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Three-dimensional effects on flag flapping dynamics
- Sankha Banerjee, Benjamin S. H. Connell, Dick K. P. Yue
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- Journal:
- Journal of Fluid Mechanics / Volume 783 / 25 November 2015
- Published online by Cambridge University Press:
- 19 October 2015, pp. 103-136
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We examine three-dimensional (3D) effects on the flapping dynamics of a flag, modelled as a thin membrane, in uniform fluid inflow. We consider periodic spanwise variations of length $S$ (ignoring edge effects), so that the 3D effects are characterized by the dimensionless spanwise wavelength ${\it\gamma}=S/L$, where $L$ is the chord length. We perform linear stability analysis (LSA) to show increase in stability with ${\it\gamma}$, with the purely 2D mode being the most unstable. To confirm the LSA and to study nonlinear responses of 3D flapping, we obtain direct numerical simulations, up to Reynolds number 1000 based on $L$, coupling solvers for the Navier–Stokes equations and that for a thin membrane structure undergoing arbitrarily large displacement. For nonlinear flapping evolution, we identify and characterize the effect of ${\it\gamma}$ on the distinct flag motions and wake vortex structures, corresponding to spanwise standing wave (SW) and travelling wave (TW) modes, in the absence and presence of cross-flow respectively. For both SW and TW, the response is characterized by an initial instability growth phase (I), followed by a nonlinear development phase (II) consisting of multiple unstable 3D modes, and tending, in long time, towards a quasi-steady limit-cycle response (III) dominated by a single (most unstable) mode. Phase I follows closely the predictions of LSA for initial instability and growth rates, with the latter increased for TW due to suppression of restoring forces by the cross-flow. Phase II is characterized by multiple competing flapping modes with energy cascading towards the more unstable mode(s). The wake is characterized by interwoven (SW) and oblique continuous (TW) shed vortices. For phase III, the persistent single dominant mode for SW is the (most unstable) 2D flag displacement with a continuous parallel wake structure; and for TW, the fundamental oblique travelling-wave flag displacement corresponding to the given ${\it\gamma}$ with continuous oblique shedding. The transition to phase III occurs slower for greater ${\it\gamma}$. For the total forces, drag decreases for both SW and TW with decreasing ${\it\gamma}$, while lift is negligible in phase I and II and comparable in magnitude to drag in phase III for any ${\it\gamma}$.
Flapping dynamics of a flag in a uniform stream
- BENJAMIN S. H. CONNELL, DICK K. P. YUE
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- Journal:
- Journal of Fluid Mechanics / Volume 581 / 25 June 2007
- Published online by Cambridge University Press:
- 22 May 2007, pp. 33-67
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We consider the flapping stability and response of a thin two-dimensional flag of high extensional rigidity and low bending rigidity. The three relevant non-dimensional parameters governing the problem are the structure-to-fluid mass ratio, μ = ρsh/(ρfL); the Reynolds number, Rey = VL/ν; and the non-dimensional bending rigidity, KB = EI/(ρfV2L3). The soft cloth of a flag is represented by very low bending rigidity and the subsequent dominance of flow-induced tension as the main structural restoring force. We first perform linear analysis to help understand the relevant mechanisms of the problem and guide the computational investigation. To study the nonlinear stability and response, we develop a fluid–structure direct simulation (FSDS) capability, coupling a direct numerical simulation of the Navier–Stokes equations to a solver for thin-membrane dynamics of arbitrarily large motion. With the flow grid fitted to the structural boundary, external forcing to the structure is calculated from the boundary fluid dynamics. Using a systematic series of FSDS runs, we pursue a detailed analysis of the response as a function of mass ratio for the case of very low bending rigidity (KB = 10−4) and relatively high Reynolds number (Rey = 103). We discover three distinct regimes of response as a function of mass ratio μ: (I) a small μ regime of fixed-point stability; (II) an intermediate μ regime of period-one limit-cycle flapping with amplitude increasing with increasing μ; and (III) a large μ regime of chaotic flapping. Parametric stability dependencies predicted by the linear analysis are confirmed by the nonlinear FSDS, and hysteresis in stability is explained with a nonlinear softening spring model. The chaotic flapping response shows up as a breaking of the limit cycle by inclusion of the 3/2 superharmonic. This occurs as the increased flapping amplitude yields a flapping Strouhal number (St = 2Af/V) in the neighbourhood of the natural vortex wake Strouhal number, St ≃ 0.2. The limit-cycle von Kármán vortex wake transitions in chaos to a wake with clusters of higher intensity vortices. For the largest mass ratios, strong vortex pairs are distributed away from the wake centreline during intermittent violent snapping events, characterized by rapid changes in tension and dynamic buckling.