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摘要: 固体界面的不稳定性增长是武器内爆物理、惯性约束聚变点火和材料动力学行为研究等工程和科学领域中的关键物理过程, 它可能导致固体表面出现轻重介质相互侵入、物质微喷射乃至湍流混合等复杂的物理现象, 对其发展规律的研究具有重要的意义. 本文在第二、三章分别介绍了对固体界面Richtmyer−Meshkov (RM) 和Rayleigh−Taylor (RT) 不稳定性发展规律的研究现状, 总结了现有的不稳定性增长理论模型并探讨了其局限性. 与纯流体界面不稳定性增长所不同的是, 由于固体材料存在强度, 其界面不稳定性发展过程中扰动增长的部分能量会被强度这一耗散因素转化为晶格的热能, 从而降低扰动增长速率乃至抑制扰动增长, 因此如何衡量动态加载过程中固体的强度效应成为认识其不稳定增长规律的关键. 另一方面, 由于固体界面不稳定性的发展结果蕴含着材料的本构、状态方程等信息, 人们提出可以应用不稳定性的增长规律来获取材料的动态屈服强度、校验材料的高压本构模型和抑制不稳定性的增长, 本文在第四章中对这部分内容进行了介绍, 并指出建立能精确描述不稳定性增长现象与材料性质间“关联性机制”的理论模型是成功进行应用的关键. 在这些基础上, 本文于第五章展望了该领域存在的机遇和挑战.Abstract: The growth of interface instability in solids is a critical phenomenon affecting various fields of engineering and science, including implosion physics, inertial confinement fusion ignition, and the dynamic behavior of materials. This instability can lead to complex phenomena such as the interpenetration of light and heavy media at solid surfaces, material micro-jetting, and turbulent mixing, highlighting the significance of understanding its underlying mechanisms. This paper reviews the current research status of Richtmyer−Meshkov (RM) and Rayleigh−Taylor (RT) instabilities at solid interfaces in Chapters 2 and 3. We summarize existing theoretical models of instability growth and discuss their limitations. Unlike the instability growth observed in pure fluid interfaces, solid materials possess inherent strength, which enables some of the energy from perturbation growth to be transformed into lattice thermal energy through dissipative mechanisms. This energy conversion reduces the rate of perturbation growth and may even suppress the development of instabilities. Consequently, understanding the effects of material strength under dynamic loading conditions is crucial for comprehending instability growth behavior. Moreover, the outcomes of solid interface instability are indicative of various material properties, including constitutive relationships and equations of state. Researchers have proposed that instability growth can be leveraged to determine the dynamic yield strength of materials, validate high-pressure constitutive models, and mitigate instability growth. Chapter 4 focuses on this aspect, emphasizing the need to establish a theoretical model that accurately describes the “correlation mechanism” between instability phenomena and material properties for effective applications. Building on these foundations, Chapter 5 explores future opportunities and challenges in this field.
