49

tl 4

v

D%, ftia'l

EFFECT OF SWIRL ON TURBULENT JETS IN DUCTED STREAMS

by

Harmon Lindsay Morton

Under the Sponsorship of:

General Electric Ccmpany

Allison Division of General Motors Company

GAS TURBINE LABORATORY

REPORT No. 95

December 1968

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Cambridge, Massachusetts

ii.

ABSTRACT

An experimental and theoretical program of research has been

carried out to determine the effect of swirl on the mixing of a

turbulent jet in a ducted stream. The term swirl is used to

describe a flow pattern within the jet where mean streamlines are

spirals. It has been found experimentally that the mixing rate of

a turbulent jet with a surrounding stream can be increased by a

factor of three due to the presence of swirl. Three dimensionless

parameters, one of which is the jet nozzle to test section dia-

meter ratio, have been used to describe the experimental results.

The well-known integral technique has been used to predict

the mean flow field of the turbulent jet in a ducted stream.

Velocity profiles and turbulent eddy viscosity have been evaluated

from turbulent free jet data. The increased mixing due to swirl

has been explained in part by the adverse pressure gradient along

the jet centerline associated with a decaying swirl and also by

an increase in the magnitude of the Reynolds shear stresses within

the jet. The calculation procedure divides the flow into two

regions - one before the jet attaches to the wall and one after.

Two possible correlations giving first order effects of swirl

strength on local eddy viscosity are found to give good results

for both the ducted jet and the free jet (a jet exhausting into a

still fluid). Comparison of data with theory for the axial

position of jet attachment is good for all cases studied.

MITLibraries 77 Massachusetts Avenue Cambridge, MA 02139 http://libraries.mit.edu/ask

DISCLAIMER NOTICE

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111

iv.

ACKNOWLEDGMENTS

The author gratefully acknowledges the suggestions and guidance of

his thesis committee: Professor Philip G. Hill, thesis supervisor, whose

advice and previous work in jet flows without swirl served as a basis

for this study; Professor T. Y. Toong, Professor Joseph L. Smith, and

Professor David G. Wilson. The author is also indebted to Professor

Edward S. Taylor, Director of the Gas Turbine Laboratory, who surved

as an unofficial committee member and offered many valuable suggestions

during the course of the work.

Thanks are given to all the members of the GTL staff for their help

and the many enjoyable coffee breaks shared together. Two people who

must be mentioned are Mr. Thorwald Christensen who helped with the

design and construction of the experimental apparatus and Mrs. Lotte

Gopalakrishnan who typed the manuscript.

The author gives special acknowledgment to his wife, Gail, who

spent many hours typing the rough draft and drawing the final figures.

V.

TABLE OF CONTENTS

page

Abstract ii

Acknowledgments iv

Table of Contents V

List of Figures vii

Nomenclature X

I. INTRODUCTION 1

A. Object of this study 1

B. Effect of swirl on the turbulent free jet 1

C. Basic equations 5

D. Existing theories 9

II. EXPERIMENTAL PROGRAM 14

A. Preliminary remarks 14

B. Test Apparatus 16

C. Measurement of m 17

D. Measurement of M 18

E. Measurement of H 21

F. Measurement of wall static pressure 22

G. Measurement of mean velocity and pressure profiles 22

H. Experimental results 24

I. Effect of L/D 27

J. Effect of d/D 27

K. Applications 28

III. ANALYSIS 30

A. Preliminary remarks 30

B. Velocity profiles

C. Turbulent shear stress

D. Ducted jet equations

E. Initial conditions

F. Comparison of prediction with experimental results

IV. SUMMARY AND CONCLUSIONS

V. RECOMMENDATIONS FOR FURTHER STUDY

REFERENCES

APPENDIX A - Calculations of Reynolds shear stress distrib

APPENDIX

TABLE I

TABLE II

TABLE III

TABLE IV

in the turbulent free jet

B - Equations of motion for a swirling turbulent jet

in a ducted stream

Equations before jet attachment

Equations after jet attachment

Integrals of ducted jet velocity profiles

Integrals of free jet velocity profiles

vi.

page

31

33

35

38

39

42

45

46

ut ion

48

52

60

62

66

69

vii.

LIST OF FIGURES

1. Mean Axial Velocity Profile in a Turbulent Free Jet

2. Mean Tangential Velocity Profile in a Turbulent Free Jet

3. Spreading Rate of a Turbulent Free Jet

4. Mass Entrainment Rate of a Turbulent Free Jet

5. Half-Velocity Thickness of Turbulent Free Jet

6. Half-Velocity Thickness of Turbulent Free Jet

7. Reynolds Shear Stress in the Turbulent Free Jet

8. Eddy Viscosity in the Turbulent Free Jet

9. Schematic Diagram of Test Apparatus

10. Test Section

(a) 6-1/2 inch Diameter

(b) 3-7/16 inch Diameter

(c) Jet Nozzle

11. Swirl Generators

12. Probes

13. Instrumentation and Traversing Mechanism

14. Thrust Balance

15. Determination of Jet Thrust

16. Torque Balance

17. Calibration of + 0.05 psi Pressure Transducer

18. Calibration of + 0.20 psi Pressure Transducer

19. Calibration of + 2.00 psi Pressure Transducer

20. Wedge Probe Calibration Curve

21. Sphere-Static Probe Calibration

viii.

22. Wall Pressure Distribution

(a) No Swirl

(b) Series A

(c) Series B

(d) Series C

(e) Series D

(f) Series E

(g) Series F

23. Axial Velocity Profiles

(a) Series A; x/D = 1-1/2

(b) Series A; x/D = 2-1/2

(c) Series A; x/D = 3-1/2

(d) Series A; x/D = 5-1/2

(e) Series D; x/D = 1-1/2

(f) Series D; x/D = 2-1/2

(g) Series D; x/D = 3-1/2

(h) Series D; x/D = 5-1/2

24. Tangential Velocity Profiles

(a) Series A; S = 0.106

(b) Series A; S = 0.190

(c) Series D; S = 0.106

(d) Series D; S = 0.190

25. Axial Position of Jet Attachment

26. Axial Position of Jet Attachment

27. Qualitative Hot-Wire Anemometer Measurements

28. Effect of Diameter Ratio - No Swirl

ix.

29. Effect of Diameter Ratio - Weak Swirl

30. Effect of Diameter Ratio - Strong Swirl

31. Effect of Wall Boundary Layer on Wall Pressure Distribution

32. Wall Boundary Layer Prediction - Moses

33. Effect of Diameter Ratio - Weak Swirl

34. Velocity Near the Wall

35. Jet Excess Velocity at Centerline

36. Maximum Tangential Velocity

37. Predicted Departure from Self-Preservation

38. Swirl Generator Pressure Drop vs. Percentage Reduction ofJet Attachment Length

x.

NOMENCLATURE

A Coefficients in Tables I and II

B Coefficients in Tables I and II

b Half-velocity thickness

c Value of n where maximum tangential velocity occurs

C Free jet velocity profile integrals (Table IV)

C Wall friction coefficient, T /T p U0

Cf Wall friction coefficient, T p W2

d diameter of jet nozzle

D Duct diameter

f(n) Dimensionless axial velocity profile for free jet,

U/U (Figure 1)

F Jet thrust (equation 23)

F(n) Function giving departure of axial velocity profile from

self-preservation (equation 28)

g(n) Dimensionless tangential velocity W/W (equation 30)

H Angular momentum parameter (equation 20)

H12 Wall boundary layer shape factor

L Length of duct

m Total mass flow per unit area (equation 18)

M Momentum parameter (equation 19)

p Pressure

P p + p vt 2

P Initial stagnation pressure of secondary stream

r Radial distance

R Radius of flow area

xi.

R D/2

RT U 6/VT

Rey Reynolds number

S Free jet swirl parameter, 2H/Fd

U Mean axial velocity

V Mean radial velocity

W Mean tangential velocity

U'1 Axial velocity fluctuation

v' Radial velocity fluctuation

wl Tangential velocity fluctuation

w Primary mass flow ratep

x Axial distance

X 0Axial position of jet attachment

Y WI/U

6 2.6 b

A Wall boundary layer thickness

Al Wall boundary layer displacement thickness

n r/6

UO/U

v Kinematic viscosity

VT Eddy viscosity (equations 31 and 32)

& Axial velocity profile shape parameter (equation 27)

p Density

T Wall friction in axial directionw,x

T WWall friction in tangential direction

Value of pu'v' at distance A from wall

Ducted jet velocity profile integrals (Table III)

Subscripts

c Characteristic value

i Initial value

j Maximum value at a given axial location

o Value at edge of shear region

w Value at wall

TA

0

xii.

1.

I. INTRODUCTION

A. Object of this Study

Application of experimental and theoretical information to practical

devices such as jet pumps, combustion chambers, and thrust augmentors

has stimulated considerable research in the area of turbulent jet mixing.

Extensive bibliographies of work in this field prior to 1956 have been

given by Forstall and Shapiro and also by Krzywoblocki . A more recent

bibliography has been prepared by Seddon and Dyke . Previous experiments

indicate that some of the factors which influence the mixing region of a

turbulent jet are the flow velocity outside the mixing region 1, the ratio

of the jet fluid density to that of the surrounding fluid , and swirl5$

that is, a flow field where mean streamlines are spirals. The object of

this study is to examine, both theoretically and experimentally, the

effect of swirl on the turbulent jet immersed in a ducted stream. The

fundamental problem, as with any turbulent shear flow, is one of relating

the turbulent Reynolds stresses to the mean flow field.

B. Effect of Swirl on the Turbulent Free Jet

The effect of swirl on a much simpler turbulent jet flow has already

been determined. Recent experiments have indicated that swirl has a

considerable effect on the turbulent free jet. Rose6 used a hot wire

anemometer to measure the mean velocity field and turbulence intensities

in a turbulent free jet issuing from a long, rotating pipe. At a pipe

rotation speed of 9500 rpm, he found that the spreading rate of the

half-velocity thickness, that is, the radius where the mean axial velocity

is one-half its centerline value, was 1.7 times that of the non-swirling

turbulent free jet.

2.

Kerr and Fraser5 studied the effect of swirl strength on jet spreading,

mass entrainment, and mean centerline velocity decay. Swirl was created

by turning vanes located in an annular region of the jet nozzle and the

mean velocity field was measured using a three dimensional pitot probe.

They found that at a fixed axial position the mass entrainment and the

half-velocity thickness vary linearly with a dimensionless swirl para-

meter. This parameter was defined as the ratio of the torque necessary to

produce the swirl to the product of jet thrust and nozzle diameter.

Numerical values of the swirl parameter varied from 0.0 to 0.53.

Chigier and Chervinsky7 produced a swirling free jet by introducing

air both axially and tangentially at a fixed angle into a chamber located

just before the jet nozzle. The degree of swirl was determined by the

ratio of the two flow rates. The mean velocity and static pressure fields

were measured with a 5-hole spherical impact probe. They found that the

swirl parameter defined by Kerr and Fraser5 could also be used to correlate

their data and that of Rose6 even though the nozzle diameters of the three

investigations varied from 13.5 mm to 100 mm.

Craya and Darrigol made an extensive investigation of both isothermal

and heated swirling turbulent free jets with swirl parameters of 0 to 0.79.

Measurements were made of both the mean and fluctuating kinematic and

thermal fields. They found that in the region from 0 to 20 nozzle dia-

meters, the kinematic and thermal expansion of a jet with strong swirl is

much more rapid than in a jet without swirl. In addition, it was discovered

that the presence of swirl caused a noticeable increase in the intensity

of U'2 over its no swirl value, and that the energy spectrum of u' revealed

a K-5/3 zone as usual.

3.

A comparison of the results of these free jet studies indicates quite

consistent findings for the effect of swirl on the turbulent free jet.

Figure 1 shows mean axial velocity profile data for the turbulent free jet

both with and without swirl. These data indicate that beyond about ten

diameters from the orifice the mean axial velocity assumes a self-preserv-

ing form which is independent of the manner in which swirl is generated

as well as the strength of the swirl. Figure 2 gives the available data

on the mean tangential velocity profile in the turbulent free jet.

Although the data do not clearly indicate a single profile, there appears

to be no easily discernible trend with increasing swirl strength. The

assumption of self-preservation for the mean tangential velocity profile

seems justifiable with the observed scatter in the data being attributed

to experimental error. The effect of swirl on the turbulent free jet is

more evident in the data on half-velocity thickness as shown in Figure 3

and the data on mass entrainment as shown in Figure 4. It can be seen

that for a swirl parameter of 0.53, which is the largest value studied

experimentally by Kerr and Fraser5 , the half-velocity thickness and the

mass entrainment rate are more than 3-1/2 times their no-swirl value.

It is possible to identify two differences between swirling and non-

swirling jet flows. One difference is the adverse axial pressure gradient

which can be created by a decaying swirl. The other is the influence of

the tangential velocity profile on the turbulent velocity fluctuations and

thus on the turbulent Reynolds stresses. The measurements of Craya and

Darrigol indicate that this latter effect is a significant one. The

Reynolds shear stresses u'v' and v'w' can be computed for the self-preserv-

ing turbulent free jet by integrating the axial and tangential momentum

4.

equations, respectively, with respect to radius. The integrations are

carried out in Appendix A for the downstream region of the jet (x/D>l0)

where W /U is small compared with unity, and thus pressure forces are

vanishingly small. The result for u'v' is:

where q is the dimensionless radius r/6, and 6 is arbitrarily chosen as

2.6 times the half-velocity thickness. The dimensionless velocity profile

f(n) is that given by the solid line in Figure 1.

