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TECHNICAL PAPERS

Nonlinear Free-Surface and Viscous-Internal Sloshing

[+] Author and Article Information
Daniel T. Valentine1

Department of Mechanical and Aeronautical Engineering,  Clarkson University, Potsdam, NY 13699-5725clara@clarkson.edu

Jannette B. Frandsen

Department of Civil and Environmental Engineering,  Louisiana State University, Baton Rouge, LA 70803frandsen@lsu.edu

1

To whom correspondence should be addressed.

J. Offshore Mech. Arct. Eng 127(2), 141-149 (Apr 10, 2003) (9 pages) doi:10.1115/1.1894415 History: Received April 10, 2003

This paper examines free-surface and internal-pycnocline sloshing motions in two-dimensional numerical wave tanks subjected to horizontal excitation. In all of the cases studied, the rectangular tank of liquid has a width-to-depth ratio of 2. The first set of results are based on an inviscid, fully nonlinear finite difference free-surface model. The model equations are mapped from the physical domain onto a rectangular domain. Case studies at and off resonance are presented illustrating when linear theory is inadequate. The next set of results are concerned with analyzing internal waves induced by sloshing a density-stratified liquid. Nonlinear, viscous flow equations are solved. Two types of breaking are discussed. One is associated with a shear instability which causes overturning on the lee side of a wave that moves towards the center of the container; this wave is generated as the dominant sloshing mode recedes from the sidewall towards the end of the first sloshing cycle. The other is associated with the growth of a convective instability that initiates the formation of a lip of heavier fluid above lighter fluid behind the crest of the primary wave as it moves up the sidewall. The lip grows into a bore-like structure as it plunges downward. It falls downward behind the primary wave as the primary wave moves up the sidewall and ahead of the primary wave as this wave recedes from the sidewall. This breaking event occurs near the end of the first cycle of sloshing, which is initiated from a state of rest by sinusoidal forcing.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 9

Internal sloshing motion for NF=0.25. The onset of the second type breaking of the primary sloshing mode is shown. (a) At t=3.4 the streamlines (solid lines) correspond to 0⩽ψ⩽0.3, Δψ=0.015. The isopycnals (dashed lines) correspond to, from bottom to top, θ=0.3, 0.5, and 0.7, respectively. (b) At t=3.8 the streamlines (solid lines) correspond to −0.02⩽ψ⩽0.2, Δψ=0.01 with positive values near the center of the tank. The isopycnals (dashed lines) correspond to, from bottom to top, θ=0.3, 0.5, and 0.7, respectively.

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Figure 8

Internal sloshing motions for NF=0.375. The onset of breaking of the second type is illustrated. At t=3.4 the streamlines (solid lines) correspond to −0.045⩽ψ⩽0.015, Δψ=0.003. The isopycnals (dashed lines) correspond to, from bottom to top, θ=0.3, 0.5, and 0.7, respectively.

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Figure 1

Lower-left corner details of grid for internal wave study

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Figure 2

Free-surface elevation at the left wall in horizontally excited tank off resonance; ωh∕ω1=1.3; (a) κh=0.0036 and (d) κh=0.072; dashed line second-order solution; solid line, numerical solution. The corresponding wave phase plane and spectra of the numerical model: (b), (c) linear solution; (e), (f) nonlinear solution.

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Figure 3

Free-surface elevation at the left wall in horizontally excited tank at resonance; ωh∕ω1=1; (a) κh=0.0014 and (d) κh=0.014; dash-dot line, linear solution; dashed line, second-order solution; solid line, numerical solution. The corresponding phase-plane of the numerical model: (b) linear solution; (d) nonlinear solution.

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Figure 4

Initial condition of the density-difference ratio field for the internal waves simulations. The isopycnals shown are, from bottom to top of the pycnocline, 0.1<θ⩽0.9, Δθ=0.1.

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Figure 5

Internal sloshing for NF=1 case. (a) Time histories of the streamfunction at (x,z)=(1.45,0.25) (the large amplitude) and at (x,z)=(0.05,0.25) (the small amplitude). (b) The corresponding spectra.

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Figure 6

Spatial variations of internal sloshing for NF=1. (a) The streamline pattern at t=2; −0.004⩽ψ⩽0.004, Δψ=0.00038. Note that ψ is positive in the center of the tank. (b) Isopycnal contours at t=8.75; for the three isopycnals shown, from lower to the upper, θ=0.3, 0.5, and 0.7, respectively.

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Figure 7

Internal sloshing motion for NF=0.425. The onset of the first type of breaking is illustrated. (a) At t=4 the streamlines (solid lines) correspond to −0.12⩽ψ⩽0.2, Δψ=0.01 with negative values near the center of the tank. The three isopycnals (dashed lines) correspond to, from bottom to top, θ=0.3, 0.5, and 0.7, respectively. (b) At t=4.4 the streamlines (solid lines) correspond to −0.056⩽ψ⩽0.016, Δψ=0.005 with negative values near the center of the tank. The three isopycnals (dashed lines) correspond to, from bottom to top, θ=0.3, 0.5, and 0.7, respectively.

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