第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分_第1页
第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分_第2页
第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分_第3页
第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分_第4页
第四章 跨音速定常小扰动势流混合差分方法及隐式近似因式分_第5页
已阅读5页,还剩107页未读 继续免费阅读

下载本文档

版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领

文档简介

第四章跨音速定常小扰动势流混合差分方法及隐式近似因式分解法

chapter4TheMixedFiniteDifferenceMethod(FDM)forVelocityPotentialFunctionofSteadySmallPerturbationandImplicitApproximateFactorDecompositionMethods主要内容:maincontents混合差分解法MixedPDMethod小扰动方程及小扰动激波差分式

Smallperturbationequationandsmallperturbationrelationshipforshockflow小扰动速势差分方程

Thefinitedifferentialequationofsmallperturbationpotentialfunction边界条件及边界条件的嵌入Theinitialconditionandboundarycondition线松弛迭代解法Linearrelaxationiterationmethod升力翼型的跨音速小扰动势流差分方法FDmethodofvelocitypotentialfunctionforsmallperturbation隐式近似因子分解法ApproximatefactordecompositionmethodAF1方法AF1methodAF2方法AF2method

方法比较Comparisonofthemethod重点:Focus混合差分方法MixedFDMethod

难点:Difficulty隐式近似因子分解法ImplicitApproximatefactorydecomposition第四章跨音速定常小扰动势流混合差分方法及隐式近似因式分解法

chapter4TheMixedFiniteDifferenceMethodforVelocityPotentialFunctionofSteadySmallPerturbationandImplicitApproximateFactorDecompositionMethods跨音速流:局部超音区与亚音速同时存在的流场Transonicflow:Localsupersonicflowandsupersonicflowexistsmeantime偏微分方程:混合型方程ThePDE:Mixedtypeequation混合差分方法:用不同的差分方程求解跨声速流场

MixedFinitedifferencemethodistosolvetransonicflowwithdifferentFDMs混合型方程及流场:采用迭代方法求解,求解之前不知道方程的类型MixedEquationandflowfield,theiterativemethodisusedbecausethetypeoftheequationisunknownbeforeitwassolved小扰动方程:小马赫(0.6~1.4)流过薄而微变的叶片(机翼或叶栅)时全速势方程可简化为小扰动方程

Smallperturbationequation(SPE):whenmachnumberissmall(ie0.6~1.4)thefullvelocitypotentialequationcanbesimplifiedtoSPE混合差分:用混合差分格式求解小扰动方程MixedFDM:TosolveequationusingMFDM混合差分和松弛迭代法求解全速势方程MixedFDMandRelaxationiteration:Tosolvefullvelocitypotentialequation.优缺点:Advantage/disadvantage

跨音速松弛法---速度快,有效Transonicrelaxationmethodfasterefficient

时间推进法:适用范围广Timematchingmethods,widelyusage

近似因子分解法:快速Approximatefactordecomposition:faster

多层网格法:收敛性好Multi-gridtechnique:goodconvergence4.1跨声速小扰动速度势方程Equationoftransonicsmallperturbationvelocitypotentialfunction跨声速气流绕过薄翼的情况Forthecaseoftransonicflowpassathinairfoil二维平面速势方程2Dvelocitypotentialequation气流绕过薄翼适用范围:亚、跨、超音速无旋流动Suitable

case:subsonic,transonic,supersonic

irr-rotational

flow.将流动分解为两部分:未经扰动的流动、扰动流动To

decomposetheflowintounperturbedflowand

perturb

flow未经扰动的流动就是无穷远前方来流Flow

atunperturbedfieldsisfarfieldflow扰动运动速度势可以用表示。速度可以用表示Potentialfunctionofperturbationflowis,perturbationvelocitycomponents两部分的合速度势Thetotalvelocitypotentialfunction代入速势方程可得小扰动速度应满足的方程Substitutetheequationandthenthesmallperturbationeq.求得速度场之后,可以得到压强及压强系数为Thepressureandpressurecoefficientcanbeobtainedfromthefollowingequations.再用等熵流动的关系式可得到其他参数

