Numerical and Engine Cycle Analyses of a Pulse Laser ...

Trans. JSASS Space Tech. JapanVol.1, pp.9-16, 2003

Numerical and Engine Cycle Analyses of a Pulse Laser Ramjet Vehicle

By Hiroshi Katsurayama, Kimiya Komurasaki, Ai Momozawa and Yoshihiro Arakawa

The University of Tokyo, Tokyo, Japan

(Received December 6th, 2002)

A preliminary feasibility study of a laser ramjet SSTO has been conducted using engine cycle analysis.

Although a large amount of laser energy is lost due to chemically frozen flow at high altitudes, the laser

ramjet SSTO was found to be feasible with 100 MW laser power for 100 kg vehicle mass and 1 m2 vehicle

cross-section area. Obtained momentum coupling coefficient, Cm, was validated by means of CFD. As a

result, the engine cycle analysis was under-estimating Cm. This would be because of the strong unsteady

energy input in the actual heating process and the spatially localized pressure on the afterbody.

Key Words: Laser Propulsion, Laser Ramjet, SSTO

Nomenclature

A : cross-section area of a vehicleAL : cross-section area of laser beam

C.A.R. : capture area ratioCd : drag coefficientCm : momentum coupling coefficientCp : specific heat at constant pressureCs

v : specific heat at constant volumefor species s

Csv,v : specific heat at constant volume

for species s for vibrational energyDCJ : Chapman-Jouguet velocity

e : energy per unit massEL : total laser energyEB : blast wave energy

(the sum of pressure and kinetic energy)f : focusing f number of an afterbody mirrorF : thrustg : acceleration of gravity

H : flight altitude of a vehicleh : enthalpy per unit massj : mass diffusion flux

M : Mach numbermv : vehicle massmp : air or propellant mass flow ratePL : laser power

p : static pressureq : heat fluxR : gas constant per unit massV0 : explosion source volumeS : maximum cross-section area of a vehicleT : static temperaturet : time

U : vehicle speedu, v : axial, radial velocity components

c©2003 The Japan Society for Aeronautical and Space Sciences

ws : mass rate of production of species sper unit volume

r, θ, z : cylindrical coordinatesγ : specific heat ratio

∆efs : chemical potential energy of species s

at T = 0Kε : structure coefficient

ηd : diffuser efficiencyηB : blast wave efficiencyλ : payload ratio

πd : total pressure ratioρ : densityτ : viscous stress tensor

Subscriptss : speciest : stagnation condition∞ : free-stream property

1. Introduction

There is a strong demand for frequent deliver of pay-loads to space at low cost. A pulse laser ramjet vehiclecould satisfy this demand: The payload ratio would beimproved drastically because energy is provided from alaser base on the ground to the vehicle and atmosphericair can be used as the propellant. In addition, once alaser base is constructed, the only cost is electricity.

The pulse laser ramjet vehicle shown in Fig.1 will beable to achieve SSTO by switching its flight mode. Ini-tially, the air inlet is closed to prevent a blast wave fromgoing upstream beyond the inlet. Air is taken in andexhausted from the rear side of the vehicle. This flightmode is called “pulsejet mode.” When ram-compressionbecomes available as vehicle velocity increases, the in-let is opened and the flight mode is switched to “ramjetmode.” Finally, when the vehicle cannot breathe suffi-cient air at high altitude, the flight mode is switched to

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Trans. JSASS Space Tech. Japan Vol.1 (2003)

PulseLaser

Blast wave

Laser plasma

Thrust

InletInflow

Fig. 1. Pulse laser ramjet vehicle.

“rocket mode.”In any flight mode, gas-breakdown occurs by focusing

a transmitted laser beam. The front of the producedplasma absorbs the following part of laser beam and ex-pands in the form of a laser supported detonation (LSD)wave.1) This expansion induces a blast wave, and theblast wave imparts thrust to a nozzle wall.

