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Article(id=1243880212249821551, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243880211004113260, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2024.10.002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1713888000000, receivedDateStr=2024-04-24, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774493752776, onlineDateStr=2026-03-26, pubDate=1729353600000, pubDateStr=2024-10-20, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774493752776, onlineIssueDateStr=2026-03-26, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774493752776, creator=13701087609, updateTime=1774493752776, updator=13701087609, issue=Issue{id=1243880211004113260, tenantId=1146029695717560320, journalId=1240685776644648972, year='2024', volume='28', issue='10', pageStart='1463', pageEnd='1632', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774493752479, creator=13701087609, updateTime=1774494057708, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243881491290243855, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243880211004113260, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243881491290243856, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243880211004113260, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1472, endPage=1485, ext={EN=ArticleExt(id=1243880212509868402, articleId=1243880212249821551, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Interactions and influence factors of a NACA0018 sail array, columnId=1241023037940748650, journalTitle=Journal of Ship Mechanics, columnName=Hydrodynamics, runingTitle=null, highlight=null, articleAbstract=

Sail-assisted navigation is one of the effective ways to reduce carbon dioxide emissions in ship industry. In this study, a sail array consisting of six hard sails with a cross section of NACA0018 airfoil was designed. The lift and resistance of each sail were analyzed by numerical simulation under three conditions: upwind, crosswind and downwind. By comparing with the single sail case, the law of inter-sail interference on sail propulsion performance was explored. The confidence level of several independent geometric parameters on the propulsion performance of the sail array was evaluated by combining the experimental design method with ANOVA. The results show that the significance level of the independent parameters is highly dependent on the apparent wind direction. By changing sail layout and using non-uniform angle of attack, the propulsion performance can be better than that of single attack angle. This study is helpful to find the main parameters affecting the thrust performance of a multi-sail propulsion system and support for its global manipulation.

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风帆助航是船舶领域减少二氧化碳排放的有效方式之一。本文设计了一种由六面横截面为NACA0018翼型的硬帆组成的风帆阵列，采用数值仿真的方法，在迎风、侧风和顺风三种工况下分析了每面帆的升、阻力特性，通过与单帆对比，探究了帆间干扰对风帆推进性能的影响规律。将实验设计法与方差分析法相结合，评估了多个独立几何参数对风帆阵列推进性能的影响水平。结果表明，独立参数的显著性水平高度依赖于表观风向，并且通过改变风帆布局和采用非一致攻角，可获得比单一攻角时更好的推进性能。该研究有助于寻找影响多帆推进系统推力性能的主要参数，为多帆推进的全局化操纵提供数据支持。

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通讯作者，E-mail：
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曾青松（1991-），男，博士，助理研究员，通讯作者，E-mail：

蔡薇（1969-），女，博士，教授，E-mail：。

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蔡薇（1969-），女，博士，教授，E-mail：。

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journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Fig.12, caption=Comparison of thrust performance between the case in DOE and the basic case, figureFileSmall=3pgC2oFZiStDG1G6+qcV0g==, figureFileBig=cbf8PMQaUIrjvT24wR97FQ==, tableContent=null), ArticleFig(id=1243880221892526805, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=图12, caption=试验分析中各工况与基本工况的推力性能对比, figureFileSmall=3pgC2oFZiStDG1G6+qcV0g==, figureFileBig=cbf8PMQaUIrjvT24wR97FQ==, tableContent=null), ArticleFig(id=1243880222009967328, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.1, caption=

Design-of-experiment factors of the sails

, figureFileSmall=null, figureFileBig=null, tableContent=
参数标识内容符号/单位水平1水平2水平3
F1帆1攻角 α_S1/°0.8*base1base11.2*base1
F2帆2攻角 α_S2/°0.8*base2base21.2*base2
F3帆3攻角 α_S3/°0.8*base3base31.2*base3
F4帆4攻角 α_S4/°0.8*base4base41.2*base4
F5帆5攻角 α_S5/°0.8*base5base51.2*base5
F6帆6攻角 α_S6/°0.8*base6base61.2*base6
F7帆间距（x轴向） D x/m0.8*base7base71.2*base7
F8帆间距（y轴向） D y/m0.8*base8base81.2*base8
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风帆的试验设计参数

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参数标识内容符号/单位水平1水平2水平3
F1帆1攻角 α_S1/°0.8*base1base11.2*base1
F2帆2攻角 α_S2/°0.8*base2base21.2*base2
F3帆3攻角 α_S3/°0.8*base3base31.2*base3
F4帆4攻角 α_S4/°0.8*base4base41.2*base4
F5帆5攻角 α_S5/°0.8*base5base51.2*base5
F6帆6攻角 α_S6/°0.8*base6base61.2*base6
F7帆间距（x轴向） D x/m0.8*base7base71.2*base7
F8帆间距（y轴向） D y/m0.8*base8base81.2*base8
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Grid parameters and calculated results of CL, CD for each case

