S.J.D. D'Alessio

A vorticity model for viscous flow past a cylinder

Abstract A mathematical model is proposed for the steady two-dimensional flow of a viscous incompressible fluid past a cylinder which incorporates the details of the structure of the vorticity in this case where its behaviour is known. The model is constructed to be consistent both with boundary-layer theory for sufficiently large Reynolds numbers and with the asymptotic solution at large distances from the cylinder. The governing Navier-Stokes equations are transformed to a set of equations which we refer to as the modified Navier-Stokes equations and are then solved numerically. Solutions have been obtained for the cases of flow past a stationary and a rotating circular cylinder and flow past an inclined elliptic cylinder. Good agreement is found with existing results.

Film flow over heated wavy inclined surfaces

The two-dimensional problem of gravity-driven laminar flow of a thin layer of fluid down a heated wavy inclined surface is discussed. The coupled effect of bottom topography, variable surface tension and heating has been investigated both analytically and numerically. A stability analysis is conducted while nonlinear simulations are used to validate the stability predictions and also to study thermocapillary effects. The governing equations are based on the Navier-Stokes equations for a thin fluid layer with the cross-stream dependence eliminated by means of a weighted residual technique. Comparisons with experimental data and direct numerical simulations have been carried out and the agreement is good. New interesting results regarding the combined role of surface tension and sinusoidal topography on the stability of the flow are presented. The influence of heating and the Marangoni effect are also deduced.

The nuclear and aerial dynamics of the Tunguska Event

Abstract A mathematical-physical characterization of an atmospheric “explosive” event—commonly called the Tunguska Event of 1908—has been formulated. Emphasis is placed upon the aerial dynamics and the nuclear energy released in the gas cap of the meteor as it passed through the atmosphere. The results obtained are consistent with the dominant phenomena observed for the Tunguska Event suggesting therefore a plausible reconstruction of the physical processes associated with this unusual event.

Intrusive gravity currents

Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormacks method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.

A Parameterization of the Roughness Length for the Air-Sea Interface in Free Convection

The response of the upper ocean to the parameterization of the roughness length z0 on the air side of the air-sea interface is studied using a one-dimensional mixed-layer model. In particular, it is shown that in the free convection limit when both the wind speed and the friction velocity approach zero, the familiar Charnock formula for the momentum roughness, which relies solely on wind generation, can be modified to account for contributions arising from the thermally generated turbulence. Therefore, a new parameterization is proposed for the momentum roughness length which extends the Charnock formula down to zero friction velocity. The value of a parameter which enters in the new formulation is determined by making use of exsisting free convection surface flux parameterizations. The effect of the new parameterization on the model performance is tested using data from the ocean weathership station Papa (OWS P), and data from the Long-Term Upper-Ocean Study (LOTUS) experiment. Simulations were carried out using a recently developed one-dimensional, second-order, turbulence closure scheme over diurnal as well as seasonal time scales. The findings suggest that the new momentum roughness parameterization improves the overall agreement between the observed and simulated sea-surface temperature (SST).

Stability of differentially heated flow from a rotating sphere

We present results on the flow of a thin fluid layer over a rotating sphere having a surface temperature that varies in the latitudinal direction. The fluid is taken to be viscous, incompressible and Newtonian while the flow is assumed to possess both azimuthal and equatorial symmetry. The governing Navier-Stokes and energy equations are formulated in terms of a stream function and vorticity and are solved subject to no-slip boundary conditions. An approximate analytical solution for the steady-state flow has been derived and is compared with numerical solutions to the steady and limiting unsteady equations. For small Rayleigh numbers these solutions are found to be in close agreement. However, as the Rayleigh number is increased noticeable differences occur. A numerical solution procedure is presented and a linear stability analysis has been conducted to predict the onset of instability. Good agreement between the theoretical predictions and the observed numerical simulations was found. Differentially heated flow of a thin fluid layer from a rotating sphere has been investigated.A numerical solution procedure for solving the steady and unsteady equations has been proposed.An approximate analytical solution has been derived.A linear stability analysis has estimated a theoretical value for the onset of instability.Good agreement was found between numerical, analytical and theoretical results.

Solutions for Initial‐Boundary‐Value Problems Representing Gravity Currents Arising from Variable Inflow

In this article, we report on theoretical and numerical studies of models for suddenly initiated variable inflow gravity currents in rectangular geometry. These gravity currents enter a lighter, deep ambient fluid at rest at a time-dependent rate from behind a partially opened lock gate and their subsequent dynamics is modeled in the buoyancy-inertia regime using ½-layer shallow water theory. The resistance to flow that is exerted by the ambient fluid on the gravity current is accounted for by a front condition which involves a non-dimensional parameter that can be chosen in accordance with experimental observations. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through an opening of fixed area which is suddenly opened under a lock gate at one end of a large rectangular tank. The fluid in the lock is subjected to a (possibly) time varying pressure applied uniformly over its surface and the finite movement of the free surface is accounted for. Finding this time-dependent inflow velocity, which will then serve as a boundary condition for the solution of the shallow-water equations, involves solving forced non-linear ordinary differential equations and the form of this velocity equation and its attendant solutions will, in general, rule out finding self-similar solutions for the shallow-water equations. The existence of self-similar solutions requires that the gravity currents have volumes proportional to, where and t is the time elapsed from initiation of the flow. This condition requires a point source of fluid with very special properties for which both the area of the gap and the inflow velocity must vary in a related and prescribed time-dependent manner in order to preserve self-similarity. These specialized self-similar solutions are employed here as a check on our numerical approach. In the more natural cases that are treated here in which fluids flow through an opening of fixed dimensions in a container an extra dimensional parameter is introduced thereby ruling out self-similarity of the solutions for the shallow-water equations so that the previous analytical approaches to the variable inflow problem, involving the use of phase-plane analysis, will be inapplicable. The models developed and analyzed here are expected to provide a first step in the study of situations in which a storage container is suddenly ruptured allowing a heavy fluid to debouch at a variable rate through a fixed opening over level terrain. They also can be adapted to the study of other situations where variable inflow gravity currents arise such as, for example, flows of fresh water from spring run-off into lakes and fjords, flows from volcanoes and magma chambers, discharges from locks and flash floods.t±±e 0

The effects of density extremum and rotation on the onset of thermal instability

This paper addresses the onset of Be´nard convection on a rotating horizontally confined layer of water near the temperature of maximum density that is heated from below. A quadratic relation between temperature and density is assumed near the density extremum. A linear stability analysis is employed to determine the critical conditions for the onset of thermal instability. The resulting eigenvalue problem is numerically solved by expanding the amplitudes of the temperature and velocity perturbations in a truncated eigenfunction and power series. The validity of the principle of exchange of stabilities is proved analytically for a certain case and numerically investigated in general. Plots of the marginal stability curves as well as the variation of the critical Rayleigh number with other dimensionless parameters which naturally arise in the problem are also presented and discussed.

On the aerial dynamics of the Swift-Tuttle comet

Abstract Nonlinear dynamical equations describing the atmospheric dynamical interaction of comet Swift-Tuttle—possibly due to approach the Earth in 2126—are formulated, and mass ablation, velocity reductions and flight times are evaluated. These calculations suggest that the atmosphere offers little resistance to such a massive hypersonically travelling body. In addition, an analytical formulation is also found which is shown to agree well with the detailed nonlinear computational results.

Comet induced nuclear fusion in the atmosphere

Abstract The extent of fusion energy release as a result of the entry into the atmosphere of a massive high-speed body is considered here. Based on our analysis involving combined nuclear and aerial dynamics, we show that the release of significant quantities of fusion energy is highly improbable.