用三维雷诺时均模型模拟小尺度组合桩柱上的波浪力
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摘要
波浪是海岸、海洋工程最重要的动力因素之一。海上结构物所承受的波浪荷载通常是结构物设计的控制荷载,它对工程的造价、安全度及寿命起着重要作用。因此,研究海浪对结构物的作用是很重要的。海岸和海洋工程中,把特征尺度D与波浪的特征波长L的比值满足D / L≤0.15的结构物称为小尺度结构物,小尺度桩柱的组合结构在实际工程中应用十分广泛。
本文采用时变雷诺方程作为控制方程,并用k ?ε湍流模型来封闭雷诺方程,建立了模拟波浪与结构物相互作用的三维数学模型。采用能模拟波浪破碎的流体体积函数(VOF)方法跟踪波动自由表面;为了避免在网格内采用斜面表示自由表面时的复杂的几何计算,采用“施主-受主”算法计算网格内的流体体积函数值,并在网格内用水平或垂直的平面重构自由表面。采用能消去造波板二次反射的可吸收式数值造波机进行数值造波;在开边界处,采用了适合本模型的海绵层吸收方法。
雷诺方程的求解采用两步投影法:第一步,不考虑压强梯度对速度场的贡献,即在动量方程中略去压力梯度项,且时间项采用向前差分,得到中间速度场;第二步将此速度场向无散度的空间投影,得到满足连续性方程的流场。积分压强得到小尺度结构物上的惯性力,利用速度场计算得到小尺度结构物上的粘性力。为了验证模型对流场和自由面的模拟以及小尺度桩柱波浪力计算的准确性,本文用数学模型模拟了(1)孤立波在直立墙前的爬高过程(2)小尺度孤立柱上受到的波浪力,用解析解和实验结果进行了验证,数值结果与解析解和实验结果符合较好。
利用本文建立的数学模型,对不同波浪场中的小尺度孤立柱所受的波浪力进行了模拟,并且和Morison公式的计算结果进行了比较,两个结果比较接近。最后,对波浪流过纵向和横向排列的两个柱以及2×2的柱群的过程进行了数值模拟,得到了不同情况下该种结构物的水波动力学特性及每个柱的干扰系数。计算方法和计算结果对小尺度结构物的设计施工有一定的参考价值。
Water wave is a key hydrodynamic factor in offshore and coastal engineering, and wave forces acting on the structures are usually the control load for design. It is necessary to study the interaction between water waves and coastal structures. In coastal and ocean engineering, the structure with the ratio of its diameter to the wavelength being less than 0.15 is regarded as small-diameter structure, and the structures of arrays of small-diameter cylinders are widely used in practice. Three dimensional numerical model based on the Unsteady Reynolds Averaged Navier-Stokes (URANS) equations which being closed with k ?εturbulence model has been developed to simulate the interaction between water waves and coastal structures in this study. The VOF (Volume of Fluid) method is used to track the free surface. An active generating-absorbing numerical wave paddle is used to generate waves, and an active sponge layer is introduced at the open boundaries to absorb waves.
Two-steps projection method is used to calculate the URANS. Firstly, the effect of pressure on velocity is ignored. Secondely, the temp velocity is projected into a none-divergent space to calculate the real veolocity. The inertia force is obtained by integrating the pressure and the drag force is calculated by using the veolocity field.
In order to validate the numerical model, two examples were simulated by using the numerical model, given as follows: (1) solitary wave run-up on a vertical wall (2) wave forces on a solitary small-diameter cylinder. The numerical results of velocity fields, free surface and wave forces agree with the analytical and experimental results very well.
Wave forces on solitary small-diameter cylinder in different wave fields were simulated, and the results were compared with the results obtained by using the Morison equation, the two results are similar. Finally, the interaction between water waves and arrays of small diameter cylinders were simulated. The hydrodynamic properties of the structures and the interference coefficients of every cylinder were analysed.
