﻿ 规则波与流相互作用的数值模拟与不确定度分析
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 哈尔滨工程大学学报  2021, Vol. 42 Issue (2): 172-178  DOI: 10.11990/jheu.201909081 0

### 引用本文

YAO Shun, MA Ning, DING Junjie, et al. Numerical simulation and uncertainty analysis of wave-current interaction with regular waves[J]. Journal of Harbin Engineering University, 2021, 42(2): 172-178. DOI: 10.11990/jheu.201909081.

### 文章历史

1. 上海交通大学 海洋工程国家重点实验室, 上海 200240;
2. 上海交通大学 船舶海洋与建筑工程学院, 上海 200240

Numerical simulation and uncertainty analysis of wave-current interaction with regular waves
YAO Shun 1,2, MA Ning 1,2, DING Junjie 1,2, GU Xiechong 1,2
1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
2. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Abstract: The law of uniform current effects on the characteristics of regular waves and its uncertainty is investigated for wave-current interactions. Firstly, relevant tests of the interaction of regular waves and currents were conducted in the circulating water channel at Shanghai Jiao Tong University. Numerical simulation based on Reynolds-Averaged Navier-Stokes (RANS) equations was then carried out to solve the problem of regular-wave-current interaction. Finally, the simulation results were used to perform the uncertainty analysis. It is concluded that wave heights are low and wave peaks and troughs are flat in the co-current case. However, completely opposite phenomena are observed in waves with countercurrents. The waves are blocked when the velocity of the countercurrent increases to 0.6 m/s. Moreover, the simulated results are more sensitive to grid size than time steps in both no-current and co-current cases. Furthermore, compared with the no-current case, the influence of grid size and time step are clearer in the co-current cases.
Keywords: hydrodynamics    numerical methods    regular wave    uniform current    wave-current interaction    wave blocking    numerical wave flume    uncertainty analysis

1 数值模型及验证 1.1 试验安排

1.2 数值模型

 $\rho \frac{{\partial {u_i}}}{{\partial t}} + \rho {u_j}\frac{{\partial {u_i}}}{{\partial {x_j}}} = - \frac{{\partial p}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left[ {\mu \left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right)} \right] + \rho {g_i}$ (1)

 $\begin{array}{l} \rho \frac{{{\rm{d}}{{\bar u}_i}}}{{{\rm{d}}t}} + \rho {{\bar u}_j}\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}} = - \frac{{\partial \bar p}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left[ {\mu \left( {\frac{{\partial {{\bar u}_i}}}{{\partial {x_j}}} + \frac{{\partial {{\bar u}_j}}}{{\partial {x_i}}}} \right)} \right] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\partial }{{\partial {x_j}}}\left( { - \rho \overline {u{_i^\prime}u{_j^\prime}} } \right) + \rho {g_i} \end{array}$ (2)

 $- \rho \overline {{u{_i^\prime}u{_j^\prime}}} = {\mu _t}\left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right) - \frac{2}{3}\rho {\tau _{ij}}k$ (3)

 $\phi = \sum\limits_{n = 1}^\infty {{\varepsilon _{\rm{s}}}} {\phi _n}$ (4)
 $\eta = \sum\limits_{n = 1}^\infty {{\varepsilon _{\rm{s}}}} {\eta _n}$ (5)

n=2，5时，代入自由表面条件泰勒展开式中，即可得到弱非线性的二阶和五阶Stokes波面方程和势函数。本文数值模拟的研究对象为Stokes五阶波，在软件中选择波浪类型并输入指定波高和周期，即可得到目标波浪。试验及模拟工况如表 1所示，其中C为流速，H为波高，L为波长，T为周期，EXP和NUM分别表示对该工况进行试验或者数值模拟。每个工况重复3组，每组试验或者模拟时长为50 s。

1.3 数值模型验证

 Download: 图 4 工况R1数值模拟结果的验证 Fig. 4 Verification of simulation results of case R1

2 结果及不确定度分析 2.1 结果分析

 Download: 图 5 数值水池中波浪自由表面情况 Fig. 5 The water surface of waves in the numerical wave flume

 Download: 图 6 波高时历数值模拟结果 Fig. 6 Simulation results of the water surface elevations

2.2 不确定度分析

 Download: 图 8 不同网格对应的波高时历曲线 Fig. 8 Water surface elevations simulated with different grids

