AN EXPERIMENTAL STUDY OF THE
BOUNDARY-LAYER CHARACTERlSTSCS FOR
WGeFHASE CGASuLEQWB SPRAY) FLOW

OVER A CERWR CYLENDER

fins-ls Cm- (fim- Dangm @‘5 pk. D.
MECEIGAN SMTE UKEVERSE'FY
Hwoid E. Wright
1966

 

 

 

LIBRARY

THE ‘ Michigan State
5'5 University

This is to certify that the ’

thesis entitled

AN EXPERIMENTAL STUDY OF THE BOUNDARY—LAYER
CHARACTERISTICS FOR TWO-PHASE ( GAS-LIQUID
SPRAY) FLOW OVER A CIRCULAR CYLINDER

presented by

Harold E. Wright

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mechanics

Department of Metallurgy, Mechanics
and Materials Science

24f) 2 J )1

/'
x q . :' ,. I I .'
H \‘4/' (zit/{J ’31, flL'C’l: k '\ ~
Major professor
Lawrence E. Malvern

I? c. ”is
\~’MA&'1—\

.-,; . ,,- W /
( I ~ ‘ ‘ (— /L 1/ ' 5
J U «l/l I I ’f I
Date " “a U~~ J 1 » k. L
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§

0-169

 

ABSTRACT

AN EXPERIMENTAL STUDY OF THE BOUNDARY-LAYER
CHARACTERISTICS FOR TWO-PHASE (GAS-LIQUID
SPRAY) FLOW OVER A CIRCULAR CYLINDER

by Harold E. Wright

This research is concerned with the transverse flow of a two—phase
(air-water spray) mixture over a right circular cylinder for a range of
gas Reynolds numbers from 1 X lO'+ to 2 X 105, with mass ratios of water
to air less than 0.1. The particular area of investigation is the
region of the flow field where the boundary-layer separates from the
cylinder.

This research includes an experimental investigation of the flow
field in the region of the cylinder and some of its boundary-layer
properties. As a result of the investigation two experimental tools
for two-phase boundary-layer studies were successfully developed: A
technique for the measurement of the pressure profile was developed,
utilizing liquid-filled pressure lines from which a small amount of
water is injected into the liquid boundary layer. For the case con-
sidered, a maximum error of 5.2 per cent was found at the point of
pressure minimum in comparison with a standard method in single—phase
flow. The second technique developed was a method for visually
observing the point of boundary-layer separation. A mixture of
distilled water and wetting agent was injected into the liquid boundary

layer, which promoted the formation of small gas bubbles, whose motion

disclosed the point of boundary-layer separation. The results of these
observations agreed with those interpreted from the observed pressure
profiles.

The pressure profiles were measured for the case of two-phase flow
for a gas Reynolds number of 5.6M x lOL+ and three different water
nozzle pressures. In each case the point of pressure minimum shifted
downstream approximately 6 degrees while the separation point shifted
downstream 14 degrees when compared to single-phase flow at the same
gas Reynolds number. The rate of growth of the liquid-boundary—layer
thickness was observed to be high in the region between the point of
pressure minimum and the separation point. A bubble of water was
observed to be located downstream of the separation point, and vortices
were observed. It was by these vortices that water was discharged
from the cylinder. It is believed that the presence of the relatively
thick liquid boundary layer in the region of the separation point and
the presence of the water bubble provided the external flow with a
contour sufficiently different from a cylinder to produce the observed
reduction in the slope of the pressure profile in this region.

Two analytical models were considered: the first assumed a
laminar boundary-layer system, while the second considered turbulent
flow. Neither analytical model gave good agreement with experimentally—

observed separation points.

AN EXPERIMENTAL STUDY OF THE BOUNDARY-LAYER
CHARACTERISTICS FOR TWO—PHASE (GAS—LIQUID

SPRAY) FLOW OVER A CIRCULAR CYLINDER

by

Harold E.¢Wright

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics
and Materials Science

1966

ACKNOWLEDGMENTS

I gratefully acknowledge my indebtedness to Dr. Max Scherberg of
the Aerospace Research Laboratories of the United States Air Force,
who suggested the problem and gave valuable guidance and counsel
throughout the project.

I wish to express my sincere graditude to my major professor,
Dr. L. E. Malvern for his continued encouragement, patience, sugges-
tions and technical guidance throughout my entire Ph.D. program. I
am grateful to my research professor, Dr. J. E. Lay, for his many
suggestions and technical advice on problems encountered in the
research. I hereby acknowledge my appreciation to the other members
of my guidance committee, Dr. W. A. Bradley, Dr. J. F. Foss and
Dr. R. H. Wasserman.

I am grateful to the United States Air Force for the opportunity
to pursue this research. In particular, I am indebted to the Air
Force Institute of Technology (AFIT) and the Aerospace Research
Laboratories who provided facilities and instrumentation. I want
to thank Mr. W. W. Baker, laboratory technician of AFIT for his
assistance throughout the experimental phase of the research.

Finally, I wish to thank my wife and children for the sacrifice

they made in order for me to pursue this research.

ii

CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . .

LIST OF

LIST OF

FIGURES O O O O O O O I O O O O O O C O O O O O

TABLES .

NOMENCLATURE . . . . . . . . . . . . .

CHAPTER

I.

II.

III.

IV.

INTRODUCTION 0 O O O O O O I O O O O O O O O O

l. 1 General Discussion and Purpose .
l. 2 Some Theories of Boundary- Layer Behavior
in Two— Phase Flow .

ANALYSIS . . . . . . . . . . . . . . . . . . . .

2.1 General Description and Objective .
2. 2 Gas Boundary- Layer Model for Both Liquid and
Gas in Laminar Flow . . . . . . .

2. 3 Turbulent Gas Boundary— Layer Model for Liquid
in Laminar Flow and Gas in Turbulent Flow .

EXPERIMENTAL METHODS AND APPARATUS .

General Description and Objective . . .

The Quality of the Test Apparatus . . . . .

The Inlet Screens and Diffuser . . . . .

The Water- —Spray System . . . . . .

The Wind- Tunnel Test Section . . . . . .

The Wind- Tunnel Fan and By- -Pass Spray
Nozzle Assembly . . . . .

.7 Butterfly Valve and Outlet Diffuser . . .

.8 Summary of Test Facility . . . . . . . .

3.9 The Test Specimen and Control Panel .

000003036000
mmszl—J

000.)

EXPERIMENTAL PROCEDURE . . . . . . . . . . . . .

u.1 Determination of the Pressure Profile . . .
H.2 Determination of the Water Droplet Velocity

and Droplet-Size Distribution . . . . . .
”.3 Measurement of the Liquid-Film Thickness .

iii

PAGE

ii

ix

19

25

25
26
27
28
28

28
29
29
29
32
32

36
38

CONTENTS (contd.)
CHAPTER PAGE
IV. EXPERIMENTAL PROCEDURE (contd.)

u.u The Determination of the Ratio of the Mass

of Water Flow to Mass of Air Flow . . . . . . . . 39
4.5 Measurement of the Liquid-Film Velocity . . . . . . Ml
4.6 The Determination of the Boundary-Layer

Separation Point . . . . . . . . . . . . . . . . H2

V. NUMERICAL PROCEDURES . . . . . . . . . . . . . . . . . . 44

The Scope of the Numerical Procedures . . . . . . . H4
Polynomial Fitted to Experimental Data . . . . . . 44
The Runge-Kutta Numerical Procedure . . . . . . . . nu
Numerical Procedures for the Laminar

Gas Boundary-Layer Model . . . . . . . . . . . . 50
5.5 Numerical Procedure for the Turbulent
Gas Boundary-Layer Model . . . . . . . . . . . . 51

U‘IU'IU'IU'!
router-1

VI. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 52

Preliminary Remarks . . . . . . . . . . . . . . . 52
Wind- Tunnel Evaluation in Single- -Phase Flow . . . . 53
Experimental Results in Two— Phase Flow . . . . . . 57
Results of Analytical Investigation . . . . . . . . 68
Summary of Results . . . . . . . . . . . . . . . . 71

0105010103
(fl-CODMl-J

VII. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . 75
APPENDIX

A GRAVITATIONAL EFFECTS . . . . . . . . . . . . . . . . . 79

B FIGURES . . . . . . . . . . . . . . . . . . . . . . . . 81

C TABLES . . . . . . . . . . . . . . . . . . . . . . . . . 131

D BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . 143

iv

LIST OF FIGURES

FIGURE PAGE
1 Boundary Layer Quantities for Gas Boundary Layer . . . 82
2 Schematic Drawing of AFIT lO"XIO" Wind Tunnel . . . . 83
3 AFIT lO"XlO" Wind Tunnel . . . . . . . . . . . . . . . 84
4 Test Section of AFIT lO"XIO" Wind Tunnel . . . . . . . 85
5 Schematic of Test Cylinder . . . . . . . . . . . . . . 86
6 One and One-Half Inch Diameter Test Cylinder . . . . . 87
7 Schematic Drawing of Pressure Measuring System . . . . 88
8 Pressure Measuring System . . . . . . . . . . . . . . 89
9 Calibration Curve for Statham Pressure Transducer

with 14 Volts DC Applied Voltage . . . . . . . . . . 9O

10 Pressure Correction for Change in Elevation Between
Pressure Tap on Test Cylinder and Transducer vs
Angle of Cylinder Rotation, Calibration Constant
equal 1.840 Millivolt Per Inch Water Pressure . . . 91

ll Grid Wire Layout for Droplet Velocity Ud

Measurement . . . . . . . . . . . . . . . . . . . . 92
12 Apparatus for Measurement of Liquid Film

Thickness . . . . . . . . . . . . . . . . . . . . . 93
13 Liquid Film Thickness vs Probe Voltage, ¢ = 90°,

Pw = 20 pSig o o o o o o o o o o o o o o o o o o o 0 9'4
14 Droplet Capture Tube Apparatus . . . . . . . . . . . . 95
15 (a) One-Half Inch Capture Tube

(b) Three—Eighths Inch Capture Tube . . . . . . . . . 95

I6 Schematic Drawing of Schlieren Apparatus . . . . . . . 96
I7 Boundary-Layer Separation in Single-Phase Flow

with Forward Stagnation Point at Top of
Cylinder, Reynolds Number 5.6ux10“ . . . . . . . . . 97

V

FIGURE

18

19

2O

21

22

23

24

25

26

27

28

29

3O

31

32

LIST OF FIGURES (contd.)
PAGE
Runge—Kutta Subroutine . . . . . . . . . . . . . . . . 98
Computer Flow Chart for Runge—Kutta Subroutine . . . . 99
Computer Program for Equation (2.15) . . . . . . . . . 100
Computer Program for Equations (2.25) and (2.26) . . . 101

Average Nusselt Number vs Reynolds Number for
Single—Phase Flow (Wind Tunnel Evaluation) . . . . . 102

Pressure Coefficient vs ¢, 1.5 Inch Diameter
Cylinder, Single—Phase Flow . . . . . . . . . . . . 103

Pressure Coefficient vs ¢, 1.5 Inch Diameter
Cylinder for Single-Phase Flow, Re = 5.64X101+ . . . 104

Hysteresis Effects in Pressure Measuring System,
Single-Phase Flow, Re = 5.64X10l+ . . . . . . . . . . 105

Pressure on Cylinder vs ¢ for 1.5 Inch Diameter
Cylinder Single—Phase Flow, Re = 5.64X10L+ . . . . . 106

Difference Between Actual and Observed Pressure,
Observed Pressure Measured by Water Injection
Method . . . . . . . . . . . . . . . . . . . . . . . 107

Pressure vs 0 for Gas Reynolds Number equal
5.64X10”, Nozzle Water Pressure equal 20 psig . . . 108

Pressure Coefficient vs 0 for Two-Phase Flow Over
1.5 Inch Diameter Cylinder, Nozzle Water Pressure
equal 15 psig, Red = 522, Gas Reynolds Number
equal 5.64X10” . . . . . . . . . . . . . . . . . . . 109

Pressure Coefficient vs ¢ for Two-Phase Flow Over
1.5 Inch Diameter Cylinder, Nozzle Water Pressure
equal 20 psig, Red = 458, Gas Reynolds Number
equal 5.64xlo1+ . . . . . . . . . . . . . . . . . . . 110

Pressure Coefficient vs 0 for Two—Phase Flow Over
1.5 Inch Diameter Cylinder, Nozzle Water Pressure
equal 25 psig, Red = 380, Gas Reynolds Number
equal 5.6uxlol+ . . . . . . . . . . . . . . . . . . . 111

Per Cent of Observations vs Fraction of Standard

Deviation, Gas Reynolds Number equal 5.64X10“,
Nozzle Water Pressure equal 15 psig . . . . . . . . 112

vi

FIGURE

33

34

35

36

37

38

39

4O

41

42

43

LIST OF FIGURES (contd.)

Per Cent of Observations vs Fraction of Standard
Deviation, Gas Reynolds Number equal 5.64X10”,
Nozzle Water Pressure equal 20 psig . . . . . .

Per Cent of Observations vs Fraction of Standard
Deviation, Gas Reynolds Number equal 5.64X10”,
Nozzle Water Pressure equal 25 psig . . . . . . .

Liquid Boundary-Layer Thickness vs ¢ for Gas
Reynolds Number equal 5.64X10”, Nozzle Water
Pressure equal 15 psig . . . . . . . . . . . . . .

Liquid Boundary-Layer Thickness vs ® for Gas
Reynolds Number equal 5.64X10”, Nozzle Water
Pressure equal 20 psig . . . . . . . . . . . . . .

Liquid Boundary—Layer Thickness vs i for Gas
Reynolds Number equal 5.64X10”, Nozzle Water
Pressure equal 25 psig . . . . . . . . . . . . . .

Liquid-Gas Interface Velocity vs ¢ for Gas
Reynolds Number equal 5.64XIO“, Nozzle Water
Pressure equal 15 psig . . . . . . . . . . . . .

Liquid-Gas Interface Velocity vs a for Gas
Reynolds Number equal 5.64X10”, Nozzle Water
Pressure equal 20 psig . . . . . . . . . . . . .

Liquid-Gas Interface Velocity vs ¢ for Gas
Reynolds Number equal 5.64X10”, Nozzle Water
Pressure equal 25 psig . . . . . . . . . . . . . .

Boundary-Layer Separation for Gas Reynolds
Number equal 5.64x10“ and Water Pressure
equal 20 psig at Spray Nozzle . . . . . . . . . .

Water Spray Distribution Milliliters Per Minute
Along Axis of Test Cylinder, for 1/2 Inch
Collecting Tube, Nozzle Nr. 2116, Gas Reynolds
Number equal 5.64Xl01+ and Water Pressure at
Spray Nozzle equal 20 psig . . . . . . . . . . .

Coordinate Plot of Water Spray Distribution Over
Test Segment of Cylinder in Milliliters Per Minute

for One-Half Inch Capture Tube, Gas Reynolds Number
equal 5.64X10”, Water Pressure at Spray Nozzle equal

20 psig and Average Mass Ratio of Water to Air
equal 0.045 . . . . . . . . . . . . . . . . . . .

Vii

PAGE

113

114

115

116

117

118

119

120

121

122

123

FIGURE

44

45

46

47

48

49

50

51

52

LIST OF FIGURES (contd.)

Water Vortex Separating from Liquid Film and
Indicating Direction of Vortex Rotation . .

Series of Vortices Extending into Liquid

Boundary Layer . . . . . .

Boundary-Layer Separation for Gas Reynolds Number
equal 3.55X10q and Water Pressure equal 20 psig
at Spray Nozzle . . . . . . . . . . . . . . . . .

Boundary-Layer Separation for Gas Reynolds Number
equal 1.O4><105 and Water Pressure equal 20 psig
at Spray Nozzle . . . . . . . . . . . . . . . . .

Velocity Pressure vs ¢ for Gas
equal 5.64X10”, Nozzle Water
15 psig . . . . . . . . . .

Velocity Pressure vs ¢ for Gas
equal 5.64X10“, Nozzle Water
20 pSig O O O O O O O O O 0

Velocity Pressure vs O for Gas
equal 5.64X10“, Nozzle Water

Reynolds Number
Pressure equal

0 O O C I O O O

Reynolds Number
Pressure equal

Reynolds Number
Pressure equal

25 psig . . . . . . . . . . . . . . . . . . . . .

Boundary-Layer Separation on a Vertical Cylinder in
Horizontal Two—Phase Flow, Gas Reynolds Number
equal 5. 6|+xlol+ I O O O O O O O I O O O O O I O O

Boundary-Layer Separation on a Vertical Cylinder in
Horizontal Two-Phase Flow with Dye Injected into
Liquid Boundary Layer, Gas Reynolds Number
equal 5.64X10l+ . . . . . . . . . . . . . . . . .

viii

PAGE

124

124

125

125

126

127

128

129

130

TABLE

10

11

LIST OF TABLES

Heat Transfer Data . . . . . . . . . . . .
Velocity Profiles . . . . . . . . . . . . . .
Velocity Pressure in Inches Water, with Test
Cylinder Removed, at Test Station in AFIT
10"X10" Vertical Wind Tunnel, Run No 1 .
Velocity Pressure in Inches Water, with Test
Cylinder Removed, at Test Station in AFIT
10"XIO" Vertical Wind Tunnel, Run No 2
Velocity Pressure in Inches Water, with Test
Cylinder Removed, at Test Station in AFIT
10"X10" Vertical Wind Tunnel, Run No 3 . .
Velocity Pressure in Inches Water, with Test
Cylinder Removed, at Test Station in AFIT
10"X10" Vertical Wind Tunnel, Run No 4 . .
Velocity Pressure in Inches Water, with Test
Cylinder Removed, at Test Station in AFIT
10"X10" Vertical Wind Tunnel, Run No 5 . .
Droplet Velocity Data . . . . . . . . . . . .

