Table Of ContentDOCTORAL DISSERTATION SERIES
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UNIVERSITY
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DEGREE PUBLICATION NO, pi
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Thy UNIVERSITY MICROFILMS
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ANN A IIO I • MICHIGAN
STUDY OF LAMINAR SWIRLING PIPE FLOW
by
Lawrence Talbot
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy in the
University of Michigan
Committee in charges
Professor W. W. Hagerty, Chairman
Associate Professor R. C. F. Bartels
Professor R. A. Dodge
Professor A. S. Foust
Associate Professor P. F. Chenea
ACKNOWLEDGMENT
The author wishes to take this opportunity to ex
press his very deep appreciation to Professor W. W. Hagerty,
teacher and friend, whose continued encouragement and as
sistance made this work possible; to Professor P* F. Chenea
for his valuable assistance on both the theory and experi
ment; to Professor R. A. Dodge for his many kind criticisms
and helpful advice; to Professor R. C. F. Bartels for his
help on mathematical portions of this work, and to Professor
A* S. Poust for many suggestions*
The author wishes to thank Mr* M* V* Kaufman for
his assistance in the design and construction of the equip
ment; Mr* J. B. Cline for his help in the analysis of the
motion picture data; and Messrs. J. R. Sellars and M* A*
Brull for their aid in' performing calculations on the dif
ferential analyser*
The author is Indebted to Professor B* L. Briksen
for his aid in securing funds to construct the equipment
and for his continued kindness during the course of thii
work*
11
TABLE OP CONTENTS
Pag*
ACKNOWLEDGMENT....................................... 11
CHAPTER I THEORETICAL INVESTIGATION................. 1
1. Introduction............................... 1
2. Formulation of the Basic Equations.......... 2
3. Exact Solution to ths Swirl Equation........ 5
4* Approximate Solution to tho Swirl Equation.•• 11
5* Calculation of tho Perturbation Velocities... 15
6. Example Problem........ 22
CHAPTER II EXPERIMENTAL INVESTIGATIONS............. 27
7. Design of the Experiment.......... 27
6. Experimental Apparatus ............ 30
9« Stability of the Swirl. ............ 34
10. Velocity Measurement Technique. ........... 36
11. Experimental Results and Comparison with
Theory........................... 40
12. Conclusions. ................. 43
13 • Recommendations.................. 44
APPENDICES
A• Calculation of an Eigenvalue•••••........... 46
B. Derivation of the Momentum Integral Equation. 46
C. Sample Computation of Drop Velocity 50
D. Specimen Data and Computations for Decay Rate 54
BIBLIOGRAPHY 57
LIST OF FIGURES
Page
1. Coordinate System and Velocity Notation......... 59
2. Differential Analyzer Circuit.................. 60
3a First Harmonics of Swirl............. . •• 61
3b Second Harmonics of Swirl......... 62
4. Decay Parameter vs. p ........ 63
5. Calculated Axial Velocity Distribution......... 64
6* Variation of K(r) with r...................... 65
7. Axial Velocity Perturbation vs. z.............. 66
6. Radial Velocity Perturbation vs. z............. 67
9. System Flow Diagram........................... 66
10. Constant Head Tank Mounting. ............ 69
11. Interior of Constant Head Tank................. 70
12. Entrance Chamber...... ........................ 71
13. Detail of Entrance Chamber.......... 72
14. Detail of Fixed Pipe Joints.................... 73
15. Detail of Pipe Entrance........... %
16. Bearing Detail................................ 75
17. Section of Bearings........................... 76
16. Swirl Section and Drive........................ 77
19. Motor Drive and Variable Speed Control.......... 76
20. Vater Chamber and Test Section.............. 79
21. Exit End of System....................... 60
22* Stable and Unstable Swirls.... ............... 61
iv
Page
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23. ” 3ma vs* R......... d2
24* Schematic of Motion Picture Camera Set-up....... S3
25* Specimen Picture of Droplets.......... 64
26. Specimen Picture of Droplets*................... #5
27• Experimental Values ^ vs. R.................... 36
23. Experimental Swirl Decay Data................... 67
LIST OF TABLES
I. Specimen Data.................................. 54
II. Experimental Values for yS ...... 56
v
CHAPTER I
THEORETICAL IHVBSTIQATIOH
1. Xn*r9<to9tt9B
The problem treated here is that of determining
the resulting fluid motion obtained when a ewirl is super
imposed upon Poieeuille flow in a straight round pipe.