To evaluate the distribution of u'v' for the swirling turbulent free

jet, it is necessary to know db/dx in the downstream region as a function

of swirl strength. It appears that one possible correlation for the

value of db/dx in the downstream region of the swirling jet could be:

jb =(Qg4 + 0.30 S (2)

j X

This correlation, however, does not depend entirely on local properties

since the swirl parameter, S, contains the nozzle diameter in its de-

nominator. Another possible correlation which specifies the downstream

spreading rate entirely in terms of local properties would be:

d b 0 94 0.19 d (3)dx b

If a correlation for db/dx is substituted into equation (1) above, the

resulting correlation for u'v' can be used along with the momentum

integral equation, the angular momentum integral equation, and the moment

of momentum integral equation, which are given in Appendix B, to calculate

b/d versus x/d. The results of this calculation using both equations (2)

5.

and (3) are shown in Figure 5 and 6 along with the available data. The

virtual origin of the predicted values has been placed at x = 0, but the

uncertainty of its axial position is at least one nozzle diameter.

Figure 7 shows the effect of swirl strength on the u'v' distribution as

computed from equations (1) and (2). The effect of swirl strength on

eddy viscosity as defined by

T ~ L/cr (4)

has been computed from equations (1) and (2) and is shown in Figure 8.

Figures 7 and 8 indicate that the computed effect of swirl on the

Reynolds shear stress is a significant one which can increase their

magnitude by a factor of three.

C. Basic Equations

The Reynolds equations of motion for a steady, axisymmetric turbulent

flow are:

Continuity

+0

Axial momentum

Radial momentum

+ (7,W7U)+r TX Y-

6.

Tangential momentum

Since (vt 2/U2 - u'2/U2 ) < 0.1 except at the very edges of the jet, the

normal stress term is neglected in the axial momentum equation. We can

rewrite the equations of motion in dimensionless form by referring all

velocities to a characteristic velocity, Uc, and all linear dimensions

to a characteristic length, L. The dimensionless variables become:

0= U/U

= V/Uc

= W/U

I = x/L

= r/L

4 ~pThP0)/pU,

where P is a reference pressure which is independent of x and r. The

characteristic velocity, Uc, is chosen so that the dimensionless velocity,

A, does not exceed unity and the characteristic length, L, is chosen so

that the dimensionless derivative, aL/a2, also does not exceed unity.

The dimensionless equations of motion become:

a0 t _+--X V )=x a(5)

A + _ _ A u/' L 2 V' 6

7.

A A A A

A ,- + V --- +712- ---\A-/-1e a

C_

A ,.

V A - (8V\/ IW)UW

X r4r42

These equations of motion can be simplified considerably by re-

cognizing that the turbulent jet is a free turbulent shear flow of a

boundary layer nature for which the characteristic width, 6, is much

less than the characteristic length, L.

Equation (5) indicates that V has the same order of magnitude as 6,

which is much less than one. Equation (7) can be simplified by examin-

ing the order of magnitude of its terms. It is clear that the terms on

the right hand side of equation (7) are important only in o far as

they affect the axial pressure gradient term in equation (6) which has

an order of magnitude of 5. Using equation (7) to express the order of

magnitude of D:

-=- V Vl*2.

It is seen that only the term of order W2 on the right hand side of

equation (7) is important when compared with the order of magnitude of

the terms in equation (6), and even that term becomes unimportant as W

becomes much less than U. Thus equation (7) reduces to the condition of

radial equilibrium for the case of the swirling turbulent jet.

8.

The u'w' term in equation (8) is a Reynolds stress which represents

the flux of angular momentum due to the fluctuating velocity field. The

total angular momentum flux is given by the expression:

e(U~W+ 1~).2 7Tr-j r

The experimental results of Kerr and Fraser5 indicate that the ratio

u'w'/UW is less than eight per cent, and thus the u'w' term in equation

(8) is quite small in comparison with the other terms in that equation

and can be neglected.

These simplifications allow the equations of motion for the swirling

turbulent jet to be written as:

( U r) + \ r(9)3X )

/ (r

-- r4

(10)

+ VIZ(11)

L 3VV+ V C) (fA/WVr)- __

(12)

Equations (9) through (12) are the basic equations of motion used to

describe the swirling turbulent jet in this investigation.

9.

D. Existing Theories

Turbulent jets and wakes represent a class of flows known as free

turbulent boundary layers and are of a boundary layer nature with a charac-

teristic width which is much less than their characteristic length. The

integral technique which has proved so useful in the case of the turbulent

wall boundary layer is also equally useful in predicting the turbulent

jet. Since all experimental data indicate that the turbulent jet is

approximately self-preserving, the mean axial velocity profile can be

completely described in terms of the centerline excess velocity, U, and

the half-velocity thickness, b. One of the two equations necessary to

predict U and b is obtained by integrating the axial momentum equation

across the entire mixing region - the mcmentum integral equation. There

are several theories available in the literature which furnish the second

equation necessary to predict the mean axial velocity profile. In all

cases, the proposed second equation contains at least one arbitrary

constant whose value is chosen so that the theory agrees with the well-

documented case of the turbulent free jet. The theories are then used to

predict the more complicated case of the turbulent jet in a streaming

flow with a pressure gradient.

An excellent review of the existing theories for predicting the mean

flow field in turbulent jets and wakes is presented in the section of

Fluid Mechanics of Internal Flow9 written by B. G. Newman. In this review

Newman found that for the turbulent jet three methods give satisfactory

agreement with experimental data provided that the free stream velocity

is less than 0.3 times the jet nozzle velocity. A summary of these

three methods is presented here.

10.

Abramovich10 applies an intuitive argument to the mixing region

between two uniform flows of different velocities, U and (U + U ), to

arrive at an equation for spreading rate in terms of those two velocities.

He reasons that the spreading rate should depend on the level of

turbulence in the mixing region which is proportional to the velocity

difference U .

db -

Since the spreading rate is the same if the velocities of the two uniform

streams are interchanged, the spreading rate should also be given by

d b - u-

d X UO+ Uj)

Thus the most general expression for spreading rate of the mixing region

between two uniform flows of different velocities is:

db Fi Ul

Expanding this relation as a power series:

j L I U! L

dx A + A0 + - -

For the case where U = 0 there is no mixing region and db/dx = 0; and

thus A = 0. If only the first order term is retained,

a DI ~ U.0 (13)

dxAI ~ ULU.& = Aj

11.

This analysis is extended to the case of the turbulent jet by assuming

that U may be thought of as the jet centerline excess velocity. For the

two-dimensional turbulent free jet Al = 0.0425. Thus the momentum integral

equation and equation (13) are solved simultaneously to give U and b as

functions of x in the method of Abramovich.

The integral moment-of-momentum equation or the integral energy

equation can be used along with the momentum integral equation to predict

U and b for a turbulent jet in a streaming flow provided the Reynolds

stresses can be specified in terms of the mean flow field. One method

of specifying the Reynolds shear stress u'v', which forms the basis of

the theory by Hill1 1 ' 12, is to assume the eddy viscosity Reynolds number

S= Us b

is a constant throughout the entire flow field. The value of the eddy

viscosity Reynolds number is chosen to agree with that of the turbulent

free jet. For the two dimensional turbulent free jet

R - 3 3 (14)

while for the round turbulent free jet with no swirl

- = 45 (15)

Another method of specifying the turbulent Reynolds stress u'v' which

forms the basis of theories by Bradbury13 and also Gartshore14 allows

the eddy viscosity Reynolds number to vary with distance downstream. The

variation of RT is based on Townsend's large eddy hypothesis16 which was

originally proposed to explain the differences in eddy viscosity Reynolds

12.

numbers found in the various turbulent self-preserving flows. These

theories allow the value of RT to approach that of the small perturbation

wake as the strain rate ratio given by

C- U /ax

at a typical point in the outer region of the jet approaches zero. The

expression for RT given by Gartshore14 for the two-dimensional case is:

= .077 - (16)

The expression for RT given by Bradbury13 for the two-dimensional case

is:

= 0.062 CexpThis expression can be simplified since the strain rate ratio is typically

less than 0.20 to:

Oro 5 3 ](17)

Equations (16) and (17) are the same except for the numerical values of

the two arbitrary constants in each one. Bradbury and Riley15 show that

for a two-dimensional turbulent jet issuing into a parallel moving air-

stream, the value of RT as measured with a hot-wire anemometer changed

from 33 to 24 while the predicted change was from 33 to 19.

Newman9 has compared the predictions of these three methods with

13.

data available in the literature. As expected from the measurements of

Bradbury and Riley 5, the data for the turbulent jet lie in between the

prediction based on a constant RT and that allowing RT to vary with x.

The method of Abramovich appears to agree more closely with the theory

based on a constant value of RT. All three theories agreed satis-

factorily with the data with no one appearing to be significantly better

than the others.

14.

II. EXPERIMENTAL PROGRAM

A. Preliminary Remarks

The objective of the experimental program was to investigate the

effects of free stream velocities and axial pressure gradients on the mean

flow field of the incompressible, swirling, turbulent jet. The test

section was a circular duct with a constant diameter of 6.5 inches and

a length of 70 inches. The jet nozzle diameter was 1.2 inches. The

working fluid was air. It was possible to vary the relative importance

of the effects of free stream velocity and free stream pressure gradient;

however, both effects were present in all test runs.

It is possible to define three flow constants for the purpose of

describing experimental data. The first constant, m, is the mass flow

per unit area.

Tr R = \ jU 27r r (18>Jo

The thrust per unit area and the angular momentum flux per unit area are

reduced only slightly in magnitude due to wall friction as x/D varies

from 0 to 10, and their values at x/D = 0 make convenient descriptors of

the flow. The quantity M can therefore be defined by the expression

r R2 M f-) )+)l U,'+- 2 rdr (19)

where P is the stagnation pressure of the secondary stream and the

integral is evaluated at the initial plane of the test section. The

angular momentum flux per unit area, H, is defined by the expression

15.

T P2 H =- wwW & 2 Tr r2-dr (00(20)

where again the integral is evaluated at the test section initial plane.

The three constants m, M, and H and the test section diameter, D, can be

combined to form two dimensionless parameters:

/M H(piV)B/2 MD

The parameter m/(pM)1/2 can be interpreted physically as the ratio of a

mass averaged velocity to a thrust averaged velocity, and its value

should normally lie between 0 and 1. A value of zero corresponds to the

case of a jet exhausting into a duct whose far downstream end is closed,

and thus there is no net mass flow. A value of m/(pM)1/2 approaching

unity corresponds to the case where the excess thrust of the jet is

negligible in comparison with the total thrust of the entire flow and

most closely resembles the case of the turbulent jet in a streaming flow

with no free stream pressure gradient. The parameter m/(pM)1/2 is

uniquely related to variables proposed by Curtet and Craya and also by

Spalding 8, and was used by Hilll1, 12 to completely describe the mean

flow field of the non-swirling ducted turbulent jet whose nozzle dia-

meter to duct diameter ratio was small in comparison with unity.

Wall pressure data have been taken for six series of cases in each

of which m/(pM)1/2 was held fixed and H/MD was varied to reveal the effect

of swirl. The six values of m/(pM)1/2at which wall pressure data have

been measured are: 0.62, 0.57, 0.54, 0.495, 0.44, and 0.39. In addition,

detailed profiles of mean axial velocity and mean tangential

16.

velocity have been measured at seven downstream stations for m/(pM)1/2

equal to 0.62 and 0.495 with the effect of swirl determined by varying

H/MD. Finally, the effects of L/D and d/D on wall static pressure have

been measured for a few well-chosen cases in order to determine whether

m/(pM)1/2 and H/MD are sufficient to completely specify the mean flow

field for the enclosed swirling turbulent jet.

B. Test Apparatus

The test section geometry is that of an ordinary constant diameter

jet pump. A schematic diagram of the test apparatus is shown in

Figure 8. The primary air flow was supplied by an oil-free, two-stage,

piston type compressor capable of supplying 450 c.f.m. at 125 psi. A

steady primary air flow of 0.097 pounds per second was obtained by

bleeding some of the compressor air out to the atmosphere so that the

compressor operated continuously. The secondary flow entered directly

from the atmosphere through a radial inlet. The secondary flow rate,

and thus, the parameter m/(pM)1/2 , was controlled by an adjustable end

plate which was situated in front of the downstream end of the test

section as shown in Figure 9. Annular vanes of constant chord and angle

similar to those used by Kerr and Fraser5 were located inside the jet

nozzle to generate the swirl. The four swirl generators used, which are

shown in Figure 11, each had eight vanes with a hub to tip ratio of 0.42

and vane angles of 6-1/2, 15, 30, and 38 degrees. The traversing

mechanism shown in Figure 13 was driven by a small variable speed d.c.

motor and could travel a maximum distance of 6.1 inches. A threaded rod

with twenty threads per inch controlled the position of the probe mount

and made 12 revolutions for every one revolution of the ten turn potentio-

meter used to measure radial position.

17.

