NDLI: Shallow-water solutions for gravity currents in non-rectangular cross-area channels with stratified ambient

Content Provider	Springer Nature Link
Author	Ungarish, Marius
Copyright Year	2014
Abstract	We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $$x$$ . The current is of constant density, $$\rho _c$$ , and the ambient has a linear stable stratification, from $$\rho _{b}$$ at the bottom $$z=0$$ to $$\rho _{o}$$ at the top $$z=H$$ . The cross-section of the channel is given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . We develop a one-layer shallow-water (SW) formulation, and we implement it for the solution of a gravity current of fixed volume released from a lock (we focus on a “heavy” bottom current with $$\rho _{c}\ge \rho _{b}$$ ). The dependent variables are the position of the interface, $$h(x,t)$$ , and the speed (averaged over the area of the current), $$u(x,t)$$ , where $$t$$ is the time. The non-rectangular cross-section geometry enters the formulation via $$f(h)$$ and integrals of $$f(z), z f(z)$$ and $$z^2 f(z)$$ , where $$f(z) = f_1(z) + f_2(z)$$ is the width of the channel. For a given geometry $$f(z)$$ , the free input parameters are (1) the height ratio $$H/h_0$$ of ambient to lock; and (2) the stratification parameter $$S = (\rho _{b}- \rho _{o})/(\rho _{c}- \rho _{o})$$ . In general, the SW equations of motion are a hyperbolic PDE system which we solve by a finite-difference method. However, we show that the initial motion displays a “slumping” stage with constant speed of propagation; this can be calculated exactly by the method of characteristics. An analytical solution for the long-time self-similar propagation is also presented for $$S=1$$ and the power-law $$f(z) = bz^\alpha $$ cross section profile. Solutions are presented for various stratification and typical geometries: power-law ( $$f(z) = b z^\alpha $$ , where $$b, \alpha $$ are positive constants), power-law B ( $$f(z) = b( H-z)^\alpha $$ ), trapezoidal, and circle. In general, the increase of the stratification parameter $$S$$ causes a reduction of the speed of propagation, but the details depend on the geometry of the cross section. We also show that, upon a simple transformation, the solutions for the bottom (“heavy”) current can be used for the prediction of top (“light”) currents The present solution is a significant generalization of the classical analysis of the gravity current in a stratified ambient problem (including the “intrusion” situation when $$S=1$$ ). The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $$f(z) = $$ const., in the wide domain of cross-sections covered by the new model. The present solution is reduced to the non-stratified (homogeneous) ambient counterpart by setting $$S=0$$ . The flow under consideration is relevant to environmental and geophysical applications like ventilation of tunnels and discharges in ducts and rivers.
Starting Page	793
Ending Page	820
Page Count	28
File Format	PDF
ISSN	15677419
Journal	Environmental Fluid Mechanics
Volume Number	15
Issue Number	4
e-ISSN	15731510
Language	English
Publisher	Springer Netherlands
Publisher Date	2014-08-22
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Gravity current Intrusion Nonrectangular cross-section Stratified Earth Sciences Environmental Physics Hydrology/Water Resources Mechanics Hydrogeology Oceanography
Content Type	Text
Resource Type	Article
Subject	Environmental Chemistry Water Science and Technology

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