Stationary force production:experimental and theoretical investigations

APOLLONOV V V

( Prokhorov General Physics Institute，Russion Academy of Sciences，Moscow 119991，Russia)

1 Introduction

Up to now the possibility of using a laser engine to launch light satellites into orbit looks like a very attractive idea for world wide spectrum of researchers［1-9］. The solution of problems considered in Ref.［3］is still of current interest. This is an increase in the efficiency—the coupling coefficientJrof using laser radiation( the ratio of the propulsion to the radiation power) by several times and the prevention of the shock damage of the apparatus，which appears when high-power repetitively pulsed laser radiation with low repetition rates is used. For example，forJr～0.3 kN/MW( this value is typical for an air-jet laser engine) ，the mass of 200 kg，and the acceleration of 10g，the required laser power should be ～60 MW( the energyQ～100 g in the TNT equivalent，f～100 Hz) ，and the power of a power supply should be 0.5 -1 GW. However，it seems unlikely that such a laser will be created in the near future. In our experiments，Jr～1 kN/MW( obtained experimentally) and 3 -5 kN/MW( estimated，special conditions) ，which allows us to reduce the laser power by a factor of 3 -10. A power of 10 -15 MW can be obtained already at present with the help of gas-dynamic lasers by using the properties of repetitively pulsed lasing with high repetition rates and methods for power scaling of lasing［10，11］.

To solve these problems，it was proposed to use repetitively pulsed radiation withf～100 kHz，the Optical Pulsating Discharge( OPD) ，and the effect of merging of shock waves produced by the OPD［12-14］. The merging criteria were confirmed in experiments［15］. The OPD is laser sparks in the focus of repetitively pulsed radiation，which can be at rest or can move at high velocities［16-20］. The high frequency repetitively pulsed regime is optimal for continuously-pumpedQ-switchedhigh-power lasers. In this case，the pulse energy is comparatively small and the stationary propulsion is possible.The aim of our paper is to verify experimentally the possibility of using laser radiation with a high pulse repetition rate to produce the stationary propulsion in a laser engine.

2 Experiments

In the model considered in Ref. ［12 -14］，the pulsed and stationary regimes are possible. Fig. 1 explains the specific features of these regimes. An OPD is produced at the focus of a lens on the axis of a gas jet flowing from a high-pressure chamber or an air intake to a cylindrical reflector. The shock waves generated by the OPD merge to form a quasi-stationary wave-high-pressure region between the OPD and reflector.

Fig.1 Scheme of the experiment side( a) and front( b) view.

As a result，the propulsionFrappears. In a cylindrical reflector，the coupling coefficient is maximal，Jr= 1.1 N/kW［13］，as for a plane explosion［21］. In the pulsed regime，the OPD is produced by trains of laser pulses. A narrow jet of diameterDj～0.3Rd［13］，which is smaller than the reflector diameterDr，carries a plasma out from the OPD region，which is necessary for the efficient formation of shock waves. Here，Rd= 2A5(q/p0)1/3is the dynamic radius of a spark，q( in J) is the laser pulse energy absorbed in a spark，andp0( in Pa) is the gas pressure. The propulsion acts during a pulse train，whose duration is limited by the air heating time. The hot atmospheric air is replaced by the cold air during the interval pulses. In the stationary regime，gas continuously arrives to the reflector from the bottom，by forming a jet over the entire section.In the experiments of this regime，we haveDj～2Rd～3 mm，which is comparable with the reflector diameterDr～5 mm.

The scheme of the experiment is shown in Fig.1. The OPD was produced by the radiation from a pulsed CO2laser. The pulse duration is ～1 μs，the duration of the front peak is 0.2 μs. The pulse repetition rate is varied from 7 to 100 kHz，the pulse energy is 0.1 -0.025 J. The peak power is 300 -100 kW，the average power of repetitively pulsed radiation is 600 -1700 W，and the absorbed power is ηW( η≈0.7) . Fig.2 shows the shapes of the incident pulse and the pulse transmitted through the OPD region. Note that for a short pulse duration and high power，η ～0.95. Because the radiation intensity at the focus is lower than the optical breakdown threshold in air，the argon jet was used. The lengthlof sparks along the flow was ～0.5 cm.