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图 1 典型的流动不稳定性现象. (a) 平面激波作用下空气/SF6界面的演化(Collins & Jacobs 2002), (b) 水杯中轻重流体间的混合, (c) 火箭发动机推进剂喷注与燃烧, (d) 大气中环流的云涡
图 3 高爆炸药 (HE) 产生的平面冲击波加载到初始有正弦型扰动的样品前 (a) 和加载后 (b) 界面的几何构型示意图 (Buttler et al. 2012)
图 4 冲击波加载到两个理想弹性固体分界面后, (a) 界面无量纲扰动周期T/T0随材料密度比ρ2/ρ1变化,不同符号代表不同材料性质组合, 实线代表式 (5) 的理论预测 (α = 1.55); (b) 振荡振幅η (圆圈)、v0T/2π (三角形)、最大振荡速度 (正方形) 随振荡周期T的变化, 实线代表经典情况 (G = 0) 时扰动速度随T的演化
图 5 理想弹性固体在冲击加载后界面振荡周期T/T0随材料密度比ρ2/ρ1变化. 实线代表数值结果, 长虚线代表式 (8) 的理论预测, 短虚线代表式 (5) 理论预测, 点划线通过在复平面内定位极点计算极点频率得到
图 6 PAGOSA模拟得到的Cu界面RM扰动增长结果. (a) 界面随时间演化图, 黄色为Cu材料, 黑色为气体, 从上至下四组图像的初始扰动分别为kη0 = 0.125, 0.18, 0.22和0.4; (b) 无量纲的尖钉极值振幅 (实心符号)、气泡极值振幅 (空心符号) 随无量纲尖钉极值增长速度的变化, 虚线代表CD = 0.24时式 (13) 理论预测
图 7 (a) 真空与弹塑性固体界面RM扰动增长问题初始构型示意图; (b) 模拟得到的尖钉极值振幅随极值增长速率变化图, 实线代表Cp = 0.29时式 (11) 理论预测
图 12 PBX9501炸药爆轰驱动Cu靶后 (a) 殷建伟模拟得到的尖钉相对自由面速度随时间发展结果, (b) Prime模拟 (实线) 和Piriz 理论预测 (虚线) 得到的尖钉绝对速度随时间发展结果
图 13 炸药爆轰加载下柱面金属的RM扰动增长研究. (a) 实验和计算构型示意图, (b) 钢壳内爆过程中的实验结果照相图像, (c) 扰动振幅随时间发展的数值和实验结果对比图, (d) 钢壳内爆过程中数值模拟图像
图 15 基于PHELIX平台开展的磁驱柱面RM扰动增长研究. (a) 实验构型示意图, (b) 锡靶内爆过程中质子照相图像, 其内界面预制有不同的初始扰动 (kη0 = 0.1, 0.2, 0.3), (c) 锡靶内爆过程中扰动振幅随时间的变化, 实线代表Flag程序模拟结果, 实心点代表质子照相实验结果, (d) Flag程序模拟得到的锡靶界面扰动增长图像 (kη0 = 0.3)
图 17 炸药爆轰加载钢体球壳后的界面扰动增长结果. (a) 实验回收得到的样品照相图像, 上方为外层球壳, 下方为内层球壳, h0代表球壳的初始厚度; (b) 钢体球壳在两个不同时刻的伽玛断层成像结果, 上方为t = 2 μs时刻, 下方为内半径汇聚到20 mm的时刻; (c) 数值模拟得到的1/8球壳随时间演化示意图, 其中t = 0 μs图像代表初始计算构型示意图; (d) 球壳内界面速度随时间演化图, 其中黑线代表实验结果, 红线和蓝线分别代表采用Glushak和Mises弹塑性本构模型的计算结果
图 18 物质界面处RT扰动发展示意图. (a) 物质界面处RT扰动发展几何构型示意图, (b) PBX-9404炸药爆轰加载Al板的实验装置示意图, (c) Al板界面RT不稳定性发展过程示意图
图 19 弹性平板RTI增长临界波长与平板厚度比值λc/h随
$ \sqrt{{G}/{{p}}_{0}} $ 变化. 其中实线代表Plohr和Sharp (1998) 理论结果, 两条虚线分别代表Miles (1966) 和Lebedev (Lebedev et al. 1993) 理论预测, 图中数据点为Swegle和Robinson (1989) 对铝 (棱形) 和钨 (圆圈) 材料的数值模拟结果, 误差符号代表Dimonte等(1998)的实验结果
图 20 弹性固体/固体界面RT扰动增长过程中 (a) 无量纲扰动振幅
