A mathematical framework for the analysis of particle–driven gravity currents

Abstract

Recent studies have modelled the flow of particle-driven gravity currents over horizontal boundaries using either shallow-water equations or simple 'box' (integral) models. The shallow-water equations are typically integrated numerically, whereas box models admit analytical solutions. However, the theoretical validity of the latter models has not been fully established. In this paper a novel mathematical technique is developed which permits the derivation of analytical solutions to the shallow-water model for gravity current motion. These solutions, confirmed by comparison with the results of numerical integration, are in good agreement with experimental observations. They also indicate why the simplified box models have been so successful. Moreover, they reveal how the internal dynamics of particle-driven flows are different from gravity currents arising solely due to compositional density differences. While compositionally driven gravity currents, which have a fixed density difference between the intruding and ambient fluids, may be modelled using similarity solutions to the governing equations, particle-driven gravity currents do not possess such solutions because their density is progressively reduced by particle sedimentation. Instead the new analysis determines how their behaviour progressively diverges from the similarity solution. By a change of independent variables, it is possible to develop convergent series expansions for each of the dependent variables which characterize the motion. It is suggested that this approach may find application to a number of other problems in which the dynamics are initially governed by a simple dynamical balance which is progressively lost as extra physical effects begin to influence the motion.

Gravity currents: a personal perspective

“…Alternatively, Huppert & Simpson (1980) introduced the concept of a 'box model', which considers the simple model that results from assuming the current to evolve through a series of equal-area rectangles, or equal-volume cylinders, as appropriate, with no variations of any properties in the horizontal. (An integral justification of this approach is given in the Appendix of Harris, Hogg & Huppert 2001). With the assumption that the ambient is sufficiently deep that the Froude number is constant, this approach leads to the relationship…”

“…It must, on the other hand, be possible to obtain asymptotic solutions to (4.2) in a perturbative sense. Such a technique is constructed, in part, in Harris et al (2001).…”

“…1/2 , where g o is the initial reduced gravity of the system (Harris et al 2001). This is an example of the interesting problem which can be stated in general as the nonlinear partial differential equation 2) where N 1 and N 2 are nonlinear operators in some spatial coordinate x and time t. For ε ≡ 0 there is a similarity solution to (4.2) and a special value of x, x * say, such as that at the nose of the current, which increases continually with time, of the form x * = f (t).…”

Stability of gravity currents generated by finite-volume releases

“…In the case of particle-driven currents however, particle sedimentation means that the density difference between the fluids is no longer constant in time, but declines progressively and the motion is not selfsimilar. Significant insight into the dynamics of these flows can be gained by deriving asymptotic series about the self-similar solution to obtain the deviations from the self-similar flow which describe the particle-driven currents (see, for example, Hogg et al 2000;Harris, Hogg & Huppert 2001). Although the regime in which these expansions are valid is usually limited, the first-order asymptotic functions derived using this approach provide valuable information about the structure of the solutions of the particle-driven currents and the way in which their evolution differs from the homogeneous currents of the same initial excess density.…”

The inertial dynamics of thin film flow of non-Newtonian fluids

“…To model faster flows, potentially with wave effects, we must resolve the dynamics of both the fluid thickness and a measure of horizontal momentum [2,10], we used η andū in (1)-(2). For example, Harris et al [11] modelled particle driven gravity currents using shallow water equations that resolve the dynamics of both the fluid thickness and the mean lateral velocity. However, such modelling of essentially dissipative flows, albeit dissipative via turbulence, by the laminar inviscid foundation of shallow water equations appears a contradiction that demands resolution.…”