"• ~II

A STUDY OF

JET EJECTOR PHENOMENA

By HA RD COPChia-An Wan .• ,--

Research Report No. 57 November 1964

U.ý- .-

L) C

JAPRDDE, SRA B

E R J PUHIYS IT CM'a•G • I- S Sr I PPI S

A STUDY OF

JET EJECTOR PHENOMENA

By

Chia-An Wan

Research Report No. 57 November 1964

Conducted For

OFFICE OF NAVAL RESEARCH

Under

CONTRACT NONR 978(03)

By

The Aerophysics Department

Mississippi State University

Reproduction in whole or in part is permitted

for any purpose of the United States Government.

LIST OF ILLUSTRATIONS ------------------------------------------ iv

LIST OF TABLES ------------------------------------------------- vi

LIST OF SYMBOLS ------------------------------------------------ vii

INTRODUCTION --------------------------------------------------

EXPERIMENTAL APPARATUS AND TECHNIQUES -------------------------- 3

THEORETICAL CONSIDERATIONS ------------------------------------- 5

DISCUSSION AND CONCLUSIONS ------------------------------------- 15

BIBLIOGRAPHY -------------------------------------------------- 21

ILLUSTRATIONS -------------------------------------------------- 24

TABLES -------------------------------------------------------- 51

'4

"r

Ii .iii

i---

II

1.TLJ15TRATIONS

Figure Page

1 Jet Ejector, Nomenclature and Flow Regimes ------------ 24

2 Descriptive Layout of Apparatus ------------------------ 25

3a Thrust Force Measuring Unit ---------------------------- 26

3b Jet Flow Unit, Nozzle, and Supporting Rig -------------- 26

4 Extension for the Nozzle Exit -------------------------- 27

5 Pitot Tube Probe and Traversing Mechanism -------------- 27

6 Velocity-Measuring Setup ------------------------------- 28

7 Thrust Forces of Nozzles ------------------------------ 29

8 Thrust Forces of Jet Ejector, A = 105.5 ------------- 30

9 Effect of the Length Ratio on Augmentation Factor ..... 31

10 Effect of Gap Ratio on Augmentation Faccor ------------- 32

11 Comparison of Theoretical and Experimental PotentialVelocity ---------------------------------------------- 33

12 Comparison of Theoretical and Experimental Thrust

Augmentation Factor ------------------------------------ 34

13 Comparison of Theoretical and Experimental VelocityRatios ------------------------------------------------ 35

14 Comparison of Theoretical and Experimental PressureReduction Ratio at Initial Plane ----------------------- 36

15 Theoretical Boundary of the Turbulent Mixing Zone andComparison of Theoretical and Experimental Thicknessof Boundary Layer of the Walls ------------------------- 37

16 Comparison of Theoretical and Experimental Dimensionsof Laminar Core and Turbulent Mixing Zone -------------- 38

iv

FigurE Page

17 Comparison of Theoretical and Experimental VelocityProfiles in Inlet Length of EjectorAA=Z'.+,,V 1.O --- 39

18 Comparison of Theoretical and Experimental Velocity

Profiles at Outlet of Ejector -------------------------- 40

19 Effect of 2I on m ---------------------------------- 40AA.

20 Velocity Profiles, u = 11.7 ------------------------- 41

21 Velocity Profiles, 'a. = 16.0 ------------------------- 42

22 Velocity Profiles, AZ, = 47.0 ------------------------- 43

23 Velocity Profiles, ,W = 105.5 ------------------------ 44

24 Velocity Profiles. ,,& = 29.4 ------------------------- 45

25 Variations of Centerline Velocity Along the Length ofEjector ----------------------------------------------- 46

26 Comparison of Theoretical and Experimental CenterlineVelocity Ratio of Ejectors and That of a Free Jet 47

27 Comparison of Theoretical and Experimental InverseCenterline Velocity Ratio of Ejectors and That of aFree Jet ---------------------------------------------- 48

28 Effect of Area Ratio on k ------------------------------ 49

29 Variations of Pressure Along the Length of Ejectors --- 50

v

IP'AIT • 'C

Table Page

1 The Dimensions of the Extensions of Nozzles ------------ 51

2 The Dimensions of the Ejectors ------------------------- 51

3 Axial Locations Where Velocity Distributions WereMeasured ---------------------------------------------- 52

vi

SYMBOLS

A Cross section area of exit of nozzle, in 2

Ae Cross section area of ejector, in2

D Diameter of exit of nozzle, in

De Diameter of ejector, in

k Slope of -2U vs 'AD curve

1 Length of gap, in

L Length of ejector, in

m Inverse of exponent related to velocity profile at outlet ofejector, defined in Equation 2

n Exponent related to velocity profile, defined in Equation 36

p Local pressure, lb/in2

Pa Pressure of the atmosphere, lb/in2

PO Pressure at exit of nozzle, lb/in2

Pl Pressure at initial plane, lb/in2

P2 Pressure at outlet of ejector, lb/in2

Reynolds Number,

R Radius of exit of nozzle, in

Re Radius of ejector, in

r Distance measured in radial direction from ejector centerline,in

rc Radius of the laminar core, in

T Thrust force of jet ejector with no gap, lb

Tg Thrt'st force of jet ejector with gap, lb

vii

_ _ _ _ _ _ __ _ _ _ _ _ _I

_ _ _ _ _ _ _ __ _ _ _ _ _ _ _ I!