Droplet Size Distribution and Reynolds
Number Data . . . . . . . . . . . . . . . .

Laminar Boundary Layer Data . . . . . . . . .

Turbulent Boundary Layer Data . . . . . . . .

ix

PAGE

132

133

134

135

136

137

138

139

140

141

142

CE

3‘ D‘l

Re

Red

R69

NOMENCLATURE

area

pressure coefficient

particle diameter (liquid or solid)
cylinder diameter

6*/6 boundary layer dimensionless shape factor for
turbulent flow

average heat transfer coefficient
gas thermal conductivity

constant distance measured from the gas-liquid interface
to an arbritrary point in the external flow field

number of frames (photographic)

film speed in frames per second

number of observations

average gas Nusselt number based on cylinder diameter
pressure

water pressure at spray nozzle

static pressure

velocity pressure

heat flow per unit time

cylinder radius

gas Reynolds number based on cylinder diameter

droplet Reynolds number based on droplet diameter and gas
properties (Red = d (U0° - Ud)/V)

USO/v, Reynolds number based on momentum thickness and gas
properties

3':

NOMENCLATURE (contd.)
distance
surface temperature
gas temperature at infinity
velocity inside of gas boundary layer in x direction
particle (solid or liquid) velocity at infinity
velocity at outer edge of liquid film
velocity at edge of gas boundary layer
velocity of gas at infinity
nonsteady velocity disturbance in x direction
nonsteady velocity disturbance in y direction
nonsteady velocity disturbance in z direction

coordinate along the interface between the liquid film and
the gas boundary layer

coordinate normal to interface and measured from interface
6 2 Re

(I? 733 dimensionless gas boundary-layer thickness

difference between arithmetic mean value and observed value
of droplet population

U
l - 6—, dimensionless velocity ratio
5
gas boundary-layer thickness
liquid boundary layer thickness

gas boundary-layer displacement thickness defined by
Equation (2.4)

intensity of turbulence

Y/6, dimensionless coordinate normal to the surface (interface)

_ gas boundary-layer momentum thickness defined by Equation (2.5)

dimensionless parameter for laminar flow

dynamic viscosity of gas

xi

000

T0

NOMENCLATURE (contd.)
kinematic viscosity of gas
density of gas
density of gas at infinity
standard deviation defined by Equation (4.2)
shear stress at liquid—gas interface

x/R, dimensionless coordinate along the surface (interface)

xii

CHAPTER I

INTRODUCTION

1.1 General Discussion and Purpose

 

One of the first examples of artificially created two-phase flow
for the purpose of significantly increasing the attainable levels of
surface heat-transfer rates was accomplished in the nuclear power
field by addition of solid particles, such as graphite dust, to the
coolant flow (1). Following this Elperin 1961 (2) reported in the
Russian literature that, by the addition of a small amount of water
in droplet form to the coolant air, it was possible to increase the
surface heat-transfer rate by an order of magnitude in flow across a
bundle of tubes.

Until Elperin published his paper on two-phase flow the major
emphasis was the investigation of such flows in open and closed
conduits. One of the first major contributions in the field of two-
phase flow was by Martinelli et al. in 1941 (3), which was superseded
by a Lockhart and Martinelli article in 1949 (4). These publications
attempted to develop a procedure for the calculation of pressure loss
in pipes for each of the four possible modes of flow:

1. Liquid and gas in turbulent flow.

2. Liquid in laminar flow and gas in turbulent flow.

3. Liquid in turbulent flow and gas in laminar flow.

4. Both liquid and gas in laminar flow.

Later McManus 1956 (5) made an extensive experimental investi-
gation of the flow properties for a two-phase (air-water) flow in
circular tubes. Here pipe flow in horizontal flow and vertical
up-flow and down-flow configurations were considered. The investi-
gation considered a range of from one—hundred—per cent air to
one-hundred-per cent water for each pipe configuration.

The purpose of the present study is to investigate boundary
layers which are developed on a circular cylinder when subjected to
transverse flow of a two-phase mixture. The two-phase mixture is to
be composed of air as the primary fluid stream, which is conveying a
small amount of liquid (water) in droplet form.

The two-phase mixture will develop on the leading surface of the
cylinder surrounding the stagnation point two hydrodynamic boundary
layers and, for the case of a heated or cooled cylinder, two thermo-
dynamic boundary layers. These boundary layers are composed first of
a liquid film which wets the cylinder, and overlaying this film
(boundary layer) will be a two—phase mixture boundary layer. This
assumes that water droplets will flow in the gas boundary layer, so
that it is in fact a two-phase boundary layer. In general, there will
also be a thermal boundary layer within the liquid boundary layer and
one within the gas (two—phase) boundary layer, possibly extending
beyond the hydrodynamic boundary layer into the free stream. This
makes a possible total of four boundary layers, two hydrodynamic and
two thermal.

For the case of transverse flow over a cylinder the curved
surface will provide the necessary conditions for the develOpment of

a pressure gradient in the flow direction. This gradient for the case

of low—speed flow will first be negative and later be positive. One
would assume that it is possible for boundary—layer separation to

occur in the region of the positive pressure gradient. This indeed
takes place, and as a result of this separation the local heat-transfer
rate is affected. The region of flow in the neighborhood of the
separation point is the major subject of this investigation.

The present investigation was divided into three distinct parts:
First, an analytical investigation of the boundary layer was developed,
as presented in Chapter II. Two flow models were considered, both
utilizing numerical procedures for their solutions. Chapter V presents
the numerical procedures and the computer programs for the two models.
The results of the analytical investigations are given in Chapter VI.
The second part of the investigation was the collection of the re-
quired experimental data for the above computer solutions. Chapter III
presents a description of the apparatus, and Chapter IV presents the
experimental procedure, including a description of the technique
developed for the measurement of the pressure profile around the test
cylinder. The third segment of the investigation was the experimental
determination of the boundary-layer separation point. The system
developed for this observation is presented in Chapter IV, and the
results are given in Chapter VI.

Discussion of the results is given in Chapter VII. Included

there are also conclusions and recommendations for further research.

1.2 Some Theories of Boundary-Layer Behavior

 

 

in Two—Phase Flow

 

A brief history of the development of some of the boundary-layer
theories for two-phase flow will be reviewed. In order to keep the
history brief only those articles pertaining to gas-liquid droplet and
gas—solid particle flow will be reviewed.

Chiu 1962 (6), assuming laminar flow, made an analysis for the
case of two—phase (gas-solid particle) flow over a flat plate. Two
salient features of the analysis may reflect similar situations for
the problems at hand.

The first was the assumption of laminar two-phase flow. Here a
gas flow which initially is assumed laminar will remain laminar,
providing the solid particle Reynolds number remains in the neighbor-
hood of unity. The mechanism which would cause the flow field to
become turbulent was the wake produced by the particle. For this
analysis the solid particle Reynolds number was determined by the

expression

d(Um-Ud)

Red = ——;—— (1.1)

where d is the diameter of the particle, Um—Ud the velocity difference
between the gas and the particle at infinity and v the gas kinematic
viscosity. It should be observed that a velocity difference of ten
feet per second would limit the particle diameter to a few microns.
This places a rather severe constraint on the system, as in most
practical cases the liquid drOplet diameter would be much greater than

a few microns. In fact the particle momentum must be sufficiently

high in order that the particle may cross the gas streamlines and
impinge upon the cylinder.

The second item of importance was the assumption that the solid
particles did not contribute to the pressure of the system. Thus,
the system pressure was equated to the partial pressure of the gas.
For the case of small mass ratios of water to total mass flow,
negligible blockage of the tunnel may be assumed.

Tribus 1952 (7) made an analysis of the trajectories of water
drops around streamlined bodies. Here the investigation was con-
cerned with the icing of surfaces. Two significant conclusions were
made. First, the liquid catch rate is strongly controlled by the
droplet size. The larger drops would cross the gas stream lines and
impinge on the cylinder, while the smaller drops are deflected by
the gas streamlines and may flow around the cylinder without making
contact. The second conclusion was that only the section of the
cylinder near the stagnation point was wetted by the drops. The
included angle of the wetted surface is a function of droplet size.
This analysis would indicate that one should explore those systems
that provide large droplets and relatively low gas stream velocities,
if the objective is to increase the level of heat-transfer rate.

Tifford 1964 (8) made an analysis of two—phase flow (gas-liquid
spray) for the case of flow over a flat plate. Again no pressure
gradient existed, and flow separation could not take place. Because
of the assumptions that were made, the results only predicted that
the heat—transfer rate and wall shearing stress could be maximized.
Thus, for given free-stream single—phase flow conditions there exists

an optimum liquid spray rate. Here it was assumed that the flat plate

6
was isothermal, and that the velocity and temperature at the outer
edge of the liquid film remained constant.

Goldstein, Yang and Clark 1965 (9) made an analytical investi-
gation of the liquid film formed on a cylinder in two-phase flow.
This analysis was based on the assumption that the flow remains
laminar. The analysis also assumed that the cylinder liquid catch
rate was controlled by the particle trajectory. The trajectory
analysis followed the analysis as outlined by Tribus (7). In order
to solve the problem, the pressure on the cylinder was assumed to be
the same as that developed by single-phase flow (given by US = 2Uwsin¢).
The shear stress between the liquid film and the gas boundary layer
was assumed to be the same as that produced by single-phase flow over
a dry cylinder. Further, the analysis only gave results for that part
of the cylinder where the potential flow was still accelerating. Thus,
only a negative pressure gradient was experienced. This condition did
not yield a separation point.

One should note that all analyses performed to date have assumed
the flow to be laminar, providing a more accessible route to a solution.
However, an assumption of turbulent flow is well worth considering.
There are several basic reasons for assuming turbulent flow. First,
Kestin and Maeder (10) found that the rate of local heat transfer
in the region of the forward stagnation point may be doubled by

increasing the free—stream turbulence intensity 8, defined by

 

1
‘/ §'(u'2+v'2+w'2)
s: = (1.2)
U

00

 

where u', v' and w' are nonsteady velocity disturbances in the flow
field. This is especially true in single-phase flow if the initial
intensity is less than one per cent. Another reason for considering
turbulent flow is that separation of the boundary layer is changed in
turbulent flow conditions. As an example, consider a flow Reynolds
number of l X 10”. For laminar flow over a cylinder the separation
point is approximately 80 degrees from the stagnation point. For
turbulent flow and the same Reynolds number, the separation would

shift behind the 90 degree point.

CHAPTER II

ANALYSIS

2.1 General Description and Objective

 

Let us restate the purpose of the investigation and then formulate
two analytical models. The objective of this study is to investigate
the transverse flow of a two—phase (air-water spray) mixture over a
right circular cylinder for a range of gas Reynolds numbers of 1 X 10”
to 2 X 105 and mass ratios of water to air less than 10 per cent. For
single—phase flow this would be in the regime of subcritical flow and
the phenomenon of flow separation would occur with separation forward
of the 90-degree point. The particular area of investigation is the
region of the flow field where the boundary—layer separates from the
cylinder. A preliminary experimental investigation had established
the fact that the boundary layer does indeed separate.

For both analytical models, the flow at a great distance upstream
from the test specimen is assumed to be a homogeneous mixture of
saturated air and water droplets. Both the velocity of the air and
the velocity of the water droplets are assumed to be known, while only
average drOplet size distribution is obtainable.

Martinelli (3) classified twojphase flow in pipes into four
possible modes of flow:

1. Liquid and gas in turbulent flow.

2. Liquid in laminar flow and gas in turbulent flow.

3. Liquid in turbulent flow and gas in laminar flow.

4. Both liquid and gas in laminar flow.
He considered two—phase pipe—flow regimes where the conduit can be
divided into two distinct regions, the first region conveying liquid
and the second gas. Separating the regions in a gas-liquid interface
whose average surface is parallel to the direction of flow. He found
for the case of pipe flow that the first mode was realized only with
large liquid to gas mass ratios, and that third mode was impossible
to achieve and the fourth possible only for the flow in a capillary
tube. These conditions might be thought equally possible to exist
for the external flow problem. With this as a guide the following
models are presented: Section 2.2 presents the laminar gas boundary—
layer model for both liquid and gas in laminar flow. Section 2.3
presents the gas boundary-layer model for liquid in laminar flow and

gas in turbulent flow.

2.2 Gas Boundary-Layer Model for Both Liquid and

 

Gas in Laminar Flow

 

This mode of flow assumes that both the liquid film on the
cylinder and the gas boundary layer on top of the liquid film are in
laminar flow. This is a reasonable assumption when the droplet
Reynolds number is in the neighborhood of unity, according to the
results of Chiu (6) quoted in Section 1.2. We assume that the gas
boundary layer will not separate from the cylinder before the liquid
film separates, since preliminary experimental evidence indicates a

single separation point.

10

For the complete laminar flow system the following assumptions
are imposed on the solution:

1. The flow is taken to be two-dimensional, and laminar within
the boundary layer.

2. On top of the liquid film a gas boundary layer forms, which
joins the flow in the liquid film to the external flow field. The
usual boundary-layer assumptions are assumed to apply in this region.

3. Surface-tension effects on the surface of the liquid film
are neglected.

4. The effects of compressibility and of heat generated by
dissipation can be ignored for the case of low-speed flow.

5. For that region of flow over a right Circular cylinder, up
to and including the point in the flow field where separation takes
place, the effects of gravity are assumed negligible. This assumption
is supported by the results, reported in Appendix A, of a study by
the author of horizontal flow over a vertical cylinder. When dye was
injected into the boundary layer, the mean flow of the dye followed a
horizontal path up to the point of separation, indicating that the
gravitational force was negligible in comparison to the other forces
acting on the boundary layer.

6. All fluid properties will be taken as constant; e.g., the
liquid and gas are separately assumed incompressible.

7. No appreciable vaporization occurs.

8. Under certain conditions waves may be formed on the surface
of the liquid film, and splashing or bouncing may occur at this

surface. These effects are neglected.

11

9. The envelOpe formed by the liquid film which is developed on
the cylinder produces essentially a circular cylindrical surface.

With these assumptions, the von Karman integral equation and
associated boundary conditions are formulated for the outer gas

boundary layer from the elementary volume shown in Figure l:

l 2
d dUs _ dP
p EE-(IO (US-U) Udy) - p-E;— J0 Udy - To + A 5;. (2.1)

U)
Q)
('1'

“<1
ll
20

where US and U are velocity components, x is the coordinate along the
interface between the liquid film and the gas boundary layer while

y is the coordinate normal to the interface and measured from the
interface, 0 is the gas density, To the shear stress at the liquid-gas
interface and A is the constant distance from the liquid-gas interface
to a layer arbitrarily selected in the external flow. From the well—
known Prandtl boundary layer analysis, "there is negligible variation
in pressure through the thickness of the boundary layer for regions
not near the stagnation point." Hence the Bernoulli Equation can be

put into the form

dP dU

5;“0‘1

S

s d;—' (2.2)

Substituting Equation (2.2) into Equation (2.1) results in

éi‘lCL

2 dUs i
J (US-U) UdY) 'l‘ p TEX—J (US-U) dy = To (2.3)
O O

12
The boundary—layer thickness 6 is defined as the distance measured
from the liquid-gas interface to the point where the external velocity
prevails. This point is reached asymptotically. On the other hand, a
practical definition of 6 is that distance from the liquid-gas inter-
face to a point where the velocity differs by one per cent from the
external velocity. The integration limits in Equation (2.3) may now
be changed, since in the range 6 < y < 2 the term US-U in the integrands
of the left side of the equation is equal to zero. The equation can
further be written in a simpler form when the following definitions are

used for the boundary layer displacement thickness 6* and the momentum

 

thickness 0

a: 2] (1 — LL) dy (2.4)
US
6 U U
8 E f (l - ——-)--dy (2-5)
0 US US

First changing the integration limits of Equation (2.3) to run from
0 to 6 and then utilizing the definitions of Equations (2.4) and
(2.5) results in

dUs

pé—(uze) use:
dx 8 +0.. E—

= To (2.6)

Equation (2.6) is a form of von Karman's integral equation. The
only new feature of the present treatment appears in the boundary
conditions used in the evaluation of 6* and 6. Here, a boundary con-
dition is no longer given by U = 0 at y = 0 but rather U = U0 at y = 0.
Further it is not expected that 6* and 6 would have the same typical

values as found in single-phase flow.

13

The information desired from the above equation will, to some
extent, dictate the most desirable method of solution and hence the
form the equation takes on for this analysis. The major item of
information of current interest is an estimate of the boundary layer
separation point, and in particular the role of the film velocity U0
at y = 0 in altering the separation point. Curle 1962 (11) published
a comparison of the approximate methods of solution for the incom-
pressible laminar boundary—layer equations. Here one finds that any
of the standard methods of solution provides an estimate of the
separation point. Further, if detailed distribution of skin—friction
is sought, any method is satisfactory, with the qualification that
for flow which starts from a stagnation point rather than a sharp
leading edge, the Stratford-Curle method is only used downstream of
the pressure minimum.

