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CIO
HALOCLINE EROSION DUE TO WIND INDUCED STRESS
By
KENNETH K. HUMISTON
A THESIS PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF ENGINEERING
UNIVERSITY OF FLORIDA
1974
Coastal Engineering Archives
University of Florida
ACKNOWLEDGEMENTS
I wish to thank Dr. Omar H. Shemdin, Chairman of my supervisory
committee, for his encouragement and guidance in the preparation of
this thesis, and for his part, as Director of the Coastal and Oceano-
graphic Engineering Laboratory, in making the excellent internal wave
facility there available for this study. I also extend my thanks to
the other members of my supervisory committee, Dr. Robert G. Dean, Dr.
D. Max Sheppard, and Dr. Yu-Hwa Wang for their instruction and inspira-
tion during the period of my graduate learning and thesis preparation.
Special thanks are due Dr. Wang and Dr. Ronald J. Lai for their assist-
ance with the instrumentation during the experimental investigation.
Finally a word of thanks to Fena Jones and Evelyn Hill for typing,
Denise Frank for drafting, and the personnel at the Laboratory for their
cooperation and assistance in obtaining and setting up the experimental
apparatus.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES iv
LIST OF SYMBOLS vii
ABSTRACT ix
I. INTRODUCTION 1
II. THEORY 6
III. EXPERIMENTAL EQUIPMENT AND PROCEDURES 9
IV. RESULTS 18
V. DISCUSSION OF RESULTS 56
VI. CONCLUSIONS AND RECOMMENDATIONS 64
APPENDIX A DIFFUSIVITY OF N C1 IN WATER 66
a
APPENDIX B NEGATIVE BUOYANCY GENERATION AT
THE SURFACE DUE TO EVAPORATION 68
BIBLIOGRAPHY 71
BIOGRAPHICAL SKETCH 73
iii
Coastal Engineering Archives
University of Florida
LIST OF FIGURES
Schematic drawing of the facility showing the overall
dimensions and the location of the various components.
Wind velocity measurement instrumentation, pitot tube,
pace transducer, signal conditioner and voltmeter.
Hot film anemometry instrumentation; anemometer power
supply and linearizer, integrator, voltmeter, tape
recorder, and rms meter.
Hot film calibration by specific gravity, 0 21 cm/sec.
Hot film calibration by specific gravity, 0 1.5 cm/sec.
Hot film calibration by specific gravity, 0 10 cm/sec.
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Figure 13.
Density profile
wind velocity 2
Density profile
wind velocity 2
Density profile
wind velocity 2
Density profile
wind velocity 2
Density profile
wind velocity 2
Density profile
wind velocity 2
Density profile
wind velocity 2
at Station 6 for wind duration 3 hours,
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
6 for wind duration 4 hours,
12 for wind duration 3 hours,
12 for wind duration 4 hours,
12 for wind duration 5 hours,
18 for wind duration 3 hours,
18 for wind duration 4 hours,
Figure 14. Density profile at Station 6 for wind duration 3 hours,
wind velocity 2.5 mps.
Figure 15.
Density profile at Station
wind velocity 2.5 mps.
6 for wind duration 4 hours,
Figure 1.
Figure 2.
Figure 3.
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
mps.
at Station
wind velocity 3 mps.
Figure 32.
Lines at constant density for
wind velocity of 2 mps. (See
wind of 4 hour duration,
Figures 8, 10, 13)
V
Coastal Engineering Archive
University of Florida
Density profile at Station
wind velocity 2.5 hps.
Density profile at Station
wind velocity 2.5 mps.
Density profile at Station
wind velocity 2.5 mps.
Density profile at Station
wind velocity 2.5 mps.
Density profile at Station
wind velocity 2.5 mps.
Density profile at Station
wind velocity 2.5 mps.
Density profile at Station
wind velocity 2.5 mps.
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
wind velocity 3
Density profile
6 for wind duration 5 hours,
12 for wind duration 3 hours,
12 for wind duration 4 hours,
12 for wind duration 5 hours,
18 for wind duration 3 hours,
18 for wind duration 4 hours,
18 for wind duration 5 hours,
6 for wind duration 3 hours,
6 for wind duration 4 hours,
6 for wind duration 5 hours,
12 for wind duration 3 hours,
12 for wind duration 4 hours,
12 for wind duration 5 hours,
18 for wind duration 3 hours,
18 for wind duration 4 hours,
18 for wind duration 5 hours,
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
Figure 33.
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
34.
35.
36.
37.
38.
39.
40.
41.
42.
Figure 43.
Lines of constant density for wind of 3 hour duration,
wind velocity of 2.5 mps. (See Figures 15, 18, 21)
Density profiles for turbulence measurement data.
Mean velocity profiles for wind velocity 2.5 mps.
RMS velocities for wind velocity 2.5 mps.
Wind velocity vs. log z, (height z) above SWL.
One dimensional scalar energy spectra at a depth of 2 cm.
One dimensional scalar energy spectra at a depth of 10 cm.
One dimensional scalar energy spectra at a depth of 15 cm.
One dimensional scalar energy spectra at a depth of 33 cm.
Velocity distributions induced by wind over a closed basin
with a stable stratification.
Experimentally observed entrainment velocity vs. predicted
entrainment velocity.
