A Sameen

In this paper, we investigate mixing in flows dominated by bubble-type vortex breakdown. Three-dimensional Navier-Stokes equations in cylindrical polar coordinates are solved to simulate flow in a cylindrical container. We have found that in steady regime of the flow, the vortex breakdown bubble is axisymmetric and apparent non-axisymmetric features observed in experiments are artifacts of imperfections in experimental set-ups. We also find that the heteroclinic manifold joining hyperbolic points of the vortex breakdown bubble is stable in the absence of any perturbation and no chaotic advection was found within vortex breakdown bubble. This makes the vortex breakdown bubble impermeable to outer fluid and hence, the vortex breakdown bubble inhibits mixing. We conclude that symmetry is a barrier to mixing.

Non-axisymmetric vortices are ubiquitous in nature; examples include polar vortices in planets, the giant red spot in Jupiter, tornadoes and cyclones on Earth, mesoscale eddies in the ocean. Turbulent flows are furthermore known to be dominated by small- and large-scale vortex structures. Owing to the wide range of applications, knowledge of conditions under which a given vortex becomes unstable is beneficial. Here, the centrifugal instability of two-dimensional, non-axisymmetric vortices in the presence of an axial flow (w) and a background rotation (Ωz) is studied using the local stability approach. The local stability approach, based on geometric optics and similar in formulation to the rapid distortion theory [2], considers the evolution of shortwavelength perturbations along streamlines in the base flow. This approach, developed by Lifschitz & Hameiri [4], is particularly useful for base flows for which a global stability analysis is computationally expensive. A sufficient criterion for centrifugal instability in an axisymmetric vortex with (w) and (Ωz) is first derived by analytically solving the local stability equations for wave vectors that are periodic upon evolution around a closed streamline. This criterion is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities on a streamline. The criterion is then shown to be accurate in describing centrifugal instability over a reasonably large range of parameters that specify Stuart vortices and Taylor-Green vortices.

Entropically Damped Artificial Compressibility (EDAC) equation-based solution method is investigated to simulate the incompressible Navier–Stokes equations for buoyancy-driven flows. The method introduces an evolution equation for pressure, which is used to close the system of equations. The resulting parabolic system removes the need to solve the traditional Poisson's equation at each time step. The energy equation with the Boussinesq approximation and the EDAC system of equations with the low Mach number approximation is solved for the thermal convection problem. This system is discretized using a sixth-order compact difference in space and advanced in time using an explicit fourth-order Runge–Kutta scheme. To investigate the suitability of the EDAC model for buoyancy flows, two widely used benchmark problems, namely: thermal cavity and Rayleigh–Bénard problems, are simulated. The simulation results are compared against the literature data. An excellent agreement is obtained, showing the feasibility and accuracy of the EDAC method in simulating buoyancy-driven flows. The EDAC pressure equation derived from entropy balance, together with the energy equation, are shown in this paper to model thermally dominant flows accurately.

Flow past a heated/cooled circular cylinder is computationally investigated in this paper. The presence of a buoyancy force arising due to a change in density alter the vortex shedding dynamics. The governing equations, Navier-Stokes and energy equation within the Boussinesq approximation along with continuity equation are solved using a hybrid FEM-FVM technique. In this work we focus on the minimum Reynolds number at which vortex shedding occur. In hydrodynamic stability literature, this value is most often termed as the critical Reynolds number, which is approximately 47 for flow past an unheated cylinder. This demarcation between steady and unsteady regimes of the flow changes due to surface temperature. Due to an increase in surface temperature the frequency of vortex shedding is known to enhance. We show that the nondimensional shedding frequency, the Strouhal number increases with heating and has a strong dependence on Prandtl number and Richardson number of the flow, which are a measure of fluid diffusivities and buoyancy force respectively. We also discuss the regime of complete suppression of vortex shedding due to cylinder surface temperature.

We perform a direct numerical simulation of three-dimensional Navier-Stokes equations for a Rayleigh-Bénard convection system in a stationary cylinder with the top cold lid rotating. This convection system with an imposed swirl flow is a canonical problem for investigating axial vortices under unstable thermal gradients. The base flow is established by rotating the top lid and the fluid moves azimuthally along the side vertical wall into a meridional flow in the r-z plane. This forms an axial vortex core at the axis of the cylinder. This axial core, under a certain rotational Reynolds number (Re), breaks down to a vortex breakdown bubble whose dynamics are modified under thermal convection. We study the effect of rotation on varying Re for a Rayleigh number Ra = 2 × 105. The equations are formulated in such a way that the rotating-lid cylinder and Rayleigh-Bénard convection are extreme cases of the same numerical set up. From the present study, we find that as the rotational rate is increased, the system dynamics shift from a convection-dominated flow regime to a rotation-dominated regime. This shift in dynamics is quantified using the volume-averaged and time-averaged temperature, the heat flux, the thickness of the Bödewadt boundary layer and the relative Nusselt number. These quantities are shown to demarcate the convection- A nd rotation-dominated regimes, as compared to the qualitative description of flow patterns from velocity and temperature contours.