Instrumentation used to record experimental data are shown in Figure

13. Measurements of radial profiles of velocities and pressures were

recorded on a Moseley Autograf X-Y Plotter. The X-axis recorded the

voltage signal from the potentiometer located on the traversing mechanism,

while the Y-axis recorded the voltage corresponding to the quantity being

measured. The magnitudes of fluctuating voltages were measured with a

Hewlett Packard model 801 R.M.S. voltmeter. A transistorized, constant

temperature, hot-wire anemometer manufactured by Leslie T. Miller of

Baltimore, Maryland, was used to obtain qualitative data in the edge of

the swirling jet. The hot-wire anemometer produced a signal directly

proportional to the velocity being measured, and this signal was

monitored on a Tektronix type 555 dual beam oscilloscope which was

equipped with a type 1L5 spectrum analyzer plug-in unit.

C. Measurement of m

The measurement of m, the mass flow per unit cross sectional area

of the duct, was fairly straightforward. Primary and secondary mass

flows were measured separately and then added to determine wR2m.

Measurement of the primary mass flow rate was accomplished by means of

an A.S.M.E. square-edged orifice meter with an orifice diameter of one

inch and a pipe diameter of two inches. This meter was located upstream

of the primary nozzle. In the exit plane of the nozzle, the secondary

flow can be regarded as a potential flow with a constant total pressure

equal to atmospheric pressure. Under these conditions, the secondary

flow velocity can be calculated using the Bernoulli equation:

Po s r I u

The secondary mass flow is then obtained by multiplying pU 0 i times the

18.

secondary flow area /4(D2 - d2 ). Adding the two mass flow rates and then

dividing by 7 R2 gives the desired value of m.

The measurement of primary mass flow rate should be accurate to

within 0.5 per cent according to information in Aerodynamic Measurements 1 9

on the ASME orifice flowmeter. The wall pressure at X/D = 0 was

measured with a micromanometer manufactured by the R. Hellwig Co. in

Berlin, Germany, and could measure pressure differences to within 0.01 mm

of the manometer liquid which, in this case, was methanol. Since

pressure differences measured were in the range of 0.50 to 4.00 mm of

methanol, the accuracy of this pressure measurement is considered to be

within 2% and thus the secondary mass flow measurement should be accurate

to within 1%. Thus it is expected that the percentage error in measure-

ment of the value of m was at most 1%.

D. Measurement of M

Measurement of the total thrust, T, of the duct flow was accomplished

by means of a beam type thrust balance located at the downstream end of

the duct. The thrust balance, which is shown in Figure 14, was operated

as a null deflection device. A counterweight was adjusted to give zero

deflection of the beam for the case of no flow, then the flow was turned

on and the thrust measured by balancing a weight of known mass at the

moment arm which gave zero beam deflection. By applying the law of

conservation of momentum to the control volume shown at the bottom of

Figure 14, one obtains

R

T -- UP+e12 -- 2 rrdr

19.

where the integral is evaluated at the downstream end of the duct. While

the quantity M' as defined by

I T R2 M' T (21)(22

is easy to measure, it is less than the desired flow constant, M, due to

the effect of wall friction by the amount

M - M' = 8 ( ) Tj (22)

where T is the area averaged wall shear stress. This difference can be

evaluated experimentally. The quantity, M, is first evaluated in terms

of the contribution from the primary flow and the contribution from the

secondary flow.

R

7r R P.) ,2T2-ardr

d/z.

PO)+ +- r2 .d )

Thus one obtains the expression:

M = F + 2( -Po ); [I 2(D)] (23)

where F is defined by:

1r R2FfF = /2 - U 2+ , 2-nrd r0

The quantity F in equation 23 represents the contribution of the primary

20.

flow to the value of M, while the term proportional to (P - P ). gives

the contribution of the secondary flow. The value of F is determined

entirely by the jet velocity field in the exit plane of the primary

nozzle and is independent of the secondary flow rate. By keeping the

primary flow rate constant for a particular swirl generator, the

magnitude of F is also held constant. By determining the value of F for

each swirl generator at the particular primary flow rate used, one may

then calculate M for each case using equation 23. Combining equations

21, 22, and 23 and substituting r/R = 0.185 gives

where all quantities on the right hand side are measured directly.

Keeping the primary flow rate constant while successively decreasing

the secondary flow rate causes the left hand side of equation 24 to

approach the value of F for that particular primary flow rate - -

decreases as the total flow rate in the duct decreases, but F remains

constant. Figure 15 shows the right hand side of equation 24 evaluated

from the data plotted versus (P - P ). . The magnitude of F in each caseo w i

is determined by extrapolating the data to a value of (P0 - P w equal to

zero. This procedure is not valid unless the condition

(25)

holds in the limiting case of no secondary flow. An estimate of this

ratio is obtained by assuming

2- _j

21.

for the case of no secondary flow where w is the primary mass flow ratep

and C is the skin friction coefficient which is taken as 0.005. Under

this assumption one obtains

2/L w 2 0.2D F TR2 F

which gives a maximum value of 0.006 for the data considered and easily

satisfies equation (25) above. The values of F determined in this

manner for each swirl strength are given in Figure 15. Since the extra-

polation in figure 15 appears to be a linear one, the error in the value

of F determined by this procedure should be less than about 2%. Thus

the value of M determined by substituting F and measured values of

(P - Pw )i into equation (23) should have an error of less than 2% -

much less than that incurred by evaluating the integral in equation (19)

from measured pressure and velocity profiles.

E. Measurement of H

The angular momentum of the primary flow was measured directly with

the torque balance shown in Figure 16. To carry out this measurement,

the test section was removed from its stand and the torque balance placed

in front of the jet nozzle. The torque balance was used as a null

deflection device. A counterweight was first adjusted to give zero

deflection for no primary flow; then, the flow was turned on and a weight

of known mass placed on the moment arm so as to give zero deflection.

Since the soda straws in the downstream end of the torque balance remove

almost all of the angular momentum from the jet flow, the torque

necessary to keep the torque balance in static equilibrium with zero

22.

deflection is equal to fR2 H. Measurements of torque obtained in this

manner were repeatable to within 3%, and thus the experimental error for

this method should be less than 3% also. Once the value of angular

momentum had been measured for a particular swirl generator at the

desired primary flow rate, the test section could be replaced.

F. Measurement of Wall Static Pressure

The test section was instrumented with wall static pressure taps

located along a line parallel to the duct centerline. The first 15

pressure taps were spaced 1.625 inches apart while seven more further

downstream were spaced 6.50 inches apart. The pressures were measured

with a Statham unbonded strain gauge pressure transducer with a range of

+ 0.05 psi. The pressure transducer was excited with an a.c. signal of

60 Hz., and the output voltage, which was proportional to the pressure

difference being measured, was amplified and measured with a null-

balance circuit. The balancing units were calibrated in terms of pressure

difference, and this calibration is shown in Figure 17. Due to the

inherent reliability of the wall static pressure tap and the excellent

linearity of the calibration curve, the experimental error in the

measurement of wall static pressure is expected to be the same as the

average departure of the calibration points from the solid line drawn

through them in Figure 17, which is about 3%.

G. Measurement of Mean Velocity and Pressure Profiles

The mean velocity and pressure profiles were calculated from pressure

data taken with a wedge probe, a sphere-static probe, and a kiel probe

which are shown in Figure 12. Pressures were measured with either of two

23.

Statham pressure transducers having maximum ranges of + 0.20 psi and

+ 2.0 psi. The input voltage for the transducers was supplied by a

transistorized d.c. power supply and the output signal was recorded on

the y-axis of the X-Y plotter. Total pressure profiles were measured

with the kiel probe which was accurate to within 1% for relative pitch

and yaw angles as large as 40 degrees. The sphere-static probe measured

a pressure which was related to the local static and dynamic pressure by

the relation

The calibration of K versus probe Reynolds number is shown in Figure 21.

The sphere-static probe allowed measurement of the local static pressure

to within 2% of the dynamic pressurie for relative pitch and yaw angles

as large as 32 degrees. The solid line shown in Figure 21 is the result

of a quadratic least squares fit of the data shown and is given by:

k .6_150 _( ) - 0.015 ( Re /-

where Rey is the Reynolds number based on the sphere diameter of 0.14

inches. The swirl angle 0 as defined by the expression

tcme

was calculated from the measured pressure difference between two sides of

a wedge-shaped probe which had its axis of symmetry aligned with the duct

centerline. The calibration of pressure difference measured by the

wedge probe as a function of swirl angle is given in Figure 20 with

probe Reynolds number as a parameter. The solid lines in Figure 20 are

calculated from the expression

('p2 ap Uwhere

A4= 0 + 3 )

0. 217 + 4 63( o*a-62 ( F0#2

The expression above for the swirl angle was found to be accurate to

within onre degree for pitch angles as large as 15 degrees. The mean

axial and tangential velocity profiles and also the static pressure

profiles were easily calculated from the values of total pressure,

static pressure, and swirl angle obtained from the data. Radial velocities

were assumed small relative to axial velocities in the data reduction - a

condition which is justifiable except in regions of recirculation. A

pitch angle of 15 degrees would cause the measured values of U and W to

be 3-1/2% higher than their actual values. Since the dynamic pressure

can be measured to within 4% and the swirl angle to within 1 degree, the

measurement for U should be accurate to within 8%, while the measurement

fo~r W should be accurate to within 10% even for pitch angles as large as

1.5 degrees.

H. Exerimental Results

The experimental results of this investigation are shown in Figures

22 through 30. These data show the effect of swirl strength on wall

pressure, mean axial velocity profiles, and mean tangential velocity pro-

24.

25.

files. In addition, hot wire anemometer data of a qualitative nature was

recorded in the "edge" of the jet.

The effect of m/(pM)1/2 on dimensionless wall pressure, (Pw - P0)/M,

is given in Figure 22a for the case of no swirl. It is evident that, as

the value of m/(pM)1/2 increases, the initial dimensionless secondary

velocity, U0 /(M/p)1/2 increases and the dimensionless wall pressure rise

in the duct decreases. As the value of m/(pM)1/2 decreases, the initial

dimensionless secondary velocity decreases and the dimensionless wall

pressure rise in the duct increases. The axial point of jet attachment

is indicated by the sudden change in slope of the wall pressure curve.

It is seen that as m/(pM)1/2 decreases, the axial distance necessary for

the jet to reach the wall also decreases.

Figures 22b through 22g show the effect of swirl strength on the

dimensionless wall pressure distribution. In each series of data, the

swirl strength is varied while the value of m/(pM)1/2 is held constant.

It is seen that in the range of swirl strengths shown there is little

effect of swirl strength on the initial and far downstream values of

dimensionless wall pressure. There is, however, quite a significant

effect of swirl strength on the axial distance necessary for the jet to

reach the duct wall. The distance necessary for jet attachment can be

reduced by a factor of 3-1/2 due to swirl for the range of swirl strength

investigated.

Figures 23a through 23h show the effect of swirl on mean axial

velocity profiles with m/(pM)1/2 having values of 0.62 and 0.49. Except

in the region near the jet nozzle, the axial velocity profiles seem to

correspond closely to the turbulent free jet axial profile shown in

26.

Figure 1. As swirl strength is increased, the centerline jet velocity

is seen to decrease more rapidly with axial distance.

Figure 24a through 24d show the development of the mean tangential

velocity profiles with axial distance. Before jet attachment, these

profiles seem to correspond closely to the turbulent free jet tangential

velocity profile shown in Figure 2; however, after jet attachment, the

point of maximum tangential velocity moves rapidly to the outer part of

the flow field. The far downstream tangential velocity profiles

correspond quite closely to those measured by Kreith and Sonju20 for a

fully developed swirling flow in a pipe with a maximum tangential

velocity occuring at about r = 0.84 R.

The point of jet attachment can be determined to within 10% from the

wall pressure distribution data. Figure 25 shows the eff ect of m/(PM)1/2

on the point of jet attachment with swirl strength as a parameter.

Figure 26 shows a crossplot of these same data giving the effect of

swirl strength on the point of jet attachment with m/(pM)1/2 as a para-

meter.

Qualitative measurements of mean and fluctuating velocity were made

at an axial position before jet attachment with the vertical hot-wire

probe shown in Figure 12. The edge of the jet was defined by observing

the radial position at which the fluctuation intensity Q/max

abruptly decreased. The fluctuating velocity signal as well as its

spectrum were then recorded on an oscilloscope with the probe positioned

at the edge of the jet. This data is shown in Figure 27. The fluctuat-

ing velocity signal seems typical of turbulence and its spectrum indicates

that most of the intensity of the signal occurs at frequencies above ten

cycles per second.

27.

I. Effect of L/D

The effect of test section L/D on the mean flow field of the ducted

turbulent jet was investigated experimentally by measuring the wall

static pressure distributions for two different values of L/D - 10-1/2

and 18. Using a swirl generator with a vane angle of 15 degrees and a

corresponding value for 2H/Fd of 0.106, data on wall static pressure were

recorded for both values of L/D mentioned above and with m/(pM)1/2 taking

on six different values in each case - 0.62, 0.58, 0.55, 0.49, 0.44,

and 0.39. No measurable effect of test section L/D was observed using

this swirl generator. Additional wall pressure data were taken using

a swirl generator with a vane angle of 38 degrees and a value of 2H/fd

of 0.413. Values of m/(pM)1/2 of 0.62 and 0.57 were studied at both

values of test section L/D. Again, no measurable effect of test section

L/D on wall static pressure was observed.

J. Effect of d/D

The effect of the ratio of jet nozzle diameter to duct diameter was

determined experimentally using two different test sections with dia-

meters of 6-1/2 inches and 3-7/16 inches, which are shown in Figures 10a

and 10c. Since the nozzle diameter was 1.20 inches, the two values of

d/D investigated were 0.35 and 0.185. The procedure was to compare the

wall static pressure distributions of test cases having the same values

for the parameters m/(pM)1/2 and H/MD, but having different values of d/D.

