Small internal waves in sheared flows: lower bound for the vorticity transient growth and propagation bound in the parameter space

摘要

We show that transient kinetic energy growth of small traveling waves insideincompressible viscous wall flows is not a sufficient condition for theirenstrophy (the size of vorticity) growth. Historically, mathematicaldifficulties related to the wall vorticity boundary conditions left thevorticity perturbation problem open. In the last decades of the 20th century,the discovery of the transient perturbation growth and its link to thesubcritical transition to turbulence was still treated by means of energy-basedanalysis. By using the non-modal approach, we extend the work of J. L. Synge(1930s) which was an alternative way to attack the flow stability problem basedon the flow vorticity analysis instead of the classical kinetic energyanalysis. Our calculations leads to the lower bound for the algebraic enstrophygrowth as a function of the perturbation wavelength. This highlights how theenstrophy monotonic decay region inside the wavenumber - Reynolds number (Re)map is wider than that of the kinetic energy. By taking the general point ofview of internal wave propagation in a medium, we present and discuss thedispersion variability inside the stability map. By keeping fixed the flowcontrol parameter, we observe that this variability implies that inside ageneral wave packet, there will be a subset of waves that disperse aside asubset of waves that do not. In association to a situation where wave packetsundergo powerful and long-lasting transient growth, this information is key tothe understanding of a potential process of nonlinear coupling onset.