位移浅水内孤立波
|
摘要
研究两层浅水系统中的内孤立波,该系统由两层常密度不可压缩无黏性水组成。利用Lagrange坐标和Hamilton原理,推导了两层浅水系统的位移浅水内波方程,并进一步导出了两层浅水系统的位移内孤立波解。数值实验表明,位移内孤立波与经典的KdV内孤立波吻合很好,说明Lagrange坐标和Hamilton方法适用于内波分析,可以为构造内波分析的保辛方法提供一种途径。
Here it is addressed that the internal solitary wave in a water system,which consists of two layers of constant-density incompressible inviscid water.By using the Lagrange coordinate and Hamilton principle,an internal wave equation for shallow water displacement of the two-water system and the corresponding internal solitary displacement wave(DISW) are derived.The numerical tests show good agreement between the DISW and the classical KdV solitary wave,which means it is effective to use the Lagrange coordinate and Hamilton principle to analyze the internal wave problem,and thence a way to develop the symplectic method for it.
Here it is addressed that the internal solitary wave in a water system,which consists of two layers of constant-density incompressible inviscid water.By using the Lagrange coordinate and Hamilton principle,an internal wave equation for shallow water displacement of the two-water system and the corresponding internal solitary displacement wave(DISW) are derived.The numerical tests show good agreement between the DISW and the classical KdV solitary wave,which means it is effective to use the Lagrange coordinate and Hamilton principle to analyze the internal wave problem,and thence a way to develop the symplectic method for it.
引文
[1] Sutherland B R.Internal Gravity Waves[M].New York:Cambridge University Press,2009.
[2] 程友良.分层流体中内孤波的研究进展[J].力学进展,1998,28(3):383-391.(CHENG You-liang.Advances in studies of internal solitary waves in stratified fluids:A review[J].Advances in Mechanics,1998,28(3):383-391.(in Chinese))
[3] Chen B F,Huang Y J,Chen B,et al.Surface wave disturbance during internal wave propagation over various types of sea bottoms[J].Ocean Engineering,2016,125:214-225.
[4] 李家春.水面下的波浪——海洋内波[J].力学与实践,2005,27(2):1-6.(LI Jia-chun.Billow under the sea surface —internal waves in the ocean[J].Mechanics in Engineering,2005,27(2):1-6.(in Chinese))
[5] Chen M,Chen K,You Y X.Experimental investigation of internal solitary wave forces on a semi-submersible[J].Ocean Engineering,2017,141:205-214.
[6] 钟万勰,姚征.位移法浅水孤立波[J].大连理工大学学报,2006,46(1):151-156.(ZHONG Wan-xie,YAO Zheng.Shallow water solitary waves based on displacement method[J].Journal of Dalian University of Technology,2006,46(1):151-156.(in Chinese))
[7] 钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2006.(ZHONG Wan-xie.Symplectic Solution Methodology in Applied Mechanics[M].Beijing:High Education Press,2006.(in Chinese))
[8] 钟万勰,吴锋.力-功-能-辛-离散[M].大连:大连理工大学出版社,2016.(ZHONG Wan-xie,WU Feng.Force - Work - Energy - Symplecticity - Discretization [M].Dalian:Dalian University of Technology Press,2016.(in Chinese))
[9] 钟万勰,吴锋,孙雁,等.浅水机械激波[J].应用数学和力学,2017,38(8):845-852.(ZHONG Wan-xie,WU Feng,SUN Yan,et al.Shallow water mechanical shock wave[J].Applied Mathematics and Mechanics,2017,38(8):845-852.(in Chinese))
[10] 吴锋.基于位移的水波数值模拟—辛方法[M].大连:大连理工大学出版社,2017.(WU Feng.Numerical Modeling of Water Waves Based on Displacement:Symplectic Method[M].Dalian:Dalian University of Technology Press,2017.(in Chinese))
[11] 姚征,钟万勰.位移法浅水波方程解及其特性[J].计算机辅助工程,2016,25(2):1-4,13.(YAO Zheng,ZHONG Wan-xie.Solutions and characteristics of shallow water equation based on displacement method[J].Computer Aided Engineering,2016,25(2):1-4,13.(in Chinese))
[12] 钟万勰,陈晓辉.浅水波的位移法求解[J].水动力学研究与进展(A辑),2006,21(4):486-493.(ZHONG Wan-xie,CHEN Xiao -hui.Solving shallow water waves with the displacement method[J].Journal of Hydrodynamics(Ser A),2006,21(4):486-493.(in Chinese))
[13] 吴锋,钟万勰.浅水问题的约束Hamilton变分原理及祖冲之类保辛算法[J].应用数学和力学,2016,37(1):1-13.(WU Feng,ZHONG Wan-xie.The constrained Hamilton variational principle for shallow water problems and the Zu-type symplectic algorithm[J].Applied Mathematics and Mechanics,2016,37(1):1-13.(in Chinese))
[14] Wu F,Zhong W X.A shallow water equation based on displacement and pressure and its numerical solution[J].Environmental Fluid Mechanics,2017,17(6):1099-1126.
