Citation
Water drop deformation and fragmentation due to shock wave impact

Material Information

Title:
Water drop deformation and fragmentation due to shock wave impact
Creator:
Krauss, William Edward, 1928- ( Dissertant )
Leadon, B. M. ( Thesis advisor )
Clarkson, M. H. ( Reviewer )
Williams, D. T. ( Reviewer )
Irey, R. K. ( Reviewer )
Blake, R. G. ( Reviewer )
Uhrig, Robert E. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1970
Language:
English
Physical Description:
xii, 81 leaves. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Acceleration ( jstor )
Average linear density ( jstor )
Boundary layers ( jstor )
Coordinate systems ( jstor )
Diameters ( jstor )
Experimental data ( jstor )
Liquids ( jstor )
Shock waves ( jstor )
Stagnation point ( jstor )
Velocity ( jstor )
Aerospace Engineering thesis, Ph. D. ( local )
Dissertations, Academic -- UF -- Aerospace Engineering ( local )
Drops ( lcsh )
Shock waves ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Results are presented of experimental and theoretical studies of water drop deformation and fragmentation in the high velocity air stream following a normal shock wave moving into still air. Experiments were conducted in a shock tube with shock wave Mach numbers in the range from 1.6 to 3.0 and for drop diameters from .5 to 3.2 millimeters. From the experimental data, it is determ.ined that the water drop deforms axisymmetrically into an approximately ellipsoidal shape which is distorted as to maintain a nearly hemispherical frontal surface but with a concave rear surface. The data also indicate that the time for complete fragmentation of the water drop is approximately one-half of the time observed for complete reduction of the drop to a trace of mist. Simultaneous potential flow within the distorted ellipsoidal drop and mass loss by stripping of a viscous layer of surface liquid are included in a theoretical model for drop deformation and fragmentation. Although the actual drop deformation is shown to deviate from the assumed ellipsoidal shape, the general dimensional and displacement variations arc predicted by the model and the theoretical mass history of the drop agrees well with the experimental data.
Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 79-80.
General Note:
Manuscript copy.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
001033933 ( AlephBibNum )
AFB6212 ( NOTIS )
18218572 ( OCLC )

Downloads

This item has the following downloads:


Full Text








WATER DROP DEFORMATION AND FRAGMENTATION

DUE TO SHOCK WAVE IMPACT















By
WILLIAM EDWARD KRAUSS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY












UNIVERSITY OF FLORIDA
1970


























To my wife

Barbara

for her patience, understanding

and encouragement













ACKNOWLEDGEMENTS


The author is indebted to many individuals for aid and

encouragement in the completion of this dissertation. lie

wishes to express his gratitude to his fellow students and

the faculty of the Aerospace Department of the University of

Florida for their aid and suggestions during the course of

the research work. In particular, the author wishes to thank

Messrs. Tom McRae, Jerry Ward, Ron Brunsvold, Donald Dietrich,

John Fisher and George Perdue for their assistance in prepar-

ing and conducting the experiment.

Special thanks are due to his supervisory committee chair-

man, Dr. B. M. Leadon, who has served with great understanding

as a teacher, mentor, and critic. The author wishes to ex-

press his appreciation to members of his supervisory committee,

Dr. M. H. Clarkson, Dr. D. T. Williams, Dr. R. K. Irey, and

Dr. R. G. Blake.

The author also wishes to thank Mrs. Nancy Bryan, who

graciously and ably prepared this manuscript.

Finally the author wishes to thank the Martin Marietta

Corporation and, in particular, Mr. R. Dewey Rinehart, and

Mr. John Calathes who sponsored the Doctoral Research Award

which provided the financial support for completion of this

research.


iii














TABLE OF CONTENTS


Page
c


ACKNOWLEDGEMENTS

LIST OF TABLES .

LIST OF FIGURES .


NOMENCLATURE . . . . .

ABSTRACT . . . . . .

CHAPTERS

I. INTRODUCTION . . . .

II. EXPERIMENTAL INVESTIGATION

III. THEORY . . . . .

IV. DISCUSSION OF RESULTS .

V. CONCLUSIONS . . . .


APPENDICES

I. TIMING CIRCUIT SCHEMATICS. . .

II. BOUNDARY LAYER STRIPPING ANALYSIS.

III. COMPUTER PROGRAM FLOW CHART. . .

REFERENCES . . . . . . . .

BIOGRAPHICAL SKETCH . . . . . .


. . . . . . . iii


v


. . . . ix

. . . . xii





5
. . . . 1

. . . . 5

. . . . 13

. . . . 23

. . . . 2 7


. . 79

. . 81













LIST OF TABLES


I. Time Sequence for Photographs Shown in Figure 5


Table


Page













LIST OF FIGURES


Figure Page

1. Experimental Arrangement . . . . . . ... 29

2. Water Drop Injectors . . . . . . . . 30

3. Test Section . . . . . . . . . 31

4. Back Lighting Arrangement . . . .. . . . 32

5. Typical Shadow Photograph Sequences: a. D =
2.06 mm, M = 1.84; b. D = 2.74 mm, M = 2.01;
c. D = .985 mm, M = 1.90 . . . . . . .33

6. Experimental Drop Deformation Data. . . . .. 35

7. Experimental Drop Stagnation Point Displacement
Data. . . . . . . . . . . . .. 36

8. Coordinate Systems for Analysis of Experimental
Data. . . . . . . . . . . . .. 37

9. Measured Eccentricity of the Frontal Surface of
the Drop and Least Mean Square Curve Fit to the
Data. . . . . . . . . . . . .. 38

10. Measured Deformation of the Drop and Least Mean
Square Curve Fit to the Data. . . . . . . 39

11. Measured Displacement of the Stagnation Point of
the Drop and Least Mean Square Curve Fit to the
Data. . . . . . . . . . . . .. 40

12. Measured Semi-Minor Diameters of the Frontal
Surface of the Drop and of the Drop .. . . 41

13. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = .4 . . . . . . . . 42

14. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = .6 . . . . . . . . 43

15. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = .8 . . . . . . . . 44






Figure Page

16. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 1.0 . . . . . . 45

17. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 1.2 . .. . . . . 46

lS. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 1.4 . . . . . . 47

19. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 1.6 . . . . . . 48

20. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 1.8 . . . . . . 49

21. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 2.0 . . . . . . 50

22. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 2.2 . . . . . . 51

23. Measured Cross Sectional Shape of the Frontal
Surface and Calculated Cross Sectional Shape
of the Drop at T = 2.4 . . . . . . 52

24. Summary of Figures 13-23 . . . . ... 53

25. Measured Semi-Major Diameter of the Drop . . 54

26. Experimentally Determined Eccentricity of the
Drop . . . . . .. . . . . 55

27. Experimentally Determined Velocity of the
Stagnation Point of the Drop . . . . . 56

28. Experimentally Determined Normalized Mass of
the Drop . . . . . . . . . . 57

29. Coordinate Systems for Analysis of Shock Wave
Passage Over the Drop . . . . ... 5S

30. Coordinate Systems for Analysis of Drop
Deformation and Fragmentation . ... . .. 59

31. Theoretical Semi-Minor Diameter of the Drop. . 60

32. Theoretical Major Diameter of the Frontal
Surface of the Drop. . . . .. . . . 61


vii







Figure Page
33. Theoretical Normalized Mass of the Drop. .. . .62

34. Theoretical Displacement of the Stagnation
Point of the Drop . . . . . . . . 63

35. Theoretical Velocity of the Stagnation Point
of the Drop . . . . . . . . . . 64

36. Theoretical Eccentricity of the Drop . . . ... .65

37. Timing Circuit Schematic . . . . . . . 67

38. Delay Circuit Schematic . . . . . . . 68

39. Computer Program Flow Chart. . . . . . .. .78


vil 1












NOMENCLATURE


a = Dimensionless semi-minor axis of ellipsoid(a/D) ,
speed of sound (C-/`p-,)
a = Semi-minor axis of ellipsoid, speed of sound
B = Dimensionless diameter of equator (Zb)
b = Dimensionless semi-major axis of ellipsoid(b/D)
b = Semi-major axis of ellipsoid
C = Pressure coefficient
p
D = Initial diameter of drop
F = Dimensionless force (F/P Z)C D)
F = Force
f = Dimensionless function
M = Relative Mach number of drop CQ,- /)/L_

