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Article(id=1241023039480066177, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1241023036892172648, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2025.11.002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1745424000000, receivedDateStr=2025-04-24, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1773812549705, onlineDateStr=2026-03-18, pubDate=null, pubDateStr=null, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773812549705, onlineIssueDateStr=2026-03-18, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773812549705, creator=13701087609, updateTime=1773812549705, updator=13701087609, issue=Issue{id=1241023036892172648, tenantId=1146029695717560320, journalId=1240685776644648972, year='2025', volume='29', issue='11', pageStart='1663', pageEnd='1826', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1773812549088, creator=13701087609, updateTime=1773812775651, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241023987245970093, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1241023036892172648, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241023987245970094, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1241023036892172648, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1678, endPage=1689, ext={EN=ArticleExt(id=1241023041598189708, articleId=1241023039480066177, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Numerical simulation of submarine advancing under ice at different submerged depths and speeds, columnId=1241023037940748650, journalTitle=Journal of Ship Mechanics, columnName=Hydrodynamics, runingTitle=null, highlight=null, articleAbstract=

In order to investigate the hydrodynamic characteristics of submarines moving in ice regions, a mathematical model and corresponding numerical method, considering the coupling interaction of submarine, water and ice, were established based on RANS equations for level ice and brash ice conditions. The ice sheet was regarded as a no-slip wall boundary with a tangential velocity equal to the incoming flow speed, while the discrete element method incorporated with a linear elastic model was employed to simulate the motion and collision of brash ice. The SUBOFF submarine model was selected as the research object. The influence of sea ice on the resistance was numerically investigated first after mesh convergence analysis. Then further calculations were conducted on resistance components for different submerged depths and speeds under various ice conditions. And wave patterns of the SUBOFF were analyzed for free surface and brash ice condition. Numerical results show that the sea ice predominantly affects the pressure resistance, especially the wave-making component, while there are little differences in frictional resistances for different submerged depths and ice conditions. As the submarine moves under level ice, a significant decrease in pressure resistance and total resistance is observed owing to the lack of wave disturbance. In contrast, when navigating in water covered by brash ice, waves are generated. However, the random and uncertain collisions of floating ice lead to strong fluctuations in the pressure resistance and total resistance curves. And the corresponding average resistance values are slightly smaller than those in open sea due to the wave attenuation induced by brash ice. Furthermore, the submerged depth and forward speed are found to have a significant impact on the hydrodynamic characteristics of the submarine sailing in ice regions. With the increase of submerged depth, both pressure and total resistances will gradually decrease and tend to be consistent. Generally, the increase of forward speed will lead to larger resistance. However, when the submarine navigates beneath the open sea or brash ice, the pressure resistance increases at first and then decreases slightly, which makes the total resistance to grow at a slower rate.

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为了研究潜艇在冰区航行时的受力特性，本文基于流场平均运动的RANS方程，将冰盖视为切向速度等于来流速度的无滑移壁面边界，采用离散元法结合线弹性模型模拟碎冰运动以及它们之间的接触碰撞，建立了平整冰和碎冰下潜体-水-海冰多介质耦合作用的数学模型和数值求解方法。选取SUBOFF潜艇为研究对象，在网格收敛性分析的基础上，首先数值模拟和分析了海冰对潜艇阻力特性的影响。进一步计算了不同冰况下潜艇以不同潜深和航速航行时的各阻力成分，并针对自由面和碎冰条件研究了潜艇的兴波特性。研究表明，不同冰况下潜艇以不同潜深航行时所受到的摩擦阻力差异很小，海冰主要对压阻力（尤其是其中的兴波阻力）产生影响。平整冰下无法产生兴波，潜艇的压阻力和总阻力显著降低；碎冰下虽产生兴波，但浮冰间随机不确定的碰撞、翻转等使得潜艇的压阻力和总阻力曲线呈现出强烈的波动现象，且浮冰的消波作用使得相应的阻力平均值略小于自由面下的值。潜深和航速对冰区航行潜艇受力特性影响显著，随着潜深增加，各冰况下航行潜艇的压阻力和总阻力均逐渐减小并趋于一致；航速增加则普遍导致阻力上升，但在碎冰或自由面下，压阻力先增加随后略有减小，使得总阻力增速减缓。

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通讯作者，E-mail：。
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杨云涛（1991-），男，博士，副教授，通讯作者，E-mail：。

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2.张家港江苏科技大学（先进金属材料）产业技术研究院，江苏　张家港　215600, bio={"content":"

杨云涛（1991-），男，博士，副教授，通讯作者，E-mail：。

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杨云涛（1991-），男，博士，副教授，通讯作者，E-mail：。

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tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1241023039480066177, language=CN, label=图18, caption=SUBOFF以不同航速在自由面和碎冰下航行时的兴波云图, figureFileSmall=Ke4P4CK05ZUUuEKoUZKxbw==, figureFileBig=GkqGRReOtmUWa1Jkk9t0bg==, tableContent=null), ArticleFig(id=1241766164163002549, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1241023039480066177, language=EN, label=Tab.1, caption=

Principal dimensions of SUBOFF

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参数名称及符号/单位数值
总长L /m4.356
最大直径Dmax /m0.508
围壳长Lf /m0.368
围壳高Hf /m0.206
), ArticleFig(id=1241766164251082938, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1241023039480066177, language=CN, label=表1, caption=

SUBOFF潜艇主尺度

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参数名称及符号/单位数值
总长L /m4.356
最大直径Dmax /m0.508
围壳长Lf /m0.368
围壳高Hf /m0.206
), ArticleFig(id=1241766164360134845, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1241023039480066177, language=EN, label=Tab.2, caption=

Mesh parameters used in convergence analysis

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基本尺寸网格数
平整冰碎冰
1.089 m10 49220 215
0.087 m136 1521 472 024
0.058 m348 2494 106 658
0.050 m483 0346 194 588
0.045 m640 4238 470 059
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收敛性分析网格参数

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基本尺寸网格数
平整冰碎冰
1.089 m10 49220 215
0.087 m136 1521 472 024
0.058 m348 2494 106 658
0.050 m483 0346 194 588
0.045 m640 4238 470 059
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潜艇以不同潜深和航速在海冰环境中航行的数值模拟研究
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杨云涛 1, 2 , 胡官清 3 , 湛俊华 4 , 马超 5
船舶力学 | 流体力学 2025,29(11): 1678-1689
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船舶力学 | 流体力学 2025, 29(11): 1678-1689
潜艇以不同潜深和航速在海冰环境中航行的数值模拟研究
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杨云涛1, 2 , 胡官清3, 湛俊华4, 马超5
作者信息
  • 1.江苏科技大学　船舶与建筑工程学院，江苏　张家港　215600
  • 2.张家港江苏科技大学（先进金属材料）产业技术研究院，江苏　张家港　215600
  • 3.武汉第二船舶设计研究所，武汉　430205
  • 4.中国船舶科学研究中心　水动力学国家重点实验室，江苏　无锡　214082
  • 5.武汉理工大学　船舶与能源动力工程学院，武汉　430063

    • 杨云涛（1991-），男，博士，副教授，通讯作者，E-mail：。

通讯作者:

通讯作者，E-mail：。
Numerical simulation of submarine advancing under ice at different submerged depths and speeds
Yun-tao YANG1, 2 , Guan-qing HU3, Jun-hua ZHAN4, Chao MA5
Affiliations
  • 1.School of Naval Architecture and Civil Engineering, Jiangsu University of Science and Technology, Zhangjiagang 215600, China
  • 2.Industrial Technology Research Institute (Advanced Metal Materials) of Zhangjiagang, Jiangsu University of Science and Technology, Zhangjiagang 215600, China
  • 3.Wuhan No.2 Ship Design and Research Institute, Wuhan 430205, China
  • 4.National Key Laboratory of Science and Technology on Hydrodynamics, China Ship Scientific Research Center, Wuxi 214082, China
  • 5.School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
doi: 10.3969/j.issn.1007-7294.2025.11.002
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摘要
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为了研究潜艇在冰区航行时的受力特性，本文基于流场平均运动的RANS方程，将冰盖视为切向速度等于来流速度的无滑移壁面边界，采用离散元法结合线弹性模型模拟碎冰运动以及它们之间的接触碰撞，建立了平整冰和碎冰下潜体-水-海冰多介质耦合作用的数学模型和数值求解方法。选取SUBOFF潜艇为研究对象，在网格收敛性分析的基础上，首先数值模拟和分析了海冰对潜艇阻力特性的影响。进一步计算了不同冰况下潜艇以不同潜深和航速航行时的各阻力成分，并针对自由面和碎冰条件研究了潜艇的兴波特性。研究表明，不同冰况下潜艇以不同潜深航行时所受到的摩擦阻力差异很小，海冰主要对压阻力（尤其是其中的兴波阻力）产生影响。平整冰下无法产生兴波，潜艇的压阻力和总阻力显著降低；碎冰下虽产生兴波，但浮冰间随机不确定的碰撞、翻转等使得潜艇的压阻力和总阻力曲线呈现出强烈的波动现象，且浮冰的消波作用使得相应的阻力平均值略小于自由面下的值。潜深和航速对冰区航行潜艇受力特性影响显著，随着潜深增加，各冰况下航行潜艇的压阻力和总阻力均逐渐减小并趋于一致；航速增加则普遍导致阻力上升，但在碎冰或自由面下，压阻力先增加随后略有减小，使得总阻力增速减缓。

平整冰和碎冰  /  潜艇  /  阻力  /  数值模拟

In order to investigate the hydrodynamic characteristics of submarines moving in ice regions, a mathematical model and corresponding numerical method, considering the coupling interaction of submarine, water and ice, were established based on RANS equations for level ice and brash ice conditions. The ice sheet was regarded as a no-slip wall boundary with a tangential velocity equal to the incoming flow speed, while the discrete element method incorporated with a linear elastic model was employed to simulate the motion and collision of brash ice. The SUBOFF submarine model was selected as the research object. The influence of sea ice on the resistance was numerically investigated first after mesh convergence analysis. Then further calculations were conducted on resistance components for different submerged depths and speeds under various ice conditions. And wave patterns of the SUBOFF were analyzed for free surface and brash ice condition. Numerical results show that the sea ice predominantly affects the pressure resistance, especially the wave-making component, while there are little differences in frictional resistances for different submerged depths and ice conditions. As the submarine moves under level ice, a significant decrease in pressure resistance and total resistance is observed owing to the lack of wave disturbance. In contrast, when navigating in water covered by brash ice, waves are generated. However, the random and uncertain collisions of floating ice lead to strong fluctuations in the pressure resistance and total resistance curves. And the corresponding average resistance values are slightly smaller than those in open sea due to the wave attenuation induced by brash ice. Furthermore, the submerged depth and forward speed are found to have a significant impact on the hydrodynamic characteristics of the submarine sailing in ice regions. With the increase of submerged depth, both pressure and total resistances will gradually decrease and tend to be consistent. Generally, the increase of forward speed will lead to larger resistance. However, when the submarine navigates beneath the open sea or brash ice, the pressure resistance increases at first and then decreases slightly, which makes the total resistance to grow at a slower rate.

level ice and brash ice  /  submarine  /  resistance  /  numerical simulation
杨云涛, 胡官清, 湛俊华, 马超. 潜艇以不同潜深和航速在海冰环境中航行的数值模拟研究. 船舶力学, 2025 , 29 (11) : 1678 -1689 . DOI: 10.3969/j.issn.1007-7294.2025.11.002
Yun-tao YANG, Guan-qing HU, Jun-hua ZHAN, Chao MA. Numerical simulation of submarine advancing under ice at different submerged depths and speeds[J]. Journal of Ship Mechanics, 2025 , 29 (11) : 1678 -1689 . DOI: 10.3969/j.issn.1007-7294.2025.11.002
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0　引言
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极地地区蕴藏着巨大的经济、科学和战略价值，已成为全球各国竞相合作与争夺的要地。在极地资源勘探开发、科研探索甚至是军事活动中，潜体因其不受冰层限制可以在水下自由航行、适应低温极端环境等优势发挥着不可替代的作用。然而，海冰的存在改变了近冰面流场环境，所引起的特殊水动力性能不仅影响潜体的航行速度、稳定性和操纵性，还可能对潜体的结构安全构成威胁。因此，深入研究冰区潜体航行性能及其受力特性具有十分重要的意义。
很长时间以来，国内外学者对潜体水动力性能的研究主要集中在自由水域。早在上世纪六十年代，Hess和Smith[1]就基于势流理论采用面元法对无界流域中圆球、椭球等简单几何形状潜体的绕流问题进行了研究；Farell[2]进一步将势流方法应用于近水面潜体水动力性能的计算，并对自由面的影响进行了分析；此后，为了能够更高效准确地求解不同的水下潜体航行运动问题，学者们[3-4]又陆续提出了许多针对该方法的改进措施。势流理论方法计算简单高效，但是在处理流体粘性、模拟复杂流动现象等方面存在明显不足。随着计算机技术的持续进步，基于粘流理论的计算流体力学方法（CFD）迎来了蓬勃的发展与广泛应用。在潜体水动力性能的CFD研究方面，Yang等[5]通过求解湍流运动方程对高速航行的回转体在尾部形状发生改变时的绕流场进行了分析；Phillips等[6]、Bhushan等[7]和Fureby等[8]探讨了在潜体CFD计算中湍流模型的选取对模拟结果的影响；通过利用VOF技术捕捉自由面，Zhang等[9]数值模拟和分析了不同自由面和水底条件下近水面航行潜体的水动力系数；赵骥等[10]针对SUBOFF潜艇绕流问题，发展了一种基于粘势流耦合模型的CFD方法，显著提高了数值计算的效率。
迄今为止，对于潜体在自由水域航行性能的研究已经积累了丰富的经验，但在冰区环境，相关研究却相对滞后。已有的一些研究主要针对的是水面舰艇或结构物，例如：Hansen和Løset[11]针对碎冰间以及碎冰与结构物间的碰撞建立线弹性模型，采用离散元法对碎冰与船舶的相互作用问题进行了研究；Luo等[12]采用CFD-DEM耦合数值方法研究了某型巴拿马散货船在碎冰航道中的船舶阻力；Xie等[13]进一步针对带螺旋桨的商船，研究了其在碎冰航道中的自推进性能。潜体在穿越冰区时，因其特殊的结构形式和航行位置，所承受的载荷特性与水面舰艇存在显著不同，但国内外对这一领域的研究都还相对较少。最近，包括本文作者在内的一些学者[14-15]基于势流理论，采用格林函数法对无限宽广冰盖下潜体的水动力问题进行了研究，在一定程度上填补了该领域的空白；柏铁朝等[16]将冰面视为无变形的壁面，采用CFD方法对潜艇以不同潜深、航速在近冰面航行时的阻力和绕流场进行了分析。以上的这些研究主要局限于较为简单的平整冰下的潜体航行运动问题，并未考虑其它更为复杂的冰况（比如碎冰）。闫允鹤等[17]针对碎冰航道内浮出冰面的潜艇，采用CFD-DEM耦合法对其受到的冰载荷特性进行了深入分析，但他们的研究主要聚焦于潜艇露出水面的状态（与水面舰艇类似），缺乏对更常见的冰下航行情况的深入探讨。
