图1 颗粒-湍流涡-气泡的多尺度耦合机制

Fig.1 Multiscale coupling mechanisms of particles, turbulent eddies and bubbles

图2 球形颗粒湍流输送系统的控制机制的三维示意图 [26]（坐标轴分别为颗粒直径与最小湍流涡尺度比值Dp/η，颗粒与流体密度的比值ρp/ρf，以及固相体积分数Φv）

Fig.2 Three-dimensional view of the different regimes characterizing the transport of spherical particles in turbulence [26]（The phase space is defined by the particle size compared to the smallest turbulence scales Dp/η, the particle-to-fluid density ratio ρp/ρf, and the solid-phase volume fraction Φv）

图3 负载颗粒对湍流能谱结构的影响[27-28]

Fig.3 Effects of loaded particles on the entire energy spectrum of turbulent flow[27-28]

图4 基于速度梯度计算的能量耗散率云图（Reg=10800）[37]注:(a) 单相流;(b)单相流和1 mm固定颗粒;(c)单相流和3 mm固定颗粒;(d)单相流和8 mm固定颗粒

Fig.4 Specific energy dissipation rate contours computed using velocity gradient method for Reg=10,800[37] (a) fluid (b) fluid and 1 mm particle (c) fluid and 3 mm particle and (d) fluid and 8 mm particle

图5 颗粒对流场湍流调制趋势的分布图[40]

Fig.5 Mapping of the trend of particle effects on turbulence modification[40]

图6 不同时间间隔的示踪粒子速度增量的概率密度分布[47]注:(时间间隔分别为100τ/TL =1.3;2.7;5.4;11.2;22.4;44;89.3;174)

Fig.6 Probability distribution function of velocity increment calculated for time lags[47](for time lags 100τ/TL =1.3;2.7;5.4;11.2;22.4;44;89.3;174)

图7 速度幅值云图和颗粒分布图[30]（t=0起始时刻颗粒均匀分布；t=30Te当计算时间为30个涡流周期时计算停止）

Fig.7 Contours of fluid velocity magnitude and particle position[30]（at the start of simulation with particle evenly inserted and at the end of simulation when t equals 30 times eddy turnover time）

图8 惯性颗粒（a、c、e）和零加速点（b、d、f）的空间分布[55]（a和b中绘图区域的边长约为7倍的积分长度，相当于200倍的Kolmogorov尺度；c和d中绘图区域是a和b中框选区域的4倍；e和f中绘图区域是c和d中中框选区域的4倍）

Fig.8 Spatial distribution of inertial particles (a, c, and e) and zero-acceleration points (b, d, and f）[55]((a and b) The side length of plots is about 7 integral scales corresponding to 200 Kolmogorov scale. (c and d) The four-times magnification of squared region in (a and b). (e and f) The four-times magnification of squared region in (c and d))

图9 一个立方体区域内的颗粒团聚物的单独展示[56]（颗粒颜色由其所属的团聚物尺寸决定：红色：nagg<4；黄色：4≤nagg<7；绿色：7≤nagg<10；蓝色：nagg≥10）

Fig.9 Single realizations of aggregates in cubic domains[56](primary spheres colored by the size of the aggregate they are part of (red: nagg<4; yellow: 4≤nagg <7; green: 7≤nagg<10; blue: nagg≥10))

图10 气泡诱导液相波动的能谱[24]（a）竖直方向速度（b）水平方向速度

Fig.10 Spectra of bubble-induced liquid fluctuations in (a) vertical and (b) horizontal velocities[24]

图11 湍流场中气泡和湍流涡的两种不同作用机制[78]

Fig.11 Two different interaction mechanisms between turbulent eddies and droplets in the turbulence[78]

图12 基于不同能谱结构的气泡破碎频率[79]注:(SMA:谱模型A (Pope model[80]); SMB: 谱模型 B (Hinze model[81]); SIS: 惯性子区谱模型)

Fig.12 Breakage frequencies predicted by different energyspectrum models[79](SMA: Spectrum model A (Pope model[80]); SMB: Spectrum model B (Hinze model[81]); SIS: Spectrum of inertia subrange)

图13 不同的颗粒与气泡界面作用方式[91]（A:干净的气液界面；B：连接颗粒发挥稳定气液界面作用；C:轻吸附颗粒通过改变界面曲率影响毛细管力; D:连接的悬浮颗粒群传递‘离散’力; E: 非吸附颗粒改变液膜黏度; F:强吸附颗粒形成阻力边界层）

Fig.13 various interaction mechanisms of particles and bubbles at the bubble interface[91] (A: clean bubble interfilm; B: stabilisation from bridging particles; C: weakly adsorbing particles reducing capillary flow through meniscus alterations in curvature; D: bridging flocs transmitting mechanical ‘disjoining’ force; E: non-adsorbing particles altering viscosity; F: strongly adsorbing particles creating steric barrier)

图14 空气气泡在水/甘油-聚苯乙烯混合物中的运动[94]

Fig.14 Gas bubble motion in water/glycerol-polystyrene mixture[94]

图15 不同固含率的气泡聚并过程[98]

Fig.15 Bubbl coalescence at different particle loadings [98]

图16 εsgs*的轴向分布[34]：（a）搅拌桨向上的布局中r/R=0.55径向位置处；（b）搅拌桨向下的布局中r/R=0.65径向位置处

Fig.16 Vertical distribution of εsgs* for: (a) the impeller operated in (U) configuration at a radial position of r/R=0.55; (b) the (D) the impeller operated in configuration at a radial position of r/R=0.65[34]

图17 气泡下方颗粒区域示意图[105]

Fig.17 Schematic of the particle area beneath the bubble [105]