The solution of Equation (2.6) requires knowledge of the velocity
distribution in the gas boundary layer in order to evaluate 6* and 6.
A velocity distribution as outlined by Pohlhausen (l2) and cited by
Curle (11) is assumed to exist. The velocity dependence on y at any

given x is assumed to be of the form

C.‘

US‘= f(n) (2.7)

Now the boundary conditions to be satisfied by the local velocity U

are

14

3U BP 320 320
U=U0(x),pU—=-—+u(‘—2+—‘2') aty=0 (2.8)
3x 3x 3x By
and
2
U-US=O,§£J_=0,3_E=0 aty=6 (2.9)
2
3y 3y

The second boundary condition of Equation (2.8) is the Navier-Stokes
equation of motion in the x direction evaluated at y equal zero (v and

w are assumed zero at this point). The second boundary condition of
Equation (2.9) is a result of vanishing shear stress at y equal 6 and
the third condition is arbitrary as in Pohlhausen (12). Since we have
five independent boundary conditions, an approximation to the velocity
profile may be assumed as a polynomial in n with five coefficients which

are functions of x that can be determined from the boundary conditions:

U
6—-= a + bn + cn2 + dn3 + en” (2.10)
s

As will be seen in the discussion of initial conditions following
Equation (2.15), the solution is obtained in two parts: a small
neighborhood of the stagnation point where the flow is assumed to be
the same as single—phase flow and the region outside this neighborhood.
For analytical convenience U0 will be assumed independent of x in

each part of the solution with a discontinuity between the two. As

may be seen in Figures 38 to 40 there is actually a considerable
variation in the experimentally observed values of U0. But it is found
that the solution of Equation (2.15) is relatively insensitive to

variations in U0.

15

The second boundary condition of Equation (2.8) and the Bernoulli

Equation (2.2) then yield for the coefficient 0 of Equation (2.10)

C = --—-——--——— (2.11)

A s —— (2.12)

After some algebra and utilizing the definition of Equation (2.12) one

may obtain the velocity profile in the form

U UO ”0 A l U0 A
US—=Ug+2(l-U +12)” 2n2+2(U_S'-l+4) n3+
U0 A 4
--—- --— 2.13
( Us 6)n ( )
U
Setting 1 — U—-= B one obtains
S
U A A A A
@21‘B+(§+23)“‘§”2+(§‘23)”3+(B'glnb' (2.14)

It may be observed that the new parameter 8 characterizes the change
in velocity across the gas boundary layer. Substituting Equation
(2.14) into Equation (2.6), one obtains, after some algebra, the

boundary-layer equation in the form

16

 

 

 

 

 

'- H I!
U U
S S S S
9072+(295.2B-1965.6)A+ 80.4-33B+4.8 12+ 1 + l3
.91 0.8Um ( (ug)2) ( (U§)2)
d3 Us 544.328-331.282+(36.6B-45.36)A-A2
b d
(2.15)
where
_ 6 2 Re
Y - (R) ‘2—
—3i
¢ ' R
R8 : QRUoo/V

Us and U; are derivatives with respect to x

It should be observed that for the case of flow over a dry cylinder
U0 vanishes and B is equal to unity. For this case Equation (2.15)
reduces to the Pohlhausen Equation (12).

The initial value of Y at ¢ = O is unknown, but may be determined
as follows from the requirement that the right-hand side of Equation
(2.15) remain finite as ¢ + 0, where Us and U0 are zero, and the
assumption that in a small neighborhood of the stagnation point the
flow is the same as in single-phase flow so that B = l in this
neighborhood. For dY/d¢ to be finite at the stagnation point the
numerator must also vanish. The value of A to force the numerator

to vanish when Us = 0 and 8 = l is obtained from the expression
9072 - 1670.41 + 47.412 + l3 = o (2.16)

For a physical solution to the problem, A must be real and positive.

The root which satisfies this condition is

17
A = 7.052 (2.17)

This then is the initial value of A, which determines the initial value

of Y as follows. Returning to the definition of A and Y, we may

write

 

_ 6 Re
Y ’ (R) 2
Us (2.18)
- dx dx

Under the assumption of potential flow outside the boundary layer Us

is given by the expression

Us = 2Umsin® (2.19)
whence

-——43 = 2 c030 (2.20)

Evaluating Equation (2.20) for 0 equal to zero and substituting the

results into Equation (2.18) yields

A = 2Y

or (2.21)

..<
II

3.526

Equations (2.17) and (2.21) are now the initial conditions for 0 equal
to zero. Two different procedures have been used to give Us. The
first procedure assumed that the external potential flow Equation
(2.19) applies for all ¢, giving the required second derivative very

accurately. The second procedure used a measured pressure profile

18
and calculated US from the Bernoulli Equation, requiring a second
derivative from the numerical data, and did not successfully predict
separation in the cases considered. For the studies of the effect
of the choice of the constant UO the first procedure was used.
If we assume, as in single—phase flow, that separation will take
place when the shear stress at the interface vanishes, we may write

from Equation (2.14)

3U p_ A
T = U __ = (—+ 28) : O (2.22)
0 3y _ (S 6
y-O
OP
U0
A = - 128 = - 12 (1 - -”') (2.23)
US

at the separation point. (A more appropriate criterion might be the
vanishing of the shear stress at the solid surface.)

The solution of Equation (2.15) to determine that value of 0 for
which A satisfies Equation (2.23) requires values of US and its first
two derivatives.

All that remains is to patch the initial conditions, the value
of U0 and the flow properties at the edge of the boundary layer
tOgether. For this purpose the single-phase flow pattern is assumed
to exist with U0 = O and 8 = l in a small but finite neighborhood
0 < 00 of the stagnation point. The neighborhood must be large enough
that B will be positive for any values of U0 used outside of the
neighborhood. Solutions were obtained for the three cases of 00 equal

to 0.1, 0.2 and 0.3 radians in order to determine whether the solution

19

was sensitive to the choice of ¢0. It was found relatively insensitive
in this range, and $0 = 0.1 radian was used for the remainder of the
investigation. Five solutions were then obtained with the constant
value of U0 in the region ¢ > @0 chosen to be 1, 2, 3, 4 or 5 per cent
of Uco to test the effect of variations in U0 on the location of the
separation point. As the results given in Section 6.4 show, the
separation point is quite insensitive to the value of Uo chosen in

the range considered. One now seeks values of ¢ which would yield

the location of the separation point for various values of U0. The

results are presented in Section 6.4.

2.3 Turbulent Gas Boundary-Layer Model for Liquid in

 

Laminar Flow and Gas in Turbulent Flow

 

The existence of a laminar gas boundary layer was based on the
assumption that the drOplet Reynolds number was of the order unity.
This assumption required both a small droplet diameter and a small
difference between the droplet and gas velocity. For the case of real
processes this may be difficult to realize and thus Reynolds numbers
outside the regime of Stokes flow (Schlichting (13), Figure 1.5) may
occur. Under these conditions appreciable disturbances may be
generated by the wake which is formed behind the droplet. From this
an increased intensity of turbulence may be precipitated in the free
stream.

The following assumptions are imposed on the solution of the
turbulent gas boundary—layer model.

1. The gas boundary layer which joins the liquid film to the

external flow is assumed turbulent.

2O

2. Surface—tension effects on the surface of the liquid film
are neglected.

3. The effects of compressibility and of heat generated by
dissipation can be ignored.

4. The effects of gravity in the gas boundary layer are
neglected.

5. All fluid prOperties will be taken as constant.

6. No appreciable vaporization occurs.

7. Under certain conditions waves may be formed on the liquid
surface. Considering the case of a turbulent gas boundary—layer model
it is assumed that such a phenomenon would only produce an increased
intensity of turbulence.

8. The envelope formed by the liquid film produces essentially
a cylindrical surface.

9. From the results of Section 2.2 given in Section 6.4 it was
found that the presence of the velocity U0, which for that analysis
of laminar flow was assumed constant, had little effect on the rate
of growth of the boundary layer and the displacement of the separation
point. It is assumed that U0 will have negligible effect on the
growth of the boundary layer and the location of the separation point
for turbulent gas boundary layer. Under the limitations of these
assumptions a turbulent gas boundary layer is investigated with the
aid of a semi-empirical method to be discussed. The objective here
is to investigate the relationship between the boundary-layer
separation and parameters of the semi-empirical method.

From Schlichting the momentum integral equation for the turbulent

model is obtained in the form

21

To d dUS
_: ._ (U826) + 6°}: US ——- (2.24)
dx dx

'0

where 6* and 0 are defined by Equations (2.4) and (2.5) respectively.
Equation (2.24) has the same form as Equation (2.6), since they apply
to the laminar and turbulent boundary layer alike, as long as no
statement is made concerning T0. As in the case of laminar flow
several methods exist for calculating the boundary-layer prOperties.
The method best suited for a particular problem depends upon the
information desired and the available experimental data. Currently
the information pertaining to the separation point is desired.

For this problem we utilize an experimental pressure profile.
We also obtain a set of initial conditions by using the single-phase
flow solution as in Section 2.2 for a neighborhood of the stagnation
point. This would represent the minimum required information for a
meaningful solution to Equation (2.24). Von Doenhoff and Tetervin (14)
developed a method for the calculation of the turbulent boundary-layer
characteristics, requiring as minimum information a set of initial
conditions and the pressure profile. We assume that the method of
calculation by von Doenhoff and Tetervin is adequate for this study.
A more detailed discussion of the methods of calculating the turbulent
boundary—layer characteristics, which utilize either the momentum inte-
‘ gral equation or the energy integral equation, can be found in the cited
reference of Schlichting (13).

Von Doenhoff and Tetervin showed from analysis of experimental data
that the boundary-layer profile was a function of a single parameter

H (the ratio of the boundary-layer displacement thickness 6* to the

22
momentum thickness 0). They developed an empirical equation in terms

of the rate of change of H

9 ——'= — ‘—‘- 2.035(H-1.286
dx q dx T

d 4.680 H-2.975 6 d 2
e ( ) [: --3- q E] (2.25)

O

1
where q is the velocity pressure (q = E'pUsz). Equation (2.24) can
be put into a more convenient form for numerical calculations. The
introduction of H and 6* into Equation (2.24) yields

dq

_ ___l___
dx + ( 2 ) q dx _ (2.26)

.B’IS‘

Equation (2.25) and Equation (2.26) represent a system of semi-empirical
equations for the turbulent boundary layer.

The shear stress is still unknown. In single-phase flow Squire and
Young (15) have proposed the empirical formula

25.1.
T

- 5 890 lo (4 075 Re ) 2 (2 27)
0 - . glo . 8 .

where Ree is the Reynolds number based on the momentum thickness. In the
absence of experimental data for two—phase flow, we tentatively assume
that the same formula applies.

The turbulent boundary-layer properties can now be obtained from
the simultaneous solution of Equations (2.25), (2.26) and (2.27). It
should be observed that the empirical equation for dH/dx, Equation
(2.25), and the momentum equation, Equation (2.26), represent a
system of first-order differential equations, which can be solved

numerically with a step—by—step calculation once the initial conditions

23

are known. The solution for the system of equations is completed
once the separation point has been reached.

At this time a criterion for the establishment of the region of
separation is required. In single-phase flow Gruschwitz (l6) estab-
lished a criterion that separation was imminent when H increased to
a value of 1.85. Von Doenhoff and Tetervin found that single-phase
flow separation would take place for the range of values of H between
1.8 and 2.6. For the first attempt at this analysis the range of
values of H established by von Doenhoff and Tetervin was assumed
adequate to establish the region of separation. In single-phase flow
this range was not a large segment of the cylinder, since the rate of
change of H with respect to x is large in the region near the separa-
tion point. Further, the separation point in turbulent flow is not
well defined. As is reported in Section 6.4 it turns out that for
the two-phase flows investigated experimentally separation occurs in
the vicinity of H = 1.23, so that the single-phase flow criterion is
not applicable.

For the case of two-phase flow systems, which are the gas-
liquid spray type, two models of the flow field have been considered.
In these analyses a gas-liquid spray system is defined as that system
where the gas is the primary fluid which is transporting a small
amount of liquid in droplet form ( mass ratio of liquid to gas less
than 10 per cent).

The first model, Section 2.2, considered the liquid film to be
laminar as well as the gas boundary layer. A model of the gas boundary
layer was developed which permitted a finite velocity U0 to prevail at

the gas-liquid boundary layer interface. Two parameters Aand 8

24

characterized the flow. The parameter A, Equation (2.12), by virtue of
being dependent on the pressure gradient assured one that separation
would not take place in a region of a negative pressure gradient.

The parameter 8 characterized the change in velocity across the gas
boundary layer. The objective of the analysis was to determine, under
the limitation that dUo/dx = 0, what effect UO had on the displacement
of the separation point from that found in single—phase flow for the
same gas Reynolds number.

The second model which represents the case of turbulent flow in
the gas boundary layer was based on a semi-empirical model obtained
from the literature. This was made possible by the assumption that
U0 did not affect the location of the separation point. Here, the
objective of the investigation was to determine, under a condition of
turbulent flow and assuming that U0 may be neglected, if calculated
values of H would predict separation.

The above two analytical models have as their common objective
the prediction of the separation point for the case of two—phase flow
over a circular cylinder. It would not be expected that both models
would predict the same results, since they represent different types
of flow structure. At the same time both models should provide a
better understanding of the flow phenomenon. The results of the two
analytical models are compared with each other and with experimental

results in Section 6.5.

CHAPTER III

EXPERIMENTAL METHODS AND APPARATUS

3.1 General Description and Objective

 

The basic apparatus for the present investigation was developed
specifically for this study. A schematic drawing of the apparatus is
shown in Figure 2, and a series of general views of the test setup are
shown in Figures 3 and 4.

The test apparatus is essentially a low-speed wind tunnel (0 to
185 feet per second), whose test section is located in a vertical
position upstream of the fan. Both the air and the water spray pass
vertically downward through the test section into the fan and on
leaving the fan are exhausted to the atmosphere. This open wind
tunnel is composed of the following components.

1. Inlet screens

2. Water spray assembly

3. Inlet diffuser

4. Test section and test specimen
5. By—pass spray assembly

6. Fan

7. Butterfly valve

8. Exhaust diffuser

9. Control panel

25

26

The objective of this experimental investigation is to perform
the necessary diagnostic studies of the flow field, as well as perform
visual, photographic and physical measurements of the boundary layers
formed on a right circular cylinder. The results will be utilized
to predict the behavior of the boundary layers in the region of the

separation point.

3.2 The Quality of the Test Apparatus

 

Before passing to a discussion of components of the test
apparatus, let us first consider the quality. For this evaluation
we return to consideration of single-phase flow. The reason for
this is to make it possible to compare performance data for this
apparatus in single-phase flow with that which is published in the
literature. Schmidt and Wenner (17) published the results of a com-
prehensive investigation for transverse flow of air over a circular
cylinder under laminar flow conditions. Their paper included a plot

of the average Nusselt number Nu versus the flow Reynolds number,

Nu : —E— (301)

where

 

(3.2)

is the average heat transfer coefficient, Q/A the heat transfer per
unit time per unit area, A the thermal conductivity, Tw the wall
temperature and T0° the stream temperature at infinity. Also published
was a graph of pressure coefficient Cp versus Q measured from the
stagnation point, for various Reynolds numbers. By definition CP has

the form

27

S (3.3)

 

C13:

where PS is static pressure at infinity, qco velocity pressure at
infinity and P local pressure.

Kestin (10) found that an increased turbulence level affected
not only the separation point by shifting it downstream but also
materially increased the overall heat transfer rate. For cases where
results do not agree with those of Schmidt and Wenner (17) one may
conclude that the intensity of turbulence is different than their
value.

The pressure in the wake behind the cylinder is an index to
tunnel blockage. Since the flow Reynolds number is directly propor—
tional to the cylinder diameter, one would want to select the largest
possible cylinder diameter as an aid to increasing the flow Reynolds
number in order to investigate as wide a range of Reynolds numbers
as possible.

The results of this tunnel evaluation experimental data and the
conclusions drawn from a comparison of this data with that found in

the cited literature are given in Section 6.2.

3.3 The Inlet Screens and Diffuser

 

The inlet screens and diffuser were salvaged from an abandoned
low-turbulence wind tunnel. The screens were made in two sections,
each composed of a 20—mesh hardware cloth covered with cheese cloth.
The square inlet diffuser has a 30X30—inch inlet and reduces to a
lOXlO-inch outlet. This, it should be noted, determined the basic

cross-sectional area of the test section.

28

3.4 The Water-Sprsy System

 

A full-cone impactor nozzle, number 2116 manufactured by the
Spray Engineering Company, was employed to supply the water spray for
this study. The nozzle was located on the centerline of the diffuser,
between inlet screens and diffuser inlet, and directed toward the
geometric center of the test specimen. Its installation permitted
rotation in a plane including the axis of the test cylinder and also

in the plane normal to the cylinder axis.

3.5 The Wind—Tunnel Test Section

 

The wind-tunnel test section was made of plexiglass and had
internal cross-sectional dimensions of 10 X 10 inches. A series of
.005-inch diameter thermocouple wires were installed in the test
section to provide temperature histories along the test section.

The test cylinder was located 20 inches downstream from test
section inlet. Distance was selected after analysis of velocity
profiles taken with a standard Pitot tube at various stations along
the test section revealed a variation of less than one per cent in
free-stream velocity at this station. The test section wall boundary

layer at this station was approximately one-half inch thick.

