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

Two-dimensional (2D) flows can be controlled efficiently using spanwise “waviness,” i.e., a control (e.g., wall blowing and 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 three-dimensional (3D) spanwise-periodic control. Building on previous work for parallel flows, 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 suboptimal controls) for stabilization and 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: (i) wall actuation via blowing and/or suction and (ii) wall deformation. By decomposing the eigenvalue variation, we 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 wave numbers, the optimal control for flow stabilization is symmetric about the wake centerline, 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.

Figure 1

Sketch of the steady spanwise-periodic control considered in this study: (a) wall blowing and/or suction and (b) wall deformation.

Figure 2

Sketch of the flow modification (velocity isosurfaces) induced by a steady spanwise-periodic control (Fig. 1): first-order spanwise-periodic modification (red and blue; here corresponding to low- and high-speed streamwise streaks) and spanwise-invariant second-order modification (mean flow correction, green). (The spanwise-periodic component of the second-order modification is not shown.) The net linear effect of ${\mathbf{U}}_1$ on the eigenvalue is zero, $\epsilon {\lambda }_1=0$. By contrast, ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$ have a nonzero quadratic effect ${\epsilon }^2{\lambda }_2$.

Figure 3

(a) Normalized growth rate variation induced by the optimal wall control ${\mathbf{U}}_w$ for stabilization (${\lambda }_{2r}<0$) and destabilization (${\lambda }_{2r}>0$). Optimal ($k=1$) and first suboptimals ($k=2$, 3). $\text{Re}=50.$ (b) Close-up view of suboptimals. In this and subsequent figures, increasing values of $k$ are shown with thinner lines and lighter hues.

Figure 4

Effect of the optimal (a) destabilizing and (b) stabilizing wall blowing and/or suction on growth rate $\sigma$ and frequency $\omega$. Lines, sensitivity prediction; symbols, full stability analysis. $\text{Re}=50$, $\beta =1$.

Figure 5

Optimal wall control ${\tilde{\mathbf{U}}}_w$ for destabilization or stabilization: left, arrows of in-plane velocity components; right, full velocity field variation with $\theta$. $\text{Re}=50$, $\beta =1$. (a) Optimal destabilizing wall control; (b) optimal stabilizing wall control; and (c) first suboptimal stabilizing wall control.

Figure 6

First-order flow modification ${\mathbf{U}}_1={[{\tilde{U}}_1(x,y)cos\left(\beta z\right),{\tilde{V}}_1(x,y)cos\left(\beta z\right),{\tilde{W}}_1(x,y)sin\left(\beta z\right)]}^T$ (spanwise periodic) induced by the (a) optimal destabilizing, (b) optimal stabilizing, and (c) first suboptimal stabilizing wall control, at $\text{Re}=50$, $\beta =1$. Left: vector fields ${(U_1,V_1)}^T$ at $z=0$ and contours of $W_1$ at $z=\pi /\left(2\beta \right)$. Middle: vector field ${(V_1,W_1)}^T$ and contours of $U_1$ at $x=2$. Right: induced mean flow correction ${\mathbf{U}}_2^{2D}={(U_2^{2D}(x,y),V_2^{2D}(x,y),0)}^T$ (spanwise invariant), shown with vector field ${(U_2^{2D},V_2^{2D})}^T$ and contours of velocity magnitude.

Figure 7

(a) Energy density $E\left(x\right)$ of the flow modification ${\mathbf{Q}}_1$ induced by the optimal stabilizing wall control, $\beta =0.5$, 1, and 2. (b) Maximum energy density and location of the maximum as functions of spanwise wave number. $\text{Re}=50.$

Figure 8

Variation with Reynolds number. (a) Normalized growth rate variation (in absolute value), and maximum energy density, both for the flow modification induced by the optimal wall control for stabilization. (b) Wall control amplitude ${\epsilon }_s=\sqrt{{\lambda }_{0r}/\left(\right|{\lambda }_{2r}|/||{\tilde{\mathbf{U}}}_w{\left|\right|}^2)\phantom{{}^{2^2}}}$ necessary to fully stabilize the leading eigenvalue. $\beta =1$.

