-
The Method of CharacteristicsTheMethodofCharacteristics
-
2D Isentropic Compressible Flow2DIsentropicCompressibleFlow
22 MASS+MOMENTUMIRROT.
02)1()1( 222
2
2
xv
auv
yv
av
xu
au0
yu
xv
ORGANIZESOASTOSOLVEFORTHEVELOCITYDERIVATIVESDERIVATIVES
-
Partialff l
TOMATRIXSOLVEFORVELOCITYDERIVATIVES
vuvvvuu 02)1()1(22
DifferentialEqns.? dvx
vdxyvdy
xayaxa
0)1()1( 222
qdu
xvdy
xudx
xy
y,v U
x,u
Mach2NozzleFlow
-
Matrix Solution (Cramers Rule)MatrixSolution(Cramer sRule)
dvyvxu
dxdyauvavau 0
//
0/2)/1()/1( 22222
dudv
xvyv
dydxdxdy
//
00
dvdyavau
00)/1()/1( 2222
SOLVINGFOR
v/x DN
auvavau
dudxdvdy
xv /2)/1()/1(0
0
/22222v/x
dydxdxdy
auvavau
00
/2)/1()/1(
-
Denominator Ddxdyauvavau
0/2)/1()/1( 22222y,v U
dydx
y0x,u
CHARACTERISTICEQUATIONS
-
Characteristic LinesCharacteristicLines)tan(
dxdy
)tan( dxdy
M>1
-
Numerator dvdyavau
00)/1()/1( 2222y,v U
NumeratorDN
dxdyauvavau
dudx 0
/2)/1()/1(
022222
x,u
COMPATABILITYRELATIONS
dydxdxdy
00
-
Characteristics SummaryCharacteristics Summary
C+ (left running)C (leftrunning) =const.
Streamline
C (rightrunning)+ =const.
-
ReflectedExpansionFan
C+
C
d
e
g i1
3
M1
e
f
hl
/2a b c
k
2
/2j
-
ReflectedExpansionFan
C+
SOLVINGFORFLOWPROPERTIESINTHEUNIFORMANDSIMPLEREGIONS
C
1
2
GivenM =2
d
e
g i1
3
3
a
bGivenM1=2andso1 =26.38oand1 =30oandgeometry
e
f
hl
c
d
10o5o
a b ck
2
e
f
g105j h
i
-
ReflectedExpansionFan
C+
SOLVINGFORFLOWPROPERTIESINTHECOMPLEXREGION
C
1
2
d
e
g i 3
3
a
bGivenM =21
e
f
hl
c
d
GivenM1=2andso1 =26.38oand1 =30oandgeometry
10o5o
a b ck
2
e
f
g105j h
i
-
ReflectedExpansionFan
C+
SOLVINGFORTHEWAVEGEOMETRY
C
1 0 26.4 30
2 10 36.4 24.8P P i t S l ti
d
e
g i 3
3 0 46.4 20.5
a 0 26.4 30
b 5 31.4 27.2GivenM =21
PowerPointSolution
e
f
hl
5 31.4 27.2
c 10 36.4 24.8
d 0 26.4 30
GivenM1=2andso1 =26.38oand1 =30oandgeometry
10o5o
a b ck
2
e 5 36.4 24.8
f 10 36.4 24.8
g 0 36.4 24.8105j h 5 41.4 22.6
i 0 46.4 20.5
-
Variation
C+
1.Waveoriginatesatasuddenturn
C
d g i 3
GivenM =21
e
f
hlGivenM1=2
andso1 =26.38oand1 =30oandgeometry
k2O
j
-
Variation
C+
2.Wavereflectsfromajetedge
C
dg
GivenM =21
e
fh i
GivenM1=2andso1 =26.38oand1 =30oandgeometry
10o
2
3
O
10o
PowerPointSolution
-
Variation
C+
3.Wavecancelationatawall
C
gGivenM =2
1a b
e
fh i
GivenM1=2andso1 =26.38oand1 =30o
c
d
10o
2
O
= 10o
-
Design ApplicationsDesignApplications
1 Constant Mach number turn1. ConstantMachnumberturn
2. WindTunnelTypeNozzle
3 k l3. RocketTypeNozzle
-
1.ConstantMachnumberturna. Chooseinitialturngeometryabc of
angle/2b. Computereflectionofwaveasthough
f j t d f t t M
hMfromajetedgeofconstantMachM1.Shapeofjetedgegivesshapeofupperwall.