Comparisons were made with m/(pM)1/2 having values of 0.62, 0.57, 0.54,

and 0.49. For each value of m/(pM)1/2 three values of H/MD were employed

- one being the zero value corresponding to no swirl.

Typical results are shown for m/(pM) 1 / 2 equal to 0.54 in Figures 28

28.

through 30. For the case of no swirl the d/D effect manifests itself by

a difference in the magnitude of the dimensionless wall static pressure,

(P - P )/M, in the region before jet attachment; however, the axial

position of jet attachment is not affected. For the case of weak swirl

with H/MD equal to 0.005, there is a noticeable effect of diameter ratio

on the axial position of jet attachment with the faster spreading jet

corresponding to the case with the smaller value of d/D. For the case

of stronger swirl with H/MD equal to 0.031, the point of jet attachment

as indicated by the wall pressure distribution appears to be the same

for both values of d/D; however, the wall pressure reaches its maximum

value sooner for the case with the smaller value of d/D.

It is concluded that, while there is no effect of diameter ratio

on the mixing rate of a non-swirling turbulent jet in a ducted stream,

there is such an effect for the case with swirl. With values of

m/(pM)1/2 and H/MD held constant, the jet mixing rate appears to

increase as d/D is decreased.

K. Applications

The experimental results of this study indicate that swirl

significantly increases the rate of mixing of a turbulent jet with a

surrounding flow. Thus the size and therefore the weight of gas turbine

or ramjet combustion chambers can be reduced due to swirl when chemical

reaction times are much smaller than characteristic flow times and

turbulent mixing is the limiting process. In addition, it is possible

to create or sustain regions of reversed axial flow near the nozzle at

the jet centerline - a useful flow pattern in combustion applications.

A comparison of recent free jet experiments suggests that the design of

the swirl generator is quite important in determining whether central

29.

recirculation will occur. Chigier and Chervinsky reported central re-

circulation for x/d less than ten and a swirl parameter of 0.30 when

swirl was created by tangential injection. Kerr and Fraser5, however,

found no central recirculation for swirl parameters as large as 0.53

when swirl was generated by annular turning vanes in the nozzle.

Figure 38 shows how the length necessary for jet attachment in a

mixing device such as a jet pump can be reduced at the expense of

pressure loss in the swirl generator. An experimental correlation by

Mathur and Maccallum21 for a swirl generator with eight annular turning

vanes and a hub to tip ratio of 0.32 has been used to estimate pressure

drop across the swirl generator. This correlation is:

R Pn___ 2 2..7

where Un is the mass averaged velocity of the jet flow. This correlation

and the data given in figure 25 have been used to calculate the points

shown in figure 38. It is seen that when

AP 0, 0 3

which corresponds to a swirl parameter of 0.068, the length of duct

necessary for jet attachment is reduced by 30 to 4 5 % depending on the

value of m/(pM)1/2. Part of this pressure loss could be regained at the

downstream end of the mixing tube by using a radial diffuser there.

30.

III. ANALYSIS

A. Preliminary Remarks

The well known technique has been employed to predict the mean flow

field for the turbulent jet in a ducted stream. This technique results

in a system of simultaneous ordinary, non-linear differential equations

with streamwise direction as the independent variable. These equations

and their initial conditions were solved on a digital computer using

the Runge-Kutta-Merson22 integration procedure.

The turbulent jet in a ducted stream contains, in general, three

distinct flow regions.

1. The first region occurs before the jet shear layer has diffused

to the duct wall, and thus there is a secondary potential flow between

the jet fluid and the wall.

2. A second distinct region can occur when the velocity near the

wall becomes negative and there is recirculation of the secondary fluid

through the jet.

3. The third region begins at the point of jet attachment to the

duct wall. The beginning of this region is distinguished by a more

rapid rise of wall pressure than in the region before jet attachment

and acceleration of the fluid near the wall.

First order effects of the wall boundary layer upon the ducted jet

flow field were included in the analysis. The net effect of the wall

boundary layer on the rest of the flow field was taken as being equivalent

to a decrease in the duct radius equal to the value of the boundary layer

displacement thickness. The boundary layer displacement thickness and

shape factor were calculated using the method of Moses23 in the region

31.

before jet attachment. For cases where the boundary layer calculation

predicted separation, the calculation was repeated neglecting the effect

of wall boundary layer. This was done because it is very difficult to

predict the behavior of a separated wall boundary layer with any degree

of accuracy.

B. Velocity Profiles

A turbulent jet immersed in a general secondary stream is considered

self-preserving when its dimensionless velocity profile can be described

by:

U-LL 0 .(26)

U -J

where 6 is proportional to the half velocity thickness. One of the

requirements that a turbulent jet be self-preserving is that the ratio

U /U be a constant. While only a few jet flows (such as the free jet)

have a constant value of U0 /U , many are quite adequately described by

the assumption of self-preservation. Hill1 1' 12 found this to be true

for the non-swirling turbulent jet in a ducted stream. For the swirling

turbulent jet in a ducted stream, the mean axial velocity is also

assumed to be adequately described by equation (26) in the region before

jet attachment. The dimensionless axial profile is taken to be that of

the turbulent free jet show in Figure 1. The axial velocity is also

assumed self-preserving in regions of recirculation where U0 has a

negative value. After jet attachment, the axial velocity is allowed to

change shape and is described by:

U U0 f f) + (x) F(f) (27)

Uj

32.

The function F(n) is chosen arbitrarily as:

and satisfies the conditions

F(O) =0 F(l) = 0

Ft(O) = 0 F'(l) = 0

This simple choice for the function F(n) is justified later on the grounds

that the predicted values of

.U- U 0 (X)

always remain small compared with unity.

The mean tangential velocity profile is also assumed approximately

self-preserving in the region before jet attachment and its form is

given by the solid line in Figure 2. The assumed profile in Figure 2

agrees well with free jet data in the outer portion of the jet but is

generally higher than the data near the jet centerline. A mean tan-

gential profile which agreed well with free jet data near the jet center-

line, however, would have a negative value of eddy viscosity based on

the definition

ZI(V ) (29)

This anomaly results since v'w', as calculated from equation (A5) in

Appendix A, is positive, and a (1) for a curve passing through the freeDr r

jet data is also positive near the centerline. This discrepancy in the

free jet data probably is due to the difficulty in obtaining velocity

measurements near the center of a swirling jet without disturbing the

33.

flow field. Figure 2 also shows that the assumed form for the mean

tangential velocity agrees well in the inner part of the jet with the

theoretical solution of Loitsyanskii24 which assumes a constant eddy

viscosity. The experimental results of this study show that after jet

attachment the point of maximum tangential velocity moves to the outer

part of the duct. A one parameter family of profiles was used to

describe the mean tangential velocity with the parameter, c, being the

value of n at which the maximum value of W occurs. The mean tangential

velocity field was described by:

AO+ Iej+ 2 3+5= A0 +12- AA31f +A41 +< As I

where the coefficients are chosen to satisfy the condtions:

g(0) = 0 g(c) = 1.0 g(c/0.34) = 0

= 0 g'(c) = 0 g'(c/0.34 ) = 0

These conditions specify solid body rotation near the duct centerline

and a tangential velocity near the wall which increases above zero as

the parameter c increases above its free jet value of 0.34. The

expression for tangential velocity becomes:

g(n,c) = 1.68724 (D-) - 1.12401 (.) 3 + 0.49907 (n)4 - 0.06229 (D-)5 (30)c c c c

The profile in Figure 2 was computed with c equal to a value of 0.34.

C. Turbulent Shear Stress

The turbulent Raynolds shear stresses were specified in terms of

the mean flow field of the ducted turbulent jet by means of a turbulent

eddy viscosity by the expressions:

34.

- )f (31)

V'w' = -7 Y (32)

The relationship between vT/U 6 and swirl parameter, which for the

ducted jet equals 2H/Fd, was assumed to be the same as for the

turbulent free jet. In chapter I it was shown that two possible

correlations for the effect of swirl strength on vT/U 6 for the

turbulent free jet are:

H= 0.00 56 + o-031 d (33)U F

and

= 0.00856 4- 0.050 (4)

The effects of axial and tangential wall friction were also included in

the calculation. Axial wall friction was specified by:

1H 2 CT = ~u 0 d (35)

The value of C was predicted in the upstream region by the wall

23boundary layer method of Moses23 After jet attachment the value of

C 9 was held constant and equal to the predicted value at jet attachment.

The tangential wall shear stress Tw,6, is specified by a tangential shear

stress coefficient which, as a first approximation, is assumed proportional

to the square of the tangential velocity near the duct wall, W0:

35.

2-W 2 ;j 9, (36)

Data presented by Kreith and Sonju20 for the decay of a fully developed

swirling flow in a pipe yield a value for C of 0.04. Of course, the

value of Cfo probably increases as a Reynolds number based on the

distance between the wall and the point of maximum tangential velocity

decreases.

After jet attachment the effect of the wall boundary layer on the

jet flow field quickly diminishes, indicating a decreasing boundary

layer displacement thickness. An approximation for the Reynolds stress

u'v' at the edge of the wall boundary layer was utilized to predict the

development of the wall boundary layer after jet attachment. Equation

(32) was used to specify u'v' at a distance A from the duct wall by

approximating aU/ar of the jet profile near the wall as being proportional

to A. The resulting expression is:

-. 25 AA (37)

SUf I D Up R)

D. Ducted Jet Equations

In the region before jet attachment five variables are used to

describe the flow field: U, W, U0 , 6, and Pw. These variables are

determined by using:

a) continuity integral equation

b) momentum integral equation

c) moment of momentum integral equation

d) angular momentum integral equation

36.

all of which are derived in Appendix B, along with Bernoulli's equation:

r .-. g(38)

In regions of reversed flow near the wall, equation (38) was replaced by

the condition of constant wall pressure. The coefficient matrix for the

ducted jet equations before jet attachment is given in Table I. Since

the velocity profiles are assumed self-preserving in the region before

jet attachment, the $'s in the ducted jet equations have constant values.

These values are:

o, = 0.04307

02 = 0.01636

03 = 0.01280

04 = 0.10719

05 = 0.05419

06 = 0.03330

07 = 0.09515

o8 = 0.14373

09 = 0.06113

018 = 0.38658

After jet attachment six variables are necessary to describe the

flow field of the ducted jet: U, Wi, U 0 , P , E, and c. The equations

used to determine these variables are:

a) continuity integral equation

b) momentum integral equation

c) moment of momentum integral equation

d) second moment of momentum integral equation

37.

e) angular momentum integral equation

f) moment of angular momentum integral equation

All of these equations are evaluated from those given in Appendix B for

the downstream region by replacing 6 with R, the radius of the flow area.

After jet attachment the jet velocity profiles are no longer considered

self-preserving, and thus the dimensionless integrals of the velocity

profiles can now be functions of E and c, which are both functions of X:

d 0; 24 j 84 Cdx 3t dx - 3c dX

Since the relationship between C and Reynolds number based on

the distance between the wall and the point of maximum tangential

velocity is not known, the prediction of the value of c in the downstream

region becomes less reliable as c approaches unity. For the hypothetical

case of no tangential wall shear stress, the tangential velocity profile

would develop towards solid body rotation with c approaching infinity.

For this reason the moment of angular momentum equation is replaced by

the equation

dc 0 (40)dx

after c reaches a value of 0.85, which is the value observed experi-

mentally by Kreith and Sonju20 for a fully developed turbulent swirling

flow in a pipe.

The coefficient matrix for the ducted jet equations after jet

attachment is given in Table II. Values of the 4. in terms of , and c

are given in Table III and are based on the assumed axial and tangential

velocity profiles.

38.

E. Initial Conditions

To integrate the ducted jet equations of motion, initial values of

all dependent variables are required in the exit plane of the jet nozzle.

Of course, the jet fluid does not necessarily have axial and tangential

velocity profiles like those shown in Figures 1 and 2 as it leaves the

nozzle, but requires five to ten nozzle diameters to reach this self-

preserving form. When the duct to nozzle diameter ratio is large

compared with unity, the initial transition region can be ignored by

assuming that the jet fluid emanates from a point source which is

called the virtual origin. The effective position of the virtual origin

is usually within one nozzle diameter of the nozzle exit plane and for

this study has been placed in the nozzle exit plane.

If an initial value of jet width, 6, is chosen, then the correspond-

ing initial values of A and y are specified by two dimensionless para-

meters:

( )" =O (41)

/ /0 (V 2 2 z 2 2

H I(42)MD 4 R 2 2-

R0 To

where R0 is the duct radius and R is the effective radius of the flow

area due to the presence of an initial boundary layer. Knowing the

value of A, the dimensionless value of jet excess velocity, U /(M/p)1/2,

can be determined from the definition of m:

39.

J AL./2. + (43)

The initial value of dimensionless wall pressure is then determined from

the Bernoulli equation:

2

M //M/)(44)

The initial value of the independent variable x is then determined from:

(45)

The initial state of the wall boundary layer was assumed as a

first approximation to be the same for all cases. The initial value of

boundary layer shape factor was assumed to be 1.3 while the initial

value of displacement thickness was taken to be 2% of the duct radius

giving an initial value for R/R of 0.98.

