0.9

0.8

0.7

0.4

0.3

0.2

0.1

A T T T T
x
ep b o x/D=10
Ax x x/D=15 4
o fA A‘x/D= 20
oo
% 0 x/D= 25
a 4
a
Ro
°
Nd
. J
%
[A
8
& 4
*A
é 4
x
o
a
> x q
ndad
Bo
Re
a Bo
x °
(dy, = 3434) o7
L i { 1 1
0.2 04 0.6 0.8 1.0 12
r/2b,,
FIG. 22 CONCENTRATION FLUX PROFILE

57


APPENDIX V

ABSTRACTS OF PUBLICATIONS

1. "The Response of a Hot-Wire Anemometer to a Bubble
of Air in Water"

The sensitivity of peak voltage and duration of the
change in sensor voltages due to the impaction of differ-
ent size bubbles are computed and measured. Excellent
agreement between these is found for bubbles somewhat larger
than the sensor diameter and smaller than its effective
length in water streams in a range of 1.5 to 9 feet per
second. The method suggests a reliable method for sizing
bubbles in a water stream. The effects due to nondirect
hits are not treated.

2. "Bubble Formation due to a Submerged Capillary Tube in a
Quiescent and Co-Flowing Streams"

The case of bubble formation in both quiescent and
moving streams due to the injection of a constant gas flow
through a small tube is considered. Relationships predicting
the expected size and quantity of bubbles generated are
proposed. These are compared with measurements taken with
stream velocities up to 9 ft/sec, while generating gas bubbles
from 40 to 700 microns in diameter.

For the case of generation in a quiescent stream the
forces due to the virtual mass, surface tension, viscous
drag, buoyancy, and the wake formed by the preceding bubble
are accounted for. There still remains some question (only
partly answered by a comparison with measurements) as to
the proper added mass coefficient and the geometry of the
bubble previous to detachment, as well as an adequate
estimate of the interaction with a preceding bubble's wake.

The proposed model for generation in a moving stream
is in good agreement with actual measurements for co-flowing
velocities between 1 and 9 fps and capillary tubes in the
order of 1073cm in dia.

3. "Dynamic Simulation of the Turbulent Diffusion of Small
Particles"

The equation of motion of a small spherical nonrotating
particle was solved numerically in a two-dimensional homo-
geneous and isotropic turbulent field. Ensembles of solutions
were obtained for particles of different sizes and densities...
It was shown that the turbulent transport coefficient in-
creases with particle size and decreases with particle

125


WATER DEIONIZER

PIG. 2 CONSTANT HEAD TANK, WATER FILTER,DELONIZEK


RESEARCH REPORTING SERIES

Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five broad
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technology
transfer and a maximum interface in related
fields. The five series are:

1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research

4. Environmental Monitoring

5. Socioeconomic Environmental Studies

This report has been assigned to the ECOLOGICAL
RESEARCH series. This series describes research
on the effects of pollution on humans, plant and
animal species, and materials. Problems are
assessed for their long- and short-term
influences. Investigations include formaticn,
transport, and pathway studies to determine the
fate of pollutants and their effects. This work
provides the technical basis for setting standards
to minimize undesirable changes in living
organisms in the aquatic, terrestrial and
atmospheric environments.


2C n?

F(n) = 5
l+tl U, (eta) /2e1Cn

The constant of integration C follows from the condition that
£(0) = 1. This yields c=1/4. Consequently

ue (2 + U,, (xta) 2) 2
uh ee 7 (vI-11)
APPENDIX IV

Derivation of Equations (VI-19) to (VI-23)

Consider the body of water with bubbles as a "gas-water" mixed
solution, Let

%* = volume of one gas bubble

‘' = number of gas bubbles per unit volume of solution

&g = mass of gas per unit volume of solution (=pg*v, where
Pg is the gas density)

oy = mass of water per unit volume of solution (=o,,(1-"'v),
where Py is the water density)

o' = density of the solution (= g + ey)

¥g = velocity vector of gas species

Vey = velocity vector of water species + >

> et vi + yy Ww

v = velocity vector of solution (= 34)

>

Ry = mass flux vector of gas (eg v9)

> =o!

n, = mass flux vector of water ( On Vy)

D = mass diffusivity (molecular effect)

then Fick's first law gives the following relation (Bird,
Stewart and Lightfoot (1960)).

PA Pg
Rg - of (i, + a,) - e' DV (cB (A-IV-1)

g

The above equation states that the diffusion flux of gas
species is the resultant of two vectors, The first is the
flux of gas resulting from the convection of the fluid motion,
and the second is the flux of gas resulting from the diffusion

lal


0.9 ts J
Nea
0.8 Sup 4
° %,
0-7 "ve
. Yugo 4
vy
eb,
0.6 * ;
%e
Bs
1-5 4
os
u %
ay
0.8 oV d
“
Without with 8
Bubbles Bubbles oy,
0-3 ° e x/D= 10 % ]
Bw a
A a 15 an %
Ve
0.2 o . 20 4 uy
v v 25
ort 4
1 L rT 1 1 L 1
0 O.ob Qos 0.12 0.16
N
FIG. 28 NORMALIZED VELOCITY PROGILE (WITH & WITHOUT CONTAMINATION )

66


1.0

0.8

0.2

in 2)-2(0.68)
Fre (1+ 46.02 717)

$

- ° x/De 10 7
“ 15 5
A 20
o 25
r (dy = 970 pp) 4
1 1 1 1 t 1
0.02 0.04 0.06 Q08 0,10 0.12 0,14

1

FIG. 35 CONCENTRATION PROFILE

77


Particles Ey/Em Particle-carrier
stream density ratic

1250 soap bubbles 1.52 1.41
700u polystyrene beads 1.42 39
900u polystyrene beads 1.10 39

Singamsetti (1966) used sand with a geometric mean sieve diameter
of 0.068, 0.115, 0.23, 0.32 and 0.46 mm for measurement of
diffusion in a circular jet of water and obtained the following:

Particle diameter (mm) Particle Reynolds number €+/Em
0.068 0.4 1.23
0.115 1.2 1.20
0.23 7.2 1.24
0.32 18.0 1.35-1.39
0.46 33.0 1.43-1.45

The particle Reynolds number is based on the fall velocity of
the particle in a quiescent infinite fluid. Singamsetti
concluded from his measurements that (1) the linear oscillation
model for simulating the turbulent flow field is invalid for
three dimensional turbulence, (2) €y/€y approaches unity within
the Stokes range. eyo is the diffusivity of the marked fluid parti-
cle and the Stokes range is understood to be that for which

the viscous drag on the spherical particle is given by Stokes
law. (3) €y/€yo is greater than unity for particle Reynolds
number beyond the Stokes range provided that the fall velocity
is less than the rms value of the turbulent fluctuation in

the direction of fall.

Jones et al (1966) developed a linearized theory based on

the particle displacement rather than the particle velocity

to describe the motion of small particles in a turbulent flow.
They found that most of the energy of the particle fluctuation
was contained at lower frequencies (less than 3 cps) than
that of the carrier fluid (less than 10 cps). Moreover,

the heavier particles contained more tubulent energy than

the lighter one.

Majumdar and Carstens (1967) used nearly neutrally buoyant
particles in an apparatus restricting the particle motion to
a horizontal plane in a supposedly homogeneous isotropic
turbulent flow field. The particle diffusion occurs without
mean motion. The motion of individual particles was recorded
on motion picture film, which was subsequently projected for
displacement measurements. The particle displacement data
were analyzed, using the diffusion theory of continuous
movement. The values of the particle diffusivity, ey, and
particle mixing length, %,, were plotted as a function of
particle diameter, d. They concluded that

(1) if d < he,ey=tyo:

(2) if d >324, ey=0; and

(3) Ey/ey, continuously decreases in the range 1<d/%,<3.

13


ZL

Table 4 ‘Turbulent Schmidt Number

First Model
Bubble Size (Constant En and ¢, Model) (Tomotika's Solution) (Gaussian Solution)
(Mierons) v
€
Ss Ss Ss

Uy Grta) t t t

343 0.0025 1.06 1.19 1.17

527 0.0029 0.92 1.00 1.07

780 0.0030 0.90 0.98 1.00

970 0.0031 0.88 0.94 0.93

Note: In the second and third models, ey
Figure VII-17

and €, are dependent on n as shown in


o€

0.5

°o {>} Lo)
. . .
nN» w oo

peak voltages (volts)

or gt itered
PP

0°

L 1 L

1
0.2 0.4 0.6

®, unfiltered peak voltages (volts)

FIG. 5 FILTERED vs UNFILTERED PEAK VOLTAGES

10


Leibson, I., Holcomb, E.G., Cacoso, A.G. and Jacmic, J.J. "Rate
of Flow and Mechanics of Bubble Formation From Single Submerged
Orifices," AICHE J., Vol. 2, 1956, pp. 296-306.

Levich, V.G. and Kuchanov, S.I. "Motion of Particles Suspended in

Turbulent Flow," Soviet Physics Doklady, Vol. 12, No. 6, 12, 1967,
pp. 546-548.

Lin, C.C. "On a Theory of Dispersion by Continuous Movement=I,"
Proc. Nat. Academy of Sci., Vol. 46, 1960a, pp. 566-570.

Lin, C.C. "On a Theory of Dispersion by Continuous Movement-II,"
Stationary Anisotropic Process," Proc., Nat. Academy of Sci., Vol.
46, 1960b, pp. 1147-1150.

Liu, C.L. "Characteristics of Hot Wire and Hot Film Sensors for
Turbulence Measurements in Liquids", Ph.D. Thesis, School of
Civil Engr., Purdue University, W. Lafayette, Indiana, 1967.

Longwell, J.P. and Weiss, M. "Mixing and Distribution of Liquid in
High Velocity Air Streams," Ind. Engr. Chem., Vol. 45, 1953, pp.
667-677.

MacIntyre, F. “Vibrating Capillary for Production of Uniform
Small Bubbles," Review of Sci. Inst., Vol. 38, No. 7, 1967,
pp. 969.

Majumdar, H. and Carstens, M.R. "Diffusion of Particles by
Turbulence: Effect of Particle Size," Final Report G. R. No. 5,
Vol. WP00912-02ESE, C.E., Git., Atlanta, Ga., 1967.

Milne-Thomson, L.M. Theoretical Hydrodynamics, 3rd Edition,
London: MacMillan & Co., Ltd., 1955.

Pattle, R.E. "The Aeration of Liquids, Pt. II. Factors in the
Production of Small Bubbles," Trans. Inst. Chem. Engr., Vol.
28, 1950, pp. 32-37.

Ramakrishnan, S., Kumar, R. and Kuloor, N.R. "Studies in Bubble
Formation-I, Bubble Formation Under Constant Flow Conditions,"
Chem. Engr. Sci., Vol. 24, 1969, pp. 731-747.

Rennie, J. and Evans F. "The Formation of Froths and Foams above
Sieve Plates," Brit. Chem. Engr., Vol. 7, 1962, pp. 498-502.

Roberts, P.H. "On the Application of a Statistical Approach of
the Theory of Turbulent Diffusion," J. of Math. and Mech., Vol.
6, 1957, pp. 781-799.

Roberts, P.H. "Analytical Theory of Turbulent Diffusion", N.Y.U.,
June, 1960, Research Report No. HSN-2.

103


The corresponding half width, b_, and center line velocity
decay, (U_/U_)-1l, are shown in Figures 10 and 11. The
half width, &s defined in Equation (VI-4) follows the re-
lationship

Pu = 0.0952 (£ + 0.8) (vIII-1)

D

for the pitot tube measurement and

Pu = 0.0978 (£ + 0.5) (VIII-2)
D

for the hot film measurement.

The possible error due to the pitot tube as a mean velocity
measuring device is found to be negligible. The detailed
analysis of the error due to the hot film anemometer as a
mean velocity measuring device has not been investigated
here. Intuitively, a_hot film sensor will sense the total
velocity component, V1 2 2, instead of only the axial
velocity component. Re tn€ half width location the ratio
of the radial to the axial mean velocity component reaches
its maximum of about 3 percent and this will cause a posi-
tive error in the axial mean velocity reading. Further-
more, the radial and the tangential fluctuation components
will also give a positive error in the axial mean velocity
reading (Heskestad (1965)). For these reasons the coeffi-
cient of the jet widening rate given in Equation (VIII-2)
appears to be larger than the coefficient given in Equation
VIII-1). It is felt that Equation (VIII-1) should be the
more accurate of the two.

From the principle of momentum conservation, the axial
velocity component along the jet axis should decrease
hyperbolically. This was noted in Equation (VI-5).
Figure 11 shows the linear relationship of (U_/U_)-1

versus x/D. The best fit linear relationship"re&ds:

37


os § oft

FIG.

half width spreading of the velocity

concentration flux
half width spreading
° dp = 343 #
x dp = 527 #

A  4p=780 %
4,,=970 &

x/D

26 HALF WIDTH SPREADING OF THE CONCENTRATION FLUX

62


stroposcope SCHMITT TyeR AMPLIFIER OSCILLATOR

TRIGGER '

PIG, 8 ELECTRONIC INSRUMENTATION


is about 0.4 percent volume at the nozzle efflux of the jet.
It compares to the highest concentration used in the diffusion
measurements (see Table 2). The bubble size used in this
experiment is about 950 microns which is comparable to the
largest bubble size (970 microns) used for the diffusion
measurements.

Effect on the Mean Flow Properties

The mean velocity profiles in the now bubble contaminated
flow field were re-measured with the hot film anemometer.
The bubble when impacting on the sensor will change the
cooling of the sensor and hence introduce errors both in the
DC and RMS voltages. For the bubble concentrations treated
they may be neglected.

The measured velocity profiles are shown in Figure 28. They
are noted to preserve the similarity characteristics and to
practically coincide with the bubble-free profiles. The half
width spreading coefficient k, in this measurement is of 0.0961.
This value is comparable to tRe k, of 0.0978 obtained for the
single phase jet. The distance £Pom the virtual origin ob-
tained by the intersection of the half width spreading line and
the jet axis to the jet nozzle plane, a, is now 0.9D and this
value is comparable to the value of 0.5D obtained for the
single phase jet (Figure 29). The center line velocity de-
caying rate, k_, in this measurement is 0.162 which agrees well
with the sing18 phase jet measurement. However, the distance
from the virtual origin to the jet nozzle plane obtained by

the decay of the center line velocity is now 2.3D. This is
bigger than the single phase result of 0.8D (Figure 30). This
is probably due to the fact that the presence of the bubbles

in the potential core disturbs and shortens the length of the
potential core thus shifting the virtual origin toward the
upstream end.

Effect on the Turbulence Intensities

When a turbulent flow field is contaminated with foreign par-
ticles distinguishable in their size and density from the
carrier stream, the relative motion of the particles with
respect to the fluid (slip velocity) may cause extra dissi-~
pation of the turbulent kinetic energy of the turbulent

fluid and hence a decrease in the turbulent intensity
(Hetsroni and Sokolov (1969)). Plots of the measured tur-
bulence levels at x/D=10, 15, 20 and 25 for the bubble con-
taminated flow field are shown in Figure 31. For compari-
son, the fluctuation levels for the single phase jet measure-
ment (as per Figure 18) are re-plotted in Figure 31. The
following intensities. are noted at the axis of the jet.

65


Rouse, H. "Experiments on the Mechanics of Sediments Suspension,
"Proc., Fifth Intern. Cong. of Appl. Mech., Cambridge, Mass., 1938,
pp. 550-554.

Sayre, W. W. and Chang, F. M. "A Laboratory Investigation of Open-
Channel Dispersion Process for Dissolved, Suspended, and Floating
Dispersants," Geological Survey Professional Paper 433-E. 1968.

Schlichting, H. Boundary Layer Theory, McGraw Hill Book Co., 1960.

Shirazi, M. A., Chao, B. T. and Jones, B.G. "On the Motion of
Small Particles in a Turbulent Fluid Field," Nuclear Eng. Dept.
Univ. of Ill., Urbana, Ill., 1967.

Siemes, W. "Gas Bubbles in Liquids-I, the Formation of Gas Bubbles
at Upward Pointing Circular Orifices," Chem. Ing. Techn., Vol. 26,
1954, pp. 479-496.

Singamsetti, S. R. "Diffusion in a Submerged Jet," Proc. ASCE, No.
Hy-2, 1966, pp. 153-168.

Smith, F. B. "The Turbulent Spread of a Falling Cluster," Advances
in Geophysics, Vol. 6, 1959.

Soo, S. L., Ihrig, H. K. J. and El Kouh, A. F. "Experimental
Determination of Statistical Properties of Two Phase Turbulent Motion,"
J. of Basic Engr., ASME Trans., 9, 1960.

Squire, H. B. and Trouncer, B. A. J. "Round Jets in a General Stream,"
Great Britain Aero. Res. Council Rept. & Memo. No. 1974, 1944.

Stanisic, M. M. and Groves, R. N. "On the Eddy Viscosity of an
Incompressible Turbulent Flow," J. of Applied Math. and Physics,
ZAMP, Vol. 16, 1965, pp. 709-712.

Sullivan, S. L. Jr., Hardy, B. W. and Holland, C. D. "Formation of
Air Bubbles at Orifices Submerged Beneath Liquids," AIChE Vol. 10,
1964, pp. 848.

Taylor, G. I. "Diffusion by Continuous Movements," Proc. of the
London Math. Soc., Ser. 2, Vol. 20, 1921, pp. 196-212.

Tchen, C. M. "Mean Value and Correlation Problems Connected with the
Motion of Small Particles Suspended in a Turbulent Fluid," Dis-
sertation, Delft, Martinus Nijhoff, “fhe Hague, Netherlands, 1947.

oy

Tomotika, S., Proc. Roy. Soc. London, 165A, 1938, pp. 65.
Trentacoste, N. and Sforza, P. "Further Experimental Results for
Three Dimensional Free Jets," AIAA J. Vol. 5, No. 5, 1967, pp.
885-891. ‘

104


eh

3.

2.

“4.

