﻿ 非静压水波模型研究综述
 海洋与湖沼  2022, Vol. 53 Issue (4): 813-821 PDF
http://dx.doi.org/10.11693/hyhz20220200041

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#### 文章信息

MA Yu-Xiang, AI Cong-Fang, DONG Guo-Hai. 2022.

A REVIEW ON NON-HYDROSTATIC WATER WAVE MODELS

Oceanologia et Limnologia Sinica, 53(4): 813-821.
http://dx.doi.org/10.11693/hyhz20220200041

### 文章历史

A REVIEW ON NON-HYDROSTATIC WATER WAVE MODELS
MA Yu-Xiang, AI Cong-Fang, DONG Guo-Hai
Dalian University of Technology, State Key Laboratory of Coastal and Offshore Engineering, Dalian 116024, China
Abstract: Water wave numerical simulation has always been one of the important research fields of hydraulic engineering, coastal engineering, marine engineering, and physical oceanography. Among many water wave models, non-hydrostatic water wave models are favored by researchers because of its balance between computational accuracy and efficiency. After nearly three decades of development, non-hydrostatic water wave models provide important technical means for scientific research, engineering design and analysis, and marine resources development. However, developing more-efficient non-hydrostatic models and broadening the application field of the models have always been the pursuit of non-hydrostatic model researchers. In this paper, we first introduce the concept of "non-hydrostatic pressure", and then review the application of non-hydrostatic models in wave propagations and evolutions, and wave-structure interactions from the perspective of model development and application.
Key words: numerical simulation    water wave model    non-hydrostatic model    wave-structure interactions

1 非静压模型在波浪运动模拟方面的进展

Ai等(2011)Ai等(2019a)开发的非静压表面流模型抛弃了传统的交错定义变量的方式, 将垂向流速由分层网格界面移到分层网格中心定义, 同时水平流速的仍然按照交错的方式定义(图 1), 同样构建了模拟波浪运动的“完全”非静压模型。相对于其他模型而言(Stelling et al, 2003; Yuan et al, 2004; Zijlema et al, 2005; Anthonio et al, 2006; Cea et al, 2009; Wu et al, 2010; Young et al, 2010; Ma et al, 2012), 采用这种变量定义方式构建的非静压模型的最大特点是最终求解的压力Possion方程是对称正定的, 可以采用预条件共轭梯度法高效求解, 这样极大地提高了非静压模型的计算效率。Ai等(2011)基于垂向边界拟合坐标系统建立了模拟波浪运动的非静压模型(Ai et al, 2011)。该模型采用投影法, 即压力修正法, 求解不可压缩Euler方程。在水平笛卡尔网格框架和垂向边界拟合坐标系下, 提出了上述这种新颖的网格变量定义方式。通过与解析解和试验数据对比, 表明此非静压模型采用两个垂向分层就能够准确高效地模拟波浪浅化、非线性、色散、折射和绕射现象。对于线性色散关系的模拟, 采用两个垂向分层即能准确模拟(误差小于1%) kh=π的深水波浪(图 2)。为实现近岸波浪破碎爬高和强非线性波群的准确模拟, Ai等(2012)对模型进行了扩展, 采用动量守恒的计算格式离散动量方程中的水平对流项, 这使得模型能够有效地模拟包括破碎波和水跃在内的间断流动问题。同时, 再引入干湿动边界处理方法即实现了模型模拟波浪爬高的能力。数值模拟结果表明(图 3), 这样构建的非静压模型可以有效准确地模拟近岸水深变化导致的波浪破碎和爬高过程。为降低非静压模型在模拟强非线性波群时由于压力梯度项离散导致的数值误差, 保证了非静压模型可以准确地模拟强非线性波群的演化, Ai等(2014)首次在非静压模型中引入了广义垂向边界拟合坐标系统。数值模拟结果表明(Ai et al, 2014), 基于广义垂向边界拟合坐标系统建立的非静压模型可以准确地模拟畸形波浪的产生(图 4)。最近, Ai等(2019a)提出了两种半隐的非静压模型来模拟波浪运动。这两种半隐模型同样采用上述变量定义方式构建, 一种是非迭代模型, 另一种需要迭代求解。这类半隐的非静压模型与之前的模型相对比, 具有时间步长不受制于表面波波速的优点, 在模拟深水波浪问题方面更具优势。研究评估和讨论了这两个半隐模型求解线性色散关系的精度和执行效率。通过从浅水到深水的多个数值算例对这两个模型进行了验证。验证结果表明(Ai et al, 2019a), 这两个半隐模型的结果非常相似, 均与实验数据吻合良好(图 5)。然而, 迭代模型的执行效率低于非迭代模型, 求解迭代模型花费的时间是非迭代模型的1.1~2.5倍。

