I

NATIONAL ADVISORY COMMIT’lEE FOR AERONAWICS

wlur’’mll IuwolrrORIGINALLY ISSUED

May 1945 asAdvance Confidential Report L5D20

PRELIMINARY INVESTIGATION OF SUPERSONIC DIFFUSERS

By Arthur Kantrowitz and Coleman duP. Donaldson

Langley Memorial Aeronautical LaboratoryLangley Field, Va.

WASHINGTON

NACA WARTIME REPORTS are reprints of papersoriginaUy issued to provide rapid distribut.ion ofadvance research results to an authorized group requiring them for the war effort. They were pre-viously held under a security status but are now unclassified. Some of these reports were not tech-nically edited. All have been reproduced without change in order to expedite general distribution.

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3 1176013544045

NACA ACR- No. I)5D20

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

ADVAKCJ2CONFIDENTIAL REPORT

PRELIMIJT~Y INVESTIGATION OF SUPERSONIC DIFFUSERS

By Arthur lWntrowitz and Coleman duP. Donaldson

SUMMARY

The deceleration of air from supersonic velocitiesin channels has been studied. It has become apparentthat a normal shock in the diverging part of the diffuseris probably necessary for stable flow, and ways of mi~.i-mizing the intensity of this shock have been developed.The effect of’various geometrical parameters, especiallycontraction ratio in the entrance region, on the perform-ance of supersonic diffusers has been investigated.

By the use of these results, diffusers were designed.which, starting without initial boundary layer, recovered90 percent of the kinetic energy in supersonic air streamsup to a Mach number of 1.85.

INTRODUCTION

The deceleration of air from supersonic to subsonicvelocities is a problem that is encountered in the designof high-speed rotary compressors and supersonic air intakes.The efficiency of the supersonic diffusers used to accom-plishthis deceleration has an important effect on theperformance of these mschines. The present study isintended to provide information upon which to designefficient supersonic diffusers for use in cases in whichthe flow starts without initial boundary layer.

The available data on supersonic diffusers are verymeager and are reviewed by Crocco in reference 1. Thisreview indicates that, in the decelera.t$on”of air fromsupersonic velocities, the total-head losses are so largeas to impair seriously the efficiency of machines employingthis process. The experiments reported in reference 1 were”primarily designed to serve the needs of supersonic windtunnels, and therefore only diffusers starting with initialboundary layer were considered.

2 NACA ACR N,o.L5D20

FLOW IN A SUPERSONIC DIFFUSER

Stabilit .- In a Laval nozzle the gases start at alow vd y, are accelerated to the velocity of sound inthe converg& part of the nozzle, and are accelerated tosupersonic velocities in the diverging part of the nozzle.The supersonic velocities reached can be calculated approx-imately from the isentropic-mass-f’low curve of figure 1 andi+e geometry of’the nozzle. It is well known that, forshock-free I’1ow, experiment is in good agreement with thisone-dimensional isentropic theory although, since theboundary layer thickens in the diverging part of the nozzle,the Mach numbers reached may be a little lower than thevalues calculated. Two-dimensional nozzles can be designedby the Prandtl-Busemann method (reference 2) to giveessentially shock-free expansions, which can be obtainedexperimentally provided no moisture-condensation effectsare presentO

It might be supposed thatby the Prandtl-Busemann methodproper allowance were made for

the flow in a nozzle designedcould be reversed and, ifboundary-layer displacement

~hi~kness, a smooth deceleration t’hrou&h the speed ofsound obtained. A flow of this type is, however, unstablein the sense that it is unattainable in practice. Considerthat a flow of this type has been established. (Seefig, 2(a).) In this flow pattern the mass 21OW per unitarea through the throat is the maximum possible for thegiven state and velocity of the gas entering the diffuser.@ long as the flow entering the diffuser is supersonic,the entering mass flow would be unaffected by events down-stream. A transient disturbance propagated upstream fromthe subsonic region would, however, reduce the mass flowat least temporarily in the velocity-of-sound region,Thus, a disturbance would result in an accumulation ofair ahead of the throat. The perturbation of the originalisentropic flow produced by this accumulation of air wouldprevent the mass flow from returning to its initial max-imum value; thus, air would continue to accumulate aheadof the throat until the mass flo~:entering the diffuserwas reduced. In the case of a supersonic diffuser hmnersedin a supersonic stream as in the experimental arrangementdescribed later, this would necessitate the formation ofa normal shock ahead of the diffuser and, In other arrange-ments, would likewise necessitate drastic changes in theflow pattern. From the discussion of the starting ofsupersonic flows in diffusers given later, it will be seen