$ {\mathrm{\eta }}_{\pm }/{\mathrm{\eta }}_{0} $ 随无量纲时间$ \left|{\gamma }_{{\mathrm{e}}\pm }\right| $ t的演化, 实线代表式 (49) 理论预测; (b) 对于不稳定扰动增长情况, 渐进无量纲振幅随无量纲扰动波数的变化, 实线和实心点分别代表Piriz (Piriz et al. 2005)、Plohr和Sharp (1998) 理论预测,
图 21 弹塑性厚平板与理想流体界面的RT扰动发展过程中 (A = 1, Kh>>1), (a) 临界发生弹塑性转变和不稳定增长时, 无量纲初始扰动振幅
$ \widehat{\mathrm{\xi }} $ 随无量纲波长$ \widehat{\mathrm{\lambda }} $ 的变化. (b) 不同初始扰动条件下, 无量纲扰动振幅z随无量纲时间τ的演化
图 23 (a) 电磁加载“铝飞层—聚乙烯夹层—铜靶”三层套筒结构后产生的RTI现象. 前两幅图分别为5 μs和8.5 μs时的数值模拟结果, 后两幅图为X光照相结果, (b) 电磁直接加载外表面预制有扰动的套筒X光照相结果
图 24 基于FP-1平台的磁驱柱面RTI实验结果与计算结果对比, (a) 套筒内、外界面位置随时间变化, (b) 界面扰动振幅随时间变化, 其中虚线代表不含强度效应的理论预测 (公式), 实线为考虑固体强度效应后的理论预测 (公式)
图 26 气炮驱动Cu飞片撞击Ce样品后的 (a) X光照相图像, F0代表初始静态图像, F1到F5代表时间间隔为153.4 ns的动态图像, (b) 尖钉和自由面的速度剖面, 黑线代表实验PDV测速结果, 蓝线代表FLAG程序数值模拟结果 (Y = 0.13 GPa)
图 27 采用FLAG程序模拟Cu飞片撞击Ce样品得到的界面扰动图像 (t = 950 ns), 其中Ce样品采用理想弹塑性本构模型. (a) 初始计算构型示意图, (b) Y = 0.26 GPa计算结果, (c) Y = 0.13 GPa计算结果, (d) Y = 0.065 GPa计算结果
图 28 PBX9501炸药爆轰加载Cu样品后的RM扰动增长结果. (a) 采用FLAG程序模拟得到的尖钉速度随时间变化结果 (kη0 = 0.39), 不同线型代表模拟中对Cu材料预设不同的屈服强度 (Y = 0.4 ~1.6 GPa), (b) 当取Y = 0.53 GPa, 尖钉极值增长速度随初始扰动变化的模拟 (实线) 与实验结果 (红点) 对比图
图 29 (a) 基于Omega激光实验得到的6061-T6铝合金界面RT扰动增长因子随时间变化, 五个黑色圆点代表不同时刻的实验结果, 不同线型代表模拟中对材料预设不同屈服强度 (Y = 0 ~ 1.2 GPa) 的计算结果; (b) 基于VNIIEF柱面内爆实验得到的Cu界面RT扰动增长因子随时间变化, 红色圆圈代表不同时刻的实验结果, 绿色虚线和黑色实线代表分别代表取Y = 0.15和0.18 GPa时的数值模拟结果
图 30 气炮驱动 Cu界面RM扰动增长后的尖钉位移随时间演化结果, 符号代表实验结果, 线型代表对Cu材料采用不同本构模型时的数值模拟结果. (a) 采用理想弹塑性本构模型, Y = 0.1, 0.5, 1 GPa时的数值与实验结果对比图; (b) 采用ZA、JC、PTW、SGL本构模型时的数值与实验结果对比图
图 31 (a) Omega激光器准等熵驱动钒样品RT不稳定增长实验装置示意图; (b) 扰动增长因子随时间的变化, 其中方框代表Omega实验数据, 从上到下的棕色、绿色、蓝色、紫色虚线分别代表采用理想流体、SG、PTW和多尺度本构模型进行二维数值模拟的计算结果, 绿色和蓝色实线分别代表调整原始输入参数后的SG和PTW模型预测结果
图 32 (a) 爆轰加载后金属Sn样品的熔化破碎密度图像 (陈永涛等 2013); (b) 磁驱动柱面Sn样品汇聚过程中的组分分布 (t = 11.42 μs), 红色为液相, 绿色为伽马相, 蓝色为β相 (Rousculp et al. 2016)
图 34 对数坐标系下三种材料的
$\bar{\varepsilon}_p^{\max }$ 随$ k\eta _{sp}^{{\text{max}}} $ 的变化及定标关系. (a) Cu, (b) Al, (c) 钢, 图中紫色实线代表$ \overline \varepsilon _{_{\text{p}}}^{{\text{max}}} = 1.08{({\text{k}}\eta _{sp}^{\max })^{0.65425}} $ 表 1 固体界面RM扰动增长理论模型总结
