Matthew J. Grismer and Joseph M. Powers- Calculations for Steady Propagation of a Generic Ram...

8/3/2019 Matthew J. Grismer and Joseph M. Powers- Calculations for Steady Propagation of a Generic Ram Accelerator Configuration

1/7

J O U R N A L O F P R O P U L S IO N A N D P O W E RVol. 11, No. 1, January-February 1995

Calculations for Steady Propagation of a Generic RamAccelerator ConfigurationMatthew J. Grismer* and Joseph M. Powerst

University of. Notre Dame, Notre Dame, Indiana 46556

This study describes a m ethodology and gives analysis to determine the steady propagation spe ed of a projectilefired into a gaseous mixture of fuel and oxidizer. So that numerical results can be analytically verified in limitingcases, a simple model and geometry are used: the steady supersonic flow of an inviscid, calorically p erfect,ideal, reacting gas w ith high activation energy over an unconfined symmetric double w edge is considered. Typicalpredicted flow fie Ids consist of a lead inert oblique shock followed by a thick induction zone, thin reaction zone,rarefaction, and trailing oblique shock. The reaction strengthens the lead shock to the extent that the far-fieldwave angle is that of a Chapman-Jouguet (CJ) oblique detonation. Propagation speeds are found that give riseto shocks of such strength as to induce the reaction zone to be in a region that allows the combustion-inducedthrust to balance the wave drag. For a fixed heat release greater than a critical value, two steady propagationspeeds are predicted. The solution at the higher Mach number is stable to quasistatic perturbations, whereasthe solution at the low er Mach num ber is unstable. The steady solutions are superdetonative in that the projectilevelocities that give rise to force balances are greater than the C J velocity for a detonation w ave aligned normalto the direction of projectile motion. This methodology can be applied to analyze devices that have more complexgeometries, such as the ram accelerator or oblique detonation w ave engine.

IntroductionIT is possible to employ oblique shock waves to inducecombustion to generate thr ust. R ecent discussion has beenmotivated by the ram accelerator, which has been used topropel projectiles to high speeds, and the oblique detonationwave engine (ODWE), which has been proposed to propelthe National Aerospace P lane (NASP). For such devices, itis of fundamental importance to have a theory that ca n predicta steady propagation speed. To this end, a R ankine-Hugoniotanalysis that utilized a thermal explosion theory was per-formed by Powers an d Gonthier,1 and was qualitatively ver-ified in a numerical analysis by Powers et al.2 Additionally,the numerical analyses of Brackett an d Bogdanoff3 an d Yungsteran d Bruckner, 4 which model ra m accelerators powered by H2- O 2 combustion, suggest that steady velocities in the range5000-10,000 m/s may be achievable, a lthough in these studiesonly a small numb er of steady propagation speeds were foun d.In this article we briefly review some of the relevant lit-erature, describe a general methodology fo r determining th esteady propagation speed of either ra m accelerator projectilesor O D W E - p o we re d aerospace planes, pose a simple model

by comparing its predictions to a n u mb er of one- an d two-dimensional inert and reactive exact solutions. The main newcontributions of this article, in com bination with R efs. 1 and2, are 1) a numerically verified theory for the dependency ofsteady propagation speed on heat release, and 2) a numeri-cally verified theory for the far-field flowfield behavior.Review

The ram accelerator (see Fig. 1) was first tested by Hertz-berg et al.6'7 In this application, a high-speed projectile isfired into a tube filled with an unreacted mixture of combus-tible gases. In one test a 70-gprojectile was fired into a 16-m-length, 38-mm-bore tube filled in its first three stages withvarying combinations of CH4, O2, N 2, and He at a pressureof 31 bar,and in its final stage with 0.9 C2H4 + 3O2 + 5CO2at a pressure of 16 bar. A shock-induced combustion processaccelerated the projectile from an initial velocity of near 1200m/ s to a velocity of 2475 m/s (corresponding to a Mach numberM = 8.4) at the end of the tube, at which location it was stillaccelerating. Downstream pressures near 600 bar were mea-sured. The diameter of the projectile's main body was 28.9


