參考力學(xué)大會(huì)2015集ms4116

中國(guó)力學(xué)大會(huì)-王圣業(yè)*，董義道*，王光學(xué)*，+，鄧小剛*（國(guó)防科學(xué)技術(shù)大學(xué)航天科學(xué)與工程學(xué)院，湖南長(zhǎng)沙+（中國(guó)空氣動(dòng)力研究與發(fā)展中心，四川綿陽(yáng)計(jì)算流體力學(xué)（）中高階精度方法發(fā)為了廣泛應(yīng)用于實(shí)際工程計(jì)對(duì)高階精度格式進(jìn)行系統(tǒng)的驗(yàn)證與確認(rèn)十分重要。本文首先給出了一種高精度格式的驗(yàn)證方法，然后利用該對(duì)一種魯棒的一致高階精度有限差分格式進(jìn)行了驗(yàn)證，并將結(jié)果和二階格式進(jìn)行對(duì)驗(yàn)證使用的標(biāo)準(zhǔn)算例來(lái)源于2012年召開的第一屆高精度方法國(guó)際研討會(huì)，包括二維等熵管道流動(dòng)b問(wèn)題、平板層流邊界層問(wèn)題和層流三角翼。本文驗(yàn)證結(jié)果表明，一致高階格式能夠達(dá)到設(shè)計(jì)精度，達(dá)到相同誤差水平時(shí)的計(jì)算花費(fèi)小于二階L高精度方法，驗(yàn)證與確認(rèn)引等有限差分方法，采用O重構(gòu)的有限體積譜差分譜體積該研討會(huì)對(duì)不同的高精度方法制定了統(tǒng)一的方法，并提供了3類15個(gè)標(biāo)為各種高精度方法系統(tǒng)的驗(yàn)證與確認(rèn)提供了良好的2015n道流動(dòng)(C1.1)、Ringleb問(wèn)題(C1.2)、平板層流邊界層問(wèn)題(C1.4)和層流三角翼(C2.4)。法，如果空間離散誤差Error隨網(wǎng)格尺寸h減小而減小，并有Error hperrorjJjerrorjJjerTheprojectwassupportedby 圖1誤差隨網(wǎng)格尺寸的收斂關(guān) 圖2誤差和計(jì)算花費(fèi)的一般關(guān)p= Inε/ε +q(q(

rp-=In( rp-、計(jì)算外推值u21r/r (1Res(1Res2N為總計(jì)算節(jié)點(diǎn)數(shù)。為了避免收斂標(biāo)準(zhǔn)中的尺度問(wèn)題，計(jì)算節(jié)Resj由密度計(jì)加權(quán)緊致非線性格式（weightedcompact v+ v+ v v v 在計(jì)算網(wǎng)格導(dǎo)數(shù)及雅克比時(shí)采用滿足幾何守恒律的 ?E?= - - - + ?ξ j+1/2左右值采用基于原始變量的5階線性插值QLQ

=13Q

-

+

+

-

j+2QRQ

=1-

+

+90Qj-

+

j+2

1

11 17 +3 -5 ??ξ=

-12E1/2+24 8 247/2+24E9/2 ?

9 9 =11 - + -1E?

Dξ24 8 8 247/2 =1(315Q-420Q+378Q-180Q+35Q =1(35Q+140Q-70Q+28Q-5Q κ1/3的線性插值。二維帶光滑凸起的管道流動(dòng)-1.5￡x￡1.5，上邊界為y=0.8，下邊界y0.0625exp(-25x2的光滑曲線，入口處馬赫數(shù)為界采用對(duì)稱邊界條件。求解采用五套結(jié)構(gòu)網(wǎng)格32·16、64·32、128·64、256·128、512· 圖4算例1.1熵誤差分布云(一致5階WCNS計(jì)算，網(wǎng)格512·256 error exp(sexact) 明了高階格式較高的精度。同時(shí)圖5中，SBP-SAT(BC4thRingleb問(wèn)題中定義參數(shù)k和q，沿每條流線k為常數(shù)，對(duì)于內(nèi)壁面k=kmax=1.5，對(duì)于外壁面k=kmax=0.7，q代表速度標(biāo)量，對(duì)于任何固定的k，q的變化在q00.5和k之間。對(duì)于每個(gè)q，速度a，密度2γ 2

