一维CFD模拟仿真设计.docx
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一维CFD模拟仿真设计.docx
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一维CFD模拟仿真设计
CFDsimulationinLavalnozzle
SIAE090441313
Abstract
WeaimtosimulatethequasionedimensionflowintheLavalnozzlebasedonCFDcomputationinthispaper.Weconsiderthechangeofthetemperature,thepressure,thedensityandthespeedoftheflowtostudytheflow.TheanalyticsolutionoftheflowintheLavalnozzleisprovidedwhentheinputvelocityissupersonic.WeusetheMac-CormackExplicitDifferenceSchemetoslovethequestion.
Keywords:
Lavalnozzle,CFD,throatnarrow.
Contents
Abstract................................................................................................1
Introduction..........................................................................................2
Simulationofone-dimensionalsteadyflow..........................................3
Basisequations...............................................................................3
Dimensionless...............................................................................10
Mac-CormackExplicitDifferenceScheme..................................11
Boundaryconditions.....................................................................13
Reference........................................................................................13
Annex..............................................................................................14
Introduction
Lavalnozzleisthemostcommonlyusedcomponentsofrocketenginesandaero-engine,constitutedbytwotaperedtube,oneshrinktube,anotherexpansiontube.
Lavalnozzleisanimportantpartofthethrustchamber.Thefirsthalfofthenozzlefromlargetosmallcontractiontoanarrowthroattothemiddle.Narrowthroatandthenexpandoutwardsfromsmalltobigtotheend.Thegasintherocketbodybythefronthalfofthehighpressureintothenozzle,throughthenarrowthroattoescapebytherearhalf.Thisarchitectureallowsthespeedoftheairflowchangesduetochangesinthejetcross-sectionalarea,theairflowfromsubsonictothespeedofsound,untilacceleratedtotransonic.So,peopleflarednozzlecalledtransonicnozzle.SinceitwasinventedbytheSwedishLaval,alsoknownasLavalnozzle.AnalysisoftheprincipleoftheLavalnozzle.Therocketenginesofthegasflowinthecombustionchamberunderpressure,afterthebackwardmovementofthenozzleintothenozzle.Atthisstage,thegasmovementfollowtheprincipleof"thefluidmovesinthetube,thesmallcross-sectionattheflowratelargesectionallargeflowvelocity",thusacceleratingairflow.
Lavalnozzle
Whenyoureachthenarrowthroat,theflowratehasexceededthespeedofsound.Transonicfluidmovementtheynolongerfollowtheprincipleof"cross-sectionatsmallvelocity,ataflowrateofsmallcross-sectionlarge",butonthecontrarythelargercross-sectionalflowfaster.Thegasflowspeedisfurtheracceleratedto2-3km/sec,equivalentto7-8timesthespeedofsound,thuscreatingagreatthrust.TheLavalnozzlefactplayedtheroleofa"flowrateEnlargementDevice".Infact,notjustrocketengines,missilenozzleisthishornshape,sotheLavalnozzleweaponshasaverywiderangeofapplications.
Simulationofone-dimensionalsteadyflow
1.Basisequations
Asweknow,Lavalnozzleisazoomingnozzleflowchanneltonarrowfurtherexpansion.Allowstheairflowtofurtheracceleratetoreachthespeedofsoundatthethroatintoasupersonicflow.Now,wewanttosimulatethequasione-dimensionflowing.Firstly,wewillanalysisontheory.Theflowisisentropic,sowecanapplythefollowingequations.
(1)Continuityequation:
Intheflow,weneedtoconsiderthefollowingphysicalquantities.Thepression,thetemperature,thespeedofthefluidandthecross-section.TheyarerespectivelyrepresentedbyP,T,u,A.Weapplytheconservationofthemass.wewillobtainthisequation.
Andthenweget
(2)Equationofmomentum(inthedirectionoftheaxis)
Accordingtothetheoryofmomentum:
Thesimplificationofthisequationis
(3)Energyequation
Idealgasequationofstate
Risidealgasconstant,R=8.314J/g/K.
Misthemassepermole.
(4)TheequationofThermodynamics
Becausetheflowisisentropic,so
dS=0
Andweusetheequationofmomentum,wehave
Combinewithothersequations,weresultwith
Wecalleduthespeedofsound,wenoteda.