本文采用时变雷诺方程作为控制方程,并用k ?ε湍流模型来封闭雷诺方程,建立了模拟波浪与结构物相互作用的三维数学模型。采用能模拟波浪破碎的流体体积函数(VOF)方法跟踪波动自由表面;为了避免在网格内采用斜面表示自由表面时的复杂的几何计算,采用“施主-受主”算法计算网格内的流体体积函数值,并在网格内用水平或垂直的平面重构自由表面。采用能消去造波板二次反射的可吸收式数值造波机进行数值造波;在开边界处,采用了适合本模型的海绵层吸收方法。
雷诺方程的求解采用两步投影法:第一步,不考虑压强梯度对速度场的贡献,即在动量方程中略去压力梯度项,且时间项采用向前差分,得到中间速度场;第二步将此速度场向无散度的空间投影,得到满足连续性方程的流场。积分压强得到小尺度结构物上的惯性力,利用速度场计算得到小尺度结构物上的粘性力。为了验证模型对流场和自由面的模拟以及小尺度桩柱波浪力计算的准确性,本文用数学模型模拟了(1)孤立波在直立墙前的爬高过程(2)小尺度孤立柱上受到的波浪力,用解析解和实验结果进行了验证,数值结果与解析解和实验结果符合较好。
利用本文建立的数学模型,对不同波浪场中的小尺度孤立柱所受的波浪力进行了模拟,并且和Morison公式的计算结果进行了比较,两个结果比较接近。最后,对波浪流过纵向和横向排列的两个柱以及2×2的柱群的过程进行了数值模拟,得到了不同情况下该种结构物的水波动力学特性及每个柱的干扰系数。计算方法和计算结果对小尺度结构物的设计施工有一定的参考价值。
Water wave is a key hydrodynamic factor in offshore and coastal engineering, and wave forces acting on the structures are usually the control load for design. It is necessary to study the interaction between water waves and coastal structures. In coastal and ocean engineering, the structure with the ratio of its diameter to the wavelength being less than 0.15 is regarded as small-diameter structure, and the structures of arrays of small-diameter cylinders are widely used in practice. Three dimensional numerical model based on the Unsteady Reynolds Averaged Navier-Stokes (URANS) equations which being closed with k ?εturbulence model has been developed to simulate the interaction between water waves and coastal structures in this study. The VOF (Volume of Fluid) method is used to track the free surface. An active generating-absorbing numerical wave paddle is used to generate waves, and an active sponge layer is introduced at the open boundaries to absorb waves.
Two-steps projection method is used to calculate the URANS. Firstly, the effect of pressure on velocity is ignored. Secondely, the temp velocity is projected into a none-divergent space to calculate the real veolocity. The inertia force is obtained by integrating the pressure and the drag force is calculated by using the veolocity field.
In order to validate the numerical model, two examples were simulated by using the numerical model, given as follows: (1) solitary wave run-up on a vertical wall (2) wave forces on a solitary small-diameter cylinder. The numerical results of velocity fields, free surface and wave forces agree with the analytical and experimental results very well.
Wave forces on solitary small-diameter cylinder in different wave fields were simulated, and the results were compared with the results obtained by using the Morison equation, the two results are similar. Finally, the interaction between water waves and arrays of small diameter cylinders were simulated. The hydrodynamic properties of the structures and the interference coefficients of every cylinder were analysed.
引文
[1] 陶建华,水波的数值模拟,天津:天津大学出版社,2005
[2] Morison J R,et al., The Force Exerted by Surface Wave on Piles Petrol, Trans. AIME, 1950
[3] Sarpkaya T., Isaacson M., Mechanics of wave forces on offshore structures, Published by Van Nostrand Reinhold Company, New York, 1981.
