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A01-16501

AIAA 2001-0648Advanced Performance Modeling OfExperimental Laser Lightcrafts

Ten-See WangNASA Marshall Space Flight CenterHuntsville, AL

Yen-Sen Chen and Jiwen LiuEngineering Sciences, Inc.Huntsville, AL

Leik N. MyraboRensselaer Polytechnic InstituteTroy, NY

Franklin B. Mead, Jr.Air Force Research LaboratoryEdwards AFB, CA

39th AIAA Aerospace SciencesMeeting & Exhibit

8-11 January 2001 / Reno, NV

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AIAA 2001-0648Advanced Performance Modeling of Experimental Laser Lightcrafts

Ten-See Wang*NASA Marshall Space Flight Center, Huntsville, AL 35812

Yen-Sen Chenf and Jiwen Liu*Engineering Sciences, Inc., Huntsville, AL 35802

Leik N. Myrabo§

Rensselaer Polytechnic Institute, Troy, NY 12180and

Franklin B. Mead, Jr.Air Force Research laboratory, Edwards AFB, CA 93524

Abstract

A computational plasma aerodynamics model isdeveloped to study the performance of a laser propelledLightcraft. The computational methodology is based ona time-accurate, three-dimensional, finite-difference,chemically reacting, unstructured grid, pressure-basedformulation. The underlying physics are added andtested systematically using a building-block approach.The physics modeled include non-equilibriumthermodynamics, non-equilibrium air-plasma finite-ratekinetics, specular ray tracing, laser beam energyabsorption and refraction by plasma, non-equilibriumplasma radiation, and plasma resonance. A series oftransient computations are performed at several laserpulse energy levels and the simulated physics arediscussed and compared with those of tests andliteratures. The predicted impulses and couplingcoefficients for the Lightcraft compared reasonably wellwith those of tests conducted on a pendulum apparatus.

NomenclatureAp,Am matrix coefficients of transport equationsc speed of light in a vacuum.D species diffusivityev vibrational energy

eveq equilibrium vibrational energy->

F flux vectorH total enthalpy/ intensity or number of quantum levelsj emission coefficientK transfer constant between quantum levelskb Boltzmann's constantkc Coulomb constantkv thermal conductivityM molecular weightme electron massN total quantum level populationNI quantum leveln refraction indexn unit normal vector

ne electron number densityPr Prantdl numberp gas static pressurepe electron partial pressureQc non-elastic collision energy transferQr net radiative heat fluxqe electron charge—>r position vector

S source termsT gas temperature, K

Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17,U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Government purposes.All other rights are reserved by the copyright owner.

Staff, Senior Member AIAAPresident, Member AIAASenior Research Engineer, Member AIAAAssociate Professor, member AIAASenior Scientist, Member AIAA

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TeTtTvtHIV

Greek L

electron temperature, Ktranslational temperature, Kvibrational temperature, Ktime, svelocity componentsvelocity magnitudeCartesian coordinates

eo permittivity0 dependent variableT control surfaceKI laser absorption coefficient.Ae electron species thermal conductivityA/ laser wavelengthIJL viscosityQ refractive anglep densityTLT vibrational-translational relaxation time scale<rs> vibrational-translational energy relaxation time

for molecular species sTy shear stress tensorQ, control volume domainCO - chemical reaction source termcop plasma frequency

IntroductionBeamed energy propulsion was first promoted by

Kantrowitz1. Since then, a propulsion system supportedby a laser-sustained plasma has been the subject of manystudies.2"6 The main advantage gained by laserpropulsion is the low-weight system derived fromdecoupling the energy source from the vehicle, and highspecific impulse resulting in low fuel consumption. Thefirst ground and flight tests of a ground-based laserpropelled vehicle were reported in 1998,7 in which activetracking and beam control were demonstrated to 122 mon a horizontal wire and spin-stabilized free flights in thelaboratory were accomplished to altitude of 4 m. Sixmonths later, the spin-stabilized vertical free flightsoutdoors reached 30 m. These results wereaccomplished with axisymmetric vehicles which are of aspecial design in which a nosecone shaped forebody, anannular shroud, and a parabolic afterbody are the onlymajor components. A vehicle of such design is herebynamed as the "Laser Lightcraft" in this study. The layoutof a computational grid for a Laser Lightcraft is shown inFig. 1. The parabolic afterbody surface serves both as anaerospike nozzle and also the main receiving optic,whereas the annular shroud surrounds the ring focus ofthe parabolic optic. The Laser Lightcraft uses a Rocket-Based Combined Cycle (RBCC) to operate in both therocket and air-breathing modes. When operating in an

AIAA 2001-0648air-breathing mode, the specific impulse is infinity sinceno fuel is consumed. These successful tests7'8demonstrated the concept and the potential feasibility oflaunching small payload with a Laser Lightcraft.