c. Chooselowerwallshapetocancelreflectedwaveatjkl
g1 d
e
f hi
3
GivenM1
/2a
3b c
j k l
-
2.WindTunnelTypeNozzle
StraighteningSection
Subsonic
ExpansionSection
MMe
-
3.RocketEngineTypeNozzle
i
MMe
PowerPointExample
-
MatlabCodesandScriptsfortheh d f hMethodofCharacterisics
Computing and Plotting Simple
wavesComputingandPlottingSimplewaves
ComputingandPlottingComplexRegionsO h f i Otherfunctions
ApplicationExamples
-
Basic Simple WaveBasicSimpleWavefunction
[a,n,x,y]=simple(ai,ni,xi,yi,le,g)
Terminatingcharacteristic
2
3
l
N
le
Flow
1 2 3 N
Incomingwaveboundary
-
Other Simple Wave FunctionsOtherSimpleWaveFunctionsfunction
[a,n,x,y]= simpleCancel(ai,ni,xi,yi,c,x0,y0,a0,g)
Outgoingwave
boundary
23
N
1
(x0 y0)
function simpleplot(a,n,x,y,g,cl,ch)
Flow
(x0,y0)a0
1 2 3 N1 2 3 N
Incomingwaveboundary
-
Basic Complex WaveBasicComplexWavefunction
[a,n,x,y]=complex3(ai,ni,xi,yi,bc,g)
(1,1) (2,2)
(1,2)
(N,N)
bc ( , )
-
Other Complex Wave FunctionsOtherComplexWaveFunctionsfunction
[a,n,x,y]=complex3curve(ai,ni,xi,yi,bc,g)
function [a,n,x,y]=complex3free(ai,ni,xi,yi,bc,g)
function [a,n,x,y]=complex4(ap,np,xp,yp,an,nn,xn,yn,g)
function complex3plot(a,n,x,y,g,cl,ch)function
complex4plot(a,n,x,y,g,cl,ch)
-
ExtrasExtrasfunction uniformplot(a,n,x,y,g,cl,ch)
function n=nu(m,g) function m=m_nu(n,g)
function [x,y]=intercept(x1,y1,t1,x2,y2,t2)
function [x,y,a]=interceptCurve(x1,y1,t1,cf)[ ,y, ] p ( ,y , ,
)
-
Example:RocketEngineNozzlei2/)( ei M
Me
function [a,n,x,y]=simple(ai,ni,xi,yi,le,g)
function [a,n,x,y]=complex3(ai,ni,xi,yi,bc,g)
function [a,n,x,y]= simpleCancel(ai,ni,xi,yi,c,x0,y0,a0,g)
Matlab DemoArocketEngineNozzle.m
-
Example:Rocketl
iEngineNozzle
Me
clear all %Mach 2 minimum length nozzle, initial turn angle
ai=0.23 radiansc ea a % ac u e gt o e, t a tu a g e a 0. 3 ad a
sg=1.4;ai=[.00001 .0001 .001 .005 .01:.01:.23]; %28
wavesxi=zeros(size(ai));yi=ones(size(ai));ni=ai;le=-1;figure(1);clf(1);cl=1;ch=2;
[a1,n1,x1,y1]=simple(ai,ni,xi,yi,le,g);simpleplot(a1,n1,x1,y1,g,cl,ch);
[a2,n2,x2,y2]=complex3(a1(end,:),n1(end,:),x1(end,:),y1(end,:),0,g);
%complex wave reflectioncomplex3plot(a2 n2 x2 y2 g cl
ch);complex3plot(a2,n2,x2,y2,g,cl,ch);
[a3,n3,x3,y3]=simpleCancel(a2(:,end),n2(:,end),x2(:,end),y2(:,end),1,xi(end),yi(end),ai(end),g);simpleplot(a3,n3,x3,y3,g,cl,ch);
uniformplot([a1(1,1) a1(2,1) 0],[n1(1,1) n1(2,1) 0],[x1(1,1)
x1(2,1) x1(1,1)],p ([ ( , ) ( , ) ],[ ( , ) ( , ) ],[ ( , ) ( , ) (
, )],uniformplot([a1(1,end) a1(2,end) a3(2,1)],[n1(1,end) n1(2,end)
n3(2,1)],uniformplot([a3(1,end) a3(2,end) a3(1,end)],[n3(1,end)
n3(2,end) n3(1,end)],
hold off;caxis([cl ch]);colorbar;axis image
-
PrePackaged ScriptsPre PackagedScripts
ConstantMachNumberTurn Minimumlengthnozzle Wave interaction
Waveinteraction Underexpandedjet Windtunnelnozzle
4 3
2
3
2
2.5
5 10 15 20 250
1
1
1.5