F. Comparison of Prediction with Experimental Results

The mean flow field of a swirling turbulent jet in a ducted stream

has been calculated based on data obtained from the swirling turbulent

free jet. Two possible correlations for the effect of swirl on turbulent

eddy viscosity, which are given by equations (33) and (34), were found

to give good results for the turbulent free jet and have been used to

predict the turbulent ducted jet. The comparisons of experimental results

with prediction for wall pressure, mean axial velocity and mean tangential

velocity are given in figures 22, 23, and 24, respectively. The theories

based on equations (33) and (34) give the same result for the case with

no swirl, but differ slightly for the cases with swirl. The predictions

4o.

for wall pressure agree well with the data except at the strongest swirl

strength where the predicted point of jet attachment occurs sooner than

for the bxperimental data. The prediction for (P -P )/M is always morew o

negative than the data near X equals zero since the theoretical model

assumes that the jet flow emanates from a point source at the virtual

origin. The position of jet attachment is indicated on each wall

pressure distribution by a sudden change of slope. The point of jet

attachment determined from wall pressure data is compared with the

predicted point of jet attachment in figure 25. Agreement between data

and theory is good except at the strongest swirl strength where jet

attachment occurs further downstream than predicted.

Good agreement between predicted and measured values of mean axial

velocity is observed at X/D equal to 1-1/2, but the comparisons further

downstream suggest that mixing occurs slightly more rapidly than is pre-

dicted by the theory. The predicted values for mean tangential velocity

agree fairly well with the data for a swirl parameter of 0.19, but at

the smaller swirl parameter of 0.106, the data again suggest that

turbulent mixing occurs slightly faster than predicted. Comparisons of

data with prediction for the characteristic velocities U0 /(M/p)1/2,

U /(M/p) 1/2, and W /(M/p)1/2 are shown in figures 34, 35, and 36,

respectively.

The influence of the wall boundary layer on the wall pressure dis-

tribution for the case of no swirl is indicated in Figure 31. Here

predictions both including and neglecting the wall boundary layer are

compared with the data for the case where m/(pM)1/2 equals 0.618. This

comparison indicates that, if the effect of the wall boundary layer is

41.

included, a noticeable improvement in the prediction for wall pressure

is realized. Figure 32 shows the predicted behavior of the boundary

layer shape factor and displacement thickness for a typical case where

m/(pM)1/2 equals 0.570 with swirl strength as a parameter. As swirl

strength increases, the adverse pressure gradient at the wall becomes

more severe and the boundary layer develops faster towards separation.

Figure 33 compares the predicted and measured effect of diameter

ratio for the weak swirl case where H/MD = 0.0051. The theory based

on equation 34 predicts no effect and is clearly in disagreement with

the data. The theory based on equation 33 predicts an effect which is

slightly smaller than that measured but correct qualitatively.

42.

IV. SUMMARY AND CONCLUSIONS

The results of this study indicate that the presence of swirl causes

a turbulent jet to mix more rapidly with an exterior ducted stream than

it would if swirl were absent. Previous studies have indicated that this

is also the case for the turbulent free jet. It has been shown that it

is possible to successfully predict this increased rate of mixing for

both the free jet and the ducted jet by assuming that swirl causes an

increase in the value of the turbulent eddy viscosity. Two possible

correlations between eddy viscosity and swirl strength are given in

equations (33) and (34). The prediction for the point of jet attachment

agrees with experimentally measured values to within 10%, as indicated

in Figure 25, except at the strongest swirl where the jet attaches to

the wall in less than one duct diameter. After jet attachment as U /U

approaches unity, mixing takes place slightly faster than predicted.

In particular it is concluded that:

1. Three dimensionless parameters are necessary to specify the flow

field of a swirling turbulent jet in a ducted stream:

m/(pM)1/2 , H/MD, and d/D

2. Since Figure 37 shows that predicted values for the axial

velocity shape factor, F, never become large enough to be important, the

axial velocity profile can be assumed self-preserving both before and

after jet attachment when the ratio 6/b is chosen as 2.6. The dimension-

less axial velocity profile is taken to be that of the turbulent free jet

shown in Figure 1.

43.

3. The tangential velocity profile can be considered self-preserv-

ing before jet attachment with a profile like that of the turbulent free

jet shown in Figure 2. After jet attachment, the tangential velocity

profile departs from self-preservation with the point of maximum

tangential velocity moving rapidly to the outer part of the jet as

indicated by predicted values of c in Figure 37.

4. Each of two correlations for the effect of swirl on turbulent

eddy viscosity give good results for both the free jet and the ducted

jet. One correlation specifies the dimensionless eddy viscosity in

terms of the free jet swirl parameter, which for the ducted jet equals

2H/Fd:

= 0,0095: + 00.31 .H (33)

The correlation given in equation (33) allows the jet to "remember" the

diameter of the jet nozzle. Another correlation which specifies the

dimensionless eddy viscosity entirely in terms of local flow properties

is given by:

2 .0o56 + 0.050 H (34)

The correlation given by equation (34), however, does not predict a

diameter ratio effect which is shown in Figures 29, and 33. Clearly,

more experimental work is needed to improve upon equations (33) and (34).

5. Including the influence of the wall boundary layer in the

calculation procedure results in a noticeable improvement in the wall

pressure, as shown in Figure 31, but has only a slight influence on the

44.

predicted point of jet attachment.

6. As the ratio U0/U approaches unity, turbulent mixing occurs

faster than predictions based on either of equations (33) or (34).

Bradbury and Riley15 found this to be the case also for a jet with no

swirl in a constant velocity stream.

145.

V. RECOMMENDATIONS FOR FURTHER STUDY

Before any significant improvement can be made in the method of

prediction, it will be necessary to improve the correlation between

Reynolds stresses and the mean flow field. To do this will require more

experimental data on swirling turbulent flows, especially in the region

far from the jet nozzle (X/d > 15). Additional measurements of half-

velocity thickness and mass entrainment in the swirling turbulent free

jet would be helpful. More desirable would be detailed quantitative

measurements of the structure of turbulence in a swirling turbulent

jet flow. The measurement of Reynolds stresses and spectra of velocity

fluctuations in the swirling turbulent free jet would lend greater

insight into the mechanism of increased mixing due to swirl in the

turbulent jet.

46.

REFERENCES

1. Forstall, W. and Shapiro, A., "Momentum and Mass Transfer in CoaxialGas Jets", ASME Trans., Vol. 72, (J. App. Mech., Vol 17), Dec. 1950,

pp. 399-408.

2. Krzywoblocki, M. Z. V., "Jets - Review of Literature", Jet Propulsion,

Sept. 1956, pp. 760 - 779.

3. Seddon, J. and Dyke, M., "Ejectors and Mixing of Streams", Bibliography

6, Advisory Group for Aerospace Research and Development, NATO,Nov. 1964.

4. Corrsin, S. and Uberoi, M. S., "Further Experiments on the Flow and

Heat Transfer in a Heated Turbulent Air Jet", NACA Report 988, 1950.

5. Kerr, N. M. and Fraser, D., "Swirl. Part I: Effect on Axisymmetrical

Turbulent Jets", Jour. Inst. of Fuel, Vol. 38, No. 299, 1965, pp.

519-526.

6. Rose, W. G., "A Swirling Round Turbulent Jet, 1 - Mean Flow Measure-

ments", ASME Trans. (J. App. Mech., Vol. 29, Ser. E, no. 4), Dec.

1962, pp. 615-625.

7. Chigier, N. A. and Chervinsky, A., "Experimental Investigation of

Swirling Vortex Motions in Jets", ASME Trans. (J. App. Mech., Vol 34,Ser. E, No. 2) June 1967, pp. 443-451.

8. Craya, A. and Darrigol, M., "Turbulent Swirling Jet", Physics of Fluids,

Suppl. on Boundary Layers and Turbulence, 1967, pp. S197-S199.

9. Newman, B. G., "Turbulent Jets and Wakes in a Pressure Gradient" in

Fluid Mechanics of Internal Flow, ed. by G. Sovran, New York,

Elsevier Publ. Co., 1967, pp. 170-209.

10. Abramovich, G. N., The Theory of Turbulent Jets, Cambridge, Mass.,M.I.T. Press, 1963.

11. Hill, P. G., "Turbulent Jets in Ducted Streams", J. Fluid Mech.,

Vol. 22, Part 1, 1965, pp. 161-186.

12. Hill, P. G., "Incompressible Jet Mixing in Converging-DivergingAxisymmetric Ducts", ASME Trans., (J. of Basic Engineering), March

1967, pp. 210-220.

13. Bradbury, L. J. S., "An Investigation into the Structure of a

Turbulent Plane Jet", Ph.D. Thesis, Univ. of London, 1963.

14. Gartshore, I. S., "The Streanwise Development of Two-Dimensional Wall

Jets and Other Two-Dimensional Turbulent Shear Flows", Mech. Eng.

Ph.D. Thesis, McGill Univ., 1965.

47.

15. Bradbury, L. J. S. and Riley, J., "The Spread of a Turbulent Plane

Jet Issuing into a Parallel Moving Airstream", J. Fluid Mechanics,

Vol. 27, Part 2, 1967, pp. 381-394.

16. Townsend, A. A., The Structure of Turbulent Shear Flow, Cambridge,Cambridge, Univ. Press, 1956, p. 189.

17. Craya, A. and Curtet, C. R., Acad. Sci., Vol. 241, Paris, 1955,

p. 621.

18. Spalding, D. B., Seventh (Int.) Symposium on Combustion, Oxford,

England, Butterworth Scientific Publications, 1958.

19. Dean, R. C., Jr., "Aerodynamic Measurements" Cambridge, Mass.,

M.I.T. Gas Turbine Laboratory, 1953.

20. Kreith, F. and Sonju, 0. K., "The Decay of a Turbulent Swirl in a

Pipe", J. Fluid Mech., Vol. 22, Part 2, 1965, pp. 257-271.

21. Mathur, M. L. and Maccallum, N. R. L., "Swirling Air Jets Issuing

from Vane Swirlers. Part I: Free Jets", Jour. of the Institute of

Fuel, May, 1967, pp. 214 -225.

22. Fox, L., Numerical Solution of Ordinary and Partial Differential

Equations, Reading, Mass., Addison-Wesley Publ. Co., 1962, pp.24-25.

23. Moses, H. L., "The Behavior of Turbulent Boundary Layers in Adverse

Pressure Gradients", Cambridge, Mass., M.I.T. Gas Turbine Laboratory

Rep. No. 73, Jan. 1964.

24. Loitsyanskii, L. G., "The Propagation of a Twisted Jet in an Un-

bounded Space Filled with the Same Fluid", Prikladnaya Matematika i

Mekhanika, Vol. 17, 1953, pp. 3-16.

25. Goldstein, S., ed., Modern Developments in FluiqD namics, vol. 1,

Oxford, England, Oxford Univ. Press, 1938.

26. Hinze, J. 0., Turbulence, New York, McGraw-Hill, 1959.

27. Utrysko, B., "Jets Tournants en Espace Confind", Paris, Publications

Scientifiques et Techniques du Ministere de l'Air, Publ. No. 436, 1967.

APPENDIX A - CALCULATIONS OF REYNOLDS SHEAR STRESS DISTRIBUTION IN THE

TURBULENT FREE JET

The Reynolds shear stress u'v' for the turbulent free jet can be

calculated using the axial momentum equation and experimental data on

velocity profiles and rate of jet spreading. For the free jet, the

axial pressure gradient outside the mixing region is zero and the radial

pressure gradient within the mixing region is given by the condition of

radial equilibrium:

iCP

where

Integrating this equation yields:

Ps P= ri

or:

- . -- 4- d,;

Substituting this term into the axial momentum equation yields:

6 W2rI -

This equation can be rewritten as:

It is clear that when

U. 2 \A/( >iVJ

49.

the term representing the adverse axial pressure gradient associated with

the decaying swirl can be neglected. Since the term

f2~

has an order of magnitude of one, the condition for neglecting the

pressure gradient term is

It is reasonable to neglect the pressure gradient term provided that the

data used to calculate the Reynolds shear stress satisfy the above re-

quirement. Under this condition the axial momentum equation becomes:

a u 3 Li 3-Ur -- + Vr - =- -Cr'V') (Al)

Using the continuity equation to eliminate V:

Ur U rad U --UV

Integrating each term from 0 to an arbitrary radial distance R gives:

0

Integrating the second term by parts:

a U C Urd r -U(R)] rdr L-R ,

~~ [u,2 fXd S ml=RT

50.

Requiring now that the velocity profiles be self-preserving makes the

integrals functions only of n, which is a constant. The axial momentum

equation becomes:

where n = R/6.

For the case where n= 1, equation (A2) reduces to

Ui) )= 0since U and u'v' are both 0 at R = 6. Although this result was evaluated

at the edge of the jet, it must also hold true for all values of n.

Equation (A2) can thus be simplified to give:

UV \ (A3)

and is restricted to the jet region where (W /U )2 < 1

The Reynolds shear stress v'w' can be evaluated using the tangential

momentum equation.

7=

U +V 4 - (\W r) =- r V'w'

Using the continuity equation to eliminate V and integrating each term

fram 0 to R gives:

U X ~r rg r RJr R vw

51.

d UW r W r - =-R Lir r = -R v

Using again the assumption of self-preserving velocity profiles, the

tangential momentum equation becomes:

-R2

For the case where r = 1, equation (A4) reduces to:

\yf~3)=Osince W and v'w' are both zero at R = 6. This result again must hold

true for all values of r. This result as well as the result

d (u :0

can be used to simplify equation (A4) to:

V'w'

uj w.(A5)

dx u

-9 1L-. '1 0

52.

