Linear and nonlinear perturbation analysis of the symmetry breaking in time-periodic propulsive wakes

The two-dimensional and time-periodic wake flows produced by a pitching foil are investigated numerically for a fixed flapping amplitude. As the flapping frequency is increased, three regimes are identified in the time-marching nonlinear simulations. The first regime is characterized by nondeviated wake flows with zero time-averaged lift. In the second regime, the wake flow is slightly deviated from the streamwise direction and the time-averaged lift is slightly positive or negative. The third regime is characterized by larger deviations of the wake, associated with larger values of both the time-averaged lift and the thrust. The transition from the first to the second regime is examined by performing a Floquet stability analysis of the nondeviated wake. A specific method is introduced to compute the time-periodic, nondeviated wake when it is unstable. It is found that one synchronous antisymmetric mode becomes unstable at the critical frequency where deviation occurs. Investigation of its instantaneous and time-averaged characteristics show that it acts as a displacement mode translating the nondeviated wake away from the streamwise direction. Finally, it is demonstrated that the transition from the second to the third regime is linked to nonlinear effects that amplify both antisymmetric and symmetric perturbations around the foil.

Figure 1

Sketch of the pitching foil pivoted at the center of the leading edge half-cylinder.

Figure 2

Computational domain and typical mesh used for the finite-element discretization. The mesh is made of triangles, only one-tenth of them are shown in the figure.

Figure 3

Instantaneous vorticity fields for the flapping frequency $f=0.1$ (a, b), $f=0.35$ (c, d), $f=0.43$ (e, f), and $f=0.45$ (g, h) at $t=T/4$ (left) and $t=3T/4$ (right). Black and white denote negative and positive values, respectively.

Figure 4

Instantaneous vorticity fields for $f=0.43$ at $t=T/4$ showing an upward (a) and downward (b) deviation of the wake. Black and white denote negative and positive values, respectively.

Figure 5

Vorticity field of the mean flow (left) and the real part of the first Fourier component (right) for the flapping frequency $f=0.1$ (a, b), $f=0.35$ (c, d), $f=0.43$ (e, f), and $f=0.45$ (g, h). Black and white denote negative and positive values, respectively. The foil at zero incidence is shown in red.

Figure 6

Instantaneous drag versus lift coefficients over one flapping period for $f=0.35$ (solid line), $f=0.43$ (dashed line) and $f=0.45$ (dash-dotted line). The mean positions of the drag and lift coefficients are indicated by the black circle ($f=0.35$), diamond ($f=0.43$), and square ($f=0.45$).

Figure 7

Mean lift and drag coefficients as a function of the flapping frequency.

Figure 8

Evolution of the symmetric (solid) and antisymmetric (dashed) component as a function of time with $\chi =0.3$ at $f=0.43$. Simulation is started from converged deviated wake.

Figure 9

Vorticity snapshots (top) of the non deviated wake at $f=0.43$ obtained using $\chi =0.3$. Vorticity field averaged in time and first temporal Fourier component (bottom). Black and white denote negative and positive values respectively. The foil at zero incidence is shown in red.

Figure 10

Mean lift and drag coefficients as functions of the flapping frequency $f$. $\circ$, asymmetric solutions; $•$, symmetric solutions.

Figure 11

Spectrum of the Floquet multipliers at $f=0.4$. The dominant mode's Floquet multiplier, in black, is ${\mu }_1\simeq 1.046$ (a). Evolution of Floquet multiplier magnitude $\left|\mu \right|$ with the flapping frequency $f$ (b). The line $\left|\mu \right|=1$ corresponds to marginal stability of Floquet modes. The dominant mode is always real.

Figure 12

Vorticity snapshots of the dominant Floquet mode at flapping frequency $f=0.45$ and time $t=T/4$ (a) and $t=3T/4$ (b). Black and white denote negative and positive values, respectively.

Figure 13

Vorticity snapshots of (a) the base flow, (b) the Floquet mode, and (c) a zoom of their superimposition at $f=0.43$ and $t=0$. Black and white colors [in (a), (b), and (c)], and dotted and continuous lines [in (c)] denote negative and positive values, respectively.

Figure 14

Vorticity field averaged in time over one period and first temporal Fourier component at $f=0.43$. Black and white denote negative and positive values, respectively. The foil at zero incidence is shown in red.

Figure 15

(a–d) Snapshots of the vorticity field for (a, b) linear and (c, d) nonlinear perturbations at time $t=T/4$ and frequency (a, c) $f=0.43$ and (b, d) $f=0.45$. (e, f) $y$-integrated kinetic energy of the linear (dashed) and nonlinear (solid) perturbations as a function of the streamwise coordinate for (e) $f=0.43$ and (f) $f=0.45$.

Figure 16

Evolution of the nonlinear perturbations' mean kinetic energy (integrated in the transverse direction) in the streamwise direction for $f=0.43$ (a) and $f=0.45$ (b): ${\mathit{u}}^{\prime }$ (black solid), ${\mathit{u}}_s^{\prime }$ (red dashed), ${\mathit{u}}_a^{\prime }$ (blue dash-dotted).

Figure 17

Evolution of $x_c$ as a function of $f$. The trailing edge position is indicated with a dashed line.

Figure 18

Mean lift and drag coefficients for the symmetric ${\mathit{u}}_s^{\prime \prime }$ (red circles) and antisymmetric ${\mathit{u}}_a^{\prime \prime }$ (blue diamonds) components as a function of the flapping frequency $f$.

Figure 19

Vorticity isolines for two different instants $t$. Guilmineau et al. (2002) at $t=0$ (a) and $t=19T/72$ (c). Present method at $t=0$ (b) and $t=19T/72$ (d).

Figure 20

The $x$-velocity component in the transverse direction $y$ at $x=-0.6D$ for two different instants $t$ of the flapping period $T$. (a) $t=T/2$; (b) $t=7T/12$. Dütsch et al. (1998): black squares. Hosseinjani et al. (2015): blue dashed line. Present method: black solid line.

Figure 21

Norm of the antisymmetric velocity component as a function of time for increasing damping coefficients $\chi =[0.02,0.1,0.2,0.3]$ at flapping frequency $f=0.45$.