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TECHNICAL PAPERS

Numerical Investigation of Two-Dimensional Sloshing: Nonlinear Internal Waves

[+] Author and Article Information
Daniel T. Valentine

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

J. Offshore Mech. Arct. Eng 127(4), 300-305 (May 17, 2005) (6 pages) doi:10.1115/1.2073154 History: Received October 04, 2004; Revised May 17, 2005

In this paper we examine internal sloshing motions in 2-D 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 results presented are from simulations of internal waves induced by sloshing a density-stratified liquid. Nonlinear, viscous flow equations of a Newtonian, Boussinesq liquid are solved. Some of the features of the evolution of sloshing in nearly two-layer and three-layer fluid systems are described. Initially, the middle of the two layers and the center of the middle layer of the three layers are horizontal and located at the center of the tank. The two-layer cases are forced at resonance. The evolution of sloshing from rest is examined. The maximum amplitude of sloshing occurs during the initial transient. If breaking occurs, it is at the center of the container in the two-layer cases. The subharmonic forcing of a three-layer case induces a resonant response with the middle layer moving in such a way that motion is perpendicular to the isopycnals within this layer. These model problems provide some insights into the relatively complex sloshing that can occur in density-stratified liquids.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 8

Internal sloshing motion for subharmonic forcing of a nearly three-layer liquid with NF=0.125,FV=2.14. (a) At t=29 the streamlines (dashed lines) correspond to 0⩽ψ⩽0.006,Δψ=0.0005. The isopycnals (solid lines) correspond to 0.1⩽θ⩽0.9,Δθ=0.1. (b) Boundary layer details at t=29, where the streamlines (dashed lines) correspond to 0⩽ψ⩽0.0004,Δψ=0.00005. The isopycnals (solid lines) correspond to the same values of θ as in (a).

Grahic Jump Location
Figure 1

Lower-left corner details of grid for internal wave study

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Figure 2

Initial conditions of the density-difference ratio field for two sets of internal wave simulations. The isopycnals shown are, from the bottom to the top of the pycnocline, 0.1⩽θ⩽0.9,Δθ=0.1. (a) and (b) are for the nearly two-layer and nearly three-layer fluid simulations, respectively.

Grahic Jump Location
Figure 3

Internal sloshing for the NF=0.25 case. (a) Time histories of the streamfunction at (x,z)=(0.65,0.25) [P4], vertical component of velocity at (x,z)=(1.05,0.25) [P18], and forcing function [F]. (b) Lissajous figure for 0⩽t⩽18 of applied forcing function [F] versus a w component of the velocity field [P18] at (x,z)=(1.05,0.25).

Grahic Jump Location
Figure 4

Spatial variations of internal sloshing for NF=0.25,FV=2.18. (a) The streamline pattern (dashed lines) and isopycnals (solid lines) at t=8; −0.0002⩽ψ⩽0.006, Δψ=0.0005 and θ=0.25, 0.5, and 0.75 from the bottom to top of pycnocline, respectively. Note that ψ is positive in the center of the tank. (b) The streamline pattern [dashed lines]and isopycnals [solid lines] at t=15.2; 0.0⩽ψ⩽0.3, Δψ=0.03 and 0<θ<1,Δθ=0.1.

Grahic Jump Location
Figure 5

Internal sloshing for the NF=0.25,FV=5.45 case. (a) Time histories of the vertical component of velocity at (x,z)=(0.65,0.25) [P16] and the coefficient of horizontal forcing [F] in Eq. 1. (b) Lissajous figure for 0⩽t⩽18 of applied forcing function [F] versus the w component of the velocity field [P16] at (x,z)=(0.65,0.25).

Grahic Jump Location
Figure 6

Internal sloshing motion for NF=0.25,FV=5.45. An illustration of nonbreaking sloshing with near-resonance forcing. (a) At t=14.8 the streamlines (dashed lines) correspond to −0.006⩽ψ⩽0.0,Δψ=0.001. The three isopycnals (solid lines) correspond to 0.1⩽θ⩽0.9,Δθ=0.1. (b) Boundary layer details at t=14.8; the streamlines (dashed lines) correspond to −0.0007⩽ψ⩽0.0001,Δψ=0.000,05 with negative values near the center of the tank. The isopycnals (solid lines) are the same as those plotted in (a).

Grahic Jump Location
Figure 7

Internal sloshing for the NF=0.125,FV=2.14 case. (a) Time histories of the vertical component of the velocity at (x,z)=(0.65,0.25) [P16], and the coefficient of the horizontal forcing term in Eq. 1 divided by 20, [F∕20]. (b) Lissajous figure for 0⩽t⩽20 (solid line) and 20<t⩽45 (dashed line) of the w component of the velocity field [P16] at (x,z)=(0.65,0.25) versus the applied forcing function [F].

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