We employ numerical simulation to investigate the hydrodynamics of carangiform locomotion as the relative magnitude of viscous and inertial forces, i.e. the Reynolds number (Re), and the tail-beat frequency, i.e. the Strouhal number (St), are systematically varied. The model fish is a three-dimensional (3D) mackerel-like flexible body undulating with prescribed experimental kinematics of carangiform type. Simulations are carried out for three Re spanning the transitional and inertial flow regimes, Re=300 and 4000 (viscous flow), and ∞ (inviscid flow). For each Re there is a critical Strouhal number, St^*, at which the net mean force becomes zero, making constant-speed self-propulsion possible. St^* is a decreasing function of Re and approaches the range of St at which most carangiform swimmers swim in nature (St∼0.25) only as Re approaches infinity. The propulsive efficiency at St^* is an increasing function of Re while the power required for swimming is decreasing with Re. For all Re, however, the swimming power is shown to be significantly greater than that required to tow the rigid body at the same speed. We also show that the variation of the total drag and its viscous and form components with St depend on the Re. For Re=300, body undulations increase the drag over the rigid body level, while significant drag reduction is observed for Re=4000. This difference is shown to be due to the fact that at sufficiently high Re the drag force variation with St is dominated by its form component variation, which is reduced by undulatory swimming for St>0.2. Finally, our simulations clarify the 3D structure of various wake patterns observed in experiments – single and double row vortices – and suggest that the wake structure depends primarily on the St. Our numerical findings help elucidate the results of previous experiments with live fish, underscore the importance of scale (Re) effects on the hydrodynamic performance of carangiform swimming, and help explain why in nature this mode of swimming is typically preferred by fast swimmers.

We investigate numerically vortex-induced vibrations (VIV) of two identical two-dimensional elastically mounted cylinders in tandem in the proximity–wake interference regime at Reynolds number Re = 200 for systems having both one (transverse vibrations) and two (transverse and in-line) degrees of freedom (1-DOF and 2-DOF, respectively). For the 1-DOF system the computed results are in good qualitative agreement with available experiments at higher Reynolds numbers. Similar to these experiments our simulations reveal: (1) larger amplitudes of motion and a wider lock-in region for the tandem arrangement when compared with an isolated cylinder; (2) that at low reduced velocities the vibration amplitude of the front cylinder exceeds that of the rear cylinder; and (3) that above a threshold reduced velocity, large-amplitude VIV are excited for the rear cylinder with amplitudes significantly larger than those of the front cylinder. By analysing the simulated flow patterns we identify the VIV excitation mechanisms that lead to such complex responses and elucidate the near-wake vorticity dynamics and vortex-shedding modes excited in each case. We show that at low reduced velocities vortex shedding provides the initial excitation mechanism, which gives rise to a vertical separation between the two cylinders. When this vertical separation exceeds one cylinder diameter, however, a significant portion of the incoming flow is able to pass through the gap between the two cylinders and the gap-flow mechanism starts to dominate the VIV dynamics. The gap flow is able to periodically force either the top or the bottom shear layer of the front cylinder into the gap region, setting off a series of very complex vortex-to-vortex and vortex-to-cylinder interactions, which induces pressure gradients that result in a large oscillatory force in phase with the vortex shedding and lead to the experimentally observed larger vibration amplitudes. When the vortex shedding is the dominant mechanism the front cylinder vibration amplitude is larger than that of the rear cylinder. The reversing of this trend above a threshold reduced velocity is associated with the onset of the gap flow. The important role of the gap flow is further illustrated via a series of simulations for the 2-DOF system. We show that when the gap-flow mechanism is triggered, the 2-DOF system can develop and sustain large VIV amplitudes comparable to those observed in the corresponding (same reduced velocity) 1-DOF system. For sufficiently high reduced velocities, however, the two cylinders in the 2-DOF system approach each other, thus significantly reducing the size of the gap region. In such cases the gap flow is entirely eliminated, and the two cylinders vibrate together as a single body with vibration amplitudes up to 50% lower than the amplitudes of the corresponding 1-DOF in which the gap flow is active. Three-dimensional simulations are also carried out to examine the adequacy of two-dimensional simulations for describing the dynamic response of the tandem system at Re = 200. It is shown that even though the wake transitions to a weakly three-dimensional state when the gap flow is active, the three-dimensional modes are too weak to affect the dynamic response of the system, which is found to be identical to that obtained from the two-dimensional computations.

