^{1,a)}, U. Ehrenstein

^{2,b)}and F. Gallaire

^{1,c)}

### Abstract

Linear optimal gains are computed for the subcritical two-dimensional separated boundary-layer flow past a bump. Very large optimal gain values are found, making it possible for small-amplitude noise to be strongly amplified and to destabilize the flow. The optimal forcing is located close to the summit of the bump, while the optimal response is the largest in the shear layer. The largest amplification occurs at frequencies corresponding to eigenvalues which first become unstable at higher Reynolds number. Nonlinear direct numerical simulations show that a low level of noise is indeed sufficient to trigger random flow unsteadiness, characterized here by large-scale vortex shedding. Next, a variational technique is used to compute efficiently the sensitivity of optimal gains to steady control (through source of momentum in the flow, or blowing/suction at the wall). A systematic analysis at several frequencies identifies the bump summit as the most sensitive region for control with wall actuation. Based on these results, a simple open-loop control strategy is designed, with steady wall suction at the bump summit. Linear calculations on controlled base flows confirm that optimal gains can be drastically reduced at all frequencies. Nonlinear direct numerical simulations also show that this control allows the flow to withstand a higher level of stochastic noise without becoming nonlinearly unstable, thereby postponing bypass transition. In the supercritical regime, sensitivity analysis of eigenvalues supports the choice of this control design. Full restabilization of the flow is obtained, as evidenced by direct numerical simulations and linear stability analysis.

This work is supported by the Swiss National Science Foundation (NSF(CH)) (Grant No. 200021-130315) and the French National Research Agency (Project No. ANR-09-SYSC-001).

I. INTRODUCTION

II. PROBLEM DESCRIPTION AND GOVERNING EQUATIONS

III. RESPONSE TO FORCING: NOISE AMPLIFICATION

A. Base flow

B. Optimal gain

C. Direct numerical simulations

1. DNS with harmonic forcing

2. DNS with stochastic forcing

IV. SENSITIVITY ANALYSIS

A. Sensitivity of optimal gain

B. Reduction of nonlinear noise amplification

V. FLOW STABILIZATION

VI. CONCLUSIONS

### Key Topics

- Eigenvalues
- 26.0
- Flow instabilities
- 23.0
- Flow control
- 20.0
- Reynolds stress modeling
- 11.0
- Separated flows
- 10.0

## Figures

Bump geometry y = y b (x), inlet velocity profile (U, V) = (U Blasius , 0), time-dependent forcing F(t), steady volume control C, and steady wall control U c .

Bump geometry y = y b (x), inlet velocity profile (U, V) = (U Blasius , 0), time-dependent forcing F(t), steady volume control C, and steady wall control U c .

Recirculation length as function of Reynolds number. Solid line: steady-state base flow calculated in the present study. Symbols: steady state computations (+) and time-averaged recirculation length of oscillatory flow field (×) obtained by Marquillie and Ehrenstein. ^{31}

Recirculation length as function of Reynolds number. Solid line: steady-state base flow calculated in the present study. Symbols: steady state computations (+) and time-averaged recirculation length of oscillatory flow field (×) obtained by Marquillie and Ehrenstein. ^{31}

(a) Optimal linear gain at , 400, 500, and 580. (b) Variation of the maximal optimal gain with Reynolds number, and (c) frequency of this maximum.

(a) Optimal linear gain at , 400, 500, and 580. (b) Variation of the maximal optimal gain with Reynolds number, and (c) frequency of this maximum.

(a) Optimal forcing and (b) optimal response at for different frequencies ω. The real part of the streamwise component is shown. The dashed line shows the base flow separating streamline.

(a) Optimal forcing and (b) optimal response at for different frequencies ω. The real part of the streamwise component is shown. The dashed line shows the base flow separating streamline.

Spatial structure of the divergence-free Gaussian forcing (8) : (a) streamwise and (b) cross-stream components.

Spatial structure of the divergence-free Gaussian forcing (8) : (a) streamwise and (b) cross-stream components.

