thrust augmenting ejectors, ii

1698 AIAA JOURNAL VOL. 21, NO. 12

Thrust Augmenting Ejectors, Part II

Morton Alperin* and Jiunn-Jenq WutFlight Dynamics Research Corporation, VanNuys, California

The solution of the equations representing the flow of a compressible fluid through an ejector has been shownto possess two distinctly different results after complete mixing. Part I of this study reported the analysis anddescribed the performance of ejectors configured under the first solution where the flow after mixing is sub-sonic. The second solution to the ejector flow equations where the flow after mixing is supersonic provides abasis for selection of an optimal and a limiting ejector configuration for any given set of flight and injected gascharacteristics. These special ejector configurations provide very high ideal thrust augmentations while trans-lating from zero to supersonic velocities. Maps showing this ideal performance over large ranges of flight andinjected gas characteristics are presented. Consideration is given to the influence of realistic, nonisentropicoutlets as required for starting the supersonic flow (swallowing the starting shock wave) and for return of thesupersonic mixed flow to ambient pressure. The data presented are limited to ejectors having a constant areamixing duct whose cross section is equal to 25 times that of the minimal area of the exhaust flow from the gasgenerator.

Nomenclaturea = primary jet areaa* = primary jet throat area or area at ambient pressure

exhaustM = Mach numberm = mass flow rateP0 = stagnation pressureP = pressurer = entrainment or mass flow ratio (= m, / mp)S = total entropyT = temperatureT0 = stagnation temperatureU = secondary or mixed flow velocityV = primary or injected flow velocityX - duct width or areaa. •= area ratio (=J^/a7)a* = area ratio (=X2/a*)y = ratio of specific heats (Cp/Cv), =1.4 for data

presentedAP = primary j et pressure rise ( = P0p- P0oo)AS = total entropy production due to mixingAT = primary jet temperature rise (= T0p — T0oo)0 = thrust augmentationSubscriptsi = induced or secondary flowp = primary flow1,2,3 = ejector stationsoo = ambient or freestream conditions

Introduction

A COMPRESSIBLE flow analysis of the flow through athrust augmenting ejector has been described in Ref. 1

for ejectors in which the mixing process occurs in a duct ofconstant cross section. In that analysis it was shown that theflow after complete mixing has two distinctly differentcharacteristics, depending upon which solution to the flowequations is chosen. One flow is always subsonic or sonicafter mixing, regardless of the characteristics of the two flows

Received June 2, 1982; revision received Jan. 10, 1983. Copyright© 1982 by Flight Dynamics Research Corporation. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., withpermission.

Technical Director. Member AIAA.jResearch Director.

at the start of the mixing process. This situation is called thefirst solution, and the performance of ejectors designed underthis solution has been discussed in Part I.

The second solution results in a supersonic or sonic flowafter mixing, regardless of the characteristics of the two flowsat the start of mixing. This second solution, althoughpreviously described by Keenan et al.,2 has not been utilizedin the analysis or the design of thrust augmenting ejectorsprior to the disclosure by the present authors in Ref. 3. Onepossible reason for this neglect appears to be related to thefact that the second solution, under certain conditions,represents flows which violate the Second Law of Ther-modynamics. However, there do exist practical conditions atwhich the second solution represents realistically achievableflows having very desirable ejector performance, and whichdo not violate the Second Law of Thermodynamics.

Using the notation and station designation described onFig. 1, the analyses presented in Refs. 1 and 3 provide a meansfor the determination of the performance and specification ofthe geometry of thrust augmenting ejectors designed undereither the first or second solution and operating under anygiven set of flight and injected gas conditions. Under theassumption of constant area mixing, ejector geometry consistsof the cross section of the mixing duct (X2) in relation to thatof the minimum cross section (a*) of the mass flow from thegas generator, and the general shapes of the inlet and outlet.

The minimum cross section (a*), as defined in thisdocument, is either the throat area of the gas generator'sdischarge to ambient pressure when the pressure ratio issupercritical, or the exit area of the gas generator's dischargeto ambient pressure when the pressure ratio is subcritical. Thegeneral shape of the inlet and outlet are determined from therequirement for isentropic ingestion of the secondary flowand the discharge of the mixed flow through the inlet andoutlet, respectively, as described by the analysis. Thrustaugmentation is defined as the ratio of the net thrust of theejector to the net isentropic thrust of its gas generatoroperating as a free jet at the same flight condition andutilizing the same mass flow and stagnation conditions asthose of the primary jet of the ejector.

With any given set of flight and injected gas characteristics,and mixing duct size in relation to the cross section of theprimary flow, there exist certain specific ejector inlet andoutlet geometries which provide either a local maximumperformance or a limiting performance. The local maximumrefers to an ejector configuration which produces a localmathematical maximum thrust augmentation in the

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DECEMBER 1983 THRUST AUGMENTING EJECTORS, PART II 1699

relationship between thrust augmentation and the Machnumber of the secondary flow at the start of the mixing (Ml).This optimization procedure was discussed in detail in Ref. 1.

