Implicit Numerical Model of a High-Pressure Injection System

An implicit finite-difference numerical method has been developed and applied to the simulation of unsteady flow phenomena in a high-pressure injection system. A first-order one-step BSBT (backward space, backward time) scheme was used to obtain the difference analogue of the one-dimensional, elemental-volume averaged, partial differential equations governing the pressure-pipe flow. Second and higher-order implicit difference representations were employed for the ordinary differential equations simulating the pump and injector dynamics. The resultant nonlinear algebraic equations were solved by the Newton-Raphson method and a fast modified version of the Gaussian elimination procedure was used to solve the linearized equations. This was an extension of the Thomas solver to a multidiagonal system of algebraic equations. A compact, efficient and stable numerical algorithm was so obtained. The mathematical model takes into account the compressibility of the liquid fuel, the boundary shear, and also includes the simulation of possible cavitation occurrence at one or multiple locations in the injection system. No artificial viscosity has to be added to the solution in the vicinity of discontinuities induced by cavitation in the flow properties. The cavitation simulation is based on a simple mixture model of transient two-phase flow in pipes and can incorporate the effects of gaseous cavitation occurrence. Experimental values of the flow coefficients were used for the pump and injector and, for the latter, the dependence of the discharge coefficients on the needle lift and injection pressure was also taken into account. The model was tested and validated by comparing the numerical results with those of experiments carried out at the Fiat Research Center on a diesel-engine inline injection system, with a jerk-pump and an orifice type nozzle-injector.