Analysis for Steady Propagation of a Generic Ram Accelerator/ - PDF document

Analysis for Steady Propagation of 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 January 11-14, 1993 Reno, Nevada 1 Assistant Professor 2 Undergroduate Student 3 Graduate Assistant 4 Graduate Assistant

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

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.

~- -~- P~ ~-~- Reaction-inducing Inert Oblique Oblique Shock Shock ,,' (Oblique Detonation) ' ' M>l M=8.4 P=16bar 600 bar Projectile s s s s · m=70g v = 2,475 m/s Accelerator Barrel 166mm Ram Accelerator, Hertzberg, et al., 1988, 1991 oblique detonation fuel --- - . . inlet m1x1ng --- ------- --- - zone -;;;::::: - Oblique Detonation Wave Engine, Dunlap, et al., 1958

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.

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.

Generic Configurations upper cowl surface y incon1ing __.. . supersonic premixed --.. H flow L lower cowl surface 0 axis of symmetry

Non-Dimensional Model Equations Continuity: dp avi d +pa =O, t X. 1 Momentum: Energy: (- 8) - y p dP P dp ( ) ( ) dt dt = y-1 p K q 1-A. exp T , Species: Caloric Equation of State: 1 p --Aq, e= y-1 p Thermal Equation of State: P=pT.

~ x=~ ~Po/Po ~ ~ ~ ~ T~ ~ ~ Non-Dimensional Variables - p - P =-=-::- , P=f T= R - - ' ' Po Po Po/Po ,....., ,....., e - e U= U V V - - - ' ' 0 I Po ' 0 I Po ,.; p = ,.; p Po/Po - t = 'y=Y t . ' L L ,_ -: L Non-Dimensional Parameters y = 1 + ~ , 8 = E q = _ q_ Po/po ' Po/po ' Cv K=-L__ M _ Uo 0 - ,.; Po!Po ' ~ 'Y Po/Po .

Reaction Model - Simple one-step, irreversible reaction: 'A A ..... B (exothermic reaction) A = reaction progress variable - Arrhenius kinetics: rate oc exp( - Ea / RT) kinetic _ - high activation energy limit.

~ ~ K(l-A)exp(-~p) Thermal Explosion Theory Reduced Equations (assumed v. = 0): I y-1) p K q ( 1-A) exp(- ~ p) , = ( , = P(O) = P1 , A(O) = 0 . Linearize the equations: P = P1 + P' , A= "A' . where "A' << 1 . P' << 1 , Solve for the pressure perturbation P'.

~ Solution: 0 -8 1 1 P' Pi I = - + n 1 - ( y-1) q K exp p p t 2 ( ) • P 1 0 Pi 1 Solve for the thermal explosion time: (corresponds to the induction time) Induction distance: I Lead Shock 1 ~ I I I y I I x

~ Shock 1 ~Lead Flame Sheet I P1 I Calculation of Surface Forces 1. Wave Drag: 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 .

+~A sin~ Jump Relations Across Lead Shock 1 + yM5 sin -l 2 ~ Pi= + 1) M5 sin ( y 2 ~ Ui = .JY Mo (ti sin 2 ~ + cos 2 ~) Vi= .JY"Mo cos~ 1 - ti) ( where 2 A = ( 1 + y M5 sin - ( y + 1) y M5 sin 2 ~ 2 ~ ) - x (2 + (y- 1 )M5 sin 2 ~)

-1~ -!~ Flow Expansion Region Prandtl - Meyer Function: /Y+f y-1 ( 2 ) 2 v(M ) 3 = "\/ y:1 tan y+ l M 3 - 1 - tan M 3 - 1 Isentropic Relations: 1 _1 1 + y- Mi y-1 p3 - 2 ---- l + Pl y;l M~ 1 _J_ 1 + y- Mi y-1 P3 2 ------ l + y-1 M~ P1 2 Velocity Components: V3 = - M3 . f?3 sine U3 = M3 - l.Y3 11 ·yp;- 11 ·yp;- cos e '

M~ M~ Jump Relations ·Across Flame Sheet ±ill p4 _ .1 + y -i . (y+ l)M~ p3 where B = (1 + yM~) 2 -(y- l)M~ Q) x ( 2 + ( + 2 ( y-1 ) y-1 ) ~:

sin~a+S) Tail End Compression Region 1 + yM~ sin,a+e) ± {E -I (y+ l)M~ sin,a+S) where (1 + yM~ C = sin,a+s))2-(y+ l)'YM~ sin,a+S) x (2 + (Y- 1) M~

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.

~ ~ ~ Parameters Geometric: 8 = 5°, L = 0.10 m . Kinetic: k = 1.0 x 10 7 /sec, E = 1.019 x 10 6 J/kg, 1.295 x 10 6 J/kg < q < 1.704 x 10 7 J/kg. Atmospheric free-stream conditions: Po= 1.01325 x 10 5 Pa , p 0 = 1.225 kg/m 3 • Thermodynamic constants: 7 Y= 5 , R = 287 J/(kg K) , cv = 717.5 J/(kg/K).

