基于二阶全非线性Boussinesq方程的波浪及波生流模拟
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摘要
海岸带和近海区域是各种水动力因素最为复杂的地区,近岸波、流场影响水工建筑物的规划和设计,并且是构成近岸泥沙运动和污染物运动的重要原因。本文基于有限差分法,在非交错网格下建立了二维的二阶完全非线性Boussinesq方程和近岸波生流数值模型,并结合经典实验资料对该方程的波浪及近岸流模型进行了相应的验证和分析。
首先,本文对二阶全非线性Boussinesq方程的推导过程和性能作一简单的分析。基于有限差分法在非交错网格下对该波浪模型进行了数值模拟验证,同时进一步验证分析了模型中不同精度的非线性项对数值模拟结果的影响以及模型的非线性项中引入缓坡假定对数值计算结果的影响。数值离散中时间层采用混合四阶Adams-Bashforth-Moulton预报校正格式。为了消弱波浪在边界处的反射以及二次反射的发生,采用源函数造波法与海绵层来吸收波能的技术。
通过与圆形浅滩以及椭圆形浅滩典型实验结果比较表明,二阶完全非线性Boussinesq方程的二维波浪模型具有良好的适用性；模型非线性项中缓坡假定的引入以及不同精度的非线性项对二维数值模型的模拟结果影响不明显。同时在二维二阶全非线性Boussinesq方程波浪模型的基础上添加了植被阻尼层项,建立了考虑植被阻尼层消浪的模型,以模拟实际水域中植被层对波浪的影响。通过分别与带有植被层的平底地形与平直斜坡地形实验资料比较,初步验证了带有植被阻尼层项的模型的可靠性。
其次,在二维二阶全非线性Boussinesq方程波浪模型的基础上添加波浪耗散项,同时考虑了次网格湍流的影响,从而建立了近岸波生流数值模型。对规则波作用下的不同坡度、不同波高、不同周期、不同粗糙率下的波生沿岸流、圆形浅滩下的裂流以及带有沙坝缺口斜坡地形下的离岸流和沿岸流进行了数值模拟,并结合实验结果验证了所建立的近岸波生流数值模型的可靠性。
Coastal and offshore areas are extremely dynamic regions. Near-shore wave-current field affects the layout and design of hydraulic constructions and is of great importance to nearshore sediment transport and pollutant transport.In this paper, the 2-D Second-Order Fully Non-linear Boussinesq Equation numerical model and the near-shore wave-induced current mathematical model is developed in un-staggered grid with finite difference method.
First, simple analysis of the deduction and the characters of the 2-D Second-Order Fully Non-linear Boussinesq Equation are put forward. Under un-staggered grid, the 2-D Second-Order Fully Non-linear Boussinesq mathematical model are simulated. Compared with the modeled results, analysis about the influence on the results of different precision level of nonlinearity and the mild slope hypothesis in non-linear terms are given. The mixing 4-order Adams-Bashforth-Moulton is used for time difference terms. Source function for wave generation and sponge layer boundary condition are used since here they are able to damp wave energy and possible wave reflection.
Comparing with the experiments'results, the 2-D Second-Order Fully Non-linear Boussinesq Equation model has good reliability; the influence of different precision level of nonlinearity and the mild slope hypothesis in non-linear terms is not so obvious. Then, vegetation term is added into the wave model to simulate the affection on the wave that the vegetation damping brings. Comparing with the experiment information, the numerical results prove the reliability of the vegetation model.
Second, breaking term、bottom friction term and subgrid turbulent mixing term are added to the wave model to simulate the breaking phenomenon. The regular wave-induced longshore current is caculated under the circular shoal terrain and the action of regular wave with various grade、wave height、wave period, and the results are in good agreement with experiment results.
