Algebraic disturbances and their consequences in rotating channel flow transition

Abstract

It is now established that subcritical mechanisms play a crucial role in the transition to turbulence of non-rotating plane shear flows. The role of these mechanisms in rotating channel flow is examined here in the linear and nonlinear stages. Distinct patterns of behaviour are found: the transient growth leading to nonlinearity at low rotation rates Ro, a highly chaotic intermediate Ro regime, a localised weak chaos at higher Ro, and complete stabilization of transient disturbances at very high Ro. At very low Ro, the transient growth amplitudes are close to those for non-rotating flow, butCoriolis forces already assert themselves by producingdistinct asymmetry about the channelcentreline. Nonlinear processes are then triggered, in a streak-breakdown mode of transition. The high Ro regimes do not show these signatures, here the leading eigenmode emerges as dominant in the early stages. Elongated structures plastered close to one wall are seen at higher rotation rates. Rotation is shown to reduce non-normality in the linear operator, in an indirect manifestation of Taylor-Proudman effects. Although the critical Reynolds for exponential growth of instabilities is known to vary a lot with rotation rate, we show that the energy critical Reynolds number is insensitive to rotation rate. It is hoped that these findings will motivate experimental verification, and examination of other rotating flows in this light. * josesk@tifrh.res.in arXiv:1609.05459v1 [physics.flu-dyn] 18 Sep 2016 the inviscid criterion given above. Henceforth this system will be referred to simply as rotating channel flow. On neglecting the effect of end walls, one sees that the base flow is described by the familiar parabolic velocity profile [10]. Close to these walls, a secondary flow in the form of a double vortex is set up [13].This flow is characterised by two parameters, the Reynolds number Re = U 0 d/ν, and the rotation number Ro = Ωd/U 0 , where U 0 is the centreline velocity in the channel, d its half-width, Ω the rotation rate, and ν the kinematic viscosity of the fluid. It was found experimentally that the critical Reynolds number cr , below which no exponential instabilities exist, may be up to two orders magnitude lower than that of a non-rotating channel [8,11]. This critical Reynolds number shows a non-monotonic variation with the strength of rotation, and is very sensitive to it. Just past Re cr , the first unstable mode corresponds to a stationary streamwise-invariant disturbance. As we move further into the unstable part of the parameter space, we may find oblique modes that have growth rates comparable to the streamwise-invariant mode [14]. At high Ro, Taylor-Proudman behaviour sets in and these streamwise-invariant rotation modes are suppressed. The two-dimensional spanwise-invariant Tollmien-Schlichting (TS) mode can still be triggered for values of Re above its critical value 5772 [15,16]. But in the regime where both the TS mode and the rotation mode are present, the rotation mode is expected to win over due to a much larger ...

Non-normal origin of modal instabilities in rotating plane shear flows

“…This growth can trigger nonlinearities, and lead to transition to turbulence, as has been shown for PPF and PCF [29,30]. In RPPF, Jose et al [31] showed that even when the flow is linearly unstable, there exist regimes where perturbation energy can be enhanced significantly by transient algebraic growth. Besides this well-known property of non-normality, we show here that non-normality in a particular flow system can be the reason for a large decrease in the critical Reynolds number of a closely related flow.…”

Non-modal stability analysis of stratified two-phase channel flows

“…The transient energy growth was argued to be responsible for transition from laminar flow at subcritical values of the Reynolds number (Brandt 2014). In the unstable regime, it was shown that the non-modal effects can enhance perturbation energy significantly before modal behavior dominates, and therefore can be considered important for studying transition scenarios (e.g., Lucas et al, 2015, Jose et al, 2017.…”