M = Shock wave Mach number

m = Dimensionless mass of drop (r/pD3)
m = Mass of drop

p = Dimensionless static pressure (p/pL 23 2
S= Static pressure

q = Dimensionless dynamic pressure (C /1 1D 8
q = Dynamic pressure
Rh = Ratio of viscosities
R =Dimensionless radius ( R/D)
R = Radius
T = Dimensionless time (t P (/ D)
T = Dimensionless time for shock wave to pass
over drop

T = Dimensionless time for boundary layer separation
sep
t = Time after impact






u,v,w = Dimensionless velocities
U = Dimensionless velocity of fluid (LU/UP)
U = Velocity of fluid
W = Dimensionless velocity of drop

x,y,z = Dimensionless coordinates
X = Dimensionless displacement of drop (X/D)
X = Displacement of drop
SK= Dimensionless acceleration, Boundary layer
shape factor
3 = Density ratio (pz/P )
S= Dimensionless boundary layer thickness
= Eccentricity (I-cLZ/b2)I/z

= Dimensionless boundary layer coordinates
8 = Eccentric angle of ellipse
=Dimensionless cinematic viscosity of fluid( "D U)
S= Kinematic viscosity of fluid
= Viscosity of fluid
= Dimensionless coordinate
P = Density of fluid
= Dimensionless surface tension (c/P DP )
CY = Surface tension

zL = Dimensionless time for one oscillation of drop,
dimensionless shear stress
= Angle, velocity potential
SUBSCRIPTS
1 = Condition upstream of shock wave
2 = Condition downstream of shock wave
d = Drop
i = Initial
I = Interface
i,j = Computation indices
L = Stagnation Point






= Liquid, lost, local

m = Condition at maximum

o = Stagnation condition

p = Point at which pressure is acting
oo = Condition in free stream






Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


WATER DROP DEFORMATION AND FRAGMENTATION
DUE TO SHOCK WAVE IMPACT

By

William Edward Krauss

June, 1970


Chairman: B. M. Leadon
Major Department: Aerospace Engineering

Results are presented of experimental and theoretical

studies of water drop deformation and fragmentation in the

high velocity air stream following a normal shock wave moving

into still air. Experiments were conducted in a shock tube

with shock wave Mach numbers in the range from 1.6 to 3.0 and

for drop diameters from .5 to 3.2 millimeters. From the ex-

perimental data, it is determined that the water drop deforms

axisymmetrically into an approximately ellipsoidal shape which

is distorted as to maintain a nearly hemispherical frontal

surface but with a concave rear surface. The data also indi-

cate that the time for complete fragmentation of the water

drop is approximately one-half of the time observed for com-

plete reduction of the drop to a trace of mist.

Simultaneous potential flow within the distorted ellip-

soidal drop and mass loss by stripping of a viscous layer of

surface liquid are included in a theoretical model for drop

deformation and fragmentation. Although the actual drop

deformation is shown to deviate from the assumed ellipsoidal

shape, the general dimensional and displacement variations

are predicted by the model and the theoretical mass history

of the drop agrees well with the experimental data.
xii













CHAPTER I

INTRODUCTION


The deformation and fragmentation of water drops caused

by sudden exposure to a high velocity air stream currently

has important application in the field of missile and air-
craft structural and heat shield design. Rain erosion by

the impingement of rain drops on the exterior surfaces of

missiles causes severe damage to ablative heat shields and
may result in structural failure of the missile. The effects
of erosion damage can perhaps be minimized by judicious

selection of the heat shield material to withstand drop im-
pact or by suitable aerodynamic design of the missile. In

this latter approach, the missile nose cap may be designed
to generate a shock wave with a standoff distance suffi-

ciently large to permit drop fragmentation before impinge-

ment upon the heat shield. Either of these approaches for

minimizing rain erosion damage requires knowledge of rain

drop size and shape as function of time.

Fragmentation of liquid drops in high velocity gas

streams also occurs in fuel injection systems. The rates

of mixing and of combustion in such systems are significantly
enhanced by drop fragmentation.

A comprehensive literature survey was conducted on

liquid drop deformation and fragmentation. Two modes of
drop fragmentation which occur are related to Weber's num-
ber. The first of these modes occurs when the velocity of

the gas stream relative to the drop is small and corre-

sponds to a magnitude of Weber's number of from 1 to 10 or






20 times its critical value. In this mode of fragmentation,
the drop is deformed into a "bag" shape prior to fragmenta-

tion. The second mode of fragmentation occurs when the

velocity of the gas stream relative to the drop is high and

corresponds to the case of Weber's number being very much

greater than the critical value for drop fragmentation. In

this mode the drop is deformed axisymmetrically and is

fragmented by the stripping of a liquid layer from its sur-

face.

The second mode of fragmentation is of interest for the
present study; therefore, only those papers in the literature

which are relevant to the second mode of fragmentation are

discussed.

In general, the approach taken by all investigators has
been similar in that each used photographic techniques to

measure drop deformation and fragmentation in the gas stream
behind a normal shock wave in a shock tube. Quantities
measured were drop diameter, drop displacement and shock

wave Mach number. Appropriate values of liquid density,

viscosity and surface tension and of gas density and tempera-

ture were noted.

Engel conducted comprehensive studies wherein the
parameters varied were water drop size and shock wave Mach

number. These were the first studies to show that drop

fragmentation is not caused by the initial impact of the

shock wave on the drop. In addition, these were the first
studies to show deformation of the drop into a lenticular

shape and stripping of a liquid surface layer from the drop.

Engel considers several phenomena which might explain the

stripping action but reaches no clear conclusion as to the

cause of stripping. In addition, an approach which defines

drop deformation based on potential flow within the drop

is suggested.






2
Studies by Hanson, et al. were conducted on very small

drops (0.5 mm diameter) of various liquids. Shock wave Mach

number was again varied but, as in Engel's1 studies, did not

exceed approximately 1.7. These studies verified Engel's

observations on larger drops.

Nicholson and Figler3 conducted the first studies with

shock wave Mach numbers greater than 2 thereby producing a

supersonic air velocity behind the shock wave. This was

accomplished by using sub-atmospheric pressures in the

driven section of the shock tube. The stripping phenomenon

was again observed. Drop fragmentation was shown to depend

largely on the dynamic pressure which was acting on the drop.

An empirical correlation for the time required to achieve

complete drop fragmentation is presented using the data de-

veloped for this study as well as those available from Engel

and Hanson, et al.

The studies of Ranger and Nicholls4 were conducted on

water drops having diameters of .75 to 4.0 mm. Shock wave

Mach numbers from 1.5 to 3.5 in air were used to extend the

work of previous investigators. Dynamic pressures achieved

were significantly greater than those of Nicholson because

the pressure in the driven section of the shock tube for

these studies was one atmosphere. The drop was observed to

assume an ellipsoidal shape during deformation, and mass

removal was attributed to boundary layer stripping. Using

empirical relations for drop diameter and drop relative

velocity, the validity of Taylor's5 boundary layer stripping

analysis was investigated. Based on the assumption of in-

compressible flow using approximate relations for the ex-

perimental data, the authors concluded that the agreement

between Taylor's analysis and their experimental data was

encouraging.







