呂宏強(qiáng), 張 濤, 孫 強(qiáng), 陳建偉, 秦望龍(南京航空航天大學(xué) 航空宇航學(xué)院, 江蘇 南京 210016)

間斷伽遼金方法在可壓縮流數(shù)值模擬中的應(yīng)用研究綜述

呂宏強(qiáng)*, 張 濤, 孫 強(qiáng), 陳建偉, 秦望龍
(南京航空航天大學(xué) 航空宇航學(xué)院, 江蘇 南京 210016)

本文對(duì)近三十年來(lái)，國(guó)內(nèi)外對(duì)于高精度數(shù)值方法研究中的熱點(diǎn)——間斷伽遼金方法在可壓縮流數(shù)值模擬方面的應(yīng)用研究進(jìn)行了綜述。首先對(duì)間斷伽遼金方法的基本概念和特點(diǎn)作了簡(jiǎn)單介紹，然后對(duì)應(yīng)用該方法解決雙曲型及橢圓型問(wèn)題的發(fā)展歷程進(jìn)行了回顧，并重點(diǎn)梳理了其在計(jì)算流體力學(xué)領(lǐng)域可壓縮流數(shù)值模擬方面的應(yīng)用發(fā)展以及研究現(xiàn)狀，之后對(duì)該方法在對(duì)應(yīng)的網(wǎng)格技術(shù)、激波捕捉方法、湍流流動(dòng)模擬以及計(jì)算量需求方面目前仍然存在的研究難點(diǎn)和可能的發(fā)展趨勢(shì)做出了總結(jié)和分析。最后給出了間斷伽遼金方法在可壓縮流數(shù)值模擬中的若干應(yīng)用實(shí)例。

間斷伽遼金方法；高精度方法；計(jì)算流體力學(xué)；可壓縮流；彎曲網(wǎng)格

0 引 言

近些年來(lái)，高精度數(shù)值方法的研究成為計(jì)算流體力學(xué)(Computational Fluid Dynamics, CFD) 領(lǐng)域研究中的前沿?zé)狳c(diǎn)問(wèn)題之一。我們通常所說(shuō)的高精度方法是指空間精度為三階或三階以上的高精度數(shù)值格式，相比于傳統(tǒng)的空間二階精度的有限體積格式，高精度方法具有空間精度高，數(shù)值分辨率高，數(shù)值耗散小的優(yōu)點(diǎn)。

目前計(jì)算流體力學(xué)領(lǐng)域高精度方法主要可以分為三大類(lèi)：高精度有限差分(Finite Difference, FD)方法[1-3]，高精度有限體積(Finite Volume, FV)方法[4-9]和高精度有限元類(lèi) (Finite Element, FE)方法[10-17]。高精度有限差分法，通常為在結(jié)構(gòu)化網(wǎng)格下一種高效而易于實(shí)施的高精度格式，由于其計(jì)算量小，且易于達(dá)到較高數(shù)值精度的特點(diǎn)，常用于簡(jiǎn)單幾何區(qū)域的復(fù)雜流動(dòng)直接數(shù)值模擬。這一類(lèi)型的高精度格式一般只能用于結(jié)構(gòu)化的笛卡爾網(wǎng)格，對(duì)于處理復(fù)雜幾何區(qū)域則會(huì)帶來(lái)一定的困難，但近年來(lái)鄧小剛等[18]做了大量的工作將該方法推廣到復(fù)雜幾何網(wǎng)格上；高精度有限體積法是通過(guò)選取目標(biāo)單元及其周?chē)南噜弳卧鳛槟０?，?gòu)造滿足一定條件的重構(gòu)高階多項(xiàng)式來(lái)達(dá)到高階精度的目的，比較有代表性的高精度有限體積格式有：有限體積型加權(quán)本質(zhì)無(wú)振蕩格式[4-5]、高精度k-exact有限體積格式[6]和近年來(lái)的緊致高階精度有限體積法[7-9]。這類(lèi)方法理論上可以處理任意網(wǎng)格和較為復(fù)雜的幾何區(qū)域，能夠保證格式的守恒性且具有良好的數(shù)值穩(wěn)定性。然而傳統(tǒng)的高精度有限體積法的不足之處在于其模板的非緊致性，即模板不僅包含目標(biāo)單元及其有公共邊的鄰居單元，通常還需要包含其鄰居單元的相鄰單元。因此，該方法在處理邊界和三維問(wèn)題方面則存在一定困難。緊致高階精度有限體積法克服了這一問(wèn)題，不過(guò)需要采用隱式方法求解重構(gòu)方程。第三種高精度方法以間斷伽遼金方法(Discontinuous Galerkin Method，DGM)為代表，通過(guò)提高相應(yīng)單元上的解函數(shù)多項(xiàng)式的次數(shù)，增加相應(yīng)單元上解函數(shù)的自由度(Degree of Freedom, DoF)來(lái)提高空間精度，這類(lèi)方法中其他有代表性的方法還包括：譜體積方法(Spectral Volume, SV)[11-12]，譜差分方法(Spectral Difference, SD)[13-15]，通量重構(gòu)方法(Flux Reconstruction, FR)[16]和修正過(guò)程重構(gòu)方法(Correction Procedure via Reconstruction, CPR)[17]。間斷伽遼金方法具有易于處理任意網(wǎng)格和復(fù)雜幾何區(qū)域的能力，且易于實(shí)現(xiàn)高階精度，格式構(gòu)造的緊致性導(dǎo)致這類(lèi)方法更適合做大規(guī)模的并行和自適應(yīng)計(jì)算。正因?yàn)檫@些優(yōu)勢(shì)，使得間斷伽遼金方法得到了計(jì)算流體力學(xué)、計(jì)算電磁學(xué)領(lǐng)域等諸多學(xué)者的廣泛關(guān)注，成為高精度格式研究的熱點(diǎn)之一。 本文將對(duì)間斷伽遼金方法的基本思想和理論發(fā)展歷程做出概述，并重點(diǎn)介紹在可壓縮流CFD領(lǐng)域國(guó)內(nèi)外對(duì)該方法的發(fā)展以及研究現(xiàn)狀。本文也將給出部分將該方法應(yīng)用于CFD領(lǐng)域的實(shí)例。最后，對(duì)間斷伽遼金方法中仍然存在的問(wèn)題和可能的發(fā)展空間做出分析和展望。