3NACA ACR No. L5D20

that these ch~nges are irreversible (certainly in theexperimental arrangement described later and probably inmost other arrangements). It therefore appears thatisentropic deceleration through the speed of sound in,_chann8_ls~s_xnstable end unattainable in practice.

In a series of preliminary attempts to produce anapproximation to Isentropic deceleration through the speedof sound, it was found that supersonic flow could not bestarted into diffusers designed to produce this flow. Indiffusers with a larger throat arsa, the normal shockjumped from a position ahead of the diffuser to a positionin the diverging part of the diffuser. Flows of the typeshown in figure 2(b), which involve a normal shock in tinediverging part of the diffuser, were found to be stable.

Contraction rstio and losses.- ~ important part ofthe losses in e supersonic diffuse!? are associated withthe dissipation accompanying the normal shock in thediverging part of the diffuser. It is therefore importantto consider the factors that determine its intensity. Asin a Laval nozzle, the position of the shock wave iscontrolled by the back pressure on the diffuser and movesupstream as the back pressure is increased. When the backpressure forces the shock to a point close to the minimumarea of the diffuser, the shock Mach number approachesits lowest value and the associated losses are minimized.The magnitude of’these minimum losses,depends upon howmuch the air entering the diffuser is slowed up by thetime it reaches the minimum cross section. The more theentrance area of the diffuser can be contracted, the lower.the Mach number of the normal shock and the greater theefficiency of the diffuser. It is therefore valuable toconsider what determines the ma.ximuxncontraction ratiothat can be used. (Contraction ratio is defined a.stheratio of the area at the entrance of a diffuser to the areaat its minimum section. See fig. 2(b).)

In most applications, the establishment of supersonicflow is preceded by a normal shock traveling downstream.If this normal shock is to move into the diffuser at agiven entrance Msch number and thus esteblish supersonicflow, the throat of the diffuser must be large enough topermit the passsge of the mass flow in a streem tubehaving an are~ thet corresponds to the entrance area ofthe diffuser snd.a.total head that corresponds to thevalue behind a normal shock at the entrance Mach number.Thus, if the throat area has a minimum value for a given .

4“ NACA ACR No. L5D20

entrance Mach number, the Mach number at the IAroat willbe close to 1 when there is a normal shock ahead of thediffuser. An approximation to the contraction ratio thatproduces this condition can be found from conventionalone-dimensional-flow theory. The conditions after thenormal shock are known from the usual normal-shock equaticmsand.it is necessary merely to find the stream tube contrac-tion, which increases the Mach number at the throat to 1.Since the mass flow per unit area at the Mach number of 1for a given ste.gnation temperature is proportional to thetotal head, the maximum permissible contraction ratio isequal to the contraction retio that would be required foran isentropic compression to the Mach number of 1 (fromthe initial supersonic conditions) multiplied by thetotal-head ratio across the normal shock. The maximumtheoretical contraction ratio that permits starting ofsupersonic flow is computed in this way in appendix A andis shown in figure 3. If the throat area were reducedafter supersonic flow had been established or if the flowthrough the diffuser were started by temporarily inc=rea@~---the entrance Mach number to a value greater than thedesign value, a less intense shock mad lower losses couldprobably be obtained. In these cases, the.lowest limitof the shock intensity would be provided by stabilityconsiderations.

