Application of implicit Roe-type scheme and Jacobian-Free Newton-Krylov method to two-phase flow problems

A new implicit second-order accurate spatial scheme for steady-state thermal-hydraulic simulations of the two-phase two-fluid six-equation model is proposed. The new scheme is based on a Roe-type numerical flux that is formulated with the help of analytical approximate eigenvalues and eigenvectors of the two-phase system. Approximate eigenvalues and eigenvectors are obtained with a structured Jacobian matrix that is general for arbitrary Equation of State (EOS). Three issues are solved in this article: (1) A fully-implicit scheme using the backward Euler method is proposed to improve the stability and avoid the time step limit of the originally explicit scheme. The fully implicit scheme is solved with the Jacobian-Free Newton-Krylov method. (2) A second-order accurate spatial scheme is proposed by converting the first-order Roe-type numerical flux to a second-order one. The conversion is made by extending an existing procedure for single-phase flows to two-phase flows. (3) Phase appearance issue is treated with a numerical procedure, which requires little modification to the schemes. The new scheme is verified and validated with two numerical tests: the faucet flow problem and the BFBT benchmark.