When the focused beam energy strikes the shroud of aLaser Lightcraft, free electrons form to invoke InverseBremsstrahlung (IB) and the optical breakdown ensues.In order to achieve maximum intensity and to avoidundue plasma resonance, repetitive short pulses are used.As such, repetitive Pulsed Detonation Waves (PDW) aregenerated to propel the vehicle, hence the LaserLightcraft engine is also known as a PDW engine. It canbe seen that the underlying physics involved such as IBand PDW are so complicated that a simple system modelis not capable of describing the phenomena, let alonepredicts the thrust performance and the associated flowand thermal environments necessary for designing for thestructure integrity. A detailed computational plasmaaerodynamics model is therefore needed for the designand scaling of future Laser Lightcrafts.

From May to July 1999, performance data in the formof coupling coefficients were taken in a series of indoortests for the Model #200 series vehicles. These test wereperformed under a cooperative effort between RensselaerPolytechnic Institute, the Air Force ResearchLaboratory's Propulsion Directorate and NASA'sMarshall Space Flight Center (MSFC), to enhance LaserLightcraft propulsion research. These data are the first ofa kind benchmark quality data for computational modeldevelopment. Armed with these data for comparisonpurposes, a computational plasma aerodynamicsmethodology is developed as a first attempt to predictingthe thrust performance of a PDW engine propelled LaserLightcraft.

Computational Plasma Aerodynamics DevelopmentSeveral attempts have been made to analytically or

computationally model the physics inside continuouswave (CW) devices. For example, an early effort byRaizer9 assumed constant-pressure one-dimensional (ID)flow in air with heat addition by a laser. Thermalconduction was considered dominant and the radiationloss from the plasma was ignored. Kemp and Root10

later extended this ID analysis to hydrogen and toinclude thermal radiation. Molvik, Choi, and Merkleextended this problem to a two-dimensional (2D)structured-grid hydrogen flow and implemented real raytracing, but ignored thermal radiation. Jeng and Keeferdid similar analysis and added thermal radiation.Myrabo, Raizer, and Surzhikov6 later simulated theradiative-gasdynamics processes in optical dischargesmaintained in a subsonic De Laval nozzle. In the workdescribed in this paper, a multidimensional unstructured-grid computational plasma aerodynamics methodology is

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developed for both CW and repetitively pulsed wave, . . u u .... . ^ . tdevices, with emphases on nonequihbnum effects andlaser-induced physics such as laser-plasma interactions,Realistic laser absorption coefficients are used. Thegoverning equations will be described first, with theauxiliary equations that compute the nonequilibrium andlaser induced physics to follow.

The Governing EquationsTo properly describe the plasma aerodynamics

involved in laser-propelled propulsion, the time- varying

AIAA 2001-0648,. . , .surrounding surfaces, and n is a unit normal vector of F°

in the outward direction. The flux functionthe inviscid and the viscous flux vectors,

contains

Let's consider a control volume interface, e, betweencontrol volumes £ and F, with_a_ normal vector W. And,0£~0P ~ V0^*fe~' rp), where V0<? is interpolated from

transport equations of continuity, species continuity, the neighbor cells E and P. For the face e betweenmomentum, global energy (total enthalpy) and electron control volumes P and £, the diffusive flux can beenergy are formulated and written in a Cartesian tensor approximated as:form:

dt

dtd / \_ d [/ \dYi] .

d x v U i ' ~ J T x ^ '~dlT I ^1 J^~ J -*

d \(\ ^ V(\ /2/2)^xj{{ Pr )

d (2 } d (2 }_ d ( dTe\kbneTe ~kbneTeUj l ~ e ~ ~ r ~d Qc

where the shear stress TJ/ can be expressed as:

d u{ d U: 2d ukI . J K.~ " ™k o---——fyj*k

— F -ndT=$Q S^dQ,

where Q is the domain of interest, F denotes the

^imte v°lume formulation of flux integral can beevaluated by the summation of the flux vectors over eachface,

7=*<0

where k(i) is a list of faces of cell i, Fy representsconvection and diffusion fluxes through the interfacebetween cell i and j, and AFj is the cell-face area.

The convective flux is evaluated through the upwind-cell quantity by a linear reconstruction procedure toachieve second order accuracy:

_, , u £The beam energy absorption, beam energy transfer, andplasma radiation are invoked through the source terms ofthe global energy and the electron energy equations,

, , , . ? i i iwhereas the plasma species are generated through thesource terms of the species continuity equations.