(Equation (¥-1})

v,
pt 0-0952(x/D+ eA
7
7

b,
“pi = 0.0806(x/D + 0.6)

—_—0O-—— Without Syringe Tube

——— a With Syringe Tube

FIG. 13 VELOCITY

20 30
x/D

HALF WIDTH (WITHOUT SYRINGE TUBE )



0.6
L
Tm
0.4
“ 15
A 20
2p 4
° b 25
5 (a, = 780") 7
L - L j 1 1 L
0 0.02. GOL 0,06 0,08 0.10. O42 0.18
|

FIG. 42 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

85


SECTION II

RECOMMENDATIONS
This study was limited to the measure of the turbulent transport
coefficient of small contaminants of spherical shape in free
turbulent jet flows. The results give a direct measure of the
eddy diffusivity coefficient necessary to predict the dispersion
of contaminants when released in such conditions.

To make the study complete it should be extended to consider
the following cases:

1. Dispersion in stratified and disturbed (crossed, deflected,
etc) flows.

2. Dispersion of non-spherical contaminants.

3. Optimization of dispersion by control of particle size
and shape.

4. Dispersion inflow fields of different geometries.


ABSTRACT

Laboratory studies were conducted on the dispersion of gas droplets
of different sizes in turbulent water jets. The main purpose was
to determine the turbulent transport coefficient of contaminants
suspended in turbulent flows.

The experimental results were compared to measurements of diffusion
of liquid droplets in air jets as well as to a numerical analysis
based on the equations of motion of the particles themselves.

The results confirm that small particles in turbulent flows have an
increasing turbulent transport coefficient with size. The collated
experimenta] results exhibit when Reynolds analogy (in the transport
of mass and momentum) can be validly employed.

This report was submitted in (partial) fulfillment of Contract 16070

DEP under the (partial) sponsorship of the U.S. Environmental Protection
Agency.

iii


-1_ x -
[3] = 0.1626 (5 + 0.8) (VIII-3)
Up

for both the pitot tube and the hot film measurements.

Equations (VIII-1) through (VIII-3) are the empirical
results describing the axi-symmetric jet in which the
dispersion experiments will be performed. It is in-
teresting to compare these results with those for the
jet flow without the syringe tube. The results are
shown in Figures 12 through 14. The corresponding
relationships are

ge = (1+ 63.8n7) ~? (vIII-4)
m
= x -

by = 0.0806 (% + 0.6) (VIII-5)

D
ie)" = 0.1562 (¥+ 0.6) (vIII-6)
a

fo}

Equation (VIII-4) is the relationship obtained by Hinze
and Van Der Hegge Zijnen (1949) which agrees very well
with the present measurements. The curve given by Equa-
tion (VIII-4) is also shown in Figure 12.

The results show that the presence of the syringe tube
has little effect on the virtual origins, but it does
cause a faster spreading of the half width and a faster
decreasing of the axial velocity in the jet.

In table 1 the characteristics of the submerged jet
used in this experiment are compared with those of other
investigators.

Comparison with proposed Models for Mean Velocity Profiles:

Three phenomenological models were proposed in Section VI
for describing the jet mean velocity field. These resulted
in Equations, (VI-11), (VI-14) and (VI-17). The measured
mean velocity profiles will be used to calculate the
diffusion coefficient. Consequently they should be compared
to the analytical predictions.

The axial velocity profile from the first proposed model
with € constant everywhere is given in Equation .(VI-11).
The collstant coefficient U. (xta)/8e_, obtained by mini-
mizing the mean square error of the Measured data fitted
to this model is found to be 46.02. The corresponding

40


SECTION IX
TURBULENT DIFFUSIVITIES

The bubble concentration, ‘, can be obtained from the measured
concentration flux, N, as

v2 _N

b As (IX-1)

In the above A, is the effective area sampled by the sensor, as
defined in Chapter VIII, and Vp is the velocity of the bubbles.

The bubble velocity is given as Vp=U+vp, where U is the mean
velocity of the carrier stream and v, is the mean relative
velocity between the bubble and the carrier stream. Baker and
Chao (1965) suggest that the mean relative velocity between

the bubble and the mean flow of the turbulent carrier stream can
be taken as the rise velocity of the bubble in a quiescent
liquid. Based on this assumption, Vp is then a function only

of bubble size. Haberman and Morton (1953) measured rising
bubble velocities leading to the following:

Table 3 Rising Bubble Velocity

4, (Microns) v, (fps
343 0.105
527 0.186
780 0.301
970 0.397

The normalized concentration is then calculated by

vow Unt Yb ;
~~) ON Ix-
Tm N, UtYy ‘ ( )
where N/N_ is obtained from Section VIII, U and Uy, are described
by one of the three different models as given in Section VIII
and vp is obtained from Haberman and Morton's measurement. The
normalized concentration profile can thus be calculated based
on Equation (IX-2) and the corresponding diffusivities computed

with Equations (VI-25), (VI-27), or (VI-28).

First Model: (Constant Eddy Viscosity and Constant Eddy Diffusivity)
Assuming the velocity profile to be given by Equation (VIII-6) the
normalized concentration profiles can then be calculated from
Equation (IxX-2). The profiles for 343, 527, 780 and 970 micron

7


0.8 +

S85 040 ; T T T T T
Aa x9 Ao
a Eq. (1X-4) with ém/é;= 1.00
Ne 4
ery
\
° x/D=10
x 15
A 20
o 25 1
(4, = 527 4) 1

[ 1 L
0.02 0. OF 0.06 0.08 0.10 0.12

1

FIG. 37 CONCENTRATION PROFILE

80

0.14



EPA-660/3-74-004b
March 1974

Ecological Research Series



2

2 2 &

Nur u2

mean DC voltage output (true value)

fluctuation voltage output due to the turbulent
velocity fluctuation

DC voltage output at a particular frequency f
voltage drop due to an impacting bubble

Froude number = d,f ¥ga,

bubble generating frequency; frequency of turbulence
distance from the center of the sphere to the wall

heat transfer coefficients per unit length of the
wire exposed to the air and the water phase respectively

axial conductivity of a sensor; wave number
constant in Eq. (VI-17)

heat conduction coeff. of the air and of the water at
the film temperatures Tey and Teo respectively

rate of half width spreading in a jet, Eq. VI-4

decaying rate of the jet center line velocity,
Eq. VI-5

rate of concentration half width spreading, Eq. VI-22
center line concentration decaying rate, Eq. VI-23
length scale in r-direction

length scale in x-direction

length scale in @-direction

mixing length related to the momentum transport;
effective length of the sensor

mixing length related to the vorticity transport
3

0.7 ay/6

bubble concentration flux

bubble concentration flux at the jet mouth

Nusselt number for sensor exposed to air and water
phase respectively

110


EPA Review Notice

This report has been reviewed by the Environmental Protection
Agency, and approved for publication. Approval does not
signify that the contents necessarily reflect the views and
policies of The Environmental Protection Agency, nor does
mention of trade names or commercial products constitute
endorsement or recommendation for use.

a
Be


BY By 5
= a

ss vU VY NHN DW
° -H 8

'S
K
iN}

yp 0

number of bubbles impacting in a time period Tt
mass flux vector of gas

mass flux vector of water

mean pressure

measured static pressure by pitot tube

true static pressure

instantaneous pressure

fluctuation pressure

static pressure in the outside field of a jet
Prandtl number of the water

gas flow rate through a syringe tube
correlation coeff. of two fluctuating velocity
components; total electric resistance of a sensor;

Reynolds number

orifice Reynolds number of the jet, Reynolds number
of the air and water phases based on the sensor

diameter
U3 a,

v

Iv. - v,| 2
particle Reynolds numbers P ) £ py
a,

v

or

radial coordinate

gas bubble radius

electric resistance at wire temperature T,)

area of the stagnation pressure tap of a pitot tube
turbulent Schmidt number related to the bubble
diffusion; sensor sensitivity based on the traverse
time, t, of an impacting bubble

sensor sensitivity to the impacting bubbles based on
the voltage drop, &

sensor sensitivity to the velocity of the water
stream

111


82

ENERGY DENSITY
40-9 square volts

1.

ao

.
n

.

e
iw

n
0.8 1.2 1.6 2.0 2.4 2.8

frequency, KHz

FIG. 4 ENERGY DISTRIBUTION OF BUBBLE SIGNAL


SECTION X
TURBULENT DIFFUSION CHARACTERISTICS

Dependence of Turbulent Schmidt Number on the Particle
Parameter Ven

The results reported in the previous sections dealt with the
dispersion of bubbles in a liquid jet. As noted by Householder
and Goldschmidt (1969) it is possible to compare the diffusion
of particles in entirely different flow fields by using a cor-
relation based on Vo-

This is because the turbulent transport is related primarily

to the immediate structure of the turbulent flow and the
particle size, shape and density. In selecting a parameter

to make this comparison they selected the ratio of the Stokesian
response time of the particle in a carrier stream to a charac-
teristic time scale of the flow. Ahmadi (1970) extended this
idea using the turbulent time macroscale as the specific charac-
teristic time. He defined the corresponding particle parameter
as

(X=-1)

where A, is the longitudinal turbulent macroscale of the

flow fiéld. What is to be investigated is whether or not S

is a function of y,,. Knowledge of such a function would pefmit
comparison of experimental results of the previous section to
those of other investigators and an extension to other flow
fields as well.

When the particle in concern is a bubble of gas in water the
particle parameter given in Equation (X-1) simplifies to

= 2
Y.2 4
aw Sb -
Yn = (K-2)
\

which can then be re-written as

Vz2upD up 4 Vz uD 4,
= _ TOA oO (? = a a (X-3)
m ‘ox = TM
Vayu , U_/U_ and A_/D are given in Figures 31, 11 and 20,
respedtiv@lyTM An average value of »,, can be computed for each
bubble size throughout the region in’ which megsyrements were
taken. The relationship used is Vea = 7.25x10 (2) * an average
D


0.2

o x/D=10

x“ 15
4 20
25
(a, = 970 p-) 4
1. n n 1 1 1 .
0.02 0. OF a06 0.08 0.10 012 0.14
FIG. 39 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

82


Shirazi et al (1967) applied a conditional averaging to reduce

the B.B.O. equation to a stochastic integro-differential

equation with zero mean. By assuming that the particle velocity
and the turbulent velocity vectors are uncorrelated, they obtained
specific relationships between the statistical properties of

the particle velocity and the fluid velocity. Their published
results include relationships for the Lagrangian micro scale,
autocorrelation function, and diffusivity of the particles.

Ahmadi (1970) studied the dynamic behavior of a small spherical
particle in a numerically generated two dimensional turbulent
flow field and calculated the particle diffusivity as a function
of particle size and density. He found that the particle
diffusivity increases with particle size and decreases with
particle density.

EXPERIMENTAL STUDIES

Rouse (1938) performed a basic study in particle diffusion by
Measuring the concentration in the vertical direction of sand
particles in which the turbulence was generated in repeating
patterns by mechanical agitation in a cylindrical tank. He

found that for any particle size, the particle diffusivity

e,, is directly proportional to the frequency of agitation;

at low frequencies, is independent of the particle size;

and at high frequencies ey increases with an increase in particle
size.

Kampe de Feriet (1938) reported diffusion experiments by Dupuit
in a wind tunnel, where soap bubbles about 3 mm in diameter were
used and introduced at a point in the wind tunnel stream.

The concentration of the bubbles passing a plane perpendicular
to the mean flow showed a Gaussian distribution.

Kalinske and Pien (1944) performed diffusion experiments in
water flowing through a channel at a certain distance from

the bottom and side walls, the turbulent flow there being nearly
isotropic. Small drops were formed by injecting a mixture

of carbon tetrachloride and benzene of the same density as

the water. The value of the variance of the displacement

was determined from photographic records. The concentration
distribution of the droplets passing a plane perpendicular to
the flow at a distance from the point of injection was found

to be very close to Gaussian. The rough shape of the Lagrangian
correlation coefficient could be well approximated by an empirical
function.

Vanoni (1946) used sand of geometric mean sieve sizes of

0.147 mm, 0.12 mm and 0.091 mm and found that for coarse sediment
e, is less than the momentum diffusivity em and for fine

sédiment ey is greater than ¢,. The difference in the two eddy

10


d. The curves were then changed to F(K) versus K,
where K is as previously defined and F(K) =

u F(f). The resulting curves of F(K) versus

*E are shown in Figure 19.

The longitudinal macro scale of the turbulence, defined
as

lim F(K) (VIII-10)
K>0

AL =
x 2

can be calculated from Figure 19. Figure 20 shows the
plot-of A/D versus x/D. The longitudinal macro scale
increases” linearly along the jet axial direction, fol-
lowing the approximate relationship

A, x (VIII-11)
p= 0-0492 (= + 2.5)

Comparing Fig. 20 with Fig. 13 shows that the longi-
tudinal macro scale along the jet axis is approximately
half the value of the half width by. and follows the ex-
pected linear growth.

CONCENTRATION FLUX MEASUREMENT

In this experiment the efflux velocity of the turbulent
water jet was 10.83 feet per second. The generated bub-
bles were measured directly through a microscope for four
different sets of runs. When the stroboscopic illumin-
ation frequency is synchronized with the bubble generation
frequency, the generated series of bubbles will appear
stationary, Their size can then be measured hy a micro-
scope placed outside the jet tank. The corresponding
average bubble sizes were 343, 527, 780 and 970 microns
with a standard deviation of less than 15 percent.

As described in Section VII the mean concentration flux

is obtained by counting the number of bubble impactions on
a cylindrical hot film sensor. If A_ is the upstream

cross sectional area containing bubbfes ultimately touch-
ing the cylindrical hot film, the actual number of
bubbles per unit volume of the flow field at the measuring
location, and V, the mean bubble velocity at this location,
the concentration flux measured will be N=!VpAg. Theore- |
tically A, is related to the impaction coefficient (defined
as the rakio of the area containing particles touching the
collector to the frontal area of the collector) and is not
a constant value with velocity. Householder (1968) made
an extensive study of the impaction coefficient by assuming
that the particle trajectory in the vicinity of a cylind-
rical collector is governed by the particle inertial and

52


6

7 T T T T {— T

o
, eollected by Householier & Goldschmidt °
pipe and chamel (Wn = Yip)
6F ¥ Kernedy (isotropic turbulence) on +
a Brush .
St A Singassetti (axi-symmetriec jet) . . . 4
. present study ° . .
y {| 0 Householder (plane jet) ° |
4 Zynamiec simulatior ‘Atma
Ss
3F we 4
2Fr 4
vo, vo.
Ler . wei 4
0 L 1 l L ! 1
107* 1073 197? 1o7} 10° 104 10? 103

Uy

FIG. 45 DEPENDENCE OF sz ON PARTICLE PARAMETER Vv;


bubbles are respectively shown in Figures 32 to 35. It is
seen that the normalized: concentration data preserve the
similarity characteristics. (This point will be further
discussed in the next section).

Combining Equations (VIII-6) and Equation (VI-25), one obtains

v .
— = (1 + 46.0207) 72 &m/ey

mn (1X-3)

The computed concentration profiles are given in terms of n.
The corresponding turbulent Schmidt number for the bubble
diffusion, (St=€y,/ey), can be obtained by comparing Equation
(IX-3) with the data of Figures 32 to 35. The resulting
coefficients of the eddy particle diffusivities and the
corresponding turbulent Schmidt numbers for the four bubble
sizes tested are tabulated in Table 4. These were obtained by
minimizing the mean square difference between the data and
Equation (IX-3) with different exponents. The theoretical
curves given by Equation (IX-3) with the regressed coefficients
are also plotted in Figures 32 to 35. It is seen that the
general shape of the curves agrees very well with the values
computed based on the actual measurements.

Second Model: (Taylor's Vorticity Transport Theory)

Assuming that the velocity profile of the flow field is given
by Equation (VIII-8), the normalized concentration profiles
calculated with Equation (IX-2) for the 343, 527, 780 and ‘970
micron bubbles will differ from those of figures 32 to 35:
The corresponding values are shown in Figures 36 to 39.

Combining Equations (VIII-8) and Equation (VI-27) the assumed
concentration profile takes the form

1

2 a 2 2a
11.87 1 2 2,92 733 5
_ (oxof Z(11.8n) ~Fay (11. 8n) “+e=(11.8n) -zg— (11. 8n) |

|

En
€

[22.san 9a
m

° - 2a, + 3a, (ar.en?/? - 3 (1.90)?

(Ix-4)

Equation (IX-4) can be numerically integrated for various values
of n provided the exponent €,/ey is assumed to be known. .

In this manner, repeating the calculation of the concentration
profile by considering different values for en/Eys the value
for which the calculated’ concentration profile appears to best
fit the measured data can be found. This is then taken to be
the representative turbulent Schmidt number.

73


Hewlett Packard Electronic Counter, Model H22-5211B
Maximum counting rate: 300 KHz
Number of digits: 4
Time base accuracy: + 0.1%

Other Instrumentation

Other supplementary instruments were used while measuring the
fluctuation components of the velocity field, the energy
Spectrum, and the frequency of bubble generation. They are
itemized in the following:”

Bruel & Kjaer Frequency Analyzer, Type 2107

Dana DC Amplifier Model 2000

Hewlett Packard 200 CD Wide Range Oscillator
Hewlett Packard Vacumm Tube Voltmeter, Type 400 AB
Honeywell Accudata 105 Gage Control Unit

Honeywell Accudata 120 DC Amplifier

Honeywell Visicorder Oscillograph Model 2106

TSI Model 1025 Temperature Compensator

TSI Model 1125 Calibrator

General Radio Electronic Stroboscope, Type 1531-A
General Radio Strobolume, Type 1532-B

Dimco-Gray Model 172 Timer

American Optic Model 371734 Long Focusing Distance Microscope
Nikorex 35 mm Camera.

