EDCaTDRa64a3

c~ I

(Prepared under Contract No. AF 40(600)-1000 by ARO, Inc.,contract aperator of AEDC, Arnold Afr Force Station, Tenn.)

January 1964

By

R. C. BauerRocket Test Facility

ARO, Inc.

: F<OPER-TY OF U. S. AIR,I.\EDCLlBRARY

40{600) 1000

Program Element 62405184/6950, Task 695002

TECHNICAL DOCUMENTARY REPORT NO. AEDC·TDR·64·3

ARNOLD ENGINEERING DEVELOPMENT CE

AIR FORCE SYSTEMS COMMAND,

UNITED STATES AIR FORCE, .

THEORETICAL BASE PRESSURE ANALYSIS OF= ~~ AXISYMMETRIC EJECTORS WITHOUT INDUCED FLOW= ~ ;f~JJt;~~

Qualified requesters may obtain copies of this report from DOC, CameronStation, Alexandria, Va. Orders will be expedited if placed through thelibrarian or other staff member designated to request and receive documentsfrom DOC.

When Government drawings, specifications or other data are used for any purposeother than in connection with a definitely related Government procurement operation,the United States Government thereby incurs no responsibility nor any obligationwhatsoever; and the fact that the Government may have formulated, furnished, or inany way supplied the said drawings, specifications, or other data, is not to beregarded by implicatron or otherwise as in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission to manufacture,use, or sell any patented invention that may in any way be related thereto.

.1

UNCLASSIFIED

Errata AEDC-TDR-64-3*, January 1964

Please note the revisions of Figs. 16c and d, copies ofwhich are attached .

*R. C. Bauer. "Theoretical Base Pressure Analysis ofAxisymmetric Ejectors without Induced Flow." Arnold EngineeringDevelopment Center, Arnold Air Force Station, Tennessee. AEDC­TDR-64-3, January 1964. (Unclassified Report) .

UNCLASSIFIED

'.

AF - AEOCArl1O:d AFS T"ro.n

AEDC-TDR-64-3

THEORETICAL BASE PRESSURE ANALYSIS OF

AXISYMMETRIC EJECTORS WITHOUT INDUCED FLOW

By

R. C. Bauer

Rocket Test Facility

ARO, Inc.

a subsidiary of Sverdrup and Parcel, Inc.

January 1964

ARO Project No. RW2141

'.

AEDC-TDR·64·3

FOREWORD

The experimental data originating in this report were obtainedby J. H. Panesci and R. C. German. Theoretical results wereobtained from an IBM 7070 computer programmed by D. H. Hoyle.

, .

AEDC-TDR-64-3

ABSTRACT

The two-dimensional supersonic base pressure theory developedby Dr. H. H. Korst, which applies to two-dimensional systems withstraight jet boundaries, has been modified to be more applicable toaxisymmetric ejector systems. The modification consists of a newtheory for estimating the peak recompression static pressure whichis applicable to either axisymmetric or two-dimensional jets havingstraight or curved boundaries. Significantly, the recompressionmechanism is not independent of viscous effects for systems which pro­duce a non-uniform inviscid flow field.

Both the recompression theory and the modified base pressuretheory are experimentally verified for the case of isoenergetic mixingand negligible initial boundary layer. The recompression theory isshown to agree with experimental results to within ±10 percent, where­as the experimental base pressure results have a standard deviationof 6 percent with respect to the modified base pressure theory.

PUBLICATION REVIEW

This report has been reviewed and publication is approved.

t~£HActing Chief, Propulsion DivisionDCS/Research

v

Don~d~. f:t£~jk-DeS/Research

1.02.03.0

4.0

CONTENTS

ABSTRACT....NOMENCLATURE.INTRODUCTIONRECOMPRESSION MECHANISMTHEORETICAL MINIMUM CELL PRESSURERATIO OF AXISYMMETRIC EJECTOR SYSTEMSCONCLUSIONSREFERENCES . . . . . . . . . . . . . . . . .

TABLES

AEDC-TDR-64-3

Page

vix12

121415

1. Comparison of Theoretical with Experimental PeakRecompression Static Pressure. . . . . . .

2. Comparison of Theoretical with ExperimentalMinimum Cell Pressure Ratios of AxisymmetricEjectors. . . . . . . . . . . . . . . . . . .

ILLUSTRATIONS

Figure

1. Sketch of a Typical Ejector System.

17

18

19

2.

3.

4.

5.

Sketch of Experimental Configurationsa. Total Pressure Rake Installation.b. Total Pressure Probe Installation

Comparison of Theoretical with ExperimentalLines of Constant Mach Number in a Free Jet,A ne / A* = 10.848. . . . . . . . . . . . . .

Comparison of Theoretical with ExperimentalLines of Constant Mach Number in a Free Jet,A ne / A* = 25. 000. . . . . . . . . . . . .

Comparison of Theoretical with ExperimentalMeasured Total Pressure in a Free Jet

vii

2020

21

22

23

Figure Page

6. Theoretical Recompression Flow Field (FirstOrder) 0 0 • • • • • • • • • • • • • • • •

24

7. P2/Pt vs M - Two-Dimensional Shock Wave Theory. 25

8. Typical Isentropic Compression and ExpansionProcesses. . . . . . . . . . . . . . . . . 26

9. Typical Error of Approximation to Two-DimensionalShock Wave Theory. . . . . . . . . . . . . . . 27

10. Theoretical and Experimental Impingement ZoneCharacteristics

a. Inviscid Flow Conditions . . . . . . . . . 28b. Diffuser Wall Static Pressure Distribution. 28

11. Theoretical and Experimental Impingement ZoneCharacteristics

a. Inviscid Flow Conditions. . . . . . . . . . 29b. Diffuser Wall Static Pressure Distribution. . 29

12. Idealized Flow Phenomena at Impingement Pointa. Idealized Static Pressure Distribution. . 30b. Idealized Flow Phenomena. . . . . . . 30

13. Variation of Mixing Zone Characteristics withCrocco Number Squared. . . . . . . . . . . 31

14. Comparison of Theoretical with Experimental PeakRecompression Static Pressure as a Function of

Pc/Pt

15.

16.

a. Ane / A* = 3.627.b. Ane / A* = 10. 848c. Ane / A* = 25. 000 .

Comparison of Theoretical with Experimental PeakRecompression Static Pressure as a Function ofNozzle Total Pressure

a. Ane / A* = 3.627.b. Ane / A* = 10. 848c. Ane / A* = 25. 000

Comparison of Theoretical with Experimental Mini­mum Cell Pressure Ratios

a. Data from Ref. 7.b. Data from Ref. 16c. Data from Ref. 4d. Data from Ref. 4e. Data from Ref. 4

viii

323233

343435

3637383940

A*

AD

b

c

k

t.J

M

f ne

u

x

y

{3

y

AEDC-TDR-64-3

NOMENCLATURE

Area of nozzle throat

Area of diffuser

Area of nozzle exit (Fig. 1)

Width of mixing zone (Fig. 8b)

Inviscid jet boundary Crocco number. VI - Tj/Tt

Rate of growth of mixing zone

Location of mixing zone relative to inviscid jetboundary (Fig. 12)

Length along inviscid jet boundary from nozzleexit to diffuser wall

Mach number

Mach number of inviscid jet boundary

Static pressure

Base or cell pressure (Fig. 1)

Total pressure

Total pressure downstream of a normal shock wave

Static pressure on diffuser wall

Radius of diffuser

Radius of nozzle exit (Fig. 1)

Radius of total pressure probe (Fig. 2b)

Gas total temperature

Inviscid jet boundary static temperature

Velocity anywhere in mixing zone (Fig. 12b)

Velocity at inner edge of mixing zone (Fig. 12b)