Response to harmonic forcing at , ω = 0.25. (a) Time evolution of the energy of the perturbations. Dashed lines correspond to A = 2 × 10^{−6} and 3 × 10^{−6}. (b) Mean asymptotic energy in the steady-state regime as function of the forcing amplitude A. The solid line has a slope 2. (c) Time series of the streamwise perturbation velocity u ^{′} at x = 80, y = 1, for A = 10^{−7}, 10^{−6}, 10^{−5}. (d) Power spectrum of this velocity for forcing amplitudes A = 10^{−7} (dashed-dotted line), A = 10^{−6} (dashed line), A = 10^{−5} (solid line) (arbitrary unit, logarithmic scale).

Response to harmonic forcing at , ω = 0.25. (a) Time evolution of the energy of the perturbations. Dashed lines correspond to A = 2 × 10^{−6} and 3 × 10^{−6}. (b) Mean asymptotic energy in the steady-state regime as function of the forcing amplitude A. The solid line has a slope 2. (c) Time series of the streamwise perturbation velocity u ^{′} at x = 80, y = 1, for A = 10^{−7}, 10^{−6}, 10^{−5}. (d) Power spectrum of this velocity for forcing amplitudes A = 10^{−7} (dashed-dotted line), A = 10^{−6} (dashed line), A = 10^{−5} (solid line) (arbitrary unit, logarithmic scale).

Actual response to harmonic forcing F = f(x, y)e ^{ iωt } with the particular choice (8) for the spatial structure f. Solid line: actual linear gain G lin ; Symbols: linear DNS gain G DNS obtained from DNS calculations with small-amplitude forcing. The dashed line indicates for reference the optimal gain G opt (reported from Figure 3 ).

Actual response to harmonic forcing F = f(x, y)e ^{ iωt } with the particular choice (8) for the spatial structure f. Solid line: actual linear gain G lin ; Symbols: linear DNS gain G DNS obtained from DNS calculations with small-amplitude forcing. The dashed line indicates for reference the optimal gain G opt (reported from Figure 3 ).

Response to stochastic forcing at . (a) Time evolution of the perturbation energy E p . Dashed lines correspond to A = 3 × 10^{−5} and 3 × 10^{−4}. (b) Mean asymptotic energy in the steady-state regime as function of the forcing amplitude A. Time series of the streamwise perturbation velocity u ^{′} measured at y = 1 and x = 80 and 140 for (c) A = 10^{−7} and (d) A = 10^{−5}. Power spectrum of the streamwise velocity measured at y = 1 and x = 80, 100, 120, 140, for (e) A = 10^{−7} and (f) A = 10^{−5}. For reference, the thick line shows the (uncontrolled) linear gain G lin (ω) from Figure 7 (arbitrary unit, linear scale).

Response to stochastic forcing at . (a) Time evolution of the perturbation energy E p . Dashed lines correspond to A = 3 × 10^{−5} and 3 × 10^{−4}. (b) Mean asymptotic energy in the steady-state regime as function of the forcing amplitude A. Time series of the streamwise perturbation velocity u ^{′} measured at y = 1 and x = 80 and 140 for (c) A = 10^{−7} and (d) A = 10^{−5}. Power spectrum of the streamwise velocity measured at y = 1 and x = 80, 100, 120, 140, for (e) A = 10^{−7} and (f) A = 10^{−5}. For reference, the thick line shows the (uncontrolled) linear gain G lin (ω) from Figure 7 (arbitrary unit, linear scale).

Subharmonic instability occurs as a manifestation of nonlinear effects when forcing amplitude is large enough. Amplitude of the stochastic forcing: (a) A = 10^{−7}, (b) A = 10^{−5}, (c) A = 3 × 10^{−5}, (d) A = 10^{−4}. Contours of streamwise perturbation velocity, t = 2000, . The axes are not to scale.

Subharmonic instability occurs as a manifestation of nonlinear effects when forcing amplitude is large enough. Amplitude of the stochastic forcing: (a) A = 10^{−7}, (b) A = 10^{−5}, (c) A = 3 × 10^{−5}, (d) A = 10^{−4}. Contours of streamwise perturbation velocity, t = 2000, . The axes are not to scale.

