A numerical methods for 2-D flow of flapping airfoil

Alexandru Dumitrache∗1

1 Institute of Mathematical Statistics and Applied Mathematics, P.O. Box 1-24, RO-010145, Bucharest, Romania

To study the physics of the flow of flapping flight, a numerical model for the two-dimensional flow around an airfoil undergo-ing prescribed oscillatory motions in a viscous flow is described. The model is used to examine the flow characteristics andpower coefficients of a symmetric airfoil heaving sinusoidally over a range of frequencies and amplitudes.

1 Introduction

The oscillation of an airfoil (flapping) is a common form of propulsion for many flying or swimming species. The generationof thrust by an oscillating airfoil was first described in the early part of the 20th century [1], [2]. After 1990 experimentaland numerical results [3], [4], identified the Strouhal number (St ≡ A/U∞), as an important parameter for thrust generation(A is the (double) amplitude of motion). To determine the wake structure on a two-dimensional airfoil undergoing pitching,heaving, and combined motions, as the length scales of flapping flight shrink, viscous effects will play a greater role [5].

2 Problem definition

It is considered a rigid, two-dimensional airfoil undergoing prescribed, periodic, flapping motions at zero angle of attack inan incompressible, viscous flow with constant horizontal (chord-wise) free stream velocity, U∞. The flapping consists ofheaving, lagging, and pitching motions and the airfoil is in cruise conditions: level flight with constant average velocity, U∞.

The problem is formulated in a non-inertial reference frame (with axyz acceleration) fixed to the airfoil such that the x-axisis fixed parallel to the chord, and the y-axis is perpendicular to both the x-axis and the span. In the inertial frame, the X-axisis horizontal (mean flow direction) and the Y -axis is vertical. The dimensional horizontal and vertical positions of the airfoil,X0, Y0, are made non-dimensional, X0, Y0, by dividing by the chord length, c. The angular position is denoted by θ.

2.1 Kinematics. Introducing the reduced frequency, k = 2πfc/U∞, the position of the airfoil for sinusoidal motions isgiven as Y0 = h sinkτ , X0 = l sin(kτ−δ), θ = θmax sin(kτ−ε), where h ≡ Ymax/c is the dimensionless heave amplitude,l ≡ Xmax/c and τ ≡ U∞t/c is the dimensionless time, δ and ε are dimensionless phase parameter for lagging and pitching,respectively. The dimensionless heave amplitude velocity can be expressed as V0 ≡ ∂Y0/∂τ = kh coskτ .

2.2 Governing Equations. In the non-inertial frame, the non-dimensional Navier-Stokes equations are written:

∂u/∂t = −∇p− (u · ∇)u+1Re

∇2u+ axyz(t), (1)

where u is the velocity, Re is the Reynolds number and p is the pressure. Taking the curl of Equation 1 and defining thevorticity in 2-D as ζ ≡ (∇× u) · k, (k is the unit vector perpendicular to the plane of flow), results:

∂ζ/∂t = −∇ · (uζ) +1Re

∇2ζ − 2θ, (2)

As the only source of vorticity from non-inertial effects is from the angular acceleration of non-inertial reference frame, (θ),one can be made a simplification by defining vorticity to be the sum of a background vorticity, Z and a disturbance, ζ′ :ζ = Z + ζ′, where Z = −2θ. Substituting these into Equation 1 results in:

∂ζ′/∂t = ∇ζ′ +1Re

∇2ζ′, (3)

(the gradient and Laplacian of the spatially-constant quantity Z are zero).Assuming incompressibility, Equation 2 is suplemented by the continuity equation (∇·u = 0) and after defining the stream

function, ψ, by u ≡ ∇ × ψ, it can be related to ζ through the continuity equation ∇2ψ = −ζ. As with vorticity, the streamfunction and velocity are decomposed into combinations of background and disturbance terms: ψ = Ψ+ψ′ and u = U∞+u′,such that U∞ = ∇×

(Ψk

)and u′ = ∇×

(ψk

). For a constant U∞, the stream function equation can be reduced to

∇2ψ′ = ∇2ψ −∇2Ψ = −ζ + Z = −ζ′. (4)

∗ Corresponding author: e-mail: dalex@gheorghita.ima.ro, Phone: +40 21 318 2433, Fax: +040 21 318 2439

PAMM · Proc. Appl. Math. Mech. 6, 563–564 (2006) / DOI 10.1002/pamm.200610262

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Then the problem has been reduced to two analytical expressions for the background values of the vorticity and the streamfunction, and two PDE: a vorticity transport equation for ζ′ (Equation 3) and a Poisson equation for ψ′ (Equation 4). Theinteraction between these two sets of equations is in the advection term for ζ.

