Tracking stages of transition in Couette flow analytically

Abstract

The current study focuses on a transition scenario in which the linear transient growth mechanism is initiated by four decaying normal modes. It is shown that the four modes, the initial structure of which corresponds to counter-rotating vortex pairs, are sufficient to capture the transient growth mechanism. More importantly, it is demonstrated that the kinetic energy growth of the initial disturbance is not the key parameter in this transition mechanism. Rather, it is the ability of the transient growth process to generate an inflection point in the wall-normal direction and consequently to make the flow susceptible to a three-dimensional disturbance leading to transition to turbulence. Because of the minimal number of modes participating in the transition process, it is possible to follow its earlier key stages analytically and to compare them with the results of direct numerical simulation. This procedure reveals the role of various flow parameters during the transition, such as the difference between symmetric and antisymmetric transient growth scenarios. Moreover, it is shown that the resulting modified base flow of the linear process is not sufficient to produce a significant localized maximum of the base-flow vorticity (i.e. a 'strong' inflection point), and it is only due to nonlinear effects that the base flow becomes unstable with respect to an infinitesimal three-dimensional disturbance. Finally, the physical mechanism during key stages of transition is well captured by the analytical expressions. Furthermore, the vortex dynamics during these stages is very similar to the model proposed by Cohen, Karp & Mehta (J. Fluid Mech., vol. 747, 2014, pp. 30-43) according to which streamwise variation of the initial counter-rotating vortex pair is required to generate concentrated spanwise vorticity, which together with the lift-up by the induced velocity and shear of the base flow generates packets of hairpins.

Oblique and streamwise vortex paths in a plane Couette flow using a RNL system

“…These scaling factors are consistent with the theoretical analysis by Chapman [5]. More recently, Karp et al [6,7] showed that the optimal set of parameters for transition in the pCf is associated with initial disturbances, which generate the strongest inflection point in the mean flow profile when nonlinearities come into play. In the same vein, Cossu et al [8] investigated the streak breakdown in a two-dimensional parameter space with respect to the amplitudes of the low-speed streak and its secondary perturbation.…”

Optimal wavepackets in streamwise corner flow

“…In the closely related case of the flow in a square duct, Biau, Soueid & Bottaro (2008) found that the key mechanism for rapid transition is the generation of an unstable mean flow distortion that supports further exponential or algebraic growth. Along a similar train of thought, it was recently shown by Karp & Cohen (2014) that non-modal growth can lead to rapid transition in Couette flow through nonlinear interactions. The authors demonstrated that the key factor to cause a rapid breakdown is the ability of the transient growth process to induce inflexion points in the base state.…”

Cause-and-effect of linear mechanisms sustaining wall turbulence

“…In this appendix we describe the linear stability analysis of a base flow, , which is inhomogeneous in two spatial directions (e.g., Karp & Cohen 2014). At given time , we assume the following velocity field: Here the base flow is assumed parallel, steady and streamwise independent, and is the disturbance.…”