Ti Thrust force on jet flow, lb

u Axial velocity, ft/sec

U Centerline axial velocity, ft/sec

Uo Axial velocity at the exit of nozzle, ft/sec

U1 Secondar° or induced flow velocity at initial plane, ft/sec

u 2 , U2 Velocity at the outlet of ejector and its centerline value, ft/sec

U P Potential velocity beyond the edge of the jet, ft/sec

v Radial velocity, ft/sec

w Distance in radial direction from edge of the laminar core, in

W Distance in radial direction from the edge of the laminar coreto the edge of jet, in

x Axial distance from the initial plane, in

xk Length of the laminar core, in

xm Length of the modified free mixing flow regime, in

y Distance in the radial direction from the centerline of theejector to the edge of the jet, in

Yk Radius of the jet at x = xk, in

z Distance in the radial direction from the walls of the ejector, in

C<1 Induced velocity ratio at initial plane, U-

U0~O(Z Velocity ratio at outlet of ejector, U0

A& Area ratio,4

Augmentation factor without gap,-X-

Augmentation factor with gap,21.

Overall augmentation factor, TG

SDensity of air,1LbM

viii

"T Shear stress, lb/in2

8 Thickness of boundary layer of the walls of ejector, in

2r" Kinetic viscosity of air, ft 2 /sec

ix

- r- -r --'

INTRODUCTION

A jet ejectro, or jet pump as it is sometimes called, is a device inwhich a fluid is emitted from an orifice or nozzle exit into a hollow,generally cylindrical, body. Due to the shear stresses between themoving fluid and the ambient still fluid, part of the kinetic energy ofthe moving is transferred to the still fluid. In the case of a constant-pressure mixing tube, which can be made through careful design,(Reference 1), this mixing process is the primary cause of secondary orinduced flow.

However, in the case of a constant-area mixing tube, a low-pressurefield is created by the jet flow. This difference in pressure betweenthe exit plane of the nozzle and the ambient fluid upstream causes anentrainment of the ambient fluid or a secondary flow. Due to the trans-fer of energy, the secondary flow is either accelerated or pumped to a

L• higher pressure. The jet flow and its induced flow are not necessarilyof the same fluid; either of them can be air, water, steam, or theexhausted gas of a jet engine. The ejector is not necessarily circular,but can be of any shape, as required by the physical installation.

-7 The principle of a confined jet flow finds many difterent applica-tions in industry, especially the aerospace industry. An ejector canbe used as a pumo, blower-augmenter, and noise-reducer. (References 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12). In case the jet ejector is operatedwith no change in pressure it does not provide a pumping effect. How-ever, it does provide a thrust increase for the installation.

A typical constant-area jet ejector arrangement is shown schemat-ically in Figure 1. With this arrangement, the primary jet flow isI emitted from the exit of t-e nozzle into the ejector, and the velocityis almost uniformly d4 stributed throughout the cross section area of theexit of the nozzle. A subatmospheric pressure field is created, whichconsequently causes a secondary flow in the rest of the cross sectionarea of the ejector at the initial plane. If the boundary layer growthalong the contour of the faired inlet of the ejector is neglected, thesecondary velocity distributica can be considered to be constant.

The velocity of the secondary flow is less than that of the primaryflow. As aconsequence, there is a near discontinuity in the velocityprofile across the ejector in the initial plane. Downstream of thisplane, a process of turbulent mixing tends to smooth out this near dis-continuity so that, if the ejector is long enough, the mixed flow velocity

profile becomes that of fully developed turbulent pipe flow.

Since the static pressure increases downstream of the initial planeuntil, at the outlet of the ejector, it becomes equal to that of the

t•1

j

_ o

surrounding atmosphere, the static, npire.re inrreAse nor snit length %fthe ejector depends upon the total length of the ejector. The mass rateflow remains constant throughout the length of the ejector, but totalmomentum at the outlet of the ejector is less than that at the initialplane due to losses which result fi~m skin friction.

The maximum augmentation factor for a straight ejector is 2.0 ascalculated theoretically by von Karman. (Reference 6). He assumed thatthe velocity distributed was uniform at the outlet of the ejector, andthat skin friction losses were negligible. A similar calculation hasbeen performed ini this report using somewhat different assumptions.

A knowledge of the free-mixing process and pipe flows aids inunderstanding the confined-mixing process. Information concerning thosephenomena can be found in many works. (References 13, 14, 15, 16, 17,18, 19, 20, 21). The equations of motion for the flow field of aconstant-pressure mixing tube is similar to that of a free jet flow,however, the boundary conditions are different. The axial pressuregradient term in the equation of motion for a constant-area mixing tubeand the coexistence of laminar and turbulent flows make the flow fielddifficult for mathematical analysis.

For comparison purposes a satisfactory solution of the problem ofa constant-area mixing process can be obtained by approximate solutionof the Naview-Stokes equations. However, this approach requires such agreat amount of work that it becomes questionable wheLher the approxi-mate solution justifies the effort. As a consequence, this report adoptsan approach in which the author uses existing theories and empiricalformulae to solve the problem.

2

EXPERTMINTAT. APPARATIM ANfl TP4NTOITAF.

For the sake of convenience, t-o sets of experimental apparatuswere used - one for measuring the thrust forces and one for measuringthe velocity distributions. Two convergent nozzles and three constant-area ejectors were used in both measurements; with one constant-areaejector of relatively long length being used in velocity distributionmeasurements only. Compressed air used in both measurementF wassupplied by the equipment shown schematically in Figure 2. Thrustforces were measured with the device shown in Figure 3a.

In order to change the relative location of the exit of the nozzle,brass tubes were used as extensions of the nozzle exit. (Figure 4).Their dimensions are shown in Table 1.