(The term ’swirl* in the sense used here denotes motion in
concentric circles about the pipe axis) In particular,
interest is centered upon determination of the space rat#
of decay of the swirl as it is connected down the pipe,
the stability of the flow, and the interaction between the
superimposed swirl and the main Poieeuille flow.
Swirling flows such as the one described, or flows
similar to it, do at times occur in practice. A few ex
amples of engineering interest are the flows which occur
in pipe bends, in swirl atomisers and in certain types of
spray nossles. Some of the results to be obtained here
may find application in these and other engineering problems,
although no attempt will be made to connect this work with
any particular problem. Rather, emphasis is directed to
ward an analysis of the fluid motion itself, in the belief
that an understanding of the mechanism of decay of
1
2
disturbances is of fundamental interest* Furthermore, some
of the results to be obtained here are of qualitative value
in deciding certain questions of a more general nature
concerning the stability of viscous flows and the mechanism
of transition* The connection between the problem under
consideration and the problems of stability and transition
will become apparent later on, and further discussion on
this will be reserved until that time*
2. Formulation of the Basic Equations
The assumptions upon which the mathematical theory
Is based are as follows;
(a) A steady-state, rotationally symmetric swirl
is superimposed upon Poieeuille flow in a
straight round pipe*
(b) The swirl is of prescribed magnitude at
s ■ 0 in the coordinate system of Fig* 1*
The exact Havier-Stokes equations for the rota
tionally- symmetric steady motion of an incompressible fluid,
neglecting body forces, are
U + w 3k (2.01)
dr pa
uiir + u r S x *■ M » -x (v*tr-3f ) 12.02)
»r »* r R \ r*/ w.u*i
* ur%j - *• £ *7 *ur (2.03)
where R is the Reynolds number and
V * m + r / * £ (2.04)
*r « r Sr
In the above equations all velocities have been
made non-dimensional by referring them te the charseter
istic velocity Wm9 here taken to be the maximum velocity
ef Poieeuille flow. The pressure p has been made non-
dimensional by referring it to the characteristic pressure
f Wm2f where f is the density of the fluid* The independ
ent variables r and s have likewise been referred te the
characteristic length a, here taken to be the radiua of the
pipe. With this system of non-dimensional variables, the
Reynolds number is then defined
R * « (2.05)
where ^ denotes the kinematic viscosity of the fluid. A
detailed treatment of the method by which equations are
reduced to non-dimensional form may be found in Goldstein
(l)1, p. 100.
To linearise the equations of motion, the axial
velocity w is replaced by W + wv, where w* is a perturbation
^Numbers enclosed by parentheses will refer to
roferoncteess cciitteedd iinn tthhee BBiibblliiooggrraapphhyy,, uunn]less otherwise
specified.
4
and V is the main Poieeuille flew
W * 1 - r2 (2.06)
The pressure p is replaced by P + p*, where p* is a
perturbation and P is the "Peiseuille pressure” Which
is uniform across the pipe and is a linearly decreasing
function of s. If square and product terms in u, w» and
their derivatives are neglected, as well as the tests Ml, ^>v
in Equation (2.02), then one obtains the linearised equa
tions of motion
W t e _ V* m _ dk
(2.07)
** r Sr
VSJu/
(2.09)
3Jt
Note that the equality satisfied by the Peiseuille velecity
(2.10)
has been subtracted out of Eq. (2.09).
It is seen that the ”swlrl equation,” Eq. (2.06),
does not involve the perturbations u, w*, and p', and
hence Eq. (2.06) may be treated independently from the ether
