Spatial wake transition past a thin pitching plate

The spatial transition of the wake behind a thin pitching plate in the thrust regime is studied. The drag-to-thrust transition is seen to occur at a threshold pitching frequency which becomes smaller for higher pitching angle and aspect ratio. The reverse von Kármán wake shows deflected asymmetric vortex pairing in two dimensions, while a bifurcated wake with a swirling ring is the signature in three dimensions, both resulting in nearly the same scaling for the power coefficient. The wake transition in the thrust regime occurs through three spatial regions comprising the reverse von Kármán vortex street, transitional zone, and twin jet region. Near the trailing edge, reverse von Kármán shedding is marked by linear decay of the spanwise wake width and growth of secondary instability. A large volume of fluid resulting from inward horizontal acceleration renders a weak form drag in the subsequent transition zone which is followed by vertical bifurcation into twin jet formation in the far wake. The spanwise pressure gradient is seen to guide the wake compression, with a streamwise adverse pressure gradient aiding it in the near wake.

Figure 1

(a) Schematic of the physical problem showing a uniform flow past a rigid pitching plate with the relevant boundary condition. (b) Plot of the $C$-normalized spatial grid resolution in the three coordinate directions.

Figure 2

Variation of ${\overline{C}}_T$ as a function of St for varying ${\theta }_{\text{max}}$ at $\mathcal{R}=0.54$ and 2.38. A clear demarcation of the drag-to-thrust transition is observed for increasing St beyond a critical value and ${\theta }_{\text{max}}\ge {10}^{\circ }$.

Figure 3

For the selected parameter values, instantaneous isosurfaces (or contours) of ${\omega }_z$ are shown for both the drag (${\overline{C}}_T<0$) and thrust (${\overline{C}}_T>0$) regimes. A comparison made with the contours of ${\omega }_z$ for the 2D simulation data is shown.

Figure 4

(a) Time-averaged power coefficient ${\overline{C}}_p$ and root mean square of the (b) thrust $\left(C_T^{\prime }\right)$ and (c) lift $\left(C_L^{\prime }\right)$ coefficients as a function of St. The nearest ${\overline{C}}_p\left(\text{St}\right)$ scaling is shown alongside the least-squares fit.

Figure 5

Streamwise evolution of the time-averaged streamwise velocity $\overline{u}\left(y\right)$ for only the thrust cases. The corresponding streak-line visualizations of the flow are shown for the $z=0$ plane. Three distinct regions of the wake transition are indicated by the dashed lines.

Figure 6

(a)–(d) Plan view of instantaneous streak lines and $Q$ ($=1$) structure characterizing the wake evolution. Streamwise evolution of the (e) spanwise wake width and (f) normalized spanwise local disturbances exhibit distinct patterns in the three regions indicated.

Figure 7

(a)–(d) The region with high horizontal velocity ${(u^2+w^2)}^{1/2}>U_{\infty }$ is seen to be guided by a strong spanwise pressure gradient. Growth of (e) secondary and (f) primary circulation in regions 1 and 2 results in a streamwise decay of ${\omega }_z$ shown in (g).

Figure 8

Mesh sensitivity analysis results at $\mathcal{R}=2.38$, $\text{St}=1$, ${\theta }_{\text{max}}={15}^{\circ }$ for four progressively refined meshes $M_1$, $M_2$, $M_3$, and $M_4$. The (a) instantaneous $Q$ ($=1$) structures and (b) time-averaged streamwise velocities yield better resolution with increasing mesh size and become nearly identical at $M_3$ and $M_4$.

Figure 9

Comparison test results for a pair of 3D freely vibrating identical circular cylinders for a reduced velocity $U^{*}=4$, mass ratio $m^{*}=2$, and $\text{Re}=200$. The results are compared with [42] for the (a) temporal evolution of the centroids of the upstream ($y_F$) and downstream ($y_R$) cylinders and (b) isosurfaces of ${\omega }_z$.