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No. h iCells(×103) C L C D
12.83621.23360.04285
22.001011.22650.04163
31.411711.22310.04117
41.002961.22130.04097
), ArticleFig(id=1243880222282597108, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表2, caption=

网格信息及计算结果

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No. h iCells(×103) C L C D
12.83621.23360.04285
22.001011.22650.04163
31.411711.22310.04117
41.002961.22130.04097
), ArticleFig(id=1243880222374871801, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.3, caption=

Mesh discretization errors and uncertainties

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主要变量 C L C D
ϕ01.22000.04087
α1.64E-031.12E-04
p2.0572.762
S D1.754E-069.05E-08
U（ϕ1）/ϕ12.26%14.22%
U（ϕ2）/ϕ20.97%5.87%
U（ϕ3）/ϕ30.39%2.54%
U（ϕ4）/ϕ40.36%1.23%
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网格离散误差和不确定性计算结果

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主要变量 C L C D
ϕ01.22000.04087
α1.64E-031.12E-04
p2.0572.762
S D1.754E-069.05E-08
U（ϕ1）/ϕ12.26%14.22%
U（ϕ2）/ϕ20.97%5.87%
U（ϕ3）/ϕ30.39%2.54%
U（ϕ4）/ϕ40.36%1.23%
), ArticleFig(id=1243880222588781318, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.4, caption=

Lift and drag coefficients for four time steps

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主要变量 C L C D
Δt1/s1.21280.04152
Δt2/s1.22010.04128
Δt3/s1.22310.04117
Δt4/s1.22380.04110
), ArticleFig(id=1243880222689444623, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表4, caption=

四组时间步长对应的升阻力系数计算结果

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主要变量 C L C D
Δt1/s1.21280.04152
Δt2/s1.22010.04128
Δt3/s1.22310.04117
Δt4/s1.22380.04110
), ArticleFig(id=1243880222777525009, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.5, caption=

Results for temporal discretization uncertainties

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主要变量 C L C D
ϕ01.22400.04106
α-6.087E-044.957E-05
p2.8392.153
S D2.752E-061.641E-09
U（ϕ1）/ϕ12.88%3.40%
U（ϕ2）/ϕ21.05%1.64%
U（ϕ3）/ϕ30.34%0.83%
U（ϕ4）/ϕ40.14%0.40%
), ArticleFig(id=1243880222861411093, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表5, caption=

时间步长不确定性计算结果

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主要变量 C L C D
ϕ01.22400.04106
α-6.087E-044.957E-05
p2.8392.153
S D2.752E-061.641E-09
U（ϕ1）/ϕ12.88%3.40%
U（ϕ2）/ϕ21.05%1.64%
U（ϕ3）/ϕ30.34%0.83%
U（ϕ4）/ϕ40.14%0.40%
), ArticleFig(id=1243880222953685787, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.6, caption=

ANOVA analysis for CT and CH in the condition of upwind 45°

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C T C H
参数 F PSig. F PSig.
F1帆1攻角0.460.640.080.92
F2帆2攻角0.260.782.860.08*
F3帆3攻角17.382.2E-05***0.070.94
F4帆4攻角0.220.801.090.35
F5帆5攻角0.230.802.780.08*
F6帆6攻角0.250.782.350.12
F7帆间距（x轴向）0.880.433.460.05**
F8帆间距（y轴向）0.190.830.530.60
), ArticleFig(id=1243880223079514914, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表6, caption=

侧逆风45°工况下对CT和CH的参数方差分析

, figureFileSmall=null, figureFileBig=null, tableContent=
C T C H
参数 F PSig. F PSig.
F1帆1攻角0.460.640.080.92
F2帆2攻角0.260.782.860.08*
F3帆3攻角17.382.2E-05***0.070.94
F4帆4攻角0.220.801.090.35
F5帆5攻角0.230.802.780.08*
F6帆6攻角0.250.782.350.12
F7帆间距（x轴向）0.880.433.460.05**
F8帆间距（y轴向）0.190.830.530.60
), ArticleFig(id=1243880223175983908, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.7, caption=

ANOVA analysis for CT and CH in the condition of crosswind 90°

, figureFileSmall=null, figureFileBig=null, tableContent=
C T C H
参数 F PSig. F PSig.
F1帆1攻角0.910.420.640.54
F2帆2攻角0.010.990.030.97
F3帆3攻角0.300.740.170.84
F4帆4攻角0.700.510.950.40
F5帆5攻角0.230.800.430.65
F6帆6攻角0.620.541.070.36
F7帆间距（x轴向）23.812.0E-06***23.032.6E-06***
F8帆间距（y轴向）0.370.690.380.69
), ArticleFig(id=1243880223285035817, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表7, caption=

