Title: Determination of vertical turbulent diffusivities of heat in a North Florida lake /
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Title: Determination of vertical turbulent diffusivities of heat in a North Florida lake /
Physical Description: ix, 206 leaves : ill. ; 28cm.
Language: English
Creator: Steinberg, Jerry Allen, 1945-
Publication Date: 1975
Copyright Date: 1975
 Subjects
Subject: Water temperature   ( lcsh )
Turbulence   ( lcsh )
Diffusion   ( lcsh )
Mize, Lake   ( lcsh )
Environmental Engineering Sciences thesis Ph. D
Dissertations, Academic -- Environmental Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 202-205.
Statement of Responsibility: by Jerry A. Steinberg.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098159
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000150185
oclc - 02352398
notis - AAR6423

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DETERMINATION OF VERTICAL TURBULENT DIFFUSIVITIES OF HEAT
IN A NORTH FLORIDA LAKE





By

Jerry A. Steinberg


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





UNIVERSITY OF FLORIDA


1975





























S UNIVERSITY OF FLORIDA

3 1262 08666 218 5





























To my parents
Louis Steinberg and Bette King Steinberg
who have given to me,
directly and indirectly,
in ways I know
and in ways I will never comprehend,
much of the ability necessary to accomplish
what is represented here.















ACKNOWLEDGMENTS


This work was made possible in part by the Environmental

Protection Agency which provided stipends and other financial

aid and by the National Science Foundation which furnished funds

for much of the research apparatus. I gratefully acknowledge

the assistance of these two agencies.

I also appreciate the cooperation of the University of Florida

School of Forest Resources and Conservation, its director, Dr. John

Gray, and Professors J. W. Miller, Jr. and Don Post, in permitting

the use of Lake Mize in the Austin Cary Memorial Forest as the

field research site.

Many people helped accomplish the field investigation,

unfortunately all cannot be mentioned here. However, special recog-

nition should be given to Mr. Howard McGraw and Mr. Bob Batey who

provided advice, materials, the use of shop equipment and tools,

and much personal assistance; Mr. Lloyd Chesney who worked on the

design of the electronic portion of the data acquisition system;

Dr. Robert Sholtes who was instrumental in my acquiring the paper

tape punch at no cost; and Mr. Alan Peltz, who provided scuba diving

services needed to install the research tower in the lake. These

people made it possible to conduct the field experiments on a meager,

and otherwise insufficient, budget.

Dr. Robert Dean provided advice toward the development of various











analytical techniques at a time when progress was virtually nil

and this impetus was most helpful.

I would also like to acknowledge the continuous support and

assistance provided by Dr. Wayne Huber, Chairman of my supervisory

committee. His encouragement and friendship have been as valuable

as his efforts in his role as my mentor.

The remainder of my supervisory committee, Dr. Edwin Pyatt,

Dr. Omar Shemdin, and Dr. John Cornell, have also been most helpful,

and I want to express my thanks to them.

The production of this volume has required exceptional effort

from a number of people who deserve special recognition. Ms. Mary

Polinski and Ms. Donna Hagen have typewritten both intermediate and

final manuscripts; Mr. Rocke Hill and Mr. Jim Cauthorn provided

drafting services; and fellow students, Miguel Medina and Henry

Malec, gave invaluable general assistance.

;Finally, the influence and effect of my wife, Emily Gill

Steinberg, must be acknowledged, for as significant as all other

aid I have received has been, hers is the only assistance I could

not have done without. Her love, her patience, her support, her

encouragement, and her companionship have both directly and indirectly

been crucial to the fruition of this research effort.















TABLE OF CONTENTS


Page


ACKNOWLEDGEMENTS. . . . . . . . . i

ABSTRACT ... . . . . . . . iii

CHAPTER

I INTRODUCTION. . . . . . . ...... 1

II TURBULENT TRANSPORT CONCEPTS. . . . . . 6

Introduction. . . ... .......... . .. 6
Transport in Receiving Waters . . . . . 6
Advective-Diffusion Equation . . . . . 8
Introduction of Diffusion Coefficients. ... .13
Heat Transport in Lakes . . . ..... 17
Motion in Lakes . . . ..........18
Evaluation of Diffusivities . . . ... .20
Results of Previous Studies . . . . ... 23
The Problem of Convective Mixing. . . . ... 25
Averaging . . . . . . . . * 27
Goals of this Study . . . . . . .. 30

