Duncan E. Farrow

Periodically forced natural convection over slowly varying topography

Asymptotic and numerical methods are used to analyse periodically forced natural convection over slowly varying topography. This models the diurnal heating/cooling cycle in lakes and reservoirs. The asymptotic solution includes the effects of advection on the temperature. The asymptotic results are confirmed by the numerical results. The numerical results are also used to examine flow regimes where the asymptotic results break down. In particular, the presence of a vertical boundary leads to a permanent stratification in the deeper regions due to a nonlinear pumping process in the shallows. Heat transfer calculations and two limiting cases are also presented.

A numerical model for withdrawal from a two-layer fluid

This paper reports the results of several direct numerical simulations of the withdrawal of a two-layer fluid with a finite-thickness interface through a slot in the base of a finite rectangular cavity. Particular attention is paid to the role of long (basin scale) interfacial waves on the processes leading to drawdown of the interface into the slot. It is shown that these waves play an important role and can either delay or accelerate drawdown. This means that drawdown can occur over a range of Froude numbers. The results are compared with previous work for ideal flow and experimental results.

Numerical modelling of a surface-stress driven density-stratified fluid

The initial response of a density-stratified fluid in a rectangular domain to a surface stress is modelled numerically. The model is laminar, two-dimensional and non-hydrostatic. Upwelling of deep fluid at the upwind end of the fluid is critical to the subsequent evolution of the stratification. It is confirmed that upwelling is a wave process and consideration of flow at the upwind end-wall illustrates the flow structure of partial upwelling. Numerically, to ensure adequate penetration of surface stress, an increased viscosity is needed. Comparisons are made between the present numerical results and previously published experimental observations.

Coriolis effects and the thermal bar

[1] A model for the thermal bar system in the rotating frame that includes unsteady inertia is formulated. Asymptotic solutions are found to the initial value problem in the frictionless, small bottom slope limit. These solutions include inertial oscillations that are significant enough to reverse the circulation ahead of the thermal bar. These asymptotic solutions are compared with numerical solutions of the full model that includes friction. The consequences of both sets of results on the thermal bar in lakes is discussed.

Periodically driven circulation near the shore of a lake

Solutions are found for a linear model of the circulation near the shore of a lake that is subject to two diurnal forcing mechanisms. The first is the day/night heating/cooling induced horizontal pressure gradient. The second is an unsteady surface stress modelling a sea breeze/gully wind pattern. The two forcing mechanisms can oppose or reinforce each other depending on their relative phase. The interplay of different dynamic balances at different times and locations in the domain lead to complex circulation patterns especially during the period of flow reversal.

An intrusion layer in stationary incompressible fluids: Part 1: Periodic waves

Waves on a neutrally buoyant intrusion layer moving into otherwise stationary fluid are studied. There are two interfacial free surfaces, above and below the moving layer, and a train of waves is present. A small amplitude linearized theory shows that there are two different flow types, in which the two interfaces are either in phase or else move oppositely. The former flow type occurs at high phase speed and the latter is a low-speed solution. Nonlinear solutions are computed for large amplitude waves, using a spectral type numerical method. They extend the results of the linearized analysis, and reveal the presence of limiting flow types in some circumstances.

A model of the thermal bar in the rotating frame including vertically non-uniform heating

Small bottom slope, inviscid solutions are found for a model of the temperature and circulation structure of the thermal bar system. This model includes Coriolis effects, a vertically non-uniform heat input and is axisymmetric. The model also includes general topography and time dependent heating. These solutions include inertial oscillations that have a significant effect on the circulation, especially for the case when the heating is instantaneously applied.

A model for the evolution of the thermal bar system

A new framework for modelling the evolution of the thermal bar system in a lake is presented. The model assumes that the thermal bar is located between two regions: the deeper region, where spring warming leads to overturning of the entire water column, and the near shore shallower region, where a stable surface layer is established. In this model the thermal bar moves out slightly more quickly than predicted by a simple thermal balance. Also, the horizontal extent of the thermal bar region increases as it moves out from the shore.

A numerical model of periodically forced circulation near the shore of a lake

Numerical calculations for a model of the near-shore circulation in a lake subject to two diurnal forcing mechanisms are presented. The first mechanism is a heating/cooling term in the heat equation representing the daytime heating and nighttime cooling of the diurnal cycle. The second is a periodic surface stress modelling a sea-breeze/gully wind system typical of some coastal regions. The two forcing mechanisms can either act together or against each other depending on their relative phase. The numerical solutions are compared with previously published analytical solutions and used to explore the extra dynamics associated with non-linear effects (specifically advection). The latter dynamics include the formation of gravity currents and unstable density profiles leading to secondary circulation.

Thermal layer instability in a shallow wedge subject to solar radiation

A model for radiation-induced natural convection in a water-filled wedge is described in [3]. With this model there is the potential for a thermal instability [2]. Flow visualisation revealed the primary form of the instability [5], which was confirmed by numerical simulations [4]. A corresponding scaling analysis has identified the critical condition and the controlling parameters as the Prandtl number Pr, the Grashof number Gr, and the cavity aspect ratio A [6].