Particle-driven gravity currents in non-rectangular cross section channels

Abstract

We consider a high-Reynolds-number gravity current generated by suspension of heavier particles in fluid of density ρi, propagating along a channel into an ambient fluid of the density ρa. The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. The flow is modeled by the one-layer shallow-water equations obtained for the time-dependent motion which is produced by release from rest of a fixed volume of mixture from a lock. We solve the problem by the finite-difference numerical code to present typical height h(x, t), velocity u(x, t), and volume fraction of particles (concentration) ϕ(x, t) profiles. The methodology is illustrated for flow in typical geometries: power-law (f(z) = zα and f(z) = (H − z)α, where α is positive constant), trapezoidal, and circle. In general, the speed of propagation of the flows driven by suspensions decreases compared with those driven by a reduced gravity in homogeneous currents. However, the details depend on the geometry of the cross section. The runout length of suspensions in channels of power-law cross sections is analytically predicted using a simplified depth-averaged “box” model. The present approach is a significant generalization of the classical gravity current problem. The classical formulation for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.

Bidisperse Particle-Driven Gravity Currents in Power-Law Cross-Section Channels

“…As for the homogeneous and monodispersed currents propagating in channels of power-law cross-section (see Zemach & Ungarish [13], Zemach [14]), either here we can distinguish between three main stages of propagation.…”

“…This theory was verified by comparison between the SW and experimental results and a good agreement was found between the theory and the data for a V-shaped valley f (z) = z (Ungarish et al [12]). For 0 < β (1,2) ≪ 1 and for a given cross-section-width function f (z) = z α , the model becomes identical to the SW theory of monodisperse gravity currents propagating in the non-rectangular cross-section channels (Zemach [14]). This theory was supported by the experimental data and a good agreement was found between the theory and the data for a V-shaped valley f (z) = z (Mériaux et al [6]).…”

“…The present formulation is new and is an extension of the work of Zemach [14] for bidisperse particle-driven gravity currents: the problem is formulated in terms of the height h of the interface, speed u (averaged over the area of the current) and scaled volume fraction variables φ 1 and φ 2 as functions of x and t (time). The dimensionless variables and the equations of motion are modified due to φ 1 and φ 2 and additional diffusion equation is formulated.…”

On the propagation of particulate gravity currents in circular and semi-circular channels partially filled with homogeneous or stratified ambient fluid

“…Two configurations are of interest: (i) the homogeneous, with the ambient fluid of a constant density ρ a , and (ii) the stratified ambient. The formulation presented below for the homogeneous ambient fluid is based on the work of Zemach, 44 while the model for the stratified ambient fluid developed here is new. The modeling methodology has been detailed in previous papers; see the work of Ungarish 40 for a comprehensive review.…”

“…Refined two-layer models of saline GCs accounting for the channel shape were developed by Ungarish, 39 obtaining satisfactory agreement with the experiments by Monaghan et al 27 and by Marino and Thomas. 20 Further theoretical and experimental analyses have been developed by Ungarish et al 42 and Longo et al 18,19 For TCs, Monaghan et al 26 performed lock-exchange experiments in a V-shaped channel; Zemach 44 developed a one-layer theory valid for propagation within a generic non-rectangular cross section and performed numerical simulations to illustrate the results obtained; Mériaux et al 24 performed experiments in V-shaped channels and compared them successfully with the general theory; Mériaux and Kurz-Besson 23 theoretically and experimentally studied currents carrying polydisperse particles along a V-shaped valley and found a criterion of equivalence with monogranular TCs: the mass-weighted mean size of the initial distribution of particles is representative of the suspension (also in terms of runout length) provided a sufficient number of size classes are considered.…”