Under the second solution this local maximum alwaysoccurs when the secondary flow at the start of mixing is in thetransonic or supersonic range. The limiting performancealways occurs when the secondary flow at the start of mixingis subsonic (for all conditions examined to date). Itcorresponds to the situation where the total entropy of themixed flow is equal to the sum of the total entropies of theprimary and secondary flows at the start of mixing—a limitimposed by the Second Law of Thermodynamics. Obviouslythe inherent production of entropy or the irreversibility of theflow process due to phenomena such as skin friction, heattransfer across the ejector boundaries, vorticity production,and shock waves will alter the flow and performance fromthat predicted by ideal calculations at the limiting point andthe local maximum point. Specific examples of the influenceof major losses are presented in Part I,1 Ref. 3, and in thisdocument. Other losses including the external drag due to skinfriction or shock waves could have a substantial effect onoverall aircraft performance, but the evaluation of thoseeffects are intimately dependent upon the particular designand therefore are not estimated in this general study.

Ideal Performance of Second Solution EjectorsThe ideal performance of second solution ejectors is

evaluated under the assumption that the primary andsecondary flows enter the ejector isentropically from theirsource and the outlet flow discharges isentropically to am-bient pressure after complete mixing. As previously indicated,the performance of thrust augmenting ejectors operating atany given set of flight and injected gas conditions is stronglydependent upon the Mach number of the secondary flow atthe start of mixing. The existence of a local maximum at atransonic or supersonic secondary flow at the start of mixingand a subsonic limiting value of the Mach number of thesecondary flow at the start of mixing—limited by the SecondLaw of Thermodynamics—have been verified by numerouscomputations and these criteria will be utilized in describingoptimal, ideal ejector performance in the followingdiscussions.

Second Solution Ejectors with Transonic or Supersonic M1

The ideal performance of second solution thrustagumenting ejectors designed to operate with transonic orsupersonic secondary flow at the start of mixing—at theirlocal maximum performance point—is described on maps ofconstant augmentation lines plotted on surfaces of primary jetstagnation properties on Fig. 2 for flight Mach numbers of 0,0.65, and 2. For purposes of comparison, performance ateach flight speed is based upon a fixed ejector cross section(X2) in relation to the cross section of the mass flow from thegas generator (a*), i.e., a* =X2/a^ =25.

Ejectors which are stationary with respect to the freestreamvelocity component in the thrust direction must accelerate theingested fluid to transonic or supersonic speeds when designedunder this criterion. This involves a large expansion and alarge recompression for ingestion and return of the flow toambient pressure. The ideal performance of stationaryejectors requiring supersonic ingested flow at the start ofmixing may be of limited practical interest and is presentedfor observation of performance trends as a function of flightand injected gas characteristics.

As shown on the lower part of Fig. 2, the ideal thrustaugmentation of stationary ejectors designed under thetransonic or supersonic Ml criterion is quite large at lowprimary stagnation conditions and decreases with increasingprimary jet stagnation temperature and pressure. It is in-teresting to note however, that the local maximum per-formance under this criterion occurs at the upper choking

Gas Generator

(Secondary Flow)

Fig. 1 Schematic representation of ejector.

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Jet Pressure Rise,

Fig. 2 Ideal thrust augmentation, optimal second solution ejectorswith transonic or supersonic A/7; e = 25.

point when the injected gas stagnation temperature is large.At higher temperatures, the optimization procedure forejectors designed under this criterion results in nonproductiveperformance (</><!, as shown by the dashed lines), andeventually the solution vanishes, as illustrated by the cross-hatched portion of the figure, due to a deficiency of the mixedflow kinetic energy required to return the flow to ambientpressure after mixing at the upper choking point. However, itshould be noted (as dicsussed in Part I) that a trivial point(0=1) always exists at a supersonic value of M7 which canalways replace the nonproductive or "no-solution" region ofthe map for ejectors designed under this criterion.

Ideal inlets must be accelerating (converging or con-verging/diverging) and outlets may be either converging,

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1700 M. ALPERIN AND J.-J. WU AIAA JOURNAL

diverging, or converging/diverging, depending upon thestagnation properties of the primary jet as shown on the samefigure.

The center part of Fig. 2 displays an example of the per-formance of second solution ejectors operating in the mid-subsonic speed range and designed under the same criterion.The primary jet stagnation pressure varies over the rangeAP/P00=0 to 20 (P0p/Pw = 1.3283 to 21.3283) and theprimary jet stagnation temperature varies over the rangeAT/T00=-0.0845 to 10 (7^/7^ = 1 to 11.0845). Theexample in the center of Fig. 2 represents flight at a Machnumber of 0.65.

The ideal thrust agumentation achievable under this designcriterion for ejectors operating in the mid-subsonic speedrange is very low over the entire range of primary jetstagnation conditions studied. Generally, in this mid-subsonicrange of flight Mach numbers, the appearance of the map isquite similar to that of the stationary ejector (M^ =0, shownon the bottom portion of the figure), when designed underthis criterion, except for the fact that the band where Ml < 1 isnarrower, and the triangular region where P2 >P00 appears onthe upper right hand corner of the map. This appears to be aresult of the similarity of the accelerating inlets which arerequired for these two flight Mach numbers. Evidently thelarge acceleration of the secondary flow prior to mixing is notdesirable in terms of achievable performance.

