c)2001 american institute of aeronautics & astronautics or ... · lightcraft configurations...

c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-34350

AIAA 2001-3665Numerical Analyses on Pressure WavePropagation in Repetitive Pulse LaserPropulsionHiroshi Katsurayama, Kimiya Komurasaki,and Yoshihiro ArakawaUniversity of Tokyo, Tokyo, Japan

37th AIAA/ASME/SAE/ASEEJoint Propulsion Conference & Exhibit

8-11 July 2001Salt Lake City, Utah

For permission to copy or to republish, contact the copyright owner named on the first page.For AIAA-held copyright, write to AIAA Permissions Department,

1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.


AIAA-2001-3665

NUMERICAL ANALYSES ON PRESSURE WAVE PROPAGATION INREPETITIVE PULSE LASER PROPULSION

Hiroshi KATSURAYAMAf Kimiya KOMURASAKIJ and Yoshihiro ARAKAWA*University of Tokj'o, Kongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan

AbstractPressure wave propagation in a repetitive pulse air-breathing laser thruster flying at Mach 5 flow has beencomputed in order to clarify the behavior of a shock wave induced by focusing laser beams in supersonicflow, as well as to estimate the impulse acting on the thruster. As a result, the detailed process of shockreflection on the body and the mechanism for thrust generation have been analyzed. Moreover, theestimated coupling coefficient, which is the cumulative impulse per one pulse of laser energy, has beenfound to be fairly constant at a value of approximately 150 N • s/MJ, being independent of the laserenergy. Results also reveal that the coupling coefficient is sensitive to the focus location.

NOMENCLATUREC.A.R. — capture area ratioCd = drag coefficientCm = coupling coefficientD — aerodynamic dragM = Mach number5 = area relative to bodye — total energy per unit volume

or electron chargeh = flight altitude of vehiclenc — critical electron number density

for cut-offme = electron massp = static pressurep = densityq = heat fluxR = air gas constant = 287J/kgT = static temperaturet = timeu,v — axial, radial velocity componentz,r, 9 = cylindrical coordinates7 = specific heat ratio =1.4e0 = dielectric constant of vacuum?7iaser = fraction of laser energy absorbed

by plasmar = viscous stress tensorUJL = laser radiation frequency

subscriptsB — body part of vehicleinC — inner cowl part of vehicleinlet = inlet of vehicleoutC — outer cowl part of vehicleoo = freestream property

* Graduate student, Department of Aeronautics and Astro-nautics, Student Member AIAAt Associate Professor, Department of Advanced Energy,Member AIAA* Professor, Department of Aeronautics and Astronautics,Member AIAACopyright ©2001 by the American Institute of Aeronauticsand Astronautics, Inc. All rights reserved.

INTRODUCTIONLaser propulsion will serve as an alternative

propulsion system for ground-to-orbit launch sys-tems in the near future. Since energy is transmit-ted from the ground or from laser base in spaceto the vehicle and propellant is not needed duringthe air-breathing mode, the payload ratio becomeshigh. Therefore, the launching cost should be low.In addition, laser propulsion is free from fuel mix-ing and combustion problems in high speed flows.

Myrabo et al. proposed an axisymmetriclaser thruster, so-called "Lightcraft," and con-ducted flight tests with a scaled model J1'2! TwoLightcraft configurations exist: closed-inlet typeand air-breathing type. The air-breathing typeLightcraft, with a weight of 50 g, has flown upto 30 m vertically and up to 121 m horizontally byusing a repetitive pulse laser of 10 J per pulse and10 Hz.

Wang et al. computed the flow in the LaserLightcraft featuring a closed inlet J3^ Their compu-tations included detailed physical models to clar-ify the laser-plasma interactions and the propa-gation processes of Laser Supported Detonationwave (LSD) and Combustion wave (LSC) in qui-escent atmospheric air. These LSD and LSC mech-anisms were investigated in detail by RaizerJ4"^However, when the electron density reaches thecritical value for cut-off, it is not clear what per-centage of laser energy is absorbed by the plasma.Wang et al. assumed 40 % absorption and ob-tained a value for the impulse which agreed verywell with the one measured by Myrabo et al..