2/7

106 GRISMER AND POWERS: PROPAGATION OF A GENERIC RAM ACCELERATOR

fuelinjectorsobliquedetonation upper cowl surface

exhaust

Fig. 2 Envisioned oblique detonation wave engine, adopted fromDunlap et al .10tion; the temperature-sensitive reaction would be associatedwith the lead shock for faster projectile speeds, an d with adownstream shock for slower speeds. For even slower speeds,the reaction would either be downstream of the projectile ora normal detonation running ahead of the projectile wouldbe triggered. It was suggested that such a device can be scaledfo r direct launch to orbit, for hypervelocity impact studies,and for a hypersonic test facility.Another relevant propulsion device is the proposed ODWE.The idea of the ODWE ha s existed fo r decades (e.g., Dunlapet al .10 ). The hypothesized operation is as follows (see Fig.2). Supersonic air enters th e inlet. On-board fuel is injecteddownstream that mixes with the air without significant reac-tion. The mixture then encounters a downstream wedge. Theoblique shock associated with th e wedge compresses and ig-nites the mixture, generating a propulsive force. Relative toconventional air-breathing engines, Dunlap et al. cite theODWE's advantages as 1) simpler inlet diffuser design sinceth e inherently supersonic oblique detonation does not requiredeceleration to a subsonic state, 2) reduced total pressurelosses, 3) shorter combustion chamber length, 4) no ignitiondevice other than the wedge, and 5) faster flight velocities.Cited concerns are 1) the lack of static thrust , 2) uncertaintyas to whether mixing lengths are practical, and 3) uncertaintyof the process's stability.

Methodology and Model ProblemMost recent theoretical studies related to ram acceleratorsan d ODWEs8"24 have not given detailed analyses focused ondetermining a steady propagation speed, i.e., the terminalvelocity. Typically, the related problem of flow with a fixedincoming Mach num be r over a fixed geometry is examined.Emphasis is placed on characterizing the resulting flowfield,and only a small number of incoming Mach numbers ar estudied. This approach does yield the thrust force as a functionof flight speed, but in general does no t give th e upper velocitybound for the device.Here we present a general approach to predict the steadyspeed. O ne first selects a mathematical model for the fluid

incoming _supersonicpremixed flow

lower cowl surfaceFig. 3 Schematic of generic configuration.

Fig. 4 Detailed schematic for H < .geometry that retain essential features of real systems. Th egeometry, shown in Fig. 3, is a symmetric double wedge withhalf angle 9 and length L. Tw o cowl surfaces ar e placed sym-metrically about the wedge and are separated by height H.The depth of the double wedge an d cowl is taken to be infinite,and the flow is assumed to have no variation in this direction.The Cartesian coordinate system, with its origin at the leadingedge an d with th e x axis aligned with the incoming flow, isalso indicated. It is appropriate to think of a ram acceleratoras the axisymmetric analog in which th e projectile moves whileth e cowl is stationary; likewise, an aerospace plane poweredby an ODWE can be thought of as the axisymmetric analogin which th e cowl moves with th e wedge. In both scenariosone must assume that the incoming fuel an d oxidizer ar ecompletely mixed; in actuality this is more appropriate fo rthe ram accelerator than the ODWE.Analysis of the geometry of Fig. 3 leads in general to acomplicated interaction of shocks, rarefactions, and combus-tion processes as the flow passes between the projectile andcowl surface. To further simplify, w e consider only th e limitH - G O , Fig. 4. Consequently, ou r geometry bears only a


3/7

GRISMER AND POWERS: PROPAGATION OF A GENERIC RAM ACCELERATOR 107

reaches th e Chapman-Jouguet (CJ) wave angle, as will bedemonstrated in this article.Model EquationsThe dimensionless model equations are given below:

(1)

(2)d t p d X j

dt dt

e y - i pP = pT

(3)

(4)

(5)(6)