Thenintroduce

the

isentropy

relationtogetotherparameters比热比绝热指数小扰动条件下,扰动速度远小于自由来流速度

on

smallperturbationcondition,theperturbationvelocityless

than

free

stream补充条件:

Supplement

conditions来流不能接近音速incoming

flow

velocitydoes

not

approach

sonic

来流非高超声速incoming

flow

velocitydoes

not

approach

hypersonic为进一步简化扰动方程,忽略扰动速度一次项,可得到下列关系:Simplified

equation最后得到:

Final

equation应用范围:亚、超声速

Suitable

for

subsonic

and

supersonic不适用于跨声速区域:对于跨声速≈1,必须取消补充假设条件,即取消来流不能接近音速的假设,这时速势方程首项的系数一次项不能忽略

For

transonic

flow

field

(M≈1),the

supplement

condition,the

first

item

of

the

potential

function

equation

can

not

be

neglected.

跨声速小扰动方程应为:Thesmall

perturbation

equation

of

velocity

function可以证明:当M∞→1时,

It’sproved,whenM∞→1,因此跨声速条件下,小扰动方程可以写成Sothatthesmallperturbationequationattransonicflowcanbewrittenas此方程的类型取决于:Typeoftheequationdependson=B2-4AC=4(M2-1)当M<1时,<0,不存在实特征根,没有特征线,为椭圆型WhenM<1,norealeigenvalueexists,thatisnocharacterline,theequ.iselliptic.当M>1时,>0,存在两个特征根,有两条特征线,为双曲型WhenM>1,therearetwoeigenvalue,twocharacterlines,theequ.ishyperboliceq.当M=1时,=0,存在一个特征根,有一条特征线,为抛物型WhenM=1,thereisoneeigenvalue,onecharacteristicline,theequ.isparabolic

特征线(当M>1时):斜率

Theslopeofcharacteristicline特征线与x轴夹角为局部马赫角,对称于x轴。LocalMachangleistheanglebetweenvelocityvectorandthecharacteristiclinexyoqr’pq’r影响区依赖区是马赫角issocallMachangle

影响区:P点下游由两条特征线所夹的区域Influencezone:

upwindzonebetweencharacteristiclines依赖区:P点上游由两条特征线所夹的区域Dependzonedownstreamzonebetweenthecharacteristiclines扰动下的压强系数公式

Thepressurecoefficientonsmallperturbationcondition§4-2小扰动激波关系式Theshockrelationsofsmallperturbation.

等熵激波小扰动激波的熵增是三阶小量Forsmallperturbationshock,entropyincreaseisthirdorder,soitisisentropyshock。

激波的精确速度关系式:Accuratevelocityrelationofshock激波前后的速度关系式(几何关系)Velocityrelationsinfront/rear-shock即

对于直角坐标系AtCartesiancoordinates

因此sothat由能量方程可得Fromenergyequation由此得到M∞→1时的方程(跨声速中)Fromwhere,theequationwhenM∞→1,(transonicflow)超声速中Atsupersonicflow适用范围:激波前后小扰动方程,适用于等熵波

Aboveeqs.areavailableforsmallperturbationflowinfront/behindoftheshock,i.e.,iso-entropyflow§4-3跨声速小扰动速势差分方程

Smallperturbationequationfortransonicflow

混合性方程,在同一流场中不同点所用的差分方程不同。Mixedequation,differentFDEisusedforthescheme一、中心差分格式

CenteralFDEschemeflowfield

对速度势Forvelocitypotentialfunction一阶导数的差分格式Firstorderdifferenceequationisobtainedas二阶导数的差分格式Plustwoequations,andget2edorderPD二阶精度

2ndorder在超音速流中,气流参数只受上扰动游影响与下游扰动无关。Atsupersonicflow,theparametersofflowaredependentonupwindperturbationandindependentondownflowperturbation需建立迎风一侧差分格式TheupwindonesideFDschemeisneededtobuilt

取上游一侧的点构成差分格式TaketheupwindpointtoconstructFDscheme一阶精度迎风格式1storderupwindscheme二阶精度迎风格式2ndorderupwindscheme二、一侧差分格式Oneside

FDEofthederivatives三、亚音速点的差分方程Atsubsonicflowequation取网格点如图:正交等间距网格Thespacenodesareshownas中心差分格式构成的差分方程即受周围四点的影响,这是亚声速流动的特点iseffectbyaroundfourpoints,thisissubsonicfeature