理论模型 无量纲尖钉极值振幅与无量纲尖钉极值增长速度关系 斜率(拟合系数) 截距 Piriz 2008 $ {{k}}\eta _{{\text{sp}}}^{\max } = {{k}}{\eta _0} + {C_{\mathrm{p}}}\dfrac{{{\rho _0}\mathop {\eta _0^2}\limits^ \cdot }}{Y} $ $ {C_{\mathrm{p}}} = 0.29 $ $ {{k}}{\eta _0} $ Dimonte 2011 $ {{k}}\eta _{{\text{sp}}}^{\max } = 0.08 + {C_{\mathrm{D}}}\dfrac{{{\rho _0}{{{{(v}}_{{\mathrm{sp}}}^{\max })}^2}}}{Y} $ $ {C_{\mathrm{D}}} = 0.24 $ 0.08 Buttler 2012 $ {k}{\eta }_{\text{sp}}^{\mathrm{max}}={k}{\eta }_{0} + {C}_{{\mathrm{B}}} \left(\dfrac{1{-}\Delta{V/(2D)}}{1-\Delta {V/D}}\right)^{2}\dfrac{{\rho }_{\text{0}}{{\eta }_{0}^{2}}}{Y} $ $ {C_{\mathrm{B}}} = 0.29 $ $ {{k}}{\eta _0} $ Mikaelian 2013 $ {{k}}\eta _{{\text{sp}}}^{\max } = {{k}}{\eta _0} + {C_{\mathrm{M}}}\dfrac{{({\rho ^{\text{h}}} + {\rho ^{\text{l}}})\mathop {\eta _0^2}\limits^ \cdot }}{{{Y^{\text{h}}} + {Y^{\text{l}}}}} $ $ {C_{\mathrm{M}}} = 0.33 $ $ {{k}}{\eta _0} $ 殷建伟 2018 $ {{k}}\eta _{{\text{sp}}}^{\max } = {C_{\mathrm{Y}}}\dfrac{{{\rho _0}{{({{v}}_{{\mathrm{sp}}}^{\max })}^2}}}{Y} $ $ {C_{\mathrm{Y}}} = 0.3 $ 0 Chen et al. 2019 $ {{k}}\eta _{{\mathrm{sp}}}^{{\text{max}}} = \dfrac{Y}{{4\sqrt 3 kG}} + {C_{\mathrm{C}}}\dfrac{{{\rho _S} {\eta _0^2} }}{Y} $ $ {C_{\mathrm{C}}} = {{f}}(A) $ $ \dfrac{Y}{{4\sqrt 3 kG}} $ 
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表 2 固体界面 RT 扰动增长理论模型总结
模型 界面运动方程/色散关系 渐进界面扰动增长率 (t$ \to \mathrm{\infty } $) Remington2004: 任意A, 任意kh, 显含应变率 $ \begin{gathered} {\gamma ^2} + 2{k^2}{\nu _{{\mathrm{eff}}}}\gamma + k \times \tanh kh\times \left(\dfrac{{kG}}{\rho } - Ag\right) = 0 \\ {{{v}}_{{\mathrm{eff}}}} = Y/(\sqrt 6 \rho \mathop \varepsilon \limits^ \cdot ) \\ \end{gathered} $ $ \gamma = {\nu _{{\mathrm{eff}}}}{{{k}}^2}\left[{\left(1 - \dfrac{{\tanh kh\times (kG/\rho - Ag)}}{{\nu _{_{{\mathrm{eff}}}}^2{k^3}}}\right)^{1/2}} - 1\right] $ Piriz 2009: 固体/理想流体界面, A = 1, kh >> 1 $ \mathop \eta \limits^{ \cdot \cdot } = \left\{ \begin{gathered} {{k}}g(\eta + {\eta _0})- 2{k^2}G\eta /\rho \begin{array}{*{20}{c}} &{\eta \leqslant {\eta _p}} \end{array} \\ {{k}}g(\eta + {\eta _0})- \alpha Yk/\rho \sqrt 3 \begin{array}{*{20}{c}} &{\eta \geqslant {\eta _p}} \end{array} \\ \end{gathered} \right. $