1+a ,ρ=a-1,p aγ-1,J=

log

1-1 1 q 1-k 對(duì)于4個(gè)邊q=q0kmin￡k￡kmaxy>0q=q0kmin￡k￡kmaxy<0；內(nèi)壁k=kmaxq0￡q￡kk=kminq0￡q￡k。求解采用四套結(jié)64·32、128·64、256·128、512·256，網(wǎng)格采用解析方式生成，沿入口、出口按照均勻方式布置參數(shù)k差（同算例1.1），圖6為一致5階WCNS格式在網(wǎng)格512·256得到的熵誤差分布云圖。圖7Ringleb問(wèn)題(網(wǎng)格64·32 圖8算例1.2熵誤差分布云(一致5階WCNS計(jì)算，網(wǎng)格512·256表21.2熵誤差23.45E-3.09E-1.78E-7.15E-7.24E-1.74E-2.26E-4.33E-圖9算例1.2熵誤差2范數(shù)及收斂精度結(jié)果對(duì) 圖10算例1.2熵誤差2范數(shù)及計(jì)算花費(fèi)對(duì)趨近于理論精度5；而一致5階平板層流邊界層問(wèn)題寸LH和LV（圖9），經(jīng)過(guò)遠(yuǎn)場(chǎng)無(wú)關(guān)性計(jì)算后，選取LH=0.5LV=1.0。計(jì)算采用3套網(wǎng)格：圖11平板層流邊界層問(wèn)題邊界條件示 圖12平板層流邊界層問(wèn)題網(wǎng)格(176·145表31.4h2.51E-1.26E-6.28E-0.0Orderof圖13平板x=0.5處流向速度分布(網(wǎng)格45·37 圖14平板x=0.5處法向速度分布(網(wǎng)格45·37L75，夾角σ30，弦長(zhǎng)c=1.0，厚度t=0.024，參考面積：0.133974596半模)。計(jì)M￥0.3，α12.5Rec=4000。采用三套網(wǎng)格，為避免頭部出現(xiàn)奇性軸，采用C- 圖16層流三角翼幾何尺 圖17層流三角翼表面網(wǎng)格（粗網(wǎng)格算例h1.48E-7.53E-3.77E-0.0Orderof圖18層流三角翼二次渦結(jié)圖19x1.75c處渦結(jié)構(gòu)等馬赫數(shù)線（左：中網(wǎng)格；右：密網(wǎng)格翼(C2.4)E.vanderWeide,G.Giangaspero,M.Sv?rd.EfficiencyBenchmarkingofanEnergyStableHigh-OrderFiniteDifferenceDiscretization[J],AIAAJOURNAL(publishedonline21January2015)ADIGMA,AdaptiveHigher-OrderVariationalMethodsforAerodynamicApplicationsinIndustry,SpecificTargetResearchProjectofthe6thEUFrameworkProgramme.WangZ.High-orderCFDmethods:currentstatusandperspective[J].Int.J.Numer.Meth.Fluids2013;72:811-鄧小剛宗文剛張來(lái)平，等.計(jì)算流體力學(xué)中的驗(yàn)證與確認(rèn)[J].力學(xué)進(jìn)展,2007,37(2279-DengX,ZongW,ZhangL,etal.Verificationandvalidationincomputationalfluiddynamics[J].AdvancesinMechanics,2007,37(2):279-288(inChinese)WangZ.High-ordermethodsfortheEulerandNavier–Stokesequationsonunstructuredgrids[J].ProgressinAerospaceSciences43(2007)1–41DengX,ZhangH,DevelopingHigh-OrderWeightedCompactNonlinearSchemes[J],JournalofComputationalPhysics165,22–44(2000)涂國(guó)華鄧小剛毛枚良.5階非線性WCNS和WENO差分格式頻譜特性比較[J]．空氣動(dòng)力學(xué)報(bào)，2012，30(6)：709-TuG,DengX,MaoM.Spectralpropertycomparisonoffifth-ordernonlinearWCNSandWENOdifferenceschemes[J].ActaAerodynamicsSinica,2012,30(6):709-712(inChinese)LiuX,DengX,MaoM.High-orderBehaviousofWeightedCompactFifth-OrderNonlinearSchemes[J].AIAAJournal,DengX,MaoM,TuG,ZhangY,ZhangH.ExtendingWeightedCompactNonlinearSchemestoComplexGridswithCharacteristic-BasedInterfaceConditions[J].AIAAJournal,2010,48(12).DengX,WangG,TuG.ApplicationsofHigh-orderWeightedCompactNonlinearSchemeforComplexTransonicFlows:proceedingsofthe49thAIAAAerospaceSciencesMeetingIncludingtheNewHorizonsForumandAerospaceExposision,Orlando,Florida,[C].2011.LiaoF,YeZ,ZhangL.Extendinggeometricconservationlawtocell-centeredfinite[J].JournalofComputationalPhysics284(2015)419–433DengX,MaoM,TuG,LiuH,ZhangH.Geometricconservationlawandapplicationstohigh-orderfinitedifferenceschemeswithstationarygrids[J].JournalofComputationalPhysics,2011,230:1100-1115.DengX,MinY,MaoM,LiuH,TuG,ZhangH.FurtherstudiesonGeometricConservationLawandapplicationstohigh-orderdifferenceschemeswithstationaygrids[J].JournalofComputationalPhysics,2013,239:90-111.DengX,ZhuH,MinY,LiuH,MaoM.Symmetricconservativemetricmethod:alinkbetweenhighorderfinte-differenceandfinite-volumeschemesforflowcomputationsaroundcomplexgeometries:proceedingsofthe8thInternationalConferenceonComputationalFluidDynamics,Chengdu,China,[C].2014.B.Gustafsson,Math.Comput.29,THEVERIFICATIONOFUNIFORM5TH-ORDERACCURACYWCNSTHROUGHBENCHMARKCASESWANGShengye1DongYidao1WangGuangxue1,2Deng(1CollegeofAerospaceScienceandEngineering,NationalUniversityofDefenseTechnology,Changsha410073,China)(2ChinaAerodynamicsResearchandDevelopmentCenter,MianyangSichuan621000,China）AbstractInrecentyears,high-ordermethodshavedevelopedrapidlyincomputationalfluiddynamics(CFD).Forpracticalengineeringcalculations,theverificationandvalidationofthehigh-ordermethodsareveryimportant.Inthispaper,amethodforverifyingthehigh-orderaccuracyschemesisgiven,andthroughthismethod,arobustuniformhigh-orderaccuracyfinitedifferenceschemeWCNSisverifiedandtheresultsarecomparedwiththe2nd-orderMUSCLscheme.Benchmarkcasesareselectedfromthe1stInternationalWorkshoponHigh-OrderCFDMethodsheldin2012,includingInviscidflowthroughachannelwithasmoothbump,Ringlebproblem,Laminarboundaryla