Reconstruction of time-dependent concentration gradients around a KDP crystal growing from its aqueous solution

Abstract

Reconstruction of the steady three-dimensional concentration field from path-integrated two-dimensional data has been studied by the authors earlier. The extension of this approach to reconstruct three-dimensional unsteady concentration gradient field around a KDP crystal growing from its aqueous solution is examined in the present work. The experiments reported in the present work have been carried out in the mixed convection regime where inertial as well as buoyancy effects are significant. The crystal size is large enough to induce a time-dependent movement of convection currents in the growth chamber. Laser schlieren is used as the measurement technique for mapping the concentration gradients around the crystal. Projection data is in the form of a time sequence of schlieren images over an angular span of 0–360°. Images are recorded by rotating the growing crystal while the growth chamber is kept fixed. The time sequence recorded for one projection is uncorrelated to the next, creating an asynchronous data set. The technique proposed herein is an application of proper orthogonal decomposition to the image sequence. The method decouples the spatial and temporal components of the measured time-dependent data. The spatial modes, in turn, are ordered across all projections, thus facilitating three-dimensional reconstruction over the entire physical domain. The algorithm proposed integrates principles of tomography with proper orthogonal decomposition. It has been validated with simulated as well as experimental data. The results show that the first few modes contain much of the information of the time-dependent growth process. The differences in the nature of temporal fluctuations of the concentration gradient field in regions close to the growing crystal against those in the bulk are clearly revealed. The concentration gradient field is found to be purely unsteady near the crystal–solution interface, whereas the level of temporal fluctuations decreases as one moves towards the bulk of the solution. The reconstructed POD modes reveal an overall axisymmetry of the concentration gradients field in the growth chamber while a slight loss of symmetry was revealed in the vicinity of the growing crystal.