34


004

003

002

Em
Um (+a)

001

(0)

U

Lov Model

(constant eddy viscosity)
Second Model (Eq. I-16)

Third Model (Eq. I-18)

1

1
202

1 L =e
04 206 08 el
t

012,

014

FIG. 15 DISTRIBUTION OF EDDY VISCOSITY BASED ON DIFFERENT MODELS

46


hence

Ay A,D
vee
& xta

_ 2x

1 3a

In such a manner, Equation A-II-6 becomes

2 1d
£° + nff' - ' = = -II-
n gf! + = ay (nh) 0 (A-II-7)

Writing the equation of continuity in terms of f(n) and g(n),
the function g(n) in Equation A-II-7 can then be eliminated
and Equation A-II-7 simplified to

_ FE" _ 1 gyy2 yg F_ _4 “tte
TG (EF) + om P= ag nh) (A-II-8)

where

n
F(n) = | n £(n) dn
°

Equation A-II-8 after integration yields

FF'_ sf ["
fe)

n £(n) dn (A-II-9)

where the constant of integration must be zero, since h(n) is
finite at n=0. Writing in the original notation, Equation
A-II-9 becomes

n

xr 1 U | U
srr dn n(3-) (vI-3)
pun, 7 On ° Un

Furthermore, the similarity of the flow field requires that

D._ oD = = = .
d= l= sae consequentely n = 575 0r © n(xta). Now letting

b.. be the value of r where the velocity is half the maximum at
e¥ch section and kp the corresponding value of n, then

b, = ky (xta)

119


TS

0.05

0-8

0.03

0.02

0.01

80 100 120
«x (th

FIG, 19 ENERGY SPECTRUM ALONG CENTER LINE OF

140 169

THE JET

180


E
(volta)

14

13

12

11

FIG, 16 SENSOR CALIBRATION CURVE

qT q y T T T
L pos
[— % 7
if ©. pre-calibration 4
/ »  post-calibration

iT L L L L. L

2 30 4 5 6 ? 8

U (fps)


0.2

b> x O-

45
20
) 25 1
(4, = 780) 4
i L _L 1 n i
0,02 0,04 0,06 0,08 O00 0.12 O14

q

FIG. 34 CONCENTRATION PROFILE

76


Tv TT
Em/Ex = 0.93
0.65}
FL
Tm
O.bE
L o x/D=10
“ 15
A 20
O.2- 4
o 25
F(a = 970 2) 1
4 L. L 1 1 1
0 0.02 0.04 0.06 008 010 0.12 . O14

q

FIG. 43 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

86


Ty

Author

Keuthe
(1935)

Squire & ITrouncer
(1944)

Hinze & Van Der

Hegse Zijnen
(1948)

Albertson et al
(1948)

Keascy & Weller
(1349)

Corrsin & Uberoi
(1950)

Porstall & Gaylord
(1955)

Trentacoste &
Sforza (1966)

With

Syringe
Present Tube
Study Without

Syringe

Tube

Table 1

Half Width
Spreading
Coefficient Mp
(see £q.I-4)

0.068

0.0906

0.0826

0.0956
0.0888
0.085
0.0920
0.093

0.0952
0.0978

0.0806

Comparison of Various Axi-symmetric Turbulent Jet Measurements

Virtual Origin
Based on the
Half Width
Spreading

44D

3.6

0.6D

-1.2D

0.80
0.5D

0.6D

Center Line
Vel. Decaying
Rate Xy

(See Eq. M-5)

0.1506

0.1563

0.1612

0.1722

0.148

0.161

0.143

0.1626
0.1626

0.1562

Virtual Origin Potential

Based on the
Center Line

Vel. Decaying

-1.2D

0.8D
0.8D

0.6D

Core Reynolds
Length Number
6.19D -

- 6.7(10")
6.20 2-10(10")
6.0D 5.5(10")

- acioty
7.0D -

- 4.5(10")

sot

- 4&.5(10")

- 4, 5(10")

Remark

Air Jet

Air Jet

Air Jet

Air Jet

Air Jet

Air Jet

Water Jet

Air Jet

Water Jet
(Pitot Tube)
(Hot Film)

(Pitot Tube)


SECTION XVI

APPENDICES

APPENDIX I
Derivation of Equation VI-1

The averaged steady Navier Stokes equation in cylindrical
coordinates for an axi-symmetric turbulent jet, assuming

ur=U+u, VisVtv, Wow, P'=P +p, ao 25
ca 0 oot 0, reads
—~ Mer ~ Ma ,
rae 17907
aU ou 1 ap 3 2 U
- Im—— + Ve ee a (- =
x-component Us> + Ve_ 2 ox t Ox (-u* + vy =)
13s — aU
+ 35 r(-uv + v 3x (A-I-1)
- OV, yes Ley tL 3Vv.
r-component Us + Vay 5 Or +x (-uv + v 3x
13 2 Wy _ 1l1,;2, Vv a
+Say r(-v" + Vay rs (-w" + vz) (A-I-2)

where U', V', W' are instantaneous velocities in the axial,
radial and tangential directions respectively and U, V, and W
are respectively the mean velocities and u, v, w are the
fluctuation components.

The length scales Lye Ly and Lg for the x, r and 6 directions

respectively may be defined. Then L, Ly (since the range
. mao << il

: ty, ty
of mixing in axial direction is much larger than in the radial
direction). Letting uy, uy and ug be the corresponding velocity
scales and assuming that u,; and ug have the same order of
magnitude, the equation of mass conservation then gives the
following relationship

u
v '
x

fF
D |o

vv

rate

Furthermore, a fluctuation velocity scale Ug such that

wiv v2 wwe vue

NI

115


concentration profile as

i (xtra) DU, Un n
= oe {- |. n { | ng) an] anh (VI-21)
fo} m

Let by denote the value of r where the concentration is one
half SE the maximum, k, denote the corresponding value of
n then

‘m

(% + 3)

by _
pe Gt

h (VI-22)

The equation (VI-22) states that the concentration half width
increases linearly along the axial direction.

Furthermore v

m_ D 7
7 = Ay ta’ and for T/A,=ky then

= X43)

=ky (5 + 5) (vI-23)

Equation (VI-23) states that the inverse of the center line
decay of the concentration is also linearly proportional to
axial distance,

124


EPA 660/3-74-004b
March 1974

TURBULENT DIFFUSION IN LIQUID JETS: FINAL REPORT

Strong C. Chaung and Victor W. Goldschmidt
Engineering Experiment Station
School of Mechanical Engineering

Purdue University
Lafayette, Indiana 47907

Project No. 16070 DEP
Program Element 1BA025

Project Officer

George R. Ditsworth
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330

Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $1.70


SECTION XIV

PUBLICATIONS

The following is a list of publications which resulted due
(in part or fully) to the sponsorship of Grant 16070DEP

by The Water Quality Office, EPA. Abstracts are presented
in Appendix Vv.

1.

S. C. Chuang and V. W. Goldschmidt "The Response
of a Hot Wire Anemometer to a Bubble of Air in
Water" Symposium on Turbulence Measurements in
Liquids, University of Missouri, Rolla, Missouri,
September 1969.

S.C. Chuang and V. W. Goldschmidt "Bubble
Formation Due to a Submerged Capillary Tube in
Quiescent and Co-Flowing Streams" ASME, J. Basic
Engineering, Vol. 92, D, No. 4, December 1970,
pp 705-711.

G. Ahmadi and V. W. Goldschmidt "Dynamic
Simulation of The Turbulent Diffusion of
Small Particles" Hydrotransport I, First
Internatiénal Conference on The Hydraulic
Transport of Solids in Pipes, University of
Warwick, Sponsored by The BHRA Cranfield,
Bedford, England, September 1-4, 1970.

G. Ahmadi and V. W. Goldschmidt "Analytical
Prediction of The TurbulentDiffusion of Small
Particles," Fluid DynamicsSymposium, McMaster
University, Hamilton, Ontario, August 25-27,
1970.

The following articles are currently under the review and
pre-publication process:

5.

V. W. Goldschmidt and S. C. Chuang "Energy Spectrum
and Turbulent Scales in an Axi-Symmetric Water Jet"

Submitted to’ the ASME.

V. W. Goldschmidt, M. K. Householder; G. Ahmadi and
S. C. Chuang "Turbulent Diffusion of Small Particles

Suspended in Turbulent Jets" Submitted for consi-
deration at a forthcoming Symposium on Two Phase
Flows.

107


The slopes of the corresponding plots in Figure 27 depend
upon the size of the bubbles. The following results are
noted

_ 1/2 _ x (VIII-21)

for 4d, = 343u (NU/N,) = 29.2(= + 0.8)
_ 1/2 _ x (VIII-22)

a, = 527u (N,/Nq) = 30.5(5 + 0.8)
_ 1/2 _ x (VIII-23)

a, = 780u (NL/N,,) = 31.5(= + 0.8)
(VIII-24)

= 1/2 _ x
d_ = 970u (NL/N,,) 32.515 + 0.8)

EFFECT OF BUBBLE CONTAMINATION ON THE CARRIER STREAM

When a turbulent flow field is contaminated with foreign
particles the characteristic of the flow field itself will
probably be changed.

Soo et al (1960) carried out measurements on the diffusion
of helium gas in air and in suspensions which contained
0.0005to 0.003 volume fraction of glass beads of diameter
less than 0.01 inches in a horizontal pipe. They found
that the presence of the solid particles did not affect the
diffusion in the fluid phase.

Kada and Hanratty (1960) investigated the effect of parti-
cles on the diffusion characteristics of turbulence. They
concluded that foreign particles with volume concentration
less than 2.5 percent do not appear to have a large effect
on the diffusion rate unless there is an appreciable average
slip velocity between the solid particles and the fluid.

Brandon (1968) investigated the effect of glass beads on the
longitudinal turbulent intensity and the friction coefficient
of a pipe flow. He concluded that (1) there is no detectable
effect of the solids on the longitudinal turbulent intensity
for particle sizes ranging from 20 to 60 microns and solid
sphere volume fraction of about 0.003, (2) the friction

factor increases with particles size from 20 up to 170 microns
and remains constant for sizes up to 560 microns for volume
fraction less than 0.01.

Hetsroni and Sokolov (1969) measured the longitudinal turbulent
intensity in an aerosol contaminated two dimensional jet with
aerosol consisting of oil droplets 13 microns in size. They
reported that the intensity of turbulence is lowered by the
particles and that the suppression of turbulence is pro-
portional to the concentration of the droplets.

In this section the effect of the bubbles on the characteristics
of the flow field is considered both in the mean flow property,

fluctuation components, momentum flux deviation and the

energy spectrum. For the case treated the bubble concentration

64


in shape and have the maxima at r/b, about equal to

+ 0.5 to + 0.8. The measured turbulent levels reveal
that there is a trend of increasing turbulence level
along the axial direction.

The results of air jet turbulence level distribution
measured by Corrsin (1943) for x/D = 10 and 20 are also
shown in Figure 18. It is seen that the turbulence level
obtained from the present measurement is higher than
Corrsin's results.

ENERGY SPECTRUM AND TURBULENT SCALES

Experimentally the energy spectrum is obtained_by taking

the ratio of the mean square voltage response e2 ata
particular frequency band centered at a _frequenéy £ to the
total mean square fluctuation voltage _2 of the sensor out-
put divided by the effective band widtfi of the filter (Curtis
(1960)). Energy spectra were obtained from measurements per-
formed at stations with x/D=10, 20, and 30 along the jet axis.
The resulting curves,are plotted in a normalized form as F(K)
versus K such that { F(K)dK = 1 where K is the wave number and
is defined as K = 2nf/U, where f is the frequency and U is the
mean velocity at the measurement location. Figure 19 shows
such plots. The procedure for making energy spectrum measure-
ments was the following:

a. Determined the characteristics of the wave
analyzer used (General Radio Type 1900-A
Wave Analyzer) by feeding in a known fre-
quency sine wave signal and measuring the
output of the wave analyzer at various
center frequencies,/ .2. The, attenuation

factor,.which is defingda as e2af/e*, was
then calculated and found to/Be §qual to 5.17.

b. The turbulent signal was then fed into the wave
analyzer and the output at various center fre-
quencies |2 was measured. With it ee was cal-

culated. fhe normalized energy spectrum F (£)
is then defined as e2/(5 17e2 Where ,2 is the

total fluctuation enérgy and was simultaneously
measured through a Briel and Kjaer Wave Analyzer.

c. The measured data were then plotted as F(f) versus
£. The total area under the F(f) curve should be
equal to unity. If not, another correction factor
is obtained and the values of F(f) corrected through
this factor.

50


Eckert, E. R. G. and Drake, R. M. Jr. Heat and Mass Transfer, McGraw
will, Inc., N. Y., 1959.

Bversole, W. G.; Wagner, G. H. and Stackhouse, E., "Rapid Formation
of Saf popbtes in Liquids," Ind. Engr. Chem., Vol. 33, 1941, pp.
1459- .

Flora, J. J. J. and Goldschmidt, V. W. "Virtual Origin of a Free
Plane Turbulent Jet" AIAA Journal Vol. 7, No. 12, 12, 1969, pp.
2344-2346.

Folsom, R. G. "Review of the Pitot Tube", Trans. ASME, Oct., 1956,
pp. 1450.

Forstall, W. and Gaylord, E. W. "Momentum and Mass Transfer in a
Submerged Water Jet," J. of Applied Mech., ASME Trans., Vol. 22,
No. 2, 1955.

Frenkiel, F. N. “Turbulent Diffusion of Mean Concentration Dis-
tribution in a Flow Field of Homogeneous Turbulence," Advance in
Applied Mech., Academic Press, Inc., 1953, pp. 61-107.

Frielander, S. K. "Behavior of Suspended Particles in a Turbulent
Fluid" AIChE J. Vol. 3, No. 3, 9, 1957, pp. 381-385.

Goldschmidt, V. W. "Measurement of Aerosol Concentrations with a
Hot Wire Anemometer," J. of Colloid Science, Vol. 20, No. 6, 1965,
pp. 617-634.

Goldschmidt, V. W. and Eskinazi, S., "Two-Phase Turbulent Flow in
a Plane Jet," Journal of Applied Mechanics, Vol. 33, Series E, No.
4, Dec., 1966, pp. 735-747.

Goldschmidt, V. W. and Householder, M. K. "Measurement of Aerosols
by Hot Wire Anemometry," AFOSR Final Report, AFOSR No. 68-1492, July
1968, pp. 134-152.

Goldschmidt, V. W. and Householder, M. K., "The Hot Wire Anemometer .-
as an Aerosol Droplet Size Sampler," Atmospheric Environment, Vol.
3, Nov., 1969, pp. 643-652.

Haberman, W. L. and Morton, R. K. "An Experimental Study of Bubbles
Moving in Liquids," ASCE Trans. Paper No. 2799, 1953, pp. 227-252.

Heskestad, G. "Hot Wire Measurements in a Plane Turbulent Jet," J.
Of Applied Mech. Dec., 1965, pp. 721-739.

Hetsroni, G. and Sokolov, M. "Distribution of Mass, Velocity and

Intensity of Turbulence in a Two Phase Turbulent Jet," Dept. of
Nuclear Sci. and Mech. Engr., Technion, Israel Institute of Tech-

nology, Haifa, Israel, 1969.

Hinze, J. O. Turbulence McGraw Hill Co., Inc. 1959.

lol


TRAVERSING
MECHANISM

JET TANK

SUPPORTING
FRAME

FIG.

3

PHOTOGRAPH

2k

OF

JET TANK


Second Model: (Taylor's Vorticity Transport Theory)

If the velocity profile is calculated according to Taylor's
modified vorticity transport theory and the eddy diffusivity is
expressed as

€
—  Y¥ 2 e247 2 (WL -
uGeray — “y| dn & (vI-26)

then substitution of Equations (VI-14) and (VI-26) into Equation
(VI-21) yields a solution of V/‘y, in the form:

a €
2 2047/2 3,45) { _m
v : n* gl/2 Sa, 42, gent ’2* ant] CS
— =; exr[ | an* i22n 7TAln** + 5 13 {Fy
mt ° 9a
-3 3/2 _ 73,43 -
za, + 3ayn* mal (VI-27)
where n* is as defined in Equation (VI-15).
Third Model: (Reichardt's Inductive Theory)

The velocity profile based on Reichardt's inductive theory is
given in Equation (VI-17). Substituting now Equation (VI-17) into
Equation (VI-21) and expressing U,(xta) in terms of the eddy vis-
cosity as given in Equation (VI-7), the final solution for the
concentration profile reads:

i)

v - i
— = exp} - = Kn?» (VI-28)
TMm Q Y -
Hereafter, Equations (VI-25), (VI-27) and (VI-28) will be referred
to as the first, second and third proposed models respectively.
To obtain the dependence of the dimensionless parameter t,/ey,
which is defined as the turbulent Schmidt number, S;, on the con-
taminant particle size is the major objective of this work. Its
value will. be found from measured concentration flux distributions
and equations (VI-25), (VI-27) or .(VI-28)..

20


Valentin, F. H. H. "Absorption in Gas-Liquid Dispersions" Spon's
Chemical Engr. Senis, London, 1967.

Vanoni, V. A. "Transportation of Suspended Sediment by Water"
Trans. ASCE., Vol. III, 1946, pp. 67-133.