四、超声速点的差分方程FDEforsupersonicflow当计算点为超音速(M大于1)时,方程为双曲线型Whenlocalsupersonicflowappear,theequationis

hyperbolic存在依赖区(上游马赫锥内部)Thedependencezoneexists,(upmachcore)对y的差分可以用中心格式Thecenturialdifferenceisusedforthederivativewithspedtoy对x的差分要用迎风格式UpwindschemeisusedforX-direction显示格式:差分式取,而不用线法Explicitscheme每次都用i网格线上的已知值,可以从左到右逐点计算Theknownvalueisusedtocalculatethevalueateverynodesequently隐式格式:利用当前网格线上的值构筑差分方程Implicitscheme:usingpresentvaluetoconstructFDE

具有三个未知量(在网格线i上)

Wherethereare3unknownpoints显式比隐式方便Explicitlyschemeismoreconvenientthanimplicitscheme显式格式稳定区域小Thestabilityzoneofexplicitissmallerthanthatofimplicitly稳定性和收敛性

Stabilityandconvergence收敛性:当步长趋于零时,差分方程解趋于微分方程解Convergence:whensteplengthtendstozero,thesolutionofthePDFtendstothesolutionofPDE稳定性:差分误差在传播过程中有界且逐渐减小Stability:theerrorislimitedordecreased对波动方程(双曲型):稳定性条件是差分方程依赖区不小于微分方程的依赖区Forviberation

Eq,thestabilityconditionisthatthedependentzoneofPDElessthanthatofPDE对超声速势函数

Forpotentialvelocityfuction

差分方程依赖区半顶角ThehalfconicalangleThedependentzoneoftheFDE微分方程的半顶角theangleofthedependentzone差分方程稳定条件为对于跨声速势流,不满足稳定条件,因为Fortransonicflow,thestabilityconditionisnotsatisfied跨声速势流不能用显示格式

sotransonicpotentialfunctioncannotsolvewithexplicitmethod隐式格式的依赖范围大于微分方程的依赖范围ThedependentzoneofimplicitschemeisgreatthanthatofPEDJ+1JJ-1双曲方程差分采用一侧隐式格式Forhyperbolicequation,onesideimplicitlyschemeisused五、音速点的差分方程Thefinitediffenceatsonicpoints

当M=1时,方程为抛物性,存在一族特征线WhenM=1,theequationisparabolic,thereexistaseriesofcharacteristline速度势方程化为potentialequationbecome

Subsonic采用差分方程可以写成UsingFDE六、速度判别式Velocitycriticalcondition

四种情况:

Fourcases

亚声速sub亚声速sub

超声速supe超声速super

亚声速sub超声速super

超声速super亚声速subsupersupersonicairfoilⅠⅡⅢ:过渡连续

continuallychanges

Ⅳ:出现激波参数不连续theshockappears,parametersarediscontinous

Ⅲ:有音速线存在Thereexistssonicpoints逐点判别:根据系数进行判别Judgeaccordingtothecoefficientof情况的值的值Ⅰ>0>0亚-亚声速subsonicⅡ<0<0超-超声速supersonicⅢ>0<0亚-超声速sonicⅣ<0>0超-亚声速subsonic中心差分一侧差分A

(i,j)点性质对应的差分方程any亚音速subsonic超音速supersonic音速点sonic差分方程形式PDEform七.跨声速小扰动激波的差分方程

PDEfortransonicsmallperturbationshockflow激波处:速度由超声速过渡到亚声速Atshock,theflowtransferfromsupersonictosubsonic激波前流场均匀(近似)

Infrontoftheshock,theflowisuniformsupersonicflowi,ji-1ii+1j+1i-1ji,j+1i+1,j+1i+1,ji-1shock激波后流场均匀(近似)Aftertheshock,theflowisalsouniform差分方程(跨声速小扰动方程的差分形式)FDE(Transonicsmallperturbationflow)对无旋流动(无旋条件)Conditionofirrotationalflow

其差分形式ItsFDform

考虑了无旋条件的扰动速度差分方程Afterconsideringtheirrotatationalconditionthesmallperturbationequationbecomes讨论:discussion:

跨声速区小扰动激波差分方程与小扰动激波关系相同八、超音速点差分方程的人工粘性

artificial

viscous

for

supersonicFDE速势方法假设了流场均为等熵流The

velocitypotentialmethodassumethattheflowisiso-entropy导致流场间断解不唯一(可由亚-超,也可由超-亚)Itleadsto

non-unique

solution如果采用迎风格式(单侧差分),则只适合压缩突跃(由超-亚),不可能出现膨胀解。Continuous

solution,if

theupwindschemeisused,thesolutiononlysuitableforcompressiblesharpincrease(shock),notsuitableforsharpdecrease.超声速点差分方程(迎风格式)FDE

of

thepotentialequationatsupersonicflow原因:采用1阶迎风格式1st

order

upwind

scheme应用当地M数改成相对应的微分方程UsinglocalMachnumberMtorewritethePDEthen其中类似于跨音速小扰动粘性流方程中的粘性项。称为人工粘性

Where

issimilarastheviscousformofsmallpertubationequation,socalleditartificialviscous差分方程的解只含压缩突跃,即激波(是熵增过程)PDEonlyincludescompressedshapechange(wheretheentropycreases)不可能产生膨胀突跃(即熵减过程)Notsuitableforexpandingshapechange(whereentropydecreases)4.4边界条件及其嵌入

EmbedingofBoundaryconditions一、边界条件(BoundaryCondition)1.物面:无粘,无穿透条件onwallnonormalvelocity对于翼型(叶栅),设物面方程为,则定常流动边界条件即:若翼型上下表面可表示为则速度分量可写成

上表面的边界条件为BConupsurfaceis其中,,为扰动速度Where,istheperturbationvelocitycomponents对于薄翼型Forthinwing小迎角下,时ForsmallAOA,when故上表面(onupsurface)或写成orbewrittenas

同理,对于下表面meantimeforlowerside综合上下表面可以写成以下小扰动方程翼型上下表面边界条件Considerupperandlowersideofairfoil,thesmallperturbationssatisfyfollowingcondition2.库塔条件(后缘边界条件)Kuttacondition(trailingedgecondition)上下表面流线在后缘尖点平滑汇合thestreamlinesonupsideandLower-sidesmoothlysinksattrailingedge在受气动载荷时,速度势在后缘不连续,形成间断面。Undertheaerodynamicloads,velocitypotentialfunctionattractingedgeisdiscontinuous在这条间断面上必须满足Onthediscontinuitysurface,whatmustsatisfyis。。后上下c

(1)上下压强相等

the

pressureonup

andlowersideofairfoilisequal

(2)速度方向相同,大小不同

the

direction

ofvelocityareconsistent,butthevalueofthevelocityisnotequal

小扰动条件下,因此上述方程可写成:

forsmallperturbation,aboveequationscanbewrittenas:

经间断面速度势变化称为环量

through

the

section

surface

the

velocity

potential

function

changes

is

circulation.3.远场条件Farfieldcondition用有限远代替无限远场,扰动速度势的近似条件为:usinglimitedfarfieldreplacetherealfarfieldperturbationvelocitypotentialfunctionBCcanbewrittenas:二、边界条件的嵌入Embedingoftheboundarycondition边界点上速度势应同时满足边界条件和速势方程OnboundarythevelocitypotentialfunctionsatisfyboththeBCandthepotentialEq.1.物面边界嵌入Embedingofwallboundarycondition翼型上表面Ontheairfoilsurface将速势拓延到边界的另一侧(i,j-1)Extendthevelocitypotentialfunctiontoothersideofboundary即Or边界点的中心差分Thecentraldifferenceonboundary利用边界条件得到:UsingBCthenget2.库塔条件的嵌入EmbeddingofKuttacondition增加新方程使上下表面上相同,即Additionalnewequationtomakeconsistentonupandlowersurface3.远场条件的嵌入Embeddingoffarfieldcondition根据具体问题特点建立运动场的计算方法Tofoundthecomputationmethodaccordingtothecharacterofcertainproblem对于自由绕流,运动速度为,自由来流的速度势为forafreeflowaroundtheairfoil,thefarfieldvelocityis,andthevelocitypotentialfunctionoffreeflowis扰动速度势应满足Thereforetheperturbationvelocitypotentialsatisfy§4.5线松弛迭代解法