1 間斷伽遼金方法基本概念

高精度間斷伽遼金方法是有限元方法的一種，該方法的基本思路，是利用分段連續(xù)的多項(xiàng)式空間來(lái)近似表達(dá)偏微分方程組的解。以一個(gè)典型的一階雙曲守恒型方程系統(tǒng)為例：

將計(jì)算域劃分為互不重疊的單元集合Ωv=∪kΩvk，定義Φh,p是單元Ωvk上直到p階的多項(xiàng)式函數(shù)張成的函數(shù)空間，p≥0且為整數(shù)，設(shè)單元內(nèi)守恒變量的近似Uh∈Φh,p。在每個(gè)單元內(nèi)，對(duì)方程兩邊同時(shí)乘以測(cè)試函數(shù)φh，在計(jì)算域內(nèi)積分，進(jìn)行分部積分整理后，得到原方程(1)的弱解形式為：

式(2)中，對(duì)于邊界通量H=F(Uh)·n，與有限體積方法中的處理方法類(lèi)似，可以采用一個(gè)相容的數(shù)值通量來(lái)代替，而在邊界上對(duì)守恒變量不做連續(xù)性要求。如此便得到了p階間斷伽遼金方法的離散格式。

因此間斷伽遼金方法結(jié)合了有限元方法和有限體積方法的優(yōu)點(diǎn)：在單元內(nèi)部同傳統(tǒng)的連續(xù)有限元方法一樣，使用多項(xiàng)式逼近來(lái)獲得高階精度；在單元邊界上借鑒有限體積法，通過(guò)解決Riemann問(wèn)題來(lái)實(shí)現(xiàn)逆風(fēng)格式。實(shí)際上，當(dāng)p=0時(shí)，間斷伽遼金方法即退化為傳統(tǒng)的有限體積方法。

除高精度外，間斷伽遼金方法還有很多其他吸引人的特點(diǎn)：

1) 能夠保持單元平均值意義下守恒性，最重要的是具有良好的穩(wěn)定性和收斂性[19]；

2) 通過(guò)改變插值多項(xiàng)式的階數(shù)，很容易延拓到高階(p>2)，并且允許不同的單元采用不同的階數(shù)，即p-adaptivity[20]；

3) 能夠處理復(fù)雜的幾何外形和物理邊界條件，甚至可以直接處理含有懸掛點(diǎn)的網(wǎng)格[21-22]，因此極易實(shí)現(xiàn)網(wǎng)格自適應(yīng)，即h-adaptivity[20-23]。另外方法本身適用于各種類(lèi)型的網(wǎng)格；

4) 利用該方法進(jìn)行計(jì)算時(shí)，具有緊致性，單元只與相鄰單元有數(shù)據(jù)交換，很容易實(shí)現(xiàn)大規(guī)模并行計(jì)算且并行效率很高；

因此，高精度間斷伽遼金方法在計(jì)算流體力學(xué)領(lǐng)域得到廣泛的嘗試。

2 間斷伽遼金方法的歷史和現(xiàn)狀

2.1 國(guó)際發(fā)展與現(xiàn)狀

在應(yīng)用間斷伽遼金方法處理雙曲型方程的研究理論方面，1973年，Reed和Hill[10]在關(guān)于求解中子輸運(yùn)方程(時(shí)間無(wú)關(guān)的線性雙曲型方程)問(wèn)題的論文中首次在間斷伽遼金方法中引入了逆風(fēng)格式。1982年，Chavent和Salzano[24]首先在間斷伽遼金方法中引入Godunov數(shù)值通量求解了非線性雙曲型問(wèn)題，將間斷伽遼金方法從求解線性問(wèn)題延伸到求解非線性雙曲型問(wèn)題。在20世紀(jì)末，Cockburn和Shu等[25-30]對(duì)用間斷伽遼金方法和顯式時(shí)間積分方法求解非線性雙曲型問(wèn)題的研究取得了重大突破，成功地建立了著名的龍格-庫(kù)塔間斷伽遼金(RKDG)方法。最初始的RKDG有限元方法采用Shu和Osher[31]提出的顯式TVD二階龍格-庫(kù)塔格式，隨后他們將該方法在時(shí)間和空間上都發(fā)展到高階精度。同一時(shí)期，Allmaras[32]和Giles[33]采用二階精度間斷伽遼金方法求解了二維歐拉方程，他們將van Leer的moments限制器從一維線性波動(dòng)方程拓展到二維歐拉方程，在每個(gè)單元內(nèi)部計(jì)算單元平均和單元梯度平均值從而對(duì)單元變量進(jìn)行線性重構(gòu)。

而應(yīng)用間斷伽遼金方法求解橢圓型方程或求解NS方程的粘性項(xiàng)則存在相當(dāng)?shù)睦щy，諸多學(xué)者對(duì)此展開(kāi)了研究。Arnold[34]和Wheeler[35]于20世紀(jì)80年代在間斷伽遼金方法中引入了內(nèi)罰函數(shù)(interior penalty)，這種方法隨后被廣泛應(yīng)用于求解擴(kuò)散問(wèn)題。1999年，Oden和Babuska[36]等提出了一種求解擴(kuò)散問(wèn)題新格式，其優(yōu)點(diǎn)是沒(méi)有引入額外的過(guò)渡變量，但理論上該格式必須在2階以上才穩(wěn)定。21世紀(jì)初，Bassi和Rebay[37-40]提出了經(jīng)典的混合方法(mixed formulation)，將二階方程寫(xiě)成多個(gè)一階方程的形式，然后采用間斷伽遼金方法對(duì)一階系統(tǒng)進(jìn)行數(shù)值離散。Bassi和Rebay提出的第一種混合方法(BR1格式)計(jì)算模板較大，不緊致，且采用隱式方法計(jì)算時(shí)穩(wěn)定性會(huì)受到影響。為了克服這些缺點(diǎn)，他們又在BR1格式基礎(chǔ)上進(jìn)行了修改，得到了穩(wěn)定緊致的BR2格式。Cockburn和Shu[41-42]則在同一時(shí)期對(duì)混合方法思想進(jìn)行了一般化分析，得出了當(dāng)?shù)亻g斷伽遼金方法(Local Discontinuous Galerkin methods, LDG)。但是在多維度數(shù)值計(jì)算時(shí)，當(dāng)?shù)亻g斷伽遼金方法同樣存在模板大、不緊致的問(wèn)題，于是之后Persson和Peraire[43]對(duì)該方法進(jìn)行了修改，提出了緊致間斷伽遼金方法(Compact Discontinuous Galerkin methods, CDG)，在保留當(dāng)?shù)亻g斷伽遼金方法優(yōu)點(diǎn)的同時(shí)使得該格式緊致。Arnold等[44]也引入了內(nèi)罰函數(shù)和混合方法的統(tǒng)一分析框架，對(duì)各種格式進(jìn)行誤差分析。近年來(lái)，由Liu 和Yue[45-46]提出的一類(lèi)直接間斷伽遼金方法 (Direct DG, DDG) 逐漸受到了學(xué)者的關(guān)注，DDG 方法的導(dǎo)出過(guò)程不需要引入臨時(shí)變量將原有的二階偏微分方程分解為一階偏微分方程組，而是直接基于DG方法的弱形式構(gòu)造單元界面處的粘性數(shù)值通量。