105


ze

HOT FILM

ANEMOMETER

20-30 DB

AMPLIFIER

BAND PASS

FILTER

SCHMITT

TRIGGER

ELECTRONIC

COUNTER

rey

FIG. 7 BLOCK DIAGRAM OF ELECTRONIC INSTUMENTATION



oU oU 19 13 —
vues Se - a SPL fo

Ox v or Pp ox ror (ruv) (A-I-5
Lap, 3 2, v-w
= Pye wiawWw .
p or ZV ft r 0 (A-I-6)

Equation A-I-6 can be integrated with respect to r to give
z i v-w

+ + =

Pp ov p dr 7 Po (A-I-7)

where P, is the pressure at the same section but outside the
turbulent region. Equation A-I-7 states that the pressure in
a jet flow field is only a function of r and is independent
of x. With such equation (A-I-5) simplifies to

au aw_ ol 8
Use + V Se = pe bE OF xz) (11I-1)
where
Oyy =~ puV

APPENDIX II

Derivation of Equations (VI-3) to (VI-5)

Dividing Equation (VI-1) by v2 one obtains

ro
U eC) U Vv a U 13 xx
wT Gita =e GO Fey 6 a
uS Ox TS LC Yr or ous (A-II-1)
where U, is the jet efflux velocity. The conservation of
momentum requires that
“ 2
| rv dr co) = Constant (A-II-2)
° °

Now denoting

qe
12.
oP:
=
\°

= £(n) =

|
K
cs
3

m

117


9S

—_—_—
mean

1,12

1,08

1.04

1.00

0.96

0.92

T T T T— T

° aie 970 microns

a x 780 J
A 527
8 1
9 o 343
. o 4
o
x =
a 2) x
°
A
al x -
o
L L a1 L L
10 20 30

FIG.

x/D

21 VARIATION OF THE INTEGRATED CONCENTRATION FLUX
ALONG THE AXIAL DIRECTION


€ instantaneous DC voltage output of the total signal

E instantaneous DC voltage output of the turbulent
signal

xX numerical constant related to mixing length

) anf

Ue Up particle parameter, defined in Equation (X-1) and
(X-5), respectively

Ay longitudinal macroscale

E(£) energy spectrum in terms of frequency

F(K) energy spectrum in terms of wave number

F(n) [Ranz my ,

£(t) half length of the wire exposed to the air bubble

£(n) U/U,

g(n) V/U,,

g(w) Fourier coefficient of E(f)

h(n) 5, -/ 002

I(t) instantaneous current across the sensor

K(n) var

r(y,t) electric resistivity per unit length at location y
and at time t

Ty (t) instantaneous bubble temperature

TY, (y-t) wire temperature at position y and time t

vy) velocity in a flow field possessing gradient

o(&) D/ (x+a)

We) U,/T5

¥ volume of one bubble

tT particle Response time (general)

T. Stokesian Particle Response time

114


74

LOOP T T i: ea | mn i oy ~
ty
P23
OA
A a
0.9}- o PN, Fy 4 06,0272 7241-06)
a
a
0.8F * my ° A
x
0.7- 5
0.6F 4
P.
Pn
0.5- 5
O.4E °o |
0.35 e x/D=10 4 aN
x x/D=15
A x/D=20°
0.2F 4
@ x/D=25
0.1 - (a= 343 pt) 4
1 1 1 L AL 1
() 0.02 0.04 0.06 0.08 O10 O12 O14
q
FIG. 32 CONCENTRATION PROFILE


photographic techniques and hot film anemometer sampler).

The use of the laser system is discussed in a separate report.
The hot film anemometer system was selected in this first

phase of the work as the bubble concentration sampler primarily
because of its availability. The method described herein was
originally developed by Goldschmidt (1965) and then modified
by Goldschmidt and Householder (1969) for the diffusion
measurement of oil aerosols in a two dimensional air jet.

The applicability of the hot film anemometer for the bubble

in water measurement was first described by Hsu et al (1963)
and then by Delhaye (1969) and Chuang and Goldschmidt (1969).
The principles and the techniques for the application of the
hot film sensor to bubble concentration measurement are
described in the following.

Principle

The mean concentration of the bubbles at a certain position of
an axi-symmetric turbulent jet can be obtained by knowing the
mean concentration flux and mean velocity component at that
location. The mean concentration flux is measured by counting
the number of bubbles impacting on a hot film sensor. The
heated sensor of a hot film anemometer when placed in a
turbulent water stream contaminated with gas bubbles will
suffer unsteady cooling dueto the carrier stream velocity
fluctuations and the low heat capacity gas bubbles which
impact on the sensor. The response of the hot film anemometer
to a bubble in a laminar water stream has been extensively
discussed and is given in Chuang and Goldschmidt (1969).,

(see section XVI).

By discrimination, either through frequency filters or amplitude
discriminators, the decrease of the cooling signal due to

the impaction of the gas bubbles may be isolated and measured.
Since the characteristic frequency of the bubble impaction
signal is above 1 KHz and most of the turbulent energy of the
carrier stream is below 1 KHz, frequency discrimination was
adopted.

The possible errors due to the uncertainty in the impaction
coefficient caused by the curving pathlines around the
cylindrical sensor are noted in Section VIII.

Frequency Discrimination

In order to use the hot film anemometer as a bubble sampler

in a turbulent flow field, it is necessary to frequency dis-
criminate the decrease of the cooling signal from the impacting
bubble. The electronics used to accomplish this are shown in
Figure 7. The band pass filter was set at the analyzing
frequency where the impacting signal has the highest energy
density. Discrimination is then possible provided that most

of the energy due to the turbulent cooling of the fluid lies

at values of frequency below that selected. It is desired,

26


With t defined, the values of y,, in Equation (X-5) can be
calculated for various measureménts. These are shown in
Table 5.

'
Figure 45 shows the relationship now between S -1 and v,,.
Better agreement is now noted. However the deviation bé-
tween the circular jet measurements and the two dimensional
flow field measurements (together with the theoretical two-
dimensional prediction) is still noted.

In comparing the results of the present bubble diffusion
measurements with the other diffusion measurements in circu-
lar jets with solid particles as contaminants (i.e. sediment
in Singamsetti's and glass beads in Brush's), it is seen that
the agreement is very good except for a point in Brush's data.
This interesting result could be explained in the following
conjecture. For a relatively small bubble moving in a turbu-
lent liquid stream the bubble shape will remain approximately
spherical due to the surface tension effect. If this is the
case, the bubble can be considered as a rigid particle. The
density effect of the particle has been accounted for by the
particle parameter y,. Consequently the, measured results
unified through yy, give a comparable S - ¥,, relationship
as shown in Figuré 45. Such results iéaa to_the conclusion
that with v1, so defined the dependence of S on the particle
parameter vE is universal for measurements tn similar flow
fields regafdless on whether the particle is heavier or lighter
than the_qarrier stream. In other words the relationship be-
tween S and ~, is the same for similar flow fields for
bubbles “in water; solid particles in water or even aerosols
in air streams.

One point yet unclarified, as noted previously, is that a two-
dimensional and a three~dimensional flow field, appear to
give different trends of S - 3. This dilemma could be
clarified by extending the experimental study to larger pp,
domain, such as measuring the diffusion of larger aerosols

in both circular and plane air jets to see whether there
exists a large deviation in between these two.

The relationship for § “1 versus j,, as just described is ori-
ginated form a rather intuitive background. The theoretical
basis for the universality of such relationship is rather
weak, and although confirmed by experiments, it deserves
further study.

92


SECTION IIT

INTRODUCTION

A prediction of the concentration distribution of scalar
contaminants (such as heat, dye, particles or bubbles) introduced
in a turbulent flow field requires a knowledge of the corres-
ponding turbulent transport coefficients. This work concerns
itself with the transport of discrete contaminants suspended

in turbulent flows. The specific questions being answered are:
Is the coefficient describing the turbulent transport of sus-
pended particles equal to that of momentum? And if it is not,
how does it depend on the size of the particle-contaminant?

The specific set of experiments reported herein are for the
dispersion of gas bubbles in a turbulent water jet. They were
performed simultaneously with a separately funded study on

the dispersion of liquid droplets in air jets. In sections IX
and X the results are compared and generalized. It is shown
that for contaminants with a certain size and specific gravity
range it is not possible to assume equality of the turbulent
fluid and particle transport coefficients. The results correlate
sufficiently well to permint an estimate of the actual turbulent
transport coefficient for a given contaminant to carrier stream
density ratio, contaminant size, and characteristics of the
turbulent flow.

The stated objectives of this work were twofold. One to
establish an effective method of tracking contaminants in a
liquid jet, including the use of a laser system. Two to
determine the diffusion due to turbulence of contaminants of
different size and density. The second objective was fully
met. This report deals directly with it. On the other hand
the first objective was only partly met. It will be reported
on in a separate report. The second objective was met through
the research of Dr. Strong C. Chuang. His doctoral thesis was
completed on August 1970. The research into the use of a laser
as a sampling device was conducted by Mr. Charles H. Tinsley as
part of his M. S. studies at Purdue University. His Master's
Thesis is currently (May 1971) in the final write-up stages.
Mr. -Tinsley's research and study program is presently being
financed through university funds.


0.2

o x/D=10 —_
¥ 15
A 20
o 25

(4, = 5274) 4
al al 1 L Lt 1

0,02 0.04 0.06 008 0.10 0.12 0,14

n

FIG. 33 CONCENTRATION PROFILE

75


o ° x
1.0 Aw x 4 oT +r 7 1 :
oO °
a °
o KA Eq. (IX-4) with €,/€y = 0.98
L XN A : miCy 9
xDo
~
a
0.8 ° -
wv
8
L. a J
A
x x
0.6 4b Rg AN,
()
Le
fob
Tm
O4F
q eo x/D=10 4
x 15
A 20
0.2 Fr 4
a 25
= (a,= 780 2) -
4 J 4 1 1 n
0 0.02 0,.04 0.06 a08 aio 0.12 0.14
FIG, 38 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

81


No.

38
39
40
41
42
43
44
45

Concentration
Concentration
Concentration
Concentration
Concentration
Concentration
Dependence of

Dependence of

Profile
Profile
Profile
Profile
Profile

Profile

t
-1

St

(With
(With

(With

-(With

(With

(With

viii

Bubble
Bubble
Bubble
Bubble
Bubble

Bubble

Rising Vel.
Rising Vel.
Rising Vel.
Rising Vel.
Rising Vel.

Rising Vel.

Corr.)
Corr.)
Corr.)
Corr.)
Corr.)

Corr.)

Page
81

82
83
84
85
86

s_! on Particle Paramenter Vor se ee + 89

on Particle Parameter Vp ee ew ew ee OA


= = iy

<

velocity vector of water

Weber number

instantaneous tangential velocity
tangential fluctuating velocity
axial coordinate

potential core distance of the jet

axial distance between dynamic and static taps
of a pitot tube

fluid density, water density, gas density

_v
0, (1-'¥)
Reynolds stress
Maximum quantity of (.)

r/ (xt+a)

(o?/ 2) 71730

(x+a) /D

mean concentration

concentration at efflux of the jet
instantaneous concentration

fluctuating component of the concentration
tangential coordinate

kinematic viscosity

eddy diffusivity of momentum, bubble and marked
fluid particle respectively

mass diffusivity (molecular effect)

113


0.9

0.8

0.7

0.6
&
Po
0.5
0.4
ie)
x
NX
N
0.3 o x/D=10 A YX
x x/D=15 ° _~
A x/D=20 ‘
0.2 4
o x/D=25
0-1 (ay= 343)4) 4
1 1 L 1 1 1
0 0.02 ° 004 0.06 0.08 0.10 012 O18

q

FIG. 36 CONCENTRATION PROFILE

79


SECTION XV

GLOSSARY OF SYMBOLS

ALA, numerical constants

A intercepting area of the sensor

a virtual origin of the jet, defined by equation VI-4
and VI-5

ay 1 agra coefficients given in Eg. (VI-15)

b characteristic jet width; temperature coefficient
of the electric resistivity of the sensor

by half width of the velocity profile

b, half width of the concentration flux profile

by half width of the concentration profile

cy drag coefficient

Cc, specific heat of the sensor

c correction factor in pitot tube measurement

cy numerical constant related to the vorticity transport
mixing length

cy numerical constant related to the scalar quantity
transport mixing length

D jet nozzle diameter
bubble diameter

a 4/4;

a; inside diameter of the syringe tube

qd, outside diameter of the syringe tube

ds 4/4;

ay sensor diameter

E mean DC voltage output (measured)

e peak voltage drop in the sensor due to an impacting

P bubble

109


Stokes range. For the present and the other two circu-

lar jet measurements where the particle Reynolds numbers

are in the order of 10% to 10° another particle parameter has
to be defined. Let this be

T (X-5)

where tT is the actual particle response time. An expression
is then sought for T.

Neglecting the potential force, nonlinear effect and Basset
term, the B.B.O. Equation takes the form

2

7a
tT 43

a _ 3
6 ob 'p aE (Mp) = Cp Goal y- ve

2
|

lt a
+ 354 pee aElYp - Vel

(X-6)
where v, and vg are respectively the particle and fluid

velocitfes. Assuming one dimensional steady flow such that
dv

mes = 0, Equation (X-6) simplifies to
3¢,
a D 2 = -
a Ive - vel + 3p Iv, - vgl% = 0 (X-7)
2d, (+P +1)
‘b Pe

Note that in Stokesian motion Cp=24/Re =24v/| vp - vg|dp. The
expression for the Stokesian response Eime, Ts, as defined in
Equation (X-4) follows from Equation (X-7).

Consider first the case where the particle Reynolds number is

in between 2 and 500. In this Reynolds number range, Bird,
Stewart and Lightfoot (1960) suggested that

18.5 18.5

Cp = R 375 ~ TV “Ve a, 375
&p { v : } (X-8)
Equation (X-7) then simplifies to
3/5
a _ ill iv _ vy JI/5 . _
aE Iv, vel + Ive Vel ty) (X-9)

8/5 5°p
4a,07° (258 +1)

90


1.0

0.9

0.8

0.7

0.6

ele

0.5

0.4

0.3

0.2

0.1

58

q T T T T T
84 .
AR o x/D=10
ry * 15 4
ck
A 20
A
4 o 25
& 4
°
A
&
oA 4
x
is)
& 4
x
°
A
®
o
i, 4
4a
°
®
Ao 4
“FP
AX
o
4 4
g« °
o
My
Ao °
(a, = 5272) °o 7
n n 1 l n
0.2 0.4 0.6 0.8 1.0 1.2
r/2b,,
FIG. 23 CONCENTRATION FLUX PROFILE


viscous forces only. He further assumed that the stream
flow field is not affected by the presence of the parti-
cles. He obtained a solution for two extreme cases. The
first case is when the Reynolds number is extremely low
such that the stream is solely viscous flow; and the
second case is when the Reynolds number is very high and
the stream is assumed non-viscous. He went further by
empirically extrapolating two solutions into the middle
Reynolds number range by fitting a model simultaneously
satisfying the two extreme cases. It is attempted here,
by applying such results, to predict the possible cor-
rection on the intercepting area A_ at the different
locations of the flow field. In this work the range of
the carrier stream velocity in the axi-symmetric water

jet within which the bubble concentration flux measurements
are performed is from 1.4 to 7 feet per second, and the
corresponding range of the collector Reynolds number is
from 24 to 120 based on sensor diameter. Based on the
reported results the maximum difference in the impaction
coefficient would then be less than 2 percent. This error
is within the accuracy of the concentration flux measuring
system and any correction in the impaction coefficient

was neglected.

If the impaction coefficient is then assumed constant

in the whole flow field and if furthermore the partial

hit effect is similar throughout, then for e/e, fixed

the intercepting area, A_,within which every bubble hitting
the sensor will give a signal perceptible by the meas-
uring system will be constant. e and e_ are as defined

in the Appendix; for all sets of measurfments e/e_ was
fixed to be 0.15. P

The absolute concentration flux was measured by two dif-
ferent means. First the flux over a whole cross section
was measured at the jet nozzle by counting the number of
bubbles generated per unit time. This was achieved by
snychronizing the bubble generation frequency with a
stroboscope. Second the actual concentration flux meas-
urements were integrated over a whole cross sectional
area assuming an impaction coefficient and neglecting
partial hit effects. The actual wire sampling area was
6.5x1074cm2.° A count of 200/min. on the counter yould
then cgrrespond to a concentration flux of 4.8x10° counts
per ft“ per second.

54


relationship as given in Equation (VI-5). The Reynolds stress
o i> expressed as 6 = -puv, where u and v are respectively
thE fluctuation velocity components in the x and r directions.
In this derivation the "kinematic" and "dynamic" virtual ori-
gins have been assumed to be equivalent. The experimental
results for the particular jet will confirm this. In general
(see Flora & Goldschmidt (1969)) this néed not be true. In
analogy with the coefficient of viscosity in Stokes' law for
laminar flows, Boussinesq introduced the kinematic eddy vis-
cosity, Ent for the Reynolds stress in turbulent flows by
putting

ou

fo = Pen Or (VI-6)

Note that €_ accounts only for the turbulent transport. The
molecular transport is neglected since it is approximately three
orders of magnitude smaller than the transport due to turbulence.
Substituting Equation (VI-6) into Equation (VI-3), the relation
between ¢, and U/U, reads:

n
U
en =- Un ih an " G (VI-7)
UL, (xta) d U

1 an ‘

An assumption concerning ¢€_ has to be made in order to calculate
the velocity distribution.

First Model: (Constant Eddy Viscosity)

Prandtl's mixing length hypothesis leads to a simplified expres-
sion for the eddy viscosity (Schlichting (1960)) as follows:

— 92 |3U
ey = g z|

In the above %£ is the mixing length, a length scale related to
the size of the fluid lump in the turbulent flow field. If it
is further assumed that the dimensions of the lumps of fluid
which move in a transverse direction during turbulent mixing are
of the same order of magnitude as the width of the mixing zone
and that the mixing length is proportional to the half width,
by of the mixing zone then

En = Xby Uy (VI-8)
Here x denotes a dimensionless constant to be determined exper-
imentally. From conversation of momentum, Schlichting (1960)
showed that for a turbulent circular jet

by _, (VI-9)

and
~ ycl
Un x

16


constant eddy viscosity, e_, is then equal to 0.0027U
(x+a) and the resulting velocity profile reads: m

we = (1 + 46.02n7)7? (vIII-7)
m

The similar regression procedure applied to the gecond
proposed model yields a numerical coefficient (C“/2)7~ /3
equal’ to 11.8 and a velocity profile given by

U__ l+2v2 (11.8n) 9/7 + a2 (11.8n)2 1363V2_ (11.8n) 9/2
a1

Un 3/3 6123673
(VIII-8)
Similarily, the third proposed model gives:
2
=e -73.04n (VIII-9)

3q|¢

The analytical curves given in Equations (VIII-7) through
(VIII-9) are plotted together with the measured data in
Figure 9. It is seen that the velocity profile given by
Equation (VIII-7) fits the experiment data best of all.

As discussed in Chapter VI, the corresponding eddy viscos-
ities «| for the second and third models are not constant
in the Plow field. They depend on n as described in
Equation (VI-16) and (VI-18), respectively. Figure 15
depicts the dependence of e«, on n in this particular jet
flow measurement as inferred for the three different models.

FLUCTUATION VELOCITY MEASUREMENT

Since the velocity components in a turbulent jet flow field
are random functions of space and time, it would be necessary
to measure moments and cross moments of all orders to under-
stand the whole statistics of the flow field. In this study
only the rms value of the axial fluctuation component was
measured,

The sensor was calibrated in the potential core of the water
jet. The calibration curve is shown in Figure 16. From
Figure 16 the corresponding sensitivity S_, the slope of the
calibration curve which is directly measu¥ed from the cali-
bration curve, can be obtained. It is shown in Figure 17.

