AIAA 2002-3778Computational Performance Estimationof Laser Ramjet Vehicle

Hiroshi Katsurayama, Kimiya Komurasaki,and Yoshihiro ArakawaUniversity of Tokyo, Tokyo, Japan

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38th AIAA/ASME/SAE/ASEEJoint Propulsion Conference & Exhibit

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38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit7-10 July 2002, Indianapolis, Indiana

AIAA 2002-3778

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

AIAA 2002-3778

COMPUTATIONAL PERFORMANCE ESTIMATIONOF LASER RAMJET VEHICLE

Hiroshi KATSURAYAMA∗, Kimiya KOMURASAKI†, and Yoshihiro ARAKAWA‡

University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan

AbstractThe momentum coupling coefficient of a laser ramjet vehicle is calculated by CFD and an engine cycleanalysis. In addition, the fraction converted to pressure and kinetic energy of a laser energy is estimatedby CFD. In supersonic flights, the momentum coupling coefficient by CFD is about 1/3 of that of theengine cycle analysis mainly due to chemical frozen loss. In order to prevent the large chemical frozenloss at high altitudes, it is suggested that the laser power should be optimized according to the flightaltitude.

NOMENCLATURE

A = cross section of a vehicleC.A.R. = capture area ratioCd = drag coefficientCm = momentum coupling coefficientCp = specific heat at constant pressureE = total energy per unit volumeEL = laser energyEB = the pressure and kinetic energy

converted from the laser energyF = thrustg = acceleration of gravityH = flight altitude of the vehicleh = chemical enthalpyj = mass diffusion fluxM = Mach numbermv = vehicle massm = mass flow ratePL = laser powerp = static pressureq = heat fluxR = gas constantr0 = explosion source radiusS = maximum cross section of the vehicleT = static temperaturet = timeU = vehicle speedu, v = axial, radial velocity componentz, r, θ = cylindrical coordinatesγ = specific heat ratioηd = diffuser efficiencyηL = the fractional absorption

of the laser energyηB = the fraction converted to

the pressure and kinetic energyof the laser energy

∗Graduate student, Department of Aeronautics and Astro-nautics, Student Member AIAA†Associate Professor, Department of Advanced Energy,Member AIAA‡Professor, Department of Aeronautics and Astronautics,Member AIAACopyright c©2002 by the American Institute of Aeronauticsand Astronautics, Inc. All rights reserved.

πd = total pressure ratioρ = densityτ = viscous stress tensorsubscriptsi = inlets = speciest = stagnation condition∞ = freestream property

INTRODUCTION

In the past decade, alternatives to conventionalchemical rocket launch system have been sought inorder to reduce the launch cost. Especially, thereare strong demand to frequently deliver payloadsto a space station at a low cost. A pulsed-laserpowered ramjet vehicle will be able to satisfy thisdemand: Since energy is provided from a laser basein ground or space to the vehicle and the atmo-spheric air is used as a propellant, the payloadratio is improved drastically. In addition, once alaser base is constructed, the cost is only electricitycharges.Myrabo et al. proposed an air-breathing pulsed-

laser vehicle, so-called “Lightcraft,” and con-ducted flight tests with a scaled model.1) Theirlatest model, with additional solid ablative propel-lants, recorded the launch altitude of 121-meters.2)

The Lightcraft consists of a nosecone forebody, aparabolic afterbody, and a cowl, as shown in Fig.1.When a high-power pulsed laser beam is focusedby the afterbody mirror, air-breakdown occurs atthe ring focus on the cowl. The front of producedplasma absorbs the following part of laser pulseand expands in the form of Laser Supported Det-onation wave (LSD).3) This expansion induces ablast wave. The Lightcraft gains main thrust bythe blast wave sweeping on the afterbody.Wang et al.4) computed the flow field in the

Lightcraft featuring a closed inlet. By incorporat-ing a detailed laser-plasma interaction model, theyinvestigated the propagation processes of LSDwaves and estimated the momentum coupling co-

1American Institiute of Aeronautics and Astronautics

High speed air

Plasma

Shock wave

Pulsed laser

Axisymmetry

Fig.1 Pulsed-laser powered vehicle

efficient Cm, which is the ratio of cumulative im-pulse to one pulse laser energy.

In the present paper, Cm of a pulsed-laser ram-jet vehicle is estimated by an engine cycle analysisand by CFD. Then, estimated Cm by these twomethods are compared.

