Multifidelity kinematic parameter optimization of a flapping airfoil

We construct a multifidelity framework for the kinematic parameter optimization of flapping airfoil. We employ multifidelity Gaussian process regression and Bayesian optimization to effectively synthesize the aerodynamic performance of the flapping airfoil with the kinematic parameters under multiresolution numerical simulations. The objective of this work is to demonstrate that the multifidelity framework can efficiently discover the optimal kinematic parameters of the flapping airfoil with specific aerodynamic performance using a limited number of expensive high-fidelity simulations combined with a larger number of inexpensive low-fidelity simulations. We efficiently identify the optimal kinematic parameters of an asymmetrically flapping airfoil with various target aerodynamic forces in the design space of heaving amplitude, flapping frequency, angle of attack amplitude, and stroke angle. Notably, it is found that the angle of attack can significantly affect the magnitude of aerodynamic forces by facilitating the generation of the leading-edge vortex. In the meanwhile, its combination effect with the stroke angle can determine the attitude and trajectory of the flapping airfoil, thus further affect the direction of the aerodynamic forces. With the influence of the streamwise in-line motion, the asymmetrical vortex structures emerge in the wake fields because the streamwise velocities of shedding vortices are different in the upstroke and downstroke. Furthermore, we conduct the kinematic parameter optimization for a three-dimensional asymmetrically flapping wing. Compared to the two-dimensional simulations, we further investigate the flow induced by the vortex ring and its unsteady effects on the vortex structure and aerodynamic performance.

Figure 1

The sketch of the flapping airfoil. (a) The airfoil travels along a single stroke line with sinusoidal components as a function of time. This line is defined by the stroke angle $\beta$. (b) As the airfoil moves forward at a constant velocity, the stroke line translates, resulting in a skewed-harmonic trajectory for the airfoil. The $\alpha \left(t\right)$ denotes the instantaneous effective angle of attack of the flapping airfoil.

Figure 2

(a) Thrust coefficient and (b) lift coefficient under five meshes with different resolutions, the kinematic parameters are $f=0.25, h=1, {\theta }_0={30}^{\circ }$.

Figure 3

The sketch of the multifidelity framework. The close view of the two meshes and the corresponding vorticity fields with the kinematic parameters ($f=0.25, h=1, {\theta }_0={30}^{\circ }$) are shown in the left row. The multifidelity surrogate model correlating the pitching amplitude ${\theta }_0$ with time-averaged thrust coefficient $\overline{C_T}$ is shown in the right row, where std is the standard deviation.

Figure 4

(a) The time-averaged thrust coefficient response surface interpolated by 60 high-fidelity samples. (b) The absolute error between the interpolation surface and the predictions of the multifidelity framework constructed by 20 high-fidelity samples and 156 low-fidelity samples.

Figure 5

Correlation of time-averaged thrust coefficient between the high-fidelity (${\overline{C_{\text{T}}}}^{\text{HF}}$) versus (a) the low-fidelity (${\overline{C_{\text{T}}}}^{\text{LF}}$) and multifidelity (${\overline{C_{\text{T}}}}^{\text{MF}}$) with different number of training samples: (b) 10 high-fidelity samples; (c) 30 high-fidelity samples; (d) 60 high-fidelity samples.

Figure 6

Comparison between the single-fidelity and multifidelity frameworks: (a) the averaged squared error; (b) the variance of the squared error. The parameter plane is constructed by fixing the flapping frequency with 0.15 and varying the heaving and pitching amplitudes, the averaged value and variance of the squared error of the time-averaged thrust coefficient vary with the increase of the number of high-fidelity samples, and the number of low-fidelity samples is 156.

Figure 7

The flow diagram of the surrogate-based optimization framework.

Figure 8

The time-averaged thrust coefficient contours on the slices of the surrogate model of the symmetrically flapping airfoil: (a) three slices with various frequencies; (b) three slices with various heaving amplitudes; (c) three slices with various pitching amplitudes. The scaled variable is defined by $\frac{x-x_{\text{min}}}{x_{\text{max}}-x_{\text{min}}}$ thus it ranges from 0 to 1.

Figure 9

The flapping trajectory, aerodynamic forces, and instantaneous vorticity fields of the optimal case in the first symmetrically oscillating optimization; the optimal kinematic parameters are $f=0.2722, h=1.0274, {\theta }_0=0.2325$.

Figure 10

The flapping trajectory, aerodynamic forces, and instantaneous vorticity fields of the optimal case in the first asymmetrically oscillating optimization; the optimal kinematic parameters are $f=0.1660, h=1.1166, {\alpha }_0=0.6271, \beta =0.9972$.

Figure 11

The contours on the slice of the surrogate model of the asymmetrically flapping airfoil with fixed flapping frequency ($f=0.15$) and heaving amplitude ($h=1.25$): (a) time-averaged thrust coefficient $\overline{C_{\text{T}}}$; (b) time-averaged lift coefficient $\overline{C_{\text{L}}}$.

Figure 12

The optimal trajectories and force vectors of the seven optimization cases: (a) $\beta$ = 1.5338, birdlike; (b) $\beta$ = 1.7322, neutral; (c) $\beta$ = 1.8559, neutral; (d) $\beta$ = 2.2813, turtlelike; (e) $\beta$ = 2.3840, turtlelike; (f) $\beta$ = 1.6346, neutral; (g) $\beta$ = 1.3801, birdlike.

Figure 13

The instantaneous vorticity fields of the seven optimization cases at $t=\frac{1}{4}T,\frac{2}{4}T$.

Figure 14

The straight Naca0015 wing used for the three-dimensional simulation.

Figure 15

In the three-dimensional trajectory optimization, the squared errors of the three surrogate models: the single-fidelity model, the multifidelity model with two-dimensional low-fidelity samples, and the multifidelity model with three-dimensional low-fidelity samples. The number of two-dimensional low-fidelity samples is 1200 while the number of three-dimensional low-fidelity samples is 500.

Figure 16

(a) The flapping trajectory and corresponding aerodynamic force vectors; (b) and (c) the vortex structures identified by $Q$ criterion $(Q=2)$ under different viewing angles at $t=0.35\mathrm{T}$; (d) the spanwise vorticity contour plot on the symmetry plane at $t=0.35\mathrm{T}$; (e) and (f) the vortex structures identified by $Q$ criterion $(Q=2)$ under different viewing angles at $t=0.95\mathrm{T}$; (g) the spanwise vorticity contour plot on the symmetry plane at $t=0.95\mathrm{T}$. The red arrows denote the direction of the vortices and the yellow line denotes the wing chord.

Figure 17

Ellipsoidal wing used for the code validation.

Figure 18

The code validation of the boundary data immersed method. (a) Lift coefficient; (b) thrust coefficient.

Figure 19

The mesh convergence validation. (a) Lift coefficient; (b) thrust coefficient.