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Gravity currents in non-rectangular cross-section channels: Analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation
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10.1063/1.4790796
/content/aip/journal/pof2/25/2/10.1063/1.4790796
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4790796

Figures

Image of FIG. 1.
FIG. 1.

Schematic description of typical channels with non-rectangular cross-section.

Image of FIG. 2.
FIG. 2.

Schematic description of the gravity current system. (a) Side view. (b) Cross-section of channel. Here () is the width of the channel. In the analysis denotes the area occupied by the ambient fluid, = (, ) is the area occupied by the current, and is the total area.

Image of FIG. 3.
FIG. 3.

Sketch of the power-law cross-section () = α for α = 0.5, 1, 2 (red, blue, and green). Here = 2.

Image of FIG. 4.
FIG. 4.

Profiles of and vs. for various values of . = 20, () = 1/2 for t= 2(2)20.

Image of FIG. 5.
FIG. 5.

As above, for () = .

Image of FIG. 6.
FIG. 6.

As in Fig. 4 , for () = 2.

Image of FIG. 7.
FIG. 7.

Numerical results: behavior of the nose speed vs. for () = and various values of : 1.1, 2, and 20. Note the constant speed during the initial (slumping) stage.

Image of FIG. 8.
FIG. 8.

Sketch of curved-trapezoidal cross-sections () = ( + α)/( + 1) for α = 0.5, 1, 2 (red, blue, and green). Here = 1, = 2.

Image of FIG. 9.
FIG. 9.

Profiles of and vs. for various values of . = 20, () = (1 + )/2 for = 2(2)20.

Image of FIG. 10.
FIG. 10.

View from above of the current in trapezoidal cross-section () = (1 + )/2 channel, for = 2, for = 2(2)20.

Image of FIG. 11.
FIG. 11.

Analytical results for the slumping stage of the current for () = α: Nose height and speed as functions of 1/. α = 2 (upper line, online light blue), α = 1 (red), α = 0.5 (green), α = 0 (lowest line, online blue; this is the classical rectangular case).

Image of FIG. 12.
FIG. 12.

Analytical slumping speed for trapezoidal () = ( + )/( + 1), where = 0 (upper line, red online), = 1 (blue online), = 5 (black). The lowest line (green) is for the classical rectangular cross-section.

Image of FIG. 13.
FIG. 13.

Circle cross-section .

Image of FIG. 14.
FIG. 14.

Analytical slumping results for circle segment , 0 ⩽ and = 20: nose height and nose speed as functions of 1/.

Image of FIG. 15.
FIG. 15.

Channel with the V-shape valley cross-section, α = 1.

Image of FIG. 16.
FIG. 16.

Analytical slumping results for V-shape configuration with () = α and = 0.5: nose height and nose speed as functions of 1/. α = 0.5 (lower line, online blue), α = 1 (green), α = 2 (upper line, online red).

Image of FIG. 17.
FIG. 17.

Comparison between analytical similarity solutions (5.1) (dots) and numerical results of the lock-release current (line) for the cross-sections () = α, α = 0.5, 1, 2 (lower, middle and upper lines; online blue, red, and green).

Image of FIG. 18.
FIG. 18.

Slumping speed comparisons: experiments of Marino and Thomas 4 (rectangle symbols) and of Monaghan 3 (dot symbol), present SW prediction (solid line), and the steady-energy-conserving (dashed line). Here = 1, () = α.

Image of FIG. 19.
FIG. 19.

Comparison of the experimental results of Monaghan 3 (symbols) and SW prediction (line) for = 1, () = .

Tables

Generic image for table
Table I.

Similarity solution coefficients for () = α cross-sections.

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/content/aip/journal/pof2/25/2/10.1063/1.4790796
2013-02-22
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Gravity currents in non-rectangular cross-section channels: Analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4790796
10.1063/1.4790796
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