Model of Droplet Impingement Based on Least-Squares Solution of Proper Orthogonal Decomposition Basis Matrices

A reduced order model of the surface profiles of droplets impinging on a flat surface is presented based on axisymmetric, transient computational fluid dynamics (CFD). The free surfaces resulting from the volume-of-fluid simulations were interpolated in polar coordinates and arranged as rectangular matrices (time versus space). Proper orthogonal decomposition was then used to expand the data into sets of temporal and spatial basis vectors, which were truncated beyond diminishing singular values. The reduced model is a general linear combination of constant matrices and dimensionless parameters that, when combined, recreate the temporal and spatial basis vectors for each case. The constant matrices were determined with a least-squares solution to the overdetermined linear combinations. To predict a new case, the initial Reynolds, Weber, and Ohnesorge numbers were combined with the calculated constant matrices to determine the new basis vectors, which were used to create the new free surface profile. A new case predicted by the model was validated using a CFD simulation. The single maximum error between the CFD profile and the general linear model was approximately 9% of the initial droplet diameter. The root-mean-squared error for the entire droplet motion was approximately 2% of the initial droplet diameter.