Table Of ContentAPPROXIMATE ANALYSES INTERRELATING PRESSURE DISTRIBUTION
AND AXISYMMETRIC BODY FORM
En-Iun
University of lows
LI 8 K aHY
A dissertation submitted in partial fulfillment of the require
ments for the degree of Doctor of Philosophy, in the
Department of Mechanics and Hydraulics, in
the Graduate College of the State
University of Iowa
February, 19-50
ProQuest Number: 10991959
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This dissertation is hereby approved as a credit
able report on an engineering project or research carried
out and presented in a manner which warrants its acceptance
as a prerequisite for the degree for which it is submitted.
It is to be understood, however, that neither the Department
of Mechanics and Hydraulics nor the dissertation advisor is
responsible for the statements mad© or for the opinions
expressed.
Dissertation Advisor
ACKNOVDIDGMTS
Grateful acknowledgments are due to Dr, John S* LfcNown for
his continuous advice and helpful suggestions and to Dr. Hunter House*
Director of the Iowa Institute of Hydraulic Hesearch for his general
encouragement and support. Gratitude is also expressed to Hr. Emmett VL
Laursen for the special assistance which he so generously provided* to
Ifessrs. Wen Heieh and Yu Ch h Soong for their assistance in carrying
out the relaxation routines# and to Miss Leona Amelon for her untiring
efforts in the preparation of this dissertation.
Hie investigation was sponsored by the Office of Kaval
Research under Contract H8onr-S00.
ill
TABLE OF CONTENTS
Page
INTRODUCTION .................................... 1
THEORETICAL ANALYSIS ............................. 4
Preliminary Considerations ....................... 4
Relaxation Method ....................... 7
Mathematical Analysis .......................... 14
RESULTS AND DISCUSSIONS............................ 29
A Symmetrical Profiles............... 29
G. UnaynaoBtrlcal Profiles............... SI
C Cavitation Pocket Form...................... S3
D. Profile with a linear Pressure Gradient ...... 36
E. Applications.............................. 37
CONCLUSIONS..................................... 39
REFERENCES ..................................... 59
iv
TABLE CF FIGURES
Figure Page
I* Cylindrical coordinate system ........... 6
2. Characteristics of relaxation elements.
(a) General case, (b) Specific case encountered ...... 8
S. Relaxation pattern for a regular net . . . . . . . . . . . . 10
4. Evaluation of velocity at boundary, (a) Velocity
cDisponente. (b) StreaEKfunetlon gradient ...............11
5. Sketch showing velocity relationship for a
slender body ...............................22
8. Characteristics of assumed linear velocity distribution . . 24
7. Symmetrical profiles with velocity distribution a * c * 0 41
8. Symmetrical profiles with velocity distribution a * c * b/Z 42
9. Symmetrical profiles with velocity distribution a * c » b 43
10. Symmetrical profiles with velocity distribution a • c * 1.2b 44
11. Summary plot for symmetrical profiles .................. 45
12. Unsymmetrical profiles with 1 » -1.0.................... 48
13. Unsyimaetrical profiles with X * -0.4.................... 47
14. UnsyiMetrical profiles with X» -0,2 ........... 48
15. Unsymetricai profiles with X « 0 . 49
16. Cmyranetrical profiles with X » +0.2 . 50
17. Summary plot for unaymmetrical profiles . . . . . . . . . . 51
18. Coarse net for the cavitation pocket around
a hemispherical head K « 0.2 62
v
Figure Page
19. Final net for the cavitation pocket around
a hemispherical head K » 0.2 63
20. Silhouette photograph for the vapor pocket around
a hemispherical head K « 0.2 64
21. Comparison of th© vapor pockets and pressure
distributions for hemispherical head K * 0.2
(Relaxation and experiment) ........................ 66
22. Comparison of the vapor pockets and pressure
distributions for hemispherical head K ® 0.2
(Mathematical and experiment) ................ 66
23. Comparison of the vapor pockets and pressure
distributions for 2*1 ellipsoidal head Z » 0.2 67
24. Profile with an assumed linear pressure gradient...... 68
vi
LIST CF TABLES
Table Pane
ft*X
I. Values of 1 Pn(Ml) . 28
'
•**«
vii
1
INTRODUCTION
Usually the main interest in the design of an immersed body
form moving in a fluid medium is the reduction of the resistances
encountered* in effect, the reduction of the drag coefficient. The
portion of the total resistance known as form drag can be varied since
It is dependent on the pressure distribution which is a function of the
body shape. Separation of the flow from the boundary will result in a
low pressure on the rear of the body and therefore will have a major
influence on the form drag. Although no satisfactory methods have been
developed for the prediction of the point of separation in either the
laminar or turbulent boundary layer, various pressure distributions can
be compared qualitatively for separation effect. Two methods are pro
posed herein that allow the design of a body form to a predetermined
pressure distribution and thereby control of the probable separation
point.
Similarly cavitation around a body can be prevented by design
ing the shape for a pressure distribution having a maximum negative
pressure less than the vapor pressure. The same methods may be used in
the solution of this problem.
The past analyses have had in common the approach in which the
body form is assumed and the pressure distribution determined. A basic
assumption which is always made is that the Reynolds number of the flow
is so large that the flow is essentially irrotational. For a given
axisysBaetric body* von Karman developed a method in which a series of
2
line sources and sinks was assumed# the strengths of 'which were deter
mined by the condition that the body is necessarily a stream surface in
a parallel field of flow [1# 2], Kaplan [3] presented a more rigorous
mathematical analysis of the potential flow around a body of revolution.
A series solution of the velocity potential function was obtained by
expressing the profile of the body in a power series in the elliptical
coordinate system.
Recently Southwell developed a numerical method of solving
differential equations* known as the relaxation method, which may be
used to solve the Laplacian equation for a given boundary condition.
However* this method does not give the general function of the Laplacian
solution, but furnishes only numerical values of either the velocity
potential or the stream function* at a large number of points throughout
the region. It is a serai graphical method and has its major achievement
in solving the difficult problem of free stream lines [4, 5].
In this dissertation Southwell's relaxation method is applied
in finding the shape of a cavitation pocket having a known pressure
distribution by trial and error. In a recent publication Birkhoff has
also suggested this method [6]. Although the method proved successful*
it was not considered wholely satisfactory because of the number of
times the boundary had to be modified, and the time required to liquidate
all the residues in each trial in the entire region.
A mathematical analysis lias also been developed for irrota-
tional flow around slender bodies in which either the body form or the
pressure distribution can be specified. This analysis consists of