[15] 钟万勰,高强,彭海军.经典力学辛讲[M].大连:大连理工大学出版社,2013.(ZHONG Wan-xie,GAO Qiang,PENG Hai-jun.Classical Mechanics-Its Symplectic Description[M].Dalian:Dalian University of Technology Press,2013.(in Chinese))
[16] 吴锋,钟万勰.基于祖冲之类方法具有保辛性[J].计算力学学报,2015,32(4):447-450.(WU Feng,ZHONG Wan-xie.The Zu-type method is symplectic[J].Chinese Journal of Computational Mechanics,2015,32(4):447-450.(in Chinese))
[17] Wu F,Zhong W X.On displacement shallow water wave equation and symplectic solution[J].Computer Methods in Applied Mechanics and Engineering,2017,318:431-455.
[18] Liu P,Li Z L,Luo R Z.Modified (2+1)-dimensional displacement shallow water wave system:Symmetries and exact solutions[J].Applied Mathematics and Computation,2012,219(4):2149-2157.
[19] Liu P,Fu P K.Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions[J].Chinese Physics B,2011,20(9):090203.
[20] Liu P,Lou S Y.A (2+1)-dimensional displacement shallow water wave system[J].Chinese Physics Let-ters,2008,25(9):3311-3314.
[21] Wu F,Yao Z,Zhong W X.Fully nonlinear (2+1)-dimensional displacement shallow water wave equation[J].Chinese Physics B,2017,26(5):054501.
[22] Kinnmark I.The Shallow Water Wave Equations:Formulation,Analysis and Application[M].Berlin:Springer,1986.
[23] Kim D H,Lynett P J.Dispersive and nonhydrostatic pressure effects at the front of surge[J].Journal of Hydraulic Engineering,2011,137(7):754-765.
[2] 程友良.分层流体中内孤波的研究进展[J].力学进展,1998,28(3):383-391.(CHENG You-liang.Advances in studies of internal solitary waves in stratified fluids:A review[J].Advances in Mechanics,1998,28(3):383-391.(in Chinese))
[3] Chen B F,Huang Y J,Chen B,et al.Surface wave disturbance during internal wave propagation over various types of sea bottoms[J].Ocean Engineering,2016,125:214-225.
[4] 李家春.水面下的波浪——海洋内波[J].力学与实践,2005,27(2):1-6.(LI Jia-chun.Billow under the sea surface —internal waves in the ocean[J].Mechanics in Engineering,2005,27(2):1-6.(in Chinese))
[5] Chen M,Chen K,You Y X.Experimental investigation of internal solitary wave forces on a semi-submersible[J].Ocean Engineering,2017,141:205-214.
