蘇華山，陳從平，趙美云，高振軍，余 萬，張揚軍

（1. 三峽大學(xué)水電機(jī)械設(shè)備設(shè)計與維護(hù)湖北省重點實驗室，宜昌 443002； 2. 三峽大學(xué)機(jī)械與動力學(xué)院，宜昌 443002）

泵輪軸向振動條件下高速液力耦合器特性

蘇華山，陳從平※，趙美云，高振軍，余 萬，張揚軍

（1. 三峽大學(xué)水電機(jī)械設(shè)備設(shè)計與維護(hù)湖北省重點實驗室，宜昌 443002； 2. 三峽大學(xué)機(jī)械與動力學(xué)院，宜昌 443002）

針對泵輪軸向振動條件下高速液力耦合器特性問題，基于RNG k-ε模型、流體體積法（volume of fluid，VOF）兩相流模型、動網(wǎng)格技術(shù)、壓力隱式算子分裂（pressure-implicit with splitting of operators，PISO）算法和變時間步長法對液力耦合器泵輪在軸向振動條件下的內(nèi)流場進(jìn)行數(shù)值模擬，通過試驗完成對模型的準(zhǔn)確性驗證。分析液力耦合器流道內(nèi)部兩相流動規(guī)律以及受力特性，結(jié)果表明：與徑向振動相比，相同振幅條件下的軸向振動對循環(huán)圓內(nèi)流量脈動和泵輪、渦輪轉(zhuǎn)矩影響較大；額定轉(zhuǎn)速越高，其泵輪、渦輪轉(zhuǎn)矩脈動幅值、軸向力波動范圍越大；振動頻率越大，泵輪、渦輪轉(zhuǎn)矩偏差越大；軸向振動幅值越大，泵輪渦輪轉(zhuǎn)矩波動范圍越大。從減小轉(zhuǎn)矩波動范圍和軸向力的角度控制軸向竄動值不應(yīng)超過0.04 mm較為合適。

計算機(jī)仿真；可視化；模型；液力耦合器；軸向振動；氣液兩相流

蘇華山，陳從平，趙美云，高振軍，余 萬，張揚軍. 泵輪軸向振動條件下高速液力耦合器特性[J]. 農(nóng)業(yè)工程學(xué)報，2017，33(7)：51－57.doi：10.11975/j.issn.1002-6819.2017.07.007 http://www.tcsae.org

Su Huashan, Chen Congping, Zhao Meiyun, Gao Zhenjun, Yu Wan, Zhang Yangjun. Characteristics of high speed hydraulic coupler under pump wheel axial vibration conditions[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2017, 33(7): 51－57. (in Chinese with English abstract)doi：10.11975/j.issn.1002-6819.2017.07.007 http://www.tcsae.org

0 引 言

液力傳動因具有高效、輕載啟動、自動過載保護(hù)、柔性傳動等優(yōu)點，已被廣泛應(yīng)用于大慣量機(jī)械設(shè)備中解決啟動困難問題[1-3]。液力傳動部件與介質(zhì)的流-固耦合作用特性決定了液力傳動的品質(zhì)，而實際中因傳動部件如葉輪、軸瓦等的安裝、磨損等缺陷易使系統(tǒng)發(fā)生振動，進(jìn)而使流場的流動狀態(tài)發(fā)生改變[4-7]，動力輸出惡化。

目前已有學(xué)者對液力耦合器平穩(wěn)運行狀態(tài)下其的內(nèi)部流場動力學(xué)特性進(jìn)行了研究[8-20]，得到了泵輪、渦輪內(nèi)部氣液兩相的壓力和速度分布等內(nèi)部流動特性，并分析損失來源，得到液力耦合器優(yōu)化設(shè)計方案。然而，前人研究并未考慮泵輪輸入軸振動對液力耦合器性能的影響。實際中因葉輪安裝的不完全對中性、載荷擾動等易使泵輪輸入軸發(fā)生振動，進(jìn)而導(dǎo)致軸承振動，噪聲大并伴有異響，情況嚴(yán)重時會導(dǎo)致液力耦合器葉片斷裂、軸瓦失效、軸承卡死等事故[7]。因此，研究振動條件下液力耦合器葉輪內(nèi)部兩相流動特性，探索振動對液力耦合器內(nèi)外特性影響極為必要。本文首先將液力耦合器正常運行條件下的泵輪轉(zhuǎn)矩特性試驗曲線與數(shù)值計算結(jié)果進(jìn)行對比，驗證數(shù)值計算方法對液力偶合器流場計算與性能預(yù)測的可行性和準(zhǔn)確性；在此基礎(chǔ)上，采用數(shù)值計算方法重點研究軸向振動條件下液力耦合器轉(zhuǎn)矩以及流場變化規(guī)律，以期為液力耦合器設(shè)計、安裝和故障診斷提供參考。

1 研究模型

1.1 計算模型

本文研究的液力耦合器葉片采用直葉片徑向分布形式，液力耦合器流道模型如圖1，模型參數(shù)如表1所示。液力耦合器泵輪振動方向包括軸向和徑向如圖 2所示，振動位移Y(m)、振動速度v（m/s）、振動加速度G(m/s2)與振幅A和角速度ω的換算關(guān)系如下：

式中角速度ω=2πf，rad/s；f為頻率，Hz ；A振幅，m；α相位角，rad；t為時間，s。

圖1 液力耦合器模型Fig.1 Model of hydrodynamic coupling

表1 液力耦合器模型參數(shù)Table1 Hydraulic coupling model parameters

圖2 葉輪振動示意圖Fig.2 Schematic diagram of impeller vibration

1.2 模型建立

根據(jù)實際流域建立液力耦合器的流道模型和網(wǎng)格模型如圖3、圖4所示?？紤]振動時計算域發(fā)生振動液力偶合器流動計算域?qū)凶兓?，在使用FLUENT進(jìn)行數(shù)值計算時需要使用到動網(wǎng)格技術(shù)。由于流域邊界變化主要發(fā)生在泵輪與渦輪交界處，設(shè)置分界面將完整模型劃分為3個獨立的網(wǎng)格模型，并設(shè)置分界面為interface，通過分界面?zhèn)鬟f數(shù)據(jù)[21-23]。

圖3 流道模型Fig.3 Channel model of hydraulic coupling

圖4 模型網(wǎng)格Fig.4 Grid model

泵輪和渦輪流道部分采用結(jié)構(gòu)化六面體網(wǎng)格，單元數(shù)量少，計算速度快，結(jié)果可靠。邊界運動域采用非結(jié)構(gòu)網(wǎng)格以便FLUENT軟件實現(xiàn)邊界運動域計算，同時考慮到邊界運動域計算域體積相對較小，進(jìn)行網(wǎng)格劃分的時候，為了提高計算精度和效率，選擇較小的網(wǎng)格尺寸比例因子實現(xiàn)網(wǎng)格加密，以保證邊界運動域網(wǎng)格足夠精細(xì)[24-26]。

模型相關(guān)設(shè)置如表 2所示，考慮到流體體積法（volume of fluid，VOF）模型難以收斂，耗時較長，計算量大等特點，采用變時間步長法，設(shè)置柯朗數(shù)（Courant number）等于 0.25[27-28]，即保證其計算精度同時縮短計算所需時間。

表2 模型參數(shù)Table2 Model parameters

1.3 試驗測試系統(tǒng)

為了得到液力耦合器的特性曲線，選用充液量q分別為 40%、60%和 80%的工況點進(jìn)行樣機(jī)性能試驗，試驗臺主要由電動機(jī)、增速齒輪箱、測量裝置、減速齒輪箱，齒輪泵以及其他零部件所組成，設(shè)計試驗臺簡圖如圖5a所示，試驗臺實物圖如圖5b所示，試驗臺主要針對樣機(jī)進(jìn)行試驗，由于液力耦合器額定轉(zhuǎn)速較高，需通過齒輪箱增速連接泵輪輸入端，渦輪輸出需通過減速齒輪減速再接齒輪泵加載。考慮到液力耦合器振動主要原因[10]是由不平衡與裝配時不對中所引起的，因此為減小振動的發(fā)生，對泵輪與渦輪的動平衡進(jìn)行校核。液力耦合器與連接件不對中，也是導(dǎo)致振蕩因素之一，因此對增速箱-液力耦合器-減速箱進(jìn)行重新找正，具體數(shù)據(jù)見表3和表4，通過校核使不對中和不平衡到達(dá)許用范圍[7]。同時采用高精度電渦流傳感器來對轉(zhuǎn)子的軸向、徑向位移進(jìn)行檢測，確保轉(zhuǎn)矩測量過程中液力耦合器在技術(shù)要求范圍內(nèi)工作，通過上述檢測和校核方法即認(rèn)為可忽略液力耦合器振動影響，同時采集液力耦合器同一工況條件 10組數(shù)據(jù)取平均值，即得到文中試驗數(shù)據(jù)[29-30]。

圖5 樣機(jī)試驗Fig.5 Prototype test

表3 旋轉(zhuǎn)組件校核前后的數(shù)據(jù)Table3 Data before and after verification of rotating components

2 計算結(jié)果及分析

2.1 數(shù)值與試驗對比

為了驗證 CFD混合模型和流體體積法（volume of fluid，VOF）模型的準(zhǔn)確性，首先通過臺架試驗測量了液力耦合器正常工作條件（即不對中和動平衡在技術(shù)要求許可范圍內(nèi)）、充液量q為40%、60%和80% 時液力偶合器同步工況附近的泵輪轉(zhuǎn)矩值，然后將CFD計算的相應(yīng)工況點的數(shù)值與試驗結(jié)果進(jìn)行比較，結(jié)果如圖6所示，可見VOF計算所得傳遞扭矩大小與相應(yīng)試驗值基本吻合，二者誤差在 5%以內(nèi)，證明所建立的液力耦合器兩相流計算模型是準(zhǔn)確的，同時可以發(fā)現(xiàn)，基于混合模型計算的結(jié)果誤差大于10%，因此后續(xù)對振動條件下液力耦合器數(shù)值計算均采用VOF模型計算，以提高結(jié)論的可靠性。

2.2 流場計算結(jié)果及分析

數(shù)值計算中設(shè)定轉(zhuǎn)速比i= 0.94、泵輪轉(zhuǎn)速nB=n0（設(shè)定n0=10 000 r/min），對式（2）取振幅A=0.02 mm、相位角α=π/2、f= f0（旋轉(zhuǎn)基頻f0=1 047.2 Hz），即振動速度隨時間作正弦變化，另通過用戶自定義函數(shù)描述泵輪軸向和徑向的運動速度，通過式（1）、式（3）即可得到軸向和徑向位移和加速度。

循環(huán)圓面 A（即液力耦合器的軸面，左側(cè)為泵輪流道、右側(cè)為渦輪流道，圖 7中上部為循環(huán)圓外徑、下部為循環(huán)圓內(nèi)徑）內(nèi)兩相分布如圖7a所示（紅色部分為氣相，藍(lán)色部分為液相），泵輪軸向和徑向振動兩相界面分布基本與無振動條件相分布規(guī)律相同，而泵輪徑向振動導(dǎo)致泵輪兩相界面分布處沿徑向方向氣液混合現(xiàn)象增加，氣相與液相分界面清晰度降低，同時由于徑向運動導(dǎo)致渦輪產(chǎn)生小流量脈動，交界面向渦輪流道傾斜的幅度較小。軸向振動條件下因振動方向與流道內(nèi)液體相對速度相同，對液流產(chǎn)生擾動，泵輪軸向振動導(dǎo)致泵輪兩相界面明顯泵輪內(nèi)的氣液分界面向泵輪出口處移動，且交界面向渦輪流道傾斜的幅度增加，開始出現(xiàn)流量脈動幅度增加的現(xiàn)象。

圖7 A面流場分布Fig.7 Flow distribution of A surface

循環(huán)圓面A內(nèi)壓力分布如圖7b所示。泵輪軸向和徑向振動兩相界面分布基本與無振動條件壓力分布規(guī)律相同，離心力占主導(dǎo)地位，壓力隨著半徑增加而增大。但泵輪徑向振動導(dǎo)致泵輪內(nèi)液體沿徑向運動速度增加，泵輪外緣壓力增加。泵輪軸向振動導(dǎo)致泵輪壓力在兩輪交界處有明顯凸起，且交界面向渦輪流道傾斜的幅度增加，流量脈動導(dǎo)致壓力脈動。

循環(huán)圓面A內(nèi)流線分布如圖7c所示（紅色部分為氣相，藍(lán)色部分為液相），泵輪與渦輪間的滑差較小，環(huán)流形態(tài)為小環(huán)流。因計算給定振動速度相對于液流在離心力作用下環(huán)流速度相比較小，故軸向、徑向振動條件下流線分布與無振動條件基本相同。

2.3 不同振動方向條件輸出特性曲線

設(shè)定式（2）中A=0.02 mm，α=π/2，f=f0，截取3個整周期的外特性數(shù)據(jù)繪制曲線如圖 8所示（為方便比較文中渦輪力矩、軸向力、徑向力均設(shè)定計算方向相反）。設(shè)定泵輪（輸入）扭矩大小TB約等于B0+ABsin(ω1t+α1)，渦輪（輸出）扭矩大小TT約等于T0+ATsin(ω2t+α2)?？梢钥闯鰺o振動條件下泵輪和渦輪力矩大小相等方向相反，且脈動幅值較小。徑向振動條件下B0略大于T0，波動幅值A(chǔ)1約AT的3倍且α1=α2+π/2，周期ω相同。與無振動條件相比，徑向振動條件渦輪力矩脈動幅值A(chǔ)T大小與無振動條件大致相等，B0減小4%，T0減小4.5%即代表傳遞效率下降。

圖8 不同振動方向動態(tài)轉(zhuǎn)矩曲線Fig.8 Dynamic torque curve under different vibration directions

徑向振動條件下徑向振動條件下MB略大于MT，波動幅值A(chǔ)B近視與AT相等且α1=α2無相位差，周期ω相同。與無振動條件相比，徑向振動條件渦輪力矩脈動幅值A(chǔ)2大小與無振動條件大致相等，B0減小約5%與T0減小約6.5%。與徑向條件規(guī)律相同，軸向振動會導(dǎo)致輸入和輸出力矩減小，即做功能力下降，且相同振幅A下軸向振動對做功能力影響比徑向大。

產(chǎn)生原因為徑向振動條件下擾動方向垂直于氣液分界面此時擾動雖然能在耦合器泵輪內(nèi)部形成較大的波峰，但是受泵輪內(nèi)離心力影響，波峰會快速減小，因此渦輪內(nèi)流量脈動較小即力矩變化較小。而軸向振動條件下振動方向與環(huán)流方向相同或者相反，因此會有效加強(qiáng)流量脈動造成較大的擾動從而引起渦輪力矩較大的波動。

2.4 不同轉(zhuǎn)速條件輸出特性曲線

為研究軸向振動對脈動影響，設(shè)定較大振幅A=0.04 mm、α=π/2、f=a、不同轉(zhuǎn)速條件外特性曲線如圖9a所示。從圖9a中可以不同轉(zhuǎn)速下B0近似與n02成正比，這與理論計算結(jié)果相同，且隨著轉(zhuǎn)速增加，泵輪渦輪轉(zhuǎn)矩波動幅值增加。nB=2 500 r/min時幅值A(chǔ)B約為5 N·m，nB=5 000 r/min時幅值A(chǔ)B約為7 N·m，nB=10 000 r/min時幅值A(chǔ)B約為10 N·m。即AB增加比例小于對應(yīng)轉(zhuǎn)速增加比例。產(chǎn)生原因為轉(zhuǎn)速較低時環(huán)流速度較小，軸向振動對力矩變化起主導(dǎo)作用。當(dāng)轉(zhuǎn)速較高時環(huán)流速度較大，環(huán)流和軸向振動共同影響下，也是導(dǎo)致0～0.0003 s內(nèi)力矩波動的一個重要因素。

圖9 不同轉(zhuǎn)速條件特性曲線Fig.9 Characteristic curves under different rotation speeds

不同轉(zhuǎn)速條件軸向力曲線如圖9b所示，可見軸向力的大小隨著轉(zhuǎn)速增加而增加，其波動幅值基本保持在1 000 N左右，根據(jù)式（3）可計算得到其加速度大小，進(jìn)而可求解得到速度大小。產(chǎn)生該現(xiàn)象的原因是高轉(zhuǎn)速比時環(huán)流較弱，殼體上作用的軸向振動對軸向力變化起主導(dǎo)作用。

不同轉(zhuǎn)速條件徑向力曲線如圖9c所示，徑向力的大小隨著轉(zhuǎn)速增加，波動幅值略有增加，因軸向振動方向與徑向力方向垂直故整體上軸向振動對徑向力影響較小。

2.5 不同頻率相同振動位移條件輸出特性曲線

取α=π/2，A=0.02 mm，不同頻率相同振幅條件下外特性曲線如圖10所示，可以看出f=0.5f0時泵輪力矩波動幅值約為2.1 N·m，渦輪力矩波動幅值約為1.5 N·m。f=f0時泵輪力矩波動幅值約為1.3 N·m，渦輪力矩波動幅值約為1.2 N·m，f=2f0時泵輪力矩波動幅值約為1 N·m，渦輪力矩波動幅值約為0.6 N·m?？梢钥闯稣駝宇l率越大，泵輪轉(zhuǎn)矩波動范圍增加且泵輪渦輪振幅偏差增加。

圖10 不同振動周期條件動態(tài)轉(zhuǎn)矩曲線Fig.10 Dynamic torque curve under different vibration periods

2.6 不同振幅條件輸出特性曲線

取α=π/2、f=f0，計算得到轉(zhuǎn)矩隨時間變化曲線如圖11a所示，從圖11a中可以振幅A增加則轉(zhuǎn)矩振幅AB增加，B0下降，特別是當(dāng)振幅A=0.05 mm時，液力耦合器轉(zhuǎn)矩波動范圍超過 30%且轉(zhuǎn)矩存在明顯跌落，這表明振幅達(dá)到一定程度對液力耦合器流動擾動作用明顯增加，即過大的軸向振幅造成液力耦合器流量脈動幅值急劇增加，做功能力降低。

軸向力隨時間變化曲線如圖 11b所示，振幅A=0.04 mm條件下液力耦合器力矩波動不超過0.5%，因軸向力波動主要由軸向振動引起的，故此時軸向力脈動與振動加速度成正比。當(dāng)振幅A=0.05 mm時液力耦合器軸向力輸出不太穩(wěn)定，結(jié)合圖11a轉(zhuǎn)矩跌落情況可以表明此時液流流量脈動較大，液力耦合器內(nèi)部環(huán)流形態(tài)發(fā)生較大改變。

振幅A分別取0、0.01、0.02、0.03、0.04、0.05、0.06 mm，通過擬合得到泵輪力矩幅值A(chǔ)B與振幅A、泵輪力矩平均值B0隨A關(guān)系如圖11c所示，從圖中可以看出，隨著振幅增加轉(zhuǎn)矩波動幅值減小，即此時振動對液力耦合器流場影響較小。A＞0.04 mm時轉(zhuǎn)矩波動范圍曲線急劇增加，且存在明顯的降幅。而振幅A≤0.02 mm時，液力耦合器轉(zhuǎn)矩波動不超過2%且轉(zhuǎn)矩降幅較小。對于該液力耦合器來說，從控制液力耦合器內(nèi)部穩(wěn)定狀態(tài)及減少其轉(zhuǎn)矩波動及軸向力的角度，軸向允許的波動幅值小于 0.02 mm較為合適，即對應(yīng)軸向竄動值小于0.04 mm。

圖11 不同振幅條件特性曲線Fig.11 Characteristic curves under different amplitudes

3 結(jié) 論

1）振動會導(dǎo)致傳遞轉(zhuǎn)矩下降，且軸向振動對傳遞轉(zhuǎn)矩影響較大、對徑向力影響較小。

2）旋轉(zhuǎn)基頻附近，隨著振動頻率的增加，泵輪、渦輪轉(zhuǎn)矩脈動幅度增加，且泵輪轉(zhuǎn)矩脈動幅度增加量大于渦輪。

3）振動會導(dǎo)致泵輪、渦輪轉(zhuǎn)矩產(chǎn)生波動，其脈動幅值隨著振動振幅增加而增加，且當(dāng)振幅＞0.04 mm時轉(zhuǎn)矩脈動幅值急劇增加，振幅≤0.02 mm時轉(zhuǎn)矩脈動幅值變化不太明顯。從控制液力耦合器內(nèi)部穩(wěn)定狀態(tài)及減少其轉(zhuǎn)矩波動及軸向力的角度，軸向允許的波動幅值小于0.02 mm較為合適，即對應(yīng)軸向竄動值小于0.04 mm。

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Characteristics of high speed hydraulic coupler under pump wheel axial vibration conditions

Su Huashan, Chen Congping※, Zhao Meiyun, Gao Zhenjun, Yu Wan, Zhang Yangjun
(1.Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, China Three Gorges University, Yichang443002,China; 2.College of Mechanical & Power Engineering of China Three Gorges University, Yichang443002, China)

