensional analysis. More recently, Brewster and Juniper [24] employed a continuous adjoint approach to obtain the shape gradients of the eigenvalues related to the cylinder wake first instability, and analyze the effects of the local deformations on the flow instability.…”