For diffusers in which the geometry (particularlythe throat area) cannot be ’varied and in which the super-sonic flow cannot be started by temporarily increasing theentrance Mach number, the minimum-loss diffusion occurswith the shock just downstream from the minimum section.The Mach number preceding s~.cha shock (with isentropicflow assumed) can be found from the computed contractionratio (fig. 3) and equation (2) of appendix A. The total-heed loss across. a normal shock at this Mach number(equation (4), appendix A) is then an approximation tothe minimum losses (with boundary-layer losses neglected)in a supersonic diffuser subject to the foregoing startingrestrictions. The performances of diffusers obtained inthis way are given in figure ~.

It should be pointed out that these theoreticalconsiderations are derived with the tacit assumption thatconditions in a plane perpendicular to the axis of thechannel are constant; that is, one-dimensional flow isassumed. For example, the occurrence of oblique shocksat the entrance of 8 diffuser would sllghtly alter theseconditions ; in particular, the normal shock in the diverging

ITACAACR NO. L5D20 5

part of the diffuser would have a somewhat red edMintensity and the theoretical efficiency would. e some-

what higher. It is considered, however, that the generalfeatures would not be much altered by the departures fromon&-dimensional flow that would occur. in diffusers suchas those discussed in the experimental part of this report.

EXPERIMENTAL TECHNIQUE

In order to investigate experimentally the propertiesof constant-geometry supersonic diffusers, the apparatusshown schematically in figure ~ was desidmed ~d con-structed. The settling chamber was connected to a supplyof dry compressed air controlled by a valve in such a waythat the chamber pressure could be held constant at anydesired value. The air left the chamber through inter-changeable two-dimensional nozzles .ths,twere designed togive parallel flow, at various desired Mach numbers. Thefeather-edge tip of the diffuser (fig. 6) was held in thecenter of the supersonic jet at the exit of the nozzle.The experimental arrangement was designed to study theoperation of supersonic diffusers that started withoutinitial boundary layer. This condition was studied fortwo reasons: (1) It is the simplest defined boundary-layer condition to obtain experimentally, and (2) it isconsidered to approximate more closely than any other theboundary=layer conditions that occur at the entrance tosupersonic diffusers used in compressors. A long sub-sonic diffuser cone behind the supersonic diffuser tipwas provided to complete the diffusion process. The valvebehind the cone was used to control the back pressure inthe subsonic portion of the diffuser and an orifice wasused to measure the mess flow through the diffuser. Thesurface in the supersonic diffuser tips was machinedsteelj whereas the cone in the subsonic portion was rolledand finished heavy sheet steel.

In order to compsre the efficiencies of the variousdiffuser combinations tested, two quantities were required:(1) the percentage of the total head that the diffuserrecovered and (2) the entrance M8ch number at which thediffuser atteined this recover”y.

‘ Because the losses In well=de.signednozzles are sm8.11,the,absolute pressurechamber WS.S assumed to be the total head

supersonicin-the settlingbefore diffusion.

6 NACA ACR No. L5D20

This pressure was measured with a large mercury manometer.The total ’head after diffusion can be assumed equal to thestatic pressure at th’eend of the subsonic diff’user conewithout appreciable error, inasmuch as the kinetic energyat the end of the cones was of the order of 0.16 percentof the entering kinetic energy. A mercury manometer wasused to measure the difference between the total headsbefore and after diffusion. ‘J!hesetwo measurements weresufficient to determine the percentage of total headrecovered.

The mass flow per unit ar~”a”andthe stagnation con-ditions are sufficient to determ~e the Mach number atany point. (See equation (2), appendix A. ) The Machnumber at which a diffuser was operating was determinedby measuring the mass flow through the diffuser, whichhad a known entrance are~, and by measuring the settlingchamber pressure and temperature that correspond tostagnation conditions.

Two other observations were made. The pressure justinside the supersonic tip of the diffuser was measured tomake sure that the shock had passed down the diffuser andthat supersonic flow existed in the contracting portion.The flow in the nozzle and into the diffuser was observedwith a schlieren system to check visually whether theshock had entered the diffuser.