At supersonic flight speeds, the use of transonic orsupersonic secondary flow at the start of mixing becomes ofpractical interest, since in most cases the inlet need be onlyconverging or diverging and the difficulty of providing a sonicthroat is avoided. Under these conditions the beneficial ef-fects of high temperature injected gas and ram compressioncombine to provide very acceptable performance.

The upper part of Fig. 2 is a map of isoaugmentation linesfor ejectors designed under the second solution with thetransonic or supersonic M1 optimization criterion andtranslating at a Mach number of 2. Contrary to therelationship illustrated for subsonic flight speeds, the idealthrust augmentation increases with increasing primary jetstagnation temperature and a compressive inlet (Pl >P00) andan accelerating outlet (P2>P00) are required for ideal per-formance over most of the range studied. The optimalaugmentation occurs at the upper choking point (M2 = 1) inmost regions of the map. In the region representing low-temperature injected gas, the optimal thrust augmentationoccurs at mixed flow Mach numbers larger than 1.0, but this

/ Primary Thrust = Its Ram Dra<

region also provides very poor performance, as indicated.Thus at this speed, ejectors have their most favorablecharacteristics when driven by high temperature gasgenerators operating at their upper choking point. It shouldbe noted that the narrow band where M7 < 1, represents aconvergent/divergent inlet which is required to compress thefreestream to a subsonic value of Ml to achieve the optimalperformance. Since the performance when M1 - 1 within thisband is only degraded slightly from the optimal value shownon the map, the narrow band with convergent/divergent inletscan be replaced with convergent inlets without noticeablealteration to the appearance of the map.

Effect of Translational MotionThe influence of flight Mach number on the performance of

thrust augmenting ejectors can be illustrated more effectivelyby mapping isoaugmentation lines on a surface of flight Machnumber vs injected gas pressure ratio, for a fixed primary jetstagnation temperature ratio. Maps of this type were preparedfor second solution ejectors at two temperatures ratios. Thefirst was that corresponding to a laboratory condition wherethe primary jet is derived from stored, compressed air. Forthat case the stagnation temperature of the pressurized air isassumed equal to the ambient temperature. The secondtemperature ratio selected for mapping was thatcorresponding to a moderate temperature in view of materialslimitations imposed by conventional model fabricationtechniques for sea level laboratory testing. For that case thestagnation temperature is assumed to be 816°C (1500°F) atsea level conditions.

Cold Primary Gas EjectorsFigure 3 presents the ideal thrust augmentation of an

ejector, assuming the geometry of the ejector is that derivedfrom the second solution with optimal transonic or supersonicsecondary flow at the start of mixing. Under the conditionsshown on Fig. 3, the thrust augmentation achievable at highflight Mach numbers is small. The assumption that theprimary gas stagnation temperature is equal to that of theambient temperature represents a cooled primary gas, since

AT_T0p-T0oo when (1)

I 2 3 4 5 6 7 8 9 10 I I 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, P0p/Poo

Fig. 3 Ideal thrust augmentation, optimal second solution ejectorswith transonic or supersonic M7; «„, = 25, T0p = T^.

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, P^/P^

Fig. 4 Ideal thrust augmentation, optimal second solution ejectorswith transonic or supersonic A/7; «„, = 25, T^/T^ = 3.7.

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As a result of the cooled primary jet gas, the primary jetstagnation pressure must be higher than that of the freestreamstagnation pressure if the net thrust of air-breathing gasgenerators are to be positive (Vpm >UQO). The linerepresenting V ^ = U^ is marked as "primary thrust = its ramdrag." As indicated, performance at high freestream Machnumbers is limited by the available primary jet momentumcorresponding to the low-temperature injected gas. However,large thrust augmentations are achievable at lower trans-lational speeds.

The geometries of the inlet and outlet are depictedschematically on Fig. 3, assuming isentropic flow through theinlet and outlet. Since M2 > 1, Pl <P00, and P2 <P00 over theentire map, the inlets are all accelerating and the outlets aresupersonic diffusers.

Heated Primary Gas EjectorsThe ideal thrust augmentation of ejectors operating with

heated primary injected gas at a stagnation temperature ratioof 3.7 [equivalent to 816°C (1500°F) at sea level], anddesigned under the second solution at the optimal transonic orsupersonic Ml is shown on a surface of flight Mach numbervs primary jet pressure ratio on Fig. 4. Since the temperatureratio is fixed, the corresponding temperature at high altitudewill be much smaller. For example, at an altitude of 11 km(36,000 ft) the stagnation temperature would be only 528 °C(982°F).

Under the conditions stated on Fig. 4, the performance ofejectors designed under this criterion is divided by a band inwhich the maximum thrust augmentation is 1.00023. Theright portion of this band surrounds the line in whichP1=P2=P00 for the unchoked mixed flow (M2>1). Theperformance inside the left portion of this band is mostly"nonproductive" (</><!, dashed lines) or "no solution"(shaded area) exists and can be replaced by the trivial point(/> = &= 1, A5=/* = 0, as discussed earlier. Comparison of thisband and a similar band which exists on the optimum firstsolution shown in Part I of this document1 indicates a closerelationship. As illustrated on Fig. 4, ejectors operating belowthis band have characteristics which are similar to those withcold primary jet injection (Fig. 3). These ejectors generallyrequire accelerating inlets for ingestion of subsonic freestreamflow and supersonic diff users as outlets. Ejectors operatingabove this band generally require compression inlets foringestion of supersonic freestream flow and supersonic(accelerating) nozzles to discharge the mixed flow.