Normalized sensitivity of optimal gain to base flow modification in the streamwise direction, , at and frequencies ω = 0.05, 0.15, …0.55. The vertical dashed line is the base flow separatrix. The axes are not to scale.

Normalized sensitivity of optimal gain to base flow modification in the streamwise direction, , at and frequencies ω = 0.05, 0.15, …0.55. The vertical dashed line is the base flow separatrix. The axes are not to scale.

Sensitivity of optimal gain to control at and frequencies ω = 0.05, 0.15, …0.55. (a) Normalized streamwise component of the sensitivity to volume control, . Black circles indicate the location of volume control (x, y) = (75, 3.5) discussed in the text and in Figure 12 . The axes are not to scale. (b) Normalized sensitivity to wall control, , rescaled for each frequency by the largest point-wise L ^{2} norm on the wall . This maximal value is shown by symbols in the inset (where the solid line is an indicative fit through the data). The grey region shows the streamwise extension of the bump. The dashed line is the base flow separatrix.

Sensitivity of optimal gain to control at and frequencies ω = 0.05, 0.15, …0.55. (a) Normalized streamwise component of the sensitivity to volume control, . Black circles indicate the location of volume control (x, y) = (75, 3.5) discussed in the text and in Figure 12 . The axes are not to scale. (b) Normalized sensitivity to wall control, , rescaled for each frequency by the largest point-wise L ^{2} norm on the wall . This maximal value is shown by symbols in the inset (where the solid line is an indicative fit through the data). The grey region shows the streamwise extension of the bump. The dashed line is the base flow separatrix.

Variation of the optimal gain at when applying at (x, y) = (75, 3.5) a steady volume control of amplitude C x in the streamwise direction. (a) Prediction from sensitivity analysis (SA, red solid line) and nonlinear controlled base flows (NL, blue symbols) at ω = 0.25. The main plot is in logarithmic scale, the inset in linear scale (the sensitivity is a straight line). (b) G opt (ω) for C x = 0 (thick solid line), C x = −0.01 (thin solid line), and C x = −0.02 (dashed line).

Variation of the optimal gain at when applying at (x, y) = (75, 3.5) a steady volume control of amplitude C x in the streamwise direction. (a) Prediction from sensitivity analysis (SA, red solid line) and nonlinear controlled base flows (NL, blue symbols) at ω = 0.25. The main plot is in logarithmic scale, the inset in linear scale (the sensitivity is a straight line). (b) G opt (ω) for C x = 0 (thick solid line), C x = −0.01 (thin solid line), and C x = −0.02 (dashed line).

Variation of the optimal gain at when applying vertical wall blowing/suction at the bump summit. (a) Prediction from sensitivity analysis (SA, red solid line) and nonlinear controlled base flows (NL, blue symbols). The main plot is in logarithmic scale and shows that varies exponentially with flow rate. In linear scale (inset), the sensitivity is a straight line. (b) Reduction of G opt (ω) with flow rates W = −0.010, −0.035, −0.100.

Variation of the optimal gain at when applying vertical wall blowing/suction at the bump summit. (a) Prediction from sensitivity analysis (SA, red solid line) and nonlinear controlled base flows (NL, blue symbols). The main plot is in logarithmic scale and shows that varies exponentially with flow rate. In linear scale (inset), the sensitivity is a straight line. (b) Reduction of G opt (ω) with flow rates W = −0.010, −0.035, −0.100.

Effect of wall suction on harmonic response. Upper line and symbols (reported from Figure 7 ) show the actual gain in the uncontrolled case; lower line and symbols are for wall suction at the bump summit with flow rate W = −0.035. Solid lines: linear results G lin ; symbols: G DNS from DNS calculations with small-amplitude harmonic forcing.

Effect of wall suction on harmonic response. Upper line and symbols (reported from Figure 7 ) show the actual gain in the uncontrolled case; lower line and symbols are for wall suction at the bump summit with flow rate W = −0.035. Solid lines: linear results G lin ; symbols: G DNS from DNS calculations with small-amplitude harmonic forcing.

