ON THE ANALATICAL SOLUTIONS AND NUMERICAL VERIFICATIONS OF THE TWO-PHASE WATER FAUCET PROBLEM

摘要

The one-dimensional water faucet problem is one of the benchmark problems originally proposed by Ransom for two-phase flow studies. The test problem consists of a vertical pipe, which is initially filled with a uniform column of liquid moving with a prescribed initial velocity and an annulus of gas sitting still. At the top boundary, liquid is supplied at the same velocity as the initial velocity, and the bottom of the pipe opens to the ambient. Due to the gravity effect, the liquid column accelerates and becomes thinner as it descends. With certain simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. This test problem and its analytical solutions have been widely used for the purposes of code assessment, benchmark and numerical verifications. In our previous study, it was used for the mesh convergence study of a high-resolution spatial discretization scheme. It was found that, at the steady state, an expected second-order spatial accuracy could not be achieved when compared to the existing analytical solutions. A further investigation showed that the existing analytical solutions do not actually satisfy the commonly used two-fluid single-pressure two-phase flow equations. In this work, we will demonstrate the derivation of an extension of Ransom's transient solutions with the assumption of separate water and gas pressures and a new set of analytical solutions for the steady-state conditions with the single pressure assumption. The transient analytical solutions are compared to the numerical results with a first-order and a high-resolution spatial discretization schemes. The steady state analytical solutions are used for mesh convergence studies, from which expected second-order of accuracy is achieved for the 2nd order scheme.