Thin-layer models for gravity currents in channels of general cross-section area,a review

We present a brief review of the recent investigations on gravity currents in horizontal channels with non-rectangular cross-section area (such as triangle, \(\bigvee \)-valley, circle/semi-circle, trapezoid) which occur in nature (e.g., rivers) and constructed environment (tunnels, reservoirs, canals). To be specific, we discuss the propagation of a gravity current (GC) in a horizontal channel along the horizontal coordinate x, with gravity g acting in the \(-z\) direction, and y the horizontal–lateral coordinate. The bottom and top of the channel are at \(z=0,H\). The “standard” problem is concerned with 2D flow in a channel with rectangular (or laterally unbounded) cross-section area (CSA). Recent investigations have successfully extended the standard knowledge to the channels of CSA given by the quite general \(-f_1(z)\le y \le f_2(z)\) for \(0 \le z \le H\). This includes the practical \(\bigvee \)-valley, triangle, circle/semi-circle and trapezoid; these geometries may be in “up” or “down” setting with respect to gravity, e.g., \(\bigtriangleup \) and \(\bigtriangledown \). The major objective of the extended theory is to predict the height of the interface \(z=h(x,t)\) and the velocity (averaged over the CSA) u(x, t), where t is time; the prediction includes the speed and position of the nose \(u_N(t), x_N(t)\). We show that the motion is governed by a set of simplified equations, called “model,” that provides versatile and insightful solutions and trends. The emphasis in on a high-Reynolds-number current whose motion is dominated by buoyancy–inertia balance; in particular a GC released from a lock, which also contains general effects such as front and internal jumps (shocks), and reflected bore. We discuss two-layer, one-layer, and box models; Boussinesq and non-Boussinesq systems; compositional and particle-driven cases; and the effect of stratification of the ambient fluid. The models are self-contained, and admit realistic initial and boundary conditions. The governing equations are amenable to analytical solutions in some special circumstances. Some salient features of the buoyancy-viscous regime, and the estimate for the length at which transition to this regime takes place, are also presented. Some experimental support to the theory, and open questions for further investigations, are also mentioned. The major conclusions are (1) The CSA geometry has significant influence on the motion of the GC; and (2) The new theory is a useful, very significant, extension of the standard two-dimensional GC problem. The standard current is just a particular case, \(f_{1,2} =\) constants, among many other covered by the new theory.