THE PERFORMANCE OF A NEW IMMERSED BOUNDARY METHOD ON SIMULATING UNDERWATER LOCOMOTION AND SWIMMING

Abstract

We report the benchmark results of a new Immersed Boundary Method (IBM) incorporated into Direct Numerical Simulation (DNS) of a pitching panel, representing fishlike swimming, using foam-extend-3.2. The panel is flat and thin, and it has a triangular (convex) trailing edge, similar to that seen in the caudal fin of some fish. The accuracy of the solver is verified by comparing four cases of bluff body wake simulations with reported experimental and numerical studies. For example, the structure of the mean wake compared well with that obtained using PIV in a companion experiment. The differences in thrust generation and propulsion efficiency of square and convex thin panels are examined to identify the effect of trailing edge shape using proper orthogonal decomposition. The effect of Reynolds number is also evaluated by comparing the wake at Reynolds numbers of 2,000 and 10,000. INTRODUCTION Many biological species have evolved to develop efficient propulsive systems for swimming that also exhibit high speed and maneuverability (Sambiley, 1990; Sumich & Morrissey, 2004). Understanding the mechanics of aquatic propulsion has attracted the attention of many researchers over the years. Such explorations can yield valuable information on designing energy efficient and fast systems with high maneuverability and stealth that match and possibly surpass the performance of biological species. In this regard, it is often useful to focus on simple systems that can be used to study fundamental aspects of swimming performance. For instance, the implications of the trailing-edge shape of a tail fin as well as its orientation and movement (that is, harmonic fin motion) on the formation of vortex structures, wake dynamics and thrust generation plays a dominant role in understanding physics of fish-like swimming (Van Buren et al., 2016). The oscillating motion of a NACA 0012 airfoil was studied by Triantafyllou et al. (1991) as a representative of swimming motion by fish, which demonstrated a maximum propulsive efficiency of 25% for the Strouhal number St = 0.25− 0.35, where St = fpc/U , fp is the frequency of oscillation, c is the fin characteristic length, and U is the swimming speed. The extensive study by Buchholz & Smits (2006) on the wake of a pitching rigid rectangular panel at moderate Reynolds numbers (Re =Uc/ν) revealed that the flow is dominated by horseshoe-like structures. The aspect ratio of the panel was identified to impact the wake, and thus, the propulsive performance efficiency, which ranged from 9%− 21%. Green & Smits (2008) investigated the distribution of pressure on the pitching panel, which revealed that the favorable streamwise pressure gradient that is present over most of the panel reversed near the trailing edge. There are many experimental challenges, however, in determining detailed surface pressure distributions in unsteady wakes, and it is even more difficult to determine the instantaneous shear stress distributions. In contrast, computational fluid dynamics (CFD) can be used to give insight into the stress distributions, and characterize the wake and identify implications of the wake dynamics on thrust generation. CFD simulations can also be helpful in providing insight into wake structures, their formation and interactions, and the effects of Reynolds number. Numerical simulations of Blondeaux et al. (2005) on a pitching foil showed vortex-loop (chain-like) structures dominate the wake at Re = 1000, as found experimentally by Green et al. (2011). Blondeaux et al. (2005) used a distinctly developed CFD solver based on the Immersed Boundary Method (IBM). Jantzen et al. (2014) also used an IBM-based CFD solver to evaluate the wake of pitching rigid rectangular panels at Re = 300. This study provided details on the vortex formation process in the wake of rectangular panels of aspect ratios 2 and 4, revealing that the Reynolds number influences the formation frequency and length of leading edge vortices, and that higher aspect ratios result in early detachment of leading edge vortices. The IBM is routinely used in fluid flow simulations as an alternative to the boundary-fitted method due to its lower computational cost. The IBM formulation in foam-extend3.2 uses a discrete forcing approach based on a weighted least square approximation to impose boundary conditions independent of the actual boundaries. This approach can capture the boundary as a sharp interface, which eliminates the issue of smearing. It also mitigates the distortion of cells around the moving boundaries, while alleviating errors that arise from transformation of curvilinear grids (Lundquist et al., 2009). See also Iaccarino & Verzicco (2003) and Mittal & Iaccarino (2005). Jasak et al. (2014) incorporated the IBM forcing approach into OpenFOAM as part of foam-extend-3.2. Case studies included flows over fixed bodies. In particular, a high Re flow case using the k− ε model showed a lack of stability for extreme mesh refinements. Şentürk et al. (2016) extended this work to consider the (a) 2D flow over a stationary circular cylinder, (b) 2D wake of a transversely