Numerical Simulations Of The Instabilities And Transitions In The Wake Of A Flapping Foil

In recent years,the development of micro air vehicles(MAVs)and small unmanned under-water vehicles have led to a growing interest in aerodynamics and hydrodynamics of the flapping wings operating at low Reynolds numbers,particularly when the wings engage in flapping motion-s.It is well known that in the nature,fish,birds and insects flap their tails or wings in a so-called stroke plane employing a combination of rotation with respect to the wing-body junction and pitch-ing to the spanwise axis to produce forward thrusts,with the characteristics of high efficiency,low energy consumption,high cruise speed,low noise and excellent maneuverability.These unique locomotive abilities of natural animals have inspired people by providing with new cencepts for the design of aircrafts or marine vehicles.As a reasonable simplification,the flapping motion of a foil has attracted attentions from the researchers in the community of biomimetic hydrodynamics.In this thesis,we study nunerically the instbilities and transitions in the wake of a flapping foil.First we study the wake dynamics of low Reynolds number flows around a two-dimensional NACA0015 airfoil with fixed angles of attack(AoA).The AoAs we have considered can be cat-egorized into two groups,exhibiting different flow modes,such as the“leading-edge vortex”at high AoAs and the "separation vortex" modes at low AoAs.The Power Spectral Density(PSD)is used to explain the characteristics of the near flow fields around the foil.We find that the wake transits to chaotic flow regime by successive period-doubling bifurcations and various incommen-surate bifurcations.To predict the secondary instabilities,or three-dimensional transitions of the wake,at fixed moderate angles of attack,we examine the evolution of the periodic base flow in response to infinitesimal perturbations using the Floquet stability analysis.The wake topologies of the base flow are compared to the different unstable modes,to understand the different under-lying physics of the flow transitions for the two groups of AoAs.A good agreement between the Floquet analysis and the 3D DNS is achieved,indicating that the linear stability dominates the three-dimensional transitions.It is interesting to find that the critical Reynolds numbers for the onset of three-dimensionalities are in the range of[159.7,234.2]by re-defining the Reynolds num-ber according to the width of the flow wake.These values are very close to that of a bluff body,evidencing that the wake of an airfoil,at sufficiently large AoAs,i.e.after the stall,follows the same route of transitions with that of a bluff body.Second,we study the three-dimensional instabilities of the flow past a periodically flapping foil with pitching and heaving motions considered respectively.By fixing the Reynolds number at Re=1700 and varying the flapping frequency and amplitude,we identify three key dynamical features of the wake:first,the transition from Benard-von Karman(BvK)vortex streets to reverse BvK vortex streets,and second,the symmetry breaking of this reverse BvK wake leading to a deflected wake,and a further transition from two-dimensional(2D)wakes to three-dimensional(3D)wakes.The drag-thrust tansition has also been identified,which is slightly different from the Bvk to reverse BvK transition,while accords well with the optimal Strouhal numbers observed in the nature for flying or swimming animals.Further more,for the pure heaving foil,we also study the sequential emergences of different instability modes,and analyse them by using POD and DMD method.More interestingly,we have studied the correlation between wake transitions and propulsive efficiency for the flapping foil.Plotting the contours of propulsive efficiency in the same frequency-amplitude parametric space,in which we identify the wake transition boundaries,we find a coinci-dence that the efficiency maximum agrees well with the 2D-3D transition boundary.The compar-ison between the pure pitching and the pure heaving foils shows that the 2D-3D transition occurs earlier for the pure heaving one than that of the pure pitching one.In addition,we report that the efficiency for the pure heaving foil peaks more closely to the wake deflection boundary than that of the pure pitching foil.Quantitively,we address that the maximum efficiency for a pure pitching foil is 15.6%,and that of a pure heaving foil is 17%,indicating that the pure heaving foil has a slightly better propulsive performance than that of the pure pitching foil at least for the currently studied Reynolds number Re=1700.