 图 1 非静压模型新颖的变量定义方式(Ai et al, 2011) Fig. 1 Variables definition for the non-hydrostatic model developed by Dalian University of Technology (Ai et al, 2011) 注: i, j, k分别为x, y, z方向的网格索引; u, v分别为水平流速和垂向流速; q为非静压项

 图 2 线性波波速的计算结果与解析解的对比(Ai et al, 2011) Fig. 2 Comparison of wave celerity between model results and analytical solution (Ai et al, 2011)

 图 3 波浪爬高破碎过程(红色实线: 模型结果; 圆点: 实测值)(Ai et al, 2012) Fig. 3 The process of wave runup and breaking (red solid line: model results; dot: experimental data) (Ai et al, 2012)

 图 4 畸形波的计算结果与实测数据的对比(红色实线: 模型结果; 圆形: 实测值)(Ai et al, 2014) Fig. 4 Comparison of the freak wave between model results and experimental data (red solid line: model results; circle: experimental data) (Ai et al, 2014)

 图 5 聚焦点处三维深水聚焦波的计算结果与实测数据的对比(红色虚线: 非迭代模型结果; 蓝色实线: 迭代模型结果; 圆形: 实测值)(Ai et al, 2019a) Fig. 5 Comparisons of the time histories of the free-surface elevation at the focusing position among the two model results and experimental data (red dash line: non-iterative model results; solid line: iterative model results; circle: experimental data) (Ai et al, 2019a)

2 非静压模型在波浪-结构物相互作用模拟方面的进展

 图 6 三角形与四边形混合网格(Ai et al, 2017) Fig. 6 A hybrid-grid of triangular and rectangular cells (Ai et al, 2017)

 图 7 受水下结构物影响的波面时间序列的计算结果与实测数据的对比(蓝色实线: 模型结果; 圆形: 实测值)(Ma et al, 2019) Fig. 7 Comparisons of the time histories of free-surface elvations between model results and experimental data (blue solid line: model results; circle: experimental data) (Ma et al, 2019)