In the following chapters are reported the results of

similar measurements which were being started as the Ranger
4
and Nicholls paper appeared in the literature. This is

followed by an analytical treatment of simultaneous deforma-

tion and fragmentation of the drop and by a comparison of

the experimental results with the analysis.












CHAPTER II

EXPERIMENTAL INVESTIGATION

Experimental Apparatus


The experimental arrangement used for this investiga-

tion is shown in Figure 1. A stream of water drops was in-
jected by means of hypodermic needles into a shock tube test

section which contained photographic windows. Figures 2 and

3 show, respectively, the water drop injectors and the test
section. The drops fell vertically into the test section

which was open to the atmosphere. Initial pressure in the
test section was one atmosphere. The basic shock tube as

described by Clemons was modified by increasing the driver

length to 36 inches. This was necessary to ensure that there

was sufficient time for complete drop fragmentation prior to

the arrival of the expansion wave.

Both helium and air were used as driver gases. A model

600 Honeywell Strobnar photographic flash unit was used to

back light the drops as is shown in Figure 4. A model ID

TRW image converter camera was employed to photograph the
drops after shock wave impact. The image converter camera

was electronically triggered by a platinum film velocity
8
gauge located a known distance from a fiducial marker
scribed on the test section window. In addition, one window
contained a millimeter grid which was used during data re-
duction to establish initial drop diameter. The experi-

mental procedure consisted of obtaining a history of the
deformation and displacement of an individual drop by taking

a series of shadow photographs at various time intervals

5







after the shock wave impinged upon the drop. The timing

sequence was set by a combination of TRW trigger delay gen-
9 10
erators, Techtronix oscilloscopes,0 and delay circuits as

shown in Appendix I.

Data Reduction

The individual photographs were first examined with a

Gaertner optical micrometer to establish a history of the

displacement and major diameter variation of the drop. The

time at which the first picture was taken was established by

calculating the time required for the shock wave to travel

from the velocity gauge to the drop whose initial position

relative to the fiducial marker was recorded on the first

photograph. The time interval between subsequent photographs

was established by the intervals manually set into the timing

circuit. Following initial examination of the photographs

with the optical micrometer, the photographs were enlarged

with a Beseler Vu-Lyte III opaque projector and measurements

were made to determine the shape of the windward surface of

the drop.

Data Analysis and Results

The results discussed here are for experiments that

cover the shock wave Mach number range of MS = 1.6 to 3.0

in air with drops having diameters in the range D = 0.5 to

3.2 millimeters. Deformation and displacement data obtained

from photographs were reduced in terms of dimensionless dis-

placement, XL, dimensionless semi-major diameter, b, dimen-

sionless semi-minor diameter, a, and dimensionless time, T.

Typical image converter camera photographs of shock wave-

water drop interactions are shown in Figure 5. The dimen-

sionless and corresponding real times at which the exposures

were made are shown in Table I. Figure 5a, exposure 5,

Figure 5b, exposure 3 and Figure 5c, exposure 2 show the drops







at approximately their maximum deformation which, from Table I,

is seen to occur in the time interval T = 1.4 to 1.8. Figure

5b exposure 5 and Figure 5c exposure 4 show similar stages of

drop fragmentation and indicate that while time T is similar,

actually, in real time, the smaller drop shatters much more

rapidly than the larger drop.

Two stages of development can be observed in the deforma-

tion and fragmentation process. During the first stage,

deformation by flattening and curving is the primary phenome-

non observed with little or no fragmentation in the form of

mist visible. Deformation into a curved axisymmetrical

ellipsoid is the result of the external pressure distribution

on the drop. The second stage is characterized primarily by a

surface stripping action which is due to the shearing action

of the external flow field. This shearing action rapidly re-

duces the drop to a cloud of finely divided particles which

will be termed "micromist." The second stage is well estab-

lished by T = .20.

From Figure 5 it is seen that a well defined wake is

formed behind the drop during early stages of fragmentation.

The shape of this wake is similar to that developed behind a

hypersonic blunt body where the flow converges because of

strong lateral pressure gradients to form a recompression neck

region several body diameters downstream from the rear stagna-

tion point. This phenomenon was also observed by Ranger and
4
Nicholls. Since the liquid surface layer which is continu-

ously being stripped from the drop is able to follow the

streamline pattern of the wake, it is reasonable to conclude

that the stripped liquid is composed of very fine water parti-

cles, a micromist. If large pieces of liquid were being

stripped from the drop, the inertia of these pieces would pre-
clude their following the stream lines and a significantly







different wake shape would be observed in the photographs.

Figure 5c and other similar photographs which show considerable

portions of the wake indicate that fragmentation is a continu-

ous process which ultimately reduces the entire drop to a

micromist cloud.

Taking the time for complete drop fragmentation to be

that to leave only a trace of mist, as was done by Engell and

other investigators, curves of the deformation of the drop

and the displacement of the drop were obtained and are shown

in Figures 6 and 7, respectively, as functions of dimension-

less time, T. These figures also include a band which indi-

cates the range of experimental data available in the literature.

It is seen that the data of this study agree well with those of

other investigators. Figure 6 indicates that complete frag-

mentation of the drop occurs at approximately T = 5.0 as was

also found by others.

Examination of the enlargements of the photographs ob-

tained with the opaque projector indicated that significant

irregularities occur in the windward surface of the drop at

dimensionless times between approximately T = 2.0 and T =

2.5 and suggests that a breakup criterion other than reduction

of the drop to a trace of mist might be identified. Measure-

ments made on the windward surface of the drop in the various

photographs using the parameters as shown in Figure S indicate

this surface is approximately that of an axisymmetrical

ellipsoid,the eccentricity of which varies with time as shown

in Figure 9. These data were fitted with a mean fifth degree

curve of least mean square deviation expressed by

= 3.432T-6.515T + 4.656T3
4 5
-1.477T + .1753T (II-1)

Figure 9 shows the eccentricity varies from 0 to a maxi-

mum of .59 and returns to 0. The frontal surface changes from







that of a sphere to that of an axisymmetric ellipsoid and re-
turns to the spherical surface shape. The measured maximum
diameter of the drop and the measured displacement of the
stagnation point of the drop were fitted with least mean square
curves given by
B = 1 + 2.615T-.6776T2-.0971T3 (11-2)
and
X .409T2 + .317 T3 (11-3)
L
and are shown in Figures 10 and 11, respectively.
The fitted curve for the displacement, Equation (II-3),
also satisfies the known conditions at T = 0 that the dimen-
sionless velocity of the drop is zero and the dimensionless
acceleration is .818. The dimensionless acceleration is, of
course, readily calculated for a sphere with a known pressure
distribution. Details of this calculation as well as a method
for determining the pressure distribution on axisymmetrical
ellipsoidal bodies are presented in Chapter III.
The dimensionless mass, m, of the drop and the thickness,
2ad, of the drop were estimated by dividing the drop into
annular elements of fixed thickness y and fixed radius y
and applying Newton's Second Law to each element. The dimen-
sionless force acting on an element is given by
F = 27y a y 6p (11-4)
where p is the pressure acting on the element at radius y .
The dimensionless mass of the cylindrical element is given by
m = 47y !yxd (11-5)
At time T the dimensionless location of the element with re-
spect to the fiducial marker is given by
X = X + tZ (11-6)
p L >p
which in terms of the eccentricity and equation for an
ellipse can be written as

X = XL + b (l- (Ll-l-(yp/b) (11-7)







