We investigate analytically the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects. The complete inviscid problem is also solved as an initial value problem with a density perturbation. We show that the kinetic energy of the disturbances grows algebraically at early times, contrary to the wellknown algebraic decay at time tending to infinity. This growth can persist for arbitrarily long times in response to sharp enough initial perturbations. The simplest in our three-stage study is a model problem forced by a buoyancy perturbation in the absence of background stratification. A linear growth with time is obtained in the vertical velocity component. This model provides an analogy between the transient mechanism of kinetic energy growth in a twodimensional density-varying flow and the lift-up mechanism of the three-dimensional constant density flow. Next we consider weak stable background stratification. Interestingly, the lowest order solution here is the same as that of the model flow. Our final study shows that a strong background stratification results in a sub-linear growth with time of the perturbation. A framework is thus presented where two-dimensional streamwise disturbances can lead to large transient amplification, unlike in constant density flow where three dimensions are required.

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Transient growth of perturbations by a linear non-modal evolution is studied here in a stably stratified bounded Couette flow. The density stratification is linear. Classical inviscid stability theory states that a parallel shear flow is stable to exponentially growing disturbances if the Richardson number (Ri) is greater than 1/4 everywhere in the flow. Experiments and numerical simulations at higher Ri show however that algebraically growing disturbances can lead to transient amplification. The complexity of a stably stratified shear flow stems from its ability to combine this transient amplification with propagating internal gravity waves (IGWs). The optimal perturbations associated with maximum energy amplification are numerically obtained at intermediate Reynolds numbers. It is shown that in this wall-bounded flow, the threedimensional optimal perturbations are oblique, unlike in unstratified flow. A partitioning of energy into kinetic and potential helps in understanding the exchange of energies and how it modifies the transient growth. We show that the apportionment between potential and kinetic energy depends, in an interesting manner, on the Richardson number, and on time, as the transient growth proceeds from an optimal perturbation. The oft-quoted stabilizing role of stratification is also probed in the non-diffusive limit in the context of disturbance energy amplification.

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 ...

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