3.6 The Wind—Tunnel Fan and By-Pass Spray Nozzle Assembly

 

The wind tunnel was driven by a one-hundred-horse-power motor
and two-stage fan assembly. Since performance characteristics of the
fan strongly depend on the condition of air at the fan inlet, it was
found that these conditions must remain constant if one desired to
maintain the same test-section gas Reynolds number for both single

and two—phase flow. This was accomplished by the installation of a

29

secondary spray system located downstream of the test section,
providing an air-water mixture to the fan for those runs when single-

phase flow conditions were desired at the test section.

3.7 Butterfly Valve and Outlet Diffuser

 

A butterfly valve was located in the fan discharge duct. This
permitted the test section velocity to be varied from 0 to 185 feet
per second. The air—water mixture next passed through a straight
duct (diffuser) and then exhausted to the atmosphere outside the

laboratory.

3.8 Summary of Test Facility

 

In operation the test facility had both desirable and undesirable
qualities. The major objection to the system was an excessive noise
level of the apparatus. Some vibrations of the test section were
also observed. On the plus side there was a condition such that the
static pressure, in the test section and all duct work up to the fan,
was below atmospheric pressure. Under this condition all leakage

including water was inboard.

3.9 The Test Specimen and Control Panel

 

The calibration of the wind tunnel was based on the analysis of
data collected for a right circular cylinder in single-phase flow.
The average heat transfer coefficient and the pressure coefficient
for the cylinder were analyzed.

The determination of the average heat-transfer coefficient
required a test cylinder capable of being maintained at some constant

wall temperature above the free-stream temperature. A 1.5—inch

30

diameter test cylinder was fabricated; it is shown in schematic form
in Figure 5, and in a photograph in Figure 6. The cylinder is
composed of five basic sections: a copper test section, two copper
guard sections and two bakelite end sections. The test section was
provided with six heating elements on 60-degree centers, six thermo-
couples similarly spaced, and three pressure taps. Each of the two
copper guard sections were also provided with six heating elements
and equally spaced thermocouples. The two bakelite end sections
serve a dual purpose: first, to reduce the conduction of heat to the
tunnel walls; second, to provide facilities for mounting the test
cylinder in the wind tunnel.

The maintenance of a constant wall temperature for the test
cylinder was accomplished by providing each heater with a regulated
power supply to control the power input to each of the six heaters
in the test specimen and the twelve heaters in the two guard sections.
The regulated power supply consisted of a variac for each heater and
a master variac for each of the three cylinder sections. A 24—point
recording potentiometer was utilized to measure the output of the
18 thermocouples. The power input to each of the 18 heaters was
measured by a wattmeter through a switching arrangement. Pressure
profile for the cylinder in the case of single phase flow was deter-
mined by rotating the cylinder through two—degree increments and
observing pressure on an inclined manometer.

For the case of single-phase forced flow, Schlichting (13) shows
that under the assumption of incompressible flow and constant prOp-
erties (i.e., prOperties independent of temperature) the velocity

field is independent of the temperature field. This assumption was

31

utilized in the analysis by Tifford (8), and in that by Goldstein,
Yang and Clark (9). For the case where the free-stream Mach number is
equal to or less than 0.1 and the difference between the cylinder wall
temperature and the free—stream temperature is less than 50 degrees F,
the properties of the hydrodynamic boundary layer can, without serious
error, be determined in the absence of a temperature gradient. Under
this assumption a second copper cylinder, Figure 6 was fabricated.
This test cylinder was made from a section of copper tubing with
mounting sections fitted at each end of the tube. The cylinder was
provided with six pressure taps and provisions to rotate the cylinder
in order to make a complete pressure survey. Grid lines were
inscribed into the cylinder surface and filled with epoxy. These
lines were located every five degrees and utilized in the measurements

of the liquid film velocity.

CHAPTER IV

EXPERIMENTAL PROCEDURE

4.1 Determination of the Pressure Profile

 

Fundamental to all experimental boundary-layer investigation is
a detailed knowledge of the pressure gradient along the surface in
question. This not only represents the key to the study, but also
the minimum information that one must gather for a meaningful boundary-
layer investigation in the absence of heat transfer.

The pressure impressed on the two hydrodynamic boundary layers
is assumed constant through these boundary layers. This is equivalent
to the assumption of zero pressure gradient normal to the surface.
Thus, only the pressure gradient along the surface is to be determined.
Introduction of the information from this pressure profile into the
Bernoulli Equation yields the velocity at the location where the
boundary layer joins the external flow field.

For the required pressure measurements a system of pressure taps
0.025 inches in diameter and associated pressure lines were selected.
This tap size is approximately the size of a number 71 drill, and
represented two degrees of cylinder are on the 1.5-inch diameter test
cylinder. This now places a limit on both the concept of local pres-
sure measurements and the arc length between pressure measurements.

In order to develop the pressure profile the cylinder was rotated

through two or five degree increments. The pressure was measured

32

33

every two degrees of cylinder rotation in the region where the
boundary layer separation takes place, and at five-degree intervals
for rest of the profile.

For the case of single-phase flow the pressure determination
would present no challenge, and data would be collected with the use
of standard manometers or electrical pressure transducers, depending
on the particular application.

For the case of two-phase flow a search of the literature
(References l8, l9 and 20) revealed two basic approaches to the
measurement of pressure drop in pipe flow.

The first system was one where the transducer lines were filled
with the same liquid as that flowing in the duct. This was successful
for those cases where the flowing film thickness in the pipe was
approximately an order of magnitude larger than the pressure tap
diameter. This system, as one would anticipate, would not function
adequately for the external flow problem, since the film thickness of
the boundary layer was of the same order of magnitude or less than
the pressure tap diameter.

The second method was found to be more promising; it again
utilized liquid-filled transducer lines, but employed a technique
whereby the lines were externally pressurized. This caused a small
amount of secondary fluid to be injected through the pressure tap
into the pipe. Those cases in which the injected liquid was a small
fraction of the total liquid flow proved acceptable. This second
system was investigated and found to be a practical method of pressure

measurement along the surface of the cylinder.

34

A schematic diagram and a picture of the pressure-measuring
system is shown in Figures 7 and 8. The system is composed of two.
essential parts. First a standard differential pressure transducer
with a range of i 0.7 psi was employed; this transducer was a strain-
gage type requiring a l4-volt source. The transducer output was
measured with a portable potentiometer. The second essential item
of the system was a 0.030 1 0.0001 inch internal diameter capillary
tube 34 inches long, used to meter the secondary fluid being injected
into the boundary layer. In order to make the system function two
reservoirs were employed. One supplied the distilled water to the
capillary tube, while the other was used to flush the system with
distilled water to remove any air bubbles. A wetting agent was added
to the distilled water to yield a mixture of approximately one part
in ten thousand.

In order to test the system the pressure profile for single—phase
flow was obtained. If one is successful in obtaining an acceptable
pressure profile by injecting water (whose density is three orders of
magnitude greater than air) into an air boundary layer, one would
expect favorable results for the case of two—phase flow. This should
be a conservative approximation to the problem, since less error
would be anticipated from the injection of water into a water boundary
layer. The results of pressure measurements taken in single-phase
flow and compared to data collected by conventional methods are given
in Section 6.3.

The development of the system required the solution of the

following problems.

35

l. The control of the rate of water injection.

2. Determination of the optimum rate of water injection.

3. Elimination of hysteresis effects.

A capillary tube was selected to meter the injected water for the
injection rate in the range of one to one and a half milliliters of
water per minute. This gave an injection velocity less than 0.1 feet
per second normal to the surface and a pressure drop through the
pressure tap of 0.01 inches of water. A suitable injection rate was
determined experimentally by observing pressure at the stagnation
point, and at a location ninety degrees from this point, where the
gravity vector was first normal to the surface and then parallel to
the surface. For the case of single—phase flow, insufficient water
yielded pressure measurements less than the conventionally observed
values, while large rates of water injection yielded excess pressure
measurements. It is believed that insufficient water injection
permitted the pressure tap to be exposed to the primary fluid flow,
so that capillary action at the pressure tap produced a reduction in
pressure at the transducer. In the case of excess fluid injection, a
disturbance to the external flow field was experienced to an extent
that the gas velocity was reduced, with a corresponding pressure
rise.

In order to maintain a constant water injection rate the differ-
ential pressure across the capillary tube must remain constant. In
order to accomplish this a preliminary survey was made, providing an
approximate pressure profile. The reservoir for the capillary tube
was then raised or lowered in order to maintain a constant differential

pressure across the capillary tube.

36

It was observed that the capillary tube had a memory of past
pressure excursions impressed upon it. This was erased by installing
a shutoff valve between the pressure tap and the capillary tube. After
each measurement the valve was closed to bring the fluid to rest in
the entire system. Next, a small amount of fluid from the secondary
reservoir was injected into the boundary layer in order to replace
the liquid at the pressure tap, since it was observed that during the
shutoff period the fluid was eroded. Figure 9 is the calibration
curve for the pressure transducer.

One is now ready to measure the pressure along the surface of the
cylinder. Since the cylinder is in a horizontal position, a correction
must be made for the relative height between the pressure transducer
and the pressure tap when the cylinder is rotated. This correction is
a simple function of the cosine of the angle of rotation from the
stagnation point, the radius of the cylinder, and the calibration of
the pressure transducer. Figure 10 is the graph employed for this

required correction.

4.2 Determination of the Water Droplet Velocity and

 

 

Droplet—Size Distribution

 

For a first-order approximation to the problem at hand one could
assume that the difference between the gas velocity and drOplet
velocity approaches zero at a great distance upstream from the test
section. For this case the droplet Reynolds number would be of the
order of unity, which was a basic assumption for the laminar flow
model. The difference between gas velocity and liquid velocity and

also the droplet diameter are required for the droplet Reynolds

37

number. These values were determined experimentally by a high—speed
photographic system capable of taking pictures up to 7000 frames per
second and of so tagging the pictures that the local film speed may
be determined.

The droplet velocity used a grid of wires parallel to the cylinder
axis, installed directly upstream from the stagnation point. The wire
size and grid pattern are indicated in Figure 11.

The droplet velocity was determined by counting the number of
frames (consecutive pictures) required for a droplet to advance from
one grid wire to the next. The droplet velocity Ud in feet per second

was then determined from the expression:

Ud =%(1-S§) (9.1)

where n is the number of frames per second and m is the number of
elapsed frames for a droplet to travel a distance 8 inches. For ease
of calculations the grid—wire Spacing was one inch, and the picture
frames were tagged every one-hundreth of a second. In order to obtain
a meaningful average velocity a large number of droplets must be
observed and samples taken from various sections of the roll of film.
The determination of the droplet-size distribution was also made
with the aid of high-speed photography, using the same photographs as
for the velocity determination. The drops observed on a single frame,
which were located between adjacent grid wires, were classified into
four basic sizes. The four sizes selected were 1/4, 3/16, 1/8 and
1/16 inches when projected on a screen. The distance from the pro-

jector to the screen was adjusted to yield a magnification factor of

38

four, so that the actual droplet sizes were one fourth the size
observed on the screen.

The number of drops for a given size classification and the total
number observed per frame varied considerably from one sample to the
next. Thus in order to obtain a representative size distribution a
large number of observations must be made and the results analyzed
statistically. A large number of observations may now be defined as
that number N that would yield 68 per cent of the observations
possessing residuals a within the limits of plus or minus one standard
deviation 0, where a is the difference between the arithmetic mean

value and the observed value, and o is determined from the expression:

(4.2)

o
n
l-‘MZ
Q
2|“,

4.3 Measurement of the Liquid-Film Thickness

 

The measurement of the liquid—film thickness used a system similar
to that employed by McManus (5), who measured the liquid-film thickness
for the case of two-phase flow in a duct. The system, Figure 12, is
essentially a microscope traversing mechanism employing an electrical
circuit. An open circuit is experienced until the probe comes into
contact with the liquid film, and a change in resistance for the
circuit is indicated later when the probe makes contact with the
cylinder.

A suitable value for the potential applied between the probe and
the cylinder was determined experimentally by measuring the film
thickness at a location 90 degrees from the stagnation point, where

the gravity vector was normal to the probe. A range of voltages were

39

investigated, and it was found that the indicated film thickness was
the same for the voltage range of 5 to 7 volts. When the voltage was
in excess of 7 volts the indicated film thickness was increased,
possibly because the film was distorted by the higher voltage, or a
droplet could have been captured and retained between the probe and
cylinder. The failure of the lower voltage to indicate the same film
thickness, as say 6 volts indicates, could be caused by lack of
sensitivity of the oscilloscope employed for the project. Figure 13
is a plot of film thickness versus applied probe voltage. A value of
six volts was chosen for the film-thickness investigation.

The film thickness was measured over a range of angles (measured
from the stagnation point) from 60 to 90 degrees. Angles less than
60 degrees did not yield acceptable results because droplets impinging
on the probe caused water to run down the probe and make contact with
the cylinder.

In order to reduce the error produced by droplets striking the
probe and causing a film to run on it, a paraffin coating was painted
on the probe. A film could not form over the paraffin surface, and
the droplets which came in contact with the probe would form very
small particles and be swept away by the air. This served satisfac-

torily for angles equal to or greater than 60 degrees.

4.4 The Determination of the Ratio of the Mass of Water

 

Flow to Mass of Air Flow

 

The determination of the mass of air flow per unit of time was
accomplished by the utilization of the continuity equation (pAU =

constant). The density p was determined from the equation of state,

 

 

. . \ 1s .= .r. in r. u... “A ‘L 3,.

.. v. o . cc .3 .e r. r. .3 ..~ .1 1» F. a . L. a» :C we. L9
.5 an ,2 a. .: ... 2.. 3 1. .... 2. a. .1. J.“ L.. 2. .1 “a

. .. .1 .. Q. a “a F. . . . . .L .3 L n a: e .w a.» F. C .L

 

 

40

which required both the static pressure and temperature at the test
section. The velocity was determined with a standard Pitot tube.

The discussion of the determination of the mass of water flow
per unit of time and per unit cross-sectional area will be divided
into two parts: first, the actual measuring technique; and second,
the development of a so-called uniform spray pattern over the test
section.

The mass of water flowing per unit time was determined by placing
a capture tube, Figures 14 and 15, in the flow field. Arrangements
were provided to receive the collected water in a graduated cylinder.
For the flow profile a one—half—inch capture tube was used, with the
leading edge of the tube tapered on the inside to present a knife edge
to the flow and minimize the deflection of the streamlines around the
tube. The graduated cylinder was vented to the tunnel; in this way
air passed through the capture tube into the graduated cylinder and
back into the tunnel.

Two basic capture—tube assemblies were evaluated. First, a
straight tube 0.503 inch OD and 0.485 inch ID at the knife-edge
entrance was employed. Because one may question such a large tube
when considering the amount of liquid which may be deflected around
the tube opening, a 0.366 inch OD and 0.350 inch ID capture tube,
Figure 15, was also evaluated. After it was determined that the
capture rate per unit area and per unit of time for the two capture
tubes showed a difference of less than one per cent, the larger tube

was selected for the experimental investigation.

41

The test section at the cylinder location was traversed, and
sufficient data collected to determine the average water-flow rate
and the repeatability of the observation. The results are given in
Section 6.4.

The major problem encountered in this phase of the program was
the development of a uniform spray field. For the development of a
uniform spray field, two basic spray nozzles were considered. First,
a series of nozzles similar to the one designated in Section 3.4, and
a series of internal mixing nozzles were evaluated. These nozzles
may be characterized by the production of large drops for the solid
cone nozzle and much smaller droplets for the internal mixing nozzle.
It has already been stated that the larger drops have a greater
probability of impinging on the cylinder and thus enhancing the heat-
transfer rate. There are other phenomena to be considered in the
selection of the nozzle: Does the particle bounce from the cylinder
before it has arrived at the cylinder temperature, or does the
particle splash other fluid from the cylinder?

The final nozzle selection was a compromise between several
variables. Finally it should be noted that most nozzle specifications

are based on a spray system in a quiescent atmosphere.

4.5 Measurement of the Liquid-Film Velocity

 

 

The determination of the velocity at the outer edge of the liquid
film was accomplished by injecting dye (safranin bluish) into the
liquid film through a pressure tap. The test cylinder had lines
inscribed every five degrees on the surface starting at one of the

pressure taps and extending for 110 degrees. The dye when injected

02

into the film would disperse throughout the fluid. That part which
arrives at the film surface is swept around the cylinder at the
greatest velocity. It is the time history of this interface that is
of interest. Here high-speed color photography was utilized to
observe the progress of the interface. It was determined experi-
mentally that a camera speed of four thousand frames per second was
required for an adequate definition of the flow field. It was
assumed for this experiment that the maximum velocity occurs at the

outer edge of the liquid film.

4.6 The Determination of the Boundary-Layer

 

Sgparation Point

 

For the case of single-phase flow a schlieren apparatus,

Figure 16, was utilized to observe and photograph the boundary-layer
separation point. A photograph of the separation point is given in
Figure 17 for a flow Reynolds number of 5.64 X 10”.

For the case of two—phase flow the disturbance produced by the
water droplets made it impossible to observe the boundary—layer
separation point with the aid of the schlieren apparatus. A simple
and direct method was developed. A mixture of distilled water and
wetting agent was injected into the liquid film (boundary layer) through
a pressure tap. For example, in Figure 44 the mixture is injected at
the upper tap shown in the figure, four and a half inches from the
right wall and approximately 30 degrees from stagnation. The sepa-
ration point could then be observed visually. It was noted that
small bubbles were produced by the turbulence in the region of

separation. These bubbles were observed to move both upstream and

43

downstream in the separated liquid film downstream of the separation
point. The point where the upstream movement of the bubbles reversed
and returned in a streamwise direction was considered to be the
separation point. By this technique the separation point may be
located with an accuracy estimated to be i 2 degrees. The separation
point was also visible due to reflection of light from the cylinder,
under proper lighting conditions, since the air space under the
separated liquid film altered the characteristics of the reflected

light.