LIST OF SYMBOLS
A A constant
D Depth
E(k) Scalar energy
K Curvature
N Brunt-Vaisala Frequency
R. Richardson number
U Velocity
a A constant
b Fluctuating component of buoyancy force
g Gravity
i Subscript, i = 1, 2, 3
k Wave number
1 Length
L Mixing length
m Unit vector upwards
n Mixing frequency
p Pressure
r Separation vector
t Time
U Mean velocity
u' Perturbation velocity
ue Entrainment velocity
u* Shear velocity
u Horizontal velocity components
w Vertical velocity fluctuations
ij. Velocity spectrum tensor
a Subscript, a = 1, 2
P Angle between direction of velocity U and wave number k
E Viscous dissipation
p Density
Pa Density of air
p Density of water
po Reference density
T Shear stress
Viscosity
e Eddy viscosity
viii
Astract of Thesis Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering
Halocline Erosion Due to Wind Induced Stress
By
Kenneth K. Humiston
August, 1974
Chairman: Omar H. Shemdin
Major Department: Coastal Engineering
A laboratory investigation of the effects of wind on the depth of
formation of a halocline is presented. A wave tank measuring 24.39 m
long by 1.83 m high by 0.61 m wide was filled to a depth of 0.75 m with
water so that a continuous density stratification was obtained, the vary-
ing density being achieved by varying salt concentrations. This strati-
fied system was subjected to a wind induced stress at the surface and
the changes in the velocity and density profiles and horizontal turbulent
fluctuations were measured. It was found that on a macroscopic scale a
discontinuous density jump initially forms below the surface a short time
after the wind starts blowing, thereafter the interface is depressed by
entrainment of the more dense lower layer in the turbulent upper layer.
The rate of entrainment is dependent upon an overall Richardson number.
Comparison of the horizontal turbulent energy spectra for stratified and
unstratified situations shows a greater decreasing level of energy with
depth in the presence of a stable stratification.
Chairman
I. INTRODUCTION
A shear stress applied to a fluid boundary will induce motion
and, if the Reynolds number is high enough, turbulence. In a fluid
with a stable stratification the turbulent mixing results in the for-
mation of a layer nearly homogeneous with respect to density. This
layer extends from the surface of applied stress down to a level at
which a discontinuous jump in density marks the border between the
turbulent region and the underlying region where the original density
distribution may remain unchanged. The extent of the homogeneous layer
is dependent upon the magnitude and duration of the applied stress,
physical parameters of the fluid, and characteristics of the flow. The
formation of the discontinuous density jump inhibits further extension
of the homogeneous layer by acting as a barrier to turbulent mixing.
This phenomenon is observed in many areas of significance and
economic importance. Heavy atmospheric layers which form and are re-
sistant to mixing may enhance air pollution problems. Methane can
accumulate in high areas of a coal mine shaft where the extent of
mixing with ventilating air has an important bearing on safety
(Ellison and Turner, 1959). The degree of mixing of thermal discharges
from water cooled power plants is important to the ecology of the sur-
rounding area.
This thesis examines the formation of a discontinuous density jump
in a stably stratified body of water when a wind induced shear stress
1
Coastal Engineering Archives
University of Florida
is applied to the surface. ~n the experiments conducted here the for-
mation of a halocline is analagous to the formation of a thermocline
in nature. Varying salinity was used to achieve density stratification
rather than temperature since the latter is more difficult to control
in the laboratory situation due to the high rate of thermal diffusion
as compared to molecular diffusion. (Turner, 1968). Knowledge of the
process involved will be useful in the study of internal waves as well
as in situations such as quiescent basins and thermal discharges where
a stable stratification may be altered by climactic conditions and form
a barrier to vertical mixing.
The concept of entrainment is important in the process being studied.
The turbulent region grows as the underlying non-turbulent region becomes
entrained in it. The rate of entrainment can be described by an entrain-
ment coefficient. Morton, Taylor, and Turner (1956) were the first to
use such an entrainment coefficient in their examination of entrainment
in buoyant plumes rising through a stably stratified ambient body of
water. They found the entrainment constant to be proportional to the
axial vertical velocity of the plume. Ellison and Turner (1959) per-
formed a series of experiments on stratified flows induced by a heavy
salt solution flowing down the sloping bottom beneath a layer of fresh
water, and the spread of a surface jet of lighter fluid over a heavier
ambient fluid. They measured a characteristic velocity at the inter-
face by timing the movement of entrained sheets of fluid and found that
the ratio of the entrainment velocity to the characteristic mean flow
velocity is a function of a stability parameter having the form of a
Richardson Number, g(p p )h
Ri po Vz
where V is the characteristic velocity, p is the density of the fluid,
3
g is the acceleration of gravity and h is the thickness of the turbulent
layer and the subscript o refers to the reference fluid. Turner (1968)
did some experiments on turbulent entrainment with no mean shear to de-
termine the effects of molecular diffusivity. He found that it is not
always possible to neglect molecular effects when the stability is high
and the Reynolds number is not very large. By comparison of experiments
using salinity to experiments using temperature to obtain stratification,
he showed however, that below a critical value of the Richardson Number
the entrainment is independent of diffusivity. This value is Ri = 1,
where R. is defined
i R=
where n is the frequency of the mechanical mixing generating the turbu-
lence Ap is the difference between the densities of the turbulent and
non-turbulent regions and t is the mixing length. The results of these
experiments also show an inverse proportionality between his Richardson
number and entrainment velocity ue, namely ue~RiR. ue is the rate of
depression of the interface.
Kato and Phillips(1969) did an experiment in which they applied a
constant shear stress to the surface of an annular tank containing water
with a linear stratification. A motor in the center of the tank rotated
an annular screen at the water surface, constant shear was obtained by
mounting the motor on a bearing then restraining it from turning with a
spring, and adjusting the controls so that the tension in the spring was
constant. They observed that the mean velocity varied most rapidly near
the surface below the screen and immediately above the interface which
formed shortly after the constant shear was applied. The central region
remained relatively uniform. When the interface reached the lower quarter
of the tank the density jump had become so large that entrainment
virtually ceased, yet the fluid below the interface was set in motion
by the purely viscous stress across it, the rate of diffusion of momen-
tum by molecular viscosity being greater than that of salt by molecular
diffusion.
The Reynolds number UD/v, where U is the speed of the screen, D
the depth of the layer and v the kinematic viscosity, during these ex-
periments ranged from 5 x 103 to 2 x 104, large enough, so that the
large scale structure of the turbulence would be independent of the
molecular viscosity. The properties of the turbulence are characterized
by the friction velocity u. = T-/p and D. Their entrainment constant E
defined as the ratio of the entrainment velocity ue, to the friction ve-
locity u., is thus a function of; the fractional change in buoyancy
across the interface g-- where po is the density of fresh water, the
friction velocity.itself; and the .depth D of the interface, character-
izing the scale of the turbulence. Grouping these three variables in
a dimensionless group gives an overall Richardson number
R. = g 6p D
1o Po u.