In the present computation, the pressure wavepropagation in the air-breathing laser thruster fly-ing at Mach 5 is investigated. The shape of thethruster is almost identical to the one of the LaserLightcraft.

It is assumed that the laser energy absorbed bythe plasma is simply deposited at the focus. Adetailed laser-plasma interaction model is not in-cluded in our present computation. Still, the pres-

1American Institiute of Aeronautics and Astronautics


sure wave propagation process is expected to beprecisely reproduced at the given energy absorp-tion.

LASER LIGHTCRAFTThe Laser Lightcraft consists of a nosecone fore-

body, a parabolic afterbody, and a cowl, as shownin Fig.l. The parabolic afterbody works bothas an aerospike nozzle and as a focusing mirror.When a high-power pulsed laser beam is focusedby the afterbody mirror, air-breakdown occurs atthe ring focus on the cowl. Then, the annularshaped plasma produced near the focus absorbsthe following part of the laser pulse and thengrows. The expansion of this annular plasma gen-erates a shock wave. The shock wave strikes theafterbody and the cowl, and expands through theaerospike nozzle. The former effect exerts the di-rect impulse on the afterbody and the cowl, andthe latter effect becomes the additional impulsecaused by expansion of the pressure wave throughthe nozzle. As the enthalpy of the plasma is con-verted into thrust through these processes, therepetitively transmitted laser pulses are convertedinto repetitive thrust impulses.

COMPUTATIONAL METHOD AND GRID

Governing equationsAlthough the temperature becomes very high af-

ter air-breakdown, air is assumed as a caloricallyperfect gas in all computations.

The Navier-Stokes equations are solved in ax-isymmetric cylindrical coordinates. The equationsare expressed via an unsteady vector equation inconservative form.

OF . dG _ dM~lh

U =

M =

u^J L/J: c/vj cyivx tyi> , .QJ. ' "HT •" ~HTT "~^T~ ' "HZ" "" \ /

pvpuvpv2

roTzr

Trr_UTzr + VTrr - qr_

_"P "pupv

_e

' puF _ P^2 4

' puvm(*+\

•oTZZTzr

_urzz + vrzr - qz.

s = -r'-pv

— pUV + Tzr

— pV2 + Trr —— (e + p)v —

-po) u

•) •

TOOqr-\

(2)- urzr + t;rrr _

The total energy per unit volume is defined as

The equation of state for ideal gas being employedis,

p = pRT. (4)For the viscosity and thermal conductivity, theSutherland formulas'7' for air are used. The valuesobtained by these formulas deviate from the actualvalues over 1900 K due to the effect of chemical re-actions. To this respect, although the viscosity isabout twice as large as Sutherland formula underthe condition of 10,000 K and 1 atm,'8' this devi-ation is ignored in the present computations.

Numerical schemeA cell-centered finite volume scheme is adopted.

The inviscid flux is estimated with a AUSM-DVscheme'9' and extended to 3rd-order space accu-racy is achieved employing a MUSCL approach'10'with Van Albada's flux limiter. The viscous fluxis estimated with a standard 4th-order central dif-ference. Time integration is performed with a LU-SGS scheme.'11' In order to avoid the complicationof the boundary condition around the cowl, theflow field has been divided into four zones (Fig.l).Four points at the interface of each zone are over-lapped. At the interface, 3rd order space accuracyis maintained by using the so-called Fortified So-lution Algorithm.'12'

Computational gridFigure 2 shows the computational grids. The

configuration is almost the same as the one ofthe Lightcraft'1' proposed by Myrabo. The bodylength and its maximum radius are set to 20 cmand 6.75 cm respectively. The cowl is placed at1.25 cm above the body shoulder and covers thebody from z — 10 cm to z — 16 cm. The parabolicafterbody surface is designed to focus a laser beamon the inner cowl surface, at z — 14 cm, r — 8 cm.The grids are concentrated ahead of the forebodyto clearly capture the bow shock, and between thecowl and afterbody so as to capture the propaga-tion of the pressure wave after air-breakdown.