在上述數(shù)值格式研究成果的基礎(chǔ)上，間斷伽遼金方法迅速在氣動(dòng)可壓縮流數(shù)值模擬方面引發(fā)了廣泛關(guān)注和嘗試。如前文所述，在計(jì)算流體力學(xué)的應(yīng)用中，間斷伽遼金方法結(jié)合了傳統(tǒng)有限元方法和有限體積方法的優(yōu)點(diǎn)，因此相比于傳統(tǒng)的有限元方法，間斷伽遼金方法由于容易實(shí)現(xiàn)逆風(fēng)格式，很容易對(duì)對(duì)流項(xiàng)主導(dǎo)的流動(dòng)問(wèn)題進(jìn)行離散求解；相比于有限體積方法，高階間斷伽遼金方法達(dá)到高精度所需要的計(jì)算量大大減少。得益于這些優(yōu)勢(shì)，間斷伽遼金方法成為了目前計(jì)算流體力學(xué)領(lǐng)域極具潛力的高精度方法之一。

在相應(yīng)的網(wǎng)格技術(shù)方面，Diosady等于2007年左右對(duì)間斷多重網(wǎng)格技術(shù)及線性預(yù)處理方法進(jìn)行了研究[47-48]。同時(shí)Lubon等則對(duì)適用于間斷伽遼金方法的網(wǎng)格進(jìn)行了研究，發(fā)展了物面彎曲技術(shù)并采用間斷伽遼金方法進(jìn)行了RANS及DES數(shù)值模擬[49-51]。 2007年，F(xiàn)idkowski在其博士論文中發(fā)展了網(wǎng)格切割技術(shù)，并采用間斷伽遼金方法求解了二維Navier-Stokes方程[52]。隨后其又在間斷伽遼金方法上發(fā)展了網(wǎng)格自適應(yīng)技術(shù)及網(wǎng)格變形方法[53-55]。Oliver研究了二維自適應(yīng)高階間斷伽遼金方法，結(jié)合Spalart-Allmaras一方程湍流模型對(duì)湍流流動(dòng)進(jìn)行了數(shù)值仿真[56-58]。Li Wang在博士論文中對(duì)二維間斷伽遼金方法的氣動(dòng)優(yōu)化問(wèn)題、網(wǎng)格自適應(yīng)問(wèn)題及高階時(shí)間積分方法進(jìn)行了研究[59-62]，近年來(lái)其采用間斷伽遼金方法求解了三維RANS方程及麥克斯韋方程并得到了較好的數(shù)值結(jié)果，同時(shí)其對(duì)SUPG方法也在進(jìn)行研究[63-65]。Burgess采用自適應(yīng)間斷伽遼金方法對(duì)二維湍流流動(dòng)進(jìn)行了數(shù)值模擬，并對(duì)湍流方程加速求解技術(shù)進(jìn)行了研究[66-69]。2011到2014年，Bassi等牽頭一項(xiàng)歐盟高精度方法工業(yè)化應(yīng)用項(xiàng)目(IDIHOM)[70]，吸引了來(lái)自歐洲各地的學(xué)者，繼續(xù)推進(jìn)高階網(wǎng)格處理技術(shù)以及間斷伽遼金方法數(shù)值求解技術(shù)。

在間斷伽遼金方法的可壓縮流求解技術(shù)方面，Bassi和Rebay等于1997年采用間斷伽遼金方法求解了Euler方程和Navier-Stokes方程[37-39]之后，又對(duì)湍流模型的求解進(jìn)行了研究，于2005年采用k-ω兩方程模型求解了雷諾平均Navier-Stokes (Reynolds-Averaged Navier-Stokes，RANS)方程[40]。隨后該團(tuán)隊(duì)又將間斷伽遼金方法拓展到DES及不可壓縮流動(dòng)領(lǐng)域[71-72]。Landmann在其博士論文中發(fā)展了并行高階間斷伽遼金方法并求解了二維Navier-Stokes方程及RANS方程[73-74]。2010年左右，Persson等研究了動(dòng)網(wǎng)格技術(shù)，將間斷伽遼金方法應(yīng)用于兩相流及撲翼飛行的數(shù)值仿真[75-77]。Wang等研究了CPR-DG有限元方法并求解了二維和三維Navier-Stokes方程，最近又將其拓展到RANS-LES混合方法的求解[78-81]。Hartmann等[82-86]研究了SST兩方程湍流模型和激波捕捉技術(shù)在間斷伽遼金方法上的應(yīng)用，采用間斷伽遼金方法對(duì)二維和三維復(fù)雜外形的流動(dòng)進(jìn)行了數(shù)值求解。 值得注意的是，很多學(xué)者提出并發(fā)展了一類(lèi)重構(gòu)型和混合型的間斷伽遼金方法。這一類(lèi)方法的基本思想是在DG方法的框架下，借鑒DG方法的優(yōu)勢(shì)，通過(guò)重構(gòu)方式在原有DG解函數(shù)自由度的基礎(chǔ)上，重構(gòu)高階自由度從而達(dá)到高階精度的目的。van Leer 等提出并發(fā)展了一類(lèi)重構(gòu)DG方法(Recovery DG, RDG)[87]。Dumbser等將DG方法和高精度有限體積方法統(tǒng)一到同一PnPm框架下，提出了一類(lèi)PnPm方法[88]。之后，Luo[89-95]等在成熟的有限體積求解器基礎(chǔ)上發(fā)展了另一類(lèi)重構(gòu)間斷伽遼金方法(Reconstructed Discontinuous Galerkin Method)，并對(duì)隱式LES方法進(jìn)行了初步探究。