Numerical SchemeThe cell-centered scheme is employed to allow the

volume surfaces to be represented by the grid cellsurfaces. The transport equations can be written inintegral form as:

where the subscript u

Irepresents the upwind cell and \j/e is a limiter used toensure that the reconstruction does not introduce local

11 is used here.(Qu,$j) (and

extrema. The limiter proposed by Earth1

Defining min

assuming is computed with ¥e = 1) the scalar We, . . , ,. „ , ,associated with the gradient at cell u due to edge e is:°

mini 1, —2^L-1 (p —

if

Solution ProceduresA general implicit discretized time-marching scheme

for the transport equations can be written as below,

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AIAA 2001-0648NB , s

where M? means the neighbor cells of cell P. The highorder differencing term and cross diffusion term aretreated using known quantities and retained in the sourceterm and updated explicitly. The second term of theright-hand side (RHS) is a perturbation that comes fromthe RHS. A predictor and corrector solution algorithm isemployed to provide coupling of the governingequations. The discretized finite-volume equations forma set of linear algebraic equations, which are non-symmetric matrix system with arbitrary sparsity patterns.The preconditioned Bi-CGSTAB12 and GMRES(m)13

matrix solvers are used to efficiently solve the linearalgebraic equations.

Auxiliary EquationsThermal Non-Kquilihriiirn Energy Equations

For high temperature flows, thermal non-equilibriumstate may be important. In Landau and Teller'sderivation,14 a master equation is employed to describethe evolution of the population of quantum level M. Thismaster equation is written as:

atn

-N 2;=0

Results from the quantum mechanical solution of theharmonic oscillator are used to relate the variousquantum transition rates to one another, and then themaster equation may be summed over all quantum statesto obtain the Landau-Teller equation:

Dpes_Dt

An empirical expression is used to model theLandau-Teller relaxation time scale

s=mol.<r

s=mol.

where subscript s represents the participating species(only diatomic species are involved). To solve thisvibrational energy equation, the source term is linearizedto result in an explicit term and an implicit term. Thistreatment is important for an unconditionally stablesolution of the equation. That is,

Explicit source term = P~

Implicit source term =

?(n)

The vibrational temperature is used to influence thereaction rates of chemical reactions by assuming that therate coefficients for dissociation are functions of thegeometrical mean temperature between T and Tv.16

Non-Equilibrium Air ChemistryA point-implicit (operator splitting) method is

employed to solve the chemistry system. For thebreakdown of air, Park's multitemperature air chemistry16

is baselined in this study. This mechanism composes ofthe dissociation, NO exchange, associative ionization,charge exchange, electron-impact ionization, andradiative recombination reactions. With this mechanism,electrons are produced first by the associative ionizationprocess and more electrons are produced by the electron-impact reactions. Since the number of electrons doublesin each such event, electron density increasesexponentially, in the form of an avalanche. Thismechanism thus provides the initial electron density forignition and produces the avalanche of electronsnecessary for the subsequent optical breakdown. It canbe seen that the IB process is embedded in thedissociation, associative ionization, and electron-impactionization reactions. The reaction rates were validated16

with experiments of postshock temperatures ranging from20,000 - 60,000 K which are inline with the computedbreakdown temperatures.

Plasma TnitiationUnderstanding the mechanisms responsible for plasma

initiation (ignition) has been the subject of ongoingresearch. A spark gap, an extremely intense pulse oflaser energy striking the focal point, seeded molecules orparticles, or a retractable tungsten target placed at thefocal point7 have been used as sources of free electronsfor initiation. It has also been shown that plasma can beignited quite easily off metal surfaces.2 Likewise innumerical modeling, initial free electrons are required forplasma ignition. This means determining an initialelectron density or a threshold breakdown intensity. Forexample, Ergun described an initial electron densitythat is necessary for modeling the ignition of a hydrogenplasma initiated with a retractable tungsten wire. In thisstudy, a threshold breakdown intensity approach isdevised in anticipation that a different thresholdbreakdown intensity may be encountered for differingsurface material, target approach, and laser power. Theadvantage of this approach is that the threshold

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AIAA 2001-0648breakdown intensity can either be calibrated or measured.For example, the effect of seeds on laser breakdownintensity has been reported.2 In this study, this approachis accomplished by performing parametric studies onplasma initiation with spark time, spark region, and sparkpower. A "spark" provides a fixed amount of energy(spark energy) in a region (spark region) centered aroundthe focal point for a fixed amount of time (spark time).When the strength of a spark reaches a threshold, or thethreshold breakdown intensity is satisfied, a lasersupported combustion (LSC) or detonation (LSD) isinitiated.

