Two‐phase laminar axisymmetric jet flow: Explicit, implicit, and split‐operator approximations

An hybrid Eulerian‐Lagrangian numerical scheme is developed for a two‐phase problem and four finite‐difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second‐order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas‐phase equations has been solved by four different numerical methods: a predictor‐corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator‐splitting method. The subsystem of liquid‐phase equations is solved along the droplet trajectories by a second‐order Runge‐Kutta scheme. The computations have been made to predict the hydro‐dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference‐schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.