CFD METHODGoverning equations

Axisymmetric Navier-Stokes equaitionsare solved. Chemical reactions are treatedas finite rate reactions. The following11 species of air plasma are considered:N2, O2, NO, N, O, N+

2 , O+2 , NO+, N+, O+

and e−. The effects of thermal non-equilibriumand radiative energy transfer are not considered.Then, the governing equations are given by

∂U∂t

+∂F∂z

+1r

∂rG∂r

=∂Fv

∂z+1r

∂rGv

∂r+

Hr+S. (1)

U =

ρρuρvEρ1...ρ11

,F =

ρuρu2 + pρuv(E + p)uρ1u...ρ11u

,G =

ρvρuvρv2 + p(E + p) vρ1v...ρ11v

,

Fv=

0τzzτzruτzz+vτzr+qzj1z...j11z

,Gv=

0τzrτrruτzr+vτrr+qrj1r...j11r

,

H =

00p− τθθ00...0

,S =

0000ω1...ω11

. (2)

E and the equation of state are defined as

E =∑11

ρshs (T )− p +ρ

(u2 + v2

)2

, (3)

p =∑11

ρsRsT. (4)

0

2

4

6

8

10

12

0 5 10 15 20

r, c

m

z, cm

(a) Type A : 30◦ slope cowl (inlet is closed.)

0

2

4

6

8

10

12

0 5 10 15 20

r, c

mz, cm

(b) Type B : non-slope cowl (inlet is open.)

0

10

20

30

40

50

60

70

-60 -40 -20 0 20 40 60 80

r, c

m

z, cm

(c) Overall mesh (72,000 cells)

Fig.2 Computational meshs

hs and the transport properties are taken fromRef.(5)In the air chemical reaction model, the forward

rates of Ref.(6) are used. The backward ratesare calculated by the principle of detail balance.The chemical equilibrium constants are taken fromRef.(5).

Numerical SchemeA cell-centered finite difference scheme is

adopted. Inviscid flux is estimated with theAUSM-DV scheme7) and space accuracy is ex-tended to 3rd-order by the MUSCL approach withEdwards’s pressure limiter.8) Viscous flux is esti-mated with a standard central difference. Time in-tegration is performed with the LU-SGS9) schemewhich is extended to 3rd-order time accuracy by

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Matsuno’s inner iteration method.10) The calcu-lation is performed with the CFL number of 2 ∼20.

Computational meshFigures 2 (a) ∼ (c) show computational meshes.

The body length is set to 20 cm. The type Avehicle is almost the same as the “Label cE”Lightcraft.1) This is used for Cm validation.

The type B vehicle with non-slope cowl is usedfor the calculation of a supersonic flight to reducethe aerodynamic drag.

The mesh cells are set to be fine between thecowl and body to correctly capture a blast wave.In addition, the mesh is concentrated near the wallto resolve the viscous boundary layer. The meshwidth in the vicinity of the wall is ∆y = 80µm.

The outer boundary of the computational zoneis set far from the vehicle body to reduce the in-fluence of non-physical reflection waves from theouter boundary.

Explosion source

A explosion source model11) is employed insteadof solving complex propagation processes of LSDwave: The explosion source is modeled as a pres-surized volume centered at the laser focus. Thefocus is located at the middle on the inner cowlsurface. Since LSD processes can be consideredas isometric heating,3) the density in the source isassumed to be equal to an ambient atmosphere.

The source is assumed to be in chemical equilib-rium. The chemical composition is calculated bythe method of Ref.(12).

ηL is chosen as 0.6 to reproduce the experimen-tal data in Ref.(1). In addition, the dependencyof Cm on the source volume is investigated.

In supersonic flights, ηL is also assumed as 0.6.

ENGINE CYCLE ANALYSIS METHODAt first, the vehicle is launched from the ground

in a pulsejet mode. Air is taken and exhaustedfrom the rear side of the vehicle (Fig.3). In thismode, the thrust is estimated using Cm.

F = CmPL (5)

Cm and PL are assumed to be 250 N/MW and3.5MW, respectively.

When thermal choking by laser heating does notoccur even if air is taken from the inlet, the flightmode is switched to a ramjet mode. The thrustis computed by an engine cycle analysis assum-ing Humphrey cycle13) as indicated in Fig.(4) andFig.(5). Chemical reactions are not considered inthis analysis.

The area ratio is listed in table 1. A1 ∼ A4 areidentical to cross sections of the type B vehicle.A0/S = C.A.R. is assumed to be 0.6.

clos

e

exhaust

Shock wave

Laser heating

reflesh

Fig.3 Schematic of pulsejet mode.

shock wave

0 1 2 4Ram compression

Isentropic expansion

Diffuser Isentropic expansion

Isometric heating

3

Fig.4 Ramjet engine cycle analyses.