Thelinerelaxationiterationmethod一、非线性代数方程的迭代解法Iterativemethodfornon-linearequations跨声速小扰动速势方程是非线性的TransonicsmallperturbationequationisnonlinearPDE其差分方程为非线性代数方程,即系数是与函数值或其导数有关ItsFDEisalsononlinearequationthatisitscoefficientsarerelatedtothevariables迭代求解:

Iterationmethod把系数假设成已知量,每次求解之后再重新计算系数,再次求解直到得出收敛解为止.Assumethecoefficientareknownatfirstiteration,thenrecalculatethecoefficientsagainafteronceiteration,repeatiterationuntiltheiterationconvergences二、高阶代数方程的线松弛解法

ThelinerelaxationiterationmethodforHighorderarithmeticlinearequations

高阶线性方程组,线性化后的差分方程

Highorderlinearequations,linearizedFDE阶数为,M为网格点数,n为问题的维数.或阶数M*N*L(M,N,L为空间三坐标方向的网格点数)Theorderoflinear-algebraequationis,whereMisthenumberofthegrids,nisthenumberofdimension.Theorderoflinear-algebraequationisM*N*L,whereM,N,Larenumberofgridsincoordinatesx,yandz松弛迭代:Relaxationiteration

轮流放松流场中的的部分速势,将其假设为未知,其余部分看成已知的,利用线性方程组联立求解

Relaxatethepotentialfunctionsequently,assumethatthepresentpointisunknown,andothersareknown.松弛迭代点松弛:每次把一个点作为未知点Pointrelaxation:onlyonepointisassumedtobeunknown线松弛:每次把一条网格线上的所有点作为未知Linerelaxation:allpointononegridlineareassumedtobeunknown线松弛linerelaxationj线松弛linerelaxationi线松弛法:要求内存较多,方程组的个数减少到一维点数Linerelaxation:requiremorememorysource,thenumberofequationequalstothenumberof1Dpointsij点松弛pointrelaxation逐点松弛:要求内存较少(为线性松弛的倍),扫描流场中的各个网格点,把周围点均看成是已知点。Sequentpoint:requirelessmemoryresource,onlytimesoflineelaxation.Scanallthegridpointssequently.线松弛方程组可采用三对角矩阵快速解法Forlinerelaxationmethod,thetri-diagonalarraycanbesolvewithquickmethod.三、简单迭代和改进迭代Improvemethodofsimpleiterationmethod简单迭代:迭代公式右端的速度势全部采用前次迭代结果Simpleiteration,allparametersonrighthandareoldvalueoflastiteration.改进迭代:每次迭代都用最新速度势值代替右端项。速度判别式要用简单迭代方式计算,则会导致超临界气流计算振荡发散。Theimprovediterationmethodalwaysusesthenewestvalue,andthevelocitycriteriamustbecalculatedaccordingtothesimpleiterationway,otherwise

thedivergencewilloccuratcriticalstate.四、追赶法Thechasemethod求解三对角矩阵线性方程快速方法Itisafastmethodtosolvetri-diagonalmatrix线松弛方法求解方程组Equationforlinerelaxationmethod对于边界点:forboundarypoints上边界upboundary下边界lower

boundaryj=1,2……Ni-1,ji,ji,j-1i,j+1i+1,j对应的系数矩阵为三对角矩阵

追赶法:顺着消去,逆着带入。从上至下消去首项,从下而上代入末项。Thechasemethod:eliminatingsequently,substitutinginversely.Eliminatefromtoptodown,substitutefromdowntoup.五、初场Initialfield

初始值分布:影响收敛速度Initialfielddistributionofparameterswillinfluenceconvergence对亚音速流场:可以选全场速度势为0,即未经扰动Subsonicflowfield:globefieldcanbeputas0,thatistheflowisnotdisturbed对跨音速流场:初值选取需谨慎,合理初场能加速收敛Fortransonicflowinitialvaluemustgivencarefully,thereasonableinitialvaluemightaccelerateconvergence