CHAPTER V

NUMERICAL PROCEDURES

5.1 The Scope of the Numerical Procedures

 

The solution of Equation (2.15) and the solution of the system
of Equations (2.24) and (2.25) were obtained by the Runge—Kutta method
with the aid of a digital computer. The solutions also required
functional relationships which were derived from experimental data.
The functions were obtained by fitting a polynomial to the experi-

mental data by the technique of least squares.

 

.Ssg Polynomial Fitted to Experimental Data

An IBM Library Program No. 7.0.002 was used to obtain a polynomial
fitted to the experimental data. This program determines by the least
squares technique the coefficients of a polynomial up to and including
a fifteenth order polynomial. The program, in Fortran language for an

IBM 1620 computer, accommodates up to 100 data points.

5.3 The Runge-Kutta Numerical Procedure

 

The Runge-Kutta numerical procedure is a numerical method of
obtaining a solution to a system of first order differential equations
when certain initial conditions are obtainable. Equation (2.15) and
the system of Equations (2.24) and (2.25) meet these requirements;
i.e., they are first order differential equations of the initial value

type. Before a solution can be obtained, the differential equations

44

45

must first be put into standard form

as

. : fj(X9YI:YZ9 °'°s YN) 3 j : 132s '°°s N (501)
3

where N is the number of first order differential equations. For the
case of N = l and the value yi at the left end of the ith interval is

known, the value yi+l at the right end is calculated by the following

set of formulas.

kl = f(xi,yi)

1 1
h2 = f(Xi + E'h, yl + E'hhl)
h - f( - l h - l hh ) (5 2)
3 - X1 + E- , yl + E- 2 .
Rn = f(Xi + h, Yi + hhg)

y=yi+%(h1+2h2+2h3+hu)

where h is the width of the ith integration interval.
If there are N first order equations

d .
3%.. = fj(X.Y1.Y2. '°°a YN) ; J : 1:23 '°°s N
3

there will be N solution values, one for each of the N equations. Let

the solution function of the jth

equation (j = 1,2, °°°, N) at the
left end of the ith integration interval be yj. Then the above set

of formulas become

46

 

 

 

 

kjl = fj(x,y1,y2, "°, yN) (a)
kj2=fj[x+%—, y1+E§Ll—, y2+h:21, ~--,yN+§—§—Ifl) (b)
hja = fj(x + %3 Y1 + hilz, y2 + h:22, ..., yN + hZNZ) (c) (5.3)
hj4 - fj(x + h, y1 + hh13, y2 + hh23, ---, yN + th3) (d)

* _ h
Yj - yj + g<kj1+ 2hj2 + 2hj3 + hj4) (e)

V.
\

Here 1 5 j S N and yj’ is the new value of the y, i.e., yj,i+l° The
set of Equations (5.3) can be evaluated in sequence in such a way that
only one value of A need be used for each of the equations, say kj,

since hjl computed in (a) is used only in (b), h computed in (b) is

32
used only in (c), and so on. This assumes that the term (hjl + 2hj2 +

2k-

]3 + qu) in (e) is computed by accumulating the partial sum as each

new hji is determined. The entire system of difference equations can

then be solved by a multi-pass procedure described below.

Pass 1:
1. Compute the derivative F(J) (J = 1,2, °-°, N) for each of
the N equations, using the current X and Y(J) values. These are the

values of the hjl = fj(x,y1,y2,y3, °°-, yN) of Equation (5.3a).

F(J) = fj(X,Y(1),Y(2), ---, Y(N)) ; J = j = l,2,°°°, N

47

Pass 2:
2. Save all the current Y(J) values (i.e., the initial values
of the solution yji at the left edge of the integration step) in

another array called OLDY.
OLDY = Y(J) ; J = 1,2, -°°, N

3. Begin accumulation of the hji terms in Equation (5.3e) in an

array called PHI

PHI(J) = F(J) ; J = 1,2, ---, N

 

4. Compute the next y value, (yj + 231), needed for Equation
(5.3b).
Y(J) = OLDY(J) + .5*F(J) ; J = 1,2, °°°, N
where H = h

5. Compute the next x value needed for Equation (5.3b), x + g-

X = X + .5*H

6. Compute the derivative values F(J) (J = 1,2, °°°, N) for
each of the N equations, using the current X and Y(J) values. These

F(J) are the hjz of Equation (5.3b).
F(J) = fj(X,Y(1),Y(2), °'°, Y(N)) ; J = 1,2, °°°, N

Pass 3:

7. Add the contribution of ka = F(J) to PHI(J)

PHI(J) = PHI(J) + 2.*F(J) ; J = 1,2, °°‘, N

48
2

 

8. Compute the next y value (yj + ) needed for Equation

(5.3e)
Y(J) = OLDY(J) + .5*H*F(J) ; J = 1,2, °°°, N
Note that x already has the proper value from step 5.

9. Compute the derivative values F(J) (J = 1,2, °°', N) for
each of the N equations, using the current X and Y(J) values. These

F(J) values are the hj3 of Equation (5.3e).
F(J) = fj(X,Y(1),Y(2), °°°, Y(N)) ; J = 1,2, °°°, N

Pass 4:

10. Add the contribution of hj3 = F(J) to PHI(J)
PHI(J) = PHI(J) + 2.*F(J) ; J = 1,2, ---, N

11. Compute the next y value, (yj + hhj3) needed for Equation

(5.30)
Y(J) = OLDY(J) + H*F(J) ; J = 1,2, ---, N

12. Compute the next x value, (x + h), needed for Equation (5.3d).
Since the original x was incremented by h/2 in step 5 only h/2 need be

added to the current x.

13. Compute the derivative values F(J) (J = 1,2, °°', N) for
each of the N equations, using the current X and Y(J) values. These

F(J) values are the hi4 of Equation (5.3d).

49

F(J) = fj(X,Y(l).Y(2). °". Y(N)); J = 1.2. "°. N

Pass 5:

14. Add the contribution of hj4 F(J) to PHI(J)

PHI(J) = PHI(J) + F(J); J

H
l"
u
M
o
o
o
o
o
z

15. Compute yj*, the new value of yj, for all N equations, and

put results into the solution array y.
Y(J) = OLDY(J) + PHI(J)*H/6; J = 1,2, °-°, N

At this point all computed solution functions for N equations, at the
right end of the integration step, are in location Y(l), Y(2), -°~,
Y(N). Also, X has been incremented by H from its value before step 1.
The procedure can be repeated if integration is required across
another interval of width H (H can be changed if desired). Note that

steps 1, 6, 9 and 13 are identical and of the form
F(J) = fj(x,Y(1). Y(2), ---, Y(N)); J = 1.2. -°-. N

These steps are the only ones in which the N different equations are
specifically mentioned. Hence it is possible to write a very general
integration subroutine for an arbitrary set of N first order differ-
ential equations which implement all steps in the sequence except 1,
6, 9 and 13. A possible communication sequence between a calling
program, which contains the definition statements for any specific
set of N equations (the fj'é) and subroutine RUNGE and accompanying
flow chart is given in Figure 20 or Figure 21. Steps 1, 6, 9,

13 and any other essential steps such as the setting of initial

50

values, the punching of results, etc., are incorporated into the

calling program.

_5;4 Numerical Procedures for the Laminar Gas
Boundary-Layer Model
The computer program for the laminar gas boundary-layer model is
listed in Figures 18 and 20, where Figure 18 is the Runge-Kutta sub-
routine. Listed below is the correspondence between nomenclature in

the computer program and nomenclature used in text.

 

 

Computer Text
Nomenclature Nomenclature
N 1 (number of equations)
DB U0
XL A
X x/R = 0
Y (1) Y
Q Us/Uco
QPR Us/Um
Q2PR Ug/U.o
FUDU usug/(ug)2
A (5) coefficients of polynomial
for Us/U0°
CC the upper limit on X for which
A (5) are valid
ICNT the number of computations for
each output
H h

 

 

 

 

The results of the computer program are reported in Section 6.4.

51

5.5 Numerical Procedure for the Turbulent Gas

Boundary-Layer Model

 

The computer program for the turbulent gas boundary-layer model,

Equations (2.25), (2.26) and (2.27), is listed in Figures 18 and 21,

where Figure 18 is the Runge-Kutta subroutine.

Listed below is the

correspondence between nomenclature used in computer program and

nomenclature used in text.

 

 

 

 

 

Computer Text
Nomenclature Nomenclature
N 2 (number of equations)
T 2q/To
RE Ree
Y (1) 0/R
Y (2) H
X x/R = 0
Q q
QPR dq/dx
D. —— :_._ 2i:
m a
ICNT the number of computations for each output
A (5) coefficients of polynomial for q
C the upper limit on 0 for which the
coefficients A (5) are valid
H h
CONS=l./LOG(10) conversion of log from the base 10
to the base e

 

The results of the computer program are reported in Section 6.4.

 

CHAPTER VI

RESULTS

6.1 Preliminary Remarks

 

 

The objective of this experimental investigation is to perform
the necessary diagnostic studies of the flow field and perform visual,
photographic, and physical measurements of the boundary layer formed
on a right circular cylinder. The results will be utilized to predict
the behavior of boundary layers in the region of the separation point.

The investigation was divided into three sets of experiments.

The first set of experiments provided the data for an evaluation of
the wind tunnel under the conditions of single-phase flow. The second
set of experiments was conducted with the gas Reynolds number held in
the neighborhood of 5.6 X 10”. The objectives of this second set of
experiments were:

1. To develop a pressure-measuring technique for the case of
two-phase flow.

2. To locate the region of boundary-layer separation and
investigate the flow properties in this region.

3. To collect experimental data required for the solutions

of the analytical models.

The third set of experiments was conducted with the mass flow of water
from the spray nozzle held constant, while the gas Reynolds number was

varied from 3.55 X10” to 1.04 X 105. The objective of this third set

52

53

of experiments was to determine the effect of variations in gas
Reynolds number on the region of flow separation.

The results of these investigations are reported in the following
three sections: Section 6.2 contains the results of the wind-tunnel
evaluation in single-phase flow. Section 6.3 contains the results of
the experimental investigation. The results of the analytical

investigations are reported in Section 6.4.

6.2 Wind-Tunnel Evaluation in Single-Phase Flow

 

An evaluation of the system under the conditions of single—phase
flow was made prior to the two-phase flow investigation. The
evaluation included:

1. Determining the average Nusselt number over a range of
Reynolds numbers for the case of transverse flow of air over a heated
cylinder.

2. Measuring profiles for a range of velocities in the test
section.

3. Determining a pressure profile for a test cylinder.

The results of the heat-transfer studies are tabulated in Table l
and plotted in Figure 22, where the curve is from Hilpert (21), based
on a large number of experiments. The points marked with a plus sign
are from the present investigation. The range of Reynolds number
based on cylinder diameter was from 3.24 X 10'+ to 1.1 X 105. Runs
one through five yielded an average Nusselt number (Equation (3.1))
less than the average experimental values reported by Hilpert (21)
for corresponding Reynolds number. Runs six and seven yielded values

in excess of Hilpert's measurements. The measurements of Schmidt and

54

Wenner (17) for a 50mm diameter cylinder (1.965 inch diameter) are
also included in Figure 22 by solid circles. The data spreads for
Schmidt and Wenner and for the present investigation are similar, and
in the same range of values reported by Hilpert. Kestin (10) esti—
mated that the intensity of turbulence e for Hilpert's measurements

was in the range of 0.85 to 0.90 per cent where e (in per cent) is

 

given by e = 100 V1/3(u'2+v'2+w'2/Um, u', v' and w' are nonsteady
velocity disturbances in the flow field. Since the heat-transfer data
of the present investigation is in the range of Hilpert's, it is con-
cluded that the intensity of turbulence for the system utilized for
this investigation is also in the same range as Hilpert's, and there-
fore less than one per cent for single—phase flow.

The test cylinder was located at a point in the test section
where the variation of the free-stream velocity from the average free—
stream velocity was found to be less than one per cent. Table 2
contains the tabulated results of velocity measurements which met
this specification. These measurements were made in a plane normal
to the free-stream velocity and twenty inches downstream from the
test section inlet in the manner described in Section 3.5. Table 2
gives the maximum and minimum values of the free-stream velocity for
a series of tests in which the maximum ranged from 42 to 176 feet
per second. In run number four the maximum per cent variation in
free-stream velocity occurred. Here, it varied from 137.5 to 139
feet per second or i 0.6 per cent from the average. Tables 3 through 7
are coordinate plots of the velocity pressure measurements summarized
in Table 2. It was observed that all velocity measurements taken at

least one inch from the tunnel walls were in the free-stream velocity

55

field. Measurements taken one—half inch from the tunnel walls were
observed to be (in most cases) inside the wall boundary layer. The
maximum reduction of velocity at the one—half inch station occurred
in run number four. A reduction of six per cent was observed.

The pressure profile for a Reynolds number of 1.15 X 105 is
plotted in Figure 23. For comparison, pressure profiles by Giedt (22)
for Reynolds numbers of 1.01 X 105 and 1.4 X 105 are plotted on the
same figure. It was observed that the separation took place at the
anticipated location, while the pressure on the downstream side of
the cylinder was less than the value obtained by Giedt.

The principal cause for the pressure reduction was believed to be
tunnel blockage. From Kestin the free—stream velocity is subject to
two principal corrections, namely those due to "solid blocking" and
to "wake blocking". The solid-blocking correction for a cylinder as

given by Kestin is

— 1.. _1§.2
' 12 (100)
= 0.0185

where A denotes the area of the test section and s is the frontal
area of the cylinder. The wake-blocking correction as given by

Kestin is

_ l s

Bwb ' ‘ t'CD g”
_ l 15
' 4 (1’18) (100)

II
I
O
O
4:
co
4:

56

where CD denotes the drag coefficient for a cylinder (CD = 1.18 for
cylinder with a range of Reynolds numbers of l X 10'+ to 2 X 105,

Schlichting, page 16) Consequently

Uco = u' (l + 83b + Bwb)

u' (0.974)

0.974 u'

where u' is the measured free-stream velocity and U00 denotes the
corrected free-stream velocity. Adopting no correction introduces an
error of approximately 2.5 per cent in Reynolds number. Applying this
correction to the heat—transfer data of Figure 22 and comparing the
new results to Hilpert's curve (21) would yield the following results:
Runs one through five would show a slight improvement. Six and seven
would yield an increased divergence.

In single-phase flow, the average Nusselt number for the range of
Reynolds numbers considered was found to be in the same range of values
as those published in the cited references (References 10, 17, 21 and
22). The boundary—layer separation point agreed with that reported
in Reference 22. From this it may be concluded that two sets of
meaningful comparisons may be formulated: First, the result of the
two-phase flow studies may be compared to those for single-phase flow.
Second, the results of this investigation may be compared to the
findings of other investigators whose test facilities perform similarly

in single-phase flow.

57

6.3 Experimental Results in Two—Phase Flow

 

 

In order that we may arrive at some definite conclusions as to
the boundary—layer behavior in two-phase flow, the separation point
and pressure profile for single-phase flow are required for the same
gas Reynolds number.

For the case of single-phase flow the phenomenon of boundary-
layer separation was photographed with the aid of a schlieren
apparatus. Figure 17 is a photograph of the separation region for a
Reynolds number of 5.64 x 10”. The photograph is an axial view of
the test cylinder with the forward stagnation point located at the
top of the cylinder. Flow was observed to separate from the cylinder
at an angle of 78 i 2 degrees from the forward stagnation point. In
Figure 17 separation is observed as a light line tangent to the cylinder
at the 78-degree point and extending downstream. The quality of the
photograph is not typical of that obtained when optical glass windows
are utilized on a wind tunnel as part of the schlieren optical system.

Figure 24 is a plot of the pressure coefficient for a gas Reynolds
number of 5.64 x 10“. The separation point was indicated by the
inflection in the pressure coefficient curve. This occurred in a
region approximately 78 i 2 degrees from the forward stagnation point,
which agreed well with the results in the schlieren photograph,

Figure 17.

The results of calibration of the pressure transducer and evalua-
tion of pressure measuring technique utilizing water injection are
given in Figures 9, 25, 26, and 27. Figure 9 is a plot of the milli-
volt output of the presshre transducer versus pressure impressed on

the transducer in inches of water. For an applied voltage of 14 volts

58

the calibration constant was 1.840 millivolts per inch of water
pressure. The linearity of the transducer through the pressure range
of interest was good. Over a range of ten inches of water pressure
the maximum deviation was i 0.005 millivolts per inch of water. For
a pressure measurement utilizing a standard inclined manometer with
a least scale graduation of 0.02 inches of water pressure (estimating
the measurement to the hundreth inch of water) would require, for
equal accuracy, that the output of the transducer be read to the
nearest 0.018 millivolt. The results of the hysteresis studies of
the pressure measuring system are given in Figure 25. The pressure
profiles were obtained for the case of single—phase flow with a gas
Reynolds number of 5.64 X 10”. A pressure profile determined with an
inclined manometer and gas-filled pressure lines was assumed to be
the actual pressure profile for comparison with the measured profile.
The measured pressure profile utilized a pressure transducer, liquid
filled pressure lines and a water injection rate of 1.4 milliliters
per minute into the gas boundary layer. The measured profile took
the 90-degree point as the initial point, and measurements were made
every two degrees over the range of 90 to 0 degrees and 90 to 110
degrees. Operating the system without taking the necessary precautions
to reduce hysteresis effects resulted in a maximum error of 13.5
per cent. This occurred at a point 50 degrees from the forward
stagnation point. The same error was observed at the forward stagna-
tion point.

