Thermal and dynamic evolution of a spherical bubble moving steadily in a superheated or subcooled liquid
Authors: Legendre Dominique , Borée Jacques , Magnaudet Jacques 
Published: July 2002

The heat transfer rate and the hydrodynamic forces experienced by a single vapor bubble of variable radius moving in a superheated or subcooled liquid are studied by means of numerical simulation. For that purpose the full Navier–Stokes equations and the temperature equation are solved in a frame of reference where the bubble surface is steady. The time evolution of the bubble radius is determined by solving the energy balance at the bubble surface. The numerical method is first validated by comparing present predictions with previous asymptotic or numerical results in the case where no relative motion between the liquid and the bubble exists. Then the situation where a constant relative velocity exists is considered. Effects of the mean flow on the heat transfer rate and on the bubble radius evolution are first discussed. Two different stages are generally observed in the computations. First, the radial motion induced by the displacement of the bubble surface dominates and the bubble evolution is essentially identical to the one observed in a liquid at rest. Then the ratio between the radial velocity and the translatory velocity decreases and the heat transfer rate becomes governed by streamwise advection effects. In this second stage a substantial increase of the growth or collapse rate of the bubble is observed, compared to the case of a liquid at rest. For a growing bubble it is shown that the complete process is successively described by the analytical solutions given by Scriven [Chem. Eng. Sci. 10, 1 (1959)] and Rückenstein [Chem. Eng. Sci. 10, 22 (1959)]. The situation is much less simple for a collapsing bubble and the reasons of this increased complexity are discussed. It is found that, when the heat transfer mechanism is dominated by streamwise advection, the bubble evolution and the collapse time predicted by the simulations agree well with the experimental results obtained by Chen and Mayinger [Int. J. Multiphase Flow 18, 877 (1992)]. Based on the present results, a general correlation giving the collapse time as a function of the characteristic parameters of the problem is proposed. The second contribution of the present work concerns the hydrodynamic force experienced by the bubble. Using a general decomposition procedure, the added mass effect and the viscous contribution are separately identified. It is first shown that the added mass coefficient is strictly constant and equal to one half, whatever the Reynolds number and the relative magnitude of the radial velocity. The viscous drag is then systematically compared with the quasisteady viscous drag corresponding to the instantaneous value of the Reynolds number. In situations of boiling, effects due to unsteadiness are found to exist during the first stages of the motion if the initial Reynolds number is not very large. In contrast, for a collapsing bubble, such effects remain significant all along the process because the relative importance of viscous phenomena increases in time. In both cases it is shown that the time variations of the bubble radius may affect deeply the viscous drag force. For example, when the radial velocity is high enough, the viscous drag force is found to be identical to the one corresponding to a potential flow, even if the instantaneous Reynolds number is low. These effects are discussed with the help of two asymptotic expressions of this force derived recently by Magnaudet and Legendre [Phys. Fluids 10, 550 (1998)] for a bubble with a time-dependent radius.