基于OpenFOAM的周期性管內(nèi)流動(dòng)邊界處理方法研究

張 鵬, 凡鳳仙

(1.上海理工大學(xué) 能源與動(dòng)力工程學(xué)院,上海 200093;2.上海理工大學(xué) 上海市動(dòng)力工程多相流動(dòng)與傳熱重點(diǎn)實(shí)驗(yàn)室,上海 200093)

基于OpenFOAM的周期性管內(nèi)流動(dòng)邊界處理方法研究

張 鵬1,2, 凡鳳仙1,2

(1.上海理工大學(xué) 能源與動(dòng)力工程學(xué)院,上海 200093;2.上海理工大學(xué) 上海市動(dòng)力工程多相流動(dòng)與傳熱重點(diǎn)實(shí)驗(yàn)室,上海 200093)

為探究開(kāi)源計(jì)算流體力學(xué)軟件OpenFOAM中不同周期性邊界條件處理方法的適用性,針對(duì)圓管內(nèi)泊肅葉流動(dòng),分別在mapped,cyclicAMI,cyclic fan邊界條件下開(kāi)展數(shù)值模擬,對(duì)模擬得到的速度分布和壓力分布進(jìn)行分析.結(jié)果表明:3種邊界條件設(shè)置下,數(shù)值模擬得到的速度分布均與解析解吻合,其中mapped和cyclicAMI邊界條件下的數(shù)值解非常接近,且與解析解吻合更好;采用mapped邊界能夠得到與解析解符合的壓力場(chǎng),采用cyclic fan邊界時(shí)在模擬區(qū)域進(jìn)、出口存在很大的壓力梯度,采用cyclicAMI邊界得到的壓力則恒為0.基于此,探討了mapped邊界條件下,數(shù)值模擬結(jié)果對(duì)計(jì)算網(wǎng)格的敏感性.研究發(fā)現(xiàn):隨著徑向網(wǎng)格數(shù)目的增加,速度的計(jì)算精度先迅速提高而后趨于穩(wěn)定;邊界層層數(shù)的增加能夠有效提高近壁區(qū)速度的計(jì)算精度;數(shù)值模擬時(shí)徑向網(wǎng)格數(shù)目采用 16個(gè)以上為宜,邊界層層數(shù)不宜少于3層.

管內(nèi)流動(dòng); 周期性邊界條件; 泊肅葉流動(dòng); 計(jì)算流體力學(xué); OpenFOAM

在工程實(shí)際應(yīng)用中,不可壓縮流體的管內(nèi)流動(dòng)非常普遍.隨著計(jì)算機(jī)技術(shù)的快速發(fā)展,數(shù)值模擬技術(shù)在管內(nèi)流動(dòng)研究中得到了日益廣泛的應(yīng)用.數(shù)值模擬能夠獲得復(fù)雜流場(chǎng)的細(xì)節(jié)信息,從而成為研究流場(chǎng)內(nèi)部流動(dòng)結(jié)構(gòu)的有力手段.能源、化工、制藥、食品加工領(lǐng)域的管內(nèi)物料輸運(yùn)通常是恒定壓力梯度下的定常運(yùn)動(dòng),且管道往往長(zhǎng)達(dá)數(shù)十米以上.由于研究對(duì)象尺度很大,若對(duì)整個(gè)區(qū)域進(jìn)行計(jì)算,工作量將非常大.在數(shù)值模擬中若能采用周期性邊界條件,則可利用長(zhǎng)度小很多(0.5～2 m)的管道代替實(shí)際管道,以減小模擬尺度,從而在不失精確度的前提下,大大降低計(jì)算量[1-3].如何施加周期性邊界條件直接、有效地體現(xiàn)出管內(nèi)周期性流動(dòng)的所有特征,在目前的工程實(shí)際應(yīng)用中具有重要的現(xiàn)實(shí)意義.

近年來(lái),隨著開(kāi)源程序OpenFOAM的出現(xiàn)和發(fā)展,越來(lái)越多的科研和工程技術(shù)人員開(kāi)始使用它進(jìn)行計(jì)算流體力學(xué)(computational fluid dynamics,CFD)研究[4-6].OpenFOAM是一種基于C++語(yǔ)言的開(kāi)源CFD程序庫(kù),它在本質(zhì)上是一個(gè)預(yù)編譯的功能庫(kù),用戶可以通過(guò)命令調(diào)用這些功能庫(kù),也可以根據(jù)需要對(duì)這些功能庫(kù)進(jìn)行修改、繼承,從而開(kāi)發(fā)自己的CFD軟件,以滿足問(wèn)題的需求[7].OpenFOAM因其免費(fèi)、開(kāi)源、穩(wěn)定強(qiáng)大的底層類庫(kù),以及能夠并行計(jì)算的優(yōu)點(diǎn)受到廣大CFD用戶的青睞.由于邊界條件的設(shè)置會(huì)影響數(shù)值計(jì)算結(jié)果的準(zhǔn)確性,基于OpenFOAM對(duì)周期性管內(nèi)流動(dòng)進(jìn)行高精度模擬,需要探討不同周期性邊界條件處理方法在OpenFOAM中的實(shí)現(xiàn),并考量這些處理方法的適用性.管內(nèi)泊肅葉流動(dòng)作為一種典型的管內(nèi)流動(dòng),因其有特定的解析解,常用于數(shù)值模擬中驗(yàn)證計(jì)算結(jié)果的準(zhǔn)確性[8-10].為此,筆者以圓管內(nèi)泊肅葉流動(dòng)為研究對(duì)象,應(yīng)用開(kāi)源OpenFOAM軟件,開(kāi)展不同邊界條件處理方法下周期性流動(dòng)速度場(chǎng)與壓力場(chǎng)的數(shù)值模擬,并將所得結(jié)果與解析解進(jìn)行對(duì)比,以校驗(yàn)邊界條件設(shè)置的適用性和準(zhǔn)確性,為基于OpenFOAM系統(tǒng)研究周期性管內(nèi)流動(dòng)提供基礎(chǔ).

1 數(shù)學(xué)模型、邊界條件設(shè)置及數(shù)值計(jì)算方法

1.1 數(shù)學(xué)模型

(1)

(3)

(4)

式中:ux為流體速度,m/s;ρ為流體密度,kg/m3;ν為流體運(yùn)動(dòng)黏度,m2/s;p為壓力,Pa;t為時(shí)間,s.

1.2 邊界條件設(shè)置方法

基于OpenFOAM模擬管內(nèi)周期性泊肅葉流動(dòng)可采用mapped,cyclicAMI和cyclic fan 3種邊界類型,它們對(duì)應(yīng)于不同的邊界條件和初始條件設(shè)置.需要注意的是:OpenFOAM中的壓力是以壓力與流體密度之比的形式給出的,單位是m2/s2.

1.2.1 mapped邊界

mapped邊界的思想是從出口流出的流體以同樣的速度從進(jìn)口流入,從而形成周期性流動(dòng)狀態(tài).本文模擬中,對(duì)應(yīng)的邊界條件設(shè)置見(jiàn)表1.

1.2.2 cyclicAMI邊界

cyclicAMI邊界通過(guò)對(duì)進(jìn)口和出口的網(wǎng)格進(jìn)行耦合,構(gòu)建兩者的聯(lián)系,使流體的進(jìn)口能夠順利地捕獲出口的速度和壓力信息.該邊界條件的一個(gè)突出優(yōu)點(diǎn)是即使進(jìn)、出口網(wǎng)格數(shù)量不同,也能夠?qū)崿F(xiàn)網(wǎng)格的耦合.采用cyclicAMI邊界時(shí),邊界條件設(shè)置見(jiàn)表2.此外,需要增加抽象基類選項(xiàng)文件“fvOptions”,并在該文件中設(shè)置流體速度場(chǎng),平均速度Ubar設(shè)置為(0.01 0 0).

表2 cyclicAMI邊界設(shè)置Tab.2 CyclicAMI boundary setup

1.2.3 cyclic fan邊界

cyclic fan邊界的基本思想是在進(jìn)口設(shè)置虛擬風(fēng)扇,利用風(fēng)壓對(duì)管內(nèi)流體進(jìn)行單向強(qiáng)力驅(qū)動(dòng),使流體流動(dòng)并最終達(dá)到穩(wěn)定狀態(tài).在進(jìn)口為cyclic fan邊界時(shí),需要設(shè)置風(fēng)壓,OpenFOAM中風(fēng)壓由流體速度ux的n-1次n項(xiàng)式給出[7],即

(5)

在對(duì)進(jìn)口風(fēng)壓進(jìn)行設(shè)置時(shí),采用“n-1 (A1A2A3…An)”的格式.據(jù)此,對(duì)于本文模擬,進(jìn)口風(fēng)壓設(shè)置為“1 (0.0005 0)”.表3給出了具體的邊界條件設(shè)置.

1.3 數(shù)值計(jì)算方法

OpenFOAM包含有適用于求解可壓和不可壓流體層流和湍流模型的多種算法.本文研究的泊肅葉流動(dòng)為不可壓縮流體的穩(wěn)態(tài)流動(dòng),可采用SIMPLE算法[12]進(jìn)行求解.求解流程為:a.在有限體積法的框架下,施加初始條件和邊界條件;b.基于高斯二階線性離散格式,求解動(dòng)量方程,獲得速度場(chǎng);c.計(jì)算網(wǎng)格面的質(zhì)量通量,進(jìn)行亞松弛;d.求解壓力方程,修正質(zhì)量通量;e.在修正的壓力場(chǎng)基礎(chǔ)上,修正速度場(chǎng);f.更新邊界條件,重復(fù)上述步驟,直到滿足收斂條件,計(jì)算結(jié)束.

表3 cyclic fan邊界設(shè)置Tab.3 Cyclic fan boundary setup

2 結(jié)果與討論

2.1 不同邊界設(shè)置時(shí)數(shù)值模擬結(jié)果與解析解的對(duì)比

為驗(yàn)證數(shù)值模擬的準(zhǔn)確性,本節(jié)將數(shù)值模擬結(jié)果與解析解進(jìn)行對(duì)比分析.對(duì)于泊肅葉流動(dòng)而言,流體速度在圓管的橫截面上呈旋轉(zhuǎn)拋物面分布,流體壓力沿管長(zhǎng)線性下降,它們的表達(dá)式分別為[13]

(6)

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2.1.1 網(wǎng)格劃分

運(yùn)用OpenFOAM自有的網(wǎng)格生成程序block-Mesh與snappyHexMesh對(duì)圓管進(jìn)行網(wǎng)格劃分,如圖1所示(見(jiàn)下頁(yè)).其中,圖1(a)給出了沿管長(zhǎng)方向的網(wǎng)格劃分,圖1(b)給出了圓管出口段的網(wǎng)格情況,圖(c)為圓管中心橫截面的網(wǎng)格劃分;計(jì)算區(qū)域總網(wǎng)格數(shù)目為126 400個(gè).

2.1.2 速度場(chǎng)數(shù)值模擬結(jié)果與解析解的對(duì)比

3種周期性邊界處理方法(mapped,cyclicAMI,

圖1 圓管網(wǎng)格Fig.1 Cells on the circular pipe

cyclic fan)下得到的圓管中心截面直徑AB(見(jiàn)圖1(c))上的速度分布如圖2所示.由圖2(a)可知,在近壁面區(qū)域3種方法得到的數(shù)值模擬結(jié)果均與解析解吻合良好,而在中心區(qū)域數(shù)值解比解析解略小,并且數(shù)值解之間也有差異.為了清楚地展示中心區(qū)域的數(shù)值誤差,對(duì)-0.3≤2z/d≤0.3對(duì)應(yīng)的曲線進(jìn)行局部放大,如圖2(b)所示.可見(jiàn),采用cyclicAMI和mapped邊界條件處理方法所得到的數(shù)值解非常接近,且更接近于解析解,而采用cyclic fan邊界與解析解的差異較前兩種邊界時(shí)略大.計(jì)算表明,3種周期性邊界條件處理方法下數(shù)值模擬結(jié)果的相對(duì)誤差在-4.4%～6.8%范圍內(nèi),且相對(duì)誤差上限和下限均出現(xiàn)在近壁區(qū),這是由于近壁區(qū)速度的真實(shí)值(即解析解)接近0,較小的數(shù)值計(jì)算誤差即引起相對(duì)誤差絕對(duì)值的增大.

2.1.3 壓力場(chǎng)數(shù)值模擬結(jié)果與解析解的對(duì)比

3種周期性邊界處理方法下,數(shù)值模擬得到的沿管中心軸線的壓力分布如圖3所示.結(jié)果表明,3種方法下壓力分布各不相同.mapped邊界下,壓力分布與解析解符合很好; cyclicAMI邊界下,沿管中心軸線壓力始終為零;cyclic fan邊界下,壓力在包含進(jìn)口邊界的網(wǎng)格內(nèi)由Δp/2 (0.25 Pa)迅速上升至Δp(0.5 Pa),再由Δp線性減小至0,在包含出口邊界的網(wǎng)格內(nèi)迅速升至Δp/2 (0.25 Pa),表現(xiàn)為除進(jìn)、出口邊界網(wǎng)格內(nèi)具有很大的壓力梯度外,其余部分的壓力分布和理論解吻合良好.本文是對(duì)單相流體的泊肅葉流動(dòng)進(jìn)行數(shù)值模擬,當(dāng)將OpenFOAM用于周期性邊界條件的兩相流或多相流問(wèn)題研究時(shí),若需要考慮壓力梯度力對(duì)顆粒的影響,則應(yīng)當(dāng)對(duì)壓力進(jìn)行準(zhǔn)確預(yù)測(cè),此時(shí)應(yīng)采用mapped邊界條件設(shè)置;若不需要考慮壓力梯度力的影響,則可選用cyclicAMI和cyclic fan邊界條件設(shè)置.考慮到cyclicAMI邊界條件設(shè)置更為簡(jiǎn)便,且所得的速度場(chǎng)和解析解吻合更好,推薦采用cyclicAMI邊界條件設(shè)置.

圖2 3種周期性邊界處理方法得到 的速度分布及與解析解的對(duì)比Fig.2 Velocity profiles obtained using the three periodic boundary implementation methods and their comparison with the analytical solution

圖3 3種周期性邊界處理方法得到的 壓力分布及與解析解的對(duì)比Fig.3 Pressure profiles obtained using three periodic boundary implementation methods and their comparison with analytical solution

2.2 網(wǎng)格劃分對(duì)數(shù)值模擬結(jié)果的影響

以上數(shù)值模擬結(jié)果表明,與cyclicAMI,cyclic fan邊界相比,mapped邊界在模擬泊肅葉流動(dòng)時(shí),很好地實(shí)現(xiàn)了周期性泊肅葉流動(dòng)的速度場(chǎng)和壓力場(chǎng)特征,但其得到的速度場(chǎng)與解析解之間仍存在一定的偏差.在數(shù)值模擬過(guò)程中,計(jì)算結(jié)果與網(wǎng)格劃分情況密切相關(guān),因此本節(jié)在mapped邊界條件設(shè)置下探究網(wǎng)格劃分,如圓管徑向網(wǎng)格數(shù)目和近壁面邊界層層數(shù)對(duì)數(shù)值模擬結(jié)果的影響.

