The incompressible flow solver

The governing equations employed are the incompressible Navier—Stokes equations in Arbitrary Lagrangian Eulerian (ALE) formulation. They are written as: 1 2 3 Here, p denotes pressure, and v_a=v-w, the advective velocity vector, where v is flow velocity and w is mesh velocity and the material derivative is with respect to the mesh velocity w. Both the pressure p and the stress tensor σ have been normalized by the (constant) density ρ and are discretized in time t using an implicit time-stepping procedure. It is important for the flow solver to be able to capture the unsteadiness of a flow field. The present flow solver is built as time-accurate from the onset, allowing local time stepping as an option. The resulting expressions are subsequently discretized in space using a Galerkin procedure with linear tetrahedral elements. To be as fast as possible, the overheads in building element matrices, residual vectors, etc., should be kept to a minimum. This requirement is met by employing simple, low-order elements that have all the variables (v,p) at the same location. The resulting matrix systems are solved iteratively using a preconditioned conjugate gradient algorithm (PCG), as described by Martin and Löhner (1992). The flow solver has been successfully evaluated for both two-dimensional and three-dimensional laminar and turbulent flow problems by Ramamurti and Löhner (1992) and Ramamurti et al. (1994).

To carry out computations of the flow about oscillating and deforming geometries, one needs to describe grid motion on a moving surface, i.e. to couple the moving surface grid to the volume grid. The volume grid in the proximity of the moving surface is then remeshed to eliminate badly distorted elements. The velocity of the mesh is obtained in a manner so as to reduce this distortion. A detailed description of the various mesh movement algorithms is given in Ramamurti et al. (1994). In that study, smoothing of the coordinates was employed for the mesh movement with a specified number of layers of elements that move rigidly with the wing. In two-dimensional studies (Ramamurti and Sandberg, 2001), the grid showed that the elements at the edge of the rigid layers were very distorted after one cycle of oscillation. This is due to a residual mesh velocity that is present as a result of the non-convergence of the mesh velocity field. This will appear whether a spring-analogy or a Laplacian-based smoothing is used.

To reduce the distortion of the mesh, the coordinates at the new time xⁿ⁺¹ were obtained as a weighted average of the original grid point location at time t=0(x⁰) and the location of the point as if it moved rigidly with the body (x_rigidⁿ⁺¹): 4 where the weighting function (f) is a simple linear function based on the distance from the center of rotation r, and is given by: 5 6 and 7

The mesh velocity is then obtained using: 8

The values of r_min and r_max used in this study are 2.0 and 10.0, respectively. To reduce the computational effort, the experimental arrangement of Dickinson et al. (1999) was approximated by introducing a symmetry plane. Because of the proximity of the wing at the beginning of the downstroke and the rotation of the wing during the pronation phase, the normal component mesh velocity of the points on the symmetry plane can become non-zero. This would result in the points being pulled away from the symmetry plane. To avoid this problem, the points on the symmetry plane are allowed to glide along this plane. A similar technique has been employed for the simulation of torpedo launch from a submarine by Ramamurti et al. (1995) where the gap between the launch tube and the torpedo was small.

Symmetrical case

The flow solver described here is employed to compute the flow past the Drosophila wing undergoing translation and rotation. First, an inviscid solution was obtained using a grid consisting of 178 219 points and 965 877 tetrahedral elements. An initial steady-state solution was obtained in 1500 time steps. The unsteady solution using the prescribed kinematics (Fig. 2) is then obtained. The surface pressure on the wing is integrated to obtain the forces on the wing along the three axes (F_x, F_y, F_z). The thrust T and the drag D forces are then computed as T=-F_x and D=√(F_y²+F_z²), respectively. These forces are compared with those obtained from the experiments of Dickinson et al. (1999).

The unsteady computation was carried out for five cycles of oscillation. Fig. 3 shows the thrust and drag forces during one cycle of the wingbeat and compares the values with those of Dickinson et al. (1999). The present computations capture the peak forces well. The mean thrust force is approximately 0.318 N, and the mean thrust coefficient is 1.317. The mean drag force is 0.375 N and the drag coefficient is 1.55. The force coefficients were obtained using the following non-dimensionalization: 10 11 where and are the mean thrust and drag forces, respectively, and r₂²(s) is the second moment of the dimensionless area of the wing (0.40). The variation of these forces during the translational phase of the wing is also predicted correctly, but the magnitude of the thrust force during the downstroke is higher than that of Dickinson et al. (1999). The kinematics is symmetrical between the up- and downstroke, so the resultant force should also be symmetrical. That this is not the case may be due the mechanical play in the experimental arrangement, as suggested by M. H. Dickinson (personal communication). To understand the different mechanisms occurring during the wingbeat cycle, we can divide the cycle into two rotational and two translational phases. The rotational phase near the beginning of the downstroke (pronation) occurs between time t0 and t3 (Fig. 3A). Thrust decreases between t0 and t1, then increases until t2. This behavior can be explained by a rotational mechanism. The wing continuously rotates in a counterclockwise direction producing a circulation pointing nearly along the +y direction. Between t0 and t1, the wing is translating in the -z direction, resulting in a force pointing in the -x direction, thus producing a peak in thrust at t0. If a rotational mechanism alone were present, the thrust should continue to decrease until t3; in fact, the thrust force increases between t1 and t2. This happens after the wing changes direction at the start of each half-stroke. Dickinson et al. (1999) attribute this increase in thrust to a wake-capture mechanism in which the wing passes through the shed vorticity of the previous stroke.