Boundary conditionsThe freestream inflow condition is given as listed

in table 1.Table. 1 Flight condition.

hA<foo

PooT-LOO

20km5

8.89 x 10~2kg/m3

216.65K

e = P7-1 (3)

At the inflow boundary, all physical propertiesare fixed according to the supersonic inflow con-dition. In addition, the inflow is assumed to beparallel to the z axis.

At the body and cowl surfaces, slip and adia-batic wall conditions are adopted. Static pressureis determined by considering the curvature of thewall surface.

American Institiute of Aeronautics and Astronautics


12

10

r-, 8

E— 6^

i Axisymrneif.y r • i --i-4••••••.... j....... . . . * . . . . . . ....10 25 3015 20

z[cm]Fig.l Lightcraft configuration and divided computational zones.

35

10 15 20z [cm]

Fig.2 Computational grids.

30 35

Explosion source

When air-breakdown occurs at the focus, freeelectrons absorb the laser radiation by the InverseBremsstrahlung process and a high temperatureplasma is produced. The expansion of this plasmainduces a strong shock wave. The layer behind theshock front is ionized by shock heating, resulting inlaser absorption in the shock layer. In this process,the strong shock wave propagates to the laser inci-dent direction maintaining its intensity. This waveis referred to as LSD.'4' As the shock wave movesupstream toward the laser incident direction, thelaser energy density decreases. When the laser en-ergy density decreases to a critical value, the shockseparates from the plasma front. The plasma frontexpands with subsonic speed because the electronheat conduction and radiation heat transfer en-ables the neighboring cold layer to absorb the laserradiation. This mechanism is referred to as LSC.'5'

However, in order to clarify the behavior of LSDand LSC, many physical models must be includedin the numerical code. Still, to avoid a high com-putational cost, laser absorption processes are ne-

glected in the present computation. The energyis assumed to be deposited on one numerical celllocated at the ring focus, which is shown in Fig.l

When the electron density of the plasma frontreaches the following critical value,

nc = (5)

the laser beam cannot penetrate the plasma front.Then, the laser radiation is partially reflected orscattered off, and partially absorbed at the plasmafront.'3' Unfortunately, this mechanism has notbeen clarified yet, and it can be said that it isvery difficult to determine the portion of laser en-ergy that is absorbed by the plasma. In the com-putations of Wang et al.'3', it is assumed that??iaser = 40% during this critical plasma reso-nance. The impulse estimated by their compu-tation agreed very well with the experiments byMyrabo et al..

In the present computation, it is assumed thatthe plasma absorbs half the energy of one pulselaser energy,

??laser - 50%. (6)


c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

The laser energy absorbed by the plasma is sim-ply deposited on the focus. Although this ap-proach cannot assess the validity of ryiasen the pres-sure wave propagation process is expected to beprecisely reproduced at the given energy absorp-tion.

RESULTS AND DISCUSSIONInital flow field

At first, steady state flow before laser-focusingis obtained. Figure 3 shows the pressure contours.The Capture Area Ratio^13! calculated by the fol-lowing equation is a considerably low value, 31.3%.

~i A r> _./i.rt. —pv - dsr________jj ——— —— - —

//s pv • ds (7)

The Smiet is the area between the cowl leading edgeand the body shoulder. The Soo is the circle areacalculated by taking the radius of the cowl leadingedge. With respect to C.A.R., this Lighter aft con-figuration may be more suitable to higher Machnumber flow than M^ = 5.