2.2.1 徑向網(wǎng)格數(shù)目的影響

保持圓管軸向網(wǎng)格和近壁面邊界層不變,改變橫截面直徑方向的網(wǎng)格數(shù)目,考察徑向網(wǎng)格數(shù)目對(duì)于數(shù)值模擬結(jié)果的影響.用N表示徑向網(wǎng)格數(shù)目,定義為不計(jì)邊界層時(shí)沿截面直徑方向的網(wǎng)格數(shù)目,圖4給出了N=8,12,16,20,24時(shí)圓管中心截面網(wǎng)格情況.在N不同的情況下,數(shù)值模擬得到的圓管中心截面直徑AB上的速度分布如圖5(a)所示,對(duì)圖5(a)中-0.4≤2z/d≤0.4對(duì)應(yīng)的曲線進(jìn)行局部放大,如圖5(b)所示.可見(jiàn),隨著徑向網(wǎng)格數(shù)目的增加,速度分布更接近于解析解.在N=8,12,16,20,24時(shí),近壁區(qū)(2z/d<-0.8或2z/d>0.8)相對(duì)誤差絕對(duì)值最大值分別為9.7%,6.8%,4.4%,3.1%,2.5%;其余區(qū)域(-0.8<2z/d<0.8)相對(duì)誤差絕對(duì)值最大值則分別為7.3%,4.5%,2.2%,1.8%,1.6%.根據(jù)數(shù)值模擬結(jié)果,徑向網(wǎng)格數(shù)目采用 16個(gè)以上為宜.

圖4 不同徑向網(wǎng)格數(shù)目下圓管中心截面網(wǎng)格劃分Fig.4 Cells at the cross section of the circular pipe generated with different diametric cell numbers

2.2.2 邊界層層數(shù)的影響

保持軸向網(wǎng)格劃分和徑向網(wǎng)格數(shù)目N=12不變,考察邊界層層數(shù)對(duì)于數(shù)值模擬結(jié)果的影響.圖6給出了邊界層層數(shù)n分別為0,1,2,3時(shí)圓管中心截面的網(wǎng)格情況.在圖6對(duì)應(yīng)的邊界層層數(shù)情況下進(jìn)行數(shù)值模擬,得到的速度分布如圖7(a)所示(見(jiàn)下頁(yè)),對(duì)圖7(a)中-0.4≤2z/d≤0.4對(duì)應(yīng)的曲線進(jìn)行局部放大,見(jiàn)圖7(b)(見(jiàn)下頁(yè)).對(duì)數(shù)值計(jì)算誤差進(jìn)行分析可知,雖然邊界層層數(shù)越多,數(shù)值模擬所得到的速度場(chǎng)越接近解析解.然而邊界層層數(shù)主要影響近壁區(qū),而對(duì)其余區(qū)域的影響很弱.例如,n=0和n=1時(shí),近壁區(qū)相對(duì)誤差絕對(duì)值高達(dá)18.9%,而n=2和n=3時(shí),近壁區(qū)相對(duì)誤差絕對(duì)值分別為7.0%和4.4%.因此,為保證較高的計(jì)算精度,應(yīng)在邊界層層數(shù)不低于3的情況下進(jìn)行數(shù)值模擬.

圖5 不同徑向網(wǎng)格數(shù)目下的速度分布Fig.5 Velocity profiles in cases with different diametric cell numbers

圖6 不同邊界層層數(shù)下圓管中心截面網(wǎng)格劃分Fig.6 Cells at the cross section of the circular pipe generated with different boundary layers

圖7 不同邊界層層數(shù)下的速度分布Fig.7 Velocity profiles in cases with different boundary layers

