Three-dimensional instabilities for the flow around a heaving foil

This paper investigates the three-dimensional instabilities of the flow past a periodically heaving airfoil. By comparison with a pitching foil [Deng et al., Phys. Rev. E 92, 063013 (2015)], here we present distinctive characteristics for the heaving foil, particulary regarding its Floquet modes. By increasing the frequency (Sr), or equivalently decreasing the amplitude ($A_D$) along the marginal stability curve in the ($\mathrm{Sr},A_D$) phase space, the critical Floquet mode emerges sequentially as A, quasiperiodic (QP), and B. It is interesting to note that both modes A and B are synchronous with the base flow, in contrast to the quasiperiodic mode QP. To further investigate the instability across the marginal curve, we fix the frequency at $\mathrm{Sr}=0.187$, of which the critical Floquet mode is located in the synchronous regime, while varying $A_D$ around the critical point. We find that the dominant mode switches from mode A to mode B, while mode QP never becomes critical as we increase $A_D$. We note that mode S, a subharmonic mode, can also be unstable, which, however, is not physically realizable, because the magnitude of its Floquet multiplier is always smaller than that of mode B. We have also studied the influence of various Reynolds numbers at the same critical point on the marginal stability curve, with the results resembling that by varying the amplitude $A_D$.

Figure 1

Schematic view of the foil profile, which is symmetric, heaving with the peak-to-peak amplitude $A$. $D$ denotes the thickness of the foil, which is $0.15c$ for a NACA0015 airfoil.

Figure 2

$A_D$ vs. $\mathrm{Sr}$ map for $\mathrm{Re}=1700$. Red (dash-dot-dot) line: the curve of marginal three-dimensional stability or the transition boundary from 2D to 3D flows, with computational points labeled as stars and dots for further discussion. Blue (dotted) line: transition between BvK and reverse BvK wakes. Green (dash-dot) line: transition from symmetric to asymmetric wakes with respect to the wake center line. The shaded area corresponds to the ${\mathrm{Sr}}_A=0.2-0.4$ interval.

Figure 3

Variation of Floquet multiplier magnitude $\left|\mu \right|$ with wave number $\beta$ at $\mathrm{Re}=1700$ for different (Sr, $A_D$): (a) ($0.157,2.118$), (b) ($0.187,1.431$), (c) ($0.200,1.303$), (d) ($0.215,1.200$), (e) ($0.225,1.150$), (f) ($0.235,1.105$), (g) ($0.250,1.042$), (h) ($0.275,0.955$), (i) ($0.300,0.884$), and (j) ($0.375,0.747$). The line $\left|\mu \right|=1$ corresponds to a neutral stability line.

Figure 4

Time histories of $\left|\mu \right|$ for the Floquet modes. (a) (Sr, $A_D)=(0.200,1.303)$, (b) (Sr, $A_D$) = (0.215, 1.200), and (c) (Sr, $A_D$) = (0.225, 1.150) correspond to Fig. 3, 3 and 3, respectively.

Figure 5

Variation of Floquet multiplier magnitude $\left|\mu \right|$ with wave number $\beta$ at $\mathrm{Re}=1700$ and $\mathrm{Sr}=0.187$ for different $A_D$: (a) 1.420, (b) 1.430, (c) 1.431, (d) 1.432, (e) 1.433, and (f) 1.435. The line $\left|\mu \right|=1$ corresponds to neutral stability of the Floquet modes.

Figure 6

[(a)–(e)] Instantaneous streamwise perturbation vorticity for mode A of the flow past a heaving foil at $\mathrm{Re}=1700$ for (Sr, $A_D)=(0.187,1.433)$ and $\beta =6$. Blue (dark gray) and red (light gray) colors denote negative values and positive values, respectively, for continuous contour levels from $-6$ to 6. Spanwise vorticity contours of the base flow are also shown with solid and dashed lines for positive and negative values, respectively. (f) Time variation of $\left|\mu \right|$.

Figure 7

[(a)–(i)] Instantaneous streamwise perturbation vorticity for mode S of the flow past a heaving foil at $\mathrm{Re}=1700$ for (Sr, $A_D)=(0.187,1.433)$ and $\beta =16$. Blue (dark gray) and red (light gray) colors denote negative values and positive values, respectively, for continuous contours levels from $-6$ to 6. Spanwise vorticity contours of the base flow are also shown with solid and dashed lines for positive and negative values, respectively. (j) Time variation of $\left|\mu \right|$.

Figure 8

[(a)–(e)] Instantaneous streamwise perturbation vorticity for mode B of the flow past a heaving foil at $\mathrm{Re}=1700$ for (Sr, $A_D)=(0.187,1.433)$ and $\beta =32$. Blue (dark gray) and red (light gray) colors denote negative values and positive values, respectively, for continuous contours levels from $-6$ to 6. Spanwise vorticity contours of the base flow are also shown with solid and dashed lines for positive and negative values, respectively. (f) Time variation of $\left|\mu \right|$.

Figure 9

[(a)–(p)] Instantaneous streamwise perturbation vorticity for mode QP of the flow past a heaving foil at $\mathrm{Re}=1700$ for (Sr, $A_D)=(0.187,1.432)$ and $\beta =29$. Images are set at one base-flow period apart. Blue (dark gray) and red (light gray) colors denote negative values and positive values, respectively, for continuous contours levels from $-6$ to 6. Spanwise vorticity contours of the base flow are also shown with solid and dashed lines for positive and negative values, respectively. (q) Time variation of $\left|\mu \right|$.

Figure 10

Isosurfaces of the streamwise vorticity fields rebuilt from the Floquet modes for $\mathrm{Re}=1700$, (Sr, $A_D$) = (0.187, 1.433): (a) mode A at $\beta =6$, (b) mode S at $\beta =16$, and (c) mode B at $\beta =32$. Blue (dark gray) and red (light gray) colors denote, respectively, the negative values and positive values of the vorticity.

Figure 11

Variation of Floquet multiplier magnitude $\left|\mu \right|$ with wave number $\beta$ at (Sr, $A_D$) = (0.187, 1.432) for different Re: (a) 1680, (b) 1685, (c) 1690, (d) 1700, (e) 1705, (f) 1708, (g) 1710, and (h) 1720. The line $\left|\mu \right|=1$ corresponds to neutral stability of the Floquet modes.