 图 8 结构物附近的流场 Fig. 8 Flow field around the plate

 图 9 波浪与浮式沉箱的相互作用 Fig. 9 Wave interaction with a floating caisson
3 结论

 刘桦, 何友声, 2000. 河口三维流动数学模型研究进展. 海洋工程, 18(2): 87-93 DOI:10.3969/j.issn.1005-9865.2000.02.016 吴亚楠, 武贺, 周庆伟, 等, 2021. 基于Boussinesq波浪模型的港池波浪数值模拟与泊稳分析. 海洋通报, 40(3): 301-308 林鹏程, 刘忠波, 刘勇, 2021. 基于Boussinesq数值模型的波浪速度垂向分布模拟研究. 海洋湖沼通报, 43(4): 7-15 饶永红, 刘忠波, 梁书秀, 等, 2021. 双层Boussinesq模型非线性波浪模拟研究. 水道港口, 42(5): 614-622 DOI:10.3969/j.issn.1005-8443.2021.05.008 赖锡军, 曲卓杰, 周杰, 等, 2006. 非结构网格上的三维浅水流动数值模型. 水科学进展, 17(5): 693-699 DOI:10.3321/j.issn:1001-6791.2006.05.017 AI C F, DING W Y, 2016. A 3D unstructured non-hydrostatic ocean model for internal waves. Ocean Dynamics, 66(10): 1253-1270 DOI:10.1007/s10236-016-0980-9 AI C F, DING W Y, JIN S, 2014. A general boundary-fitted 3D non-hydrostatic model for nonlinear focusing wave groups. Ocean Engineering, 89: 134-145 DOI:10.1016/j.oceaneng.2014.08.002 AI C F, DING W Y, JIN S, 2017. A hybrid-grid 3D model for regular waves interacting with cylinders. Journal of Hydraulic Research, 55(1): 129-134 DOI:10.1080/00221686.2016.1212943 AI C F, JIN S, 2010. Non-hydrostatic finite volume model for non-linear waves interacting with structures. Computers & Fluids, 39(10): 2090-2100 AI C F, JIN S, 2012. A multi-layer non-hydrostatic model for wave breaking and run-up. Coastal Engineering, 62: 1-8 DOI:10.1016/j.coastaleng.2011.12.012 AI C F, JIN S, LV B, 2011. A new fully non-hydrostatic 3D free surface flow model for water wave motions. International Journal for Numerical Methods in Fluids, 66(11): 1354-1370 DOI:10.1002/fld.2317 AI C F, MA Y X, DING W Y, et al, 2021a. An efficient three-dimensional non-hydrostatic model for undular bores in open channels. Physics of Fluids, 33(12): 127111 DOI:10.1063/5.0073241 AI C F, MA Y X, DING W Y, et al, 2022a. Three-dimensional non-hydrostatic model for dam-break flows. Physics of Fluids, 34(2): 022105 DOI:10.1063/5.0081094 AI C F, MA Y X, YUAN C F, et al, 2018. Semi-implicit non-hydrostatic model for 2D nonlinear wave interaction with a floating/suspended structure. European Journal of Mechanics-B/Fluids, 72: 545-560 DOI:10.1016/j.euromechflu.2018.08.003 AI C F, MA Y X, YUAN C F, et al, 2019a. Development and assessment of semi-implicit nonhydrostatic models for surface water waves. Ocean Modelling, 144: 101489 DOI:10.1016/j.ocemod.2019.101489 AI C F, MA Y X, YUAN C F, et al, 2019b. A 3D non-hydrostatic model for wave interactions with structures using immersed boundary method. Computers & Fluids, 186: 24-37 AI C F, MA Y X, YUAN C F, et al, 2021b. A three-dimensional non-hydrostatic model for tsunami waves generated by submarine landslides. Applied Mathematical Modelling, 96: 1-19 DOI:10.1016/j.apm.2021.02.014 AI C F, MA Y X, YUAN C F, et al, 2022b. An efficient 3D non-hydrostatic model for predicting nonlinear wave interactions with fixed floating structures. Ocean Engineering, 248: 110810 DOI:10.1016/j.oceaneng.2022.110810 ANTHONIO S L, HALL K R, 2006. High-order compact numerical schemes for non-hydrostatic free surface flows. International Journal for Numerical Methods in Fluids, 52(12): 1315-1337 DOI:10.1002/fld.1225 BRADFORD S F, 2005. Godunov-based model for nonhydrostatic wave dynamics. Journal of Waterway, Port, Coastal, and Ocean Engineering, 131(5): 226-238 DOI:10.1061/(ASCE)0733-950X(2005)131:5(226) CANTERO-CHINCHILLA F N, CASTRO-ORGAZ O, DEY S, et al, 2016. Nonhydrostatic dam break flows. I: physical equations and numerical schemes. Journal of Hydraulic Engineering, 142(12): 04016068 DOI:10.1061/(ASCE)HY.1943-7900.0001205 CASULLI V, 1999. A semi-implicit finite difference method for non-hydrostatic, free-surface flows. International Journal for Numerical Methods in Fluids, 30(4): 425-440 DOI:10.1002/(SICI)1097-0363(19990630)30:4<425::AID-FLD847>3.0.CO;2-D CASULLI V, CHENG R T, 1992. Semi-implicit finite difference methods for three-dimensional shallow water flow. International Journal for Numerical Methods in Fluids, 15(6): 629-648 DOI:10.1002/fld.1650150602 CASULLI V, ZANOLLI P, 2002. Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Mathematical and Computer Modelling, 36(9/10): 1131-1149 CEA L, STELLING G, ZIJLEMA M, 2009. Non-hydrostatic 3D free surface layer-structured finite volume model for short wave propagation. International Journal for Numerical Methods in Fluids, 61(4): 382-410 DOI:10.1002/fld.1961 CHANG T J, CHANG K H, KAO H M, 2014. A new approach to model weakly nonhydrostatic shallow water flows in open channels with smoothed particle hydrodynamics. Journal of Hydrology, 519: 1010-1019 DOI:10.1016/j.jhydrol.2014.08.030 CHOI D Y, WU C H, YOUNG C C, 2011. An efficient curvilinear non-hydrostatic model for simulating surface water waves. International Journal for Numerical Methods in Fluids, 66(9): 1093-1115 DOI:10.1002/fld.2302 CHOI D Y, YUAN H L, 2012. A horizontally curvilinear non-hydrostatic model for simulating nonlinear wave motion in curved boundaries. International Journal for Numerical Methods in Fluids, 69(12): 1923-1938 DOI:10.1002/fld.2676 DONG G H, FU R L, MA Y X, et al, 2019. Simulation of unidirectional propagating wave trains in deep water using a fully non-hydrostatic model. Ocean Engineering, 180: 254-266 DOI:10.1016/j.oceaneng.2019.03.037 HE D B, MA Y X, DONG G H, et al, 2020. Predicting deep water wave breaking with a non-hydrostatic shock-capturing model. Ocean Engineering, 216: 108041 DOI:10.1016/j.oceaneng.2020.108041 HE D B, MA Y X, DONG G H, et al, 2022. A numerical investigation of wave and current fields along bathymetry with porous media. Ocean Engineering, 244: 110333 DOI:10.1016/j.oceaneng.2021.110333 HU P X, WU G X, MA Q W, 2002. Numerical simulation of nonlinear wave radiation by a moving vertical cylinder. Ocean Engineering, 29(14): 1733-1750 DOI:10.1016/S0029-8018(02)00002-1 KANG A Z, LIN P Z, LEE Y J, et al, 2015. Numerical simulation of wave interaction with vertical circular cylinders of different submergences using immersed boundary method. Computers & Fluids, 106: 41-53 KIM D H, LYNETT P J, 2011. Dispersive and nonhydrostatic pressure effects at the front of surge. Journal of Hydraulic Engineering, 137(7): 754-765 DOI:10.1061/(ASCE)HY.1943-7900.0000345 KIRBY J T, 2017. Recent advances in nearshore wave, circulation, and sediment transport modeling. Journal of Marine Research, 75(3): 263-300 DOI:10.1357/002224017821836824 LAI Z G, CHEN C S, COWLES G W, et al, 2010. A nonhydrostatic version of FVCOM: 2. Mechanistic study of tidally generated nonlinear internal waves in Massachusetts Bay. Journal of Geophysical Research, 115(C12): C12049 LI Y S, LIU S X, YU Y X, et al, 1999. Numerical modeling of Boussinesq equations by finite element method. Coastal Engineering, 37(2): 97-122 DOI:10.1016/S0378-3839(99)00014-9 LI Y S, ZHAN J M, 1998. Three-dimensional finite-element model for stratified coastal seas. Journal of Hydraulic Engineering, 124(7): 699-703 DOI:10.1061/(ASCE)0733-9429(1998)124:7(699) LI Y S, ZHAN J M, 2001. Boussinesq-type model with boundary-fitted coordinate system. Journal of Waterway, Port, Coastal, and Ocean Engineering, 127(3): 152-160 DOI:10.1061/(ASCE)0733-950X(2001)127:3(152) LIN P Z, 2006. A multiple-layer σ-coordinate model for simulation of wave-structure interaction. Computers & Fluids, 35(2): 147-167 LIN P Z, LI C W, 2002. A σ-coordinate three-dimensional numerical model for surface wave propagation. International Journal for Numerical Methods in Fluids, 38(11): 1045-1068 DOI:10.1002/fld.258 MA G F, FARAHANI A A, KIRBY J T, et al, 2016. Modeling wave-structure interactions by an immersed boundary method in a σ-coordinate model. Ocean Engineering, 125: 238-247 DOI:10.1016/j.oceaneng.2016.08.027 MA G F, SHI F Y, KIRBY J T, 2012. Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Modelling, 43/44: 22-35 DOI:10.1016/j.ocemod.2011.12.002 MA Q W, WU G X, TAYLOR R E, 2001a. Finite element simulation of fully non-linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. International Journal for Numerical Methods in Fluids, 36(3): 265-285 DOI:10.1002/fld.131 MA Q W, WU G X, TAYLOR R E, 2001b. Finite element simulations of fully non-linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation. International Journal for Numerical Methods in Fluids, 36(3): 287-308 DOI:10.1002/fld.133 MA Y X, YUAN C F, AI C F, et al, 2019. Comparison between a non-hydrostatic model and OpenFOAM for 2D wave-structure interactions. Ocean Engineering, 183: 419-425 DOI:10.1016/j.oceaneng.2019.05.002 MIGLIO E, QUARTERONI A, SALERI F, 1999. Finite element approximation of quasi-3D shallow water equations. Computer Methods in Applied Mechanics and Engineering, 174(3/4): 355-369 MIGNOT E, CIENFUEGOS R, 2009. On the application of a Boussinesq model to river flows including shocks. Coastal Engineering, 56(1): 23-31 DOI:10.1016/j.coastaleng.2008.06.007 MOHAPATRA P K, CHAUDHRY M H, 2004. Numerical solution of Boussinesq equations to simulate dam-break flows. Journal of Hydraulic Engineering, 130(2): 156-159 DOI:10.1061/(ASCE)0733-9429(2004)130:2(156) OISHI Y, PIGGOTT M D, MAEDA T, et al, 2013. Three-dimensional tsunami propagation simulations using an unstructured mesh finite element model. Journal of Geophysical Research: Solid Earth, 118(6): 2998-3018 DOI:10.1002/jgrb.50225 PEROT B, 2000. Conservation properties of unstructured staggered mesh schemes. Journal of Computational Physics, 159(1): 58-89 DOI:10.1006/jcph.2000.6424 RIJNSDORP D P, ZIJLEMA M, 2016. Simulating waves and their interactions with a restrained ship using a non-hydrostatic wave-flow model. Coastal Engineering, 114: 119-136 DOI:10.1016/j.coastaleng.2016.04.018 STELLING G, ZIJLEMA M, 2003. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. International Journal for Numerical Methods in Fluids, 43(1): 1-23 DOI:10.1002/fld.595 WANG C Z, WU G X, 2010. Interactions between fully nonlinear water waves and cylinder arrays in a wave tank. Ocean Engineering, 37(4): 400-417 DOI:10.1016/j.oceaneng.2009.12.006 WU C H, YOUNG C C, CHEN Q, et al, 2010. Efficient nonhydrostatic modeling of surface waves from deep to shallow water. Journal of Waterway, Port, Coastal, and Ocean Engineering, 136(2): 104-118 DOI:10.1061/(ASCE)WW.1943-5460.0000032 XING Y, AI C F, JIN S, 2013. A three-dimensional hydrodynamic and salinity transport model of estuarine circulation with an application to a macrotidal estuary. Applied Ocean Research, 39: 53-71 DOI:10.1016/j.apor.2012.10.003 YOUNG C C, WU C H, 2010. Nonhydrostatic modeling of nonlinear deep-water wave groups. Journal of Engineering Mechanics, 136(2): 155-167 DOI:10.1061/(ASCE)EM.1943-7889.0000078 YUAN H L, WU C H, 2004. An implicit three-dimensional fully non-hydrostatic model for free-surface flows. International Journal for Numerical Methods in Fluids, 46(7): 709-733 DOI:10.1002/fld.778 ZHAO M, CHENG L, TENG B, 2007. Numerical simulation of solitary wave scattering by a circular cylinder array. Ocean Engineering, 34(3/4): 489-499 ZHONG Z Y, WANG K H, 2009. Modeling fully nonlinear shallow-water waves and their interactions with cylindrical structures. Computers & Fluids, 38(5): 1018-1025 ZIJLEMA M, STELLING G S, 2005. Further experiences with computing non-hydrostatic free-surface flows involving water waves. International Journal for Numerical Methods in Fluids, 48(2): 169-197 DOI:10.1002/fld.821 ZIJLEMA M, STELLING G, SMIT P, 2011. SWASH: an operational public domain code for simulating wave fields and rapidly varied flows in coastal waters. Coastal Engineering, 58(10): 992-1012 DOI:10.1016/j.coastaleng.2011.05.015