Since X, < and b are known functions of time, T, as given by
L'
equations (II-l), (11-2), and (II-3), and since y is known
for any element, the acceleration of the element at any time,
T, can be found by differentiating equation (11-7) twice with
respect to T. Making the substitutions noted and performing
the differentiation, the dimensionless acceleration, X of
the element is given by
X = X + (bE + 2bE + bE)(l-?)
+2bE( -)/ -b2 E( 2-)/b (II-8)
bE 2_1)/ 2 2 )2/
( 1))/ + b( )
2 1
where E = (1- )2

S=/l-(y /b)9 12
Applying Newton's Second Law and using equations (11-4), (11-5),
and (11-7), the dimensionless thickness of the element is
given by
ad = .p/2 X (11-9)
d p
The dimensionless total mass of the drop is readily found by
summing the masses of the elements as found by equation (II-5).
Figure 12 presents the dimensionless semi-minor axis of
the drop, ad, and the measured semi-minor axis, a, of the sur-
face visible in the photographs which is hereafter called the
frontal surface. The water drop itself is deformed into a
curved ellipsoid which has the observed frontal surface but
a rearward surface which is increasingly concave. The axis
of the axial section of this curved ellipsoid is calculated
from the data. The semi-minor axis ad is calculated as indi-
cated above, and the semi-major axis bd is measured along the
locus of centroids of the cylindrical segments whose lengths
have been calculated and plotted in Figures 13-24.
Figure 25 presents the semi-major axis of the drop as a
function of dimensionless time, T, and Figure 26 presents the
eccentricity of the drop as a function of T also. Comparison
of Figures 10 and 25 shows the semi-major axis of both the






frontal surface and the drop reach maximum values at approxi-

mately T = 1.5. Figure 24 shows the cross section of the for-

ward portion of the frontal surface and the deformation of the

water drop as determined by the solution of equations (11-4)

through (II-9) for various values of T. This figure shows

the water drop deforms into an axisymmetric body which is

crescent shaped, or curved elliptical, in cross section. The

semi-minor axis of the curved ellipse decreases with time as

mass is lost from the drop.

Figure 27 presents the velocity of the stagnation point

of the drop. The velocity of the stagnation point is given

by

WL = .81T + .951T2 (II-10)

which is directly obtained by differentiation of equation

(11-3).

Figure 28 presents the dimensionless mass of the drop

normalized to its initial mass and shows that the mass of the

drop is reduced to approximately 2 percent of its initial mass

by time T = 2.5 and indicates that for all practical engineer-

ing purposes drop breakup is complete at approximately this

time. This result indicates the drop is fragmented in approxi-

mately one half the time for breakup as found by using the

criterion that the drop is reduced to only a trace of mist as

observed on photographs.

Experimental Error

During the experiment, shock tube driver pressures were

measured within -3.5 percent with a Heise model C-55950 pres-

sure gauge, temperatures within -1.5 percent and velocity of
+ 8
the shock wave within -3.0 percent. Time intervals between

photographic exposures were established by use of the time

delay circuits in the Tektronix oscilloscopes which have an






accuracy of -2 percent of full scale and the time delay cir-

cuits in the TRW trigger pulse generators which have an
9
accuracy of 0.01 percent of full scale.

Repeated measurements made with the optical micrometer

indicated that distances between two points, e.g., between

the fiducial marker and the forward stagnation point of the

drop, could be measured within -3.5 percent. Comparison of
identical measurements made on the photographs using the

optical micrometer with those made on the enlargements of the

photographs indicate that the latter measurements were accu-

rate to within 17 percent.

The standard deviations associated with each set of ex-

perimental data which were fitted with least mean square

curves are shown in Figure 9, 10 and 11 and are .041, .095,

and .112 for the eccentricity, the semi-major diameter of the

front surface, b, and the displacement of the stagnation point,

XL, respectively. Since the standard deviation is the most

probable expected error, it is indicative of the error due to

curve-fitting the data. The standard deviation of the dis-

placement curve, Figure 11, and the associated equation,

(II-3), are of particular interest since equation (II-3) is

differentiated to find the velocity and acceleration of the

stagnation point of the drop. The maximum standard deviations

for the velocity and the acceleration are calculated to be

.124 and .146 respectively. Therefore, based on the average

acceleration of the drop over the period T = 0 to T = 2.5,

the calculated mass of the drop is accurate to within
-10 percent.













CHAPTER III

THEORY
General
For convenience the analysis is divided into two parts.
In the first part, the passage of the shock wave over the

drop, the velocity and displacement of the water drop are

mathematically characterized assuming the flow remains at-

tached to the drop as the shock wave passes over it. No

fluid motion within the drop is assumed to occur. This lat-

ter assumption appears to be justified, based on the photo-

graphic data of Engel. In addition, for the conditions of
this study the natural period, 2 for steady oscillation of

the drop under surface tension as given by Rayleigh,1

7 = 27 26-n(n-n)(n+2) 2

is at least three orders of magnitude greater than the time
-1
TB = (2U)1
for the shock wave to pass over the drop.

The second part of the analysis considers drop deforma-

tion and fragmentation and begins when the shock wave reaches
the rearmost point on the surface of the spherical drop. The

flow field around the drop and the boundary layer are assumed

to be fully established. Considering the time required for
the boundary layer to become established as approximately
that required for boundary separation to take place, it is
12
found from Schlichting that

T sep .1963e
sep \
The maximum dimensionless separation time for the conditions
-2
of this study is T = 1.34 x 10 This is also the time
sep







required for the shock wave to move approximately a quarter

of a body diameter; thus, the assumption of a fully estab-

lished boundary layer appears to be reasonable.

Potential flow within the drop is assumed, and deforma-

tion of the drop is assumed to occur in a manner similar to

that suggested by Engel. The analysis here, however, is

assumed not to be limited to small changes in the drop radius.

Stripping of a liquid layer from the surface of the drop as

suggested by Taylor5 is also assumed. These two flow mech-

anisms are combined in a model which undergoes simultaneous

drop deformation and mass loss. Axial symmetry is assumed

throughout the analysis.

The analyses are developed in terms of non-dimensional

variables and reduced to numerical solution by the standard

forward marching technique used on high speed electronic

computers. The necessity for computer solution is discussed

as the analysis is developed.

Passage of Shock Wave Over the Drop

The coordinate systems used to define the position of

the shock wave, the drop velocity, and the drop displacement

are shown in Figure 29. The force acting on the drop at

time T is:

4)Cr)


o
where q2' C K2 and are functions of T. K2 is a

parameter which governs the difference between the pressure
ahead of and behind the shock wave as it moves over the drop

and is given by

K2 = 1.166 (1-MS2) p1/q2CP-






Since the dimensionless mass of the drop is
m -
6
the drop acceleration can be found from Newton's Second Law
provided the parameters under the integral sign in equation
(III-1) are known functions of time. Since this is not the
case, a forward marching numerical solution is employed wherein
the increment O is taken sufficiently small such that
q2, Cpm K2 and ;9 can be assumed with small error to be con-
stant within the interval 0 to +Z_ A substantial cumu-
lative error may be incurred with this process, of course, but
this is not investigated here because the objective is only to
estimate the drop deformation and displacement which occur
during the shock passage time. The drop acceleration is then
found to be:




C ETCLCJ-VJ fC-K,),5)K -Z8SI ,14 32 (111-2)



The drop velocity and displacement are found in the
usual manner by integration with the initial conditions
CC = W = XL = 0 at T = 0. Using A to set the com-

putation interval, the ,corresponding time interval is:


XL -.Scosad>
X-L =X (111-3)


and the velocity and displacement of the drop center are

given, respectively, by:


+2 ( .J. I






and


X C W AT (111-5)


The maximum pressure coefficient is given by



Cpm \. IM 1-- --A -l3l (III-6)


where for subsonic flow C is unity. C is approximated
pm pm
by a linear increase from C = 1.0 to C = 1.8 for
pm pm
.7 MR < 1.3 and by Cpm = 1.8 for MAR > 1.3. The value
of C = 1.8 is typical of that found experimentally by
pm
numerous investigators.