After the flow passes through the bow shock,the Mach number is reduced to 2.5 at the inlet.Figure 4 shows the pressure contours and velocityvectors near the inlet. An oblique shock wave isgenerated at the cowl leading edge. After passingthrough this shock wave, the flow above the cowlis accelerated by a series of expansion waves whichoriginate from the outer cowl surface. Between thecowl and the body, a series of compression wavesoriginates from the inner cowl surface. On thebody shoulder, the flow separates from the bodysurface. A shock wave originates at this separationpoint.

Figure 5 shows the pressure contours and ve-locity vectors from the vicinity of the cowl end toz = 35 cm. The wake caused by this separationspreads till z = 27 cm on the z axis. The velocitydirection in the wake is opposite to the main flowdirection.

The axial aerodynamic drag can be decomposedinto the body part, DB? the inner cowl part, AnC>and the outer cowl part, A>utc- The DB» AnC andA>utc are 605 N, -88 N, 141 N, respectively andthe total drag is 655 N. Although D[nc works asthe positive force, the effect is small because mostof the surface is parallel to the axial direction. Thedrag coefficient, Cd, is 0.37 if the representativearea is taken as the area which is calculated byprojecting the body and the cowl into the axialdirection.

Fig.3 Pressure contours in the steady state flowfield.(max. = 180818 Pa, min. = 408 Pa, mere. = 3608 Pa)

Fig.4 Pressure contours and velocity vectors nearthe inlet.

Fig.5 Pressure contours and velocity vectorsfrom the vicinity of the cowl end to z = 35 cm.

Laser induced shock propagationThree cases of laser-focusing with 200, 400 and

600 J energy are computed. The focus is locatedon the cowl surface. Since 77iaser = 50% is assumed,the net deposited energy is half the value of laserenergy. Figures 6(a) ~ (f) show the pressure con-tours and velocity vector plots at t = 0.5 ~ 15.0/.iswith 400 J laser energy.

At t = 0.5//S, just after the laser-focusing, theinduced shock wave is reflected on the cowl sur-face, and the shape of shock wave becomes featuresa half-torus shape. The distributions of physicalproperties in this region encompassed by the shockwave still resemble the self-similar solution of apoint source J141

At t = 2.5//S, each of the legs of the shock wavebegins to propagate with different speed; One of



Shock wave legpropagating downstream

(a) t = 0.5 //s(max. = 3743926 Pa, min. = 403 Pa, incre. = 74870 Pa)

(d) t = 7.5 //S(max. = 1488152 Pa, min. = 1153 Pa, incre. = 29740 Pa)

(b) t = 2.5 p.s(max. = 934008 Pa, min. = 405 Pa, incre. = 18672 Pa)

e) t = 10.0 /is(max. = 1207828 Pa, min. = 1399 Pa, incre. = 24129 Pa)

(c) t - 5.0 /ZS(max. = 696236 Pa, min. = 411 Pa, incre. = 13917 Pa)

(f) t = 15.0 us(max. = 647684 Pa, min. = 1400 Pa, incre. = 12926 Pa)

Fig.6 Pressure contours and velocity vectors at t = 0.5 ~ 15.0 ^s in the case of laser-focusing on thecowl with 400 J energy.



(a) t = 20 jus(max. = 3550556 Pa, min. = 1400 Pa, incre. = 70983 Pa)

(e) t = 60 fj,s(max. = 103193 Pa, min. = 157 Pa, incre. = 3613 Pa)

(b) t = 25 fJS(max. = 637040 Pa, min. = 1400 Pa, incre. = 12712 Pa)

(f) t = 70 jJ,S(max. = 86716 Pa, min. = 3 Pa, incre. = 3616 Pa)

(b) t = 30 ^S(max. = 540184 Pa, min. = 1421 Pa, incre. = 10775 Pa)

(g) t = 100 p,S(max. = 46978 Pa, min. = 200 Pa, incre. = 3612 Pa)

(c) t = 40 fJS(max. = 321745 Pa, min. = 1736 Pa, incre. = 6400 Pa)

(h) t = 300 /is(max. = 46978 Pa, min. = 47 Pa, incre. = 3615 Pa)

(d) t = 50 ^S(max. = 134052 Pa, min. = 1962 Pa, incre. = 3577 Pa)

Fig.7 Pressure contours at t = 20 ~ 300 //s in the case of laser-focusing on the cowl with 400 J energy.



them moves upstream against the inflow, and theother moves downstream with higher speed sinceit rides on the flow.

