A review of the study on coherent structures in turbulence by the clustering method
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摘要: 湍流的无序性要求对拟序结构的研究必须从统计的角度出发, 而聚类连通法 (clustering method) 则是实现拟序结构与统计方法深度融合的有力工具. 该方法是一种基于数据的流动特征提取方法, 它将每个连通域, 即单个拟序结构作为一个统计样本. 此外, 其衍生的基于连通域空间重叠的时空追踪方法可以进一步研究这些结构的时空演化, 该方法将每个拟序结构从生成到消亡的演化过程也视为统计样本, 从而实现了对拟序结构运动学特征和动力学过程的统计刻画. 本文回顾了聚类连通法的发展历程并着重介绍了人们采用该方法在雷诺切应力结构、速度条带和能量级串方面取得的重大进展, 这些结果表明该方法极大拓展了人们基于传统的逐点统计方法对湍流的认识, 因而具有很大的潜力. 最后, 对该方法在湍流中的应用给出了建议和展望.Abstract: The intrinsic randomness of turbulence requires that the study of coherent structures has to be from the statistical point of view. The data-based clustering method is a powerful tool to realize the deep integration of coherent structures and statistics. It makes each continuous region, i.e., an individual structure, as a statistical sample. In addition, the spatial-temporal tracking method based on the spatial overlap of individual structure between consecutive snapshots makes the evolution of individual structure during its lifetime also as a statistical sample. Thus, both the kinematics and dynamics of coherent structures can be statistically described. We reviewed the history of this method and introduced the progress on Reynolds stress structures, velocity streaks, and cascades with emphasis. These results suggest that the clustering method extraordinarily extended our understanding of turbulence compared with the traditional point-wise statistics. Further prospects are also provided for future research.
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图 2 槽道湍流中提取的涡簇(上)和雷诺应力结构(下). 结构以离壁面距离着色, 坐标以黏性长度δν为单位 (Lozano-Durán 2015)
图 3 拟序结构从生成到消亡的示意图, (b)为用节点、边和图表示的(a)中拟序结构的演化 (Lozano-Durán & Jiménez 2014a)
图 4 (a) Reτ ≈ 4200的槽道湍流中瞬时的涡簇和(b)雷诺应力结构. (b)中暖色为上抛, 冷色为下扫. 结构以离壁面距离着色 (Lozano-Durán & Jiménez 2014a)
图 5 (a) Reτ ≈ 4200的槽道湍流中构成一个图的涡簇随时间的演化. (b)涡簇演化构成的1个graph, 实线代表分支, 红色和蓝色虚线分别代表结构的合并和分裂. (c)一个主分支中上抛结构从生成到消亡的演化过程 (Lozano-Durán & Jiménez 2014a)
图 6 涡簇体积的概率密度函数, 其中虚线表示ymin+ = 20及ymax = ymin. 等值线分别包含20%, 40%, 60%和80%的数据. 有阴影的等值线为另一阈值的结果, 包含40%的数据 (del Álamo et al. 2006)
图 7 对ymax+ > 100的壁面附着涡簇作条件平均之后涡簇中心xy截面上(
$\{u'\},\{v'\}$ )的矢量图. 阴影为网格点属于涡簇的概率密度函数, 分别包含50%和75%的数据. 实线表示$\{u'\}$ = 0 (del Álamo et al. 2006)
图 8 对壁面附着且ymax+>100的涡簇做条件平均之后的平均流场. 其中红色和蓝色等值面分别为
$\{u'\}^+$ = 0.3和−0.1. 箭头代表($ \{v'\},\{w'\}$ ), 黑色代表网格点属于涡簇的概率密度函数等值面, 绿色等值面为平均流场速度梯度的判别式 (del Álamo et al. 2006)
图 9 Reτ = 2000的槽道湍流中一个超大尺度上抛. 坐标轴以槽道半高h为单位 (Lozano-Durán et al. 2012)
图 10 壁面附着结构流向和展向相对位置的概率密度函数. (a)上抛之间, (b)下扫之间, (c)涡簇之间, (d)上抛相对于下扫, (e)涡簇相对上抛, (f)涡簇相对下扫 (Lozano-Durán et al. 2012)
图 11 (a)网格点属于上抛、下扫和涡簇的概率密度函数等值面; (b) 瞬时的上抛、下扫和涡簇; (c) 平均流向脉动速度, 红色和蓝色分别表示
