A framework about flow measurements by LDA–PDA as a spatio-temporal average: application to data post-processing

Abstract

Phase Doppler anemometry (PDA) is a well-established technique to study two-phase flows and its principles are also used in laser Doppler anemometry (LDA) for measurements of fluid velocity. Raw measurements of individual particle data require post-processing to obtain useful and consistent information (moments of velocity, particle concentration and flux, velocity autocorrelation, etc). This is called in this paper the reconstruction of statistical information. In the 1970s, several basic algorithms to perform the statistical reconstruction were developed for LDA measurements (such as the transit time method, the inverse velocity method, etc). With the advent of PDA, the scientific community developed reconstruction algorithms to obtain mean variables of the dispersed phase. All these basic algorithms were expounded as unconnected methods, following independent threads not integrated into a general framework. Assuming that the PDA works under ideal conditions (all particles that cross the probe volume are validated), this paper provides a general formulation and fully systematizes a large set of previous statistical reconstruction methods. In this paper, the statistical reconstruction of both the dispersed and the continuous phase is unified: the continuous phase post-processing emerges as the same reconstruction method of the dispersed phase. The general framework proposed offers many advantages. First, some previous calculation methods of particle concentration turn out to be particular cases of this general formulation. Second, it provides an easy way to deduce unbiased estimators of any statistical parameter of the flow. Third, a wide set of new post-processing methods are proposed to be tested by any member of the scientific community. In the fourth place, the generalized integral method to compute the particle concentration also gives information about the probe volume geometry and two new auto-calibration algorithms are proposed: the integral calibration method and the cross-section integral calibration method. Finally, a physical interpretation of the statistical reconstruction process is provided: it is a spatio-temporal averaging of the detected particle data, and some of the algorithms used are related to the Eulerian–Eulerian mathematical description of multiphase flows