Starting Flow Past an Airfoil and its Acquired Lift in a Superfluid

We investigate superfluid flow around an airfoil accelerated to a finite velocity from rest. Using simulations of the Gross-Pitaevskii equation we find striking similarities to viscous flows: from production of starting vortices to convergence of airfoil circulation onto a quantized version of the Kutta-Joukowski circulation. We predict the number of quantized vortices nucleated by a given foil via a phenomenological argument. We further find stall-like behavior governed by airfoil speed, not angle of attack, as in classical flows. Finally we analyze the lift and drag acting on the airfoil.

Figure 1

Generation of circulation: (a) a cartoon showing the starting vortex produced in a viscous fluid. (b) The phase field around the airfoil potential. By counting phase jumps around the airfoil the value of the circulation can be obtained. A quantized vortex is visible behind the airfoil’s trailing edge. (c) Left hand: the density field in the full computational box. The density is rescaled by the superfluid bulk density, length scales are expressed in units of $\xi$, quantized vortices are shown as red dots. A closer view of airfoil is shown on the right. Relevant airfoil parameters are labeled and the vortex is circled in red.

Figure 2

Vortex emission: (a) vortices nucleated at the tail and top of a Joukowski airfoil having $\alpha =15°$, $\tilde{\lambda }=0.1$. Here the width of the foil scales like $\tilde{\lambda }L$ (see the Supplemental Material [18]). The computational box size is $1024\xi \times 1024\xi$. Tail number reflected by color, top nucleation by octagon mark. (b) Top three frames are snapshots of the density field for $U_{\infty }=0.260c$ and $L=325\xi$; here $n_{\mathrm{tail}}=2$. Bottom frame has $U_{\infty }=0.345c$ and $L=200\xi$. This foil nucleates thrice from tail before nucleating uncontrollably from top; right shows closer view of top vortices.

Figure 3

Nucleation predictions: (a) plot of tail and top nucleation numbers in $U_{\infty }-L$ parameter space for $\alpha =15°$. Predictions are stripes in background, white area signifies predicted top nucleation. All predictions used a cutoff distance of $A=0.55\xi$ from the foil. Simulation data are circled in white. (b) Values of $\mathrm{\Delta }{n}^2$ calculated for each simulation in (a) with their average plotted vs $L/\xi$. Error bars are standard deviation of the mean for the $U_{\infty }/c$ values in (a).

Figure 4

Viscous vs superfluid flight/stall: (a) flight of foil in a viscous fluid at a low angle of attack. (b) Stall at high angle of attack. (c) Stall in a superfluid at low angle of attack. (d) Flight at high angle of attack.

Figure 5

Evolution of lift and drag: (a) nondimensional lift (dotted line) and drag (solid line) experienced by the airfoil throughout simulation with $U_{\infty }=0.29c$, $L=325\xi$, $\alpha =15.0°$, and $\tilde{\lambda }=0.1$. Inset shows an exaggerated density field around the airfoil. Included are the integration contours for computing the force. (b) Nondimensional lift (dotted line) and drag (solid line) experienced by the airfoil using ${\mathbf{u}}_I$ and $\rho (u_I)$. A grid is overlaid to demonstrate the quantization of the lift, the steps coincide with vortex nucleation. Lift and drag were not computed on a contour if a vortex was within $8\xi$.