In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.

中文翻译：

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接近相似时间

在最近的一篇文章中，Ball and Huppert（J. Fluid Mech。，874，2019）引入了一种新颖的方法来确定特征时间尺度，从给定的初始条件开始，通过数值积分获得与给定时间相关的非线性微分方程的相似解收敛到实际解的特征时标。在本文中，我们将此方法应用于描述固定体积传播重力流的一系列不同偏微分方程，并修改技术以适用于数值解收敛于振荡的情况，适用于在雷诺数大。我们研究所有这些情况下的收敛特性，包括不同的初始几何形状如何影响两种解决方案的一致率。注意，流动不再是单向的几何形状花费更长的时间收敛。引入了一种时移相似度解决方案的方法，以提高相似度解决方案给出的近似值的准确性，并且还提供了所有时间内百分比不一致的上限。