Effects of the spatial resolution of planar PIV on measured turbulence multi-point statistics
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摘要:
粒子图像测速（Particle Image Velocimetry, PIV），尤其是平面PIV，因其技术成熟且能提供瞬时流场，被广泛应用于湍流统计特性实验研究。PIV测量的空间分辨率（实验测量中所能解析的最小流场尺度，由图像处理时问询域窗口大小决定）不能准确分辨湍流中的小尺度运动，导致PIV测量获得的湍流多点统计量的定量信息（如速度结构函数、湍流耗散率等）包含一定误差。为定量分析这一误差，本文采用空间滤波模型模拟PIV测量对单点速度矢量的作用，进而推导出PIV测得的速度结构函数及湍流耗散率随滤波尺度的变化规律。为检验该模型，通过模拟示踪粒子在基于均匀各向同性湍流直接数值模拟（DNS）得到的湍流速度场中的运动及PIV系统对其的成像，获得了计算合成的PIV粒子图像，采用标准PIV算法对该图像进行处理，获得了速度场并计算得到了相关的湍流统计量，将其与基于真实DNS数据计算得到的统计量对比，获得了定量的误差信息，并与模型的预测结果进行了对比。研究结果表明：本文建立的模型反映了PIV测量中空间分辨率对湍流统计量的影响。在冯·卡门旋流系统产生的近似均匀各向同性流场中进行了平面PIV测量，针对测量得到的速度结构函数与K41理论结果的误差，采用基于本文模型得到的结论进行了分析与校正。本文工作为PIV测量得到的湍流统计量（尤其是小尺度多点统计量）给出了一个基于理论指导的修正方案。
Abstract:Particle Image Velocimetry (PIV), in particular planar PIV, has been widely implemented in the experimental study of statistical characteristics of turbulence because of its maturity in technology and capability to provide instantaneous flow fields. The spatial resolution of PIV measurement, i.e., the smallest scale of the flow field resolvable by PIV, is determined by the size of the Interrogation Window (IW) during image processing. Hence, small-scale turbulence fluctuations might not be accurately resolved, which leads to deviations in measured turbulence multi-point statistics, such as velocity structure functions, turbulent dissipation rate and so on. To quantify this deviation, we model the effect of PIV measurement on the instantaneous single-point velocity vectors as spatial filtering, which allows the change of the velocity structure functions and the turbulent dissipation rate measured by PIV with filter size to be derived. To check these predictions, synthetic PIV image pairs were generated based on simulated tracer particle motions following the velocity fields from direct numerical simulation (DNS) of isotropic turbulence, which were then processed by a standard PIV algorithm. Turbulence statistics obtained from such measured velocity fields were then compared with those from the exact DNS data to evaluate quantitative deviations. The results show that our model captures the effect of spatial resolutions on turbulence statistics in PIV measurement. Experimentally, planar PIV measurements were carried out in the center of a von Kármán swirling flow device, where the turbulence is nearly homogeneous and isotropic. The deviation between measured velocity structure functions and the K41 theory was also analyzed and corrected using the aforementioned model prediction. This work provides a theoretical guidance for examing turbulence statistics measured by PIV, especially multi-point statistics at small scales.
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图 6 模拟PIV得到的湍流耗散率DNS数据及二阶和四阶拟合结果
Fig. 6 The variations of DNS turbulent dissipation rate obtained from the PIV calculations and the second- and fourth-order fitting curves
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图 1 空间任意两点速度差对问询域中心点速度差的贡献
Fig. 1 The effects of velocity differences of two arbitrary points on the measurement of velocity differences of two interrogation windows
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图 2 基于DNS数据模拟得到的PIV合成图像及速度矢量
Fig. 2 A pair of PIV synthetic images using DNS data and diagram of velocity vectors
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图 3 利用DNS数据模拟PIV测量所得二阶纵向速度结构函数及其绝对误差
Fig. 3 The longitudinal structure function and its absolute errors obtained from the DNS data used to simulate PIV measure-ments
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图 4 绝对误差δDLL(r)在惯性区的平均值随分辨率的变化及拟合结果
Fig. 4 The trend of the mean values of absolute errors δDLL(r) of second-order structure function in the inertial region with the resolution of PIV and its fitting
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图 5 二阶纵向速度结构函数及其绝对误差
Fig. 5 The trend of the longitudinal second-order structure function and its absolute errors
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图 7 冯·卡门旋流系统流场示意图及实验布置图
Fig. 7 Schematic diagram of flow field and experimental arrangement of von Kármán swirling system
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图 8 实验中随机误差修正前后对结构函数的影响
Fig. 8 The influence of random errors on longitudinal second-order structure function before and after correction in the experiments