In order to make a test, the nozzle was brought upto design speed by increasing ”the pressure in the settling~h~ber PO to some value that was held constant. through-out the test. The throttling valve behind the diffusercone was open and the shock passed down the diffuser, ifthe contraction ratio permitted, and stopped at some placein the diffuser cone. The throttling valve was thenslowly closed, thus increasing the pressure at the end ofthe cone .pf and pushing the shock upstream to lower and

lower Mach numbers. When the shock had been moved apstreamas far as possible, that is, just downstream from theminimum section of the diffuser, pf reached. its maximum

value. Although pf was increased during this process,

the mass flow through the diffuser was not affected becausethe flow was supersonic into the diffuser tip. When thevalve was closed farther, the shock wave passed the minim-um section and suddenly moved out in front of the diffuser.

NACA ACR ??0. LbD2~ 7

The mess flow immediately @ropped (and continued to dropes the valve was closed farther) and the pregsure insidethe diffuser tip immediately jtiped to R subsonic value.

,. The results of a typical test tie,presented in fig-ure 7. The breaks in the ,?nass-flowand tip-pressurecurves give an excellent indication of ‘when the diffuserwas operating at maximum efficiency .&d when it failedto act as a supersonic diffuser. The slight change inmass flow while the diffuser was operating was ‘due to thefact that the pressure in the settling chembe.rvariedslightly from the beginning to the end of the test run.The curves indicate that a given diffuser may have anyvalue of total-head recovery, up to a certain maximum,depending upon the position of the shock. Therefore, theobvious method of comparing the performance of a numberof diffukers is to compare their maximum recoveries.

RESULTS AND Discussion

The primary design parameter of a supersonic diffuseris its contraction ratio, which determines the minimumMach number at which the supersonic diffuser operates andthe amount of compression that the entering air undergoesbefore it must negotiate the normal shock. If the con-traction retio of a diffuser is increased, the minimumMach number at which it operates theoretically increasesas shown in figure 3. The minimum Mach numbers at whicha number of diffusers were observed to operate and theMach numbers at which they first failed to operate areshown in figure 3. The points so plotted give excellentagreement with the theoretical contractim-ratio curve.

As was pointed ●ut prev~ously, the effect of contrac-tion ratio upon the performance of a supersonic diffusershould be approximately as shown in figure .!+.The observedperformances of three diffusers with different contractionratios are plotted in figure 8. The effect of contractionratio is very similar to the approximate theoreticalresults shown in figure .4. The indicated disc~epencybetween experimental and theoretical results is probablychiefly due to losses In the subsonic portion of thediffuser.

After the contraction ratio of a supersonic diffuserhas been fixed according to the minimum Mach number at

‘8 .,NACA ACR No. L5D20

which it must operate, two other parameters - the entr”mce -cone angle and the exit-cone angle - may be considered;

Owing to the difficulty of measuring the exactentrance angles on the small diffusers tested, the dataevaluating the effect of the entrance-cone angle are notconsidered quantitative and are not presented herein.The trend observed, however, was that the larger theentrance-cone angle, the better the performance of thediffuser. Further experiment is needed to determine theoptimum entrance-cone angles although, for the threediffusers of figure 8P the entrance-cone angles areprobably so close to the optimum that no large gain inrecovery could be expected from e change in this parameter.In the diffusers tested, the internal shape was faired ina smooth curve between the en”trance cone and the exit cone.The “curve was close to a circular arc end started very nearthe leading edge of the entrance cone.

Two diffusers of equal contraction ratio and entrance-cone angle but different exit-cone angle were tested. Theperformances of the two diffusers with exit-cone anglesof ~~ and 30 are plotted in fi.gurq9. The diffuser withan exit-cone angle of 30 was found to give consistentlyhigher recoveries. As is pointed out in reference j, theboundary layer is thick after a normal shock and thereforethe pressure recovery in thesubsonic cone must be slowto prevent separation. The slightly different shspe ofthe performance curve of these diffusers when comparedwith the other diffusers reported (fig. 8) may be due tothe fact that, although the two diffusers correspondclosely to each other except for exit-cone angles, theydo not correspond to the other three diffusers.

The total-head recoveries messured in the experimentswere transformed into energy efficiencies. The energyefficiency q is defined as the percentage of availablekinetic energy recovered in the diffusion process or thekinetic energy of an expansion from the pressure at restafter diffusion .pf to the pressure at the entronce of

the diffuser pe divided by the kinetic energy of an

expansion from the initial chamber pressure p. tO Pg.