Since the assumed primary jet stagnation temperature ratiois only 3.7, which is not very high for supersonic flight,relatively low ejector thrust augmentation can be expected athigh flight speeds as illustrated on Fig. 4 and the top portionof Fig. 2.

Limiting Second Solution Ejectors (Subsonic M7, AS = 0)As described in Refs. 1 and 3, the ideal change of total

entropy decreases from some positive value to negative valuesas the Mach number of the secondary flow at the start ofmixing decreases below its lower choking point or below 1,when the second solution is utilized. The point where the totalentropy change becomes zero is a limit under the Second Lawof Thermodynamics and therefore the range of smaller valuesof secondary flow Mach numbers is not considered in thisdocument. Maps have been prepared for ejectors operatingunder this limiting condition. Figure 5 illustrates the variationof thrust augmentation on a surface of primary jet stagnationproperties for this type of ejector having area ratios (a*) of 25and flight Mach numbers of 0, 0.65, and 2.

The lower portion of Fig. 5 illustrates the thrustaugmentation achievable by ideal second solution ejectors atthe limiting point, operating at rest or thrusting normal to theundisturbed stream. Ideal thrust agumentations are very largeover the entire .practical range of primary jet stagnationconditions and decreases slightly with increasing stagnation

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Jet Pressure Rise, AP/P^,

Fig. 5 Ideal thrust augmentation, limiting second solution ejectors;a, =25.

pressure. It is important to observe that the ideal thrustaugmentation increases with increasing primary jet stagnationtemperature over most of the range studied, as illustrated onFig. 5.

The inlet geometry for stationary ejectors designed underthis limiting criterion is always convergent, since the flowmust be accelerated from zero velocity only to a subsonicspeed prior to mixing. However, since P2 <P00 over the entiremap the outlet is either converging (supersonic discharge) orconverging/diverging (supersonic diffuser for subsonicdischarge) to achieve isentropic return to ambient pressure.

Second solution ejectors with secondary flow Machnumbers at the limit point where the total entropy change iszero, display very excellent performance in the mid-subsonicrange of flight speed, particularly with heated primary in-jection. The center portion of Fig. 5 is a map of idealisoaugmentation lines on a surface of primary jet stagnationcharacteristics, for second solution ejectors designed underthis criterion. "As shown, the ideal thrust augmentationachieves very large values over the entire range of primary jetcharacteristics, with maximum performance at the lowstagnation pressure and high stagnation temperature region.Ideal thrust augmentation increases with increasing primaryjet stagnation temperature at virtually all regions of the map.Thus the best performance is achieved with a ramjet-type gasgenerator, although excellent ideal thrust augmentation isavailable with turbojet- or turbofan-type gas generators.

The ejector geometry required for ideal flow patterns is alsoshown schematically on the same portion of Fig. 5. The inletis required to accept freestream flow and to provide for theachievement of the limiting subsonic secondary flow Mach

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I 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, Pop/Poo

Fig. 6 Ideal thrust augmentation, limiting second solution ejectors;a.=25,r,. = 7..

\ \ \ \ \ \ \ \ \ \ \ \I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20

Primary Nozzle Pressure Ratio, Pop/P^,

Fig. 7 Ideal thrust augmentation, limiting second solution ejectors;

number at the start of mixing. In the mid-subsonic flightspeed range this requires either converging (accelerating) ordiverging (diffusing) inlets, since the secondary flow at thestart of mixing is subsonic throughout the ingestion process.Since P2 <P00 over the entire map, the outlet required forreturn of the supersonic mixed flow to ambient pressure maybe a converging supersonic diffuser (at high primary nozzlepressure ratios), or a converging/diverging supersonic dif-fuser (at lower primary nozzle pressure ratios).

As flight speed increases to supersonic values, the per-formance of second solution ejectors with subsonic secondaryflow at the start of mixing continues to display excellent idealthrust agumentation over most of the practical range of in-jected gas characteristics.

The top part of Fig. 5 is a map of isoaugmentation lines ona surface of primary jet stagnation characteristics, forlimiting second solution ejectors operating at a flight Machnumber, of 2. The advantage previously observed, resulting

from high temperature primary gas injection at supersonicspeeds, is very evident in ejectors designed under thiscriterion. As illustrated, the maximum ideal performanceoccurs at high stagnation temperatures and low primary jetstagnation pressure ratios, with a slow decrease in per-formance resulting from increasing primary jet pressureratios.

Due to the requirement for isentropic compression fromsupersonic to subsonic flow at the inlet, it is necessary thatthis element of the ejector be a convergent/divergentsupersonic diffuser. As an alternative, a shock wave systemcan be employed to achieve the required flow at the start ofmixing with a compromise in performance. Design of suchinlets is not uncommon, since conventional engine inlets forsupersonic aircraft pose the same problem. The compromisedperformance encountered by the use of nonisentropic inlets isvery acceptable as discussed earlier in Part I of thisdocument.1 Outlets required to return the mixed supersonicflow to ambient pressure are very simple and consist primarilyof divergent supersonic nozzles since P2>P00 except forinjected gas conditions corresponding to the top right cornerof the map where P2<P00 and convergent supersonic dif-fuser s are required.