APPENDIX B - EQUATION OF MOTION FOR A SWIRLING TURBULENT JET IN A

DUCTED STREAM

The equations of motion of a swirling turbulent jet in a ducted

stream can be obtained from the basic conservation equations given in

section I-C. These conservation equations are:

Continuity:

a ( UrY) + (VV)

Mcmentum:

U r a + Vr

Angular momentum:

Ut rz 22 w + Vr (W r) = - (2VW')

Radial equilibrium:

(c r

--14

+ V - (B4)

r

The integral conservation of mass equation is obtained by integrating

equation (Bl) from 0 to the duct radius, R, which is allowed to be a

function of x for generality. Carrying out the integration yields:

g Urdr -Vr\ = -uoPThis equation can be rearranged to give:

d (U-u)rJr R2 duo dxdx 0 d2 x x

(Bl)

3r ( r L'v') (B2)

(B3)

+ r - + V'2 ) -=

53.

or equivalently:

. + d* = U R (BS)

The momentum integral equation is obtained by integrating equation

(B2) with respect to radius from 0 to the edge of the jet 6. This

integration yields:

5 u rdr - 8rX 2)0

rdr P)rdr d(r=7)

where -2

Integrating the second term by parts gives:

2 S U2)Ur Jr 00S ) r dr

p,0

+2 rdJr S(r LkV* )

Substituting the condition of radial equilibrium and rearranging gives:

dxiA (U-Uo)2 rr d U, (U-t,)r r + X S (u-u0)rd r

Jrrjdr 7v,)

Examining the last term on the left hand side:

2- '5d (7. W 2rj2.2

Thus the momentum integral equation becomes:

6 UOr dr

s2 dj

so, aa- (

2Un 10 r

54.

~(Ujz 2~

d~ x

+ 0+

+ (dx (U O Uj

zU0

+ U& 24 d2O

(B6)= - ad(r

The moment of momentum integral equation is obtained by multiplying

equation (B2) through by r and then integrating each term from 0 to 6.

This procedure results in the equation:

DUL rdrj> , . Uvd40DX

T'7) r Jr

0'r d (rJ~)

Integrating the second term by parts gives:

rZc r + Udr rdr, 6 33

d(FP.

-a} ( ,- P )r2d r = rd

This equation can be rearranged to give:

d ( U -u ) r + d UO(x- xo) r 2 r

+3 dU (-U)rJZ dx '

+S(U-Uo)dr 7 ( u-U.)r dr,0 0

S,33PW -Py j

r

g0

Ut

(r uv)

$

+ 164S D57

a LkTF

(Wx (

P1w +

55.

The condition of radial equilibrium permits evaluation of the term in-

volving radial pressure distribution.

PwE-P\1 zA ~ 1Yd WrQ 74r w- 5d=r Sd;r WZdr

0 dx h om e q X bom

2 d

Thus the moment of mometum integral equation becomes:

/ Pw+3 dx i

+ -)dx

2- L~j3~ UO

2K 3

+S (U-LUo)dr[((-U) rr = -g r d(rW~F)

For the case where the dimensionless axial velocity profile

(U - U )/U is a function of x as well as y = r/S, a higher moment of

the axial momentum equation is required. This equation is obtained by

multiplying (B2) through by r2 and integrating each term from 0 to 6.

& rdr-2 r r r

+4- P () r 3 jr =- r24(r UIvi)X /0 0

(BT)

56.

Integrating the second term by parts gives:

2 SU r3dr + 2~ Lrrf rd,

+8 (K)4 dx ,

m A( P~;P) r 3dr =j aJrLr/

This equation can be rearranged to give:

(uw-0)2r) dr + I O( ,)A + d) u-u)r(L(- 0)'J dU% .)~d

r+4-2 (U-u>) rde r,

u2+ 4+4 +

( PPd-~ )radr

0r2d(ru ' )

The condition of radial equilibrium can again be used to evaluate the

integral involving radial pressure distribution.

Sy ( Pw~P r3dr = - r~c rgWd

- 4 gd(r4)j

w 2. r 3 j r

Thus the second moment of momentum integral equation becomes:

r8dr

aId '

C

2

- 9 CU

SP3, ,

- I d 84 d x Jo

4 + d 5

+2 (u- u,)rdfr a

a

(U -u,)rdr+

(w 2 )

dX

(r 0)

The angular momentum integral equation is obtained by integrating

equation (B3) from 0 to 6.

S" Wr2 Jr

Using the continuity equation

0 r

+Vr a (Wr)dr =

to eliminate V gives:

(r )-( m

Integrating the second term by parts gives:

U DCs' _ rZcr =rcr (r2 v~TE')

or

6.8 Uw\ dr =d x

S

-~s d(r2QvW')

The left hand side of this relation can be rearranged to give:

d 8 6 s V(dux d x2 +o W u-J w r .-d(,' )

57.

5dx

(B8)

=- d

A ' ( U ?-dx

=S r2d

d zvlwlIr

(Wr) dri r Nd

58.

Thus the angular momentum integral equation is:

d (0 U ) + j (U,W 83<)=- drva' (9)

d 0

One last equation is necessary when the dimensionless tangential velocity

profile W/W is a function of x as well as y = r/6. This equation can

be a first moment of the angular momentum equation, (B3). Multiplying

equation (B3) through by r and integrating from 0 to 6 gives:

W r3dr - d(Wr)rS rd = - rA(r v )

Integrating the second term by parts yields:

or

orrd guw cdr + SWCr" r d r -S r(e +" ')d 0 0

This equation can be rearranged to appear as:Lta-u ') r~d + S, WrA r - r d rr

+ Wr dr(@-u)rd 1 +- o WrdC =f - dax rv)0

59.

Thus the moment of angular momentum integral equation is

(Blo)

Equations (B5) through (Blo) are the equations of motion used to

predict the turbulent jet in a ducted stream for this study. After the

position of jet attachment to the duct wall, the variable 6 in these

equations must be replaced by R, the duct radius, which can also be a

function of x/D.

6o.

TABLE I

EQUATIONS BEFORE JET ATTACHMENT

Form of Equations:

A dU A A dPa + AA + "+ A + =B

U dx a2dx 6 dx a4dx PU2dx a

Continuit y E

All =

A 1 2 =

A1 3 =

Al 4 =

A1 5 =

Bl -

x

1

quation: a = 1

+ 2 (6/R) 2c4

4 *4 (6/R) 2

0

0

2X dRR dx

Momentum Integral Equation: a = 2

A 2 1

A 2 2

A2 3

A2 4

A2 5

B2

= 2 *5 +

= 2 $4 +

= 2 *5 +

= - Y 8

2

3 X *4 + X2 _ y2 821x2

2 X $4 - Y 2 *8

= 0

Angular Momentum Integral Equation: a = 3

A 3 1 = 2 y

A3 2 = Y 7/(46 + X 07)

A 3 3 = 3 y

61.

A 3 4 = 1

A 3 5 = 0

B 3 = 0

Mcment of Mcmentum Integral Equation: a = 4

A41 = 2 02 + 7 X1 + 03 + X2 _- 2 2 *92 3 3

A 42 = 2 01 + 3

A4 3 = 2 02 + 2 03 + 3 X *1 - Y2 o9

A4 4 = - 2Y 9

A4 5 -

= (VT 18B4 = ( ) --

Bernoulli Equation: a = 5

A 51 = X2

A 52 = X

A 5 3 = 0

A 5 4 = 0

A 5 5 = 1

B5 = 0

62.

TABLE II

EQUATIONS AFTER JET ATTACHMENT

Form of Equations:

A dU

U +dx A2 dx

Continuity Equation: a

All = 2 * + A

A 1 2 = 1

A1 3 = 0.53333

A14 = 0

A1 5 = 0

A 16 = 0

A dP-=A + A -+ Aa5 dw+ A = Bdd3 dx d4hdx pU 2 dx a6dx a

Bl 2 -2 (2 04 + X B1~ Rdx (~+x

Mxmentum Integral Equation: a = 2

A2 1 = 2 45 + 3 A 04 + X2 _2 082

A2 2 = 2 04 + - A2

A2 3 = 32 C3 - 64 C4 + 32 C5 + O.2666T X + 0.40634 C

A2 4 = - Y 08

A25 - 2

A I2 [1.42329 + 2.528632 1.20293 O.78990 0.872606A26 - - 2 3 C8

3 5C6

C 8

0.31128 + 0.050868 0.003231

C9 CIO Cil

63.

B2 = - (2 05 + 2 X 04 - Y2 08) f2I

Angular Momentum Integral Equation: a = 3

A3 1 = 2 y (06 + X 07)

A3 2 = Y 07

= o.16069 _ 0.04996 0.01613 0.00151A 33 = y4 5

A 3 4 = 06 + X 0

A3 5 = 0

1.68724 C 3 3.37203 C 5 1.99628 C6 0.31145 CA36 = Y - 2 + -4 5 + - 6

C__ ___

+ y .16 0.14988 _ 0.06452 0.0075C2 C4 C5 C 6

+___.2 + 0.56202 _ 0.28520 + 0.03895

C2 C4 C 5 C 6

B3 = - (46 + A 07)

2 C f, [1.68724 1.12401 +0.49907 0.0622

2R c C3 C4 C5

Moment of Mamentum Integral Equation: a = 4

A4 1 = 2 02 + X 01 + 03 + 1 X2 Z y2 0923 3

A42 = + 1i +x2 3

A43 = 36 C4 - 70.4 C 5 + 34.6667 C6 + 0.27736 E + 0.15238 X

A4 4 - - 2 . 9

3

64.

A46 = Y2 1.138712 2.1674- 1.052565- 0.702132 + 0.785344

3 c3 c5 c6 c7 C8

0.282984 + o.046629 0.002981

C 9 c 1 0 c11 I1 _ y2 09 X2 C1. VT 018

B4 = - (2 02 + 3 X 01 + 2 03 - Y2 09 _ + ( -

Second Moment of Momentum Integral Equation: a = 5

A51 = 2 014 + 2 10 + 4 X 015 + 1 X2 _ l2 01612

A 5 2 = 3 015 + X

A53 = 40 C5 - 76.8 C6 + 37.3333 C7 + 0.20041 & + 0.09524 x

1A 5 4 = - - Y 016

A5 5 =

1 2 [ 0.948926 1.896476 0.935610 o.63192A56 = T [ C3 c5 c6 7

+ 0.71395 _ 0.2594o + 0.043047 - 0.002771

C8 C9 clo cili

x2 CB5 = - (2 014 + 4 O10 + 4 X 015 - y2 016) - 2R

R 016 2R

4 04 VT

Moment of Angular Momentum Equation: a = 6

A61 = y (2 01, + 013 + X 012)

A6 2 = Y 012

F=.2214o 0.089012 0.032263 _ 0.00336A6 3 = - + 224 30

A6 = C3 C4 Cl

A64 = 011 + X 012

A 6 5 = 0

65.

1.68724 C4 3.37203 C6 1.99628 C7 0.31145 C1

A66 = -L 4

+ [ 0.10713 0 .108996 0.048396 + 0.00581

C2 C4 C5 C6

+ .337448 0.481719 0.249536 + 0.03460

C2 C4 C5 C6

VT)[ 1 1.68724B6 = dR (3 11 + 2 13 + 4 x $12) + y( 3 $ - 8

R dx *1' R 'ujiRL c

+ 1.12401 0.49907 + 0.06229 L 2Fe 1.68724 - 1.12401

C3 C4 c5 2R c C3

+ 0.49907 _-.62+ 4 0.06291

66.

TABLE III

INTEGRALS OF DUCTED JET VELOCITY PROFILES

U-U

U dn= C 2 +

U-U

U 0)2 T2 d= C22Uj

U-U n U-U

(u 1 ) dn o ( U 0j 0 j

= Ci 9 + t (0.53333 C1 -

0 .15238

+ 32 & (C4 - 2 C5 + C6 ) + 0.11082 2

i dn 1

1.33333 C4 + 1.6 C5 - 0.53333 Cd)

+ 0.05572 E2

1 U-U$4 =j ( U ) n dn = Cl +

0 j

1U-U

5 Ujo ( 0)2 n dn = C2 10 j

0.26667 &

+ 32 (C3 - 2 C4 + Cs) + 0.20317 2

1.68724 C3 1.12401 C5 0.49907 C6

*6 J 0) (74)n 2 dn +0 j j c c

0.06229 C7 0.16069 .4996 0.01613 _ 0.0015135 c 0 3 94 5

07 ( )n2an = 0.42181 _ 0.18734 + 0,07130 _ 0.00779

o j C C C

.1 )2 dn = 0-711695 _ 0.632158 + 0.240586 + 0.13165o j C2 C4 C5 C6

0.124658 +0.03891 0.005652 + 0.000323

1

*1

01

*2

*3 =10

1l U-U

c 8 c 9 c 10

(wL)2 n~2 dl = 0.569356 - 0.5 + 0.210513 +0.117022____ 4 5 +

j c-

0.112192 + 0.035373 _ 0.005181 + 0.000298

C7 C8 C9 c1

1 U-U41io = Jo (S ) n dn

0 j( U0 ) nl dn1 = C20 + 0.26667 Ci

o uj

+ 0.03555 2

1.68724 C4. 1.12401 C6 0.49907 07n3 an = c C3 C4C C3 CL

0.06229 C8

C 5

0.10713 0.036332 + 0.012099 _c C3 C4

dn = 0.337448 - o.160573 + 0.062384 - 0.006921C- C. -

J ) UU 0.56241 (C1-C4) ( ." i a- C

0.22480 (ci - 06) 0.083178 (Ci - 07) 0.008899 (C1 - C8 )

C3 C4

+ .1142 0.05268 + 0.020164 _ 0.002207

c C3 C4 C5

414 u j ( 0)2 n3 dn = C23 + 32 & (C5 -2 C6 + 07) + 0.06465 C20 1

U-U

(U 0) n3 dn = C3 + 0.09524 Ej1

9 0

67.

f100

U-U

Uj 0) (ww.