Fig.2 Oscillograms of laser pulse( 1) and radiation pulse transmitted through the OPD( 2) for f =50 kHz.

The model of a rocket with a laser engine was a duralumin cylinder of diameter ～8 cm，length～26 cm，and weight 1.1 kg，which was suspended on four thin wires of length 1.1 m and capable of moving only in the axial direction. A reflector( replaceable cylindrical attachment) was mounted on the chamber end. Laser radiation was directed to the chamber through a lens with a focal distance of 17 cm. The argon jet was formed during flowing from a high-pressure chamber through a hole with the diameter of 3 -4 mm. The jet velocityvwas controlled by the pressure of argon，which was delivered to the chamber through a flexible hose. The force produced by the jet and shock waves was imparted with the help of a thin( diameter ～0.2 mm) molybdenum wire to a weight standing on a strain-gauge balance( accurate to 0.1 g) . The wire length was 12 cm and the block diameter was 1 cm.

The sequence of operations in each experiment was as follows. A weight fixed on a wire was placed on a balance. The model was slightly deviated from the equilibrium position( in the block direction) ，which is necessary for producing the initial tension of the wire( ～1 g) . The readingFmof the balance was fixed，then the jet was switched，and the reading of the balance decreased toF1. This is explained by the fact that the rapid jet produces a reduced pressure( ejection effect) in the reflector. After the OPD switching，the reading of the balance becameF2.The propulsionFrproduced by the OPD is equal toF1-F2. The pressure of shock waves was measured with a pressure gauge whose output signal was stored in a PC with a step of ～1 μs. The linearity band of the pressure gauge was ～100 kHz. The gauge was located at a distance of ～5 cm from the jet axis( see Fig.1) and was switched on after the OPD ignition(t=0) . The pressure was detected for 100 ms.

Let us estimate the possibility of shock-wave merging in the experiment and the expected values ofFrandJr. The merging efficiency depends on the parameters ω =fRd/c0andM0=v/c0(M0＜1) ，wherec0is the sound speed in gas. If the distance from the OPD region to the walls is much larger thanRdand sparks are spherical or their lengthlis smaller thanRd，then the frequencies characterizing the interaction of the OPD with gas are:

For ω ＜ω1，the shock waves do not interact with each other. In the range ω ＜ω1＜ω2，the compression phases of the adjacent waves begin to merge，this effect being enhanced as the value of ω approaches ω2. In the region ω ＜ω2，the shock waves form a quasi-stationary wave with the length greatly exceeding the length of the compression phase of the shock waves. For ω ＜ω0，the OPD efficiently( up to ～30%) transforms repetitively pulsed radiation to shock waves.

In the pulsed regime，the value ofM0in formula ( 1) corresponds to the jet velocity. Because shock waves merge in an immobile gas，M0≈0 in formulas (2) and (3) . The frequenciesf=7 -100 kHz correspond toRd=0.88 -0.55 cm and ω=0.2～1.7. Therefore，shock waves do not merge in this case. In trains，where the energy of the first pulses is greater by a factor of 1.5 -2 than that of the next pulses( ω≈2) ，the first shock wave can merge. The propulsion produced by pulse trains isFr=JrηW=0.3 N，whereJr=1.1 N/kW，η =0.6，andW≈0.5 kW.

In the stationary regime forM0～0.7，the shock wave merge because ω ＞ω2( ω =1.8，ω2≈1.3) . A quasi-stationary wave is formed between the OPD and the cylinder bottom. The excess pressure on the bottom is δp=p-p0≈25 -50 kPa，and the propulsion isFr≈π(Dr2 -Dj2) δp/4 =0.03 -0.06 kg.

3 Measurement results

3．1 Control measurements

Fig.3 Pressure pulsations produced by the OPD for v=300 m/s( without reflector) ，f=7 kHz，W=690 W( a) ; f =100 kHz，W= 1700 W( b) ，and f=100 kHz，the train repetition rate φ=1 kHz，W=1 000 W，the number of pulses in the train N=30( c) ; the train of shock waves at a large scale，parameters are as in Fig.3( c) ( d) .