正侧风90°工况下对CT和CH的参数方差分析

, figureFileSmall=null, figureFileBig=null, tableContent=
C T C H
参数 F PSig. F PSig.
F1帆1攻角0.910.420.640.54
F2帆2攻角0.010.990.030.97
F3帆3攻角0.300.740.170.84
F4帆4攻角0.700.510.950.40
F5帆5攻角0.230.800.430.65
F6帆6攻角0.620.541.070.36
F7帆间距（x轴向）23.812.0E-06***23.032.6E-06***
F8帆间距（y轴向）0.370.690.380.69
), ArticleFig(id=1243880223394087728, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.8, caption=

ANOVA analysis for CT and CH in the condition of downwind 135°

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C T C H
参数 F PSig. F PSig.
F1帆1攻角2.230.133.530.05**
F2帆2攻角0.560.580.080.92
F3帆3攻角0.350.714.460.02**
F4帆4攻角0.580.570.050.95
F5帆5攻角0.400.680.740.49
F6帆6攻角1.270.301.510.24
F7帆间距（x轴向）0.840.441.220.31
F8帆间距（y轴向）11.942.5E-04***2.390.11
), ArticleFig(id=1243880223532499765, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=CN, label=表8, caption=

侧顺风135°工况下对CT和CH的参数方差分析

, figureFileSmall=null, figureFileBig=null, tableContent=
C T C H
参数 F PSig. F PSig.
F1帆1攻角2.230.133.530.05**
F2帆2攻角0.560.580.080.92
F3帆3攻角0.350.714.460.02**
F4帆4攻角0.580.570.050.95
F5帆5攻角0.400.680.740.49
F6帆6攻角1.270.301.510.24
F7帆间距（x轴向）0.840.441.220.31
F8帆间距（y轴向）11.942.5E-04***2.390.11
), ArticleFig(id=1243880223683494720, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1243880212249821551, language=EN, label=Tab.A, caption=

The orthogonal array for the L27(38)

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RunF1F2F3F4F5F6F7F8
111111111
211112222
311113333
412221112
512222223
612223331
713331113
813332221
913333332
1021231231
1121232312
1221233123
1322311232
1422312313
1522313121
1623121233
1723122311
1823123122
1931321321
2031322132
2131323213
2232131322
2332132133
2432133211
2533211323
2633212131
2733213212
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L27（38）的正交表

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RunF1F2F3F4F5F6F7F8
111111111
211112222
311113333
412221112
512222223
612223331
713331113
813332221
913333332
1021231231
1121232312
1221233123
1322311232
1422312313
1522313121
1623121233
1723122311
1823123122
1931321321
2031322132
2131323213
2232131322
2332132133
2432133211
2533211323
2633212131
2733213212
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NACA0018风帆阵列帆间干扰及影响因素研究
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曾青松 1, 2 , 蔡薇 1, 2
船舶力学 | 流体力学 2024,28(10): 1472-1485
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船舶力学 | 流体力学 2024, 28(10): 1472-1485
NACA0018风帆阵列帆间干扰及影响因素研究
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曾青松1, 2 , 蔡薇1, 2
作者信息
  • 1.武汉理工大学　绿色智能江海直达船舶与邮轮游艇研究中心，武汉　430063
  • 2.武汉理工大学　海南研究院，海南　三亚　572024

    • 曾青松（1991-），男，博士，助理研究员，通讯作者，E-mail：

    蔡薇（1969-），女，博士，教授，E-mail：。

通讯作者:

通讯作者，E-mail：
Interactions and influence factors of a NACA0018 sail array
Qing-song ZENG1, 2 , Wei CAI1, 2
Affiliations
  • 1.Green & Smart River-Sea-Going Ship, Cruise and Yacht Research Center, Wuhan University of Technology, Wuhan 430063, China
  • 2.Hainan Institute of Wuhan University of Technology, Sanya 572024, China
出版时间: 2024-10-20 doi: 10.3969/j.issn.1007-7294.2024.10.002
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摘要
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风帆助航是船舶领域减少二氧化碳排放的有效方式之一。本文设计了一种由六面横截面为NACA0018翼型的硬帆组成的风帆阵列，采用数值仿真的方法，在迎风、侧风和顺风三种工况下分析了每面帆的升、阻力特性，通过与单帆对比，探究了帆间干扰对风帆推进性能的影响规律。将实验设计法与方差分析法相结合，评估了多个独立几何参数对风帆阵列推进性能的影响水平。结果表明，独立参数的显著性水平高度依赖于表观风向，并且通过改变风帆布局和采用非一致攻角，可获得比单一攻角时更好的推进性能。该研究有助于寻找影响多帆推进系统推力性能的主要参数，为多帆推进的全局化操纵提供数据支持。

