Response of spray number density and evaporation rate to velocity oscillations

A theoretical investigation of the effect of gas velocity oscillations on droplet number density and evaporation rate is presented. Oscillations in gas velocity cause a number density wave, i.e. an inhomogeneous, unsteady variation of droplet concentration. The number density wave, as it propagates downstream at the mean flow speed, causes modulation of the local evaporation rate, creating a vapour wave with corresponding oscillations in equivalence ratio. The present work devises an analytical formulation of these processes. Firstly, the response of a population of droplets to oscillations in the gas velocity is modelled in terms of a number density wave. Secondly, the formulation is extended to incorporate droplet evaporation, such that an analytical expression for the evaporation rate modulation is obtained. Subsequently, the droplet 1D convection-diffusion transport equation with the calculated evaporation source term is solved using an appropriate Green’s function to determine the resulting equivalence ratio perturbations. The dynamic response of equivalence ratio fluctuations to velocity oscillations is finally characterized in terms of a frequency-dependent transfer function. The aforementioned analytical approach relies on a number of simplifying approximations, nevertheless it was validated with good agreement against 1D Euler-Lagrange CFD simulations.