On the Dynamics of a Torus Cloud Employing the Van Wijngaarden Ansatz and the Gilmore Bubble Dynamics Equation

The picture of the cloud cavitation consists of three parts, which are at the same time the three phases of the bubble: (i) birth, i.e. bubble nucleation (cf. Groß and Pelz, 2017), (ii) midlife, i.e. transition from sheet to cloud cavitation (cf. Pelz et al., 2017), and (iii) death, i.e. cloud collapse (cf. Buttenbender and Pelz, 2012). To model cavitating flows, these three phases have to be analysed separately. The present work analyses the dynamics of cloud cavitation in a compressible flow, focusing on part (iii) mentioned above. Recent experiments highlight, that a common cloud geometry for cavitation clouds is a horseshoe. With the Helmholtz vortex theorem in mind, the horseshoe is artificially completed to be a generic torus shaped cloud. Following van Wijngaarden, the mixture of cavitation bubbles and liquid inside the cloud is treated as a continuous medium, i.e. a homogenous model with the share of vapour α and the dimensionless single bubble radius R. Hence, the bubble radii are a function of the radial position inside the cloud only, R = R(r,t). The flow outside the cloud is modeled by a potential flow. The excitation of the cloud is carried out dynamically by applying a pressure history at infinity. With the Gilmore equation, the resulting system of partial differential equations is a parabolic system. This is due to the compressibility of the flow, which is taken into account. Previous investigations (cf. Buttenbender and Pelz, 2012) considered an incompressible flow, resulting in a hyperbolic system. The resulting pressure coefficient C_p and the bubble radii R inside the cloud are highlighted and analysed. They vary in time t and position r and depend on the pressure excitation of the cloud C_P∞, the Mach number Ma, the Reynolds number Re and the Weber number We.