 ${\left\| {\left. {{\varepsilon _{{\rm{G21}}}}} \right\|} \right._2} = {[\frac{1}{N}\sum\limits_{i = 1}^N {({\eta _{{\rm{G2i}}}} - } {\eta _{{\rm{G1i}}}})^2}{]^{1/2}} = 0.000{\kern 1pt} 263$ (6)
 ${\left\| {\left. {{\varepsilon _{{\rm{G32}}}}} \right\|} \right._2} = {[\frac{1}{N}\sum\limits_{i = 1}^N {({\eta _{{\rm{G3i}}}} - } {\eta _{{\rm{G2i}}}})^2}{]^{1/2}} = 0.000{\kern 1pt} 859$ (7)

 $\left\langle {{R_{\rm{G}}}} \right\rangle = \frac{{{{\left\| {\left. {{\varepsilon _{{\rm{G21}}}}}\;\; \right\|} \right.}_2}}}{{{{\left\| {\left. {{\varepsilon _{{\rm{G32}}}}}\;\; \right\|} \right.}_2}}} = 0.305$ (8)

RG〉∈(0, 1)，因此模拟结果单调收敛，可采用广义Richardson外推法估算准确度阶数pG和误差δREG*

 ${p_{\rm{G}}} = \frac{{\ln ({{\left\| {\left. {{\varepsilon _{{\rm{G32}}}}} \;\;\right\|} \right.}_2}/{{\left\| {\left. {{\varepsilon _{{\rm{G21}}}}}\;\; \right\|} \right.}_2})}}{{\ln ({r_{\rm{G}}})}} = 3.419$ (9)
 $\delta _{{\rm{R}}{{\rm{E}}_{\rm{G}}}}^* = \frac{{{{\left\| {\left. {{\varepsilon _{{\rm{G21}}}}} \;\;\right\|} \right.}_2}}}{{r_{\rm{G}}^{{p_{\rm{G}}}} - 1}} = 0.000{\kern 1pt} 116$ (10)

 ${C_{\rm{G}}} = \frac{{r_{\rm{G}}^{{p_{\rm{G}}}} - 1}}{{r_{\rm{G}}^{{p_{{\rm{G,est}}}}} - 1}} = 2.270$ (11)

 $\delta _{\rm{G}}^* = {C_{\rm{G}}}\delta _{{\rm{R}}{{\rm{E}}_{\rm{G}}}}^* = 0.000{\kern 1pt} 263{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{m}}$ (12)

CG远小于或远大于1时，网格尺寸引起的数值模拟不确定度为：

 ${U_{\rm{G}}} = \left[ {9.6{{(1 - {U_{\rm{G}}})}^2} + 1.1} \right] \times \left| {\delta _{{\rm{R}}{{\rm{E}}_{\rm{G}}}}^*} \right|$ (13)

 ${U_{\rm{G}}} = (1 + 2\left| {1 - {C_{\rm{G}}}} \right|) \times \left| {\delta _{{\rm{R}}{{\rm{E}}_{\rm{G}}}}^{\rm{*}}} \right|$ (14)

 Download: 图 9 不同时间步长对应的波高时历 Fig. 9 Water surface elevations simulated with different time steps

 ${\left\| {\left. {{\varepsilon _{{\rm{T21}}}}} \right\|} \right._2} = {\left[ {\frac{1}{N}\sum\limits_{i = 1}^N {({\eta _{{\rm{T2i}}}} - } {\eta _{{\rm{T1i}}}}{)^2}} \right]^{1/2}} = 0.000{\kern 1pt} 207$ (15)
 ${\left\| {\left. {{\varepsilon _{{\rm{T32}}}}} \right\|} \right._2} = {\left[ {\frac{1}{N}\sum\limits_{i = 1}^N {({\eta _{{\rm{T3i}}}} - } {\eta _{{\rm{T2i}}}}{)^2}} \right]^{1/2}} = 0.002{\kern 1pt} 18$ (16)

 $\left\langle {{R_{\rm{T}}}} \right\rangle = \frac{{{{\left\| {\left. {{\varepsilon _{{\rm{T21}}}}} \;\;\right\|} \right.}_2}}}{{{{\left\| {\left. {{\varepsilon _{{\rm{T32}}}}} \;\;\right\|} \right.}_2}}} = 0.095{\kern 1pt} 0$ (17)

