一维气液两相漂移模型全隐式AUSMV算法研究
|
摘要
气液两相漂移模型显式AUSMV(advection upstream splitting method combined with flux vector splitting method)算法的时间步长受限于CFL(Courant-Friedrichs-Lewy)条件,为了提高计算效率,建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS(flux vector splitting)格式构造连续方程和运动方程的对流项数值通量,AUSM格式构造压力项数值通量.离散控制方程是非线性方程组,采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题,结果分析表明:全隐式AUSMV算法,色散效应小,无数值震荡,计算精度高.在压力波波速高的条件下,可以显著提高计算效率,耗散效应小.
The time step of the explicit AUSMV(advection upstream splitting method combined with flux vector splitting) algorithm is limited by the CFL(Courant-Friedrichs-Lewy) conditions. To improve computational efficiency, an implicit AUSMV algorithm was proposed for the gas-liquid two-phase drift flux model. The numerical flux of convective terms in the continuity equations and motion equations was set up with the AUSM scheme plus the FVS(flux vector splitting) scheme, while the numerical flux of pressure terms in the motion equations was built with the AUSM scheme. The nonlinear dynamical discrete governing equation system was solved numerically with the 6th-order Newtonian method and the numerical Jacobian matrix. The classical test examples were simulated, which involved the Zuber-Findlay shock tube problem and the variable mass flow problem with complex slip relation. The numerical results show that, the implicit AUSMV algorithm has small dispersion effects, no numerical oscillation and high computational accuracy. Under the condition of high pressure wave velocity, the algorithm has superior calculation efficiency with low dissipation effects.
The time step of the explicit AUSMV(advection upstream splitting method combined with flux vector splitting) algorithm is limited by the CFL(Courant-Friedrichs-Lewy) conditions. To improve computational efficiency, an implicit AUSMV algorithm was proposed for the gas-liquid two-phase drift flux model. The numerical flux of convective terms in the continuity equations and motion equations was set up with the AUSM scheme plus the FVS(flux vector splitting) scheme, while the numerical flux of pressure terms in the motion equations was built with the AUSM scheme. The nonlinear dynamical discrete governing equation system was solved numerically with the 6th-order Newtonian method and the numerical Jacobian matrix. The classical test examples were simulated, which involved the Zuber-Findlay shock tube problem and the variable mass flow problem with complex slip relation. The numerical results show that, the implicit AUSMV algorithm has small dispersion effects, no numerical oscillation and high computational accuracy. Under the condition of high pressure wave velocity, the algorithm has superior calculation efficiency with low dissipation effects.
引文
[1] ZUBER N, FINDLAY J A. Average volumetric concentration in two-phase flow systems[J]. Journal of Heat Transfer, 1965, 87(4): 453-468.
[2] BOURé J A. Wave phenomena and one-dimensional two-phase flow models[J]. Multiphase Science and Technology, 1997, 9(1): 63-107.
[3] EVJE S, FL?TTEN T. Hybrid flux-splitting schemes for a common two-fluid model[J]. Journal of Computational Physics, 2003, 192(1): 175-210.
[4] EVJE S, FL?TTEN T. On thewave structure of two-phase flow models[J]. SIAM Journal on Applied Mathematics, 2006, 67(2): 487-511.
[5] LIOU M S, STEFFEN C J. A new flux splitting scheme[J]. Journal of Computational Physics, 1993, 107(1): 23-29.
[6] KITAMURA K, SHIMA E. Towards shock-stable and accurate hypersonic heating computations: a new pressure flux for AUSM-family schemes[J]. Journal of Computational Physics, 2013, 245: 62-83.
[7] CHANG C H, LIOU M S. A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM+-up scheme[J]. Journal of Computational Physics, 2008, 227(1): 840-873.
[8] EVJE S, FJELDE K K. Hybrid flux-splitting schemes for a two-phase flow model[J]. Journal of Computational Physics, 2002, 175(2): 674-701.
[9] EVJE S, FJELDE K K. On a rough AUSM scheme for a one-dimensional two-phase model[J]. Computers & Fluids, 2003, 32(10): 1497-1530.
[10] NIU Y Y, LIN Y C, CHANG C H. A further work on multi-phase two-fluid approach for compressible multi-phase flows[J]. International Journal for Numerical Method in Fluids, 2008, 58(8): 879-896.
[11] NIU Y Y. Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD[J]. Journal of Computational Physics, 2016, 308: 389-410.
[12] KITAMURA K, NONOMURA T. Simple and robust HLLC extensions of two-fluid AUSM for multiphase flow computations[J]. Computers & Fluids, 2014, 100: 321-335.
[13] EVJE S, FL?TTEN T. Weaklyimplicit numerical schemes for a two-fluid model[J]. SIAM Journal on Scientific Computing, 2005, 26(5): 1449-1484.
[14] EVJE S, FL?TTEN T. CFL-violating numerical schemes for a two-fluid model[J]. Journal of Scientific Computing, 2006, 29(1): 83-114.
[15] COLONIA S, STEIJL R, BARAKOS G N. Implicit implementation of the AUSM+ and AUSM+-up schemes[J]. International Journal for Numerical Method in Fluids, 2014, 75(10): 687-712.
[16] ONUR O, EYI S. Effects of the Jacobian evaluation on Newton’s solution of the Euler equations[J]. International Journal for Numerical Method in Fluids, 2005, 49(2): 211-231.