Mean asymptotic energy of the perturbations vs. forcing amplitude, at . Open symbols: without control; Filled symbols: with vertical wall suction at the bump summit (flow rate W = −0.035). (a) Harmonic forcing at ω = 0.25 (circles) and ω = 0.35 (triangles); (b) stochastic forcing.

Mean asymptotic energy of the perturbations vs. forcing amplitude, at . Open symbols: without control; Filled symbols: with vertical wall suction at the bump summit (flow rate W = −0.035). (a) Harmonic forcing at ω = 0.25 (circles) and ω = 0.35 (triangles); (b) stochastic forcing.

Flow restabilization at in direct numerical simulations with steady vertical wall suction at the bump summit (flow rate W = −0.035). (a) Energy of the perturbations (calculated with the final steady-state as reference base flow). (b) Streamwise velocity of the total flow at (x, y) = (80, 1). The subcritical flow, stationary for t < 0, is perturbed from t = 0 with stochastic forcing of amplitude A = 3 × 10^{−4}, and control is turned on at t = 1000.

Flow restabilization at in direct numerical simulations with steady vertical wall suction at the bump summit (flow rate W = −0.035). (a) Energy of the perturbations (calculated with the final steady-state as reference base flow). (b) Streamwise velocity of the total flow at (x, y) = (80, 1). The subcritical flow, stationary for t < 0, is perturbed from t = 0 with stochastic forcing of amplitude A = 3 × 10^{−4}, and control is turned on at t = 1000.

Global linear eigenspectrum at of the uncontrolled flow and of the flow controlled with vertical wall suction at the bump summit with flow rate W = −0.015, −0.025, −0.035, −0.040.

Global linear eigenspectrum at of the uncontrolled flow and of the flow controlled with vertical wall suction at the bump summit with flow rate W = −0.015, −0.025, −0.035, −0.040.

Sensitivity analysis of the most unstable eigenvalues at . (a) Sensitivity of the growth rate of modes 1–9 (Kelvin-Helmholtz branch) to vertical wall control. The dashed line shows the bump summit location. (b) Effect of vertical wall control at the bump summit, as predicted by sensitivity analysis. Red solid lines indicate a flow rate W = −0.005. The lower panel is a close-up view of eigenvalues 1–3, comparing sensitivity analysis (SA, red solid lines) and linear stability analysis results for nonlinear base flows controlled with W = −0.001 and −0.002 (NL, blue circles).

Sensitivity analysis of the most unstable eigenvalues at . (a) Sensitivity of the growth rate of modes 1–9 (Kelvin-Helmholtz branch) to vertical wall control. The dashed line shows the bump summit location. (b) Effect of vertical wall control at the bump summit, as predicted by sensitivity analysis. Red solid lines indicate a flow rate W = −0.005. The lower panel is a close-up view of eigenvalues 1–3, comparing sensitivity analysis (SA, red solid lines) and linear stability analysis results for nonlinear base flows controlled with W = −0.001 and −0.002 (NL, blue circles).

(a) and (b) Flow restabilization at in direct numerical simulations with steady vertical wall suction at the bump summit (flow rate W = −0.035). Same notations as Figure 16 . The supercritical flow is naturally unsteady, no perturbation is added, and control is turned on at t = 1000. Dots correspond to the times of snapshots in Figure 20 .

(a) and (b) Flow restabilization at in direct numerical simulations with steady vertical wall suction at the bump summit (flow rate W = −0.035). Same notations as Figure 16 . The supercritical flow is naturally unsteady, no perturbation is added, and control is turned on at t = 1000. Dots correspond to the times of snapshots in Figure 20 .

Flow restabilization in the supercritical regime, , in DNS with steady vertical wall suction at the bump summit (flow rate W = −0.035): contours of vorticity of the total flow at t = 0, 500, 1000… 2500. The black dot shows the location of the point (x, y) = (80, 1) where the velocity signal of Figure 19 is recorded. The axes are not to scale.

Flow restabilization in the supercritical regime, , in DNS with steady vertical wall suction at the bump summit (flow rate W = −0.035): contours of vorticity of the total flow at t = 0, 500, 1000… 2500. The black dot shows the location of the point (x, y) = (80, 1) where the velocity signal of Figure 19 is recorded. The axes are not to scale.

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