Myrabo et al. proposed a pulse laser vehicle, named“Lightcraft,” and conducted flight tests with a scaledmodel.2) Their latest model, with additional solid ab-lative propellants, recorded the launch altitude of 71meters.3) Wang et al. computed the flow field in theLightcraft in the pulsejet mode.4)

The objectives of this paper are to analytically exam-ine the performance of a laser ramjet vehicle in super-sonic flights and to study the feasibility of SSTO launchby the vehicle. A simple engine cycle analysis is con-ducted along with a CFD simulation.

2. Momentum Coupling Coefficient and BlastWave Efficiency

In pulse laser propulsion, the momentum coupling co-efficient Cm is commonly used as a performance indi-cator. Cm is the ratio of cumulative impulse to pulsedlaser energy and defined as

Cm =

∫ t

0Fdt

EL. (1)

Laser energy absorbed in a gas is distributed into blastwave energy EB, chemical potential energy and radiationenergy. EB is defined as

EB =∫ [

ρet+r (T ) +ρ

(u2 + v2

)

2

−ρ0et+r (T0)−

ρ0

(u2

0 + v20

)

2

]dV, (2)

where

ρet+r =11∑s=1

ρs

[∫Cs

v (T ) dT −∫

Csv,v (T ) dT −∆ef

s

].

(3)

The subscript 0 indicates the properties before laser inci-dence and et+r is the sum of translational and rotationalenergy. Cs

v and Csv,v are taken from Ref. 5). Cm would

be a function of EB/EL because only EB contributes tothrust. Therefore, we introduce the blast wave efficiencyηB defined by

ηB =EB

EL. (4)

3. Engine Cycle Analysis

3.1. Analysis methodIn the pulsejet mode, thrust was estimated using the

measured data of Cm as2)

F = CmPL , Cm = 100N/MW. (5)

In the ramjet mode, thrust was calculated by an en-gine cycle analysis method assuming the Humphrey cy-cle6) as indicated in Figs.2 and 3. Area ratios at Points

Shock wave

0 1 2 4Ram compression

Isentropic expansionIsentropic

expansion

Isometric heating

3

InletNozzleAir Exhaust

Fig. 2. Laser ramjet engine cycle.

Volume

0 4

Ram

compression

Isentropic expansion

Isometric heating

Pre

ssur

e

1

2

3

Fig. 3. Humphrey cycle with additional isentropic expansion#1 → #2.

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H. Katsurayama et al.: Numerical and Engine Cycle Analyses of a Pulse Laser Ramjet Vehicle

Table 1. Area ratios.

S A0/S A1/S A2/S = A3/S A4/S

1m2 0.6 0.38 0.75 1

#0 ∼ #4 in Fig.2 are listed in Table 1. A0 is defined as

C.A.R. =

∫∫Sinlet

ρv · ds∫∫S

ρv · dsand (6)

A0 = C.A.R.× S. (7)

From Point #0 to #1, air is ram-compressed. The totalpressure ratio and total temperature are:

πd =pt1

pt0=

[1 + (1− ηd)

γ − 12

M20

]− γγ−1

and (8)

Tt1 = Tt0, (9)

where ηd and γ were assumed as 0.97 and 1.4, respec-tively. Then, M1 was calculated by solving the followingequation by the Newton-Rapson method:

[2 + (γ − 1)M2

1

] γ+12(γ−1)

M1

= πdA1

A0

[2 + (γ − 1)M2

0

] γ+12(γ−1)

M0. (10)

Here, ρ1, T1 and p1 were calculated using M1, pt1 andTt1. From #1 to #2, the air is isentropically expanded.Physical properties at #2 were calculated by Eqs. (8)∼(10) with πd = 1. From #2 to #3, the air is isometricallyheated. Physical properties at #3 were calculated as

ρ3 = ρ2, u3 = u2, T3 = T2 +ηBPL

Cpmpand

p3 = ρ3RT3, M3 = u3/√

γRT3. (11)