Distance from nozzle exit plane parallel to nozzlecenterline (Fig. 2)

Distance perpendicular to nozzle centerline (Fig. 2)

Inviscid jet boundary recompression turning angle

Ratio of specific heats

ix

AEDC·TDR·64·3

SUBSCRIPTS

exp

max

mIll

theo

Non-dimensional mixing zone coordinate (Refs. 7.8. and 9; Fig. 12b)

Non-dimensional location of inner edge of mixingzone (Fig. 12b)

Non-dimensional location of stagnating streamline(Fig. 12b)

Non-dimensional location of inviscid jet boundary(Fig. 12b)

Nozzle exit wall angle relative to nozzle centerline

Experimental

Maximum

Minimum

Theoretical

x

AEDC-TDR-64-3

1.0 INTRODUCTION

The ejector research program conducted in the Rocket Test Facility(RTF), Arnold Engineering Development Center (AEDC), Air Force Sys­tems Command (AFSC), has been primarily of an experimental nature.The purpose of the overall research program is to define the variousfactors which influence the performance of axisymmetric ejector sys­tems. The results are presented in Refs. 1 through 6. Based on theseexperimental results, simple empirical methods were developed forestimating the performance of axisymmetric ejector systems. In thisreport, the two-dimensional base pressure theory for isoenergetic mix­ing and negligible initial boundary layer, developed by Korst (Ref. 7), isextended to improve the applicability of this theory to predict the mini­mum cell pressure ratio produced by axisymmetric ejector systems.

The two-dimensional supersonic base pressure theory developed byKorst consists of two independent theories which, when solved simul­taneously, yield the base pressure. The criterion for the simultaneoussolution is that the total pressure on the dividing streamline in the mix­ing zone must equal the peak recompression static pressure in the im­pingement zone. In Ref. 7 these theories were developed for two­dimensional mixing along a uniform inviscid jet flow field. The totalpressure on the dividing streamline in the mixing zone is determinedby applying overall momentum and conservation of mass flow relationsto the mixing zone using a velocity distribution derived by simplifyingand solving the basic equations for turbulent flow. The peak recom­pression static pressure is determined by turning the inviscid flow fieldparallel to the impingement surface by means of a two-dimensional shockwave.

In Ref. 7 it is shown that, for axisymmetric ejector systems, Korst ' stwo-dimensional supersonic base pressure theory predicts minimum cellpressure ratios greater than experimental values (1.60 < Mj < 3.50) withthe deviation increasing as the area ratio of the driving nozzle increases.The basic assumptions of the theory are reviewed (Ref. 8) with regardto their compatibility with the conditions that are produced by an axisym­metric ejector system. The results of that investigation of interest hereare as follows:

1. The theoretical total pressure on the dividing streamli:n~

agrees with the measured peak recompression staticpressure if it is assumed that the stagnation processis isentropic and that the mixing is two-dimensional.

Manuscript received December 1963.

1

A E DC· T DR·64·3

2. The theoretical peak recompression static pressureobtained by applying two-dimensional shock wave theorybased on a free-jet Mach number determined by the ratioPel Pt is shown to disagree seriously with the measuredpeak recompression static pressure.

These conclusions have also been experimentally verified by dataobtained at the AEDC. In addition to the above-mentioned considera­tions, the axisymmetric effects on the characteristics of a mixing zonehave been theoretically investigated in Refs. 8 and 9. Application of thetheory presented in these references to the typical ejector systemsstudied in Refs. 1 through 6 shows that the axisymmetric effects on mix­ing are negligible for these systems.

The above discussion leads to the conclusion that the major sourceof error in the application of Korst' s two-dimensional supersonic basepressure theory to axisymmetric ejector systems is in the method usedfor predicting the peak recompression pressure. An empirical methodfor estimating the peak recompression, based on relatively limited ex­perimental data (2.57 < Mj < 3.4l), is presented in Ref. 8. The correctedbase pressure theory is then shown to predict very accurately the mini­mum cell pressure ratio of a limited number of axisymmetric ejectorsystems. In this report, a theoretical method of estimating the peakrecompression pressure is developed, and the modified base pressuretheory is applied to a wide range of axisymmetric ejector systems(1.60 < Mj < 7.4).

The recompression theory developed in this report is applicable toany system involving the impingement of a free jet on a solid surface.

2.0 RECOMPRESSION MECHANISM

The basic premise of this analysis is that the recompression staticpressure distribution is a function of the combined inviscid flow fieldand the mixing zone flow field. The following discussion treats thetheoretical inviscid flow field and the superposition of a mixing zone.

2.1 THEORETICAL INVISCID JET FLOW FIELD

A typical axisymmetric ejector system is shown in Fig. 1. TheinvisCid flow field of the jet emanatiI:!:g from the nozzle exit can bedetermined by applying the method of characteristics solution to the

2

AEDC-TDR-64-3

general potential flow field equations using the known nozzle exit flow

conditions and the cell pressure ratio (::). This method is adequately

covered in Ref. 10 for both isentropic (irrotational) and rotational (totalpressure gradient) flow. From the practical point of view, the assump­tion of isentropic flow simplifies the flow field calculation; however, thisobviously is a questionable assumption if the nozzle exit flow is knownto be highly rotational because of the contour of the nozzle. Even if thenozzle flow is isentropic, the jet flow field cannot be treated in generalas isentropic flow because of the presence of a shock wave referred toherein as the jet boundary shock wave. The jet boundary shock wave isformed by the coalescence of infinitesimal compression waves whichresult from the curvature of the axisymmetric constant pressure jetboundary (Ref. 11). The strength (change in entropy) of the jet boundaryshock wave increases with jet radius. Fortunately, for most ejectorsystems, the jet impinges on the diffuser wall before the boundary shockwave can develop significant strength and therefore may be treated asisentropic. This is also indicated by the fact that the location of theinviscid jet boundary can be accurately calculated neglecting the boundaryshock wave.

2.2 EXPERIMENTAL VERIFICATION OF THEORETICAL INVISCID FLOW FIELD

On the basis of the previous discussion the jet flow field is assumedto be isentropic and is theoretically treated by the method presented inRef. 12. To verify this theoretical method, lines of constant Mach num­ber in the free jets of two ejector systems .were experimentally deter­mined from total pressure measurements and the assumption of isen­tropic flow. Thus, the total pressure anywhere in the flow field wasassumed to be equal to the nozzle plenum total pressure. Since a totalpressure probe in a supersonic stream measures the total pressuredownstream of a normal shock wave, the ratio of measured total pres­sure to nozzle plenum total pressure can be used to determine the free­stream Mach number.

The two ejector systems tested used 18-deg, half-angle conicaldriving nozzles having area ratios of 10. 848 and 25. 000, which aredescribed in Ref. 4. The diffuser diameter for both configurations was10. 19 in. The total pressures in the internal portion of the jet flow fieldwere measured by a fixed rake (Fig. 2a). and the driving nozzle wasaxially translatable relative to the fixed rake; thus measurements couldbe obtained at various axial stations. A complete description of the rakeand movable nozzle system is presented in Ref. 6. The driving fluid wasunheated air at a temperature of about 80°F.

3

AEDC·TDR·64·3

The theoretical flow field, based on the assumption of isentropicflow, is shown in Figs. 3 and 4 to agree well with the experimentalresults for the internal portion of the free jet. The theoretical flowfield was calculated by the IBM 7070 computer. The experimentalresults presented in Figs. 3 and 4 were obtained by cross-plotting theoriginal experimental data in such a manner as to obtain lines of con­stant Mach number.