[4] 赵德廷,海底水平圆管正向波浪力系数( C D, C m)测试研究,中国海上曲气(工程),Vol.1,No.5, 1989
[5] Miyata H., Nishimura S., Finite difference simulation of nonlinear ship waves, J. Fluid Mech., 1985, 157: 327-357
[6] Yamada F.,Takikawa K., Numerical models with Reynolds equation based energy dissipation for plunging breakers on a uniform slope, Coastal Engineering, 1999, 41: 247-267
[7] Bai Y., et. al., 近岸海域富营养化研讨会论文集,2002, 天津
[8] 林明森,陶建华,三维溃坝问题的数值模拟研究,水利学报.1996(7): 67-74
[9] 谢伟松,有自由表面粘性水动力学问题的数值模拟研究,天津大学博士学位论文,1999
[10]王金瑞,波浪运动的数值模拟,华北水利水电学院学报,1996,17(1): 13-20
[11]齐鹏,王永学,非线性波浪时域计算的三维耦合模型,海洋学报,2000,22(6): 102-109
[12] Ramaswamy B., Numerical simulation of unsteady viscous free surface flow, J. Comp. Phys., 1990, 39, 396-430
[13]Brorsen M., Larsen J., Source generation of nonlinear gravity waves with the boundary intergral equation method, Costal Engineering, 1987, 11, 93-113
[14] Wei G., Kirby J. T., Sinha A., Generation waves in Boussinesq models using a source function method, Coastal Eng., 1999, 36: 271-299
[15]钦文婷,Boussinesq 方程数学模型的改进及其工程应用,天津大学硕士学位论文,天津大学,天津,2003
[16] 高学平,曾广冬等,不规则波浪数值水槽的造波和阻尼消波,海洋学报,2002,24(2): 127-132
[17] Kawasaki K., Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater, Coastal Engineering, 1999, 41: 201-223
[18] Larsen J., Dancy H., Open boundaries in short wave simulations – a new approach, Coastal Eng. 1983, 7: 285-297
[19]刘长根,用 URANS 方程模型模拟非线性水波与海工结构物的相互作用,天津大学博士学位论文,天津大学,天津,2003
[20] Launder B.E. , Spalding D.B., The numerical computation of turbulent flows, Computer mehhods in Applied Mechanics and Engineering, 1974, 3: 269-289
[21] YOUNGS D. L., Time-dependent multi-material flow with large fluid distortion, Numerical Methods for Fluid Dynamics[M]. Academic, New York. 1982: 273-285.
[22] HIRT C. W.,NICHOLS B. D., Volume of fluidmethod for the dynamics of free boundaries [J].Journal of Computational Physics.1981,39:201-225
[23] F. Ursell, R. G. Dean, Y. S. YU, Forced small-amplitude waters: a comparison of theory and experiment, Fluid Mechanics, 1959: 33-52.
[24] Chorin A.J., A numerical method for solving incompressible viscous flow problems, J. of Comp. Phys., 1967, 2
[25] Chorin A. J., Numerical Solution of the Navier-Stokes equations, Math. Comp., 1968, 22
[26] C. L. Merkel, M. Athavale, Time-accurate unsteady incompressible algorithms based on artificial compressibility, AIAA 1987
[27] Temam R., The approximation solution of the Navier-Stokes equations by fractional method (I), Archiv. Ration. Meth. Anal., 1969, Vol 32
[28] Temam R., The approximation solution of the Navier-Stokes equations by fractional method (II), Archiv. Ration. Meth. Anal., 1969, Vol 33
[29] Eisenstat S. C., Elman H. C., Schultz M. H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Num. Anal., 1983, 20: 88-102
[30] Saad Y., Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 1981, 37: 260-281
[31] Saad Y., Preconditioned Krylov subspace methods: an overview, Comp. Fluid Dynamics review, 1995
[32] Fenton, J.D., Rienecker, M.. M., A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. Journal of Fluid Mechanics, 1982, 118: 411-443
[33]Maiti S., Sen D., Computation of solitary waves during propagation and runup on a slope, Ocean Engineering, 1999, 26: 1063-1083
[34]刘长根,陶建华,用时变雷诺方程模型模拟孤立波与半圆型防波堤的相互作用,应用数学与力学,2004, 25(10): 1023-1032
[35]刘长根,陶建华,用 N-S 方程模型模拟孤立波在各种防波堤堤前的爬高,水动力研究与进展,2004, 19(4): 484-493
[36] CHORIN A. J., On the convergence of discrete approximations of the Navier-Stokes equations[J]. Math. Comp. 1969, 232: 341-353
[37] Bushnell M. J., Forces on Cylinder Arrays in Oscillating Flow.[J]OTC Paper, 1977, No.2903, Houston, Tx.
