計(jì)算流體力學(xué)第7章 粘性流動(dòng)數(shù)值計(jì)算

第七章網(wǎng)格設(shè)計(jì)GridDesign網(wǎng)格是流場(chǎng)計(jì)算的基礎(chǔ)Itisthebasisofcalculatingflowfield7-1幾何方法構(gòu)筑葉柵通道網(wǎng)格Geometrymethodtoconstructgrids兩種方法：TwoMethods:1.物理空間構(gòu)筑曲線網(wǎng)格，變換到計(jì)算空間的正交網(wǎng)格，計(jì)算在正交網(wǎng)格中進(jìn)行。Toconstructcurvemeshesinphysicspace，thentransfertoorthogonalgridtocomputeincomputationalspace2.直接在物理平面內(nèi)構(gòu)筑網(wǎng)格并求解，此方法比較容易變化網(wǎng)格密度以適應(yīng)參數(shù)梯度。Directlyconstructgridsinphysicplaneandsolvetheequationsinphysicplane，thismethodmakesiteasytochangedensityofgrids.網(wǎng)格邊界分別平行于求解域邊界或與邊界相適應(yīng)。Thegridboundaryparalleltotheboundaryofflowfield.通道內(nèi)X方向間隔相等?X=constKeepconstantgridgapinsidechannel通道進(jìn)口前Infrontofchannel通道出口后BehindthechannelY方向網(wǎng)格間距常數(shù)（Euler方程計(jì)算）ThegridgapsinYdirectionkeptconstant幾何關(guān)系如下Geometryrelationship按指數(shù)規(guī)律伸展?X?Xchangesinpowerlaw變換關(guān)系導(dǎo)數(shù)Transformderivatives變換后的方程Transferredequation變換后在平面內(nèi)的控制方程為Theequationintransformedplane二.局部加密的葉柵通道網(wǎng)格Themeshincascadetunnelwithlocalrefining翼型或葉柵頭部附近加密Refiningatleadingedgeofairfoilandcascade葉柵通道出現(xiàn)激波時(shí)，在激波位置加密Refiningwhenshockappearsincascadechannel壁面附近邊界層Refininginboundarylayerofthewall例：example通道內(nèi)X方向均勻UniformgridsinXdirectionofcascadetunnel通道前與第一個(gè)網(wǎng)格寬度,其余采用Keepthewidthoffirstmeshassameasthatofinchannelmeshonothersgrids

i=2，3，4,I下游類似Thesamewayusedfordown-streammeshesY方向網(wǎng)格劃分Gridsinydirection

即y方向按指數(shù)規(guī)律變化為ACG邊（下）

為上邊界

Thatispowerlawusedinydirection

一般要求第一個(gè)間隔<0.015，可調(diào)節(jié)J數(shù)來(lái)保證Thefirstgridgapis0.015,itcanbeensurebyadjustingthetotalnumberofgridJ在計(jì)算平面內(nèi)，各個(gè)相鄰網(wǎng)格點(diǎn)的間距

incomputationalplane,thegridgapisusedequal§7-2幾何方法構(gòu)造c形網(wǎng)格ConstructionofC-Typemeshusinggeometricmethod

對(duì)外流問(wèn)題（如NACA-0012翼型繞流）通常可構(gòu)筑C形網(wǎng)格

Foroutflowquestion,usuallyconstructctypegridsyLx將前緣至后緣流向劃分50個(gè)點(diǎn)，按2次曲線（拋物線分布）Discretechordwiseinto50points,using2ndparaboliccurve將坐標(biāo)向左平移0.00125，則NACA-0012的翼向y坐標(biāo)Movingcoordinates0.00125in–direction,thentheycoordinatescanbedeterminedby用兩組拋物線構(gòu)筑c型網(wǎng)格Construct

themeshusingtwoserialofparaboliccurves§7-3保角變化法

Conformaltransformation一、基本概念basicconcept（一）在(x,y)平面內(nèi)兩族網(wǎng)線若存在

on(x,y)plane,theserialcarvesasfollows則兩族網(wǎng)正交

thenthetwocurvesareorthorgal(二)：用復(fù)數(shù)代表(x,y)平面

Definethe(x,y)planewithacomplexvariety兩族曲線用復(fù)函數(shù)表示為

twocurvescanbedenoteswithacomplexfunction(三)正交坐標(biāo)系稱為Theorthogonalitycoordinateexpressedas兩平面的關(guān)系therelationshipbetweentwocoordinatescanbeexpressedas在平面內(nèi)兩線族inplanethecurvesare彼此關(guān)系可寫成

