Flapping dynamics of a flag in a uniform stream
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- 22 May 2007
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Fluid Mechanics
- Vol. 581, 33-67
- https://doi.org/10.1017/s0022112007005307
Abstract
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.Keywords
This publication has 36 references indexed in Scilit:
- Heavy Flags Undergo Spontaneous Oscillations in Flowing WaterPhysical Review Letters, 2005
- Numerical simulations of a filament in a flowing soap filmInternational Journal for Numerical Methods in Fluids, 2003
- Mechanics of nonlinear short-wave generation by a moored near-surface buoyJournal of Fluid Mechanics, 1999
- Dynamic Response of Cables Under Negative Tension: an Ill-Posed ProblemJournal of Sound and Vibration, 1994
- Stable and unstable nonlinear resonant response of hanging chains: theory and experimentProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 1993
- Physical condition for absolute instability in inviscid hydroelastic couplingPhysics of Fluids A: Fluid Dynamics, 1992
- Dynamical symmetry breaking and chaos in Duffing’s equationAmerican Journal of Physics, 1991
- Universal properties of dynamically complex systems: the organization of chaosNature, 1988
- Numerical Solution of the Navier-Stokes EquationsMathematics of Computation, 1968
- Dynamics of flexible slender cylinders in axial flow Part 1. TheoryJournal of Fluid Mechanics, 1966