Effect of Translational MotionIsoaugmentation lines were plotted on a surface of flight

Mach number vs primary nozzle pressure ratio for secondsolution ejectors designed under the limit point (AS = 0) andoperating with fixed temperature ratio injected gas and0^=25. Temperature ratios representing cold gas(Top/Too = 1) and heated gas (T0plT^ = 3.7) were utilized.

The situation where T0pIT^ = 1 represents gas that is storedin a pressure vessel and cooled below its isentropic com-pression temperature. Thus it is not comparable to flightconditions, but is considered useful for purposes oflaboratory investigations. The situation where T0p/T(X = 3.1represents injected gas with heat added after compression.The temperature ratio choseri represents 816°C (1500°F) atsea level, a temperature which is easily achievable withmaterials used in conventional model fabrication.

Cold Primary Gas EjectorsFigure 6 illustrates the ideal limiting performance of second

solution ejectors with cold, pressurized primary gas(T0p = T00). It is extremely interesting to note that under theseconditions, the thrust augmentation is very high over most ofthe range of flight conditions and pressure ratios illustratedon Fig. 6. This is due to the fact that under the conditionwhere T0p = Tx, the secondary flow stagnation temperature ishigher than that of the primary fluid (except during stationaryoperation, where the stagnation temperatures are equal) andthe influence of heat addition previously discussed as beingbeneficial to ejector performance is also operative at highflight speeds in this reverse situation. As indicated andpreviously discussed, cold primary gas is not representative ofhigh flight Mach number/low primary nozzle pressure ratioconditions due to the absence of net injected momentum whenair-breathing gas generators are considered.

Heated Primary Gas EjectorsFigure 7 illustrates the ideal performance of limiting second

solution ejectors with heated primary gas injection. Theprimary gas is assumed to have a temperature ratio of 3.7,which is equivalent to a stagnation temperature of 816°C(1500°F) at sea level conditions, and 528 °C (982°F) at analtitude of 11 km (36,000 ft). Temperatures encountered inhigh speed propulsion systems can be considerably larger thanthese, therefore the temperature ratio of 3.7 can not representa realistic situation for high speed flight.

As illustrated on Fig. 7, the ideal thrust augmentation isvery large over most of the range of conditions depicted,

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despite the limitation of primary jet stagnation temperatureutilized for this map. It is important to recognize that at afreestream Mach number of 2 for example, the freestreamstagnation temperature ratio is 1.8. Thus the value ofAr/roo = 1.9 at the conditions utilized for Fig. 7 is a smallvalue compared to those which exist in modern propulsionsystems during supersonic flight. Therefore the advantageattributed to the injection of hot gas (equivalent to thoseencountered in propulsion systems during high speed flight) isnot clearly evident on Fig. 7.

As can be observed on Fig. 7, subsonic discharge occursonly during subsonic flight with relatively low primary nozzlepressure ratios. Thus in that region of the map outlets aresupersonic convergent/divergent diff users. For flight Machnumbers above about 1.5, the appropriate outlets are ac-celerating supersonic nozzles. However, at supersonicfreestream Mach numbers the inlets required for isentropicingestion are convergent/divergent supersonic diffusers asrequired to compress the freestream flow to the limitingsubsonic flow.

Nonisentropic Second Solution Ejector OutletsThe flow after complete mixing in second solution ejectors

is always supersonic or sonic. Therefore, as in supersonicwind tunnels, special attention must be given to the startingproblem. In addition, it is essential to provide an outletgeometry capable of maintaining the supersonic flow with theproper characteristics at the conclusion of mixing and with anefficient return to ambient pressure.

First consider the starting problem. Supersonic wind tunneldesign practice indicates that the region downstream of thesupersonic section must have a cross section which is suf-ficiently large to accommodate the mass flow when a normalshock wave exists in the supersonic section. The minimumarea capable of "swallowing" the starting shock wave (Xs) isa function of the Mach number (M2) and the area of themixing section (X2) as described in Ref. 4 for example. Thisrelationship is

y + l

INote:X, = minimum starting area

2(T-7)

X,

l2+(y-l)M22 2yM2

2-(y-l) (2)

This equation represents monotonically decreasing values ofXS/X2 from a value of 1, as M2 increases above 1. Thus anygiven value of XS/X2 between 0 and 1 determines a minimumsupersonic value of M2 which can satisfy the startingrequirement. Therefore, the starting area ratio can be con-sidered as one of the controlling factors for establishment ofthe second solution flow.

As discussed earlier there exist three isentropic outletconfigurations required to return the mixed flow to ambientpressure for ideal ejector flows, as sketched on Figs. 8a-c.With consideration of the starting problem, these con-figurations encompass four discrete fixed geometry outletsdescribed on Figs. 8d-g.

Obviously, if P2 is greater than Pm and since M2 is greaterthan or equal to 1, the outlet is a diverging nozzle and nostarting problem exists since X3 is always greater than X2 orXs. Thus, the isentropic oulet represents a configurationwhich requires no special starting adjustment and appears asillustrated schematically on Fig. 8d.