That means enough seed electrons are producedthrough the likes of associative ionization reactions suchthat the subsequent optical breakdown (ignition) ispossible. It is found that an ignition is sustainable whenthe energy absorbed by the plasma reaches 15% of thatdelivered. Spark times computed based on that criterionrange from 0.4 ^s to 1.2 jis. As expected, lower laserpulse energies require longer spark times for ignition. Itis also reasonable to assume that the plasma initiates in a(spark) region enclosed by the two outer laser rays and anarc with an origin at the focal point (in a 2D sense), andthe (spark) radius of that region is determined empiricallyas 1 mm. In actuality, the spark region is a torusgenerated by a pie-shaped cross-section rotated about thevehicle axis. Finally, the spark power is also empiricallydetermined as 15% of that of the incident laser - itstheoretical maximum.

Laser RadiationGeometric optics is used to simulate the local

intensity of the laser beam, which is split into a numberof individual rays. In the presence of absorption, the localintensity of each ray follows the Beer's law:

The electron is the only plasma species that absorbsthe laser energy. The absorption coefficient of the CO2laser radiation, corrected for stimulated emission in the

18single ionization range, is approximated by the formula:

-1/3!

KlCO ='5.72/?g

2ln[27(r/104)4/3pg-1/J](7V104)7'2'

A ray may change its propagating direction due to theinhomogeneous refractive index within the hot plasma.The index of refraction is taken from Edwards andPleck:19

n - 1-mec2n

The refracted angle is associated with the refractive indexthrough the Snell's law:20

where ni and n2 represent the refractive indexes for twodifferent control volumes, and 0i and 02 are the incidentand refracted directions with respect to the normaldirection of the interface between two volumes.

Non-equilibrium Plasma RadiationTreatment of radiative heat transfer of the plasma is

different from that of the laser and the solution of theradiative transfer equation (RTE) is required. Neglectingtransients and assuming a non-scattering medium, thecomplete RTE becomes

d/(j,Q)ds

In equilibrium gases, the electronic energy-levelpopulations are determined as a function of a uniquelydefined equilibrium temperature according to aBoltzmann distribution, and K andy are related accordingto Kirchhoffs law as = K l ( s ) 9 where I b ( s ) isblackbody intensity determined by Planck's function. Innon-equilibrium flows, however, all these simplerelations no longer apply, whereas the non-equilibriumabsorption and emission coefficients must be determinedby non-equilibrium populations of each energy level andby the transitions among various energy levels.

Air plasma includes the following atomic andmolecular species: O, N, NO, N+, O+, NO+, N2, O2, N2

+,O2

+, and e" which all contribute to the non-equilibriumradiative heat transfer. To determine K and j for thesespecies, the following four radiative transitions must betaken into account: atomic line transitions, atomic bound-free transitions, atomic free-free transitions, andmolecular transitions. The NEQAIR21 and LORAN22

codes provide detailed information on these transitions.In this study, the LORAN code is used to calculate the airplasma radiative properties for its relatively smalldatabase.

With the determination of K and 7, the RTE can besolved by either deterministic or stochastic approach.Unlike the computational plasma aerodynamicsgoverning equations, RTE is an integral differentialequation and numerical treatment is, thus, different from

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those for differential equations. Currently, there areseveral methods available for solving non-equilibriumradiative heat transfer which include the Monte Carlomethod, P-l method, quadromoment method,discrete ordinates method (DOM), discrete transfermethod (DIM),27 etc. The Monte Carlo method isaccurate but too time-consuming for practicalapplications. The P-l and quadromoment methods areonly accurate for an optically thick medium. The DOMand DTM are mathematically simple and can provideaccurate results for all optical ranges if the discretedirection number is reasonable large. Thus, the DOM andDTM are suitable for modeling radiation with aparticipating medium. For the Laser Lightcraftapplication, the high temperature region where non-equilibrium radiation is prominent is usually smallcompare to the entire computational domain. If theDOM is applied, RTE must be solved for the entire flowdomain, thus, much of the computational time is wastedin regions where radiation is not important. On the otherhand, with the DTM, RTE can be solved in a designatedarea. The more efficient DTM is selected to solve theRTE.