We carry out fluid–structure interaction simulations of self-propelled virtual swimmers to investigate the effects of body shape (form) and kinematics on the hydrodynamics of undulatory swimming. To separate the effects of form and kinematics, we employ four different virtual swimmers: a carangiform swimmer (i.e. a mackerel swimming like mackerel do in nature); an anguilliform swimmer (i.e. a lamprey swimming like lampreys do in nature); a hybrid swimmer with anguilliform kinematics but carangiform body shape (a mackerel swimming like a lamprey); and another hybrid swimmer with carangiform kinematics but anguilliform body shape (a lamprey swimming like a mackerel). By comparing the performance of swimmers with different kinematics but similar body shapes we study the effects of kinematics whereas by comparing swimmers with similar kinematics but different body shapes we study the effects of form. We show that the anguilliform kinematics not only reaches higher velocities but is also more efficient in the viscous ( Re ∼102) and transitional ( Re ∼103) regimes. However, in the inertial regime ( Re =∞) carangiform kinematics achieves higher velocities and is also more efficient than the anguilliform kinematics. The mackerel body achieves higher swimming speeds in all cases but is more efficient in the inertial regime only whereas the lamprey body is more efficient in the transitional regime. We also show that form and kinematics have little overall effect on the 3-D structure of the wake (i.e. single vs double row vortex streets), which mainly depends on the Strouhal number. Nevertheless, body shape is found to somewhat affect the small-scale features and complexity of the vortex rings shed by the various swimmers. * a max : tail-beat amplitude a ( z ) : amplitude envelope A : width of the wake A m : oscillation amplitude BCF : body/caudal fin c : damping factor c cr : critical damping factor CF∗ : mean force coefficient CFo : mean non-dimensional force in the axial direction CP∗ : mean power coefficient CPo : mean non-dimensional power CT∗ : mean thrust coefficient CTo : mean non-dimensional thrust CT : computed tomography D : cylinder diameter D ( t ) : drag EBT : elongated body theory f : tail-beat frequency f c : oscillation frequency fn : natural frequency fo : non-dimensional tail-beat frequency F : force F : force vector exerted on the virtual swimmer's body by the fluid FSI : fluid–structure interaction h : lateral displacement of fish body ![Graphic][1] : lateral velocity of fish body h max : maximum displacement of the tail h ( z,t ) : lateral excursion of the body at time t and location z HCIB : hybrid Cartesian immersed-boundary IB : immersed boundary k : wave number KC : Kuelegan–Carpenter number L : total body length LC–FSI : loose coupling fluid–structure interaction LL : lamprey swimming like a lamprey LM : lamprey swimming like mackerel m : mass of the virtual swimmer ML : mackerel swimming like a lamprey MM : mackerel swimming like a mackerel MPG : miles per gallon M red : reduced mass n : normal vector the surface nj : j th component of the normal vector p : non-dimensional pressure PIV : particle image velocimetry P side : power losss due to lateral undulations of the fish body Q T : rotation matrix of the non-inertial frame relative to the inertial frame ra : non-dimensional position in the inertial frame rr : non-dimensional position relative to the non-inertial frame r.m.s. : root mean squared Re * : mean Reynolds number based on swimming speed U* Reo : non-dimensional viscosity or Reynolds number based on characteristic velocity Uo S : symmetrical parts of the velocity gradient SC–FSI : strong coupling fluid–structure interaction St * : mean Strouhal number based on swimming speed U * Sto : Strouhal number based on the characteristic velocity Uo t : time T : mean thrust T : period of a tail-beat cycle ua : non-dimensional Cartesian absolute velocity vector of the fluid in the inertial reference frame uc : velocity of the non-inertial frame or the center of mass u m : maximum oscillation velocity un : fluid velocity normal to the body ur : non-dimensional Cartesian relative velocity vector of the fluid in the non-inertial reference frame ut : fluid velocity vector tangential to the body U : mean swimming speed U * : mean non-dimensional swimming speed Uo : characteristic (tether) velocity U red : reduced velocity VIV : vortex-induced vibration x c : position vector of center of mass non-dimensionalized by L x c : location of the center of the cylinder z : axial direction measured along the fish axis from the tip of the fish's head α : under-relaxation coefficient Γ( t ) : a dynamically evolving surface η* : mean efficiency (velocity over power) ηf : Froude efficiency λ : wavelength ν : kinematic viscosity ρ : fluid density τij : viscous stress tensor ω : angular frequency ξ : damping coefficient Ω : angular velocity [1]: /embed/inline-graphic-3.gif

We employ numerical simulation to investigate the hydrodynamic performance of anguilliform locomotion and compare it with that of carangiform swimming as the Reynolds number (Re) and the tail-beat frequency (Strouhal number, St) are systematically varied. The virtual swimmer is a 3-D lamprey-like flexible body undulating with prescribed experimental kinematics of anguilliform type. Simulations are carried out for three Reynolds numbers spanning the transitional and inertial flow regimes, Re=300, 4000 (viscous flow), and ∞ (inviscid flow). The net mean force is found to be mainly dependent on the tail-beat frequency rather than the tail-beat amplitude. The critical Strouhal number, St^*, at which the net mean force becomes zero (constant-speed self-propulsion) is, similar to carangiform swimming, a decreasing function of Re and approaches the range of St numbers at which most anguilliform swimmers swim in nature (St∼0.45) only as Re increases. The anguilliform swimmer's force time series is characterized by significantly smaller fluctuations above the mean than that for carangiform swimmers. In stark contrast with carangiform swimmers, the propulsive efficiency of anguilliform swimmers at St^* is not an increasing function of Re but instead is maximized in the transitional regime. Furthermore, the power required for anguilliform swimming is less than that for the carangiform swimmer at the same Re. We also show that the form drag decreases while viscous drag increases as St increases. Finally, our simulations reinforce our previous finding for carangiform swimmers that the 3-D wake structure depends primarily on the Strouhal number.