Dynamics of Vapor Bubble in a Variable Pressure Field

Abstract

This paper presents analytical and numerical results of vapor bubble dynamics and acoustics in a variable pressure field. First, a classical model problem of bubble collapse due to sudden pressure increase is introduced. In this problem, the Rayleigh—Plesset equation is treated considering gas content, surface tension, and viscosity, displaying possible multiple expansion—compression cycles. Second, a similar investigation is conducted for the case when the bubble originates near the rounded leading edge of a thin and slightly curved foil at a small angle of attack. Mathematically the flow field around the foil is constructed using the method of matched asymptotic expansions. The outer flow past the hydrofoil is described by linear (small perturbations) theory, which furnishes closed-form solutions for any analytical foil. By stretching local coordinates inversely proportionally to the radius of curvature of the rounded leading edge, the inner flow problem is derived as that past a semi-infinite osculating parabola for any analytical foil with a rounded leading edge. Assuming that the pressure outside the bubble at any moment of time is equal to that at the corresponding point of the streamline, the dynamics problem of a vapor bubble is reduced to solving the Rayleigh-Plesset equation for the spherical bubble evolution in a time-dependent pressure field. For the case of bubble collapse in an adverse pressure field, the spectral parameters of the induced acoustic pressure impulses are determined similarly to equivalent triangular ones. The present analysis can be extended to 3D flows around wings and screw propellers. In this case, the outer expansion of the solution corresponds to a linear lifting surface theory, and the local inner flow remains quasi-2D in the planes normal to the planform contour of the leading edge of the wing (or screw propeller blade). Note that a typical bubble contraction time, ending up with its collapse, is very small compared to typical time of any variation in the flow. Therefore, the approach can also be applied to unsteady flow problems.