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Stability And Transition In Shear Flows

A study of optimal temporal and spatial disturbance growths is carried out for three-dimensional viscous incompressible fluid flows with slippery walls. The non-modal temporal stability analysis is performed under the framework of normal velocity and normal vorticity formulations. A Chebyshev spectral collocation method is used to solve the governing equations numerically. For a free surface flow over a slippery inclined plane, the maximum temporal energy amplification intensifies with the effect of wall slip for the spanwise perturbation, but it attenuates with the wall slip when perturbation considers both streamwise and spanwise wavenumbers. It is found that the boundary for the regime of transient growth appears far ahead of the boundary for the regime of exponential growth, which raises a question on the critical Reynolds number for the shear mode predicted from the eigenvalue analysis. Furthermore, the eigenvalue analysis or the modal stability analysis reveals that the unstable region for the shear mode decays rapidly in the presence of wall slip, which is followed by the successive amplification of the critical Reynolds number for the shear mode and ensures the stabilizing effect of slip length on the shear mode. On the other hand, for a channel flow with slippery bounding walls, the maximum spatial energy amplification intensifies with the effect of wall slip in the absence of angular frequency, but it reduces with the wall slip if the angular frequency is present in the disturbance. Furthermore, the maximum spatial energy disturbance growth can be achieved if the disturbance excludes the angular frequency. Furthermore, it is observed that the angular frequency plays an essential role in the pattern formation of optimal response. In addition, the pseudo-resonance phenomenon occurs due to external temporal and spatially harmonic forcings, where the pseudo-resonance peak is much higher than the resonance peak.

Non-modal stability analysis in viscous fluid flows with slippery walls

: It is shown that the Orr-Sommerfeld equation governing the stability of any mean shear flow in an unbounded domain which has finite energy under some Galilean transformation has a continuous spectrum. This result applies to both the temporal and spatial stability problems. Formulae for the location of this continuum in the complex wave speed plane are given. The continuous spectrum and corresponding eigenfunctions are calculated for two samples problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, are discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

The continuous spectrum of the Orr-Sommerfeld equation. Part 1: The spectrum and the eigenfunctions

Mixing in numerous medical and chemical applications, involving overly long microchannels, can be enhanced by inducing flow instabilities. The channel length is thus shortened in the inertial microfluidics regime due to the enhanced mixing, thereby making the device compact and portable. Motivated by the emerging applications of a lab on a compact disk based microfluidic devices, we analyze the linear stability of rotationally actuated microchannel flows commonly deployed for biochemical and biomedical applications. The solution of the coupled system of Orr-Sommerfeld and Squire equations yields the growth rate and the neutral curves for the Coriolis force-driven instability. We report on the existence of four different types of unstable modes (Type-I to Type-IV) at low rotation numbers. Furthermore, Types-I and II exhibit competing characteristics, signifying that Type-II can play an important role in the transition to turbulence. Type-III and Type-IV modes have relatively lower growth rates, but the associated normal velocity has an oscillatory nature near the center of the channel. Thus, we infer that Types-III and IV might cause strong mixing locally by virtue of strong velocity perturbation in proximity to the various point depths. Moreover, the situation is reliable if the channel is too short to allow for the amplification of Types-I and II. We quantify the potential of all the unstable modes to induce such localized mixing near an imaginary interface (near a hyphothetical interface) inside the flow using the notion of penetration depth. This study also presents an instability regime diagram obtained from the parametric study over a range of Reynolds numbers, rotation numbers, and streamwise and spanwise wavenumbers to assist the design of efficient microchannels. Further insight into the mechanism of energy transfer, drawn from the evaluation of the kinetic-energy budget, reveals how the Reynolds stress first transfers energy from the mean flow to the streamwise velocity fluctuations. The Coriolis force, thereafter, redistributes the axial momentum into spanwise and wall-normal directions, generating the frequently observed roll-cell structures. A qualitative comparison of our predictions with the reported experiments on roll-cells indicates a good agreement.

Rotational instabilities in microchannel flows

We investigated immiscible displacement in a channel considering regular surface roughness at walls. A color-fluid LBM code is developed and validated against the Hagen–Poiseuille flow before it is employed for simulating the displacement process. The dimensionless roughness height and roughness spacing ratio are defined to characterize the surface roughness. The simulation results show that the presence of surface roughness obviously impels a finger formation in a channel. The impelling effect is more significant at larger roughness heights and medium roughness spacing ratios. The critical capillary number and viscosity ratio of a finger formation is reduced with increasing roughness height. The obvious effect of wettability on the finger development in a smooth channel is attenuated in rough channels. The attenuation magnitude increases with increasing roughness height.

LBM Investigation of Immiscible Displacement in a Channel with Regular Surface Roughness

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 initi...

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2015.0267

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

How rotation affects transition to turbulence of pressure-driven flow in a channel is investigated. Even at extremely low rotation rates, optimal perturbations are asymmetric about the centerline. Subcritical transition features are significant even in regimes of exponentially growing instabilities.

/pdf/algebraic-disturbances-and-their-consequences-in-rotating-4nkb7gvoyy.pdf

Algebraic disturbances and their consequences in rotating channel flow transition

Optimal energy growth in a stably stratified shear flow

Viscosity stratification can create new instabilities in a shear flow, or act to stabilise it. This talk will examine mechanisms of such instabilities, and situations where they can occur. It will also contrast these with instabilities in density-stratified flows, and examine the combination of the two. The aim of this talk is to invite future research on viscosity stratification in the context of the Sun or Earth’s interior.

Sharath Jose

Papers

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Algebraic disturbances and their consequences in rotating channel flow transition

Optimal energy growth in a stably stratified shear flow

Instabilities in viscosity and density stratified flow

Effects of equatorially-confined shear flow on MRG and Rossby waves