We consider an inertial (large Reynolds number) gravity current (GC) released from a lock of length \(x_0\) and height \(h_0\) into an ambient fluid of height \(Hh_0\) which propagates along a channel with permeable floor and/or walls. The cross section (CS) of the container is defined by the general \(-f_1 (z) \le y \le f_2 (z)\) for \(0 \le z \le Hh_0\), while the top and the bottom are at \(z = Hh_0\) and \(z=0\) (z is the physical coordinate measured from the bottom of the channel). We use Boussinesq assumption and formulate the shallow-water equations of motion. These equations include a parameter \(\lambda\) which reflects the ratio between the typical time of propagation of the current over a length \(x_0\) to the typical time of the drop of the interface due to porous boundary over a height \(h_0\). To solve the problem we employ the method of finite-differences. It is demonstrated that initially, the effect of a porosity of the floor or the walls on the distance of propagation of the GC is insignificant, but the “slumping” stage is absent. During the advanced stages, the porosity slows down the current and decreasing the effective distance of propagation. We illustrate the methodology for flow in symmetric channels with typical power-law CSs \(f_1(z) = f_2(z) = 0.5(c+bz^{\alpha })\), where \(b,c, \alpha\) are non-negative constants and show that for the identical characteristics of the porous material, the presence of the porous floor has more significant effect on the velocity of the propagation of the current than the presence of the permeable walls in all tested forms of containers. Special attention is given to triangular CS channels \(f_{1}(z)=f_{2}(z) = bz\) with permeable walls. We show that in such containers the volume of the current decreases like \(\exp \left( -\sqrt{1+\frac{1}{b^2}}\lambda t\right)\) (where t is time) and for \(Hh_0 \gg 1\) its run-out length can be expressed by a compact analytical formula.

我们考虑的惯性（大雷诺数）重力电流（GC）从长度的锁定解除\（X_0 \）和高度\（H_0 \）到高度的环境流体\（Hh_0 \） ，其传播连同可渗透的信道地板和/或墙壁。该容器的横截面（CS）的一般定义\（ - F_1（Z）\文件Y \文件F_2（Z）\）为\（0 \文件ž\文件Hh_0 \），而顶部和底部位于\（z = Hh_0 \）和\（z = 0 \）（z是从通道底部测量的物理坐标）。我们使用Boussinesq假设并制定了浅水运动方程。这些方程式包含参数\（\ lambda \）它反映了电流在长度\（x_0 \）上的典型传播时间与在高度\（h_0 \）上由于多孔边界导致的界面下降的典型时间之间的比率。为了解决这个问题，我们采用了有限差分法。已经证明，最初，地板或墙壁的孔隙率对GC传播距离的影响不明显，但是没有“塌陷”阶段。在进阶阶段，孔隙度会减慢电流并降低有效传播距离。我们举例说明了典型幂律CS为\（f_1（z）= f_2（z）= 0.5（c + bz ^ {\ alpha}）\）的对称通道中的流动方法，其中\（b，c，\ alpha \）是非负常数，并且表明对于多孔材料的相同特性，在所有测试形式的容器中，多孔底板的存在比可渗透​​壁的存在对电流传播速度的影响更大。特别注意具有可渗透壁的三角形CS通道\（f_ {1}（z）= f_ {2}（z）= bz \）。我们证明了在这样的容器中，电流的大小会像\（\ exp \ left（-\ sqrt {1+ \ frac {1} {b ^ 2}} \ lambda t \ right）\）一样减少 （其中t是时间），对于\（Hh_0 \ gg 1 \），其跳动长度可以用一个紧凑的解析公式表示。