NDLI: Gravity currents in non-rectangular cross-section channels: extensions to non-Boussinesq systems

Content Provider	Springer Nature Link
Author	Zemach, T.
Copyright Year	2015
Abstract	We consider the propagation of a gravity current of density $$\rho _c$$ from a lock length $$x_0$$ and height $$h_0$$ into an ambient fluid of density $$\rho _a$$ in a horizontal channel of height $$H$$ along the horizontal coordinate x. The bottom and top of the channel are at $$z = 0, H$$ , and the cross-section is given by the quite general $$f_{1}(z)\le y \le f_{2}(z)$$ for $$0 \le z \le H$$ . When the Reynolds number is large, the resulting flow is governed by the parameters $$\rho _{c}/\rho _{a}$$ , $$H^{*} = H/h_{0}$$ and $$f(z)=f_{1}(z)+f_{2}(z)$$ . We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness $$h$$ and speed $$u$$ of the current. The results cover in a continuous manner the range of light $$\rho _{c}/\rho _{a} \ll 1$$ , Boussinesq $$\rho _{c}/\rho _{a} \approx 1$$ and heavy $$\rho _{c}/\rho _{a} \gg 1$$ currents in a fairly wide range of depth ratio in various cross-section geometries. We obtain analytical solutions for the initial dam-break stage of propagation with constant speed, which appears for any cross-section geometry, and derive explicitly the trend for small and large values of the governing parameters. For large time, $$t$$ , a self-similar propagation is feasible only for heavy currents in channels with $$f(z)=bz^{\alpha }$$ cross-sections or for light currents in channels of $$f(z) = (H-z)^\alpha $$ , with $$t^{(2+2\alpha )/(3+2\alpha )}$$ . The methodology is illustrated for non-Boussinesq flow in typical geometries: power-law ( $$f(z) = z^{\alpha }$$ and $$f(z) = (H-z)^{\alpha }$$ , where $$\alpha $$ is positive constant), trapezoidal and circle (full or sector). The present approach is a significant generalization of the classical non-Boussinesq gravity current problem. 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 this new model.
Starting Page	975
Ending Page	998
Page Count	24
File Format	PDF
ISSN	15677419
Journal	Environmental Fluid Mechanics
Volume Number	15
Issue Number	5
e-ISSN	15731510
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-01-04
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Gravity currents Non-Boussinesq Non-rectangular cross-section 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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