Two-Size Moment Multi-Fluid Model: A Robust and High-Fidelity Description of Polydisperse Moderately Dense Evaporating Sprays

Abstract

High fidelity modeling and simulation of moderately dense sprays at relatively low cost is still a major challenge for many applications. For that purpose, we introduce a new multi-fluid model based on a two-size moment formalism in sections, which are size intervals of discretization. It is derived from a Boltzmann type equation taking into account drag, evaporation and coalescence, which are representative of the complex terms that arise in multi-physics environments. The closure of the model comes from a reconstruction of the distribution. A piecewise affine reconstruction in size is thoroughly analyzed in terms of stability and accuracy, a key point for a high-fidelity and reliable description of the spray. Robust and accurate numerical methods are then developed, ensuring the realizability of the moments. The model and method are proven to describe the spray with a high accuracy in size and size-conditioned variables, resorting to a lower number of sections compared to one size moment methods. Moreover, robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebra thanks to a tailored overall strategy. This strategy is successfully tested on a difficult 2D unsteady case, which proves the efficiency of the modeling and numerical choices.

Simulation of reactive polydisperse sprays strongly coupled to unsteady flows in solid rocket motors: Efficient strategy using Eulerian Multi-Fluid methods

“…The choice of the case depends on the value of the ratio m (k) /n (k) . More details on the reconstruction and on its mathematical properties are given in [31]. The affine reconstruction is an interesting alternative to the exponential technique as it offers a significant computational gain when processing the inversion step.…”

“…It is developed in the spirit of other developpements presented in [36] where the spray is described using moment methods for which size-dependent evaporation laws can be treated robustly. More details on the quadrature based method dedicated to kinetic equation are given in [31]. As an asset of the new scheme, we present the strategy to integrate simultaneously the average sectional terms, which are the drag force and the heat transfer.…”

Statistical modeling of the gas–liquid interface using geometrical variables: Toward a unified description of the disperse and separated phase flows

“…multi-fluid models, where the size phase space is discretized into sections. Then moments up to order two can be used in each section [16,17], B. quadrature-moment methods such as QMOM, DQMOM or EQMOM [18,19], consider the NDF as a sum of Dirac-delta functions, potentially extended to kernels, C. high order moments with continuous reconstruction developed in [20,21,22,23,24]. At each step, a continuous NDF maximizing the Shannon entropy is reconstructed from the high order moments [25], with a complete coverage of the whole moment space.…”

High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

“…An alternative method consists in deriving an Eulerian moment model from the Williams-Boltzmann Equation. In this approach, a differential system of a finite set of moments of the NDF is closed by expressing the velocity NDF conditioned in size as a function of the known velocity moments [24,30,31,18,40]. For the modeling of the size distribution, three possible approaches can be used: 1-Multi-fluid models also called sectional methods (see [47,27] and references therein), where the size phase space is discretized into intervals called sections using the conservation of size moments up to two. 2-Method of moments with interpolative closure [16], essentially used for soot modeling and simulation, provides a closure of negative as well as fractional moments from the integer ones through a logarithm Lagrangian interpolation.…”

“…where St(S) = θS is the Stokes number and R S is the dimensionless evaporation rate, assumed to be constant: R S (S) = −K. We refer the readers to the following articles and references therein [27,23,10], showing that such a mesoscopic approach is capable of describing droplet heating, coalescence and break-up and two-way coupling. The high dimension of the phase space makes its direct discretization too costly for complex applications.…”