2.2 國(guó)內(nèi)發(fā)展與現(xiàn)狀

國(guó)內(nèi)間斷伽遼金方法研究起步相對(duì)較晚，但在近十幾年得到了越來(lái)越多的科研工作者在應(yīng)用研究方面的關(guān)注。目前廈門(mén)大學(xué)、南京航空航天大學(xué)、上海理工大學(xué)、北京航空航天大學(xué)、西北工業(yè)大學(xué)、中國(guó)空氣動(dòng)力研究與發(fā)展中心、北京應(yīng)用物理與計(jì)算數(shù)學(xué)研究所等學(xué)校和研究機(jī)構(gòu)都對(duì)間斷伽遼金方法開(kāi)展了相關(guān)研究。2005年，蔚喜軍和張鐵[19]采用RKDG有限元方法求解了二維可壓縮Euler方程，并與差分方法的計(jì)算結(jié)果進(jìn)行了比較，證明了間斷伽遼金法的高精度特性和在處理復(fù)雜邊界問(wèn)題上的優(yōu)勢(shì)。隨后趙國(guó)忠等將該方法拓展到拉格朗日坐標(biāo)系下并求解了二維氣動(dòng)方程組[96-98]。近年來(lái)，邱建賢、朱君等將著名的加權(quán)本質(zhì)無(wú)振蕩(Weighted Essentially Non-Oscillatory, WENO)格式作為限制器應(yīng)用于間斷伽遼金方法，并求解了兩相流等流動(dòng)問(wèn)題[99-104]。2006年左右，呂宏強(qiáng)等采用間斷伽遼金方法求解了面接觸彈性流體動(dòng)力潤(rùn)滑問(wèn)題，最高階數(shù)達(dá)到13階，并成功應(yīng)用了hp-adaptivity技術(shù)，用極少的計(jì)算代價(jià)得到了高精度的數(shù)值結(jié)果[105-106]。近年來(lái)，其團(tuán)隊(duì)研究了針對(duì)高階間斷伽遼金方法的高階彎曲網(wǎng)格生成方法、網(wǎng)格自適應(yīng)方法，以及基于間斷伽遼金方法和Moro[107]的修正S-A模型的RANS和DES求解方法，并將CFD領(lǐng)域的間斷伽遼金方法應(yīng)用于時(shí)域電磁場(chǎng)數(shù)值模擬領(lǐng)域[108-113]。2010年左右，陳二云等將間斷伽遼金方法應(yīng)用于彈尾超聲速噴流計(jì)算問(wèn)題及氣動(dòng)聲學(xué)問(wèn)題中[114-116]。同時(shí)，閻超、于劍、姜振華等對(duì)間斷伽遼金方法中的間斷捕捉和Navier-Stokes方程進(jìn)行了研究，并采用Baldwin-Lomax零方程湍流模型求解了二維流動(dòng)問(wèn)題[117-122]。郝海兵、李喜樂(lè)等也對(duì)高階間斷伽遼金方法的限制器及Baldwin-Lomax零方程湍流模型進(jìn)行了研究，求解了二維流動(dòng)問(wèn)題及三維Euler方程[123-124]。賀立新、張來(lái)平[125-130]等發(fā)展了DG/FV混合方法，以較少的內(nèi)存需求得到了與間斷伽遼金方法相同的求解精度。程劍和楊小權(quán)等[131-132]人成功地實(shí)施了一類(lèi)新型直接間斷伽遼金方法用于求解粘性可壓縮NS和RANS方程，取得了很好的計(jì)算效果。

總的來(lái)說(shuō)，目前國(guó)際上對(duì)于間斷伽遼金方法的理論分析、高效求解方法，以及相應(yīng)的網(wǎng)格技術(shù)(如物面高階擬合、自適應(yīng)方法、動(dòng)網(wǎng)格、變形網(wǎng)格方法等)的研究都很廣泛。并且，很多當(dāng)前的研究工作都已發(fā)展到了三維湍流流動(dòng)模擬階段，許多正在向RANS-LES混合方法以及一些跨學(xué)科方向(如計(jì)算電磁學(xué)、多學(xué)科優(yōu)化、計(jì)算聲學(xué))邁進(jìn)。

3 間斷伽遼金方法的研究難點(diǎn)及挑戰(zhàn)

盡管間斷伽遼金方法已經(jīng)取得了相當(dāng)?shù)某晒?，但目前該方法的研究中仍存在著許多難點(diǎn)：

1) 高精度間斷伽遼金方法一般應(yīng)用于較為稀疏的網(wǎng)格，但是稀疏網(wǎng)格對(duì)于復(fù)雜幾何外形的表達(dá)精度會(huì)對(duì)模擬結(jié)果的精度產(chǎn)生很大的影響，而在湍流流動(dòng)模擬中，該方法對(duì)網(wǎng)格質(zhì)量更為敏感，高效通用的高階物面擬合以及高階網(wǎng)格處理技術(shù)是間斷伽遼金方法的研究難點(diǎn)之一；

2) 高階間斷伽遼金方法在處理間斷問(wèn)題時(shí)(例如激波)一般需使用限制器抑制數(shù)值振蕩，然而在實(shí)際應(yīng)用中，特別是定常流動(dòng)的求解計(jì)算過(guò)程中，使用限制器會(huì)影響殘差的收斂性，造成殘差收斂困難；

3) 采用高階間斷伽遼金方法數(shù)值求解帶湍流模型的RANS仍然是一個(gè)值得繼續(xù)深入探索的問(wèn)題，目前的相關(guān)研究顯示，湍流模型會(huì)導(dǎo)致迭代的穩(wěn)定性和魯棒性不強(qiáng)；

4) 高精度間斷伽遼金方法中的數(shù)值積分策略以及隱式時(shí)間格式的采用較同樣網(wǎng)格量下的傳統(tǒng)方法會(huì)產(chǎn)生較大的計(jì)算量及內(nèi)存存儲(chǔ)需求且通常需要求解大型的稀疏線性系統(tǒng)。

很多研究者對(duì)這些問(wèn)題做出了努力并且提出了一些較為有效的解決方案，但仍然存在很大的優(yōu)化和進(jìn)步空間。

3.1 適用于間斷伽遼金方法的網(wǎng)格技術(shù)

由于間斷伽遼金方法的高階精度特點(diǎn)，在使用該方法進(jìn)行計(jì)算時(shí)，如果采用過(guò)密的網(wǎng)格，將會(huì)帶來(lái)巨大的計(jì)算量和冗余的精度。然而采用稀疏網(wǎng)格進(jìn)行計(jì)算時(shí)，傳統(tǒng)的分段線性網(wǎng)格無(wú)法對(duì)彎曲的邊界進(jìn)行精準(zhǔn)的表達(dá)。Bassi[37]、Krivodonova[133]等在采用間斷伽遼金方法進(jìn)行流場(chǎng)的數(shù)值模擬時(shí)都發(fā)現(xiàn)，物面的表述精度會(huì)對(duì)模擬結(jié)果的精度產(chǎn)生非常大的影響，甚至?xí)鹩?jì)算無(wú)法收斂的問(wèn)題。于是利用高階多項(xiàng)式來(lái)精準(zhǔn)表達(dá)物面幾何信息的方法被引入，并且被證明對(duì)計(jì)算結(jié)果的改善十分有效[133]。Lubon等[51]則發(fā)展了適用于三維四面體網(wǎng)格單元的壁面彎曲修正方法。 但是彎曲網(wǎng)格的方法并不總是有效，例如存在厚度很小的邊界層網(wǎng)格時(shí)，彎曲的物面有可能與外層網(wǎng)格交叉，產(chǎn)生負(fù)體積。為了解決該問(wèn)題，Landman[74]博士發(fā)展了多層四邊形網(wǎng)格彎曲的方法避免網(wǎng)格單元出現(xiàn)負(fù)體積。Persson[134]等采用拉格朗日固體平衡方程使邊界彎曲信息向外單元傳播，通過(guò)全局變形達(dá)到平衡。Li Wang等則通過(guò)CAPRI方法得到三維物體的真實(shí)物面信息，然后根據(jù)線彈性理論求解各向同性線彈性方程得到全局變形后的網(wǎng)格點(diǎn)坐標(biāo)[64]。呂宏強(qiáng)等發(fā)展了基于求解線性彈性方程的網(wǎng)格高階彎曲方法[135]。秦望龍等[136]則采用了網(wǎng)格結(jié)塊的方法，將多個(gè)網(wǎng)格單元聚合成高階有限元單元，利用高階網(wǎng)格單元來(lái)對(duì)物面進(jìn)行擬合，并且不會(huì)出現(xiàn)交叉和重疊。

盡管上述方法都被證明十分有效，但這些方法的復(fù)雜程度都較高，通用性也有待檢驗(yàn)，更通用、更魯棒、更簡(jiǎn)單高效的高階網(wǎng)格處理方法，仍然處在發(fā)展之中。 另一方面，諸如傳統(tǒng)CFD方法中的動(dòng)網(wǎng)格、重疊網(wǎng)格、滑移網(wǎng)格及變形網(wǎng)格等網(wǎng)格技術(shù)由于前述的物面擬合的難點(diǎn)，這些方法在間斷伽遼金方法中的應(yīng)用存在額外的困難，相關(guān)研究也并不多。

3.2 間斷問(wèn)題的處理與激波捕捉

由于間斷伽遼金方法在單元交界面上對(duì)變量不做連續(xù)性強(qiáng)制要求，并且采用數(shù)值通量來(lái)包容間斷，所以實(shí)際上該方法可以不加處理地解決一些含有弱間斷解的問(wèn)題[20]。但是當(dāng)變量的強(qiáng)間斷落入單元之中時(shí)，插值多項(xiàng)式函數(shù)無(wú)法對(duì)其進(jìn)行準(zhǔn)確的高階表達(dá)，進(jìn)而引發(fā)數(shù)值振蕩甚至發(fā)散。為此，在間斷伽遼金方法中引入適當(dāng)?shù)南拗破魇怯斜匾摹?/p>

Shu等采用當(dāng)?shù)赝队暗姆椒ㄒ种茢?shù)值振蕩[26-28]。Luo和Xia[90-92]等采用WENO和HWENO格式結(jié)合重構(gòu)間斷伽遼金方法對(duì)間斷問(wèn)題進(jìn)行了數(shù)值求解。邱建賢、朱君[99-104]等也將WENO格式作為限制器應(yīng)用于間斷伽遼金方法。

另一類(lèi)人工黏性的思想也被引入。Persson[138]等采用級(jí)數(shù)展開(kāi)思想捕捉間斷區(qū)域，通過(guò)在流動(dòng)變量間斷的單元內(nèi)添加人工黏性求解了激波問(wèn)題。在其基礎(chǔ)上，Barter[139]等引入了基于偏微分方程(PDE-based)的人工黏性激波捕捉方法，Nguyen[140]等采用速度的散度來(lái)表征流場(chǎng)的壓縮特性，從而確定單元內(nèi)部所需添加的人工黏性的數(shù)值大小。此外，Bassi和Rebay[141]等在2009年提出了一種新的激波捕捉人工黏性添加方法。該方法通過(guò)檢測(cè)壓力梯度添加人工黏性，人工黏性函數(shù)則通過(guò)考慮單元間無(wú)黏數(shù)值通量跳躍和單元內(nèi)無(wú)黏數(shù)值通量法向分量得出。

然而上述諸多對(duì)數(shù)值振蕩進(jìn)行抑制的方法，都會(huì)或多或少地帶來(lái)更多的數(shù)值耗散和色散，引起數(shù)值精度的損失，抑制振蕩的效果也不一而足，在實(shí)際應(yīng)用中，特別是定常流動(dòng)的求解計(jì)算過(guò)程中，使用限制器還會(huì)破壞殘差的收斂性，造成殘差收斂困難。該方面的研究空間依然很大。

3.3 湍流流動(dòng)數(shù)值模擬

盡管湍流流動(dòng)的機(jī)理和理論仍處在研究當(dāng)中，但是經(jīng)過(guò)研究者的長(zhǎng)期摸索，已發(fā)展出了一系列被廣泛應(yīng)用的湍流模型，例如一方程Spalart-Allmaras湍流模型、兩方程k-ω湍流模型及兩方程SST模型。而在實(shí)際計(jì)算中，湍流方程的計(jì)算變量在網(wǎng)格分辨率不夠的單元內(nèi)會(huì)出現(xiàn)或短暫出現(xiàn)非連續(xù)項(xiàng)。如前文所述，對(duì)于間斷伽遼金方法，過(guò)多的網(wǎng)格將帶來(lái)巨大的計(jì)算量，此時(shí)間斷伽遼金方法對(duì)于網(wǎng)格質(zhì)量也更加敏感。而由于在單元內(nèi)通過(guò)多項(xiàng)式插值實(shí)現(xiàn)高精度格式，間斷伽遼金方法又很難對(duì)非連續(xù)項(xiàng)進(jìn)行表述。這會(huì)導(dǎo)致數(shù)值求解的振蕩，從而影響計(jì)算收斂甚至引發(fā)發(fā)散。