The fluctuation level is then defined as ve?/S where ve*

is the rms value of the fluctuation voltage output. The
axial turbulent levels were measured at x/D = 10, 15, 20,

and 25. The data are shown in Figure 18. It is seen that
the turbulent levels at each section are approximately similar

45


6h

ut
wt J |
Un ;
0.2 L 4
a / 4
L fa % °° (7
A , o *
I i ~~ x/D=#10 Corrsin \
alka ——~ x/D=20 Corrsin 4 1
& © *x/D=10 a
r /. a x/D=15 ofNo |
of ° A. x/D=.20 o Kn 6
0.24 L 2/4 0 (x/D=25 aX. a
/ ° \S
ofa . n t 1 1 oN
2.0 -1.0 oO 1.0 2.0
r/by

FIG. 18 TRANSVERSE

DISTRIBUTION OF TURBULENCE INTENSITIES


The corresponding theoretical curves calculated with the best
fit coefficients of e,/ey are shown in Figures 36 to 39.

In order to see how sensitive the plots given in Figures 36
to 39 are to a change in St the curves of Equation (IX-4)with
S,=0.8, 1.0 and 1.4 are also included in Figure 36. The
numerical values of €,/ey leading to a best fit are shown in
Table 4.

In comparing the general shape of the theoretical curves with
the measurements, it is seen that the theoretical curves show
a greater curvature at small values of n. The general shape
of the curves does not agree with the measured concentration
profiles very well. The same deviation (between measured
values and model) was observed in the velocity profiles as
shown in Figure 9.

Third Model: (Reichardt's Inductive Theory)

The data can also be analyzed considering the third model
proposed. Assuming that the velocity profile is given by
Equation (VIII-9), the corresponding concentration profiles
calculated from Equation (IxX-2) for the 343, 527, 780 and

970 micron bubbles are shown in Figures 40 to 43. From
Equations (VIII-9) and VI-28) the assumed concentration profile
takes the form

v fm 2
— = Exp} - 73.04 —n (IxX-5)
vim . ey

The corresponding turbulent Schmidt number ep/e, leading to a
best fit, as shown by the plots of Figures 40 to 43 are tabulated
in Table 4. The calculated curves show a smaller curvature near
the center line region and under predict near the edge of the
jet. The general shape of the analytical curve does not agree
very well with the measured data. The same deviation has been
observed in the velocity profile when calculated using this

model as shown in Figure 9.

The comparison of the analytical and semi-empirical curves
given by the above models indicates that the first model seems
to fit the measured concentration profile best. Hence it is
concluded that the particle diffusivity and the corresponding
turbulent Schmidt number obtained from the first model should
be the most reliable of the three. No thorough statistical
Measures were sought to compare the predictions made by the
three models. An attempt at using a simple regression dnalysis
did not lead to a better interpretation of the results.

This was primarily due to the spread of the data.

78


diffusivities is due to differences in the possible combinations
of u and v as compared to the combinations of v and Y , thus
leading to a difference between momentum transfer and sediment
transfer ( u and v are respectively the longitudinal and lateral
velocity fluctuations and Y is the fluctuation of the sediment
concentration). Vanoni argued that during acceleration the
relative velocity between the sediment and the fluid must be
greater than the fall velocity in the quiescent fluid and thus
the sediment is carried a shorter distance than assumed. This
tends to make ey less than ep.

Carstens (1952) solved the equation of motion of a sphere in
an oscillating fluid, omitting the gravitational force. For

a motion of the oscillating fluid specified by x=x9 sin wt,

he showed that the motion of the sphere may be described by
Xs=Xso sin (wt+¢), in which ¢ is a phase difference. By
assuming such an oscillation to be prevailing in a sediment-
laden flow field, he concluded that for sediment particles in
water ey/em can never be greater than 1. He developed a mathe-
matical’ expression for the ratio ¢e y/Em for an oscillating
spherical particle in a liquid flow field. It reads:

fy _ (143N/2)? + L(3N/2) (24n) 1°

en 5c 1 2
(s — + z + any” + [(3N/2) (14N) ] (II-1)
where N = /8V/(W/d)_, an oscillation parameter and w is circular
frequency.

Ismail (1952) performed experiments with suspended sediment in
a closed channel. He obtained ¢,/e,=1.5 for 0.1 mm sand and
1.3 for 0.16 mm sand. Brush et al (1965) applied corrections
to the fall velocity term and showed that the values would be
reduced to 1 and 0.6, respectively.

Longwell and Weiss (1953) assumed the velocity fluctuations to
be sinusoidal functions of time, and by solving the equation
of motion with certain simplifications they showed that the
eddy diffusivity of liquid drops in a turbulent gas stream
should be less than the eddy diffusivity of the marked gas
particles.

Laursen (1958) measured the velocity distribution and the
sediment concentration distribution for 0.04 mm and 0.1 mm sands.
Without correcting the concentration by accounting for the actual
fall velocity, and assuming Karman's K to be a constant and

that the turbulence characteristics were unchanged by the
particles, he concluded that ¢,/e,>1.

Smith (1959) investigated the expansion of a falling cluster of
particles in an isotropic homogeneous turbulent flow field. He

11


€s

0.8

OW

FIG,

x .
Dp 7 0-0092(F + 2.5)

x/D

20 LONGITUDINAL MACRO SCALE ALONG THE

TURBULENT JET AXIS

30


CONTENTS

Section
I Conclusions
Ir Recommendations
III Introduction
Iv Statement of The Problem
Vv Literature Review
Vi Theoretical Background
VII Experimental Set-up and Measurement
Principles
VIII Experimental Results
IX Turbulent Diffusivities
xX Turbulent Diffusion Characteristics
XI Significance of Results
XII Acknowledgements
XIII References
XIV Publications
xv Glossary of Symbols
XVI Appendices

15
21

35
71
87
95
97
99
107
109

115


x/D With Bubbles Without Bubbles Percent Decrease

10 0.220 0.225 2
15 0.275 0.285 3
20 0.318 0.336 5
25 0.305 0.35 12

It is seen that the fluctuation levels are attenuated due to
the bubble contamination. This is especially so at large
axial distances.

The result seems reasonable since the relative motion of the
contaminants would dissipate more kinetic energy and hence damp
out the turbulent fluctuations. At a relatively large x/D

the range of mixing, and hence the amount of energy dissipated,
will be greater, and consequently the deviation between the

two sets of turbulent fluctuation measurements (with and
without the bubbles) becomes greater. This result, as dis-
cussed previously, has been observed by Hetsroni and Sokolov
(1969).

CONCENTRATION PROFILES

It must be noted that the measured quantities were the
concentration flux, N, and the mean velocity, U. From these
the corresponding half widths, by and b, were determined.
The actual concentration profiles,‘, are obtained from the
measured N and U as discussed in the following section.

When looking at figures such as Figure 26 it must be noted
that although the concentration flux half width always grows
slower than the velocity half width, the concentration half
width need not.

70


and checked with the hot film anemometer. The pitot tube used
has an outside diameter of 3/32 inch and a dynamic tap opening
of 1/32 inch. Two lines of 4 mm inside diameter tygon tubings
were connected from the pitot tube to a differential manometer.
The indicating fluid of the manometer was Meriam No. D-8325 fluid
with a specific gravity equal to 1.75.

The speed of reading depends on the length and the diameter
of the pressure passages inside the probe, the size and the
length of the pressure tubes to the manometer, and the dis-~
placement volume of the manometer. The time constant in this
system was found to be in the order of a few seconds.

The mean velocity measurement was also checked by a TSI 1217-20w,
(Thermo Systems Inc., 2500 Cleveland Ave., N. Saint Paul,
Minnesota 55113) quartz coated hot film and TSI 1050 anemometer
with TSI 1025 Temperature Compensator in series. The axial
fluctuation velocity components were also measured with the

same hot film sensor.

BUBBLE GENERATION

Generation of a more or less uniform series of air bubbles is
possible when supplying a constant air pressure to a small
tube submerged in the water. The intent was to generate
bubbles and introduce them upstream of the jet mouth in order
to permit the study of their undisturbed motion in the
turbulent jet. The objective was fulfilled with the system
shown in Figure 1. A long glass tube, drawn to a small diameter
nozzle at the tip was introduced at the base of the jet tank.
Use of a constant pressure nitrogen source (filtered and
carefully regulated) resulted in bubbles being generated
slightly upstream of the jet mouth.

The sizes of the syringetubes used were very small when compared
with the jet nozzle. The outside diameter of the capillary
tubes used was less than.120 microns. (The ratio of the tube
cross sectional area to the jet nozzle opening area is
correspondingly less than 1074.) The fragile nature of the
glass nozzles caused frequent breakage. The smallest one used
had a tip inside diameter of 13.6 microns, and the largest

one used had a tip inside diameter of 96 microns.

An analytical and experimental study of bubble formation due
to a submerged capillary tube in quiescent and co-flowing streams

has been made, and published (Chuang and Goldschmidt (1970)).
An abstract is included in the Appendix of Section XVI.

CONCENTRATION FLUX MEASURING SYSTEM

Several techniques for measuring the concentration of the
bubbles were considered (such asa laser sampling device,

25


89

" T ms T T —
44° 2 0 2%
r a®"8qy pada
o*# aéoa
L o4 3 3 oA
F.) A, baat wa. :
0.3 s20 4 Gee cAD
L. ao eA? 2 f] g
go PAF at atte :
L A * a Foy o, on? i
a) ° ° 8°
L ea a4 ; eo a
e
. ae 4 2A 64
A 3 1
a o 3 2 a
rey ° ° e °
a eo A %45 0
0.2 ° a
: . eo 4
} b @ x/D=19 o os °
y ° a x/D=1 \ ° 4
oo 4 A ee with bubbles
r 8 1) =2 J A o |
e xX, e2
» 4 5 sa 7 yg
5 a 4 a a 4
| 1 4° © x/D=10 °
o x/D=15 | without o4 8 4
2, ° A x/D=20 { bubbles ae
1 Loy o x/D=25 $ 6° (Ol
ray of 1
2 R n ! a t %m o
-2.0 -1.0 ie) 1.0 2.0
t/d,

FIG. 30 VELOCITY DISTRIBUTION ALONG JET AXIS (WITH & WITHOUT BUBBLE CONTAMINATION)


Half Width Dependence

The values of b. versus x/D for four different bubble sizes
are plotted in BR dimensionless form as b /D versus x/D

in Figure 26. Figure 26 indicates that Phe concentration
flux half width spreads linearly along the axial direction
from a virtual origin at 0.8D from the jet nozzle plane.
Coincidentally, this coincides with the virtual origin

of the momentum flux. The plots of Figure 26 give:

for a, = 343u bi/p = 0.0650(x/D + 0.8) (VIII-13)
a, = 527u bi/D = 0.0683(x/D + 0.8) (VIII-14)
a, = 780n b,/D = 0.0708 (x/D + 0.8) {VIII-15)
a, = 970u b/d = 0.0735(x/D + 0.8) (VIII-16)

From the above four expressions it is seen that the rate

of spread of the concentration flux half width consistently
increases when the bubble size increases. Qualitatively
this implies that larger size bubbles diffuse faster.

To evaluate the diffusion coefficients a careful quan-
titative analysis accounting for all the measured data
will be done in the next Section.

Center Line Concentration Flux Decay

From the similarity requirement and the principle of mass
conservation, the following relations must be true

©
Nb? ~ | Nrdr = Constant (VIII-17)
mn °

Furthermore, from the form of equations (VIII~13) to
(VIII-16)

2 2

N. b’ ~ N. x” = Constant (VIII-18)
mon TMm

and hence
Na V2 (VIII-19)

The center line decay of the concentration flux is thus
plotted as (N /N,,) 1/2 versus x/D in Figure 27, where N,
is defined as’,

2n | Surdr
°
wD
4

~

N= a, = 82 [ wear (VIII-20)
°

fe} s

61


Third Model: (Reichardt's Inductive Theory)

Another solution is obtained by applying Reichardt's inductive
theory and transforming Equation (VI-1) into a diffusion type of
equation (Hinze (1959)). The theory yields an exponential solution:

Dos Kn (VI-17)

Substituting Equation (VI-17) into Equation (VI-7) the corresponding
eddy viscosity becomes

€
mTMm i- UW

It is worthy to note that the above three models have all been
used in the literature, together with other solutions not reported
herein. Equations (VI-11), (VI-14) and (VI-17) result from
different models used to obtain an expression relating the Reynolds
stress to the mean flow properties. Which one of the listed models
is more suitable can only be determined by a comparison with
accurate measurements of the mean velocity and the Reynolds stress
or by an up to now unavailable rigorous formulation for the eddy
viscosity itself. For comparison purposes all of the models
described will be used in the discussion that follows.

SCALAR QUANTITY TRANSPORT IN AN AXI-SYMMETRIC JET

The equation of transport for a scalar quantity in an axi-symmetric
jet (Appendix IV) reads:

v v —
ve + va =- i - (x vy) (VI-19)
where y is the fluctuation component of the scalar quantity. The
above is limited, again, to (1) steady mean motion, (2) an axi-
symmetric flow field, (3) a narrow turbulent region in the radial
direction when compared with the axial direction, and (4) neglect
of molecular transport.
Assuming that the turbulent transport of ‘ can be described by
analogy to Fick's law, using a coefficient of diffusivity Ey, the
following expression is postulated:
— av
| (VI-20)

- v= ey 35

Again, as for €m, €y doesn't account for the molecular effects. By
substituting Equation (VI-20) into Equation (VI-19) and invoking
conservation of ‘, the distribution of the scalar quantity becomes
(Appendix IV):

18


SECTION VIII

EXPERIMENTAL RESULTS

Mean velocity profiles in subsonic axi-symmetric jets have
been widely investigated, for instance Kuethe (1935),
Corrsin (1943), Squire and Trouncer (1944), Hinze and Van
Der Hegge Zijnen (1948), Albertson et al (1948), Keagy and
Weller (1949), Corrsin and Uberoi (1949), Trentacoste and
Sforza (1967) in air jets and Binnie (1942), Forstall and
Gaylord (1955), and Liu (1967) in water jets. In general,
there is a satisfactory agreement among the experimental
results obtained by the various investigators. For axial
coordinate x greater than the length of the potential core
x (as defined in Appendix II), the following are observed:
(f) radial distributions showing good similarity in con-
secutive sections, (2) a linear spread of the jet, and (3)
a hyperbolic decrease of the jet center-line velocity with
distance from some apparent source. These three character-
istics have been already noted in Section VI.

The jet flow field used in this experiment is somewhat dif-
ferent from the ordinary jet in that there is a bubble
generating capillary tube in the center of the jet nozzle.
Hence the well known characteristic of the jet mean flow
field obtained by the previous workers probably is not
applicable to this specific jet flow. An actual flow field
measurement becomes a necessity.

MEAN VELOCITY PROFILE MEASUREMENT AND ITS CHARACTERISTICS

Measured Results

The axial velocity profiles were measured with a pitot
static tube and checked with a hot film anemometer. The
efflux velocity of the jet was 10.83 feet per second.

The corresponding Reynolds number was 45,000, based on
orifice diameter. The velocity profiles were measured

at x/D = 10, 15, 20, 25 and 30. Figure 9 shows a plot of
U/U_, versus n, where n is the dimensionless variable defined
in Appendix II as r/(xta). The assumed similarity is con-
firmed. The predicted curves shown in Figure 9 will be
further discussed in a later part of this section.


TABLE 2

Total Bubble Concentration of the Experiments

No (EFETux bubble rate,I/7sec.)

Diffusion Efflux Volume

Experiments by Stroboscope by Hot Film Concentration
Measurement Measurement (percent)

343u case 9.64x10° 11.13x103 0.048

527u case 6.26x10° 5.93x107 0.116

780u case 4.24x103 4.02x10° 0.252

970u case 3.41x107 2.97x10° 0.390

The concentration flux was measured at x/D=10, 15, 20 and
25. All the data were obtained in a form of N versus r.

The data so obtained were normalized in terms of Nae

Check for Bubble Coalescence and/or Breakage

If there is neither bubble breakage nor bubble coalescence,
the conservation of mass requires that at each successive
section the following relation must be satisfied:

o Nrdr=M

(VIII-12)

Equation (VIII-12) was calculated by graphical integration
for four sets of measurements. Figure 21 is the plot of
M normalized by the mean of the four section measurements.
It shows that at four different sections deviated by

less than 6 percent from the mean.

As a 6 percent devia-

tion is probably due to the inaccuracy of the measurements,

M can be considered to be essentially constant.

This sug-

gests the absence of coalescence and/or breakage of the

bubbles. \

Transverse Profiles of the Concentration Flux

In order to check the similarity of the concentration
flux profiles, the measured data were normalized and

plotted as N/N_(r/b_.), where N_ is the maximum value of

the concentratton ftux profile and b

is the half width

of the concentration flux profile dePined as the radial

distance from the center line to the point where the
The

concentration flux is one-half the maximum value.
resulting plots for four bubble sizes are shown in Fig-
ures 22 through 25. All of these figures indicate that
the similarity characteristics of the concentration flux

are preserved.

35


SECTION VII

EXPERIMENTAL SETUP AND MEASUREMENT PRINCIPLES

The main purpose of the experimental part of this work is the
measurement of the concentration distribution of bubbles in

an axi-symmetric jet. The basic setup employs a submerged
axi-symmetric water jet as the flow field, a series of constant
size bubbles generated through a capillary tube placed at

the center of the mouth of the jet as the contaminant, and a
hot film sensor as the bubble concentration flux sampler.

The experimental equipment and the measurement procedures are
described in more detail in the following sections.

JET FLOW PIELD
Axi-symmetric Turbulent Water Jet

A schematic diagram of the experimental setup is shown in
Figure 1. The system consists of a constant head tank, a

jet tank, a reservoir, a pump and a water filter. The constant
head tank has an adjustable elevation permitting changes of

the jet efflux velocity. In addition, two interchangeable
nozzle contracting sections, of 1/4 and 1/2 inch diameter,
permit further variation of the efflux velocity of the submerged
jet. The maximum jet velocity reached with the non-metallic
1/4 HP pump and a Cuno Model SS-8 super flow water filter is

20 feet per second. This corresponds to a Reynolds number
based on the jet nozzle diameter in the order of 8x10‘.