[38] Sarpkaya T., Cinar M., Ozkaynak S., Hydrodynamic Interference of Two Cylinders in Harmonic Flow, [J]OTC Paper, 1980, No.3775, Houston, Tx.
[39]陶建华等,长江口深水航道二期整治工程建筑物结构设计波浪要素推算,研究报告,2000年 8 月～11 月。
[40]大连工学院海洋工程研究所海工实验室,规则波作用下群桩波浪力的实验研究报告[R]., 大连:1985 年 2 月
[41]大连工学院海洋工程研究所海工实验室,金塘大桥桥梁基础波浪力第二次计算研究报告, 大连:2005 年 2 月
[42]张玲,小直径桩柱波浪荷载的计算分析及程序实现,中国海洋大学硕士学位论文,中国海洋大学,青岛,2004
[43] 张玲,王爱群,关于小直径垂直桩柱结构的波浪力研究,海洋湖沼通报,2004(3): 90-98
[44]王爱群,徐立论,小直径组合桩的波浪力实验分析,海洋湖沼通报,2002(1): 9-17
[45] 吴望一,《流体力学》,北京大学出版社,1983
[2] Morison J R,et al., The Force Exerted by Surface Wave on Piles Petrol, Trans. AIME, 1950
[3] Sarpkaya T., Isaacson M., Mechanics of wave forces on offshore structures, Published by Van Nostrand Reinhold Company, New York, 1981.
[4] 赵德廷,海底水平圆管正向波浪力系数( C D, C m)测试研究,中国海上曲气(工程),Vol.1,No.5, 1989
[5] Miyata H., Nishimura S., Finite difference simulation of nonlinear ship waves, J. Fluid Mech., 1985, 157: 327-357
[6] Yamada F.,Takikawa K., Numerical models with Reynolds equation based energy dissipation for plunging breakers on a uniform slope, Coastal Engineering, 1999, 41: 247-267
[7] Bai Y., et. al., 近岸海域富营养化研讨会论文集,2002, 天津
[8] 林明森,陶建华,三维溃坝问题的数值模拟研究,水利学报.1996(7): 67-74
[9] 谢伟松,有自由表面粘性水动力学问题的数值模拟研究,天津大学博士学位论文,1999
[10]王金瑞,波浪运动的数值模拟,华北水利水电学院学报,1996,17(1): 13-20
[11]齐鹏,王永学,非线性波浪时域计算的三维耦合模型,海洋学报,2000,22(6): 102-109
[12] Ramaswamy B., Numerical simulation of unsteady viscous free surface flow, J. Comp. Phys., 1990, 39, 396-430
[13]Brorsen M., Larsen J., Source generation of nonlinear gravity waves with the boundary intergral equation method, Costal Engineering, 1987, 11, 93-113
[14] Wei G., Kirby J. T., Sinha A., Generation waves in Boussinesq models using a source function method, Coastal Eng., 1999, 36: 271-299
[15]钦文婷,Boussinesq 方程数学模型的改进及其工程应用,天津大学硕士学位论文,天津大学,天津,2003
[16] 高学平,曾广冬等,不规则波浪数值水槽的造波和阻尼消波,海洋学报,2002,24(2): 127-132
[17] Kawasaki K., Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater, Coastal Engineering, 1999, 41: 201-223
[18] Larsen J., Dancy H., Open boundaries in short wave simulations – a new approach, Coastal Eng. 1983, 7: 285-297
[19]刘长根,用 URANS 方程模型模拟非线性水波与海工结构物的相互作用,天津大学博士学位论文,天津大学,天津,2003
[20] Launder B.E. , Spalding D.B., The numerical computation of turbulent flows, Computer mehhods in Applied Mechanics and Engineering, 1974, 3: 269-289
[21] YOUNGS D. L., Time-dependent multi-material flow with large fluid distortion, Numerical Methods for Fluid Dynamics[M]. Academic, New York. 1982: 273-285.
