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The hydrodynamics in fish school are studied through simulating the flow over two wiggling hydrofoils in diagonal arrangement by using the immersed boundary method, both in-phase and anti-phase wiggling motions are considered in this study, the effect of the longitudinal spacing on the hydrodynamic performance has been investigated. It is revealed that, when the longitudinal spacing is smaller than body length, the diagonal formation is helpful to the follower but harmful to the leader; when the longitudinal spacing is larger than body length, the opposite effect is occurring; and a significant enhancement of the propulsive performance is obtained when the longitudinal spacing is optimized.

The diamond performance of fish school has been studied by biologists and physicists for several decades [

In this study, the flow over two wiggling hydrofoils in diagonal arrangement has been simulated using immersed boundary method [

In this paper, we consider the hydrodynamic interactions between two diagonal hydrofoils in a two-dimensional incompressible viscous flow, as shown in

where y_{0} is the lateral coordinate of the midline along the hydrofoil, λ is the wavelength, c is the phase velocity, x1 is defined as x_{1} = (x − x_{0})/L, x_{0} is the position of the leading edge of hydrofoil, L is the length of the hydrofoil, A(x_{1}) is the amplitude along the hydrofoil, coefficients a_{0} = 0.02, a_{1} = 0.0825, a_{2} = 0.1625, are

calculated from the data of steadily swimming motion of fish [_{x} = G_{x}/L, the lateral spacing D_{y} = G_{y}/L.

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

where u is the velocity, p is the pressure, the Reynolds number Re = LU_{0}/𝜈, 𝜈 is the viscosity, and f is the Eulerian force density. A simple immersed boundary method [

The time-averaged thrust coefficient, the time-averaged power coefficient and propulsive efficiency for each hydrofoil are calculated as follows:

where F_{1} and F_{2} are the x-component and y-component of the Lagrangian force, respectively.

For the whole two-hydrofoil system, the coefficients and efficiency are defined as follows:

where

The parameters in the current work are defined as follows: U_{0} = 1.0, L = 1.0, Re = 200.0. To generate the reversed Von-Kármán vortices and thrust, the wiggling parameters are set as λ = 1.0, c = 5.0, both in-phase and anti-phase motions are considered. The separation distances are set as: D_{x} = 0.0 - 3.0, D_{y}_{ }= 0.3 - 0.4, the values of lateral spacing selected here are similar to the optimal distance in the theory of Weihs [

The performance of in-phase wiggling hydrofoils, such as the thrust coefficient, the power coefficient and the propulsive efficiency, has been shown in _{x} < 1.0), the downstream hydrofoil has a significant enhancement of the thrust coefficient, but the upstream hydrofoil has the reduction, compared to the single hydrofoil, and so as the propulsive efficiency, although the time-averaged power coefficient of the hydrofoils is larger than that of single foil in some cases. This

may arise because the vortices generated by the leader are captured by the follower, and merged into the wake of the follower, as shown in _{x} = 0.0), the thrust coefficients for each hydrofoils are smaller than that of a single hydrofoil, and the wake is weak, as shown in _{x} > 1.0), the thrust coefficient of the upstream body is larger than that of the downstream body, and so as the propulsive efficiency and the time-averaged power coefficient, since the follower has experienced the jet flow produced by the leader, as shown in

For the whole in-phase system, as shown in Figures 2(d)-(f), the thrust coefficient and propulsive efficiency are smaller than that of a single hydrofoil when D_{x} = 0.0, 0.25, but larger when D_{x} = 0.5, 0.75. The best propulsive efficiency is obtained when D_{x} = 0.5, although the more power is required. When the longitudinal spacing is larger than body length (D_{x} > 1.0), the propulsive performance of the whole two-hydrofoil system is similar to that of a single hydrofoil.

The propulsive performance of anti-phase wiggling hydrofoils has been shown in _{x} < 1.0), since the vortices generated by the former are also captured and merged into the wake of the latter, as shown in _{x} = 0.0) are better than that for a single one, and a strong wake is generated by the paratactic system, as shown in _{x} > 1.0), the propulsive performance of the upstream hydrofoil is better than that of the downstream hydrofoil, since the downstream body has experienced the jet flow produced by upstream one, as shown in

For the whole anti-phase system, a significant enhancement of thrust coefficient has been obtained when D_{x} = 0.0, 0.25, compared to a single hydrofoil, and

so as the propulsive efficiency (except the case of D_{x} = 0.25, D_{y} = 0.3 which is similar to a single one), although the increased power is required. The best performance is obtained by two paratactic hydrofoils (D_{x} = 0.0), and no better performance than that of a single hydrofoil is achieved by the anti-phase system when D_{x} ≥ 0.5.

In the present work, the flow over two wiggling hydrofoils in diagonal formation has been simulated using the immersed boundary method, both the in-phase and the anti-phase wiggling motions are investigated. It is found that, whether the hydrodynamic beneficent obtained depends on the longitudinal spacing between individuals, when the longitudinal spacing is smaller than body length (D_{x} < 1.0), the downstream hydrofoil has the increased thrust and efficiency while the upstream hydrofoil is dragged, when the longitudinal spacing is larger than body length (D_{x} > 1.0), the upstream hydrofoil obtains the increased thrust and efficiency but the downstream is dragged. For the whole two-hydrofoil system, if wiggles in-phase, the best performance of the largest thrust coefficient and propulsive efficiency is gained when the longitudinal spacing equals to half of the body length (D_{x} = 0.5), but if wiggles anti-phase, the paratactic formation is beneficial to obtain the highest thrust coefficient and propulsive efficiency. The results obtained here may be helpful to understand the hydrodynamics in fish school, although the two dimensional model used in this study is simplified, and the three dimensional model will be made in the 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 the Hydrodynamic Per- formance of Two Wiggling Hydrofoils in Diagonal Arrangement. Journal of Applied Mathematics and Physics, 5, 31-38. http://dx.doi.org/10.4236/jamp.2017.51005