III METHODOLOGY OF DIFFUSIVITY EVALUATION . . .. .32

Introduction.. .. ...... .. . . . 32
Flux-Gradient Method. ... . . . . . 33
Evaluation of Heat Flux and Gradient Terms. . 35
Evaluation of Solar Flux Term ;.. . . ... 37
McEwen's Method .......... . .... 38
Analytical Representation of Temperature
Profiles . . . . . . . . . 41
Cayuga Lake . . . . . ... . . . 44
Correlation of Fluctuations Method. . . .. .56

IV SELECTION OF A TURBULENCE MEASURING DEVICE. . 58

Introduction. ... .. .. . . . . . 58
Equipment Requirements. . . . . ...... 59
Hot-Wire, Hot-Film Anemometers . . . ... 60
Laser-Doppler Velocimeters . . . . ... 62
Rotating Element Velocity Meters. . . . ... 63
Acoustic or Ultrasonic Devices. . . . ... 65
Lagrangian Methods. . . . . . . . 68
Electromagnetic Flowmeters. . . . . ... 70
Other Devices . . . . . . * * 71
v











TABLE OF CONTENTS (continued)


Page


CHAPTER


Price Comparisons ... . . . . . . .
Requirements of the Lake Mize Study. . . .
Selection of the Electromagnetic Flow Meter. .

V DESIGN OF THE DATA ACQUISITION SYSTEM . . .

Introduction . . . . . . . . .
Data Requirements for Flux-Gradient Method .
Data Requirements for Correlation of
Fluctuations Method . . . . . . .
Other Data Requirements . . . . . .
Aggregate Requirements . . . . . . .
Data Recording Requirements . . . . .
Additional Data Acquisition Needs . . . .


Commercial Equipment . . .
Design of Data Acquisition Sysl
Velocity Measurement . . .
Temperature Measurement . .
Solar Radiation . . . .
Cup Anemometers . . . .
Wind Vane . . . . .
Relative Humidity Sensor . .
Velocimeter Calibration. .
Wind Anemometers and Direction
Solar Pyranometer . . .
Relative Humidity . . .
Thermistors . . . .
Punch Paper Tape Recorder. .


tem. . . .








Indicator


VI ORGANIZATION OF THE FIELD STUDY AND DATA
REDUCTION TECHNIQUES. . . . .


Introduction.. . . . . . . . .
Selection of Lake Mize as the Research Site.
The Instrument Installation . . . .
Data System Operating Procedure . . .
Data Reduction . . ... . . . .


. . 105











TABLE OF CONTENTS (continued)


Page

CHAPTER

VII RESULTS OF ANALYSIS. ... . . . .. 121

Introduction ..... . . . .. . . .121
Flux-Gradient Analysis ........... .122
Inadequacy of the Flux-Gradient as Typically
Applied. ... . . . . . . .. 122
Detailed Examination of Convective Mixing. . 125
Manual Calculations of Diurnal Values of
Diffusivities. . . . . . . . 132
Correlation of Fluctuations. . . . ... 137
Comparison of Flux-Gradient and Correlation
of Fluctuations Results. . . . .. . 156
Variations of Diffusivities with.Time and
Depth . . . ... . . . . . 168
Comparison with Other Lakes . . . ... 181

VIII SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR.
FUTURE RESEARCH. . ... . . . . 184

Summary . . . . . ... ... . 184
Conclusions. . . . . . . . . 185
Suggestions for Future Research. . . .. .187

APPENDIX A FORMAT OF STORED DATA.. ... .189

APPENDIX B DATA FOR DIFFUSIVITY CALCULATIONS. 194

LITERATURE CITED . . . . . ... .202

BIOGRAPHICAL SKETCH. .......... . 206










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


DETERMINATION OF VERTICAL TURBULENT DIFFUSIVITIES OF HEAT
IN A NORTH FLORIDA LAKE

By

Jerry A. Steinberg

August 13, 1975


Chairman: Wayne C. Huber
Major Department: Environmental Engineering Sciences


Knowing or predicting the distribution of water quality parameters

in water bodies is essential to efficient management of water-as.a

valuable natural resource. Theoretical concepts which describe the

transport of substances throughout the water have been developed, and

the effects of turbulence have been accounted for by the semiempirical

approach of relating turbulent diffusive flux to the average gradient of

concentration through coefficients of diffusivity. Because of the

significance of vertical temperature distributions in lakes and reser-

voirs, this research has focused on evaluating vertical diffusivities

of heat in a deep Florida lake.

