Wake topology and hydrodynamic performance of low-aspect-ratio ...

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326 H. Dong, R. Mittal and F. M. Najjar(a)yzx(b)zyxTV 1 TV 1V 1TV 1TV 2 TV 2V 3V 12V 2TV 2(c)yxzR 2R 3V V 12 1V1y 0V 2R 1(d)2TV 2 TV 1V 12TV 2TV 1V 3–20 2 4xFigure 9. Vortex topology for AR =2.55 foil at the phase where the foil is at the centre ofits heaving motion. (a) Perspective view. (b) Side view. (c) Topview.(d) Spanwise vorticitycontour plot on spanwise symmetry plane.to convect it along its axis and this mechanism then explains the convection of thevortex rings at an oblique angle to the wake centreline. The experimental visualizationsof Buckholtz & Smits (2006) for a pitching panel also show a bifurcation in the wakevery similar to what is seen here. It should also be pointed out that in figure 9(c)which is a top-view, we see a spanwise narrowing of the vortex rings as they convectdownstream. This is clearly due to the induction of one set of tip-vortices on theadjacent set as explained in the previous paragraph. Spanwise narrowing has alsobeen observed in the simulations of Blondeaux et al. (2005a, b) for a rectangularpitching–heaving foil as well as the experiments of Buckholtz & Smits (2006) for apitching plate. In contrast, the experiments of von Ellenrieder et al. (2003) do notshow any clear spanwise narrowing and the reason for this disagreement is not clear.Also in agreement with Blondeaux et al. (2005a, b) and Buckholtz & Smits (2006),we do not observe a distinct vortex loop associated with the trailing-edge as wasobserved by von Ellenrieder et al. (2003).
Low-aspect-ratio flapping foils 3272y0–20 2 4xFigure 10. Contours of mean streamwise velocity for the AR =2.55 case along the spanwisesymmetry plan. Contour levels in this plot are the same as those in figure 7.The last distinct feature that requires explanation is the vortex contrails that areseen extending from the upstream end of one loop to the downstream end of theprevious loop. If we consider the ones extending from vortex V 12 to vortex V 2 infigure 9(a), it is clear that these structures are segments of the tip-vortices TV 1 whichare induced by the V 2 vortex to wrap around itself in a counterclockwise direction.Figure 10 presents contours of the mean streamwise velocity along the spanwisesymmetry plane for this case. From these plots, we observe that the wake develops abifurcated jet with a relatively large inclination angle to the wake centreline. For thiscase, the inclination angle of each of the two branches of the wake from the symmetryline is about 16 ◦ . This angle is estimated by drawing a straight line through localmaxima in mean streamwise velocity as indicated in figure 10. It is also worth pointingout that within statistical uncertainty, the mean wake is symmetric about the wakecentreline. Thus, end effects tend to symmetrize the wake which is strongly asymmetricfor the two-dimensional case at this Strouhal number. A closer examination showsthat there is in fact a single streamwise directed jet in the very near wake whichbifurcates about two chord-lengths downstream of the trailing edge. The inclinationof the jet leads to a reduction in the streamwise momentum flux into the wake andthis is connected with the lower thrust production of this finite-span flapping foil ascompared to the two-dimensional foil.The spanwise structure of these mean jets is examined by plotting contours of themean velocity at several streamwise planes in figure 11. Figure 11(a) shows that thejet has a double-lobed structure in the near wake which clearly indicated that evenin the very near wake, the influence of the tip vortices cannot be neglected. Furtherdownstream at x/a x = 2, we see in figure 11(b) the transverse bifurcation in the wake.In addition, each branch of the jet actually has two separate lobes and these lobestend to merge further downstream (figure 11c). With regard to this merging it bearspointing out that the apparent intensification of the jets around x/a x = 6 which issuggested by figure 10 is in fact not an intensification, but rather, a manifestation ofthe merging of the two lobes of the jet onto the spanwise symmetry plane.The preceding discussion therefore explains the processes which lead to the formationof all the significant topological features in the wake of the finite-aspect-ratio
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