Ejectors of various lengths were attached to the primary nozzle asshown in Figure 3b. Care was taken to insure that the ejector andprimary nozzle were aligned axially. The dimensions of the ejectorsare shown in Table 2.

By using these nozzles and ejectors, the area ratio of the jetejector could be varied from 4.0 to 105.5.

The primary nozzle flow and the secondary entrained flow could becalculated from velocity profiles measured at various locations in theejector by means of a pitot-static tube. This tube was attached to amicrometer-type mechanism which allowed it to be positioned at anydesired location along the diameter of the ejector. (Figure 5). Velo-cities were measured by means of a U-tube water manometer.

Profiles were measured at the distances downstream of the initialplane as shown in Table 3.

All velocity profiles and static pressure were measured when thenozzle and ejector were positioned horizontally. (Figure 6). Allthrust force measurements were made with the apparatus positionedvertically.

Thrust forces of the nozzles were firsc measured at various gaugepressures as shown in Figure 7. These measurements -:ere "Ien repeatedwith the ejector attached and were plotted against gauge cressure.Typical curves are shown in Figure 8. At an arbitrary gauge pressure of35 p.s.i. the augmentation factor was obtained by dividing the thrustof the jet uith ejector by the thrust of the jet without ejector. Theeffect of the length ratio on augmentation factor is shown in Figure 9.

3

_I 4n

The thrust forces of the Jet ejectors at different gaps weremeasured. These forces were divided by the thrust cf the jet ejectorswith no gap in order to obtain the gap augmentatioo factor (p , andwere similarily divided by the thrust of the jet in order to obtain theoverall augmentation factor, Qr-. 0 and (.Pr are plotted againstgap ratio in Figure 10. Gap is the distance between the nozzle exitplane and the initial plane of the ejector. The initial plane isdefined as the plane of the ejector where the faired inlet of theejector ends and the constant-area portion of the ejector begins. Gapis defined as negative when the exit of the nozzle is inserted into theejector, as positive when the exit is away from the ejector, and as zerowhen the nozzle exit plane coincides with the initial plane of theejector.

4

THEORETICAL CONSIDERATIONS

Several assumptions are made in the theoretical considerations.They are:

1. The flow is incompressible, hence the static pressure isconstant throughout the initial plane within the ejectorand the pressure at the outlet of the ejector is equal tothat of the surrounding atmosphere,

2. The velocity at the exit of the nozzle is uniformly distributed,i.e., the velocity profile is constant.

3. The secondary flow velocity is uniformly distributed at theinitial plane

4. There is no pressure change in the radial direction.

5. The flow field is of an axisymmetric case.

A. Thrust Augmentation Factor-and Velocity Ratios

A method similar to that used by von Karman (Reference 6) was usedin calculating the thrust augmentation factor and the velocity ratios.It was also assumed that,

1. The velocity distribution at the exit of the ejector followsthe relation; u.

2. Skin friction losses are negligible,

3. The walls of the nozzle are infinitely thin at the initial plane.

4. The velocities at the nozzle exit remain unchanged by the instal-lation of the ejector.

The equation of continuity may be written as;

PAU.+ e(Ae-A) U, =Po raa 7

5

-- now C -

After dividing throughout by p AUJ, one may write the equation as2 •d

(1)

Using the assumption that

(2)

equation 1 may be rewritten as

(3)

The momentum equation may be writt:en as.• •~' (euo" ± P'o) + (Ae-A)(efU,a+1-p,)J'e(PLua2+-pa) zir~ d ,•z,

where -r0 and according to Bernoulli's theory

The momentum equation may be rewritten as

Integrating the term on the right-hand side results in

(.5)

Solving the velocity ratios from equations 3 and 5 form =4,

and *-) + J115 At [AJ._,.

(7)

i6

mmAA.

and----

If 1L. is murch reatpr t-hAn 1 .0, ama,,.nse n 6 afd 7 ma.y be --mple d

rewritten as

4 40

(8)

ando( q ~'z4. ou 81an 80+• 0•to,-Ls• • -;f0

(9)

The thrust augmentation factor, , is defined by the formula

PA Uoy"Carrying out the calculation, one has for the thrust augmentation factor,

5 At(C<X(10)

It is somewhat arbitrary to define the thrust augmentation factoras the ratio of the actual thrust to the product of the mass and thevelocity of the jet instead of to its total impulse. However, ingeneral, the difference is not large.

In Figure 12 the thrust augmentation factor, Cf , is shown.Velocity ratios 0(1 and a(Z are plotted in Figure 13, showing compari-son witi, the experimental measurements.

The pressure reduction may be obtained by using equation 4

= to [ 40]

This equation has been plotted in Figure 14.

B. Structure of the Flow Field in the Inlet Lenath of the Elector

The structure of the flow field at the inlet of the ejector is verycomplicated (Figure 1). As the compressed air is emitted from the nozzle

7

I

into the constant-area ejector, there is a field of subatmosphericpressure created. Consequently, a secondary flow of constant velocityis formed at the inlet of the ejector. Disregarding the faired portion ofthe inlet, the boundary layers at thie wall are infinitely thin at theinitial plane. Normal boundary layer growth occurs downstream from theinitial plane. Disregarding the flow of the jet, the boundary layerskeep growing, until they meet at a point on the axis of the pipe. Untilthis happens, there is a core of fluid practically uninfluenced byviscous effects in which the total head may be :onsidered constant.This problem of inlet flow of a ciiu] ar pipe had beea investigated byGoldstein (Reference 22), Langhaar (Reference 23), Schiller (Reference24), and Talbot (Reference 25).

Due to the existence of the jet flow, the edge of the wall boundarylayers will not meet at a point on the axis of the pipe; instead theyjoin the boundary of the jet.

The flow field in the inlet portion of the ejector, disregardingthe primary jet flow, becomes similar to the case studied by Schiller.Schiller suggested that the wall boundary layer thickness, S , isrelated to the velocities, U1 and Up, by the following relationship,

S- l e-RJ & ( U-4A

where Re is the radius of the circular tube, U1 is the constant velocityof the potential core at the initial plane and U is the velocity in thepotential core which is a function of x. P

The velocities have been related to the axial coordinate as

Re -". (2

where R _ eUi is the Reynolds Number, and

UI (13)

The function J has been given by Schiller as

15 ±1I 5 s-i 1

63 _2 -YL 48T5Vz I V15Wzsivn 3 3S i (14)SF--,

and has been simplified by Goldstein to the following form which hasbeen plotted in Figure 11,

1L6"'4-2rL '; 37d/._7 Si_ z_,_4_-

(15)

For the sake of simplicity, one can approximate the curve ofeqiation 15 •'y

O.O o oo'• S3 +oo5*zs +o.oozos3. (16)

Equation 16 is plotted in Figure 11 for comparison with equation 15.

Since the thickness of the wall boundary layer, 8 , is an explicitfunction of Up (equation 11), Up is an implicit function of X(equation 12). Consequently, 8 is a function of %also. W/Re isplotted against the nondimensional variable in Figu

The variation of the radius of the mixing zone can be estimated bya straight line. Starting with the equation given by Helmbold (Refer-ence 26), for a mixing tube of constant pressure,

-17Z3 + o.4•7(I-(-,) + 0 .2 !5 (-1 (17)

where Yk denotes the radius of the mixing zone at the end of the laminarcire of the jet flow. Hence, " V-

where xk, as given by Helmbold, expressing the length of the laminarcore, is

-I - -,(18)

Thus, _X

R (19)

By equating the radius of the mixing zone (Re - ), one can findthe location where the wall boundary layer joins the boundary of themixing zone.

9

- - _ __ -. - •~~w- -- - . .- .. .. 4,.-••-_-

The primary flow, except in the turbulent boundary layer at thewall of the nozzle, emerges as a laminar jet with uniform velocitywhich, beginning at the rim of the nozzle, becomes increasingly turbulentdownstream as the mixing of primary and secondary flow spreads inward.The length of the laminar core has been investigated by many authors.For a free jet, according to Faris (Reference 15), it is 4.3 nozzlediameters, which compares favorably with Keuthe's 4.44 nozzle diameters(Reference 14), and Davies' 4.35 nozzle diameters (Reference 17); butfor confined jet flow, it depends on area ratio.

For the confined Jet flow of the present case, the dimensions ofthe core could be computed by the following formulae derived from theresults of Szablesaki's calculation by Helmbold (Reference 26), validfor the case of a confined jet with constant pressure acrobs the ejector.The length of the laminar core in terms of nozzle radius is

(18)

The radius of the mixing zone at the distance x~-xk is

.*.= 1. 723 + o.07 (1_0(1) + O.:Z5 (1-a%)2.

A comparison of the lengths of the laminar core is presented inFigure 16.

C. Velocity Distribution Near the Inlet of the Ejector

The velocity distribution of a confined jet flow has been Investi-gated by many authors, such as Cruse and Tontini (Reference 1), Helmbold(References 26, 27, 28) and Squire and Trouncei (Reference 29).

Since the flow is almost parallel and the pressure is constant overany cross section area of the ejector, the simplified form of theboundary layer equations is justified here. The equation of motion is

ULk J U- +-rAt=-16 P _ reax • r x er

(21)

Following the apparent viscosity theory of Prandtl (Reference 21),the shearing stress, " , may be expressed as r)u

(22)

10

where • the so called virtual kinematic viscosity, could he written as

(23)

)j is a constant, and Y is the radius of the boundary of the jet beforethe boundary meets the edges of the boundary layers of the walls of theejector. 'max, in the present report, is the centerline velocity U andamin is the potential velocity Vp outside'the boundary of the spreadingof the jet.

Thus, the shearing stress, following the terminology used in thisreport, could be written as

(24)

Substituting the expression for the shearing stress into theoriginal equation of motion (equation 21), one has

CLa r, i5 t.19 Yr t- ) r-x r r t ,atr'()25)

It seems reasonable to assume the radial velocity component, v,disappears at the outer potential flow zone. Since the axial velocitycomponent is a function of x only, the equation of motion, within thepotential flow zone, can be simplified to

d X C1%(26)

Consequently, the equation of motion can be rewritten as

(27)

As suggested by Helmbold, the axial velocity component, u, may bewritten as

J '(28)

where A)x) U (X)-- L ).

!1

~----~½-.- ~ ~ - ~ - -

I

Substituting equation 28 into equation 27, one has

(U~+*~fl(p' +&u'f + ALf X)+ irANUf~

= U•Uo t (y/29)

where

and

If the law of similarity holds,

and I > d(Cfy

The boundary conditions are as follows:

at r o: =oI hence fx =0

T 0 (30)

at r = yx)fc-o ~f =o

hence i (31)

As a consequence,

at r= 0)(u-P,a uX(U P,+ .U) .polo + • ,Cy(x.*4, u=;r•,--Vo(•- + r•;)

or

(39?)

12

At r Y

or r'r ( J)J=O. (33)

Helmbold had, for the sake of simplicity, suggested the function

r 4-(34)

This function satisfies the boundary conditions of equation 33 and ofequations 30 and 31.