A complex data acquisition system was designed, built, and installed

at Lake Mize, Florida, to gather field data to be used to evaluate

diffusivities. The investigationoinvolved measuring both the actual

turbulent motion responsible for the transport, and the change in heat

content as a result of the motion. Emphasis was also given to making


viii











observations rapidly enough to enable the study of lake motion due

to convective cooling.

The results indicate that motion in the lake was at most times

too low to be detected by the velocity measuring equipment used.

Other results point out the need for closely spaced temperature

measurements near the surface in order to adequately define the con-

vective cooling process. Detailed analysis of convective cooling

revealed that it may be described by the semiempirical concepts pro-

vided that suitably short averaging intervals are used. Finally,

diffusivities found in Lake Mize are much smaller in the upper layers

than those observed in other, larger lakes, but in the lower strata

values are more nearly equivalent, and in most cases, two orders of

magnitude ,greater than values for molecular motion.














CHAPTER I

INTRODUCTION


The problem of determining the distribution of a substance in

a water body is often encountered as an aspect of managing the water

as a resource. The ever-increasing need for water for industrial,

recreational, and potable uses, among others, requires that the

management of it be given more attention. For example, these needs

may dictate that available water be reused several times and often

essentially shared by successive users as it progresses through the

hydrologic cycle. Thus, one important facet of water resources

management is determining the fate of substances introduced into

receiving waters as discharges from some user. Another important

aspect is to understand and predict the behavior of naturally occurring

physical phenomena in fresh water and marine environments. Both

man-caused and intrinsic variations in physical, chemical, and

biological water quality are important because of the effects they

may have on other uses.

In order to obtain better knowledge of variations in water quality,

it is necessary to better understand the hydrodynamics of receiving

water bodies. This is because both artificial and naturally occurring

substances are transported by the movement of the water. By defining

.this-motion, a description of the levels of substances existing at











various times and places can be obtained. Knowing or predicting

these levels is valuable for purposes of planning uses of waterways,

administering regulations aimed at maintaining minimum standards of

water quality, and otherwise managing the resource in a way which

provides optimum usefulness.

Most water motion existing in nature is turbulent, that is, much

of the movement is characterized by randomness which is usually

described in terms of temporal and spatial averages. To describe

this turbulent flow,equations of motion, continuity, and conservation

of mass, among others, may be used to express the existing hydrodynamic

phenomena. The equations are used to develop a model for making

predictions of concentrations, and included in the model are usually

coefficients which relate to the turbulent nature of the water motion.

The lack of information regarding these coefficients is usually a

serious impediment to the application of the hydrodynamic concepts

to engineering and management problems. In their most common form,

the coefficients are called diffusion coefficients or merely

diffusivities and may be functions of direction and location and the

type of transported quantity (i.e., heat, mass, or momentum). The

focus of this study is to present a better understanding of diffusivities,

especially diffusivities of heat in the vertical direction.

In particular,transport of heat in lakes and reservoirs is a

subject which has received much emphasis for a variety of reasons.

The intrinsic relationship between temperature and density makes the

vertical distribution of temperature important because it reveals the

structure of existing thermalstratification. In reservoirs the





3




vertical distribution and transport of heat affect the withdrawal

operations of the impoundment due to considerations of thermal water

quality downstream. Another example of the importance of evaluating

heat transport is the need to solve problems of recirculation of

heated discharges. Because of the widespread use of lake and reservoir

waters for cooling in many industrial processes, especially power

production, this matter has received much attention.

The particular problem of transport in relatively deep,

quiescent water bodies like most lakes and many reservoirs is one

which has received much attention and is continuing to receive

attention. This is primarily due to the occurrence of stratified

conditions which develop in lakes and reservoirs annually. In many

cases the stratification exists with such stability that vertical

transport throughout the entire depth is greatly restricted, and the

effect on water quality is significant. Because mixing between upper

layers (epilimnion) and lower layers (hypolimnion) is restricted,

undesirable accumulations of chemical and biological substances can

take place in the hypolimnion, and effluents discharged into the

stratified waters will move to depths of equivalent density and tend

to concentrate there. An increased knowledge of the magnitudes and

variability with time and depth of the diffusion coefficients will

provide improved application of the turbulent transport concepts.

This work is a report of a study of vertical turbulent transport

of heat in a small, deep sinkhole lake in North Central Florida.