Beesuse no extarnel work is done, the whole process ofexpansion and difi’usion is a throttling process and thestagnation tem.pereture .To is the same after diffusion

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NACA ACR NO. L5D20 9

as in the settling chamberiefficiency may be written

“[’””2CP T~ -

The equation for the energy

/’pe R/9p“{)1T—Pt.

CPThe symbols are define”d in appendix B.” When ~ = 3.5,

(1)

where M is the Mach number of the flow entering thediffuser.

The efficiencies obtained by equation (1) are comparedin figure 10 with the typical efficiencies (converted toefficiency as defined in equation (l))of the work previouslydone with supersonic diffusers presented by Crocco inreference 1, the efficiency of a normal shock (combinedwith compression to rest without further loss), and theapproximate maximum theoretical efficiency for constant-geometry diffusers previously derived. Figure 10 showsthat the normal-shock efficiency may be exceeded and thatenergy recoveries of over ~0 percent can ‘be obtained upto a Mach number of 1.8~; thus, the results presented forsupersonic diffusers in reference 1 are far too conservativefor diffusers that have no initial boundary layer.

CONCLUDING REMARKS

An investigation of the deceleration of air in channelsfrom supersonic to subsonic velocities was conducted. Achannel flow involving the shock-tree decelerati~n of agas stream through the local speed ot sound was found tobe unstable. A stable flow probably involves a normalshock in the diverging part of the diffuser. The losses

..

10 NACA ACR No. L5D20

involved inthe throatMach number

/“””this normel shock can be minimized bj making

area as small as possible f’or a given entrance. The maximum contraction ratio-that permits

starting of supersonic flow at a given entrance Machnumber has been calculated and checked very closely byexperiment.

With the use of these results, diffusers were designedwhich,starting without initial boundary leyer, recoveredover 90 percent of the kinetic energy in supersonic airstreams up to a Mach number of 1.85.

Langley Memorial Aeronautical LaboratoryNational Advisory Committee for Aeronautics

Langley Field, Va.

‘-. . . . ...,

NACA ACR No. L5D20 11

APPENDIX A

CALCULATION OF MAXIMUM PEwTsSIBLE CONTRACTION RATIO

It can be shown that the.mass flow per unit area atMach number M is

(2)

where the symbols are defined in appendix B.

The Isentropic arena-conttiactionratio from a Machnumber M to.the local velocity of sound is then

(pv)M=~

(PV)M(3)

where pV is computed from equation (2).

When air crosses a shock wave, its stagnationtemperature is unchanged; hence, the reduction in possiblemass flow per unit area, from equation (2) and the perfect“gas law, is proportional t~ the total-head loss across,the shock. The total-head ratio p3/p@, across a normal

shock wave can be shown to be

.“

(4)

Multlplylng equetion (4) by expression (3) gives themaximum contraction retio that permits supersonic flow ta.start in “adiffuser. This quantity is plotted in figure 2.

I

.,

12

1 1

APPENbIX B

111 1 ■ m mm I mm,nu

NACA ACR Noo L5D20

SYMBOLS

Y ratio of specific heat at constant pressure tospecific heat at constant volume

P density

a velocity of sound

v velocity

M Mach number. .

Cp specific heat at constant pressure

R gas constant

T efficiency

P~ pressure at”entrance of diffuser

Pf pressure at rest after diffusion

Po initial chamber pressure

P3 total head after normal shock wave

pd pressure ,at internal leading edge of supersonicdiffuser (see fig. 7)

Md design Mach number of supersonic diffuser; thatis, minimum starting Mach number of diffuserwith given contraction ratio .“

T entrance angle of diffuser (see f’lg.6)

0 exit angle of diffuser ,*

b, C dimensions used in fig, 2

CR contraction ratio (see fig. 2(b))

. .

/ “(,

NACA ACR NO. L5D20

s passage area

T temperature

The subscript o refers ‘to‘initial

13

.

stagnation conditions.