Faris (Reference 15), in studying the free jet mixing, used thevelocity distribution suggested by Cornish (Reference 13):

W- (35)

where n is an empirical function of the axial coordinate, x variesbetween 1.0 and 2.0, and w½ denotes the w where {4- =½"

Equation 35 should be changed so that it can be relevant andcompared with the measurements of the present investigation. Equation35 becomes

U-U-O(36)

it- Ujiwhere wk denotes the w where U== Equation 36 isplotted in Figure 17 for various values of n and is compared withexperimental measurements.

D. Velocity Distribution in the Aft Part of the Ejector

If the ejector is long enough, the flow eventually becomes fullydeveloped turbulent pipe flow. That is, after the mixing process ofthe primary and secondary flows in the inlet length of the mixing tubeand the transition processes, a fluid particle will enter a flow regimewhere the velocity profile is unchanged with respect to the axial dis-tance. This velocity profile can be expressed as a function of the radialcoordinate

( sV, (38)

13

ii

where *=Re-Y- , and m is a functica of the area ratio, A, , and thelength ratio, L/D, or L/D

Experimental measurements and Equation 38 are compared in Figure18. The variation of m with respect to .L/ is plotted in Figure19.AA

14

DIjt:XSSUUiN ANO CQOCLUSIONS

A. Discussion:

The experimental measurements for optimum thrust augmentation factor,velocity ratios, and pressure reduction ratios for the initial plane atvarious area ratios are compared with theory in Figures 12, 13, and 14.The discrepencies between experimental measurement and theory aresmaller for the low area ratios than for the high area ratios. (Figure12). This phenomenon can be explained as follows: frictional losseswere neglectedt in deriving the formula, but in reality, skin frictionlosses were quite influential. For low area ratios, a shorter ejectorwas adequate to produce the optimum thrust augmentation factor; but forhigh area ratios, a longer ejector was needed. Consequently, frictionallosses uere larger for the high area ratios than for the low area ratios.The uptimum thrust augmentation factor obtained in the present investiga-tion was 1.50 when)AI was 105.5.

Experimental results are compared with theory for the velocity ratios,04 , and tKO , in Figure 13. The comparison between theory and

experiments for the pressure variation with respect to area ratio ispresented in Figure 14. The augmentation factor assumed its maximumvalue when the gap ratio was 0.8, as can be seen in Figure 10.

It seems reasonable to divide the flow field in the ejector intothree regimes. (Figure 1). The first regime, wnere the mixing processof the free jet is almost -etained, may be denoted as a modified free-mixing regime. Like the ca~e of a free jet, a laminar core is attachedto the exit of the nozzle. Unlike the case of a free jet, a potentialflow parallel to the axis of the ejector surrounds the laminar core. Anear discontinuity in velocity exists at the rim of the exit of thenozzle. This near discontinuity smoothes out downstream due to theviscous shearing stresses. The mixing zone "survives" the laminar coreand the potential zone downstream, and eventually becomes fully developedturbulent pipe flow. (Figure 1). The wall boundary layer starts at therim of the inlet of the ejector and continues to grow downstream. Theedge of the wall boundary layer meets the boundary of the mixing zoneat a location which can be computed theoretically or measured experimentally.

Theoretically,-2- is 11.0 forAL = 16.0, 15.0 for). = 29.6, 18.0for)-L = 46.7, and 32.6 focL = 105.5. Experimentally, this location wasfound to be close tc the theoretical location (Figure P;). ;-- is notonly a function of the area ratio, but is also affected by the lengthratio, L/De. Schiller, in his theory of inlet length flow of circu)larpipes, did not specify the length of the pipe; however, he implied thatthe pipe length was so great that the conditions at the inlet portion

15

_-JAW..

was no onnier a function of the length of the pipe. The ejectors of thepresent case were not long enough to fulfill the conditions in whichSchiller's theory would be entirely applicable. The length of theejector from the initial plane up to the location where the edge of thewall boundary layer joins the edge of the jet flow spread is denoted asthe modified free-mixing flow regime.

As mentioned above, the flow field eventually becomes fully developedturbulent pipe flow if the ejector is long enough. A fully developedturbulent pipe flow, as denoted in this report, has a velocity profilewhich may be expressed by a fraction power formula,

LL_

where m is a positive real number, z is the distance measured from thewalls, and Re is the radius of the pipe.

The flow field in the ejector, after the potential flow vanishesand before the flow field becomes fully developed turbulent pipe flow,is denoted as the transition regime. The portion of the ejector, fromthe location where the velocity profile becomes that of a turbulent pipeflow to the end of the ejector, is denoted as the pipe flow regime.However, since the velocity profiles only asymptotically approachedthose of fully developed turbulent pipe flow, the length of the transi-tion flow regime could not be distinctly defined.

The measured dimensions for the laminar core and mixing zone arecompared with theory in Figure 16. The dimensions of a confined jet arequite different from those of a free jet (Reference 15). In the presentcase, the dimension of the laminar core is a function of the initialvelocity ratio,cO(t ; consequently, it is a function of area ratio AA-There exists a close correlation between experiment and theory of thesedimensions, (Figure 16). Also, Figure 16 shows close agreement betweenexperiment and theory for the boundary of the mixing zone.

Measured potential velocity is compared with theory in Figure 11.The experimental results for the growth of the wall boundary layer arecompared with theory in Figure 15. In these two figures, the axialdistance was converted into the nondimensionalized product of ( 4j )an Reynolds number where I was based on the radius of theejector Re, and the induced secondary flow velocity UI. In Figure 11,the potential velocity was plotted as the ratio of the difference betweenthe local potential velocity, U , and the induced secondary velocity, UI,to the induced secondary veiocity, UI.