The jet propulsionsFjandFrand the excess pulsation pressure δp=p-p0were measured for the model without the reflector. We considered the cases of the jet without and with the OPD. The jet velocityvand radiation parameters were varied. Forv=50，100，and 300 m/s，the propulsion produced by the jet wasFj=6，28，and 200 g，respectively，and the amplitude of pulsations was δp=5 ×10-4，2 ×10-3and 3 ×10-2kPa. The OPD burning in the jet did not change the reading of the balance. This is explained by the fact that the OPD is located at a distance ofrfrom the bottom of a high-pressure chamber，which satisfies the inequalityr/Rd＞2，when the momentum produced by shock waves is small［3，22］. As follows from Fig.3，pulsations δp(t)produced by the OPD greatly exceed pressure fluctuations in the jet.

3．2 Stationary regime

The OPD was burning in a flow which was formed during the gas outflow from the chamber through a hole(Dj=0.3 cm) to the reflector(Dr=0.5 cm) ( see Fig.4) . Because the excess pressure on the reflector bottom( the angle of inclination to the axis is ～30°) was ～50 kPa( see above) ，to avoid the jet closing，the pressure used in the chamber was set to be equal to ～200 kPa. The jet velocity without the OPD wasv=300 and 400 m/s，Fj=80 and 140 g. The OPD was produced by repetitively pulsed radiation withf=50 and 100 kHz and the average powerW≈1200 W( the absorbed power wasWa≈650 W) . Within several seconds after the OPD switching，the reflector was heated up to the temperature more than 100 ℃.

Fig.4 Reflector of a stationary laser engine

Fig.5 illustrates the time window for visualization of shock waves with the Schlieren system in the presence of plasma. Before 7 μs，the plasma is too bright relative to the LED source，and all information about the shock wave is lost. At 7 μs，the shock wave image could be discerned under very close examination. By 10 μs，the shock wave is clearly visible in the image. However，the shock wave has nearly left the field of view at this time. A technique was needed to resolve the shock waves at short timescales，when plasma was present.

Forf=50 kHz andv=300 m/s，the propulsion isFr=40 g，and forv=400 m/s，the propulsion is 69 g; the coupling coefficientJris about 1.06 N/kW. The propulsionFris stationary because the criteria for shock-wave merging in front of the OPD region are fulfilled. Downstream，the shock waves do not merge. One can see this from Fig. 5 demonstrating pressure pulsations δp(t) measured outside the reflector. They characterize the absorption of repetitively pulsed radiation in the OPD and the propulsion. Forf= 50 kHz，the instability is weak ( ±5%) and forf=100 kHz，the modulation δp(t) is close to 100%. The characteristic frequency of the amplitude modulationfa≈4 kHz is close toc0/(2H) ，whereHis the reflector length. The possible explanation is that at the high frequencyf，the plasma has no time to be removed from the OPD burning region，which reduces the generation efficiency of shock waves. The jet closing can also lead to the same result if the pressure in the quasistationary wave is comparable with that in the chamber. Thus，repetitively pulsed radiation can be used to produce the stationary propulsion in a laser engine.

Fig.5 Pressure pulsations δp produced upon OPD burning in the reflector with Dr =0.5 cm，H=4/6 cm，v=400 m/s，Dj=0/3 cm for f=50 kHz，W=1 300 W( a) and f=100 kHz，W=1 200 W( b) ，( c) .

3．3 Pulsed regime

To find the optimal parameters of the laser engine， we performed approximately 100 OPD starts. Some data are presented in Tab. 1. We varied the diameter and length of the reflector，radiation parameters，and the jet velocity( from 50 to 300 m/s) . Forv=50 m/s，the ejection effect is small，forv=300 m/s≈c0，this effect is strong，while forv≈100 m/s，the transition regime takes place. In some cases，the cylinder was perforated along its circumference to reduce ejection. The OPD was produced by radiation pulse trains，and in some cases-by repetitively pulsed radiation. The structure and repetition rate of pulse trains were selected to provide the replacement of the heated OPD gas by the atmospheric air. The train duration was ～1/3 of its period，the number of pulses was 15 or 30 depending on the frequencyf. The heating mechanism was the action of the thermal radiation of a plasma I［23］，the turbulent thermal diffusivity with the characteristic time of ～300 μs［24］and shock waves.

