ALBERT

Description: The linear stability of variable viscosity, miscible core-annular flows is investigated. Consistent with pipe flow of a single fluid, the flow is stable at any Reynolds number when the magnitude of the viscosity ratio is less than a critical value. This is in contrast to the immiscible case without interfacial tension, which is unstable at any viscosity ratio. Beyond the critical value of the viscosity ratio, the flow can be unstable even when the more viscous fluid is in the core. This is in contrast to plane channel flows with finite interface thickness, which are always stabilized relative to single fluid flow when the less viscous fluid is in contact with the wall. If the more viscous fluid occupies the core, the axisymmetric mode usually dominates over the corkscrew mode. It is demonstrated that, for a less viscous core, the corkscrew mode is inviscidly unstable, whereas the axisymmetric mode is unstable for small Reynolds numbers at high Schmidt numbers. For the parameters under consideration, the switchover occurs at an intermediate Schmidt number of about 500. The occurrence of inviscid instability for the corkscrew mode is shown to be consistent with the Rayleigh criterion for pipe flows. In some parameter ranges, the miscible flow is seen to be more unstable than its immiscible counterpart, and the physical reasons for this behaviour are discussed. A detailed parametric study shows that increasing the interface thickness has a uniformly stabilizing effect. The flow is least stable when the interface between the two fluids is located at approximately 0.6 times the tube radius. Unlike for channel flow, there is no sudden change in the stability with radial location of the interface. The instability originates mainly in the less viscous fluid, close to the interface. © 2007 Cambridge University Press.

Description: We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the ‘stable stratification’ of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh–Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood numberAtshow growth rates scaling asAt1/3close to the critical radius, andAt1/2further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vortex, consistent with recent work which found Gaussian profiles to be stable. For sharp gradients, it is argued that wave interaction can be supported due to the presence ofquasi-modes. Probably for the first time, a quasi-mode which decays exponentially is shown to interact with a neutral wave to give exponential growth in the combined case. We finally study the nonlinear stages using viscous direct numerical simulations. The initial exponential instability of light-cored vortices is arrested due to a restoring centrifugal buoyancy force, leading to stable non-axisymmetric structures, such as a tripolar state for an azimuthal wavenumber of 2. The study is restricted to two dimensions, and neglects gravity.

Description: During an attempt to work on a stratified flow problem envisaged as a sequel of the paper by Sameen & Govindarajan (2007), it was found that the original paper contained errors in §§ 3.4 and 4.3 due to a factor of iα, which was inadvertently missed in two places in the code (i) in the buoyancy term due to the use of vertical velocity and streamfunction interchangeably, and (ii) in the apportionment between kinetic and potential energy in the Gmax calculation. Because of this, there were significant differences in the effect of Grashof number on stability. Figure 1 is the modified figure 9 of the original paper, for Pr =7 and ΔT = 25 K. The Poiseuille–Rayleigh–Bénard mode appears at Gr = 39.12 and is seen not to merge with the Poiseuille mode, unlike the conclusion made earlier. This modification applies at any Prandtl number from 10−2 to 102. The corrected versions of figures 17 and 21, showing Gmax contours for different Pr at Gr = 0 and different Gr for Pr = 1, are plotted in figures 2 and 3, respectively. The large growth reported at β = 0 was thus erroneous. The other main conclusions of the paper, that Prandtl number changes transient growth qualitatively, but not the least stable eigenmode, whereas viscosity stratification, which has a huge impact on exponential growth/decay, does not change transient growth much, remain the same. The secondary instabilities also remain unchanged. The stability equations (3.2) to (3.4) in the paper should read (for explanation, please refer to Sameen & Govindarajan 2007) (1)(2)(3)

Description: The Maxey-Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey-Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the unified transform method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as a stretched exponential at short time while slowing it down to a power-law relaxation, , at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity , whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of raindrops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a raindrop can be grossly overestimated when history effects are not accounted for. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover, our results indicate that the Basset history can affect dynamics significantly. © 2019 Cambridge University PressÂ.

Description: Double-diffusive density stratified systems are well studied and have been shown to display a rich variety of instability behaviour. However double-diffusive systems where the inhomogeneities in solute concentration are manifested in terms of stratified viscosity rather than density have been studied far less and, to the best of the authors' knowledge, not in high-Reynolds-number shear flows. In a simple geometry, namely the two-fluid channel flow of such a system, we find a new double-diffusive mode of instability. The instability becomes stronger as the ratio of diffusivities of the two scalars increases, even in a situation where the net Schmidt number decreases. The double-diffusive mode is destabilized when the layer of viscosity stratification overlaps with the critical layer of the perturbation. © 2011 Cambridge University Press.

