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This study introduces novel physics-based scaling laws to estimate the propulsive performance of synchronously pitching foils in various schooling configurations at Re=4000. These relations are derived from quasi-steady lift-based and added mass forces. Hydrodynamic interactions among the schooling foils are considered through vortex-induced velocities imposed on them, constituting the ground effect. Generalized scaling equations are formulated for cycle-averaged coefficients of thrust and power. These equations encompass both the pure-pitching and induced velocity terms, capturing their combined effects. The equations are compared to computational results obtained from two-foils systems, exhibiting foil arrangements over a wide range of parameter space, including Strouhal number (0.15 \leq St \leq 0.4), pitching amplitude (5 deg \leq \theta_0 \leq 14 deg), and phase difference (0 deg \leq \phi \leq 180 deg). The individual contributions of pure-pitching and induced velocity terms to propulsive performance elucidate that solely relying on the pure-pitching terms leads to inadequate estimation, emphasizing the significance of the induced velocity terms. Validity of the approach is further assessed by testing it with a three-foil configuration, which displays a collapse. This indicates that the scaling laws are not only applicable to two-foils systems but also extend their effectiveness to multi-foil arrangements.

The correspondence of secondary spanwise structures and pressure gradient is numerically evaluated for a foil, performing heaving and pitching motion, at a range of phase offsets (90$^\circ$ $\le \phi \le$ 270$^\circ$) and reduced frequency (0.32 $\le St_c \le$ 0.56). The Reynolds number is $Re =$ 8000. The wake is shown to be dominated by secondary hairpin-like structures that are formed due to an elliptic instability prompted by the paired primary and secondary leading edge vortex ($LEV$). The weaker secondary $LEV$ undergoes a core deformation, resulting in streamwise vorticity outflux across the span of the foil, and hence, the growth of hairpin-like structures. Evaluating pressure gradients on the surface of the foil reveals a unique fundamental measure to quantitatively characterize the growth of these coherent structures. Their dominant presence can be directly linked to the growth of the secondary $LEV$ formed due to the large-scale interactions under localized adverse pressure gradients. These promote a streamwise flow compression in neighboring regions of the primary $LEV$. This association also presents a vivid consistency across a range of kinematics. Therefore, this correspondence provides a novel procedure to investigate the mechanisms involved in the formation of secondary structures in the wake of an oscillating foil.

Three-dimensional wake of an oscillating foil with combined heaving and pitching motion is numerically evaluated at a range of chord-based Strouhal number (0.32 \le Stc \le 0.56) and phase offset (90 deg \le \phi \le 70 deg) at Re = 8000. The changes in \phi and Stc reflect a unique route of transition in mechanisms that govern the origin of spanwise instabilities and growth of secondary wake structures. At lower Stc, heave dominated kinematics demonstrates a strong secondary leading edge vortex (LEV ) as the source of growing spanwise instability on the primary LEV , followed by an outflux of streamwise vorticity filaments from the secondary LEV . With increasing heave domination, the origin of stronger spanwise instability is governed by a counter-rotating trailing edge vortex (TEV ) and LEV that leads to growth of streamwise secondary structures. A decreasing heave domination ultimately coincides with an absence of strong LEV undulations and secondary structures. The consistent transition routes are represented on a phase-space map, where a progression of spanwise instability and growth of secondary structures becomes evident within regimes of decreased heave domination. The increasing strength of circulation for the primary LEV , with increasing Stc, provides a crucial reasoning for this newly identified progression.

Several methods have been proposed to characterize the complex interactions in turbulent wakes, especially for flows with strong cyclic dynamics. This paper introduces the concept of Fourier-Averaged Navier-Stokes (FANS) equations as a framework to obtain direct insights into the dynamics of complex coherent wake interactions. The method simplifies the interpretations of flow physics by identifying terms contributing to momentum transport at different timescales. The method also allows for direct interpretation of non-linear interactions of the terms in the Navier-Stokes equations. By analysing well-known cases, the characteristics of FANS are evaluated. Particularly, we focus on physical interpretation of the terms as they relate to the interactions between modes at different timescales. Through comparison with established physics and other methods, FANS is shown to provide insight into the transfer of momentum between modes by extracting information about the contributing pressure, convective, and diffusive forces. FANS provides a simply calculated and easily interpreted set of equations to analyse flow physics by leveraging momentum conservation principles and Fourier analysis. The method is applicable to flows with complex cyclic waveforms, including broadband spectral energy distributions., Comment: 28 pages, 23 figures. Submitted to the Journal of Fluid Mechanics

