Cavitation Analogy to Gasdynamic Shocks: Model Conservativeness Effects on the Simulation of Transient Flows in High-Pressure Pipelines

Abstract

A comparison between conservative and nonconservative models has been carried out for evaluating the influence of conservativeness on the prediction of transient flows in high-pressure pipelines. For the numerical tests, a pump-line-nozzle Diesel injection system was considered because the pipe flow presented interesting cases of cavitation. The validity of a conservative model in the simulation of cavitating transient flows was substantiated by the comparison between computed pressure time histories and experimental results at two pipeline locations in the injection system. Although nonconservative models can assure satisfactory accuracy in the evaluation of the wave propagation phenomena, they introduce fictitious source terms in the discretized equations. Such terms are usually negligible, but can play a significant role in the presence of acoustic cavitation, i.e., pressure-wave-induced cavitation, producing errors in the pressure-wave speed prediction. A theoretical analysis based on unsteady characteristic lines was carried out, showing that the cavitation desinence is a shock gas-dynamic-like event, whereas cavitation inception is a supersonic expansion. The Rankine–Hugoniot jump conditions were applied to evaluate the shock wave speed in the presence of cavitation. Analytical relations to calculate the flow property variations across the cavitation-induced discontinuities were also derived. A previously published analytical expression of the sound speed in a homogeneous two-phase flow model was also derived from the eigenvalues of the Euler flow equations for the two distinct phases and a comparison was made with Wallis’ formula, which is commonly applied to cavitating flow simulation in transmission lines. Finally, a novel algorithm for calculating the shock speed, as is predicted by nonconservative models, was presented and applied to Burgers’ equation, pointing out the contribution of internal fictitious fluxes in the shock-speed wrong estimation.