Keywords:Polydisperse spray High order Eulerian multi-fluid model Adaptive quadrature for coalescence integrals Solid propellant combustion Aluminum oxide droplets CEDRE code a b s t r a c tThe accurate simulation of polydisperse sprays undergoing coalescence in unsteady gaseous flows is a crucial issue. In solid rocket motors, the internal flow depends strongly on the alumina droplet size distribution, which spreads up with coalescence. Yet solving for unsteady two-phase flows with high accuracy on the droplet sizes is a challenge for both modeling and scientific computing. As an alternative to Lagrangian approaches, a wide range of Eulerian models have been recently developed to describe the disperse liquid phase at a lower cost, with an easier coupling to the gaseous phase and with massively parallel codes. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations by separately solving fluids of size-sorted droplets, the so-called sections. The existing one size moment method, which describes the size distribution with one size moment per section, provides simple and fast resolution for coalescence. On the other hand, a two size moment method has been suggested to reduce the number of sections but it lacks an efficient coalescence resolution method. After introducing a new strategy for two size moment coalescence, the two methods are compared on various configurations in a research code and an industrial-oriented code, in order to conclude on computational accuracy and cost. Then the paper aims at describing the most efficient approach for multi-dimensional unsteady and eventually coalescing rocket chamber simulations. Its objective is threefold: first, to validate the two size moment method by comparing simulations to reference solutions and dedicated experimental measurements conducted at ONERA, second to study the efficiency and robustness of both methods, third, to draw some firm conclusions about the necessity to use the one size moment or two size moment method to simulate solid propellant alumina sprays. We finally perform the first simulations of coalescence in realistic 2D boosters with the two size moment method, implemented in the industrial-oriented code CEDRE.

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International audienceIn this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydisperse sprays strongly coupled to unsteady gaseous flows. In solid propulsion, aluminum particles are included in the propellant to improve the global performances but the distributed combustion of these droplets in the chamber is suspected to be a driving mechanism of hydrodynamic and acoustic instabilities. The faithful prediction of two-phase interactions is a determining step for future solid rocket motor optimization. When looking at saving computational ressources as required for industrial applications, performing reliable simulations of two-phase flow instabilities appears as a challenge for both modeling and scientific computing. The size polydispersity, which conditions the droplet dynamics, is a key parameter that has to be accounted for. For moderately dense sprays, a kinetic approach based on a statistical point of view is particularly appropriate. The spray is described by a number density function and its evolution follows a Williams-Boltzmann transport equation. To solve it, we use Eulerian Multi-Fluid methods, based on a continuous discretization of the size phase space into sections, which offer an accurate treatment of the polydispersion. The objective of this paper is threefold: first to derive a new Two Size Moment Multi-Fluid model that is able to tackle evaporating polydisperse sprays at low cost while accurately describing the main driving mechanisms, second to develop a dedicated evaporation scheme to treat simultaneously mass, moment and energy exchanges with the gas and between the sections. Finally, to design a time splitting operator strategy respecting both reactive two-phase flow physics and cost/accuracy ratio required for industrial computations. Using a research code, we provide 0D validations of the new scheme before assessing the splitting technique's ability on a reference two-phase flow acoustic case. Implemented in the industrial oriented CEDRE code, all developments allow to simulate realistic solid rocket motor configurations featuring the first polydisperse reactive computations with a fully Eulerian method