If P2 is less than P^, but the mixed flow is compressed toP^ and results in a smaller area than X2, but still larger thanXs, then again no starting problem exists, as illustratedschematically on Fig. 8e. For a realistic estimate of ejectorperformance under this condition, the system of obliqueshock waves shown on Fig. 8e is included in the calculation.Obviously the length of the outlet section is a function of M2,P2 /P^, and the assumed wave pattern.

Isentropic Outlets | Fixed Geometry Outlets | Simple Adjustable Outlets

Fig. 8 Schematic outlets for second solution ejectors.

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I 1.60o1 1.55E? 1.50<^ 1.45

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^_

Simple Adjustable Outlets/-m —1 1 1 ~t~~~->

Fixed Geometry Outlets/

5£ '# 1

---»^

fs

\

^

t""~-1

Primary Jet Pressure Rise,

Fig. 9 Influence of outlet wave losses on performance of limitingsecond solution ejectors; a+ = 25, Mx = 0.65, T0 = Tw.

When the second solution isentropic outlet has a minimumarea (either a sonic throat or its exit area) requirement whichis smaller than Xs and results in low supersonic or subsonicvelocities in order to return the mixed flow to P^,, it is im-possible to avoid outlet shock waves if the starting problem isconsidered. These situations result in two types of outlets asillustrated schematically on Figs. 8f and 8g.

If the flow properties are such that a normal shock wave atXs will result in an exit pressure downstream of the shockwave which is larger than P^, then the final compressionrequired to return the flow to P^ can be accomplished by asystem of oblique shock waves, as illustrated on Fig. 8f, if theminimum starting area (Xs) is used as the exit area. As will bediscussed, this situation dominates ejectors which are trans-lating at low subsonic to low supersonic flight speeds,especially when the primary jet stagnation temperature ishigh.

In some instances a normal shock wave at Xs will result in apressure downstream of the shock wave which is smaller thanP^. In that case a weak normal shock wave at the minimumstarting area followed by a subsonic diff user is required toreturn the mixed flow to ambient pressure. As will be shown,this situation, described schematically on Fig. 8g, applies toejectors that are translating at subsonic flight speeds with lowprimary stagnation pressures and temperatures.

When the isentropic outlet is a convergent supersonicdiffuser as illustrated on Fig. 8b, the outlet can usually bereplaced by outlets 8e and 8f, as far as the starting problem isconcerned. Similarly outlet 8c can be replaced by outlets 8fand 8g. The situation where the mixed flow is choked (M2 = 1)and P2 <P00 can be considered as a special case of the outlettype of Fig. 8g when X2 =XS and the outlet wave loss is zero.

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With outlet configurations as depicted on Figs. 8d-g, thethrust of the ejector can be evaluated by conventionalmethods utilized for supersonic jets. Figures 8d, 8e, and 8grepresent configurations in which the exit pressure is equal tothe ambient prressure. In these cases the net ejector thrust issimply

^ej = (^/ + ^p)(^exit-t/oo) (3)

However, for ejector outlet configurations, similar to thoseshown on Fig. 8f, the final flow after the pressure returns toambient is unknown and the ejector thrust must be evaluatedin terms of its exit conditions as follows:

Fej =( (4)

Thrust augmentation is then defined as the ratio of theejector's net thrust as given by Eq. (3) or (4) to that of theprimary jet expanded isentropically to ambient pressure or

F——— q

Fixed Geometry OutletsUsing the above method, thrust augmentations were

calculated for two-dimensional fixed geometry outlets of thetype illustrated on Fig. 8 with realistic shock waves forlimiting second solution ejectors translating at a Machnumber of 0.65 and assuming that T0p = T^ and a* = 25, as aparticular example representing easily achievable laboratoryconditions. Results shown on Fig. 9 indicate significantperformance degradation compared to the ideal limitingsecond solution ejector as a result of this starting requirement.

Despite the adverse influence of the starting requirement,the performance of fixed geometry ejectors is considerablybetter than that of the ideal; optimal first solution ejector (seePart t), and second solution ejectors with transonic orsupersonic secondary flow at the start of the mixing (seebottom line of Fig. 2) at the identical conditions.

Calculations to include a large range of flight Machnumbers and all types of fixed geometry outlets illustrated onFig. 8 are presented on Fig. 10 for conditions which areidentical to those of Fig. 6 (a* = 25 and T0p = 7^). These fixedgeometry outlets provide excellent thrust augmentation overvirtually the entire range of flight speeds from the stationarycase to supersonic speeds despite the use of cold injected gasand consideration of the outlet wave losses.

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, Pop/Pco

Fig. 10 Performance df limiting second solution ejectors with fixedgeometry outlets; «„, = 25, T0p = T00.

The discussion of the influence of outlet wave losses on theperformance of the second solution ejectors will be basedupon the limiting second solution ejector criterion becausethis limiting point is uniquely determined from the parametersin the mixing duct and is independent of outlet losses. On theother hand, the procedure to determine the local optimalperformance of the second solution ejector with transonic orsupersonic M1 depends on the consideration of outlet wavelosses.

Simple Adjustable OutletsThe simple adjustable outlet consists of a flat surface (in a

two-dimensional ejector) on either side of the outlet, capableof very small rotation only, as illustrated on Figs. 8f, 8h, and8i. The concept can conceivably be generalized to axisym-rrietric situations. In the starting configuration these surfacesare adjusted to provide the required minimum starting area(similar to Fig. 8f; however, the wave pattern is more complexthan that shown on Fig. 8f if the length of the rotating sur-faces remains unchanged). The surfaces can then be rotated toreduce the outlet area. The reduction of outlet area results in areduction of the Mach number and an increase of the staticpressure at the exit section. Obviously, the Mach numberinside the exit section is still supersonic. When the increase ofstatic pressure is sufficient to return the mixed flow to am-bient pressure, the external starting oblique wave system willbe eliminated during the outlet area adjustment, and the wavepattern associated with the cruise configuration will appear asshown on Fig. 8h, which has a supersonic exit flow. When theincrease of static pressure is not sufficient to return the mixedflow to ambient pressure, the external starting oblique wavesystems will require larger wave angles, due to the decreasingMach number, and finally form a normal shock wave at theexit section during the outlet area adjustment as illustrated onFig. 8i. Since the final compression of the mixed flow is ac-complished by a normal shock wave, the discharged flow issubsonic. The final flow at ambient pressure can be deter-mined from the outlet wave pattern and the flow parametersat station 2. The net ejector thrust and thrust augmentationcan then be calculated as indicated by Eqs. (3) and (5).

The performance of limiting second solution ejectors withsimple adjustable outlets and shock wave patternscorresponding to those illustrated on Figs. 8h and 8i is alsoillustrated on Fig. 9 as discussed earlier. As indicated on Fig.9, the achievable thrust augmentation of the ejectors withsimple adjustable outlets indicates a recovery of most of thewave losses due to the starting requirement when applicable.Some degradation due to the outlet wave losses can still beobserved as the difference between the thrust augmentation ofthe simple adjustable outlet and the isentropic outlet ejectors.

Where applicable, the simple adjustable outlet offers apractical solution to the nonisentropic outlet for secondsolution ejectors. The applicable region for simple adjustableoutlets with cold gas injection can be visualized by recognitionof the fact that outlet wave patterns similar to the illustrationon Fig. 8f can be advantageously adjusted to a reduced area(smaller than the starting area) for improved performance.Thus, for example, the simple adjustable outlet is applicableover the wide region shown in Fig. 10 where the outlet wavepattern is similar to that of Fig. 8f. As shown on Fig. 10, forcold primary gas ejectors, the region of applicability for thesimple adjustable outlet is restricted to relatively high primarynozzle pressure ratios at low freestream Mach numbers.However, with larger primary jet stagnation temperatures,this restriction is diminished and the region of applicability ofthe simple adjustable outlet extends to lower primary jetpressure ratios.

Figure 11 illustrates the performance of ejectors with fixedand simple adjustable outlets on a pressure/temperature mapfor second solution ejectors under the limiting criterion at afreestream Mach number of 0.65 and a* =25. The solid linesare isoaugmentation lines at the cruise configuration for the

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0:40)

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2 3 4 5Primary Jet Pressure Rise,

1.33 3 4 5 6Primary Nozzle Pressure Ratio, P

Fig. 11 Performance of limiting second solution ejectors with outletwave losses; «„, = 25, M^ = 0.65.

simple adjustable outlet while the dashed lines areisoaugmentation lines for the fixed geometry outlets. Theperformance of ejectors with isentropic outlets configuredand operating under corresponding conditions has beenillustrated on the lower left corner of the center portion ofFig. 5. As can be observed on Fig. 11, the region of ap-plicability for the simple adjustable outlet is significantlyextended to lower primary nozzle pressure ratios as theprimary jet stagnation temperature is increased. Both thefixed and the simple adjustable outlets provide excellentperformance at high temperatures with a considerable ad-vantage in the use of the simple adjustable outlet in its regionof applicability, as shown on Fig. 11.

Figure 12 displays the performance of simple adjustableoutlet ejectors (solid lines) in their cruise configuration, intheir region of applicability, and the fixed geometry ejectors(dashed lines) in the remaining regions. The performancedisplayed on Fig. 12 is presented as isoaugmentation lines on asurface of freestream Mach number vs primary nozzlepressure ratio for a primary jet stagnation temperature ratioof 1 and a* =25. The influence of outlet wave losses can beobserved by comparison of Figs. 12 and 6, and the advantageof the simple adjustable outlet compared to the fixedgeometry outlet can be observed by comparison of Figs. 12and 10.

Figure 13 illustrates the ejector performance and con-figurations with fixed and simple adjustable outlets and withheated primary gas injection, under conditions identical tothose displayed on Fig. 7. A comparison of Figs. 12 and 13indicates that the region of applicability for the simple ad-justable outlet is widened as a result of the injection of heatedprimary gas. In other words, the subsonic freestream velocity,low primary pressure ratio region, which requires an outletthroat and a subsonic diffuser for starting, is considerablynarrowed. It also is important to note that, as shown on Figs.12 and 13, in the region of high supersonic freestream Machnumbers (M00>1.3) the outlet wave losses become unim-portant. However, it is also important to note that althoughthe outlet wave losses become unimportant, the major lossesduring supersonic freestream motion occur at the inlet.

