Global modes with multiple saddle points

Benoît Pier (1) , Nigel Peake (2)

1 LMFA - Laboratoire de Mecanique des Fluides et d'Acoustique
2 DAMTP - Department of Applied Mathematics and Theoretical Physics

Significant progress has been made towards understanding the global stability of slowly-developing shear flows. The WKBJ theory developed by Patrick Huerre and his co-authors has proved absolutely central, with the result that both the linear and the nonlinear stability of a wide range of flows can now be understood in terms of their local absolute/convective instability properties. In many situations, the local absolute frequency possesses a single dominant saddle point in complex X-space (where X is the slow streamwise coordinate of the base flow), which then acts as a single wavemaker driving the entire global linear dynamics. In this paper we consider the more complicated case in which multiple saddles may act as the wavemaker for different values of some control parameter. We derive a frequency selection criterion in the general case, which is then validated against numerical results for the linearised third-order Ginzburg–Landau equation (which possesses two saddle points). We believe that this theory may be relevant to a number of flows, including the boundary layer on a rotating disk and eccentric Taylor–Couette–Poiseuille flow.

Mots clés: complex saddle points; absolute instabilities; global modes

Domaines: Mécanique des fluides [physics.class-ph]

Benoît Pier, Nigel Peake. Global modes with multiple saddle points. European Journal of Mechanics - B/Fluids, 2015, 49, pp.335-344. ⟨10.1016/j.euromechflu.2014.03.006⟩. ⟨hal-01084705⟩