风帆推进  /  数值仿真  /  实验设计  /  方差分析

Sail-assisted navigation is one of the effective ways to reduce carbon dioxide emissions in ship industry. In this study, a sail array consisting of six hard sails with a cross section of NACA0018 airfoil was designed. The lift and resistance of each sail were analyzed by numerical simulation under three conditions: upwind, crosswind and downwind. By comparing with the single sail case, the law of inter-sail interference on sail propulsion performance was explored. The confidence level of several independent geometric parameters on the propulsion performance of the sail array was evaluated by combining the experimental design method with ANOVA. The results show that the significance level of the independent parameters is highly dependent on the apparent wind direction. By changing sail layout and using non-uniform angle of attack, the propulsion performance can be better than that of single attack angle. This study is helpful to find the main parameters affecting the thrust performance of a multi-sail propulsion system and support for its global manipulation.

wind-assisted propulsion  /  numerical simulation  /  design of experiment  /  analysis of variance
曾青松, 蔡薇. NACA0018风帆阵列帆间干扰及影响因素研究. 船舶力学, 2024 , 28 (10) : 1472 -1485 . DOI: 10.3969/j.issn.1007-7294.2024.10.002
Qing-song ZENG, Wei CAI. Interactions and influence factors of a NACA0018 sail array[J]. Journal of Ship Mechanics, 2024 , 28 (10) : 1472 -1485 . DOI: 10.3969/j.issn.1007-7294.2024.10.002
正文
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0　引言
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风帆助航是一种古老的驱动方式，由于其采用了清洁能源——风能，在目前“双碳”目标的驱动下，风帆助航技术重新获得了人们的重视[1]。不同于木帆船时代的传统风帆，现代风帆的设计采用了新技术、新材料与新思想，并且多用作船舶的辅助动力，高效风帆装备的综合节能效果可达到15%以上[2]，因而越来越多地应用于现代船舶上。中国船级社也在2020年发布了《船用硬质翼面帆评估与检验指南》[3]，进一步推动了风帆助航的实践应用。
风帆可分为软帆和硬帆，通常将涤纶等材质的帆面视为软帆，常见于体育竞技类帆船，而金属或碳纤维等材质的帆则归为硬帆，常见于商用船舶。Viola等[4]基于风帆游艇，对软帆和硬帆系统进行了风洞对比试验，研究发现帆的软硬材质对帆的升阻力和帆面压力分布有显著影响，且软帆帆面上的压力差更大。Domínguez-Brito等[5]为一艘海洋无人帆船设计了两种碳纤维刚性翼帆，经过测试表明，两种设计都能够保证船舶在几乎所有横摇角时的稳性。事实上，对于大多数商船来说，硬帆由于其操作便利且易于维护而逐渐被青睐，本文也将以翼型硬帆作为研究对象。
