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

Abstract

We consider a high-Reynolds-number Boussinesq gravity current propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity g′) is released from a rectangular lock (of length x0 and height h0), 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 xN(t)=Kt2∕3 (where K is a constant and t is time from release). The height h(x,t) and velocity u(x,t) display a similarity shape of the variable y=x∕xN(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=x0∕(g′h0)1∕2 to the drainage time tB (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 behavior: xN(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 λt. 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.