2.3 Boundary Conditions. The airfoil is a no-slip surface so that the total vorticity at the airfoil, ζa, can be related to thetotal stream function. Thus ζ′a = −∂2ψ′

∂n2 |airfoil − ∂2ψ∂n2 |airfoil − Z . The appropriate boundary conditions at infinity are that

the velocity equals the free stream velocity, ψ′|∞ = 0 and the fluid is irrotational in the inertial frame,[∇× ψ′k

]= 0.

3 Numerical implementation

The governing equations are discretized using finite differences on a conformal map. Before discretization, the governingequations are transformed from physical x, y space to computational ξ, η space. Stepping in time is performed using a second-order Runge-Kutta scheme. Boundary conditions suffer an appropriate transformation.

The total force on the airfoil is a combination of pressure and viscous forces. The viscous force is found from the shear stresson the airfoil, τs, which is related to the tangential velocity, us. After a numerically integration are found the average forcecomponents, FX , FY and moment, M , in the inertial frame, having a number of steps per flap. The input power coefficient,Pi is the combination of the power needed to heave, lag and pitch the airfoil: Pi = FY V0 + FXU0 + Mθ. The outputpower coefficient, Po, from thrust is Po = −FXU∞ (positive when thrust is produced and negative for drag). Integrating theinstantaneous power yields the total work required (input) to flap the airfoil, Wi, and the work done in propelling it (output),W0. The thrust efficiency, η, is defined as the ratio of total work out to total work in: η = −W0/Wi .

Fig. 1 Output power for k = 2 and k = 5.333 over one heaving cycle.

After validation of the code, the present numerical model was used to simulate a heaving airfoil with frequencies rangingfrom 2 < k < 10 and maximum heave velocities from 0.8 < kh < 1.5 at Re = 500. For kh = 1.0, asymmetric solutionsare observed. For kh ≥ 1.2, aperiodic, quasi-periodic, and asymmetric solutions are common. Figure 1 shows the horizontal(output) power components for two different simulations with kh = 1.0, k = 2.0 (low efficiency) and k = 5.333 (highefficiency). There is a strong correlation between heaving efficiency and the separation between the heaving frequency for agiven profile and the frequency resulting in the maximum spatial amplification of this profile.

4 Conclusion

A numerical study for two-dimensional flow around an airfoil undergoing prescribed oscillatory motion in viscous flow isdeveloped. The model is used to examine the flow characteristics and power coefficients of a symmetric airfoil heavingsinusoidally over a range of frequencies and amplitudes. Both periodic and aperiodic solutions are found. For a given Strouhalnumber, the maximum efficiency occurs at an intermediate heaving amplitude. Above the optimum frequency, the efficiencydecreases similarly to inviscid theory.

References

[1] R. Knoller, Die Gesetze des Luftwiderstaades. Plug- und Motortechnik 3, 1–7, (1909).[2] A. Betz, Ein Beitrag zur Erklarung des Segel-fluges. Zeitschrift fur Flugtechnik und Motor- Luftschiffahrt 3, 269–272, (1912).[3] K. D. Jones, C. M. Dohring and M. F. Platzer, Experimental and computational investigation of the Knoller-Betz effect, AIAA Journal

37 (7), 1240–1246, (1998).[4] G. S. Triantafyllou, M. S. Triantafyllou and M. A. Grosenbaugh, Optimal thrust development in oscillating foils with application to

fish propulsion, J. Fluid Struct. 7, 205–224, (1993).[5] P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University

Press, (1996).

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Section 10 564