PHYSICAL REVIEW E 91, 043017 (2015) Three-dimensional transition after wake deflection behind a flapping foil Jian Deng * Department of Mechanics, Zhejiang University, Hangzhou 310027, People’s Republic of China C. P. Caulfield BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, United Kingdom and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Received 19 November 2014; revised manuscript received 8 April 2015; published 24 April 2015) We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: B´ enard-von K´ arm´ an (BvK) vortex streets; reverse BvK vortex streets; and deflected wakes. Imposing a three-dimensional spanwise periodic perturbation, we find that the resulting Floquet multiplier |μ| indicates an unstable “short wavelength” mode at wave number β = 30, or wavelength λ = 0.21 (nondimensionalized by the chord length) at sufficiently high flow Reynolds number Re = Uc/ν ≃ 600, where U is the upstream flow velocity, c is the chord length, and ν is the kinematic viscosity of the fluid. Another, “long wavelength” mode at β = 6(λ = 1.05) becomes critical at somewhat higher Reynolds number, although we do not expect that this mode would be observed physically because its growth rate is always less than the short wavelength mode, at least for the parameters we have considered. The long wavelength mode has certain similarities with the so-called mode A in the drag wake of a fixed bluff body, while the short wavelength mode appears to have a period of the order of twice that of the base flow, in that its structure seems to repeat approximately only every second cycle of the base flow. Whether it is appropriate to classify this mode as a truly subharmonic mode or as a quasiperiodic mode is still an open question however, worthy of a detailed parametric study with various flapping amplitudes and frequencies. DOI: 10.1103/PhysRevE.91.043017 PACS number(s): 47.20.Ky, 47.27.Cn, 47.63.M− I. INTRODUCTION Extensive work on the flow behind a circular cylinder [1–7] indicates that wakes of two-dimensional bluff bodies undergo transition to three-dimensional flow and eventually turbulence through a sequence of mode emergence. For a circular cylinder, the transition from a two-dimensional B´ enard-von K´ arm´ an (BvK) vortex street to three-dimensional flow occurs with the onset of the first three-dimensional mode, named “mode A,” at a critical Reynolds number about 190. The spanwise wavelength of mode A is three to four cylinder diameters, and mode A appears to be an instability of the primary vortex cores [3]. A further transition to “mode B ” shedding occurs as the Reynolds number is increased to 230–250 with a shorter spanwise wavelength of approximately one diameter of the circular cylinder, which is conjectured to scale on the vorticity thickness of the braid shear layer [3,4]. Another theoretically possible mode is the quasiperiodic “mode QP,” identified by Floquet stability analysis [8]. Investigations have also been conducted on the flow fields around bluff bodies with noncircular sections, such as those around bluff elongated cylinders [9] and stalled airfoils [10]. For a cylinder of square cross section, “mode S ” was found to be critical within 150 < Re < 225, but only after the other modes had already undergone transition, and therefore may not be observed experimentally. Mode S was first found to be sub- harmonic [11], with a period double that of the base flow. It was then discovered that mode S has a complex Floquet multiplier * Corresponding author: zjudengjian@zju.edu.cn with real part negative and a small imaginary component, and thus it appears to repeat every second cycle [8]. It is noted that a real harmonic mode cannot be physically realized unless the Z 2 spatiotemporal symmetry in the wake is broken. This can be achieved in two ways: (i) in geometries without reflection symmetry, such as a circular cylinder with a tripwire placed adjacent to the cylinder but not on the symmetry plane [12], inclined flat plates [13], inclined square cylinders [14], and stalled airfoils [10]; (ii) by a transversely oscillating cylinder which can also change the spatiotemporal symmetry of the two-dimensional wake [15,16]. At high oscillation amplitudes, the wake takes on the “P + S ” configuration, with a pair of vortices on one side of the wake and a single vortex on the other side for each oscillation cycle. As a result of the asymmetry about the center line, a real subharmonic “mode C” instability emerges, or more specifically for oscillating cylinders two subharmonic modes, “SL” and “SS ,” appear, with long and short wavelengths respectively. Efforts have been made to distinguish subharmonic modes from quasiperiodic modes based on the behavior of the relevant Floquet multipliers [17,18]. An apparent difference between quasiperiodic and subharmonic wake instability modes is that the latter do not generically arise if the two-dimensional state has Z 2 spatiotemporal symmetry. The most direct way of seek- ing subharmonic modes is thus to seek an asymmetric wake. Most previous studies focused on drag wakes or the classic BvK vortex street. It is unknown whether a similar transition route exists in a thrust wake. As a well-known thrust wake, the wake behind a flapping airfoil has attracted much attention [19,20] because it acts as a simplified model of an aquatic propulsor [21,22]. The majority of these studies were 1539-3755/2015/91(4)/043017(8) 043017-1 ©2015 American Physical Society