so it takes much more CPU time and memory so it is expensive to due with. It is often the case that researchersassume that the thermal behavior inside a bubble is adiabatic or isothermal. This makes easier programming and1
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CAV2001:sessionB6.002quicker simulation. But the results of those conventional methods (adiabatic and isothermal model) are not goodenough in several cases. Therefore, making a simple model that gives good agreement with DNS but requiresless CPU time and memory is a big problem to simulate cavitating flows. Yongliang (1995) proposed anempirical model of pressure difference to the change of cavitation bubble size (density). Matsumoto (1998) usedswitching model to predict the processes of an oscillating bubble. Prosperetti (1991) studied the polytropicmodel 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 theway, there is, at least, a method that can avoid using bubble Dynamics equation (as well as the model of thermalbehavior inside a bubble). Alajbegovic (1999) used a method to compute bubble number density directlywithout taking care of bubble Dynamics equation. But this method is not in the scope of our consideration. Theaim of this work is to make a model of thermal behavior inside a oscillating bubble that is easy to use, requiresless computational time and memory but gives results reliable (herein comparing to DNS results).2 Simulation MethodsOn 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 bubblemotion with respect to the liquid compressibility and the phase change at the bubble wall (Fujikawa (1980)). Inthe present model, the liquid phase is solved using the Rayleigh-Plesset equation and the constitutive equation ofthe 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. RrGrTKRRPdtd13313(1)According to the order estimation written in a paper of Prosperetti (1988), we propose the temperaturegradient model at the bubble-water interface expressed below. 2/10DtTTrTsbRr(2)Where Tbis average temperature of bubble. This temperature is calculated by assuming that gas insidebubble behaves itself as ideal gas.
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