Fig.6 Pressure pulsations δp in the OPD produced by pulse trains with φ=1.1 kHz，f=50 kHz，W=720 W，N=15，v=300 m/s，Dr =1.5 cm，H=5 cm，Dj =4 mm，and F=4.5 g.

The propulsionFrwas observed with decreasing the reflector diameter and increasing its length. The OPD burned at a distance of ～1 cm from the reflector bottom. One can see from Fig.6 that the shock waves produced by the first high power pulses in trains merge. Forf=100 kHz，the pulse energy is low，which is manifested in the instability of pressure pulsations in trains. As the pulse energy was approximately doubled at the frequency of 50 kHz，pulsations δp(t) were stabilized. The OPD burning in the reflector of a large diameter(Dr/Rd≈4) at a distance from its bottom satisfying the relationr/Rd≈3 did not produce the propulsion.

Tab. 1 presents some results of the measurements. One can see that the coupling coefficientJrstrongly depends on many parameters，achieving 1 N/kW in the stationary regime and 0.53 N/kW in the pulsed regime.

Tab．1 Experimental conditions and results

At present，the methods of power scaling of laser systems and laser engines，which are also used in laboratories，are extensively developed［10，25］. Let us demonstrate their applications by examples. We observed the effect when the OPD produced the ＇negative＇propulsionFtis 97 g( see Tab.1) ，which corresponds to the deceleration of a rocket. The value ofJrcan be increased by approximately a factor of 1.5 by increasing the pulse energy and decreasing their duration down to ～0.2 μs. An important factor characterizing the operation of a laser engine at the high-altitude flying is the efficiencyImof the used working gas. The valueIm=0.005 kg/( N·s)can be considerably reduced in experiments by using a higher-power radiation. The power of repetitively pulsed radiation should be no less than 10 kW. In this case，Frwill considerably exceed all the other forces. The gas-dynamic effects that influence the value ofFr，for example，the bottom resistance at the flight velocity ～1 km/s should be taken into account.

Thus，our experiments have confirmed that repetitively pulsed laser radiation produces the stationary propulsion with the high coupling coefficient.The development of the scaling methods for laser systems，the increase in the output radiation power and optimization of the interaction of shock waves will result in a considerable increase in the laserengine efficiency.

4 Impact of thermal action

A Laser Air-jet Engine ( LAJE) uses repetitively pulsed laser radiation and the atmospheric air as a working substance［1-3］. In the tail part of a rocket，a reflector focusing radiation is located. The propulsion is produced due to the action of the periodic shock waves produced by laser sparks on the reflector. The laser air-jet engine is attractive due to its simplicity and high efficiency. It was pointed out in papers［26］that the LAJE can find applications for launching space crafts if ～100 kJ repetitively pulsed lasers with pulse repetition rates of hundreds hertz are developed and the damage of the optical reflector under the action of shock waves and laser plasma is eliminated. These problems can be solved by using high pulse repletion rates(f～100 kHz) ，an optical pulsed discharge， and the merging of shock waves［12，13］. The efficiency of the use of laser radiation in the case of short pulses at high pulse repetition rates is considerably higher. It is shown in this paper that factors damaging the reflector and a triggered device cannot be eliminated at low pulse repetition rates and are of the resonance type.

Let us estimate the basic LAJE parameters: the forces acting on a rocket in the cases of pulsed and stationary acceleration，the wavelength of compression waves excited in the rocket body by shock waves，the radiusRkof the plasma region( cavern)formed upon the expansion of a laser spark. We use the expressions for shock-wave parameters obtained by us. A laser spark was treated as a spherical region of radiusr0in which the absorption of energy for the time ～1 μs is accompanied by a pressure jump of the order of tens and hundreds of atmospheres.This is valid for the LAJE in which the focal distance and diameter of a beam on the reflector are comparable and the spark length is small. The reflector is a hemisphere of radiusRr. The frequencyfis determined by the necessity of replacing hot air in the reflector by atmospheric air.

