Simulation of Cavitating Flows by the Space-Time CE/SE Method

The newly developed method of Space-Time Conservation Element and Solution Element, or the CE/SE method for short, is employed to study the complex cavitation phenomenon in this paper. The tenet of the CE/SE method is treating space and time as one entity, and the calculation of flow properties is based on the local and global space-time flux conservation. As a contrast to the modern upwind schemes, no Riemann solver is used, thus the logic of the present scheme for cavitating flows is much simpler. The numerical examples reported in this paper include (1) onedimensional internal cavitating pipe flows; and (2) twodimensional exteranl cavitating flows around a NACA0015 airfoil. While we focused on the numerical method in this paper, a simple two-phase homogeneous equilibrium cavitation model was employed in the simulation. We first demonstrated the capability of the CE/SE method to capture contact discontinuities in cavitating fluids, using a hydraulic shock problem. Then we presented the simulation results for cavitating internal pipe flows and external airfoil flows. The numerical results compared favorably with the experimental data and analytical solution. INTRODUCTION Cavitation is a very complex trwo-phase phenomemon. When local flow pressure drops under the vapor pressure of the fluid, bubbles may be formed. The collapse of these bubbles will generate high pressure waves propagating through the pipe, which will lower the performance of the system, produce vibration and noise, and even cause damage to the device surfaces [6]. Numerical simulation of cavitating flows poses unique challenges both in modeling the physics and in developing a robust numerical methodology. The numerical difficulty of simulating cavitation comes from the fundmental features of the tow-phase flow. One of the typical features is the rapid change in the acoustic velocity of the flow. When the void fraction is zero, the fluid is liquid and almost imcompressible. Therefore the acoustic velocity is very high. On the other hand, when the void fraction is 1.0, the fluid is vapor and compressible. In between, the acoustic velocity changes remarkbly and can be as low as 20m/s [12]. Hence the desired solver must be able to handle compressible and imcompressilble flows simutaniously. Another feature that cause numerical instability is the tremendous density difference between gas the liquid. The numerical scheme should be robust enough to deal with these enormous instability problems. Various numerical methods, such as the method of characteristics [8] and high resolution upwind schemes for capturing the interface of two phase flows [11], have been developed to study cavitating flows. In this paper, we employ a simple homogenious cavitation model and concentrate on the numerical scheme. We employed the CE/SE method originally developed by Chang [1]. The method is substantially different in both concept and methodology from the traditional methods. It enforces both local and global flux conservation in space-time domain and has remarkable ability to resolve discontinuity interfaces. The rest of this paper is organized as follows. In Section 2, the essence of the CE/SE method will be illustrated. In Section 3, we discuss the theoretical model of the onedimensional pipe flows and the simulation results. In Section 4, we present numerical results of twodimensinal cavitating flows around NACA0015 airfoil, using ustructured triangular mesh. We then give the concluding remarks. 3. THE CE/SE METHOD In this section, we give a brief description of the CE/SE method. For one-dimensional case, the governing equations can be written in the form of ) , ( 2 1 q q s x f t q m m m = ∂ ∂ + ∂ ∂ , (2.1) where m equals 1 and 2, representing the continuity equation and momentum equation, respectively. The source term ) , ( 2 1 q q sm is a function of q1 and q2. Let x1 (= x) and x2 (= t) be the coordinates of a two-dimensional Euclidean space E2. Thus Eq. (2.1) becomes ) , ( 2 1 q q sm m = • ∇ h (2.2) in which the current density vector is ( ) m m m q f , = h . By using Gauss’ divergence theorem in the space-time domain E2, it can be shown that Eq. (2.1) is the differential form of the integral conservation law: