JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow

Abstract The Jacobian-free Newton-Krylov (JFNK) method with an efficient physics-based preconditioner is applied for the numerical solution of the one-dimensional drift-flux model with the closure constitutive equations. Additional closure correlations, including the flow pattern dependent heat transfer correlations and the flow pattern independent kinematic constitutive correlation, are used to close the governing equations of one-dimensional drift-flux model. The governing equations have been discretized using the first-order upwind method for spatial discretization and the fully implicit method for temporal discretization. An efficient physics-based preconditioner derived from the semi-implicit solution of governing equations is used in the JFNK method to improve the efficiency and numerical stability. The numerical verification and code validation have been performed for subcooled boiling two-phase flow in a vertical tube. By comparing with the other methods (JFNK method without preconditioner, the Broyden method and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method), the preconditioning JFNK method shows the robustness and a good computational efficiency. Moreover, by comparing the numerical simulation results with the experimental results, it is found that the JFNK method with a physics-based preconditioner shows the good accuracy for the numerical simulation for one-dimensional boiling two-phase flow.

[1]  Glen Hansen,et al.  Parallel multiphysics algorithms and software for computational nuclear engineering , 2009 .

[2]  William J. Rider,et al.  CONSISTENT METRICS FOR CODE VERIFICATION , 2002 .

[3]  A. Ashrafizadeh.  Investigation of a Jacobian-free Newton-Krylov solution to multiphase flows , 2014 .

[4]  Katherine J. Evans,et al.  Implementation of the Jacobian-free Newton-Krylov method for solving the first-order ice sheet momentum balance , 2011, J. Comput. Phys..

[5]  I. Mudawar,et al.  Universal approach to predicting two-phase frictional pressure drop for adiabatic and condensing mini/micro-channel flows , 2012 .

[6]  Afshin J. Ghajar,et al.  A flow pattern independent drift flux model based void fraction correlation for a wide range of gas–liquid two phase flow , 2014 .

[7]  Andrzej A. Wyszogrodzki,et al.  An implicitly balanced hurricane model with physics-based preconditioning , 2005 .

[8]  Henk A. Dijkstra,et al.  The application of Jacobian-free Newton-Krylov methods to reduce the spin-up time of ocean general circulation models , 2010, J. Comput. Phys..

[9]  M. Ishii.  Thermo-fluid dynamic theory of two-phase flow , 1975 .

[10]  D. Mavriplis.  An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers , 2001 .

[11]  Haihua Zhao,et al.  Applications of high-resolution spatial discretization scheme and Jacobian-free Newton–Krylov method in two-phase flow problems , 2015 .

[12]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[13]  David E. Keyes,et al.  Nonlinearly Preconditioned Inexact Newton Algorithms , 2002, SIAM J. Sci. Comput..

[14]  M. Ishii,et al.  Thermo-Fluid Dynamics of Two-Phase Flow , 2007 .

[15]  Dana A. Knoll,et al.  Jacobian-Free Newton-Krylov Methods and Physics-based Preconditioning for Problems in Computational Geophysics (Invited) , 2009 .

[16]  Dana A. Knoll,et al.  An Implicit Nonlinearly Consistent Method for the Two-Dimensional Shallow-Water Equations with Coriolis Force , 2002 .

[17]  Haihua Zhao,et al.  Application of Jacobian-free Newton–Krylov method in implicitly solving two-fluid six-equation two-phase flow problems: Implementation, validation and benchmark , 2016 .

[18]  M. Ishii,et al.  One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes , 2003 .

[19]  K. V. Moore,et al.  RETRAN: a program for one-dimensional transient thermal--hydraulic analysis of complex fluid flow systems. Volume I. Equations and numerics. Final report. [PWR and BWR] , 1977 .

[20]  Rongpei Zhang,et al.  On linearization and preconditioning for radiation diffusion coupled to material thermal conduction equations , 2013, J. Comput. Phys..

[21]  Glenn E. Hammond,et al.  Application of Jacobian-free Newton–Krylov with physics-based preconditioning to biogeochemical transport , 2005 .

[22]  M. Z. Podowski,et al.  TWO-PHASE FLOW DYNAMICS , 1992 .

[23]  A. Ashrafizadeh,et al.  A Jacobian‐free Newton–Krylov method for thermalhydraulics simulations , 2015 .

[24]  Rolf Walder,et al.  A Jacobian-free Newton-Krylov method for time-implicit multidimensional hydrodynamics , 2015, 1512.03662.

[25]  Jorge Nocedal,et al.  Theory of algorithms for unconstrained optimization , 1992, Acta Numerica.

[26]  William J. Rider,et al.  Physics-Based Preconditioning and the Newton-Krylov Method for Non-equilibrium Radiation Diffusion , 2000 .

[27]  Ronald B. Morgan,et al.  Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations , 2000, SIAM J. Matrix Anal. Appl..

[28]  Neil E. Todreas,et al.  COBRA IIIcMIT-2 : a digital computer program for steady state and transient thermal-hydraulic analysis of rod bundle nuclear fuel elements , 1981 .

[29]  N. Zuber,et al.  POINT OF NET VAPOR GENERATION AND VAPOR VOID FRACTION IN SUBCOOLED BOILING , 1974 .

[30]  N. Zuber,et al.  Average volumetric concentration in two-phase flow systems , 1965 .

[31]  Haihua Zhao,et al.  Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with Jacobian-free Newton–Krylov method , 2016 .

[32]  I. Mudawar,et al.  Universal approach to predicting two-phase frictional pressure drop for mini/micro-channel saturated flow boiling , 2013 .

[33]  B. Krasnopolsky,et al.  Application of the Jacobian-Free Newton-Krylov method for multiphase pipe flows , 2015 .

[34]  Saeid Abbasbandy,et al.  Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method , 2003, Appl. Math. Comput..

[35]  H. Kretzschmar,et al.  The IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam , 2000 .

[36]  Andrzej A. Wyszogrodzki,et al.  An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows , 2003 .

[37]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[38]  D. Knoll,et al.  Jacobian-Free Newton-Krylov Discontinuous Galerkin (JFNK-DG) Method and Its Physics-Based Preconditioning for All-Speed Flows , 2007 .

[39]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .