Flapping dynamics of a flag in a uniform stream.

*(English)*Zbl 1124.76011The authors consider two-dimensional flag problem in an unbounded fluid domain. The effects of Reynolds number, structure-to-fluid mass ratio, and nondimensional bending rigidity on the system stability are examined. For the case of low bending frequency, the authors study the influence of the above mass ratio on the flag flapping.

At first, the linear analysis is performed to show how the relevant non-dimensional parameters of the system influence the onset of flapping instability. The structural tension in this theoretical stability analysis is modelled as a function of Reynolds number by using boundary-layer theory, and no attempt is made to model wake effects in the linear analysis. To examine the full nonlinear flapping problem, including wake effects and variable tension in the body, the authors develop a coupled fluid-structure direct simulation (FSDS) of Navier-Stokes equations and geometrically nonlinear structural equations. A systematic series of FSDS runs is performed to understand the trends of stability and response for the flag flapping problem. The simulations corroborate the stability criteria identified in the linear analysis: the destabilizing effect of increasing structure-to-fluid mass ratio, where a lesser proportion of the inertia is convecting with the flow as added mass; and a stabilizing effect of higher bending rigidity and lower Reynolds number, both of which increase the structural restoring force. For the case of very low bending rigidity and low-to-moderate Reynolds numbers (\(Re=100 \rightarrow 5000\)), the increase in the mass ratio is found to transfer the system through three distinct regimes: (I) fixed-point stability, (II) limit-cycle flapping, and (III) chaotic flapping; each with representative wake characteristics. Parametric stability is explained with a nonlinear softening spring model. The chaotic flapping response occurs as a breaking of the limit cycle by inclusion of 3/2 superharmonic components. This occurs when the increased flapping amplitude exhibits the flapping Strouhal number \(St \simeq 0.2\) in a neighborhood of the natural vortex wake.

At first, the linear analysis is performed to show how the relevant non-dimensional parameters of the system influence the onset of flapping instability. The structural tension in this theoretical stability analysis is modelled as a function of Reynolds number by using boundary-layer theory, and no attempt is made to model wake effects in the linear analysis. To examine the full nonlinear flapping problem, including wake effects and variable tension in the body, the authors develop a coupled fluid-structure direct simulation (FSDS) of Navier-Stokes equations and geometrically nonlinear structural equations. A systematic series of FSDS runs is performed to understand the trends of stability and response for the flag flapping problem. The simulations corroborate the stability criteria identified in the linear analysis: the destabilizing effect of increasing structure-to-fluid mass ratio, where a lesser proportion of the inertia is convecting with the flow as added mass; and a stabilizing effect of higher bending rigidity and lower Reynolds number, both of which increase the structural restoring force. For the case of very low bending rigidity and low-to-moderate Reynolds numbers (\(Re=100 \rightarrow 5000\)), the increase in the mass ratio is found to transfer the system through three distinct regimes: (I) fixed-point stability, (II) limit-cycle flapping, and (III) chaotic flapping; each with representative wake characteristics. Parametric stability is explained with a nonlinear softening spring model. The chaotic flapping response occurs as a breaking of the limit cycle by inclusion of 3/2 superharmonic components. This occurs when the increased flapping amplitude exhibits the flapping Strouhal number \(St \simeq 0.2\) in a neighborhood of the natural vortex wake.

Reviewer: Peter A. Velmisov (Ul’yanovsk)