[22] HIRT C. W.,NICHOLS B. D., Volume of fluidmethod for the dynamics of free boundaries [J].Journal of Computational Physics.1981,39:201-225
[23] F. Ursell, R. G. Dean, Y. S. YU, Forced small-amplitude waters: a comparison of theory and experiment, Fluid Mechanics, 1959: 33-52.
[24] Chorin A.J., A numerical method for solving incompressible viscous flow problems, J. of Comp. Phys., 1967, 2
[25] Chorin A. J., Numerical Solution of the Navier-Stokes equations, Math. Comp., 1968, 22
[26] C. L. Merkel, M. Athavale, Time-accurate unsteady incompressible algorithms based on artificial compressibility, AIAA 1987
[27] Temam R., The approximation solution of the Navier-Stokes equations by fractional method (I), Archiv. Ration. Meth. Anal., 1969, Vol 32
[28] Temam R., The approximation solution of the Navier-Stokes equations by fractional method (II), Archiv. Ration. Meth. Anal., 1969, Vol 33
[29] Eisenstat S. C., Elman H. C., Schultz M. H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Num. Anal., 1983, 20: 88-102
[30] Saad Y., Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 1981, 37: 260-281
[31] Saad Y., Preconditioned Krylov subspace methods: an overview, Comp. Fluid Dynamics review, 1995
[32] Fenton, J.D., Rienecker, M.. M., A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. Journal of Fluid Mechanics, 1982, 118: 411-443
[33]Maiti S., Sen D., Computation of solitary waves during propagation and runup on a slope, Ocean Engineering, 1999, 26: 1063-1083
[34]刘长根,陶建华,用时变雷诺方程模型模拟孤立波与半圆型防波堤的相互作用,应用数学与力学,2004, 25(10): 1023-1032
[35]刘长根,陶建华,用 N-S 方程模型模拟孤立波在各种防波堤堤前的爬高,水动力研究与进展,2004, 19(4): 484-493
[36] CHORIN A. J., On the convergence of discrete approximations of the Navier-Stokes equations[J]. Math. Comp. 1969, 232: 341-353
[37] Bushnell M. J., Forces on Cylinder Arrays in Oscillating Flow.[J]OTC Paper, 1977, No.2903, Houston, Tx.
[38] Sarpkaya T., Cinar M., Ozkaynak S., Hydrodynamic Interference of Two Cylinders in Harmonic Flow, [J]OTC Paper, 1980, No.3775, Houston, Tx.
[39]陶建华等,长江口深水航道二期整治工程建筑物结构设计波浪要素推算,研究报告,2000年 8 月～11 月。
[40]大连工学院海洋工程研究所海工实验室,规则波作用下群桩波浪力的实验研究报告[R]., 大连:1985 年 2 月
[41]大连工学院海洋工程研究所海工实验室,金塘大桥桥梁基础波浪力第二次计算研究报告, 大连:2005 年 2 月
[42]张玲,小直径桩柱波浪荷载的计算分析及程序实现,中国海洋大学硕士学位论文,中国海洋大学,青岛,2004
[43] 张玲,王爱群,关于小直径垂直桩柱结构的波浪力研究,海洋湖沼通报,2004(3): 90-98
[44]王爱群,徐立论,小直径组合桩的波浪力实验分析,海洋湖沼通报,2002(1): 9-17
[45] 吴望一,《流体力学》,北京大学出版社,1983