大多數(shù)將間斷伽遼金方法應(yīng)用于湍流流動(dòng)模擬的學(xué)者，都對(duì)湍流模型方程進(jìn)行了修正，以使得計(jì)算過(guò)程能夠穩(wěn)定和魯棒。但是這些修正的合理性和有效性都需要更深層次的理論與實(shí)際檢驗(yàn)，新的修正方法或者模型也有待提出。近年來(lái)，一大批學(xué)者也正在嘗試將間斷伽遼金方法拓展到分離渦模擬(DetachedEddySimulation,DES)、大渦模擬(LargeEddySimulation,LES)及直接數(shù)值模擬(DirectNumericalSimulation,DNS)計(jì)算當(dāng)中。

3.4 如何降低存儲(chǔ)需求以及計(jì)算量

盡管高性能計(jì)算設(shè)備的發(fā)展日新月異，但是常規(guī)的高性能設(shè)備仍然無(wú)法滿足大規(guī)模的間斷伽遼金方法的計(jì)算量要求。

高精度間斷伽遼金方法中通常采用數(shù)值積分策略來(lái)求解積分項(xiàng)，需要在每個(gè)單元內(nèi)逐個(gè)求取積分點(diǎn)上的數(shù)值并求和。但是隨著近似多項(xiàng)式階數(shù)的提高，積分點(diǎn)的數(shù)目迅速增加，高階間斷伽遼金方法的計(jì)算量也迅速提高，一些學(xué)者為此提出了積分無(wú)關(guān)方法[145]，大大減少了由于積分點(diǎn)數(shù)量帶來(lái)的計(jì)算量。

在采用顯式方法進(jìn)行計(jì)算時(shí)，僅需存儲(chǔ)右端殘值

項(xiàng)，存儲(chǔ)量為(Ndegr×Netol)×Nelem，其中Ndegr是單元變量的自由度個(gè)數(shù)，Netol是方程個(gè)數(shù)，Nelem是網(wǎng)格單元數(shù)。但是在采用高精度方法求解大尺度流動(dòng)問(wèn)題時(shí)，顯式時(shí)間積分方法的時(shí)間步長(zhǎng)比傳統(tǒng)顯式方法的時(shí)間步長(zhǎng)還要小，極大影響了收斂速率。因此，我們只能選擇穩(wěn)定性不受時(shí)間步長(zhǎng)限制的隱式時(shí)間格式，很多學(xué)者對(duì)隱式間斷伽遼金方法進(jìn)行了研究[48,90,146-147]。而采用隱式方法計(jì)算時(shí)，不僅需要保存右端殘值項(xiàng)，往往還需對(duì)雅可比矩陣進(jìn)行保存，該矩陣規(guī)模為(Ndegr×Netol×Nelem)2。可見(jiàn)雅可比矩陣的存儲(chǔ)規(guī)模遠(yuǎn)大于右端殘值項(xiàng)。

4 間斷伽遼金方法的應(yīng)用舉例

4.1 三維帶凸起管道流動(dòng)

采用三維帶凸起管道流動(dòng)對(duì)三維歐拉方程程序進(jìn)行精度驗(yàn)證。該問(wèn)題為內(nèi)流問(wèn)題，來(lái)流條件為Ma∞=0.5。管道的長(zhǎng)、寬和高分別為3、0.5 和0.8，管道下表面x=-1.5到x=1.5之間存在凸起，該凸起的函數(shù)描述為y=0.0625e-25x2。采用歐拉方程進(jìn)行計(jì)算，管道底面設(shè)置為滑移邊界條件，兩側(cè)設(shè)置為對(duì)稱邊界條件，其余面設(shè)置為特征邊界條件。文中采用四套連續(xù)加密的網(wǎng)格進(jìn)行數(shù)值計(jì)算，網(wǎng)格點(diǎn)數(shù)由疏到密分別為6×3×2、11×5×3、21×9×5和41×17×9(圖1)。網(wǎng)格均采用結(jié)塊二階網(wǎng)格進(jìn)行數(shù)值計(jì)算。

(a) Grid 1 (b) Grid 2 (c) Grid 3 (d) Grid 4

圖1 三維帶凸起管道流動(dòng)計(jì)算網(wǎng)格
Fig.1 Mesh for three-dimensional tube with protuberance computation

圖2、圖3和圖4分別給出了四套網(wǎng)格上不同階數(shù)情況下計(jì)算得到的密度云圖，壓力云圖及馬赫數(shù)云圖結(jié)果。橫向?qū)Ρ葹榫W(wǎng)格加密，縱向?qū)Ρ葹殡A數(shù)提高?？梢园l(fā)現(xiàn)，計(jì)算結(jié)果隨著網(wǎng)格加密或者計(jì)算階數(shù)的提高變得越來(lái)越光滑。在相對(duì)稀疏的網(wǎng)格上采用高階方法即可得到較好的數(shù)值結(jié)果。

表1給出了本算例的數(shù)值精度計(jì)算結(jié)果。由于本算例為等熵流動(dòng)，與二維圓柱繞流算例一樣，文中采用熵增的L2誤差作為誤差計(jì)算的標(biāo)準(zhǔn)。從計(jì)算結(jié)果可以看出，該算例誤差為O(hp+1)，基本達(dá)到了預(yù)期的精度。圖5三維帶凸起管道流動(dòng)問(wèn)題精度測(cè)試給出了本算例的數(shù)值誤差-網(wǎng)格尺寸曲線及數(shù)值誤差-自由度曲線，可以看出，本文發(fā)展的三維DG有限元方法計(jì)算程序基本達(dá)到了格式的預(yù)期計(jì)算精度。采用高階方法進(jìn)行計(jì)算，數(shù)值誤差下降的速率比低精度更快。在同等自由度情況下對(duì)光滑流場(chǎng)進(jìn)行計(jì)算時(shí)，采用高階方法產(chǎn)生的數(shù)值誤差比低階方法更小，驗(yàn)證了高階方法在光滑流場(chǎng)計(jì)算中的優(yōu)勢(shì)。圖6三維帶凸起管道流動(dòng)流場(chǎng)計(jì)算結(jié)果為最密的網(wǎng)格上計(jì)算得到的馬赫數(shù)云圖及Z=0截面的壓力系數(shù)Cp的分布?？梢钥闯?，流場(chǎng)的計(jì)算結(jié)果較為光滑，截面得到的壓力系數(shù)分布較為對(duì)稱，驗(yàn)證了文中三維DG歐拉方程計(jì)算程序的可靠性。

