The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases

Grundy, R. E.; Rottman, James W.

doi:10.1017/s0022112085001975

Cited by 69 publications

(79 citation statements)

References 12 publications

(16 reference statements)

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“…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 .…”

supporting

confidence: 81%

Stability of gravity currents generated by finite-volume releases

Mathunjwa

Hogg

2006

J. Fluid Mech.

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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 .

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confidence: 81%

Stability of gravity currents generated by finite-volume releases

Mathunjwa

Hogg

2006

J. Fluid Mech.

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“…The behavior in this latter phase, in which the flow is governed by a balance between buoyancy and inertial forces, is more fully discussed in Grundy and Rottman. 2 After sufficient time, the flow will enter a final phase, in which it is governed by a buoyancy-viscous balance that is well described by shallow-water theory.…”

Section: Introductionmentioning

confidence: 99%

Vortical motion in the head of an axisymmetric gravity current

et al. 2006

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A series of experiments that examine the initial development of an axisymmetric gravity current have been carried out. The experiments highlight the growth of a ring vortex that dominates the dynamics of the gravity current's early time propagation. In particular, the experiments show three distinct stages of early time development that have previously been described as the "initial phase" of a gravity current. The first phase of the early time development is dependent on the fractional depth of the lock release, followed by a secondary phase wherein the frontal speed is approximately constant and a third phase of reducing speed. The second phase of the gravity current's propagation comes to an abrupt end with the breakdown of the ring vortex at a clearly defined position. All of the experimental results show the development of a complex flow field where the generation and collapse of a ring vortex dominate the gravity current's early time propagation. The complexity of the flow field and the dependence of the propagation speed on the presence of the ring vortex in the head of the gravity current highlights the unsuitability of shallow-water modeling for axisymmetric lock releases at early times.

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“…A thorough analysis of the phase portraits has shown that unlike in the Boussinesq regime investigated by Grundy and Rottman [26] and Gratton and Vigo [25], the shallow-water equations for non-Boussinesq regimes do not systematically admit similarity solutions. For these similarity solutions to exist, the front point P must be a singular point of the phase portrait.…”

Section: Discussionmentioning

confidence: 96%

“…In this context, seeking analytical solutions is of paramount importance both for gaining insight into the flow dynamics and for testing numerical models. Various solutions have been worked out using different techniques: the method of characteristics [14,52,66], the hodograph transformation [13,20], and self-similar solutions [25,26]. Similarity solutions are here of particular relevance in the present context since they embody the inertia-buoyancy balance, which is anticipated to be the driving mechanism in the flow dynamics when the influence of the initial conditions becomes negligibly small.…”

Section: Introductionmentioning

confidence: 99%

Existence and features of similarity solutions for non-Boussinesq gravity currents

Ancey

Cochard

Rentschler

et al. 2007

Physica D: Nonlinear Phenomena

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In this paper, the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions. The current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density. Unlike earlier investigators, we assume that the ambient fluid exerts no significant resisting action on the current, and therefore the flow depth is expected to drop to zero at the front in the absence of friction. In this context, the shallow-water equations are highly appropriate for computing the mean velocity and flow depth of the current. The boundary condition imposed at the front leads to technical mathematical difficulties. Indeed, unlike in the Boussinesq case, no regular solution to the shallow-water equations satisfies the downstream condition, but when the flow is supercritical at the channel inlet, it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution, which represents the head behavior. To better understand this result and make sure that the result is physically relevant, we consider the Navier-Stokes equations within the high-Reynolds-number limit. Approximate similarity solutions can be worked out, which support our earlier analysis on the shallow-water equations. While the flow body is self-similar and weakly rotational, the head is not self-similar, but tends toward a self-similarity shape at long times. It is characterized by a strong vorticity, a straight free surface, and a nonuniform velocity profile, which becomes flatter and flatter with time.

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The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases

Cited by 69 publications

References 12 publications

Stability of gravity currents generated by finite-volume releases

Stability of gravity currents generated by finite-volume releases

Vortical motion in the head of an axisymmetric gravity current

Existence and features of similarity solutions for non-Boussinesq gravity currents

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