Propulsion performance of a two-dimensional flapping airfoil with wake map and dynamic mode decomposition analysis

In the present study, we use the dynamic mesh method based on the radial basis function interpolation for the two-dimensional simulation of harmonically oscillating NACA0015 airfoil. Under various flapping frequencies, heaving and pitching amplitudes, the observed wake flows can be divided into seven types, including the Bénard-von Kármán (BvK) vortex street, the reversed BvK (RBvK) vortex street, the $1P$ wake, the $mP$ wake, the $2P+mS$ wake, the $2S+mS$ wake, and the $mS$ wake, where $m$ is around 4 and $xS+yP$ signifies $x$ single vortices and $y$ vortex pairs shedding per oscillation period. Then we have constructed two phase diagrams of the wake types in terms of the flapping frequency, heaving and pitching amplitudes. Importantly, we have combined the propulsion performance of the flapping airfoil with the wake map and found that $\alpha \left(\frac{T}{4}\right)$, the angle of attack at $t=T/4$, can determine the wake type: negative value corresponding to drag-dominated wakes, while positive value corresponding to thrust dominated flow wakes. With the increase of $\alpha \left(\frac{T}{4}\right)$, the wake transforms from the $mP$ to $2S+mS$ then to RBvK and eventually to $1P$ wake. Furthermore, the coherent structure analysis and spectral analysis are conducted for all the types of wakes by using dynamic mode decomposition. And there is a positive correlation between the strengths of vortices shedding at $i$ times flapping frequency and the modulus of the $i\text{th}$ dynamic mode decomposition mode, which can further reveal the differences among different types of wakes.

Figure 1

The sketch of the flapping foil.

Figure 2

The sketch of the two-dimensional and three-dimensional meshes: (a) the holistic view of the two-dimensional mesh; (b) the close view of the two-dimensional mesh; (c) the surface mesh of the airfoil in the three-dimensional mesh.

Figure 3

The gird convergence results for the two-dimensional flapping airfoil with kinematic parameters ($f=0.25$, $h=1$, ${\theta }_0={30}^{\circ }$): (a) thrust coefficient; (b) lift coefficient.

Figure 4

(a) the effects of the radius $R$ of the computational domain and (b) the effects of the simulation time-step ($dt$) for the two-dimensional flapping airfoil with kinematic parameters ($f=0.25$, $h=1$, ${\theta }_0={30}^{\circ }$).

Figure 5

The mesh validation for 3D flapping airfoil with the kinematic parameters ($f=0.3$, $h=1$, ${\theta }_0=5^{\circ }$): (a) the effects of the spanwise mesh size with fixed aspect ratio ($AR=1$); (b) the effects of the aspect ratio with fixed spanwise mesh size ($dz=0.02c$).

Figure 6

The instantaneous streamwise vorticity field of three specific cases at $t=8$ T: (a) $f=0.25$, $h=1$, ${\theta }_0={30}^{\circ }$, St = 0.5 (RBvK wake); (b) $f=0.3$, $h=1$, ${\theta }_0=5^{\circ }$, St = 0.6 (asymmetric wake); (c) $f=0.4$, $h=1$, ${\theta }_0=5^{\circ }$, St = 0.8 (beyond the parameter range in the current study). The isosurfaces are drawn transparent on the level of the spanwise vorticity equals $\pm 2$, and the isosurfaces of streamwise vorticity with values of $\pm 0.1$ are further drawn to evaluate the intensity of 3D instability.

Figure 7

The comparison between the 2D and 3D thrust coefficient results: (a) $f=0.25$, $h=1$, ${\theta }_0={30}^{\circ }$ (RBvK wake); (b) $f=0.3$, $h=1$, ${\theta }_0=5^{\circ }$ (asymmetric wake); (c) $f=0.4$, $h=1$, ${\theta }_0=5^{\circ }$ (beyond the parameter range in the current study).

Figure 8

The vorticity contours of the seven types of wakes: (a) the BvK wake ($f=0.15,h=0.375,\theta ={50}^{\circ }$); (b) the RBvK wake ($f=0.275,h=1,\theta ={20}^{\circ }$); (c) the asymmetric wake ($f=0.30,h=1,\theta =5^{\circ }$); (d) the $mP$ wake ($f=0.06,h=1,\theta ={30}^{\circ }$); (e) the $2P+mS$ wake ($f=0.10,h=1,\theta =5^{\circ }$); (f) the $2S+mS$ wake ($f=0.18,h=1,\theta ={30}^{\circ }$); (g) the $mS$ wake ($f=0.20,h=1,\theta ={50}^{\circ }$). The middle line is the equilibrium position of the heaving motion and the box contains the wake of one period.

Figure 9

The wake maps and propulsion efficiency isolines: (a) Group 1; (b) Group 2.

Figure 10

The development of vortices in $mP$ ($f=0.06,h=1,\theta ={30}^{\circ }$) wake in one period.