Finally, the air was again isentropically expanded from#3 to #4, and thrust was calculated as the following:

F = mp (u4 − u0) + A4 (p4 − p0) . (12)

As the vehicle reaches high altitudes, the mass flowrate taken from the inlet decreases due to the decreasein air density. In this calculation, the flight mode isswitched to the rocket mode just before thermal chokingoccurs in the ramjet mode. The inlet is closed and hy-drogen propellant is injected between #1 and #2. Thepropellant is laser-heated from #2 to #3 and the flow isassumed to choke thermally at #3. Since the energy offlow before heating is negligibly small compared to thelaser energy input, the following relations are derivedfrom the energy conservation law and the equation ofstate:

T3 =ηBPL

mp

[2

Cp (γ + 1)

]and (13)

p3 =mp

A3

√RT3

γ. (14)

From #3 to #4, isentropic expansion was assumed, andthrust was calculated by Eq.(12).

Vertical launch trajectories are calculated by solvingthe following equation of motion by the 4th order Runge-Kutta scheme:

mvdU

dt= F − 1

2ρ∞U2SCd −mvg (15)

Herein, flight conditions were decided by tracing the tra-jectory. Cd is the function of M in Ref. 7). The payloadratio was estimated when the vehicle reached the firstcosmic velocity, 7.91 km/s:

λ =mv(t=0) −

∫mpdt

1− ε

mv(t=0)(16)

Although ε is about 0.25 to achieve SSTO using aSCRamjet engine,7) the structure weight of the laserramjet vehicle can be reduced due to its simple struc-ture. In this calculation, ε is assumed as 0.1.3.2. Computed trajectory and payload ratio

Figure 4 shows the altitude and Cm vs. Mach num-ber diagram, which is calculated under the conditionstabulated in Table 2. The mode is switched from thepulsejet to the ramjet at M = 2.0 and H = 7 km. Cm ofthe ramjet mode has the maximum value of 185 N/MWand then gradually decreases with the altitude due tothe decrease in air mass flow rate. The mode is switchedfrom the ramjet to the rocket at M = 8.7 and H = 36km. In the rocket mode, Cm is 30 N/MW independentof M and H. Figure 5 shows the payload ratios forPL = 113, 300 and 500 MW. This indicates that thelaser ramjet SSTO is feasible with PL & 100MW for100 kg vehicle mass and 1 m2 vehicle cross-section area.

0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

180

200

H, k

m

Flight Mach number

Cm

, N

/MW

H

Cm

Pulsejet

Ramjet Rocket

Cm

Cm

H

Fig. 4. H, Cm vs. M diagram.

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Table 2. Calculation conditions in the engine cycle analysis.

mv (t = 0), kg PL, MW ηB, % mp(rocket mode), kg/s

100 500 40 1

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300Puls

ejet

Ramjet

Rocke

t

PL=50

0MW

PL=30

0MW

PL=113MW

U, m

/s

Time, s

λ=0.67 λ=0.55 λ=0.30

Fig. 5. Flight velocity vs. time diagram.

4. CFD Analysis

4.1. Governing equations and numerical schemeAxisymmetric Navier-Stokes equations are solved

with finite rate chemical reactions. The fol-lowing 11 species of air plasma are considered:N2, O2, NO, N, O, N+

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

e−. Thermal non-equilibrium effect and radiative en-ergy transfer are not considered. Then, the governingequations are given by:

∂U∂t

+∂F∂z

+1r

∂rG∂r

=∂Fv

∂z+

1r

∂rGv

∂r+

Hr

+ S, (17)

U =

ρρuρvρeρ1...ρ11

, F =

ρuρu2 + pρuv(ρe + p)uρ1u...ρ11u

, G =

ρvρuvρv2 + p(ρe + p) vρ1v...ρ11v

,

Fv =

0τzzτzruτzz+vτzr+qzj1z...j11z

, Gv =

0τzrτrruτzr+vτrr+qrj1r...j11r

,

H =

00p− τθθ00...0

, S =

0000w1...w11

. (18)