REFERENCES

1. Crocco, Luigi: Gallerie aerodynamiche per alte velocit’a.L~Aerotecnica, vol. XV, f’asc. 3, March 1935, PP. 2.3~-275 and vol. XV, fast. 7 and 8, July and Aug. 1935;PP~ 735-778.

2. Busemann, A.: Gasdynamik. Ha,ndb. d. Experirnentalphy s.,Bd. IV, 1. Teil, Akad. Verlegsgesellschaft m. b. H.(Leipzig), 1931, pp. 421-431 and 4-4-7-4.4.9.

~. Donaldson, Coleman duP.: Effects of Interactionbetween Normal Shock and Boundary Layer. NACA CBNo. 4.A27, 1944.

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Mach number, M

Figure l.- Isentropic-mass-flow ratio as a function of Mach number. ~

Mass flow measured through area S at Mach number M: Po, density: &ao, speed of sound at initial stagnation conditions.

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NACA ACR NO. L5D20 Fig. 2a,b

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~—Supersonic flow Subsonic flow

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NATIQNIILAOVISOR~COMMITFEE FOR AERONAUTICS

(a) Reversed Laval nozzle with isentropic flow (unstable).

TT———.

c— —

~JSupersonic flow Subsonic flow

———

Shock

(b) Stable supersonic diffuser flow. (For circulardiffuser> . C!R, where CR is contraction

ratio. )

Figure 2.- Flow in a converging-diverging diffuser.

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Figure 4..- Theoretical performance by approximate method of five supersonic diffusers! wwith maximum theoretical contraction ratios correspondi~g to minimum starting or

design Mach numbers of 1.0, 1.4, 1.8, 2.2, and 2.6. mpVertical lines indicate design .I Mach numbers.&

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Chamber control valve

Variable two-dimensional nozzlesCalibrated orifice

LI

\Diffuser exit cone

Screens Valve

Settling-c<amberPressureP. Interchangeable circular diffuser tip(see fig. 6)

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Figure 5.- Schematic diagram of apparatus used to test supersonic diffusers.

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NACA ACR No. L5D20 Fig. (j

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NATIONAL AOVISORYCOMMlllEE FOR AERONAUTICS

Figure 6.- Interchangeable circular diffuser tips for which performancesare shown in figures 8 and 10. These different tips were screwed intoa permanent cone having an exit angle of 3°. r, entrance-cone angle.

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1.0 1.2 1.4 I*6 1*8 2*o 2.2 2JI 2*6 2.8 3*o

Entrance Mach number, M

Figure 8.- Performance of the three diffusers shown in figure 6with exit-cone angles of 3°. r, entrance-cone angle.

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NATIONALAOWX)RYCOMMKIEE FORAERONAUTICS

9.- Effect of exit-cone angle on the performance of two supersonic difequal contraction ratios and entrance-cone angles. 0, exit-cone angle

fusers.

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Figure 10. - Comparison on an energy basis of diffusers testedwith theoretical results and with results of former experiments,Experimental points are from same tests as thoseshowninfigure 8.

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CONFIDENTIAL OTD- 8841 MVHJOM

none AUTHORfS): Kantrowitz, A.; Donaldson,; Coleman, DuP. ORIGINATING AGENCY: National Advlsorv Committee for Aeronautics. Washlneton. D. C.

<W10. AGENCY NO.

ACR-L5D20

Majp« E&an TJ.S. | Eng. PAOES

23 iturxnuTtoNS

diagr, graphs

An investigation of the deceleration of air in channels from supersonic to subsonic veloci- ties was conducted for use in high-speed rotary compressors and supersonic air intake design. A stable flow probably involves a normal shock in diverging part of diffuser. Losses caused by this shock are reduced by making throat area as small as possible for given entrance Mach number. Several geometrical features of diffusers were investigated. Using these results, very efficient diffusers were designed.

DISTRIBUTION: Request copies of this report only from Originating Agency DIVISION: Aerodynamics (2) SECTION: internal Flow (4) ATI SHEET NO.: f:-2-4-1

SUBJECT HEADINGS: Induction systems - Diffusors (51601); Diffusors, Supersonic - Pressure recovery (30205)

1_ Air Materiel Command

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