Drop Deformation and Fragmentation
The coordinate systems used in this part of the analy-
sis are shown on Figures 8 and 30. The drop is assumed to be
an axisymmetric ellipsoid which has been distorted so that the
front surface is spherical in shape. The (X,Y) coordinate
system is fixed in space while the z, y-coordinate system
is body-fixed and 5 ,? are the coordinates fixed in the
boundary layer. Assuming that potential flow exists inter-
nally in the drop, let the components of velocity of any
liquid particle in the body-fixed coordinate system be
u = -2f (III-7a)
v = fy (III-7b)
w = fz (III-7c)
where f is an undetermined function of time and is zero at
T = 0. Based on the assumption of potential flow, there
exists a velocity potential


S = C- 4Z) (III-8)






such that

-2z S (III-9a)


S C (III-9b)

S 2(III-Yc)

Equations (111-7) and (111-9) clearly satisfy the continuity
equation assuming no mass'loss from the drop. Taking the term
2 2 2
S(u + v w ) to be small and integrating the equation
of motion the pressure is found to be given by


p = +C(T) (III-10)
P -T

Applying equation (III-10) at the stagnation point, a = -ad'
and at the equator, y = b of the drop, the following re-
lationship is obtained:

S P (III-11)
C +.5 b+

where / p is the pressure difference between the points
-0 = -7 and cZ = '/2 on the spherical surface of the drop.
All parameters in equation (III-11) are unknown functions of
time; hence, appeal is again made to a numerical solution
where the computation interval is sufficiently small to assure
that the equation







is valid with negligible error. The function f may therefore
be constructed as


~j 1(_ (111I-12)
-1 3l 0 -'- 1?




18

Since +2fad is the velocity of the point (-a,o,o) on the sur-
face of the axisymmetric ellipsoid,


(III-13)


S-
nZ b- b;


a new value for the semi-minor axis, ad, at the end of the com-
putational interval is readily determined from:


(CL-)L COC8L- C1 Z, .)


(III-14)


The pressure difference, \ p required for the solution
of equation (III-12) is given by:


(III-15)


where q2, Cpm and K3 are functions of time. K3 is defined
and discussed later in this chapter.
From Figure 30, the semi-major diameter, bd, of the drop
is the arc length along the center line of the drop and is
given by


. -z 45 E )
S1S7b.4 ,l 2 -a 2304


(111-16)


where

E- l b,- Co: .


'. =- CbzcPm CK )





The mass of the drop at time, T, is given by


L-= M -t rMl
rrnm* ~AY'


(III-17)


where Am. is the mass loss from the drop as determined by a
5
modified version of Taylor's boundary layer stripping analy-
sis which is developed in Appendix II. The mass loss, L m.,
is given by




/ b-z ,-Cz ('_ PW, _I Iz"Z
z.m; -Q.z7vb,:) K4( isv-.,v/J


(III-18)


The radius of the frontal surface of the drop is given by


bc = C.75 r/Tactkj


(III-19)


Equations (III-14), (III-16), (III-17) and (III-19) de-
fine the mass and approximate shape of the drop. Since the
shape of the drop is assumed to be fixed during the calcula-
tion interval, the acceleration of the center of gravity and
the stagnation point of the drop is the same and is given by
equation (AII-20). Hence, the velocity of the stagnation
point, WL, is readily found by equation (III-4) and the dis-
placement, XL, by equation (III-5).


-.O0 6 5 7T1b,, Ck( fLi-, ) ATL







Alternatively, as in the analysis of the experimental
data, the velocity and displacement of the stagnation point
can be found by considering a cylindrical element of mass
which contains the stagnation point.
The force acting on the cylindrical element is given by
(see Figure S)


F= 27/n 2p


(111-20)


The mass of the element is


m --4~T h C


(III-21)


By Newton's Second Law the acceleration of the element is


.= 5 WCI)


(III-22)


The velocity of the stagnation point, WL, is now readily
found by equation (III-4) and the displacement, XL, by equation
(III-5).
The pressure difference, A p, in equation (III-22) is the
difference of the pressures acting on the upstream and down-
stream sides of the drop. The pressure on the upstream side
of an axisymmetric ellipsoidal drop is given by


t = P0-t-%zC n-0


k~I3 $l M\N4'
92~ )(\L ~C-~


(III-23)


where it has been assumed that the local pressure on the
ellipsoid is the same as that found on a sphere by applying
a tangent cone of constant vertex angle to both the ellipsoid'





and the sphere. The local pressure on the downstream side of
the drop is given by



P pm P tCp m 3 (III-24)


K1 and K3 are parameters which occur if the relationships for
pressure distribution on the upstream and downstream sides of
a sphere considering both subsonic and supersonic flow are
written, respectively, in the general forms


Cp CPC\- K,) ) (III-24a)
and


Cp =Cpm K (III-24b)


13
K1 for supersonic flow is unity, and for subsonic incom-
1 1
pressible flow K1 = 2. On the rearward side of the drop the
pressure is taken to be constant and independent of 0 .
For supersonic flow K3 = 0, and for subsonic incompressible
flow K3 = -.5. Transonic flow is assumed to occur between
M = 0.7 and M = 1.3. In this regime a linear transition
R R
from the subsonic incompressible to the supersonic values of

K1 and K3 is assumed; K1 and K3 are thus given by








2.4



where MR is the Mach number of the air stream relative to the
drop.




22

The equations for deformation and fragmentation of the

drop are solved by a forward marching numerical technique em-

ploying a time interval sufficiently small that negligible

error is introduced in any given time interval by assuming

time dependent variables are constant during that time inter-

val.

Theoretical results and experimental data are compared

in the next chapter.












CHAPTER IV
DISCUSSION OF RESULTS

The numerical solutions for displacement of the drop in-

dicate that for the conditions of this study the maximum time

for the shock wave to pass over the drop is 1.375 Aseconds

or a dimensionless time of .0505. The displacement of the
drop during this time period is .009 body diameters, which is

not measurable on photographs. The major axis is calculated

to have changed less than .0003 percent. Thus the assumption

made herein that the drop remains spherical during this period

is justified, and the observations of Engell and of Ranger and

Nicholls that the drop exhibits no measurable displacement or

distortion during shock passage are confirmed.

For the analysis of the deformation and fragmentation of

the water drop a relatively simple analytical model was

sought which would approximate the observed displacement of
the stagnation point of the drop, show a variation of the semi-
major diameter of the frontal surface of the drop similar to

that observed in the experimental results and show a change

in mass of the drop approximately the same as that found from
the experimental data. In addition, the analytical model

should reasonably determine the velocity of the stagnation
point of the drop and the semi-minor and semi-major axes or
the eccentricity of the drop.
The theoretical model employed in the analysis is based

on the assumptions of potential flow within the drop and mass

flow from the drop by the stripping of a viscous surface layer

of liquid at the equator of the drop which is taken to co-

incide with a point 90 degrees from the stagnation point

23






on the spherical frontal surface of the drop.

Calculated results of the theory are shown for the cases

of shock wave Mach numbers, M of 1.6 and 3.0 which cover

the range of the experimental data of this study.

The assumption of potential flow within the drop elimi-

nates viscous flow considerations from the distortion and

accounts for mass continuity during distortion of the drop

analysis. While this assumption appears to be desirable in

this first attempt to model simultaneously drop deformation

and fragmentation, it is seen by comparison of analytical and

experimental results for the semi-minor diameter, ad, shown in

Figures 31 and 12, respectively, that the model semi-minor

diameter does not decrease initially as rapidly as observed

in experiment. This indicates that the assumed velocity po-

tential for the liquid flow within the drop is inexact. As

will be seen, however, this simple potential is quite adequate

for the principal purpose of the theory, that is, the pre-

diction of the mass history of the drop.

