章大海,王文顥,石凡奇,郝木明

(中國(guó)石油大學(xué)化學(xué)工程學(xué)院,山東青島266580)

質(zhì)量比對(duì)二自由度圓柱渦激振動(dòng)影響的計(jì)算研究

章大海,王文顥,石凡奇,郝木明

(中國(guó)石油大學(xué)化學(xué)工程學(xué)院,山東青島266580)

利用Fluent平臺(tái)的用戶(hù)自定義程序(UDF)以及動(dòng)網(wǎng)格模型,實(shí)現(xiàn)圓柱運(yùn)動(dòng)方程的一種迭代求解算法,對(duì)二自由度彈性支承圓柱體在一定約化速度下的渦激響應(yīng)進(jìn)行數(shù)值模擬;探討不同質(zhì)量比對(duì)渦激響應(yīng)升力的影響。研究表明:采用的迭代求解算法能對(duì)彈性支承圓柱渦激振動(dòng)做出合理預(yù)測(cè);質(zhì)量比對(duì)渦激響應(yīng)的升力影響顯著,不僅低質(zhì)量比圓柱產(chǎn)生的振幅更大,低質(zhì)量比較高質(zhì)量比能產(chǎn)生更大的升力系數(shù),且升力相對(duì)橫向位移的“相位突跳”現(xiàn)象對(duì)應(yīng)的約化來(lái)流速度U*更大;圓柱運(yùn)動(dòng)軌跡從最初的弧型轉(zhuǎn)變至“8”字型,而后“8”字型逐漸消失并轉(zhuǎn)變?yōu)樗涡巍?/p>

渦激振動(dòng);流固耦合;質(zhì)量比;數(shù)值模擬

渦激振動(dòng)對(duì)長(zhǎng)輸管道、高聳結(jié)構(gòu)、輸電線(xiàn)路等設(shè)備易產(chǎn)生疲勞破壞,其研究方法主要為基于相似原理的實(shí)驗(yàn)研究和基于數(shù)值計(jì)算的模擬研究。Williamson[1-2]、Sarpkaya[3]和Gabbai[4]等已對(duì)渦激振動(dòng)進(jìn)行了相關(guān)綜述;Williamson等[5-9]以低質(zhì)量比的振動(dòng)系統(tǒng)為研究對(duì)象,通過(guò)大量的實(shí)驗(yàn)研究發(fā)現(xiàn)質(zhì)量-阻尼聯(lián)合參數(shù)m*ζ對(duì)振動(dòng)幅值有明顯的影響,當(dāng)渦激振動(dòng)發(fā)生在空氣中,即m*ζ較大時(shí),振幅曲線(xiàn)只有“初始支”和“下支”兩支響應(yīng),而在低質(zhì)量比的水中時(shí),振幅曲線(xiàn)出現(xiàn)三支響應(yīng),也就是經(jīng)典的三支曲線(xiàn):初始支、上支和下支,且在低質(zhì)量比情形發(fā)現(xiàn)一種對(duì)應(yīng)最大橫向振幅的“2T”模態(tài)。計(jì)算流體力學(xué)在渦激振動(dòng)領(lǐng)域已成為一種主要的研究方法,而基于RANS方程的雷諾時(shí)均方法更是被大量研究者所采用。Bahmani等[10]用渦量-流函數(shù)法處理N-S方程,模擬層流情況下的圓柱渦激振動(dòng),驗(yàn)證了該方法的有效性;潘志遠(yuǎn)[11]利用RANS方法研究了圓柱體橫向自激振動(dòng)與受迫振動(dòng)的機(jī)理并精確預(yù)測(cè)了圓柱體的振動(dòng);黃智勇等[12]對(duì)二自由度圓柱渦激振動(dòng)進(jìn)行了數(shù)值模擬,發(fā)現(xiàn)在質(zhì)量比低于3.5時(shí),二自由度圓柱的橫向振幅比限制流向振動(dòng)的單自由度時(shí)的橫向振幅大。林琳等[13]比較了彈性支承圓柱與固定圓柱渦激振動(dòng)的尾流,分析了振動(dòng)對(duì)流體力系數(shù)、壁面壓力等參數(shù)的影響;董婧等[14]通過(guò)數(shù)值模擬研究了低雷諾數(shù)下圓柱體振動(dòng)的影響參數(shù)及尾渦結(jié)構(gòu)與氣動(dòng)力的關(guān)系,并觀察到“拍”和“相位開(kāi)關(guān)”等現(xiàn)象。章大海等[15]數(shù)值模擬研究了彈性支承圓柱體的渦激振動(dòng)響應(yīng),實(shí)現(xiàn)了單自由度振子方程的簡(jiǎn)單迭代求解。筆者在文獻(xiàn)[15]的基礎(chǔ)上,進(jìn)一步研究二自由度彈性支承圓柱體在一定約化速度范圍內(nèi)的渦激響應(yīng)振幅、頻率、升阻力系數(shù)、瀉渦模式、運(yùn)動(dòng)軌跡等,分析不同質(zhì)量比對(duì)渦激響應(yīng)升力的影響。

1 計(jì)算模型

1.1 幾何模型

圖1為采用二自由度彈性支承圓柱振動(dòng)系統(tǒng)簡(jiǎn)化物理模型。采用與文獻(xiàn)[15]相同的幾何模型,圓柱直徑D,圓柱中心距離上、下及左邊界的距離為10D,距離右邊為20D,以保證尾渦的充分發(fā)展;總網(wǎng)格數(shù)約1.8×104個(gè)。考慮到圓柱運(yùn)動(dòng)時(shí),周邊網(wǎng)格的不規(guī)則重構(gòu),在圓柱周?chē)贾?D厚度的結(jié)構(gòu)化網(wǎng)格,并讓其隨圓柱一起運(yùn)動(dòng),以保證圓柱附近流場(chǎng)的計(jì)算精度,如圖2所示。具體的邊界條件、流體物性等參數(shù)的設(shè)置參閱文獻(xiàn)[15]。

圖1 幾何模型示意圖Fig.1 Model sheme of elastically mounted cylinder

圖2 圓柱附近網(wǎng)格Fig.2 Grid near cylinder

1.2 結(jié)構(gòu)動(dòng)力學(xué)方程

對(duì)于圖1中的雙自由度(2-DOF)彈簧-質(zhì)量-阻尼系統(tǒng),其控制方程為

其中,x、y、m、c和k分別為圓柱順流向位移、橫向位移、圓柱質(zhì)量、結(jié)構(gòu)阻尼系數(shù)和系統(tǒng)剛度系數(shù);FD(t)和FL(t)為圓柱所受順流向和橫向流體力,進(jìn)行無(wú)量綱化后可得

式中,U∞為來(lái)流速度;fn為系統(tǒng)固有頻率;D為圓柱直徑;m*為質(zhì)量比(m*=m/md,md為圓柱排開(kāi)的水的質(zhì)量);ζ為阻尼比;ccr為系統(tǒng)臨界阻尼,;CL、CD分別為升、阻力系數(shù);ρ為流體密度。式中導(dǎo)數(shù)是關(guān)于無(wú)量綱時(shí)間τ的求導(dǎo)。

1.3 計(jì)算流程

二自由度圓柱的運(yùn)動(dòng)須將順流向的方程(3)和橫流向的方程(4)同時(shí)進(jìn)行求解,先將式(4)簡(jiǎn)化為如下形式:

據(jù)(8)～(11)利用迭代思想就可進(jìn)行二自由度渦激振動(dòng)的求解。編寫(xiě)代碼時(shí),給定初始速度為0 m/s,在每時(shí)間步只須提取流體力FL、FD,根據(jù)方程(8)～(11)求解圓柱的運(yùn)動(dòng)速度,根據(jù)Fluent里的用戶(hù)自定義函數(shù)(User-Defined Functions),通過(guò)Define_CG_Motion宏將圓柱的運(yùn)動(dòng)速度返回給Fluent主程序,Fluent給UDF返回流體力,實(shí)現(xiàn)流體與結(jié)構(gòu)控制方程的耦合求解。

2 二自由度圓柱的渦激響應(yīng)

圖3為計(jì)算結(jié)果與實(shí)驗(yàn)數(shù)據(jù)的對(duì)比,采用約化速度表達(dá)式U*=U∞/fnwD進(jìn)行處理[5],圓柱直徑D=0.0381m,質(zhì)量比m*=2.6,約化速度U*=3～12。

圖3 圓柱渦激振動(dòng)響應(yīng)曲線(xiàn)Fig.3 Amplitude and frequency response as a function of reduced velocity

由圖3可知,橫向振幅能明顯分辨出初始分支、上端分支、下端分支。最大橫向振幅出現(xiàn)在U*=6.4處,振幅值約為y/D=1.22,最大流向振幅發(fā)生在U*=6,幅值約為x/D=0.2。橫向、流向振幅變化趨勢(shì)與Williamson[7]的實(shí)驗(yàn)結(jié)果大體相符。圖3(b)清楚地顯示了頻率鎖定現(xiàn)象。在隨著U*的增大,橫向振動(dòng)頻率由符合Strohal規(guī)律,突然轉(zhuǎn)變?yōu)殒i定在系統(tǒng)在水中自然頻率fnw附近,在U*=12出現(xiàn)回歸Strohal規(guī)律的振動(dòng)分量,鎖定現(xiàn)象開(kāi)始減弱?？梢钥闯?數(shù)值計(jì)算與實(shí)驗(yàn)結(jié)論一致,本文中采用的迭代算法是有效的。實(shí)驗(yàn)中沒(méi)有出現(xiàn)最大橫向振幅y/D=1.55,誤差為20%。計(jì)算采用二維模型以及RANS方法,主觀上沒(méi)有考慮流場(chǎng)的三維特性和湍流的隨機(jī)性,而文獻(xiàn)[12]表明渦激振動(dòng)瀉渦的三維特性以及隨機(jī)性對(duì)振幅均有影響;要實(shí)現(xiàn)渦激振動(dòng)的三維計(jì)算并模擬出隨機(jī)特性,所花費(fèi)的時(shí)間與資源將會(huì)成倍增加[11]。另外,采用光滑圓柱進(jìn)行計(jì)算,忽略了表面粗糙度的影響,與實(shí)驗(yàn)情況不符,而圓柱的表面粗糙度會(huì)使剪切層分離點(diǎn)提前,能夠增大渦激振幅[16],造成計(jì)算誤差。由此可見(jiàn),本文中的算法是可取的。

圖4給出了無(wú)量綱位移x/D、y/D,阻力系數(shù)CD和升力系數(shù)CL隨時(shí)間變化的曲線(xiàn)。當(dāng)U*=3.0時(shí),圓柱體的振幅較小;隨著約化速度增加,振幅、升力系數(shù)曲線(xiàn)發(fā)生明顯的差拍振動(dòng),圓柱體的振幅、升阻力系數(shù)幅值明顯增大;U*=6時(shí),振幅和阻力系數(shù)幅值繼續(xù)增大,升力系數(shù)減小,差拍振動(dòng)現(xiàn)象消失,發(fā)生鎖定現(xiàn)象;在U*=12時(shí),振幅減小,鎖定現(xiàn)象開(kāi)始消失,振動(dòng)重新出現(xiàn)服從Strohal規(guī)律的分量?？梢钥闯?圓柱體振動(dòng)響應(yīng)與文獻(xiàn)結(jié)論相符,本文中算法是可取的。

圖4 不同約化速度下圓柱的渦激響應(yīng)時(shí)程及對(duì)應(yīng)頻譜Fig.4 Amplitude and fluid force response and corresponding spectrum under different U*

圖5給出圓柱一個(gè)振動(dòng)周期內(nèi)(U*=6.4)渦量等值線(xiàn)圖清晰地捕捉到了2T瀉渦模式。Williamson等[5-9]實(shí)驗(yàn)結(jié)果表明:低質(zhì)量比彈性支撐圓柱渦激振動(dòng)中與振動(dòng)響應(yīng)初始分支對(duì)應(yīng)的尾渦為2S模式,而與上端分支對(duì)應(yīng)的尾渦則為2T模式。本文結(jié)果與之相符,也驗(yàn)證了計(jì)算方法的有效性。圖6給出圓柱質(zhì)心的運(yùn)動(dòng)軌跡,從U*=3的弧型轉(zhuǎn)變至U*=6的“8”字型,與文獻(xiàn)[17]相符,而后“8”字型逐漸消失,至U*=10時(shí),本文中算法計(jì)算的軌跡呈水滴型。

3 質(zhì)量比對(duì)升力系數(shù)的影響

研究不同質(zhì)量比對(duì)升力、相對(duì)橫向位移相位角的影響。分別進(jìn)行質(zhì)量比m*為2.6和8.63的圓柱渦激振動(dòng)數(shù)值計(jì)算,提取升力系數(shù)與橫向位移時(shí)程,計(jì)算升力系數(shù)的均方根值,通過(guò)Hilbert變換計(jì)算出升力相對(duì)橫向位移的實(shí)時(shí)相位。計(jì)算結(jié)果如圖7～8所示。

圖5 圓柱體單個(gè)振動(dòng)周期內(nèi)流場(chǎng)尾渦Fig.5 Vorticity magnitude contours within a vortex shedding period

圖6 圓柱質(zhì)心軌跡Fig.6 Cylinder centroid trajectory

圖7 振幅時(shí)程曲線(xiàn)和升力系數(shù)的實(shí)時(shí)相位Fig.7 Typical time traces of cylinder displacement and phase

圖8 不同質(zhì)量比下升力系數(shù)及相位角Fig.8 CLrmsand phase angle variation with U*

圖8(b)中誤差條帶表示相位值在0°～180°變化,圖8由升力系數(shù)時(shí)程給出升力系數(shù)均方根值,升力系數(shù)實(shí)時(shí)相位由Hilbert變換獲得。Williamson[5-9]指出,升力系數(shù)CL均方根值會(huì)隨約化速度由小到大變化,最大值出現(xiàn)在初始支與上支的轉(zhuǎn)換區(qū)域,在此之后會(huì)急劇下降。升力系數(shù)相對(duì)圓柱位移的相位Φ在初始支和上支為0°,即升力與位移同相,然后“突跳”為180°,即升力與位移反相。本文中得到的兩種質(zhì)量比下的升力系數(shù)均方根值變化規(guī)律都與實(shí)驗(yàn)規(guī)律相符,均先增大再減小,m*=8.63給出的升力均方根值峰值在U*=3.5出現(xiàn),為CLrms=1.15,在U*=4.5即大幅下降至CLrms=0.4;相位的“突跳”區(qū)域出現(xiàn)在U*=3.5～4(圖8(b));m*=2.6的CLrms峰值2.5在U*=5出現(xiàn),比m*=8.63的升力系數(shù)更大,相位突跳在U*=6～7出線(xiàn),比m*=8.63情況出現(xiàn)得更晚。

4 結(jié) 論

(1)迭代求解算法能夠?qū)椥灾С袌A柱渦激振動(dòng)做出合理的預(yù)測(cè)。

(2)質(zhì)量比對(duì)渦激響應(yīng)的升力影響顯著,不僅低質(zhì)量比產(chǎn)生的振幅更大,低質(zhì)量比較高質(zhì)量比能產(chǎn)生更大的升力系數(shù),且升力相對(duì)橫向位移的“相位突跳”現(xiàn)象對(duì)應(yīng)的約化來(lái)流速度U*更大。

(3)圓柱運(yùn)動(dòng)軌跡從最初的弧型轉(zhuǎn)變至“8”字型,而后“8”字型逐漸消失并逐漸轉(zhuǎn)變?yōu)樗涡汀?/p>

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(編輯 劉為清)

Calculation study on the influence of mass ratio on VIV of a 2-DOF cylinder

ZHANG Dahai,WANG Wenhao,SHI Fanqi,HAO Muming
(College of Chemical Engineering in China University of Petroleum,Qingdao 266580,China)

An iterative solution algorithm was developed to solve the cylindrical motion equation using Fluent UDF codes and dynamic mesh method,through which the numerical simulation of the vortex-induced vibration(VIV)of a two degree of freedoms(2-DOF)elastically mounted cylinder at a certain reduced velocity.And the influence of different mass ratios on the VIV lift was analyzed.It is found that the iterative solution algorithm can reasonably predict the VIV of elastically mounted cylinder.And the influence of the VIV lift is significant.Compared to high mass ratio cylinders,low mass ratio cylinders can excite higher amplitude,larger lift coefficient,and the phase angle reverse corresponds with a larger reduced velocity U*.The trajectory of the cylinder transfers from the initial arc figure to"8"figure,and finally to a water drop figure.

vortex-induced vibration;solid-fluid interaction;mass ratio;numerical simulation

O 237

:A

章大海,王文顥,石凡奇,等.質(zhì)量比對(duì)二自由度圓柱渦激振動(dòng)影響的計(jì)算研究[J].中國(guó)石油大學(xué)學(xué)報(bào)(自然科學(xué)版),2017,41(3):169-175.

ZHANG Dahai,WANG Wenhao,SHI Fanqi,et al.Calculation study on the influence of mass ratio on VIV of a 2-DOF cylinder[J].Journal of China University of Petroleum(Edition of Natural Science),2017,41(3):169-175.

1673-5005(2017)03-0169-07doi:10.3969/j.issn.1673-5005.2017.03.021

2016-09-22

山東省自然科學(xué)基金項(xiàng)目(ZR2012EEQ011)

章大海(1978-),男,副教授,博士,研究方向?yàn)榱黧w耦合與CFD的工業(yè)應(yīng)用。E-mail:dhzhang@upc.edu.cn。