Axisymmetric three-dimensional gravity currents generated by lock exchange

Roberto Inghilesi,Federico Roman,Vincenzo Armenio,Claudia Adduce,Valentina Lombardi

doi:10.1017/jfm.2018.500

Abstract

Unconfined three-dimensional gravity currents generated by lock exchange using a small dividing gate in a sufficiently large tank are investigated by means of large eddy simulations under the Boussinesq approximation, with Grashof numbers varying over five orders of magnitudes. The study shows that, after an initial transient, the flow can be separated into an axisymmetric expansion and a globally translating motion. In particular, the circular frontline spreads like a constant-flow-rate, axially symmetric gravity current about a virtual source translating along the symmetry axis. The flow is characterised by the presence of lobe and cleft instabilities and hydrodynamic shocks. Depending on the Grashof number, the shocks can either be isolated or produced continuously. In the latter case a typical ring structure is visible in the density and velocity fields. The analysis of the frontal spreading of the axisymmetric part of the current indicates the presence of three regimes, namely, a slumping phase, an inertial–buoyancy equilibrium regime and a viscous–buoyancy equilibrium regime. The viscous–buoyancy phase is in good agreement with the model of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), while the inertial phase is consistent with the experiments of Britter (Atmos. Environ., vol. 13, 1979, pp. 1241–1247), conducted for purely axially symmetric, constant inflow, gravity currents. The adoption of the slumping model of Huppert & Simpson (J. Fluid Mech., vol. 99 (04), 1980, pp. 785–799), which is here extended to the case of constant-flow-rate cylindrical currents, allows reconciling of the different theories about the initial radial spreading in the context of different asymptotic regimes. As expected, the slumping phase is governed by the Froude number at the lock’s gate, whereas the transition to the viscous phase depends on both the Froude number at the gate and the Grashof number. The identification of the inertial–buoyancy regime in the presence of hydrodynamic shocks for this class of flows is important, due to the lack of analytical solutions for the similarity problem in the framework of shallow water theory. This fact has considerably slowed the research on variable-flow-rate axisymmetric gravity currents, as opposed to the rapid development of the knowledge about cylindrical constant-volume and planar gravity currents, despite their own environmental relevance.

Highlights

The spreading of three-dimensional, unconfined gravity currents is a primary concern in many environmental problems in hydraulics, coastal dynamics, oceanography and meteorology
Less is known for axially symmetric gravity currents generated by a variable-volume flow of dense fluid even in the easiest case, i.e. constant inflow
The existence of characteristic frontal spreading regimes governed by the dynamical balance of some of the terms in Axisymmetric 3-D gravity currents generated by lock exchange the shallow water equations was suggested by Fay (1969) and successively by Hoult

Summary

Introduction

The spreading of three-dimensional, unconfined gravity currents is a primary concern in many environmental problems in hydraulics, coastal dynamics, oceanography and meteorology. Hallworth, Huppert & Ungarish 2001; Huppert 2006; Ungarish 2009; Dai & Wu 2016), refers to constant-volume flows, i.e. currents generated by the release of a constant volume of a dense fluid This is because of the relative feasibility of field or laboratory experiments and the availability of numerical and analytical similarity solutions for the associated initial value problem in the context of shallow water theory Less is known for axially symmetric gravity currents generated by a variable-volume flow of dense fluid even in the easiest case, i.e. constant inflow This particular type of flow has rarely been investigated because laboratory experiments are often prone to being dominated by hydraulic shocks and, shallow water theory does not provide continuous analytical solutions of the similarity equations for the associated initial value problem (Grundy & Rottman 1986). In the case of axially symmetric, constant volume gravity currents, the hydrostatic balance between the buoyancy and the inertial term can be expressed as:

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