On gravity-driven flow

This thesis investigates three areas of scientific interest under the umbrella of gravity- driven flow. In the first problem we analyse the effect of curved boundaries on the evolution of finite-release low-Reynolds-number exchange flow. From the equations of motion in an annular geometry we derive the governing nonlinear PDE for the evolution of the density interface. Systematically varying the initial conditions leads to the recognition that there are different routes to equilibrium. Asymptotic analysis of squeezing and draining limits allows us to find analytical solutions with good agreement to the numerical solutions. Finally, we show the time taken to reach equilibrium can vary by two orders of magnitude. We relate these time scales to the application of displacing drilling lubricant by cement in a horizontal oil well. In the second problem we develop a simplified theory relating the downstream structure of horizontally propagating turbulent gravity currents to uniform source conditions. We constrain the downstream solutions by conserving the flux of horizontal buoyancy and momentum. With an experimentally motivated ansatz that the down- stream horizontal velocity and buoyancy structure is either i) entirely linearly stratified ii) consists of a well-mixed uniform lower region overlain by a linearly stratified region, we can relate the upstream conditions to the downstream conditions as a function of source Froude number, downstream gradient Richardson number, and a shape factor φ. These solutions lead to global constraints on the entrainment flux and energy dissipation of transitioning currents. In the third problem we present a series of new numerical simulations on two- dimensional gravity currents propagating down a rigid inclined boundary. From the data generated by the simulations we can estimate the rate of entrainment E of these currents, and present this data over a range of angles θ. Furthermore, we show that around 10% of the buoyancy, and 15% of the momentum flux, is transported by turbulent fluctuations. We also show that the Richardson number Ri of these currents remains marginally stable Ri ≈ 0.25 and is independent of angle. Finally we show that the coefficient of drag cd experienced by these currents is constant and approximately cd ≈ 0.01.