Numerical computations of two-dimensional unsteady sprays for application to engines

The problem of the numerical computation of the penetration and vaporization of two-dimensional unsteady sprays is considered. This topic is of interest in the field of modelling of combustion in direct injection, internal combustion engines with self-ignition (Diesel), or spark ignition (stratified charge). The spray equation, which is the governing equation and, in this application, exhibits six independent variables, is numerically solved using an upwind difference scheme which is first-order accurate and explicit in time. The problem is set up both as a boundary value problem and an equivalent initial value problem. The boundary value formulation is preferable for applications. A study of the numerical error of the initial value problem is undertaken by comparing the numerical solution with an analytical one. The analytical solution is obtained by discretizing the spray, reducing the spray equation to a set of ordinary differential equations, solving these equations in closed form, and reconstructing the solution for the entire spray by integrating the contributions from the individual, discretized spray parcels. It is concluded that the selected numerical method yields sufficiently accurate results in reasonable computational time. However, the practical usefulness of the model is presently limited due to the simplifying assumptions that were made in order to concentrate on the formulation of the problem, the method of solution, and computational time and accuracy.