圖2 四套網(wǎng)格上的計(jì)算密度云圖Fig.2 Density contours on four mesh types

圖3 四套網(wǎng)格上的計(jì)算壓力云圖Fig.3 Pressure contours on four mesh types

圖4 四套網(wǎng)格上的計(jì)算馬赫數(shù)云圖Fig.4 Mach number contours on four mesh types

表1 三維帶凸起管道流動(dòng)問(wèn)題的精度驗(yàn)證Table 1 Accuracy analysis for three-dimensional tube with a protuberance

(a) 數(shù)值誤差—網(wǎng)格尺寸

(b) 數(shù)值誤差—自由度

(a) 馬赫數(shù)云圖

(b) 截面壓力系數(shù)分布

4.2 二維圓柱繞流模擬

4.2.1 高雷諾數(shù)圓柱繞流非定常DES模擬

計(jì)算來(lái)流參數(shù)為Ma∞=0.2，Re=3×106，第一層網(wǎng)格高度為8×10-6，y+≈1，物理時(shí)間步長(zhǎng)無(wú)量綱后取值為Δt*=Δt×a/D=0.2。計(jì)算區(qū)域X方向[-8D,15D]，Y方向[-8D,8D]，物面布點(diǎn)僅有36個(gè)，網(wǎng)格數(shù)總量為1105，其中結(jié)構(gòu)網(wǎng)格834個(gè)單元。在此雷諾數(shù)下可以認(rèn)為圓柱尾部分離流動(dòng)已經(jīng)變?yōu)橥牧鞣蛛x。圖7為圓柱繞流計(jì)算網(wǎng)格。

(a)

(b)

圖8為升力系數(shù)、阻力系數(shù)隨時(shí)間的變化，都表現(xiàn)出了良好的周期性。因?yàn)槎S圓柱沒(méi)有三維圓柱展向流動(dòng)，所以非定常流場(chǎng)穩(wěn)定以后變化幅值為固定值。

如圖9所示，將五階精度(DG_p4)結(jié)果取七個(gè)周期計(jì)算結(jié)果平均后的平均壓力系數(shù)、摩擦系數(shù)分布與Achenbach[151]的實(shí)驗(yàn)結(jié)果進(jìn)行了對(duì)比，兩者雷諾數(shù)略有差別，但總體趨勢(shì)吻合程度較好；與Travin[152]基于有限體積法的三維DES結(jié)果和Nyugen[153]基于DG的二維DES和二維URANS的結(jié)果進(jìn)行了對(duì)比，兩者計(jì)算的的網(wǎng)格量分別為41萬(wàn)、1.8萬(wàn)。Nyugen雖然也采用了DG方法，但是只達(dá)到了P2階數(shù)，也就是三階空間精度，本文則達(dá)到了五階空間精度。因?yàn)槔字Z數(shù)較高時(shí)，圓柱會(huì)存在展向間的流動(dòng)，二維和三維計(jì)算結(jié)果會(huì)有所差異。但是相比Nyugen的結(jié)果，本文結(jié)果在尾部流動(dòng)區(qū)域結(jié)果更趨向合理。摩擦系數(shù)和三維DES計(jì)算結(jié)果也十分接近，尾部流動(dòng)分離區(qū)域模擬結(jié)果也更加接近實(shí)驗(yàn)值。即使在稀疏網(wǎng)格下，隨著階數(shù)提高精度增加，高階間斷有限元法依然可以對(duì)分離流動(dòng)做到極佳的模擬。對(duì)比Nyugen的三階精度計(jì)算結(jié)果，本文通過(guò)提高階數(shù)的方法，相比其提高網(wǎng)格數(shù)帶來(lái)的精度提高效果要更加顯著。

(a) CL～t

(b) CD～t

(a)

(b)

從圖10中也可以看出，在p1時(shí)候，尾跡區(qū)沒(méi)有分離，在進(jìn)入p2以后,流場(chǎng)發(fā)生了巨大的改變，開(kāi)始出現(xiàn)明顯的周期性的脫落渦。體現(xiàn)了精度提高對(duì)流場(chǎng)模擬結(jié)果的改善是巨大的。并且隨著階數(shù)的繼續(xù)提高，脫落渦的耗散也被降低，在遠(yuǎn)場(chǎng)區(qū)域也越發(fā)明顯。

圖11為二維URANS計(jì)算結(jié)果，采用的是SA模型。在采用RANS模型進(jìn)行計(jì)算時(shí)，因?yàn)橥牧髂Ｐ驮?，抑制了尾部的分離流動(dòng)，未形成周期性的渦街，整體流動(dòng)轉(zhuǎn)變?yōu)槎ǔＡ鲃?dòng)。而在采用二維DES時(shí)，可以對(duì)圓柱繞流的分離進(jìn)行準(zhǔn)確的捕捉，說(shuō)明在二維情況下，對(duì)長(zhǎng)度尺度的修改依然是有效的，有效提高了對(duì)分離流動(dòng)的模擬。

(a) 1階DG

(b) 2階DG

(c) 3階DG

(d) 4階DG

圖11 二維URANS計(jì)算結(jié)果Fig.11 Numerical result obtained using two- dimensional URANS

4.2.2 低雷諾數(shù)網(wǎng)格自適應(yīng)圓柱繞流模擬

我們采用非結(jié)構(gòu)自適應(yīng)網(wǎng)格方法對(duì)卡門(mén)渦街問(wèn)題進(jìn)行了模擬，遠(yuǎn)場(chǎng)來(lái)流Ma=0.1,α=0,Re=150。初始網(wǎng)格如圖12所示，計(jì)算域僅包含1114個(gè)單元，并且物面僅有12個(gè)網(wǎng)格點(diǎn)。