Volume

0 4

Ram

compression

Isentropic expansion

Isometric heating

Pre

ssur

e

1

2

3

Fig.5 Humphrey cycle(with additional isentropic expansion 1 → 2).

Table 1 Aera ratios of Type B vehicle.S A0/S A1/S A2/S = A3/S A4/S

201cm2 0.6 0.38 0.75 1

Between the location 0→1, air is ram-compressed. The total pressure ratio and totaltemperature are the following,

πd =pt1

pt0=

(1 + (1− ηd)

γ − 12

M0

)− γγ−1

,(6)

Tt1 = Tt0. (7)

ηd and γ are assumed as 0.97 and 1.4, respectively.Then, M1 is calculated by solving the following

equiation by Newton-Rapson method.

(2 + (γ − 1)M2

1

) γ+12(γ−1)

M1

3American Institiute of Aeronautics and Astronautics

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

Mach number

Cd

Fig.6 Cd used for the trajectory calculation.

= πdA1

A0

(2 + (γ − 1)M2

0

) γ+12(γ−1)

M0. (8)

The other phsyical properties at the location 1are calculated by M1, pt1 and Tt1.

Between 1→2, air is isentropically expanded toprevent thermal choking by laser heating at thethroat. The physical properties at location 2 arecalculated by the equations (6)∼ (8) with πd = 1.

Between 2→3, the air is isometrically heated.A3 is assumed to be equal to A2. The physicalproperties at location 3 are calculated by massconservation law and energy conservation law.

ρ3 = ρ2, u3 = u2, T3 = T2 +ηLPlaser

Cpm,

p3 = ρ2RT2, M3 = u3/√

γRT3. (9)

Finally, air is again isentropically expanded be-tween the location 3→4, and the thrust is calcu-lated as the following,

F = m (u4 − u0) + A4 (p4 − p0) . (10)

The trajectory of the vehicle is calculated bysolving the following motion equation by 4th orderRunge-Kutta scheme.

mvdU

dt= F − 1

2ρ∞USCd −mvg (11)

Herein, mv is 50 g, and Cd is shown in Fig.6.The flight condition is decided automatically by

tracing the trajectory.

RESULTS AND DISCUSSION

Code validation

In order to validate the model, Cm is comparedwith the experimental data under the followingcondition: EL = 400J ,r0 = 1.5mm and ηL = 60%.The estimated Cm is agreed with the experimentaldata, as listed in table 2.

0

5000

10000

15000

20000

0 200 400 600 800 1000

Thr

ust [

N]

Time [µs]

Fig.7 Thrust history (r0 = 1.5mm)

-1000

-500

0

500

1000

1500

2000

2500

3000

0 50 100 150 200

total

inle

t

cowl

body

Thr

ust [

N]

Time [µs]

Fig.8 Each thrust received by body, cowl andclosed inlet (r0 = 1.5mm)

Table 2 Cm in an quiescent atmosphere.Vehicle name Cm (N/MW)

Label E (Myrabo1)) 100Type A (present) 96

The history of the axial thrust is shown in Fig.7.After the positive thrust maintains till 125 µs, thenegative thrust continues till 900 µs. After 1000µs, the thrust almost is equal to be zero.Figure 8 shows the each thrust received by body,

cowl and closed inlet till 200 µs. After the explo-sion source bursts at t = 0µs, the shock wave ex-pands suddenly with weaking the intensity. There-fore, the thrust received by the cowl decreases fast.The thrust received by the closed inlet and the

afterbody decreases slower than that of the cowl.Figures 9 (a) ∼ (c) show the propagation pro-

cesses of the shock wave. The shock wave startsto sweep on the afterbody at t = 45µs. The shockwave propagates beyond the middle of the after-body at t = 100µs. Then, the shock wave leavesthe afterbody tail at t = 190µs.

Effect of r0

In order to estimate the fraction converted topressure and kinetic energy of the laser energy, thefollowing efficiency is introduced.

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(a) At t1 = 45µs(pmax = 5.98atm, pmin = 0.86atm, dp = 0.26atm)

(b) At t2 = 100µs(pmax = 3.49atm, pmin = 0.73atm, dp = 0.14atm)

(c) At t3 = 190µs(pmax = 3.56atm, pmin = 0.54atm, dp = 0.15atm)Fig.9 Pressure contours in quiescent atmosphere.