The dependent variables in Eqs. (1-6) are the density p, theCartesian velocity component v , - , the pressure P, the internalenergy e, the temperature T, and the reaction progress var-iable A. The independent variables ar e time t and the Carte-sian position coordinate #,-. Th e dimensionless parameters ar eth e ratio of specific heats y, th e kinetic rate constant K, theheat of reaction q, and the activation energy . Here th ematerial derivative (d/dt) = (d/dt) + v,(d/djt,).Equations (1-3) express conservation principles for mass,momenta, and energy, respectively. Equation (4) is a speciesevolution equation that incorporates an Arrhenius depletionmodel. Equations (5) and (6) are caloric and thermal equa-tions of state. A single, first-order, irreversible, exothermicreaction is employed, A -> B. The reaction progress variableA ranges from zero before reaction to unity at complete re -action. Species mass fractions, YA, YB, are related to thereaction progress variable by the formulas, YA = 1 - A,YB = A.In Eqs. (1-6) pressure, density, and temperature are scaledso their preshock values ar e unity; velocities are scaled by anumber closely related to the preshock acoustic speed. Thelength of the projectile L is chosen as the reference lengthscale. The reference time is of the same order of magnitudeas the time for a fluid particle to traverse th e length of theprojectile. In terms of dimensional (indicated by the notation"""") variables, parameters, an d preshock ambient conditions

Here, M0 is the freestream Mach number that is required inth e initial preshock conditions. T he dimensional parametersare q the heat of reaction, E the activation energy, R the gasconstant for the particular fluid, cv th e specific heat at constantvolume, and k th e kinetic rate constant. W ith this scaling,initial preshock conditions ar e specified as p = 1, u = VyM(),v -0, P = 1, T = 1, e = II(y - 1), and A-0.

Numerical AnalysisA numerical analysis of Eqs. (1-6) was performed using ane w code25 based on the Roe method. In brief, the code usesan explicit Roe scheme5 and fractional stepping to integrateth e equations in a g eneralized, curvilinear coordinate system.

The integration has second-order spatial accuracy in smoothregions of the flowfield an d first-order temporal accuracy. Inthe implementation of the R oe scheme, all eigenvalues andeigenvectors of the generalized flux Jacobian matrices wereobtained analytically, resulting in an efficient and robust code.The second-order spatial accuracy was obtained using a ver-sion of the higher-order total-variation-diminishing (TVD)schemes for Roe averaging suggested by Chakravarthy andOsher.26The code works in a finite volume sense, requiring fluxboundary conditions on all sides of the dom a i n. At the inflow,al l fluxes were specified based upon th e chosen inflow qua n-tities. At the projectile surface, slip wall bounda r y fluxes werespecified by setting the transverse component of the velocity(in the curvilinear space) to zero. The only nonzero compo-nents of the transverse flux then involved th e pressure at theprojectile surface. This was estimated by using one-dimen-sional Riemann invariants and the nearest cell centered qua n-tities, as suggested by Dadone an d Grossman.27 Lastly, thenonreflective boundary conditions suggested by Thompson28were specified at the outflow. This method utilizes the char-acteristic formulation of the equations to remove reflectionsfrom th e outflow boundary. Basically, outflowing character-istics ar e calculated with upwind differencing, while any in-flowing characteristics are set to zero. The bounda r y at thetop of the computational domain was treated as a wall, bu tthis had no effect on the solutions presented here since in allcases th e far-field oblique detonation exited through th e rearboundary w ithout touching the top of the domain.One-dimensional inert and reactive test cases were used toverify the m e t hod. First, th e classical R iemann shock-tubeproblem was considered (see, e.g., Hirsch,29 pp. 204-211)for a calorically perfect inert ideal gas [y = ? , R = 287 J/(kgK)]. Figure 5 gives instantaneous pressure, density, andparticle velocity profiles at t j= 39 ms. Dimensional initialconditions were specified as P = 105 Pa, p = 1 kg/m 3 , u =0 m/s for 0 m < x < 5 m, and P = 103 Pa, p = 10"2 kg/m 3 ,u = 0 m/s for 5 m < x < 10 m. In Fig. 5, a slightly different


4/7


90

80i7060

peak=99.5

10 20 30 40Nondimensional time,T/Ti^

50

Fig. 6 Unsteady ZND detonation (/ = 1.6, 0 = 50, q - 50, y1.2).