~ ~ ~ ~ Velocity (m/s) 2000 2500 3000 0.030 Net Thrust, 0.020 200 Numerical 0.010 0.000 0 -0.010 .... -0.020 -200 -0.030 ~._._._~._._~ 5 6 7 8 9 10 M 0 , Mach Number Velocity (m/s) 4000 5000 6000 7000 0.20 2000 ,,-.-, Net Thrust, ca -e- q = 18.13, q = 1.50x106 J/kg c -A- q = 16.32, q = 1.35x106 J/kg 0 ....... Rankine- (/.) q = 14.51, <I= 1.20x106 J/kg 5 1 000 0.10 :::s'"Ij 8 ~ ....... Hugoniot ....... Q I c 0 z --- - ------- 0 "-"' .... (/.) z z -0.10 ~ .... -1 ooo a "-"' .... - 0. 2 0 '-'-'-'-'-~-l.-.J'-'-'-._._,_._._._~_._._._,_-'-= - 2 0 0 0 10 12 14 16 18 20 22 M 0 , Mach Number

~ ~ Pressure on Wedge Surf ace I x (m) 0.00 0.05 0.10 15 1500 q = 18.13 q = 1.5 x 10 6 J/kg ,.-.... M 0 = 17.53 ---- ] (stable) 0 •.-I CZl 10 Q.) s • .-I Q I ~ 0 _...._ z M 0 = 13.4 "-" 500 - (unstable) 0 0 -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

· er-1): ~.:.,-,.,-~-·eft=· *-~ ~-·<:· lil·-·-~ Pressure traces on wedge surface. k=1x10A7 Ea=3550 1500000.0 .-----..-----.-----.-----.----..------..------.-----, o Mach 13.4 Inert o Mach 13.4 Q = 1.5x1 OA6 Mach 17.53 Inert 1000000.0 Mach 17.53 Q = 1.5x1 OA6 - ro - a.. (l) '- :J en en (l) '- a.. 500000.0 0.0 L-----L---...1----.l.-----l-----'----..._ _ _____. __ ____. 0.15 0.20 0.00 0.05 0.10 x {m)

~ _._._._._._._._._._._~.1.1.J'-'-'- ~ ~ ~ ~ ~ ~ ~ ~ q, Heat of Reaction (J/kg) 1.300X10 6 1.310X10 6 1.320X10 6 1.330X10 6 10 9 3000 ~ "8 z ('I) ........ Rankine- 0 (") ....... .... .c Hugoniot '< 8 ,,--.. Analysis 2500 '-' 0 7 6.__._.__._._._._._..._._._...l....1...J._._._...__._.__._._._._._ .............. .............. _._.__. 15.90 16.10 15.60 15.70 15.80 16.00 q, Heat of Reaction (Non-Dimensional) q, Heat of Reaction (J/kg) 1.40x10 6 1.50x1o 6 1.60x10 6 1. 70x10 6 20 18 quasi-stable 6000 lo-c C1) "8 z ........ 1 6 0 0 ....... .... ..c:: Numerical u '< c;rj ,,--.. Analysis 14 c;I) 0 '-' 12 4000 10 16 17 19 20 21 18 q, Heat of Reaction (Non-Dimensional)

=~-'-'-.L.-.J =~j,_.L.-'-~ x (m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.40 0.04 Quasi-Stable Configuration 17.53, u M 0 = 0 = 5,965 mis -.. 0.30 ca 18.13, q = 1.5 x 10 6 J/kg q = ~ 0 ....... rl:l 5 "<l 0.02 s .§ 0.20 Reactant mass fraction Q '-" I § (1-A.) contours 6 :>.. 0.10 0. 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 13.4, u -.. 0.30 M 0 = 0 = 4,560 m/s ca ~ 0 ....... 18.13, q = 1.5 x 10 6 J/kg q = rl:l 5 "<i 0.02 s 8 ....... 0.20 Q '-" I ~ 0 6 :>.. 0.10 0. 00 L.dSsm~RSl- 0. 0 0 0.2 0. 0 0.4 0.6 0.8 1 .0 1 .2 x (Non-Dimensional)

-1-L_._.L.J =~~=~ ~="'"',._L-L-_._._,0.0 x (m) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.40 0.04 Quasi-Stable Configuration - 17.53, u 0.30 M 0 = 0 = 5,965 mis ~ s:: ·- 0 q = 1.5 x 10 6 J/kg q = 18.13, C'IJ 5 0.02 - '-<: t ·- s 0.20 s a 6 '-"' Pressure (P) contours e 0 >-. 0.10 0.2 0.4 0.6 0.8 1 .0 1 .2 x (Non-Dimensional) x (m) 0.00 0.0 2 0.04 0. 06 0.08 0. 10 0 . 12 0.40 0 . 04 Unstable Configuration - 13.4, u ..- 0.30 M 0 = 0 = 4,560 mis ro ·- 0 0 q = 1.5 x 10 6 Jlkg q=l8.13, en 0 <!) '-<: i s ·- 0.02 3 0.20 a I '-"' 0 Pressure (P) contours 0 z _... >-. 0. 10 0 . 00 loi.:IC~ 0 . 00 0 .0 0 .2 0. 4 0. 6 0.8 1 .0 1 .2 x (Non-Dimensional)