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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
MISSING PAGE(S)
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