Computation of Low Reynolds Number Aerodynamic Characteristics of a Flapping Wing in Free Flight Dominic D. J. Chandar1 and M. Damodaran2 School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 1 2
Graduate Student; E-mail:
[email protected] Associate Professor; E-mail:
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1 Introduction It is well known from literature that unsteady motions of airfoils and wings create aerodynamic forces suitable thrust and lift forces which propel and sustain insect and bird flight. Most of the experimental studies carried out by Freymuth[1], Jones et al.[2], Lai and Platzer[3] and numerical computations by Liu and Kawachi[4], Wang[5], Lewin and Haj-Hariri[6], have focused on static flapping or active flight while neglecting the dynamics of forward flight. In an experiment by Vandenberghe et al.[7], the problem of forward flight has been discussed in detail with more emphasis on the dynamic motion of a wing undergoing plunging motion. The transition between the state of rest and free flight for a flapping wing is shown to take place at a critical reduced frequency of plunging. Numerical computations in two-dimensions following the lines of Vandenberghe et al.[7] have been performed by Alben and Shelly[8], Chandar and Damodaran[9]. Three-dimensional computations on free flight highlighting the effect of wing rotation modes can be found in Chandar and Damodaran[10]and is extended in the present paper to understand the effect of outer boundaries on trajectories due to flapping motion in two-dimensions and analyze the effect of combined wing rotation on forward flight in three-dimensions.
2 Computational Model The motion of a wing is governed by the unsteady incompressible Navier Stokes equations coupled with the dynamical equations of motion. The complete description of the numerical method, implementation of the boundary conditions, interpolation on the overlapping mesh and convergence studies, can be found in Henshaw [11], Chandar and Damodaran [9]. However for the sake of completeness, an outline of the numerical method is provided here. The time dependent incompressible Navier Stokes Equations are given by,
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∂u ∇p + (u.∇) u + = ν∆u + F ∂t ρ ∇.u = 0
(1) (2)
Here u is the velocity vector, p is the pressure, F is a vector of external forces per unit volume, ν is the kinematic viscosity and ρ is the fluid density. These equations are discretized in space on a system of overlapping meshes. An implicit multi-step method is used for time stepping and second order differences are used for spatial discretization. The pressure is obtained by solving a pressure Poisson equation using the Bi-Conjugate Gradient Stabilized method from PETSc(Portable Extensible Toolkit for Scientific Computation) [12]. The acceleration of the body is computed using aerodynamic forces FA and torques R T = dΩ (r − xcm ) × dFA where r is any point on the body and xcm is the center of mass. This appears in the boundary condition for pressure in the pressure equation. The Navier-Stokes equations are solved numerically with this boundary condition and a new set of forces are obtained. This procedure is repeated over a period of time till the final time of computation. When the meshes move rigidly with the body, the solutions in the region of overlap are interpolated using a Lagrange interpolation formula. The non-dimensional quantities of interest are the Reynolds number Re = Vp c/ν and the Strouhal number St = ωch0 /Vp , where Vp is the maximum plunge velocity, c the mean aerodynamic chord, ω the flapping frequency, and h0 the maximum plunge amplitude.