The scale-space energy density function is defined as the derivative of the two-point velocity correlation as, where is the spatial coordinate of interest and is the separation vector. The function describes the turbulent kinetic energy density of scale at a location and can be considered as the generalization of the spectral energy density function concept to inhomogeneous flows. In this work, we derive the scale-space energy density function transport equation for compressible flows to develop a better understanding of scale-to-scale energy transfer and the degree of non-locality of the energy interactions. Specifically, the effects of variable-density and dilatation on an energy cascade are identified. It is expected that these findings will yield deeper insight into compressibility effects on canonical energy cascades, which will lead to improved models (at all levels of closure) for mass flux, density variance, pressure-dilatation, pressure-strain correlation and dilatational dissipation processes. Direct numerical simulation (DNS) data of mixing layers at different Mach numbers are used to characterize the scale-space behaviour of different turbulence processes. The scaling of the energy density function that leads to self-similar evolution at the two Mach numbers is identified. The scale-space (non-local) behaviour of the production and pressure dilatation at the centre-plane is investigated. It is established that production is influenced by long-distance (order of vorticity thickness) interactions, whereas the pressure dilatation effects are more localized (fraction of momentum thickness) in scale space. The analysis of DNS data demonstrates the utility of the energy density function and its transport equation to account for the relevance of various physical mechanisms at different scales.

We study the centrifugal instability of non-axisymmetric vortices in the presence of an axial flow (w) and a background rotation (Ωz) using the local stability approach. Analytically solving the local stability equations for an axisymmetric vortex with w and Ωz, growth rates for wave vectors that are periodic upon evolution around a closed streamline are calculated. The resulting sufficient criterion for centrifugal instability in an axisymmetric vortex is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities and their derivatives with respect to the streamfunction on a streamline. The new criterion for non-axisymmetric vortices, which converges to the exact criterion of Bayly (Phys. Fluids, vol. 31, 1988, pp. 56-64) in the absence of background rotation and axial flow, is validated by comparisons with numerically calculated growth rates for two different anticyclonic vortices: the Stuart vortex (specified by the concentration parameter ρ, 0 < ρ ≤ 1) and the Taylor-Green vortex (specified by the aspect ratio E, 0 < E ≤ 1). With no axial velocity and finite background rotation, the criterion predicts a lower and an upper threshold of |Ωz| between which centrifugal instability is present. We further demonstrate that the criterion represents an improvement over the criterion of Sipp & Jacquin (Phys. Fluids, vol. 12, 2000, pp. 1740-1748). Finally, in the presence of both axial velocity and background rotation, the criterion is shown to be accurate for large enough ρ and E.

Taylor-Couette flow is a canonical flow to study Taylor-Görtler (TG) instability or centrifugal instability and the associated vortices. TG instability has been traditionally associated with flow over curved surfaces or geometries. In the computational study, we confirm the presence of TG-like near-wall vortical structures in two lid-driven flow systems, the Vogel-Escudier (VE) and the lid-driven cavity (LDC) flows. The VE flow is generated inside a circular cylinder by a rotating lid (top lid in the present study), while the LDC flow is generated inside a square or rectangular cavity by the linear movement of the lid. We look at the emergence of these vortical structures through reconstructed phase space diagrams and find that the TG-like vortices are seen in the chaotic regimes in both flows. In the VE flow, these vortices are seen when the side-wall boundary layer instability sets in at large Re. The VE flow is observed to go to a chaotic state in a sequence of events from a steady state at low Re. In contrast to VE flows, in the LDC flow with no curved boundaries, TG-like vortices are seen at the emergence of unsteadiness when the flow exhibits a limit cycle. The LDC flow is observed to have transitioned to chaos from the steady state through a periodic oscillatory state. Various aspect ratio cavities are examined in both flows for the presence of TG-like vortices. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)'.

We demonstrate the specific non-Boussinesq roles played by various fluid properties in thermal convection by allowing each of them to possess, one at a time, a temperature dependence that could be either positive or negative. The negative temperature dependence of the coefficient of thermal expansion hinders effective thermal convection and reduces the Nusselt number, whereas the negative dependence of fluid density enhances the Nusselt number. Viscosity merely smears plume generation and has a marginal effect on heat transport, whether it increases or decreases with temperature. At the moderate Rayleigh number examined here, the specific heat capacity shows no appreciable effect. On the other hand, the conductivity of the fluid near the hot surface controls the heat transport from the hot plate to the fluid, which suggests that a less conducting fluid near the bottom surface will reduce the Nusselt number and the bulk temperature. © EPLA, 2009.

A global stability study of a divergent channel flow reveals features not obtained hitherto by making either the parallel or the weakly non-parallel (WNP) flow assumption. A divergent channel flow is chosen for this study since it is the simplest spatially developing flow: the Reynolds number is constant downstream, and for a theoretical Jeffery-Hamel flow, the velocity profile obeys similarity. Even in this simple flow, the global modes are shown to be qualitatively different from the parallel or WNP. In particular, the disturbance modes are often not wave-like, and the local scale, estimated from a wavelet analysis, can be a function of both streamwise and normal coordinates. The streamwise variation of the scales is often very different from the expected linear variation. Given recent global stability studies on boundary layers, such spatially extended modes which are not wave-like are unexpected. A scaling argument for why the critical Reynolds number is so sensitive to divergence is offered. © 2010 Springer-Verlag.