一般应选用与流场相近的速度势分布Usuallyselectanearsolutionofpotentialfunction可以用相近的亚声速计算结果Theapproximatelysubsonicresultcanbeused六、收敛标准Criterionofconvergence所有点相邻两次计算所得的速势差别的最大值Themaximumdifferencebetweentwoimmediatevicinityiterative可以用与初始比值判别收敛

theratioofcurrentandinitialcanbethecriterion七、超松弛法Superrelaxationmethod加速速度势函数的修正步伐Toacceleratetheconvergence

超松弛

superrelaxation

弱松弛weakrelaxation

八、加密网格法Meshrefinemethod计算精度增加,计算网格数增加Toincreasetheprecision,toincreasethemeshNo.问题复杂度增加TheincreaseofcomplexitytoincreasethemeshNo.计算机时与网格总点数以正比增加computationtimeincreaseasthemeshNo.采用疏密结合的方法可以减少计算时间

Usingcoarse/finemeshmaydecreasecomputationtime加密网格法:先用疏网格数算初始场,加密之后获得精确解meshesrefiningmethod多重网格:先疏后密、再疏;交替使用疏密相间的网格multiplegrid*§4-6绕升力翼型的跨声速小扰动势流差分计算方法FDMforpotentialfunctionoftransonicsmallperturbationflowaroundairfoil一、绕升力翼型的跨声速小扰动方程势流的差分方程Theequationforpotentialfunctionoftransonicsmallperturbationflowaroundairfoil4—7隐式近似因式分解法的基本思想

ThebasicconceptofApproximateDecompositionMethod求解速度势方程的快速收敛解法ItisafastworkingmethodforPotationequationSLOR是显式迭代方法,因此收敛慢SLORisfullexplicitmethod,thusitworksslowly全隐式松弛算法:每次迭代中流场中的任意一点能受到它的依赖区中全部其点的影响

Fullimplicitrelaxationmethod,anypointinflowfieldcanbeinfluencedbyallotherpointsADI(AlternatingDirectionImplicit)交替方向隐式迭代,分为AF1和AF2UsingAF1andAF2基本差分算子:Somebasicfinitedifferencecalculator迎风差分(前差)upwindFD顺风差分(后差)backward/rearwardFD二阶中心差分:2ndordercentralFD二阶一侧迎风差分:upwind2ndonesideFD位移算子:displacement(FD)calculator用位移算子表示差分算子

TheFDexpressedusingdisplacementcalculators差分算子位移:ThedisplacementofFDcalculator差分算子的分解与组合:ThedecompositionandcombinationofFDs差分方程可以用算子表示TheexpressionofFDEusingFDcalculatorL代表未经松弛的差分算子

FDcalculatorrelaxation松弛差分算子N,第n次迭代的修正值为TheFDcalculatorofrelaxationiteration,thecorrectionofnthiteration.算子表达式:

Calculatorexpression当松弛迭代收敛时,,即两者相同Whentheiterationconverged,bothcalculatorarethesame

.

当时,表示其不是差分方程的解,因此表示差分方程满足微分方程的程度。When,denotesthesolutionofFDEisnotthesolutionoftheoriginalPDE,thereforedenotesthedegreeofhowdosetheFDEsatisfythePDE.差分松弛迭代算子的选取原则TheprincipleforseleetingFDcalculator

便于求解,线性,有快速解法convenienceforsolvingequation,linearmethod,fastsolver

稳定,能达到收敛标准stable

高效,N尽可能接近L。

higherefficiency,NapproachingL差分算子的用途:可以清晰的显示差分方程的结构UsageofthePDcalculator,itmakestheFDEsimpleandevident近似因式分解的基本思路ThebasicconsiderationofapproximatedecomposeLaplace方程的差分格式(简单迭代法)TheFDschemeofLaplaceequation(simpleiterationmethod)