The pressure profile for a case where the water injection into
the gas boundary layer was turned off between each pressure measure—

ment to remove the hysteresis effect is given in Figure 26. Variations

59

between the measured pressure and actual pressure are given in
Figure 27. The data was obtained for a single—phase gas Reynolds
number of 5.64 X 101+ and a water injection rate of 1.4 milliliters
per minute. Pressure measurements were made every five degrees
starting at the forward stagnation point and extending 110 degrees
around the test cylinder. Similar measurements were made at the
110-degree point and returning to the stagnation point. A maximum
error of 5.2 per cent was observed at 75 degrees. This occurred in
the region of the pressure minimum which is also the region of maximum
free—stream velocity. A 1.4 per cent error was observed at the
forward stagnation point.

The performance characteristics of the pressure—measuring
technique for the case of two-phase flow are exhibited in Figure 28.
Here, the difference between local pressure on the cylinder and the
atmospheric pressure is plotted against the angle (distance) measured
from the forward stagnation point. The system was operated at gas
Reynolds number of 5.64 X 10”. Water-nozzle pressure was maintained
at 20 psig and a water—injection rate of 1.4 milliliters per minute
was employed.

The curve of Figure 28 represents two sets of data. The first
set of data was collected by starting at the forward stagnation point
and making pressure measurements every four degrees until the pressure
at the l40—degree point had been measured. The rotation of the
cylinder was reversed and each pressure measurement repeated. The
second set of data was collected by observing the pressure at the
forward stagnation point and then indexing to 2 degrees. From this

location on around the cylinder pressure measurements were made every

60
four degrees until the pressure had been measured at the 138-degree
point. The direction of cylinder rotation was again reversed and all
measurements repeated. The general scatter of data was small, with
the largest variations in pressure occurring at 26, 46 and 86 degrees.
The average difference for these three points was 0.14 inch of water
pressure. After reviewing the method of reading angle of cylinder
rotation and analyzing the data, it was concluded that at these three
stations an incorrect angle was observed for cylinder rotation.
Excluding these three data points, an average pressure variation (at
a repeated data point) of 0.02 inches of water was observed. This
represents the estimated accuracy of pressure measurements. The
pressure at the forward stagnation point was —0.714 inches of water.
For the case of single-phase flow a pressure of «0.72 inches of water
was observed when employing the water injection technique. The actual
pressure at the forward stagnation point for single-phase flow was
-0.73 inches of water. This represents a 2 per cent increase in pres-
sure at the forward stagnation point.

For the case of two—phase flow the pressure coefficients for a
constant gas Reynolds number of 5.64 X 10“ are plotted in Figures 29,
30 and 31. Nozzle water pressures for the above curves were 15, 20
and 25 psig respectively with the 20 psig curve taken as the base
value. In each case the pressure at the forward stagnation point
was higher than the pressure observed at the corresponding point for
single—phase flow with the same gas Reynolds number. The magnitude
of pressure coefficient at the point of minimum value (which is the
region of maximum free—stream velocity) was in all cases less than

the corresponding value in single-phase flow. The location of the

61

region where the pressure coefficient passes through its minimum

value was observed, in all cases, to have shifted downstream approxi-
mately 6 degrees. Also, the slope of the pressure coefficient curve
downstream from the point of its minimum value is less than the
corresponding slope for the case of single-phase flow. As a result

of these two effects the separation point was observed to have shifted
downstream approximately 14 degrees. The distance along the cylinder
between the separation point and the region where the wake pressure
approaches constant value has increased in all three cases, with the
15 psig case indicating the greatest disturbance.

The determination of the droplet Reynolds number [Red = Eigfggél)
requires an estimate of the droplet diameter d and the droplet
velocity Ud. High—speed photography (4000 frames per second) was
utilized to obtain a gross value of the droplet diameter, and an
average value for the velocity. It should be observed that the study
included only those droplets the camera could see.

The droplet velocity, the average velocity of the water leaving
the nozzle, the air velocity at infinity and the volumetric flow rate
of the nozzle are given in Table 8 for a constant gas Reynolds number
of 5.64 X 101+ and nozzle water pressures of 15, 20 and 25 psig.

From Table 8 it is observed that an increase in water nozzle
pressure increases the volumetric flow rate and the average liquid
velocity at the nozzle. Accompanying this increase in water pressure
an additional decrease in the difference between the air velocity and

the droplet velocity is observed. This suggests that the droplet

size is also decreasing as nozzle pressure increases. All this tends

62

to decrease the droplet Reynolds number for increased water nozzle
pressures.

The scatter in droplet velocity data was i 5 per cent without
establishing correlation between droplet size and droplet velocity.
The range of droplet diameters observed was 0.015 to 0.0625 inches.

In order to determine a representative size distribution of
those droplets which were photographed it was found necessary to
take 40 photographic samples for each flow condition. This provided
approximately 68 per cent of the film samples with a variation in
total number of drops per observation less than one standard deviation.
Figures 32, 33 and 34 are plots of per cent of observations versus
fraction of standard deviation. The nozzle pressures were 15, 20 and
25 psig respectively and the gas Reynolds number was 5.64 X 101+ for
each run. The above figures are similar and indicate a large fluc-
tuation in droplet pOpulation. The droplets were classified into four
size groups (1/16, 3/64, 1/32 and 1/64 of an inch) and an equivalent
droplet diameter was calculated. The equivalent diameter is a droplet
diameter that would provide the same mass flow and same total number
of drops. Table 9 is a summary of the results of the survey. The
important result of the survey was the determination of the drOplet
Reynolds number. The drOplet Reynolds number decreased as the nozzle
pressure increased. This was brought about by a reduction in droplet
size and decrease in the velocity difference that exists between the
gas and the droplet as the water pressure was increased. This is in
agreement with the pressure coefficient curves of Figures 29, 30 and
31. These curves were ordered with the 15 psig curve indicating the

maximum disturbance.

63

The variation in the measured liquid boundary-layer thickness
versus probe voltage is given in Figure 13, all measured at 90 degrees
from the stagnation point. The measured boundary—layer thickness
appeared to be independent of probe voltage in the range of 4.5 to
7 volts. Here a 3 per cent variation in liquid boundary—layer thick-
ness was observed. For a probe voltage of 10 volts there was a 12
per cent increase in thickness. A probe voltage of 4 volts indicated
a 20 per cent reduction in liquid boundary—layer thickness. For all
subsequent boundary—layer measurements a probe voltage of 6 volts was
employed.

Curves of liquid boundary-layer thickness versus angle (distance)
measured from the forward stagnation point are plotted in Figures 35,
36 and 37 for water nozzle pressures of 15, 20 and 25 psig respec-
tively. For the above curves, the gas Reynolds number was held at
5.64 X 10+ and each data point represents an average of twenty or more
observations. For a typical data point a scatter in the measurements
of 0.002 inches was observed. This apparent large scatter in the
measurements was attributed to two major factors. The first of these
was the presence of vibrations in the system. Coupled with this it
was found by observing the liquid boundary layer with the aid of high—
speed photography, that waves were formed on the liquid surface as a
result of droplets striking the surface. As a result of the vibrations
and the presence of waves it was difficult to determine the exact
location of the film surface.

For angles less than 60 degrees an added complication was the
problem of water running down the probe and making contact with the

cylinder. Boundary—layer thickness measurements less than 0.002 inches,

54

which are typical values for angles less than 60 degrees, would be
subject to serious error. It is doubtful if the present technique of
measurement could be employed in that region.

The three curves of Figures 35, 36 and 37 are all similar,
indicating a rapid growth of the liquid boundary layer in the regions
of the pressure minimum and the separation points.

The velocity U0 at the outer edge of the liquid boundary layer
was determined with the aid of high—speed color photography. A
filming speed of 4000 frames per second was required to adequately
define the flow field. Plots of the velocity U0 versus angle (distance)
measured from the forward stagnation point are given in Figures 38,
39 and 40. The gas Reynolds number of 5.64 X 10l+ was maintained for
the three runs and the water nozzle pressure was 15, 20 and 25 psig
respectively.

The above curves are all similar and a reduction in the velocity
U0 was observed for all cases in the region of the separation point.
Two significant observations were made during the evaluation of the
photographic film. It was observed that waves were formed on the
cylinder and traveled around the cylinder at a constant velocity.

The range of velocity of the waves was from 1.82 to 3.40 feet per
second. These waves were generated by droplets striking the surface
of the liquid film. When the droplets would strike the liquid sur—
face it was also observed that part of the liquid film would splash
from the cylinder. This fact was established by the presence of dye
in the fluid leaving the cylinder.

The phenomenon of boundary-layer separation can be observed in

the photograph of Figure 41, for Reynolds number 5.64-X10“ and water

65

nozzle pressure 20 psig. The region of boundary—layer separation was
made visible by the injection of a mixture of distilled water and
wetting agent into the liquid boundary layer at the upper pressure
tap shown in Figure 41. The presence of the wetting agent promoted
the formation of gas bubbles in the separated liquid film. Within
the separated liquid film in a region downstream of the separation
point some of these bubbles are observed to move in a direction
parallel to the axis of the cylinder, while some of the bubbles were
observed to move upstream and some downstream in the liquid film.
The point where the upstream movement of the bubbles reversed and
returned in a streamwise direction was considered to be the separation
point. This location was in agreement with the pressure profiles of
Figures 29, 30 and 31. Both of these techniques yielded separation
point with an accuracy estimated to be i 2 degrees. Three important
pieces of information were obtained from this photograph.

The line forming the separation region spans the wetted length
of the cylinder and was located between 90 and 95 degrees from the
forward stagnation point. The water spray flow distribution along
the axis of the cylinder is plotted in Figure 42, and the water spray
distribution over the test segment of the cylinder is shown in the
coordinate plot of Figure 43. The first important observation can
now be made by considering water spray flow distribution and the line
of separation. The observation is that the separation region is not
strongly affected by the water spray distribution for mass ratios of
liquid to gas less than 8 per cent. Further it was observed that a
liquid film would not form on the cylinder in a region near the end

of the cylinder for the case of low rate of water spray, particularly

66

if the surface was contaminated with oil or wax film. This may be
observed at the right end of the cylinder.

The second important observation was the ability of the liquid
film to transport the mixture of distilled water and wetting agent
laterally along the cylinder. This was Characterized by the appear-
ance of soap bubbles along the complete length of the cylinder when
the mixture was injected through a pressure tap located four and one
half inch from the right wall of the tunnel, approximately 30 degrees
from stagnation point. The rotation in the liquid boundary layer was
indicated by the shedding of streamwise vortices. There was no set
pattern for the shedding of the vortices even though a pair was
observed to be leaving the cylinder in Figure 41. The vortex in
Figure 44 can be observed to be rotating so that fluid on the air
side of the liquid film is moving from the wall toward the center of
the cylinder. The rotation of the fluid in the region of separation
can be observed in Figure 45 as a series of vortices whose axis of
rotation is in the streamwise direction. The vortices may be similar
to the Taylor-Goertler vortices (23) which are caused by three-
dimensional disturbances in the flow along a concave wall. Such a
shedding of vortices was not observed in horizontal flow over a
vertical cylinder (Appendix A).

The last observation was the existence of a bubble of water
located downstream of the separation line Figure 41. The bubble
extended the length of the wetted portion of the cylinder and appeared
as a raised section on the cylinder. From this bubble vortices were
shed in random order. Presence of the water bubble and the relatively

thick liquid boundary layer in the region of the separation point

67

provided a contoured surface for the gas boundary layer to flow over.
The contoured surface presented an Opportunity for the gas stream to
flow against a reduced pressure hill. This was reflected in the
decreased lepe (as compared to single—phase flow) for the pressure
curve of Figures 29, 30 and 31, for that region downstream of the
pressure minimum.

The third set of experiments was conducted in order to determine
the effect that a change of gas Reynolds number would have on the
separation region. A photograph of the boundary-layer separation
region for a gas Reynolds number of 3.55 X 101+ and a constant water
nozzle pressure of 20 psig is given in Figure 46. The boundary layer
is similar to those of Figures 41 and 45. A vortex was observed to
have removed the turbulent fluid downstream of the separation point in
the foreground of the photograph. The low gas velocity (48 feet per
second) permitted a liquid film to be formed along the complete length
of the cylinder. This was a result of droplets crossing the gas
streamlines upstream of the cylinder, thus covering the complete cross
section of the test area. At higher velocities this did not occur and
only the central core of the test section was under the condition of
two-phase flow.

A photograph of the boundary-layer separation region for a gas
Reynolds number of 1.04 X 105 and a constant water nozzle pressure of
20 psig is given in Figure 47. At this gas velocity of 140 feet per
second only the central section of the cylinder was wetted. The
boundary-layer separation region has shifted downstream approximately

5 degrees, and the bubble of the liquid film has shifted downstream

68

from the region of separation. This gave an appearance of an

irregular separation line.

6.4 Results of Analytical Investigation

 

The laminar gas boundary—layer investigation of Section 2.2 was
to determine what effect the velocity 00 had in altering the separation
point (U0 is the interface velocity between the gas and the liquid
boundary layer, which for this analysis was assumed to be constant).
Equation (2.15) was first solved numerically with U0 = 0 and B = l
with the external velocity expressed by Us = 2Uwsin¢. This was
accomplished with the aid of an IBM 1620 computer utilizing a modified
form of the program listed in Figures 18 and 20. This provided a
check for the computer program, since the separation point for single-
phase flow over a cylinder is well established for this external
velocity. The solution also provided initial data for the turbulent
gas boundary-layer analysis.

Equation (2.15) was then solved with the same flow assumed to
prevail in the external flow field for a range of values of U0 up to
and including 5 per cent of Uco except in a small neighborhood of the
forward stagnation point where U0 was kept zero and B unity as
explained at the end of Section 2.2. The results are listed in
Table 10.

For a value of U0 equal to 5 per cent of Um, which corresponds
to the maximum observed value of U0, the calculated separation point
shifted downstream approximately one degree. The observed separation
point shifted downstream 14 degrees for a gas Reynolds number of

5.64 X 10”. The analysis only accounted for 7 per cent of the total

69
displacement of the separation point. It was observed that UO had
very little effect on the boundary—layer prOperties upstream of the
pressure minimum. For the external flow used, given by Equation
(2.19), A is zero at 6 equal to 90 degrees, while at the point
¢ = 1.5 radians (86 degrees) the maximum variation in Y from the
single—phase flow value was 0.1 per cent for the range of values of
U0 considered. It may be concluded from the results shown in Table
10 that the solution of Equation (2.15) is not sensitive to U0 up-
stream of the point of pressure minimum, while it is somewhat sen—
sitive to U0 downstream of the point of pressure minimum. For
two—phase flow and a gas Reynolds number of 5.64 X 10l+ the distance
between the observed point of pressure minimum and the observed
separation point was increased 8 degrees over that of single—phase
flow. Thus Equation (2.15) predicted only 12.5 per cent of the
observed increase in the distance of the separation point from the
point of pressure minimum.

The turbulent boundary-layer investigation was to determine
whether the region of boundary-layer separation could be predicted
by the use of Equations (2.25), (2.26) and (2.27). The equations
required for solution the input of actual velocity pressure infor-
mation, viz., the derivative of the velocity pressure with respect
to 0, and initial conditions.

The pressure information contained in Figures 29, 30 and 31 was
plotted in terms of the velocity pressure q = %'9”32° The plots of
q versus the angle (distance) measured from the forward stagnation
point are given in Figures 48, 49 and 50 for nozzle water pressures

of 15, 20 and 25 psig respectively. The velocity pressure for

7O

single-phase flow at the same gas Reynolds number is plotted as a
broken line. The plots for two-phase flow show negative values of q
near the stagnation point. Since q = %—pUs2 and only real velocities
are permitted, this restricts q to positive values. To accomplish
this, single-phase flow was assumed in the region of the stagnation
point. Each of the curves were divided into sections as indicated in
the above figures. A polynomial fitted to the section of data along
with an overlap of sections was accomplished with a Fortran computer
program as referred to in Section 5.2.

Coefficients of the polynomials become the coefficients of the
array A (5) of the computer program Figure 21 and C represents the
maximum value of 0 for which a given set of coefficients was valid.
In order to calculate the initial value of the momentum thickness 0
and the momentum Reynolds number (Ree), single-phase flow was assumed
to exist for values of ¢ less than 0.2 radians, with the external
flow given by Equation (2.19) in this region only. A numerical
solution for the above equation was obtained with the aid of an
IBM 1620 computer utilizing the computer program of Figures 18 and 21
for a range of e from 0.2 radians to 1.96 radians. The results are
summarized in Table 11.

The range of values of 0, Ree and to/2q are in agreement with
results that would be predictable from analysis of data by Squire and
Young (15), which indicates the computed data is reasonable. But
the value of H did not increase to a value in the range 1.8 to 2.6
that would predict separation according to the criterion that von
Doenhoff and Tetervin established for single—phase flow. The cal—

culated values of H in the experimentally-established separation

71

region were near 1.23 (i 0.01 in the cases observed). The results

of the computer solution of Equations (2.25) and (2.26) are reasonable
for values of q less than the maximum q, where H decreases with
increasing 0 because first term in Equation (2.25) is negative.