At t — 5.0^s, the shock wave strikes the after-body and the entire inflow is compressed by theshock.

At t = 7.5^s, the shock wave leg propagat-ing downstream appears on the afterbody. Att = 7.5 ~ 15^s, this shock wave leg sweeps theafterbody and the pressurized region on the af-terbody becomes wide. By these processes, theLightcraft obtains the main thrust. During thissweep, a regular reflection of shock appears on theafterbody and the pressure at the reflection pointreaches maximum at each instant. The regular re-flection transits to Mach reflection as reported inmany experiments and computationsJ15^ On theother hand, the shock wave propagating upstreamstays near the inlet because the shock wave prop-agation speed relative to the body becomes equalto the inflow speed. Due to a similar reason, theshock wave near the cowl end also stays there.

Figures 7(a) ~ (h) show the pressure contoursat t = 20 ~ 300/xs. At t = 20/is, the shock wave legreaches the afterbody tail. Although the pressureat this point rapidly increases because of the ax-isymmetrical shock focusing, the thrust level doesnot increase so much. The reason is that the incli-nation of the afterbody tail to the axial directionis small and the pressurized region by this shock-focusing is narrow. This peak of pressure staysnear the afterbody tail for a while, shown in 7(b).After times larger than t = 25/zs, the height of thetriple point gradually increases and the Mach stempropagates downstream.

On the other hand, the shock wave near the in-let gradually begins to be blown downstream anddecays to the pressure waves.

The small Mach stem observed at t = 50/^swould be generated by the re-concentration ofpressure waves blown from the inlet. The shockwave attaching at the cowl end separates at t =70/xs and then is blown downstream. At t — 100 ~300/zs, the flow field almost returns to the steadystate.

Influence of focus locationTo examine the influence of the focus location

on the flow field, the laser beam is assumed tobe focused at the point (z — 0.14 cm, r = 6 cm)which is located in the middle of the flow channel,regardless of the afterbody shape.

With 200 J laser energy, the transition of theflow field is basically the same as the case of laser-focusing on the cowl. The thrust becomes largerthan the one of the case of laser-focusing on thecowl with the same energy. The reason for this isthat a stronger shock wave is induced at the pointcloser to the afterbody, as shown in Fig.8.

With 400 J laser energy, the shock wave is spatout from the inlet, as shown in Fig.9, because thestrong shock wave can propagate upstream with-out disturbance by the cowl. This shock wave actsas drag by sweeping the forebody.

In the case of laser-focusing on the cowl, thisshock spit-out does not occur although the laserenergy is 600 J. Therefore, in order to prevent theshock spit-out of an actual process involving LSD,the cowl shape needs to be optimized. For exam-ple, the inclination of cowl to the inflow should besteeper than the one used in the present computa-tion.

Thrust and Coupling coefficientFigure 10 shows the axial thrust histories of the

case of laser-focusing on the cowl by 200 , 400 and600 J. After laser-focusing, during initial severalmicro seconds, the net thrust is negative due to theaerodynamic drag in all cases. Since the inner cowlsurface near the focus is parallel to the z axis, theinitially strong shock wave, generated on the fo-cus, does not act as thrust. When the shock wavestrikes the afterbody, the thrust rapidly increases.The Larger the laser energy, the higher the thrustjump and the earlier it occurs. The thrust has amaximum value while the shock wave leg sweepsthe afterbody. After this time, the thrust grad-ually returns to the value of the drag. The du-ration, while a positive thrust is maintained, in-creases with the the laser energy.