RT〉∈(0, 1)，结果也是单调收敛，采用广义Richardson外推法估算准确度阶数pT和误差δRET*

 ${p_{\rm{T}}} = \frac{{\ln ({{\left\| {\left. {{\varepsilon _{{\rm{T32}}}}} \;\;\right\|} \right.}_2}/{{\left\| {\left. {{\varepsilon _{{\rm{T21}}}}} \;\;\right\|} \right.}_2})}}{{\ln ({r_{\rm{T}}})}} = 6.792$ (18)
 $\delta _{{\rm{R}}{{\rm{E}}_{\rm{T}}}}^{\rm{*}} = \frac{{{{\left\| {\left. {{\varepsilon _{{\rm{T21}}}}} \;\;\right\|} \right.}_2}}}{{r_{\rm{T}}^{{p_{\rm{T}}}} - 1}} = 0.000{\kern 1pt} 021{\kern 1pt} 7$ (19)

pT, est同样取2，时间步长修正因子：

 ${C_{\rm{T}}} = \frac{{r_{\rm{T}}^{{p_{\rm{T}}}} - 1}}{{r_{\rm{T}}^{{p_{{\rm{T,est}}}}} - 1}} = 9.53$ (20)

 $\delta _{\rm{T}}^* = {C_{\rm{T}}}\delta _{{\rm{R}}{{\rm{E}}_{\rm{T}}}}^{\rm{*}} = 0.000{\kern 1pt} 207{\kern 1pt} {\kern 1pt} {\rm{m}}$ (21)

 ${U_{\rm{T}}} = (1 + 2\left| {1 - {C_{\rm{T}}}} \right|)\left| {\delta _{{\rm{R}}{{\rm{E}}_{\rm{T}}}}^*} \right| = 1.034\% {{\bar H}_1}$ (22)

 ${U_{{\rm{SN}}}} = \sqrt {U_{\rm{G}}^2 + U_{\rm{T}}^2} = 1.495\% {{\bar H}_1}$ (23)

 $\delta _{{\rm{SN}}}^{\rm{*}} = \delta _{\rm{G}}^* + \delta _{\rm{T}}^{\rm{*}} = 0.000{\kern 1pt} 470{\kern 1pt} {\kern 1pt} {\rm{m}}$ (24)

 ${U_{\rm{V}}} = \sqrt {U_{{\rm{SN}}}^2 + U_{\rm{D}}^2} = \sqrt {{{1.495}^2} + {{2.5}^2}} \% {{\bar H}_1} = 2.913\% {{\bar H}_1}$ (25)

 $E = {\left\| {{\eta _{{\rm{EXP1,}}x{\rm{ = 4m}}}} - {\eta _{{\rm{NUM1,}}x{\rm{ = 4m}}}}} \;\;\right\|_2} = 1.425\% {{\bar H}_1}$ (26)