As previously indicated, the data presented on Figs. 12 and13 represent ejector performance under laboratory conditionsand do not represent realistic flight conditions at high speedsdue to the assumed primary gas stagnation temperatures.

DiscussionIt has been shown that the performance of thrust

augmenting ejectors depends strongly upon the selection ofthe inlet and outlet configurations in view of the flight andinjected gas characteristics. Traditionally, ejectors have beendesigned with converging inlets and diverging outlets. Thisstudy has indicated that designs of this type are appropriate

I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, Pop/P<x>

Fig. 12 Performance of limiting second solution ejectors with outletwave losses; «„ = 25, T0 = T00.

I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20Primary Nozzle Pressure Ratio, P0p/Peo

Fig. 13 Performance of limiting second solution ejectors with outletwave losses; «„ = 25, T^/T^ = 3.7.

for first solution ejectors operating at low subsonic flightspeeds, and for second solution ejectors operating atsupersonic flight speeds. However, it should be recognizedthat, the inlets and outlets of second solution ejectorsoperating at supersonic flight speeds perform the reversefunction from similarly shaped inlets and outlets of firstsolution ejectors operating at low subsonic flight speeds.Further, it has been shown that the combination of com-pression inlets (for deceleration of the freestream flow priorto mixing) and high temperature injected gas dominate op-timal ejector design for high speed flight.

The very high ideal performance predicted by the analysisof second solution ejectors has stimulated an investigationinto the feasibility of achievement of the required flows. Sincethe flow after complete mixing is supersonic (or sonic), itappears essential to consider the problem of starting, whichcorresponds to the problem encountered in starting supersonic

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wind tunnels. Results of that study indicated that in thesubsonic or low supersonic flight speed range a startingproblem does exist. Outlet configurations required forstarting (swallowing the starting shock wave) have beendescribed, and ejector performance has been evaluated forejectors with those nonisentropic outlet flows (withgeometries which are fixed at the starting configuration).Maps were presented for ejector performance over the ap-propriate range of flight and injected gas characteristics.

After swallowing the starting shock wave and establishingsupersonic flow in the mixing section, it is possible tominimize the wave losses by a simple adjustment of the outletconfiguration (except for low flight speed, with low primaryjet stagnation pressure and temperature). In a two-dimensional ejector, the adjustment of the outlet can beaccomplished by a small rotation of the outlet plates. Thissimple adjustable outlet has been described and the thrustaugmentation achievable by ejectors in that configuration hasalso been mapped over appropriate ranges of flight and in-jected gas characteristics. It has been shown that even in therange of conditions where the starting problem exists, the

performance of ejectors with nonisentropic outlets, includingthe influence of wave losses, is not seriously degraded by thewave losses.

AcknowledgmentsThe information contained in this paper was acquired in

part as a result of contractual support from the Air ForceOffice of Scientific Research and the Air Force FlightDynamics Laboratory.

References^Iperin, M. and Wu, J. J., "Thrust Augmenting Ejectors, Part

I," AIAA Journal, Vol. 21, Oct. 1983, pp. 1428-1436.2Keenan, J. H., Neumann, E. P., and Lustwerk, F., "An In-

vestigation of Ejector Deisgn by Analysis and Experiment," Journalof Applied Mechanics, Vol. 17, No. 3, 1950, pp. 299-309.

3 Alperin, M. and Wu, J. J., "High Speed Ejectors," AFFDL-TR-79-3048, May 1979.

4Liepmann, H. W. and Roshko, A., Elements of Gasdynamics,John Wiley and Sons, Inc., N.Y., 1957, pp. 131-136.

From the AIAA Progress in Astronautics and Aeronautics Series...

INJECTION AND MIXING IN TURBULENT FLOW—v. 68

By Joseph A. Schetz, Virginia Polytechnic Institute and State University

Turbulent flows involving injection and mixing occur in many engineering s i tuat ions and in a variety of na tura lphenomena. Liquid or gaseous fuel injection in jet and rocket engines is of concern to the aerospace engineer; themechanical engineer must estimate the mixing zone produced by the injection of condenser cooling water in to a waterway;the chemical engineer is interested in process mixers and reactors; the civil engineer is involved wi th the dispersion ofpol lu tan ts in the atmosphere; and oceanographers and meteorologists are concerned wi th mixing of f luid masses on a largescale. These are but a few examples of specific physical cases t ha t are encompassed w i t h i n the scope of th is book. Thevolume is organized to provide a detailed coverage of both the available experimental data and the theoretical predictionmethods in current use. The case of a single jet in a coaxial stream is used as a baseline case, and the effects of axial pressuregradient , self-propulsion, swir l , two-phase mixtures , three-dimensional geometry, transverse injection, buoyancy forces,and viscous-inviscid interact ion are discussed as variations on the baseline case.

200 pp. , 6 x 9 , ill us., 5/7.00 Mem., 527.00 List

TO ORDER WRITE: Publications Order Dept., AIAA, 1633 Broadway, New York, N.Y. 10019

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thrust augmenting ejectors, ii

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