Two-layer shallow-water dam-break solutions for gravity currents in non-rectangular cross-area channels

摘要

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel 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)≤y≤f_2(z) for 0≤z≤H. We develop a two-layer shallow-water (SW) formulation, and we implement it for the solution of the dam-break problem. The dependent variables are the position of the interface, h(x,t), and the velocity (averaged over the area of the current), u(x,t). The non-rectangular cross-section geometry enters the formulation via f(h) and integrals of f(z) and zf(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 of the lock-release problem are: (i) the height ratio H/h_0 of ambient to lock; and (ii) the density ratio of 'light' to 'heavy' fluids, R. We show that the dam-break problem is amenable to solution by the method of characteristics, but various complications (internal jumps, critical restrictions) appear when the return flow in the ambient is significant; these features are not captured by a one-layer shallow-water model. The role of the nonstandard geometry in the detection and calculation of the internal and reflected jumps, and on the dam-break flow field, is elucidated. The methodology is illustrated for Boussinesq flow in typical geometries: power-law (f(z)=bz~a, where b,a are positive constants), trapezoidal and circle (full or sector); we solved for full-depth (H=1) and part-depth (H > 1) lock-release configurations. The approach seems to work well: in all the tested cases, the solution was obtained with simple mathematical and numerical tools (a Runge-Kutta integrator and a secant-method solver); the insights are compatible with our previous understanding of the problem; and the standard (rectangular) case is recovered in the appropriate limit (i.e. when the side-boundaries of the trapezium cross-section are vertical or far apart). A comparison of the speed of propagation with available experiments is also presented, and the agreement is satisfactory. The present solution is a significant generalization of the classical twolayer gravity-current dam-break 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 the new model.