Weapplythecontinuityequation
WedefinedtheMachnumber
Ifwehavetherelationas
Wehavethefigure
So
M>1,supersonic
IfdA<0,wehavedu>0.
IfdA>0,wehavedu<0.
M<1,subsonic
IfdA<0,wehavedu>0.
IfdA>0,wehavedu<0.
Thisisthereasonwhythisarchitectureallowsthespeedoftheairflowchangesduetochangesinthejetcross-sectionalarea,theairflowfromsubsonictothespeedofsound,untilacceleratedtotransonic.
Wehavetheconsequenceasfollows
ThenwereplacePandTinthisequation.Theconsequencewillbecome
Tosimplify
Inthisequation,thevariableisthemuchnumber,asthespeedoftheflowisfromsubsonictosupersonic,sowecansupposethatthereexistacriticalsectionwhereMequalto1.Then
Figure
Thissectioniscallednarrowthroat.Thesamemethod,wecanobtain
Figure
Weknowthesectioninnarrowthroatisminimum.
wecanjudgethatthefunctionattainsthemaximumornot
2Dimensionless
CombiningCFDwithone-dimensionflowtheory,wemakethevariablesdimensionless.Accordingtotheconditioninitialwhichisgiven.Wenote
Weusethevariabledimensionlesstorepresenttheequations.Andtheequationshavethefollowingchanges
(1)Continuityequation
(2)Equationofmomentum
(3)Energyequation
3.Mac-CormackExplicitDifferenceScheme
ThenweusetheMac-CormackExplicitDifferenceScheme,theprincipalofthistheoryisusingthesurroundingpointstopresentdifferentialpartsofapointandweconsiderthequestionwithonedimension.Thedistancebetweentwopointsish.
Sowecanusetwopointsadjacenttopresentthedifferentialparts.
Usingthismethod,wemakeanestimationandcorrecttheerror.
Estimation
correcttheerror
Intermediatevalue
Thentheequationhasthefollowingchange
Atthemomentt,wewillknowthevalueinthewholeplan.
Andwedefine
4.Boundaryconditions
Hyperbolaequationhastwocharacteristicslines.Whenoneofthecharacteristicslinesentertheflowzone.Weadmitaparametertobefixed,otherwisewhenoneofthecharacteristicsgoouttheflowzone,weadmitaparametertobeavariabledependsthetime.Applyingthistheory,wecandeterminetheboundaryconditions.
Reference:
[1]章利特,高铁瑜,夏庆锋,徐廷相.拉瓦尔喷管的准一维定常流动.中国科技论文在线。
[2]王平,昌平,柏松.基于CFD数值模拟的拉瓦尔喷管流场分析.航空计算技术2012年7月第42卷第4期。
[3]王如根,瑞贤,全通.基于实际发动机拉瓦尔喷管的流场分析.99学术会议空军工程学院飞机推进系统实验室。
[4]周文祥,黄金泉,周人治.拉瓦尔喷管计算模型的改进及其整机仿真验证.航空动力学报2009年11月第24卷第11期。
[5]王克印,韩星星,晓涛,耀鹏,吉潮.缩扩型超音速喷管的设计与仿真.中国工程机械学报2011年9月第9卷第3期。
Annex1:
Figure1程序:
x=0:
0.1:
5;a=1.398+0.374*tanh(0.8*x-4);plot(x,a)
Figure2程序:
gama=1.33;
M=0:
0.01:
2;
A=(1./M).*((1+(gama-1).*M.^2./2)./((gama+1)./2)).^((gama+1)./(2.*(gama-1)));
Xlabel('variableM');
ylabel('variableA');
Plot(M,A)
Figure3程序:
gama=1.33;
M=0:
0.01:
2;
T=((gama+1)./2)./(1+(gama-1.).*M.^2./2);
P=(((gama+1)./2)./(1+(gama-1).*M.^2./2)).^(1./(gama-1));