The unsteady hydrodynamics of two in-phase pitching foils arranged in side-by-side (parallel) configurations is examined for a range of Strouhal number and separation distance. Three distinct vortex patterns are identified in the Strohual number-separation distance phase maps, which include separated wake, merged wake, and transitional-merged wake. Furthermore, a novel model is introduced based on fundamental flow variables including velocity, location, and circulation of dipole structures to quantitatively distinguish vortex patterns in the wake. The physical mechanism of wake merging process is also elucidated. When an oscillating foil experiences the jet deflection phenomenon, secondary structures shed from the primary street traverse in the other direction by making an angle with its parent vortex street. For parallel foils, secondary structures from the vortex street of the lower foil interact with the primary vortex street of the upper foil under certain kinematic conditions. This interaction triggers the wake merging process by influencing circulation of coherent structures in the upper part of the wake. It is unveiled that merging of the wakes leads to enhancements in propulsive efficiency by increasing thrust generation without a significant alteration in power requirements. These are attributed to the formation of a high-momentum jet by the merged vortex street, which possesses significantly larger circulation due to the amalgamation of the vortices, and major alterations in the evolution of leading edge vortices. Thus, flow physics that are thoroughly explored here are crucial in providing novel insights for future development of flow control techniques for efficient designs of bio-inspired underwater propulsors.

This study explored the aerodynamics of a new multi-stage co-axial vertical-axis wind turbine based on bio-inspiration from natural swimming habit of fish. The turbine was formed from a conventional straight-bladed vertical-axis turbine (VAWT) with a small inner rotor, also of three blades. The azimuthal and radial locations of the inner rotor were varied. Using numerical simulations, performance of the proposed new design was evaluated over a range of tip-speed ratios. The preliminary results identified a 600% increase in power output for multi-stage VAWTs at tip-speed rations TSR = 0 - 3, and a substantial drop in power coefficient at TSR > 3.0. The wake dynamics analyses revealed that the increase was due to interactions between the blades of one rotor and the other. This reduced the unsteady separation from the outer rotor which produced most of the power. A detailed parametric study was also completed, which showed the implications of geometric and kinematic details on the performance of the proposed multistage VAWT.

This study was inspired from the swimming habits of red nose tetra fish that prefer side-by-side configurations and exhibit changes to their synchronization mid-swimming. Using numerical simulations, alterations to the unsteady wake dynamics imposed by abrupt changes in the phase angle between two pitching side-by-side foils were examined at Reynolds number of 4;000 and Strouhal number of 0:50. Four hybrid modes were considered in thisstudy with two modes representing an abrupt phase change by pi during the 20th cycle. The other two modes represented a simplified case of burst-and-coast swimming, in which there was a brief (2 oscillation periods) suspension of oscillations before imposing a change in phase angle. In all cases, the foils initially performed out-of-phase pitching, and then they started their in-phase motion by either performing the upstroke or down-stroke first. This kinematic change resulted in the formation and growth of a secondary vortex street in between two primary streets, which enabled and maintained a split wake configuration. Furthermore, the phase switching altered the pressure levels on the top and bottom surfaces of both foils to almost similar levels, which attributed to a reduction in the side-force. The growth rate of the secondary vortex street remained consistent for all four hybrid modes.

The wavelength of undulatory kinematics of fish is an important parameter to determine their hydrodynamic performance. This study focuses on numerical examination of this feature by reconstructing the real physiological model and kinematics of steadily swimmning Jack Fish. We perform three-dimensional numerical simulations for flows over these models composed of the trunk, and dorsal, anal, and caudal fins. Moreover, we prescribe the carangiform-like motion for its undulation for a range of wavelengths. Undulation with larger wavelengths improves the hydrodynamic performance of the carangiform swimmer in terms of better thrust production by the caudal fin, lower drag production on the trunk, and reduced power consumption by the trunk. This coincides with the formation of stronger posterior body vortices and leading-edge vortices with more circulation on the caudal fin. The real kinematics of Jack Fish surpasses the performance of those with prescribed motion owing to the flexibility of the caudal fin., Comment: 11 pages, 9 figures