对于大型远洋船舶，单个风帆已无法满足助航需求，由多面风帆组成的风帆阵列将逐渐成为趋势。通常情况下，风帆阵列由于存在帆间干扰，会降低彼此的助推效果。Li等[6]研究了三种叶栅帆在不同间距和不同风向下的推力系数，他们建议增加帆距以减少由于帆间干扰而产生的负面影响。Lee等[7]选取了由三面翼帆组成的风帆阵列，计算并优化了襟翼长度以获得更好的空气动力学性能，他们研究发现，由于帆的相互作用，风帆阵列的平均推力系数比单帆小21%~43%。在Bordogna等[8]对一组Flettner转筒帆的研究结论中，建议将两个转筒尽可能远离，以最大限度地减少帆间干扰的负面作用。Guo等[9]对不同攻角下双翼面翼型风帆的间距影响进行了数值研究，发现当帆间距/弦长比小于5时，两翼型之间的不利干扰是不可忽略的。
然而，通过合理的设计或操作，实现多帆之间的有利干扰是完全可能的。在Domínguez-Brito等[5]的研究中，通过反向调节两面伺服机翼的攻角，可产生有利干扰，进而避免前翼设计中经常出现的动力不稳定问题。Wang等[10]发现对于自动俯仰翼，通过适当选择几何和运动参数，可以使两面翼形帆之间产生有利干扰，进而提高系统效率。李森茂等[11]通过在翼形主帆之后增加襟翼帆，良性的帆间相互作用可提高升力系数和升阻比。
虽然目前对多帆阵列的帆间干扰已有一定程度的研究，但对多帆系统影响参数的重要性水平缺乏综合的评价。李臣等[12]研究了翼帆缝隙相对宽度和襟翼偏转角对双元素翼帆的气动特性的影响，讨论了这两种参数对翼帆的影响。然而，风帆阵列的特征参数远不止这两个，每个风帆的攻角、风帆纵向和横向的间距等，均是影响帆间干扰和助航能力的独立参数。若对每一个独立参数进行枚举，将产生巨大的样本总体，这其中存在许多无效的计算，会耗费计算资源。为了采用尽可能少的计算量识别重要的自变参数，将实验设计（design of experiment，DOE）[13]与方差分析（analysis of variance，ANOVA）[14]两种方法结合起来将是一种有效途径。
本文设计一个由六面截面形式为NACA0018的翼形风帆组成的风帆阵列，应用计算流体力学（Computational Fluid Dynamics，CFD）计算风帆的升力和阻力，并换算成船舶的推力和侧倾力，分析风帆阵列的速度场和压力场，探讨风帆相互作用的物理规律，并采用DOE结合ANOVA的方法，评估不同风向下影响参数的显著性水平。
1　计算模型
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本章将介绍仿真的几何模型、数值仿真方法、初始工况和实验设计工况。
1.1　几何模型
为了提高数值计算效率，在能够揭示多帆干扰主要物理规律的前提下，提出以下假设：
（1）采用二维模型。虽然风帆边缘的三维流动对计算结果有一定影响，但二维流场仍能揭示多帆阵列区域的主要流动规律，为研究帆间干扰物理规律提供有价值的参考；
（2）不考虑船体、桅杆、锁具等对风帆的影响。虽然船体的存在使风帆的推进系数发生改变[15]，但其数量级不足以导致帆间干扰的物理规律发生本质性变化。
鉴于NACA系列翼形有大量可用的验证数据，本文选取了剖面编号为NACA0018的几何模型。其升力（L）和阻力（D）的定义及其在x、y方向上的投影如图1所示。
图1是基于风帆定义的坐标系（x-O-y），为了方便计算船舶推力，定义船首朝+y方向，α为攻角，β为帆角，α+β即为表观风向。升力（L）与风向垂直，阻力（D）沿着风向，则风帆的助航推力（T）、侧倾力（H）可分别表示为
式中，T指向船首为正，H指向右舷为正。本文所研究的六帆阵列的布置方案如图2所示，坐标系x0-O0-y0为方便计算域的划分而设立，C是帆的弦长，船的每侧各设计三面帆，其中帆1、3、5的中点偏离y0轴2.5C的距离，帆2、4、6的中点偏离y0轴0.5C。帆3和4的中点在-x0轴上，帆1和2向+y0方向偏移2C，帆5和6向-y0方向偏移2C。
推力系数（CT）和倾侧力系数（CH）分别定义如下：
式中，ρ为空气密度，V为表观风速。
1.2　数值仿真方法
（1）求解器与网格
模拟中流场为非定常不可压缩流体，基于雷诺平均的Navier-Stokes（RANS）方程来模拟流场的运动，采用SST （Menter） k-ω模型来处理湍流。网格划分和仿真计算通过商业求解器Star-CCM+（v16.02）来完成。采用重叠网格技术，可在不改变背景网格的情况下任意改变风帆攻角。运用边界层网格来处理贴体粘性流动，并保证无量纲首层网格厚度y+<1。对风帆周围及尾流区域进行加密处理，如图3所示。
（2）计算域与边界条件
为了消除计算域边界对仿真计算的影响，边界应尽量远离风帆区域。在本文中，计算域由一个半圆和一个矩形组成，其中半圆的半径为28C，矩形的长边为60C，如图4所示。这里，半圆形边界有利于来流方向的设置，通过速度分量的控制可实现来流0°~180°方向的调节，以模拟风帆可能遇到的不同工况。半圆形区域的圆心与坐标系x0-O0-y0的原点重合。
