包振忠,秦國(guó)良,耿艷輝,和文強(qiáng)

(西安交通大學(xué)能源與動(dòng)力工程學(xué)院,710049,西安)

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譜元法應(yīng)用于渦聲傳播問(wèn)題的研究

包振忠,秦國(guó)良,耿艷輝,和文強(qiáng)

(西安交通大學(xué)能源與動(dòng)力工程學(xué)院,710049,西安)

為了滿足計(jì)算氣動(dòng)聲學(xué)對(duì)低色散、低耗散高精度數(shù)值離散格式的需求,將高精度譜元法結(jié)合聲比擬理論應(yīng)用于求解氣動(dòng)聲學(xué)問(wèn)題。以偽聲壓的時(shí)間二階導(dǎo)數(shù)作為非齊次波動(dòng)方程的聲源項(xiàng),空間離散采用譜元法,時(shí)間離散應(yīng)用隱式Newmark法,并在外邊界采用C-E-M吸收邊界條件,求解了由兩個(gè)相距為2r0的等環(huán)量點(diǎn)渦組成的同向旋轉(zhuǎn)渦對(duì)的發(fā)聲問(wèn)題。旋轉(zhuǎn)渦對(duì)的不可壓縮流場(chǎng)通過(guò)復(fù)位勢(shì)理論獲得,聲源由流場(chǎng)量計(jì)算得來(lái),并將數(shù)值結(jié)果與應(yīng)用多級(jí)匹配展開(kāi)法得到的解析解進(jìn)行比較,可得數(shù)值解與分析解吻合較好。研究結(jié)果表明:應(yīng)用高精度譜元法進(jìn)行空間離散時(shí),每波長(zhǎng)的網(wǎng)格數(shù)為11時(shí)可達(dá)到很高的精度;網(wǎng)格數(shù)一定的情況下,時(shí)間步長(zhǎng)越小得到的數(shù)值解與分析解之間的誤差就越小;另外,證明了將偽聲壓對(duì)時(shí)間的二階導(dǎo)數(shù)作為聲源項(xiàng),能夠高精度求解不可壓縮流動(dòng)引起的氣動(dòng)聲學(xué)問(wèn)題。

計(jì)算氣動(dòng)聲學(xué);譜元法;聲比擬理論;吸收邊界條件;有限元;同向旋轉(zhuǎn)渦對(duì)

隨著各種交通工具速度的快速提升,輻射的氣動(dòng)噪聲也急劇增大。氣動(dòng)噪聲不僅引起環(huán)境污染甚至還會(huì)造成結(jié)構(gòu)的疲勞和破壞,會(huì)對(duì)人的生理和心理構(gòu)成嚴(yán)重影響。隨著計(jì)算流體力學(xué)(CFD)的飛速發(fā)展,數(shù)值模擬被廣泛應(yīng)用于工程實(shí)踐,但傳統(tǒng)的CFD方法無(wú)法滿足氣動(dòng)聲學(xué)問(wèn)題的研究,計(jì)算氣動(dòng)聲學(xué)(CAA)對(duì)計(jì)算格式提出了比較苛刻的要求[1-2]。由于聲波在傳播過(guò)程中衰減很慢且聲學(xué)量相對(duì)于流動(dòng)量在量級(jí)上要小得多,這就要求數(shù)值方法本身必須是高精度數(shù)值方法,其數(shù)值噪聲要盡量小,避免聲波脈動(dòng)量被數(shù)值誤差所掩蓋,即該高精度方法具有極低的數(shù)值色散和耗散以確保能夠高精度模擬聲波的長(zhǎng)距離傳播。另外,傳統(tǒng)的CFD所求解的區(qū)域較小,數(shù)值模擬總在有限的區(qū)域內(nèi)進(jìn)行,聲波在傳播過(guò)程中具有低耗散特性相較于流場(chǎng)的數(shù)值模擬,聲場(chǎng)的數(shù)值模擬需要更多的計(jì)算花費(fèi)。為減少計(jì)算量,聲波在計(jì)算域開(kāi)放邊界必須能無(wú)反射出入,采用適當(dāng)?shù)奈者吔鐥l件[3]才能達(dá)到這一效果。

湍流產(chǎn)生的噪聲大都起源于流動(dòng)中的渦結(jié)構(gòu),因此在計(jì)算氣動(dòng)聲學(xué)中常使用渦源作為人工聲源來(lái)分析噪聲產(chǎn)生的機(jī)理[4]。雙渦流動(dòng)結(jié)構(gòu)是復(fù)雜流場(chǎng)中的常見(jiàn)現(xiàn)象,大型民機(jī)增升裝置后緣襟翼側(cè)緣所卷起的旋渦即為典型的雙渦流動(dòng)[5]。Manfred等用有限元法求解了不可壓縮N-S方程和非齊次波動(dòng)方程,結(jié)合聲比擬理論將流場(chǎng)信息插值到聲場(chǎng)網(wǎng)格中,得到了雙渦流動(dòng)發(fā)聲的數(shù)值解[6-7]。Zhu等采用高階有限差分格式結(jié)合黏聲分離法對(duì)雙渦流動(dòng)發(fā)聲問(wèn)題進(jìn)行了數(shù)值模擬[8-9]。Farshchi等采用線性和非線性黏聲分離技術(shù),并使用高階色散保持格式進(jìn)行數(shù)值離散,求解了雙渦流動(dòng)發(fā)聲問(wèn)題,結(jié)果表明,線性計(jì)算在保持計(jì)算精度的條件下大幅降低了計(jì)算時(shí)間[10]。Ekaterinaris發(fā)展了黏聲分離技術(shù),并用高階迎風(fēng)格式對(duì)控制方程進(jìn)行了離散求解,驗(yàn)證了數(shù)值格式具有較好的數(shù)值精度[11-12]。

本文基于Chebyshev譜元法對(duì)控制方程進(jìn)行空間離散,時(shí)間離散應(yīng)用隱式Newmark法并結(jié)合聲比擬理論從流場(chǎng)中提取聲源,對(duì)同向旋轉(zhuǎn)渦對(duì)的發(fā)聲問(wèn)題進(jìn)行了數(shù)值模擬,并研究了數(shù)值精度的影響因素[13-14],為更好地求解計(jì)算氣動(dòng)聲學(xué)問(wèn)題提供了一定的理論依據(jù)。

1 控制方程

自1952年Lighthill提出聲比擬理論以來(lái),Lighthill方程

(1)

在氣動(dòng)聲學(xué)問(wèn)題中得到廣泛應(yīng)用,其中Tij為L(zhǎng)ighthill應(yīng)力張量,在絕熱無(wú)黏的情況下表達(dá)式為

Tij=ρuiuj

(2)

p′=pi+pa

(3)

由不可壓縮連續(xù)性方程和動(dòng)量方程可得偽聲壓,滿足條件[16]

(4)

將式(4)代入式(1)可得

(5)

式(5)提供了一種只用偽聲壓的二階時(shí)間導(dǎo)數(shù)即可作為聲源來(lái)計(jì)算聲壓的方法,該方法等價(jià)于文獻(xiàn)[9]中所提的黏聲分離法,初始條件為

(6)

2 數(shù)值算法

對(duì)邊界條件的處理是聲波傳播的一個(gè)關(guān)鍵問(wèn)題,由于計(jì)算機(jī)資源和區(qū)域的限制,所以在數(shù)值模擬波動(dòng)問(wèn)題時(shí)需要對(duì)求解區(qū)域進(jìn)行邊界截?cái)?。由于截?cái)嘣谶吔缣帟?huì)引起波的反射,影響數(shù)值求解區(qū)域,為了減小波的反射,本文采用C-E-M吸收邊界條件[17]

(7)

結(jié)合吸收邊界條件,式(5)相應(yīng)的Galerkin變分形式為

?pa,w∈H1(Ω),t∈[0,T]

(8)

式中:ΓI為邊界;w為檢驗(yàn)函數(shù)。

小學(xué)語(yǔ)文閱讀文章的篇幅比較短，大多是幾段，每一段也由幾句話組成。從文章整體上看，完整的文章由段落組成，段落則由句子組成，句子由詞語(yǔ)或者文字組成。因此，培養(yǎng)學(xué)生的綜合閱讀能力，需要關(guān)注詞句基礎(chǔ)訓(xùn)練。

2.1 空間離散

(9)

式中:M、C、K、S分別為質(zhì)量矩陣、阻尼矩陣、剛度矩陣和源項(xiàng),具體離散過(guò)程和表達(dá)式可參見(jiàn)文獻(xiàn)[18]。

2.2 時(shí)間離散

式(9)在時(shí)域上的求解一般有隱式和顯式兩種逐步積分途徑,顯式逐步積分方法一般是條件穩(wěn)定的,隱式方法是絕對(duì)穩(wěn)定的?？紤]到算法的穩(wěn)定性,本文選擇隱式Newmark方法,遞推公式為

(10)

3 數(shù)值算例

3.1 數(shù)值模型

圖1 同向旋轉(zhuǎn)渦對(duì)配置

在不可壓縮、無(wú)黏的條件下,渦對(duì)產(chǎn)生的流場(chǎng)可用復(fù)位勢(shì)函數(shù)來(lái)表示,即

(11)

式中:z=x+iy=reiθ;b=r0eiωt。

由式(11)可得計(jì)算聲源時(shí)需要的流場(chǎng)量

(12)

聲壓波動(dòng)的分析解[18]可通過(guò)匹配漸近展開(kāi)法獲得,即

(13)

式中:k=2ω/c0;J2(kr)、Y2(kr)分別為第1、2類二階貝塞爾函數(shù);ψ=2(ωt-θ)。

計(jì)算區(qū)域選為400 m×400 m的方形區(qū)域,將源項(xiàng)區(qū)域限定在中間范圍的200 m×200 m區(qū)域?？偣?jié)點(diǎn)數(shù)Nd=(NmNx+1)(NnNy+1),旋轉(zhuǎn)半徑r0=1 m,環(huán)量Γ=1.005 31 m2/s,聲速c0=1 m/s,波長(zhǎng)λ≈39 m,Mr=0.079 6。本文在模擬中取均勻單元,單元內(nèi)的插值節(jié)點(diǎn)數(shù)為2,在全部區(qū)域的網(wǎng)格都為均勻網(wǎng)格。

3.2 數(shù)值結(jié)果

圖2給出了式(5)中的源項(xiàng)在t=100 s時(shí)的云圖和三維圖,其中時(shí)間步長(zhǎng)Δt=0.1 s,網(wǎng)格間距Δx=3.64 m,此時(shí)每波長(zhǎng)的網(wǎng)格數(shù)約為11。由圖2可看出四極子源項(xiàng)的特征,且源項(xiàng)區(qū)域足夠大,使得截?cái)嗵幵错?xiàng)的幅值遠(yuǎn)小于最大值,僅為其0.25%;原點(diǎn)處具有奇異性,源項(xiàng)梯度特別大。為了消除這一奇異性,Manfred等采用Scully渦核模型[19],Ekaterinaris等采取了對(duì)原點(diǎn)附近的源項(xiàng)進(jìn)行截?cái)嗟霓k法[12]。本文采用文獻(xiàn)[12]的方法,即r/r0≤1.5時(shí)不存在源項(xiàng)。

(a)源項(xiàng)云圖

(b)源項(xiàng)三維圖圖2 源項(xiàng)分布圖

同向旋轉(zhuǎn)渦對(duì)在240 s時(shí)產(chǎn)生的聲壓場(chǎng)如圖3所示。由圖3可看出,產(chǎn)生的聲場(chǎng)為旋轉(zhuǎn)的雙螺旋形結(jié)構(gòu),吸收邊界條件的應(yīng)用可吸收外向聲波,不影響聲場(chǎng)中解的精度,聲波傳遞出去基本沒(méi)有引起反射。

(a)同向旋轉(zhuǎn)渦對(duì)聲場(chǎng)云圖

(b)同向旋轉(zhuǎn)渦對(duì)聲場(chǎng)三維圖圖3 240 s時(shí)同向旋轉(zhuǎn)渦對(duì)聲場(chǎng)

為了比較不同網(wǎng)格間距對(duì)數(shù)值結(jié)果的影響,分別計(jì)算了Δx=9.52,6.45,3.92,2.82 m、Δt=2.0 s時(shí)的聲場(chǎng),聲壓沿x正半軸方向的變化情況如圖4所示。隨著傳播距離的增加,聲壓總體呈現(xiàn)衰減的趨勢(shì)。Δx=9.52 m時(shí),由于其網(wǎng)格間距太大,與其他節(jié)點(diǎn)數(shù)時(shí)的解有明顯偏差,其他聲壓型線整體基本一致,幅值和相位都沒(méi)有太大偏差。

圖4 不同網(wǎng)格間距時(shí)x方向的聲壓型線

Δx=3.64 m、不同時(shí)間步長(zhǎng)時(shí)聲壓沿x正半軸方向的變化曲線如圖5所示。由圖5可看出,在靠近原點(diǎn)附近數(shù)值解與分析解差別較大,這是由消除原點(diǎn)奇異性對(duì)源項(xiàng)的截?cái)嗨鸬?。另?隨著時(shí)間步長(zhǎng)的減小,聲壓型線與分析解之間的差別也越來(lái)越小,但是幅值仍稍微低于分析解,可能和譜元算法在計(jì)算離散波動(dòng)方程時(shí)的耗散有關(guān)。色散誤差除了在Δt=2.0 s時(shí)較大,其余時(shí)間步長(zhǎng)均很小,與分析解吻合較好。具體譜元方法求解波動(dòng)方程及數(shù)值精度的相關(guān)因素分析可參照文獻(xiàn)[20]。