Figure 2 shows the constant head tank, water filter and pump.

A deionizer, also shown in Figure 2, is used to treat the
water introduced into the flow system.

The entire flow system is mounted on a level steel frame
structure. Figure 3 shows a photograph of the jet tank. It
is rectangular in shape to permit optical inspection of the
flow field. The top edge of the side wall serves as an over-
flowing weir so as to maintain a constant water level inside
the jet tank. The side gutter which is attached to the outside
of the tank wall collects the overflow water and returns it

to the reservoir. The overall dimensions of the tank are
18"x18"x25" in height. The jet tank provides a turbulent jet
flow field with an axial distance of up to 40 jet diameters
when the 1/2 inch nozzle is in use. A traversing mechanism is
included to be able to position the measuring probe in the
jet. As designed, it allows variations in position in the
longitudinal, transverse and vertical directions.

Velocity Measuring System

The mean velocity component of the turbulent water jet was
measured with a straight-leg Prandtl-type Pitot-static tube

21


for 10<x/D<25, computed along the axis. The resulting re-
lationship between the inverse of the turbulent Schmidt

number and the particle parameter ,, for the four bubble sizes
is shown in Figure 44. The dynamic’ numerical prediction of
Ahmadi (1970) and the data for the pipe and open channel ex-
periments collected by Householder and Goldshmidt (1969) are
also included. In addition, results of two other jet diffusion
experiments by Singamsetti (1966) and Brush (1962), one plane
jet diffusion experiment by Householder (1968) and one “iso-
tropic" turbulent diffusion experiment by Kennedy (1965) are
included in Figure 44. The results of Householder include

one questionable data point currently under a separate study.

From Figure 44, it is seen that the data obtained from the
circular jet measurements (i.e. Singamsetti, Brush and present
study) seem to fall below the other measurements and Ahmadi's
prediction.

Such deviation suggests that the relationship between S, and
v, May have different trends, one for the reported pipe;
channel and plane jet measurements (which follow the theore-
tical curve predicted by Ahmadi) and another for the circular
jet measurements.

The difference between these may be either due to an essential
difference in their geometries or to some inherent difference

in the specific experiments reported. The predictions of Ahmadi,
as well as the selection of ,, assumed that the particle Rey-
nolds number (which according’ to Ahmadi is defined as

R, = 2Vu2 J) is in the order of one or smaller. This condi-

tiSn was satisfied for Householder's experiment and most of the
data gathered by him. Unfortunately it was not for the current
measurements nor for the other reported for circular jets (all
of which has values of Ro in the order of 10°-10~).

It is desirable to attempt to find another particle parameter.
Hopefully a university relationship could be found between Sy
and this new particle parameter

Dependence of Turbulent Schmidt Number on the Particle Parame-
ter
T

Recalling the definition of Vp, one can decompose it as

—= 2 to} =

2d 2 2
~36u°"%b (“pp _ 36u _
Yo = “TK sev (Get DT Ets (x4)

Where T_ is the Stokesian response time of the particle. It

is quité obvious then that the particle parameter ),, is meaning-
ful in describing the particle diffusion only for the cases
where the particle Reynolds number lies within the

88


0.9
0.8
0.7
9.6

0,5

0.4 © x/D=10 1
x 15
0.3 ry 20
o 25
0.2 - ° 30
4a0
O1 -
L i L 1 - 1 1 1
() 0.04 0.08. 0.12 0016

1
FIG. 12 NORMALIZED VELOCITY PROFILE (WITHOUT SYRINGE TUBE)

42


0.3

0.2

0.1

© x/D=10
x x/D=15
A x/D=20

5 x/D=25

(a= 343 4)

n i 1 1 I 1

0.02 0.04 0.06 0.08 aio ai2

FIG. 40 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

83


1.0

0.9

0.8

0.7

0.4

0.3

0.2

0.1

T T ~T T T
ox,
onTM o x/D=10
FAG x 15 4
°
(N= 3 A 20
”
a 25
3 4
Qo
“ay
°
op 4
4o
A
x? |
A
°5
A 4
"4
&
4 4
x
Ox
L 4a 4
Ro
°°
°
vk
. a0 4
“ha
oA
o 4
- (dp = 970 wh) 8 7
l 1 L _t L
0.2 0.4 0.6 0.8 1.0 1.2

r/2b,

FIG. 25 CONCENTRATION FLUX PROFILE
60


o x/D=10

x 15
A 20
o 25

(a, = 527 1)

bo bnre eens cee

n n !
0,02 0,04 0,06 0, 08 020 0.12

1

FIG, 41 CONCENTRATION PROFILE
(With Bubble Rising Vel. Corr.)

84

0.14


Ch f
se tM Ta0-fovt+} (A-Iv-7)

It is noted that in obtaining the above expression the
solenoidality of the flow field has been assumed.

In steady state axi-symmetric turbulent jet flow, letting ‘'=‘t+y,
and neglecting the molecular effect, the final form of the
governing equation for the bubble diffusion becomes

v v

+V =-

@
o

U

|
|

& (x WY) (vI-19)

|
*
of
K
RIF

Similar to Appendix II the similarity requirement gives

tne DL

To 2 xXta

vy

D
— = A, se k(n)
Yo 2 xta
where n= and k(n) = _
xta 1 Tm

Assuming that the origin of the apparent source for concentration
diffusion is the same as that for the momentum diffusion, the
combination of Equations (VI-19),(VI-20) and (A-II-3) will give
the following relationship

e dk

Y
o. Gray an (A-Iv-8)

d ae ll a
“fg (kK) +o ar GF an (n

After eliminating g(n) by using the equation of continuity then

~24 my 2h @ (, —*Y ak
n dn “nh Gy [' 0, Gea) an

where F(n) is as defined in Appendix II. Integrating the above
equation and noticing that the constant of integration is zero

ey

dk
F(n) k(n) = - u, Ceray 1 an

The second integration will give the expression for the

123


density... The results are in agreement with available
measurements.

4. "Analytical Prediction of the Turbulent Diffusion of Small
Particles"

A comparison of various measurements has consistently
suggested an unexplained increase in the turbulent transport
of small suspended particles with size. A comparison with
the momentum turbulent transport coefficient suggests an
increase of the inverse of the turbulent Schmidt number (or
the ratio of mass to momentum transport coefficients) from
values close to 1, for zero-size particles, to values as
large as 5, for larger particles.

Based on the particle equations of motion, and postula-
ting certain relationships between the statistics of the
particle velocity fluctuations and the turbulent carrier
phase, a prediction of the particle transport coefficient
is made. The analysis predicts the same increasing trend
of turbulent transport coefficient with particle size. By
limiting the study to a particular range of particle Reynolds
number good agreement is noted with the experimental data.

5. "Energy Spectrum and Turbulent Scales in an Axi-Symmetric
Water Jet":

The time micro and macroscales are computed from
measured energy spectra in the fully developed region of a
circular jet. The corresponding longitudinal scales are
approximated using Taylor's frozen turbulence hypothesis.
The microscale is found essentially constant whereas the
macroscale, although increasing linearly alorig the axis,
does not show similarity in the total field.

6. "Turbulent Diffusion of Small Particles Suspended in Turbulent
Jet"

The dispersion of liquid droplets in plane two dimen-
sional air jets has been measured and compared with similar
results in the literature. From careful measurements of
the concentration flux profiles the corresponding turbulent
transport coefficient is determined. The results show that
within most ranges of interest turbulent transport increases
with particle size.

The results are not only compared with each other, but also
with an analysis based on the assumption of nonrotating

spherical rigid particles acted upon by viscous drag given
by Stokes Law. The approximate analysis predicts the same

trend as noted in the experiments.
U.S. GOVERNMENT PRINTING OFFICE: 1974— 546-319:391

126


to the unfiltered input e, was made for this band pass setting.
Figure 5 exhibits the expected linearity. Similar results at
other band pass settings had been found by Householder (1968
for the same equipment.

The output of the filtered and the un-filtered cooling signal
on a sensor in a bubble contaminated turbulent water stream
is shown in Figure 6. The upper trace is the filtered signal
and the lower trace is the unfiltered signal. It is seen
that the low frequency turbulent signals have been properly
filtered.

Counting

The components of the system are a hot film anemometer, a
tunable band pass filter, an AC amplifier, a Schmitt trigger,
and an electronic counter. Due to the active components of
the filter, the output is oscillatory as shown in Figure 6.

To eliminate erroneous multiple counts from a single bubble
impaction, the decaying oscillatory signal was fed to a Schmitt
trigger, the output of which was just longer in duration than
the oscillatory signal. The trigger, in turn, when provided
with an adjustable gate, was used for amplitude sampling.

The output from the Schmitt circuit was fed to an electronic
counter which recorded the rate of bubbles hitting the sensor
and causing a decrease of the cooling signal larger than the
trigger level of the Schmitt circuit.

Equipment

The block diagram of the system is shown in Figure 7. The
specific components used in this system are shown in Figure 8.
Their specifications follow:

TSI Constant Temperature Anemometer, Type 1050
Frequency response: 0-400 KHz
Output noise level: corresponding to less than 0.02%
turbulent intensity
Stability: drift less than 2 microvolts per °F and less
than 20 microvolts in 24 hours.
TSI Model 1217-20w Quartz Coated Cylindrical Hot Film
Sensing size: 0.002" diameter, 0.02" long
Relative frequency response: 40 KHz
Nominal resistance: 4-8 Ohms
Hewlett Packard Model 450A Amplifier
Frequency response: 20+ 1/2 db, 5 Hz to 1 meg. Hz
Kron-Hite Band Pass Filter, Model 310C
Frequency range: 20 Hz to 200 KHz
Attenuation slope: 24 db per octave
Tektronix Oscilloscope, Type 564 Storage Scope with Type
2B67 Time Base
Gate width: adjustable from 10 micro sec. to 50 sec.
Trigger level: adjustable from 0.002 to 160 volts

29


SECTION I

CONCLUSIONS

1. Turbulent diffusion measurements of four different bubble
sizes i.e., 343, 527, 780, and 970 microns were made. The
measured concentration profiles of the bubbles in the axi-
symmetric water jet revealed that similarity at successive
sections was preserved and that the profiles had the same
virtual origin as the momentum diffusion. This result
supported the original postualte that the contaminant is
diffused in a manner similar to that of momentum.

2. The particle diffusivities and the corresponding turbulent
Schmidt numbers were obtained by matching the measured data
with different models. Three models based on phenomenological
theories of turbulence were used to analyze the diffusion.

The first one, which assumed constant eddy viscosity and
constant particle eddy diffusivity, gave the best agreement
with the measured data. The results showed that in general
the contaminant diffuses faster than the momentum and that

the rate of diffusion increases with an increase of particle
size.

3. In order to compare the measured results with others, a
particle parameter, pp, originally presented by Householder and
Goldschmidt (1968) and later modified by Ahmadi (1970), was
adopted as a characterizing parameter. The present data suggest
that the particle parameter pp is suitable for describing the
particle motion only within the Stokesian range. When the
particle Reynolds number is outside this range another modified
parameter va is defined in which the particle response time is
correctly predicted from the simplified B.B.O. Equation in the
corresponding Reynolds number range.

4, The relationship between the turbulent Schmidt number and
the characteristic parameter, pp, is constant at low pp values.
When wp is large, there seem to be different trends according
to the geometry of the flow fields.


th

al

6. r 7

5eP
4e
3. F
Uo
vu," 0.1562(x/D + 0.6)
2.7r

———D—— Without Syringe Tube

1. .
_——— With Syringe Tube

= L 4. 1 n

() 10 20 30

x/D
FIG, 14 VELOCITY DISTRIBUTION ALONG JET AXIS (WITHOUT SYRINGE TUBE)



TABLES

No.

1 Comparison of Various Axi-symmetric Turbulent Jet
Measurements

2 Total Bubble Concentration of the Experiments

3 Rising Bubble Velocity

4 Turbulent Schmidt Number

5 Calculation of s;* versus Ve

ix


ge

oe”

3.

d
“p= 0.0952(x/D+ 0.8)--Pitot Tube

————_ Pitot Tube Measurement

—-o-— Hot Film Measurement

10 20 30
x/D

FIG. 10 VELOCITY HALF WIDTH (WITH SYRINGE TUBE)



superimposed on the bulk convective flow. The conservation
of mass for the gas species reads

det
ww “Roe -Iv-
sty Ay 0 (A-IV-2)

Substituting Equation A-IV-1 into A-IV-2, and noting that
g + a, = p' ¥, then

Be, 13 ' eg.
xe + Veop val {e Dv (54) (A-IV~3)

After changing notation, the above becomes

a (eg vs) . Vig) 9 . wr, ve Pg
—E st ['%, v)v] =v {(e4 +p, (1- ») "(Saar

(A-Iv-4)

singe the bubble concentration is very low (in the order of
in volume) y'v << 1 and Pg <<Pig- Hence Pgt' v +), {l-y;' vw) ~Pyy

or OTOL, and Equation A-IV-4 can be approximated by

9( v') > ports
g . 1 =v: g
—i—+V [(egs we) =v {o,27 awa (A-IV-5)

In the present case, the bubble size is relatively small

such that the rising bubble velocity due to the buoyancy
effect is less than the carrier stream velocity, consequentely
Yas and %, will be of the same order of magnitude and then

Og g<<Py, Vays The following relation is then approximately true

1 2 + 1 al
y = ®g Vg Ow Vw x Ow Vw ~ aa
pt p' w

From the above expression, Equation A-IV-5 further simplifies
to

alegre ) >
—f- + 9 (Cogr' 0%) = V+4DV(pg¢'¥) (A-IV-6)

In the above pg and v are constant scalars, so they can be
cancelled out.” Finally Equation A-IV-6 becomes

122


Thus from Equations (VI-8), (VI-9) and (VI-10), the kinematic
eddy viscosity (employing Prandtl's mixing length hypothesis)
becomes a constant value over the whole flow field of a turbulent
jet. Consequently Equation (VI-7) can be solved for U/U, . One
useful form of the solution (Appendix III) reads:

U,, (xta) 2 72

= l+—je— 1 (vI-11)
TMm

ce
|

m
Second Model: (Taylor's Vorticity Transport Theory)

Equation (VI-7) states that if U_(x+a) is constant then e

would be a function of n. Anothér form of the solution, Without
adopting the intuitive assumption given in Equation (VI-8), is
obtained using Taylor's vorticity transport theory (Schlichting
(1960)). Introducing an eddy viscosity based on the mixing
length of the vorticity transport, Loe such that

22
— _w |9U -
En = 3 32 | (VI-12)
and
ay = C,, (xta) (VI-13)

and substituting then Equation (VI-12) and (VI-13) into Equation
(VI-7), Tomotika (1938) obtained the approximate solution

or = 1 - a, (nt)?/? + ay (ne)? - agin) 97? (VI-14)
where 2-1/3
n* = (&
a, = 2v2
3v3 (vI-15)
= il
32 > BL
_ 1363V2
az =o
6123V3

The corresponding eddy viscosity based on Equation (VI-7) becomes

& 2/3 |d_u
satay - Gh 7? |b a |= B27 — Saynt/2s 3a, one)?
m
- 2 a, (ne) 7/2

17


showed that the rate of expansion of the cluster decreases as
the terminal fall velocity increases.

Brush (1962) studied the diffusion of glass beads in a submerged
axi-symmetric jet in order to compare the characteristics of the
diffusion of sediment with those of momentum in free turbulent
shear flow. The efflux velocity was maintained at 20 feet per
second to lessen the relative importance of the fall velocity

of the individual particles. He obtained the corresponding
particle turbulent Schmidt number for different particle sizes.
The following results were obtained:

Particle mean diameter (mm) €y/Eq
0.55 0.15
0.32 0.50
0.19 1.00

Ho (1964) found that the fall velocity in an oscillating fluid
was significantly lower than in a still fluid. Reasonable
values of the settling velocity in an oscillating fluid were
found to be predictable from numerical solutions of the equation
of motion.

Goldschmidt and Eskinazi (1966) measured the concentration
distribution of aerosol droplets in a plane jet of air by

means of a hot wire anemometer. It was found that the turbulent
diffusivity of the contaminant particles ¢, varies with the
location in the flow field and that an avetage value for the
turbulent diffusivity of the droplets (with an average size of
3.3 microns) in a plane jet of air is 0.03U,b,, where U, is the
maximum velocity and b, is the half width of the two difensional
jet. A comparison of the velocity and concentration profiles at
various sections of the flow field shows that although the
concentration profiles have spread more than the velocity
profiles, their rate of diffusion is lower. The average ratio
of ey/e, (inverse of the particle turbulent Schmidt number)

was found to be 0.91.

Baker and Chao (1965) measured the relative velocity of bubbles
in a vertical turbulent conduit. The size of the bubbles used
ranged from 760 to 7,000 microns. They concluded that the
bubble relative velocity in a turbulent water stream is similar
to the rise velocity of single bubbles through a quiescent
liquid.

Kennedy (1965) measured the dispersion of 1,250 micron soap
bubbles, and 700 and 900 micron polystyrene beads in a grid
generated turbulent air flow. The following results were

obtained:

12


10
11

12
13
14

15

16
17
18

19

FIGURES

Schematic Diagram of Experimental Equipment . . .
Constant Head Tank, Water Filter, Deionizer ...
Photograph of Jet Tank. ..... 2... +2. eee
Energy Distribution of Bubble Signal. ......
Filtered vs. Unfiltered Peak Voltage. ......
Filtered and Unfiltered Signals .........
Block Diagram of Electronic Instrumentation...
Electronic Instrumentation. ........4.4464-.
Normalized Velocity Profile (With Syringe Tube) .
Velocity Half Width (With Syringe Tube) .....

Velocity Distribution Along Jet Axis (With
Syringe Tube) . . 2... 2 1. ee ee ee ee ee

Normalized Velocity Profile (Without Syringe Tube).

Velocity Half Width (Without Syringe Tube). ..

