Stable states and mechanism of self-propulsion of two tandem filaments in viscous flow

Abstract

Fish schooling behavior contains very interesting but complicated hydrodynamics. A simplified and classic fish schooling model was simulated and analyzed in the present study, by using two self-propelled flexible filaments in a tandem arrangment. The two filaments plung harmonically in the vertical direction and are free to move in the horizontal direction in an incompressible viscous fluid flow. The simulations were carried out by using an efficient immersed boundary projection method, where the additional momentum forcing is calculated in an implicit way and an approximate factorization procedure is applied to solve the incompressible Navier-Stokes equations. The coupling between the fluid flow and the flexible structures is realized through the additional momentum forcing. In the immersed boundary projection method, both the pressure and the momentum forcing are considered as the Langrangian multipliers to satisfy the continuity constraint and the no-slip condition at the immersed boundary, respectively. To ensure a long period simulation, a non-inertial frame, which is fixed to the leading edge of the upstream filament, was adopted in the Navier-stokes equations and the structure motion equation, by introducing a translational acceleration in the motion equations. In this way, it significantly reduced the computational cost by avoiding a large compuational domain. Two-dimensional simulations were performed in the present study. The Reynolds number based on the plunging motion was fixed at 200, while the different initial horizontal and vertical distances between the two filaments were considered in simulations. The stable states of the two tandem filaments and the mechanism of the vortex interactions were mainly concerned. The numerical results reveal that three stable states are formed with different horizontal distances and vertical distances between the two filaments, i.e., long-distance tandem state, short-distance tandem state and parallel state. For the long-distance tandem state, the upstream filament performs like the single-filament case and the motion of the downstream filament always tends to be convergent with different horizontal and vertical distances. The downstream filament was found to move through the upstream shedding vortices, and was barely affected by the offset in the vertical direction. For the short-distance tandem state, the shear layer of the upstream filament is merged with the shear layer of the downstream one before it is shed from the filament. This effect generates a strong vortex shedding, and consequently the propulsive force is enhanced with the cruising velocity increased as well. As a result, phase gaps of motions and drag forces between the upstream and downstream filaments arise. For the paralle state, when the initial horizontal distance of the two filamens is small while the vertical offset is large enough, the influence of the upstream vortex on the downstream filament is decreased. So the downstream filament moves forward and the parallel state is formed. In this case, the shear layers of two filaments shed synchronously and strong vortices are formed in the wake. Meanwhile the windward aera are about twice as compared with the tandem states, thus the cruising velocity is reduced. In summary, as a result of the interactions between upstream vortices and downstream vortices/structures, the forward speed of the parellel state is the slowest, and that of the short-distance state is the fastest, while that of the long-distance state is middle among the three stable states.