Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes and their dynamics

Abstract

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared to traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged.

Conditional moment methods for polydisperse cavitating flows

“…First, extending the QBMM method to additional R and Ṙ quadrature points (or moments, equivalently) was found numerically unstable for most bubble cavitation problems. One approach to addressing this specific problem is introducing a recurrent neural network to correct the quadrature points and weights [48], though we do not discuss it further here. Second, we will see that the closure errors are most strongly associated with N Ro , and thus focus on the influence of this parameter.…”