Beyond the point of maximum q this term becomes positive so that H
increases again. But because of the decreased slope of the pressure
curve observed in two-phase flow as compared to single-phase flow (see
Section 6.3), the contribution of the first term of Equation (2.25) in
the region downstream of the pressure minimum is small, so that the
calculated H does not increase as much as in single—phase flow. This
discussion is all in the confines of the assumption that U0 (the
velocity at the outer edge of the liquid film) did not materially

affect the region of separation.

6.5 Summary of Results

 

As a result of an evaluation of the experimental apparatus under
the conditions of single-phase flow, it was found that the average
Nusselt number for the range of Reynolds numbers considered was in the
same range of values as those published in the cited references
(References 10, 17, 21 and 22). The boundary-layer separation point
agreed with that reported in Reference 22. From this study it was
concluded that the intensity of turbulence e for the case of single—
phase flow was less than one per cent for the range of Reynolds numbers
considered. As a result of determining the system characteristics in
single-phase flow it may now be possible to compare the findings in
the two-phase flow studies with those of single-phase flow and also
with the findings of other investigators whose system possesses similar

characteristics in single-phase flow.

72

A system for measuring the pressure distribution around the test
cylinder was developed. The system was evaluated in single-phase flow
with a gas Reynolds number of 5.64 X 10L+ by comparison with a standard
system, which was assumed to be correct. An error of 1.4 per cent in
pressure measurement at the stagnation point was observed while the
maximum error of 5.2 per cent occurred at the point of pressure
minimum. These results were considered acceptable, since this repre—
sented a severe test of the system, because of the injection of water
(whose density is three orders of magnitude greater than air) into the
air boundary layer.

Three pressure profiles were measured under conditions of two—phase
flow with a fixed gas Reynolds number of 5.64 X 101” and water pressures
of 15, 20 and 25 psig respectively, at the spray nozzle. The following
observations were made in regards to these profiles: The droplet
Reynolds numbers were found to be 522, 450 and 380 for the respective
water nozzle pressures. It was observed that the pressure profile for
the 15 psig water pressure and droplet Reynolds number of 522 (which
was the maximum observed droplet Reynolds number) experienced the
greatest deviation from single-phase flow. The maximum excursion in
pressure occurred (i.e., the least pressure at the point of pressure
minimum) for this profile. The same profile required the greatest
distance of travel from the point of pressure minimum to the point
where the near constant wake pressure was observed. For all three
pressure profiles the point of minimum pressure shifted downstream
approximately 6 degrees from that observed in single-phase flow, while
the separation point shifted downstream approximately 14 degrees. The

three pressure profiles experienced a reduction in slope as compared

73
to single-phase flow, at the same gas Reynolds number, between the
point of pressure minimum and the point where the near constant'wake
pressure was observed. Possibly this is a result of the observed
rapid growth in the liquid boundary-layer thickness from the point of
pressure minimum to the separation point and the presence of a bubble
of water which was observed just downstream of the separation point.
These two conditions could well have presented a new contour for the
external flow field. This would be observed in the pressure measure-
ments since it is coupled to the external velocity US through the
Bernoulli Equation. The velocity U0 (at the outer edge of the liquid
boundary layer) presented somewhat similar profiles for the three pressure
measurements. The important feature here was the observation that the
maximum value occurred in the neighborhood of 50 degrees from the
stagnation point. This was well upstream from the point of maximum
value of US. The average value of U0 at its maximum point was 2.8 feet
per second and decreased to approximately 1.7 feet per second at the
90 degree point, the nominal separation point.

The boundary—layer separation point was made visible by the in—
jection of a mixture of distilled water and wetting agent into the
liquid boundary layer. The separation point observed in this manner
agreed with that interpreted from the pressure profiles. For the
vertical downflow of a two-phase mixture over a horizontal cylinder a
bubble of water was observed downstream of the separation point.

Water was shed from this bubble in the form of vortices. The location
of the point of boundary-layer separation was found to be insensitive
to the water spray distribution for ratio of mass of water to air less

than 0.08. For the case where the gas Reynolds number was varied from

74

3.55 X 10” to 1.04 X 105 the point of boundary—layer separation
experienced a displacement similar to that found in single-phase flow
for the same range of gas Reynolds numbers.

The first analytical model, which assumed a laminar boundary-
layer system, where the interface velocity U0 was assumed constant
outside the neighborhood of the forward stagnation point, had as its
objective the determination of the separation point. It was found
that the calculated boundary—layer thickness from a point in the
neighborhood of the stagnation point up to the point of pressure
minimum was insensitive to U0 which was assumed constant for a range
of values of Uo/Um up to and including 0.05. However, it was found
that the calculated boundary—layer thickness from the point of pressure
minimum to the separation point was somewhat sensitive to U0 and for a
value of UO/Um = 0.05 the separation point shifted downstream approxi-
mately 1 degree. This only accounted for 12.5 per cent of the change
in distance between the point of pressure minimum and the separation
point when compared to single—phase flow.

The second model, which assumed a turbulent gas boundary layer and
U0 = 0, was intended to predict that separation would occur for a value
of the parameter H between 1.8 and 2.6, the range of H established
empirically by von Doenhoff and Tetervin for separation in single-phase
flow. But with the experimentally observed pressure profile, this
range of H was not attained, and separation actually occurred at a

point where the value of H was 1.23 (i 0.01 for those cases observed).

The result of a non—zero BUD/3x was not investigated in either

model.

CHAPTER VII
SUMMARY AND CONCLUSIONS

For the case of two—phase flow (air-water spray) a technique for
measuring the pressure profile was developed. The two-phase flow
pressure-measuring system was evaluated in single—phase (air) flow
with a gas Reynolds number of 5.64 X 10” by comparison with a standard
system, which was assumed correct. An error of 1.4 per cent in the
pressure measurement was observed at the stagnation point while the
maximum error of 5.2 per cent occurred at the point of pressure
minimum. These results were considered acceptable, since this repre—
sented a severe test of the system, because of the injection of water
(whose density is three orders of magnitude greater than air) into
the air boundary layer.

Pressure profiles for two—phase flow with a gas Reynolds number
of 5.64 X 10L+ were measured. As a result of addition of water spray
to the air flow, the separation point was observed to have shifted
downstream approximately 14 degrees for the same gas Reynolds number.
The minimum pressure was reduced and its location shifted downstream
approximately 6 degrees. The slope of the pressure coefficient curve
in the region between the point of minimum value and the location of
the wake pressure was less than the slope for the corresponding region

in single-phase flow with the same gas Reynolds number. This reduction

75

76

in slope permits a greater distance of flow against a positive pressure
gradient before separation takes place.

The region of liquid boundary—layer separation was made visible
by the injection of a wetting agent into the liquid boundary layer.

The region of separation observed in this manner falls within the
region predicted by the inflection points on the pressure profiles.

It was observed that the point of separation for a given gas Reynolds
number was not strongly affected by variations in water spray distri-
bution (mass ratio of water to air less than 8 per cent). The region
of separation moved with changes in gas Reynolds number. This response
to change in Reynolds number was similar to that experienced in single-
phase flow.

Photographs of the region of separation revealed the presence of
vortices in the separated liquid flow layer for the cases of vertical
downflow over a horizontal cylinder. The vortices were observed to
have an axis of rotation in the streamwise direction. The shedding
of vortices from the region of separated liquid flow was observed to
occur in random order. No such vortices were observed in horizontal
flow over a vertical cylinder.

In the closing remarks of Section 2.3 it was indicated that the
two analytical models would not be expected to predict the same results,
since they represent different types of flow structure. The first
analytical model failed to predict a sufficient displacement of the
separation point. The observed displacement occurred eight degrees
further downstream from the point of pressure minimum as compared to
single-phase flow for the same gas Reynolds number. The analytical

model only predicted 12.5 per cent of this value UO/Um = 0.05, which

77

was found to be a reasonable value of U0 for those cases considered.
The second analytical model failed to predict separation under the
assumption that separation would occur when H was bounded between 1.8
and 2.6. The observed separation occurred when H increased to a value
of 1.23 (i 0.01 in those cases observed). The first analytical model
predicted separation prior to the observed condition while the second
analytical model did not indicate the fulfillment of the assumed
separation criterion.

As a result of the experimental investigation a successful
technique was developed for measuring the pressure profile of a
surface in two-phase flow. A system employing liquid-filled pressure
lines and injection of a small amount of liquid into the liquid boundary
layer gave reasonable repeatability and a minimum of hysteresis effect.
It was found that the pressure at a given point would vary 1 0.01 inch
of water when measured from ascending values and then descending values
of ¢. These results compare favorably to similar observations made in
single-phase flow with an inclined manometer utilizing air filled
pressure lines. The rate of water injection, which was determined
experimentally, was 1.4 milliliters per minute. The flow rate should
be determined experimentally for each new application or range of gas
Reynolds numbers.

A system for the visual observation of the region of boundary-
layer separation was developed. This was accomplished by the injection
of a small amount of mixture, consisting of distilled water and wetting
agent, into the liquid boundary layer from one of the pressure taps.
The point of boundary—layer separation and some of the properties of

the separated liquid flow were made visible by the injection process.

78

The separation point observed by this process agreed with that
interpreted from the observed pressure profiles.

The observed pressure profile and the observed thickening of
the liquid boundary layer in the region between the point of pressure
minimum and the separation point indicate that the external flow
field sees an object of shape sufficiently different from the cylinder
that neither of the analyses based on a circular cylinder gives a
reasonable prediction of separation.

It is concluded that for the case of vertical downflow of a two-
phase (air-water spray) mixture over a circular cylinder one may expect
the point of boundary—layer separation for the range of Reynolds
numbers considered (3.55 x lO'+ to 1.04 x 105) to be approximately
lu degrees downstream from the point of separation in single-phase
flow at the same Reynolds number.

Further studies are suggested along several lines: (a) obtain
the boundary—layer properties in the region of the forward stagnation
point, (b) determine the effect of droplet size on the boundary—layer
properties, (c) investigate the phenomenon of droplet bouncing and
fluid splashing from the liquid film, (d) study the origin and prop-
erties of the vortices, (e) determine what effect the addition of a
streamline section to the back side of the cylinder would have on
separation and (f) investigate the properties of the liquid bubble
located downstream of the separation point. An investigation including
an analysis of the liquid boundary layer is recommended as this might

yield improved separation prediction.

APPENDIX A
GRAVITATIONAL EFFECTS

The experimental apparatus was so designed that the test cylinder
was in a horizontal plane and experiencing downward flow of a two-
phase mixture; under these conditions the gravitational force was in
the free—stream direction. The objective of the phase of the investi-
gation reported in this appendix was to determine if the gravitational
force may be neglected when considering the behavior of the boundary
layer up to the point of flow separation.

The boundary—layer separation point was first observed and photo-
graphed for the vertical downflow over a horizontal cylinder (Figure Ml).
A test cylinder was then installed in a vertical position in a hori—
zontal wind tunnel. The free-stream velocity was adjusted to give the
same gas Reynolds number (5.6u x 10“) used in the vertical wind tunnel
and the mass ratio of water to air was adjusted to the same range of
values (approximately 0.05). Under these test conditions two sets of
observations were made. The first utilized the technique of injecting
a mixture of distilled water and wetting agent into the liquid boundary
layer and the second required the use of dye injection into the boundary
layer.

The results of the injection of the distilled water and wetting
agent are shown in Figure 51. The injection was at the middle of the

cylinder and 10 degrees downstream from the forward stagnation point.

79

8O

Separation was observed to take place in the same neighborhood as that
of the horizontal cylinder and vertical downflow. The significant
change here was the shedding of water from the region of separated
flow. A water bubble similar to that observed for the horizontal
cylinder but located at the separation point was present and it was
from this bubble that water left the cylinder. For the vertical
cylinder it was observed that the primary method of water shedding

was by gravity flow down the bubble, which was parallel to the cylinder
axis. There was some evidence of water leaving the cylinder in the

. gas streamwise direction, but this was only a small fraction of the
water flow.

The second set of experiments utilized a dye injected into the
liquid boundary layer using the same pressure tap for the dye injection.
It was observed that the dye on entering the liquid boundary layer
would diffuse in a radial direction from the pressure tap, which meant
that some of the dye was transported upward and some in a downward
direction. The mixture of dye and water would then flow around the
cylinder in a nearly horizontal plane to the separation point (Figure
52 is a typical photograph of the process), and the mixture then would
turn and flow down the cylinder within the liquid bubble.

From these two experiments one concludes that the effect of
gravity is negligible in the boundary—layer flow and has negligible
effect on the location of the separation point, but once the flow
separates from the cylinder gravitational forces are not negligible.
The investigation did not consider the possible case of the vertical

upflow of a two-phase mixture.

APPENDIX B

FIGURES

81

82

External Flow

 

Gas Boundary Layer

9.51 $

Liquid Boundary Layer

1 I

FIGURE 1 Boundary Layer Quantities for Gas Boundary Layer

83

 
  
 

 

A Wind Tunnel Inlet Screen Assembly

- . I'ffuser

 

Elbow with Turning ’anes
Auxiliary Water Spray Assembly
Two-Stage Pan

Electric Motor

H a: 6) 'fl m

Fan Ofitlétifiuct and Butterfly Valve Assembly

FIGURE 2 Schematic Drawing of AFIT 10"Xl0" Wind Tunnel

 

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- A Wind Tunnel Inlet Screen Assembly

 

HIGH?!

 

i°x O I{Lg . 3. 9 - - I ffuser

Elbow witfiummg 'anes
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Two-Stage Fan

Electric Motor

Fan outlet Duct and Butterfly Valve Assembly

FIGURE 2 Schematic Drawing of AFIT 10"X10" Wind Tunnel

 

 

 

 

 

 

 

 

 

 

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85

 

FIGURE '4 Test Section of AFIT 10" X 10" Wind Tunnel

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Cement for Securing Heaters

Schematic of Test Cylinder

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A One and one—half inch diameter test cylinder
used for wind-tunnel evaluation

B One and one-half inch diameter test cylinder
used for boundary—layer studies

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Pressure Transducer

Test Cylinder

FIGURE 8 Pressure Measuring System

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FIGURE 11 Grid Wire Layout for Droplet Velocity Ud Measurement

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26

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27

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98

SUBROUTINE RUNGA
M=M+l

GO TO (10,11,12,13,1u),M
RETURN

DO 20 J=l,N
OLDY(J)=Y(J)

DO 21 J=l,N

PHI(J)=F(J)

DO 22 J=l,N
Y(J)=OLDY(J)+.5*H*F(J)
X=X+.5*H

RETURN

DO 23 J=l,N
PHI(J)=PHI(J)+2.*F(J)

DO 24 J=l,N
Y(J)=OLDY(J)+.5*H*F(J)
RETURN

DO 25 J=l,N
PHI(J)=PHI(J)+2.*F(J)

DO 26 J=l,N
Y(J)=OLDY(J)+H*F(J)
X=X+.5*H '

RETURN

DO 27 J=l,N
PHI(J)=PHI(J)+F(J)

DO 28 J=l,N
Y(J)=OLDY(J)+PHI(J)*H/6.
M=0

RETURN

END

FIGURE 18 Runge-Kutta Subroutine

CALLING PROGRAM

 

 

INITIALIZE
X,Y(l),Y(2),---,Y(N)

99

SUBROUTINE

 

 

 

 

AJ PASS 1
DO NOTHING, RETURN
r] TO CALLING PROGRAM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ESXESAIE PASS 2
EXECUTE STEPS
F(l)9F(2)a".aF(N) 2 3 H 5
F(J)=fj(X,Y(l),Y(2),---,Y(N)) ’ ’ ’
STEP 6
EVALUATE EXECEIESSTEPS
F(l),F(2),"',F(N) 7 8
P<J>=fj(x,Y(1),Y(2>,---,Y<N)) ’
EIXEEAIE PASS S
EXECUTE STEPS
F(l)’F(2)"°"F(N) 10 11 12
F(J)=fj(X,Y(l),Y(2),"',Y(N)) ’ ’
STEP 13
EVALUATE PASS 5
EXECUTE STEPS
F(l),F(2),---,F l” 15
F(J):fj(X’Y(l))Y(2)g...,Y(N)) ,

 

 

 

 

 

 

 

 

 

 

NEW SOLUTION VALUES
INITIAL VALUES FOR THE
NEXT INTERVAL ARE
Y(l),Y(2),°°'.Y(N)

 

 

FIGURE 19 Computer Flow Chart for Runge—Kutta Subroutine

100

C C RUNGA-KUTTA SOLUTION OFR DIFF EQNS WRIGHT
DIMENSION Y(5),F(5),OLD(5),PHI(5),OLDY(5),A(5)
READ,N

6 READ, H,XMAS,ICNT
PUNCH 100,H,XMAX,ICNT

100 FORMAT(2HH=,F5.3,4X,5HXMAX=,F6.3,uX,uHINT=12)
READ,X,Y(l),XL,DB,CC
PUNCH lOl,X,Y(l),XL,DB,CC

101 PORMAT(5E1u.6)

M=O
nu K=ICNT
33 IF(X-XMAX)3,6,6

3 CALL RUNGA
IF(M)1.2,1

52 READ,CC,(A(L),L=l,5)

1 IF(X-CC)51,51,S2

51 Q=A(1)+((((A(5))*X+A(u))*X+A(3))*X+A(2))*X
QPR=A(2)+(((H.*A(5))*x+3.*A(u))*X+2.*A(3))*X
Q2PR=2.*A(3)+(12.*A(5)*x+6.*A(u))*x
XL=Y(l)*QPR
B: 1. - DB/Q
FUDU=Q*Q2PR/(QPR*QPR)
BB=9072.+(292.5*B—1965.6)*XL
C=(80.u-33.*B+u.3*PUDU)*XL*XL
D=(l.+FUDU)*XL*XL*XL
AA=(((-XL+39.6*B-u5.36)*XL+B*(54A.32-331.2*B))*Q)
F(l)=.8*(BB+C+D)/AA

GO TO 33
2 K=K—l
IP(K)3,u,3
u PUNCH l02,X,Y(l),XL
102 FORMAT(F6.3,2(El5.8))
GO TO nu

FIGURE 20 Computer Program for Equation (2.15)

1.01

C C RUNGA-KUTTA SOLUTION OFR DIFF EONS WRIGHT
DIMENSION Y(S),F(5),OLD(5),PHI(5),OLDY(5),A(5)
CONS=1./LOG(1O.)