Since the major diameter, B, of the frontal surface of

the drop is directly related to the semi-minor diameter as is

shown by equation (III-19) for mass of the drop, it is ex-

pected that the analytically determined major diameter, B,

will exhibit a less rapid initial increase than that observed

for B in the experimental data. Comparison of Figures 10 and-

32 show this to be the case. This comparison also indicates

that the maximum analytically determined values of the major

diameter, B, are within 10 percent of the maximum experi-

mentally measured values. Further, the dimensionless time

at which the calculated major diameter, B, reaches a maximum

is within 15 percent of the experimentally observed time at

which B reaches a maximum. The differences between experi-

mentally observed and theoretical values of ad and B during
d _







latter phases of fragmentation of the drop are attributed to
assumptions made in the boundary layer stripping analysis and
are discussed later in this chapter.

Figure 33 presents the calculated mass of the drop nor-
malized to the initial mass of the drop. Comparison of this
figure with Figure 2S for the experimentally determined nor-

malized mass of the drop indicates good agreement between the

observed and calculated values. From these figures it is also

seen that the calculated mass of the drop is somewhat less

than the experimentally observed value when the mass of the

drop is reduced to approximately 20 percent of its original

mass. From Figure 31 it is seen that at approximately T =

1.7 the drop is becoming very thin and clearly the viscous

boundary layer is no longer small compared to the thickness

of the drop. Hence the theoretical model which is based on
the assumption of a thin boundary layer is inaccurate. This

deterioration of the model for mass loss is also reflected
through equation (III-19) in the behavior of the major diam-
eter of the frontal surface as shown in Figure 32 for

T > 1.8 to 2.0. Similarly, the semi-minor diameter, ad, is

also affected through equations (111-16) and (II-12).

Figure 34 presents the theoretical displacement of the
stagnation point of the drop. Comparison of Figure 34 and 11

indicates that the theoretical model predicts smaller dis-

placements during the breakup process than ai observed ex-
perimentally. At T = 2.5, however, the experimentally

observed displacement falls between the calculated values

for the range of shock wave Mach number of this study. The

difference between the experimentally observed and the cal-

culated displacement is attributed primarily to the assump-

tion made in the analysis that the maximum pressure coeffi-

cient varies linearly with the relative Mach number,MR, in








the transonic regime and is constant in the subsonic and super-
sonic regimes as given by equation (III-6).

This or a similar assumption as to the variation of the

local pressure coefficient, C on a sphere was necessary

since no experimental data on pressure distributions around

spheres or hemispheres in the transonic regime appear to be

available.

Figure 35 presents the theoretical velocity of the stag-

nation point of the drop. Since the velocity is the time rate

of change of the displacement, it is to be expected from the

slope of the displacement curves in Figure 34 that the pre-

dicted velocity will be larger than that experimentally ob-

served. Comparison of Figures 27 and 35 show this to be the

case with the calculated velocity being approximately 25 per-

cent greater than the experimentally determined velocity at

T = 2.5. This difference is also attributed to the assumed

linear variation of the maximum pressure coefficient as dis-

cussed in the preceding paragraph.

Figure 36 presents the calculated time variation of the

eccentricity of the drop. Comparison of this figure with

Figure 14 indicates similar timewise variation of the eccen-

tricity. While both the theoretically predicted and experi-

mentally determined eccentricities approach unity between

T = .6 and T = .7, the experimentally determined value in-

creases more rapidly than does the analytically predicted

value. This is due to the relatively slower decrease in the

analytically predicted variation in the semi-minor diameter,

ad, of the drop than is observed in the experimental results

as was discussed earlier in this chapter.













CHAPTER V

CONCLUSIONS

This investigation of deformation and fragmentation of

a water drop due to shock wave impact has produced several

new findings from both the experimental and theoretical re-

sults.

The primary experimental data which are measurements of

the displacement and the major diameter of the deformed water

drop are in good agreement with those of other investigators.

Analysis of the experimental data shows that the water

drop is deformed into an axisymmetric shape which has a

nearly spherical ellipsoidal frontal, or windward, surface

and a concaved rear surface.

Determination of the mass of the drops as a function of
time from the experimental data permits a new criterion to

be used to establish the time for complete fragmentation or

breakup of the drop. Using the time at which the mass of

the drop is reduced to approximately 1 percent of its origi-

nal mass, it is found that drop breakup occurs at approxi-

mately T = 2.5. This is approximately one-half of the time

identified by other investigators who used the criterion that

breakup time is that at which the water drop is observed to

be reduced to a trace of mist on photographs of water drop

fragmentation. Using this criterion the time for complete

breakup of the water is typically given as T = 5.0.

The approximate analytical model developed in this

study indicates simultaneous deformation and mass loss from

a water drop can be treated using a simple theoretical model.







It is concluded from a comparison of theory and experiment

that the classical Taylor's boundary layer stripping analysis

must be modified in order to obtain a realistic prediction of

the mass history of the water drop. The modification proposed

herein (Appendix II) permits the viscous profile to extend

through the entire drop as it becomes thin.

It is further concluded that while the assumption of

potential flow within the drop is satisfactory for an approxi-

mate theoretical model as was used herein, a more exact theory

of the internal flow should be developed.

It is also concluded that the general lack of experi-

mental data on the pressure distributions on spheres, hemi-

spheres and ellipsoids in the transonic and low supersonic

flow regime will restrict the accuracy of calculations on the

displacement, deformation and fragmentation of water drop

which are imparted with shock waves moving in the Mach number

range of approximately 1.6 to 4.0.

Finally it is concluded from the good agreement between

the experimentally and theoretically determined mass of the

drop that the mass calculation is relatively insensitive to

the precise form of the potential flow function used to cal-

culate deformation of the drop.







29






















z
w

z








P-4

w





prq








30




























0




rnq











Cq




I-,t







31

































z

E-4
Q





E-4









P4'







32























E-
z





0


z
-4I







E-4
























0



U,



00






E-r
SII2


ca)


















02
cq










02








00

EII
U)


cdLO


I-I













TABLE I
TIME SEQUENCE FOR PHOTOGRAPHS SHOWN IN FIGURE 5


Dimensionless
Time, T
5a Exposure; D = 2.06 mm.

0

.460

.985

1.140

1.455


Real Time, t,
A seconds


0

44

94

109

139


5b Exposure; D = 2.74 mm.


0

.220

1.155

2.060

2.980


0

30

124

224

324


Figure

1

2

3

4

5

Figure

1

2

3

4

5

Figure

1

2

3

4

5


0

55

85

125

195


5c Exposure; D = .985 mm.

0

1.200

1.850

2.740

4.260
























PRESENT STUDY

__DATA RANGE -
RANGER AND NICHOLLS4

3
** *

/ I I I Ix
-/ S


,s3
I- 0 S
D I S I -





L ------- oft




0 2 3 4 5
DIMENSIONLESS TIME, T = t U2/D


FIGURE 6.


EXPERIMENTAL DROP DEFORMATION DATA







































* PRESENT STUOY

DATA RANGE -
RANGER AND NICHOLLS4,
ENGEL1


- 0*


e I


2 3
OIMENSIONLESS TIME,


FIGURE 7.


T = t/ U2/D


EXPERIMENTAL DROP STAGNATION POINT
DISPLACEMENT DATA


30.0



20.0





10.0

8.0

















bd
MEASURED
ON 1.



















-Ay



ad








FIGURE 8. COORDINATE SYSTEMS FOR ANALYSIS
OF EXPERIMENTAL DATA












































0- -EI
I r=
U,
CD _














OpO
iI
C 1N)


U-3
0 I- E0Cio

C 0 1
L0 0






C,.
I / CpO9 C











0 0
cc cm/ -






E- oE-4



































8 1 + 2.615T .6776T2

.0971T3


r = .095


1 2

DIMENSIONLESS TIME, T = t/3 U2 / D


FIGURE 10. MEASURED DEFORMATION OF THE DROP AND
LEAST MEAN SQUARE CURVE FIT TO THE DATA


0 00
0
0o o o0


0 0
0 0


3






IcI

uf
m



cm

w

C,


0






40































6.0

-X = .409T2 .317T3
LL
<-D

I-







o = .112
C113
4.0 -

C30



- 2.0

o






DIMENSIOILESS TIME, T = t,3 U2/


FIGURE 11. MEASURED DISPLACEMENT OF THE STAGNATION POINT
OF THE DROP AND LEAST MEAN SQUARE CURVE FIT
TO THE DATA
















1.6 -




1.4




1.2




1.0 MS = 3.0



gu .8 -







w Ms =3.0


M = 1.6


.2 -

ad

0 I I
0 .5 1.0 1.5 2.0 2.5
OIMENSIONLESS TilE, T = t 3U2/D

FIGURE 12. MEASURED SEMI-MINOR DIAMETERS OF THE FRONTAL
SURFACE OF THE DROP AND OF THE DROP


























SCALE: --- .lb


FIGURE 13. MEASURED CROSS SECTIONAL SHAPE OF THE
FRONTAL SURFACE AND CALCULATED CROSS
SECTIONAL SHAPE OF THE DROP AT T = .4


















SCALE: --- .lb


FIGURE 14. MEASURED CROSS SECTIONAL SHAPE OF THE
FRONTAL SURFACE AND CALCULATED CROSS
SECTIONAL SHAPE OF THE DROP AT T = .6


















SCALE: -- .Ib


FIGURE 15. MEASURED CROSS SECTIONAL SHAPE OF THE
FRONTAL SURFACE AND CALCULATED CROSS SECTIONAL
SHAPE OF THE DROP AT T = .8















SCALE: -- .lb


FIGURE 16. MEASURED CROSS SECTIONAL SHAPE OF THE
FRONTAL SURFACE AND CALCULATED CROSS
SECTIONAL SHAPE OF THE DROP AT T = 1.0















SCALE: --- .1b


FIGURE 17. MEASURED CROSS SECTIONAL SHAPE OF THE FRONTAL
SURFACE AND CALCULATED CROSS SECTIONAL SHAPE
OF THE DROP AT T = 1.2





47






SCALE: .lb












































FIGURE 18. MEASURED CROSS SECTIONAL SHAPE OF THE FRONTAL
SURFACE AND CALCULATED CROSS SECTIONAL SHAPE
OF THE DROP AT T = 1.4













SCALE: -- .1b


FIGURE 19. MEASURED CROSS SECTIONAL SHAPE OF THE FRONTAL
SURFACE AND CALCULATED CROSS SECTIONAL SHAPE
OF THE DROP AT T = 1.6













SCALE: -- .lb


FIGURE 20. MEASURED CROSS SECTIONAL SHAPE OF THE FRONTAL
SURFACE AND CALCULATED CROSS SECTIONAL SHAPE
OF THE DROP AT T = 1.8













SCALE: -- .1b


FIGURE 21. MEASURED CROSS SECTIONAL SHAPE OF THE FRONTAL
SURFACE AND CALCULATED CROSS SECTIONAL SHAPE
OF THE DROP AT T = 2.0















SCALE: --- .lb


FIGURE 22. MEASURED CROSS SECTIONAL SHAPE OF THE
FRONTAL SURFACE AND CALCULATED CROSS
SECTIONAL SHAPE OF THE DROP AT T = 2.2




















SCALE: -- .lb


FIGURE 23. MEASURED CROSS SECTIONAL SHAPE OF
TIE FRONTAL SURFACE AND CALCULATED
CROSS SECTIONAL SHAPE OF THE DROP
AT T = 2.4




53


U.,
I I
I-


i,


.i




















2.5






- 2.0


C-



SS = 3 .0
1. M = 1.6



1 0 -



o 5 -
.5J







0 .5 1.0 1.5

DIMENlSIONLESS TIME, T = t3 U2/O


FIGURE 25. MEASURED SEMI-MAJOR DIAMETER OF THE DROP

































IS = 3.0
M = 1.6


FIGURE 26.


.5 1.0 1.5 2.0 2.5
DIMENSIOHLESS TIME, T = t/ U2i'

EXPERIMENTALLY DETERMINED ECCENTRICITY OF
THE DROP

















































1-

5-
=
-i
LJ

C=)
LJ


0C-














-.------I I I I


0 .5 1.0 1.5 2.0 2.5

DIMENSIONLESS TIME, T = t3 U2/D



FIGURE 27. EXPERIMENTALLY DETERMINED VELOCITY OF THE

STAGNATION POINT OF THE DROP




































MS = 3 .0


MS = 1 .6


.5 1.0 1.5 2.0

DIMENSIONLESS TIME, T = t/3 U2/0


FIGURE 28.


EXPERIMENTALLY DETERMINED NORMALIZED MASS
OF THE DROP





58





0
o






0








CO






0
Co












0
t--
t
-I I .

1 E

\ w
\ ^





\* ^-- -^
*~ -- ?^ - ~^ ~ ~
/V \ ^
















/ \











0 0
H









M<
CZ


HO


O)
0M


0 Q



---.
u Q

/
























.5





.4




i.3 -


2-










MS =3.0


0 ---- I <
0 .5 1.0 1.5 2.0 2.5

IOlENSIONLESS TIME, T = tS U2/O


THEORETICAL SEMI-MINOR DIAMETER OF THE DROP


FIGURE 31.






61
























c.4 M- =3.0



? 3 -
c---






S2 M =1.6






1


0 .5 1.0 1.5 2.0 2.5

DIMENSIONLESS TIME, T = t/ U2/D


FIGURE 32. THEORETICAL MAJOR DIAMETER OF THE FRONTAL
SURFACE OF THE DROP





























.8




.6 -\ M = 3.0



6 =1.6
4




.2





0 .5 1.0 1.5 2.0 2.5
DIMENSIONLESS TIME, T t U2/\


FIGURE 33. THEORETICAL NORMALIZED MASS OF THE DROP




































MS = 3.0


Mg =1.6


FIGURE 34


.5 1.0 1.5 2.0 2.5
DIMENSIONLESS TIME, T = tP U2/

THEORETICAL DISPLACEMENT OF THE STAGNATION
POINT OF THE DROP



































MS =3.0


MS = 1.6


FIGURE 35.