EB =p− p0

γ − 1+

ρ(u2 + v2

)−ρ0

(u2

0 + v20

)2

.(12)

ηB =∫EBdV

EL. (13)

The subscript 0 indicates the values before theexplosion. The integral is conducted over thewhole computational space.

In the case of r0 = 1.5mm, since the tempera-ture in the explosion source is about 20,000K, andthe ionization degree is about 25 %, large frac-tion of laser energy is consumed by chemical re-actions in the explosion source. Table 3 shows ηB

at t = 0µs, t = 10µs and t = 100µs. ηB increases

0

5

10

15

20

25

30

35

0

50

100

150

200

250

300

0 2 4 6 8 10Mach number

H

Cm

[N

/MW

]

Pulse jet Ramjet

Cm

H [k

m]

Fig.10 Mach number vs. Altitude and Cm

diagram.

by 18.8 % from t = 0µs to t = 10µs due to theenergy conversion from chemical enthalpy. Aftert = 10µs, the rate of this energy conversion be-comes slow.In the case of r0 = 5.5mm, the temperature in

the explosion source is about 5,000 K. Then, theenergy consumed by chemical reactions is smallerthan that of r0 = 1.5mm. However, the differencein ηB is only 3.7 % at t = 10µs. This result showsthat Cm is not so sensitive to r0.

Table 3 ηB, %r0, mm Cm ηB (0µs) ηB (10µs) ηB (100µs)1.5 96 16.1 34.9 36.25.5 114 29.4 38.6 40.8

Trajectory and Cm by engine cycle analysis

Figure 10 shows the Mach number vs. altitudeand Cm diagram. The mode switch occurs at M =2.4 and H = 8.7 km.For the supersonic flow conditions for CFD, the

following points are chosen from this trajectory.

Table 4 CFD simulation points of the trajectoryM H, km p∞ × 10−2, atm Cm, N/MW5 20 5.5 1368 30 1.1 83

Conditions for supersonic flights

Figures 11 (a) and (b) show the pressure con-tours at M = 5 and H = 20km of Type A and Bvehicle. Since the oblique shock wave is generatedfrom the tip of the cowl, the density at the focusof Type B vehicle is larger than that of Type A ve-hicle. Aerodynamic drag is large due to the wakegenerated from the body shoulder. Cd is listed intable 5. Since Cd of Type A vehicle is twice aslarge as that of Type B vehicle, the Type A couldnot produce a positive net thrust.

5American Institiute of Aeronautics and Astronautics

(a) Type B vehicle.(pmax = 1.8atm, pmin = 6× 10−3atm, dp =

5.1× 10−2atm).

(b) Type A vehicle.(pmax = 1.8atm, pmin = 2× 10−3atm, dp =

9.0× 10−2atm).

Fig.11 Pressure contours of steady flow fileds atH = 20km and M = 5 .

The mi and C.A.R of Type B vehicle are listedin table 6. The C.A.R. is defined as,

C.A.R. =

∫∫Sinlet

ρv · ds∫∫S∞

ρv · ds. (14)

Table 5 Aerodynamical drag.H,km M Cd, N (Type B) Cd, N (Type A)20 5 0.26 0.5330 8 0.25 —-

Table 6 mi of Type B vehicleH, km M mi, kg/s C.A.R., %20 5 1.40 53.130 8 0.55 61.6

Explosion in Supersonic Flow fieldSince Cm is not so sensitive to r0, r0 is decided

so that the average temperature in the explosionsource is about 20,000K. r0 and the density at thefocus, ρf , are listed in table 7

Table 7 r0 and ρf in the supersonic flow.H, km M r0mm ρf × 10−1kg/m3

20 5 4.5 1.5730 8 8.2 0.36

Figures 12 (a) and (b) show the pressure con-tours at t = 4µs and t = 10µs after explosion atH = 30km and M = 8. The blast wave sweeps

(a) At t = 4µs.(pmax = 2.57atm, pmin = 6.1× 10−2atm, dp =

0.13atm)

(b) At t = 10µs.(pmax = 5.36atm, pmin = 4.1× 10−2atm, dp =

0.27atm)

Fig.12 Pressure contours after explosion atH = 30km and M = 8.

on the afterbody without being spat out from theinlet.Figure 13 shows the thrust histories. the thrust

of M = 8 case is lower than that of M = 5 case,mainly owing to the small mass flow rate. Theblast wave speed of M = 8 case is faster thanthat of M = 5 case because of the small ambientpressure.