0.00

1

412 = fl0

0)13 = fl0

&1~) n3WIf

(w ) n danvi

415 = j0

c c c

c c

68.

16 ( )2 3 dyi = 0.474463 - 0.474119 + 0.187122 + 0.10532o j C2 C4 C5 C6

0.101993 + 0.032425 0.004783 + 0.000277

C7 C8 C9 clo

0 0.562413 _ 0.224802 0.083178 o.oo8899017 n ~ dn +

o c C4 C

1U-U

018 = j .U ( 0) dn = C18 + 0.53333 ,0 j

69.

TABLE IV

1C0 =f f n di

C2 = f n2

03 f fii 3

04

C4 =f f fi0

C5 = f f 5

07 = if i60

C7 = f 1 f n60

C7 = f 8 fn

C9 = fof n9

1

Cin = *1.Co 0

Cii = o

di

di

dn

= 0.10719

= 0.04307

= 0.02101

= 0.0118o4

= 0.007142

= 0.004587

dn =

di

f i1 0 di

f ni1 di

012 = l f i 12

C13 = f f n 1 3

0

C18 = fe

di

di

f dn

Cig = il f dn fn fii

0.003081

= 0.002144

= 0.001534

= 0.001123

= 0.000838

= 0.000636

= 0.000489

= 0.38658

dni = 0.01280

C20 = f4 fn di fn if dni = 0.00566

C 2 1 = fl f2 n dn0

C22 = f 12 I2 dri0

023 = f 1 if2 TI3 ~0

0.05419

= 0.01636

= 0.006283

70.

1.0 -

.9 A

0 CORRSIN AND USEROI.8 ROSE

7 0 VAN DER HEGGE ZIJNEN

A ROSE ( rotating pipe - S = 0.102)

U .6 $ CHIGIER AND CHERVINSKY (tangentialinjection - S= 0.207)

.5

.4

.3 A 0

.2 A 0

0 .2 .4 .6 .8 1.0 1.2 14 1.6 1.8 2.0 22 2.4 2.6 2.8r/b

1. MEAN AXIAL VELOCITY IN A TURBULENT FREE JETFIGURE

I I I I I I I I

DATA OF ROSE1.0 S X/D

9 0 0.102 9<9 0.102 15

.8 \

DATA OF CHIGIER AND CHERVINSKY.7--

0 0S X/D

0 0.102 8.3

~ / -p 4 0.208 6.2

W /0 0.208 10.0

5 10.4 0

. ASSUMED PROFILE CONSTANT EDDY VISCOSITY

.2 0( c 0.34) SOLUTION OF LOITSYANSKII

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0r/b

MEAN TANGENTIAL VELOCITY IN A TURBULENT FREE JET

l- I i I i" li No 1" 0

I I .1 1.01 ---- I

I I I 1 0

FIGURE 2.

REFERENCE

o KERR & FRASER

0 ROSE

A CHIGIER &CHERVINSKY

0 0

a

cp

U

.1

I

.2

3. SPREADING RATE OF A TURBULENT FREE JET

.6 1-

6 9 U

.5 Fx/d

11.7

U nd/Vn

.4 1.

15

15

1.4 X 105

1.5 X 104

2.8 X 105

bx

3 F

0

.2

0

I I

.3

S

I

.4 .5a I i

0

FI GURE

U U I I I

0

0

0

0REFERENCE

KERR & FRASER

o ROSE

O CHIGIER &CHERVINSKY

a

.2

S

FIGURE 4. MASS ENTRAINMENT RATE

OF A TURBULENT

1.2 L

1.05-

0

.8L0

0.2

0

x/d

11.7

15

15

Uin d/ V

1.4 x io5

1.5 X 10)

2.8 X 105

a

.1I

.3

I.14

I.5a a

I I I Ii

onk.-01 x

%No*

000% c 6EJE

*%.00

FREE JET

2.0

1.6

DATA OF ROSE

S = 0.102

1.2 -

b 1 .0 .00

d.8 -

6 - f 7d= 0.00278 + 0.01 S

--- q d = 0.00278 + 0.0163 S )o U. 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

x /d

FIGURE 5. HALF VELOCITY THICKNESS OF TURBULENT FREE JET

2.41

2.2

2.0

0 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16

x /d

FIGURE 6. HALF VELOCITY THICKNESS

OF TURBULENT FREE JET

---

DATA OF CHIGIER SWIRLAND CHERVINSKY PARAMETER

0.067- e 0.117

. 0.208

--. 0 d= 0.00278 + 0.01 S

--- d = 0.00278 + 0.0163 S ( )

I I 2 I I I L

1

bd

I.4

1.2

1.0

8[

4

2

1-4 i .

8

I I I I I - I I I I I I I0.5

..-- COMPUTED FROM EQU

0.3

o HOT WIRE ANEMOMET

-0. MEASUREMENTS OF C

x/d = 20Un d/v = 17,0

0.1S= 0

S =0.0

0

. so

ATIONS 182

ER

ORR SIN

00

- I I0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

r /b

IN THE TURBULENT FREE JET

o.04

0.03

I IU v

U-2

0.02

0.01

I I

REYNOLDS SHEAR STRESSFIGURE 7.

a g I I I I

COMPUTED FROM EQUATIONS I 82

0.5

0.4

0.3

0.2

0.1

8 0.0

a i I I a

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.4

rib

2.6

VISCOSITY IN THE TURBULENT FREE JET

.07 1-

7/t

U j.8

.o6

.05

. o4

.03

.02

.01

FIGURE 8. EDDY

ADJUSTABLE END PLATE

erzz zzzz zzer zzz zzzz zzww wwww mz I

REMOVABLE SWIRL GENERATOR

THIN TRANSITION RING

70"

FIGURE 9. SCHEMATIC DIAGRAM OF TEST

SECONDARYAIR FLOW

4PRIMARY

AIR FLOW I6.51.2"

9"

-,w o m h - -. -- II II wTwIdli -1

.-.- Z,-,- j"Aa

APPARATUS

(a) 6 1/2 INCH DIAMETER

F IGURE 10. TEST SECTION

(b) 37/16 INCH DIAMETER

(c) JET NOZZLE

TEST SECTIONFIGURE 10.

FIGURE 11. SWIRL GENERATORS

"THS- 32I Ir 4 40 I4 8 16

PRO BES12.FI GURE

FIGURE 13. INSTRUMENTATION AND TRAVERSING

MECHANISM

----44-

-- I

II--a

THRUSTBALANCE

BALANCE

I-

PU

T

DUCT

--/*

THRUST

I

FIGURE 14.

U U U

O.N+ I

++ o 0

4- .4a

.030

.028

.026

.024

.022

.020

.018

.016

.014

.012

.010

.008

.006

.004

.002

0

0

0

a

00

4'0+0

0

A

0

0

- U d/V =

.n

SYMBOL

0. '

+

a 0

I

1

120,000

F(psi)

0.03140.03140.03100.02940.02600.0200

a2

2H /Fd

00.0680.0340.1060.1900.413

a3

PRIMARYMASS FLOW

(LB/SEC)

0.0990.1000.0970.0970.0980.097

a4

(P - P )

(mm. of methanol)

FIGURE 15. DETERMINATION OF JET THRUST

M

0

0

co'

E-4(.'j

I

5

I I I

M

M

APPLIED TORQUE

2

ob pU2

SODA STRAWS

FOR STATIC EQUILIBRIUM:

APPLIED TORQUE = puw

7r R2 H

+ uIw) 2'r 2 dr

FIGURE 16.

pU,WI J-

I

TORQUE BALANCE

220 -00

200

180

16o -

140 -

120

S100 -

80

60 0 INCLINED MANOMETER

O MICROMANOMETER40-

20

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

PRESSURE DIFFERENCE(INCHES OF WATER)

FIGURE 17. CALIBRATION OF t 0.05 PSI

PRESSURE TRANSDUCER

6

VOLTAGE MEASURED ON

5 -X-Y RECORDER

'4OUTPUT

m.v.3 .

2

INPUT SIGNAL 6 VOLTS D.C.

1 -SLOPE = 66.8 m.v./psid

0 1.0 2.0 3.0

PRESSURE DIFFERENCE(INCHES OF WATER)

FIGURE 18. CALIBRATION OF 0.20 PSI PRESSURE TRANSDUCER

1 is wo - im

- 0111.1

0.4

INPUT VOLTAGE = 4.00 VOLTS

0.3

OUTPUTM.v.

0.2

SLOPE = 4.48 m.v./psid

0.1

0 1.0 2.0PRESSURE DIFFERENCE(INCHES OF WATER)

FIGURE 19. CALIBRATION PRESSURE TRANSDUCEROF 2.00 PSI

1.1

1.0

0.9

00.8

0.7

tP 0.6 .

~pU2

0.5

Rey = 2370o.4 Rey = 1440

Rey = 7800.3

0.2 .

PITCH ANGLE = 0 DEGREES

0.1 .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16YAW ANGLE(DEGREES)

FIGURE 20. WEDGE PROBE CALIBRATION CURVE

w i wml*m -- ,

0.5

- - - -E - U U U U ~U I I U I U U

I I I I I I w

0.4P - P

LpU2

0O 0

00

O.31-

- a a a I I II I I I I I I - -

103

I I I I I I I

PROBE REYNOLDS NUMBER

SPHERE-STATIC PROBE

104

1 9 1a 0

21.FIGURE CALIBRATION

........ ..

0

00

0.10 L0

0

+

4.

oA<>

AA aaa

mDATA (p)1" 2

A o.6180 0.573+ 0.543< o.492o o.144oDO 0.389o 0.307

I

3

A

K>0

II I

1 2 14x/D

5 6 8

DISTRIBUTION - NO SWIRL

.40 0

0

.301-

.20 L

0 0 0-

p p0M

0 0 -0

0

0

+

+

+

4.-

AA

o o

0

-. 10

0

A

WALL PRESSUREFIGURE 22 a.

U I U U U I I

.30 m H (D)I-

(pM)"12 MD S

0BASED ON EQ. 3 0 .618 0 0 165,000

BASED ON EQ. 34 4 0.613 o.oo4T o.o68 163,000.20 .D0 0.620 0.0132 0.190 149,000

0 0.615 0.0295 0.413 129,000

.10 -

P - P 00V 0

0 00 o o

- o 0

-. 10

0 1 2 3 4 5 6 T 8x/D

FIGURE 22 b. WALL PRESSURE DISTRIBUTION - SERIES A

Owow"Am - - -,. -4- -, . . Mi- -+--

I I I I I I I I

U U U U I U U

.30 ..

.20 .

. 10.S..10---

BASED ON EQ. 33Pw -Pm 0- BASED ON EQ. 34

0 s p0 (pM)"12 MD v

o / 0 0.573 0 0 160,000

-p '4 0.572 .0050 .068 158,ooo-. 10 0.573 .0140 .190 145,000

0 0.569 .0314 .413 125,000

0 1 2 3 4 5 6 7 8x/D

FIGURE 22 c. WALL PRESSURE DISTRIBUTION - SERIES B

I I 9 I I I I

.30

.20 . / oI

.3.0,

BASED ON EQ. 33

w o / 0 - - - BASED ON EQ. 34

/ /m H (p) D0 -/ o --- -- S -

( {pM)"12 MD

o / 0 0.543 0 0 156,000

4 0.534 .0052 .068 155,000

-. 10 D 0.540 .o146 .190 142,000

0 0.540 .0322 .413 124,000

0 1 2 3 4 5 6 7 8x/D

FIGURE 22 d. WALL PRESSURE DISTRIBUTION - SERIES C

M Immorm-

.30

.20

BASED ON EQ - 33

.10 - - BASED ON EQ. 34

p - p 0 / (M)IlIM 0 m (M)II

(pM)"1 2 MD P

o

0 .492 0 0 153,000

o o 0 4 0.498 .0054 .068 152,000

0 0.501 .0151 .190 139,000< 0.495 .0336 .413 121.000

EFFECT OF WALL BOUNDARY LAYER NEGLECTED

0 1 2 3 4 5 6 7 8x/D

FIGURE 22 e. WALL PRESSURE DISTRIBUTION - SERIES D

I U I I I I U

.30

.20-

- P BASED ON EQ. 33M - - - BASED ON EQ. 34

.10

0 H (0 Mt)HI

0 O 0 (pM)" 2 MD v

Ile 0 0 0 0 0.440 0 0 148,ooo

-o0.438 .0057 .o68 148,oo

O 0.451 .0158 .190 137,000-.10 --

EFFECT OF WALL BOUNDARY LAYER NEGLECTED

I_. I. _ _I _I I I I

0 1 2 3 4 5 6 7 8x/D

FIGURE 22 f. WALL PRESSURE DISTRIBUTION - SERIES E

waft 0- No",

.30 -

.20

0

0 - BASED ON EQ. 33

0- -- - BASED ON EQ. 34

.10 . /

pw Po /0 / / DP m H (.) IZ

(pM)'" 2 M D v0 . 0 0.389 0 0 147,000

0 0 0 0 4 0.387 .0059 .068 145,000

0 0.404 .0162 .190 134,000

-. 10 EFFECT OF WALL BOUNDARY LAYER NEGLECTED

0 1 2 3 4 5 6 7 8x/D

FIGURE 22 g. WALL PRESSURE DISTRIBUTION - SERIES F

I -MANow"wom -- . -- - - -.-, -

w Imp 1 11 11 M I 11, 111

I I I I U I U I

3.4

3.2

3.0 4

2.8

2.6

2.4

2.0

U

p

1.2

1.0

0.8

0.6

0.4

0.2

-Dm H pD

(pM) MS

0 0.62 0 0 165,000

0 4 0.62 .0072 .106 150,000

0

0

x/D = 1 1/2

0

\ BASED ON EQ. 33

- -BASED ON EQ. 34

0 \

'3

r/R

0 .2 .3 .4 .5 .6 .7 .8 .9

PROFILES - SERIES A

I I i I I I I I

..