Figure 9

Decomposition into amplification and stabilization: (a) gain $G^2=\left|\right|{\mathbf{Q}}_1{\left|\right|}^2/\left|\right|{\mathbf{U}}_w{\left|\right|}^2$ from wall control to resulting flow modification and (b) growth rate reduction induced by unit flow modification (same data as Fig. 3, normalized by the response norm $\left|\right|{\mathbf{Q}}_1{\left|\right|}^2$ rather than the control norm $\left|\right|{\mathbf{U}}_w{\left|\right|}^2$). Optimal ($k=1$) and first suboptimal ($k=2$) stabilizing wall control. $\text{Re}=50.$

Figure 10

The 3D contribution ($--$, interaction between 3D fields ${\mathbf{Q}}_1$ and ${\mathbf{q}}_1$) and 2D contribution ($-·-$, interaction between 2D fields ${\mathbf{Q}}_2^{2D}$ and ${\mathbf{q}}_0$) to the total growth rate variation (solid line) induced by (a) the optimal ($k=1$) and (b) first suboptimal ($k=2$) stabilizing wall control ${\mathbf{U}}_w$. $\text{Re}=50.$ See text for details.

Figure 11

Normalized effect on frequency of the optimal stabilizing wall control ${\mathbf{U}}_w$. $\text{Re}=50.$

Figure 12

Normalized growth rate variation for several optimal stabilizing wall controls ${\mathbf{U}}_w$: full 3D optimal, in-plane optimal, normal optimal, tangential optimal. Also shown is the variation for the tangential wall control equivalent to the optimal wall deformation $R_1$ (see Sec. 5). $\text{Re}=50$, $\beta =1$.

Figure 13

Optimal stabilizing ${\tilde{\mathbf{U}}}_w$ normalized to 1: (a) full optimal 3D control, (b) in-plane optimal control, and (c) normal optimal control. $\text{Re}=50$, $\beta =1$.

Figure 14

(a) Normalized growth rate variation induced by the optimal wall deformation $R_1$ for stabilization (${\lambda }_{2r}<0$) and destabilization (${\lambda }_{2r}>0$). Optimal ($k=1$) and first suboptimals ($k=2$, 3). $\text{Re}=50.$ (b) The effect of wall deformation via the eigenmode's boundary condition is much smaller than via the base flow modification: evaluation of (52) with ${\tilde{\mathbf{q}}}_1$ (solid line), and difference between the evaluations with ${\tilde{\mathbf{q}}}_{1,\mathrm{def}}$ and ${\tilde{\mathbf{q}}}_1$ (dashed line). See text for details.

Figure 15

(a) Optimal wavy cylinder for stabilization. Radius $R\left(\theta \right)=R_0+\epsilon {\tilde{R}}_1\left(\theta \right)cos\left(\beta z\right)$ shown here at $z=0$, and with the amplitude $\epsilon =0.023$ that just brings the flow back to marginal stability. (Dashed line: straight cylinder $R=R_0$.) (b) Optimal wall deformation for stabilization ($\left|\right|{\tilde{R}}_1\left|\right|=1$), and equivalent tangential actuation. $\text{Re}=50$, $\beta =1$.

Figure 16

Effect of the optimal stabilizing wall deformation $R_1$ on growth rate and frequency. Line, sensitivity prediction; symbols, 3D stability analysis (circles $•$, wavy cylinder; squares $\blacksquare$, equivalent tangential blowing and/or suction). $\text{Re}=50$, $\beta =1$.

Figure 17

First-order flow modification ${\mathbf{U}}_1={[{\tilde{U}}_1(x,y)cos\left(\beta z\right),{\tilde{V}}_1(x,y)cos\left(\beta z\right),{\tilde{W}}_1(x,y)sin\left(\beta z\right)]}^T$ (spanwise periodic) induced by the optimal stabilizing wall deformation (or equivalent tangential wall control) at $\text{Re}=50$, $\beta =1$. Left: vector fields ${(U_1,V_1)}^T$ at $z=0$ and contours of $W_1$ at $z=\pi /\left(2\beta \right)$. Middle: vector field ${(V_1,W_1)}^T$ and contours of $U_1$, at $x=2$. Right: induced mean flow correction ${\mathbf{U}}_2^{2D}={[U_2^{2D}(x,y),V_2^{2D}(x,y),0]}^T$ (spanwise invariant), shown with vector field ${(U_2^{2D},V_2^{2D})}^T$ and contours of velocity magnitude.