(a) The global view

(b) The local view

圖13為p4在初始網(wǎng)格上得到的卡門(mén)渦街渦量圖，從圖中可以看出由于初始網(wǎng)格比較稀疏，得到的渦量云圖不是很光滑并且渦的量級(jí)都比較小，我們可以認(rèn)為在稀疏網(wǎng)格條件下，即使采用高階格式依然得不到高精度的流場(chǎng)數(shù)值解，主要原因在于大尺度的網(wǎng)格導(dǎo)致的數(shù)值耗散。

(a) The global view

(b) The local view

圖14給出了網(wǎng)格自動(dòng)加密后的卡門(mén)渦街渦量云圖。由于在邊界層和渦街區(qū)域的網(wǎng)格自動(dòng)加密，數(shù)值耗散降低，數(shù)值結(jié)果變得非常光滑。并且應(yīng)當(dāng)注意到，網(wǎng)格自適應(yīng)過(guò)程中生成了包含大量懸掛點(diǎn)、尺度差別懸殊的非結(jié)構(gòu)網(wǎng)格，但是結(jié)果并沒(méi)有對(duì)間斷伽遼金方法的穩(wěn)定性和精度產(chǎn)生嚴(yán)重的影響，證明了間斷伽遼金方法能夠有效地處理具有復(fù)雜形狀、質(zhì)量參差不齊的網(wǎng)格。

(a) The global view

(b) The local view

圖15為隨時(shí)間變化卡門(mén)渦街脫落，網(wǎng)格自動(dòng)在渦局部進(jìn)行加密，在渦量很小的地方放粗的動(dòng)態(tài)結(jié)果。對(duì)比圖16可以看出，由于卡門(mén)渦街是一個(gè)準(zhǔn)定常問(wèn)題，流場(chǎng)周期性變化，網(wǎng)格自適應(yīng)后總量變化不大，驗(yàn)證了自適應(yīng)方法的可行性。

(a)

(b)

(c)

(d)

圖16 迭代過(guò)程中網(wǎng)格數(shù)量變化Fig.16 Number of elements during the iteration

圖17為物面加密后升力系數(shù)和阻力系數(shù)隨時(shí)間的變化，在表2中與參考文獻(xiàn)[154]進(jìn)行了對(duì)比，可以看出加入自適應(yīng)方法后，即使在很稀疏的初始網(wǎng)格條件下，依然可以得到高精度的數(shù)值解結(jié)果。

圖17 升力與阻力系數(shù)波動(dòng)Fig.17 Variation of the lift and drag coefficient

CLamplitudeCDamplitudeCDmeanInitialmesh0．480．0201．21Adaptivity0．530．0271．33Ref．[154]0．520．0261．32

5 結(jié) 論

本文總結(jié)了自20世紀(jì)末期以來(lái)間斷伽遼金方法在可壓縮流數(shù)值模擬中的應(yīng)用研究進(jìn)展。首先，從間斷伽遼金方法的基本概念出發(fā)，集中介紹了間斷伽遼金方法在可壓縮流計(jì)算中的國(guó)內(nèi)外研究歷史和現(xiàn)狀；其次，較為詳細(xì)地列舉分析了間斷伽遼金方法在實(shí)際應(yīng)用中面臨的挑戰(zhàn)和困難；最后，給出了數(shù)個(gè)典型的算例，展示了間斷伽遼金方法在高精度、網(wǎng)格自適應(yīng)等方面的優(yōu)勢(shì)。

經(jīng)過(guò)近二十年的發(fā)展，間斷伽遼金方法數(shù)值求解Euler方程、N-S方程方面有了顯著的進(jìn)步，出現(xiàn)了大量成功的嘗試和令人鼓舞的結(jié)果，高階情況下其數(shù)值結(jié)果表現(xiàn)出來(lái)的高精度特性尤其令人印象深刻。然而高階離散帶來(lái)的高度非線性也顯著增加了迭代求解方面的難度和對(duì)彎曲網(wǎng)格精度方面的要求。目前高效、魯棒的求解復(fù)雜流場(chǎng)情況下高階離散非線性系統(tǒng)是研究的熱點(diǎn)問(wèn)題。相信隨著相關(guān)技術(shù)的進(jìn)一步發(fā)展，高階間斷伽遼金方法會(huì)成為復(fù)雜流場(chǎng)高精度數(shù)值模擬領(lǐng)域的有力工具。

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Applications of discontinuous Galerkin method in numerical simulations of compressible flows: A review

LYU Hongqiang*, ZHANG Tao, SUN Qiang, CHEN Jianwei, QIN Wanglong
(College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)

In this paper, we give a review on the international and domestic applications of the promising high-order method(HOM), the discontinuous Galerkin method (DGM), in the numerical simulation of compressible flows over the last three decades. A brief introduction of the basic concepts and attributes of the DGM is given first. Then a historical survey on the DGM’s applications in solving hyperbolic and elliptical equations is presented, mainly concentrating on its development and research status in the field of computational fluid dynamics (CFD). Existing challenges and possible trends in the aspects of corresponding mesh technologies, shockwave capturing methods, turbulence simulation, and computational resource requirement are concluded and analyzed as well. Several examples of its applications in the simulation of compressible flows are provided at last.

discontinuous Galerkin method (DGM); high-order methods; computational fluid dynamics (CFD); compressible flows; curved mesh

0258-1825(2017)04-0455-17

2017-03-23;

2017-06-03

國(guó)家自然科學(xué)基金(11272152); 航空基金(20152752033)

呂宏強(qiáng)*(1977-), 山東萊陽(yáng)人，教授，研究方向：高精度數(shù)值模擬，飛行器優(yōu)化. E-mail: hongqiang.lu@nuaa.edu.cn

呂宏強(qiáng), 張濤, 孫強(qiáng), 等. 間斷伽遼金方法在可壓縮流數(shù)值模擬中的應(yīng)用研究綜述[J]. 空氣動(dòng)力學(xué)學(xué)報(bào), 2017, 35(4): 455-471.

10.7638/kqdlxxb-2017.0051 LYU H Q, Zhang T, Sun Q, et al. Applications of discontinuous Galerkin method in numerical simulations of compressible flows: A review[J]. Acta Aerodynamica Sinica, 2017, 35(4): 455-471.

V211.3

A doi: 10.7638/kqdlxxb-2017.0051