28

Plasma ResonanceThe plasma frequency is described as a property of a

space-charge-neutral plasma28'29 by which the motion ofthe electrons in specific electrostatic oscillations ischaracterized. The plasma frequency a>p of theelectrostatic oscillation of the electrons is defined as

(Op2 = neqe /meeo

It is seen that plasma frequency is a function only of theelectron density. The electrostatic oscillations cause aresonance of the plasma to incident (laser)electromagnetic waves of the same frequency, which arethen totally reflected.29 A critical electron density abovewhich the laser beam is totally reflected can then beobtained by equating the frequency of the incident laserwith that of the plasma.

Once the local electron density reaches the criticalvalue, a "total reflection" condition has to be applied tothe incident beam since the plasma becomes "opaque"and no beam can penetrate through the plasma front. Thisis achieved by assuming a total absorption at the plasmafront and then a redistribution of the energy. Zhu notedthat during the critical plasma resonance, the photons ofthe long-wavelength infrared radiation of a COi laser arepartially reflected or scattered off diffusely throughelastic collisions, and partially absorbed by the electrons(dominant) and/or some plasma particles throughnonelastic collisions. The absorbed photons are thenpartially released diffusely through radiative transitions,

AIAA 2001-0648and partially transferred to heavy gases through non-radiative transitions. Although the percentage of each isnot theoretically known, a lumped energy conversionpercentage during the critical plasma resonance can becalibrated with the test results. In this study, a 40%energy conversion is used. That translates to a 60%incident beam energy loss through elastic collisions andradiative transitions.

Experimental SetupThe 10 kW Pulsed Laser Vulnerability Test System

(PLVTS) CO2 laser at the High Energy Laser SystemTest Facility, White Sands Missle Range, NM, was usedto provide the beam energy for Laser Lightcraft impulseexperiments. The CO2 laser delivered up to 800 J single-pulses. The laser pulse energy was measured with acalorimeter and the uncertainty of the delivered energywas estimated to be ± 10 J. Several variations of thebasic Laser Lightcraft design (Model 200 series) similarto those described in Ref. 7 and 8 were examined and thetest results of the 6061-T6 all aluminum Model #200-3/4vehicle and Model A vehicle are chosen for this study.The pulse width was 18 us for Model #200-3/4 tests and30 [is for Model A tests. The impulse measurements wereconducted with a pendulum apparatus. This techniqueemployed a velocimeter coil, which was used in previouswork to determine the impulse imparted to a flat plateusing the PHAROS III laser at Naval ResearchLaboratory.31 The Laser Lightcraft was suspended andsuitably weighed before being subjected to a single pulseof energy from the PLVTS laser. The uncertainty in theimpulse measurement is estimated to be 1% or better.Details of the laser, measurement technique, andpendulum apparatus can be found in Refs. 7 and 8.

Computational Grid (GenerationFigure 1 shows the layout of a computational grid (for

Model #200-3/4). As described in Refs. 7 and 8, theLaser Lightcraft vehicle consists of a forebody (nose), anannular shroud, and a PDW engine (parabolic optic).Only half of the grid shown in Fig. 1 is actually solveddue to the axisymmetric formulation. An eight-zonestructured grid was generated first using a grid generationcode called UMESH.32 The grid is then fed to anauxiliary program such that an one-zone unstructured-grid is constructed for the actual computation.Quadrilateral elements are used such that the wallboundary layer can be computed if needed. High griddensity is used in the inner shroud region for capturingthe optical breakdown and pulsed detonation waveprocesses.

Results and DiscussionA series of computations have been performed for

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average laser pulse energies of 75, 100, 150, 200, 300and 400 J for Model #200-3/4, and 400, 600, and 800 Jfor Model A. LSD occurs in all cases with maximumMach number reaching 2.8. LSC then ensues as theplasma front expands, and the Mach number decreases tosubsonic value. Figure 2 shows the computedtemperature contours and laser beam traces at an elapsedtime of 18 jiis (the pulse width) and an average laser pulseenergy of 400 J. It can be seen that the laser beamreflects specularly on the optical surface and focuses ontoa focal "point" on the shroud where the breakdown of airstarts. Ten laser rays are plotted for clarity, whereas twohundred rays are actually used in the computation.

It is probably best to recount the phenomenon of anoptical detonation wave, which is a detonation wavegenerated by the optical breakdown of a medium with anincident laser beam, at this moment. This phenomenon

18was put forward by Raizer in detail in his book.Essentially, a strong shock wave is generated in theregion where the laser radiation that produces the plasmais absorbed and heat is deposited very rapidly. Thisshock wave travels through the medium, heating andionizing it such that the medium becomes capable ofabsorbing more laser radiation. The laser energy isdeposited in successive layers of a medium adjoining thefront of the shock wave subjected to the laser radiation.These layers themselves thus become energy sources thatmaintain the shock wave. Hence, the shock wave movesalong the optical channel in the opposite direction to thelaser beam. Thus, during the initial stages the shock waveis maintained by the laser beam and does not decay.