左侧的圆弧设置为速度入口（Velocity Inlet），风帆表面设置为无滑移壁面（No-slip Wall），计算域的所有其他边界都设置为压力出口（Pressure Outlet），参考压力设置为零。
1.3　初始工况
作为对比工况，首先计算了单帆的情况，选取表观风向为45°、90°和135°，分别代表侧逆风（S1）、正侧风（S2）和侧顺风（S3）三种风向，如图5所示。
选择攻角范围为5°~90°，以5°间隔形成系列计算方案，在升力或阻力变化剧烈的区域将适当增加额外的攻角。雷诺数是表征风帆流场特性的重要参数，理论上应选择与实船尺度相同或相近的数值，然而，实尺寸的风帆动力数据极为缺乏，本文结合NACA0018机翼已公布的数据资料，选择气动性能数据较为丰富的Re=5.0×105作为本文主要的雷诺数取值，这样有利于计算结果的校核。值得指出的是，目前对于仿真结果的验证通常在低雷诺数工况（模型尺度）下进行，以此为依据尚不足以说明CFD的方法在高雷诺数工况（实船尺度）仍然可靠，但这涉及尺度效应的诱发机理，暂不在本文的讨论范围之内。
在本文的系列计算中，选取风帆的弦长（C）为1 m，在雷诺数为5.0×105时对应的表观风速为7.579 m/s。设计六帆阵列，图2中所标识的风帆位置与尺寸是一种典型的设计方案。由于在航行过程中保持所有风帆的攻角一致是最直观、简单的操作方法，所以作为对比，在对风帆阵列的研究中首先探讨所有风帆的攻角都相同的情况。在六帆阵列的模拟中，采用了与单帆情况相同的三种表观风向。
1.4　试验设计法
为了寻找影响风帆阵列综合助航性能的重要影响参数，本文选取了八个独立自变量，即六面帆的攻角、x轴向间距与y轴向间距（假设风帆阵列在其所构成的矩形空间内均匀分布），如表1所示。
经过初步计算发现，独立变量与基础值（base）的偏差达到20%时，就可以产生有效差异且可避免数值突变，因此，就以差异20%为变化幅度来生成三个参数水平，其中基准数“baseX”（X=1，2，3，…，8）的具体值将在下文确定。
若想枚举设计空间中的所有案例，则需要6561（=38）个计算方案才能确定每个因素的影响程度。然而，通过使用实验设计（DOE），则仅需符合L27（38）正交表的27个实验案例就可获得统计学上可靠的显著性水平[16]。本文根据Eendebak等[17]和Schoen等[18]的研究设计了正交L27（38）表，如附录A所示。
2　数值校核与验证
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数值计算是对物理模型的一种近似，在采用系列数值仿真之前应校核其不确定性。此外，还应通过与试验数据的对比来验证仿真结果的可靠性。
2.1　数值校核
通过使用双精度格式和足够的迭代次数，可使舍入误差和迭代误差的影响忽略不计。因此，与网格疏密和时间步长相关的离散误差，将成为数值误差的主要部分[19]。采用Eça[19]提出的方法对网格离散化和时间步长依赖性进行评估，如式（3）所示。
式中，SRE是最小二乘法的误差估计，ϕi是目标变量，是ϕi和预估的真值ϕ0之间的误差。ϕ0、a和p将在对ϕi进行曲线拟合后确定，ϕ0表示网格加密达到上限（即物理上的连续状态）时的真值，p的值表示收敛的量级，hi表示网格密度，其中hi =1表示分辨率最高，i为计算案例序号。通常，公式（3）需建立在至少四组样本点的基础之上，数值仿真的不确定度（U（ϕi））可由公式（4）进行计算。
当0.5≤ p <2.1时，系数F=1.25，其他情况F=3，SD是标准差。
（1）网格依赖性分析
本节设计了四组不同加密程度的网格来评估网格离散的依赖性，参照凸显数值变化但不过度增加计算量的原则，选取加密因子，选择攻角α=15°、雷诺数Re=500 000的工况作为加密基础工况。在网格生成过程中，按照加密因子等比例改变网格生成的参照值来控制网格的疏密，且x0与y0方向将按照相同的加密因子进行加密，风帆壁面边界层网格层数及首层网格厚度将保持不变，据此生成了四组网格，风帆附近网格对比如图6所示。
各工况的网格信息、升力系数CL、阻力系数CD如表2所示。
根据公式（3）和公式（4）计算各组网格不确定度比例（U（ϕi）/ϕi），结果如表3所示，可以看到第三组网格CL和CD的不确定度分别为0.39%和2.54%。目前，数值仿真计算在船舶工程应用中的不确定度水平约为0~4%[20-21]，上述第三组网格计算结果在此范围内。而相比于第四组网格，第三组网格消耗的计算资源更少，因此在后续系列计算中，将参照第三组网格的参数设置进行网格划分。
（2）时间步长依赖性分析
本节探讨了时间步长对仿真结果的影响，与上节类似，选取基础变化因子，在Courant数小于1的基础上，确定四组时间步长：Δt1=0.0080 s，Δt2=0.0057 s，Δt3=0.0040 s和Δt4=0.0028 s，其升力与阻力系数计算结果如表4所示。