Let us estimate the excess of the peak valueFmof the repetitively pulsed propulsion over the stationary forceFsupon accelerating a rocket of massM. It is obvious thatFs=Ma，where the accelerationa=10 -20g0≈100 -200 m·s2. The peak value of the repetitively pulsed propulsion is achieved when the shock wave front arrives on the reflector. The excess pressure in the shock wave( with respect to the atmospheric pressurep0) produces the propulsionFj(t) and accelerationaof a rocket of massM. The momentum increment produced by the shock wave is:

Here，Fais the average value of the propulsion for the timetaof the action of the compression phase of the shock wave on the reflector，andFm≈2Fa.By equating δPito the momentum increment δPs=F/f=aM/fover the period under the action of the stationary propulsionFs，we find:

The value of Δ，as shown below，depends on many parameters. The momentum increment per period can be expressed in terms of the coupling coefficientJas δPi=JQ，whereQ［J］is the laser radiation energy absorbed in a spark. The condition δPi=δPsgives the relation:

between the basic parameters of the problem:W=Qfis the absorbed average power of repetitively pulsed radiation，andJ≈0.0001 - 0.0012 Ns/J［3，13，26］.The action time of the compression phase on the reflector ista～Rc/v，wherev≈k1c0is the shock-wave velocity in front of the wall(k1～1.2) andc0≈3.4×104cm/s is the sound speed in air. The lengthRcof the shock wave compression phase can be found from the relation:

Here，his the distance from the spark centre to the reflector surface andRd≈2.15(Q/p0)1/3is the dynamic radius of the spark( distance at which the pressure in the shock wave becomes close to the air pressurep0) . In this expression，Rdis measured in cm andp0in kPa. The cavern radius can be found from the relation:

The final expression ( 8) corresponds to the inequalityr0/Rd＜0.03 -0.1，which is typical for laser sparks(r0is their initial radius) . Let us find the range ofp0where the two conditions are fulfilled simultaneously:the plasma is not in contact with the reflector surface and the coupling coefficientJis close to its maximum［3，22，26］. This corresponds to the inequalityRk＜h＜Rd. By dividing both parts of this inequality byRd，we obtainRk/Rd＜h/Rd＜1，or 0.25 ＜h/Rd＜1. As the rocket gains height，the air pressure and，hence，h/Rddecrease. If we assume that at the start(p0=100 kPa) the ratioh/Rd=1，wherehandRdare chosen according to ( 2) ，then the inequality 0.25 ＜h/Rd＜1 is fulfilled forp0=100 -1.5 kPa，which restricts the flight altitude of the rocket by the value 30 -40 km(h=const) .

The optimal distancehsatisfies the relationh/Rd≈0.25bi，wherebi≈4 -5. By substitutingh/Rdinto (7) ，we find the length of the shock-wave compression phase and the time of its action on the reflector:

Wheres1From this，by using the relation Δ =Fm/Fa= 2/(Fta) ，we find:

Of the three parametersQ，W，andf，two parameters are independent. The third parameter can be determined from expression ( 6) . The conditionsl/f～taand Δ≈1 -2 correspond to the merging of shock waves［12］.

The important parameters are the ratio oftato the propagation timetz=L/cmof sound over the entire rocket lengthL(cmis the sound speed in a metal) and the ratio oftzto 1/f. For steel and aluminum，cm= 5.1 and 5.2 km/s，respectively. By using (10) ，we obtain:

Here，Lis measured in cm andcmin cm/s.Expression (12) gives the energy:

From the practical point of view，the most interest is the caseU＞1，when the uniform load is produced over the entire lengthL. IfU＜1，the acceleration is not stationary and the wavelength of the wave excited in the rocket body is much smaller thanL. If alsocm/f＜L，then many compression waves fit the lengthL. The caseU≈1 corresponds to the resonance excitation of the waves. Obviously，the caseU≤1 is unacceptable from the point of view of the rocket strength.