$ \gamma = \sqrt {{{k}}g} $Piriz 2014: 固体/黏性流体界面, 任意A, kh >> 1 $ \begin{gathered} \mathop \eta \limits^{ \cdot \cdot } + \dfrac{{2\mu {k^2}}}{{{\rho _2} + {\rho _1}}}\mathop \eta \limits^ \cdot = \\ \left\{ \begin{gathered} {{Ak}}g(\eta + {\eta _0})- 2{k^2}G\eta /({\rho _2} + {\rho _1})\begin{array}{*{20}{c}} &{\eta \leqslant {\eta _p}} \end{array} \\ {{Ak}}g(\eta + {\eta _0})- \sqrt 3 Yk/({\rho _2} + {\rho _1})\begin{array}{*{20}{c}} &{\eta \geqslant {\eta _p}} \end{array} \\ \end{gathered} \right. \\ \end{gathered} $ $ \gamma = \sqrt {\dfrac{{{\mu ^2}{k^4}}}{{{{({\rho _2} + {\rho _1})}^2}}} + A{{k}}g} - \dfrac{{\mu {k^2}}}{{({\rho _2} + {\rho _1})}} $ Piriz 2019: 固体/理想流体界面, 任意A, 任意kh $ {\left(\dfrac{{kg}}{{{\gamma ^2}}}\right)^2} - 1 = \left(\dfrac{{kg}}{{{\gamma ^2}}} + 1\right)\dfrac{{1 - A}}{{1 + A}}[\dfrac{{kg}}{{{\gamma ^2}}} + \coth kh] $ $ \begin{gathered} {\gamma _{1,2}} = \pm \sqrt {\dfrac{{2A{{kg}}}}{{1 + A + (1 - A)\coth kh}}} \\ {\gamma _{3,4}} = \pm \sqrt { - kg} \\ \end{gathered} $
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4 附录 表3: 符号-物理量对照表
符号 对应物理量 符号 对应物理量 η 界面扰动振幅 G 剪切模量 η0 界面初始扰动振幅 Y 屈服强度 η0* 界面在受到冲击波加载后刚开始进行不稳定增长时的振幅 μ 动力学黏性系数 $ \eta _{{\text{s}}p}^{\max } $ 尖钉的极值振幅 ν 界面两侧流体的平均黏性系数 λ 界面扰动波长 νeff 固体的等效晶格粘度 λ0 界面初始扰动波长 μ0 真空磁导率 λc RT不稳定增长的临界波长 I 电流强度 k 扰动波数 γ 界面扰动增长率 kc RT不稳定增长的临界波数 γ0 格林艾森系数 A Atwood数 cv 等容比热容 ρ0 材料初始密度 σ 表面张力系数 ρ 冲击波作用后的材料密度 c0 零压下的体波声速 Fy 作用于界面法向方向上的力 Dij 应变率张量 ω 振荡频率 Sij 偏应力张量 T 振荡周期 ΔV 冲击波经过后物质界面的速度增量 p 压力 D 冲击波的传播速度 ε 应变 R 套筒外半径 $ \mathop \varepsilon \limits^ \cdot $ 应变率 h 平板的厚度 C 数值拟合常数 g 重力加速度
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