Implementation of Bubble Dynamic Effects by Coupling the Gilmore Equation With Fluid Dynamic Equations Using the Method of Characteristics

In fluid power systems, performance as well as system dynamics are strongly influenced by the presence of bubbles — especially for low system pressures. While the static effect of dissolved air (especially the volume fraction of dissolved air) on the bulk modulus has been extensively investigated in the past, in hydraulics, the dynamic effects due to bubble dynamics have been neglected entirely. Thereby, the dynamic characteristics of the bubbles influence the compressibility of the disperse fluid and, as a consequence, the speed of sound in the mixture and the hydraulic system as a whole. In order to account for the bubble behavior in hydraulic simulation models, the present paper investigates a method for coupling bubble dynamics equations, such as the Gilmore or the Rayleigh-Plesset equation, with the fluid dynamic equations and their subsequent solution using the method of characteristics. Regarding the modeling, special attention is put on the distributed bubble nuclei sizes, since bubbles of the exact same size are unnatural and cannot be observed in reality. Since a dilute mixture — i.e. a small void fraction — is assumed, bubble-bubble interaction is neglected in this study. To account for the polydispersity, a discretized lognormal distribution for equilibrium bubble sizes is considered. In order to evaluate the discretization interval needed, case studies of different numbers of bubble size classes are presented and their results evaluated. Thereby, the question about the least required numbers of homogeneous bubble clusters shall be answered, as to reduce the computational effort that is needed. Using the method described in this paper, the profound effect of the bubble dynamics and the bubble size distribution on the fluid system dynamics is elaborated.