................. 2000 pts, 3 pts L v 4--------- 4000pts, 5 pts L v 4-----16000 pts, 18 pts LU-ZND peak pressure

40 60Nondimensional time.T/

80 100

Fig. 7 Steady ZND detonation (/ = 1.8, 0 = 50, q = 50, y = 1.2).values, and distance has been scaled by the tube length, 10m. The domain was broken into 20 0 uniform cells; no meshadaptation was used. The numerical method does an excellentjo b of matching th e exact solution. Th e weakest agreementis near th e contact discontinuity, which is apparent in the first"step" of the density curve. Shock discontinuities, present inall th e variables' profiles, are captured well in only a fewpoints.A more stringent test of the method was considered next.It is well-known that for certain values of q,0, and overdrive/, one-dimensional detonation solutions of Eqs. (1-6), knownas Zeldovich-von Neumann-Doering (ZND) detonations, ar eunstable. Here, overdrive is defined as/ = (D/D CJ )27 whereD is the speed of a piston-supported traveling detonation, an d

parameter is the time f1/2 for a fluid particle to cross th e half-reaction zone length.Figure 6 shows the peak pressure at the front of the det-

onation vs time for an unstable detonation (/ = 1.6, O =50 , q = 50, y = 1.2). The numerical method predicts a peakpressure that oscillates about that predicted by the steadyZ ND solution. The amplitude an d period of the oscillationagree well with previous solutions.33 Figure 7 shows the sameplot for a stable detonation (/ = 1.8, = 50, q = 50, y =1.2). The solution exhibits initial oscillations; these da m pe n,and the peak pressure correctly approaches a long-time valueclose to the independently-predicted ZND peak pressure. Us-in g more points in the computational domain resulted in thenumerical solution more closely approaching the ZND peakpressure. A num be r of cases even closer to the stability bo und-aries were examined, and in each the method correctly pre-dicted the steady or unsteady behavior .Results

For the unconfined double wedge, steady propagation speedswere sought that gave rise to a force balance as the heat releaseparameter q was varied, 11.908 < q < 13.456. Other param-eters were held constant at y = 7/5, 0 = 5 deg, 0 = 12.32,and K = 9179. Fo r presentation of results dimensional pa -rameters were chosen as P () = 1.01325 x 105 P a, p () = 1.225kg/m3, k = 2.64 x 107 s"1, E = 1.019 x 106 J/kg, L = 0.1m, R = 28 7 J/(kgK), cv = 717.5 J/(kgK), 0.985 x 106 J/kg^ q < 1.113 x 106 J/kg. These values were chosen after asearch for parametric values that would give rise to a forcebalance. Their specific value is thus primarily motivated by adesire to illustrate our methodology rather than a goal ofprecise prediction of a real physical system. A common 199x 99 fixed computational grid was used in all cases. Doublingthe grid resolution in both directions yielded results with onlyslight quantitative changes. Convergence to steady state wastypically achieved in about 5000 time iterations. At this pointthe unsteady residuals, ? = E/l, 2/11 \V Jk - y l\/NM,were all near 10 ~ 6. Here, and ty % are, respectively, theaverage an d local residual at time tn = nkt in either p, u, v,P , or e + (u 2 + v2)/2. Also, / an d j ar e spatial indices, Nan d M are the total number of points in the x an d y directions,respectively, an d n is a temporal index. A typical simulationrequired about three h of computing time on an IBM RS/6000 P O W E R s ta ti on 350 having 64 Mb RAM and rated at18.6 MF lops.Th e CJ Mach number (Af (,c7) for a detonation with a leadshock normal to the flow direction is determined solely by qan d y30 ; for the parameters listed 4.275 < M I ] C J < 4.517. Forcases attempted in which M (} < M ()C/, a normal detonationwould form an d propagate forward in the domain until it hitth e inflow boundary. This corresponds to the detonation at-


5/7


I- q = 0.985 MJ/kg- q = 0.992 MJ/kg- q = 0.998 MJ/kg- q=1.014 MJ/kg

MQ, Freestream Mach NumberFig. 8 Net thrust force vs Mach number for varying heat release.

q, Heat Release (nondimensional)12.4 12.6 12.8 13

statically stable branch

statically unstable branch

steady projectile Mach number normal CJ detonation Mach number

1 1.02 1.04 1.06 1.08 1.1q, Heat Release (MJ/kg)

Fig. 9 Bifurcation diagram for steady-state speed vs heat of reaction.