首先,本文对二阶全非线性Boussinesq方程的推导过程和性能作一简单的分析。基于有限差分法在非交错网格下对该波浪模型进行了数值模拟验证,同时进一步验证分析了模型中不同精度的非线性项对数值模拟结果的影响以及模型的非线性项中引入缓坡假定对数值计算结果的影响。数值离散中时间层采用混合四阶Adams-Bashforth-Moulton预报校正格式。为了消弱波浪在边界处的反射以及二次反射的发生,采用源函数造波法与海绵层来吸收波能的技术。
通过与圆形浅滩以及椭圆形浅滩典型实验结果比较表明,二阶完全非线性Boussinesq方程的二维波浪模型具有良好的适用性；模型非线性项中缓坡假定的引入以及不同精度的非线性项对二维数值模型的模拟结果影响不明显。同时在二维二阶全非线性Boussinesq方程波浪模型的基础上添加了植被阻尼层项,建立了考虑植被阻尼层消浪的模型,以模拟实际水域中植被层对波浪的影响。通过分别与带有植被层的平底地形与平直斜坡地形实验资料比较,初步验证了带有植被阻尼层项的模型的可靠性。
其次,在二维二阶全非线性Boussinesq方程波浪模型的基础上添加波浪耗散项,同时考虑了次网格湍流的影响,从而建立了近岸波生流数值模型。对规则波作用下的不同坡度、不同波高、不同周期、不同粗糙率下的波生沿岸流、圆形浅滩下的裂流以及带有沙坝缺口斜坡地形下的离岸流和沿岸流进行了数值模拟,并结合实验结果验证了所建立的近岸波生流数值模型的可靠性。
Coastal and offshore areas are extremely dynamic regions. Near-shore wave-current field affects the layout and design of hydraulic constructions and is of great importance to nearshore sediment transport and pollutant transport.In this paper, the 2-D Second-Order Fully Non-linear Boussinesq Equation numerical model and the near-shore wave-induced current mathematical model is developed in un-staggered grid with finite difference method.
First, simple analysis of the deduction and the characters of the 2-D Second-Order Fully Non-linear Boussinesq Equation are put forward. Under un-staggered grid, the 2-D Second-Order Fully Non-linear Boussinesq mathematical model are simulated. Compared with the modeled results, analysis about the influence on the results of different precision level of nonlinearity and the mild slope hypothesis in non-linear terms are given. The mixing 4-order Adams-Bashforth-Moulton is used for time difference terms. Source function for wave generation and sponge layer boundary condition are used since here they are able to damp wave energy and possible wave reflection.
Comparing with the experiments'results, the 2-D Second-Order Fully Non-linear Boussinesq Equation model has good reliability; the influence of different precision level of nonlinearity and the mild slope hypothesis in non-linear terms is not so obvious. Then, vegetation term is added into the wave model to simulate the affection on the wave that the vegetation damping brings. Comparing with the experiment information, the numerical results prove the reliability of the vegetation model.
Second, breaking term、bottom friction term and subgrid turbulent mixing term are added to the wave model to simulate the breaking phenomenon. The regular wave-induced longshore current is caculated under the circular shoal terrain and the action of regular wave with various grade、wave height、wave period, and the results are in good agreement with experiment results.
引文
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[2]Witting J M. A unified model for the evolution of nonlinear water waves. Journal of Computed Physics,1984,56:203-236.
[3]Madsen P A, Murray R, S&rensen O R. A new form of the Boussinesq equations with improved linear dispersion characteristics. [J]Coastal Engineering,15:371-388.
[4]Madsen P A, S&rensen 0 R. A new form of the Boussinesq equations with improved linear dispersion characteristics, part 2. A slowly-varying bathymetry. Coastal Engineering, 1992,18:183-204.
[5]Nwogu 0. An alternative form of the Boussinesq Equations for near shore wave propagation. Journal of Waterway, Port, Coastal Engineering,1993,119(6):618-638.
[6]Schaffer H A, Madsen P A. Further enhancements of Boussinesq-type equations. Coastal Engineering,1995,26:1-14.
[7]张永刚.波浪的折射、绕射及反射的数值模拟：[博士学位论文]大连：大连理工大学,1995.
[8]Beiji S, Nadaoka K. A formal derivation and numerical model of the improved Boussinesq Equations for varying depth. Ocean Engineering,1996,23(8):691-704.
[9]林建国,邱大洪,邹志利.新型Boussinesq方程的进一步改善.海洋学报,1998,20(2)：113-119.
[10]Zhao M, Teng B, Cheng L. A new form of generalized Boussinesq Equations for varying depth. Ocean Engineering,2004,31:2047-2072.
[11]Wei G, Kirby J T, Grilli S T, and Subramanya R. A fully nonlinear Boussinesq model for surface waves. I:Highly nonlinear, unsteady waves. Journal of Fluid Mechanics, 1995,294:71-92.
[12]邹志利.高阶Boussinesq水波方程.中国科学(E辑),1997,27(5)：460-473.
[13]邹志利.高阶Boussinesq水波方程的改进.中国科学(E辑),1999,29(1)：87-96.