The velocity profiles at different axial locations are shown inFigures 20-2& A comparison of the theoretical and experimental velocityprofiles forAA- - 29.6 are shown in Figure 17 for the modified free-•tixing flow regime. Unlike the case of a free jet flow, the value ci n

16

JL

... , o1 n .. J. .4... .4n o-u -an- _f I n... U..#- x,,anve~4~A ? 0 anvA

approached 5.0, as can be seen in Figure 17e.

The centerline velocities for different area ratios are showndimensionally in Figure 25. They are shown nondimensionally and comparedwith the centerline velocities for free jet flow in Figure 26. Thecenterline velocity for free jet flow is inversely proportional to theaxial distance. (Reference 21). The inverse of the centerline velocityratio, V , plotted against the axial distance ratio, -x shows alinear variation. (Figure 27). For the confined jets in this report,the experimental value of these quantities could be approximated bystraight lines. The formulae representing the straight lines are shownin Figure 27. The slope of these lines, denoted by k, increases witharea ratio,,,," , as shown in Figure 28. Asa. approaches infinity k tendstoward 0.1362, which is the corresponding slope for a free jet. There-fore, the free jet flow might be interpreted as a confined jet flow withvequal to infinity.

The velocity profiles at the outlets of the ejectors were measuredand plotted nondimensionally in Figure 18. The data are compared withtheory. It can be seen that the m values varied proportionally to thelength ratio L/D, and inversely proportionally to the area ratio, andthat the m value varies proportionally to the ratio of the length ratioL/D to the area ratio ,AA.t L L (Figure 19). This phenomenon isexplained as follows: the ratio, V , could be interpreted as anindication of relative importance of the frictional forces in the flowfi-ld; i.e., skin frictional forces per unit momentum issued from thenczzle, and this ratio is the inverse of Reynolds Number based onsignificant variables to be determined. If the skin friction is relative-ly important, the velocity profile is fuller than that for a flow fieldwhere skin friction is not relatively important. The uncertainty raisedherein is an interesting problem for future research.

Unlike the case of a free jet, the pressure was not constant in theflow field, as was measured and plotted in Figure 29.

B. Conclusions:

It is the author's belief that the mixing process within the ejectoris an interchanging of momentum and energy among each stratum of the flowfield. As soon as the ambient air is entrained into the ejector, theinterchanging of momentum and energy occurs. Particles or groups ofparticles in the flow field which have possessed less momentum and energywill recieve some amount of external momentum and energy, and on theother hand, particles or groups of particles which are at higher energyand momentum level will lose some. It is beyond the scope of this report

17

iN

d

to survey exactly the causes and consequences of this interchange ofenergy and momentum.

a. The Structure of the Flow Field in the Inlet Length of theEjector.

1. The jet flow emerges at the exit of the nozzle as a constantvelocity profile, and the secondary induced flow velocity is alsoconstant throughout the rest of the cross section area at the initialplane. There is a near aiscontinuity if velocity at the rim of the exitof the nozzle. This near discontinuity in the velocity profile issmoothed gradually downstream.

z. A laminar core starts at the exit of the nozzle, and itsradius decreases linearly with respect to the axial coordinate until itbecomes a point on the longitudinal axis of the ejector. The length ofthe laminar core was about 6 nozzle diameters for the rresent experiment.Flow within the core is essentially uninfluenced by thte confinement ofthe jet flow. The laminar core in the region of the turbulent-mixingflow can be described by the formula derived by Helmbold.

3. The jet flow starts mixing with the surrounding potential flowat the exit of the nozzle. The potential flow is uninfluenced by theviscous effects of the walls beyond the wall boundary layer. The mixingzone spreads outward linearly until it meets the boundary layer of theejector. The radius of the mixing zone can be expressed as

[t73+o47I()+ o.zSOi-o(4 -I 1+

4. The wall boundary layer of the ejector is infinitely thin at theinitial plane and it grows until meeting the boundary of the turbulentmixing zone. The growth of the wall boundary layer can be described interms of potential flow velocity

Re UJO

5. The velocity profile within the wall boundary layer of theejector assumes the parabolic shape U.L. _

UIP

18

'A

6. Beyond the region of turbulent mixing the potential flowvelocity can be described by the following equation:

7. The velocity profile within the turbulent-mixing region can beexpressed by Cornish's extension of Cole's two-dimensional wake law,

u- "cos ( u--o 5)]U-U W

where n varies from 1.0 to 5.0 beyond the core region and up to thelocation where the boundary of the turbulent mixing meets the edge ofthe boundary layer of the walls. For this equation to be applicablebeyond the laminar core region, w has to be changed to r and w½ to rk.

b. Centerline Velocity of the Ejector.

Centerline velocity is inversely proportional to the axialdistance and can be expressed as u= I

Uo CONSTANT + ( )A

for each value of area ratio.

c. Velocity Profile at the Outlet of the Ejector.

Velocity profile at the outlet of the ejector can be expressedas _

where m is a positive real number.

d. Velocity Ratios and Augmentation Factor.

The augmentation factor, q , and the velocity ratios, .(<and cKZ , can be expressed in terms of area ratio.

1 1 rQ< I Z(•) -0 [, A,,- to.,., so 1z-

19

4U511 VOJ- I

CK2, D 2t),.0J,.- .

