3. Geometry, boundary conditions, initial conditions, and wing kinematics

In Figure 1, we present an illustration of the overlapping grid system layout used to conduct this parametric study. In the figure, c is the wing's chord, and h is the heaving amplitude of the heaving wing. The background grid (BG) extends 4.0 c away from the wing's leading edge (LE), 10.0 c away from the wing's trailing edge (TE), 2.0 c away from the left and right wing's tips (LH-WT and RH-WT, respectively), and 4.0 c + h away from the point of maximum thickness of the upper and lower surfaces. This overlapping grid system layout corresponds to the instant, when the wing is in the mid position of the heaving cycle (as illustrated in Figure 1). In this manuscript, all the base units are expressed in the international system.

Figure 1. Left: computational domain layout in the xy plane. Right: computational domain layout in the zy plane. The figure is not to scale.

In all the cases studied, a rectangular wing with an aspect ratio AR equal to 2 was used. The cross-section of the wing is an ellipse, with a corresponding major axis a = 0.25 and a minor axis b = 0.025. Therefore, the wing's chord c is equal to 2 � a = 0.5.

The initial conditions used for all the heaving wings simulations are those of a fully converged solution of the corresponding fixed wing case. In Figure 1 (left), the left boundary of the BG corresponds to an inflow boundary condition (u = (1.0, 0.0, 0.0), ∂n^ p ¼ 0), and the top, bottom, and right boundaries of the BG are outflow boundaries (velocity extrapolated from the interior points). In Figure 1 (right), all the boundaries of the BG correspond to outflow conditions. On the wing surface (which is a moving body), we impose a no-slip boundary condition for moving walls (u ¼ G � x, G � y, G � <sup>z</sup>). The rest of the boundaries is interpolation boundaries, where we used a nonconservative Lagrange interpolation scheme. The Reynolds number (defined as Re = U � c/ν) is equal to 250 for all the simulations.

In all the simulations conducted, we assumed that the wing is undergoing pure heaving motion, wherein the wing cross-section heaves in the vertical direction (or y axis in Figure 1) and according to the following function,

$$y(t) = h \times \sin\left(2 \times \pi \times f \times t + \phi\right) \tag{15}$$

where y(t) is the heaving motion (and is defined positive upwards), h is the heaving amplitude, f is the heaving oscillating frequency, φ is the phase angle of the heaving motion (0 in this case), and t is the physical time.