[6] 钟万勰,姚征.位移法浅水孤立波[J].大连理工大学学报,2006,46(1):151-156.(ZHONG Wan-xie,YAO Zheng.Shallow water solitary waves based on displacement method[J].Journal of Dalian University of Technology,2006,46(1):151-156.(in Chinese))
[7] 钟万勰.应用力学的辛数学方法[M].北京:高等教育出版社,2006.(ZHONG Wan-xie.Symplectic Solution Methodology in Applied Mechanics[M].Beijing:High Education Press,2006.(in Chinese))
[8] 钟万勰,吴锋.力-功-能-辛-离散[M].大连:大连理工大学出版社,2016.(ZHONG Wan-xie,WU Feng.Force - Work - Energy - Symplecticity - Discretization [M].Dalian:Dalian University of Technology Press,2016.(in Chinese))
[9] 钟万勰,吴锋,孙雁,等.浅水机械激波[J].应用数学和力学,2017,38(8):845-852.(ZHONG Wan-xie,WU Feng,SUN Yan,et al.Shallow water mechanical shock wave[J].Applied Mathematics and Mechanics,2017,38(8):845-852.(in Chinese))
[10] 吴锋.基于位移的水波数值模拟—辛方法[M].大连:大连理工大学出版社,2017.(WU Feng.Numerical Modeling of Water Waves Based on Displacement:Symplectic Method[M].Dalian:Dalian University of Technology Press,2017.(in Chinese))
[11] 姚征,钟万勰.位移法浅水波方程解及其特性[J].计算机辅助工程,2016,25(2):1-4,13.(YAO Zheng,ZHONG Wan-xie.Solutions and characteristics of shallow water equation based on displacement method[J].Computer Aided Engineering,2016,25(2):1-4,13.(in Chinese))
[12] 钟万勰,陈晓辉.浅水波的位移法求解[J].水动力学研究与进展(A辑),2006,21(4):486-493.(ZHONG Wan-xie,CHEN Xiao -hui.Solving shallow water waves with the displacement method[J].Journal of Hydrodynamics(Ser A),2006,21(4):486-493.(in Chinese))
[13] 吴锋,钟万勰.浅水问题的约束Hamilton变分原理及祖冲之类保辛算法[J].应用数学和力学,2016,37(1):1-13.(WU Feng,ZHONG Wan-xie.The constrained Hamilton variational principle for shallow water problems and the Zu-type symplectic algorithm[J].Applied Mathematics and Mechanics,2016,37(1):1-13.(in Chinese))
[14] Wu F,Zhong W X.A shallow water equation based on displacement and pressure and its numerical solution[J].Environmental Fluid Mechanics,2017,17(6):1099-1126.
[15] 钟万勰,高强,彭海军.经典力学辛讲[M].大连:大连理工大学出版社,2013.(ZHONG Wan-xie,GAO Qiang,PENG Hai-jun.Classical Mechanics-Its Symplectic Description[M].Dalian:Dalian University of Technology Press,2013.(in Chinese))
[16] 吴锋,钟万勰.基于祖冲之类方法具有保辛性[J].计算力学学报,2015,32(4):447-450.(WU Feng,ZHONG Wan-xie.The Zu-type method is symplectic[J].Chinese Journal of Computational Mechanics,2015,32(4):447-450.(in Chinese))
[17] Wu F,Zhong W X.On displacement shallow water wave equation and symplectic solution[J].Computer Methods in Applied Mechanics and Engineering,2017,318:431-455.
[18] Liu P,Li Z L,Luo R Z.Modified (2+1)-dimensional displacement shallow water wave system:Symmetries and exact solutions[J].Applied Mathematics and Computation,2012,219(4):2149-2157.
[19] Liu P,Fu P K.Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions[J].Chinese Physics B,2011,20(9):090203.
[20] Liu P,Lou S Y.A (2+1)-dimensional displacement shallow water wave system[J].Chinese Physics Let-ters,2008,25(9):3311-3314.
[21] Wu F,Yao Z,Zhong W X.Fully nonlinear (2+1)-dimensional displacement shallow water wave equation[J].Chinese Physics B,2017,26(5):054501.
[22] Kinnmark I.The Shallow Water Wave Equations:Formulation,Analysis and Application[M].Berlin:Springer,1986.
[23] Kim D H,Lynett P J.Dispersive and nonhydrostatic pressure effects at the front of surge[J].Journal of Hydraulic Engineering,2011,137(7):754-765.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |