Article(id=1320037436680425972, tenantId=1146029695717560320, journalId=1316781260334530699, issueId=1320037383882519364, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1005-0329.2025.12.005, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1742745600000, receivedDateStr=2025-03-24, revisedDate=1760976000000, revisedDateStr=2025-10-21, acceptedDate=null, acceptedDateStr=null, onlineDate=1792651050013, onlineDateStr=2026-10-22, pubDate=1767110400000, pubDateStr=2025-12-31, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1792651050013, onlineIssueDateStr=2026-10-22, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1792651050013, creator=13701087609, updateTime=1792651050013, updator=13701087609, issue=Issue{id=1320037383882519364, tenantId=1146029695717560320, journalId=1316781260334530699, year='2025', volume='53', issue='12', pageStart='1', pageEnd='118', issueExtLink='null', onlineDate='null', pubDate='1767110400000', pubDateStr='2025-12-31', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1792651037426, creator='13701087609', updateTime=1792656182714, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1320058964839093145, tenantId=1146029695717560320, journalId=1316781260334530699, issueId=1320037383882519364, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1320058964839093146, tenantId=1146029695717560320, journalId=1316781260334530699, issueId=1320037383882519364, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=38, endPage=46, ext={EN=ArticleExt(id=1320037436869169653, articleId=1320037436680425972, tenantId=1146029695717560320, journalId=1316781260334530699, language=EN, title=Study on noise reduction mechanism of aerodynamic noise in axial flow fans based on multiple combined noise reduction structures, columnId=null, journalTitle=Fluid Machinery, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To address the aerodynamic noise issue in axial flow fans caused by tip leakage flow,a noise reduction structure combining “bionic serrations-suction surface tip winglet modification-multi‑scheme grooving” was designed.Using the SBES(stress‑blended eddy simulation) turbulence model coupled with the FW-H acoustic analogy method,the noise reduction mechanisms of each structure were analyzed.The results show that the combination of a biomimetic wavy leading‑edge serration with a suction side tip winglet modification can reduce the overall sound pressure level by 4.5dB.Further combining this with a tip longitudinal airfoil ribbed slot structure leads to a reduction of 7.9dB in the overall sound pressure level.The biomimetic leading‑edge serration-suction side tip winglet modification reduces the discrete noise peaks by disrupting the periodic shedding of leakage vortices,optimizes the flow uniformity at the tip,and suppresses abrupt local pressure fluctuations.The combined structure of biomimetic leading‑edge serrations,tip longitudinal airfoil ribbed slots,and suction side tip winglets accelerates vortex energy dissipation by dynamically dividing the leakage flow paths.In experiments,the prototype demonstrated significant noise reduction effects,and the observed phenomenon of “low‑frequency noise energy shifting to high frequencies” was consistent with the numerical simulation results,verifying the reliability of the proposed method.This study provides a reference for the optimization design of low‑noise axial flow fan tips.

, authors=Yichen YE1, Yuan HE2, Zhunyuan SHAO2, Canxing WANG1, 2, authorsList=Yichen YE, Yuan HE, Zhunyuan SHAO, Canxing WANG, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1320037440472076814, articleId=1320037436680425972, tenantId=1146029695717560320, journalId=1316781260334530699, language=CN, title=基于多种组合降噪结构对轴流风机气动噪声的降噪机理研究, columnId=1317484287230853622, journalTitle=流体机械, columnName=设计计算, runingTitle=null, highlight=null, articleAbstract=

针对轴流风机因叶尖泄漏流导致的气动噪声问题,设计“仿生锯齿-吸力面叶顶小翼改型-多方案开槽”降噪结构,采用应力混合涡模拟(SBES)湍流模型与FW-H声类比方法耦合,分析各结构的降噪机理。结果表明:仿生波浪前缘锯齿与吸力面小翼改型组合可使总声压级降低4.5dB;在此基础上,结合叶顶纵向翼型肋板槽结构后,总声压级降低了7.9dB;叶顶前缘仿生锯齿-吸力面小翼改型通过扰动泄漏涡的周期性脱落,降低了离散噪声峰值,优化了叶顶流动均匀性,抑制了局部压力脉动的突变;叶顶前缘仿生锯齿-叶顶纵向翼型槽肋板-吸力面叶顶小翼组合结构则通过动态分割泄漏流路径,加速了涡能量耗散。试验中样机降噪效果显著,观察到的“低频噪声能量向高频转移”现象与数值模拟结果一致,验证了文中方法的可靠性。研究可为低噪声轴流风机叶顶优化设计提供参考。

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(a)原型机  (b)方案1结构

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(a)原型  (b)仿生锯齿-叶顶改型

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基于多种组合降噪结构对轴流风机气动噪声的降噪机理研究
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叶一辰 1 , 何源 2 , 邵准远 2 , 王灿星 1, 2
流体机械 | 设计计算 2025,53(12): 38-46
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流体机械 |设计计算 2025 , 53 (12) : 38 -46
基于多种组合降噪结构对轴流风机气动噪声的降噪机理研究
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叶一辰1, 何源2, 邵准远2, 王灿星1, 2
作者信息
  • 1浙江大学 航空航天学院,杭州 310027
  • 2南安普敦大学 工程与物理科学学院,南安普敦 SO17 1BJ
Study on noise reduction mechanism of aerodynamic noise in axial flow fans based on multiple combined noise reduction structures
Yichen YE1, Yuan HE2, Zhunyuan SHAO2, Canxing WANG1, 2
Affiliations
  • 1School of Aeronautics and Astronautics,Zhejiang University,Hangzhou 310027,China
  • 2Faculty of Engineering and Physical Sciences,University of Southampton,Southampton SO17 1BJ,UK
出版时间: 2025-12-31 doi: 10.3969/j.issn.1005-0329.2025.12.005
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针对轴流风机因叶尖泄漏流导致的气动噪声问题,设计“仿生锯齿-吸力面叶顶小翼改型-多方案开槽”降噪结构,采用应力混合涡模拟(SBES)湍流模型与FW-H声类比方法耦合,分析各结构的降噪机理。结果表明:仿生波浪前缘锯齿与吸力面小翼改型组合可使总声压级降低4.5dB;在此基础上,结合叶顶纵向翼型肋板槽结构后,总声压级降低了7.9dB;叶顶前缘仿生锯齿-吸力面小翼改型通过扰动泄漏涡的周期性脱落,降低了离散噪声峰值,优化了叶顶流动均匀性,抑制了局部压力脉动的突变;叶顶前缘仿生锯齿-叶顶纵向翼型槽肋板-吸力面叶顶小翼组合结构则通过动态分割泄漏流路径,加速了涡能量耗散。试验中样机降噪效果显著,观察到的“低频噪声能量向高频转移”现象与数值模拟结果一致,验证了文中方法的可靠性。研究可为低噪声轴流风机叶顶优化设计提供参考。

SBES湍流模型  /  叶顶间隙  /  吸力面叶顶小翼改型  /  仿生锯齿  /  气动噪声  /  涡破碎  /  降噪机理

To address the aerodynamic noise issue in axial flow fans caused by tip leakage flow,a noise reduction structure combining “bionic serrations-suction surface tip winglet modification-multi‑scheme grooving” was designed.Using the SBES(stress‑blended eddy simulation) turbulence model coupled with the FW-H acoustic analogy method,the noise reduction mechanisms of each structure were analyzed.The results show that the combination of a biomimetic wavy leading‑edge serration with a suction side tip winglet modification can reduce the overall sound pressure level by 4.5dB.Further combining this with a tip longitudinal airfoil ribbed slot structure leads to a reduction of 7.9dB in the overall sound pressure level.The biomimetic leading‑edge serration-suction side tip winglet modification reduces the discrete noise peaks by disrupting the periodic shedding of leakage vortices,optimizes the flow uniformity at the tip,and suppresses abrupt local pressure fluctuations.The combined structure of biomimetic leading‑edge serrations,tip longitudinal airfoil ribbed slots,and suction side tip winglets accelerates vortex energy dissipation by dynamically dividing the leakage flow paths.In experiments,the prototype demonstrated significant noise reduction effects,and the observed phenomenon of “low‑frequency noise energy shifting to high frequencies” was consistent with the numerical simulation results,verifying the reliability of the proposed method.This study provides a reference for the optimization design of low‑noise axial flow fan tips.

SBES turbulence model  /  tip clearance  /  suction side tip winglet modification  /  biomimetic serration  /  aerodynamic noise  /  vortex breakdown  /  noise reduction mechanism
叶一辰, 何源, 邵准远, 王灿星. 基于多种组合降噪结构对轴流风机气动噪声的降噪机理研究. 流体机械, 2025 , 53 (12) : 38 -46 . DOI: 10.3969/j.issn.1005-0329.2025.12.005
Yichen YE, Yuan HE, Zhunyuan SHAO, Canxing WANG. Study on noise reduction mechanism of aerodynamic noise in axial flow fans based on multiple combined noise reduction structures[J]. Fluid Machinery, 2025 , 53 (12) : 38 -46 . DOI: 10.3969/j.issn.1005-0329.2025.12.005
叶顶间隙内的复杂流动对轴流风机的气动性能和噪声具有显著影响。叶顶泄漏流引发的复杂涡结构是轴流风机气动噪声的主要来源之一。现有的降噪方案主要集中于单一结构的叶顶改型设计,包括叶片前缘和尾缘锯齿、叶顶小翼、叶顶开槽等,但其对泄漏涡的调控机理及组合结构的降噪效果仍待进一步探索。
叶片尾缘锯齿的降噪设计通过抑制尾缘湍流及湍流-叶片相互作用来降低噪声。LYU等[1]通过风洞试验,验证了蛇形锯齿相比传统锯齿具有更显著的全频段降噪优势,为仿生锯齿的应用提供了依据。NING等[2]的试验结果表明,尖锐锯齿可抑制螺旋桨尾缘噪声,但需权衡气动性能的损失。SANTAMARIA等[3]采用低阶数值方法分析无人机螺旋桨尾缘锯齿形状的影响,发现方波锯齿在小弦长叶片中降噪效果最优。乔渭阳等[4]系统总结了仿生锯齿在前缘降噪中的应用,指出锯齿形状是优化气动噪声的关键因素之一。
叶顶小翼的构型通过抑制泄漏涡强度和优化间隙流动实现降噪。张龙新等[5]对比了压力面与吸力面小翼构型,发现吸力面小翼可降低对旋风机总声压级,验证了气动与声学性能的协同优化。隽智辉等[6]研究了叶顶小翼安装位置对轴流风机性能的影响,通过数值模拟与试验分析,探讨了不同安装位置对风机气动性能与噪声特性的作用机制,为优化叶顶小翼设计提供了理论依据。然而,传统叶顶小翼在叶片与小翼的弯折连接区域易形成新的高压力脉动区,进而影响降噪效果。
叶顶开槽是另一种用于优化气动性能与噪声的方案。孔闯[7]采用DDES/FW-H方法计算了USI7风机的噪声,分析其与试验值偏差的原因,并基于混合模型计算和分析了包括叶顶开槽在内的多种降噪措施的效果。杨天康等[8]运用LES方法,模拟了叶顶加装小翼并开设斜槽结构后噪声的变化情况。然而,目前的开槽研究多局限于传统的直线形单槽或双槽,其参数固定,未能实现动态调控与对比,且降噪效果主要依赖局部小范围的涡破碎,因此,降噪效果仍有较大提升空间。
本文针对传统叶顶降噪技术中结构单一、参数孤立优化的局限性,提出“仿生锯齿-吸力面叶顶小翼改型-多方案开槽”组合降噪结构,采用应力混合涡模拟(stress‑blended eddy simulation,SBES)模型耦合FW-H方程的高精度数值模拟方法,对比分析各方案在频域范围内的降噪机理及效果,以探索轴流风机降噪的更优途径。
计算气动声学(computational aeroacoustics,CAA)的数值模拟方法主要包括直接数值模拟(direct numerical simulation,DNS)、半经验方法和CAA混合方法三类。其中,DNS通过直接求解Navier-Stokes方程精确解析湍流与声场的耦合过程,但由于计算成本极高,仅适用于低雷诺数或简单几何结构的基础研究;半经验方法通过试验与理论推导,构建简化的数学模型,建立湍流脉动与声场的关联,可快速预测宽频噪声及声源区域,但其精度依赖于湍流模型,且无法捕捉瞬态细节。目前工程应用的主流方案为CAA混合方法,其核心思路是将计算流体力学(computational fluid dynamics,CFD)与声学求解器分步耦合:首先通过大涡模拟(large eddy simulation,LES)或分离涡模拟(detached eddy simulation,DES)等方法获取非定常流场,并提取声源项(如Lighthill应力张量),再利用FW-H(Ffowcs Williams-Hawkings)方程等积分方法计算远场声压,从而兼顾计算效率与精度。FW-H方程如下:
1c02∂2p'∂t2-∇2p'=∂∂t[ρ0vnδ(f)]-∂∂xi[Liδ(f)]+∂2∂xi∂xj[TijH(f)]
大量研究表明[9],风机气动噪声主要表现为偶极子声源特性。CAA混合方法有效避免了直接求解大范围声学网格时的数值耗散问题。该方法虽需克服声源提取误差、耦合界面相位失真等挑战,但借助自适应网格、时间降阶模型等优化技术,已在风机噪声、高速列车噪声和航空工业噪声等复杂工程场景中展现出显著优势。
为兼顾多个轴流风机气动噪声模拟方案的精度与计算效率,文中选用SBES混合湍流模型,耦合FW-H声类比方法。SBES模型由MENTER[10]提出,该模型通过屏蔽函数动态融合的雷诺平均应力(RANS)与LES,在主流及边界层等近壁区域采用RANS模型(如SST k-ω),而在流动分离、尾迹等大尺度涡主导区域,则切换至LES模式。相较于LES要求全域网格满足y+≤1,SBES通过混合策略减少了近壁区网格数量,使总网格量降低约40%,显著节省了多方案对比的计算成本。BROWN等[11]将SBES模型与其他RANS-LES模型在工业结晶釜数值模拟中进行了对比,发现SBES方法精度较高,与试验数据的一致性更好;王英杰等[12]基于SBES方法对压气机叶片动态载荷进行了高精度计算,结果也表明与试验数据很吻合。然而,目前采用SBES耦合FW-H方法计算气动噪声的研究仍较少。
基于USI-7标准风机噪声试验数据[13],对比了改进型脱离涡模拟(improved delayed detached eddy simulation,IDDES)、尺度自适应模拟(scale‑adaptive simulation,SAS)、SBES三种混合RANS-LES模型的声压频谱预测能力。对该风机进行了网格划分,网格总量为1 146万;边界条件设置为质量流量入口(对应体积流量为0.65 m3/s)、压力出口(98 370 Pa),转速为3 000 r/min;非定常计算时间步长为5×10-5 s,声源采样频率分辨率为6.25 Hz。所获得的声压频谱结果如图1所示。
IDDES模型在低频段(<1 000 Hz)的声压级预测值较试验结果偏低10~20dB,尤其在250 Hz倍频处的离散噪声峰值偏差显著。SAS模型在中低频段(100~300 Hz)的趋势与试验吻合较好,但在高频段(>1 000 Hz)整体偏低5dB~15dB。SBES模型在全频段误差最小(低频偏高3dB~5dB,高频误差<3dB),离散噪声峰值(500~1 000 Hz)偏差小于2dB,总声压级误差仅为1.5dB。
基于以上3种非定常湍流模型的对比结果,文中选用SBES作为后续目标风机气动噪声数值模拟的计算方法。
选取某轴流风机作为研究对象,该风机叶轮直径500 mm,轮毂直径200 mm,叶片数7片,叶片后缘为圆弧形锯齿,叶顶间隙为10 mm。叶轮转速1 300 r/min,设计流量2.53 m3/s,试验总声压级为73.3dB(A)。风机的几何模型如图2所示。
为验证第1节中提到的方法在文中的计算精度,对原型风机分别进行LES与SBES数值模拟对比。模拟试验环境的计算域见图3。采用压力入口和压力出口作为边界条件,并使用滑移网格法模拟旋转域。首层网格高度为0.04 mm(y+=1),边界层设置10层网格。网格单元总数为1 029万,其中,旋转域网格数量为832万,静止域网格数量为197万。
在计算原型机叶轮转过6圈的非定常流场后,分别耦合FW-H方程使叶轮再转过6圈,以计算气动噪声。计算的最高噪声频率为10  000  Hz,采样频率分辨率Δf=3.7  Hz。图4示出数值计算结果与试验值的对比。其中,SBES方法得到的总声压级为73.7 dB(A),LES方法为74.9 dB(A)。
在工程应用的关键频段(100~3  000  Hz),SBES方法计算的1/3倍频程声压级曲线与试验结果的吻合更好(误差<3dB),避免了全频段解析带来的不必要计算量。尽管采用LES耦合FW-H方程计算的结果也显示出与SBES方法相似的波动趋势,但该网格数量和质量对纯LES而言精度仍显不足,这是导致LES计算结果不如SBES方法的重要原因。另一方面,SBES方法计算的风机流量随时间波动值与试验流量的偏差在0.4%~1.4%之间(见图5),计算精度较为理想。
高速飞行的昆虫(如蝉和蜻蜓等)翅膀表面具有波浪状凸起结构,如图6所示。PELLEGRINO等[14]发现这种凸起结构可有效降低阻力。基于该波浪形状,文中设计了一种前缘仿生锯齿,该锯齿从92%径向叶高处延伸至叶顶。针对传统吸力面小翼尖端及弯折处易形成新的周期性噪声源的问题,文中提出一种新型吸力面叶顶改型结构。该改型将叶顶周向宽度扩展至原型叶顶宽度w的2倍,并以光滑曲线过渡至叶片表面,其波浪形轮廓由正弦曲线定义,波长λ为叶轮直径的2%,波幅A为叶片厚度w。该设计旨在通过在叶片前缘引入周期性速度扰动,干扰叶片表面湍流边界层的发展,从而减小叶片表面压力脉动和尾迹涡强度,最终降低气动噪声。
吸力面叶顶小翼改型如图7所示。整体叶顶改型的几何结构如图8所示,由此在原型风机基础上形成一种新型降噪组合结构,命名为方案1。方案1的叶顶间隙tc与原型风机保持一致。
改型后的旋转域及叶片网格划分如图9所示。对旋转域叶顶区域进行了网格加密,以确保锯齿部分具备较高的网格质量。其中,旋转域网格数量为1 045万,总网格数量为1 287万。
(1)频谱曲线。
采用与第2节相同的SBES耦合FWH气动噪声计算方法,计算得到方案1构型的总体声压级为69.2dB,相比原型风机降噪4.5dB。获得的噪声频谱曲线如图10所示。与原型机相比,方案1在2 000~4 000 Hz范围内的离散噪声及5 000~10 000 Hz范围内的宽带噪声均有所降低。
(2)叶片表面声压脉动时均值。
图11示出原型与组合结构叶顶表面及附近声压脉动时均值对比。可以看出,原型机叶片表面声压脉动波动最大的区域主要集中在叶尖及整个叶顶区域。叶尖前缘锯齿可显著降低叶片前缘尖端的表面声压脉动。对于叶顶区域的声压脉动,基于小翼改型叶顶的尖端及其后约2/3弦长区域仍维持在较高水平;但在叶尖至其前约1/3弦长区域内,表面声压脉动的时均值显著降低,从而可抑制宽频噪声的产生。因此,如何进一步抑制整个叶顶表面的声压脉动,仍需深入考虑。
叶尖泄漏涡作为轴流风机气动噪声的核心声源之一,其演化特性直接影响噪声的频谱分布与能量特征。为分析涡结构,采用Q准则。Q准则的数学定义基于速度梯度张量的分解,公式如下:
Q=12(Ω2-S2)
式中,Ω为旋转率张量;S为应变率张量。
原型与方案1等值面的涡结构(Q=2×106 s-2)如图12所示。原型风机的叶尖区域存在集中分布的管状泄漏涡,其周期性脱落引发了强烈的离散噪声与宽频噪声。
采用前缘仿生锯齿-吸力面叶顶小翼改型(方案1)后,流动结构得到显著优化:前缘仿生锯齿对泄漏涡进行扰动切割,促使其破碎为多尺度小涡,这些小涡沿锯齿方向向叶片后缘发展并继续破碎,而不是像原型叶片那样大量附着在整个片前缘区域。与此同时,泄漏涡整体的湍动能降低,低频涡能量更快向高频耗散。此外,小翼的流线型设计抑制了叶顶区域的局部压力梯度突变,削弱了压力脉动的强度与传播效率。叶顶改型结构还延缓了吸力面流动分离,减少了二次涡的生成规模及其相互作用强度。值得注意的是,方案1中叶顶泄漏涡发展至叶片后缘的涡核尺度B比原型涡核尺度A的半径更小,且叶片尾缘上段部分的分离涡也因前缘顶部仿生锯齿的存在而得到间接改善。
这种“涡破碎-能量转移-脉动抑制”的协同作用,使得离散噪声峰值显著降低,5 000 ~ 10 000  Hz范围内的宽频噪声能量得到衰减,最终实现了总声压级的系统性下降,印证了仿生锯齿-吸力面叶顶小翼改型组合结构在噪声优化中的有效性。
从另一方面来看,现有叶顶改型组合结构虽能抑制泄漏涡的整体强度,但泄漏流仍倾向于沿叶顶间隙的固定路径集中发展,且被仿生前缘锯齿打碎后的部分分离涡会附着在叶顶表面。因此,针对叶顶区域仍可进一步优化设计,通过动态分割泄漏流和分离涡,实现额外的降噪效果,例如采用叶顶开槽等设计。基于此,后续研究在结合吸力面小翼的改型叶顶上,进一步设计了开槽组合结构,以探究其降噪效果与作用机理。
为了进一步降低改型叶顶表面的声压脉动,在仿生锯齿-吸力面叶顶小翼改型的基础上,设计了2种带微小翼型槽肋板的结构。这两种结构的灵感来源于昆虫翅膀和鲨鱼皮表面的褶皱形态,这类结构有助于昆虫在高速飞行时减小阻力和降低噪声(见图13[15]),从而提高捕食成功率。

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2025年第53卷第12期
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doi: 10.3969/j.issn.1005-0329.2025.12.005
  • 接收时间:2025-03-24
  • 首发时间:2026-10-22
  • 出版时间:2025-12-31
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  • 收稿日期:2025-03-24
  • 修回日期:2025-10-21
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    1浙江大学 航空航天学院,杭州 310027
    2南安普敦大学 工程与物理科学学院,南安普敦 SO17 1BJ
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2种不同金属材料的力学参数
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属数
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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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