Furthermore, since the upper layer is nearly homogeneous, a salt balance
computation gives
ap 2 az o
where ( i) is the initial density gradient, so that
az 0
R g( ) D2
10o 2po u*
Thus the entrainment coefficient is a function of the Richardson Number,
E e = f(Rio)
E* 0u
5
Employing the same functional relationship as Turner, ueaRi-1, Kato
and Philips proposed the empirical relationship
E = ue = 2.5 PO u*2
u g 6p D '
where the numerical constant is stated to be uncertain to within 30
per cent.
II. THEORY
The mechanisms involved in halocline (or thermocline) erosion can
be described in terms of the turbulent energy equation for horizontally
homogeneous turbulence (see Phillips,1969)
uat at {w'(p/po + 1 U2) =__- (1)
where u. is the fluctuating component of the velocity, w' is the vertical
fluctuating velocity, p is pressure, U is the mean velocity, b' is the
fluctuating component of the buoyancy force and E represents viscous dis-
sipation. The subscript i = 1, 2, 3 and subscript a = 1, 2 specifying
velocity components. The first term on the right is the rate of transfer
of energy from the mean flow to turbulence by the work of the Reynolds
stresses. The second term on the left represents the convective redis-
tribution of turbulent energy in physical space by the turbulence itself.
In the presence of a halocline some of the energy imparted to turbulence
would be redistributed in the region of the interface where turbulent
eddies entrain fluid from the more dense lower layer and mix it with the
turbulent upper layer. This increases the density of the upper layer
and therefore increases the potential energy of the mean buoyancy field.
The transfer of kinetic to potential energy is represented by the covari-
ance between the fluctuations in vertical velocity and buoyancy in Equa-
tion (1). The continued entrainment of fluid from the lower layer results
in the erosion of the interface at a rate referred to earlier as the
entrainment velocity ue. Referring again to the results of Kato and
Phillips(1969), this entrainment velocity can be shown to be a func-
tion of an overall Richardson number characterizing the stability of
the system.
Turbulence is frequently described in terms of spectral quantities.
Phillips(1969, chapter 6) shows how the scalar energy spectrum, E(K),
E(K) = f ii(t ) d S(K) (2)
E(K) 1
can be obtained from the velocity covariance tensor of the velocity
field at a given instant, where ..ij() is the wave number spectrum
tensoror the Fourier transform of the covariance with respect to the
separation vector between the two points under consideration. Philips
then postulates that the energy density of turbulence is reduced in the
presence of a stable stratification. This is a consequence of a reduc-
tion in the covariance between vertical velocity fluctuations which re-
duces the apparent eddy viscosity defined as
Pe(z) = A pi cos2B P33(t, o=-K'*(Z), Z) dk (3)
4- 4
where A is numerical constant and is the angle between K and U. This
results in a reduction in the Reynolds stress gradient
dTr d2U
dz e dz (4)
and the stress itself if = 0 at some point in the flow, so for a
given velocity gradient the energy flux T- is also decreased. Since
the energy flux is of the order u'3/Z (Tennekes and Lumley, 1972) the
energy density of the turbulence is also reduced, reducing the eddy
viscosity still further.
8
For this postulate to be correct the turbulent energy spectrum in Eq.
(2) should show a lower level in stratified flow when compared to the
spectrum in an unstratified flow, all other conditions being equivalent.
For isotropic turbulence the longitudinal contribution to the total
kinetic energy of the turbulence is just one third of that value, so
that the above postulate could be verified with one dimensional velocity
measurements. A major aim of the experimental study described here is
to verify the postulate proposed above.
III. EXPERIMENTAL EQUIPMENT AND PROCEDURES
The experiments were conducted in the internal waves tank, 24 meters
long by .6 meters wide and 1.22 meters high, with a wind section above the
water .95 meters wide and .6 meters high (Figure 1). A complete description
of the tank was given by Sheppard, Shemdin, and Wang (1973). The tank was ini-
tially filled to a level of 37.5 cm with fresh water, then filled slowly
from the bottom with .02% salt water to bring the level to 75 cm. From
this an approximately linear profile was obtained by dragging a board
the length of the tank, at the interface between the salt and fresh water.
The board was positioned perpendicular to the length of the tank and the
interface, so that as it moved along the interface the turbulent eddies
which formed behind the board caused mixing between the two layers. This
process was continued until the desired degree of mixing was obtained.
The density profiles were measured initially with a single electrode
conductivity probe (Gibson and Schwarz, 1963) but due to the low frequency
drift, calibration was impossible, so the probes were modified by inser-
tion of a length of tygon tubing into each probe so that samples of water
could be siphoned from the wave tank at any level. The density could then
be measured with a hydrometer to an accuracy of four decimal places, al-
though the procedure is rather tedious.
Once the desired density profile was set, wind was produced by pull-
ing air through the tank with an axial flow variable pitch fan, manufactured
by Joy, and an integral part of the wave tank unit. Three fans settings
were used over the course of the experimentation, corresponding to wind
velocities of 2, 2.5, and 3 meters per second. Wind velocity profiles were
SOUTH
END
NORTH.
END
AIR FILTE
AIR IN'
HYC
r ----
Fig. 1. Schematic drawing of the facility showing the overall dimensions and the location of the
various components.
measured above the water surface with a pitot static tube, manufactured
by United Sensor and Control Corp., a Pace Model P90D differential pres-
sure transducer, a Disa Type 52B30 True Integrator signal conditioner and
a Darcy Model 440 digital multimeter (Figure 2).
For each wind velocity, observations were made for wind durations
of 3, 4, and 5 hours. In each case density profiles were measured before
alteration by the wind, for the dynamic situation with the wind blowing,
and after the wind was turned off allowing sufficient time for internal
oscillations to die out. The tank was refilled after each run in an at-
tempt to recreate the initial density structure as accurately as possible.