ρe and the equation of state are defined as

ρe =11∑s=1

ρshs (T )− p +ρ

(u2 + v2

)

2, (19)

hs (T ) =∫

Csv (T ) dT + RsT + ∆ef

s and (20)

p =11∑s=1

ρsRsT. (21)

Here, thermo-chemical properties, transport proper-ties and chemical equilibrium constants are taken fromRef. 5). Rate coefficients of chemical reactions are takenfrom Ref. 8)

Inviscid flux is estimated with the AUSM-DVscheme9) and space accuracy is extended to the 3rd orderby the MUSCL approach with Edwards’s pressure lim-iter.10) Viscous flux is estimated with a standard centraldifference. Time integration is performed with the LU-SGS11) scheme, which is extended to the 3rd order timeaccuracy by Matsuno’s inner iteration method.12) Thecalculation is performed with the CFL numbers of 2 ∼20.4.2. Computational mesh and flight conditions

Figures 6 shows the computational mesh. The typeA vehicle has a pulsejet configuration that is almost thesame as the “Label E” Lightcraft.2) Computed Cm iscompared with the measured data for code validation.The type B vehicle is used for the ramjet mode. Cd oftype B is reduced by half compared to type A.

Mesh cells are set to be fine between the cowl and thebody to correctly capture blast waves. In addition, themesh is concentrated near the wall to resolve the viscousboundary layer. The outer boundary of the computa-tional zone is set far from the vehicle body to reduce theinfluence of non-physical reflection waves from the outerboundary.

Mesh convergence was checked with doubly fine cells.The result is listed in Table 3, where ∆rmin is a minimumcell width. Since the difference was only 3 %, 72,000mesh cells were used in this computation.

For ramjet flights, combinations of H and M listed inTable 4 were adopted.

Table 3. Mesh convergence of Cm at H = 20 km.

Cell number ∆rmin, µm Cm, N/MW

72,000 87.5 66

288,000 43.7 64

Table 4. Ramjet flight conditions.

H, km M p∞, atm ρ∞, kg/m3

20 5 0.055 0.089

30 8 0.012 0.018

( 12 )


��

� � ��

��

��

Forebody

Cowl

Afterbody

Laser

(a) Type A: Inlet closed, 30◦ slope cowl.

��

� � ��

��

��

(b) Type B: Inlet open, non-slope cowl.

��

��

��

��

(c) Overall mesh (72,000 cells)

Fig. 6. Computational mesh.

4.3. Explosion source modelAn explosion source model13) was employed instead of

solving complex propagation processes of the LSD wave:An explosion source was modeled as a pressurized vol-ume centered at a laser focus. A blast wave is drivenby the burst of this source. The focus is located at themiddle on the inner cowl surface. Since the LSD pro-cess is considered to be isometric heating,1) density inthe source is invariant during the heating process. Thegas in the source is assumed to be in chemical equilib-rium, and the chemical composition is calculated by themethod of Ref. 14).

In order to estimate the source volume and ηB, LSDwave propagation was analytically calculated. Figure7 shows the calculation model: The LSD wave is theplane wave which propagates in the laser channel, andthe shape of the LSD front is a line. The physical proper-ties on the LSD front are uniform. In addtion, the physi-cal properties in the region ionized by the LSD wave areinvariant. EB is calculated by accumulating the blast

Afterbody

AL

Cowl

rd

Laser

Z

Plasma

Axis of symmetry

LSD front

rf

Zd

Fig. 7. Model of LSD propagation.

wave energy behind the LSD front at each time step.Physical properties behind the LSD front are calcu-

lated from one-dimensional detonation relations:15)

p2 =p1 + ρ1D

2CJ

γ2 + 1, (22)

ρ2 =(γ2 + 1) ρ2

1D2CJ

γ2 (p1 + ρ1D2CJ)

, (23)

