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AGARDograph No 163

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Supersonic Ejectors Edited by

J.J.Ginoux

DISTRIBUTION AND AVAILABILITY O N BACK COVER

AGARD-AG-163

NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT

(ORGANISATION DU TRAITE DE L'ATLANTIQUE NORD)

AGARDograph No. 163

SUPERSONIC EJECTORS

Edited by

J.J.Ginoux

Von Karman Institute for Fluid Dynamics 72 Chaussee de Waterloo

Rhode-St-Genese Belgium

The material in this book is an updated version of the Lectures given during Short Courses organized at the Von Karman Institute in April 1968 and March 1969.

THE MISSION OF AGARD

The mission of AGARD is to bring together the leading personalities of the NATO nations in the fields of science and technology relating to aerospace for the following purposes:

- Exchanging of scientific and technical information;

- Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture;

- Improving the co-operation among member nations in aerospace research and development;

- Providing scientific and technical advice and assistance to the North Atlantic Military Committee in the field of aerospace research and development;

- Rendering scientific and technical assistance, as requested, to other NATO bodies and to member nations in connection with research and development problems in the aerospace field.

- Providing assistance to member nations for the purpose of increasing their scientific and technical potential;

- Recommending effective ways for the member nations to use their research and development capabilities for the common benefit of the NATO community.

The highest authority within AGARD is the National Delegates Board consisting of officially appointed senior representatives from each Member Nation. The mission of AGARD is carried out through the Panels which are composed for experts appointed by the National Delegates, the Consultant and Exchange Program and the Aerospace Applications Studies Program. The results of AGARD work are reported to the Member Nations and the NATO Authorities through the AGARD series of publications of which this is one.

Participation in AGARD activities is by invitation only and is normally limited to citizens of the NATO nations.

Published November 1972

629.7.047.2:533.6.011.5

$

frinted by Technical Editing and Reproduction Ltd Harford House, 7 - 9 Charlotte St. London. W1P 1HD

PREFACE

Two Short Courses were organized on "Ejectors" at the von Karman Institute for Fluid Dynamics in April 1968 and March 1969, respectively, in which 72 scientists and engineers from seven NATO nations participated.

They were partly supported by AGARD and directed by Dr H.Uebelhack, Assistant Professor at VKI, now with Domier System in Friedrichshafen, Germany.

The objective of these Short Courses was to present a state-of-the-art review of the significant progress which has been made in the past few years in the design of high performance ejectors.

AGARD has felt it appropriate to publish the Short Courses as an AGARDograph in an updated version, with emphasis on supersonic ejectors. A small number of copies were originally printed at the time of these courses, but numerous subsequent requests for these notes were received and the demand could not be satisfied.

The two first parts, by Dr H.Uebelhack, cover the classical one-dimensional inviscid analysis and design methods for ejector systems with second throat diffusers. In the third part, Professor A.L.Addy presents an ejector flow model, developed at the University of Illinois, which significantly departs from the one-dimensional analyses. Mr D.Taylor, Assistant Manager, Facility Support Branch, Engine Test Facility of ARO, Inc., discusses in the fourth part the ejector design for a variety of applications. Finally, Dr C.E.Peters, Research Engineer at the Engine Test Facility of ARO, Inc. analyses ducted mixing and burning of co-axial streams.

The editor is most grateful to these lecturers who, in spite of a very heavy workload, have kindly agreed to revise and update their original notes.

Jean J.Ginoux Professor at VKI and

Brussels University. Editor

iii

Lecturers

Lecture Series Director Dr H.Uebelhack Dornier System Friedrichshafen Germany.

Professor A.L.Addy Mechanical and Industrial Engineering

Department University of Illinois at Urbana-Champaign Urbana, Illinois 61801 USA

Mr D.Taylor Assistant Manager Facility Support Branch Engine Test Facility ARO, Inc. Tennessee 37389 USA

Dr C.Peters Research Engineer Engine Test Facility ARO, Inc. Tennessee 37389 USA

Editor

Dr J.J.Ginoux Professor at VKI and Brussels University Head of VKI High Speed Department

CONTENTS

Page

PREFACE iii

LECTURERS iv

ONE-DIMENSIONAL INVISCID ANALYSIS OF SUPERSONIC EJECTORS by H.T.Uebelhack 1

ANALYSIS AND DESIGN METHOD FOR EJECTOR SYSTEMS WITH SECOND THROAT DIFFUSERS

by H.T.Uebelhack 17

THE ANALYSIS OF SUPERSONIC EJECTOR SYSTEMS by A.LAddy 31

EJECTOR DESIGN FOR A VARIETY OF APPLICATIONS by D.Taylor 103

ANALYSIS OF DUCTED MIXING AND BURNING OF COAXIAL STREAMS by C.E.Peters 165

ONE-DIMENSIONAL INVISCID ANALYSIS

OF SUPERSONIC EJECTORS

by

H.T.Uebelhack

Lecture Series Director Dornier System

Friedrichshafen, Germany

CONTENTS

Page

NOTATION 4

1. INVISCID ONE-DIMENSIONAL EJECTOR ANALYSIS 5 1.1 Conservation Laws S 1.2 The Supersonic-Saturated Regime 6 1.3 The Supersonic Regime 6

1.4 The Mixed Flow Regime 7

2. FRICTION 8

3. DIFFUSERS 8

4. MIXING CHAMBER LENGTH 8

5. EFFECT OF TEMPERATURE 9

6. EJECTOR DESIGN AND OPTIMIZATION 9

REFERENCES 10

FIGURES 11-16

a

A

Cf

D

f , q , z

F

L

HI

M

P

R

T

u

7

P

NOTATION

speed of sound

cross section

friction coefficient

diameter

dimensionless functions of M

thrust

mixing chamber length

mass flow

Mach number

pressure

gas constant

temperature

velocity

specific heat ratio

mass flow rate

density

Subscripts

the reference stations are shown in Figure 1

refers to the critical condition (M = 1)

Superscripts

a prime (') is used for the primary jet

a double prime (") is used for the secondary jet

ONE-DIMENSIONAL INVISCID ANALYSIS OF SUPERSONIC EJECTORS

H.T.Uebelhack

1. INVISCID ONE-DIMENSIONAL EJECTOR ANALYSIS

Experiments on ejectors have shown that a wide range of operating conditions of an ejector can be described by fundamental conservation laws. In this analysis the geometry of the ejector is considered known (Fig. 1) and the calculation yields a set of equations for pj,/p4 , p'd/p4 and the mass flow rate n .

One distinguishes in general two regimes of ejector operations which are termed by Fabri1 as the supersonic regime and the mixed flow regime. The flow pattern in the ejector for these regimes is shown in Figure 2. The supersonic regime is characterized by a certain part of the flow in the mixing chamber (which is acting at the same time as a diffuser) being supersonic over the whole cross section. In this regime the exit pressure p4 does not influence the performance characteristics (p'0,Po,i").

This regime can be subdivided into: (i) the supersonic-saturated regime, where Mj is sonic,

(ii) the actual supersonic regime, 1 > M2' > 0.3 , (iii) the supersonic regime with low secondary flow 0.3 > M'2' > 0 (or n * 0). In this regime viscous effects

cannot be neglected and the present analysis gives poor results when n approaches zero. The limit between (ii) and (iii) which is indicated by M2' ~ 0.3 is rather arbitrary. This value has been found as an average in a series of experiments.

In the mixed flow regimes there exists a subsonic region ofthe flow pattern between the secondary flow settling chamber and the exit of the diffuser (mixing chamber). The exit pressure p4 thus influences the pressure pJJ in the secondary settling chamber. The value of pJJ increases when p4 is increased and vice versa (for p. = constant). Above a certain value of p'2' the flow in the primary nozzle will separate and thus cause the thrust of the nozzle to decrease and influence the whole characteristic performance. This regime, however, is of little practical interest and will not be discussed here.

In the following analysis, stagnation enthalpies of the primary and secondary jet are assumed to be equal.

1.1 Conservation Laws

Continuity Equations The mass flow in a duct is given by

, aOPo FT* P* . ,w a

11 - M T 7 P T M ( M * ) ' J

(1)

where

is a dimensionless function of the Mach number and the specific heat ratio only (Fig.3). It has been found useful for numerical calculation to introduce dimensionless functions of M# .

Momentum Equation The momentum equation can be represented by the balance of jet thrusts in different sections of the ejector.

6

The vacuum thrust is defined by

F = p A + p u 2 A = p 0 A ( l + 7M2) Po

or F = p0Af(M*)

where

'*) = (i + M i ) ( 1 - ^ T M i ) The quantity F can also be expressed in terms of the mass flow m , using Equation (2).

A A M . . . f(M*)

The ratio

where

.1/(7-1) p

and ( - 1 represents U + \) Po Using these relations in the above equation yields

F = Po A*Z(M#) Po

7 + 1 F = a^mZ(M4)

27

(3)

,1/(7-1) f(M*) = (1 + M h (1 - . 1 - 1 MJ . (4)

F = Po T - - V ( M * ) = p0A* = - . (5) A * q(M*)

f (MJ / 2 \ ! / (7-l)

Z(M,) = M* + T J - (7) M

*

IN,

1.2 The Supersonic-Saturated Regime

The performance characteristics in this regime can be determined from the condition of sonic velocity at Station 1 for the primary jet and Station 2 for the secondary jet only.

The relation between the mass flow rate

Introducing the mass flow rate n = m"/m' and the condition M^e = 1 , Zj = 2 , Equation (10) takes the form

/-2

The mass flow rate n can also be expressed by Equation (1),

rh" p"A'' q(M") m p0 A, q(M*,)

Solving this equation for PQ/P0 and noting that q(M^,) = 1 yields

Po 1 A" Po " A i

Equations (11) and (13) represent the solution for the supersonic regime. They relate PQ/PJ, and (i through the common variable M^e .

