The New Approach for Inviscid Gravity Currents
Date Issued
2005
Date
2005
Author(s)
Ruo, An-Cheng
DOI
zh-TW
Abstract
To analyze the motion of gravity currents, a common approach is to solve the hyperbolic shallow water equations (SWE) together with the boundary conditions at both the current source far upstream and the current front at the downstream margin. The use of the front condition aims to take the resistance from the ambient fluid into account, however, which is absent in the SWE. In the present study, we re-derive the SWE by taking the ambient resistance into account, and end up with the so-called modified shallow water equations (MSWE) in which the ambient resistance is accounted for by a nonlinear term, so that the use of the front condition becomes unnecessary. These highly nonlinear equations are approximated by perturbation expansions to the leading order, and the resultant singular perturbation equations are solved by an inner-outer expansion. For constant-flux and constant-volume gravity currents, the outer solutions turn out to be exactly the same as the previous ones obtained by solving SWE with the front condition. The inner solution gives both the profile and the velocity of the current head and also leads to the recovery of the front condition while is in a more general form. The combination of inner and outer solutions gives a composite solution for the whole current, which was called by Benjamin (1968) a “formidably complicated” task. To take the turbulent drag on the current into account, we introduce the semi-empirical Chezy drag term into the MSWE and end up with a result comparable with experimental data. The result implies that the ambient resistance is contributed mainly by the inviscid form drag, and the viscous drag dominates only when the density ratio is small. Furthermore, the MSWE can be extended for three-dimensional viscous currents, while will become more complicate that present analytical approach may not be feasible.
Subjects
重力流
異重流
淺水波方程式
波前條件
形狀阻力
移動性邊界
潰壩
貼合展開法
gravity current
shallow water equations
front condition
form drag
moving boundary
Dam break
method of matched asymptotic expansion
Type
thesis
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