Numerical simulation of spray coalescence in an Eulerian framework: Direct quadrature method of moments and multi-fluid method

Abstract

The scope of the present study is Eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation. The fundamental mathematical description is the Williams spray equation governing the joint number density function f (v, u; x, t) of droplet volume and velocity. Eulerian multi-fluid models have already been rigorously derived from this equation in Laurent et al. [22]. The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox [24] to the Williams spray equation. Both the multi-fluid method and DQMOM yield systems of Eulerian conservation equations with complicated interaction terms representing coalescence. In order to focus on the difficulties associated with treating size-dependent coalescence and to avoid numerical uncertainty issues associated with two-way coupling, only one-way coupling between the droplets and a given gas velocity field is considered. In order to validate and compare these approaches, the chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with sprays having various size distributions, ranging from smooth ones up to Dirac delta functions. The second key feature of the paper is a thorough comparison of the two approaches for various test-cases to a reference solution obtained through a classical stochastic Lagrangian solver. Both Eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the Lagrangian solver. The third key point of the study is a detailed description of the limitations associated with each method, thus giving criteria for their use as well as for their respective efficiency.

Multi-Water-Bag Model And Method Of Moments For The Vlasov Equation

“…The method of moments, presented in [5,6], consists in taking the first moments of the equation that we have to solve in order to reduce the number of variables. The system is closed by assuming that the distribution function is made of water-bags.…”

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

“…, N . The following formula shows that evaporation, when the flux at zero size is zero, is only a translation of the abscissas and such an approach can be seen as equivalent to DQMOM [19]):…”

“…This approach has been widely used and has shown to be efficient in numerous cases (see example [19] and references herein). While quite accurate, its main drawback is the coupling of an Eulerian description for the gaseous phase to a Lagrangian description of the disperse phase, thus offering limited possibilities of vectorization/parallelization.…”

“…Moreover, this approach leads to serious numerical instabilities thus preventing its use for the treatment of an evaporating spray, since during the computation, a log-normal distribution function might not be reconstructed from the moment set dynamics. Another solution is high order moment methods, either quadrature method of moment (QMOM) where the dynamics of moments are evaluated after closing the source terms using quadrature methods (see [38] for example and references therein), or Direct Quadrature Method of Moment (DQMOM) [34] wherein equations are directly written on the quadrature weights and abscissas which describe the reconstructed distribution function having the same moments [19]. Such methods have proved to be very efficient in a number of configurations, such as agglomeration, sintering, coagulation-fragmentation...…”

“…However, they encounter two sets of difficulties. First, they are not able to accurately predict the evaporating flux in general (see [19] for a precise statement) since they fail to reproduce the number density disappearing flux at zero droplet size, which is a point-wise value to be reconstructed from the set of moments. Consequently, further studies have been undertaken and have lead to a proper treatment of the evaporation term.…”