By using the expressions obtained above，we estimate Δ，U，andRkfor laboratory experiments and a small-mass rocket. We assume thatbi=4，J=5 ×10-4N s/J，ands1=1.4 ×10-5. For the laboratory conditions，M≈0.1 kg，Rr≈5 cm，L=10 cm，anda=100 m·s-2. The average value of the repetitively pulsed propulsionFIPis equal to the stationary propulsion，FIP=Fs= 10 N; the average power of repetitively pulsed radiation isW=FIP/J=20 kW，and the pulse energy isQp=W/f. We estimate the frequencyf，hence，Qp≈Qfor the two limiting cases.

At the start，p0≈100 kPa and the cavern radiusRkis considerably smaller thanRr. Here，as in the unbounded space，the laser plasma is cooled due to turbulent thermal mass transfer. ForQp＜0 J，the characteristic time of this process is 2 -5 ms［8，9］，which corresponds tof=500 -200 Hz. IfRk～Rr(p0＜10 kPa) ，the hot gas at temperature of a few thousands of degrees occupies the greater part of the reflector volume. The frequencyfis determined by the necessity of replacing gas over the entire volume and is ～0.5c0/Rr-850 Hz. Let us assume for further estimates thatf= 200 Hz，which givesQp=100 J. We find from (7) and (8) that Δ =74 andU= 3.5. This means that the maximum dynamic propulsion exceeds by many times the propulsion corresponding to the stationary acceleration. The action time of the shock wave is longer by a factor of 3.5 than the propagation time of the shock wave over the model length. Forp0= 100 and 1 kPa，the cavern radiusRkis 2.5 and 11.6 cm，respectively.

5 Dynamic resonance loads

Let us make the estimation for a rocket by assuming thatM≈20 kg，Rr≈20 cm，L=200 cm，anda=100 m·s-2. The average repetitively pulsed propulsion isFIP=Fs= 2 000 N，the average radiation power isW=4 MW，forf=200 Hz，the pulse energy isQp=20 kJ，Δ =12.6，U=1，Rk=14.7 and 68 cm(p0=100 and 1 kPa) ，andFm=25.6 kN.One can see that the repetitively pulsed acceleration regime produces the dynamic loads on the rocket body which are an order of magnitude greater thanFs. They have the resonance nature because the conditionU～1 means that the compression wavelengths are comparable with the rocket length. In addition，as the rocket length is increased up to 4 m and the pulse repetition rate is increased up to 1 kHz，the oscillation eigenfrequencycm/Lof the rocket body is close tof( resonance) .

Thus，our estimations have shown that at a low pulse repetition rate the thermal contact of the plasma with the reflector and strong dynamic loads are inevitable. The situation is aggravated by the excitation of resonance oscillations in the rocket body.

These difficulties can be eliminated by using the method based on the merging of shock waves.Calculations and experiments［28］have confirmed the possibility of producing the stationary propulsion by using laser radiation with high pulse repetition rates.The method of scaling up for the output radiation power is presented in the Ref.［10］.

6 Conclusion

Thus，our experiments have confirmed that repetitively pulsed laser radiation produces the stationary propulsion with the high coupling coefficient. The development of the scaling methods for laser systems，the increase in the output radiation power and optimization of the interaction of shock waves will result in a considerable increase in the laser-engine efficiency.

The author would like to acknowledge the valuable contributions made to the “Impulsar”program by Katorgin B I，Baturin Yu M，Tishcenko V N，Grachev G N，Gulidov A I，Smirnov A L，Sobolev A V，Smirnov M I，Kijko V V，Vagin Yu S，Egorov A B and Suzdal＇tsev A G. Please verify that (1) all pages are present，( 2) all figures are acceptable，(3) all fonts and special characters are correct，and(4) all text and figures fit within the margin lines shown on this review document. Return to your MySPIE ToDo list and approve or disapprove this submission. 7005-37.

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