The Buoyancy–Drag model is a simple model, based on ordinary differential equations, for estimating the growth of a turbulent mixing zone at an interface between fluids of different density due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy–Drag model are needed to obtain correct results. In a recent paper, Youngs & Thornber (2019), the results of three-dimensional turbulent mixing simulations were used to construct the modifications required to represent the evolution of the overall width of the mixing zone due to single-shock Richtmyer–Meshkov mixing evolving from narrowband initial random perturbations. The present paper extends this analysis to give separate equations for the bubble and spike distances (the depths to which the mixing zone penetrates the dense and light fluids). The data analysis depends on novel integral definitions of the bubble and spike distances which vary smoothly with time. Results are presented for two pre-shock density ratios, ${\rho }_1∕{\rho }_2=3\mathrm{and}20$. New insights are given for the variation of asymmetry of the mixing zone with time. At early time, values of the spike-to-bubble distance are very high. The asymmetry greatly reduces as mixing proceeds towards a self-similar state. For the overall (integral) mixing width, W, the Buoyancy–Drag model gives satisfactory results at both density ratios using the same parameters. However, for the bubble and spike distances the behaviour is very different at the two density ratios. The method used to analyse the data provides a new way of estimating the self-similar growth exponent $\theta$ ($W\sim t^{\theta }$). The values obtained are approximate because of the difficulty in running the three dimensional simulations far enough into the self-similar regime. Estimates of $\theta$ are consistent with the theoretical value of 1/3 given by the model of Elbaz & Shvarts (2018). The corrected form of the Buoyancy–Drag model gives accurate fits to the data for $W,h_b\mathrm{and}{h}_s$over the whole time range with $\theta$ =1/3 for ${\rho }_1∕{\rho }_2=3$ and $\theta =0.35$ for ${\rho }_1∕{\rho }_2=20$.

浮力-阻力模型是基于常微分方程的简单模型，用于估计由于Richtmyer-Meshkov和Rayleigh-Taylor不稳定性而在不同密度的流体之间的界面处湍流混合区的增长。混合过程的早期非常依赖于初始条件，需要对浮力-阻力模型进行修改才能获得正确的结果。在Youngs＆Thornber（2019）的最新论文中，使用了三维湍流混合模拟的结果来构造所需的修改，以表示由于单冲击Richtmyer-Meshkov混合演变而引起的混合区总宽度的演变来自窄带初始随机扰动。本文扩展了这一分析，以给出气泡和峰值距离（混合区穿透稠密和轻质流体的深度）的单独方程式。数据分析取决于气泡和尖峰距离的新颖积分定义，它们随时间平滑变化。给出了两个震前密度比的结果，${\rho }_{\mathrm{1个}}∕{\rho }_2=3\mathrm{和}20$。对于混合区的不对称性随时间的变化给出了新的见解。在早期，尖峰到气泡距离的值非常高。随着混合向自相似状态发展，不对称性大大降低。对于整体（整体）混合宽度W，使用相同参数的两种密度比率下的浮力-阻力模型均给出令人满意的结果。但是，对于气泡距离和尖峰距离，两种密度比率下的行为有很大不同。用于分析数据的方法提供了一种估算自相似增长指数的新方法$\theta$ （$\mathrm{w\; ^}〜{Ť}^{\theta }$）。所获得的值是近似值，因为很难将三维模拟运行到足够接近自相似状态的程度。估计$\theta$与Elbaz＆Shvarts（2018）模型给出的理论值1/3一致。浮力-阻力模型的校正形式可以准确拟合以下数据$\mathrm{w\; ^}，H_b\mathrm{和}{H}_s$在整个时间范围内 $\theta$ = 1/3 ${\rho }_{\mathrm{1个}}∕{\rho }_2=3$ 和 $\theta =0。35$ 对于 ${\rho }_{\mathrm{1个}}∕{\rho }_2=20$。