6. Conclusions and perspectives

disappear (as in vortex ring VR2 in Figure 5). During a complete heaving cycle, two VRs are

In Figure 7, we present the wake topology for a drag-producing case (which corresponds to case 3DH-3 in Table 4). It is clear from this figure that the wake topology is very different from the one of the thrust producing case. In this case, as the vortex loops are convected downstream, they do not morph into vortex rings. Instead, they keep their original shape, and as they are convected, they diffuse. We can also observe that the wake height is very compact, in comparison to that of the thrust production case (as depicted in Figure 8). Finally, notice how the flow induced by each vortex loop is inclined in the same direction of the wing's travel direction, resulting this in a momentum surfeit linked to drag production. The momentum

Figure 7. Vortex wake topology at the beginning of the upstroke for case 3DH-3 (t = 5.0). Heaving parameters: Re = 250,

St = 0.2, and k = 1.570795 (drag-producing wake).

formed, one at the end of the upstroke and the other one at the end of the downstroke.

132 Flight Physics - Models, Techniques and Technologies

In this manuscript, we studied the unsteady aerodynamics of heaving rigid wings. The laminar incompressible Navier-Stokes equations were solved in their velocity-pressure formulation using a second-order accurate in space and time finite-difference numerical method, and to efficiently deal with moving bodies, we used overlapping structured grids. To study the dependence of the aerodynamic forces and wake topology on the wing kinematics, many simulations were conducted at different values of Strouhal number and at two reduced frequency values (low and high oscillating frequency).

The simulations show that the wake of thrust producing, rigid heaving wings is formed by two sets of interconnected vortex loops that slowly convert into vortex rings as they are convected downstream. It is also observed that the vortex rings are inclined with respect to the freestream flow, whereas for thrust producing configurations, the angle of inclination of the vortex rings is in the same direction of their travel, and for drag-producing configurations, the angle of inclination of the vortex rings is opposite to the direction of their travel. The presence of thin contrails that link the vortex loops is of interest; these structures are segments of the wing-tip vortices, and as the vortex loops are convected downstream, they become weaker and ultimately disappear. In general, the observed structures are qualitatively similar to those observed in the experiments by Parker et al. [32] and Von Ellenrieder et al. [33] and the numerical simulations of Dong et al. [34] and Blondeaux et al. [35].

From the force measurement study, two different behaviors were observed for the average thrust coefficient ct and maximum lift coefficient <sup>b</sup>cl. It seems that the reduced frequency <sup>k</sup> (and hence the oscillating frequency) plays an important role in the vortex generation and shedding frequency, therefore, on the aerodynamic forces. Thus, LEV convection and separation introduce a frequency dependence into the results. This provides a mechanism of optimal selection of heaving frequency (in the sense of propulsive efficiency) as discussed by Wang [29], Guerrero [31], and Young and Lai [36]. It is worth mentioning that the results presented in the previous references were obtained for two-dimensional airfoils. The results presented in this manuscript extend these observations to three-dimensional wings.

Finally, for the limited range of St and k values studied and the simplified wing geometry and heaving kinematics covered in this study, all the qualitative and quantitative results presented are in close agreement with the experimental observations of Rohr and Fish [12], Triantafyllou et al. [13], Nudds et al. [14], Taylor et al. [15], and Parker et al. [32]; this supports the hypothesis that "flying and swimming animals cruise at a Strouhal number tuned for high power efficiency" [15].

The results presented in this manuscript are limited to laminar flow; nevertheless, they provide an excellent insight into the wake signature of the unsteady aerodynamics of heaving wings. We envisage to extend the current study to higher Reynolds numbers and turbulent cases and use more realistic wing geometries and kinematics. Finally, in this manuscript, we did not cover propulsive efficiency and optimal frequency selection, but we hope to address these issues in future studies.