M O = 8.4q= 1.014 MJ/kgP O = 1 bar

Mo = 7.1q= 1.014 MJ/kgP0=l bar

x ( m )Fig. 12 Pressure contours for statically unstable steady configura-tion.

M 0 = 7.1q= 1.014 MJ/kg

x ( m )Fig. 13 Product mass fraction contours for statically unstable steadyconfiguration.(q = 11.993), there is a balance of combustion-induced thrustand drag such that Fnet = 0. This occurs at Af0 = 7.95. Forlarger values of heat release, there are two distinct Machnumbers for which Fnet = 0. A perturbation in the Machnumber for the steady solution at the lower Mach numberresults in a net force that tends to accelerate the projectileaway from the equilibrium Mach number. Consequently, thisis a statically unstable equilibrium. In the same manner, theequilibrium solution at the higher Mach number is staticallystable to such perturbations.These results are summarized in the bifurcation diagramshown in Fig. 9, where equilibrium Mach numbers M 0 vs heatrelease q (and q) ar e plotted. The lower branch is unstable,


6/7


x (nondimensional)0 0.2 0.4 0.6 0.8 1 1.2 1.4

: MO = 8.4 -* ^J M0 = 7.1/ \^ s \

L X ' N X -\ i x % ^ . , _ - - ,V - l " " ' ' ' / * \

7 inert (exact), F = - 1944 N/m -/ a inert (numerical), Fnet = -1942 N/ni\ _-.'----- reactive (stable), Fnet - 0N/m I

0 0020040060080.1 012014x ( m )

Fig. 14 Pressure traces on wedge surface.q, Heat Release (nondimensional)

12.2 12412612813 132134

exact CI solutionnumerical prediction

111 102104106108 1.1

q, Heat Release (MJ/kg)Fig. 15 Numerically predicted far-field wave angle and analyticallypredicted C J wave angle vs heat release #; M 0 varied to maintain forcebalance, 5.5 < M 0 < 8.5.occurs off the leeward wedge surface far downstream, re-sulting in a F net < 0. As the Mach num be r is increased, th ereaction moves forward onto th e wedge, eventually reachinga point at which the wave drag of the projectile is balancedby th e thrust due to chemical reaction. Increasing M0 pastthis equilibrium point moves the reaction closer to the wedgeapex on the leeward side, resulting in positive Fnet. IncreasingA /o still further pushes the reaction over the apex an d ontothe front of the wedge. The pressure increase due to chemicalreaction on the front of the wedge is then balanced by theresulting higher pressure on the leeward side of the wedge.Figure 14 shows plots of the pressures along the lines ofsymmetry an d projectile surface for the stable an d unstablecases of interest. Here there are about 90 grid points distrib-uted on the wedge surface. As a verification of the code'sability to predict two-dimensional flows, plots of the exact

th e reaction so as to evolve into a C J oblique detonation.This result gives numerical confirmation to ideas pu t forth byChernyi.34 As discussed in detail in many sources (see, e.g.,R efs. 14, 16, and 17), the C J oblique detonation is the uniqu eoblique detonation that has at the end of the reaction zone avelocity component normal to the shock front that is locallysonic. For each of the cases in which a force balance existedas plotted in Fig. 9, the far-field shock wave angle /3 wa sgraphically measured an d compared to the analytically knownCJ wave angle ficj. These results are plotted in Fig. 15. It isseen that th e analytic prediction is in agreement with th enumerical prediction within th e limits of accuracy (shown byth e error bars) with which th e numerically predicted anglecan be determined. It was also noted that in cases in whichth e incoming Mach num be r did not give rise to a force bal-ance, th e far-field behavior was that of a C J oblique deto-nation.Discussion

