We consider a high-Reynolds-number Boussinesq gravity current propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity $g^{\prime }$) 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 $x_N\left(t\right)=K{t}^{2∕3}$ (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 $y=x∕x_N\left(t\right),y\in [0,1]$ ($x$ 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 $\lambda$ which represents the ratio of the typical propagation time $T=x_0∕{\left(g^{\prime }{h}_0\right)}^{1∕2}$ 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 behavior: $x_N\left(t\right)$ is shorter, the profile of $u\left(y\right)$ is deflected to lower values at the nose, and $h\left(y\right)$ is reduced mostly at the tail. The deviation from the basic similarity solution increases like $\lambda t$. 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 $\lambda$, 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.

我们考虑在可渗透水平边界上方的通道中传播的高雷诺数Boussinesq重力流。电流（降低的重力$G^{\prime }$）从矩形锁（长度为 $X_0$ 和高度 $H_0$），并且预计在调整（下降）阶段后将进入相似阶段，我们将在此处着重使用薄层浅水模型。经典的分析自相似传播解决方案预测，电流的长度由下式给出：$X_{ñ}（Ť）=ķ{Ť}^{2∕3}$ （其中K是一个常数， $Ť$是发布的时间）。高度$H（X，Ť）$ 和速度 $ü（X，Ť）$ 显示变量的相似形状 $ÿ=X∕X_{ñ}（Ť），ÿ\in [0，\mathrm{1个}]$ （$X$是从锁的后壁测量的物理坐标）。该解决方案在分析重力流问题中非常有用，但由于进入多孔底部的排水作用而无效。在这里，我们通过开发关于基本解的扰动（渐近）展开来扩展经典相似性（基本）解。扩展使用小参数$\lambda$ 代表典型传播时间的比率 $Ť=X_0∕{（G^{\prime }{H}_0）}^{\mathrm{1个}∕2}$ 排水时间 ${Ť}_{乙}$（多孔底部的给定属性）。扰动项可以解析地计算，并且我们给出一阶校正的结果。与经典相似行为相比，这提供了有关多孔边界影响的有用见解：$X_{ñ}（Ť）$ 较短， $ü（ÿ）$ 偏向鼻子的较低位置，并且 $H（ÿ）$减少大部分在尾巴。与基本相似性解决方案的偏差增加，例如$\lambda Ť$。此外，我们表明，排水影响对于减小从惯性（无粘性）到粘性状态的过渡长度很重要。我们比较了解析渐近前导解和数值有限差分结果的各种数值$\lambda$，并找到了出色的定性协议和公平的定量协议。我们希望高阶术语会提高新解决方案的准确性，但是这项工作并未执行此额外的扩展。