Experimental flow field measurements in the vicinity of the jetboundary were made with a single total pressure probe, approximatelyaligned with the flow direction as shown in Fig. 2b. The jet flow fieldin this region cannot be assumed to be irrotational (constant total pres­sure) because of the mixing zone and the jet boundary shock wave.Therefore, the measured total pressure cannot be used to determineMach number without making further assumptions concerning the natureof the actual flow field or by measuring the static pressures. Staticpressure measurements in this portion of the flow field are impractic­able by conventional methods because of the non-uniformity of the flowfield. To avoid these difficulties, the ratio of the experimentally meas­ured total pressure to the nozzle total pressure is compared in Fig. 5with the theoretical total pressure that the probe should sense. Thiscomparison shows that the theoretical location of the jet boundary shockwave and the flow conditions upstream of the boundary shock wave agreewell with the experimental results. It is also shown in Fig. 5 that in theregion between the jet boundary shock wave and the inviscid jet boundary,the theoretical flow field is in error because of the previously mentionedpresence of the mixing zone and the nonisentropic nature of the jetboundary shock wave. For the particular ejector systems used in theseexperiments, the theoretical maximum total pressure loss through thejet boundary shock wave is less than 5 percent. Therefore, the mixingzone is the primary cause of the disagreement between the theory andexperiment in this region for these particular configurations.

In summary, it can be concluded that the jet flow field, out to theimpingement shock wave, is accurately predicted by applying the methodof characteristics solution to the general isentropic. flow equations exceptin the region of the mixing zone.

2.3 THEORETICAL RECOMPRESSION STA TIC PRESSURE DISTRIBUTION

The flow field approaching the diffuser wall is rotational in the mix­ing zone portion and non-uniform in the inviscid portion. The turningof such a flow field parallel to a wall can be treated theoretically (neg­lecting boundary layer) to determine the static pressure distribution

4

AEDC-TDR-64-3

along the wall by considering small increments in the approaching flowto be uniform at some average Mach number, flow direction, and totalpressure level. The impingement shock wave can be treated as two­dimensional for most systems of practical interest since it is always ata large radius from the centerline of the nozzle. A sketch of a repre­sentative first-order theoretical recompression flow field is presentedin Fig. 6. This is referred to as a first-order theory since the weakeror second-order interaction waves are neglected. The calculation pro­cedure involves the following:

1. The pr.essure PH is produced by the turning (two­dimensionally of the average flow conditions in region

1 through the angle [13; 131J .2. The average flow in region 2 turns through the angle

[( 131; 13 2) - oJ in order to satisfy the interaction

conditions with the downstream flow from region l.The interaction conditions are: equal static pressures(P

21= P

22) and parallel flow. Thus, the angle 01 can be

determined by a trial and error procedure. The magni­tude of the angle 01 is typically about 1 deg or less,depending on the size of the region. Therefore, the pres­sure P2l is determined by an isentropic compression fromPH through the angle 01 •

3. The pressure P31 is produced by an isentropic compressionfrom. P

21through the angle °1 • The pressure P

32is pro­

duced by an isentropic compression from P22

through theangle °1 • When second-order compression and expansionwaves are neglected, P

32= P3I •

4. The pressure P4I is produced by an isentropic compres­sion from P31 through the angle °2 • The pressure P42 isproduced by an isentropic compression from P32 throughthe angle °2 • The pressure P43 is produced by the turningof the average flow in region 3 through the angle 02 in sucha manner as to satisfy the previously stated interactionconditions. When' second-order waves are neglected,P 43 = P 42 = P 41 •

5. The pressure PSI is produced by an isentropic compressionfrom P

41through the angle 82 • The pressureps2 is produced

by an isentropic compression from P42 through the angle °2 •

The pressure Ps 3 is produced by isentropic compressionfrom P

43through the angle °2 • When second-order waves

are neglected, PSI = P S2 = P S3 •

5

A E DC· T D R-64-3

Steps 1 through 5 treat the theoretical recompression of the mixingzone portion of the approaching flow field. The theory shows that thestatic pressure distribution along the diffuser wall is of an increasingnature from Pu to PSl - produced by a series of isentropic compressions.It is also shown that the approaching flow does not turn parallel to thediffuser wall. initially. but through the angle (f3 - 0). The flow is thenturned parallel to the diffuser wall by an isentropic compression throughthe angle o.

The theoretical recompression of the inviscid portion of the approach­ing flow (constant total pressure and increasing Mach number) can betreated in a similar manner.

6. The pressure PM is produced by the turning of the average

flow conditions in region 4 through the angle [f33 ~ f34 + 03J

in such a manner as to satisfy the interaction conditionswith the downstream flow from region 3.

The interaction conditions require the downstream flow from region 3to expand isentropically through the angle 03 to the pressure P

63because

of the characteristics of the two-dimensional shock wave theory shown,for a typical case, in Fig. 7 (decreasing downstream static pressurewith Mach number). Therefore, the pressure P62 is produced by anisentropic expansion from PS2 through the angle 03 • The pressure P61

is produced by an isentropic expansion from Ps 1 through the angle 03 •

When second-order waves are neglected, P64 = P 63 = P62

= P61

•

7. The pressure P71

is produced by an isentropic expansionfrom P

61through the angle 03 • In a similar manner the

pressures P72 ' Pn ' and P74

are determined. When second­order waves are neglected. P74 = P73 = P72 = P71 •

8. The above procedure is repeated for regions 5, 6, 7, etc.

Steps 6 through 8 show the theoretical diffuser wall static pressuredistribution to be of a decreasing nature from Ps 1. This analysis alsoshows that the approaching flow does not turn parallel to the diffuserwall, initially, but through the angle (f3 + 0). The flow is then turnedparallel to the diffuser wall by an isentropic expansion through theangle o.

The error in this theory caused by neglecting the second-orderwaves can be estimated in the following manner. The flow interactionrequirements which determine the impingement shock wave are equalstatic pressures and flow direction. However, the Mach numbers inthe two regions produced by the interaction will be different because

6

A E DC- T D R-64-3

of the differences in the total pressure levels. As an example, theMach number in the region where P21 exists may be 1. 60 and the Machnumber in the region where P

22exists may be 2.6, depending on the

size of the flow field increment. If 01 = 1. 0 deg, then from Fig. 8for an isentropic compression,

~ 1.051P21

andP 32 l.071=P 22

Since P21

= P22

, second-order waves are necessary to satisfy the inter­action conditions. Neglecting second-order waves results in an errorof +1. 91 percent in P32 relative to Pn • The difference in Mach numberused in this example is relatively l8.rge; a typical Mach number differ­ence would be O. 25, and the corresponding error in P

32relative to P

31

would be 0.42 percent. The curves in Fig. 8 show the relative insensi­tivity of the pressure ratio with Mach number for Mach numbers greaterthan 1. 1. Thus, the first-order theoretical recompression analysis isaccurate for at least that portion of the recompression static pressuredistribution from the stagnating streamline to a point slightly beyondthe peak static pressure location.

For typical flows considered, the theoretical peak recompressionstatic pressure is P

S1and is shown in step 5 to be equal to PS3 ' The

pressure Ps 3 is determined by the average flow conditions in the vicinityof the inner edge of the mixing zone (region 3). Since the initial turningangle of the mixing zone portion of the approaching flow field is (f3 - 0)

and since the initial turning angle of the inviscid portion of the approach­ing flow field is(f3 + 0), then the streamline along the inner edge of themixing zone must turn through the angle f3. Thus, the end of the com­pression process and beginning of the expansion process is determinedby the location of the inner edge of the mixing zone. As a result, thepeak recompression static pressure is equal to the static pressure down­stream of the impingement shock wave required to turn the flow alongthe inner edge of the mixing zone parallel to the diffuser wall.