Figure 11 shows the case of laser-focusing at themiddle point of the flow channel and the case oflaser-focusing on the cowl with 200 J energy. Theduration, in the case of laser-focusing at the mid-dle point of the flow channel, is about twice aslong as the one of laser-focusing on the cowl, be-cause the shock wave is induced at a point closerto the afterbody than the one of laser-focusing onthe cowl. Accordingly, the thrust is larger and thethrust jump also occurs earlier than those of laserfocusing on the cowl. On the case of laser-focusinon the middle point of the flow channel, the sec-ond thrust peak appears near 5//s. This peak is becaused by the pressure increase around the after-body surface behind the body shoulder, as shownin Fig. 12. This pressure increase disappears aftera few micro sec.

Figure 13 shows the coupling coefficients esti-mated in the computations by Wang et alJ3^, theexperiments by Myrabo et alJ1'2' and the presentcomputations. The coupling coefficient is definedas the following,

The cumulative impulseOne pulse laser energy (8)

In the present computation, the cumulative im-pulse is calculated as the time integral of the force



increment from the aerodynamic drag. The cou-pling coefficients in the cases of laser-focusing onthe cowl are fairly constant at about 150 N • s/MJ,being independent of the laser energy or slightlydecreasing with laser energy. This tendency is thesame as the computations by Wang et al. and ex-periments by Myrabo et al.

In the case of laser-focusing at the middle pointof the flow channel, the coupling coefficient isabout 2.5 times as large as the one in the caseof laser-focusing on the cowl. Therefore, the cou-pling coefficient is found to be sensitive to the focuslocation.

The computed shock wave is stronger than theactual one because chemical reactions are ne-glected here. In order to improve the accuracy ofthe impulse and coupling coeffcient estimation, itis necessary to include the energy loss mechanisms,chemical reaction and thermal radiation.

Fig.8 Pressure contours and vector plots at t = 1/^s in the case of laser-focsuing at the middlepoint of the flow channel with 200 J energy,(max. = 740595 Pa, min. = 405 Pa, incre. = 14804 Pa)

Fig.9 Pressure contours and vector plots at t =30 fj,s in the case of laser-focsuing at the middlepoint of the flow channel with 400 J energy,(max. = 1207828 Pa, min. = 1399 Pa, incre. = 24129 Pa)

8000

7000

6000

5000

4000

3000

2000

1000

0

-mnn

r-. : : ——— 200 J laser energy. : " : \ . : : . : ......... 400 J laser energy :

f \ : ........... 600 J laser energy :»• I \. . . ' • ' • ' ' . . : . : :.

\ \ : . ' . : : ' : : . . . : : . • - ' • ' • ]

-llV- - N ^ ; ; : . : : : : :• j \ • ; : : : -\

— i — i — i — i — i — i — i — i — i — i — . — • — . — • — i — i — i — i — i — i — , — i — i — i — i — . — i — i — uJ

10 20 40 50 6030Time [y s]

Fig. 10 Thrust histories in the cases of laser-focusingon the cowl.

8000

7000

6000

_ 5000

£ 4000

jE 3000

2000

1000

0

-1000

Foucs is on the cowl (200 J)..... FOCUS is at middle point of flow channel (200 J)

10 20 40 50 6030Time [p s]

Fig. 11 Thrust histories in the cases of laser-focsuingon the cowl and at the middle point of the flow channelwith 200 J.

Fig. 12 Pressure contours and vector plots at t5 (j,s in the case of laser-focsuing at the middlepoint of the flow channel with 100 J energy,(max. = 771276 Pa, min. = 1399 Pa, incre. = 15398 Pa)



-•— Laser-focusing on the cowl (present)A Laser-focusing at the middle point of flow channel (prsent)

••&••> Wang et al. (computation)A Myrabo et al. (experiments)_________________

400

350

300

250

200

150

100

50100 150 200 250 300 350 400 450 500 550 600

One pulse laser energy [J]

Fig. 13 Coupling coefficients.SUMMARY

The shock wave leg propagating downstreamsweeps the afterbody, and provides the mainthrust to the body. The shock wave propagatingupstream stays near the inlet because the speedof the shock wave relative to the body becomesequal to the inflow speed. This shock wave is fi-nally blown downstream and decays to pressurewaves.

