AE 5347/MAE 4322
Rocket Propulsion
04(4) - Nozzle Design
Numerical solution : 2-D, Steady, IrrotationalMethod of Characteristics (ZH 16.3)
Basic Equations
Governing Equations
Characteristic/Compatibility Equations
Characteristic equation
22 2 2 2- - 2 - 0
-
, (through the flow field)
x y y
y x
avu a u v a v uvu
y
u v o
a a a u vV
tan (Mach Lines)dy
dx
Compatibility equation
2 2 2 2 2- 2 - - - 0 (along Mach line)
u a du uv u a dv a v y dx
4 4 2 2
4 - 4 2 - 2
4 4
- 4 - 4 -
4 2 2 2
- 4
2
2
1
2
2
-
Computational Equation for Steady Two-Dimensional Irrotational
Supersonic Flow
- -
- -
,
,
-
2 - ( - )
-
-
y x y x
y x y x
Q u R v T
Qu R v T
T S x x Q u R v
T
Q u a
R uv u v
S x x
- 1 - 1
2 2
-1
-1
2 2 2
- 1 - 1 - 1
2 4 2 4 2 4
1 4-
tan
1sin
; ; (predictor)
; ; (predictor)
; ; (corrector)2 2 2
Qu R v
V u v
v
u
a V
VM
a
M
u u v v y y
u u v v y y
u u v v y yu v y
u uu
1 4 1 4- - ; ; (corrector)
2 2 2
v v y yv y
2 2
2 2
2
-
Finite Defference Equations for Two-dimensional Irrotational
Supersonic Flow
- 0
tan
-
2 - -
+ or - denotes and characteristic, respectively
y x
Q u R v S x
Q u a
R uv u a
a vS
y
C C
Note For 2-D planar isentropic flowThe Compatibility Equations can be integrated to
Standard Solution Techniques Available For:Interior Points
Wall Boundary Points
Pressure Boundary Points
Wave Cancellation Surface
Shock Intersection
- -
: - =constantReimann invarients
: =constant
streamline inclination angle
Prandtl-Meyer function
C K
C K
1 1
1 1
- 1 1
1 1
: -
- -
:
- - -
C const K
or
C const K
or
Thus along
K
K
IV line
C
C
Interior point
3
3
3 3 1 2
3 3 1 2
3 3 1 2
3
2
- 3 3 - 1 1
3 3 2 2
3 - - 1 2 1 2
3 - - 1 2 1 2
3 - - 1 2 1 2
-1
3 -
-
:
: - -
2 -
1 1 1 -
2 2 2
. 2 - - -
1 -
2
C K
C K
add K K K K
K K K K
sub K K K
K
K
K
K K
3 1 2- 1 2 1 2
1 1- - -
2 2K K
Solid Boundary
3 1
3
3
3 3 1 1
3 3 3
1 1 3
3 3
( ) C Characteristic
(1) ;
(2)
-
( ) C Chara
A
K K
known K
K
B
3 1
3
3
3 3 1 1
3 3 3
1 1 3
- 3 3
cteristic
(1) ; - -
(2) -
-
K K
known K
K
K
K
C
K
K
CC
C
1
3
3
1
Wave Cancellation
3 1
3
3 1
3 3 1 1
3 1 3 1
- 3 3 1 1
-1
- -
since
-
thus right running characteristics are uniform (PM flow)
Set
K K
K
K
K
K
CC
3
1
Free-Jet Boundary
3 1
3
3
3 1
-
- - 3 3 1 1
33
30
3 - 3 1 1 3
3 3
+
3 3 1 1
(A) C Characteristic
(1) ;
(2)
- -
-
(B) C Characteristic
(1) ; - -
(2)
K K
Mpknown
p
K
K
K K
3
3
32
30
3 3 1 1 3
- 3 3
-
Mpknown
p
K
K
C
C
K
K
CC
1
1
3
3K
K
Intersection of characteristic & shock wave
3
13
3
1 1 1
3
33 3 3 3
3
3
3
?
3
Data:M
, &
Iteratiave Procedure
(1) Assume
(2) Oblique shock solution:
, , , ,
(3) calculate -
(4) Converge
(5)
M
pM f M
K
K
p
K
K
Minimum length nozzle
-
- -
@C: 0
along ac:
since a is origin of PM Exp. Wave
or
( )
2
c
c a
a a
a
a
a
a
a
c c M M
M w w
M w
w w
Mw
w
K
K
f M
K
:Max
Nozzle design: Me=2.4
1
1
0
-01
0
0
0
- -
-
18.3752
assume 0.375
along a1: 0.375
0.375 ( )
0.375
PT-1 0.75
- 0
1
2
a
Mw
ab
PM Exp
K K
K K
K K
0
0
-
10.75 0.375
2
1 1 - 0.75 0.375
2 2K K
0 0 0
-
0 0
0 0
-
0 0
-
PT-2
3.375 3.375 6.75
- 3.375 -3.375 0
1 1 6.75 3.375
2 2
1 1 - 6.75 3.375
2 2
K
K
K K
K K
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