rho=(P./T).*(((gama-1)./2)./(1+(gama-1).*M.^2./2)).^(gama./(gama-1));
x=[T;P;rho]';
y=[M;M]';
plot(M,T,M,P,M,rho);
subplot(221);plot(M,T);xlabel('variableM');ylabel('variableT');
subplot(222);plot(M,P);xlabel('variableM');ylabel('variableP');
subplot(2,2,3:
4);plot(M,rho);xlabel('variableM');ylabel('variablerho');
模拟程序(未完成):
M1=1.5;%input('来流马赫数M1=');
P1=47892.4;%input('来流气体压强P1=');
rho1=1.222;%input('来流气体密度rho1=');
gama=1.4;%input('比热比gama=');
R=8.314;%input('气体常量R=');
C=1.5;%input('科朗数C=');
T1=293;%input('来流气体温度T1=');
a1=sqrt(gama*R*T1);
V1=M1*a1;
L=10;%input('喷管长度L=');
I=300;%input('等分步数I=');
N=1000;%input('时间步数N=');
d_t=0;
e=0;
delta_x=1/(I-1);
A1=1.398-0.347*tanh(4);
A=zeros(I,1);V=zeros(I,N);rho=zeros(I,N);T=zeros(I,N);
ahead_V=zeros(I,N);ahead_rho=zeros(I,N);ahead_T=zeros(I,N);
ahead_der_V=zeros(I,N);ahead_der_rho=zeros(I,N);ahead_der_T=zeros(I,N);
der_V=zeros(I,N);der_rho=zeros(I,N);der_T=zeros(I,N);
ave_der_V=zeros(I,N);ave_der_rho=zeros(I,N);ave_der_T=zeros(I,N);
delta_t=zeros(I,N);
a=zeros(I,N);
e=zeros(I,N);
forn=1:
N
ahead_V(1,n)=M1;
ahead_rho(1,n)=1;
ahead_T(1,n)=1;
V(1,n)=M1;
rho(1,n)=1;
T(1,n)=1;
end
fori=1:
I
V(i,1)=(1.5+1.09*i*delta_x)*sqrt(T(i,1));
rho(i,1)=1-0.3146*i*delta_x;
T(i,1)=1-0.2314*i*delta_x;
A(i,1)=(1.398+0.347*tanh(0.8*(i-1)*delta_x*L-4));
end
forn=1:
N
fori=2:
(I-1)
der_V(i,n)=-V(i,n)*(V(i+1,n)-V(i,n))/delta_x-1/gama*((T(i+1,n)-T(i,n))/delta_x+T(i,n)/rho(i,n)*(rho(i+1,n)-rho(i,n))/delta_x);
der_rho(i,n)=-rho(i,n)*(V(i+1,n)-V(i,n))/delta_x-V(i,n)*(rho(i+1,n)-rho(i,n))/delta_x-rho(i,n)*V(i,n)*(log(A(i+1,1))-log(A(i,1)))/delta_x;
der_T(i,n)=-V(i,n)*(T(i+1,n)-T(i,n))/delta_x-(gama-1)*T(i,n)*((V(i+1,n)-V(i,n))/delta_x+V(i,n)*(log(A(i+1,1))-log(A(i,1)))/delta_x);
a(i,n)=sqrt(gama*R*T(i,n))/a1;
delta_t(i,n)=C*delta_x/(V(i,n)+a(i,n));
end
d_t=min(delta_t(i,n));
fori=2:
1:
(I-1)
ahead_V(i,n+1)=V(i,n)+der_V(i,n)*d_t;
ahead_rho(i,n+1)=rho(i,n)+der_rho(i,n)*d_t;
ahead_T(i,n+1)=T(i,n)+der_T(i,n)*d_t;
ahead_der_V(i,n)=-ahead_V(i,n+1)*(ahead_V(i,n+1)-ahead_V(i-1,n+1))/delta_x-1/gama*((ahead_T(i,n+1)-ahead_T(i-1,n+1))/delta_x+ahead_T(i,n+1)/ahead_rho(i,n+1)*(ahead_rho(i,n+1)-ahead_rho(i-1,n+1))/delta_x);
ahead_der_rho(i,n)=-ahead_rho(n+1)*(ahead_V(i,n+1)-ahead_V(i-1,n+1))/delta_x-ahead_V(i,n+1)*(ahead_rho(i,n+1)-ahead_rho(i-1,n+1))/delta_x-ahead_rho(i,n+1)*ahead_V(i,n+1)*(log(A(i,1))-log(A(i-1,1)))/delta_x;
ahead_der_T(i,n)=-ahead_V(i,n+1)*(ahead_T(i,n+1)-ahead_T(i-1,n+1))/delta_x-(gama-1)*ahead_T(i,n+1)*((ahead_V(i,n+1)-ahead_V(
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