T2 = p2/R2/ρ2, (24)

γ2 =∑

s

ρs2

ρ2

Csv (T2) + Rs

Csv (T2)

, (25)

v2 = c2 =√

γ2p2

ρ2, (26)

v1 = DCJ and (27)

h2 =11∑s=1

[∫ T2

0

Csv (T ) dT + RsT2 + ∆ef

s

](28)

= h1 +12

(D2

CJ − v22

)+

PL/AL

ρ1DCJ, (29)

where subscripts 1 and 2 denote the states in front of andbehind the LSD, respectively. Cs

v are taken from Ref. 5).The velocities refer to the coordinate relative to the LSDwave. Since the laser beam is focused cylindrically, thecross-section area of the laser beam is

AL = 2πrdrf − rd

f. (30)

Here, rf and rd are the positions of the focus and thedetonation wave front, respectively. f = rf/zd is 3.6.2)

The history of PL was taken from Ref. 16). h2 and γ2 arecalculated by solving iteratively Eqs. (21) ∼ (26) withthe chemical equilibrium calculation. Then, the locationof the LSD wave is calculated by

drd

dt= −DCJ. (31)

At p = 1 atm, the LSD wave is not sustained at thelaser intensity below approximately 1MW/cm2.17,18) Inthe present computation, the laser absorption is assumedterminated when the laser intensity on the LSD wave de-cays to this threshold. Figure 8 shows the histories of

( 13 )


laser intensity, fractional laser absorption and ηB. Re-sulting fractional absorption and ηB were 60 % and 27%, respectively. Table 5 shows the source volume deter-mined using this ηB.

The blast wave is driven at t0 = 0 µs by the burstof the source volume. Fractional absorption is simplyassumed invariant for any atmospheric density becausethe LSD threshold is unknown in reduced atmosphericdensities.4.4. Computed results of the pulsejet mode

Figure 9 shows the pressure contours. After the burstof the explosion source, the shock wave reaches the af-terbody at t = 45 µs and sweeps the afterbody. It leavesthe afterbody tail at t = 190 µs. The thrust history isshown in Fig.10. Positive thrust is kept until t = 125 µsand then negative thrust continues until t = 1000 µs. Att > 1000 µs, thrust is almost zero.

Since computed Cm agreed well with the experimen-tal data listed in Table 6, this computational code withseveral physical models was found reproducing experi-mental data.4.5. Computed results of the ramjet mode

Figure 11 shows the pressure contours in the ramjetmode. The shock wave sweeps the afterbody withoutbeing spat out from the inlet. Figure 12 shows the thrusthistories. The thrust at H = 30 km is smaller than thatat H = 20 km due to the small mass flow rate and lowηB. Computed Cm is listed in Table 7.

0.1

1

10

100

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

Time, µs

Lase

r in

tens

ity o

n th

e LS

D w

ave,

MW

/cm

2

Laser intensity

on the LSD

wave

Fracti

onal lase

r abso

rptio

n

LSD threshold

ηB

Ene

rgy

frac

tion,

%

Fig. 8. Histories of EB and laser intensity at H = 0 km.

Table 5. Explosion source.

H, km EL, J ηB (t0), % V0, cm3

0 400 26.8 17.4

300 18.8 44.4

20 400 17.2 51.3

500 16.1 55.6

300 12.8 94.6

30 400 12.8 94.6

500 13.2 111.5

(a) At t = 45 µs

(pmax = 6.71 atm, pmin = 0.77 atm, dp = 0.30 atm)

(b) At t = 100 µs


(b) At t = 190 µs


Fig. 9. Pressure contours after an explosion with EL = 400J,H = 0 km and M = 0.

� ��

� � ��

�

� ��

��

� � ��

��

� � ��

��

� � ��

Time, µs

Thr

ust,

N

Cow

l

Afterbody

Total

Fig. 10. Thrust history in the pulsejet mode (Type A).