The position of the wing at the beginning of the downstroke is shown in Fig. 4A. The chord in this case of symmetrical rotation is aligned with the x axis at this instant. We found a separation bubble attached to the leading edge during the interval t0-t1. Particles were released from a rake of rectangular grid of points in a plane 0.8 mm away from the bottom surface of the wing. Using the instantaneous velocity field, the positions of these particles were obtained by integrating the velocity at these rake points in both the positive and negative velocity directions until the length of the traces exceeded a specified length or the particles ended on a solid boundary or exited the computational domain. These instantaneous particle traces are colored according to the magnitude of velocity at that location. Fig. 5 shows the leading edge vortex with vorticity oriented in the +y direction. This leading edge vortex was created at the end of the preceding upstroke. This vortex is located below the wing and is rotating in the counterclockwise direction, which can be seen from the velocity vectors shown in Fig. 6A. A possible explanation for the increase in thrust between t1 and t2 is that the wing moving through this wake benefits from the shed vorticity. As the wing moves through this vortex during the downstroke, it produces a stagnation region at the bottom of the wing near the z′ axis (Fig. 5), resulting in an increase in thrust.

At the beginning of the downstroke (t1), the flow separates at the leading edge and reattaches on the bottom of the wing as shown in Fig. 6A. As the wing continues to move down, between t1 and t2, the separation point of this bubble moves back along the wing chord (Fig. 6). During the interval t2-t3, we observe a trailing edge separation bubble forming (see Fig. 7A-C). A similar separation region forms at the wing tip, as can be seen in Fig. 7D-F. Fig. 4B shows the position of the wing at t=12.5 s. Thrust production reaches a local minimum around this instant (Fig. 3). In Fig. 7A,D, a large recirculation region can be seen in the wake of the wing. This separated flow from the wing tip and trailing edge will result in a higher pressure on the upper surface of the wing and, hence, a reduction in the thrust. During the interval t1-t3, the magnitude of the translational acceleration of the wing decreases while that of the angular acceleration increases (symmetrical phase, Fig. 8).

Between t2 and t3, the magnitude of the translational acceleration is large enough to overcome the rotational effect and, when the angular acceleration becomes large enough, the rotational mechanism dominates, resulting in the observed reduction in thrust until t3. Between t3 and t4, the translational effect should result in a constant thrust because the translational acceleration is almost constant during this period. The rotational effect produces an increase in thrust between t3 and t4, with a plateau in the middle, which occurs when the trailing edge vortex is shed. Similar trends are observed during the supination phase prior to the beginning of the upstroke (t4-t5) and at the beginning of the upstroke (t5-t7).

Fig. 9 shows the instantaneous traces or streaklines of particles released 3.0 cm above or 3.5 cm below the wing in a plane parallel to the wing and near the leading edge. In Fig. 9A, a wing tip vortex can be seen, but no leading edge vortex is visible above the wing surface. A leading edge vortex spinning in the counterclockwise direction is found below the wing surface (Fig. 9B). Particle traces near the mid stroke are shown in Fig. 10A. At this instant, the wing rotation axis z′ is aligned with the body coordinate z (see Fig. 1B). Here, we can see the beginnings of the leading edge vortex on the upper surface of the wing that is also shown by the velocity vectors in the xy plane at z=20 cm (Fig. 10B). Fig. 10C shows the pressure contours on the upper surface of the wing. The pressure is non-dimensionalized with respect to the dynamic head,½ρ U_t², where ρ is the density of the mineral oil (880 kg m^-3) used in the experiments of Dickinson et al. (1999). A region of constant pressure is present from the root of the wing up to approximately 60% of the span and extending to the trailing edge. Fig. 11 shows the particle traces prior to the end of the downstroke. Here, the leading edge vortex is clearly seen on the upper surface of the wing.

Grid-refinement study

To assess the sensitivity of our computational results, we carried out a grid-refinement study. The resolution of the grid in the vicinity of the wing was doubled. The computations were carried out using a grid consisting of approximately 238×10³ points and 1.3×10⁶ tetrahedral elements. The time step was halved for this computation. The computed thrust forces are shown in Fig. 12. It can be seen that the agreement between the two analyses is very good; even the coarse grid produces adequate resolution.

