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