Marius Ungarish

An introduction to gravity currents and intrusions

Introduction Classification The Navier-Stokes equations Non-stratified ambient currents Shallow-water (SW) formulation for high-Re flows Motion of the interface and the continuity equation One-layer model A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation The steady-state current and nose jump conditions Benjamins analysis Jump condition Box models for 2D geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Two-layer SW model Introduction The governing equations Boussinesq system in dimensionless form Jumps of interface for H < 2 Energy and work in a two-layer model Axisymmetric currents, SW formulation Governing equations A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation Some comparisons Box models for axisymmetric geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Effects of rotation Axisymmetric case Rotating channel Buoyancy decays: particle-driven, porous boundary, and entrainment Particle-driven currents Axisymmetric particle-driven current Extensions of particle-driven solutions Current over a porous bottom Axisymmetric current over a porous bottom Entrainment Non-Boussinesq systems Introduction Formulation Dam-break and initial slumping motion The transition and self-similar stages Summary Lubrication theory formulation for viscous currents 2D geometry Axisymmetric current Current in a porous medium II Stratified ambient currents and intrusions Continuous density transition Introduction The SW formulation SW results and comparisons with experiments and simulations Dam break Critical speed and nose-wave interaction Similarity solution The validity of the inviscid approximation Axisymmetric and rotating cases SW formulation SW and NS finite-difference results The validity of the inviscid approximation The steady-state current Steady-state flow pattern Results Comparisons and conclusions Intrusions in 2D geometry Introduction Two-layer stratification Linear transition layer Rectangular lock configurations Cylindrical lock in a fully linearly-stratified tank Similarity solution Non-symmetric intrusions Intrusions in axisymmetric geometry Introduction Two-layer stratification Fully linearly-stratified tank, part-depth locks Box models for 2D geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Box models for axisymmetric geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Lubrication theory for viscous currents with S = 1 2D geometry Axisymmetric geometry Energy Introduction 2D geometry Axisymmetric geometry SW equations: characteristics and finite-difference schemes Characteristics Numerical solution of the SW equations Navier-Stokes numerical simulations Formulation A finite-difference code Other codes Some useful formulas Leibnizs Theorem Vectors and coordinate systems

On hindered settling of particles of different sizes

Abstract The flow of a non-dilute fluid suspension is considered in which the dispersed phase consists of particles or droplets of different sizes. A phenomenological two-phase flow theory is formulated for both continuous and discrete distributions of particle sizes and illustrated by considering the batch settling of such a mixture. The volume fractions and particle distribution functions are determined, as well as the composition of the sedimentary layer.

Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder: A numerical and an analytical study

This paper presents a combined numerical and analytical study of the impulsive axisymmetric spin-up from rest of an isothermal liquid metal in a closed cylinder. The motion of the liquid is caused by the action of a low-frequency, low-induction rotating magnetic field, whose magnetic Taylor number is in the range (0.01–0.9) Tacr3D with Tacr3D given by Grants and Gerbeth [“Linear three-dimensional instability of a magnetically driven rotating flow,” J. Fluid Mech. 463, 229 (2002)]. The computations were performed for cylindrical containers of aspect ratios (diameter/height) R equal to 0.5, 1, and 2. The numerical simulations are compared with the predictions of an analytical model, valid for small Ekman numbers E extending a former work by Ungarish [“The spin-up of liquid metal driven by a rotating magnetic field,” J. Fluid Mech. 347, 105 (1997)]. The first phase of the motion from rest is an initial adjustment: the inviscid fluid begins to rotate due to the externally forced azimuthal acceleration, and co...

The effects of rotation on axisymmetric gravity currents

Axisymmetric gravity currents in a system rotating around a vertical axis, that result when a dense fluid intrudes horizontally under a less dense ambient fluid, are studied. Situations for which the density difference between the fluid is due either to compositional differences or to suspended particulate matter are considered. The fluid motion is described theoretically by the inviscid shallow-water equations. A ‘diffusion’ equation for the volume fraction in the suspension is derived for the particle-driven case, and two different models for this purpose are presented. We focus attention on situations in which the apparent importance of the Coriolis terms relative to the inertial terms, represented by the parameter [Cscr ] (the inverse of a Rossby number), is not large. Numerical and asymptotic solutions of the governing equations clarify the essential features of the flow field and particle distribution, and point out the striking differences from the non-rotating case (Bonnecaze, Huppert & Lister 1995). It is shown that the Coriolis effects eventually become dominant; even for small [Cscr ], Coriolis effects are negligible only during an initial period of about one tenth of a revolution. Thereafter the interface of the current acquires a shape which has a downward decreasing profile at the nose and its velocity of propagation begins to decrease to zero more rapidly than in the non-rotating situation. This relates the currents investigated here to the previously studied quasi-steady oceanographic structures called rings, eddies, vortices or lenses, and may throw additional light on the dynamics of their formation. The theoretical results were tested by some preliminary experiments performed in a rotating cylinder of diameter 90 cm filled with a layer of water of depth 10 cm in which a cylinder of heavier saline fluid of diameter 9.4 cm was released.

