cav2001_sessionB6.002.doc

It is often the case that researchers assume that the

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so it takes much more CPU time and memory so it is expensive to due with. It is often the case that researchers assume that the thermal behavior inside a bubble is adiabatic or isothermal. This makes easier programming and 1
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CAV2001:sessionB6.002 quicker simulation. But the results of those conventional methods (adiabatic and isothermal model) are not good enough in several cases. Therefore, making a simple model that gives good agreement with DNS but requires less CPU time and memory is a big problem to simulate cavitating flows. Yongliang (1995) proposed an empirical model of pressure difference to the change of cavitation bubble size (density). Matsumoto (1998) used switching model to predict the processes of an oscillating bubble. Prosperetti (1991) studied the polytropic model to calculate the proper polytropic index used to describe the thermal behavior in bubble. Anyway, Matsumoto (1999) concluded that this model is well defined only in the framework of a linear theory. By the way, there is, at least, a method that can avoid using bubble Dynamics equation (as well as the model of thermal behavior inside a bubble). Alajbegovic (1999) used a method to compute bubble number density directly without taking care of bubble Dynamics equation. But this method is not in the scope of our consideration. The aim of this work is to make a model of thermal behavior inside a oscillating bubble that is easy to use, requires less computational time and memory but gives results reliable (herein comparing to DNS results). 2 Simulation Methods On solving the bubble motion, the following assumptions are employed: (1) Gases inside the bubble and the surrounding liquid move maintaining spherical symmetry. (2) Gases inside the bubble obey the perfect gas law. (3) The vapor, mist generation and diffusion of non-condensable gas in liquid are neglected. In the DNS, the full conservation equations for mass, momentum and energy in gas are solved numerically. The motion of the liquid phase is estimated by solving the first-order approximate equation for the bubble motion with respect to the liquid compressibility and the phase change at the bubble wall (Fujikawa (1980)). In the present model, the liquid phase is solved using the Rayleigh-Plesset equation and the constitutive equation of the pressure inside a bubble is proposed. The following relation at the bubble-water interface (Prosperetti (1988)) is used to consider thermal effect on the pressure inside a bubble. R r G r T K R R P dt d 1 3 3 1 3 (1) According to the order estimation written in a paper of Prosperetti (1988), we propose the temperature gradient model at the bubble-water interface expressed below. 2 / 1 0 Dt T T r T s b R r (2) Where T b is average temperature of bubble. This temperature is calculated by assuming that gas inside bubble behaves itself as ideal gas.
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