On measuring the joint probability density function of three-dimensional velocity components in turbulent flows

This paper presents the underlying theory to directly measure the joint probability density function (JPDF) of turbulent velocity fields. The raw data for these measurements can come from either reconstructed tomographic particle image velocimetry (PIV) volumes or holographic PIV volumes in 3D or from planar PIV images in 2D suitably pre-processed to satisfy the conditions required as per the analysis presented here. The underlying theory has been applied to 2D and 3D Gaussian turbulence velocity fields to illustrate its ability in determining mean velocities, standard deviations of the turbulent velocity fluctuations and correlation coefficients between the turbulent velocity fluctuation components. These examples have demonstrated that the mean velocity can be estimated to very high accuracy, i.e. to better than 1% even when estimating the 2D and 3D JPDF with the simplified equation that does not account for particle size. The turbulence velocity fluctuations through their standard deviations can be measured with a typical error of 1.5% and the correlation coefficients with a typical error of 2.5% when estimating the JPDF with the result that accounts for the particle size. This is the first technique that allows the estimation of 3C-3D turbulent velocity statistics via the direct measurement of the JPDF without any detrimental effects due to the high values of velocity gradients that are typically encountered in turbulent shear flows because it is not restricted by the requirement of uniform velocity within the measurement volume.