For the model considered, this article has shown that a forcebalance an d consequent terminal velocity is achieved whenthe reaction zone length scale is of the same order of mag-nitude as the projectile's geometric length scale. The trendsof 1) the variation of net thrust with Mach number fo r fixedheat release and 2) the variation of steady propagation speedwith variable heat release ar e consistent with those of Refs.1 and 2 that used methods that were not as quantitativelyaccurate as that of the present article.Th e results also demonstrate that th e C J condition ha s onlylimited relevance in this problem. As the normal C J velocityis independent of reaction kinetics and the steady flight speedis dependent on kinetics, the two are clearly independent.Th e only significance of the normal C J Mach n u mb er is as anoverly conservative lower bound fo r flight speeds. Further-more, as the C J oblique detonation angle is predicted in thefar field fo r cases with and without force balances, it plays norole in the determination of the steady flight speed.The results presented here suggest, but do not prove, theflow is stable, since the code is time accurate and the nu-merical metho d has demonstrated capability of capturing one-dimensional instabilities. O ur resolution yielded between threean d four cells within th e reaction zone; one-dimensional sim-ulations of piston-driven detonations indicate this may be suf-ficient, bu t that more points in the reaction zone would bedesirable. Interestingly, th e far-field C J oblique detonationappears stable, although its one-dimensional piston-drivenequivalent is known to be unstable. The oblique detonationstability question may be better addressed by a linearizedanalysis in combination with higher resolution numerical sim-ulations.


7/7


tigation of Oblique Detonation R amjet-in-Tube Concepts," Journalof Propulsion an d Power, Vol. 5, No. 3, 1989, pp . 276-281.4Yungster, S., and Bruckner, A. P., "Computational Studies of aSuperdetonative R am Accelerator Mode," Journal of Propulsion andPower, Vol. 8, No. 2, 1992, pp. 457-463.3Roe, P. L., "Approximate R iemann Solvers, Parameter Vectors,an d Difference Schemes," Journal of Computational Physics, Vol.43, No. 2, 1981, pp. 357-372.6Hertzberg, A., Bruckner, A. P., and Bogdanoff, D. W., "RamAccelerator: A New Chemical Method fo r Accelerating Projectilesto Ultrahigh Velocities," AIAA Journal, Vol. 26, No. 2, 1988, pp.195-203.7Hertzberg, A., Bruckner, A. P., and Kn o wlan , C., "ExperimentalInvestigation of Ra m Accelerator Propulsion Modes," Shock Waves,Vol. 1, 1991, pp. 17-25.

8Yungster, S., Eberhardt, S., and Bruckner, A. P., "NumericalSimulation of Hypervelocity Projectiles in Detonable Gases," AIAAJournal, Vol. 29, No. 2, 1991, pp. 187-199.9Yungster, S., "Numerical Study of Shock-W ave/Boundary-LayerInteractions in Premixed Combustible Gases," AIAA Journal, Vol.30, No. 10, 1992, pp. 2379-2387.10 Dunlap, R., B re hm, R. L., and Nicholls, J. A., "A PreliminaryStudy of the Application of Steady-State Detonative Combustion toa Reaction Engine," Jet Propulsion, Vol. 28, No. 7, 1958, pp . 451-456.nC a mbie r , J. L., Ade l ma n, H., and Menees, G. P., "NumericalSimulations of Oblique Detonations in Supersonic C ombus t ionChambers," Journal of Propulsion an d Power, Vol. 5, No. 4, 1989,pp . 483-491.12 Cambier, J. L., Ade l ma n, H., and Menees, G. P., "NumericalSimulations of an O bl ique Detonation W a ve Engine," Journal ofPropulsion and Power, Vol. 6, No. 3, 1990, pp . 315-323.13 Bruckner, A. P., Kn o wlen , C., Hertzberg, A., and Bogdanoff,D . W . , "Operational Characteristics of the Thermally Choked R amAccelerator," Journal of Propulsion and Power, Vol. 7, No. 5, 1991,pp . 828-836.14 Pratt, D. T., Humphre y, J. W., and Glenn, D. E., "Morphologyof Standing O blique Detonation Waves," Journal of Propulsion andPower, Vol. 7, No. 5, 1991, pp. 837-845.15 Bogdanoff, D. W., "Ram Accelerator Direct Space Launch Sys-tem: Ne w Concepts," Journal of Propulsion and Power, Vol. 8, No.2, 1992, pp. 481-490.16Powers, J. M., and Stewart, D. S., "Approximate Solutions forO blique Detonations in the Hypersonic Limit," AIAA Journal, Vol.30, No. 3, 1992, pp. 726-736.17Powers, J. M ., and Gonthie r , K. A., "Reaction Zone Structurefo r Strong, W e ak O verdriven, and W eak Underdriven O blique Det-onations," Physics of Fluids A, Vol. 4, No. 9, 1992, pp. 2082-2089.18Grismer, M. J. , an d Powers, J. M., "Comparisons of NumericalO blique Detonation Solutions with an A symptotic Benchmark," AIAAJournal, Vol. 30, No. 12, 1992, pp . 2985-2987.19Pepper, D. W ., and Brueckner, F. P., "Simulation of an Oblique