Viscous effect

To the study the effects of viscosity, a laminar viscous computation was carried out, for Re=120. Because of the lack of information on the transition to and the presence of uncertainties in turbulence modeling, laminar flow was assumed first. Fig. 13 shows the time history of the thrust and drag forces for the inviscid and the viscous cases. The finer mesh employed in the grid-refinement study (see above) was used for this computation, and the mesh size near the leading and trailing edge of the wing was approximately 0.02. It is clear that viscous effects are minimal, and the thrust and drag forces are dominated by translational and rotational mechanisms. Hence, inviscid computations were carried out to study the effect of phasing between the translational and rotational motions.

Effect of phasing

Fig. 14 compares the forces produced when the rotational motion precedes stroke reversal (`advanced' case) with those of Dickinson et al. (1999). Again, the agreement with the experimental results is good. In this case, the peak in the thrust force is achieved prior to the beginning of the downstroke at t=10.76 s and is approximately 0.56N compared with a value of 0.47N for the symmetrical case (see Fig. 3A). This can be explained by the rotational mechanisms discussed above. The rotational effect diminishes prior to the beginning of the downstroke, producing a negative thrust of 0.2N. The thrust then increases until t=11.98s. During this period, the wing moves through the wake created during the upstroke, as in the symmetrical case, resulting in a high pressure on the bottom of the wing. The velocity vectors near the leading edge are shown in Fig. 15. During this period, both the translational and rotational accelerations are in phase (Fig. 8). The peak thrust is approximately 0.48N compared with a value of 0.28N for the symmetrical case. Thereafter, the combined effect of rotational and translational motions produces reduced thrust until a second peak arises due to the rotational motion at t=14.23s, prior to the beginning of the upstroke. The mean thrust force for one wingbeat cycle is approximately 0.312N, and the mean thrust coefficient is 1.291. The mean drag force is 0.457N and the drag coefficient is 1.89.

In the `delayed' case, where wing rotation is delayed with respect to stroke reversal, the rotational motion does not produce any thrust prior to the beginning of the downstroke (Fig. 16A). The mean thrust force for one wingbeat cycle is approximately 0.206N and the mean thrust coefficient is 0.854. The mean drag force is 0.457N and the drag coefficient is 1.496 (Fig. 16B). In the initial period following stroke reversal, the rotational effect continues to produce a negative thrust. Fig. 17A,B shows the velocity vectors near the leading edge. The leading edge vortex from the upstroke is not present after t=12.05 s. In this case, the high pressure on the bottom of the wing together with the orientation of the wing cause a reduction in thrust. Subsequently, the combined translational and rotational mechanisms result in an increase in thrust. At t=12.8 s, we observe a plateau region in the thrust (Fig. 16A). During this period, the presence of a trailing edge vortex on the upper surface (Fig. 17C,D) increases the pressure on the upper surface of the wing, resulting in a temporary loss of thrust; when this vortex leaves the trailing edge, thrust increases again.

Fig. 18 shows the magnitude of velocity in the xz plane at y=10 cm at the beginning of the downstroke for the three cases of wing motion in the wake created by the wing. Velocities are greatest for the advanced case and smallest for delayed rotation. The wing moving through the higher-velocity fluid therefore produces an additional thrust in the advanced rotation case, whereas the wing for the delayed case intercepts the flow at an angle that produces negative thrust. Similar velocity fields can be seen in the particle image velocimetry data of Dickinson et al. (1999).

Mechanical aspects of the flapping wing

The results of the present study were then used to derive the input forces, moments, power requirement and efficiency for the creation of a robotic fly. First, the forces on the wing were integrated to a particular spanwise location from the root. The forces were obtained by marking this location and then computing the force contribution of all the surface elements on the wing up to this location. This location was then tracked using the prescribed rigid body motion. The elements contributing to the force up to this location are recomputed as the surface mesh is regenerated due to the motion. Three different spanwise locations were chosen: quarter span, half span and three-quarter span of the wing. Fig. 19 shows their force contributions compared with the total contribution of the wing and half the total thrust force generated by the wing for the symmetrical rotation case. It is clear from this figure that nearly half the thrust is generated by the outer 25% of the wing.

Moments about the wing root in the wing coordinate system (x′,y′,z′) were then computed from the moments about the fixed body coordinates and the prescribed kinematics of the wing (Fig. 20). The moment about the wing rotation axis z′ (M_z′) is nearly zero throughout the cycle, implying that there is no torsional load on the system. The variation of moment about the x′ (=x) axis (M_x′) is anti-symmetrical because only one wing is considered for the moment computation.

The power input to the wing P_in is computed by: 12 where F is the force vector and W_wing is the velocity of the surface of the wing. Fig. 21 shows the instantaneous power requirement for one wing for the three phases of wing rotation. The mean power required is 0.024 W for the symmetrical case, 0.039 W for the advanced case and 0.024 W for the delayed case. The mean thrust for the symmetrical case is 0.318 N; values for the advanced and delayed cases are 0.312 N and 0.206 N, respectively.