Velocity profiles were measured with hydrogen bubble wire and tracer
dye, but these methods proved inadequate due to the fluctuating salinity.
The hydrogen bubble wire failed because bubbles too large formed in re-
gions of high salinity while regions of low salinity saw no bubbles at
all. The tracer dye method was inadequate due to buoyancy effects.
Because of the failure of the velocity profile measurements, it was
decided to run more experiments to obtain velocity profiles with a hot
film anemometer system. A Thermo-systems, Inc. hot film sensor, model
1233 NACL was mounted on a vertical traverse mechanism anchored firmly at
the top and bottom to reduce vibrations. The sensor was moved vertically
with a pulley-cable setup. The anemometry system consisted of a Thermo-
systems model 1051 power supply and indicator, and a model 1055 linearizer.
The output was channeled simultaneously through the Disa integrator and
Darcy DVM to obtain the mean voltage directly, and through a Thermo-systems
correlator, model 1015C. The correlator amplified the signal by a factor
of 10 to reduce any subsequent interference by noise when recorded on a
Midwestern Instruments tape recorder. The recorded signal was analysed
I i3 ii' II'C'-U&ri'f i iie a .
+ .r ,r r h
r7
^ ~---
.- -. .. .. Is
O LL
* ^ tf -. y
i^ -- -"-
QO' '^ ^ .
1?
.nar '
-*ci~s;F; *TSi~; i Z .
Wind velocity measurement instrumentation, pitot tube,
pace transducer, signal conditioner and voltmeter.
Figure 2.
Cli~-t-~T C-l. P-P-t--l. CC -.--~/--~ ~
I
.." "'F"W low In
Hot-film anemometry instrumentation; anemometer power
supply and linearizer, integrator, voltmeter, tape
recorder, and rms meter.
I
Figure 3.
E .61 toa
on a Thermo-systems 1060 rms meter, and the Weston-Boonshaft and Fuchs
series 711 spectrum analyser. Figure 3 shows the anemometry instrumen-
tation.
Calibration of the velocity sensor was carried out in a plexiglass
tow tank. Calibrations were done in water of different salinities in
order to determine the dependence of response on salinity. Figure 4 shows
the calibrations at various salinities. Bubbles sometimes formed on the
tip of the sensor, probably due to disolved gases coming out of solution.
It was found that the tendency for bubbles to form at the sensor tip in-
creased with exposure time to the salt water, so immediately prior to
the data taking run the overheat ratio was reduced and a calibration curve
was obtained in water of specific gravity 1.0070, which is close to that
expected in the top layer during a run. The calibration is displayed in
Figures 5 and 6.
it) LO 0 LO
Cn d
SIOA 3
*
C
10
L
4.-
-
O
o
0
E
-o
E
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ICC
u -
i '.;^
16
4,
'4-
u
U
a-
0
,0 C
tt
E
O C
O
*r-
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S *- CL
Un
LA-
SIOA '3
4*
17
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N
>.1
u
'4J
S-
u
4)
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u Ea
EE
4-) 1
~0
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o
--
=5
LL
(D in
S41oA '3
IV. RESULTS
The results presented are from two phases of experimentation.
In the first phase density profiles (density, p vs. depth, z) were measured
to determine the depth of formation and rate of lowering of the density
jump, or "interface" between the homogeneous upper layer and the stably
stratified bottom layer, and are necessary for the determination of the
Richardson number. In the second phase velocity profiles (mean velocity,
U vs. depth, z) were measured for both the stratified and unstratified
situations to show the differences in flow distribution. At the same time
turbulent fluctuations u' were recorded to determine the horizontal tur-
bulent energy distribution (u12 vs. depth, z).
Nine separate data taking runs were performed in the first phase,
three runs of different wind duration for each of three different wind
velocities. Density profiles were measured simultaneously at three loca-
tions along the tank, 8 meters, 15 meters, and 22.5 meters from the upwind
end of the tank, designated stations 6, 12, and 18,respectively. The
profiles show the position of the interface and comparison of two profiles
under wind of the same velocity but different durations g.i.ve the entrain-
ment velocity. Figures 7 through 31 display the density profiles;each
Figure shows the profile of the starting conditions, a "steady state"
profile at a designated time after the wind was started, and a final pro-
file taken after the wind was stopped and internal oscillations had ceased.
Comparison of the corresponding profiles at the three positions along the
tank gives the horizontal density structure of the internal setup, illus-
1.005 1.010 1.015 1.020
Specific Gravity
Figure 7.
Density profile at Station 6 for wind
duration 3 hours, wind velocity 2 mps.
44 v.*
0'
1.000
1.025
Legend:
Initial
A Final
Steady State
1.010 1.015
Specific Gravity
i 020
1.025
Density profile at Station 6 for wind
duration 4 hours, wind velocity 2 mps.
E
0 40
>
1.005
Figure 8.
-4O
E
S40
01 I I L I A
1.000 1.005 1.010 1.015 1.020
Specific Gravity
Figure 9.
Density profile at Station 12 for wind
duration 3 hours, wind velocity 2 mps.
1.025
Legend:
Initial
A Final
Steady
1.015 1.020 1.025
Figure 10.
Specific Gravity
Density profile at Station 12 for
duration 4 hours, wind velocity 2
State
E
3 40
1.005
1.010
wind
mps.
Legend:
Initial
A Final
Steady
State
E
S40-
30-
20-
10-
0
1.000 1.005 1.010 1.015 1.020
Specific Gravity
Figure 11. Density profile at Station 12 for wind
duration 5 hours, wind velocity 2 mps.
1.025
1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity
Figure 12.
Density Profile at Station 18 for wind
duration 3 hours, wind velocity 2 mps.
Legend:
Initial
A Final
Steady State
1.010
1.015
1.020
1.025
Specific Gravity
Figure 13. Density profile at Station 18 for wind
duration 4 hours, wind velocity 2 mps.
60
50
E
S40
30
20
10
1.005
Legend:
Initial
A Final
Steady State
S40 -
30-
20-
10-
01
1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity
Figure 14. Density profile at Station 6 for wind
duration 3 hours, wind velocity 2.5 mps.
Legend :
Initial
Final
Steady State
E
S 40
>>
1.005
1.010
1.015
1.020
Figure 15.
Specific Gravity
Density profile at Station 6 for wind
duration 4 hours, wind velocity 2.5 mps.
1.025
t
Figure 16.
1.010 1.015 1.020 I.C
Specific Gravity
Density profile at Station 6 for wind
duration 5 hours, wind velocity 2.5 mps.
r-IP
OL- I I I
1.000 1.005 1.010 1.015
Specific Gravity
Figure 17.
Density profile at Station 12 for wind
duration 3 hours, wind velocity 2.5 mps.
1.020
1.025
Figure 18.
I I I n
1.010 1.015 1.020 1.0
Specific Gravity
Density profile at Station 12 for wind
duration 4 hours, wind velocity 2.5 mps.
E
401
Legend:
Initial
A Final
a Steady
1.015
Specific Gravity
Figure 19.
Density profile at Station 12 for wind
duration 5 hours, wind velocity 2.5 mps.
State
1.005
1.010
1.020
1.025
SY1k _
60F-
50
40-
[L'I
1.000
1.005
Legend:
Initial
A Final
Steady
1.010
1.015
i".
1.020
Figure 20.
Specific Gravity
Density profile at Station 18 for wind
duration 3 hours, wind velocity 2.5 mps.
State
1.025
Legend:
Initial
A Final
q Steady State
0 1 I 1 1
1.000 1.005 1.010 1.015 1.020
Specific Gravity
1.025
Figure 21.
Density profile at Station 18 for wind
duration 4 hours, wind velocity. 2.5 mps..
Legend :
Initial
A Final
a Steady State
1.015
1.020
Figure 22.
Specific Gravity
Density profile at Station 18 for wind
duration 5 hours, wind velocity 2.5 mps.
. 40
)0 1.005
1.010
1.025
1.005 1.010 1.015 1.020
Specific Gravity
Figure 23. Density profile at Station 6 for wind
duration 3 hours, wind velocity 3 mps.
4
40
Figure 24.
1.010 1.015 1.020 1.(
Specific Grovity
Density profile at Station 6 for wind
duration 4 hours, wind velocity 3 mps.
36
- 40
:1%
- -- -
\.
\
V
Legend:
Initial
A Final
a Steady State
1.005
Figure 25.
1.010
1.015
1.020
1.025
Specific Gravity
Density profile at Station 6 for wind
duration 5 hours, wind velocity 3 mps.
37
Legend:
Initial
A Final
Steady State
1.010 1.015
Specific Gravity
1.020
Figure 26.
Density profile at Station 12 for wind
duration 3 hours, wind velocity 3 mps.
70
60
1.025
t
Figure 27.
1.010 1.015 1.020 1.(
Specific Grovity
Density profile at Station 12 for wind
duration 4 hours, wind velocity 3 mps.
Legend:
Initial
A Final
a Steady
01 I
1.000 1.005
Figure 28.
I I I _-
1.010 1.015 1.020 1.025
Specific Gravity
Density profile at Station 12 for wind
duration 5 hours, wind velocity 3 mps.
N~
State
Legend:
Initial
A Final
a Steady State
(no data)
\%N
1.005
Figure 29.
1.010
1.015
1.020
Specific Grovity
Density profile at Station 18 for wind
duration 3 hours, wind velocity 3 mps.
50
E
340
30
1.025
i
Figure 30.
1.010 1.015 1.020 I.
Specific Gravity
Density profile at Station 18 for wind
duration 4 hours, wind velocity 3 mps.
--
Legend:
Initial
A Final
Steady
State
S40-
30- \
10-
0
O --
1.000 1.005 1.010 1.015 1.020 1.025
Specific Gravity
Figure 31. Density profile at Station 18 for wind
duration 5 hours, wind velocity 3 mps.
treated for two situations in Figures 32 and 33 as plots of lines of con-
stant density in a two dimensional tank where points with the same den-
sity are connected by straight lines for easier identification, although
the actual lines of constant density are nonlinear.
The initial density profile for the second phase of experimentation
was determined in the same way and is shown in Figure 34, along with a
stepwise approximation to the interface for the profiles after the wind
had been blowing for 1 and 5 hours. The density of the top layer and
the position of the interface were measured; below the interface the
density profile was assumed unchanged, which is a reasonable assumption
based on the data from the initial experiments.
Figure 35 is the mean velocity profile for the unstratified situ-
ation, as well as the stratified case for two different times during
the run. This illustrates the change in the flow which takes place as
the interface is eroded. Figure 36 is the rms velocity profile, {(u'2)2
vs. z} for the same situations as Figure 35, but does not show much more
than a general trend of decreasing magnitude just below the surface.
A logarithmic plot of height above the water surface vs. mean wind
velocity is shown in Figure 36. This was used to determine the shear
velocity in air u, directly, and the shear velocity in water from
a
(Shemdin, 1972)
U*2 P 2
w w w
W 'W W
The turbulent velocity fluctuations u' were analyzed on the spectrum
analyzer. The output is plotted in Figures 38 through 41 as the turbulent
energy spectra in volt2-sec vs. frequency. Figures 38, 39, 40 and 41 are
the spectra at depths of 2 cm, 10 cm, 15 cm, and 33 cm, respectively. Each
figure shows the difference in the horizontal component of the turbulent
0)
I
I n
I 0)s-
I --- 3
o
I I G)
0 t- o
C*
I-- 0
4* r--
*n cc
c -
cm
I r -
0 0 0 0 0
O- -
jI-
J^l ____I I I-\---
ISE
.0
clr
Ic o
CD)
4--
I 1 ->
W v.