The estimated coupling coefficient in the caseof laser-focusing on the cowl is fairly constant atabout 150N • s/MJ, being independent of the laserenergy.

The coupling coefficient is found to be sensitiveto the focus location.

REFERENCES[1] Myrabo, L.N., Messitt, D.G., and Mead,F.B.Jr., "Ground and Flight Tests of a Laser Pro-pelled Vehicle," AIAA Paper 98-1001, Jan., 1998.[2] Mead, F.B.Jr., Myrabo, L.N., and Messitt,D.G., "Flight and Ground Tests of a Laser-Boosted Vehicle," AIAA Paper 98-3735, 1998.[3] Wang, T.S., Chen, Y.S., Liu, J., Myrabo, L.N.,and Mead, F.B.Jr., "Advanced Performance Mod-eling of Experimental Laser Lightcrafts," AIAAPaper 2001-0648, Jan., 2001.[4] Raizer, Y.P., "Breakdown and Heating of Gasesunder the Influence of a Laser Beam," SovietPhysics USPEKHI, Vol.5, No.8, Mar.-Arp., 1966,pp.650-673.March-April, 1966.[5] Raizer, Y.P., "Subsonic Propagation of a LightSpark and Threshold Condition for the Mainte-nance of Plasma by Radiation," Soviet PhysicsJETP, Vol.31, No.6, Dec., 1970, pp.1148-1154.[6] Raizer, Y.P., and Tybulewicz, A., "Laser-Induced Discharge Phenomena," Studies in So-viet Sience, Edited by Vlases, G.C., and Pietrzyk,Z.A., Consultants Bureau, New York, 1977.

[7] Tannehill, J.C., and Anderson D.A., Pletcher,R.H., "Computational Fluid Mechanics and HeatTransfer, 2nd ed.," Tayler & Francis, Levittown,PA, 1997.[8] Gupta, R.N., Yos, J.M., Thompson, R.A.,and Lee, K.P., "A Review of Reaction Rates andThermodynamic and Transport Properties for an11-Species Air Model for Chemical and ThermalNonequilibrium Calculations to 30000 K," NASARP-1232, 1990.[9] Wada, Y., and Liou, M.S., "A Flux SplittingScheme with High-Resolution and Robustness forDiscontinuities," NASA TM-106452, 1994.[10] Anderson, W.K., Thomas J.L., and Leer B.V.,"Comparison of Finite Volume Flux Vector Split-tings for the Euler Equations," AIAA Journal,Vol.24, No.9, Sep. 1986. pp.1453-1460.[11] Jameson A., and Yoon, S., "Lower-Upper Im-plicit Schemes with Multiple Grids for the Eu-ler Equations," AIAA Journal, Vol.25, No.7, July.1987. pp.929-935.[12] Fujii, K., "Unified Zonal Method Based on theFortified Solution Algorithm," Journal of Compu-tational Physics, Vol.118, 1995. pp.92-108.[13] Messitt,D.G., Myrabo, L.N., Jones, R.A., andNagamatsu, H.T., "Computational vs. Experi-mental Performance of an Axisymmetric Hyper-sonic Inlet for Laser Propulsion," AIAA Paper 91-2547, June, 1991.[14] Sedov, L.I., "Similarity and DimensionalMethods in Mechanics, 10th ed.," CRC Press,1993.[15] Dewey B.M., McMillin, D.J., and Classen,D.F., "Photogrammetry of Spherical Shocks Re-flected from Real and Ideal Surfaces," Journal ofFluid Mechanics, Vol.81, Pt.4, 1977. pp.701-717.


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