Mathematical models for turbulent round jets based on “ideal” and “lossy” conservation of mass and energy

Abstract

We propose mathematical models for turbulent round atomized liquid jets that describe its dynamics in a simple but comprehensive manner with the apex angle of the cone being the main disposable parameter. The basic assumptions are that (i) the jet is statistically stationary and that (ii) it can be approximated by a mixture of two fluids with the phases in local dynamic equilibrium, or so-called locally homogeneous flow (LHF). The models differ in their particular balance of explanatory capability and precision. To derive them we impose partial conservation of the initial mass and energy fluxes, introducing loss factors again as disposable parameters. Depending on each model, the equations admit explicit or implicit analytical solutions or a numerical solution in the discretized model case. The described variables are the the two-phase fluid's composite density and velocity, both as functions of the distance from the nozzle, from which the dynamic pressure is calculated. Keywords Mathematical Modeling, Two Phase Fluid, Locally Homogeneous Flow, Statistically Stationary State IntroductionThe range of applications involving atomizing liquid jets forming two-phase fluid flows is still large. The complexity of the atomizing process, involving numerous physical phenomena and many variables, ranging from the conditions inside the nozzle (or some generating source) to the interaction between the atomization process and the environment into which the jet is penetrating, all account for numerous challenges in physical and mathematical modeling. Notwithstanding, several such models have been attempted to describe different aspects of the jets in this regime. For example, differential equations for a fuel jet's tip penetration distance as a function of time [1, 2, 3]; models for the gas entrainment rate in a full-cone spray [4]; and a one-dimensional model for the induced air velocity in sprays [5]. None of these models is sufficient by itself as explained below. In this study we propose three original related 1D mathematical models, so-called "energy jet models", for the macroscopic dynamics of a turbulent round jet ensuing from a circular nozzle into a stagnant fluid. This kind of jets serves as a basis for many industrial processes in modern manufacturing industry [6]. An advantage of our models over other analytical 1D models is that the simplest case of the "ideal energy jet" has a single experimentally measurable parameter (the jet half-angle θ) while it maintains reasonable predictive power and gives theoretical understanding that allows it to analytically calculate other physical quantities of interest. Moreover, the herein reported other extended models class, the "lossy energy jet" models, apply an energy conservation approach with simple turbulence and energy dissipation models, resulting in increased accuracy. Compared to the present study, past models either lack a description of the density or liquid fraction of the spray, make unrealistic assumptions or introduce parameters unavailable experimentally...