Stability of gravity currents generated by finite-volume releases

Abstract

We generalize the linear stability analysis of the axisymmetric self-similar solution of gravity currents from finite-volume releases to include perturbations that depend on both radial and azimuthal coordinates. We show that the similarity solution is stable to sufficiently small perturbations by proving that all perturbation eigenfunctions decay in time. Moreover, asymmetric perturbations are shown to decay more rapidly than axisymmetric perturbations in general. An asymptotic formula for the eigenvalues is derived, which indicates that asymptotic rates of decay of perturbations are given by t −σ where 0 < σ < 1 4 as the Froude number decreases from √ 2 to 0. We demonstrate that this formula agrees closely with numerically calculated eigenvalues and, in the absence of azimuthal dependence, it reduces to an expression that improves on the asymptotic formula obtained by Grundy & Rottman (1985). For two-dimensional (planar) currents, we further prove analytically that all perturbation eigenfunctions decay like t −1/2 .

Entraining gravity currents

“…The long-time solution of this shallow water model of an instantaneous release of a dense fluid is a 'buoyancy-inertial' similarity solution, where, in terms of the reduced gravity g and a constant volume per unit width V, both dependent on the initial conditions, the current length x f varies as (g Vt 2 ) 1/3 , current velocity u ∼ (g V/t) 1/3 and current height h ∼ (V 2 /g t 2 ) 1/3 . The existence of this buoyancy-inertial regime has been verified by laboratory experiments (Rottman & Simpson 1983) and emerges as an attracting similarity solution to the underlying model equation (Mathunjwa & Hogg 2006). However, substituting these scalings into the mass equation for an entraining gravity current (2.1) and (2.4), we find that the terms ∂h/∂t and ∂(hu)/∂x scale as (V 2 /g t 5 )…”

“…This similarity solution represents the asymptotic solution of the motion from lock-release and other initial conditions 504 C. G. Johnson and A. J. Hogg (see, e.g., Hogg 2006;Mathunjwa & Hogg 2006). We seek the perturbation to this solution due to the effects of entrainment.…”

Unsteady draining of reservoirs over weirs and through constrictions

“…We may analyse the progressive approach of the draining dynamics through a constriction to the similarity solution (4.4), by computing the linear stability of the latter. Following Grundy & Rottman (1985) and Mathunjwa & Hogg (2006), we introduce a small perturbation to the solutions for the height and velocity fields, denoted byH(x) andŨ (x) respectively, so that…”

“…We may analyse the progressive approach of the draining dynamics through a constriction to the similarity solution (4.4), by computing the linear stability of the latter. Following Grundy & Rottman (1985) and Mathunjwa & Hogg (2006), we introduce a small perturbation to the solutions for the height and velocity fields, denoted by and respectively, so that where is a small ordering parameter and is to be determined. On substitution into the governing equations (2.1 a ) and (2.1 b ) and after linearisation, we find that The boundary conditions represent no flow at the back wall, , and the linearised drainage condition is given by Straightforwardly, we note that is a solution, with and ; this corresponds to the introduction of a shift of the temporal origin, which, as discussed above, does not change the similarity solution.…”

“…In addition, is a solution with and ; this corresponds to a translation of the -axis. Both of the these solutions are inevitably found in the study of the stability of similarity solutions and do not reveal the linear stability properties (Mathunjwa & Hogg 2006). Instead we seek non-trivial solutions to (4.14)–(4.15), which determine the exponent, , as an eigenvalue.…”