2.4 SIMPLIFIED THEORETICAL RECOMPRESSION STA TIC PRESSURE DISTRIBUTION

The first-order theoretical recompression static pressure distribu­tion involves a trial and error procedure to determine 8 and a scaledrawing of the flow field to determine the shock wave arrangement. Therecompression static pressure distribution can be approximated by ignor­ing the actual impingement shock wave configuration and the flow fieldinteractions. The procedure is to determine the flow conditions-approach­ing the diffuser wall at each point and to turn each streamline parallel to

7

AEDC-TDR-64-3

the wall through an oblique shock. Two types of error exist in thissimplified analysis. One error is associated with the orientationof the flow and the other is associated with the magnitude of thestatic pressure. The orientation error is not significant if the angle,relative to the diffuser wall, of the approaching flow field is approxi­mately equal to the Mach angle of the flow downstream of the impinge­ment shock wave. This is usually the case for many ejector systems.The error in the magnitude of the static pressure can be estimated inthe following manner. Consider the streamline approaching the diffuserwall that has the same flow conditions as the average flow in region 2(Fig. 6). Let P3 be the static pressure downstream of the two­dimensional sho"ck wave required to bend this streamline parallel to the

wall. The turning angle would be ({31; (32). The static pressure P3can be approximated by considering that this streamline turns by means

of a two-dimensional shock wave through the angle [{31 ~ {32 - 01J and

then isentropically compresses through the angle 01

to the static pres­sure p;. The error in p; relative to P3 for 01 = 1 deg is shown to bevery small in Fig. 9 for typical flow conditions. In steps 2 and 3(section 2.3) the static pressure P32 is shown to be determined by thesame process as that used to determine p;. Thendore,P3 = P; = P 32 = P 31 ' Thus, the error in the magnitude of the static pres­sure obtained by this simplified analysis is on the order of that producedby neglecting second-order waves (section 2.3).

2.5 SUPERPOSITION OF MIXING ZONE ON INVISCID FLOW FIELD

The theoretical recompression of a known approaching flow fieldis presented in sections 2.3 and 2.4. However, for axisymmetric jets,the mixing takes place in a region of non-uniform flow both alone andacross the mixing zone. The greatest flow field variation is in thedirection across the mixing zone, as shown in Figs. lOa and lla.These flow conditions violate the basic assumptions of uniform flow andconstant static pressure made in Ref. 7 to theoretically determine thevelocity distribution in the mixing zone. Therefore, a direct super­position of this theoretical mixing zone velocity profile on the theoreticalinviscid flow field of an axisymmetric jet is unjustifiable. However, aspreviously stated in this report, only the peak. recompression static pres­sure is of importance in Korst's supersonic base pressure theory. Insection 2.3 it is theoretically shown that the peak recompression staticpressure is produced by the turning, parallel to the diffuser wall, of theflow field along the inner edge of the mixing zone.

A comparison of experimental recompression static pressuredistribution with the simplified theoretical distribution based on the

8

AEDC-TDR·64-3

isentropic inviscid flow field is presented in Figs. lOb and lIb. Thetheoretical distribution is shown to be in error due to neglecting themixing process. However, the theoretical distribution is shown toagree reasonably well with the experimental data in the region down­stream of the apparent mixing zone, thus verifying the simplifiedtheoretical analysis.

The region of maximum influence of the mixing process relativeto the inviscid flow field can be estimated from an idealized concep­tion of the recompression flow phenomena. Based on the results pre­sented in Figs. lOb and lIb, an idealized recompression flow modelwas developed as shown in Figs. l2a and b. The region of influenceof the mixing zone on the recompression static pressure distributioncan be determined in the following manner:

a. Impingement location of the inner edge of the mixing zone

For turbulent mixing

b-- = c

r ne

k

sin f3 (r:e) (1)

(2)

Substituting Eq. (2) into Eq. (1) yields

~X3 k c ( .tj )rne . sin f3 fne

(3)

Values for k and c (from Abramovich, Ref. 13) are derived in

Ref. 9 and presented in Fig. 13. Values for --P- and f3 arene

obtained from the theoretical inviscid jet boundary.

b. Impingement location of stagnating streamline

The thickness of the mixing zone at the impingement pointof the stagnating streamline is

by geometry

b[

.t.=c_l__

r ne cos f3J (4)

(~)sin f3

by definition (Ref. 9)

(5)

k = (6)

9

AEDC-TDR-64-3

Substituting Eqs. (4) and (6) into Eq. (5) and solving1'1x2 • dfor -r- ylel Sne

(7)

c. Location of outer edge of mixing zone

The equation for the thickness of the mixing zone is

r;!;- = c [:~e - (-~:2) cos (3 - ~nx:J (8)

By geometry, ~ can be approximated by the followingTn e

equation

~:: ~ [0 - k) C~J - (~:,) sinll] c,:f) (9)

Substituting Eq. (8) into Eq. (9) and solving for 1'1 Xl yields-!ne

(I-k) c (~.) - [(l-k) c + tan{3J (~) cos{3

[(l - k) c + tan ~ J (10)

The location 7J s of the stagnating streamline is determined bythe condition that the total pressure of the stagnating stream­line equals the peak recompression static pressure.

The theoretical mixing zone width determined by Eqs. (3). (7), and(10) is shown in Figs. lOb and 11b to agree very well with the apparentexperimental width of the mixing zone. It is also shown in Figs. lOband llb that the theoretical recompression static pressure determinedby the inviscid flow conditions along the inner edge of the mixing zoneagrees in magnitude with the experimental peak recompression staticpressure.

2.6 EXPERIMENTAL VERIFICATION OF THE THEORETICAL PEAK

RECOMPRESSION STATIC PRESSURE

In section 2.3, it was theoretically shown that the peak recompres­sion static pressure is determined by turning, parallel to the diffuserwall, the inviscid flow along the inner edge of the mixing zone. Theo­retical peak recompression static pressures are compared with experi­mental values in Figs. 14 and 15. The theoretical values were obtained

10

A E DC- T D R·64-3

by using the method of characteristics solution to the general potentialisentropic flow equations to determine the inviscid flow field, based on

the experimental (::) and Eq. (3) to determine the location of the inner

edge of the mixing zone. The experimental data were obtained fromthree ejector systems geometrically described in each figure. The

minimum cell pressure ratio (:~) for each configuration was varied

by varying the total pressure ~Pd level of the driving nozzle. Theresulting variation of (Pc/Pt) with Pt is shown in Fig. 16 and is due to aReynolds number effect described in Ref. 4. The Reynolds numberdetermines the character of the nozzle boundary layer which then in­fluences the free-jet mixing process, thus producing a new equilibriumcell pressure. The nozzle boundary layer influences the jet mixingprocess in two ways.

1. The velocity profile in the mixing zone varies in shapewith distance from the nozzle exit. However, theasymptotic profile is the same as the undisturbed mix­ing profile.

2. The mixing zone has a finite thickness at the nozzleexit and, consequently, is thicker, at any point, thanthe undisturbed mixing zone.

Theoretically, the two characteristics of the mixing zone, which areinvolved with the peak recompression static pressure, are k and b. Therelative location (k) is a function of the velocity profile in the mixing zone.The width (b) of the mixing zone at the point of impingement is a func­tion of the thickness of the mixing zone at the nozzle exit. For mostejector systems of practical interest, the length of the mixing zone issufficient to allow the velocity profile to transform into the undisturbedprofile. Thus, the relative location (k) is the same as for the undis­turbed case. However, the width of the mixing zone (b) for mostejector systems of practical interest will be thicker than the undisturbedcase and will produce a lower peak recompression static pressure thanthe undisturbed case. This is experimentally verified in Figs. 15b andc by comparing the theoretical peak recompression static pressure foran undisturbed mixing zone with the experimental values obtained at lowtotal pressure levels (low Reynolds number). The comparison in Fig. 15shows that the theoretical peak recompression static pressure deter­mined by the flow conditions along the inner edge of the mixing zoneagrees to within 10 percent of the experimental values of the maximumtotal pressure level. Included in Figs. 14 and 15 is the peak recompres­sion static pressure that would exist if there were no mixing. This is therecompression static pressure produced by the turning of the inviscid jetboundary streamline parallel to the diffuser wall.

11

AEDC-TDR-64-3

Since the major factor which determines the peak recompressionstatic pressure is (Pc/Pd, theoretical peak recOlnpression static pres­sures are compared with experimental values in Fig. 14 as a functionof (pc/Pt). These are the same experimental values as presented inFig. 15. Included in Fig. 14c are experimental peak recompressionstatic pressures obtained with a small amount of bleed flow into thecell region. The bleed flow caused the cell pressure ratio to increaseto a new equilibrium value and a new peak recolllpression static pres­sure. The theoretical peak recolllpression static pressures determinedby the flow conditions along the inner edge of the mixing zone are shownin Fig. 14c to agree very well with the new experimental values. Fur­ther experimental verification of the recompression theory is presentedin Table 1, in which the theory is compared with data obtained fromRef. 8.

From the previous discussion it can be concluded that the peak re­compression static pressure determined by the flow conditions along theinner edge of the mixing zone agrees with the experimental results towithin ±10 percent for those cases in which the nozzle exit boundary layercan be neglected.

3.0 THEORETICAL MINIMUM CELL PRESSURE RATIO

OF AXISYMMETRIC EJECTOR SYSTEMS

Korst (Ref. 7) has developed the two-dimensional, supersonic basepressure theory for negligible initial boundary layer. When trying toapply this theory to an axisylllmetric ejector system, most of the basicassumptions (discussed in Ref. 8) are clearly violated. Also in Refs. 8and 9 it is shown that the total pressure on the dividing streamline is,for most practical ejector systellls, independent of axisymmetric effects.The non-uniformity of the inviscid flow would be expected to produce asignificant deviation frolll the two-dimensional value of the total pressureon the dividing strealllline; however, in Ref. 8 and indirectly by the datain this report no appreciable deviation was noted. A possible explana­tion of this is the following. The non-uniforlllity of the inviscid flowfield distorts the mixing zone iIi such a manner as to cause the totalpressure on the dividing strealllline to be greater than the two­dimensional value (Ref. 8). The dividing strealllline may then experi­ence a normal shock before stagnating on the diffuser wall (Ref. 17),thus reducing the total pressure. These two processes tend to canceleach other, resulting in a total pressure level close to the two-dimensionalisentropic recompression pressure.

12

AEDC-TDR-64-3

As previously stated, the major source of error associated withthe application of the two-dimensional base pressure theory to anaxisymmetric ejector system is the assumption of uniform inviscidflow in the determination of the peak static pressure in the impinge­ment zone. This error in Korst' s base pressure theory has beeneliminated by using the 'recompression theory developed herein. Theresulting base pressure theory is referred to as the "modified" basepressure theory.

A comparison of theoretical minimum cell pressure ratios withexperimental data is presented in Fig. 16 for various axisymmetricejector systems. The theoretical values were obtained by using thebase pressure theory (isoenergetic, neglecting initial boundary layer)presented in Ref. 7 modified by the theory developed in this report forestimating the peak recon;wression static pressure. Included inFigs. 16a and b are theoretical minimum cell pressure ratios usingthe unmodified two-dimensional theory. As shown in Figs. 16a throughe, the modified theory agrees very well with experimental results,except for the two ejector configurations in Figs. 16d and e which wereusing "bell" type nozzles. Theoretically, these two ejector configura­tions were treated as having isentropic nozzles with uniform exit flowconditions; this, of course, is not correct since the total pressure andvelocity vary non-uniformly across the exit. This is believed to be thereason for the major portion of the difference between the theoreticaland experimental values obtained for these two configurations.

All of the data presented in Fig. 16 were obtained from ejector sys­tems using air (y = 1.4) as the driving fluid and having nozzle exit flowangles of 0, 15, and 18 deg. In Table 2 theoretical minimum cell pres­sure ratios are compared with experimental values (Ref: 14) for anejector system using a conical nozzle having a half angle of 7. 58 deg.This system was operated using both air (y = 1.40) and helium (y = 1.66)

as the driving fluids; the theoretical minimum cell pressure ratios areshown to agree very well with the experimental values.

In summary, the theoretical results were compared with experi­mental values obtained from 34 axisymmetric ejector systems, neg­lecting "bell" ejector systems, having the following characteristics:

y 1.4 and 1.66

1.00 ~ ~ ~ 25.00A*

1.46 ~ ~ ~ 150.36A*

o deg ~ One ~ 18 deg

13

The use of the modified theory resulted in rnH1l111UHl cell pressure ratiovalues which deviated from the experiInental values by a maximum of-11. 8 to +25, 2 percent, with a ~;tal1dard deviation of 5,5 percent(Ref. 15),

The modified base pressure theory also applies directly to thetwo-dimensional case with a curved jet boundary as compared with thesimpler theory derived by Karst for straight jet boundaries. The modi­fied base pressure theory shows, in general, that the minimum cellpressure ratio produced by an axisymmetric ejectqr system is a func­tion of the thickness of the mixing zone ,-it the free-jet boundary impinge­ment point, unlike the two-dimensional back step case which has auniform inviscid flow field.

4,0 CONCLUSIONS

Based on the results of this study, the following conclusions arereached:

1. The peak recompression static pressure of an axisym­metric jet is approximately determined by the inviscidflow conditions along the inner edge of the mixing zone.

2. The peak recompression static pressure and thereforethe minimum cell pressure ratio of an axisymmetricejector 'system is, in general, a function of the thick­ness of the mixing zone at the free-jet boundary impinge­ment point.

3. The minimum cell pressure ratio of the axisymmetricejector systems investigated can be estimated to within6 percent (standard deviation) through an applicationof Korst' s isoenergetic, supersonic base pressure theory,modified by the recompression theory presented in thisreport. These results were verified for jet boundaryMach numbers from 1. 6 to 7.4.

14

AEDC·TDR·64·3

REFERENCES

1, Barton, D. L. and Taylor, D. "An Investigation of Ejectorswithout Induced Flow - Phase 1." AEDC-TN-59-145,December 1959.

2. Taylor, D., Barton, D. L., and Simmons, M. "An Investiga­tion of Cylindrical Ejectors Equipped with TruncatedConical Inlets - Phase 1." AEDC-TN-60-224, March 1961.

3. German, R. C. and Bauer, R. C. "Effects of Diffuser Lengthon the Performance of Ejectors without Induced Flow. "AEDC-TN-61-89, August 1961.

4. Bauer, R. C. and German, R. C. "Some Reynolds NumberEffects on the Performance of Ejectors without InducedFlow." AEDC-TN-61-87, August 1961.

5. Bauer, R. C. and German, R. C. "The Effect of Second ThroatGeometry on the Performance of Ejectors without InducedFlow." AEDC-TN-61-133, November 1961.

6. German, R. C., Panesci, J. H., and Clark, H. K. "ZeroSecondary Flow Ejector-Diffuser Performance UsingAnnular Nozzles." AEDC-TDR-62-196, January 1963.

7. Korst, H. H., Chow, W. L., and Zumwalt, G. W. "Researchon Transonic and Supersonic Flow of a Real Fluid at AbruptIncreases in Cross Section." ME Technical Report 392-5,University of Illinois.

8. Zumwalt, G. W. "Analytical and Experimental Study of theAxially-Symmetric Supersonic Base Pressure Problem. "Ph. D. Thesis, ME Dept., University of Illinois, 1959,MIC 59-4589, University Microfilms, Inc., Ann Arbor,Michigan.

9. Bauer, R. C. "Characteristics of Axisymmetric and TwoDimensional Isoenergetic Jet Mixing Zones. II AEDC-TDR­63-253, December 1963.

10. Ferri, A. Elements of Aerodynamics of Supersonic Flows.The Macmillan Company, New York, 1949.

11. Lord, W. F. "On Axi-symmetrical Gas Jets, with Applica­tions to Rocket Jet Flow Fields at High Altitudes. "R & M No. 3235.

15

A E DC- T D R-64-.3

12. Moe, Mildred M. and Froesch, B. Andreas. lIThe Computationof Jet Flows with Shocks." STL-TR-0000-00661, May 1959.

13. Abramovich, G. N. "Turbulent Jets Theory." AD 283 858,August 3, 1962.

14. Hale, J. W. "Comparison of Diffuser-Ejector Performancewith Five Different Driving Fluids." AEDC-TDR -63 -207,October 1963.

15. Merriman, M. Method of Least Squares. John Wiley & Sons,Inc., New York, 1915.

16. Foster, R. M. "The Supersonic Diffuser and Its Application toAltitude Testing of Captive Rocket Engines." AFFTC-TR­60-1, January 1960.

17. Goethert, B. H. Base Flow Characteristics of Missiles withCluster Rocket Exhausts." Aerospace Engineering,March 1961.

16

.....-J

TABLE 1

COMPARISON OF THEORETICAL WITH EXPERIMENTAL PEAK RECOMPRESSION STATIC PRE~SURI:

y = 1.4

~ ~ene Nozzle Pc

~'d twmax

)

Error,

A* r n e P t Pc theo,Remarks

deg Type c exp percent

1.0 2.00 0 Isentropic 0.05283 3.06 2.77 -9.48 Data from Ref. 8

1. 1065 1. 843 0 " 0.04149 3.18 3.56 +10.67 "1. 568 1. 550 0 " 0.02274 3.68 4.05 +10.05 "1. 8562 1.440 0 " 0.01501 4.45 4.86 +9.21 "

>mo().-lo;u.0­l>.

W

t--'00

TABLE 2

COMPARISON OF THEORETICAL WITH EXPERIMENTALMINIMUM CELL PRESSURE RATIOS OF AXISYMMETRIC EJECTORS

~ Ane ene, (::t, (::)theoError,

y RemarksA* A* deg percent

22.25 10.71 7. 58 1.4 0.0011 0.000985 -10.46 Data

22.25 10. 71 7.58 1. 66(He) 0.00068 0.000712 +4.71 fromRef. 14

73.06 17.51 18 1. 40 0.00045 0.00045026 +0.0578

»mon,-lo;:u,0­J:>,..w

I-'co

Pt

/2""=!eJT £/// ..t c.

Diffuser Wall

ImpingementShock Wave

Streamline

Zone

~

Fig- 1 Sketch of a Typical Ejector System

»mCJn-1CJA),0­.,.W

AEDC·TDR·64·3

Total Pressure Rake

To Manometer Board

x-----.... --lilP- -------

Diffuser \

Inviscid Jet~ /'"Boundary /,y

y /'

t .../'_--_---I'-.........-........~~

a. Total Pressure Rake Insta lIation

To ManometerDiffuser '\

Inviscid Jet~ ./Boundary >'

y ./

Pc t // r p

Pt I-4.:--- \---I11lo'1lO- --lilP- X-----1t---

Total Pressure Probe

LMovable "Nozzle "

.......

b. Total Pressure Probe Installation

Fig.2 Sketch of Experimental Configurations

20

An8

A*10.848

AD

A*65.25

Pc -3- = 0.5331 x 10Pt

'Y 1. 40

Diffuser Wall Location for Experiment

~((/(/~(//~(//~(//(~/~;>f//~~/f//f/~~//i//i//i//i((!

~ 6.17-+---~

t-'

yl'ne

I I I I I I I

2.0 11-1-+-1---+--+-1--1-Pc

ConicalI-'- Nozzle

18 degI •

I I = 4.04 •___________Theoretical

""""' -t--+--I--..L. (Isentropic)_

~-.J ---O--_Experimental

»mo()1

-lo;:0

10.848

4.03.0

Centerline of Nozzle and Diffuser

2.0

(X/rne

)

Fig.3 Comparison of Theoretical with Experimental Lines of Constant Mach Number in a Free Jet, Ane/A*

! ! !I I !I I !1 : ! I' I! , !I ! !II I

! I :........ 1.0o 0

0­.b..to>

Ano

A* 25.000ADA* =150.36

Pc -3- = 0.2486 x 10P t

y = 1. 40

»mon-Io;:0

0­J:>,.

W

2.01 I I I

Pc

tvrv

(r:e)I r

Theoret·(

1ca1Isentrop' )_ 1C +--

--O-__Exp .er1menta1

o 1.0 2.0

(Xli' )nE'

3.0 4.0

Fig.4 Comparison of Theoretical with Experimental Lines of Constant Mach Number in a Free Jet, Ane/A* 25.000

AEDC·TDR·64·3

0.03

0.02

exp

0.01

Ane = 10.848A* -

KBne = 18 deg -

'Y = 1.4 -I\. y-Inviscirl

r p = 2.2194rne -

Theory,;\ p'

Pc

~:= 0.5343 x 10-3 -

rnviscid Jet Pt_Boundary

-1J"~-Location~I b.,

_/ y-Ex~er;imental,~ R' t exp

/ t\ ,P~ , I

/ \ I I : I

/- ~--r-::-+-o-_.O-Q

if-_ 1tE 1'.0-1 1I

Jet BouldarJ shIck/I Wave Location

dr

3.83.6

= 0.291 x 10-3

3.4

Pressure in a Free Jet

3.22.82.6

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

(X/rne )

Ane = 25.000

~A*

"-Bne = 18 deg

~Inviscid

Inviscid Jet LTheory, l' = 1.4Boundary ~~ Pt

,r

Location -.Pt

----E- = 2.2194r ne

2.4

o1.6

3.0

(X/rne )

Comparison of Theoretical with Experimental Measured Total

or

Fig.5

23

N~

Reference Line

Diffuser Wall

Slip Line (Vortex Sheet)

Fig.6 Theoretical Recompression Flow Field (First Order)

~

mon-1o;;0

'".....w

AEDC-TDR-64-3

~ J~\ M,Pt,'Y +-'"

\~

I~ 'Y = 1.40I

II>

:- T I--')1= 1. 22 & t 20 deg

:-- .

r-~,."" " """""""""""""'" ___ .L-

1\

\ '\~'" M-I~i"--!

0.12

0.10

0.08

0.06

0.04

0.02

o3.0 4.0 5.0 6.0 7.0

M

8.0 9.0 10.0 11.0

Fig.7 P2/ Pt vs M - Two-Dimensional Shock Wave Theory

25

I:'.:lO':l

P2

PI

1.2

1.1

1.0

0.9

I I I I I

To 11.2J2 .J • 11.182 __1 I I~

.. 1. 4 _~~::::::::::+-~_+-'-j_I-'lJI-L-~-+-+-+--~-t-t'Y-t-r1-1:~~.l-~S\,on;l;.-l:eS -,coli'\' "I

\'\'C::";'~~-+_-j-I--:LLl---t--++++-r-II-r=~~sentl:'? I I I ,I' I +--_de\; -ZL..l- P2 -11-

'" 1- - M1 ,P1 ~~ degLIJ-~-f-+-+-r-I-=t::::~ _ ~,e--e.-~ '",,, It---...:--;

,--+--"/-

Ml' PI /.

fj~Et:t~:~t~~J;f±i~=EI"j/-Jr-----t---ji-+-I I L - --- 1 deg

/;.' -j--. <5 .. l-de r»r»~)"'2";";T7"7I ~ g IS

ell troP:icLe-J--,j+--l--t-+-t-i--r-l

I--- xpallSi.oh----... .,- '"'"----- '

~-~

i

~

mon,-io;U

0­J:>.,W

0.81.0 2.0 3.0

Mach Number, M1

4.0 5.0

Fig. 8 Typical Isentropic Compress ion and Expans ion Processes

""""',

t'V-J

P 2

PI

16 . 0 .---.,------,r--"""""T--,---,---.--.,..--.,---r----r--,---,---,---,---,--,----,---..,----,.---,---.-----,

I I

14.0 E I I

1 2 . 0 i!i--+--I-----1f-----l-_+_

10. 0 lI--l---+--+--+-+--I---I--f--1f--+--+--j--.:I'-\---l--.j--

b. 0 lI---I--+--I---+---t---+--+--I---+---I--~.-+-=..::,.:::.::..:..;.::.~.::;_:=r::==-:r=_:::..

6 . 0 1l--+--l---I-----1f-----l--+--+--+-

I IY I I I I I I I I I -1 21 deg

4.0 Iil---I----+--+--+-n-1- I I I I I I I I I I I I I I I

.....

2.0 • I I I I I I I I I I I I I I I I I I I I I I

»mo\1-io;;0

0­.j:>.

W

11. 010.09.08.07.06.05.04.03.02.01.0

Mach Number, M1

Fig.9 Typical Error of Approximation to Two-Dimensional Shock Wave Theory

o ' , r I ! I , , , ! , I I ! ! It! ! ! I , ,

o

AEDCTDR-64-3

0.5

_1 I IJet BoundaryShock WaveLocation~-

,,~ I-I- " I

~

YI ""~

V I--- I

M V:,.....--

)~Inviscid Jet ----Boundary ~ V

Location\--Jl>. (3 ~/

-1- -"1'-iI-I

5.0

4.0

1.0 1.5

29

28

27(3, deg

26

25

24

23

22

212.0

(X/r )ne

a. Inviscid Flow ConditionsG. 0 .---.--.--.--,---.,..:..-.:.--:r-:..::::......r~~:.:.:..:..;..:.:.:..:.:..::.--..--r--y--,..---,

18 deg

2.0

= 3.627

= 1.453

- 0.0106

'Y = 1.4

A*

Ane

1.51.0

5.0Inviscid Jet ~heoretical

f----Boundary - :=t-- (SimPtifi~d)-+_. -+--+--+---+e ne =

Location~ I L_~Experlmental

-9: 1"\4. 0 I---t---t--+--Ia/-f-I ~

I ~c;f --!-- ~"'"--+---+-+---+

~-+----+----+" I l ~- -0---03.0 :..< - I Jet Boundaryi'-...,

~ I Shock Wave I~

/ / 'r-- ---1--, L,oca t,i on I ~PL=:_j--.!----+;----+1---~

2. 0 I--+--/i I/

I.......--lnnel! Ed~e oJ Mixing Zone~ rI I I I Idr lL ~nvilciJ FIol Fi'eld

I Theoretical Mixing I

I Zone Wi dth --t---t---t--+--+--+--+--+--+--+--I-, I I

0'-~_J.-...._l.._J._..-!.._J-_!.._..l_.....J.._J-.....J.._J..._-J-_J_....I

0.5

1.0

(X/I' )ne

b. Diffuser Wall Static Pressure Distribution

Fig. 10 Theoretical and Experimental Impingement Zone Characteristics

28

A EDC- TDR·64·3

36

34

32

30 f',. deg

28

26

244.03.02.0

I I IJet BoundaryShock Wave~W-- :--- M

"-'" ;-...

t3......:'-..;

J......,

......,-YI ...

Inviscid Jet :..-'.,."I

Bou7darr-----(CJ

6.01.4

7.0

9.0

8.0

M

(X/rne )

a. Inviscid Flow Conditions

513 x 10-3

4

48

8 deg

0.848

4.03.02.0

T 1Ane

1=

\ A*

e = 1:'--Theoretical

neInviscid Jet yBoundary

(Simplified) _ 'Y = 1.

Location---..".,. \ I I'D= 2.

\ l'ne

I~Pc

= O.-Pt

I \ IA Jet Boundary

/:f-}q~OCk Wave -

f1 I S:imental1--1-.' 0........ I

>1 1- +-+'0-- --o~I I. I I

I-f- I - --I IP-e- ~

I I Inner Edge ofp. 1__Mixing Zone -

itl:1 'I I,dT l . I. I

.£J-.J:f 1-'-"'" n v 1. s C 1. d

I'Flow Field

Theoretical Mixing I I II, Zone Widtho

1.4

2.0

6.0

8.0

4.0

10.0

12.0

14.0

(X/rne

)

b. Diffuser Wall Static Pressure Distribution

Fig. 11 Theoretica I and Experimental Impingement Zone Characteristics

29

A E DC- T D R-64-3

1.0

....1

I-'\;I 'LTheoretical

-'I1 '\:Inviscid Jet=ld /,Boundary

1/1 ~Experimental-- { "

ffil~' I ~~_I____ -.Jf... -"'Xl + L> X2 ~6X3 .I I I r I I I I '" I

X

a. Idea Iized Static Pres sure Distribution

6X

ImpingementShock Wave

1 (1 + erf T)2

b. Ideal ized F low Phenomena

Fig. 12 Idealized Flow Phenomena at Impingement Point

30

· .'

r--;"'--:--..

J+=C (Abr~moViCh'S Method, Refs, 9 and 13)-r-t+-- I:--..I --I--I""-L I ~I --"--'" --k (Ref. 9) ~ :......

~-....... ........-~

.~

\'\

-

I

:>mon-io;u

'".l>.,W

k

0.20

0.10

0.40

0.30

0.60

0.50

o1.11.00.90.80.70.40.30.2 0.5 0.6

2Ca·

J

Fig.13 Variation of Mixing Zone Characteristics with Crocco Number Squared

0.1oo

0.20

0.10

0.30

c

Wt-""

AEDC·TDR·64·3

Ane - 3.627--rt Theory- Inv iscidA*

" Jet Boundary ene = 18 deg'" I---

I' 'Y = 1.4

I~Pt. 1'~ ~ = 2.48m1n ~ r--~~orY-Inner""'''''' (e

EdgeOf~Mixing Zone ........1.........1

"'--l ~XPW--° P 1 1_"t ~

~ max~___

- 1°--:--.,. 1"'\.

I~

14.0

13.0

12.0

11.0

(::tax 10.0

9.0

8.0

7.0

6.0

5.00.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36

3.627Ane

A*- 10.848 ..

).58

= 18 deg

0 . .16O - ').,)-0.500.48

- ""' ....... II-/f--i'-i-1--_ r

n' - 1.4

IC--I- L~-+--+--+--+-::: -.... - 2.48Theory-Inviscid -.;. ....... __ r ne

..........._+-_Jet Boundary --:-.! I I

'I --~.:L_

----~__ I r-TheorY-Inner EdgeI---+----+-'---.......::""-_c+~-of Mix ing Zone 4-_--+-_-4-_-4-_---1

- i"--~L I

° ¥ °1C--...1f---f---+--+----+--f-- 00,-1----+_--+-0_ P t ~I=--Experiment I ma~

1--+---+---+--+--+---+---+--+--1--1----11--1----+.-

Ptmi~6. 0 1-_!m-.,.",I,_~_,.l,..._.l."",.,.",I,......~_,.l,..........l."",.,.",I,__..;;,;,;;.._.l."",......l

0.44 0.46

16.0

15.0

14.0

13.0

(::tax 12.0

11. 0

10.0

9.0

8.0

7.0

(::) x 10::

b. Ane/A* "" 10.848

Fig. 14 Comparison of Theoretical with Experimental Peak Recompression

Static Pressure as a Function of pc/Pt

32

A EDC- T D R·64·3

."-Pt

t--t--+---t--+-+-+--+--+---+--+--+- if min

o ~1--+--+---IC-1J--i~I-,"","""""',I---~-+----+---+--+--~-+---lI-n-_-1Bleed into

~Experiment ~ Cell Region

.............~:-.:

0.30

2.48

= 25.000

0.28

'Y = 1.4

ene

.., 18 deg

Ane

0.260.240.220.200.18

~o~Ptma/xTheOryJ.J:~~~o of Mixing Zone ........ ,

o ...... I ........

1--~-t---1I-----+--+--+--+-+--+- A*:--.., 1/Theory-Inv1scict---+--~"..( I Jet Boundary --+-

I--+--+---I'I-'-:-II--t--+----+--+--+_~, r D =

I---+----l--+---l---~ r............ ne,......

16.0

15.0

14.0

13.0

12.0

11. 0

(::tax10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

c. Ane/A* = 25.000

Fig. 14 Concluded

33

AEDC-TDR-64-3

30 40 50 60 70 80 90

Pt , psia

a. Ane/A* = 3.627 Ane 10.84!:lA*

I T~ I I I. ! dI I Ii eory- onsel A

Jet Boundary ne~ 3.627I I· I I A*~ ~ I Theory-Inner Edge

of Mixing Zone Bne 18 deg\=

':\/ . 'Y ~ 1.4

cii\ I'D= 2.48

r ne

~~- -,-,-,----\ ~riment

0,____ 0

I5.0

20

13.0

12.0

11. 0

10.0

(::tax 9.0

8.0

7.0

6.0

= 18 deg

/

~ IJ'TheorY-Inviscid BneI---l---+----+-/..~~ Jet Boundary -f--f--+

I---+---I~__/// 'l_L_I_l...... r n 'Y ~ 1. 4, T I r 2.48.11 I ,............ r ne

1--+-/ I I I I .............:--I----+-,--{~ Theory-Inner Edge ...... _

of Mixing Zone

o 00 0 l~-L-_O-l-_-I-_O 0 0

.1 ~Experiment7. 0 r--0--+--+--+-~--=-'----'r-'-t--'--t--+--t--+--+--+---1

6. 0 B-...Ir",--""",-~_I....-~-...I._"""''''''"--J,_'''-'''-''~_'''''-""",*",__20 100 200 300

16.0

15.0

14.0

13.0

12.0

(::)max11. 0

10.0

9.0

8.0

Pt' psia

b. Ane/A* = 10.848

Fig.15 Comparison of Theoretical with Experimental Peak RecompressionStatic Pressure as a Function of Nozzle Total Pressure

34

r

AneA*

25.000

8ne = 18 deg

0­.t>-o

W

»mCJ()

-ICJ;u

500

1.4'Y

400

e. Ane/A* = 25.000

Fig. 15 Cone luded

200 300

Pt' psia

100

I J I I I I :D = 2.48........ 1 l-t=t-T -I------"---__ne

I-, +'2 'd~ Theory-Inviscl,/

Jet BoundalY I,/-/

I I I , ,/_L

I Theor~ Inner EdgeI.of Mixing Zone/

I I I I II

/

I0I.

~:'-00

// ....-L..---

1

~ --~Experiment7~I

~";J--J--/

V

~/In-Bleed into

leer Rriol. 1 I I

0-l-

I I

5.0

3.040

7.0

6.0

4.0

8.0

11.0

12.0

13.0

15.0

14.0

16.0

(p) 10.0

P: max 9.0wC)l

A -neene' deg, A* -

\ 0 1.0 0

\ 0 1. 232 0\

-\

/). 1. 700 0

\ 0 1.928 0

\ ------ Theory-Two Dimensional-

Theory-Modified\ Recompression

\

'\,~~ 0

.....

"""--' .... -- ~ r--."...... ~ ... .....

'" ' ...... ~r--::::.....' ... ......~

...""' ...

-......:-.......... """r-o -r--"--

~~

~ ~ .............

.... , ........

~K r... ...... ~~""'"

~ ....... r--.i'-.....

~-. r--_L':>r-...::.r-_ ...<: ....

I'-. -- .......~r-......r--

<>

<>

AE DC· TDR·64-3

1.0

0.1

0.011.0 2.0 3.0 4.0

a. Data from Ref. 7

Fig. 16 Comparison of Theoretical with Experimental Minimum Cell Pressure Ratios

36

...

0.10

0.010

0.0010

AEDC-TDR-64·3

I I I I I I I II I

- __ Theory-Two Dimensional

Theory-ModifiedRecompression

0 Experimental - Ref. 16

Ane 4.235-- =

A*

~,e = 15 degne

'Y = 1.4

I I I.... """- ... I I If\. ...

j :..- Ref. 17'\.. .........

h r<.K t-...~ I"r-- ,...

f't)\

I\r--.0

0.00011.0

b. Data from Ref. 16

Fig. 16 Continued

37

100

A An

lne 3.627, 21. 45,

A*= =

A*

~B = 18 degne

>-, <: LTheoretica1a..... U"'

...J....... ""Q /J........ .............. 'U....o .d...."'{

" .... rTh,eoretica1'"\tlcP-

IA Anne

= 5.070, = 30.27,-A* A*Bne

= 18 deg

/I.

~"'lr--"t:\ A An/- ne = 25.000, = 150.36,-'! ~ A* A*I'~&. ...... '¢. B = 18 degIf:.. ne

-~~............006-_ -_.t'-.~

LTheoretica1

A E DC· T D R-64-3

0.01

(Pc) 0.001P t min

0.000110 50

Pt

, psia

c. Data from Ref. 4

Fig. 16 Continued

38

100 500

l AEDC-TDR-64-3

a., I'.I I I

rTheoretical, I'

'~rc\~ Ane

= 10.962,AD ~

18.33, Bile = 19 deg

0..., A* A*

" ....,'..,.

,~

".~Theoretical _

fbn,

u, :ra.~ ,.o--.cr ,.L.Joo-"!J..., .n--

...-r-~p- -0---

IlAne = 10.848,AD

= 65.25, B = 18 degA* A* ne f-

A AD

1/-~ = 23.684, = 128.19,A* A*

.......Bile = 0 deg (Bell)

~po -I--_.A

""0..._ '-c,..P-

t.'Theoretica1

0.004

0.001

0.0001

0.0000410 50 100

P t , psia

500 1000

d. Data from Ref. 4

Fig, 16 Continued

39

AEDC·TDR·64·3

0.01

I I I AID I I I I-- 3.627.~ -"- 7.66,

A*

I I- 18 deg

Theoretical

I I I I

Theoretical

= 23.30, e = 18 degne

0.001

0.0001

I-------+-l..(---'.r-!\.--+-+--+--+--+--+-+-----+----t-- Theoret i c a 1-

'''ttY".-..x , A -...... ~ .'K;; I"'-' I I I I I

L~~ = 5.070, -- = 10.81, ene = 18 deg

>--~-<>-_.~~ At* I I I I A*

1------+-----;.-+---+---+ ne AD\ V-- --- = 10.848,.~~/ A* A*

""0...... ~ -0-lo-~Theoreticalt--LJ -

II------+---+---+----+-+--+-+ Ane ADJ.. " --- - 23.684, -- = 45.79,

F........,."'O"'-"'::-=-:n::-..-.-t,-',-t-r::---o--'..;;C~-::.~_;,-+--+-I·-+A* A*1..J""I,.Joo",~ J e

ne= 0 deg (Bell)

II-----+---+---+----+-R"fl-I""'H-t---+-p. -1---+--+--+-+---+-1

'10-"''0.."'0.

Theoretical

I I I I I

..

10 20 30 100

P t , psia

e. Data from Ref. 4

Fig. 16 Concluded

40

200 300