Figure 11

The development of vortices in $2P+mS$ ($f=0.10,h=1,\theta =5^{\circ }$) wake in one period.

Figure 12

The development of vortices in $2S+mS$ ($f=0.18,h=1,\theta ={30}^{\circ }$) wake in one period.

Figure 13

The development of vortices in $2S$ (RBvK, $f=0.275,h=1,\theta ={20}^{\circ }$) wake in one period.

Figure 14

The $mS$ ($f=0.18,h=1,\theta ={50}^{\circ }$) wake.

Figure 15

The wake maps and isolines of $\alpha \left(\frac{T}{4}\right)$: (a) Group 1, (b) Group 2.

Figure 16

The vorticity contour of seven types of wakes at $t=T/4$. (a) The $mP$ wake $[f=0.6,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=-0.163],\overline{C_T}=-0.11$, Max($C_L$) = 0.23; (b) the $mS$ wake $[f=0.18,h=1,{\theta }_0={50}^{\circ },\alpha \left(\frac{T}{4}\right)=-0.026],\overline{C_T}=-0.08$, Max($C_L$) = 0.57; (c) the $2P+mS$ wake $[f=0.14,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=0.198],\overline{C_T}$ = 0.13, Max($C_L$) = 0.65; (d) the $2S+mS$ wake $[f=0.18,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=0.323],\overline{C_T}$ = 0.26, Max($C_L$) = 1.24; (e) the RBvK wake $[f=0.30,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=0.559],\overline{C_T}$ = 1.91, Max($C_L$) = 7.88; (f) the $1P$ wake $[f=0.30,h=1,{\theta }_0=5^{\circ },\alpha \left(\frac{T}{4}\right)=0.996],\overline{C_T}$ = 0.95, Max($C_L$) = 14.49; (g) the BvK wake $[f=0.15,h=0.05,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=-0.476],\overline{C_T}$ = $-0.46$, Max($C_L$) = 1.66.

Figure 17

The correlation between the pressure distribution and aerodynamic forces: (a) the $2S+mS$ wake $[f=0.18,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=0.323]$, $\overline{C_T}$ = 0.26, Max($C_L$) = 1.24; (b) the RBvK wake $[f=0.30,h=1,{\theta }_0={30}^{\circ },\alpha \left(\frac{T}{4}\right)=0.559]$, $\overline{C_T}$ = 1.91, Max($C_L$) = 7.88; (c) the $1P$ wake $[f=0.30,h=1,{\theta }_0=5^{\circ },\alpha \left(\frac{T}{4}\right)=0.996]$, $\overline{C_T}$ = 0.95, Max($C_L$) = 14.49. The left column is the time-vary lift and thrust coefficients in one period, the middle column is the pressure contour at $t=T$/4, and the right column is the pressure contour at $t=3T$/4.

Figure 18

The validation of the DMD analysis: (a) relative error; (b) growth rate; (c) mode modulus.

Figure 19

The vorticity contour of the first four modes of the seven types of wakes: (a) the BvK wake ($f=0.15,h=0.375,\theta ={50}^{\circ }$); (b) the RBvK wake ($f=0.275,h=1,\theta ={20}^{\circ }$); (c) the $1P$ wake ($f=0.30,h=1,\theta =5^{\circ }$); (d) the $mP$ wake ($f=0.06,h=1,\theta ={30}^{\circ }$); (e) the $2P+mS$ wake ($f=0.10,h=1,\theta =5^{\circ }$); (f) the $2S+mS$ wake ($f=0.18,h=1,\theta ={30}^{\circ }$); (g) the $mS$ wake ($f=0.20,h=1,\theta ={50}^{\circ }$).

Figure 20

The modulus and the stability analysis of the modes for the six types of wakes: (a) the RBvK wake ($f=0.275,h=1,\theta ={20}^{\circ }$), $\epsilon$ = 0.270; (b) the $1P$ wake ($f=0.30,h=1,\theta =5^{\circ }$), $\epsilon$ = 0.367; (c) the $mP$ wake ($f=0.06,h=1,\theta ={30}^{\circ }$), $\epsilon$ = 0.059; (d) the $2P+mS$ wake ($f=0.10,h=1,\theta =5^{\circ }$), $\epsilon$ = 0.079; (e) the $2S+mS$ wake ($f=0.18,h=1,\theta ={30}^{\circ }$), $\epsilon$ = 0.125; (f) the $mS$ wake ($f=0.20h=1,\theta ={50}^{\circ }$), $\epsilon$ = 0.046.

Figure 21

The validation of the combination of DMD modes and virtual force calculation: (a) the drag coefficient, (b) the lift coefficient.

Figure 22

The (a) drag coefficient and (b) lift coefficient of antisymmetric, symmetric, and all DMD modes in one period, which are calculated only using the velocity field.