A Reduced-order Model of Heat Transfer Effects on the Dynamics of Bubbles

The Rayleigh-Plesset equation has been used extensively to model spherical bubble dynamics, yet it has been shown that it cannot correctly capture damping effects due to mass and thermal diffusion. Radial diffusion equations may be solved for a single bubble, but these are too computationally expensive to implement into a continuum model for bubbly cavitating flows since the diffusion equations must be solved at each position in the flow. The goal of the present research is to derive reduced-order models that account for thermal and mass diffusion. We present a model that can capture the damping effects of the diffusion processes in two ODE’s, and gives better results than previous models. INTRODUCTION A continuum model that couples the Rayleigh-Plesset equation for bubble dynamics with the equations of continuity and momentum (van Wijngaarden 1968, 1972) has been used extensively in the computation of bubbly cavitating flows. Recent examples include Shimada et al. (1999),Wang (1999), Colonius et al. (2000) and Preston et al. (2002). A significant limitation is the use of a polytropic approximation to account for the expansion and compression of the gas bubble interior and an effective liquid viscosity to account for damping of the bubble radial Address all correspondence to this author. motion due to heat transfer (Prosperetti et al. 1988, Kameda & Matsumoto 1996). The correct treatment of the thermal effects requires the solution of the radial energy equation in each bubble and the surrounding liquid; however this is an expensive computation. Previous models that account for thermal diffusive effects include the models of Prosperetti (1991) valid near either the isothermal or adiabatic limits. These models work well in the limits for which they were intended, but are not accurate for behavior between the two limits. Storey & Szeri (2001) developed a model that switches between isothermal and adiabatic behavior depending upon relative timescales. While this approach yielded good estimates of peak bubble temperatures during bubble collapse, it is unable to correctly capture attenuation of bubble rebounds due to thermal damping effects. Lertnuwat et al. (2001) proposed a model that estimated the thermal energy flux out of the bubble by using an average bubble temperature and an estimation of the thermal penetration length. This seems a reasonable approach near the adiabatic limit, but is clearly not reasonable when the thermal penetration length approaches or even exceeds the bubble radius. We propose an alternative thermal model that is able to capture thermal damping effects over a wide range of applications. The thermal model requires only one additional ODE to be integrated alongside the Rayleigh-Plesset equation. The accuracy of the thermal model is tested by comparing the model response 1 Copyright  2002 by ASME of a single forced bubble to a computation in which the full energy equation in the bubble interior is solved. Results show that the proposed thermal model produces closer agreement to full computations than previous models, particularly cases where the amount of attenuation of bubble rebounds is important. Finally, we present an extension of the thermal model that incorporates mass diffusion of vapor in the bubble. Preliminary results indicate that the mass transfer model is also able to capture the extent of the initial expansion and the attenuation of the bubble rebounds very well. The mass transfer model requires one additional ODE to be integrated. THERMAL MODEL The thermal model is based on the simplified set of equations of Prosperetti et al. (1988) for a gas bubble with the internal pressure assumed to be spatially uniform. This assumption enables the derivation of the following ordinary differential equation for the internal bubble pressure, d p dt 3γ R poD R ∂T ∂y y 1 p dR dt (1) which is coupled to the Rayleigh-Plesset equation 1 for the motion of the liquid, R d2R dt2 3 2 dR dt 2 4 ReR dR dt 2 WeR p p∞ t (2) The variables in the above equations have been nondimensionalized as R R R o, T T T o , p p ρ LR 2 o ω 2 o , while y r R t is the radial coordinate chosen to fix the bubble wall at y 1. The dimensionless gas diffusivity, Reynolds number and Weber number are given respectively as D k ρ oc pR 2 o ω o, Re R 2 o ω o ν L and We ρ LR 3 o ω 2 o S , where ω o is the bubble natural frequency. The non-dimensional initial internal bubble pressure is computed from equilibrium of Eq. (2) as, po p∞o 2 We, where p∞o is the non-dimensional ambient pressure. The ordinary differential Eqs. (1) and (2) are typically closed by the radial energy equation for the temperature distribution in the bubble, that is coupled directly to Eq. (1) through the temperature gradient at the bubble wall. We focus on ways to estimate the temperature gradient at the bubble wall without solving the energy equation. From linear analysis of the energy equation in the frequency domain, we can write, ∂T̂ ∂y y 1 ω Ψ ω ˆ̄ T ω (3) where the transfer function Ψ ω is, 1The thermal model can readily be used with other forms of the RayleighPlesset equation that include effects of liquid compressibility. ω / D θ /π 10 10 10 10 10 10 10 10 10 0 0.1 0.2 θ → 0 θ → π / 4 ω / D α 10 10 10 10 10 10 10 10 10 10 10 10 10 α → 5 α → (ω / D ) 1/2 Figure 1. Magnitude, α, and phase, θ of the transfer function, Ψ αeiθ, versus ω D. Ψ ω iω D coth iω D 1 1