In Equation (11) Z2 is determined by a chosen M^e (0 < M^2 < I). The primary nozzle exit Mach number determines Z'2 . The secondary jet cross section at e is given by Ae' = A2q(M2) (continuity) and thus Ag = A3 Ajl is known. The area ration Ae/A', determines the primary Mach number Ml^ e and Ze .

From Equation (13) p0'/p0 i s found directly. The whole supersonic regime can be determined by repeating this procedure for all subsonic Ml^ 2 .

In the diagram in Figure 5 both the supersonic and supersonic-saturated regime are represented by straight lines passing through the origin of the two coordinates (p0/p4,Po7p4), i-e. there is no dependence on p4 . The slope of these straight lines, pjj/pj,, is a function of pi and the actual geometry, A2/A! and A3/A, , only.

1.4 The Mixed Flow Regime Increasing the exit pressure p4 of the ejector reduces the length of the fully supersonic part of the flow field

in the mixing chamber until it breaks down (Fig.2.3). Then the flow in the exit of the primary nozzle and in the center of the mixing chamber is still supersonic whereas the flow near the mixing chamber walls is subsonic. For most cases, the supersonic jet attaches to one side of the wall (Coanda effect). The secondary flow stagnation pressure in this regime is a function of the exit pressure p4 .

The characteristics of this mixed flow regime can be determined by the continuity and momentum equations applied between Stations 2 and 4.

The momentum equation (no friction) yields

p0A'2f(M;2) + p X f ( M ; 2 ) = PonA4f(M*4) . (14)

The continuity equation reads

m' + m" = m4

or p0A'1q(M; i)(l + p) = p04A4q(M*4) . (15)

Together with the definition of /u ,

" = S" = r 7 ^ q ( M ^ ' 02a) m p0 A,

the Equations (14) and (15) display the relation between p0/p4 and PQ/P4 for a certain mass flow rate. Here also M+2 is the parameter relating p'o/p'0 and n . To solve the above equations an arbitrary value for pj, is assumed, p'o then is found from Equation (12a). Equation (14) can be solved for p04f(M*4) (introducing the same value of M^j). In the same manner p04q(MA4) is found from Equation (15). The exit Mach number MA4 is then determined by

P0 4f(M,-- / 2 \l/(7-D p04 q ( M , j - ( x h ) Z(M"'-

The function Z(M#) is double valued. The subsonic solution has to be chosen in this case. The resulting procedure to determine p0 4 , pJJ/p4 and p0/p4 is straightforward.

8

If this calculation is repeated for various M^2 the whole characteristic line for the mixed flow regime and for a constant ii can be determined (Fig.5).

2. FRICTION

In the supersonic regimes, friction does not greatly influence the ejector performance since the distance over which the secondary flow is accelerated to supersonic speed is rather short (of the order of two diameters).

In the mixed flow regime, however, friction at the mixing chamber walls influences the characteristic line of this regime. If friction is considered in the theoretical calculation. Equation (14) should read

F'2 + , F ^ ' - F F = F4 , (17)

where FF can be represented approximately by

pL M2 dx FF = C f 7 P A -2 D

In many cases it is justifiable to simplify Equation (18) by introducing the exit Mach number M4 . Figure 7 shows the effect of friction on the characteristics of the mixed flow regime.

In other words, the pressure losses due to friction reduce p4 and therefore increase both pressure ratios p 0 /p 4 and P0/P4 The characteristic line of the mixed flow regime is shifted as shown in Figure 7.

3. DIFFUSERS

In the preceding calculations the effect of the subsonic diffuser was not taken into account. The best possible influence of a subsonic diffuser following Station 4 has already been given in the calculations in Section 1.4, where p 4 and p 0 4 were determined through M^4 . In the ideal case, p s would equal p 0 4 (100% diffuser efficiency). In most practical applications MA4 varies in the narrow range between 0.45 and 0.65 only. Assuming a mean exit velocity of M# 4 = 0.55 and a diffuser efficiency of 75%, we would get an improvement in pressure ratios of 15%. This value was found as a mean value in many experiments. The improvement of the mixed flow characteristic is shown in Figure 8 for a p = constant line.

The pressure recovery in the mixing chamber can still be increased by using a second throat (Fig. 1). Here, the important parameters are the contraction ratio A3/A3< and the position of the second throat with respect to the primary nozzle exit, X/D3 .

It was found3 that an additional improvement in pressure recovery of the order of 30% (p 0 /p 4 ,p 0 /p 4 ) can be realized in the transitional regime between mixed flow regime and supersonic regime (Fig.8) by a second throat. The influence of a second throat will be discussed in detail and a calculation method will be presented in the following chapter of this Agardograph.

4. MIXING CHAMBER LENGTH

The effect of the mixing chamber length on the ejector characteristics is shown in the experimental curve in Figure 9. The influence of friction has already been shown in Figure 7. Using schlieren photography, one can observe that at transition from the supersonic to the mixed flow regime a minimum length of the mixing chamber is required in order to get approximately a uniform flow and the exit pressure p4 . Figure 10 shows the typical pressure distribution in the mixed flow regime, very close to transition to the supersonic regime.

From Figure 10 it can be concluded that for this particular ejector configuration and mass flow rate the optimum mixing chamber length is (L/D) o p t = 12 . For larger values of L/D , the exit pressure decreases due to friction. At lower L/D than (L/D) o p t it is not possible to reach (p 4 ) o p t . The supersonic regime breaks down sooner. This effect is more drastic than the effect of friction (Fig.9). It is therefore advisable to choose a L/D > (L/D) o p t .

The main parameters which determine the optimum length of an ejector are: (i) the area ratio A3 /A, which determines the initial Mach number. At higher Mach numbers (or higher

A3/A,) the shock waves angles are smaller and the whole supersonic flow pattern (i.e., at transition to the mixed flow regime) is stretched out. A larger L/D is required.

(ii) the mass flow rate p . Larger mass flow rates smooth the above-mentioned effect. This is, however, an upper limit of pi for the addition of a second throat.

(iii) the nozzle outlet angle. Larger nozzle outlet angles provoke a stronger interaction of the primary jet with the secondary jet and the wall. The resulting shock waves are stronger and steeper. A lower (L/D)o p t is therefore to be expected.

5. EFFECT OF TEMPERATURE

The ejector performance characteristics were derived in Section 1 for equal stagnation enthalpies of both jets. In the case of different stagnation enthalpies the energy equation has to be used in order to determine T 0 4 at the exit.

In the calculation for the mixed flow regime a complete mixture of the two jets must be assumed.

In calculating the characteristics of the supersonic regime the additional assumption of no heat exchange between Stations 2 and e is necessary.

The simplest way to show the temperature influence is to look at the supersonic-saturated regime.

From Equation (I) it follows that

p o A . . in = f= x constant .

V T o

Thus, for different T 0 , Equation (9) yields u = ^ lo

A; P ^ T ; ;

fn A2' pi (19)

The characteristics derived for equal stagnation temperatures TJ, = TJJ can be used when the parameter p is replaced by pTjJ/Tj, . The calculation of the supersonic regime yields the same result.

In References 4 and 5 the theoretical calculations and experimental verification are presented. Figure 11 shows the influence of TjJ/Tj, on the supersonic regimes.

6. EJECTOR DESIGN AND OPTIMIZATION

The given data for the design of an ejector are mostly the total pressure of the secondary gas p , its mass flow rh" and the ambient pressure p s . The design problem is to determine and to optimize the geometry of the ejector and the mass flow rate n .

The form of the equations in Section 1 shows that an optimization of the ejector geometry, i.e. to find the most economic geometry for the data given above, cannot be found easily.

In many cases, however, the mass flow rate is limited or fixed to a rather narrow range. The calculations there-fore have to be performed for one or two values of n only. Figure 5 shows that the optimum operational conditions are at transition from the supersonic to the mixed flow regime. The point of transition should be considered as the design point. It has to be recalled that it is displaced when a second throat or a subsonic diffuser is added.

It can also be concluded from the equations in Section I that the parameter A4/A', (respectively A3/A',) has the greatest influence on the position of the transition point.

An easy approach to the desired optimization is to choose two limits for p and to calculate the characteristics ju = constant. If the best combination of p and A4/A", is found, the geometry can still be improved by varying Aj/A', and by adding a supersonic-subsonic diffuser.

The primary mass flow rh' is determined by p . The total pressure p'0 at transition then determines the primary throat cross section A', and thus the whole ejector geometry.

10

The problem of the optimization of ejectors is discussed in detail in Reference 6. In the case of very low required pressure ratios PQ/P 5 a two stage configuration might be considered. The characteristics for a two stage ejector are developed and compared with the single stage ejector characteristics in Reference 7.

Finally Reference 8 should be mentioned; this contains a list of almost all papers on ejectors published before 1965.

REFERENCES

1. Fabri, J. Paulon, J.

2. Lukasiewicz, J.

3. Uebelhack, H.

4. Leistner, G.

5. Le Grives, E. Fabri, J.

6. Calvet, P.

7. Loser, H.

8. Seddon, J. Dyke, M.

Theorie et experimentation des ejecteurs supersoniques air-air. ONERA NT 36, 1956.

Supersonic Diffusers. ARC R & M 2501, 1946.

Supersonic Air-Air Ejectors with Second Throat Diffuser. VKI TN 28.

Experimented und theoretische Untersuchungen an einem Hochtemperatur-Uber-schallejektor mit zylindrischer Mischkammer. Ph.D. Thesis, T.H.Darmstadt, 1966.

Divers regimes de melange de deux flux d'enthalpies d'arret diffirentes. ONERA T P 4 1 1 , 1966.

Performances d'ejecteurs supersoniques air-air a milangeur cylindrique. Revue Generale de Thermique, Jan. 1965.