Detonation W a ve Scramaccelerator for Hypervelocity Launchers,"Computers and Computing in Heat Transfer Science and Engineering,edited b y W . Na ka ya ma and K. T. Yang, CR C Press, Boca R a t o n ,FL, 1993, pp. 119-137.20Wilson, G. J., and Sussman, M. A., "Computation of UnsteadyShock-Induced C ombustion Using Logarithmic Species ConservationEquations," AIAA Journal, Vol. 31, No. 2, 1993, pp. 294-301.21 Kruczynski, D. L., and Nusca, M. J. , "Experimental and Com-putational Investigation of Scaling Phenomena in a Large CaliberR am Accelerator," AIAA Paper 92-3245, July 1992.22 Sinha, N., York, B. J., Dash, S. M., Dra bc z uk, R., andRolader, G. E., "Progress Towards the De ve l opme nt of TransientR am Accelerator Simulation as Part of the U.S. A ir Force A r m a m e n tDirectorate Research Program," AIAA Paper 92-3248, July1992.23 Ahuja, J . K., and Tiwari, S. N., " A P a ra me tric Study of Shock-Induced Combustion in a Hydrogen-Air System," AIAA Paper 94 -0674, Jan. 1994.24Li, C., Kailasanath, K., and Oran, E. S., "Detonation StructuresBehind O b liq u e Shocks," Physics of Fluids, Vol. 6, No. 4, 1994, pp.1600-1611.25Grismer, M . J., "Numerical Simulations of Steady and U n st ead yO bl ique Detonation P h e n o m e n a with Application to Propulsion,"Ph.D. Dissertation, U n i v . of Notre Dame, Notre Dame, IN, 1994.26Chakravarthy, S. R., and Osher, S., "Computing with High-Resolution Upwind Schemes for Hyperbolic Equations," Lectures in

Applied Mathematics, Vol. 22, 1985, pp. 57-86.27Dadone, A., and Grossman, B., "Characteristic-Based, RotatedUpwind Scheme for the Euler Equations," AIAA Journal, V ol . 3 0,No . 9, 1992, pp. 2219-2226.28Thompson, K . W . , "Time Dependent B ounda ry Conditions forHyperbolic Systems," Journal of Computational Physics, Vo l . 68, No.1, 1987, pp. 1-24.29Hirsch, C., Numerical Computation of Internal an d External Flows,Volume 2: Computational Methods for Inviscid an d Viscous Flows,W i ley , Ne w Y o r k , 1990.30Fickett, W., a nd Da vis , W . C., Detonation, U n i v . of California

Press, Berkeley, CA, 1979.31Lee,H. L, and Stewart, D. S., "Calculation of Linear DetonationInstability: O ne -Dime ns iona l Instability of Plane Detonation," Jour-nal of Fluid Mechanics, Vo l . 216, 1990, pp. 103-132.32Fickett, W . , a nd Wood, W . W., "Flow Calculations fo r PulsatingO ne-Dimensional Detonations," Physics of Fluids, Vo l . 9, No. 5,1966, pp. 903-916.33Bourlioux, A., M aj d a, A. J., and Roytburd, V., "Theoreticalan d Numerical Structure for Unstable One-Dimensional De tona -tions," Journal of Applied Mathematics, Vo l . 51, No. 2, 1991, pp .303-343.34 Chernyi, G. G., "Supersonic Flow Past Bodies with Formationof Detonation and Combustion Fronts, "Problems of Hydrodynamicsan d Continuum Mechanics, Society for Industrial and Applied M at h -ematics, Philadelphia, PA, 1969, pp. 145-169 (English edition).

Matthew J. Grismer and Joseph M. Powers- Calculations for Steady Propagation of a Generic Ram...

Documents

Transcript of Matthew J. Grismer and Joseph M. Powers- Calculations for Steady Propagation of a Generic Ram...