Reducing the skin-friction drag of a turbulent boundary-layer flow with low-amplitude wall-normal blowing within a Bayesian optimization framework

A Bayesian optimization framework is developed to optimize low-amplitude wall-normal blowing control of a turbulent boundary-layer flow. The Bayesian optimization framework determines the optimum blowing amplitude and blowing coverage to achieve up to a 5% net-power saving solution within 20 optimization iterations, requiring 20 direct numerical simulations (DNS). The power input required to generate the low-amplitude wall-normal blowing is measured experimentally for two different types of blowing device and is used in the simulations to assess control performance. Wall-normal blowing with amplitudes of less than 1% of the free-stream velocity generate a skin-friction drag reduction of up to 76% over the control region, with a drag reduction which persists for up to $650{\delta }_0$ downstream of actuation (where ${\delta }_0$ is the boundary-layer thickness at the start of the simulation domain). It is shown that it is the slow spatial recovery of the turbulent boundary-layer flow downstream of control which generates the net-power savings in this study. The downstream recovery of the skin-friction drag force is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

Figure 1

Schematic of the computational domain. The control region is highlighted in pink.

Figure 2

(a) Streamwise evolution of the friction coefficient as a function of the Reynolds number for the canonical and controlled turbulent boundary-layer flows. (b) Mean and fluctuating streamwise velocity profiles of canonical turbulent boundary-layer flows at ${\text{Re}}_{\theta }=1000$ (black lines) and ${\text{Re}}_{\theta }=1410$ (blue lines). Solid lines correspond to the present results and the dashed lines to the reference data of Schlatter and Örlü [33].

Figure 3

An example of using Bayesian optimization to locate a local minimum on a 1D toy problem. The black curve is the true objective function, black solid circles are the observed points, the dark blue curve is the posterior mean $\mu$, and the blue shaded area is the posterior uncertainty $(\mu \pm \sigma )$.

Figure 4

Schematic of the boundary condition of the wall-normal velocity in the control region for the Bayesian optimization studies.

Figure 5

(a) Streamwise evolution of the skin-friction coefficient as a function of ${\text{Re}}_{\theta }$ for all cases. The thick solid lines corresponds to the three cases of interest in this study: Case 0 (black), Case 5 (red), and Case 13 (blue). (b) Streamwise evolution of ${\text{Re}}_{\delta }$ for Case 0, Case 5, and Case 13. The region of wall-normal blowing control is highlighted in pink.

Figure 6

Instantaneous visualizations of the streamwise velocity in the streamwise-spanwise plane at $y^{+}\approx 10$ for Case 0, Case 5, and Case 13. Dark blue corresponds to a zero velocity and red correspond to $U_{\infty }$.

Figure 7

3D instantaneous visualizations of the enstrophy field for Case 0, Case 5, and Case 13, colored by the wall-normal velocity component.

Figure 8

Streamwise evolution of the FIK identity contributions for Case 0 (a), Case 5 (b), and Case 13 (c).

Figure 9

Streamwise evolution of the four terms of the spatial development contribution of the FIK identity for Case 0 (a), Case 5 (b), and Case 13 (c).

Figure 10

(a) Picture of the Visaton K 50 miniature speaker used to generate wall-normal blowing. (b) schematic circuit diagram for the miniature speaker setup.

Figure 11

(a) Instantaneous wall-normal velocity and input power to the speaker driven at 4 V, 440 Hz at 50% duty cycle. (b) Time-averaged wall-normal velocity versus power per unit area from 1.5–5 V over 400–500 Hz at 50% duty cycle. All data taken at the center of the miniature speaker at $y/d\sim 5$.

Figure 12

(a) Streamwise evolution of the friction coefficient as a function of ${\text{Re}}_{\theta }$ for all the cases. The thick solid lines corresponds to the three cases of interest in this study: Case 0 (black), Case 6 (blue), and Case 8 (red). (b) Streamwise evolution of ${\text{Re}}_{\delta }$ (based on the boundary-layer thickness) for the uncontrolled case, Case 6 and Case 8. The blowing section is highlighted in pink.

Figure 13

Shape of the mean (left) and covariance (right) of the objective function (net-power saving) as a function of the three parameters $C_B,\alpha$, and $N_B$ for the first Bayesian optimization (top) and combined with the extra data from the second Bayesian optimization (bottom). Net-power saving is shown as positive.

Figure 14

Shape of the mean (left) and covariance (right) of objective function (net-power saving) as a function of the three parameters $C_B,\alpha$, and $N_B$ for the second Bayesian optimization (top) and combined with the extra data from the first Bayesian optimization (bottom). Net-power saving is shown as positive.