对四组时间步长计算结果的不确定性分析结果如表5所示。
对比表3和表5可知，基于目前的设置，时间步长的不确定度要低于网格的不确定度。在时间步长为Δt3时，CL和CD的不确定度相对于数值计算而言都达到了相当的精度（<1%）。因此，在接下来的系列计算中，将应用相同的时间步长Δt3=0.0040 s。
2.2　数值计算的验证
数值仿真的计算结果需与对应试验结果进行对比来验证其可靠性。本节选取了Eggert等[22]对单翼型NACA0018在Re=250 000时升力系数随攻角（α）变化的实验结果，在相同条件下进行了仿真计算，两者结果对比如图7所示。
从图7可知，数值仿真（CFD）与试验数据接近，其误差主要来自于实验环境与测量仪器的干扰，但从计算结果来看，数值仿真可获得与实验值较为吻合的结果。
除此之外，在Re=500 000时计算得到NACA0018的升力与阻力系数随攻角的变化，还与Drela[23-24]利用XFOIL系统的结果进行了对比，如图8所示。
由图8可知，在攻角小于10°时，计算结果与XFoil吻合较好，在攻角大于10°时误差稍大。这是由于流体运动变得复杂且有流体分离的趋势造成的，但其数据量级和变化趋势仍然相同。经过对比，可认为本文的数值计算结果是相对可靠的。
由于几何模型的适度简化以及湍流近似模型等的采用，数值仿真计算不可避免地带来一定程度的误差，但数值仿真的精度不会对风帆流场干扰的物理机理造成本质性的改变。因此，在数值校核与验证中采用的数值方法将沿用至本文后续系列计算中。
3　结果与分析
收起
本章给出了六帆等攻角时的CT和CH的系列计算结果（计算所得升力L与阻力D根据公式（1）换算为推力T和侧倾力H），并与单帆工况进行比较，通过实验设计分析和自变量方差分析，确定了影响多帆推力和相互作用的主要因素。
3.1　六帆等攻角工况
在实际航行中，风帆攻角（α）理论上是任意的，但在一定表观风向（α+β）下，总存在一个攻角区间使风帆阵列的综合推力最大。在本文中，针对单帆和六帆阵列，分别探讨了在侧逆风45°、正侧风90°和侧顺风135°下的最佳攻角范围，计算结果如图9所示。
由图9可知：
侧逆风45°时，当攻角α≤25°时，此时未发生流动分离，风帆尾流干扰较小，所以六帆阵列的平均推力和侧倾力与单帆工况相近；当攻角α>25°时，帆间干扰明显，导致六帆阵列的平均推力系数相对于单帆减小了40%~60%。风帆推力系数CT在α=15°~20°达到最大（约0.65），然后在α=45°时降为零。因此，对于侧逆风45°工况，保持风帆攻角小于20°有利于保证助航效果。
正侧风90°时，六帆阵列的平均推力相较于单帆在α≤25°时减少约20%，在α>25°时则减少约50%。虽然六帆工况的推力系数降低，但此时侧倾力系数同样低于单帆工况，这在一定程度上弱化了帆间干扰的不利影响。推力系数CT在α=22°时出现局部最小值后，在α=35°~40°达到最大值（单帆约1.9，六帆平均值约1.2），因此，对于正侧风90°工况，保持风帆攻角α=35°~40°为佳。
侧顺风135°时，当攻角α≤25°时，六帆阵列的推力和侧倾力表现与单帆相近；在α>25°大攻角下则降低了约50%，但却表现出更“稳定”的推进效果。对于六帆阵列，α=35°~70°时，CT在1.25附近趋于稳定（指推力曲线平稳不随攻角改变而改变）；单帆在此攻角区间则表现出“强劲”的推进效果，并在α=50°时推力系数达到峰值（CT=3.37）。因此，对于侧顺风135°工况，维持风帆攻角α=35°~70°可获得良好的推进效果。
为了揭示风帆阵列中各帆的空气动力学特征，并探索帆间干扰规律，图10列出了三种表观风向下单帆和六帆阵列的速度与流线分布云图（Re=500 000，α=35°），与此同时，将三种表观风向下每面帆的推力和侧倾力系数与单帆进行对比，如图11所示。
由图10和图11可以得出：
在侧逆风45°时，帆1和帆2对CT和CH的贡献较其他帆更大，且在攻角α≤30°时大于同工况下单帆的推力系数。与之相反，帆4和帆6对推力和横倾力上的贡献极小。从物理意义上讲，帆1和帆2在侧逆风45°条件下正对来流，不受其他帆的尾流影响，并且由于帆3和帆4的存在，其尾流的发展受到限制，反作用力增加了其上下表面的压力差，进而增大了推力系数。帆4和帆6几乎完全位于其他风帆的尾部伴流区，所受到的不稳定来流影响抑制了其推力性能。
在正侧风90°时，无论是对于CT还是CH，帆1、帆3、帆5均占主导地位，且贡献排序为帆1>帆3>帆5，且帆1的性能与单帆相近。同时从图10（b）中可以看出，帆1周围的流场特征与单帆类似。帆2、帆4、帆6性能相近，其中帆4的贡献最小，因为这些风帆都处于上游风帆的尾流区域，来流中充满扰动，且来流攻角发生变化，其推力性能因此未能充分发挥。
在侧顺风135°时，受干扰较少的帆1、帆3、帆5对CT和CH的贡献较明显，与侧逆风45°类似，完全处于尾流的帆对推力性能的贡献甚微。
