Galerkin force model for transient and post-transient dynamics of the fluidic pinball

We propose an aerodynamic force model associated with a Galerkin model for the unforced fluidic pinball, the two-dimensional flow around three equal cylinders with one radius distance to each other. The starting point is a Galerkin model of a bluff-body flow. The force on this body is derived as a constant-linear-quadratic function of the mode amplitudes from first principles following the pioneering work of Noca (On the evaluation of time-dependent fluid-dynamic forces on bluff bodies. PhD thesis, California Institute of Technology, 1997), Noca et al. (J. Fluids Struct., vol. 13, issue 5, 1999, pp. 551–578) and Liang & Dong (Virtual force measurement of POD modes for a flat plate in low Reynolds number flows. AIAA Paper 2014-0054). The force model is simplified for the mean-field model of the unforced fluidic pinball (Deng et al., J. Fluid Mech., vol. 884, 2020, p. A37) using symmetry properties and sparse calibration. The model is successfully applied to transient and post-transient dynamics in different Reynolds number regimes: the periodic vortex shedding after the Hopf bifurcation and the asymmetric vortex shedding after the pitchfork bifurcation comprising six different Navier–Stokes solutions. We foresee many applications of the Galerkin force model for other bluff bodies and flow control.