A COMPARISON STUDY OF DIFFERENT CAVITATION MODELS ON THE DEVELOPMENT OF 2-D CAVITATING FLOWS

In the present study, three transport models for liquid/mass fraction developed by Singhal et al. [1], Zwart et al. [2], and Schnerr & Sauer[3], respectively, were employed for computing the cavitating flow phenomena in several 2-D hydrofoils. For each case, the computational results were compared to each other for different models. It is interesting to find that in some cases, the results agree with each other quite well; however, in other cases, they differ significantly. Furthermore, the existence of non-condensible gas in water can play an important role in the development of cavitating flow. In this paper, the cavitation characteristics, such as its size, location, and physical properties, are presented and discussed. Finally, some features on the cavitation simulation using these models are concluded. INTRODUCTION Flow cavitation is a vital, yet complicated, phenomenon in many engineering applications. For some examples, cavitation is an important factor in the design of propeller and impeller, as it could induce many detrimental effects, such as noise, structural damage, and severe efficiency reduction. In the area of biochemical engineering/biotechnology, it is known that cavitation results in the generation of hot spots, highly reactive free radicals, and turbulence associated with liquid circulation currents, which can result in the intensification of various physical/chemical operations [4]. Cavitation is also related to valves fractures of a mechanical heart cavitation and could result in damage of blood cells and increase the risk of thromboembolic complications found in patients implanted with a mechanical heart [5]. Cavitation has been one of the most important issues which need be considered in designing and operating a pump, especially when the rotating speed of pump increases because it could cause severe damage on the surface of flow channels and affect stability of system [6]. With the advancement of computational capability, rapid progress has been made it possible to reveal cavitation physics in various forms by computational fluid dynamics. Various cavitation models have been proposed in the past few decades. Roughly speaking, they can be divided into three main categories: interface tracking models, density-pressure coupling models, and transport models for liquid/mass fraction [7]. Among them, the type of models in the last category appears to be the most popular one in numerical simulations since the first one is difficult to take care of the detached cavity and the second one does not have the capability to couple with the transport phenomenon. The transport models are most suitable for large-scale cavitation, such as sheet cavitation that often occurs on hydrofoils and propellers. These models include two-way interactions between the phases. The homogeneous mixture concept has been the popular focus in the development of two-phase modelling. In this concept, the contents of individual cells are assumed to be uniform. Recently, the mixture model has been advanced rapidly with various mass transfer formulae. It is commonly assumed in the mixture model that the diffusion balance is maintained for both phases and no velocity/temperature differences between two phases for the entire computing domain. These two phases are thus regarded as one mixed pseudo-fluid. Because of this fact, the mixture model is also referred as the diffusion model. Under the premise of diffusion balance, the mixture model often incorporates with various mass transfer mechanisms derived from either bubble dynamics or vaporization-condensation process. Singhal et al. [1] proposed a mixture model based on the reduced Rayleigh-Plesset equation. They considered various first-order effects including phase change, bubble dynamics, turbulent pressure fluctuations, and non-condensable gases and thus named it “the full cavitation model. ” Later, Huuva derived a similar formulation in which the vapor mass fraction was replaced by the vapor volume fraction [8]. The model proposed by Zwart et al. [2] (Zwart-Gerber-Belamri model) is based on the simplified Rayleigh-Plesset equation. This cavitation model assumes large vapour bubbles of equal size, which do not affect each other. However, the expressions for condensation and evaporation are not symmetric. In particular, the evaporation expression takes into account that, as the vapour volume fraction increases, the nucleation site density must decrease accordingly. They proposed that the total interphase mass transfer rate per unit volume was determined by the bubble density numbers and the mass change rate of a single bubble. The study conducted by Schnerr and Sauer [3] (SchnerrSauer model) appears to be the first which leads to the conclusion that the mass transfer is mainly driven by the pressure differential. Their formulation of mass transfer rate is able to simulate the cyclic formation of cavitation cloud, the formation of re-entrant jet, and so on. In all models, empirical coefficients must be given before the cavitation physics can be determined. In fact, a common problem of mass transfer models is that they rely on empirical parameters which need to be tuned for specific flow problems and are not known a priori. Recently, some studies on the influence of these coefficients on predicting cavitating flows have been conducted. Employing the Zwart-Gerber-Belamri model [2] as an example, Liu et al. [9] investigated their influence on the pump cavitation performance prediction and carried out experiments to validate the computational results. For the mass transfer model using the vaporization-condensation mechanism, it is assumed that the pressure in the cavity keeps unchanged and the mass change rate is proportional in some way to the difference of liquid and vapor pressure. Based on these concepts, several models have been proposed in the literature. Saito et al. [10] proposed a model according to the theory of evaporation/condensation on a plane surface. Okita and Kajishima [11] formulated a mass transfer in the form of volume fraction. The model proposed by Kunz et al. [12] is somewhat complicated. The derived the mass transfer which was based on two different strategies for creation and destruction of liquid. The transformation of liquid to vapor was assumed to be proportional to the amount by which the pressure was below the vapor pressure. The transformation of vapor to liquid, however, was based on a third order polynomial function of volume fraction. In the present study, three mixture models were employed for computing the cavitating flow phenomena in several 2-D hydrofoils. They are the full cavitation model, the Zwart-GerberBelamri model, and the Schnerr-Sauer model. The commercial code FLUENT was employed for the study and all these three models are available in the code. Some particular turbulence models were used. The details are described in the next section. THEORETICAL FORMULATION AND COMPUTATIONAL STRATEGY Physical problem under consideration We consider two-dimensional viscous liquid flow past various hydrofoils. Figure 1 depicts the typical flow physics. The angle of attack of the incident flow is . A cavitation bubble is formed somewhere on the airfoil. Considering the cavitation physics, we can express the two-dimensional governing equations as follows, 0 ) (       m m m t u   (1)