ABSTRACT Multi-scale statistics are used to analyse the flow structure of wake flow in the boundary layer of a wind turbine array. Experimentally, a wind turbine array is tested with X-type hot-wire anemometry, providing a velocity signal at discrete locations downstream of the array along the centreline of the centre turbine. Based on the Markov property, the turbulent cascade can be taken as a stochastic process in scale, for which an underlying Fokker-Planck equation and its Kramers-Moyal coefficients are assigned. The first two terms of the Kramers-Moyal expansion (drift and diffusion coefficients) are estimated directly from the measured data by an optimisation procedure, which includes reconstruction of the joint probability density functions via short-time propagator. To quantify the accuracy of estimated the Fokker-Planck equation for describing the turbulent cascade process, the validity of a fundamental law of nonequilibrium thermodynamics named integral fluctuation theorem is verified. The results highlight that multi-scale analysis separates the stochastic cascade into universal and non-universal portions with respect to physical location downstream of the rotor. In addition, the Kramer-Moyal coefficients reveal the impact of a specific generation mechanism of turbulence and its large and small scale motions. Velocity-intermittency quadrant method is used to characterise the flow structure of the wake flow. Multifractal framework presents the intermittency as a pointwise Hölder exponent. The relationship between large and small scales in wake flow is considered by quantifying the impact of the small scales on the large scales in terms of the pointwise Hölder condition. A negative correlation between the velocity and the intermittency is shown at the hub height and bottom tip, whereas the top tip regions show a positive correlation. The second and fourth quadrants are dominant downstream from the rotor. The pointwise results reflect large-scale organisation of the flow and velocity-intermittency events corresponding to a foreshortened recirculation region near the hub height and the bottom tip. A linear regression approach based on the Gram-Charlier series expansion of the joint probability density function is used to model the contribution of the second and fourth quadrants arriving at an excellent agreement between the model and the experiment. The model shows the best fit with the correlation of 0.9864.

中文翻译：

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风力涡轮机阵列边界层后的多尺度/分形过程

摘要 多尺度统计用于分析风力机阵列边界层尾流的流动结构。在实验上，风力涡轮机阵列使用 X 型热线风速测量法进行测试，沿着中心涡轮机的中心线在阵列下游的离散位置提供速度信号。基于马尔可夫特性，湍流级联可以被看作是规模上的随机过程，为此分配了一个潜在的福克-普朗克方程及其克莱默斯-莫亚尔系数。Kramers-Moyal 扩展的前两项（漂移和扩散系数）是通过优化程序直接从测量数据估计的，其中包括通过短时传播器重建联合概率密度函数。为了量化描述湍流级联过程的福克-普朗克方程的估计精度，验证了非平衡热力学基本定律积分涨落定理的有效性。结果突出显示，多尺度分析将随机级联根据转子下游的物理位置分为通用部分和非通用部分。此外，Kramer-Moyal 系数揭示了湍流的特定生成机制及其大小尺度运动的影响。速度-间歇象限法用于表征尾流的流动结构。多重分形框架将间歇性表示为逐点 Hölder 指数。通过在逐点 Hölder 条件方面量化小尺度对大尺度的影响，考虑尾流中大尺度和小尺度之间的关系。在轮毂高度和底部尖端显示速度和间歇性之间的负相关，而顶部尖端区域显示正相关。第二和第四象限在转子下游占主导地位。逐点结果反映了流动和速度间歇事件的大规模组织，对应于轮毂高度和底部尖端附近缩短的再循环区域。使用基于联合概率密度函数的 Gram-Charlier 级数展开的线性回归方法对第二和第四象限的贡献进行建模，从而在模型和实验之间达到极好的一致性。该模型显示最佳拟合，相关系数为 0.9864。