Untersuchungen an ein- und zweitstufigen Uberschallejektoren mit zylindrischer Mischkammer. Ph.D. Thesis, T.H.Darmstadt, Germany, 1965.

Ejectors and Mixing of Streams. RAE Library Bibliography No. 252.

I I

NOZZLE 3 3 '

AIR SETTLING CHAMBER

SUPERSONIC DIFFUSER (SECOND THROAT)

MIXING CHAMBER

Fig. 1 Diagram of a supersonic ejector and reference stations

FIG.2.1 SUPERSONIC REGIME NORMAL SHOCK WAVE IN THE DIFFUSER

PC PPPx=* -i A

SUPERSONIC PORTION

FIG. 2.2 SUPERSONIC REGIME CLOSE TO TRANSITION

FIG. 2 3 MIXED FLOW REGIME

/0>^>c FIG. 2 4 MIXED FLOW REGIME SUPERSONIC JET

ATTACHED TO ONE SIDE

FIG. 2 5 MIXED FLOW REGIME WITH SEPARATION IN PRIMARY NOZZLE

Fig.2 Flow patterns in a supersonic ejector. Rising exit pressure from top to bottom

12

f(M*) "(Mm)

1.2

8

4

q ( M m )

Jtoti

5.0

Z(M)

4.0

3.0

2.0

1.0

.4 .8 1.2 1.6 2.0 2.4 /V.

Z ( M * )

0 .4 .6 12 1.6 2.0 2.4 M +

Fig.3 Dimensionless functions of M#

Fig.4 Supersonic regime

13

Fig.5 Ejector performance characteristics

w* 2

1.0

0.8

0.6

0.4

02

EXPERIMENTAL A / J .

/V

&

* 2

o o A

o A

1 A4

A j ' 6 2 5 1

Am

A]

A / /

* = 2.78

0

v ^

o s f / r

i ^ S ^

o

G > a ^

^ O

>REF (1)

.3

0 0.01 0.02 0.03 0.04

Fig. 6

0.05 0.06

Supersonic regime

0.07 006 0.09 Po ' Po

14

P "

.3

.2 EXPERIMENTS A L/D = 13

a L/D = 17

Fig.7 Effect of friction influence

10 P_ P4

Pc p*

5

4

.3

2

.1

0

Po 1 . p~ WITH \

s SUBSONIC \ DIFFUSER \

SUBSONIC \ \ DIFFUSER \ }

ANL SECON

THRO

3 \ ,D \

A \

A M 4' ,

A 2

\ A 1

y^

6.25

2.76

^

tf

P<

Fig.8 Influence of a subsonic diffuser and a second throat

15

6

4

2

A-,. -Tr- =6.25 A i

A ' l A ' I

U P b * A

- =2.78

S^/y* S . / *

A * ' S

X

/ f P EXPERIMENTAL ^ 0 L /D * 6

A L /D= 7 0 L /D = 6 V L / D = 9

10 12 Pa'

Fig.9 Influence of diffuser length

2 4 6 8 10 12 14

Fig. 10 Pressure distribution in the mixing chamber, mixed flow regime

76

16

M

0.5 - A1 -

0.4

0.3

0 2

01

0

M j m . 2 *

0.0 PQ_

Fig. 11 The influence of TQ/TJ, on the supersonic and supersonic-saturated regime, from Reference 4

17

ANALYSIS AND DESIGN METHOD

FOR EJECTOR SYSTEMS WITH

SECOND THROAT DIFFUSERS

by

H.T.Uebelhack

Lecture Series Director Dornier System

Friedrichshafen, Germany

18

19

CONTENTS

Page

NOTATION 20

1. INTRODUCTION 21

2. THE EJECTOR CHARACTERISTICS DIAGRAM AND THE

INFLUENCE OF A SECOND THROAT DIFFUSER 21

3. FLOW MODEL 22

4. ANALYSIS 22

5. THE SECOND THROAT CONTRACTION RATIO 24

6. THE EJECTOR STARTING PRESSURES 24

7. DESIGN PROCEDURE 25

8. COMPARISON WITH EXPERIMENTS 25

REFERENCES 26

FIGURES 26-30

20

a

A

D

h

L

m

M

P

T

x

y

7

e

M

p

NOTATION

speed of sound

cross-section

diameter, drag

step height

diffuser length

mass flow

Mach number

pressure

temperature

longitudinal coordinate

lateral coordinate or step coordinate

specific heat ratio

separation angle

mass flow ratio

density

Subscripts

*

stagnation conditions

before separation

behind separation

behind reattachment

diffuser outlet section, Figure 3

throat section

second throat section

nozzle outlet section

separation, starting (ejector)

Superscripts

a prime refers to primary jet a double prime refers to secondary jet

21

ANALYSIS AND DESIGN METHOD FOR EJECTOR SYSTEMS WITH SECOND THROAT DIFFUSERS

H.T.Uebelhack

1. INTRODUCTION

A second throat diffuser reduces the driving pressure as well as the suction pressure of an ejector system by up to 30% as compared to the diffuser system with a constant cross-section diffuser. In order-to benefit from this improvement in performance the improved starting characteristics of a second throat ejector must be known. Only then can the whole ejector geometry, in particular the diameters in the various sections, be adapted to certain requirements.

The performance characteristics (i.e. Pp/p4 as a function of pj,/p4 and M) can be determined by inviscid one-dimensional theories1,2 and in the lower secondary mass flow regime by additional considerations of viscous interactions3. The difficulty in calculating the improved performance characteristics of a second throat ejector system consists in determining the pressure integral over the contraction part of the supersonic (second throat) diffuser which is required in the momentum balance.

Efforts have been made in the past decade at AEDC, Tullahoma, Tennessee, USA, to develop prediction methods for the zero secondary flow ejector systems4,5. Extensive experimental studies at VKI, Rhode-Saint-Genese, Belgium, on this problem have revealed two features of the second throat ejector system which can be used in developing a prediction method for these systems:

(i) The ramp angle of the contraction portion of the supersonic diffuser is of negligible importance to the overall ejector characteristics. Angles up to 90 were investigated. The 90 step type contraction, therefore, can be used in defining a flow model for the second throat system. Results of studies of the supersonic separated flow field in front of s teps 6 - 8 , in particular the drag of the step, can be introduced into the analysis.

(ii) The starting pressure ratio (p'0/p4)s is not affected by small secondary mass flows. The starting pressure ratio which is at the same time the pressure ratio for the design and operation of a second throat ejector system, can therefore be determined for zero secondary mass flow. This ratio remains constant for secondary mass flow ratios up to n ~ 0.25, which is also the limit for second throat diffuser operations.

In the following paragraphs a flow model based on the above observations will be defined which permits the determination of the starting and operating characteristics of a second throat ejector system. The working equations will be derived briefly and the design procedure is described. The limits of application are discussed and comparisons with experiments are made.

2. THE EJECTOR CHARACTERISTICS DIAGRAM AND THE INFLUENCE OF A SECOND THROAT DIFFUSER

The representation of the ejector performance characteristics will be the same as the one utilized in the one-dimensional analysis in Chapter 1 of this Agardograph. For a better understanding of the final results, the influence of different diffuser shapes should be briefly demonstrated in a schematic performance characteristic diagram.

In Figure 1 the performance characteristics for an arbitrary ejector system and one mass flow ratio, n , is shown. The branch marked 0 represents the supersonic regime. The exit pressure has no infleunce on the secondary pressure pj| in this operational mode. The mixed flow regime, where the secondary pressure is usually highly influenced by the exit pressure is represented by the branches 1, 2 and 3 for various diffuser shapes. The curve 1 would be typical for a constant cross-section diffuser without, and the curve 2 with, a subsonic (divergent) diffuser part. A contraction of the diffuser pipe will act as a supersonic diffuser and will deliver a characteristic line shown as curve 3. The branch marked 4 would be the mixed flow regime with separation in the driving nozzle, an operational mode of merely academic interest.

In the following representation only the intersection points between the mixed flow and the supersonic regime will be shown for various mass flow ratios, n . Those starting points give at the same time design and operational condition. For given geometry and mass flow they represent an optimum (minimum pJJ) operation.

22

3. FLOW MODEL

As mentioned earlier the flow model used in the following calculation is based on two features of second throat diffusers which have mainly been found experimentally:

(i) the angle of the contraction part of a second throat diffuser has no influence on the overall starting and operational conditions of the system. The experimental results of flows over rectangular front steps can thus be introduced in the momentum balance during the analysis.

(ii) the starting pressure ratio of the driving gas is for second throat systems, in contrast to constant cross-section diffusers, independent of the secondary mass flow. The analysis to determine the driving pressure of a second throat system carried out for zero secondary mass flow, therefore, is also valid for operations with secondary mass flow up to n ca 0.25 .

In addition the following assumption and methods will be introduced for the calculation of the starting character-istics of a second throat system:

The calculations of the base pressure p^ , the initial pressure p, , initial Mach number M, , and finally the step pressure integral are carried out successively.

(iii) The base pressure pJJ can be determined in several different ways: (a) By means of Korst's base pressure theory, i.e. the combination of the shear layer characteristics with

an appropriate reattachment criterion (Ref.3). (b) Higher secondary mass flows are treated by the one-dimensional ejector theory presented in Chapter 1

of this Agardograph. (c) Conical nozzles of 10 to 20 half-angle produce a base pressure (with zero mass bleed) which is very

close to the static pressure of an isentropic expansion from stagnation conditions to the diffuser area ratio, Ad/A* (Fig.2).

(d) Contoured nozzles of zero outlet angle produce an appreciably lower base pressure which can be taken from Figure 2 for various geometries (experimental results).

(iv) The jet boundary Mach number is determined by the isentropic two-dimensional expansion from stagnation to base pressure at zero secondary flow.

(v) The flow angle of the jet boundary is given by a two-dimensional Prandtl-Meyer expansion at the exit of the nozzle.

(vi) The initial pressure p, and the initial Mach number \S\, are then calculated by the two-dimensional oblique shock equations. The geometrical parameter, A,j/Ae , which determines the relative length of the jet boundary, must be less than four in order to avoid curvature of the jet boundary due to the axisymmetric nature of the flow. In practical applications A(j/Ae is not larger than two. The step pressure integral will be calculated by the experimentally obtained relation7'8

. f t * J Pih

= 1.1 M . (1)

4. ANALYSIS

In the following analysis the major aim is to determine the one-dimensional supersonic Mach number in the section after the contraction. The pressure recovery then can be determined by normal shock equations.

Apart from two-dimensional considerations in the region between the nozzle exit plane and the contraction plane, the calculation is strictly one-dimensional.

23

Conservation Laws

The momentum equation applied over a control volume (Fig.3) between the nozzle exit (subscript e) and the second throat contraction (subscript **) yields

p j ^ ( 1 + 7 M | . ) + 4 A d ' A e - f m % Po A* Po A*

s t J p p ,h p 0

A d ; A** = f S S - ^ ^ S d + T M l U ) . (2)

A* Po** Po A*

the equation having been normalized by p'0A*. The calculation will be carried out as stated in Section 3 for zero secondary mass flow.

Equating the mass flows in the primary nozzle throat section and in the second throat section yields

u** _ P* A+ a* P** A**

The ratio u ^ / a # in this equation represents the "star" Mach number in the second throat section, ( M A ) ^ . Equation (3) can be further transformed into

(3)

(M*)** = ? P* Po Po** A* Po Po** P** A** ' (4)

where p'0 is the density in the primary settling chamber (pressure pj,) and p 0 * * P0**/RT0 . The density ratios can be expressed by the local Mach number through the energy equation and isentropic flow relations

P j P ' & $

2 \ 7 - p n * *

P** -O*2^-*-) 1/(7-0 (5) The state equation yields for adiabatic flow, T0 = Tj , ,

Po**/Po = Po**/Po

Introducing Equations (5) and (6) into Equation (4), we get

(6)

2 y/(7-D / 7 _ i y/(7-D K ^ Mi

/ 2 y/w-u / 7 - 1 v A** Po**

(7)

The relation between the Mach number MAA in the second throat and the "star" Mach number (M^)+ + based on the critical speed of sound, a*, is

11/2 (M*) I 7+ 1

7 - 1 . + 2 I (8)

7 - 1 M ^ j

Introducing Equation (8) into Equation (7) and solving for A ^ p J , ^ / A # p J , yields

Po** A ^ _ 2 \ l / ( 7 - D / T - l ' 1 + M**

1/(7-0 Po A # \ y + 1

When Equation (9) is introduced into Equation (2) and P**/p0** is replaced by

7 - . / 1 + 2 7 + 1 V" ( 7 - DMJ,

1/2 (9)

* * . - f i - f 7 (.^n.) 1 \(T-D/7 MJ (10) Po** V 2 the right-hand side (RHS) of Equation (2) becomes a function of the Mach number in the second throat section only:

RHS = 1 + / _ _ , \ ( 7 - 0 / 7 / 2 \* / (7 - l ) /

7 - l \ l K f l ) y - 1 7 + 1

1 + (7-DMJ 1/2

(l+7Mi*)-(2a)

This form of Equation (2a) enables one to determine the Mach number M*A , which itself permits the calculation of the ratio of the total pressures, P0**/Po T h e M a c h number M4 in the exit section and the pressure ratio p 0 **/P 4

24

can then be calculated by normal shock equations. This finally enables one to calculate the starting pressure ratio of the second throat ejector system, (PQ/P4)S by

(Po/PA = Pot Po P4 P

(11) 0 * *

5. THE SECOND THROAT CONTRACTION RATIO

Experiments have shown that the decrease of the ejector starting pressure, (pj,/p4)s of a second throat system is approximately proportional to the second throat contraction ratio A ^ / A j (Refs.9,10). It is therefore desirable to know and to work with the lowest value of J ^ ^ / A J for which starting is possible.

The limit of the contraction ratio, A ^ / A j , can be calculated by one-dimensional flow equations and the following argument:

When starting the ejector system (increasing pj,/p4) a normal shock wave moves downstream.

/ / / / / / / / / / / /

M r f>/ M < l

/ / / / / / / / l / / / / / J J / y / / / / / / / / ? / - /

The mass flow in the system must be swallowed by the second throat. The limiting contraction of the second throat is thus dictated by the necessary re-acceleration of the subsonic flow behind the shock wave to sonic velocity. A further decrease of the second throat section would cause choking and would make it impossible to start the second throat system. The contraction limit turns out to be a function of the supersonic Mach number before the second throat only:

- i V / 2 / 27 V/Cr- i ) / 2 1 \ " 2 / 7 - 1 1 \i/(7-D 1 + r 1 f l - 7-~ \ . (12) *** = (i^iy

2(ji_\^- l ) (,, 2 _Lv2 A 7-1 IV Ad \ 7 + l / V 7 + 1 / V 7 - 1 My V 27 M2J , 7 + 1 / \ 7 + l / V 7 - l M y V" 27

For Mach numbers Mj approaching infinity the contraction ratio approaches asymptotically to A#A/A,j = 0.6 for 7 = 1.4 .

The lower limit of the contraction ratio /V^/A,- was also determined experimentally9'11. It is shown in Figure 4 as a function of the nozzle exit area ratio Ae/A# . Because of the three-dimensionality of the flow during the starting process (flow separation, oblique shock waves), this limit was found to be lower than that obtained from a one-dimensional calculation. For high expansion rates (Ae/A* > 10) it is practically constant and takes the value A^ /A d = 0.47 .

6. THE EJECTOR STARTING PRESSURES

The ejector starting pressures pj,/p4 and (pj,/p4)s were computed by the one-dimensional theory and by the present method (outlined in Section 3) for two second throat contraction ratios: that given by Equation (12) and the experimental limit of Figure 4. The starting secondary pressure ratio pJJ/p4 has been determined by the one-dimensional theory.

In Figure 5, the starting pressures of the constant cross-section diffuser and the second throat diffusers are plotted as a function of the diffuser inlet area ratio Aj/.A* and compared. It is evident from this figure that it is not worthwhile to use a second throat diffuser at area ratios smaller than Ad/AA = 8 . At area ratios above Ad/A* = 20 the benefit (i.e. the reductions of the starting pressure) obtained by a second throat diffuser remains approximately constant.

Figures 6 and 7 show the ejector starting pressures for various geometries in the ejector characteristics diagram and Figure 8 represents a composite of these diagrams. They enable the choice of a tentative ejector geometry (Ad/A*, Ae/j\* and A^/Aj) which meets certain prescribed requirements (pj,/p4, PQ/P4, /-") The exact calculation of the ejector characteristics of the chosen geometry can then be carried out as indicated in Sections 3 and 4.

25

7. DESIGN PROCEDURE

After fixing the ejector geometry (Ag/fi^, ^/.A*, A^/A^) which meets the presented requirements by the use of Figure 8 and the calculation procedure of Sections 3 and 4, and after fixing the mass flow ratio /i , the size of the throat section A* will be determined. The other geometric parameters are then given by the above area ratios.

Restrictions: It is evident that the prediction method developed (Sections 3 and 4) is only valid for area ratios Aj/Ae < 4 .

Larger area ratios yield a flow field which can no longer be treated with two-dimensional equations as suggested in Section 3. However, in practical ejector design problems area ratios A,j/Ae > 4 are not required.

The distance between the ejector driving nozzle and the location of the second throat contraction part should be designed to be adjustable. A too long or too short distance bteween them might produce hazards in ejector operations by either breakdown of the flow field or shock wave interference.

8. COMPARISON WITH EXPERIMENTS

Comparisons of the results of the present theory with experimental data are made in Figures 6, 7 and 9. Figure 6 shows excellent agreement of the experimentally obtained starting pressure ratios with the predicted ones at a contraction ratio of .A^/Ad = 0.68 . The experimental starting pressure ratios of the constant cross-section diffuser system of Figure 6 are by about 5% higher than the values predicted by the one-dimensional theory.

The contraction of the second throat was realized by a step in this case. Figure 7 shows a comparison between calculated and measured starting pressure ratios for an ejector system with a nozzle of 15 half-angle and a 12 ramp as diffuser contraction. Here again excellent agreement between the predicted and the measured second throat starting pressure ratios can be observed. The experimental starting pressure ratios of the constant cross-section diffuser system are again 5% higher than the predictions of the one-dimensional theory.

The difference between the experimental starting pressure ratios and those obtained by the inviscid one-dimensional theory can be attributed to the effect of friction, which has been ignored in the theory. The presence of friction shifts the characteristics of the mixed flow regime and thus the starting pressure ratio to higher values of P Q / P 4

In the case of a second throat diffuser system the diffuser inlet Mach number is immediately reduced and friction becomes less important. The experimental values, therefore, correspond with those predicted by the present theory, which also ignores wall friction.

The starting pressures of the second throat systems are compared with the starting pressure of the constant area diffuser system in Figure 9.

The lines indicate the theoretical predictions of the reduction in starting pressure ratio, i.e. the starting pressure ratios of the second throat system divided by the starting pressure ratios of the constant cross-section ejector.

Both circles represent experimental values obtained in the present study (experimental starting pressure ratio of the second throat system over theoretical starting pressure ratio of the constant cross-section ejector. The theoretical starting pressure ratios of the constant cross-section ejector have been taken as a common reference here since the experimental starting pressure ratios of the constant cross-section ejector were influenced by wall friction and are 5% too high).

The two AEDC data are the ratios of measured starting pressure ratios (squares) and the lines are again the ratio of the two calculated values (second throat over constant cross-section). In both the experiments and the theoretical calculation p4 is the total pressure at the diffuser exit (a subsonic diffuser was used in the experiments).

The geometric parameters are indicated at each data point. The very good agreement between the experiments and the present theory confirms again that all important factors have been taken into account.

26

REFERENCES

1. Fabri, J., Siebstrunck, R.

2. Uebelhack, H.T.

3. Chow, W.L., Addy, A.L.

4. Panesci, J.H., German, R.C.

5. German, R.C., Panesci, J.H.

6. Zukoski, E.E.

7. Uebelhack, H.T.

8. Uebelhack, H.T.

9. Taylor

10. Uebelhack, H.T.

11. Oiknine, C , et al.

Supersonic Air Ejectors, Advances in Applied Mechanics. Vol.5, 1958.

One-Dimensional Inviscid Analysis of Supersonic Ejectors. Chapter 1 of this Agardograph.

Interaction between Primary and Secondary Streams of Supersonic Ejector Systems and Their Performance Characteristics. AIAA Journal, Vol.2, No.4, 1964.

An Analysis of Second Throat Diffuser Performance for Zero Secondary Flow Ejector Systems. AEDC TR 63-249.

Improved Method for Determining Second Throat Diffuser Performance of Zero Second-ary Flow Ejector Systems. AEDC TR 65-124.

Turbulent Boundary Layer Separation in Front of a Forward Facing Step. AIAA Journal, Vol.5, No. 10, 1967.

Theoretical and Experimental Investigation of Turbulent Supersonic Separated Flows over Front Steps. 4. Jahrestagung DGLR, Nr. 71-076, October 1971.

Turbulent Flow Separation ahead of Forward Facing Steps in Supersonic Two-Dimensional and Axisymmetric Flows. VKI, TN54, 1969.

Ejector Design for a Variety of Applications, von KarmSn Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, Short Course on Ejectors, March 17-21, 1969.

Supersonic Air-Air Ejectors with Second Throat Diffuser. VKI TN 28, August 1965.

Etude des Ejecteurs Supersoniques. Congrs de I'Association Francaise des Ingdnieurs et Techniciens de I'Aeionautique et de 1'Espace, November 1969.

P-xit

P'~

STARTING PRESSURE RATIOS

0 /Peti t

Fig. 1 Ejector characteristics diagram. Influence of subsonic and supersonic diffuser parts

27

1r-r ONE-DIM EXPANSION P/R.

EXPERIMENTS CONTOURED NOZZLES

-10

AEDC -DATA, REF CONICAL NOZZLES

Fig. 2 Ejector base pressures

* ' ( ** -* . )

(**%,*)*.

L.

SpdA

fox* **A.U'**)A*

..J

(t* 4

28

1.0

A*0 Ad

.75.

.50

ONE DIMENSIONAL STARTING LIMIT

STARTING POS/BLE

.30-

STARTING IMPOSSIBLE

to 20 30 40 Ad/A ^

Fig.4 Experimental starting limit

20

10

ONE-DIM. THEORY Ad/Ae = 1.5

NORMAL SHOCK RECOVERY

2ND THROAT ONE-DIM. CONTRACTION

UMIT.A

29

I Pjs

.3

EXPERIMENTS

THEORY

/u. = IS M - -10 4 L - OS

- - E T -- p -- v -

-_ o -

A^lAd' r i f

' A d / A m M 2S

Ae/Am, = 16

NO SEC. THROAT A / A d . .68

;o fsv-u Fig.6 Ejector starting characteristics

(f). <

EXPERIMENTS AA.'.20 A*.'.15

'M. - .10 /A.' .05 M." 0

- . --ET-- -

- V -o-

THEORY

i r >t(//j4^75 A / A * . 12.5

A * * / A d * .67

15* NOZZLE

NO SEC. THROAT

A * * / A d * 6 7

10 (Po'/P*),

Fig.7 Ejector starting characteristics

30

(Po) P4S

.4

.3

-?

.;

0

1 1 i

A d / A . * 9 A e / A * . 6 . 2 S

- A * / A d = l .68 .47

I 0 y

.05^,

0

1 1 1

Ad/A * . 1 6 Ae/A * ' 9

A * . / A d 1 .66 .47

. 1 5 ^

0 5 ^ -

0 1 1 Li

I I I I A d / A * =25 A ' / A * *16

A * * / A d =1 .63 7

\ iS*-"""

\ _ ^ 5 _ _ ^

WJSO

1 1 1

Ad/Arrr.36 A t / A n - 2 5

A * * / A d = 1 .64 47

1 i 1

t j ^ - \

_Jg^

_J75

05

! 1 1

\

0 \

10 75 20 25 (Po/P+ln

Fig.8 Composite of ejector starting characteristics

(P0lp4)**s fWs

1.0

8

6

, i

2

0

1 AEDC EXPERIMENTS, REF.

PRESENT EXPERIMENTS rH0RY

sJOZZLE HALF ANGLE = 15 A . / A m = I2.S

A K K f A d m 67 O AmjAm 16

A x x f A d ' .68 , 8

n A e l A * A X K I A d = .5

Ae /Am = 25 A * * / * d = S

18

to 20 30 40 SO A d J A A

Fig.9 Experimental and theoretical ejector starting pressure reduction

31

THE ANALYSIS OF SUPERSONIC EJECTOR SYSTEMS

by

A.L.Addy

Mechanical and Industrial Engineering Department University of Illinois at Urbana-Champaign

Urbana, Illinois 61801, USA

32

FOREWORD

These notes served as the basis for a series of lectures presented under NATO sponsorship at the von Karman Institute for Fluid Dynamics, Brussels, Belgium. April 1968.

The important contributions of Professors H.H.Korst and W.L.Chow, University of Illinois at Urbana-Champaign, to the understanding and analysis of supersonic ejector systems are herewith acknowledged.

33

CONTENTS

Page

NOTATION 35-37

INTRODUCTION 39

1. SUPERSONIC EJECTOR SYSTEM CHARACTERISTICS 39 1.1 Performance Characteristics 40 1.2 Flow Regimes 41 1.3 Recompression Within Ejector Systems 42 1.4 Methodology of Ejector System Performance Analysis 43

2. "ZERO" FLOW REGIME 43 2.1 Flow Model 43 2.2 Primary Flow Field 44 2.3 Mixing Component 44 2.4 Recompression Criteria 45

3. "SMALL" SECONDARY FLOW REGIME 47 3.1 Flow Model 47 3.2 Primary-Secondary Flow Fields 48 3.3 Mixing Component 48 3.4 Solution Criterion 49

4. "MODERATE TO HIGH" SECONDARY FLOW REGIME 49 4.1 Row Model 50 4.2 Inviscid Flow-Fields Analysis 50 4.3 Solution Criteria 52 4.4 "Downstream" Flow-Field Analysis 54 4.5 Two-Stream Mixing Correction 56

4.6 Shroud-Wall Boundary Layer 57

5. EJECTOR THRUST EVALUATION 57

6. EJECTOR FLOW MODEL IMPLEMENTATION 58

7. AREAS WARRANTING FURTHER INVESTIGATION 58

TABLE I 60

FIGURES 61-77

APPENDIX I - A Literature Review of Ejector Systems and Related Topics 78-84

APPENDIX II - Method of Characteristics Analysis for the Supersonic Axisymmetric Primary Flow Field

1. BASIC EQUATIONS 85

2. FIELD POINTS 86

3. AXIS POINTS 87

4. BOUNDARY POINTS 88 4.1 Constant Pressure Boundary 88 4.2 Non-Constant Pressure Boundary 88

5. PRIMARY FLOW-FIELD ANALYSIS 89 5.1 Calculation Sequence 89 5.2 Wave Coalescence 89

34

Page

6 INITIAL PRIMARY NOZZLE CHARACTERISTIC 89 6.1 Sonic Nozzle 90 6.2 Uniform Supersonic Nozzle 90 6.3 Conical Supersonic Nozzle 90 6.4 A Compression at the Nozzle Exit 90

FIGURES FOR APPENDIX II 91-94

APPENDIX III - Constant-Pressure Turbulent Mixing Analysis

FIGURES FOR APPENDIX III 98-99

REFERENCES 100-101

35

NOTATION

A - *

A

A . B . C

C

Cq

f ( ) i

i . n

M

M*

P

q

R

T

'gross

u , v , V x - V

w

xm

x , y

X, Y

X, R

vectorial area in ejector control volume analysis coefficients in equation defining the shroud wall contour

U /U m a x = {1 + [ 2 / ( 7 - DM 2 ]}- 1 ' 2 , Crocco number

generalized bleed coefficient. Reference 16

function of ()

momentum flux, Reference 16

iteration indices

Mach number

{ [ ( 7 + l)M2 /2] /[ l + ( 7 - 1)M2 /2]) I / 2 , Mach star

absolute pressure

mass flow rate, Reference 16

gas constant or radius

absolute temperature

gross thrust force

velocity components

velocity vector

mass flow rate

mixing length

intrinsic coordinate system

reference coordinate system

cylindrical coordinate system

Barred Symbols

P .

Pb

P i .

Pos

Pw

X

w

Pa/Pop

Pb/PoP

Pis/Pop

P0s/P0p

Pw/P0p

X/R l p

Ws^r 'RsTosl^ /Wp^p 'RpTop] 1

Greek Symbols

a

7

5

sin_1(l/M), Mach angle

Cp /Cv , ratio of specific heats

increment

turning angle

36

6* two-stream mixing region displacement thickness

e arbitrarily small positive quantity

T) ay/x , homogeneous coordinate

7jm dimensionless displacement of the mixing region

6 streamline angle (positive, ccw)

A Tn/o a - stagnation temperature ratio

p density or radius of curvature

a mixing region similarity parameter

sp U / U a , velocity ra t io

\p angle of reattachment. Reference 16

Subscripts

a conditions along the jet boundary c choked condition in secondary stream

d discriminating streamline

e ejector exit section i inviscid flow

j jet boundary streamline

/ limiting streamline or limiting initial secondary flow Mach number

m turbulent mixing

ma conditions for minimum secondary flow area

mtn conditions at the minimum secondary flow area

p primary stream

Op primary stream stagnation state

lp primary stream conditions at initial ejector section s secondary stream

Os secondary stream stagnation state

Is secondary stream conditions at initial ejector section shk shock location

x,y conditions upstream and downstream of a normal Shockwave

w conditions along the shroud wall

BO break-off point

IE ideally-expanded flow

OE over-expanded flow

i single-stream mixing

u two-stream mixing

37

Functions

P * =- (7.M*) = l - < l Z i > M *

2

( 7 + 1)

p (7 .M*)

F(7,M*) W

(7 + 1) 1 -

7/(7-0

( 7 - 1) M *: -I/(7-D

* T 1 r

1/2 I PA = M" ( 7 + 1)

1/2 , - ^ l V 2 ( 7 + 1)

1/(7-0

= two-stream mixing region displacement thickness function. See Appendix III.

l,(rj) = mixing integral function. See Appendix III.

erf (T?) = 4 " T ^ ^

38

34

THE ANALYSIS OF SUPERSONIC EJECTOR SYSTEMS

A.L.Addy

INTRODUCTION

In ejector systems, "pumping action" is achieved through the controlled interaction and mixing of a high-velocity and high-energy stream with a lower-velocity and lower-energy stream within a duct; the simplicity of such systems has resulted in their wide-spread usage. Unfortunately, this simplicity does not carry over to the design and analysis of the flow phenomena within these systems. As a consequence of the problems involved, the design and perform-ance evaluation of ejector systems has developed as a combination of scale-model studies, empiricism, and theoretical analyses applicable only to simplified configurations. This approach, although useful for the design and performance evaluation of certain configurations, has the disadvantage of failing to provide a more general insight into the signifi-cance and influence of system parameters and operating conditions on overall ejector performance. As a result of more sophisticated applications of ejector systems along with stringent performance requirements, the development of more general analytical methods for predicting their detailed performance characteristics is essential. This objective, clearly hampered by the complexity of the flow phenomena and the breadth of potential configurations, can be accomplished only by a detailed study and modeling of the flow phenomena throughout the operating regimes of an ejector system.

The objective herein is to present and discuss the ejector flow model and its implementation1-4* that was developed at the University of Illinois at Urbana-Champaign for the performance analysis of "supersonic"ejector systems. This flow model along with its current implementation represents a significant departure from one-dimensional analyses5 - 9 . In this model the flow phenomena within an ejector system are delineated on the bases of the predominant flow mechanisms which occur within the various operating regimes. In this framework, it is then possible to establish overall ejector performance characteristics and to represent these characteristics in the unified form of "characteristic performance surfaces". These surfaces serve as a qualitative basis for understanding the overall performance of an ejector system and as a quantitative basis for making judgments regarding the signifi-cance and influence of system parameters and variations in the operating conditions.

In the sections that follow, a general topical organization has been selected which will present:

1. An overview of the operating characteristics of ejector systems. 2. Qualitative aspects of the ejector flow model in relation to the various flow regimes. 3. The quantitative implementation of the ejector flow model. 4. The performance evaluation of ejector systems. 5. A brief discussion of problem areas, their significance and possible future investigations.

The material presented in the following sections is based principally on investigations-)- conducted at the University of Illinois at Urbana-Champaign and is supplemented, where appropriate, by brief discussions of the related work of others. Since only selected references are cited, a more complete compilation of recent work and other approaches in the analysis of ejector systems is included in Appendix I.

1. SUPERSONIC EJECTOR SYSTEM CHARACTERISTICS

To establish a basis for the detailed modeling and performance analysis of supersonic ejector systems, a qualitative discussion of the performance and nature of such systems is given in this section. Emphasis has been placed on defining the general functional relationships describing the performance of these systems and how their form is dependent on the internal flow phenomena.

* The numbered References will be found on p.121 (i.e. after Appendix III of this paper). t Partially supported by NASA Research Grants NSG-13-59 and NGRI4-00S-032. Computer studies in connection with these

investigations were conducted using the IBM 7094 and 360/75 Systems, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois.

40

A representative ejector configuration and the associated notation is shown in Figure 1. The primary stream is assumed to be supplied from the stagnation state (Pop-Tnp) through a sonic or supersonic nozzle, and the secondary stream is from the stagnation state (POS-TQS)- The secondary and primary streams begin their mutual interaction at their point of confluence at the primary nozzle exit. This interaction, as well as the mixing between the streams, continues to the shroud exit where they are discharged to the ambient pressure level Pa .

1.1 Performance Characteristics

The objective of any ejector analysis is to establish, for a given configuration and working media, the perform-ance characteristics of the system. In general, the mass-flow characteristics of an ejector system can be represented functionally by:

W = f (P 0 s .Pa) . ( L I )

where the "reduced mass flow ratio" (W) is chosen as the dependent variable so that the influence of the stagnation temperature ratio is essentially removed from the functional relationship.

An alternate formulation oJ[ the pumping characteristics in terms of the initial secondary-stream Mach number (M,s), the static pressureratio (Pj s) of the secondary stream at the point of confluence of the two streams, and the ambient pressure ratio (Pa) is given in functional form by;

Mis = f (Pis .P a ) - (1-2)

This selection of variables, although less obvious, is convenient for performing the numerical calculations involved in the theoretical ejector analysis to be described.

In addition to establishing the functional form of the pumping characteristics, another variable of interest is the shroud wall pressure distribution given by:

Pw = f (W,P 0 s .P a ,X) . (1.3)

After establishing the above functional relationships, the thrust characteristics of a system can then be deter-mined. In practice, this is accomplished by considering the contributions in the axial direction of the entering momentum fluxes of the primary and secondary streams and the integrated shroud-wall pressure distribution.

/. 1.1 Three-Dimensional Performance Surfaces

The functional relations, (1.1) and (1.2), characterize^the "pumping" characteristics of an ejector system and represent surfaces in the spaces described by the triples (W, PQS, Pa) and (Mis, Pj s , Pa).

The pumping characteristics of a typical ejector system in terms of the variables (W, Pu s , Pa) are shown in Figure 2. This surface clearly delineates the flow regimes wherein the mass-flow characteristics are independent or dependent on the ambient-pressure level. These flow regimes merge together along the "break-off curve" and in principle, this condition serves to uniquely define this curve.

To the left of the "break-off curve", the_mass-flow characteristics are independent of Pa and the surface is cylindrical with its generator parallel to the Pa-axis . For this regime, the mass-flow characteristics can be represented by:

W = f(Pos) (1-4)

when Pa < (Pa)go To the right of the "break-off curve", the surface is three-dimensional in nature and extends from the spatial "break-off curve" to the plane where W = 0 ; hence,

W = f(P 0 s ,P a) (1.5)

when Pa > (P a)Bo -

In principle, the "break-off curve" represents a simultaneous solution of the functional relationships (1.4) and (1.5). However, the "break-off curve" also has a phenomenological interpretation based on the recompression of the flow within the ejector shroud. Points on the "break-off curve" are determined by the condition that transition from independence to dependence, and vice versa, on the ambient pressure ratio will occur at the maximum values of Pa to which the flow can recompress. Obviously, the locus of the recompression states defining the "break-off curve" is strongly dependent on the degree of mixing between the streams and the ejector geometry, principally the shroud length-to-diameter ratio.

An alternative representation of the pumping characteristics,^ terms of the variables (Mi,s, Pis, Pa) is given in Figure 3. For this surface, there are direct counterparts to the Pa-independent and Pa-dependent flow regimes of the W-surface.

41

1.1.2 Two-Dimensional Parametric Curves The three-dimensional performance surfaces of Figures 2 and 3 have their principal value in presenting an

overview of the performance characteristics of typical ejector systems. In theoretical analyses or experimental programs, it is generally more convenient to consider two-dimensional parametric representations of these operating surfaces. These parametric curves usually represent nothing more than intersections of the performance surfaces with various planes corresponding to constant values of the respective variables.

The more useful of the possible parametric representations of the mass-flow characteristics, from an experi-mental standpoint, are obtained by intersecting the W-surface by planes of constant Pa , Figure 4(a), and planes of constant P0s , Figure 4(b). Another mteresting and useful parametric curve can be obtained by intersecting the W-surface by a plane where Pus = Pa , Figure 4(c). The latter situation corresponds to inducting the secondary fluid at ambient conditions and then discharging the ejector to the same ambient conditions.

More convenient, from the standpoint of the theoretical jinalysis, are intersections of the M]s-surface by planes of constant P. s , Figure 5(a), and planes of constant Pa , Figure 5(b).

In actuality, the theoretical analysis establishes the ejector performance characteristics first in planes of constant P[S by varying the value of M|s until the appropriate solution criteria are satisfied for the various flow conditions that can exist in this plane. The next step in establishing the overall system performance is to select another value of P ] s and repeat the calculations; this procedure is repeated until the overall ejector performance surfaces are established.

1.2 Flow Regimes

The flow regimes occurring withinan ejector system can be categorized according to whether the mass-flow characteristics are Pa-indcpendent or Pa-dependent. In addition, a further subdivision within these flow regimes can be made on the basis of the predominant governing flow mechanisms. These subdivisions are not as clearly defined as in the former case since the change in the predominant flow mechanisms is gradual and in essence continuous. A brief discussion of the general flow phenomena and the basic flow regimes within ejector systems follows.

When the mass-flow characteristics are independent of Pa , flow conditions are established within the ejector system which effectively "seal-off the secondary flow from the ambient conditions. For "small" flow rates, this is accomplished by an oblique shock system within the primary stream that is located at the shroud wall. As the secondary flow rate is increased, an operating condition will be reached where the oblique shock system can no longer be sustained at the shroud wall and as a consequence,_the primary stream "breaks" away from the wall. For the mass-flow characteristics to remain independent of Pa , the secondary stream must_then accelerate until it "chokes" inside the ejector shroud. After this condition occurs, increasing the value of Pis or Pus results in the progressive upstream movement of the "choking" point until the secondary stream "chokes" at the point of confluence of the two streams, i.e., Mjs = 1 .

When W depends on both (Prjs'Pa)* ' ' l e secondary stream does not "choke" inside the ejector shroud. For this condition, the secondary flow is initially accelerated and then decelerated and diffused as a result of the boundary condition imposed by the ambient pressure at the ejector shroud exit.

The "choked" and "unchoked" regimes correspond respectively to the Pa-independent and Pa-dependent regimes previously discussed. Thus, the "break-off curve" separates regimes of significantly different flow phenomena within the ejector shroud.

1.2.1 "Zero" Secondary Flow The typical variation of Pus with PP , when W =_0 , is shown in Figure 6; this curve represents the inter-

section of the W-surface or MIs-surface with the plane W = 0 . This flow condition corresponds to the well-known "internal base-pressure problem" which has been the subject of extensive and continuing investigations10"26.

For the portion of the curve where Prjs is a constant, the flow phenomena, Figure 7, within the ejector is essentially governed by the entrainment of fluid due to mixing along the primary boundary and the recompression-shock system resulting from the local interaction between the primary stream and the shroud wall. The recom-pression system establishes the level of mechanical^nergy required for the entrained fluid topass through the recompression zone. Thus, the requirement that W = 0 uniquely establishes the value of P0s and correspondingly the flow field to the terminus of the recompression zone. Although variations in Pa do not influence P0s until the "break-off point is reached, such variations do influence the shroud-wall pressure distribution.

The Pa-independent portion of the flow regime is sustained until the value of Pa is sufficiently large so that the recompression mechanism is modified. When this occurs, the primary stream "breaks away" from the wall and the system then operates in the Pa-dependent mode. For this mode, the primary stream is recompressed to the

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ambient pressure at the shroud exit, and the condition that W = 0 is maintained by a combined mechanism of entrainment and backflow within the ejector.

1.2.2 "Small" Secondary Flows

For Pa-independcnt operation, this regime effectively spans that part of the operating characteristics where transition of the governing flow phenomena is from an entrainment-recompression-shock mechanism to essentially an inviscid-interaction mechanism between two-streams that are distinct. This transition occurs when the recom-pression-shock mechanism can no longer be sustained at the shroud wall. As would be expected, the extent of this regime and point of transition are strongly dependent on the ejector geometry.

The flow phenomena within this regime are essentially the same as described for the zero-flow case with the following exceptions. In this case, the secondary stream has a non-zero, though small, component of velocity in the axial direction. This component of velocity has a modifying effect on: (i) the primary stream, (ii) the entrain-ment at the primary-secondary boundary, and (iii) the recompression-shock mechanism at the shroud wall.

1.2.3 "Moderate to High" Secondary Flows

This regime is characterized by the primary and secondary streams remaining essentially distinct although mixing locally along their mutual boundary. In contrast to the other flow regimes, the secondary stream is no longer "sheltered" by the primary stream and as a consequence must interact with the primary stream so that the operating conditions, e.g., (Pr-SiPa). imposed on the ejector are satisfied. The interaction can satisfy the prescribed operating conditions in either of two ways. In one case, Figure 8(a), the secondary stream is "choked" within the shroud as a result of interacting with the primary stream thus establishing the mass-flow characteristics of the system. Adjustment to the prescribed ambient pressure level is then made downstream of the "choking" point without effect-ing the mass-flow characteristics. In the other case. Figure 8(b), the secondary-primary streams interact so that "choking" does not occur within the shroud but rather the prescribed ambient-pressure conditions are satisfied by the secondary stream at the shroud exit. The mass-flow characteristics are then determined as a consequence of this interaction.

1.3 Recompression Within Ejector Systems As an example of the recompression phenomena, consider two ejector systems with shrouds of significantly

different lengths that are operating at the same values of (W, P 0 s) and discharging into a region where (Pa) is very small.

For the long-shroud ejector, the mixing between the streams will be nearly complete and the flow at the exit will be approximately uniform and at a supersonic Mach number. In contrast, the mixing between the streams in the short-shroud ejector will necessarily be incomplete and the two streams will remain essentially distinct through-out the ejector. Consequently, the flow at the exit plane will be highly non-uniform and both streams will discharge supersonically.

If Pa is increased, the flow will initially recompress external to the shroud; however, with further increases in Pa a recompression shock system will move inside the ejector shroud and be located such that the boundary condi-tion imposed at the exit by the ambient pressure level will be satisfied. The recompression in the ejector with the longer shroud is essentially accomplished by a normal shock in the mixed stream. In the short-shroud ejector, the same recompression must be accomplished by a combined mechanism consisting of a normal shock in the secondary flow and an oblique shock in the primary stream. Clearly, the latter mechanism is less effective in recompressing the flow than the former.

With further increases in P a , the recompression shock system moves further upstream in the ejector shroud. This movement in the short-shroud ejector system can only continue, without influencing the mass-flow character-istics, until the recompression shock in the secondary stream is located at the secondary stream's "choking" point. This situation essentially corresponds to a reversible subsonic recompression of the secondary stream to the ambient pressure at the shroud exit; any increase in Pa above this value, the "break-off point, will necessarily result in a reduction of the mass-flow rate and the secondary stream becoming subsonic throughout the duct. The ejector system then operates in the ambient-pressure dependent regime.

On the other hand, the long-shroud ejector system can recompress to higher values of Pa as a result of the more complete mixing between the two streams. Eventually,however, increasing Pa will move the recompression shock system upstream and out of the well-mixed flow and into the region where the two streams are essentially distinct as was the case for the short-shroud ejector. A part of the recompression now takes place at this location with final recompression to the ambient pressure level occurring in the remainder of the ejector shroud. Again the "break-off point occurs at the value of Pa corresponding to the recompression shock being located at the "choking" point of the secondary stream. As for the short-shroud ejector, this value of Pa defines the "break-off point since an increase in the ambient-pressure ratio above this value would result in a readjustment of the system's mass-flow characteristics.

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The qualitative aspects of this recompression phenomena* for these systems are depicted in Figures 9(a) and 9(b); the influence of the recompression on the location of the "break-off curve is shown in Figure 9(c). It should be noted from these figures, that in the flow regime where the mass-flow_characteristics are unaffected by variations in the ambient pressure ratio, that the shroud wall pressure distribution P w can have significant variations with P a . In contrast, a unique shroud wall pressure distribution corresponds to each set of values of (W, Pus,Pa) in the Pa-dependent flow regime.

In thrust augmentation applications, the ejector shroud-wall pressure distribution must be determined in addition to the "pumping" characteristics. The strong linking between the shroud-wall pressure distribution and the recom-pression mechanism necessitates that this mechanism bo included as an integral part of the performance analysis of ejector systems. Due to the complexity of the flow phenomena, this is a difficult, if not impossible, task for ejectors with long nonconstant-area shrouds. Fortunately, the analysis of short-shroud ejector systems, which are more practical for this particular application, is tenable.

1.4 Methodology of Ejector System Performance Analysis A single-model approach to the analysis of ejector systems is not possible because of the dependence of the

governing flow mechanisms on the system geometry and the operating conditions. Instead, their analysis must be based on a multiple-component flow model which adequately identifies and describes the predominant flow mechanisms within the various regimes. Although posing no conceptual problems, it is not known a priori when these regimes occur, as well as when transition between regimes occurs; in fact, these factors must be included as an integral part of the overall ejector analysis.

The flow regimes (Sections 1.2.1-3) are modeled according to their respective governing flow mechanisms. In principle, these models are then applied, subject to the imposed boundary conditions and the applicable internal flow solution criteria, to determine the detailed ejector performance. In practice, a direct method of analysis for specified operating conditions is impractical for the following reasons:

(i) The flow regime corresponding to the imposed operating conditions is not known a priori and, hence, must be determined.

(ii) The applicable solution criteria and means for satisfying the imposed boundary conditions must also be determined.

As a result, the performance analysis is based on an indirect method which simply consists of evaluating a system's performance in the various regimes, subject to the applicable solution criteria. This approach, in essence, establishes the overall ejector performance surfaces for a given system. Specific operating conditions are then located on this surface.

The ejector flow model, as specialized for each of the flow regimes, and it's implementation will now be considered.

2. "ZERO" FLOW REGIME

For this regime, the flow mechanism consists of: the flow entrained along the primary stream boundary due to the viscous mixing between the stream and an essentially quiescent fluid, the interaction of the "nearly" inviscid supersonic primary stream with the shroud wall, and their interdependence. Korst, Chow, et a l . 1 0 " 1 5 , have studied this problem in detail in the course of their basic investigation of separated flow problems. Carriere, Sirieix, Delery, and Hardy 1 6 - 2 3 have considered modifications to the basic analysis proposed by Korst, et al., in an attempt to improve the agreement between theory and experiment for the axisymmetric case. Their work, as well as the work of others2 5 , 2 7 , has generally been concerned with modifying the way in which the recompression of the entrained fluid is treated. This is a logical approach to take since the recompression mechanism is a predominant factor that is not well understood. In general, however, these investigations have not resulted in significant changes in the flow model originally proposed, but rather, have resulted in modifications to the "recompression criterion" and some details in the method of analysis. A discussion of current approaches and analyses related to this problem is given by Korst15.

2.1 Flow Model

The general flow situation and applicable notation is shown in Figure 10(a). The axisymmetric supersonic primary stream is assumed to expand into a region at constant pressure P-,. This constant pressure region is assumed to exist up to the point where the jet impinges on the shroud wall; at this point, an oblique shock wave exists so that the boundary condition imposed by the local wall slope is satisfied. This flow field is defined as the "corresponding inviscid jet", (Ref. 13).

* An analogy, which is more apparent for the short-shroud ejector, can be drawn between the behavior of an ejector system and a converging-diverging nozzle operated under varying back-pressure conditions.

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The entrainment as a result of the mixing between the primary stream and the quiescent fluid along their mutual boundary is assumed to take place at constant pressure. For the present analysis, the mixing component13 , is assumed to be represented by the two-dimensional turbulent mixing of a uniform stream and a quiescent fluid. On this basis, the mixing region is considered to be defined by the flow conditions along the jet boundary and within an intrinsic coordinate system which is displaced relative to the jet boundary. The mixing region, thus defined, is then localized relative to the "corresponding inviscid jet boundary" by superimposing the mixing region on the jet boundary, in a two-dimensional sense, while satisfying the integral continuity and momentum relationships.

The linking between the "corresponding inviscid je t" and the mixing region is accomplished by the recom-pression mechanism and the conservation of mass and energy within the "wake" region. The "recompression criterion" identifies streamlines within the mixing region which have sufficient mechanical energy to recompress to the high-pressure region downstream of the jet-wall interaction.

The implementation of this flow model as it applies to the analysis of ejector systems will now be discussed.

2.2 Primary Flow Field

The primary flow field is analyzed by the Method of Characteristics for irrotational axisymmetric flow. The primary nozzle flow conditions are assumed to be known along the_initial left-running characteristic emanating from the nozzle corner (X] p , Ri p ) , and the flow expands to a pressure, P-,, which is constant along the jet boundary. These conditions are then sufficient to determine the resulting flow field.

Since the general interior primary flow-field analysis is applicable here, as well as for the other flow regimes, the flow-field analysis will not be discussed in detail here but rather in Appendix II. The only specializing condition for this flow regime is that the jet boundary is maintained at constant pressure.

2.3 Mixing Component

The "restricted" two-dimensional constant-pressure turbulent jet mixing theory of Korst, et al., is the basis for the analysis of the mixing component. Since a general treatment of single-stream and two-stream mixing* can be given, the general aspects of this analysis are given in Appendix III while the conditions specifically applicable to this flow regime follow.

For this case, Figure 10(b), the dimensionless velocity profile,

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Similar relationships13 can also be written for the energy transfer within the mixing zone for the non-isoenergetic case, i.e., when Tb # Trjp . However, this significantly complicates the analysis of the problem under consideration and experience has shown1 3 , 1 4 that over a relatively wide range of temperatures, T b = Trjp , the influence on the theoretical "base-pressure ratio" is small. Consequently, only the isoenergetic case, Tj, = T Q P , will be considered for this regime.

If the overall wake region defined by the "corresponding inviscid jet" and the shroud wall is now considered, the conservation of mass within this region requires that the net flow of the entrained fluid out of this region must be zero. If the d-streamline* has just sufficient mechanical energy to "escape" from the wake but yj = y: , the problem then becomes one of finding the value of P-, such that the d-streamline coincides with the j-streamline.

2.4 Recompression Criteria

The j-streamline can be identified solely on the basis of applying the conservation equations to the mixing zone; however, the determination of the d-streamline can only be accomplished by linking the mixing phenomena and the "corresponding inviscid jet". This is done by means of the recompression criterion.

2.4.1 Korst's Recompression Criterion

The recompression criterion of Korst, et al.13 identifies the d-streamline as the streamline which possesses just sufficient mechanical energy to recompresst from the wake pressure, P-, , to the high-pressure region downstream of the shock, P^k . The recompression pressure rise for the d-streamline, based on this recompression criterion is found from the oblique shock relations for the local conditions at the impingement point, Figure 10(c). The turning angle is given by:

6 = - ( f l p - 0 W ) (2.6)

and the jet surface Mach number, Mp , is known for the assumed value of Pj, . Hence the d-streamline pressure rise is found from:

EsM = ^0d = f(5,MD)** . (2.7) Pb Pd P

The d-streamline "Mach s t a r " t t is then given by:

MJ = (J f4- \ ) U 2 [ 1 - (Pod/Pd)^-^7"] l l2 (2-8) For the isoenergetic case, the d-streamline velocity ratio is:

* - %

The solution value of Pj, is then found when

0 < l ^ - ^ j l < e . (2.10)

for e arbitrarily small.

In practice, the solution value of Pb is found by assuming a value (Pb)- and then computing the difference (^d _ ^j)i I f

(Vd-Vj)j t l 0 (2.11)

then the next value (P b ) i + i is correspondingly assumed to be:

(Pb)i+1 (Pb)i (2 '1 2> Then convergence to the solution is both well behaved and rapid.

* Defined as the "discriminating streamline".

t This compression, though irreversible and diabatic, produces the same results as an isentropic compression from the static pressure, Pb , to what would be the stagnation pressure, Pshk , of the d-streamline.

** Convenient graphical or functional forms for numerical calculations are found in Reference 28.

t t The variable M* = U/C* has been introduced and is used throughout the analysis because its finite range, 0 < M* < [(y + l)l(y - I))"3 is convenient for the computer analysis.

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It should be noted that the intermediate results obtained in the course of the zero-flow solution can be inter-preted as solutions for "mass bleed at negligible velocities" into or out of the wake region13. For this case, the mass flow is evaluated from (2.5) or expressed in more conventional form (from Appendix III):

2 Rw Xm (A/A*) (yp.MJp) , . . , . , ... . , , W = -r { l i *?;) I I ( I ?H ) (2.13)

a, R l p R l p ( A / A * ) ( y p , M * ) 'J

Although the overall recompression pressure rise is reasonably well represented by the pressure rise correspond-ing to the oblique shock at the impingement point, it was well known at the time that the "discriminating stream-line" does not recompress to this level. Rather, it recompresses to an intermediate value between Pb and P s ^ . This situation is shown in Figure 10(d).

Nash23 defined a recompression coefficient for the two-dimensional case in an effort to better correlate the theoretical analysis with the experimental data; this correlation is not sufficiently universal and as a result has not been well accepted.

Page, et al.27 has proposed another recompression correlation based on the discriminating streamline velocity ratio and an "effective" entrainment mixing length. This correlation is shown to be applicable over a wide range of experimental conditions for plane two-dimensional supersonic flow. This approach yields results which are similar to those obtained by investigators at ONERA*. Since much of the work at ONERA was directed toward the analysis of axisymmetric ejector systems operating in the zero and "small" flow regimes, it will now be discussed in some detail.

2.4.2 The Recompression Criterion of Carriere, Sirieix, et al.

An empirical relationship called "the critical angle of reattachment" that a stream must satisfy when reattaching to a wall has been deduced from a series of experiments1 6"1 9 . This relationship was established for a uniform stream entraining a quiescent fluid at constant pressure and then subsequently reattaching to a wall to form a wake. Figure 11(a). This relationship is shown graphically in Figure 12; it has the functional form

*o " >MMi)+ * (2-14)

Carriere and Sirieix have shown 1 6 , 1 7 that the influence on the base pressure of: (1) the stream's initial boundary layer momentum thickness (5 ) and (2) mass addition (q) to the wake with a finite momentum (i) can be expressed in terms of a dimensionless generalized "bleed coefficient"**, Cq . Where Cq is defined1 7 '2 0 as

(. ,

q ' + 5**

p ,U,x p ,U 2 x (2.15)

The above factors are then considered as perturbations to the case where they are absent. The reattachment angle for the perturbed case is given by

*(MC

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Z- sfi dTJ = - - 1 C Q (2.19)

in which form the analogy to "bleed" is obvious.

The solution procedure to determine Pb given: (1) the flow geometry, (2) the initial flow conditions, and (3) Cq = 0*, proceeds in the following fashion. If a value of Pb is assumed, then M, can be calculated from the isentropic relations, and the inviscid flow field can be constructed by the Method of Characteristics (Appendix II) subject to the condition of constant pressure along the boundary. Corresponding to the value of M, along the boundary, values of a., n-,, sp,, r)m , and the mixing integrals (Appendix III) - can be determined. Using (2.19) the dimensionless coordinate, )?/, locating the /-streamline in the intrinsic coordinate system can be found. Assuming a displacement of the /-streamline normal to the local inviscid boundary, the location of the /-streamline, Figure 11(b), in the inviscid flow field coordinates (X, R) is found from

R, =

Xp + (77m - 17/) sin 8.

RP + (Vm ~ Vl) cos

(2.20a)

(2.20b)

The next step is to find where the /-streamline intersects the wall; corresponding to this location, the impingement angle of the /-streamline relative to the wall is given by

h = - ( * - * ) (2.21) where 0 p is the inviscid boundary flow angle corresponding to the /-streamline impingement point and 0W is the corresponding local wall angle.

From the angular reattachment criterion, (2.16), the value of \b can be found. The question is - does i/-/ = i/-? If nott, then another value of Pb is assumed and the foregoing calculations repeated until

0 < M i ) n - (i//)nl < e . (2.22)

Typically if l(i/'/)n (i//)nI ^ 0 , the next value (P b ) n + j is chosen correspondingly such that (Pb)n+i ^ (Pb'n

3. SMALL" SECONDARY FLOW REGIME

The secondary flow, although small in this regime, has a significant modifying effect on the primary and se