$\{u'\}^+$ = 0.5和−0.5; (d) rx = 0截面上的平均流场, 云图为$\{u'\}^+$ , 箭头代表($\{v'\},\{w'\} $ ), 虚线为(a)中概率密度函数的截面 (Lozano-Durán et al. 2012)
图 12 下扫分别与(a)下扫和(b)上抛之间的最短距离与特征长度的关系. Δ为槽道湍流中壁面分离的雷诺应力结构(ymin+ > 100), 其他符号为统计平稳均匀剪切湍流中结果. (a)和(b)中虚线的斜率分别为1和2/3 (Dong et al. 2017)
图 13 统计平稳均匀剪切湍流中, 下扫分别与(a)下扫和(b)上抛之间相对位置的三维概率密度函数等值面 (Dong et al. 2017)
图 14 对统计平稳均匀剪切湍流中上抛−下扫对做条件平均之后的流场. (a)属于上抛(绿色)、下扫(蓝色)和涡簇(银灰色)的概率密度函数等值面; (b)和(c)中云图分别为δx = 0截面上的
$\{u'\} $ 和${\partial_y u'}$ , 箭头代表($\{v'\},\{w'\} $ ), 黑线为总的平均速度$\{u'\} $ + Sδy = 0, 白色线条为(a)中的截面; (d)涡簇和平均剪切(粉红) (Dong et al. 2017)
图 15 统计平稳均匀剪切湍流中的Q2−Q4对(a) (d)以及槽道湍流中的壁面分离(b) (e)和壁面附着(c) (f)的Q2−Q4对周围的统计平均流场. 其中云图和线条与图13(a)中含义相同 (Dong et al. 2017)
图 16 Reτ = 934的槽道湍流中壁面附着最优通量结构周围的统计平均流场, 其中箭头代表(
$\{v'\},\{w'\} $ ). 透明等值面代表最优动量通量, 黄色等值面为雷诺切应力$\{u'v'\}+ $ = −2.1 (Osawa & Jiménez 2018)
图 17 Reτ ≈ 1000的槽道湍流受到展向压力梯度(dP/dz = 60dP/dx)后不同时刻Q2−Q4对周围的统计平均流场 (Lozano-Durán et al. 2019)
图 18 (a)二维平稳状态湍流边界层和(b)三维非平稳状态湍流边界层中的低速条带−高速条带−流向涡模型 (Lozano-Durán et al. 2019)
图 19 (a) δ＋ ≈ 1200 ~ 1415的零压力梯度平板湍流边界层中的u结构(u'+ > 2). 图中省略了流向长度小于δ/2的结构. (b) δ＋ ≈ 1800处的一个大尺度壁面附着u结构. 大小为(Δx, Δy, Δz) = (2.42, 0.89, 0.90)δ (Sillero 2014)
图 20 δ+ ≈ 1000的零压力梯度平板湍流边界层中壁面附着的(a)流向、(b)展向和(c)法向速度结构. 红色和蓝色分别代表正和负脉动ui' = ±1.5ui,rms. 颜色的明暗代表离壁面的距离 (Hwang & Sung 2018)
图 21 (a)全流场、高速和低速条带中的平均流向速度剖面, (b)为图(a)中的对数律指示因子 (Hwang & Sung 2019)
图 22 高(PAu)、低(NAu)速流向速度条带中的剪切常数在法向的变化 (Cheng et al. 2020a)
图 23 (a, c)正向和(b, d)逆向级串周围的统计平均速度场, 其中红、蓝和黄分别代表上抛、下扫和剪切层. (c, d)中云图表示剪切
$\partial \{\tilde u'\}/\partial y$ , 箭头代表($\{\tilde u'\}$ ,$\{\tilde v'\}$ ), 红色实线为属于级串事件的概率密度函数等值线 (Dong et al. 2020)
图 24 条件平均获得的三维(a)正级串和(b)逆级串周围平均涡量的拟涡能(黄色和蓝色). 红色代表级串事件, 箭头表示平均涡量的方向 (Dong et al. 2020)
图 25 不同滤波尺度所得结构的空间重叠示意图 (Cardesa et al. 2017)
图 26 采用不同尺度滤波之后的湍动能谱 (Cardesa et al. 2017)
图 27 不同滤波尺度滤波之后提取的三维含能结构 (Cardesa et al. 2017)
图 28 (a)生命周期的不同阶段不同滤波尺度间含能结构的体积重叠, (b)用整个生命周期中体积重叠最大值归一化了的体积重叠 (Cardesa et al. 2017)
图 29 回流结构1/8生命周期时的统计平均流场. 红色, 蓝色和绿色等值面分别表示
$\{\omega_z\} $ = 2$\left\langle {{\omega _z}} \right\rangle $ ,$\{\omega_y\} $ = 0.5$\{\omega_y\}_{\rm{max}}$ 和$\{\omega_y\}$ = −0.5$\{\omega_y\}_{\rm{max}} $ . 黑色区域为回流结构的外接长方体 (Cardesa et al. 2019)
图 30 回流结构中心xy截面上整个生命周期内的平均展向涡量 (Cardesa et al. 2019)
表 1 采用聚类连通法(基于种子填充算法)在槽道湍流中提取三维低速条带的耗时
阈值 0.1 0.5 1.0 1.5 2.0 网格量 4.0542×107 2.7366×107 1.4686×107 6.544×106 2.408×106 耗时/s 2.089 1.434 0.777 0.416 0.232 
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表 2 采用聚类连通法进行湍流拟序结构研究的代表性工作
代表作 主要贡献 Moisy & Jiménez (2004) 首次研究了三维拟序结构的性质, 并确定了阈值的选取方法 Lozano-Durán 等 (2012) 首次揭示了三维上抛、下扫和涡簇的空间构成 Hwang & Sung (2019) 首次发现高速条带中的平均速度剖面更符合对数律 Cheng 等 (2020a) 首次发现高速条带符合附着涡模型, 并揭示了高低速条带不对称的原因 Lozano-Durán & Jiménez (2014a) 首次研究了三维拟序结构的时空演化过程 Cardesa 等 (2017) 首次揭示了能量在空间−时间−尺度间的五维级串
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