Asymptotic similarity solutions for particle-driven gravity currents in non-rectangular cross-section channels of power-law form

Abstract

Abstract We consider a high-Reynolds particle-driven gravity current (GC) propagating in channel above a horizontal boundary. The bottom and the top of the channel are at z = 0 , H and its cross-section is given by − f ( z ) ≤ y ≤ f ( z ) for 0 ≤ z ≤ H , where f ( z ) is a power-law function f ( z ) = z α . We focus attention on the similarity stage of propagation of the current using a shallow-water (SW) model. While it is possible to derive the analytical similarity solutions for the homogeneous GC (for which the density of the current remains constant during the propagation), such similarity-solutions do not exist for the particle-driven GCs. We extend the similarity solutions of homogeneous GC to particle-driven currents by developing an asymptotic expansions and derive approximations of such solutions for channels of typical rectangular, triangular and parabolic forms. Comparison with the numerical solution of the SW equations shows that the leading-order asymptotic results are a good approximation in the domain of expected validity. However, out of bounds of this domain the numerical solutions systematically depart from the first-order asymptotic solutions, suggesting the need to include higher-order terms in the asymptotic series.