Inertial gravity current in rectangular channels over a porous bottom: Asymptotic solutions

Abstract We consider a high-Reynolds-number Boussinesq gravity current (GC) propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity g ′ ) is released from a rectangular lock (of length x 0 and height h 0 ), and after an adjustment (slumping) stage is expected to enter into a similarity stage, on which we focus here, using a thin-layer shallow-water model. The classical analytical self-similar propagation solution predicts that the length of the current is given by x N ( t ) = K t 2 ∕ 3 (where K is a constant and t is time from release). The height h ( x , t ) and speed u ( x , t ) display a similarity shape of the variable y = x ∕ x N ( t ) , y ∈ [ 0 , 1 ] ( x is the physical coordinate measured from the backwall of the lock). This solution, which is very useful in the analysis of gravity-current problems, is invalidated by the drainage effect into the porous bottom. Here we extend the classical similarity (basic) solutions by developing a perturbation (asymptotic) expansions about the basic solution. The expansion uses the small parameter λ which represents the ratio of the typical propagation time T = x 0 ∕ ( g ′ h 0 ) 1 ∕ 2 to the drainage time t B (a given property of the porous bottom). The perturbation terms can be calculated analytically, and we present the results of the first-order correction. This provides useful insights about the influence of the porous boundary, as compared with the classical similarity behaviour: x N ( t ) is shorter, the profile of u ( y ) is deflected to lower values at the nose, and h ( y ) is reduced mostly at the tail. The deviation from the basic similarity solution increases like λ t . In addition, we show that the drainage influence is important in reducing the transition length from the inertial (inviscid) to the viscous regimes. We compared the analytical asymptotic leading-order solution with numerical finite-difference results for various values of λ , and found excellent qualitative agreement and fair quantitative agreement. We expect that higher-order terms will improve the accuracy of the new solution, but this additional extension was not performed in this work.