改进的迭代法

improvediterationmethod松弛迭代格式

Relaxationiteration

scheme中间值由此then还原为(n)和(n-1)表达式后差分方程还原为ExpresstheFDEusing(n)and(n-1)改进的差分格式为

ImprovementofFDE或(隐式)or(ImplicitForm)引入差分算子,采用差分算子表示,并令IntroducetheFDcalculator,usingFDcalculatorexpression.则松弛迭代法的差分格式为:TheFDschemeofrelaxationiterationis线松弛迭代对应的差分格式FDSrelatedtolinerelaxationiterationis因此超松弛差分算子Thus,thesuper-relaxationFDcalculatorisN分解成两个因式的乘积,则IffactorizeNintotwofactors,then

4-8AF1方法AF1method小扰动速势方程Equationofpotentialfunctionforsmallperturbationflow其中,对亚音速小扰动。可用中心差分格式或隐式方程Where,forsubsonicpoint,thecentralschemecanbeused其中where令Let

则小扰动速势方程的隐式差分格式为Then,theimplicitFDEschemeofthesmallperturbationpotentialis分解第一项系数

Tofactorizethefirstcoefficientterm原系数origin误差error松弛差分算子N可分解成为N1和N2,为加速收敛参数,求解可分两步,

RelaxationPDEcalculatorNcanbefactorizedIntoN1andN2,thesolvingcanbedecomposeintotwosteps

代表中间结果Whereisamiddleresult交替方向隐式差分格式(ADI,or,ApproximateFactorization)

Step1:全场沿X方向线松弛,解三对角矩阵

Xdirectionlinerelaxation,tosolvetrianglematrixStep2:全场方向沿y方向线松弛,也解三对角矩阵

Ydirectionlinerelaxation,tosolveatrianglematrix全隐式格式,对亚声速区适用,称为AF1,

fullimplicitscheme,Forsubsonicit’scallAFI

对超音速点,采用迎风格式Forsupersonicpointstheupwindschemeisused对应两步

Correspondingtwostepsare1.

2.二、AFI的收敛性TheconvergenceofAFI亚音速点(中心差分格式)等价于时间相依方程(将迭代过程看成时间推进)Forsubsonic(centralPDEscheme),thesolvingprecedingisequivalenttoatime-dependentproblem设和,当与系数异号时,差分方程的解收敛于微分方程的解。andhavedifferentsign,thenthePDEconvergestwothePDE三、AF1的稳定性StabilityofAFI采用Vonneumann方法分析误差UsingVoneumannanalyticmethod代入AF1差分方程SubstituteintothePDEofAFI其中where假设,为实数Assume,isrealnumber收敛条件:Theconditionofconvergence即or稳定性条件:

stabilitycondition两个可选参数和,适当选取可以加快收敛Twoparametersandcanbechosencarefullytogetquickconvergence取得到最短迭代次数()Take,thengetminimiterationtimes对应的最佳选择是:CorrespondingopticalchoiceisAFI中所有的误差Fourier分量均可以同速度下降

AFIerrorcomponentsofFourierseriesmaydecrease超声速点的AF1差分方程等价于(4-8-12),但没有阻尼项

TheAFIforsupersonicpointitequaltoequation(4-8-12)

亚,超声速流混合问题,AF1第一个算子需四对角矩阵求逆Forsub-super-sonicmixedproblem,AFIhastosolvefour-diagonalmatrix,thustheefficiencyislower对此类流动,AF1不是最有效的方法Forsuchflow,AFIisnotthemostefficiencyone§4-9AF2方法

methodofAF2对超声速流增加时间阻尼项,取Forsupersonicflow,thetunedampingistraduced差分算子FDcalculatorwithupwindscheme对超声速点,取:Forsupersonicflowpoint等价的一阶微分方程(时间依存)Equivalent1storderPDE(timedependent)其中,为超声速阻尼项,当时,与系数同号,大小取决于

Where,issupersonicdamping,whenthecofficentofandhavethesamesignandthequantitydependson更加高效(有阻尼)ItishigherefficiencyAF2格式对亚声速流收敛性比AF1差AF2convergenceforsubsonicisworstthanAF1亚,超声速点的两步AF2格式如下:Forsub-supersonicflow,twosteps,AF2havefollowingscheme亚:(y方向线松弛x方向迎风格式)

sub

LinerelaxationinYupwindschemeinX

(x向线松弛x方向顺风格式)

温馨提示

  • 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
  • 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
  • 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
  • 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
  • 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
  • 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
  • 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。

评论

0/150

提交评论