Comparison of Cm calculatedby CFD and engine cycle analysis

Cm by CFD and engine cycle analysis are shownin Fig.14. Cm simulated by CFD is about 1/3 ofthat of engine cycle analysis.This difference would be mainly due to chemical

frozen loss. Figure 15 shows ηB and ρf/ρ(H=0km).

-600

-400

-200

0

200

400

600

800

1000

1200

1400

0 50 100 150 200

Thr

ust [

N]

Time [µs]

h=30 km,M=8, ρf/ρ(H=0km)=0.03

h=20 km,M=5, ρf/ρ(H=0km)=0.13

Fig.13 Thrust histories of h = 20km andh = 30km flow case.

6American Institiute of Aeronautics and Astronautics

0

20

40

60

80

100

120

140

4 5 6 7 8 9

engine cycle analysis

CFD

Cm

[N/M

W]

Mach Number

Fig.14 Cm calculated by CFD and engine cycleanalysis.

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

H [km]

ηB [%

]

ηB (CFD)

ηB=ηL (engine cycle analysis)

ρf/ρ

(H=

0km

)

ρf/ρ

(H=0km)

Fig.15 ηB and ρf/ρH=0km

ηB of the engine cycle analysis is equal to beηL = 60%, because chemical reactions are not con-sidered. However, ηB estimated by CFD is smallerthan 40% due to chemical frozen loss. In addition,since the chemical frozen loss increases with de-creasing ρf , the ηB decreases with the flight alti-tude.

SUMMARY

Cm of the laser ramjet vehicle is calculated byCFD and the engine cycle analysis. By the CFDcode validation, Cm is found to be not so sensitiveto the explosion source radius.

In supersonic flights, Cm by CFD is about 1/3of that of the engine cycle anaylsis mainly dueto chemical frozen loss. However, if the effect ofchemical reactions is incorporated to the enginecycle analysis, the analysis would be able to pre-dict the trajectory of the laser ramjet vehicle.

In addition, in order to prevent the large chem-ical frozen loss at high altitudes, the temperatureincrease near the focus needs to be suppressed.Consequently, the laser power should be optimizedaccording to the flight altitude.

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, 1998.(2) Myrabo, L.N.: World Record Flights of Beam-Riding Rocket Lightcraft: Demonstration of “Dis-ruptive” Propulsion Technology Flight Tests ofa Laser Propelled Vehicle, AIAA Paper 01-3798,2001.(3) Raizer, Y.P.: Laser-Induced Discharge Phe-nomena, Studies in Soviet Sience, Consultants Bu-reau, New York and London, 1977.(4) Wang, T.S., Mead, F.B.Jr., Larson, C.W.:Advanced Performance Modeling of ExperimentalLaser Lightcrafts, AIAA Paper 01-3664, 2001.(5) Gupta, R.N., Yos, J.M., Thompson R.A. andLee, K.P.: A Review of Reaction Rates and Ther-modynamic and Transport Properties for an 11-Species Air Model for Chemical and ThermalNonequilibrium Calculations to 30,000 K, NASARP 1232, 1990.(6) Park, C.: Review of Chemical-Kinetic Prob-lems of Future Flight NASA Missions, I: EarthEntries, Calculations to 30,000 K, J. Thermophys.Heat Transfer, 7 (1993), pp.385-398.(7) Wada, Y., and Liou, M.S. : A Flux SplittingScheme with High-Resolution and Robustness forDiscontinuities, NASA TM-106452, 1994.(8) Edwards, J.R. : A Low-Diffusion Flux-Splitting Scheme for Navier-Stokes Calculations,Computers & Fluids, 26 (1997), pp.635-659.(9) Jameson A., and Yoon, S. Lower-Upper Im-plicit Schemes with Multiple Grids for the EulerEquations, AIAA Journal,25 (1987), pp.929-935.(10) Matsuno, K. : Actual Numerical Accuracy ofan Iterative Scheme for Solving Evolution Equa-tions with Application to Boundary-Layer Flow,Trans. Japan Soc. Aero. Space. Sci., 38 (1996),pp.311-322.(11) Ritzel, D.V., and Matthews, K.: An ad-justable explosion-source model for CFD blast cal-culations, Proc. of 21st International Symposiumon Shock Waves, pp.97-102,1997.(12) Botton, B., Abeele, D.V., Carbonaro, M., andDegrez G.: Thermodynamic and Transport Prop-erties for Inductive Plasma Modeling, J. Thermo-phys. Heat Transfer, 13 (1999), pp.343-350.(13) Bussing, T.R.A., and Pappas, G.: An Intro-duction to Pulse Detonation Engines, AIAA Paper94-0263, 1994.

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