Velocity Distribution Along Jet Axis (Without
Syringe Tube) . 2... 2 1 ww ww ee ee et

Distribution of Eddy Viscosity Based on Different

Models. . 2. 2 2 ee ee we ee we ee ee ee ew
Sensor Calibration Curve. . .. 2. 2. 2 2 ee eee
Sensor Sensitivity Curve. ...... 2... ee

Transverse Distribution of Turbulence Intensities

Energy Spectrum Along Center Line of the Jet. . .

vi

Page
22
23
24
28
30
31
32
33
36
38

39
42
43

44

46
47
48
49
51


*

<

a a

distance from sphere center to the syringe tube orifice
time period measurement
ambient temperature of a sensor in water

film temperature of the sensor in the air and water
phases respectively

time coordinate;traverse time of a bubble; outside
diameter of a pitot tube

mean axial velocity

Mean axial velocity at the jet axis

instantaneous axial velocity

jet efflux velocity

axial fluctuating velocity

fluctuating velocity scale

radial fluctuating velocity scale

axial fluctuating velocity scale
(3

tangential fluctuating velocity

mean radial velocity; co-flowing velocity for bubble
generation; general stream velocity

general stream velocity measured by pitot tube

mean velocity of the flow field with velocity gradient
measured by the pitot tube

instantaneous radial velocity
bubble velocity

average impressed velocity due to the wake of the
detached bubble

V/(a;£)
radial fluctuating velocity
velocity vector of bubbles and water mixture

velocity vector of bubbles

112


29

db,
3" 0.0978(x/D + 0.5)
2r 7
aan
D
b, =
spi= 0.0961 (x/D+ 0.9)
1. 7
—D-—— With Contamination
—-o-— Without Contamination a
A 1 n n n n n

—

ce) 10

x/D

20 30

FIG, 29 VELOCITY HALF WIDTH (WITH & WITHOUT BUBBLE CONTAMINATION)


VELOCITY

ELEGTROMG INSTRUDENTS

ELEVATED

TANK a

z.

TRAVERSING MECHANISH

KOT FILt
PRODE

xonercome — [Alii]
rs |
ANOHETER
NITROGEN Jreeutator
TANK

JET TANK

WATER FIULTSR

PUMP

“lis

RESERVOIR:

_f~\_
{©
tate

es |

FIG. 1 SCHEMATIC DIAGRAM OF EXPERIMENTAL EQUIPMENT


SECTION IV

STATEMENT OF THE PROBLEM

It is well known that a turbulent flow field will not only
transport but also rapidly disperse any material injected in
it. This is of basic importance in problems related to pollu-
tion and dispersion of contaminants in oceans, lakes, ani
rivers as well as the atmosphere and in the transport of
slurries and sediments.

Success of attempts at predicting the extent of dispersion of
injected contaminants so far has been very limited due to the
lack of a simple and yet all-encompassing mathematical model.
Consequently most of the research to date has consisted of
experimental measurement and empirical formulation. Among
these, Householder (1968) (references are alphabetically listed
in Section XIII) measured the diffusion of liquid particles in
a two dimensional air jet. In his experiment the contaminant
particles were much heavier than the carrier stream. The pre-
sent study is essentially a continuation of Householder's work
where now the diffusion characteristics of gas bubbles in an
axi-symmetric water jet are treated. In this case the conta-
minants are much lighter than the carrier stream. The major
purpose is to determine the effect of contaminant particle
size on the diffusion characteristics, be they heavier or
lighter than the carrier stream.

The extension of Householder's measurements was in part sug-
gested by Ahmadi (1970). In his thesis Ahmadi considered
Householder's data and proposed different methods of predicting
the dependence of the turbulent transport on particle size, as
well as on particle specific gravity. The measurements reported
in this work will then fill a lack of information on the dis-
persion of particles whose specific gravity is less than one.
The work is then not only a needed extension to Householder's
review but also an added input to a bank of data to substanti-
ate analytical predictions such as Ahmadi's.

'
In this work a series of constant size Nitrogen bubbles were
generated at the center of a submerged turbulent axi-symmetric
water jet. A hot film anemometer sensor was modified to measure
the concentration flux of the bubbles by counting the number of
impactions on the sensor per unit time. The corresponding con-
centration distributions were compared to predictions of three
different analytical models, based on the phenomenological theo-
ries describing the turbulent flow. The three models are re-
spectively based on Prandtl's mixing length theory, Taylor's
vorticity transport theory and Reichardt's Inductive theory.
Finally the corresponding eddy viscosity, eddy diffusivity and
turbulent Schmidt number of the bubbles are obtained.


then, to first determine the energy spectrum of the signal for
a single bubble impaction.

By harmonic analysis a bubble impaction signal can be transformed
from a time~dependent function into a frequency-dependent
function, namely the energy density function. Let e(t) be a
random time-dependent function such that e(t)=0 for t,>t and
t2<t, where t) and t2 represent the times of the pubbte hitting
and detaching from the sensor. e(t) can then be expressed in

a Fourier Integral form:

e(t) = J g (w) ett ay (VII-1)
°
where
t ‘ t
g(w) = i f 2 elt) en t¥* at = ir f 2 e(t) (cos wt - isinwt) dt
ty ty

(VII-2)

the energy density is then defined as

og et t .
|o¢a | 2.1! ({ 2 a(t) cosueat]” + [ { 2 elt) sin wt at}? /
4n l ty Lotz
(VII-3)

The bubble impaction signal was obtained from the TSI constant
temperature anemometer and displayed on a Tektronix Type 564
Storage Oscilloscope. A photograph of the cooling signal

was then taken with a polaroid camera. The resulting photo-
graph was digitized and a numerical integration of Equation
(VII-3) was performed.

The energy density function of the cooling signal due to a
bubble of about 500 microns in diameter approaching the sensor
at 3.5 fps is shown in Figure 4. The over heat ratio of the
sensor was 1.1, the same as that used in the actual concentra-
tion flux measurements. Figure 4 shows that the bubble signal
energy is a maximum at 1.'8 KHz, This frequency is well above
the maximum frequency for the turbulence energy in an axi-
symmetric turbulent water jet (discussed in the next chapter).

It was then decided to set up a frequeccy band between 1.5 and
2.2 KHz for the concentration flux measurements in such a
manner that all frequencies below 1.5 KHz and above 2.2 KHz
were cut off.

All of the electronic components are linear in their response

to a signal with the exception perhaps of the band pass filter.
To ascertain that the band pass filter did not introduce any
non-linearity, a calibration of the filtered output ent compared

27


SECTION XII

ACKNOWLEDGEMENTS

The reported work was the result of the Ph.D. research
of Dr. Strong C. Chuang. Professor Victor W. Goldschmidt
acted as supervisor of his work.

The support of the project by the Water Quality Office
Environmental Protection Agency, and the help provided
by Mr. George Ditsworth, Project Representative is
acknowledged.

The assistance of the staff of the School of Mechanical
Engineering and its Ray W. Herrick Laboratories during

the final stages of the work reported is also gratefully
acknowledged.

97


68

-1

wv y T T t T

FIG, 44 DEPENDENCE OF sz ON PARTICLE PARAMETER V+

7 T
°
collected by Householder & Goldschmiat (7.7)
* (pipe ana channel)
6b o¥v Kennedy (isotropic tarbulence?) . 7
Oo brusa °
SF 4 Bingmasetti | extant jet) - . 7
w present study . . °
yl ° Householder (plane jet) 4
dynamic simulation (Ahmadi)
2F oe 1
ob 4
vs. Vv.
1b ° me 7
me) 1 in 1 I 4 it
1074 1073 io? 107! 10° 104 10% 10?


SECTION V

LITERATURE REVIEW

The turbulent diffusion of a marked fluid particle (a particle
of infinitesimal volume) in a homogeneous turbulent flow field
was first described by Taylor (1921) and then by Batchelor
(1949), Prenkiel (1953), Roberts (1957), Lin (1960a) and Corrsin
(1962). The relative dispersion of two marked fluid particles
was studied by Batchelor (1952), Roberts (1960), Corrsin (1962)
and Kraichnan (1966).

The complications encountered when considering diffusion in a
nonuniform turbulent shear flow have been partly overcome by
Batchelor and Townsend (1956), Lin (1960b) and Cermak (1963).

A general description of turbulent diffusion of a marked fluid
particle requires a knowledge of the Lagrangian correlation
coefficient of such a marked fluid particle. When the particles
under consideration are distinguishable in their dimensions or
density from the main stream, the analysis will then depend on
the Lagrangian velocity correlations for these particles (Fried-
lander (1957)). So far, such a velocity correlation has only
been either qualitatively discussed or quantitatively calcu-
lated. Such calculations have been through very simplified
models with questionable applicability (see for instance Tchen
(1947), Shirazi et al (1967) and Ahmadi (1970)).

ANALYTICAL STUDIES

Tchen (1947) first studied the motion of a spherical particle
in a homogeneous turbulent flow field. He extended the B.B.O.
equation (Basset, Boussinesq and Oseen Equation, see Hinze
(1959) for the general equation and the significance of each
one of its terms) to the case of a particle in a fluid with
variable velocity. Corrsin and Lumley (1956) made further
modifications in order to make the equation applicable to the
motion of a small spherical particle in a turbulent fluid.

Hinze (1959) assumed that the particle is very small so that
the non-linear term in the B.B.O. equation and the viscous drag
due to the pressure gradient are negligible. Thus, it was
possible to relate the power spectrum of the particle to the
power spectrum of the Lagrangian velocity field. Levich and
Kuchanov (1967) made a similar approximation. In addition they
neglected the unsteady Basset term (which accounts for the
effect of the previous history of the particle motion from some
original time) and solved the equation for the relative velocity
between the particle and the carrier fluid. They concluded
that the mean square of the relative velocity is always much
smaller than the mean square velocity of the fluid.


SECTION XI

SIGNIFICANCE OF RESULTS

The experiments convincingly show that the turbulent trans-
port coefficient of suspended particles is not necessarily
equal to that of momentum. As a matter of fact an increasing
eddy diffusivity coefficient with particle size is noted.
This is observed for small particles in the Stokes regime,
smaller than the turbulent macroscale of the carrier stream.

The results of Figure 44 exhibit then when a particle can
be considered to have a turbulent transport same as that
for the momentum of the carrier stream. As an example, a
grain of sand, 50 microns in diameter, when suspended in
water flowing from a plane 2 inch nozzle at about 5 ft./sec.
would have a y,, parameter of about 0.05. For such a case
the transport Foefficients (of the sand and carrier stream
momentum) would be equivalent. However, if the channel
width were to be in the order of 1/16 of an inch (then

v, * 1.5 and) the transport coefficient of the sand would
ndw be in the order of three times more than that of the
momentum of the fluid! Estimates of the corresponding
dispersion of the effluent sand should then correctly ac-
count for this somewhat startling result.

9S


Hinze, J. O, and Van Der Hegge Zijnen B. G. "Transfer of Heat and
Matter in the Turbulent Mixing Zone of an Axially Symmetrical Jet,"
Applied Sci. Research A. Mech. Heat, Vol. 1, 1949, pp. 435.

Ho, H. W. "Fall Velocity of a Sphere in a Field of Oscillating Fluid."
Ph.D Thesis, Univ. of Iowa, Iowa City, Iowa, 1964.

Householder, M. K. "Turbulent Diffusion of Small Particles in a Two-
dimensional Free Jet," Ph.D Thesis, School of Civil Engr., Purdue
Univ., W. Lafayette, Ind., 1968.

Householder, M. K. and Goldschmidt, V. W. "Turbulent Diffusion and
Turbulent Schmidt Number of Small Particles-Part I," Journal of the
Engineering Mechanics Division, EM6, December 1969, pp. 1345-1367.

Hsu, Y. Y., Simon, F. F. and Graham, R. W. "Application of Hot-wire
Anemometer for Two Phase Flow Measurements Such as Void Fraction
and Slip Velocity," ASME Winter Meeting, Philadelphia, Pa., Nov., 1963,

Ismail, H. M. "Turbulent Transfer Mechanism and Suspended Sediment
in Closed Channels," Trans. ASCEE Vol. 117, 1952, pp. 409-446.

Jones, B. G., Chao, B. T. and Shirazi, M. A. "An Experimental Study
of the Motion of Small Particles in a Turbulent Fluid Using Digital
Techniques for Statistical Data Processing," Nuclear Engr. Dept.,
Univ. of I1l., Urbana, I1l., 1966.

Kada, H. and Hanratty, T. J. "Effects of Solids on Turbulence in a
Fluid," AIChE J. Vol. 6, No. 4, 12, 1960, pp. 624-630.

Kalinske, A. A. and Pien, C. L. "Eddy Diffusion," Ind. Engr. Chem.,
Vol. 36, 1944, pp. 220.

Kampe de Feriet, J., Proc. 5th Intern. Congr., Appl. Mech., Cambridge,
Mass., 1938, pp. 352.

Keagy, W. R. and Weller A. E., Proc., Heat Transfer and Fluid
Mechanics Inst., Berkeley, Calif., 1949, pp.89.

Kennedy, D. A. "Some Measurements of the Dispersion of Spheres in a
Turbulent Flow," Ph.D. Thesis, Dept. of Mechanics, The John Hopkins
Univ., Baltimore, Maryland, 1965.

Kraichnan, R. H. "Dispersion of Particle Pair in Homogeneous
Turbulence" The Physics of Fluids, Vol. 9, No. 10, 1966, pp. 1937-
1943.

Kuethe, A. M. "Investigations of the Turbulent Mixing Regions Formed
by Jets," J. Appl. Mech., Vol. 2, No. 3, 1935, pp. A-87-95.

Kumar, A. "Bubble Formation in Fluids of Low Viscosity Under Constant
Flow Conditions," Chem. Tech., Vol. 19, 1967, pp. 78-82.

Laursen, E. M. "The Total Sediment Load of Stream," Proc., ASCE, Feb.,
1958, pp. 1-36.

102


or

b
ue X42 -
p= & Gt D (vI-4)

Equation (VI-4) states that the velocity half width increases
linearly along the axial direction

uh Ay Al D
Moreover, wT, = w(—&) = EF wa! and letting 1/A, = Kye then
UG x a
uw, = ky & +5) (VI-5)

Equation (VI-5) states that the inverse of the center line
velocity decay is also linearly proportional to the axial
direction.

Equations (VI-4) and (VI-5) are the immediate consequences of
the similarity of the jet flow field.

APPENDIX III
Derivation of Equation (VI-11)

Introducing F(n) as given in Appendix II into Equation (VI-7)

¢ F ar
mM =- a aa
Uh (xta) ne (= =)
or
En
ap GE = 2d ,1 dF

“coma a Gh
This equation can be integrated readily to give

Em

(n= - 2F) 2
U,, Gera) an

1
-5 F

The constant of integration must be zero, because at ‘n=0, F(n)
must be zero. The second integration yields

120


€9

100 T

Bo
6o
el”
N
TM ho F

a, = 343 microns

A 527
20
° 780
x 970
4 n 1 n 1 L 1
(0) 10 20 30

x/D

FIG. 27 CENTERLINE CONCENTRATION FLUX DECAY



69

FIG.

6.

5.

4,

3. 7
U 2
a2 7 7 0-1626(x/D + 0.6)
U a
n

2. 7

—p——— With Contamination

1 ———— Without Contamination

Z n n 1 1 1
TO 20 30

x/D

31 TRANSVERSE FLUCTUATION LEVEL (WITH AND WITHOUT BUBBLES CONTAMINATION)


Sayre and Chang (1968) investigated the dispersion process

in a turbulent open channel flow. They found that the longi-
tudinal dispersion process for suspended silt-size sediment
particles differs from the process for a dissolved dispersant
in that the particles tend to settle toward the slower moving
flow near the bed and eventually deposit on the bed. The
lateral dispersion of floating particles (released from an
intermittent point source at the water surface) can be
represented as a Fickian diffusion process. However, the
dispersion pattern is somewhat distorted by the effects of
secondary circulation.

Householder (1968) performed an experiment similar to that of
Goldschmidt and Eskinazi, and found the following dependence
of the particle turbulent Schmidt number on the particle size:

Particle size (microns) Ey/Eq
6.4 2.38
13.1 7.70
16.7 3.33

Householder and Goldschmidt (1969) made an extensive review
and unified the available data on dispersion of contaminants
in open channels and circular pipes. They related the turbu-
lent Schmidt number to a particle parameter, , proportional
to the Stokesian particle response and defined as

2 20
v=R( (52+) (11-2)
eva Oe

where Rg is the Reynolds number, {based on the mean flow velocity
and a characteristic width of the conduit),d is the particle
diameter, a is the characteristic dimension of the flow field
and 2p and p¢ are the density of the particle and carrier
fluid respectively. They concluded that there is a general
trend towards an increase of particle diffusion with yp, and
hence particle size.

14


SECTION XIII
REFERENCES

Ahmadi, G. "Analytical Prediction of Turbulent Dispersion of
Finite Size Particles," Ph.D Thesis, Purdue University, Lafayette,
Indiana 1970.

Albertson, M. L., Dai, Y. B., Jensen, R. A., and Rouse, H.
"Diffusion of Submerged Jets," American Society of Civil Engineers,
Vol. 74, Dec., 1948, pp. 1571-1596.

Baker, James L. L. and Chao, Bei T. "An Experimental Investigation
of Air Bubble Motion in a Turbulent Water Stream," American Institute
of Chemical Engineers Journal, Vol. II, No. 2, March, 1965, pp.
268-273.

Batchelor, G. K. "Diffusion in a Field of Homogeneous Turbulence,
I. Eulerian Analysis," Australian Journal of Scientific Research,
Series A, Vol. 2, No. 4, 1949, pp. 437-450. .

Batchelor, G. K. "Diffusion in a Field of Homogeneous Turbulence,
II. The Relative Motion of Particles," Proc. Cambridge Phi. Soc.,
Vol. 48, 1952, pp. 345-362.

Batchelor, G. K. and Townsend, A. A. "Turbulent Diffusion,"
Surveys in Mechanics, Cambridge University Press, 1956, pp. 352-399.

Benzing, R. J. and Myer, J. E. "Low Frequency Bubble Formation at
Horizontal Circular Orifices," Ind. & Eng. Chem. 47, 1955, pp.
2087-2090.