.5 1.0 1.5 2.0 2.5
DIMENSIONLESS TIME, T = t, U2/0


THEORETICAL VELOCITY OF THE STAGNATION
POINT OF THE DROP
























1.0




.8

MS = 3.0

SMs = 1.6

L- .6

-m















rIME.NSIONLESS TIME, T = t/ U2/0


FIGURE 36. THEORETICAL ECCENTRICITY OF THE DROP



































APPENDIX I





























































































W
u-i3
CJ3
=C

I- .E
-LI

C> C
LL,


XC LU Li
X X

I I


I- I-
x



I I





X -=


I- I-

CL. CL





S=m
LU LLJ
co cI












































LU





C-



-=3


CJ LcI



I I I I











-C -
(= C


I.-

c.o


>-






IL

cm

0-


in *



































APPENDIX II













BOUNDARY LAYER STRIPPING ANALYSIS


The analysis for mass loss from the boundary layer on a
5
spherical drop is attributable to Taylor. The analysis pre-
sented here follows Taylor but is more general in that it
applies to axisymmetric bodies.
Upon sudden exposure of a water drop to a high speed air
stream boundary layers will form both in the air and in the
water. The coordinate system used is shown in Figure 30 where
r is the curvilinear coordinate along the interface separat-
ing the two fluids and ? is the coordinate perpendicular to
it and is taken positive in either direction from the inter-
face. An approximate solution to this double boundary layer
problem is obtained by assuming arbitrary simple velocity dis-
tributions containing several unknown parameters which are
determined through use of Karman's momentum integral relations.
Assume that the flow is steady and incompressible and that
for the gas


Z4 (AII-1)



and for the water


Z =U, e- s -- (AII-2)



where, U1 is the local velocity in the gas outside of the
boundary layer, U is a function of F and ( 2 and cK are
constants. The steady state boundary layer momentum integral






equations are for the gas


and for the water



and for the water


J d, + 2


(AII-3)


(AII-4)


Z J <
Pa d87


where the pressure gradient in the liquid layer is given by


dp d u
d,? ~1 .. y c-


(AII-5)


At the gas-water interface the shear stresses must be equal
and therefore


a q-)
/ 2( ) -


(AII-6)


(&2 )1Rc






Applying equations (AII-1), (AII-2), and. (AII-6) and evaluat-
ing the result at <5 = 7/2 where for a sphere

duS dr


equations (AII-3) and (AII-4) become, respectively,




) O (AII-7)
4-


and


(TjT9- ) cI Z 2


= zO
UP~


(AII-8)


where 7 is the kinematic viscosity coefficient.
These equations yield directly


(AII-9)


and


(AII-10)


%-E~4


~Z






Differentiating equations (AII-1) and (AII-2) with re-
spect to > inserting the result into equation (AII-6) and
evaluating at = 0, the following equation is obtained:


-_C31 -U UI +' 4f'0


(AII-11)


Since U is the interface velocity and is expected to be very
small, equation (AII-11) can be approximated by:


I-U -


(AII-12)


and the interface velocity is


(AII-13)


The mass of water in the surface layer being swept along
by the gas stream at a distance equal to the arc length be-
tween the stagnation point and the equator is given by


m00
cT 2Tr Ji2dC1
0


(AII-14)


where uis taken to be the local velocity as given by equation
(AII-2) modified by a drop distortion velocity, Ud, due to
internal flow within the drop. Hence


4/_











-r
e -t+ UxK-Opz (All-15)




The distortion velocity Ud is assumed to be proportional
to the velocity of the fluid in the drop at the equatorial
point such that
Ud = .065v

where v is given by equation (III-7b) and .065 is an em-
pirical correlation factor. Equation (AII-14) can be written
as


00

7 ob o 5sj ? + Uo ex -i] (AII-16)




where the first integral is taken over the boundary layer
thickness, S defined such that


exp = .01


Performing the integration indicated in equation
(AII-16) there results




S2 b 065Nb rS E;) V 4 U, (AII-17)
Cj iT- --' -






Integrating equation (AII-17) over a small time interval,
A T, so that all time dependent parameters can be considered
constant with negligible error and applying equations (AII-9)
and (AII-13) to the result the mass loss from the drop is
found to be


/z -5/6 'liz
A -C,. (-)b, ,) \K \ (I-W) z AT-

.0o57/7_I SL +_ ',) (AII-18)





The local velocity, U at the equator of the drop is
approximated by




U = K^Cl VW/ (AII-19)



Equation (AII-19) is based on the assumption of isentropic
flow around the drop. This assumption is of course not
strictly valid for supersonic flow; however, since M < 1.35
R
for the present study, the entropy change is small and the
assumption of isentropic flow introduces little error. For-
subsonic incompressible flow, M < 0.7, the velocity at the
14
equator of the drop is





and for supersonic flow with MR > 1.3
R


TJ -U-W)/Cjj







In the Mach number range, .7 : IR -M 1.3 K is assumed
to vary linearly and is given by


K = l.l(2 5 -0.1875 ((M,.7l- |M -1.3|)


The velocity, W., appearing in equations (All-18) and
(AII-19) is the velocity of the drop assuming no deformation
is occurring and is given by equation (III-4) with:


L


(AII-20)
































APPENDIX III



























Yes


FIGURE 39. COMPUTER PROGRAM FLOW CHART













REFERENCES


1. Engel, O. G., "Fragmentation of Water Drops in the Zone
Behind an Air Shock," J. Res. N.B.S., 60, 245-280 (1958).

2. Hanson, A. R., Domich, E. G., and Adams, H. S., "Shock
Tube Investigation of the Breakup of Drops by Air
Blasts," Phys Fluids, 6, 1070-1080 (1963).

3. Nicholson, J. E. and Figler, B. D., "Complementary
Aerodynamic Test Techniques for Rain Erosion Alleviation
Studies," AIAA Paper, 66-766, September 1966.

4. Ranger, A. A., and Nicholls, J. A., "Aerodynamic
Shattering of Liquid Drops," AIAA Journal, 7, 285-290
(1969).

5. Taylor, G. I., "The Shape and Acceleration of a Drop in
a High Speed Air Stream," The Scientific Papers of
G. I. Taylor, edited by G. K. Batchelor, III, 247-264,
University Press, Cambridge (1963).

6. Clemons, J. F., An Experimental Investigation of Shock
Phenomena in Argon, Thesis, University of Florida
(1967).

7. Anonymous, TRW Model lD Image Converter Camera Techni-
cal Manual, TRW Instruments, El Segundo, Calif. (1966).

8. Brunsvold, R. S., The Design and Construction of a
Platinum Film Velocity Gauge, Thesis, University of
Florida (1969).

9. Anonymous, TRW Model 46A Trigger Delay Generator
Technical Manual, TRW Instruments, El Segundo, Calif.
(1966).

10. Anonymous, Type 556/R556 Oscilloscope Instruction
Manual, Tektronix, Inc., Beaverton, Oregon (1966).

11. Strutt, J. D., Lord Rayleigh, Theory of Sound, Dover
Publications, Inc., New York (1945).





80

12. Schlichting, H., Boundary Layer Theory, McGraw-Hill
Book Company, Inc., New York (1960).

13. Anonymous, Handbook of Supersonics, NAVWEPS Report
1488, 3, 66 (1961).

14. Milne Thomson, L. M., Theoretical Hydrodynamics,
Macmillan Company, New York (1968).












BIOGRAPHICAL SKETCH


William Edward Krauss was born May 12, 1928, at

Cleveland, Ohio. He was graduated from East Technical High

School in Cleveland in 1946. In September, 1950, he re-

ceived the degree of Bachelor of Mechanical Engineering from

the Ohio State University. Upon graduation he was employed

by United States Steel Corporation and then North American

Aviation, Inc. In October, 1952, he returned to the Ohio

State University and in June, 1953, he received the Master

of Science degree. He was a research associate with the

Ohio State University Research Foundation until 1954 when he

was employed by Convair in Fort Worth, Texas. From February,

1959, until the present he has been employed by the Martin

Marietta Corporation, Orlando, Florida, in various engineer-

ing staff and management positions. He is currently assigned

as division staff engineer for the Aeromechanical Division.
William Edward Krauss is married to the former Barbara

Alien Marlin and is the father of two children. He is a

registered professional engineer in Texas and Ohio and an

Associate Fellow of the American Institute of Aeronautics

and Astronautics.







This dissertation was prepared under the direction of
the chairman of the candidate's supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Engineering and to
the Graduate Council, and was approved as partial fulfill-
ment of the requirements for the degree of Doctor of
Philosophy.


June, 1970


"Al


Dean, Graduate School


Supervisory Committee:



Chairman


1
N-


.7

/I 1
-, 7 K(CGPL


R


;iri~5~Lk~~ LYL/L-~ CCC


I~T~h(