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Numerical grid and time-step dependencies of discrete droplet Lagrangian spray models are identified. The two main sources of grid dependency are due to errors in predicting the droplet-gas relative velocity and errors in describing droplet-droplet collision and coalescence processes. For reducing grid dependency due to the relative velocity effects, a gas-jet theory is proposed and applied to model diesel sprays. For the time-step dependency, it is identified that the collision submodel results in drop size variation in the standard spray model. A proposed spray model based on the gas-jet theory is found to improve the time-step independency also along with the mesh independency. The use of both Eulerian (collision mesh) and Lagrangian (radius of influence) collision models along with the gas-jet theory is found to provide mesh-independent results.

A new model to predict the velocity distribution in round jets with time-varying injection profiles has been formulated as an extension of steady jet theory. The approach introduces an effective injection velocity within the jet based on a representative response time. It is assumed that the instantaneous injection velocity affects the velocity within the jet with an exponential response function and that the response time is related to the fluid particle's residence time within the jet, consistent with the theory of translation of jet vortex rings from Helmholtz's vortex motion analysis ͓P. G. Tait, London Edinburgh Dublin Philos. Mag. J. Sci. 33, 485 ͑1867͔͒. The Helmholtz theory is also shown to reduce to the well-known velocity decay rate in the case of steady turbulent gas jets. A Duhamel superposition integral is used to determine the effective injection velocity for time-varying injection rates. The model is tested with different injection profiles and different ambient densities. The results are also compared with numerical results from a computational fluid dynamics code. The comparisons agree very well and the new model is shown to offer an efficient method to predict jet tip penetrations for unsteady jets.

Numerical grid and time-step-dependencies of Discrete Droplet Lagrangian spray models are identified. The two main sources of grid-dependency are due to errors in predicting the droplet-gas relative velocity, and errors in describing droplet-droplet collision and coalescence processes. For reducing grid-dependency due to the relative velocity effects, gas jet theory is proposed and applied to model diesel sprays. For the time-step dependency, it is identified that the collision sub-model results in drop size variation in the standard spray model. A proposed spray model based on the gas-jet theory is found to improve the time-step independency also along with the mesh independency. The use of both Eulerian (collision mesh) and Lagrangian (radius of influence) collision models along with gas-jet theory is found to provide mesh-independent results.