Onset of temporal dynamics within a low reynolds-number laminar fluidic oscillator

Abstract

We investigate the onset of temporal dynamics of a fluidic oscillator (FO) in the laminar regime at very low Reynolds number (Re=Uhi/ν, where U and hi are the velocity and channel height at inlet, and ν is the kinematic viscosity of the fluid). Both two- and spanwise-periodic-three-dimensional simulations are performed using high-order spectral element methods to characterise the flow inside the FO cavity and the outcoming jet. While the flow remains steady and symmetric for sufficiently low Re, the two-dimensional FO undergoes a symmetry-breaking Hopf bifurcation at ReH2=75.7 that results in a space-time symmetric periodic solution. The remnant space-time symmetry is later broken in a pitchfork bifurcation of limit cycles at ReP2=294.2. Taking three-dimensionality into consideration results in an early spanwise-invariance disruption at about Re3≃68 of wavelength 13hi that retards the onset of the oscillation to ReH3=82.1. This effect is little representative of actual FO implementations, which are typically much narrower than the wavelength of the spanwise instability observed at these low Re. Contrary to what happens with FOs in the turbulent oscillatory regime, the oscillation frequency of the output jet in the laminar regime is found to decrease with Re. The mechanism that drives the oscillation, however, remains the same: as the high speed flow inside the cavity attaches to one of the internal walls through the Coandă effect, the feedback channels divert part of the momentum back, which pushes the incoming flow against the opposite wall. The alternate deviation of the flow inside the cavity to one or the other side results in the periodic flipping of the output jet. The pressure contribution to net momentum at the end of the feedback channels is found to be double that of the advective momentum flux at the early time-periodic regime but both become comparable as Re is increased. The jet sweeping angle amplitude is more pronounced for the two-dimensional FO as compared to three-dimensional at a fixed given Re, the Coandă effect being only partially fulfilled in the latter case. In both cases the sweep amplitude increases with Re. The instability of the output jet, which becomes slightly chaotic already at very low Re, is responsible for triggering the cavity instability that drives the oscillation slightly earlier than it would, should outside noise be suppressed altogether.