本文针对潜体在冰下的航行问题，基于流体的RANS方程，通过将冰面设为切向速度等于来流速度的无滑移壁面来对平整冰下潜体的航行进行模拟；在拉格朗日框架下，采用离散元法结合线弹性模型实现碎冰间的相互作用，并通过流体和碎冰间的双向耦合求解实现碎冰下航行潜体的研究。以SUBOFF潜艇为研究对象，首先分析了数值方法的网格收敛性。在此基础上，对潜艇在不同冰况下，以不同潜深、航速航行时的阻力和兴波特性进行了计算分析，以考察它们对潜体航行性能的影响。
1　数学模型
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极地地区近冰面航行的潜体，受漂浮于水面海冰的影响，会呈现出不同于开敞水域的水动力特性。在采用数值方法对其进行研究时，建立合适的潜体-海冰-流体（水）多介质耦合作用数学模型是关键。海冰的主要类型包括碎冰、浮冰、平整冰（或层冰）、冰脊、冰山等，其中碎冰和平整冰是最常遇到的，本文主要对潜体在这两种海冰覆盖水面下航行时的水动力性能进行研究。
1.1　流体控制方程
潜体在水下航行时周围流动是不可压湍流，当水面为平整冰或自由面时，满足如下的RANS方程[18]
式中：xi（i=1，2，3）分别代表x、y和轴坐标分量；ui代表湍流沿xi轴方向的平均速度；gi为沿xi轴方向的质量力；为湍流沿xi轴方向的脉动速度，本文选用广泛采用的SST k-ω湍流模型[19]对湍流进行处理以使上述方程封闭；ρ和μ分别为流体的密度和动力粘性系数，它们取决于控制体中空气和水所占的比例。采用VOF法，若记控制体中水和空气体积分数分别为αw和αa，则ρ=αwρw+αaρa，μ=αwμw+αaμa（其中下标w和a分别对应水和空气相关的物理量）。αw和αa均满足如下的连续性方程
当漂浮在水面上的是碎冰时，为了能够在数值上求解考虑数量庞大碎冰粒子影响的流动问题，流体运动方程（1）和（2）需要修改为以下形式[20]
式中：ε表示控制体中流体（空气和水）所占的比例；fi表示单位体积流体受到的碎冰粒子作用力。
1.2　海冰模拟
当潜体在平整冰区航行时，冰面变形对流场的影响很小[16]，可将层冰与流体的接触面视为不可滑移的壁面。而当潜体在碎冰区航行时，由于碎冰是一种不连续固体介质，在周围流场的影响下会产生升沉、翻转、碰撞、堆积等，对它的模拟比层冰复杂得多。考虑到碎冰的尺寸较小，相互间的碰撞几乎不会出现破碎现象，本文将其作为刚体进行处理，它满足刚体运动方程[12]。
计算碎冰间的接触力是模拟碎冰运动的关键，本文采用Cundall和Strack（1979）[21]提出的一种高效、精确的线弹性接触模型来模拟碎冰间的相互作用。该模型的示意图如图1所示，通过弹簧阻尼器系统模拟碎冰间的相互作用，其中弹簧用来产生碎冰间的弹性排斥力，阻尼器则用来考虑碎冰碰撞后产生的能量损耗。采用该模型时，碎冰间的接触力可以表示为
式中：n和t分别表示碎冰接触的法向和切向；Fn和Ft分别为法向力和切向力，它们与碎冰间产生的假定法向和切向重叠量（见图2）以及相互接触碎冰的相对速度有关，表达式可参考文献[13]。
基于上述数学模型，本文基于商业软件STAR-CCM+来实现潜体在不同海冰环境中航行的数值模拟。
2　计算模型与数值模拟实现
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2.1　潜体模型
为了研究潜体在冰下航行的真实情况，本文选取SUBOFF潜艇作为计算模型。该模型详细的几何参数[22]已被美国国防高级研究署（DARPA）公布，并且作为标模被广泛用于潜体在开敞水域航行时水动力性能的研究。它的几何模型和主尺度分别如图3和表1所示。
2.2　数值模拟设置
针对上述SUBOFF潜艇模型，采用计算流体力学（CFD）方法数值离散求解其在冰下航行时周围的绕流场。当潜艇上方是层冰时，忽略冰面变形，将冰-水交界面设置为切向速度等于来流速度的无滑移壁面边界，数值求解的流场控制方程为式（1）、式（2）以及SST k-ω湍流模型。考虑到流动的对称性，为了减小计算量，数值模拟时只取一半流域，并将艇体中纵剖面所在平面设置为对称面。流场的其余边界设置为（见图4（a））：艇体表面设为不可滑移壁面；上游、下游边界分别设为速度入口和压力出口；侧面和底面设为对称面。为了避免水池边界反射的影响，采用较大计算域，其中上游边界距艇体1.5 L，下游边界距艇体3.0 L，宽度和深度为1.5 L。