4.6. Effect of chemically frozen flow lossFigure 13 shows the histories of ηB. tse is the time

when the blast wave finishes sweeping the afterbody.

( 14 )


Table 6. Cm at H = 0km.

Vehicle name Cm, N/MWLabel E (Ref. 2)) 100Type A (present) 107

(a) At t = 12 µs.

(pmax = 2.27 atm, pmin = 2.1 × 10−2 atm, dp =0.11 atm)

(b) At t = 20 µs.

(pmax = 4.63atm, pmin = 2.1 × 10−2 atm, dp =0.23 atm)

(c) At t = 38 µs.

(pmax = 4.27 atm, pmin = 2.0 × 10−2 atm, dp =0.21 atm)

Fig. 11. Pressure contours after an explosion with EL = 400 J,H = 20 km and M = 5.

Table 7. Computed Cm in the ramjet mode.

H, km M mp, kg/s EL, J Cm, N/MW

300 66.9

20 5 1.4 400 66.0

500 64.0

300 40.3

30 8 0.6 400 41.0

500 41.0

At H = 0km, ηB is increased due to energy recoveryfrom the chemical potential (recombination reactions).At t > 10µs, the recovery rate is decreased and chemicalpotential energy is frozen.

At H = 20 km and 30 km, the blast wave finishes

-500

0

500

1000

1500

2000

1 10 100

H=20km, M=5

H=30km, M=8

EL=500J

EL=400J

EL=300J

EL=500J

EL=400J

EL=300J

Thr

ust,

N

Time, µs

Fig. 12. Thrust histories in the ramjet mode (Type B).

0

5

10

15

20

25

30

35

40

45

1 10 100

H=0km, M=0

H=20km, M=5

H=30km, M=8

η ,%

B

Time, µs

tse

Fig. 13. History of ηB in the case of EL = 400 J.

Table 8. Time average of ηB from t0 to tse.

H, km M EL, J ηB, %

0 0 400 38.5

300 26.6

20 5 400 25.6

500 24.5

300 16.2

30 8 400 16.0

500 15.8

sweeping the afterbody before the recovery of ηB be-comes maximum, and a large amount of chemical energyis lost. Table 8 shows the time average ηB from t0 to tse.The dependency of ηB on EL was found to be weak.

( 15 )


Table 9. Comparison of Cm between CFD and Engine CycleAnalysis (ECA).

H, km M EL, JCm (CFD),

N/MW

Cm (ECA),

N/MW

300 66.9 60.7

20 5 400 66.0 58.4

500 64.0 55.9

300 40.3 22.9

30 8 400 41.0 22.7

500 41.0 22.4

5. Comparison between Engine Cycle Analysisand CFD

In order to validate the engine cycle analysis, Cm

was re-calculated with the same flight conditions, ve-hicle cross-section area and ηB as the CFD conditions.The result is shown in Table 9. Engine cycle analy-sis was under-estimating Cm. This would be partly be-cause of strong unsteady energy input during the cycle:Although shock heating has been taken into account,actual heating is very localized and the peak pressure isabout 10 ∼ 13 times higher than that deduced by en-gine cycle analysis. Accordingly, the Humphrey cycleefficiency of the engine cycle is under-estimated.

Another possible reason would be the pressure local-ization on the afterbody: Since shock waves are reflectedand focused on the afterbody, pressure is strongly local-ized there. This 3-D effect would contribute to large Cm

in CFD.By incorporating these effects to cycle analysis, the

feasibility of a laser ramjet SSTO would be enhanced.

6. Summary

A preliminary feasibility study of the laser ramjetSSTO has been conducted using engine cycle analysis.The results show that the laser ramjet SSTO is feasiblewith PL ≥ 100MW for 100 kg vehicle mass and 1 m2

vehicle cross-section area. The obtained Cm is validatedby means of CFD. As a result, the engine cycle analysiswas under-estimating Cm. This would be because of thestrong unsteady energy input in the actual process andspatially localized pressure on the afterbody.

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