In Fig. 2 the optical breakdown is being fed by thelaser energy and the plasma front grows. Notice the raysare allowed to bend as the index of refraction varies withthe expanding plasma. As the local electron density of aplasma front reaches the critical value such that plasmaresonance occurs, the local plasma becomes opaque andthe regular laser energy absorption is greater on otherpoints of the surface of the plasma front. This process isnonlinear and may explain the irregular shape of theplasma front. It is anticipated that eventually the laserbeam is totally reflected as the plasma resonancephenomenon spreads over the entire plasma front. Thephenomenon of plasma resonance disappears as theplasma front expands and the electron density decreases.The computed optical breakdown phenomenon appearsto agree with the description given by Raizer.18 Noticethat the spreading of the shock wave outside the opticalchannel results in gradual attenuation because there is noenergy to maintain it. Thus, as the shock wave expandsin the direction opposite to the laser beam, it is weaklymaintained by laser radiation when it is not in a state ofplasma resonance.

Figure 3 shows the electron number density contours at

AIAA 2001-0648an elapsed time of 30 jis. At this time, the laser beam isturned off, thus the shock wave is not being maintainedby the laser radiation. In fact, LSD has turned into LSCafter about 12 jas and the shock wave is separating itselffrom the plasma. The electrons linger inside the shroud,near the focal point, with a portion of the electionsmoving downward and wrapping around the shroud lip.Eventually, the portion that is moving downwards breaksoff and the major electron cloud moves towards thecenter of the shroud inner space.

Figure 4 shows the computed maximum electronnumber density history. This is achieved by searchingfor the maximum value throughout the entirecomputational domain and by recording it at every timeintegration step. As shown in Fig. 4, the higher the laserpulse energy, the earlier the rise and the higher the peakvalue. After the pulse width, the maximum electronnumber density ceases to increase for all cases andeventually levels off. Plasma resonance occurs for allcases since all peak values exceed the critical value.Figure 5 shows the computed maximum electrontemperature history. Again, the higher the laser pulseenergy, the shorter the rise time and the higher the peakvalue. Within the pulse width, the maximum electrontemperature oscillates between 2.25 to 5.3 eV. After thepulse width, the maximum electron temperature dropssharply and eventually levels off.

The computational results indicate that a pulseddetonation wave often takes place within the pulse widthafter the start command. The higher the pulse energy, theearlier the generation of a pulsed detonation wave. Partof this propagating detonation wave will hit the parabolicoptic and reflect off the surface. The reflected and non-reflected portions would combine to make a propagatingMach stem. It should be noted that the phenomenon of apropagating Mach stem on the optical surface is verysimilar to that of a propagating Mach disc inside a bellnozzle during the start-up transient.33'34 This is becausethe aerospike optic geometry is an "inside-out" version tothat of a contoured bell nozzle.

As the Mach stem moves past the end of the shroud,the part that is attached to the inner-shroud starts to wraparound the outer-shroud. As the wrap around progresses,the shock wave weakens and grows like an ellipsoid. Aforebody attaching "leg" eventually develops, as shownin the pressure contours of Fig. 6 at 190 jus. Eventually,the "leg" leaves the nose and the ellipsoid expands toinfinity.

While the propulsion physics of the Laser Lightcraft iscomputed, it needs to be noted that the goal ofperformance modeling is to compute the impulse orcoupling coefficient. And it is achieved through temporalforce integrations. Figure 7 shows the computed(cumulative) coupling coefficient history for Model

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#200-3/4. Coupling coefficient is the generated impulse(N-s) divided by the delivered average laser single-pulseenergy (MJ). It is a performance measure unique to thepulsed laser propulsion, and conceptually similar to thespecific impulse (generated thrust divided by propellantweight flow rate) of a chemical rocket. The peaks arereached at around 75 to 125 JLIS - about the time when theshock wave leaves the tip of the optical surface and theforebody attaching "leg" wraps past the shroud andtouches the rear end of the nose. The leaving shock wavecreates a vacuuming effect in the optical surface area andthe forward moving forebody attaching shock generates anegative thrust. Although both effects decrease theimpulse and the coupling coefficient drops, it isanticipated both can be mitigated through designoptimizations. The coupling coefficient will recoversomewhat once the forebody shock wave leaves the nosethrough air replenishment. Then the coupling coefficientlevels off, exhibiting slight oscillations due to response tothe transient flow environment. A marker is put on eachcurve to represent the arrival time of the pressure wave atthe outer boundary. Notice the highest peak value andfinal value occur for the 200 J laser pulse energy case.