当潜艇上方是碎冰时，由于水面并未被海冰完全覆盖，在潜艇的扰动下会产生波动。为了能够模拟出这种水面的波动，本文采用VOF法捕捉空气和水的交界面，空气域的高度取2/3L，顶部边界设为速度入口。同时，考虑到碎冰碰撞运动可能的不对称性，当模拟潜艇在碎冰下的运动时需要考虑完整的流域。为了避免水池两侧边界以及出口边界对水面波动的反射，在出口前方1倍艇长的范围内设置消波区。其余边界的设置与潜艇在层冰下航行时的边界设置相同（见图4（b））。
除了上述对流体域的离散求解之外，当研究潜艇在碎冰下航行问题时，为了考虑碎冰的存在对潜艇水动力性能的影响，还需要对碎冰运动进行模拟。不同于流体，碎冰是一种不连续介质。本文在拉格朗日框架下，采用离散元法（DEM）来实现对碎冰模拟，其中流体作用力利用CFD法计算得到的流场获得，可参考文献[23-24]。本文中碎冰的物理参数选取为：密度取900 kg/m3，法向和切向弹性刚度取60 000 N/m，摩擦系数取0.35，法向和切向恢复系数取0.2；采用三维薄圆盘模拟碎冰，直径为0.15 m，厚度为0.02 m，密集度为60%。在求得碎冰的运动之后，下一步流场的计算有两种处理方式：一种是忽略碎冰对流体的影响（即单向耦合），采用CFD直接求解流场控制方程式（1）、式（2）、式（3）以及SST k-ω湍流模型；另外一种则是结合碎冰对流体的作用力（即双向耦合），利用CFD法求解耦合流场控制方程式（3）、式（4）、式（5）以及SST k-ω湍流模型。理论上双向耦合方法更能反映实际的结构物-冰-流体耦合作用，但是以往对于水面舰艇在碎冰航道中的航行运动问题，由于单向耦合计算量较小，以及碎冰运动对流体的扰动引起的流体作用于船体的力，相比于船舶航行时碎冰直接作用在船体的力较小等原因，大多选择单向耦合进行数值模拟[12]。但是对于本文与碎冰无直接接触的潜艇航行问题（即潜艇不会受到冰载荷影响），数值模拟结果显示双向耦合与单向耦合存在显著不同。图5给出了SUBOFF潜艇在开敞水域及在碎冰下航行（采用单向耦合和双向耦合）时的阻力时历变化曲线的对比。从图中可以看出，碎冰下单向耦合结果与开敞水域下的结果基本一致，换言之，单项耦合方法无法真实反映出碎冰引起的作用在潜艇上的水动力振荡特性。因此，本文后面对潜艇在碎冰下航行的模拟均采用双向耦合方法。
2.3　网格收敛性分析
潜艇在冰下航行数值仿真的效率和精度很大程度上取决于计算网格的划分，为了确立合适的网格，需要进行网格收敛性分析。本文采用六面体网格对流体域进行离散，在潜艇周围、空气和水交界面等流动变化较为剧烈区域进行局部加密，网格示意图如图6所示。保持各部分网格的相对尺寸不变，通过选取不同的基本尺寸，生成不同数量的网格用于收敛性分析（见表2）。选择航速4 m/s、潜深1.67 m、碎冰密集度60%作为计算工况，图7给出了平整冰和碎冰下航行潜艇的平均阻力随网格数的变化曲线。从图中可以看出，随着网格数量的增加，计算结果逐渐趋于收敛。当平整冰和碎冰下的计算网格数分别超过348 249和4 106 658（对应的基本尺寸为0.058 m）时，不同网格下的阻力结果差异很小。因此，为了兼顾计算精度与效率，本文数值模拟选取基本尺寸为0.058 m的网格。
3　数值结果与分析
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3.1　海冰对航行潜艇阻力特性影响分析
潜艇在水（冰）下航行时，受到的流体阻力可以分为摩擦阻力和压阻力。为了分析漂浮于水面的海冰对潜艇阻力特性的影响，本文首先选择航速U=3 m/s、潜深h=0.6 m作为计算工况，分别对在平整冰面和碎冰下航行的SUBOFF潜艇进行数值模拟和分析。
图8给出了不同冰况下潜艇总阻力在计算稳定后的一段时历曲线，为了进行对比，图中也同时展示了水面上没有冰（即自由面）时的结果。从图中可以看出，潜艇所受到的阻力在不同冰况下呈现出不同的特征：当潜艇在平整冰或自由面下航行时，总阻力比较稳定，基本是一个不随时间变化的定常值；而当潜艇在碎冰下航行时，总阻力结果则呈现出强烈的波动现象，这可能会对潜艇的航行速度、稳定性和操纵性产生影响。图9给出了SUBOFF潜艇在碎冰下航行时，某个瞬时的冰面分布图，可以看出漂浮于水面的碎冰存在翻转、碰撞等现象。这些随机的、不确定的现象会对周围流场产生扰动，从而引起潜艇阻力的波动。除了上述阻力变化的特征不同之外，从图8可看出潜艇在不同冰况下航行时所受到的阻力大小也有所不同，即平整冰下的阻力明显小于碎冰和自由面下的阻力；碎冰下的阻力在自由面下的阻力附近波动，但其平均值要比自由面下阻力略小。
