the method of characteristics - virginia techdevenpor/aoe3114/20 - method of... · 2d isentropic...

The Method of CharacteristicsThe Method of Characteristics

2D Isentropic Compressible Flow2D Isentropic Compressible Flow

22 MASS+MOMENTUMIRROT.

02)1()1( 22

2

2

2

xv

auv

yv

av

xu

au0

yu

xv

ORGANIZE SO AS TO SOLVE FOR THE VELOCITY DERIVATIVESDERIVATIVES…

Partial ff l

TO MATRIX SOLVE FOR VELOCITY DERIVATIVES…

vuvvvuu 02)1()1(22

Differential Eqns.? dv

xvdx

yvdy

xayaxa

0)1()1( 222

qdu

xvdy

xudx

xy

y, v U

x, u

Mach 2 Nozzle Flow

Matrix Solution (Cramer’s Rule)Matrix Solution (Cramer s Rule)

dvyvxu

dxdyauvavau 0

//

0/2)/1()/1( 22222

dudv

xvyv

dydxdxdy

//

00

dvdyavau

00)/1()/1( 2222

SOLVING FOR

v/x DN

auvavau

dudxdvdy

xv

/2)/1()/1(

00

/22222

v/x…

dydxdxdy

auvavau

00

/2)/1()/1(

Denominator Ddxdyauvavau

0/2)/1()/1( 22222y, v U

dydx

y0x, u

CHARACTERISTIC EQUATIONS

Characteristic LinesCharacteristic Lines)tan(

dxdy

)tan( dxdy

M>1

Numerator dvdyavau

00)/1()/1( 2222

y, v U Numerator

DN

dxdyauvavau

dudx

0

/2)/1()/1(

022222

x, u

COMPATABILITY RELATIONS

dydxdxdy

00

Characteristics ‐ SummaryCharacteristics Summary

C+ (left running)C (left running)‐ = const.

Streamline

C‐ (right running)+ = const.

Reflected Expansion Fan

C+

C‐

d

e

g i1

3

M1

e

f

hl

‐‐/2

a b ck

2

/2j


C+

SOLVING FOR FLOW PROPERTIES IN THE UNIFORM AND SIMPLE REGIONS

C‐

1

2

GivenM =2

d

e

g i1

3

3

a

bGivenM1=2and so 1 = 26.38oand 1 = 30o

and geometry

e

f

hl

c

d

‐10o‐5o

a b ck

2

e

f

g105j h

i


C+

SOLVING FOR FLOW PROPERTIES IN THE COMPLEX REGION

C‐

1

2

d

e

g i 3

3

a

bGivenM =21

e

f

hl

c

d

GivenM1=2and so 1 = 26.38oand 1 = 30o

and geometry

‐10o‐5o

a b ck

2

e

f

g105j h

i


C+

SOLVING FOR THE WAVE GEOMETRY

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

PowerPoint Solution

e

f

hl

5 31.4 27.2

c ‐10 36.4 24.8

d 0 26.4 30


and geometry

‐10o‐5o

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. Wave originates at a sudden turn

C‐

d g i 3

GivenM =21

e

f

hlGivenM1=2

and so 1 = 26.38oand 1 = 30o

and geometry

‐k

2O

‐j

Variation

C+

2. Wave reflects from a jet edge

C‐

dg

GivenM =21

e

fh i


and geometry

10o

2

3

O

‐10o

PowerPoint Solution

Variation

C+

3. Wave cancelation at a wall

C‐

gGivenM =2

1a b

e

fh i


c

d

10o

2

O

‐= 10o

Design ApplicationsDesign Applications

1 Constant Mach number turn1. Constant Mach number turn

2. Wind Tunnel Type Nozzle

3 k l3. Rocket Type Nozzle

1. Constant Mach number turna. Choose initial turn geometry abc of

angle /2b. Compute reflection of wave as though

f j t d f t t M hMfrom a jet edge of constant Mach M1. Shape of jet edge gives shape of upper wall.

c. Choose lower wall shape to cancel reflected wave at jkl

g1 d

e

f h i

3

GivenM1

‐ /2

a3

b c

‐ j k l

2. Wind Tunnel‐Type Nozzle

Straightening Section

Subsonic

Expansion Section

MMe

3. Rocket Engine‐Type Nozzle

i

MMe

PowerPoint Example

Matlab Codes and Scripts for the h d f hMethod of Characterisics

• Computing and Plotting Simple wavesComputing and Plotting Simple waves

• Computing and Plotting Complex Regions

O h f i• Other functions

• Application Examples

Basic Simple WaveBasic Simple Wavefunction [a,n,x,y]=simple(ai,ni,xi,yi,le,g)

Terminating characteristic

2

3

l

N

le

Flow

1 2 3 N

Incoming wave boundary

Other Simple Wave FunctionsOther Simple Wave Functionsfunction [a,n,x,y]= simpleCancel(ai,ni,xi,yi,c,x0,y0,a0,g)

Outgoing wave

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

Incoming wave boundary

Basic Complex WaveBasic Complex Wavefunction [a,n,x,y]=complex3(ai,ni,xi,yi,bc,g)

(1,1) (2,2)

(1,2)

(N,N)

bc ( , )

Other Complex Wave FunctionsOther Complex Wave Functionsfunction [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: Rocket Engine Nozzlei2/)( 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: Rocket l

i

Engine NozzleMe

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

Pre‐Packaged ScriptsPre Packaged Scripts

• Constant Mach Number Turn

• Minimum length nozzle

• Wave interaction• Wave interaction

• Underexpanded jet

• Wind tunnel nozzle4

3

2

3

2

2.5

5 10 15 20 250

1

1

1.5

the method of characteristics - virginia techdevenpor/aoe3114/20 - method of... · 2d isentropic...

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