Therelationshipbetweenphysicalplaneandcomputationalplaneisfollows在z平面和平面內(nèi)，點(diǎn)與點(diǎn)相互對(duì)應(yīng)inphysicplanezandcomputationalplanethepointsrespondseachother兩平面內(nèi)的正交曲線相互對(duì)應(yīng)theorthorgalcurvesintwoplanesarecorresponding相應(yīng)的交點(diǎn)也相互對(duì)應(yīng)theintersectpointsarecorresponding二：保角變換方法生成網(wǎng)格

Gridgenerationwithconformaltransformation(isogonality)一、例一:復(fù)平面內(nèi)正交直線族

Ex1,theorthogonalitycurvesincomplexdomain代表與坐標(biāo)軸平行的正交網(wǎng)格的兩個(gè)函數(shù)

Twofunctiondenotethegridsparallelwithcoordinates另一復(fù)平面，兩平面之間的關(guān)系

Incomplexdomain,therelationshipbetweentwodomainsare

兩平面間的網(wǎng)格族是Thegridsintwodomainsare雙曲線Hyperboliccurves雙曲線Hyperboliccurves取0，0.1，0.2，0.3……，1.0時(shí)對(duì)應(yīng)的網(wǎng)格Whereisspecifiedas0，0.1，0.2，0.3……，1.0,thegridsare（二）例二：復(fù)平面內(nèi)正交線族的函數(shù)

Ex2,orthogoralcurvesincomplexplane變換關(guān)系：Transformationrelationship網(wǎng)格線族為

Thegridsare取0，0.1，0.2，0.3……，1.0時(shí)對(duì)應(yīng)的網(wǎng)格線如下圖Whenaregivenas0，0.1，0.2，0.3…thegridsin(x,y)andplaneisfollow

利用保角變換可以生成任意封閉曲體外部的網(wǎng)格Thegridsaroundanarbitraryclosebody通常用于變換的方程可以寫成Therelationusedforconformaltransformationcanbewrittenas

7-4生成網(wǎng)格的微分方程

Thepartialdifferentialequationmethodformeshgeneration適/貼體網(wǎng)格生成可以用一個(gè)微分方程邊值問(wèn)題實(shí)現(xiàn)

BodyfittedgridscanbegainedusingthesolutionofapartialdifferenceEQTTM方法（Thompson,ThamesandMartin）方法:利用求解微分方程方法生成空間區(qū)域網(wǎng)格的方法

TTMmethodisthemethodtogenerategridsusingPDE

要求requirements邊界形狀已知，可以正確描述

Boundaryasknown,canbedescribed,correct

網(wǎng)格密度可調(diào)，有伸展性（可變網(wǎng)格密度）thedensityofgridscanbeadjustedandextended附面層：尺度為~Boundarylayer:size分離流：尺度為弦長(zhǎng)CSeparatedflow:sizeC尾跡區(qū)：尺度為，弦長(zhǎng)CWakeflow:sizeC無(wú)粘區(qū)（勢(shì)流）：尺度為

Inviscousflow:size正交性：可以減少差分方程所依賴的網(wǎng)格點(diǎn)數(shù)