图10进一步给出了不同冰况下潜艇摩擦阻力和压阻力的时历曲线。对比图10（a）和（b）中摩擦阻力和压阻力的数值大小可以发现：当潜艇在平整冰下航行时，摩擦阻力远大于压阻力，占总阻力的主要部分；而当潜艇在自由面或碎冰下航行时，情况则正好相反。此外，从图中还可以发现，潜艇受到的摩擦阻力在不同冰况下的差异很小，几乎不受海冰影响；而压阻力在不同冰况下则存在显著不同，其变化特征与图8中的总阻力基本一致，即碎冰下压阻力的平均值略小于自由面下的压阻力，平整冰下的压阻力则远小于其他两种冰况。造成潜艇在平整冰下航行时压阻力较小的原因很容易解释：压阻力包含粘压阻力和兴波阻力两部分，当漂浮于水面的是平整冰时，由于无法产生兴波，不存在兴波阻力，从而使潜艇受到的压阻力大大减小。
为了分析造成潜艇在自由面和碎冰下航行时压阻力存在差异的原因，图11进一步对比了某个瞬时两者的兴波云图。可以看出，两种情况下的波面形状基本一致，但是由于碎冰具有一定的消波作用，使得碎冰下航行潜艇兴波的波面升高，相比自由面下的兴波略小（尤其是在波峰和波谷处）。图12给出了相应的对称面上的波切图对比，其中横坐标x表示纵向位置坐标，纵坐标表示波面升高。可以发现，图12中的结果进一步验证了图11得出的结论。由于波能与波幅平方成正比[25]，而兴波的能量来自于潜艇航行时对水的作用力（即兴波阻力），碎冰下航行潜艇的较小兴波波幅，就导致了图10（b）和图8中所示的平均阻力（压阻力和总阻力）小于自由面下航行潜艇所受到的阻力。图13（a）和（b）分别为潜艇在自由面和碎冰下航行时中轴线以上和以下表面的压力对比，从中可以发现：潜艇在碎冰下航行时，尾部压力相对较大，这进一步验证了碎冰下航行潜艇平均阻力较小的结论。
3.2　潜深与航速对潜体航行影响
潜艇在执行任务时，常常需要以不同的潜深、航速进行作业。下面将对SUBOFF潜艇在平整冰区和碎冰区中以不同潜深、航速航行的直航运动问题进行模拟和分析。
选取潜深分别为0.6 m、1.67 m、2.5 m和4 m，计算得到SUBOFF以3 m/s航速在平整冰区和碎冰区下航行时的总阻力时历曲线，如图14所示。从图中可以看出，对于平整冰下以不同潜深航行的潜艇，在数值计算一定时间后，总阻力结果不随时间变化；随着潜深的增加，总阻力一开始逐渐减小，但是当潜深达到一定大小后，基本不再变化。而对于碎冰下的航行潜艇，当潜深较小时，即使数值计算了一段时间，总阻力结果始终不稳定，是随时间振荡变化的。这主要是因为潜艇在下潜深度较小时会受到水面碎冰随机、不确定的碰撞、翻转等的影响。但是随着潜深的增加，这种不稳定的振荡会逐渐减弱。可以发现，当潜深达到4 m时，潜艇受到的总阻力基本不再振荡，也就是说不再受到水面碎冰的影响。
图15进一步给出了航速3 m/s时潜艇分别在平整冰和碎冰下航行时各阻力成分随潜深的变化。作为对比，图中同时也给出了潜艇在自由面下航行时各阻力成分的结果。可以发现，无论何种冰况，潜艇所受到的摩擦阻力几乎不随潜深变化，而压阻力则会随着潜深的增加逐渐降低，但是下降幅度逐渐减小，直到不再变化。相比于碎冰和自由面，潜艇在平整冰下航行所受的总阻力受潜深的影响较小。这主要是因为潜艇在平整冰下航行时不会产生兴波，压阻力中只含有粘压阻力，远小于摩擦阻力。而摩擦阻力受潜深的影响较小，因此潜艇受到的总阻力受潜深的影响也小于另外两种情况。当潜深较小时，潜艇在自由面下航行时的压阻力和总阻力，略大于其在碎冰下航行时的压阻力和总阻力，但远大于其在平整冰下航行时压阻力和总阻力。这主要是由于在不同冰况下水面兴波大小不同所引起的：潜艇在平整冰下航行时不存在兴波，在碎冰下航行时碎冰具有一定的消波作用，而在自由面下航行时兴波幅值最大（相应的兴波阻力最大）。随着潜深的增大，水面兴波逐渐减小到0（见图16），因此潜艇在不同冰况下航行所受到各种成分阻力均趋于一致。
为了分析潜艇在冰区航行时航速的影响，在固定下潜深度h=0.6 m下，分别对SUBOFF以速度为2 m/s、3 m/s、4 m/s、5.29 m/s和6 m/s的直航运动进行数值仿真。不同冰况下的各阻力成分随航速的变化曲线如图17所示。由图可知，随着航速的增加，各种冰况中潜艇的摩擦阻力均有所增加，且平整冰、碎冰以及自由面下的摩擦阻力相差较小。这说明水面的海冰对潜艇的摩擦阻力影响较小，同时也说明不同冰况下潜艇总阻力的差异主要是由压阻力引起的。从图中可以看出，当潜艇在平整冰下航行时，随着航速的增加，压阻力也呈现出增大趋势，从而总阻力不断增加。而当潜艇在碎冰或自由面下航行时，随着航速的增加，压阻力先逐渐增大，然后有所减小，这主要是由于潜艇在不同航速时艏、艉兴波的不同干扰导致的。但是由于摩擦阻力持续增加，导致总阻力仍随着航速的增加不断增大，只是增速有所减缓。图18展示了航速为2 m/s、4 m/s和6 m/s时SUBOFF潜艇在自由面和碎冰面上的兴波图形，可以发现，航速为4 m/s时潜艇在水面的兴波波幅要比2 m/s、6 m/s时的兴波波幅大，也就是说航速在4 m/s附近时，潜艇的艏、艉兴波发生了不利干扰，从而导致了压阻力在此附近出现峰值（见图17）。
