The Lagrangian particle tracking shake-the-box (STB) method provides accurate evaluation of the velocity and acceleration of particles from time-resolved projection images for high seeding densities, giving an opportunity to recover the stress tensor. In particular, their gradients are required to estimate local pressure fluctuations from the Navier–Stokes equations. The present paper describes a grid-free least-squares method for gradient and pressure evaluation based on irregularly scattered Lagrangian particle tracking data with minimization of the random noise. The performance of the method is assessed on the basis of synthetic images of virtual particles in a wall-bound turbulent flow. The tracks are obtained from direct numerical simulation (DNS) of an initially laminar boundary layer flow around a hemisphere mounted on a flat wall. The Reynolds number based on the sphere diameter and free stream velocity is 7000, corresponding to a fully turbulent wake. The accuracy, based on the exact tracks and STB algorithm, is evaluated by a straightforward comparison with the DNS data for different values of particle concentration up to 0.2 particles per pixel. Whereas the fraction of particles resolved by the STB algorithm decreases with the seeding density, limiting its spatial resolution, the exact particle positions demonstrate the efficiency of the least-squares method. The method is also useful for extraction of large-scale vortex structures from the velocity data on non-regular girds.

拉格朗日粒子跟踪振动盒 (STB) 方法可从时间分辨投影图像中准确评估粒子的速度和加速度，以获得高播种密度，从而有机会恢复应力张量。特别是，它们的梯度需要根据 Navier-Stokes 方程来估计局部压力波动。本文描述了一种基于不规则散射拉格朗日粒子跟踪数据的梯度和压力评估的无网格最小二乘法，同时最小化随机噪声。该方法的性能是基于壁约束湍流中虚拟粒子的合成图像进行评估的。这些轨迹是从安装在平坦墙壁上的半球周围的初始层流边界层流的直接数值模拟 (DNS) 中获得的。基于球体直径和自由流速度的雷诺数为 7000，对应于完全湍流尾流。基于精确轨迹和 STB 算法的准确性通过与 DNS 数据的直接比较来评估，不同的粒子浓度值高达每像素 0.2 个粒子。虽然 STB 算法解析的粒子分数随着播种密度的增加而减少，限制了其空间分辨率，但精确的粒子位置证明了最小二乘法的效率。该方法还可用于从非规则网格的速度数据中提取大尺度涡结构。通过与 DNS 数据的直接比较来评估不同的粒子浓度值，每个像素高达 0.2 个粒子。虽然 STB 算法解析的粒子分数随着播种密度的增加而减少，限制了其空间分辨率，但精确的粒子位置证明了最小二乘法的效率。该方法还可用于从非规则网格的速度数据中提取大尺度涡结构。通过与 DNS 数据的直接比较来评估不同的粒子浓度值，每个像素高达 0.2 个粒子。虽然 STB 算法解析的粒子分数随着播种密度的增加而减少，限制了其空间分辨率，但精确的粒子位置证明了最小二乘法的效率。该方法还可用于从非规则网格的速度数据中提取大尺度涡结构。