3 Results and Discussions Validation cases for the OverBlown code can be found in Chandar and Damodaran[9]-[10]. Two sets of computations are shown in this paper to address the effect of (i) location of outer boundaries on estimated aerodynamic coefficients and computed flight trajectories and (ii) wing rotations on computed flight trajectories. 3.1 Effect of Outer Boundary Locations on Computed Aerodynamic Characteristics This assessment is carried out to study the effect of the extent and location of far-field outer boundary on the computedaerodynamic characteristics and trajectories for a plunging and pitching symmetrical airfoil (10% thick). Three different extents of the outer boundary locations are considered. The width (W ) and height (H) of the computational domain for these cases are given by (a) W = 24c, H = 16c (b) W = 12c, H = 8c and (c) W = 6c, H = 4c where c is the airfoil chord. The mesh density is maintained the same in all cases and the minimum mesh spacing from the wall is 5 × 10−4 c. The airfoil undergoes a combined plunge and pitch oscillation analogous to a flapping wing as described in the next section. The plunge and pitch are governed by
Flapping Wing Computations in Free Flight
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the expressions, h = −h0 (1 − cos2πf t), θi = θ0 (1 − cos2πf t) respectively with h0 = 0.34, θ0 = 10o and f = 1.5 Hz. From the aerodynamic forces, the position of the centre of mass is then obtained by integrating Newton’s second law for a rigid body. Figures 1(a)-1(b) show the variation of the vertical and horizontal force components with spatial x-coordinate of the forward flight direction for the three cases. Computations show that these forces are either overestimated or underestimated on a smaller domain. The forces corresponding to case(b) do not differ much from that of case(a) (very large domain) hence the domain corresponding to case(b) would be an optimum choice for economic threedimensional computations. The computed trajectory corresponding to these three cases are shown in Fig. 1(c). It can be seen that the computed trajectory corresponding to case(c) has a larger wavelength compared to that of case(a) and case(b). This shows that on smaller domains, numerical errors will cause the airfoil to travel slower. 40
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Fig. 1. Comparison of aerodynamic characteristics of an airfoil in free flight (a) vertical force (b) horizontal force and (c) trajectories for different domains
3.2 Effect of Wing Rotations on Computed Trajectories Of all flapping wing computations available in literature, computing the free flight characteristics is conceivably the most interesting one. By allowing the wing to translate freely, one can get an idea of the trajectory the wings establish and hence monitor its dynamic performance. From existing computations on flapping wings, it is well known that an optimal combination of reduced frequency, amplitude and plunge-pitch phase, gives rise to thrust. But it is unknown what will happen when the body is propelled forward by virtue of this thrust. Static flapping computations have some drawbacks i.e.,(i) they might over/under-estimate the thrust due to the fact that the imposed free-stream velocity is constant. The vortices which are shed as a result of flapping have a fixed residence time within the vicinity of the airfoil or wing (ii) It is unknown whether the lift produced as a result of flapping can sustain the body’s weight. These issues are avoided when the wing is set free to move at a velocity which is determined by the aerodynamic forces. An arbitrary wing consisting of elliptical wing sections with a thickness ratio of 0.1 and a low aspect ratio of 2is considered so that the three-dimensional effects are not negligible and a high density ratio of 10 is chosen to ensure that the wing will move slowly.
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Choosing a low density ratio wing will result in large accelerations requiring the imposition of very stringent time steps for the computation. The spanwise, chord-wise direction extends along the Z-axis and X-axis respectively as shown in Fig. 2(a)-2(c). Overlapping meshes with 607,000 mesh points are generated for the wing. The wing is enclosed in a box (cartesian mesh) which has all its sides as interpolation boundaries (inner box in Fig. 2(c)). When the wing moves, this box also moves with respect to a stationary cartesian mesh (outer box in Fig. 2(c)) and the interpolation relationship between the outer and inner box is regenerated every time step. The dimensions of the outer box are decided based on the observations from the previous sub-section. The domain corresponding to case(b) is used and the extent of the domain in the span-wise direction is given by 8S where S is the wing-span. Rotations about all three axes can be specified and the resulting translational motion be obtained by integrating the rigid body equations. Presently, computations have been carried out for periodic rotations about 2 axes (X and Z) and the position of the wing being constrained to move along X-axis. These are equivalent to plunging and pitching in two-dimensions. The specified rotational motions
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Fig. 2. Free flight of a flapping wing showing (a),(b) The elliptical cross-section wing of aspect ratio 2 and (c) Mesh block boundaries