Orthorgality:maydecreasedependentgridofFDE保形性：物理平面與計(jì)算平面方程相同

Conformality:theEQinphysicplaneandcomputationalplaaresame(一)TTM方法建立物理平面與計(jì)算平面的關(guān)系TTMmethodisusedtofoundtherelationshipbetweenphysicplaneandcomputationalplane可用微分方程表示上述變換Thetransformationcanbeexpressedasfollow正確表述外邊界Definetheboundarycorrectly在計(jì)算平面內(nèi)，為了差分方程簡(jiǎn)便，一般用等間隔網(wǎng)格TobeconvenientforFD，thegridsincomputationaldomainisequalgap在平面內(nèi)構(gòu)造正交網(wǎng)格，再在平面內(nèi)生成相應(yīng)的點(diǎn)坐標(biāo)，即求逆變換方程的解Normally,constructtheorthogonalgridsinplanefirst,andthencreatethecorrespondingpointsinplane,thatistofindtheinversetransformation變換關(guān)系

導(dǎo)數(shù)關(guān)系：導(dǎo)數(shù)關(guān)系：且有andthen同理可得

canbeobtainedinthesameway最后可得逆變換方程Finallytheinversetransformationequationobtained求解此方程組可得離散表達(dá)形式Tosolveaboveequationandobtaintransformationrelationshipindiscretedform在平面代表和代表和Inplane,denotelineand在平面對(duì)應(yīng)點(diǎn),邊界點(diǎn)給出后就可以得到內(nèi)點(diǎn)上的各點(diǎn)正交性和保形性（采用Laplace變換）Orthogonalityandconfor由坐標(biāo)變換關(guān)系可知Fromcoordinatetransformation采用Laplace變換，則UsingLaplacetransformation,then反變換為inversetransformationis沒(méi)有混合導(dǎo)數(shù)項(xiàng)，具有保形性和正交性Nomixderivativesexisted采用正交，保形變換后，全速勢(shì)方程的形式不變

Usingorthogonal,conformaltransformation,thefullypotentialequationunchanged.拉普拉斯和泊松方程的變換關(guān)系式Laplacevs.poissontransformation拉普拉斯變換Laplacetransformation

Laplace變換是單值變換，變換前后的點(diǎn)一一對(duì)應(yīng)

TheLaplacetransformationisauniquevaluetransformation物理平面是單連域，則計(jì)算平面也是Thesingleconnectedfieldphysicalplaneisstillasingleconnectedfieldincomputationalplane例：機(jī)翼（翼型）

Wingasanexample極值原理：參變量的極大、極小值必定在邊界上

Extremumprinciple:Themaximumandminimumisonboudary(a,b)ba02134＞（二）泊松方程

poissonequation

可以控制網(wǎng)格密度thegriddensitycanbecontrolled

解為Thesolutionis

uu10xP=0均勻uniform0xP=2>0向下加密densedtolowersurface0xuP=-2<0向上加密densedtoupsurface用于網(wǎng)格生成的poisson方程

Poissonequationusingformeshgeneration反變換方程

Theinversetransformation

P（ξ，η）和Q（ξ，η）可以調(diào)整網(wǎng)格密度

P（ξ，η）andQ（ξ，η）mayadjustthegriddensity其中表示要求網(wǎng)格向點(diǎn)靠攏，是調(diào)整量。n，m表示要向靠攏的網(wǎng)格數(shù)量

Wheredenotesthegridswillapproachto，areadjustparameters.

n，mdenotesthenumbergridstobeclosedtopoint.另一種網(wǎng)格加密辦法是由Thomas和Middlecoff

提出的OthermethodforrefininggridsisgainedbyThomasandMiddlecoff

7-5代數(shù)法和混合法一、代數(shù)法采用幾何剖分方法，利用代數(shù)運(yùn)算生成計(jì)算區(qū)域網(wǎng)格，無(wú)需求解微分方程。等比網(wǎng)格：等差網(wǎng)格：指數(shù)網(wǎng)格：二、混合法先利用方法生成一個(gè)方向（平面內(nèi)的代數(shù)網(wǎng)格），再利用TTM方法生成另一個(gè)方向（平面）內(nèi)的網(wǎng)格，最后將所生成的網(wǎng)格聯(lián)接成一個(gè)整體網(wǎng)格對(duì)收擴(kuò)噴管：橫截面用TTM方法，而軸向橫截面用代數(shù)法。7-6動(dòng)網(wǎng)格設(shè)計(jì)非定