[17] ZENG Q L, AYDERMIR N U, LIEN F S, et al. Comparison of implicit and explicit AUSM-family schemes for compressible multiphase flows[J]. International Journal for Numerical Method in Fluids, 2015, 77(1): 43-61.
[18] ZENG Q L, AYDERMIR N U, LIEN F S, et al. Extension of staggered-grid-based AUSM-family schemes for use in nuclear safety analysis codes[J]. International Journal of Multiphase Flow, 2017, 93:17-32.
[19] 徐朝阳, 孟英峰, 魏纳, 等. 一维气液两相漂移模型的AUSMV算法研究[J]. 应用数学和力学, 2014, 35(12): 1373-1382.(XU Chaoyang, MENG Yingfeng, WEI Na, et al. Research on the AUSMV scheme for 1D gas liquid two phase flow drift flux models[J]. Applied Mathematics and Mechanics, 2014, 35(12): 1373-1382.(in Chinese))
[20] MADHU K. Sixth order Newton-type method for solving system of nonlinear equations and its applications[J]. Applied Mathematics E: Notes, 2017, 17: 221-230.
[21] FL?TTEN T, MUNKEJORD S T. The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model[J]. ESAIM: Mathematical Modelling and Numerical Analysis, 2006, 40(4): 735-764.
[2] BOURé J A. Wave phenomena and one-dimensional two-phase flow models[J]. Multiphase Science and Technology, 1997, 9(1): 63-107.
[3] EVJE S, FL?TTEN T. Hybrid flux-splitting schemes for a common two-fluid model[J]. Journal of Computational Physics, 2003, 192(1): 175-210.
[4] EVJE S, FL?TTEN T. On thewave structure of two-phase flow models[J]. SIAM Journal on Applied Mathematics, 2006, 67(2): 487-511.
[5] LIOU M S, STEFFEN C J. A new flux splitting scheme[J]. Journal of Computational Physics, 1993, 107(1): 23-29.
[6] KITAMURA K, SHIMA E. Towards shock-stable and accurate hypersonic heating computations: a new pressure flux for AUSM-family schemes[J]. Journal of Computational Physics, 2013, 245: 62-83.
[7] CHANG C H, LIOU M S. A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM+-up scheme[J]. Journal of Computational Physics, 2008, 227(1): 840-873.
[8] EVJE S, FJELDE K K. Hybrid flux-splitting schemes for a two-phase flow model[J]. Journal of Computational Physics, 2002, 175(2): 674-701.
[9] EVJE S, FJELDE K K. On a rough AUSM scheme for a one-dimensional two-phase model[J]. Computers & Fluids, 2003, 32(10): 1497-1530.
[10] NIU Y Y, LIN Y C, CHANG C H. A further work on multi-phase two-fluid approach for compressible multi-phase flows[J]. International Journal for Numerical Method in Fluids, 2008, 58(8): 879-896.
[11] NIU Y Y. Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD[J]. Journal of Computational Physics, 2016, 308: 389-410.
[12] KITAMURA K, NONOMURA T. Simple and robust HLLC extensions of two-fluid AUSM for multiphase flow computations[J]. Computers & Fluids, 2014, 100: 321-335.
[13] EVJE S, FL?TTEN T. Weaklyimplicit numerical schemes for a two-fluid model[J]. SIAM Journal on Scientific Computing, 2005, 26(5): 1449-1484.
[14] EVJE S, FL?TTEN T. CFL-violating numerical schemes for a two-fluid model[J]. Journal of Scientific Computing, 2006, 29(1): 83-114.
[15] COLONIA S, STEIJL R, BARAKOS G N. Implicit implementation of the AUSM+ and AUSM+-up schemes[J]. International Journal for Numerical Method in Fluids, 2014, 75(10): 687-712.
[16] ONUR O, EYI S. Effects of the Jacobian evaluation on Newton’s solution of the Euler equations[J]. International Journal for Numerical Method in Fluids, 2005, 49(2): 211-231.
[17] ZENG Q L, AYDERMIR N U, LIEN F S, et al. Comparison of implicit and explicit AUSM-family schemes for compressible multiphase flows[J]. International Journal for Numerical Method in Fluids, 2015, 77(1): 43-61.
[18] ZENG Q L, AYDERMIR N U, LIEN F S, et al. Extension of staggered-grid-based AUSM-family schemes for use in nuclear safety analysis codes[J]. International Journal of Multiphase Flow, 2017, 93:17-32.
[19] 徐朝阳, 孟英峰, 魏纳, 等. 一维气液两相漂移模型的AUSMV算法研究[J]. 应用数学和力学, 2014, 35(12): 1373-1382.(XU Chaoyang, MENG Yingfeng, WEI Na, et al. Research on the AUSMV scheme for 1D gas liquid two phase flow drift flux models[J]. Applied Mathematics and Mechanics, 2014, 35(12): 1373-1382.(in Chinese))
[20] MADHU K. Sixth order Newton-type method for solving system of nonlinear equations and its applications[J]. Applied Mathematics E: Notes, 2017, 17: 221-230.
[21] FL?TTEN T, MUNKEJORD S T. The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model[J]. ESAIM: Mathematical Modelling and Numerical Analysis, 2006, 40(4): 735-764.