[14]Madsen P A, Schaffer H A. Higher-order Boussinesq-type equations for surface gravity waves:derivation and analysis. Philosophical Transations of Royal Society of London Series A-Mathematical Physical and Engineering Sciences,1998,356:3123-3184.
[15]Gobbi M F, Kirby J T, Wei G, A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4. Journal of Fluid Mechanics,2000,405:181-210.
[16]Agnon Y, Madsen P A, Schaffer H A. A new approach to high order Boussinesq models. Journal of Fluid Mechanics,1999,399:319-333.
[17]Madsen P A, Bingham H B, Liu Hua. A new method for fully nonlinear waves from shallow water to deep water. Journal of Fluid Mechanics,2002,462:1-30.
[18]Madsen P A, Bingham H B, Schaffer H A. Boussinesq-type formulations for fully nonlinear and extremely dispersive waver waves:Derivation and Analysis.2003, Proc. R. Soc. Lond. A,2003,459:1075-1104.
[19]Patrick L, Philip L-F Liu, A two-layer approach to wave modeling. Proc. R. Soc. Lond. A,2004,460:2637-2669.
[20]Kennedy A B, Kirby J T, Chen Q, Dalrymple R A. Boussinesq-type equations with improved nonlinear behavior. Wave motion,2001,33:225-243.
[21]Kennedy A B, Kirby J T, Gobbi M F. Simplified higher order Boussinesq equations.1: Linear considerations. Coastal Engineering,2002,44:205-229.
[22]游涛.波浪在斜坡上的传播破碎及近岸流研究[D].天津：天津大学,2004.
[23]Longuet-Higgins M S and Stewart R W. Changes in the form of short gravity waves on long waves and tidal currents [J]. The Journal of Fluid Mechanics,1960,8(4):565-583.
[24]Longuet-Higgins M S and Stewart R W. The changes in amplitude of short gravity waves on steady non-uniform currents [J]. The Journal of Fluid Mechanics,1961,10(4): 529-549.
[25]李孟国.海岸河口水动力数值模拟研究及泥沙运动研究的应用[D].青岛：青岛海洋大学,2002.
[26]李春颖,李绍武.基于Boussinesq方程的波浪破碎模型的研究综述[J],2003,04：1-4.
[27]Woo S B, Liu P-L F. A Petrov-Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagation. Int. J. for Numerical methods in Fluids,2001,37:541-575.
[28]Shi F, Dalrymple R A, Kirby J T, Chen Q, Kennedy A B. A fully nonlinear Boussinesq model in generalized curvilinear coordinates. Coastal Engineering,2001,42:337-358.
[29]Shi F, Kirby J T. Curvilinear parabolic approximation for surface wave transformation with wave-current interaction. Journal of Computational Physics,2005,204:562-586.
[30]龙文.浅水非线性色散波数学模型及其在工程上的应用：[硕士学位论文].天津：天津大学,2000.
[31]钦文婷Boussinesq方程数学模型的改进及其工程应用：[硕士学位论文].天津：天津大学,2003.
[32]Li Y X, Liu S X, Yu J X, Lai G Z. Numerical modeling of Boussinesq equations by finite element method. Coastal Engineering,1999,37(2):97-122.
[33]S(?)rensen O R, Schaffer H A, S(?)rensen L S, Boussinesq-type modeling using an unstructured finite element technique. Coastal Engineering,2004,50(4):181-198.
[34]Kennedy A B, Kirby J T. Boussinesq modeling of wave transformation, breaking, and run-up. I:1D [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE, 2000,126(1):39-47.
[35]Madsen P A, Schaffer H A. A review of Boussinesq-type equations for surface gravity waves. In:P. L.F. Liu. Advances in coastal and ocean engineering. Singapore:World Scientific,1999:1-95.
[36]Kirby J T. Nonlinear, dispersive long waves in water of variable depth. In:J. N. Hunt. Gravity waves in water of finite depth. Southampon:Computational Mechanics Publications,1997:551-125.
[37]Kirby J T. Boussiensq models and applications to nearshore wave propagation, surf zone processes and wave-induced currents. In:C. Lakhan. Advances in coastal engineering. Amsterdam:Elsevier,2003:1-41.
[38]Gobbi M F. A new Boussinesq-type model for surface water wave propagation:(Ph. D. Dissertation). Delaware:University of Delaware,1998.
[39]Zou Z L. A new form of higher order Boussinesq equations. Ocean Engineering,2000, 27(5):557-575.
[40]Zelt J A. The run-up of nonbreaking and breaking solitary waves. Coastal Engineering, 1991,15(3):205-246.
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