Self-similar gravity currents with variable inflow revisited: plane currents

Abstract

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type tα. The resistance of the ambient fluid is taken into account by a boundary condition of the von Kármán type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ]0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical near the front, but near the source it is subcritical ([Fscr ]0 < 1) for Type I currents, and supercritical ([Fscr ]0 > 1) for Types II, III, and IV. Type I solutions have already been found by other authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss their properties. For constant-volume currents one obtains Type I solutions for any β that, when β > 2, have a ‘dry’ region near the origin. For steady inflow one finds Type I currents for O < β < ∞ and Type II and III Currents for and β, if [Fscr ]0 is sufficiently large.

Sustained gravity currents in a channel

“…Our analytical techniques have also been applied to the related problem of Boussinesq dam-break flows (Appendix B). Flows driven by sustained sources and by the collapse of a dam share many features as drawn out for single layer flows by Gratton & Vigo (1994). Here we have employed the method of characteristics to construct the height and velocity fields for both partial and full-depth dams; these solutions are of self-similar form and like those due to a sustained source, yield a simple (linear) gearing between the spatial and temporal variables.…”

“…Importantly then, the motion of the overlying, less dense layer affects the streamwise pressure gradient and because the flow is generated by a sustained source, it does not become increasingly shallow in time, which would allow the motion of the upper layer to be neglected after a sufficient time. Thus this flow configuration is not amenable to study using a single layer, hydrostatic model, in which Hoult (1972), Gratton & Vigo (1994) and Hogg et al (2005), amongst others, have calculated the flow speed as a function of the governing parameters. We also note that the flows driven by a sustained influx differ from those arising from lock release initial conditions, because they do exhibit the same progressive thinning, which ultimately enhances the effects of hydraulic resistance.…”

“…However, it will be shown that the same range of solution types exist for the non-Boussinesq regime. Equivalently these flows may be thought of as similarity solutions to the governing equations because there is a simple gearing between spatial and temporal scales (Kranenburg 1993;Gratton & Vigo 1994). The type of solution depends upon the values of three dimensionless parameters that measure the magnitude of the volume flux delivered in the channel at the source, the source Froude number and the relative excess density.…”

Hot spots and nonhydraulic effects in surface gravity flows

“…In this article we develop and analyze a two layer fluid model governing the sudden release and subsequent motion of a fixed volume of light fluid whose initial density and temperature are ρ * and T * , respectively.These fixed volume releases have served as a paradigm for many atmospheric [12] and oceanic [1] gravity currents although it is the case that many of these flows being modelled arise not from a fixed volume release but rather from variable inflow through an opening in some barrier [13,14]. We will provide some suggestions as to how such variable inflow problems might be approached but for now we consider our fixed volume as being released suddenly into a heavier ambient fluid of constant density ρ 0 > ρ * overlying a gently sloping bottom.…”

A note on exact particular solutions of the generalized shallow-water equations

“…The shallow-water equations ut + uux + hx = 0, ht + uhx + hux = 0 (SW) have been intensively studied analytically and numerically (see, e.g., [2], [4, §13.10]). Several exact analytical solutions of (SW) are known: u = x/t, h = a/t\ u = (b + 2x/t)/3, h = (x/t -b)2/9 + a/t2^3; [2, §5] (PSa,b)…”