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Journal ArticleDOI

Erratum to: Gravity currents in two-layer stratified media

A. W. Tan1, David S. Nobes1, Brian A. Fleck1, Morris Flynn1 
20 Jan 2011-Environmental Fluid Mechanics (Springer Netherlands)-Vol. 11, Iss: 2, pp 225-230

AbstractThis brief communication corrects a minor error in the presentation of the model results described in §2 of Tan et al. [1] (referred to hereafter as T.al.). Previously, we predicted that the solution of Eqs. 2.10 and 2.11 becomes unphysical for g′ 12/g′ 02 ≥ 3/4 irrespective of h′1. Upon closer inspection, we observe that model breakdown occurs at g′ 12/g′ 02 = 3/4 when h′1 0.50 but extends beyond this value when h′1 0.50. This is illustrated in Fig. 2a which shows the surface plot of Fr vs. g′ 12/g′ 02 and h′1 and is identical to Fig. 3a of T. al. with the exception of the aforementioned correction. In like fashion, Figs. 1, 2, 3 and 4 are, respectively, the corrected versions of Figs. 2, 3, 4 and 5 from T. al. (Note, that the geometric variables h′1, h1, h′2 and h2 used here have been non-dimensionalized by H , the channel depth.) As these new figures make clear, the region of parameter space where the model is valid and, by extension, where the gravity current is predicted to be supercritical, is larger than described in the original manuscript. Notwithstanding this correction, the principal conclusions of T. al. remain unchanged. In addition to calculating head loss along streamlines as we do with Eqs. 2.14 and 2.15 of T. al., one can also evaluate the global dissipation by determining the change of D = ∫ u(p + 1 2ρu + ρgz) dz from far upstream to far downstream. The expression for D, written in non-dimensional form reads

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Journal ArticleDOI
Abstract: We consider the propagation of a gravity current of density ρc at the bottom of a two-layer stratified ambient in a horizontal channel of height H, in the high-Reynolds number Boussinesq domain. The study emphasizes theoretical-analytical modeling, however, experimental and Navier-Stokes simulation data are also presented and their comparison with theory is discussed. The stratification parameters are S = (ρ1 − ρ2)/(ρc − ρ2) where ρ is the fluid density, and φ = h1R/H where h1R is the (unperturbed) ambient interface height. Here, 1 and 2 denote, respectively, the lower and upper layer and c denotes the gravity current. The reduced gravity is defined as g′ = (ρc/ρ2 − 1)g. Rigorous results are obtained for the steady-state analogue of the classical problem of Benjamin [J. Fluid Mech. 31, 209 (1968)]10.1017/S0022112068000133, in which the half-infinite gravity current has thickness h and speed U. We thereby demonstrate that the Froude number F=U/(g′h)1/2 is a function of a = h/H, S, and φ. In general, two so...

14 citations


References
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Book
28 Apr 2009
Abstract: Introduction Classification The Navier-Stokes equations Non-stratified ambient currents Shallow-water (SW) formulation for high-Re flows Motion of the interface and the continuity equation One-layer model A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation The steady-state current and nose jump conditions Benjamin's analysis Jump condition Box models for 2D geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Two-layer SW model Introduction The governing equations Boussinesq system in dimensionless form Jumps of interface for H < 2 Energy and work in a two-layer model Axisymmetric currents, SW formulation Governing equations A useful transformation The full behavior by numerical solution Dam-break stage Similarity solution The validity of the inviscid approximation Some comparisons Box models for axisymmetric geometry Fixed volume current with inertial-buoyancy balance Inflow volume change Effects of rotation Axisymmetric case Rotating channel Buoyancy decays: particle-driven, porous boundary, and entrainment Particle-driven currents Axisymmetric particle-driven current Extensions of particle-driven solutions Current over a porous bottom Axisymmetric current over a porous bottom Entrainment Non-Boussinesq systems Introduction Formulation Dam-break and initial slumping motion The transition and self-similar stages Summary Lubrication theory formulation for viscous currents 2D geometry Axisymmetric current Current in a porous medium II Stratified ambient currents and intrusions Continuous density transition Introduction The SW formulation SW results and comparisons with experiments and simulations Dam break Critical speed and nose-wave interaction Similarity solution The validity of the inviscid approximation Axisymmetric and rotating cases SW formulation SW and NS finite-difference results The validity of the inviscid approximation The steady-state current Steady-state flow pattern Results Comparisons and conclusions Intrusions in 2D geometry Introduction Two-layer stratification Linear transition layer Rectangular lock configurations Cylindrical lock in a fully linearly-stratified tank Similarity solution Non-symmetric intrusions Intrusions in axisymmetric geometry Introduction Two-layer stratification Fully linearly-stratified tank, part-depth locks Box models for 2D geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Box models for axisymmetric geometry Fixed volume and inertial-buoyancy balance S = 1, inflow volume change Lubrication theory for viscous currents with S = 1 2D geometry Axisymmetric geometry Energy Introduction 2D geometry Axisymmetric geometry SW equations: characteristics and finite-difference schemes Characteristics Numerical solution of the SW equations Navier-Stokes numerical simulations Formulation A finite-difference code Other codes Some useful formulas Leibniz's Theorem Vectors and coordinate systems

217 citations


"Erratum to: Gravity currents in two..." refers background in this paper

  • ...Broken lines for large h′1 and g′ 12/g′ 02 indicate model breakdown [2]....

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Journal ArticleDOI
Abstract: An analytical, experimental and numerical study of boundary gravity currents propagating through a two-layer stratified ambient of finite vertical extent is presented. Gravity currents are supposed to originate from a lock-release apparatus; the (heavy) gravity current fluid is assumed to span the entire channel depth, H, at the initial instant. Our theoretical discussion considers slumping, supercritical gravity currents, i.e. those that generate an interfacial disturbance whose speed of propagation matches the front speed, and follows from the classical analysis of Benjamin (J Fluid Mech 31:209–248, 1968). In contrast to previous investigations, we argue that the interfacial disturbance must be parameterized so that its amplitude can be straightforwardly determined from the ambient layer depths. Our parameterization is based on sensible physical arguments; its accuracy is confirmed by comparison against experimental and numerical data. More generally, measured front speeds show positive agreement with analogue model predictions, which remain strictly single-valued. From experimental and numerical observations of supercritical gravity currents, it is noted that this front speed is essentially independent of the interfacial thickness, δ, even in the limiting case where δ = H so that the environment is comprised of a uniformly stratified ambient with no readily discernible upper or lower ambient layer. Conversely, when the gravity current is subcritical, there is a mild increase of front speed with δ. Our experiments also consider the horizontal distance, X, at which the front begins to decelerate. The variation of X with the interface thickness and the depths and densities of the ambient layers is discussed. For subcritical gravity currents, X may be as small as three lock lengths whereas with supercritical gravity currents, the gravity current may travel long distances at constant speed, particularly as the lower layer depth diminishes.

15 citations


Additional excerpts

  • ...The expression for D, written in non-dimensional form reads D = Fr [ g′ 12 g′ 02 (h′2h1 − h′1h′2) + g′ 01 g′ 02 h′1h0 + h′2h0 − 1 2 Fr2 ( h′3 1 h(2)1 + h ′3 2 h(2)2 )] (1) The online version of the original article can be found under doi:10....

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