Schematic description of typical channels with non-rectangular cross-section.
Schematic description of the gravity current system. (a) Side view. (b) Cross-section of channel. Here f(z) is the width of the channel. In the analysis A a denotes the area occupied by the ambient fluid, A c = F(x, t) is the area occupied by the current, and A T is the total area.
Sketch of the power-law cross-section f(z) = z α for α = 0.5, 1, 2 (red, blue, and green). Here H = 2.
Profiles of h and u vs. x for various values of t. H = 20, f(z) = z 1/2 for t= 2(2)20.
As above, for f(z) = z.
As in Fig. 4 , for f(z) = z 2.
Numerical results: behavior of the nose speed u N vs. t for f(z) = z and various values of H: 1.1, 2, and 20. Note the constant speed during the initial (slumping) stage.
Sketch of curved-trapezoidal cross-sections f(z) = (d + z α)/(d + 1) for α = 0.5, 1, 2 (red, blue, and green). Here d = 1, H = 2.
Profiles of h and u vs. t for various values of t. H = 20, f(z) = (1 + z)/2 for t= 2(2)20.
View from above of the current in trapezoidal cross-section f(z) = (1 + z)/2 channel, for H = 2, for t = 2(2)20.
Analytical results for the slumping stage of the current for f(z) = z α: Nose height and speed as functions of 1/H. 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).
Analytical slumping speed for trapezoidal f(z) = (d + z)/(d + 1), where d = 0 (upper line, red online), d = 1 (blue online), d = 5 (black). The lowest line (green) is for the classical rectangular cross-section.
Circle cross-section .
Analytical slumping results for circle segment , 0 ⩽ z ⩽ H and R = 20: nose height and nose speed as functions of 1/H.
Channel with the V-shape valley cross-section, α = 1.
Analytical slumping results for V-shape configuration with f(z) = z α and Z 0 = 0.5: nose height and nose speed as functions of 1/H. α = 0.5 (lower line, online blue), α = 1 (green), α = 2 (upper line, online red).
Comparison between analytical similarity solutions (5.1) (dots) and numerical results of the lock-release current (line) for the cross-sections f(z) = z α, α = 0.5, 1, 2 (lower, middle and upper lines; online blue, red, and green).
Comparison of the experimental results of Monaghan et al. 3 (symbols) and SW prediction (line) for H = 1, f(z) = z.
Similarity solution coefficients for f(z) = z α cross-sections.
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