Analysis for Steady Propagation of a Generic Ram Accelerator/ Oblique Detonation Wave Engine Configuration

Transcription

1 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, ndiana 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

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

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.

4 nert Oblique Shock ' ' M=8.4 P=16bar Projectile m=7g v = 2,475 m/s Reaction-inducing Oblique Shock,,' (Oblique Detonation) M>l P 6 bar s s s s ---- Accelerator Barrel 166mm Ram Accelerator, Hertzberg, et al., 1988, 1991 fuel oblique detonation inlet --- m1x1ng zone ;;;::::: - Oblique Detonation Wave Engine, Dunlap, et al., 1958

5 Selected Past Work 1. Theoretical: - Brackett and Bogdanoff, 1989, (steady speeds) - Cambier, Adelman, and Menees, 1989, 199, - Pratt, Humphrey, and Glenn, 1991, - Yungster and Bruckner, 1992, (steady speed) - Powers and Stewart, Powers and Gonthier, 1992a,b, (steady speeds) - Grismer and Powers, 1992, - Pepper and Brueckner, Experimental: - Hertzberg, Bruckner, and Knowlan, 1988, 1991.

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.

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

8 Non-Dimensional Model Equations Continuity: Momentum: dp avi d +pa =O, t X. 1 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.

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

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.

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

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

13 1 Lead Shock P1 Flame Sheet 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 ) - P1] tan. 3. Combustion nduced Thrust: Fe = Fnet + Fv.

14 Jump Relations Across Lead Shock Pi= 1 + ym5 sin 2 +A -l ( y + 1) M5 sin 2 Ui =.JY Mo (ti sin 2 + cos 2 ) Vi=.JY"Mo cos sin ( 1 - ti) where A = ( 1 + y M5 sin 2 ) 2 - ( y + 1) y M5 sin 2 - x (2 + (y- 1 )M5 sin 2 )

15 Prandtl - Meyer Function: Flow Expansion Region v(m ) /Y+f -! y-1 ( 2 ) = "\/ y:1 tan y+ l M tan M 3-1 sentropic Relations: 1 _1 1 + y- Mi y-1 p Pl l + y;l M 1 _J_ 1 + y- Mi y-1 P P1 l + y-1 M 2 Velocity Components: U3 = M l.y3 f?3 yp;- cos e ' V3 = - M3 11. yp;- sine

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)

17 Tail End Compression Region 1 + ym sin,a+e) ± {E - (y+ l)m sin,a+s) where C = (1 + ym sin,a+s))2-(y+ l)'ym sin,a+s) x (2 + (Y- 1) M sina+s))

18 Numerical Analysis RPLUS Code: - developed at NASA Lewis, - based on LU-SSOR numerical scheme. Computational Grid: x 99 fixed grid. Convergence: - 5 iterations,. - residual unsteady terms had scaled values -8 < 1. x 1. Computations: - run on BM RS/6 POWERstation 35 - run time about one hour.

19 Parameters Geometric: 8 = 5, L =.1 m. Kinetic: k = 1. x 1 7 /sec, E = 1.19 x 1 6 J/kg, x 1 6 J/kg < q < 1.74 x 1 7 J/kg. Atmospheric free-stream conditions: Po= x 1 5 Pa, p = kg/m 3 Thermodynamic constants: 7 Y= 5, R = 287 J/(kg K), cv = J/(kg/K).

20 Velocity (m/s) Net Thrust, Numerical _.._..._._ M, Mach Number 1-2 ca,,-.-, c... (/.) Q c z "-"'... (/.)... z Velocity (m/s) Net Thrust, Rankine Hugoniot -e- q = 18.13, q = 1.5x16 J/kg -A- q = 16.32, q = 1.35x16 J/kg q = 14.51, <= 1.2x16 J/kg :::s'"j... z -1 ooo a "-"' -. 2 '--'---'--'---'---l...-..j'--'---'--.., , -'-= M, Mach Number

21 ,.-... ].- CZl 15 1 Q.) s.- Q z "-" Pressure on Wedge Surf ace. M = (stable) M = 13.4 (unstable) x (m).5.1 q = q = 1.5 x 1 6 J/kg 15 _..._ 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

22 Pressure traces on wedge surface. k=1x1a7 Ea= , -ro a.. -(l) '- :J en en (l) '- a o o Mach 13.4 nert lil - - Mach 13.4 Q = 1.5x1 OA6 -- <: Mach nert *-- Mach Q = 1.5x1 OA6.:::.,--,.,----- eft= er--1):. L-----L l.-----l-----' x {m).15.2

23 1 q, Heat of Reaction (J/kg) 1.3X X X X1 6 "8 z.c (') Rankine- (")... Hugoniot '<,,--.. Analysis 25 '-' 6.._.._._._._._..._._._...l J._._._...._.._._._._._...._._._._.._._.._._ J'--'--'-... _._ q, Heat of Reaction (Non-Dimensional) 2 q, Heat of Reaction (J/kg) 1.4x x1o 6 1.6x x1 6 lo-c C1) "8 z..c:: u c;rj 18 quasi-stable 1 6 Numerical Analysis 6 14 c;) '-' '<,, q, Heat of Reaction (Non-Dimensional)

24 ..4 x (m) Quasi-Stable Configuration -.. ca... rl:l 5. Q 6 :> M = 17.53, u = 5,965 mis q = 18.13, q = 1.5 x 1 6 J/kg Reactant mass fraction (1-A.) contours "<l.2 s '-" ==j,.l..---' x (Non-Dimensional)..4 x (m) Unstable Configuration -.. ca... rl:l Q 6 :> M = 13.4, u = 4,56 m/s q = 18.13, q = 1.5 x 1 6 J/kg "<i.2 s '-". L..dSsmRSl- ===--'-----'---.L.-..J x (Non-Dimensional)

25 -.3 s:: x (m) Quasi-Stable Configuration M = 17.53, u = 5,965 mis q = 1.5 x 1 6 J/kg - q = 18.13, C'J 5 - '-<: t s a s '-"' 6 Pressure (P) contours e >-..1 ====""'"""',,,._L-----L-.., x (Non-Dimensional) -..- ro.3 x (m) Unstable Configuration M = 13.4, u = 4,56 mis - en q=l8.13, q = 1.5 x 1 6 Jlkg - <!) '-<: i s.2 a.2 3 '-"' z _... >-.. 1 Pressure (P) contours. loii.:c=== l......l...j x (Non-Dimensional)

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 quasistatic 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.