for the wing follow the equation θi = θ0 (1 − cos2πf t) where θi denotes the angular position of the wing about axis ‘i’ which passes through the point P(0,0,-1) as in Fig.2(a). The Reynolds number and the Strouhal number are computed based on the maximum plunge velocity Vp = 2πθ0 Sf where S is the wing-span. Based on a flapping amplitude of 10 degrees, kinematic viscosity of 0.01cm2 s−1 and a Strouhal number of 2, based on maximum arc length traversed by the wing tip, the Reynolds number is 329. Since the wing flaps in zero free-stream velocity, a steady flow initial condition is not required to start the computation at t=0. Three different types of rotation modes are considered namely (a) rotation about X-axis (b) rotation about Z-axis and (c) combined rotation about X and Z axis. Using approximately 333 time steps per cycle, the solution is computed till the wing reaches the vicinity of the boundary. Figure 3(a)-3(b) shows the computed trajectory and forward speed corresponding to different modes of oscillation. It can be seen that between
Flapping Wing Computations in Free Flight
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Fig. 3. Computed (a) trajectory (b) forward speed for a flapping wing with different modes
cycle 4.5 and 5.5 , the combined mode of oscillation propels the wing at an average speed of 2.05 cm/s. This is about 62% of the peak plunge velocity (3.29 cm/s). Whereas rotation about X, Z axis resulted in an average speed of 0.019 cm/s and 0.0013 cm/s respectively. This result confirms the fact that pure plunging motion of the wing (rotation about X-axis) results in higher thrust compared to pure pitching (rotation about Z-axis) at the same reduced frequency. Figure 3(c)-3(d) shows the contours of vorticity magnitude about a plane passing through the mid-section of the wing and iso-surfaces respectively. The wake is partially deflected downwards which indicates lift is being produced. This is also evident from Fig. 4(a) where the time history of lift over one cycle is plotted along with the wing positions at specific times. The region from cycle = 4 to cycle= 4.5 is the downstroke and from cycle=4.5 to cycle = 5 is the upstroke. The average lift in the downstroke is about 8.32 and in the upstroke is about -5.17. Figure 4(b) shows the time history of
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Fig. 4. Computed aerodynamic forces (a) Vertical force (b) Horizontal force over one cycle of flapping
the horizontal force over one cycle. The numerical values differ from an earlier computation[10] as the pressure forces were not taken into account while calculating the total force. In this case the downstroke produces a thrust of -0.74 whereas the upstroke produces a thrust of 1.25. Hence the conditions for maximizing lift and thrust occur at different phases of the cycle. A possible solution to this problem would be to provide a phase difference between pitch and plunge.
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4 Conclusion The importance of passive form of flapping flight has been discussed based on the computational study by solving the Navier-Stokes equations on moving overlapping meshes. The effect of the location of the outer boundary on computed solutions show that on a smaller domain, the aerodynamic forces are either over- or under-estimated depending on the position of the airfoil in the oscillation cycle and that the airfoil motion is slower due to boundary interference errors. For three-dimensional wings, it has been shown that more lift is produced in the downstroke than the upstroke and that the combined flapping mode which involves simultaneous rotation about two axes yielded a high forward speed due do higher thrust. By varying the phase between different rotational motions, it might be possible to optimize both thrust and lift either during the downstroke or upstroke. Further research is aimed at analyzing the flexibility of the wing in free flight.
References 1. Freymuth, P.: Propulsive Vortical Signature of Plunging and Pitching Airfoils, AIAA J, 26, 881–883 (1988) 2. Jones, K.D., Dohring, C.M., & Platzer, M.F.: Experimental and Computational Investigation of the Knoller-Betz Effect, AIAA J, 37, 1240–1246 (1998) 3. Lai, J.C.S. & Platzer, M.F.: Jet Characteristics of a Plunging Airfoil, AIAA J, 37, 1529–1537 (1999) 4. Liu, H & Kawachi, K: Numerical Study of Insect Flight, J. Comp Phy, 146, 124–156 (1998) 5. Wang, Z.J.: Vortex Shedding and Frequency Selection in Flapping Flight, J. Fluid Mech, 410, 323–341 (2000) 6. Lewin, G.C & Haj-Hariri, H: Modelling Thrust Generation of a Twodimensional Heaving Airfoil in a Viscous Flow, J. Fluid Mech, 492, 339– 362 (2003) 7. Vandenberge, N., Zhang, J & Childress, S.: Symmetry Breaking Leads to Forward Flapping Flight, J. Fluid Mech, 506, 147–155 (2004) 8. Alben, S. & Shelley, M.: Coherent Locomotion as an Attracting State for a Free Flapping Body, Proceedings of the National Academy of Sciences, 102, 32, 11163–11166 (2005) 9. Chandar, D., and Damodaran, M.: Computational Study of Unsteady Low Reynolds Number Airfoil Aerodynamics Using Moving Overlapping Meshes, AIAA J, 46, 2, 429–438 (2008) 10. Chandar, D., and Damodaran, M.,: Computational Study of the Free Flight of a Flapping Wing in Low Reynolds Numbers, 46th Aerospace Sciences Meeting and Exhibit, Reno, NV, AIAA paper 2008-0420 (2008) 11. Henshaw, W.D.: OverBlown, A Fluid Flow Solver for Overlapping Grids, Reference Guide, UCRL-MA 134289, 1–51 (2003) 12. Balay. S., Gropp, W.D., McInnes, L.C., and Smith, B.: PETSc 2.0 Users Manual Revision 2.3.3, Argonne National Laboratory., Report AN95/11, 1–190 (2007)
