图1 二维系统内的温度场示意图[16]

Fig.1 A schematic of temperature field in a two-dimensional system

图2 二维温度场的形态分析示意图[16]

Fig.2 A schematic of morphological analysis of a two-dimensional temperature field

图3 (a) 形态相空间A-L-χ；(b) 形态区别度(或相似度)；(c)过程区别度(或相似度)示意图

Fig.3 (a)Morphological phase space A-L-χ; (b)Morphological difference (or similarity); (c)Process difference (or similarity)

图4 冲击与卸载过程中的一组温度位型图(a) t=500 ns；(b) t=750 ns；(c) t=850 ns；(d) t=1100 ns

Fig.4 A set of configurations with temperature contour in the shock loading and unloading procedure (a) t=500 ns; (b) t=750 ns; (c) t=850 ns; (d) t=1100 ns

图5 冲击波与多孔材料相互作用过程的形态学分析

Fig.5 Morphological analysis for the procedure of shock wave action on porous material

图6 A-L-χ空间的形态学分析

Fig.6 Morphological analysis in the A-L-χ space

图7 不同冲击过程中相似度与温度阈值的关系(a) DT=120; (b) DT=300

Fig.7 Process similarity versus temperature threshold (a) DT=120; (b) DT=300

图8 三个相似的斑图动力学过程

Fig.8 Three similar pattern dynamical procedures in A-L-χ space

图9 三个相似动力学过程在500 ns时的温度斑图，冲击速度和温升阈值分别为(a) 1000 m ·s-1, 92 K; (b) 1100 m ·s-1, 120 K; (c) 1200 m ·s-1, 154 K

Fig.9 Condensed temperature patterns at 500 ns in three similar dynamical procedures (a) V=1000 m ·s-1, DT=92 K; (b) V=1100 m ·s-1, DT=120 K; (c) V=1200 m ·s-1, DT=154 K

图10 温度阈值与冲击强度的关系(a) Δ=0.077; (b) Δ=0.46

Fig.10 Temperature threshold versus shock strength (a) Δ=0.077; (b) Δ=0.46

图11 温度阈值与孔隙度的关系

Fig.11 Temperature threshold versus porosity

图12 冲击与卸载过程的一组压强位形图(孔隙度Δ=0.46，初始冲击速度V=1300 m ·s-1)

Fig.12 A set of configurations with pressure contours in shock loading and unloading procedure (Δ=0.46, V=1300 m ·s-1)

图13 冲击波与多孔材料相互作用过程的形态学分析

Fig.13 Morphological analysis for the procedure of shock wave action on porous material

图14 A-L-χ相空间中的形态描述

Fig.14 Minkowski description in A-L-χ space

图15 相同孔隙度不同强度冲击过程相似度与压强阈值的关系

Fig.15 Process similarity versus Pth2 for cases with different shock strengths and same porosity

图16 两组相似的动力学过程

Fig.16 Two pairs of procedures showing similarity

图17 过程相似下压强阈值与冲击强度的关系

Fig.17 Pressure threshold versus shocking strength for similar procedures

图18 相同冲击强度下两个压强阈值之间的关系

Fig.18 Pth2(Δ2) versus Pth1 under same shock strength

图19 过程相似性下的压强阈值与孔隙度之间的关系

Fig.19 Pressure threshold versus porosity for similar procedures

图20 两个相空间描述方法的发展示意图[11, 21, 50]

Fig.20 A schematic for development of two description methods based on phase space

图21 Kn数为0.5时，(a) BGK气体的速度扰动；(b) BGK气体的温度扰动；(c) 硬球气体的速度扰动；(d) 硬球气体的温度扰动[55]

Fig.21 At Kn=0.5, (a) velocity perturbations for a BGK gas; (b) temperature perturbations for a BGK gas; (c) velocity perturbations for a hard-sphere gas; (d) temperature perturbations for a hard-sphere gas[55]