圖5 不同時(shí)間步長(zhǎng)時(shí)x方向的聲壓型線

(0,58.2)點(diǎn)處聲壓數(shù)值解和分析解隨求解時(shí)間的變化如圖6所示,原點(diǎn)附近的聲波到達(dá)該點(diǎn)后,聲壓數(shù)值解與分析解匹配很好,只有在波峰與波谷處略微小于分析解。這是由為消除原點(diǎn)奇異性對(duì)源項(xiàng)進(jìn)行截?cái)?而導(dǎo)致總體源項(xiàng)偏小所引起的。240 s處聲壓數(shù)值解與分析解的比較如圖7所示。由圖7可看出,數(shù)值解與分析解的相位、幅值都比較吻合,在分界線處沒(méi)有明顯的數(shù)值跳躍。

圖6 (0,58.2)點(diǎn)處聲壓隨時(shí)間的變化

圖7 240 s時(shí)數(shù)值解與分析解的比較

4 結(jié) 論

本文實(shí)現(xiàn)了將高精度譜元法結(jié)合聲比擬理論應(yīng)用于求解氣動(dòng)聲學(xué)的問(wèn)題,并對(duì)由兩個(gè)點(diǎn)渦形成的同向旋轉(zhuǎn)渦對(duì)聲場(chǎng)進(jìn)行了數(shù)值模擬。為了分析影響數(shù)值精度的原因,計(jì)算了不同網(wǎng)格、不同時(shí)間步長(zhǎng)情況下的聲場(chǎng),并將計(jì)算結(jié)果與用多極匹配展開(kāi)法得來(lái)的分析解進(jìn)行了對(duì)比。結(jié)果表明:應(yīng)用譜元法進(jìn)行空間離散時(shí),采用極少的節(jié)點(diǎn)數(shù)就可達(dá)到很高的精度;節(jié)點(diǎn)數(shù)一定的情況下,時(shí)間步長(zhǎng)越小得到的數(shù)值解與分析解之間的誤差越小,但花費(fèi)的計(jì)算資源越多。用數(shù)值算例驗(yàn)證了將偽聲壓對(duì)時(shí)間的二階導(dǎo)數(shù)作為聲源項(xiàng),可用來(lái)求解不可壓縮流動(dòng)引起的氣動(dòng)聲學(xué)問(wèn)題。

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(編輯 趙煒 荊樹(shù)蓉)

Investigation of Vortex Sound Propagation Using Spectral Element Method

BAO Zhenzhong,QIN Guoliang,GENG Yanhui,HE Wenqiang

(School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China)

In order to meet the requirements of low dispersive and low dissipative numerical discretization schemes in computational aero-acoustics, the spectral element method combined with Lighthill’s acoustic analogy theory for the simulation of aeroacoustic problems is investigated. Taking the second temporal derivative of pseudopressure as the source of the inhomogeneous wave equation, and space discretization using spectral element method time discretization using implicit Newmark method, under the C-E-M absorbing boundary condition the acoustic field generated by a co-rotating spinning vortex pair is solved. This co-rotating vortex pair consists of two point vortices separated by a fixed distance of 2r0with a circulation intensityΓ. The incompressible flow field is obtained using the complex potential theory, and the acoustic source terms are computed using these hydrodynamic quantities. The acoustic pressure results are evaluated by comparing them with the analytical solution obtained from the matched asymptotic expansion (MAE) method The numerical solutions are in good agreement with the analytical solutions. The results show that the spectral element method can obtain high accuracy using only eleven grids in one wavelength. In the case of the same number of grids, the smaller the time step, the smaller the error between numerical solution and analytical solution. Finally, it is proved that the aero-acoustic problems induced by incompressible flows can be solved with high accuracy by using the second temporal derivative of the pseudopressure as the acoustic source.

computational aero-acoustics; spectral element method; acoustic analogy theory; absorbing boundary condition; finite element method; co-rotating spinning vortex pair

2016-05-20。 作者簡(jiǎn)介:包振忠(1992—),男,博士生;秦國(guó)良(通信作者),男,教授,博士生導(dǎo)師。 基金項(xiàng)目:國(guó)家重點(diǎn)基礎(chǔ)研究發(fā)展計(jì)劃資助項(xiàng)目(2012CB026004)。

時(shí)間:2016-09-14

10.7652/xjtuxb201611017

O121.8;G558

A

0253-987X(2016)11-0110-06

網(wǎng)絡(luò)出版地址:http:∥www.cnki.net/kcms/detail/61.1069.T.20160914.1803.002.html