3 結(jié)論

利用開(kāi)源CFD軟件OpenFOAM對(duì)周期性管內(nèi)單相流體的泊肅葉流動(dòng)開(kāi)展數(shù)值模擬,通過(guò)將OpenFOAM中3種周期性邊界處理方法下的數(shù)值模擬結(jié)果與泊肅葉流動(dòng)速度場(chǎng)和壓力場(chǎng)的解析解進(jìn)行對(duì)比,分析了3種周期性邊界處理方法的適用性.在此基礎(chǔ)上,考察了徑向網(wǎng)格數(shù)目和邊界層層數(shù)對(duì)速度場(chǎng)的影響.通過(guò)本文研究,得到以下結(jié)論:

a. 3種邊界條件設(shè)置下,數(shù)值模擬得到的速度場(chǎng)均與解析解吻合,其中mapped和cyclicAMI邊界條件下的數(shù)值解非常接近,且與解析解吻合更好.

b. 3種邊界條件設(shè)置下,數(shù)值模擬得到的壓力場(chǎng)差別很大.采用mapped邊界時(shí),壓力與解析解十分接近;采用cyclic fan邊界時(shí),進(jìn)、出口存在很大的壓力梯度;而采用cyclicAMI邊界時(shí),壓力則恒為0.

c. 在mapped邊界條件下,隨著徑向網(wǎng)格數(shù)目的增加,速度的計(jì)算精度先迅速提高而后趨于穩(wěn)定;邊界層層數(shù)的增加能夠有效提高近壁區(qū)速度的計(jì)算精度;數(shù)值模擬時(shí)徑向網(wǎng)格數(shù)目采用 16個(gè)以上為宜,邊界層層數(shù)不宜少于3層.

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(編輯:丁紅藝)

Implementation of Boundary Conditions for Periodic Pipe Flow Based on OpenFOAM

ZHANG Peng1,2, FAN Fengxian1,2

(1.SchoolofEnergyandPowerEngineering,UniversityofShanghaiforScienceandTechnology,Shanghai200093,China;2.ShanghaiKeyLaboratoryofMultiphaseFlowandHeatTransferinPowerEngineering,UniversityofShanghaiforScienceandTechnology,Shanghai200093,China)

In order to explore the applicability of different implementation methods for periodic boundary conditions involved in the open-source computational fluid dynamics software,by the name of OpenFOAM,numerical simulations on the Poiseuille flow in a circular pipe were performed,using mapped,cyclicAMI and cyclic fan boundary conditions,respectively.The numerically obtained velocity and pressure profiles were analyzed.The results show that the velocity profiles obtained under the three boundary conditions are in agreement with the analytical solution.Specifically,the results under the mapped and cyclicAMI boundary conditions,which are very close to each other,agree better with the analytical solution.The pressure profile obtained under the mapped boundary condition is in good agreement with the analytical solution,whereas there are large pressure gradients at the inlet and outlet of the simulation domain when the cyclic fan boundary condition is adopted.Moreover,the pressure along the pipe axis is always zero under the cyclicAMI boundary condition.On this basis,the sensitivity of the numerical simulation results to the computational cell was examined.It is found that as the cell number along the pipe diameter increases,the computational accuracy of the velocity increases rapidly and then becomes stable.The increase of the number of boundary layers can effectively improve the computational accuracy of the velocity near the wall.Furthermore,more than 16 cells along the diameter and not less than 3 boundary layers are necessary for numerical simulations.

pipeflow;periodicboundarycondition;Poiseuilleflow;computationalfluiddynamics(CFD);OpenFOAM

1007-6735(2017)03-0217-06

10.13255/j.cnki.jusst.2017.03.003

2016-07-12

國(guó)家自然科學(xué)基金資助項(xiàng)目(51206113,51176128,51576130);上海市科委科研計(jì)劃(13DZ2260900)

張 鵬(1992-),男,碩士研究生.研究方向:多相流數(shù)值模擬.E-mail:zippousst@126.com

凡鳳仙(1982-),女,副教授.研究方向:多相流動(dòng)與傳熱.E-mail:fanfengxian@hotmail.com

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