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The vortex formation and organization are the key to understand the intrinsic mechanism in flying and swimming in nature. The vortex wake of dual-flapping foils is numerically investigated using the immersed boundary method. Beside the deflection of the reversed von-Kármán vortex street, an interesting phenomenon, the deflection of the von-Kármán vortex street, is observed behind the dual-flapping foils. The deflected direction is not according to the initial direction of biplane’s flapping motion. And the deflection angle is related to the difference between upward and downward deflecting velocities.

The flapping foil is a common form of propulsion for many flyers and swimmers in nature, such as birds, insects and fishes. Inspired by the applications for the micro-aerial vehicles [

The deflected vortex wake of a flapping foil is an interesting phenomenon, since the vortex street deflected to one side of the flapping foil rather than locating symmetrically along the middle-line of the flapping foil [

In this paper, the unsteady flow fields around dual-flapping airfoils are simulated by solving the 2-D viscous incompressible N-S equations based on the immersed boundary method [

In this paper, a NACA0012 airfoil is used as the contour of the flapping foil, as shown in

y = y 0 + h sin ( 2 π f t ) (1)

where y is the vertical coordinates of the flapping foil, y_{0} is the vertical coordinate of the mean position of the flapping foil, h is the flapping amplitude, f is the

flapping frequency. All the control parameters are non-dimensionalized, the spacing L = L ′ / c , the flapping amplitude h = h ′ / c , the flapping frequency k = 2 π f c / U ∞ . The lower foil flaps counter-phase with the upper foil.

The governing equations of a two-dimensional incompressible viscous flow are written as follows:

∂ u ∂ t + ∇ ⋅ ( u u ) = − ∇ p + 1 R e ∇ 2 u + f , (2)

∇ ⋅ u = 0 . (3)

where u is the velocity, p is the pressure, the Reynolds number is defined as Re = LU_{∞}/ν, ν is the viscosity, and f is the Eulerian force density. The parameters in the current work are defined as follows: U_{∞} = 1.0, c = 1.0, Re = 500.0. A simple immersed boundary method [

u ˜ − u n Δ t = − 3 2 ∇ ( u u ) n + 1 2 ∇ ( u u ) n − 1 − ∇ p n + 1 2 R e ∇ 2 ( u n + u ˜ ) (4)

u ˜ ( X k ) = ∑ X u ˜ ( x ) δ h ( x − X k ) h 2 (5)

∑ j = 1 M ( ∑ X δ h ( x − X j ) δ h ( x − X k ) Δ s h 2 ) F ( X j ) = u b ( X k ) − u ˜ ( X k ) Δ t (6)

f ( x ) = ∑ k = 1 M F ( X k ) δ h ( x − X k ) Δ s (7)

u * − u ˜ Δ t = f n + 1 2 (8)

u * * − u * Δ t = ∇ p n (9)

∇ ⋅ u * * = ∇ 2 p n + 1 (10)

u n + 1 − u * * Δ t = − ∇ p n + 1 (11)

where u ˜ , u * and u * * are the intermediate velocity values between the time step of n and n + 1, the complete details of the algorithm and the validations are available in the previous work [

In order to compare the wake of dual-flapping foils to that of an isolated foil, two values of flapping amplitude (h = 0.1 and h = 0.35) are simulated. As shown in

lected von-Kármán vortex street, was observed behind the dual-flapping foils, and the deflection angle increases as the separation distance decreases. It is very surprising since the Kármán wake of a flapping foil reported in the previous studies was not deflected [

The deflected direction was according to the initial heaving direction of an isolated foil in the previous studies [

ferent to the deflection of the wake behind an isolated flapping foil. The vortex generated in the open flapping motion also followed the vortex produced during the consecutive close flapping motion, no matter the up foil starts with an upward or downward motion initially.

recorded, the rectangular symbols represent the locations of maximum value of each mean velocity profiles of the upper foil. The deflection angle is calculated through a linear curve fit (the heaving line) which is performed through the rectangular symbols in

In order to analysis the wake deflection quantitatively, Godoy-Diana et al. proposed a dipole model of two adjacent vortices and used the effective phase velocity of a dipole to quantify the trend of wake deflection [

V Γ y = ( Γ / 2 π ξ ) sin α (13)

where Γ is the vortex circulation, ξ is the distance between adjacent vortex cores, and α is the angle between the direction of the dipole and the horizontal direction, as shown in

The vertical dipole-induced velocities of two adjacent dipoles define the trend of upward deflecting and downward deflecting, respectively. As shown in

and

The vortex formation of the wake of dual-flapping foils has been numerically studied using the immersed boundary method. Beside the deflection of the reversed von-Kármán vortex street, an interesting phenomenon, the deflection of the von-Kármán vortex street, was observed behind the dual-flapping foils. The deflected direction is not according to the initial direction of the heaving motion because the vortex generated in the open flapping motion was also pairing with the vortex produced during the consecutive close flapping motion. The deflection trend correlates with the vertical induced velocities of two adjacent dipoles, which represent the trends of upward and downward deflecting, respectively. Moreover, the deflection angle is determined by the difference between upward and downward deflecting velocities. The results may provide some physical insights for understanding the intrinsic mechanism in flying and swimming in nature, and the three-dimensional model will be considered in our further work.

This work is supported by the National Natural Science Foundation of China (grant number 11462015) and the Aeronautical Science Foundation of China (grant number 2015ZC56007).

Lin, X.J., He, G.Y., He, X.Y., Wang, Q. and Chen, L.S. (2017) Numerical Study of Mechanisms of the Vortex Formation in the Wake of Dual- Flapping Foils. Journal of Applied Mathematics and Physics, 5, 1431-1439. https://doi.org/10.4236/jamp.2017.57118