Description: The stability of a mixing layer made up of two miscible fluids, with a viscosity-stratified layer between them, is studied. The two fluids are of the same density. It is shown that unlike other viscosity-stratified shear flows, where species diffusivity is a dominant factor determining stability, species diffusivity variations over orders of magnitude do not change the answer to any noticeable degree in this case. Viscosity stratification, however, does matter, and can stabilize or destabilize the flow, depending on whether the layer of varying velocity is located within the less or more viscous fluid. By making an inviscid model flow with a slope change across the ‘viscosity’ interface, we show that viscous and inviscid results are in qualitative agreement. The absolute instability of the flow can also be significantly altered by viscosity stratification.

Description: We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100-1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien-Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable. © 2016 Cambridge University Press.

Description: Droplet-laden flows with phase change are common. This study brings to light a mechanism by which droplet inertial dynamics and local phase change, taking place at sub-Kolmogorov scales, affect vortex dynamics in the inertial range of turbulence. To do this we consider vortices placed in a supersaturated ambient initially at constant temperature, homogeneous vapour concentration and uniformly distributed droplets. The droplets also act as sites of phase change. This allows the time scales associated with particle inertia and phase change, which could be significantly different from each other and from the time scale of the flow, to become coupled, and for their combined dynamics to govern the flow. The thermodynamics of condensation and evaporation have a characteristic time scale . The water droplets are treated as Stokesian inertial particles with a characteristic time scale , whose behaviour we approximate using an truncation of the Maxey-Riley equation for heavy particles. This inertia leads the water droplets to vacate the vicinity of vortices, leaving no nuclei for the vapour to condense. The condensation process is thus spatially inhomogeneous, and leaves vortices in the flow colder than their surroundings. The combination of buoyancy and vorticity generates a lift force on the vortices perpendicular to their velocity relative to the fluid around them. In the case of a vortex dipole, this lift force can propel the vortices towards each other and undergo collapse, a phenomenon studied by Ravichandran et al. (Phys. Rev. Fluids, vol. 2, 2017, 034702). We find, spanning the space of the two time scales, and , the region in which lift-induced dipole collapse can occur, and show numerically that the product of the time scales is the determining parameter. Our findings agree with our results from scaling arguments. We also study the influence of varying the initial supersaturation, and find that the strength of the lift-induced mechanism has a power-law dependence on the phase-change time scale . We then study systems of many vortices and show that the same coupling between the two time scales alters the dynamics of such systems, by energising the smaller scales. We show that this effect is significantly more pronounced at higher Reynolds numbers. Finally, we discuss how this effect could be relevant in conditions typical of clouds. © 2017 Cambridge University Press.

Description: A vortex placed at a density interface winds it into an ever-tighter spiral. We show that this results in a combination of a centrifugal Rayleigh-Taylor (CRT) instability and a spiral Kelvin-Helmholtz (SKH) type of instability. The SKH instability arises because the density interface is not exactly circular, and dominates at large times. Our analytical study of an inviscid idealized problem illustrates the origin and nature of the instabilities. In particular, the SKH is shown to grow slightly faster than exponentially. The predicted form lends itself for checking by a large computation. From a viscous stability analysis using a finite-cored vortex, it is found that the dominant azimuthal wavenumber is smaller for lower Reynolds number. At higher Reynolds numbers, disturbances subject to the combined CRT and SKH instabilities grow rapidly, on the inertial time scale, while the flow stabilizes at low Reynolds numbers. Our direct numerical simulations are in good agreement with these studies in the initial stages, after which nonlinearities take over. At Atwood numbers of 0.1 or more, and a Reynolds number of 6000 or greater, both stability analysis and simulations show a rapid destabilization. The result is an erosion of the core, and breakdown into a turbulence-like state. In studies at low Atwood numbers, the effect of density on the inertial terms is often ignored, and the density field behaves like a passive scalar in the absence of gravity. The present study shows that such treatment is unjustified in the vicinity of a vortex, even for small changes in density when the density stratification is across a thin layer. The study would have relevance to any high-Péclet-number flow where a vortex is in the vicinity of a density-stratified interface. Copyright © Cambridge University Press 2010.