4　结论
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本文针对冰区航行潜艇，构建潜艇-流体（水）-海冰多介质耦合作用数学模型，通过采用CFD方法数值离散求解流体域，将层冰-水交界面设置为切向速度等于来流速度的无滑移壁面边界，利用离散元法求解在拉格朗日框架下建立的碎冰运动方程，分别对潜艇在平整冰和碎冰下的直航运动问题进行数值模拟研究。分析结果得到以下结论：
（1）在不同冰况下，潜艇所受到的阻力变化曲线呈现出不同的特征。当潜艇在平整冰或自由面下航行时，经过一段时间的数值模拟后，阻力结果变为一个不随时间变化的定常值；而当潜艇在碎冰下航行时，若采用双向耦合法考虑碎冰的随机、不确定的碰撞、翻转等影响，得到的阻力结果呈现出强烈的波动现象，这可能会对潜艇的航行速度、稳定性和操纵性产生影响。
（2）平整冰、碎冰以及自由面下潜艇受到的摩擦阻力相差很小，海冰的存在主要对压阻力产生影响。压阻力中的主要部分为兴波阻力，而当潜艇在平整冰下航行时，无法产生兴波和兴波阻力，摩擦阻力占主导，使得潜艇受到的总阻力远小于其它冰况；当潜艇在碎冰下航行时，尽管会在水面产生兴波，但是由于碎冰具有一定的消波作用，使得潜体的平均压阻力和总阻力相比自由面下的值要略小。
（3）通过对比不同潜深、航速下的各阻力成分发现，潜深对潜艇的摩擦阻力影响较小，对压阻力影响较大，不同冰况下潜体的压阻力和总阻力均随着潜深的增加逐渐减小，并趋于相同值。随着航速的增大，在平整冰下航行的潜艇的摩擦阻力、压阻力以及总阻力均有所增加，碎冰或自由面下航行潜艇的压阻力先逐渐增大，然后有所减小，但是由于摩擦阻力持续增加，导致总阻力仍随着航速的增加不断增大，只是增速有所减缓。
基金
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  • 国家自然科学基金资助项目(52101357)
  • 江苏省高等学校基础科学（自然科学）研究项目(21KJB580012)
  • 船舶总体性能创新研究开放基金项目(31122223)
  • 镇江市科技计划项目（基础研究专项）(JC2024016)
  • 江苏省基础研究计划资助项目(BK20251830)
  • 江苏高校“青蓝工程”资助
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2025年第29卷第11期
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doi: 10.3969/j.issn.1007-7294.2025.11.002
  • 接收时间：2025-04-24
  • 首发时间：2026-03-18
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  • 收稿日期：2025-04-24
基金
国家自然科学基金资助项目(52101357)
江苏省高等学校基础科学（自然科学）研究项目(21KJB580012)
船舶总体性能创新研究开放基金项目(31122223)
镇江市科技计划项目（基础研究专项）(JC2024016)
江苏省基础研究计划资助项目(BK20251830)
江苏高校“青蓝工程”资助
作者信息
    1.江苏科技大学　船舶与建筑工程学院，江苏　张家港　215600
    2.张家港江苏科技大学（先进金属材料）产业技术研究院，江苏　张家港　215600
    3.武汉第二船舶设计研究所，武汉　430205
    4.中国船舶科学研究中心　水动力学国家重点实验室，江苏　无锡　214082
    5.武汉理工大学　船舶与能源动力工程学院，武汉　430063

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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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