The final value of the cumulative impulse or couplingcoefficient is used to compare with that of themeasurement. Figure 8 shows a comparison of modelpredicted and experimental measured impulses for ModelA. The comparison is reasonably good although itappears that the model prediction is slightly higher thanthat of the measurement. This is because thesecomputations were performed without including the non-equilibrium radiation. Figure 9 shows a comparison ofmodel predicted and experimental measured couplingcoefficients for Model #200-3/4. The model predictedcoupling coefficients compare very well with those of themeasurements, with a slight edge going to the non-equilibrium radiation computation. The scattering of themeasured data is attributed to the measurement of thelaser energy, which varied from shot to shot (± 10 J) andan average over many pulses was used.

ConclusionsA computational plasma aerodynamics model has been

developed to study the propulsion physics of anexperimental Laser Lightcraft. The model developmentis based on a building block approach such that themodel can be improved continuously with improvedunderstanding of the physics. The model predicted laserpropulsion physics such as the optical breakdown anddetonation wave propagation agree well with thosedescribed in the literature. The model predicted couplingcoefficient for a Model #200-3/4 Laser Lightcraft and thepredicted impulse for a Model A Laser Lightcraft agreedreasonably well with those measured.

AIAA 2001-0648

A ckno wledgmentsThe authors wish to thank Sandy Kirkindall of Laser

Propulsion and John Cole of Advanced PropulsionResearch for supporting this study. The lead authorwishes to thank Chris Beairsto of the Directed EnergyTeam for the laser specifics, James Reilly of NorthEastScience and Technology and Jonathan Jones fordiscussions on plasma resonance, and Shen Zhu fordiscussions on the energy conversion processes duringthe plasma resonance.

ReferencesKantrowitz, A., "Propulsion to Orbit by Ground-Based

Lasers," Astronautics and Aeronautics, Vol. 10, No. 5,May 1972, pp. 74-76.2 Glumb, R.J., and Krier, H., "Concepts and Status ofLaser-Supported Rocket Propulsion," Journal ofSpacecraft and Rockets, Vol. 21, No. 1, 1984, pp. 70-79.3 Brandstein, A., and Levy, Y., "Laser Propulsion Systemfor Space Vehicles," Journal of Propulsion and Power,Vol. 14, No. 2, 1998, pp. 261-269.

Jeng, San-Mou, and Keefer, Dennis, "TheoreticalEvaluation of Laser-Sustained Plasma ThrusterPerformance," Journal of Propulsion, Vol. 5, No. 5,Sept-Oct, 1989, pp. 577-581.5 Molvik, G.A., Choi, D., and Merkle, C.L., "A Two-Dimensional Analysis of Laser Heat Addition in aConstant Absorptivity Gas," AIAA Journal, Vol. 23, No.7, 1985, pp. 1053-1060.6 Myrabo, L., Raizer, Y.P., and Surzhikov, ST., "LaserCombustion Waves in Laval Nozzles," HighTemperature, Vol. 33, No. 1, 1995, pp. 11-20.7 Myrabo, L.N., Messitt, D.G., and Mead, F.B., Jr.,"Ground and Flight Tests of a Laser Propelled Vehicle,"AIAA Paper 98-1001, Jan., 1998.8 Mead, F.B., Jr., and Myrabo, L.N., Messitt, D.G.,"Flight and Ground Tests of a Laser-Boosted Vehicle,"AIAA Paper 98-3735, July, 1998.9 Raizer, Y.P., "Subsonic Propagation of a Light Sparkand Threshold Conditions for the Maintenance of Plasmaby Radiation," Soviet Physics JETP, Vol. 31, Dec., 1970,pp. 1148-1154.

Kemp, N.H., and Root, R.G., "Analytical Study ofLSC Waves in Hydrogen," Journal of Energy, Vol. 3,Jan-Feb, 1979, pp. 40-49.11 Barth, T.J., "Recent Developments in High Order K-Exact Reconstruction on Unstructured Meshes,1 AIAAPaper 93-0668, 1993.

Fletcher, R., "Conjugate Gradient Methods forIndefinite Systems," Lecture Notes in Mathematics, Vol.506, 1976, pp. 73-89.13 Saad, Y., and Schultz, M.H., "GMRES: A GeneralizedMinimal Residual Algorithm for Solving Nonsymmetric

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Linear Systems," SIAM Journal of Scientific andStatistical Computing.Vol. 7, No. 3, Jan. 1986, pp. 856-869.

