Linear and nonlinear perturbation analysis of the symmetry breaking in time-periodic propulsive wakes

Abstract

The two-dimensional and time-periodic wake flows produced by a pitching foil are investigated numerically for a fixed flapping amplitude. As the flapping frequency is increased, three regimes are identified in the time-marching nonlinear simulations. The first regime is characterized by nondeviated wake flows with zero time-averaged lift. In the second regime, the wake flow is slightly deviated from the streamwise direction and the time-averaged lift is slightly positive or negative. The third regime is characterized by larger deviations of the wake, associated with larger values of both the time-averaged lift and the thrust. The transition from the first to the second regime is examined by performing a Floquet stability analysis of the nondeviated wake. A specific method is introduced to compute the time-periodic, nondeviated wake when it is unstable. It is found that one synchronous antisymmetric mode becomes unstable at the critical frequency where deviation occurs. Investigation of its instantaneous and time-averaged characteristics show that it acts as a displacement mode translating the nondeviated wake away from the streamwise direction. Finally, it is demonstrated that the transition from the second to the third regime is linked to nonlinear effects that amplify both antisymmetric and symmetric perturbations around the foil.

The flow-induced vibration of an elliptical cross-section at varying angles of attack

“…For example, Williamson and Roshko (1988) and Morse and Williamson (2009) report this mode for experimental studies of the controlled oscillation of circular cylinders, and Leontini et al (2006) clearly showed this mode for the same problem numerically. Similar spatio-temporal symmetry breaking modes have been found in other oscillating body wake flows, such as flapping foils (Jallas et al, 2017). However, perhaps the most relevant is the work from Singh and Mittal (2005), who found evidence of this mode in the wake of an elastically mounted circular cylinder.…”

Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder

“…The generalised eigenvalue problem (2.3) is solved by the Arnoldi method or by a simple inverse iteration algorithm. Finally, nonlinear unsteady Navier–Stokes equations are integrated forward in time with a second-order time scheme (Jallas, Marquet & Fabre 2017).…”

Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

“…Therefore, the global analysis method must be adapted to account for this periodicity, by use of the Floquet theorem (Floquet 1883). A global Floquet formalism has been successfully applied by Barkley & Henderson (1996) to the secondary instability of a cylinder wake around Re = 200 and more recently by Jallas et al (2017) to a study of secondary instabilities in wake flows.…”