4-- I-1
.0 ) 0
D >
43 C >
o 40-
O c 0.-
I 4- /.- c "
0 c" 0
+II- 0 *_- 0
OO J -
L Ln
*r-
-J,
0 0 0 0 0 0 0 0
(wo) -
1I
1 --
1c
1
N~=
I--%-- -- -
I i\r
Ihr 4 hr \
1.005
1.010
Specific
.-Initial
\
1.015
Gravity
Figure 34.
Density profiles for the experimental
run during which turbulence measure-
ments were taken.
04-.4
sr i
SWL
60k
50s
E
S40
N
Profile
30 1-
20 F-
10o-
AL
1.000
1.020
1.025
I I I I~L
- ii
20- 4
30
2- Legend
Fresh Water
Stratified I Hour of V
A Stratified 5 Hours of
-I O i 2 3
Mean Velocity U cm/sec
Figure 35. Mean velocity profiles for
wind velocity 2.5 mps.
U- w
80
SWL
70- "
60
50
E
-40
N
30
20
Legend:
Fresh Water
10- Stratified I Hour of Wind
A Stratified 2 Hours of Wind
0 I 2 3
RMS Velocity cm/sec
Figure 36. RMS velocities for wind
velocity 2.5 mps.
0.1 I I
120 140 160
Figure 37.
180 200 220 240 260
U, cm/sec
Wind velocity vs. log z,
(height z) above SWL.
'JU
4. -,
0.1 1.0 O0
Frequency, Hz
Figure 38.
One dimensional scalar
energy spectra at a
depth of 2 cm.
Volts 2
100
Volts2-sec
0.I 1.0 10 100
Frequency, Hz
Figure 39. One dimensional scalar
energy spectra at a
depth of 10 cm.
.01
.001
Volts2 -sec
.0001
.00001
1.0 10
Frequency Hz
Figure 40. One dimensional scalar
energy spectra at a
depth of 15 cm.
100
2
Volts -sec
0.1 1.0 10 100
Frequency Hz
Figure 41. One dimensional scalar
energy spectra at a
depth of 33 cm.
55
energy between the stratified aid unstratified situations at the same
depth. The position of the interface during the recording of the tur-
bulence was just below 33 cm, a depth which develops after 1 hour of
wind at 2.5 mps. The bandwidth for the analysis was .1 Hz for the range
.1 Hz to 1 Hz, 1 Hz for the range 1 to 10, and 10 Hz for the range 10 Hz
to 100 Hz. For 80% confidence the results in each of these three ranges
were within 70%, 20%, and 10% of the true values, respectively (Schiesser).
V. DISCUSSION OF RESULTS
In the initial phase of experimentation the density profiles (Figures
7-31) lend themselves to examination by continuity. If the original pro-
file is integrated to give the mass of a column of water of unit surface area,
the result should be the same for the profile under wind. This was found
to be untrue; in general the integral of the density profile at the down-
wind end of the tank is less than that of the initial profile, while at
the upwind end it is greater. Figures 32 and 33 illustrate the density
structure with an internal setup due to the wind stress. Determination
of rate of entrainment is thus complicated by the internal setup and re-
turn flow.
The shear-velocity above-the interface at the downwind end of Lhe
tank might normally be expected to be higher than at the upwind end due
to the boundary conditions of a tank of finite length. However, the
greater potential energy of the buoyancy field at the upwind-end of the
tank presents an unstable situation which results in a counter current
driven by buoyancy forces. Figure 42 illustrates the varied velocity
profiles which occur along the length of the tank. This figure was drawn
from observations of the movements of particles suspended in the water.
It is not a steady state situation because of the constantly changing
buoyancy forces, but it is typical. Unfortunately not enough velocity
measurements were made to verify this, but making the assumption that the
velocity gradients at the interface are of the same order of magnitude
along the length of the tank results in a Richardson number at the down-
I
-a,-
-~o
4v)
4-J
0
3:rr
-C,
>) '
()4-)
O0
u L
=3 V)
u)a)
a 0
4-
0 %0
r- w --
a) > 4-
::: 0 --
al)c
S-
LL.
cj
%.. -S
58
wind end of the tank about 500 times greater than at the upwind end. In
addition the relative shallowness of the interface and smaller density
jump across it at the upwind end would increase the entrainment coeffi-
cient of Kato and Phillips. From this it is hypothesized that the majority
of mixing takes place very near the upwind end of the tank. The evidence
is by no means conclusive, but the Richardson numbers calculated from the
velocity data taken in the second phase of experimentation support this
hypothesis.
The overall Richardson number 16 meters from the upwind end of the
tank was 200 with the interface at a depth of 33 cm, 4 hours later it was
267 with the interface at 36 cm. The shear velocity term in the overall
Richardson number was determined from Figure 37 and
u*w = (p U2a/pw)
The interface receeded 3 cm in the 4 hours between the recording of velo-
city profiles giving an experimental u^ of 2.1 x 10 cm/sec. The rela-
S-2
tionship found by Kato and Phillips predicts ue = 1.1 x 102 cm/sec.
Some differences between the results of Kato and Phillipsare to be
expected due to the differences in the scale of the experiments but the
same parameters are used in defining the Richardson number so this great
difference indicates that some other mechanism may be responsible. There
are several possibilities which will be discussed but first a word about
the quality of the data is in order.
In the first phase of the experiments an attempt was made.to reproduce the
starting density profile as consistently as possible so that comparisons
could be made between the different runs. For a given wind velocity the
position of the interface at the end of three, four, and five hours should
give an estimate of the average rate of entrainment over two periods of
-ne hour each. In general this is correct but the quantitative aspects
:F the entrainment rate is poor due to slight variations in the slope of
:ne linear approximation to the density profile, and perhaps even more
significant is the variation in curvature. For example, considering the
density profiles at station 18 for a wind velocity of 3 mps (Figures 29,
30, and 31), a greater entrainment velocity is indicated from 4 to 5
tours than from 3 to 4 hours. Due to the increasing density gradient
and depth of the interface with time, this should not be so according to
the expression for the entrainment constant. A close look at the initial
profiles reveals that the 5 hour case had a gradual curvature concave
upwards while the other two were less curved and exhibited concavity both
up and down at different levels. Mixing of a finite element of a fluid
with a profile concave upward will result in positive buoyancy for that
element (Long, 1970), consequently, it will rise into the turbulent region
and be completely mixed with the upper layer and produce a lowering of the
interface. The same element in a fluid with the concavity of the profile
downwards would have a negative buoyancy and sink back towards the inter-
face rather than being immediately mixed with the turbulent layer, thereby
slowing the entrainment. This in itself is an interesting observation,
but the presence of this anomaly makes it difficult to determine an en-
trainment constant from the data.