For future research, it would be interesting to explore thecharacteristics of ar ejector with different primary and secondaryfluids with different temperature ratios. It would also be very usefulto study the characteristics of this arrangement under dynamic conditions;i.e., with the secondary fluid flowing.

For simplification of design, future studies of the jet ejectorflow should include the possibility of the characteristics of thedevice being a function of only one parameter.

20

LBIBLIOGRAPHY

1. C'use, R. E. and Tontini, R., Research on Coaxial Jet Mixing,(Gen~ral Dynaiucrs: Convair, GD/C-62-354A), (November 1962).

2. Flugel, G., The Design of Jet Pumps, (National Advisory Committeefor Aerorautics, T. M. No. 982), (July 1941).

3. Fox, N. L., Analytical Solution for Gross Thrust Change, (DouglasCompany, Report No. SM-13881), (December 1950).

4. McClintock, F. A. and Hood, J. H., Aircraft Ejector Performance,(Journal of the Aeronautical Sciences, Vol. 13), (November 1946).

5. Reid, J., Che Effect of a Cylindrical Shroud on the Performance ofa Stationa I Convergent Nozzle, (Royal Aircraft Establishment, ReportAero. 2559), (January 1962).

6. von Karman, '., Theoretical Remarks on thrust Augmentation, (ReissnerAnniversary Volume, J. W. Edwards, Ann Arbor, Michigan), (1949).

7. Rabenneck, G. I., Shumpert, P. K., and Sutton, J. F., Steady FlowElector Research Pr,',ram, (Lockheed), (December 1960).

8. Keenan, J. H., Neumann, E. P., and Lustwerk, F., An Investi,'ation <fEjector DesiRjnb Axyaly~! and Experiment, (MIT Guided MissllfsProgSram), (jtune !9-ýt-).

9. Lockwood, R. M., Investigation of the Process of Energy TransferFrom an Intermittent Jet to an Ambient Fluid - Summary Rep_(Hiller Aircraft Corp., Report No. ARD-238), (June fl°9).

10. Wells, W. G., Theoretical and Experimental Investigation of a HiyhPerformance Jet Pump Utilizing Boundary Layer Control, (MississippiState University, Aerophysics Department, Research Report No. 30),(June 1960).

11. Wagner, F. and McCune, C. J., A Progress Report on Jet Pu2£Research,(University of Wichita, En-ineering Report No. 085), (October 1952).

12. Wood, R. D., Theoretical Eictor Performance and Comparison WithExperimental Results, (Wadc TR 54-556), (August 1954).

13. Cornish, J. J., A Universal Description of Turbulent Boundary LayerProfiles With or Without Transpiration, (Mississippi State University,Aerophysics Department, Research Report No. 29), (June 1960).

21

14. Kuethe, A. M., Investigation of the Turbulent Mixing Regions Formed-.. T-- IT 4...... AW14aA UMmn"4^a unl 9 tnz 1), (Sentemher

1935).

15. Faris, G. N., Scme Entrainment Properties of a - bulent Axi-SymmetricJet, (Mississippi S ate University, Aerophysic _,u eartment, ResearchReport No. 39), (January 1963).

16. Tollmien, W,, Calculation of Turbulent Expansion Processes, (NationalAdvisory Cummittee of Aeronautics TM 1085), (September 1945).

17. Davies, P. 0. A. L., Barrett, M. J., and Fisher, M. J., Turbulence inthe Mixing Region of a Round Jet, (Aeronautical Research Council,ARC 23728), (April 1962).

18. Pai, S. I., Fluid Dynamics of Jets, New York, D. Van Nostrand, 1954,

19. Eckert, E. R. G. and Drake, Robert K., Heat and Mass Transfer, 2ndEdition, New York, McGraw-Hill, 1959.

20. Townsend, A. A., The Structure of Turbulent Shear Flow, Cambridge,1956.

21. Schlichting, H., Boundary Layer Theory, 4th Edition, New York, McGraw-Hill, 1960.

22. Goldstein, S., Modern Developments in Fluid Dynamics, 2 volumes,Oxford, 1938.

23. Langhaar, H. L., Steady Flow in the Transition Length of a StraightTube, (Journal of Applied Mechanics), (June 1942).

24. Schiller, L., Die Entwicklung der Laminaren Geschwindigkeitsuerteilungund ihre Bedentung fUTr Zhigkeitsmessungen, (Zeitschrift fur AngewandteMathematik und Mechanik, 1922).

2i. Talbot, L., Laminar Swirling Pipe Flow, (Journal of Applied Mechanics,Vol. 21), (1954).

20. Helmbold, H. B., Contribution to Jet Pump Theory, (University ofWichita Report No. 294), (September 1957).

2-'. Helmbold, H. B., Luessen, G., and Heinrich, A. M., An ExperimentalComparison of Constant-Pressure and Constant-Diameter Jet Pumps,(University of Wichita, Engineering Report No. 147), (July 1954).

22

28. Helmbold, H. B. , Energy Transfer by Turbulent Mix, Undera Longi-tudinal Pressure Gradient, (UniverRaty of uihi..., Enovin-ezng Study.182), (August 1955).

29. Squire, H. B. and Troumer, J., Round Jet in a General Stream, Citedin Reference 1.

23

___ ~Ji

A. --

60

~-1 00

244

% rN

q-1iK

0.

••l ,0

02-v4

I4'r4

N(

25

Figure 3a. Thrust Force Measuring Unit.

Figure 3b. Jet Flow Unit, Nozzle, and Supporting Rig.

26

St,/

I

Figure 4. Extension for the Nozzle Exit.

Figure 5. Pitot Tube Probe and Traversing Mechanism.