Intrusive gravity currents in a stratified ambient: shallow-water theory and numerical results

The intrusion of a fixed volume of fluid which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically stratified ambient fluid is investigated. The density change is linear, in a restricted layer or over the full depth of the container, and locks of both rectangular and cylindrical shapes are considered. A closed one-layer shallow-water inviscid formulation is used to obtain solutions of the initial-value problem. Similarity solutions for the large-time developed motion and an approximate box model are also presented. The results are corroborated by numerical solutions of the full two-dimensional Navier-Stokes equations and comparisons with previously published experiments

A shallow-water model for high-Reynolds-number gravity currents for a wide range of density differences and fractional depths

We consider the propagation of a gravity current of density ρ c from a lock of length x 0 and height h 0 into an ambient fluid of density ρ a in a channel of height H . When the Reynolds number is large, the resulting flow is governed by the parameters ρ c /ρ a and H * = H / h 0 . We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρ c /ρ a ≪ 1, Boussinesq ρ c /ρ a ≈ 1, and heavy ρ c /ρ a ≫ 1 currents in a fairly wide range of depth ratio. We obtain analytical solutions for the initial dam-break or slumping stage of propagation with constant speed, and derive explicitly the trends for small and large values of the governing parameters. This reveals the main features: ( a ) the heavy current propagates faster and its front is thinner than for the light counterpart; ( b ) the speed of the heavy current depends little on H *, while that of the light current increases with H *; and ( c ) in the shallow ambient case ( H * close to 1) the light current is choked to move with the thickness of half-channel, while the heavy current typically moves with an unrestricted smaller thickness. These qualitative predictions are in accord with previous observations, and some quantitative comparisons with available experimental and numerical simulations data also show fair agreement. However, given the paucity of the available data, the main deficiency of the model is the unknown practical limit of applicability. For large time, t , a self-similar propagation with t 2/3 is feasible for both the heavy and light currents, but the thickness profiles display differences.

On gravity currents propagating at the base of a stratified ambient: effects of geometrical constraints and rotation

The behaviour of an inviscid gravity current released from a lock and propagating over a horizontal boundary at the base of a stratified ambient fluid is considered in the framework of a one-layer shallow-water formulation. Solutions for two-dimensional rectangular and axisymmetric geometries, with emphasis on the rotation of the latter, were obtained by a Lax-Wendroff scheme. Box-model approximations are also discussed. The axisymmetric and rotating case admits steady-state lens structures, for which approximate and numerical solutions are presented. In general, the stratification reduces the velocity of propagation and enhances the Coriolis effects in a rotating system

On the centrifugal separation of a bulk mixture

Abstract The centrifugal separation of a mixture of particles and fluid in an axisymmetric container is examined. The flow consists of three distinct regions—mixture, sediment and purified fluid—with Ekman boundary layers at the interfaces and walls. In the settling process, the mixture and pure fluid acquire retrograde and prograde rotations relative to the tank. This flow pattern, and the shape and locus of the interface which are easily determined, provide another simple means to compare mixture theory and experiment. It is shown that when the Coriolis force is important, the pure fluid layer on the “outwardly” inclined wall is not thin. Moreover the interface between the mixture and the pure fluid is not perpendicular to the centrifugal force. Both features contrast those of the gravitational Boycott effect. As a consequence, there is no obvious enhancement of settling due to geometrical configuration.

On the enhancement of centrifugal separation

We consider the two-phase flow of a suspension in a rotating cylinder with inclined endplates for which inertial and viscous effects are small. It is shown that, when the Coriolis force is dominant, flow in the core is essentially unaffected by geometry. If a fluid particle can make a complete circuit about the rotation axis, the sedimentation velocity cannot be augmented by geometrical effects as it can in gravitational settling. However, with the insertion of a complete meridional barrier to block movement around the centre, separation becomes more sensitive to the shape of the container walls. In this case, behaviour similar to that in a gravitational field is possible once again.

The spin-up of liquid metal driven by a rotating magnetic field

The paper considers the flow field during spin-up from rest of liquid metal in a cylindrical stationary cavity due to a rotating transverse uniform low-frequency magnetic field. It is assumed that the Ekman and the magnetic Reynolds numbers are small. An approximate model, based on matching of Bodewadt-type layers with an inviscid core, with possible influence from the sidewall, for laminar flow is developed. It is shown that the angular velocity in the core is a function of time only. Analytical solutions for the angular velocity and the meridional flow in the core are presented, and supplemented by finite-difference results to show the sidewall effects. The spin-down following the switch-off of the magnetic forcing, the influence of the axial variations of the magnetic field, and the relevance to turbulent flows are discussed.