Binnie, A. M. "The Turbulent Spreading of the Water Jet," Engineering,
Vol. 153, 1942, pp. 503.

Bird, Byron R., Stewart, Warren E., and Lightfoot, Edwin N. Transport
Phenomena, John Wiley & Sons, Inc., New York, 1960, pp. 192-194.

Boussinesq, J. "Theory de L'Ecoulement Tourbillant," Mem. Pres. Par.
Div. Sav. XXIII, Paris, 1877.

Brandon, C. A. "An Investigation of the Interaction of Solid Parti-
cles with Fluids in Turbulent Pipe Flow," Oak Ridge National Lab.,
Oak Ridge, Tenn., ORNL TM-2124, September, 1968.

Brooks, N. H. Discussion of "Acceleration Motion of a Spherical
Particle" by M. R. Carstens Transactions, A. G. U., Vol. 34, 1953,

pp. 947-950.

Brush, L. M. Jr. "Exploratory Study of Sediment Diffusion," J. of
Geophysical Research, Vol. 67, No. 4, 1962, pp. 1427-1433.

99


v n (xta) UL [ n oy
— = exp - | _—_— | n(=—-) dn} dn (VI-21)
_ ° eyn o «(OOM

Again, the above expression implies the similarity of the ° -
distribution. The immediate consequences of the similarity give
(Appendix IV)

b

Y_ xX a
D7 ky SG + D (VI-22)
v

fe) x a

2=,x (+2 -
Tn yD D (VI-23)

Where a is the distance from the virtual source of the concentra-
tion to the jet nozzle plane and is assumed to be equal to that

to the virtual origin of the velocity distribution. This need not
be the case, it is now assumed so strictly for convenience. °

is the efflux concentration at the jet nozzle, ‘m is the maximum
concentration at the section of consideration, by and k, are re-
spectively the values of r and n where the concentratio is half

the maximum at the section of consideration, and ky is the slope

of the linear relation between the inverted centerline concentration
and the axial distance as given in Equation (VI-23).

If e¢ and e. as well as the velocity distribution are known, then
the t distr{bution can be obtained from Equation (VI-21). Con-
versely, in the absence of a_ proper model for ¢,, but provided

e. and U/U_ are given and ‘/‘ is measured, them e, can be deter-
Mined. In the first part of Bhis section different models relating
the eddy viscosity to the mean flow properties were discussed.
These in turn gave relationships for U/U_. In a very similar man-
ner if different models were defined for e,, then a relationship
for ‘/%_ would result. A basic assumption’is now made. Irrespec-
tive of how «. or €_ might vary through the jet, it is now assumed
that the same'basis for describing e¢_ will apply toward a descrip-
tion of €,. Hence the three models of section A are now employed
to obtain’alternate formulations for vane in terms of ey and ¢,.

First Model: (Constant Eddy Viscosity and Eddy Diffusivity)

If the velocity profile is assumed as expressed in Equation (VI-7),
substituting it into Equation (VI-21), the concentration distribu-
tion becomes:
v n €
— = exp J an2f in (vI-24)
v ° ey on On
m
Assuming as in section A that En and now €. are constant across
the whole flow field then Equation (vI-24) ‘simplifies to
v € U at
a= Gy) Me 1 + meta) ,2 -2m
v TMm

(VI-25)
U € 8 en ey

R

19


0.9 T T T
0.8F
0.7
0.6F
sensitivity
(84) 0.57
volt.
tps)
O.4F
0.3F
0.2 1 _t Ll
10 11 12 13

E (volts)
FIG. 17 SENSOR SENSITIVITY CURVE

48


€6

Calculation of si versus Un

Particle Reynolds

Authors dp (sz) Number, Ts
2Vue a/v » (1073sec. ) Vy ft, U sl
Present 343 338 3052 5.28 06154 (0.203) 0-81 (1.07) 0.94
Work 527 519 8.32 12.46 06161 (06172) 2.00 (2.14) 1.09
780 768 18.07 27630 0.109 (0-147) 2.97 (4-01) 1.11
970 954 28.10 42.30 0.088 (06134) 3470 (5667) Leh
Singemsetti 68 136 3043 be68 04223 (06293) 160h (1437) 1.23
115 227 905k 13.38 0.182 (0-238) 243 (3-18) 1420
230 45h 38-20 53040 0-138 (0-180) 7-37 (9.61) 12h
320 631 73070 101.50 0-132 (0.158) 13-40 (16.00) 1.37
460 907 152420 213.00 0.092 (0-138) 19-60 (29.40) lekk
Brush 190 37h, 17-55 23.50 0-148 (0-195) 3-48 (4-58) 1.00
320 632 49290 67.00 0.132 (06158 885 (10.60) 2.00
550 1084, U,7.80 196.00 0-080 (0.160) 15-60 (31.30) 6-70



0.5 volt
per
Div.

1 volt

per
Div.

0.5 MS PER DIV.

— 18 TO 22 KHZ | sigwaL 1
LSIGNAL_t
rs BAND PASS FILTER

. » T+ anewometer

——_»_"_. _sENsor
> nn SIONAL &

SUBBLES IN THE STREAM

FIG. 6 FILTERED AND UNFILTERED SIGNALS

31


SECTION VI

THEORETICAL BACKGROUND

MOMENTUM TRANSPORT IN AN AXI-SYMMETRIC JET

Based on the assumptions that (1) the mean motion is steady,
(2) the flow field is axi-symmetric, and (3) the turbulent
region is narrow in the radial direction when compared with
the axial direction, the equation of motion of an axi-symme-
tric turbulent jet in the axial direction, (Appendix I),
reads:

a
|
+
<
ff]
Dle
Kile

3
ra (ro) (vI-1)

where x and r are respectively the coordinates in the axial and
radial directions, U and V are respectively the mean values of

the x and r components of velocity, o is the Reynolds stress,
and p is the fluid density. The corrésponding boundary condi-

tions are:

r=O: V

ie)
ree: U=O0 (VI-2)

conservation of momentum

Equation (VI-1) together with the equation of mass conservation
and the boundary conditions (VI-2) can be integrated to give
(Appendix II)

fo)

ell
n

2
ou,

ala

n
[ann Go (vI-3)
fo)

m m

where n = r/(xta). With n so defined the immediate consequences
of the similarity of the flow field will give (Appendix II)

b
u x a

Bs x4 a -4
Bak E+ % (vI-4)
U

o _ x4 a -
i k, &+ 2) (VI-5)

where a is the distance from the virtual origin to the jet nozzle
plane, U_ is the jet efflux velocity, U, is the maximum velocity
at the section of consideration, b, and k, are respectively the
values of r and n where the velocity is nit the maximum at the
section of consideration, and k, is the slope of the linear

15


Bz

1.094 R-5—5 T T r

a 4°

a ox/D= 10
0.9F 0 ° * 15 4

ap 20

a o 25
0.8F 0 4

4
O.7F +
Q
%
n
w
0.6 ° 4
%
rN
O.5Fr an 7
4
A
O.4b &x 7
> $,
x
0.3- 4
+
x
o
Ao
0.2b x a.
oe
4 °

O.1- (4, = 780) 7

i i 1 Ll i
() 0.2 0.4 0.6 0.8 1.0 1.2

r/2b,,

FIG, 24 CONCENTRATION FLUX PROFILE

59


and Uv -\uw “Ww OR u

can be defined. R is the correlation coefficient between two
fluctuation components and ranges between 0 and 1. The order
of magnitude of the various terms in Equations A-I-1 and A-I-2
is then given by

“2
3u ou. Lop. aw lle Ww -I-
Osx + Vor * "5 ox 7 ox z 3 (Fw) (A-I-3)
uw? w 4p, ue R ue
over pr UU t
x x Ply x rc
2 2
or 1 1 Py Me Rive 2x
u2 u? uw L
pu, x x Ur
= — 2
av ave. _lep_12,.-2, _ aav, we _—
UxtY ET SiO E gz (rv) x Tr (A-I-4)
2 2 2 2
order u,b ul. ve Ug Ruy Ug
2 p:
Ly Ly rc rc x x
2 2,2 2 2.2
or 1 Ap Ly u-hy Ru,Ly u,Ly
2 2,2 2 2,2
pu, u Ly uy. LL. uly

In the above expressions Ap, and Ap, represent the change of
pressure in the x and r directions respectively and the viscous
terms have been neglected. In reviewing the order of magnitude,
it is concluded that in the first equation

=
a is much smaller than é 2 (rev) and is hence neglected;

in the second equation the terms in the left hand side are at

least one order less than the terms in the right hand side

and hence are also neglected, moreover in the right hand side

of the second equation duv is one order less than the other
ox

terms and is also neglected. The final form of the equations
simplifies to

116


ve ae B+ an BE, Be BE
mos dx on 3E Bx * BE Ox (A-II-
U.
m -Ln a6 a yaar
a = Ve) =p $ G mt DBE
9.
5 = h (n)
eu,
_r _ Xta
where n = 5 9(E) and & = ==
then Equations A-II-l and A-II-2 become
2
wv, ao dt ay 2, af oye od a
VC Og avn t+ § ge) + vO a= an (mh) (A-II-4
2 2 2
4 | dnn £" = Constant (A-II-5
o* fo

Now consider more closely the term occuring in Equation A-II-5.
It appears to consist of the product of two terms, one of

which is a function of € alone. Hence similarity can be achieved
only if these functions of — are constant, i.e. W=A,$ where Aj,

is a numerical constant. Substituting this relation into
Equation A-II-4 then

$ ¢ (nf Ea + £7) + 93 (s Es - + a (nn) =0 (A-II-6

The terms in the above expression also appear to consist of
two parts, one a function of € the other a function of n alone.
Hence similarity requires that

) ae =B o° , where B is an arbitrary constant.

A solution of this differential equation reads

a)
o = sta

118


No.

20

21

22
23
24
25
26
27
28

29

30

31

32
33
34
35
36
37

Longitudinal Macro Scale Along the Turbulent
Jet Axis . 6 6 ee we ee ee ee ee ee ee

Variation of the Integrated Concentration Flux
Along the Axial Direction. ...........24.-.

Concentration Flux Profile ..........2+4.e.
Concentration Flux Profile ..........424e-.
Concentration Flux Profile ........4..2.4.e-e
Concentration Flux Profile ........2.4-e.e-.
Half Width Spreading of the Concentration Flux...
Center Line Concentration Flux Decay ........

Normalized Velocity Profile (With & Without
Contamination) . . 2. 6. ee ee ee ee ew ew ee

Velocity Half Width (With and Without Bubble
Contamination) . 2. 2. 2 2 2 2 ee ee ee ee ee ee

Velocity Distribution Along Jet Axis (With & Without
Bubble Contamination)... 2. 2. 2+ 2+ 6 6 ee ee eee

Transverse Fluctuation Level (With and Without
Bubble Contamination . . 2. 2 6 2 se ee eee eee

Concentration Profile. . . 2... 2. 6 ee ee ee ee
Concentration Profile. . . 2. 2. 2. 2 2 2 ee ee eee
Concentration Profile. . . 2... 2. 2 ee ee ee ee
Concentration Profile. . .. 2. 4 2. 2 6 «© ee ee ew
Concentration Profile. . . 2. 2. 2 6 2 2 ee ee eee

Concentration Profile. . 2... 6+ 6 ee ee ee eee

vii

56
57
58
59
60
62
63

66

67

68

69
74
75
76
77
79
80


=2%u ,

Assuming that the initial condition is IVb - Veleso
Equation (X-9) can be integrated to yield

, 3/5 =
Iv, - vel -{ 11.1 vy tt a 2/4 -8/2
8/5 (9 _B -
a7? (2 - +1) (X-10)

Defining particle response time as the value of t where
lvp-ve|/2/07 equals to e~1 = 0.368 the expression for becomes

1.6 »
ay, (2 + 1)
t = 0.0335 x —— Rf for 2<R, <500 (X-11)
( ut 9-4,0.6 p
or
t _ lai yor4 _ Yn’ _ 1.59
T —0.4, 0.4 =U = 0.4
s (A2) 4, T Ree (X-12)

Consider next the gase where the particle Reynolds number is
between 500 and 10°. In this range Cp=0.44 Equation (X-7)
then becomes

ad 0.66
a= |v - vel + 28S |v, - vy

4 (22 + 1)

2
5 |
Pe

= 0 (X-13)

With the same initial condition as in Equation (X-8), Equation
(X-13) can be integrated to yield

Iv5 - vel = { 2.88 t-
a. (22 +1)
Pr

(X-14)
b
The expression for tT then becomes
Pp
1.164, (255 + 1) 5
tT =——2 $f —___ for 500<R, <10 (X-15)

‘3 P

T
Does === X-1
T Va vp OR (x-16)


0.8)

glk o

05

0.4

03)

0.2

Ot

T T T T 7

EB * 3 SE Ye
Bh Seley rarer) ~ Be BY)

an (144602 97)

Uv, 24 t
Un e

eo

oa Px oO
8

2
Se
°
3!

0-04 oa
q

FIG. 9 NORMALIZED VELOCITY PROFILE (WITH SYRINGE TUBE)

36


SELECTED WATER 1. Report No.] 2. 3. Accession No.
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM had
4. Title : 5. ReportDate x
TURBULENT DIFFUSION IN LIQUID JETS: FINAL REPORT is age : ee: <f
8. Performing Orgavization
7. Authors) REN, ee.
Strong C. Chaung and Victor W. Goldschmidt Be e070 Le
Os OSE RTA en Engineering Experiment Station 71. Contract] Grant No.
School of Mechanical Engineering 16070 DEP
Purdue University
Lafayette, IN 47907 oo ee Bio coe

psoring Organization

15. Supplementary Notes

Environmental Protection Agency Report Number, EPA-660/3-74-004b, March 1974.

16. Abstract

Laboratory studies were conducted on the dispersion of gas droplets of different
sizes in turbulent water jets. The main purpose was to determine the turbulent
transport coefficient of contaminants suspended in turbulent flows.

The experimental results were compared to measurements of diffusion of liquid
droplets in air jets as well as to a numerical analysis based on the equations of
the particles themselves.

The results confirm that small particles in turbulent flows have an increasing
turbulent transport coefficient with size. The collated experimental results
exhibit when Reynolds analogy (in the transport of mass and momentum) can
be validly employed. _

17a, Descriptors

Bubble, Diffusion, Dispersion, Equations, Instrumentation, *Turbulence, *Turbulent
Flow.

7b. Identifiers

Bubble Sampling, Bubble Sizing, Concentration of Contaminants, Hot Film Anemometer,
Jet Flow, Liquid Jets, *Turbulent Diffusion, *Turbulent Transport.

7c. COWRR Field & Group 0g B

18, Availability Send To:

WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR:
9] WASHINGTON, D. c. 20240

Institution Environmental Protection Agency id

Mbstractor_G. R, Ditsworth



Brush, L. M., Ho, H. W., and Singamsetti, S. R. "A Study of Sediment
in Suspension," Publication No. 59 of the I. A. S. H. Commission of
Land Erosion, 1965, pp. 293-310.

Calderbank, P. H. "The Inter-dispersion of Immiscible Fluid Phases,"
Brit. Chem. Engr., Vol. 1, 1956, pp. 206-209 and 267-274.

Carstens, M. R. "Accelerated Motion of a Spherical Particle" Trans.,
Ame. Geophysical Union, Vol. 33, No. 5, 1952, pp. 713-721.

Cermak, J. E. "Lagrangian Similarity Hypothesis Applied to Turbulent
Shear Flow", J. of Fluid Mech., Vol. 15, 1963, pp. 49-64.

Chuang, S. C. and Goldschmidt, V. W. "The Response of a Hot Wire
Anemometer to a Bubble of Air in Water", Symposium on Turbulence
Measurement ‘in Liquids, 9, 1969, Univ. of Mo., Rolla, Mo.

Chuang, S. C. and Goldschmidt, V. W. "Bubble Formation Due to a
Submerged Capillary Tube in Quiescent and Co-flowing Streams",
ASME Nucleonics Symposium, May, 1970. (ASME, Paper No. 70-FE-21)

Collis, D. C. and Williams, N. J. "Two-dimensional Convection from
Heated Wires at Low Reynolds Number," J. of Fluid Mech., Vol. 6,
1959, pp. 357-384.

Corrsin, S. "Investigation of Flow in an Axially Symmetric Heated
Jet of Air," NACA W-94 ACR No. 3123, 1943.

Corrsin, S. and Uberoi, M. S. "Further Experiments on the Flow and
Heat Transfer in a Heated Turbulent Air Jet," NACA Tech. Note No.
1865, 1949.

Corrsin, S. and Lumley, J. "On the Equation of Motion for a Particle
in Turbulent Flow," Appl. Sci. Research, Vol. 6A 1956, pp. 114.

Corrsin, S. "Theory of Turbulent Dispersion", Proc. for Turb. Diff.,
Marseille, France, 1962, pp. 27-52.

Curtis, A. J. "Shock and Vibration Handbook", Edited by Harris and
Crede, McGraw-Hill, 1960, Ch. 22.

Datta, R. L., Mapier, D. H. and Newitt, D. M. "The Properties and
Behavior of Gas Bubbles formed at a Circular Orifice," Trans. Inst.
Chem. Engr., Vol. 28, 1950, pp. 14-26.

Davidson, L. "A Study of the Formation of Gas Bubbles from
Horizontal Circular Submerged Orifices," Ph.D. Thesis, Columbia U.,
1951.

Davidson, L. and Schuler, B. 0. G. "Rapid Formation of Gas Bubbles
in a Viscous Liquid," Trans. Inst. Chem. Engr. Vol. 38, 1960, pp.
144-154.

Delhaye, J. W. "Hot-film Anemometry in Two Phase Flow," Two Phase
Flow Instrumentation, Presented at llth Nat'l American Society of
Mechanical Engineering, American Institute of Chemical Engr. Heat
Transfer Conf. 8 1969, Minneapolis, Minn. pp. 58-69.

100


6€

Cc

jo

6.

5.

3.

2.

1.

vu
Tv" 0.1626(x/D+ 0.0)
“mM

Pitot Tube Measurement

Hot Film Measurement

FIG, 11 VELOCITY

20 30 4O
x/D

DISTRIBUTION ALONG JET AXIS (WITH SYRINGE TUBE)