[PDF] - Analysis for Steady Propagation of a Generic Ram Accelerator/ PDF Document

SLIDE 1

Analysis for Steady Propagation

f a Generic Ram Accelerator/

Oblique Detonation Wave Engine Configuration

J.M. Powers1, D.R. Fulton2, K.A. Gonthier3, and M.J. Grismer4

Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana prepared for the

A/AA 31st Aerospace Sciences Meeting and Exhibit

1 Assistant Professor

2 Undergroduate Student 3 Graduate Assistant 4 Graduate Assistant

January 11-14, 1993 Reno, Nevada

SLIDE 2

Support

This study was supported by the Indiana Space Consortium sponsored by NASA Headquarters.

SLIDE 3

Objective of Study

Describe a methodology to determine a steady

propagation speed of a projectile fired into a gaseous fuel and oxidizer mixture.

Perform a simple theoretical and numerical analyses

to illustrate the methodology.

SLIDE 4

Inert Oblique Shock

M=8.4 P=16bar

' ' Reaction-inducing Oblique Shock

,,' (Oblique Detonation)

Projectile ·m=70g

M>l

P~

600 bar

s s

~-

v = 2,475 m/s Accelerator Barrel 166mm

Ram Accelerator, Hertzberg, et al., 1988, 1991

fuel

-- -

. .

inlet

~-~-

m1x1ng

-- -------
-- -

zone

~-

;;;::::: -
blique

detonation

Oblique Detonation Wave Engine, Dunlap, et al., 1958

SLIDE 5

Selected Past Work

1. Theoretical:
Brackett and Bogdanoff, 1989, (steady speeds)
Cambier, Adelman, and Menees, 1989, 1990,
Pratt, Humphrey, and Glenn, 1991,
Yungster and Bruckner, 1992, (steady speed)
Powers and Stewart, 1992.
Powers and Gonthier, 1992a,b, (steady speeds)
Grismer and Powers, 1992,
Pepper and Brueckner, 1993
2. Experimental:
Hertzberg, Bruckner, and Knowlan, 1988, 1991.

SLIDE 6

Modeling Difficulties

Multi - dimensional unsteady flow field Diffusive processes:

mass diffusion,
momentum diffusion,
energy diffusion.

Complex chemistry:

multiple reactions,
multiple species,
complex chemical kinetics.

Complex wave interactions:

.

compression waves,

.

expansion waves,
combustion waves.

SLIDE 7

incon1ing __..

.

supersonic premixed --.. flow

y

Generic Configurations

upper cowl surface

H L

lower cowl surface

axis of symmetry

SLIDE 8

Non-Dimensional Model Equations

Continuity:

dp

avi

d +pa =O,

t

X.

1

Momentum: Energy:

dP P dp ( ) ( ) (- 8)

dt

y p

dt = y-1 p K q 1-A. exp T

,

Species: Caloric Equation of State:

1 p

e=

-Aq,

y-1 p

Thermal Equation of State:

P=pT.

SLIDE 9

Non-Dimensional Variables

p
~

P =-=-::- , P=f

T=

R

T~

~ ~

'

-

'

Po Po

Po/Po

,....., ,.....,

~

U=

U V V

,.; p

0 I Po '

= ,.; p

0 I Po

e - e

'

- -

'

Po/Po

~

x=~

'y=Y

L

,_-:

L

~Po/Po

t =

~

t .

' L

Non-Dimensional Parameters

~

q = _ q_ 8 = E

y = 1 + ~

, Po/po '

Po/po '

Cv

K=-L__

M _

Uo

,.; Po!Po '

0 -

~ 'Y Po/Po

.

SLIDE 10

Reaction Model

Simple one-step, irreversible reaction:

'A

A

..... B

(exothermic reaction)

A

= reaction progress variable

Arrhenius kinetics:

kinetic _ rate oc exp(

Ea / RT)
high activation energy limit.

SLIDE 11

Thermal Explosion Theory

Reduced Equations (assumed v. = 0):

I

~

= (

y-1) p

K q

(

1-A) exp(- ~ p) ,

~

=

K(l-A)exp(-~p)

,

P(O) = P1 , A(O) = 0 .

Linearize the equations:

P = P1 + P' , A= "A' .

where

P' << 1 ,

"A' << 1 .

Solve for the pressure perturbation P'.

SLIDE 12

Solution:

1

8

1

P' Pi I

= -

+ n

2 ( )

1 -

~

(

y-1) q K exp

p p t

0 Pi

P1

1

Solve for the thermal explosion time: (corresponds to the induction time) Induction distance:

y I I I I I

1 ~

Lead Shock

I

x

SLIDE 13

1. Wave Drag:

1 ~Lead

Shock

I I

P1

~

Flame Sheet

Calculation of Surface Forces

2. Net Thrust Force:

Fnet = P3 (2 Lind cos 8 - 1) tan 8

+ [P4 (2 - 2 Lind cos 0) - P1] tan 0 .