综上所述，三种风向下的风帆阵列，在当前的布置和攻角下，帆间干扰对于风帆阵列总推力系数是不利的，但对于阵列中的个别风帆，帆间干扰可在特定工况产生有益效果。因此，为了提高风帆阵列的综合推进效率，需对影响帆间干扰的独立几何参数进行重要性评估，进而通过控制重要的几何参数来改变风帆阵列的助航效果。
3.2　独立自变量的DOE和ANOVA分析
本节选取攻角α=35°作为基本工况（因其在三种表观风向下均表现出稳定的推进性能），采用L27（38）正交表对1.4节表1中的八个独立参数进行试验分析（DOE），其中赋予独立参数F7和F8的基础值2 m，即为弦长的2倍，附录A中的标识1、2、3分别表示表1中参数的三种水平，计算不同表观风向下风帆阵列的综合CT和CH，应用方差分析法（ANOVA）对各独立因素的显著性水平进行分析。方差分析法可评估组间均方和组内均方的差别，计算独立参数的显著性水平，当置信度较高时，可认为对应参数的变化对计算结果有重要影响，其具体计算原理可参考Shapiro等[14]的研究。方差分析法的结果如表6、表7和表8所示。
由表6可知，F3对CT的置信度为99%，F7对CH的置信度为95%，这说明在侧逆风45°时，帆3的攻角和帆在x轴向的间距是影响助航性能的重要因素。由表7可知，正侧风90°时，帆在x轴向的间距（F7）是影响CT和CH的唯一主导因素，此时，来流垂直于帆1~帆5连线，增加x轴向的帆间距有利于减小尾流对后方帆的负面影响。由表8可知，在侧顺风135°时，F8对CT的置信度为99%，F1和F3对CH的置信度为95%，此时，增大y轴方向的间距是增加风帆阵列总推力系数的最佳方法。
因此，结合上述分析，对于推力系数和侧倾力系数的影响因素，可以得出以下初步结论：
（1）风帆间距，无论是x轴向还是y轴向，其影响总体要比风帆攻角更加明显；
（2）各参数对CT和CH的置信度不同，说明所选参数对推力和侧倾力影响是相互独立的；
（3）影响风帆助航性能主要参数的确定依赖于具体表观风向。
在对风帆阵列进行试验分析时发现，在27种不同案例中存在比六帆攻角一致时更好的推进性能。以正侧风90°为例，其在27种情况下计算的推力系数（CT）相比于攻角一致时（基本工况）推力系数的变化百分比如图12（a）所示，此外，推力系数与侧倾力系数的比值（CT/CH）是评估助航性能的另一个指标，其结果如图12（b）所示。
由图12可知：
（1）多个案例的CT和CT/CH相较基本工况均有提高，例如案例2、3、5、6和16。特别是方案2与基本工况的帆距相同，但帆1~4采用更小的攻角，提高了风帆阵列整体的推进性能，这表明采用相同攻角并非最佳的操帆方式；
（2）大多数案例的CT和CT/CH均有下降，例如案例14的CT降低了34%，CT/CH降低了7%。因此，在实际操作中应谨慎改变风帆参数，因为基本工况本身是一个高于平均水平的样本。
（3）对于推进性能有提高的案例3、6和16，在x方向上的帆间距都增大了，再次证明了F7（x轴向帆间距）是影响正侧风90°风帆推进性能的主要因素。
4　结论
收起
本文采用CFD方法对由六个NACA0018风帆组成的风帆阵列进行了推力性能和帆间干扰研究，采用实验设计和方差分析方法对独立几何参数的置信水平进行了评估，得出如下结论：
（1）对于侧逆风45°，保持风帆攻角小于20°有利于保持助航效果；正侧风90°时，保持风帆攻角α=35°~40°为佳；侧顺风135°时，维持风帆攻角α=35°~70°可获得良好的推进效果。
（2）风帆间距对风帆阵列综合助航性能的影响总体大于风帆攻角的影响，各参数对推力系数和侧倾力系数的影响是相互独立的，并且影响风帆助航性能的主要参数依赖于表观风向。
（3）风帆阵列采用相同的攻角并非是最佳的操作方法，但在实际操作中应谨慎改变风帆参数，因为等间距同攻角的基本工况自身是一个高于平均水平的操帆方式。
在未来的研究中，将考虑船帆末梢三维流动的影响，这将进一步丰富和拓展本文的研究内容和结论。
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doi: 10.3969/j.issn.1007-7294.2024.10.002
  • 接收时间：2024-04-24
  • 首发时间：2026-03-26
  • 出版时间：2024-10-20
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  • 收稿日期：2024-04-24
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    1.武汉理工大学　绿色智能江海直达船舶与邮轮游艇研究中心，武汉　430063
    2.武汉理工大学　海南研究院，海南　三亚　572024

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