In addition to the relative magnitudes of the entrainment in the
two experiments, there is a non-uniformity in the entrainment velocity
along the length of the tank. An expression for the entrainment might
look like 2
PU*w
E(R., x) = f(x) g6pD (5)
1 Ip
onere Po is the mixed layer density and
f(x) = 1-x (6)
1 1
where 1 is the length of the tank and x is the horizontal distance from
the upwind end of the tank. It does not matter that f(x) vanished at x=l
since at the boundary u, also becomes zero. The resulting form of the
entrainment constant is 2
E(R. x) = C l-x U*w (7)
1 1 g6pD
where the proportionality constant c must be determined empirically from
S pU*3
a plot of the experimentally observed entrainment velocity vs. ( 1) ~ g)
A plot of the data from the first phase of experimentation, after elimina-
ting some of the obviously bad data points and averaging the entrainment
velocities, results in a least squares fit to a straight line with a slope
of .108 as shown in Figure 43. Equation (7) becomes
2
E(Ri, x) = .108 ('-) P u* (8)
1 g6pD
Application of this to the second phase of experimentation results in a
prediction of ue = .000198 with the interface at a depth of 35 cm, as
compared with a value of .00021 experimentally, averaged over the whole
4 hour run. The agreement is not so good over the earlier and latter
parts of the run where the experimental values are respectively higher
and lower than the predictions. The initial density profile shows some
curvature which would effect entrainment as mentioned earlier. Since neg-
ative curvature increases entrainment and positive curvature retards it,
the entrainment constant might be modified to include a curvature term K,
and Equation (8) may assume the form
E(Ri, x, K) = .108 ( x) P* (-Ka) (9)
I g6pD
.001
.0009
.0008
.0007
.0006
.0005
.0004
.0003
.0002
.0001
0
(
.002 .003 .004 .005 .006 .007 .008 .009
PU3
Ue gTp-( ) cm/sec
Figure 43. Experimentally observed entrainment
velocity vs. predicted entrainment
velocity.
I I
0-
S
0
I I I I
3 .001
.010
a i N | i |
4%-"'
where a is a constant. Since a very small curvature has a pronounced
effect on entrainment, a will be small. Approximating the curvature as
circular over a small region and obtaining a radius of curvature R =
from the initial profile it is estimated, a = .03.
The range of eddy sizes supporting the Reynolds stresses is the
same as that containing the bulk of the kinetic energy of the turbu-
lence. From the spectra it is obvious that the stable stratification
reduces the transfer of energy to the large scale turbulence, as pre-
dicted by Phillips hypothesis by reducing the Reynolds stress. The
small scale turbulence, i.e. the equilibrium and dissipation range is
at a much higher frequency than the N = / for the upper layer,
p 3Z
which would be in the neighborhood of .1 sec Near the interface,
N = 1 and a corresponding reduction in turbulence in this range is
observed just above the interface.
It would be useful to compare these turbulent energy spectra Lu
the corresponding situation with no return flow. With return flow for
the same surface shear there would be a larger horizontal velocity gra-
dient in the vertical direction which would increase viscous dissipation.
There must also be a level at which -u = 0, which would inhibit the
az
energy transfer by Reynolds stresses. Considering these two factors
we would expect a lower level of turbulent energy near the interface when
a return flow is present, and therefore a lower rate of entrainment.
Besides these two reasons for expecting slower entrainment with
return flow, there are also surface and internal setups and the shear
stresses required to maintain those setups. The viscous forces would
contribute to the energy dissipation term. The internal setup represents
a transfer of kinetic energy to potential energy within the system, but
VI. CONCLUSIONS AND RECOMMENDATIONS
The functional relationship between the overall Richardson number
and the entrainment is verified but additional factors must be consi-
dered when the applied shear stress is a function of the distance along
the surface of applied shear, and a return flow is present. The mech-
anism by which return flow modifies the entrainment is due to the shear
flow at the interface and the mean shear flow in the turbulent layer
with its associated effect on the Reynolds stresses. Due to the den-
sity structure of the internal setup, buoyancy forces play an addi-
tional role not seen in the situation with no return flow, and consi-
derations of curvature in the initial density profile indicate that
buoyancy effects are very important when the initial profile is not
linear.
The first recommendation is that in future research in this area,
less time should be spent measuring initial and final density profiles,
and more effort go into studying the dynamic situation. More accurate
measurement of entrainment rates is needed to substantiate the empirical
results presented here. The same is true for the entrainment constant
involving the curvature term, although for what was originally intended
as the object of this study it would have been better if a better method
for establishing and reproducing linear density profiles had been avail-
able, leaving the curvature effects to be studied separately.
Perfection of the conductivity probe and the associated electronics
would facilitate the collection of density data. The method used here
for measuring density profiles was so time consuming that the profile
would change during the time required for measurement, thus introducing
an additional source of error.
The effects of stratification on turbulent energy transfer are
verified in that the one dimensional scalar energy spectra presented
here show a greater decrease in turbulent energy over depth for the
stratified case. The development of the theory behind this depends
on the correlation of the vertical velocity fluctuations at two sep-
arate points so that a quantitative discussion would need such measure-
ments. Determination of the scalar energy spectrum (Equation (2)) also
requires turbulence measurements from two points with separation r, so
it is recommended that additional studies in this area should include
measurement of at least two fluctuating components at two separate
points. These measurements would eliminate the need for the isotropic
assumption, and along with the information which could be gathered
with two conductivity probes it would be possible to determine the net
local buoyancy flux.
is unlike the potential energy increase due to entrainment in that it is
recoverable, and manifests itself in the form of an internal seiche when
the surface shear is removed.