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Park, C., "Review of Chemical-Kinetic Problems ofFuture NASA Missions, I: Earth Entries," Journal ofThermophysics and Heat Transfer, Vol. 7, No. 3, 1993,pp. 385-398.

Ergun, M.A., Modeling and ExperimentalMeasurements of Laser Sustained Hydrogen Plasmas,Ph.D. Dissertation, University of Illinois at Urbana-Champaign, 1993.18 Raizer, Y.P., and Tybulewicz, A., "Laser-InducedDischarge Phenomena", Studies in Soviet Science, Editedby Vlases, G.C., and Pietrzyk, Z.A., Consultants Bureau,New York, 1977.19 Edwards, A. L. and Fleck, Jr. J. A., "Two-dimensionalModeling of Aerosol-Induced Breakdown," Journal ofApplied Physics., Vol. 50, No. 6, 1979, pp. 4307-4313.20 Modest, M. F., "Radiative Heat Transfer," McGraw-Hill, NY, 1993.21 Park, C., "Nonequilibrium Air Radiation Program:User's Manual," NASA TM-86707, 1985.22 Hartung, Lin, "Theory and User's Manual for LoranCode," NASA TM-4564, 1994.

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Hora, Heinrich, Laser Plasmas and Nuclear Energy,Plenum Press, New York and London, 1975.30 Zhu, S., USRA, MSFC, personal communication.31 Lyons, P.W., Myrabo, L.N., Jones, R.A., Nagamatsu,H.T., and Manka, C., "Experimental Investigation of aUnique Airbreathing Pulsed Laser Propulsion Concept,"AIAA Paper 91-1922, 1991.32 Chen, Y.S., "UNIC-MESH: A Model Builder & MeshGenerator, Graphical User Interface," EngineeringSciences, Inc. Internal Report, UNIC-Mesh-V.3.0, 2000.33 Wang, T.-S., "Numerical Study of the TransientNozzle Flow Separation of Liquid Rocket Engines,"Computational Fluid Dynamics Journal, Vol.1, No.3,Oct. 1992, pp. 319-328.34 Wang, T.-S. and Chen Y.-S., "Unified Navier-StokesFlowfield and Performance Analysis of Liquid RocketEngines," Journal of Propulsion and Power, Vol. 9, No.5,Sept.-Oct. 1993,pp.678-685.

24

25

26Fig. 1 The layout of a computational grid for the LaserLightcraft Model #200-3/4.

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AIAA 2001-0648

1023

102

1021

1020

Pulse Width

0.0000

Critical Value

I0.0001

Elapsed time, s0.0002

Fig. 2 Computed temperature contours and laser ray traces Fig. 4 Computed maximum electron number densityfor Model #200-3/4 pulsed by 400 J laser energy at 18 jus. history for Model #200-3/4.

Se+04

5e+04 -

1e+04

Oe+000.00000 0.00005

Elapsed time, s0.00010

Fig. 3 Computed electron number density contours for „. - ~ _, t , .^ . , 11^™ Z,* t ,, .^^ T t ^^ Fig- 5 Computed maximum electron temperature historyModel #200-3/4 pulsed by 400 J laser energy at 30 ̂ s. fof Model #200-3/4.

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AIAA 2001-0648

Fig. 6 Computed pressure contours for Model #200-3/4pulsed by 400 J laser energy at 190 jis.

0.11

» 0.10

jg 0.0913

J 0.08

C"2 0.07

0.06

0.05

0.04 I350 450 550 650 750 850

Average Laser Pulse Energy. J

Fig. 8 A comparison of the impulses for Model A.

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O 1

1 1 1 1 1 1 1 1 1 1 1 1 1 . 1 1 . 1 . 1 . 1 . 1 1 1 1 1 1 1 1 1 ., 1 1 1 1 1 1 1 1 1 1 1 1 1 1-._._._. 75 j -- - - - 100 J :----- 150 J =

—— 300 J :——— 400 J :i

, -, <______x _

- » -

i A--— ._.--—•_.—/ \ /x~.-.._._.^— '" i

0.00000 0.00100 0.00200 0.00300

Elapsed time, s0.00400

250 ^

200

v 150

100

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150 200 250 300 350 400

Average Laser Pulse Energy, J450

^. -, <^ . .j i * - i- . c r - - ^ u - x Fig. 9 A comparison of the coupling coefficients forFig. 7 Computed cumulative coupling coefficient history ^, , , ̂ ^ „,, r &

r A/T j i^o^o/^ Model #200-3/4.for Model #200-3/4.

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