3. Combustion Induced Thrust:

Fe = Fnet + Fv .

SLIDE 14

where

Jump Relations Across Lead Shock 1 + yM5 sin

2 ~

+~A

l

Pi= ( y

+ 1) M5 sin

2 ~

Ui =

.JY Mo (ti sin

2 ~ + cos

2 ~)

Vi= .JY"Mo cos~

sin~

(

1 - ti)

A =

(

1 + y M5 sin

2 ~

) 2

( y + 1) y M5 sin

2 ~

x (2 + (y- 1

)M5 sin

2 ~)

SLIDE 15

Flow Expansion Region

Prandtl - Meyer Function:

v(M )

/Y+f

!~

y-1 (

2 )

1~

2

3 =

"\/ y:1 tan

y+ l M 3 - 1 - tan

M 3 - 1

Isentropic Relations: Velocity Components: 1

_1

1 + y-

Mi y-1

p3 -

2

Pl

l + y;l M~

1

_J_

1 + y-

Mi y-1

P3

2

P1

l +

y-1 M~ 2

U3 = M3 - l.Y3

cos e '

V3 = - M3 . f?3 sine

11 ·yp;-

SLIDE 16

Jump Relations

·Across Flame Sheet p4 _ .1 + y

M~

±ill

i

p3

. (y+ l)M~

where

B =

(1 + yM~)

2 -(y- l)M~

x (

2 + ( y-1 )

M~

+ 2 (

y-1 ) ~:

Q)

SLIDE 17

where

Tail End Compression Region

1 + yM~

sin,a+e) ± {E -I

(y+ l)M~ sin,a+S) C =

(1 + yM~

sin,a+s))2-(y+ l)'YM~ sin,a+S) x (2 + (Y- 1) M~

sin~a+S)

SLIDE 18

Numerical Analysis

RPLUS Code:

developed at NASA Lewis,
based on LU-SSOR numerical scheme.

Computational Grid:

199 x 99 fixed grid.

Convergence:

500 iterations, .
residual unsteady terms had scaled values
8

< 1.0 x 10 .

Computations:

run on IBM RS/6000 POWERstation 350
run time about one hour.

SLIDE 19

Parameters

Geometric:

8 = 5°, L = 0.10 m .

Kinetic:

k = 1.0 x 107 /sec, E = 1.019 x 106 J/kg, 1.295 x 106 J/kg < q < 1.704 x 107 J/kg.

Atmospheric free-stream conditions:

~

Po= 1.01325 x 105 Pa , p0 = 1.225 kg/m3 •

Thermodynamic constants:

7

~

Y= 5 , R = 287 J/(kg K) , cv = 717.5 J/(kg/K).

SLIDE 20

0.030 0.020 0.010 0.000

0.010

~

....

0.020

~

Velocity (m/s)

2000 2500 3000

Net Thrust, Numerical

200

200
0.030 ~._._._~._._~

,,-.-,

ca

c

.......

(/.)

0.20

5

0.10 8

.......

Q

I

c

z

"-"'

....

(/.)

~ ....

z -0.10

....

5

6 7 8 9 M0 , Mach Number Velocity (m/s)

4000 5000 6000 7000

Net Thrust, Rankine- Hugoniot

e- q = 18.13, q

= 1.50x106 J/kg

A- q =

16.32, q = 1.35x106 J/kg

~

q = 14.51, <I= 1.20x106 J/kg

10

2000

1

000

:::s'"Ij

~ .......

-- - -------

z

1 ooo a

"-"'

~

0. 2 0 '-'-'-'-'-~-l.-.J'-'-'-._._,_._._._~_._._._,_-'-=
2 0 0 0

10 12 14 16 18 20 22

M0 , Mach Number

SLIDE 21

15

,.-....

]

.-I

CZl

Pressure on Wedge Surf ace

I

0.00

x

(m)

0.05

M0 = 17.53 ----

(stable)

0.10

q = 18.13

q = 1.5 x 106 J/kg

1500

~

10

Q.)

s

.-I

Q

I

~ 0

z

"-" _...._

M 0 = 13.4 (unstable)

~

500

0.2

0.0 0.2 0.4 0.6 0.8 1.0

1 .2

x (Non-Dimensional)

rise on forebody due to shock and reaction
drop on aftbody due to rarefaction
lack of crisp shock indicates more

resolution necessary

propagation speed sensitive to local pressure
trends plausible

SLIDE 22

ro

a..