APPENDIX A
DIFFUSIVITY OF NaCI IN WATER
The assumptions were made that molecular diffusion and salinity
increase from evaporation were insignificant compared to turbulent
mixing and overall salinity. The mass transfer due to molecular dif-
fusion is given by
D
NA= AB () (X XA2)
A ZX BM Al A2
BM
NA = Mass flux of salt
DAB = Molecular diffusivity
Z = Thickness of diffusion layer
XB XB2
XBM In(XB2/XBl)
XAl XA2, XB1, XB2 = Mole fractions of salt and water in
p = Density of salt
M = Molar weight of
the salt and fresh water layers,
respectively
water
salt water.
An exaggerated estimate of the diffusion can be given by assuming
XA2 = 0, therefore XB2 = 1 and DAB = 1.3 x 10-5 cm2/sec (Treyball, 1968)
which results in
1.3 X 10-5 cm2/sec (.06 g mole (009) 5.3 g
A (.1 cm)(3.1) cm G 'g mole)
= 1.113 x 10-6 g/cm2/sec
As a comparison, a 1 cm erosion of the interface across which c =
.03 g/cm would result in the transfer of .03 g of salt to the upper
layer, the ratio of this to the transport due to diffusion over one
hour is approximately 10. In the experiment the salinity jump across
the interface is less than for the above calculated diffusion, it is
therefore safe to say that molecular diffusion is at the very most one
order of magnitude smaller than the turbulent diffusion, and may be
neglected.
APPENDIX B
NEGATIVE BUOYANCY GENERATION AT THE
SURFACE DUE TO EVAPORATION
Mass transfer per unit area between a flat gas-liquid interface
is given by (Treyball, 1968)
1 PAi/t
NAMA = MAFG(In 1 PAG/t) (Bl)
where
NA = Molar flux
MA = Molecular wt = 18 Ib/lb mole
FG = Mass transfer coefficient
PAi = Vapor pressure of the liquid (A) at the
interface = 17.535 mm Hg
PAG = Partial pressure of the liquid in the gas
pt = Total pressure = 760 mm Hg.
The mass transfer coefficient is obtained by the heat-mass transfer
analogy with the empirical relationship (Treyball, 1968)
h = 0.072 G'0.6
where h is the heat transfer coefficient between an air water interface
and G' is a superficial air mass velocity. The heat transfer dimension-
less group defined by
H- Cph p 2/3
jH C Up r
(B2)
c = Heat capacity of gas
U = Velocity of gas
p = Density of gas
P = Dimensionless Pradtl number = Cpviscosity
r thermal conductivity
can be related by the heat-mass transfer analogy to the mass transfer
dimensionless group
kGPBM M 2/3
D pU 2/3
where
KG = Mass transfer coefficient per concentration
difference
PBM = Log mean partial pressure difference
M = Molecular weight of the gas
Sc = The dimensionless Schmidt number
Kinematic viscosity
Diffusivity of water vapor in air
Equating (B2) and (B3)
h
C Up
p
K P M
p 2/3 =G BM S 2/3
r pU C
The mass transfer coefficient FG is determined,
F h p 2/3 PU S 2/3
G = KG PBM G r M c
For air at 5 fps the value of FG is .0114 lb mile/hr ft2. Substituting this
into (Bl) and making the extremely exaggerated assumption that the air
is initially at zero humidity, the evaporation after one hour would be
70
.0023 grams H20/cm2. This woulc leave behind
.0023 gqH?0 .006 qNaCl) = 1.38 x 10-5 gaC
cm hr g H20 1.38 x
when the top layer is at a concentration of .006 g NaCl/gH20 as in the
experiment. The ratio of the amount of salt left behind to the concen-
tration in solution is 2 X 10-4; evaporation can therefore be neglected.
BIBLIOGRAPHY
Ellison and Turner, 1959, "Turbulent Entrainment in Stratified Fluids,"
Journal of Fluid Mechanics, Vol. 6, pp. 423-448.
Gibson, C. H. and Schwarz, W. H., 1963, "Detection of Conductivity
Fluctuations in a Turbulent Flow Field," Journal of Fluid Mechanics,
Vol. 16, pp. 357-364.
Kato and Phillips, 1969, "On the Penetration of a Turbulent Layer into
a Stratified Fluid," Journal of Fluid Mechanics, Vol. 37, pp. 643-655.
Long, R. R., 1970, "A Theory of Turbulence in Stratified Fluids,"
Journal of Fluid Mechanics, Vol. 42,.pp. 349-365.
Morton, B. R., Taylor, Sir Goeffrey, and Turner, J. S., 1956, Proceedings
of the Royal Society of London, Series A, Vol. 234, Jan. 24 Mar. 6.
Phillips, 1969, The Dynamics of the Upper Ocean, Cambridge University
Press.
Schiesser, W. E., Statistical Uncertainty of Power Spectral Estimates.
Shemdin, 0. H., 1972, "Wind-Generated Current and Phase Speed of Wind
Waves," Journal of Physical Oceanography, Vol. 2, No. 4.
Sheppard, D. M., Shemdin, 0. H., and Wang, Y. H., 1973, "A Multipurpose
Internal Wave Facility," Technical Report No. 19, Coastal and Oceano-
graphic Engineering Laboratory, University of Florida, June.
Tennekes, H. and Lumley, J. L., 1972, A First Course in Turbulence,
MIT Press.
Treyball, R. E., 1968, Mass Transfer Operations, McGraw-Hill Book Co.
Turner, 1968, "The Influence of Molecular Diffusivity on Turbulent
Entrainment Across a Density Interface," Journal of Fluid Mechanics,
Vol. 33, pp. 639-656.
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