Figure 18

Variation with Reynolds number. (a) Normalized growth rate variation (in absolute value), and maximum energy density, both for the flow modification induced by the optimal wall deformation for stabilization. (b) Wall deformation amplitude ${\epsilon }_s=\sqrt{{\lambda }_{0r}/\left(\right|{\lambda }_{2r}|/||{\mathbf{U}}_w{\left|\right|}^2)}$ necessary to fully stabilize the leading eigenmode. $\beta =1$.

Figure 19

Normalized frequency variation induced by the optimal wall actuation ${\mathbf{U}}_w$ for frequency increase (${\lambda }_{2i}>0$) and frequency reduction (${\lambda }_{2i}<0$). Optimal ($k=1$) and first suboptimals ($k=2,3$). $\text{Re}=50.$

Figure 20

Normalized effect on growth rate of the optimal (a) frequency-increasing and (b) frequency-reducing wall control ${\mathbf{U}}_w$. 3D ($--$) and 2D ($-·-$) contributions. $\text{Re}=50.$

Figure 21

Optimal (a) frequency-increasing and (b) frequency-reducing wall control ${\mathbf{U}}_w$. $\text{Re}=50$, $\beta =1$.

Figure 22

First-order flow modification ${\mathbf{U}}_1={[{\tilde{U}}_1(x,y)cos\left(\beta z\right),{\tilde{V}}_1(x,y)cos\left(\beta z\right),{\tilde{W}}_1(x,y)sin\left(\beta z\right)]}^T$ (spanwise periodic) induced by the optimal (a) frequency-increasing and (b) frequency-reducing wall control, at $\text{Re}=50$, $\beta =1$. Left: vector fields ${(U_1,V_1)}^T$ at $z=0$ and contours of $W_1$ at $z=\pi /\left(2\beta \right)$. Middle: vector field ${(V_1,W_1)}^T$ and contours of $U_1$, at $x=2$. Right: induced mean flow correction ${\mathbf{U}}_2^{2D}={(U_2^{2D}(x,y),V_2^{2D}(x,y),0)}^T$ (spanwise invariant), shown with vector field ${(U_2^{2D},V_2^{2D})}^T$ and contours of velocity magnitude.

Figure 23

(a) The 2D sensitivity of the leading eigenvalue's growth rate to 2D (spanwise-invariant) volume control (arrows show the $x$ and $y$ components, while the $z$ component is zero by definition). (b) Optimal stabilizing spanwise-periodic volume control $\mathbf{C}$ at $\text{Re}=50$, $\beta =1$. Left, right: cuts at $z=0$ and $x=2$, respectively, showing the in-plane vector field and contours of the out-of-plane component. (c) Flow modification induced by the optimal stabilizing volume control of panel $b$. Left, middle: first-order modification ${\mathbf{U}}_1(x,y,z)$ (spanwise harmonic) at $z=0$ and $x=2$, respectively, shown with in-plane velocity vector fields and contours of out-of-plane velocity. Right: induced mean flow correction ${\mathbf{U}}_2^{2D}={[U_2^{2D}(x,y),V_2^{2D}(x,y),0]}^T$ (spanwise invariant), shown with vector field ${(U_2^{2D},V_2^{2D})}^T$ and contours of velocity magnitude.

Figure 24

[(a)–(d)] The 3D eigenvalues of the uncontrolled 2D base flow: [(a), (b)] growth rate vs spanwise wave number (circles, unsteady eigenmodes $\omega \ne 0$; crosses, steady eigenmodes $\omega =0$); [(c), (d)] growth rate vs frequency (closeup of the leading eigenvalue). [(a), (c)] $\text{Re}=80$; [(b), (d)] $\text{Re}=50$. (e) Minimum distance in the complex plane $(\sigma ,\omega )$ from 3D eigenvalues (${\beta }_0>0$) to the 2D leading eigenvalue (${\beta }_0=0$). In all panels, spanwise wave numbers ${\beta }_0=0$, 0.4, and 0.6 are shown in black, red and blue, respectively.