27

,- A. I

____ __ __ A- � -

-'A

1.1

4.'

28)

I I

0.140.12

0./0

S0.12ko.io - _ _ _

= 0.2 0"

0.06 _

0.04

0.02 ... .... . .

0-/0 20 30 40 1,0 60

Figure 7. Thrust Forces of Nozzles.

29

0

30

n0~w n n l al U ) ~ l m l m -- - - ~

p-

* 0

of 4 5

+ 0

00

I: 0

INNJ

0+0 "4I4J44

311

/ 0

/11

000 -,

44'

I 0% 0Ai

00

+00

32

I

0.07 1~0 o t& /0,4 a.'

0.06 0 r.4 470 o+ U /6.0 0

SA.A= 2.9.4

0.045

JcW/4LEO-I0.03

0.02

0.0/

0 o.2 0.4 0o o. /.

Figure 11. Comparison of Theoretical

and Experimental Potential Velocity.

33

MILO J; Irw-arc.

-- v

rz4

00

34J

4j4

r-4-

~~co

____ -4-k~4j I It

.000, -,_t

mec-l0--oo -Iti4

_ _ _I 035

0.

0*

:3U)ca

4,- - - - - 0 W.

"r4

oto- U

140

0 0 cc

-4 iP.-

0 0r. .4

-. 4 41

f0

to

""400-.4

36

-4

c o

0 0

0 )

U

0 04

-o 04

37,0

0

cSa

aio* X44S 0

'-4w030

r-4E

-- 4

370

\- .. ,,{

v ca

I' " 03

ca

,-4

•03

5-a0

U.

%c 03

cc NS• 'X o

4.j .-A

04-4 .

.- ,c-a

44

XIr

38

W4

- "43

S. •

-•/-I--,,I-I-°

- '1[

I o I -• .I ' _ • , •. - C

543

0Ia~ ~ .ii\ !

","i7 ' 4=°-4

0

C,'-,'

t9 -- 60

$.4

394

2-4

~ ~ (.,l-,

'.0 -.. -

0.4 +- /Z.7 3 . o Y. /6 AR3

a /6. 0 JO. 0 4.07 #a 2V@.7 000 X.o ,70 /V0 46.7 4Z.S A/.O S,

/Pr S' (d. 4 .)( a

o.2 '

0 "

A- h0 0., 04 06 0 J /0 3/.,/

Figure 18.

Comparison of Theoretical and Experimental Velocity Profiles at Outlet

of Ejector.

00.0

za.O O 0

/-/

I47..5 00. -- Y

0.

0 -0-

0 A. .2.O J.O 4. 0

SFigure 19. Effect of8.

40

, 40

I

A•- -.. . . . . . . . . .

I~- /A 7 /r.

7- 0

Re

1.1

-2.44 J, . 7 1

Re

Z2

o i 2 . 4 .-6 .7 .

Figure 20. Velocity Profile*, JA=11.7.

41

ILII

0 "A .Ys"0D..Mrs

/0

0~C A1 - O,. ~ v

Figure 21. Velocity Profiles, U =16.0.

42

F-7- 1 1

pe

LaL

0~~.

Figure 22. Velocity P~ofiles, /1 47.0. -

43

,4. J • i iI-°- --- i_ _

SI Ieaa

.2

0 .4 2 ,3 1 , .6 .7 *J *1 /o

/"X• . 1 I I s ~ ~ 15b o ,:o o-'7"•

Re

4

8 - I(--- -r ,- ,

/2 I

I _ __ _ _

0 ._-2 , 4 .- " .6___

"W,

Figure 23. Velocity Profiles, /- 105.5.

44

g! I Ie s Ig d IItII

45

i 1 -i

x. oJ<•...•4-a

I0 0

$4-

-0

C 14

L.,

~ .Si ~ t7

46i

I 0

* 41

* 0

Ii.i

CL 41

10

v-4 CC-Sb 44

41

1__ 1_ 441

4+h 0to41

0r44

C-40

"U4.

I~ %1

/47

1 ½ 1

U

C4.

-40)

r-)

.r 0u

41 V1

IV I.0i

4* ar41~U v0.

8 41

484

_ ~ .-- -- -- - - - - - - ~ -

Ir0

04

0

0

0

49z

0* o0

of I 0I

0440

co 04U4

0)

-4.

141

50

-. --i- -

TABLE 1

THE DIMENSIONS OF THE EXTENSIONS OF NOZZLES

Nozzle No. Inside Diameter of Exits Extension Lengths(inches) (inches)

1 0.304 0.54, 1.15, 2.12, 4.13

2 0.520 0.60, 1.10, 2.10, 4.15

TABLE 2

THE DIMENSIONS OF THE EJECTORS

Ejector No. Inside Diameter No. of Short Length(inches) Ejectors (inches)

1 1.04 5 11.40

2 2.08 6 15.60

3 3.12 6 17.15

4 2.82 42.00

51

TABLE 3

I AXIAL LOCATIONS WHERE VELOCITYDISTRIBUTIONS WERE MEASUREDIm

Ejector No. Axial Distance from Initial Plane in Inches

1 0.00, 1.14, 2.14, 3.14, 5.14, 7.14, 9.24, 11.14

2 0.00, 1.16, 2.16, 3.16, 5.16, 7.16, 9.16, 11.16,13.16, 15.20

3 0.00, 1.00, 2.03, 3.01, 5.00, 9.02, 11.02, 13.00

4 At an interval of 0.25 inches for the first 12,stations, 0.50 inches for the next 28 stations,and 2.00 inches for the last 6 stations.

5

52