Hydrodynamic coupler is used for startup tool in the large inertia mechanical equipment. The incomplete neutrality of impeller installation and loading perturbation cause the input shaft of pump wheel to vibrate. Internal flow characteristics of hydrodynamic coupler are affected by the vibration of the pump wheel. And the external performance of hydrodynamic coupler is determined by its distribution of internal flow field. Therefore, it is very important to make a deep research on the distribution of internal flow field under the condition of vibration. Numerical simulation is a main way to study the internal flow field of hydrodynamic coupler. The simulation physical model was created firstly by using the software of ICEM (integrated computer engineering and manufacturing), and hexahedron and tetrahedron cells were used to partition the calculation region to generate the grids. The hexahedron was used in main channel of pump wheel and turbine. The tetrahedron was used in boundary motion region. And then the software of FLUENT was used to perform the simulation. The UDF (user-defined function) of FLUENT was used to define the parameters of dynamic mesh control, as well as the axial velocity of pump. Realizable k-ε model was used, besides, the turbulence model and the second-order upwind scheme were adopted for solving the momentum and kinetic energy equation, and the PISO (pressure-implicit with splitting of operators) algorithm was used for pressure and velocity coupling. With the pump axial moving, the boundary of the corresponding flow field would change. The dynamic mesh model was used for boundary motion domain caused by vibration. The results of numerical simulation that are calculated by different two-phase flow models were quite different. In order to obtain accurate and reliable results of numerical simulation, the numerical simulation and external characteristic experimental results were compared. It showed that the error of VOF (volume of fluid) model was less than 5%, and the error of Mixture model was over 20%. It showed that the simulation results by VOF models were more accurate and close to the experimental results. Furthermore, the external characteristics and phase distribution law of fluid coupling were also compared and analyzed under different axial vibration status. And the results indicated that the vibration of the pump wheel could make the flow pulsation increase. Under the condition of radial vibration, the disturbance direction was perpendicular to the gas-liquid interface. A larger wave crest could be formed within pump wheel. However, due to the centrifugal force in the pump wheel, the wave would rapidly decrease. Therefore, the flow pulsation in the turbine was relatively small, that was to say, the torque change was relatively small. Under the condition of axial vibration, the direction of vibration was the same or opposite to the direction of circulation. Therefore, it would effectively enhance the fluctuation of the flow pulsation and cause the larger fluctuation of turbine torque. Numerical calculation showed that the higher the rated speed, the larger the torque ripple amplitude of pump turbine and the fluctuation range of radial force and axial force. The vibration period decreased and the deviation of the torque ripple of the pump turbine was bigger. Vibration would lead to the decrease of the transmission torque, and the axial vibration had a greater impact on the transmission torque, and a smaller influence on the radial force. The vibration would cause the pump wheel and turbine torque to fluctuate, and the pulsation amplitude increased with the increase of the vibration amplitude. When the amplitude of vibration was less than 0.02 mm, the amplitude of torque was smaller. But when the amplitude of vibration was 0.04 mm, the amplitude of torque increased sharply. On that basis, the axial clearance value should not be more than 0.04 mm (the axial clearance was twice of the amplitude of vibration).

computer simulation; visualization; models; hydraulic coupling; axial vibration; two-phase flow

10.11975/j.issn.1002-6819.2017.07.007

TH137.331

A

1002-6819(2017)-07-0051-07

2016-09-27

2017-04-10

國家自然基金（51475266，51605254）；水電機(jī)械設(shè)備設(shè)計與維護(hù)湖北省重點實驗室（三峽大學(xué)）開放基金（2016KJX03）；宜昌市科技局項目（A14-302-a03）

蘇華山，男，博士，講師，主要從事流體機(jī)械內(nèi)部流動特性研究。宜昌 三峽大學(xué)水電機(jī)械設(shè)備設(shè)計與維護(hù)湖北省重點實驗室，443002。

Email：suhuashan@ctgu.edu.cn

※通信作者：陳從平，男，博士，教授，主要從事流體動力學(xué)方面的研究。宜昌 三峽大學(xué)水電機(jī)械設(shè)備設(shè)計與維護(hù)湖北省重點實驗室，443002。

Email：mechencp@163.com