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图 9 实验测量的二阶速度结构函数修正前后对比
Fig. 9 Comparison of the second-order structure function measured by experiment before and after correction
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图 10 冯·卡门旋流实验中湍流耗散率测量值及拟合结果
Fig. 10 The variations of measured turbulent dissipation rate with respect to PIV resolution in von Kármán swirling system and its fittings
表 1 高斯滤波函数对多点统计量的显式误差影响
Table 1 Explicit filtering errors of Gaussian filter function on measured turbulence multi-point statistics
多点
统计量误差形式 二阶纵向结构函数绝对误差 耗散区 $ \dfrac{{{\text{δ}} {D_{{\mathrm{LL}}}}}}{{u_\eta ^2}} = \left[ { - 28{a_2}{\zeta ^2}{{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)}^2} - 756{a_3}{\zeta ^4}{{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]{\left( {\dfrac{r}{\eta }} \right)^2} $ 惯性区 $ \dfrac{{{\text{δ}} {D_{{\mathrm{LL}}}}}}{{u_\eta ^2}} = 10{a_1}{\zeta ^2}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^2} + 140{a_2}{\zeta ^4}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^4} $ 二阶横向结构函数绝对误差 耗散区 $ \dfrac{{{\text{δ}} {D_{{\mathrm{NN}}}}}}{{u_\eta ^2}} = \left[ { - 56{a_2}{\zeta ^2}{{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)}^2} - 1512{a_3}{\zeta ^4}{{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]{\left( {\dfrac{r}{\eta }} \right)^2} $ 惯性区 $ \dfrac{{{\text{δ}} {D_{{\mathrm{NN}}}}}}{{u_\eta ^2}} = 8{a_1}{\zeta ^2}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^2} + 100{a_2}{\zeta ^4}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^4} $ 湍流耗散率比值 $ \dfrac{{\bar \varepsilon }}{\varepsilon } = 1 + 420{a_2}{\zeta ^2}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^2} + 11340{a_3}{\zeta ^4}{\left( {\dfrac{{{\text{Δ}} x}}{\eta }} \right)^4} $ 
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表 2 DNS数据库流场数据（DNS单位）
Table 2 Parameters of the DNS database (DNS unit)
泰勒
雷诺数Reλ运动黏性
系数νKolmogorov
特征长度ηKolmogorov
特征速度uη湍流
耗散率ε418 1.85 × 10−4 2.8 × 10−3 6.60 × 10−2 1.03 × 10−1
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表 3 基于DNS数据模拟PIV图像的参数设置
Table 3 Parameter settings for DNS data to simulate PIV images
模拟激光片
相对厚度Δh/η问询域窗口
相对大小Δx/η问询域
窗口大小Δx问询域窗口
之间的重合度2 3.5 48像素 × 48像素 50% 4 4.7 32像素 × 32像素 50% 6 6.6 32像素 × 32像素 50% 8 9.1 32像素 × 32像素 50% 10 11.8 32像素 × 32像素 50%
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表 4 高斯滤波作用下的二阶速度结构函数绝对误差函数形式
Table 4 The specific forms of absolute errors of second-order structure function under the role of Gaussian filtering
多点统计量 绝对误差函数形式（忽略包含系数a3的高阶项） 二阶纵向
速度结构函数d1 $ - 28{a_2}{\zeta ^2}{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)^2} $ d2 $ {{{d_1}} \mathord{\left/ {\vphantom {{{d_1}} {\left[ {10{a_1}{\zeta ^2}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^2} + 140{a_2}{\zeta ^4}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]}}} \right. } {\left[ {10{a_1}{\zeta ^2}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^2} + 140{a_2}{\zeta ^4}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]}} $ 二阶横向
速度结构函数D1 $ - 56{a_2}{\zeta ^2}{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)^2} $ D2 $ {{{D_1}} \mathord{\left/ {\vphantom {{{D_1}} {\left[ {8{a_1}{\zeta ^2}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^2} + 100{a_2}{\zeta ^4}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]}}} \right. } {\left[ {8{a_1}{\zeta ^2}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^2} + 100{a_2}{\zeta ^4}{{\left( {\frac{{{\text{Δ}} x}}{\eta }} \right)}^4}} \right]}} $
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表 5 模拟PIV得到的湍流耗散率计算值与真实值之间的相对误差
Table 5 The relative errors between the calculated values and the true value of the turbulent dissipation rate obtained by simulating PIV
Δx/η 相对误差 耗散区 惯性区 式（14） 式（22） 式（23） 式（16） 修正后再以
式（16）计算3.5 9.8% 3.7% 2.0% 2.6% 1.8% 4.7 19.6% 3.7% 2.0% 3.4% 2.0% 6.6 29.4% 3.7% 2.0% 4.3% 1.8% 9.1 41.5% 3.7% 2.0% 5.7% 1.3% 11.8 52.7% 3.7% 2.0% 7.9% 1.8%
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表 6 冯·卡门旋流PIV实验参数设置
Table 6 Parameter settings of von Kármán swirling PIV experiment
实验装置 描述 PIV系统
运行平台DynamicStudio 光源设备 Nd:Yag双脉冲固体激光器 Vlite‒200，波长532 nm，脉冲能量2 × 200 mJ。示踪粒子为聚酰胺颗粒，平均直径5 μm 数据采集
设备IMPERX B3440相机， 3388 像素 ×2712 像素，像素尺寸3.69 μm，有效像素物理尺寸12.2 μm/像素
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