图22 Kn数为0.075时的稳态方腔流[55]

Fig.22 Cavity flow in steady state at Kn =0.075

图23 DBM应力与NS应力对比[54]

Fig.23 Comparison of DBM viscous stress and NS viscous stress

图24 闵可夫斯基形态学量、不同热传导系数下SD阶段持续时间及平均热通量随时间的演化

Fig.24 A schematic of Minkowski morphological quantity, SD stage duration at different heat transfer coefficients and average heat flux

图25 闵可夫斯基形态学量、不同弛豫时间(黏性系数)下SD阶段持续时间及局域平均速度随时间的演化

Fig.25 A schematic of Minkowski morphological quantity, SD stage duration at different relaxation time (viscosity coefficient) and local average speed

图26 (a) 图 21中相畴特征尺度R随时间的演化；(b) ρth=1.70时边界长度L和非平衡特征xx分量随时间的演化；(c)不同表面张力下边界长度L和相应热力学非平衡强度D随时间的演化; (d)亚稳相分解持续时间和最大非平衡强度随表面张力系数K的变化

Fig.26 (a) Evolutions of the characteristic domain sizes R; (b) Evolutions of the boundary length L and the xx component of some TNE manifestations; (c) Evolutions of the boundary lengths L (solid curves) and the corresponding TNE intensities D (curves with solid symbols) in phase separation processes with various K; (d) Duration of the spinodal decomposition stage tSD and the maximum TNE intensity Dmax as functions of K

图27 相分离过程中几种特征量的演化

Fig.27 Four kinds of criterions to discriminate the stages of the spinodal decomposition and domain growth

图28 不同热传导系数下相畴特征尺度和熵产生率演化

Fig.28 Profiles of characteristic domain sizes and rates of entropy production at different Prandtl numbers (heat conduction coefficient)

图29 临界时间和熵产生速率与热传导系数的关系

Fig.29 Profiles of critical time tSD and the maximum entropy production rates as functions of heat conductivity

图30 不同黏性系数下相畴特征尺度和熵产生速率演化

Fig.30 Profiles of characteristic domain sizes and rates of entropy production at different viscosity coefficients

图31 临界时间和熵产生速率与黏性系数的关系

Fig.31 Profiles of critical time tSD and amplitudes of entropy production rates as functions of viscosity coefficient

图32 不同表面张力下相畴特征尺度和熵产生速率演化

Fig.32 Profiles of characteristic domain sizes and rates of entropy production at different coefficients of surface tension

图33 临界时间和熵产生速率与表面张力的关系

Fig.33 Profiles of duration time of the SD stage (tSD) and amplitudes of entropy production rates as functions of coefficient of surface tension

图34 两种熵产生速率幅值之间的关系

Fig.34 Relations between two kinds of entropy production rate amplitude

图35 耦合RTKHI系统的温度分布(a) g=0.005, u0=0.05; (b)g=0.005, u0=0.1; (c) g=0.005, u0=0.15 (图像从左到右分别对应t=100，150，200，250，300。)

Fig.35 Temperature distributions of coupled RTKHI (a) g=0.005, u0=0.05; (b)g=0.005, u0=0.1; (c) g=0.005, u0=0.15 (Images from left to right correspond to t=100, 150, 200, 250, 300, respectively.)

图36 振幅A、振幅增长率dA/dt和形态边界长度L与时间t的关系

Fig.36 Perturbation amplitude A, growth rate dA/dt, and morphological boundary length L vs time t

图37 RTKHI系统早期主要机制的判断

Fig.37 Morphological analysis of main mechanism in the early stage of an RTKHI system

图38 非平衡特征在RTKHI系统早期主要机理判断(a)~(d)和过渡点捕获(e)~(f)中的应用

Fig.38 Applications of non-equilibrium characteristics in the early main mechanism judgment (a)- (d) and transition point capture(e)- (f)