A Jacobian‐free Newton–Krylov method for thermalhydraulics simulations

Abstract

SummaryThe current paper is focused on investigating a Jacobian‐free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two‐phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two‐fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well‐known benchmarks are considered, the water faucet and the oscillating manometer.An isentropic two‐fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed.For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.

Applications of high-resolution spatial discretization scheme and Jacobian-free Newton–Krylov method in two-phase flow problems

“…However, to our best knowledge, there are no fully satisfactory solutions. A recent work [20], using the JFNK method in solving two-phase flow problems, was not able to simulate such phenomena correctly. The purposes of the linear advection test with phase appearance/disappearance presented in this subsection are two-folds: 1) to test the robustness of the JFNK method in implicitly solving the phase appearance/disappearance phenomenon caused by advection; and 2) to demonstrate the capability of high-resolution spatial discretization scheme and high-order fully implicit time integration scheme in capturing sharp discontinuities.…”

“…Under the steady-state conditions, the Ransom's solutions (Equations (18) and (19)) simply become, (20) and (21) Numerical simulations were performed using different cell numbers, from 12 to 192, and a fixed time step of 5×10 −3 s. Figure 6 shows the numerical results of void fraction distribution at 0.5 s (100 time steps). The results are very similar to those found in the literatures such as the RELAP5 validation report [25], and the work done by Coquel et al [26] and García Cascales [27].…”

“…The void fraction distribution at the steady state can be obtained from the liquid phase mass equation: (27) Comparing this set of analytical solutions (Equations (25) and (27)) with Ransom's solutions (Equations (20) and (21)), it can be found that the only difference is that in our analytical solutions the effective gravity replaces the gravity in Ransom's solutions. Performing the same mesh refinement as done previously, and using equations (25) and (27) as the reference solutions, a second-order spatial accuracy could be obtained ( Table 5).…”

“…All Mousseau's works, however, were focused on a first-order upwind spatial discretization scheme. A recent work to apply the JFNK method in solving two-phase flow problems was done by Ashrafizadeh et al [20]. The spatial discretization scheme is based on a cell centered AUSM+ method, however they were not able to correctly simulate the phase appearance/disappearance problem using the JFNK method.…”

Implicitly solving phase appearance and disappearance problems using two-fluid six-equation model

“…In his work, the momentum equation of the vanishing phase is modified to avoid singularity, and mass and energy equations are manipulated when one of the two phases is expected to vanish. In a recent study done by Ashrafizadeh et al [11], a JFNK method was used to solve one-dimensional two-phase flow problems with an implicit time integration scheme. An extended AUSM+ scheme and a phase appearance/disappearance treatment strategy, originally proposed in [12], were adapted in Ashrafizadeh's work.…”