1:0
READ,N

5 READ,H,XMAX,ICNT
PUNCH 100,H,XMAX,1CNT

100 PORMAT(2HH=,P5.3,ux,5HXMAx=,P5.3,ux,uHINT:I2)
READ,X,Y(l),Y(2),DR,C
PUNCH lOl,X,Y(l),Y(2),DR,C

101 PORMAT(5E1A.6)

M=O
Wu K=ICNT
33 IF(X—XMAX)3,6,6

3 CALL RUNGA
IF(M)l,2,l

52 READ,C,(A(L),L=l,5)

1 IF(X—C)51,Sl,52

51 Q=A(1)+((((A(5))*X+A(u))*X+A(3))*X+A(2))*X
QPR=A(2)+(((u.*A(5))*X+3.*A(u))*X+2.*A(3))*x
RE=Y(l)*DR*SQRT(Q)

I=I+l
IF(RE)55,55,56

55 PUNCH 103,RE,I

103 FORMAT(E15.8,IH)

S6 T=(5.89*CONS*LOG(H.075*RE))**2
F(l)=—((.5*Y(2)+l.)*Y(l)/Q)*QPR+l./T
P(2)=EXP(A.58*(Y(2)-2.975))*(-QPR*T/Q-2.O35*(Y(2)

l-l.286)/Y(l))

GO TO 33
2 K=K-l
IP(K)3,u,3
u PUNCH 102,X,Y(l),Y(2),RE,T
102 FORMAT(F6.3,H(E15.8))
GO TO nu

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Region
I /’( i
T
L
-2 l 1 l l
~w-—TQGUREAQW-mPressure Coefficient vs ¢, 1.5 Inch Diameter

Cylinder for Single—Phase Flow, Re = 5.54x10“

 

 

n ‘aw - 1

 

 

 

 

 

 

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Pressure Inches of Water

106

 

 

 

 

 

 

3L I I .T I
| +
l
I
———-Actual pressure
2 . . Pressure measured with
-1 . water injection systan 4
. corrected for hysteresis
9 effects, injection
: rate = 1.4 ml/min
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r a ' *
, 3 f
r
-2 _ . 0 q
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r ' ‘
_ 1 k'
.' \ V. I, O
7 ' I
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0 20 40 60 80 100

¢ Degrees

FIGURE 26 Pressure on Cylinder vs ¢ for 1.5 Inch Diameter

Cylinder Single-Phase Flow, Re = 5.611l><lOL+

 

 

 

 

 

 

kl

P(actual) - P(Observed) Inches of Water

107

 

 

 

 

 

 

 

 

 

 

 

 

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FIGURE 27 Difference Between Actual and Observed Pressure,

Observed Pressure Measured By Water Injection

Method

 

 

 

 

 

 

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1/2 0U...

CP

 

FIGURE 29

 

-——-Two—Phase Flow

--- Single-Phase Flow

‘\‘~

\ rm..—
/
\\\ I!
\

     

 

 

Pressure Coefficient vs ¢ for Two-Phase Flow Over
1:5 Inch Diameter Cylinder, Nozzle Water Pressure
equal 15 psig, Red = 522, Gas Reynolds Number
equal 5.6uxlo“

Wire..-
4

 

 

 

/.

 

 

 

.\

 

 

 

110

 

 

 

 

 

[I I T [I

-——-Two-Phase Plow

—-- Single-Phase Flow

 

\ ¢ Degrees

-—1- —-.-

)K Separation
_’//’ Region

l I l J

 

.“

- FIGURE 30 Pressure Coefficient vs ¢ for Two-Phase Flow Over

1.5 Inch Diameter Cylinder, Nozzle Water Pressure
equal 20 psig, Red = #58, Gas Reynolds Number
equal 5.5uxlo”

lll

 

 

 

 

 

 

 

 

 

 

 

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.953 \ / \
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-2 .
l I l 1,
FIGURE 31 Pressure Coefficient vs ¢ for Two—Phase F10" Over

1.5 Inch Diameter Cylinder, Nozzle Water Pressure

equal 25 psig, Red = 380, Gas Reynolds Number

equal 5.614X10l+

 

 

 

 

 

 

 

ll2

 

 

 

 

 

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FIGURE 32 Per Cent of Observations vs Fraction of Standard

Deviation, Gas Reynolds Number equal 5.GMXlO“,
Nozzle Water Pressure equal 15 psig

113

 

 

 

 

 

L / 2; so . 1+
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-2 —1 0 1 2 3
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FIGURE 33 PerCent of Observations vs Fraction of Standard
Deviation, Gas Reynolds Number equal 5.6MXlO”,
Nozzle Water Pressure equal 20 psig

11H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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FIGURE 3” Per Cent of Observations vs Fraction of Standard

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Nozzle Water Pressure equal 25 psig

 

 

 

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FIGURE 35 Liquid Boundary~Layer Thickness vs ¢ for Gas

Reynolds Number equal 5.6MXI0“, Nozzle Water
Pressure.equal 15 psig

 

 

 

 

3+"

0 1‘

 

 

 

5ino3 Inches

116

 

 

 

 

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FIGURE 36 Liquid Boundary—Layer Thickness vs ¢ for Gas

Reynolds Number equal 5.6HXlO”, Nozzle Water
Pressure equal 20 psig

 

 

 

 

 

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117

 

 

 

 

 

 

 

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FIGURE 37 Liquid Boundary-Layer Thicknessuvs O for Gas
Reynolds Number equal 5.6quO“,;Nozzle Water
Pressure equal 25 psig ,
-—._L.- ”—_7L..._-_ _.._

 

 

 

 

 

 

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; Distance Along Axis of Test Cylinder

FIGURE H2 Water Spray Distribution Milliliters Per Minute Along
Axis of Test Cylinder, for l/2 Inch Collecting Tube,
Nozzle Nr. 2116, Gas Reynolds Number equal 5.6HXlO”
and Water Pressure at Spray Nozzle equal 20 psig

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FIGURE HM Water Vortex Separating from Liquid Film
and Indicating Direction of Vortex Rotation

 

FIGURE H5 Series of Vortices Extending into Liquid
Boundary Layer

125

 

FIGURE H6 Boundary-Layer Separation for Gas Reynolds
Number equal 3.55X10“ and Water Pressure
equal 20 psig at Spray Nozzle

 

FIGURE M7 Boundary—Layer Separation for Gas Reynolds
Number equal 1.0UX105 and Water Pressure
equal 20 psig at Spray Nozzle

126

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FIGURE 52 Boundary—Layer Separation on a Vertical Cylinder in
Horizontal Two—Phase Flow with Dye Injected into
Liquid Boundary Layer, Gas Reynolds Number equal
5.6ux10“

APPENDIX C

TABLES

131

132

TABLE 1

HEAT TRANSFER DATA

 

 

Run A Tw Tm v‘,° + _ +
Nr. verage Free Free Re Nu
Cylinder Stream Stream
1 122.2 92.5 lug 1.1 x105 224
2 115.2 79 135 9.18x10“ 214
3 118.5 85.5 137 9.1 x10“ 221
u 124 86.5 107 7.13x10” 198
5 135 88 107 7.13x10” 195
5 135 90 H8 3.2uxio” 130
7 125 92 H8 3.2ux10” 133

 

 

 

 

 

 

 

 

T Reynolds Number Re and Average Nusselt number Nu
are based on cylinder diameter D.

133

TABLE 2

VELOCITY PROFILES

 

Free Stream

0.5 Inch From Wall

 

 

 

Run

NI“ v v- v V-
max min max min
ft/sec ft/sec ft/sec ft/sec
l #2 42 H2 40
2 91 90.5 90.5 88
3 125 123.5 129.5 121.5
9 139 137.5 139 I30
5 176 175 176 171

 

 

 

 

 

 

.j‘

 

 

134

TABLE 3

VELOCITY PRESSURE IN INCHES WATER, WITH TEST CYLINDER REMOVED,

AT TEST STATION IN AFIT lO"XlO" VERTICAL WIND TUNNEL

 

 

 

 

 

 

 

 

RUN NO. I r‘
_ .'
11
0.38 0.38 .9
+ + + + + + + + + + + r
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 I
i.
\
r—i
_l_~
o 38 o 35 E Bro
- £28
'6: + + + + + + + + + + + -.-:o
9 J08
0.38 0.38 0.38 0.38 0.38 O 38 O_3§_.9_3§__9_33__ _ rac)o:
+]_"—> l
0.35 0.35 If
+ + + + + + + + + + + -
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
+£"4—
2 I
JL.
L=e lO" 0%

 

 

Pb = 29.325" Hg
Tm = 78°F

 

lo"

 

135

TABLE 4

VELOCITY PRESSURE IN INCHES WATER, WITH TEST CYLINDER REMOVED,

AT TEST STATION IN AFIT lO"XlO" VERTICAL WIND TUNNEL

 

RUN NO. 2
1.72 1.67
+ + + + + + + + + + +
1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 l
34
\
v—l
_ (V
1.62 1.63 4530i
(D'UQ)
+ + + + + + + + + + + 9515
o-HE
HN>3GJ
1.72 1.72 1.72 1.72 1.72 1 73 1 73 1.73 1.73 ~1c>m
__”________-_____”___“____"__1
+1"—>
1.72 1.65
+ + + + + + + + + + +
1.72 1.72 1.72 1.73 1.73 1.73 1.73 1.73 1.73
+ l” +
2

 

 

 

7K—

10"

Pb = 29.39" Hg
Tm 2 90°F

JL

 

 

 

136

TABLE 5

VELOCITY PRESSURE IN INCHES WATER, WITH TEST CYLINDER REMOVED,

AT TEST STATION IN AFIT 10"X10" VERTICAL WIND TUNNEL

 

 

 

 

2 1/2"_“

I._

Test
linder
Removed

ll'
€32,

.1

 

 

RUN NO. 3
TT'
3.19 3.19
+ + + + + + + + + + +
3.20 3.20 3.21 3.22 3.22 3.22 3.23 3.23 3.23
3.06 3.10
:0 ++ + + + + + + + ++
H
3.23 3.23 3.23 3.23 3.24 3.23 3.24 3.25 3.25
__________._-__..____________1
+-r'+
3.25 3.22
+ + + + + + + + + + +
3.25 3.25 3.25 3.26 3.27 3.28 3.28 3.28 3.28
+ I:
2
.JL
%= 10" =1

 

Pb = 29.3u" Hg

Tco = 90°F

 

137

TABLE 6

VELOCITY PRESSURE IN INCHES WATER, WITH TEST CYLINDER REMOVED,

AT TEST STATION IN AFIT 10"X10" VERTICAL WIND TUNNEL

RUN NO. 4 f'

 

4.10 4.10

 

++ + + + + + + + ++

.:
[—1
O
.I:
H
O
4:
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I

 

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j
|
I
I
I
I
I
I
I

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I
I
I
I
I
I
l
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l
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4.00 4.00
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4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00

+£"4—
2

 

 

 

 

 

 

 

F: 1o" {4

- 29.32" Hg
83°F

~—3 *0
8 U”
n I

‘_- .L.-' in}

138

TABLE 7

VELOCITY PRESSURE IN INCHES WATER, WITH TEST CYLINDER REMOVED,
AT TEST STATION IN AFIT 10"XlO" VERTICAL WIND TUNNEL

RUN NO. 5

 

6.76 6.76
+ + + + + + + + + + +
6.76 6.76 6.76 6.76 6.76 6.76 6.76 6.76 6.76

_J

2 1/2"

I

 

 

 

 

 

 

 

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(DU 0)
: BIC >
o + + + + + + + + + + + "-I o
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6.76 6.76 6.76 6.76 6.76 6.76 6.76 6.76 6.76
—> i" <—
2
JL
L: 10" >{

Pb = 29.325" Hg
Too = 78°F

139

TABLE 8

DROPLET VELOCITY DATA

 

 

 

 

 

 

 

 

 

n °
Pw Um Ud Ud' Ud Q UL
psig ft/sec ft/sec ft/sec ft/sec ml/min ft/sec
15 76 43.0 43.4 41.1 1675 34.0
20 76 44.8 45.3 43.8 2000 40.5
25 76 50.3 51.8 48 2270 45.0

 

 

Pw Water pressure at spray nozzle
Ud Average droplet velocity
Maximum observed droplet velocity

Ud Minimum observed droplet velocity

0 Water volumetric flow rate at spray nozzle

(milliliters per minute)

UL Average water velocity at spray nozzle

 

 

140

TABLE 9

DROPLET SIZE DISTRIBUTION AND REYNOLDS NUMBER DATA

 

Maximum a/o Droplet size classification

 

less than in per cent of total
Pw n m unity observed droplets d' Red
Per

Cent a/o 1/16 3/64 1/32 1/64

 

15 40 3036 65 0.79 5.8 9.1 29.9 56.2 0.031 520
20 40 4085 70 0.98 2.2 9.1 24.8 63.9 0.029 458

25 40 4390 65 0.76 2.5 8.8 32.5 56.2 0.030 380

 

 

 

 

 

 

 

 

 

 

 

 

Water pressure at spray nozzle
n Number of observations
a/o Fraction of standard deviation

d' Equivalent droplet diameter (the droplet diameter which would
yield the same numbers of droplets and mass rate of flow)

Red Droplet Reynolds number based on equivalent diameter

m Total number of observed droplets

 

 

141

TABLE 10

LAMINAR BOUNDARY LAYER DATA

 

 

 

0
radians Uo/U” Y A B
0 0 3.526 7.052 1
0.01 0 3.53289 7.06543 1
0.1 0 3.53662 7.03791 1
0.2 0 3.56821 6.99417 1
0.3 0 3.62181 6.92010 1
1.5 0 7.97245 1.12789 1
1.8 0 14.73692 -6.62234 1
1.88 0 21.00751 -l2.78518 1
0.11 0.01 3.53357 7.02444 0.95445
1.5 0.01 7.97011 1.12756 0.9949
1.8 0.01 14.53323 -6.60392 0.99487
1.88 0.01 20.78130 -12.64750 0.99475
0.11 0.02 3.52763 7.01262 0.90891
1.5 0.02 7.96801 1.12727 0.98997
1.8 0.02 14.49424 ~6.58624 0.98973
1.88 0.02 20.57323 -12.52088 0.98950
0.11 0.03 3.52064 6.99873 0.86336
1.5 0.03 7.96614 1.12700 0.98496
1.8 0.03 14.45669 -6.56918 0.98460
1.88 0.03 20.38086 -12.40380 0.98425
0.11 0.04 3.51211 6.98178 0.81781
1.5 0.04 7.96450 1.12677 0.97995
1.8 0.04 14.42054 -6.55275 0.97946
1.88 0.04 20.20219 -12.29506 0.97900
0.11 0.05 3.50132 6.96032 0.77227
1.5 0.05 7.96311 1.12657 0.97494
1.8 0.05 14.38575 -6.53694 0.97433
1.88 0.05 20.03561 -12.19368 0.97376

 

 

 

 

 

 

 

 

 

142

TABLE 11

TURBULENT BOUNDARY LAYER DATA

 

 

 

 

 

 

 

 

 

Fw 0

psig radians 9 H Ree Qq/To
15 0.20 0.001281 1.40 -- --
15 0.24 0.00090 1.37129 9 89
15 1.32 0.00247 1.22373 118 250
15 1.36+ 0.00251 1.22373 125 254
15 1.60 0.00375 1.23100 178 284
15 1.64TT 0.00399 1.23255 188 288
15 1.88 0.00572 1.24465 254 315
20 0.20 0.001281 1.40 -- --
20 0.24 0.00090 1.37129 9 89
20 1.28 0.00230 1.22449 111 244
20 1.321 0.00241 1.22472 118 249
20 1.60 0.00381 1.23244 179 284
20 1.6u++ 0.00406 1.23413 189 289
20 1.88 0.00593 1.24759 258 316
25 0.20 0.001281 1.40 -- --
25 0.24 0.00093 1.37364 12 102
25 1.28 0.00238 1.22923 114 247
25 1.32+ 0.00251 1.22927 121 251
25 1.60 0.00390 1.23707 181 285
25 1.6LI++ 0.00416 1.23883 191 290
25 1.88 .0.00594 1.25131 257 316

Re = 5.6ux10“

Pw Water pressure at spray nozzle

+ Region of pressure minimum

++ Region of observed boundary-layer separation

 

 

 

hf;
;
5
E
' .1
APPENDIX D E?
BIBLIOGRAPHY
7..

 

 

143

10.

11.

12.

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