-

FIGURE 23 of AXIAL VELOCITY

2.6 m H D

(pM)" 2 MD v

o o.62 0 0 165,000

2.2 4 o.62 .0072 .lo6 150,000

02.0

00

1.8 0 x/D= 2 1/2

0U 1.6

( ).42 BASED ON EQ. 33

- - - BASED ON EQ. 34

1.2 '3

1.0

0 .8 '3 44 1

o.6 N ''

0.4 00

0.2 . 0

I I I I I I

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

FIGURE 23b. AXIAL VELOCITY PROFILES - SERIES A

II I I

m(pM)"1 2

0 0.62

4 0.62

H

MD

0

.0072

S p) 2 D

v

0

.106

165,000

150,000

1. 8

U

p 1.14

1.0

0.]

0.4

x/D = 3 1/2

0l 0

0 BASED ON EQ. 33

0

00

4 44

m 0

0 0

0 0 0 -

0.21i-

Ia I I I I I I I I0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

23 c. AXIAL VELOCITY

2.6

2.4

2.2

2.0

PROFILES - SERIES AFIGURE

H D

(pM)1 2

0 0.62

40 0.62

MD

0

.0072

S

0.lo6

165,000

150,000

(

- x/D = 5 1/2

BASED ON EQ. 33

000000 o

w 0-

I I I I I I a I I i

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

23 d. AXIAL VELOCITY

2.4

2.2

2.0

U

(!.)112

p1.4

1.2

1.0

0.6

0.2

--

-

.

0.8

PROFILES - SERIES AFIGURE

3.4

3.2

I I I

)I12D

3,000

3,000

r4 R

r /R

0 .1 .2 .3 .4 .5 .6 -T .8 .9

AXIAL VELOCITY PROFILES - SERIES D

3.0

2.8

2.6

2.4

2.2

2.0

rn H S(pM)"1 2 MD

0 0 0.49 0 0 15

i 0.49 .0084 .106 140

0

- x/D =1 1/2

-- BASED ON EQ. 33

. ---- BASED ON EQ. 34

. <3

. \

0

0 0 0 4

- O 000

U

M)112p

1.8

1.6

1.14

1.2

1.0

0.8

0.6

0.4

0.2

FIGURE 23 e.

1 I I I I I I

m(p M)11 2

0 0.49

4 0.49

H

MD

0

.0084

M.~ )112D

S p'I

0

.106

153,000

143,000

2.2

2.0

U

(!)112 1.6p

1.4

1.2

1.0

0.8

0.6

0.4

0.2

.0

- x/D = 2 1/20

0 BASED ON EQ. 33

- '4 - --0 BASED ON EQ. 34

- '1O

. \4 -1

Oi

00 0'

\ 0 '0- -0

o '1

- x O O0

.. ... .. .-

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

PROFILES - SERIES D

2.8

2.6

4

I

AXIAL VELOCITYFIGURE 23 f.

2.4 m H p D

(pM)11 2 MD v2.2 .-

0 0.o49 0 0 153,000

2.0 .4 0.49 .oo84 .106 143,000

1.8

1.6 x/D = 3 1/2

0

U 1.4

(m- BASED ON EQ. 33

P 1.2 00

1.0 0

00.044

o.6 00g 0

0.4 0

0.2 .-

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

SERIES DAXIAL VELOCITY PROFILES -FIGURE 23 g.

I I I * * U I 3 5

1.4 .(M)112

1.3 (PM)" 2 MD _

0 o.49 0 0 153,000

1.2 .< 0.49 .0084 .106 143,000

1.1

1.0 .x/D = 5 1/2

0.9

-- BASED ON EQ. 33

0.8

)120.7 00000 000

o.6

0.5 0

0.3

0.2

0.1

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

PROFILES - SERIES D

I I I I II I i

FIGURE 23 h. AXIAL VELOCITY

U U U 5 5 5 - .

S0.620.6 - (pM)l 2

H- - 0.0072

0.5 w S o.io6

pM = 150,000

0.4 -

x/D

wo 1 1/2

()2 0.3 V 2 1/2

o 3 1/2

0.2 - BASED ON EQ. 33

0.2 -/-BASED ON EQ. 34

0.1/0

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r/R

24 a. TANGENTIAL VELOCITY

1 9 v IfI

PROFILESFIGURE

m = 0.62

0.6 (pM)" 2

H-- 0.0132

MD

0.5 S 0.19

()It?-D 1499000p

0.4 -x/D

w AM)112 0 1 1/2

p V 2 1/20.3 0 3 1/2

BASED ON EQ. 33

--- - BASED ON EQ. 34

0.2 0 0O

OV

0.1 0 0

0 .1 .2 .3 .4 .5 .6 .7- .8 .9

r/R

TANGENTIAL VELOCITY PROFILES2 4 b.FIGURE

--.6-- = 0.492(pM)" 2

H - 0.0084

0.5 MD

S = o.lo6

M.D .= 143,0000.14 p

x/D

w- 1 1/2(M) 1 12 0.3 V 2 1/2

p0 3 1/2

0.2 ._ BASED ON EQ. 33

- - BASED ON EQ. 34

0.1 /

0/R

0 .1 .2 .3 .4 .5 .6 .7 .8 .9

r /R

VELOCITY PROFILES24 c. TANGENTIALFIGURE

I I I I

m(pM)112

H

MD

S

Mp

)1122LI

0.50

0.0151

= 0.19

139,000

x/D

. 011/2 -

V 2 1/2

0 3 1/2

BASED ON EQ. 33

__ BASED ON EQ. 34

r7

1.2 .3 .4 .5 .6b .7 .8

r/R

VELOCITY PROFILES

o.6

0.5 L

w(M)112

p0.3

0.2

0.1

.9

24 d. TANGENTIALFIGURE

I I I I U

DATA

0

0

S

0.00o.o680.190o.413

- 30 Helmbold's

No Swirl Data

XoD 0

00

0~ 000 OS 0

00 -

.*to

0

-1

INCLUDING EFFECT OFWALL BOUNDARY LAYER

- - - - NEGLECTING EFFECT OFWALL BOUNDARY LAYER

.1 .2

- -

m(pM)" 2

I.3 .4 .5

FIGURE 25. AXIAL POSITION OF JET

0

0

0a

.6

I I I I II

I II I

ATTACHMENT

I U

m(pM)"

2

a o.62

* 0.57

3 4 0.'49

A 0.39

AU

2 4.. S

D A U

01 .2 .3 -4

2HFd

26. AXIAL POSITION OF JET ATTACHMENT

9 I

FIGURE

I I I I

1.2

1.0

0.8D. C.VOLTS

0.6

0.4

0.2

0 .1 .2 .3 .4 .5 .6 .7 .8 .9r/R

HOT WIRE SIGNAL

0.2 v./cm.

20 ms./cm.

SPECTRUMCENTER FREQ = 50 Hz.DISPERSION =

10 Hz./cm.(ZERO AT RIGHT)

0.01 v./cm.

OSCILLISCOPE TRACE AT r/R = 0.43

FIGURE 27. QUALITATIVE HOT WIREANEMOMETER MEASUREMENTS

0.2

RMSf0LTS

0.1

. D.C. VOLTS

0 RMS VOLTS

A -

0- o Co o - 0.6200 0 0 (p My/a

-0 S = 0.19

A 0 x/) = I60

A66 66ee 0 0 0 0SI I I I I I I I

v I I I I

U I I I I I I U

0 0

0

m(pM)"1

2

A 0.543

0 0.539

I

5

H

MD

00

I

6

dD

M1D1

0.185 156,ooo

0.349 280,000

I7 8

- NO SWIRLOF DIAMETER RATIO

P -Pv 0

m

.20

.10

0

-. 10

0

00

0

0

A

0

A

0

0

A

0a

0A

0A6

0A

0A

0A

a

1

I

2

x/D

I

3

I

I I I I I I I I

0 A

I

87

EFFECTFIGURE 28

U U U I I I I I

0 0 0

0

m(pM)" 2

A 0.534

0 0.540

I5

x/D

H

MDdD

(M)112p D

.0052 0.185 155,000

.0054 0.349 279,000

I6

a IT

OF DIAMETER RATIO

P -PV 0

M

.20

.10

0

-. 10

0

00

0

)

0

0A

0o 0

R 6 A4(0

0I1

I2

I3

I14

I 9 9 I i I I I

O l

8

29. EFFECTFIGURE - WEAK SWIRL

U U I I I U U

00 A

a

4

0

m(pM)" 2

A 0.540

0 0-540

r/DI

5

A 0

H

MD

0

dD

A

(1.)112D

p

0.032 0.185 124,000

0.031 0.349 240,000

I

6 7

Ii i

i8

- STRONGOF DIAMETER RATIO

0

0A0

p -pV 0

0

A

.20

.10

0

-. 10

0

0

0 00a

a

1

a

2

a

3

FIGURE

Ia a 2

I II I I i i

8

30. EFFECT SWIRL

I U I

m o.618(pM)1" 2

.20 H

MD

(M)1I22. :165,000

p p

.10 - 000

P -P

EFFECT OF WALL BOUNDARY LAYER INCLUDED

0 0 EFFECT OF WALL BOUNDARY LAYER NEGLECTED

-. 10-

I Ip

0 1 2 3 4 5 6 7 8

x/D

FIGURE 31. EFFECT OF WALL BOUNDARY LAYER

ON WALL PRESSURE DISTRIBUTION

U U I

I

- -- aR~ - - - -

0.4130.190

0.068

sm 0

r = 0.57(pM)" 2

I

0.57

I 0 a

U I I U U

m(pM)"12

0.413

I0.190H

0.068

s:0

x/DII

0.5I

1.0 1.5I

2.5I

2.0

32. WALL BOUNDARY LAYER PREDICTION -MOSES

I

.14

.12

.10

.0

.02

2.6

2.41-

2.2

2.0

1.e

1.6

1.4

0

I 9 9

I v i I 0

FIGURE

U U U U U I U

BASED ON EQ. 33

.20 BASED ON EQ. 34

00

.10

0

a MD0 m H pd

6 6 0.572 0.0050 o.o68 158,000 0.185

0 0.570 0.0052 0.034 282,000 0.349

I II I I -

01 2 3 4 5 6 7 8

x/D

FIGURE 33. EFFECT OF DIAMETER RATIO - WEAK SWIRL

I I- -,, - . 1. -

I II I i i i

.6

.5

.4

.3

.2

.1

DATAm

(p M)" 2

BASED ON EQ. 33

BASED ON EQ. 34

0 o.620 o.49 0

- 0

0 * 0-- --

U 0 -0 0

- S 0. 106

a

2 3 4

I

5

x/D

FIGURE 34. VELOCITY NEAR THE WALL

- -

00

p0o 0

-m 0 0

0- S =0-

.6

.5

.3

.2

.1

0 1A

-

a

10. 1

1.0 ____

6 1

0.1 1 a I I a

D ATA S0 0 BASED ON EQ. 33

10.

=pmm 0. 4 9 -

(M)I2

-0

1.0 -j

N!

0.10 1 2 3 4 5

x /D

FIGURE 35. JET EXCESS VELOCITY AT CENTERLINE

1.0 = 0.62A (pM)"2a

0

0

0.1 Wj

(MlI 0

.01I I

DATA So o.io6 BASED ON EQ. 33

A 0.1901.0

- =0.49(pM)l"a

0j

0.1 .

00

.01 I I I0 1 2 3 4 5

x/D

FIGURE 36. MAXIMUM TANGENTIAL VELOCITY

I I I

.03 -

s o .413.02 m

0.190

01

-.01--

-. 02

-. 03

m) = 0.57 = 2.6(p0MY/p b

1.0

.8

.6 c 0.413 0.19 S = 0.068

.4

.2

0 1 2 3 4 5

x/D

FIGURE 37. PREDICTED DEPARTURE FROMSELF- PRESERVATION

I I I I

I I I ~1 I I I U

DATA m(pM)0a

n Ac

0

0.9

0.8

0.7

o.6

0.5

0.4

0.3

0.2

0.1

I I I I I I I I I

V. 2

0.570.49

d = 0.185D

USING PRESSURE DROP

CORRELATION OF REF. 21

A P

TpnI I I I I1 I11I1

0.1

I I I I I lII II

1.0

FIGURE 38. SWIRL GENERATOR PRESSURE DROP vs PERCENTAGEREDUCTION OF JET ATTACHMENT LENGTH

A x.

.01

I

I