(l)

'-

:J

en en

(l)

'-

a..

Pressure traces on wedge surface.

k=1x10A7 Ea=3550 1500000.0 .-----..-----.-----.-----.----..------..------.-----, 1000000.0 500000.0

Mach 13.4 Inert

lil·-·-~

Mach 13.4 Q = 1.5x1 OA6

~-·<:·

Mach 17.53 Inert

*-~

Mach 17.53 Q = 1.5x1 OA6

~.:.,-,.,-~-·eft=·

· er-1):

0.0 L-----L---...1----.l.-----l-----'----..._

_ _____. __ ____.

0.15 0.20 0.00 0.05 0.10

x {m)

SLIDE 23

~

"8

z

.c

~

~ ~

lo-c C1)

"8

z

..c:: u

c;rj

~ ~

10 9 8 7

q, Heat of Reaction (J/kg)

1.300X106 1.310X106 1.320X106 1.330X106 3000 ~

('I)

........

Rankine-

(")

.......

Hugoniot

....

'<

,,--..

Analysis

~

2500 '-'

6.__._.__._._._._._..._._._...l....1...J._._._...__._.__._._._._._

..............

_._._._._._._._._._._~.1.1.J'-'-'-

..............

_._.__.

15.60 15.70 15.80 15.90 16.00 16.10 q, Heat of Reaction (Non-Dimensional)

q, Heat of Reaction (J/kg)

1.40x106 1.50x1o6 1.60x106

1.70x106 20 18

quasi-stable

6000

~

1 6

........

Numerical

.......

....

'<

Analysis

,,--..

14

~

c;I)

'-'

12 4000 10 16 17 18 19 20

21 q, Heat of Reaction (Non-Dimensional)

SLIDE 24

..

ca

~ 0

.......

rl:l

5

.§

Q

I

§

6

:>..

..

ca

~ 0

.......

rl:l

5

8

.......

Q

I

~ 0

6

:>..

0.00 0.40 0.30 0.20 0.10 0.00 0.40 0.30 0.20 0.10

x (m)

0.02 0.04 0.06 0.08 0.10 0.12 0.04

Quasi-Stable Configuration M0 = 17.53, u

0 = 5,965 mis

q =

18.13, q

= 1.5 x 106 J/kg

"<l

Reactant mass fraction

0.02 s

'-"

(1-A.) contours

=~j,_.L.-'-~

0. 0 0

0.2 0.4 0.6 0.8 1.0 1.2

x (Non-Dimensional)

x (m)

0.02 0.04 0.06 0.08 0.10 0.12 0.04

Unstable Configuration M0 = 13.4, u

0 = 4,560 m/s

q =

18.13, q

= 1.5 x 106 J/kg

"<i

0.02 s

'-"

0.

00 L.dSsm~RSl-

=~-'-'-.L.-.J

0. 0 0

0.

0.2 0.4 0.6 0.8

1 .0 1 .2 x (Non-Dimensional)

SLIDE 25

x (m)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.40 0.04

Quasi-Stable Configuration

0.30

M0 =

17.53, u

0 = 5,965 mis

~ s::

q

= 1.5 x 106 J/kg

·-

q = 18.13,

C'IJ

5

'-<: t

s

0.20 0.02 -

·-

a

s

6

'-"'

Pressure (P) contours

e

>-.

0.10

~="'"',._L-L-_._._,0.0

0.2 0.4 0.6 0.8 1 .0 1 .2 x (Non-Dimensional)

x (m)

0.00 0.02 0.04

0.06 0.08 0.10 0 .12

0.40

0 .04

Unstable Configuration

0.30

M0 = 13.4, u

0 = 4,560 mis

..-

ro

q

= 1.5 x 106 Jlkg

·-

q=l8.13,

en

<!)

'-<: i

s

0.20 0.02 3

·-

a

I

'-"'

Pressure (P) contours

z

_...

>-.

0. 10

0 .00 loi.:IC~

=~~=~

1-L_._.L.J

0 .

00

0 .0 0 .2

0. 4
0. 6

0.8 1 .0 1 .2

x (Non-Dimensional)

SLIDE 26

Conclusions

1. The interaction of kinetic length scales with geometric

length scales are important for determining steady propagation speeds.

2. Bifurcation phenomena observed for equilibrium

Mach numbers:

high Mach number solutions stable to quasi-

static perturbations.

low Mach number solutions, unstable.
3. Near the bifurcation point, increasing heat release

increases steady propagation speed

4. Qualitative agreement exists between theoretical and

numerical results.

5. Higher resolution required for quantitative accuracy.