Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves

Bridges, Thomas J.; Donaldson, Neil M.

doi:10.1063/1.2750541

Cited by 11 publications

(9 citation statements)

References 22 publications

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“…In some cases, it is advantageous to keep the pressures in the equations. For example, Bridges & Donaldson [8] in their study of the criticality of two-layer flows provide an appendix on the inclusion of the lid pressure in the calculation of uniform flows. In the next sections, we will derive simplified models based on certain additional assumptions on wave amplitude, wavelength and fluid depth.…”

Section: Governing Equationsmentioning

confidence: 99%

A Boussinesq system for two-way propagation of interfacial waves

Nguyen

Dias

2008

Physica D: Nonlinear Phenomena

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The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion. A Boussinesq system for two-way propagation of interfacial waves in a rigid lid configuration is derived. In most cases, the nonlinearity is quadratic. However, when the square of the depth ratio is close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. The propagation as well as the collision of solitary waves and/or fronts is studied numerically.

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Section: Governing Equationsmentioning

confidence: 99%

A Boussinesq system for two-way propagation of interfacial waves

Nguyen

Dias

2008

Physica D: Nonlinear Phenomena

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“…Here this coupled problem is considered with a shallow-water rigid-lid model for the two-layer fluid. There has been a vast amount of work on the two-layer model in open systems, and papers that have influenced the current theory are [2,9,10,11,12,15,17,21,22,26,27,28,29,32,34,35,37,41,42]. However the principal interest here is in closed systems.…”

Section: Introductionmentioning

confidence: 99%

Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing

Ardakani

Bridges

Turner

2015

Journal of Fluids and Structures

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Abstract. Numerical and analytical results are presented for fluid sloshing, of a two-layer inviscid, incompressible and immiscible fluid with thin layers and a rigid lid, coupled to a vessel which is free to undergo horizontal motion governed by a nonlinear spring. Exact analytical results are obtained for the linear problem, giving the natural frequencies and the resonance structure, particularly between the fluid and vessel. A numerical method for the linear and nonlinear equations is developed based on the high-resolution f-wave-propagation finite volume methods due to BALE, LEVEQUE, MITRAN AND ROSSMANITH (2002) (SIAM J. Sci. Comput. 24, 955-978), adapted to include the pressure gradient at the rigid-lid, and coupled to a Runge-Kutta solver for the vessel motion. The numerical simulations in the linear limit are compared with the exact analytical solutions. The coupled nonlinear numerical solutions with simulations near the internal 1 : 1 resonance are presented. Of particular interest is the partition of energy between the vessel and fluid motion.

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“…For example, the theory of criticality for two layer fluids (e.g. Bridges & Donaldson 2007) is manifestly more interesting, and the theory in this paper can be suitably generalised to generate a theory for the emergence of dispersion in that setting by modulating the two-layer uniform flow. Another example is finiteamplitude criticality of Stokes waves (Bridges & Donaldson 2006).…”

Section: Discussionmentioning

confidence: 99%

Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow

Bridges¹

2014

J. Fluid Mech.

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A new theory for the emergence of dispersion in shallow-water hydrodynamics in two horizontal-space dimensions is presented. Starting with the key properties of uniform flow in open channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux, and mass density. The theory gives a new perspectivefrom the viewpoint of hydraulics -on how and why key shallow water models like the Korteweg-de Vries equation and Kadomtsev-Petviashvili equations arise in the theory of water waves.

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Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves

Cited by 11 publications

References 22 publications

A Boussinesq system for two-way propagation of interfacial waves

A Boussinesq system for two-way propagation of interfacial waves

Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing

Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow

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