3 结论

1) 均匀顺流使规则波波高降低，传播速度加快，波峰和波谷与无流工况相比更平坦；逆流对规则波的影响与顺流的影响相反，逆流流速过大时，规则波会被阻隔。

2) 无流和顺流工况规则波波高时历曲线的比较误差都小于对应的确认不确定度，数值模拟结果得到确认，构建的数值波浪水池具有可靠性。

3) 无流及顺流情况下规则波的数值模拟结果对网格尺寸依赖程度大于时间步长，实际模拟时计算资源有限情况下应尽量满足网格精度要求。

4) 相比无流工况，顺流加剧了规则波波高时历曲线计算结果对网格尺寸和时间步长的依赖程度。

 [1] 胡捍红. 考虑波流干扰的深水畸形波数值模拟研究[D]. 上海: 上海交通大学, 2012. HU Hanhong.Numerical simulation of rogue waves in deep water considering wave/current interaction[D].Shanghai: Shanghai Jiao Tong University, 2012. (0) [2] 贾岩, 尹宝树, 杨德周. 东中国海浪流相互作用对水位和波高影响的数值研究[J]. 海洋科学, 2009, 33: 82-86. JIA Yan, YIN Baoshu, YANG Dezhou. A numerical study of the influence of wave-current interaction on water elevation and significant wave height in the East China Sea[J]. Marine science, 2009, 33: 82-86. (0) [3] GAO Ningbo, YANG Jianmin, LI Xin, et al. Wave forces on horizontal cylinder due to nonlinear focused wave groups[C]//Proceedings of the 25th International Ocean and Polar Engineering Conference. Hawaii, USA: ISOPE, 2015: 674-678. (0) [4] LIU Shengnan, ONG M C, OBHRAI C, et al. Numerical simulations of regular and irregular wave forces on a horizontal semi-submerged cylinder[C]//Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. Trondheim, Norway: OMAE, 2017. (0) [5] FERNANDO P C, GUO Junke, LIN Pengzhi. Wave-current interaction at an angle 1:experiment[J]. Journal of hydraulic research, 2011, 49(4): 424-436. DOI:10.1080/00221686.2010.547036 (0) [6] QI Wengang, LI Changfei, JENG Dongsheng, et al. Combined wave-current induced excess pore-pressure in a sandy seabed: flume observations and comparisons with theoretical models[J]. Coastal engineering, 2019, 147: 89-98. DOI:10.1016/j.coastaleng.2019.02.006 (0) [7] SOLTANPOUR M, SAMSAMI F, SHIBAYAMA T, et al. Study of irregular wave-current-mud interaction[C]//Proceedings of the Coastal Engineering Conference. Seoul, Korea: ICCE, 2014: 01613782. (0) [8] WEI C X, ZHOU D C, OU J P. Experimental study of the hydrodynamic responses of a bridge tower to waves and wave currents[J]. Journal of waterway, port, coastal and ocean engineering, 2017, 143(3): 04017002. DOI:10.1061/(ASCE)WW.1943-5460.0000381 (0) [9] JIANG Xuelian, ZOU Qingping, ZHANG Na. Wave load on submerged quarter-circular and semicircular breakwaters under irregular waves[J]. Coastal engineering, 2017, 121: 265-277. DOI:10.1016/j.coastaleng.2016.11.006 (0) [10] ITTC. 7.5-03-01-01, ITTC-recommended procedures and guidelines[S]. 2017. (0) [11] ROACHE P J. Perspective: a method for uniform reporting of grid refinement studies[J]. Journal of fluids engineering, 1994, 116(3): 405-413. DOI:10.1115/1.2910291 (0) [12] 吴乘胜, 邱耿耀, 魏泽, 等. 船模阻力数值水池试验不确定度评估[J]. 船舶力学, 2015, 19(10): 1197-1208. WU Chengsheng, QIU Gengyao, WEI Ze, et al. Uncertainty analysis on numerical computation of ship model resistance[J]. Journal of ship mechanics, 2015, 19(10): 1197-1208. DOI:10.3969/j.issn.1007-7294.2015.10.004 (0) [13] ZHU Renchuan, YANG Chunlei, MIAO Guoping, et al. Computational fluid dynamics uncertainty analysis for simulations of roll motions for a 3D ship[J]. Journal of Shanghai JiaoTong University (Science), 2015, 20(5): 591-599. DOI:10.1007/s12204-015-1666-z (0) [14] 邓磊, 彭弘宇. 小水线面双体船耐波性能CFD不确定度分析[J]. 中国舰船研究, 2016, 11(3): 17-24. DENG Lei, PENG Hongyu. Uncertainty analysis in CFD for SWATH motions in regular head waves[J]. Chinese journal of ship research, 2016, 11(3): 17-24. DOI:10.3969/j.issn.1673-3185.2016.03.004 (0) [15] SILVA M C, VITOLA M A, EÇA L, et al. Numerical uncertainty analysis in regular wave modeling[J]. Journal of offshore mechanics and arctic engineering, 2018, 140(4): 041101. DOI:10.1115/1.4039260 (0) [16] 柏君励, 马宁, 顾解忡. 畸形波的数值模拟及不确定度分析[J]. 上海交通大学学报, 2018, 23(4): 475-481. BAI Junli, MA Ning, GU Xiechong. Numerical simulation of focused wave and its uncertainty analysis[J]. Journal of Shanghai Jiao Tong University (Science), 2018, 23(4): 475-481. (0) [17] 丁俊杰, 马宁, 顾解忡. 循环水槽多层孔板消波装置开发及消波特性数值模拟[J]. 上海交通大学学报, 2020, 54(1): 52-59. DING Junjie, MA Ning, GU Xiechong. Development of a wave-absorbing device with multiple porous plates for circulating water channel and simulation on its performance[J]. Journal of Shanghai JiaoTong University, 2020, 54(1): 52-59. (0) [18] PISTIDDA A, OTTENS H, ZOONTJES R. Using CFD to assess low frequency damping[C]//Proceedings of the ASME 201231st International Conference on Ocean, Offshore and Arctic Engineering. Rio de Janeiro, Brazil: OMAE, 2012: 657-665. (0) [19] 马宁, 丁俊杰, 顾解忡, 等. 多层变角度开孔折弯板透水消波装置: 中国, CN109100113A[P]. 2018-12-28. MA Ning, DING Junjie, GU Xiechong, et al. Multilayer variable angle perforated bending plate water-permeable wave absorbing device: CN, CN109100113A[P]. 2018-12-28. (0)