Specifically, mathematical methods are presented which express the

transport but require the evaluation of diffusion coefficients.











The usefulness and propriety of the concept of diffusion coefficients

are considered. In addition, special attention is directed to the

phenomenon of convective mixing due to surface cooling in an effort

to learn whether or not the notion of diffusivities is compatible

with this type of turbulent water motion. Because the usefulness of

the transport equations is often limited by the lack of knowledge

about the diffusivities, the evaluation of these by two conceptually

different methods is described and demonstrated. One method entails

the assessment of the effects of vertical turbulence by monitoring

the changes in heat content of the lake, and the other method seeks

to measure the turbulent motion in the lake by directly monitoring

vertical water velocity. The study requires a large amount of field

data collection; and themeasurement of velocity requires the use of

a relatively new type of velocimeter because of the very low levels

.of motion encountered. A complex and highly automated data acquisition

system which is capable of fulfilling the rigorous data requirements

is developed.and discussed. A key feature of the system is its

ability to record data that can be read directly into a digital computer

for.subsequent analysis.

The results of the analysis of the data collected at the lake

intermittently over a period of eight months are given. Use of

diffusivities during periods of convective mixing is shown to be

hazardous because of the nebulous definition of the temperature

gradient under these conditions. Included are computations showing

the magnitudes and variability of the diffusivities over both short

and long time intervals and at various depths; however, the extreme






5




quiescence of the experimental lake makes extrapolation to other

conditions difficult. Suggestions for future research and guidelines

for necessary data acquisition are provided.















CHAPTER II

TURBULENT TRANSPORT CONCEPTS



Introduction


The general direction of this study is presented in this chapter.

Included is a discussion of the various types of studies in which turbulent

transport has been modeled using solutions to fundamental equations. After

a development of the general equation and accompanying theory for defining

transport due to turbulent diffusion, the presentation is narrowed to con-

sideration of the transport of a specific quantity, heat, in lakes.

Summaries of causes of lake motion and schemes for evaluating turbulent

diffusivities are given along with a review of earlier efforts to evaluate

the diffusivities.. Specific problems encountered in these previous efforts

are recounted. The latter portions of the chapter present arguments for

accepting the usefulness of semiempirical descriptions of turbulent trans-

port. The final section discusses the efforts of this study to support

those arguments.



Transport in Receiving Waters


Knowing the distribution of substances or heat in a water body can be

of great importance. The concentrations of various water quality param-

eters throughout a region of the water body may be due to a nearby

municipal or industrial discharge or due to normal physical conditions











which cause spatial variations. Regardless, it is often necessary from a

management point of view to be able to know or predict the levels which

exist or might exist at various locations and times. Historically, much

attention has been given to making such determinations, and in most

instances use has been made of hydrodynamic relationships which describe

the transport of substances in the water in terms of the dynamic and

physical characteristics of the liquid and the substances being transported.

Glover (1964) described the dispersion of sediment in a flowing stream

using solutions to the convective diffusion equations for turbulent flow.

The same type of mathematical approach has been used to predict distri-

butions of nutrients in lakes and reservoirs (Chen, 1970), and Munk (1966)

studied the vertical variations of temperature and salinity in the oceans.

Studies of oxygen content in natural waters have been concerned mostly

with reaeration processes in rivers and streams, and vertical variations

in oxygen levels have been disregarded or found to be negligible in all

but a few studies. The work of Eloubaidy and Plate (1972) is one excep-

tion. Numerous investigations have been directed toward finding specific

point concentrations of various pollutants which have been introduced into

receiving waterways. Attention has been given to estuaries (Pyatt, 1964),

lakes (Liggett and Lee, 1971), rivers (Cleary and Adrian, 1973), and

reservoirs (Morris and Thackston, 1969).

In addition to studies of substance concentration, a myriad of

attention has been given to spatial and temporal distributions of heat in

natural and impounded waters. The study by Dake and Harleman (1966)

focused on the temperature distributions in lakes and the work of Huber

(Huber and Harleman, 1968) involved predicting temperatures in reservoirs.











In most of these efforts the essential approach was to use advective

diffusive principles of transport which were modified and in most cases,

simplified to accommodate the particular situation. Some of the.works

cited above adopt and use mathematical expressions which include prior

assumptions regarding the behavior of turbulent diffusion terms without

calling attention to the significance of those assumptions. Others of

the works take care to stress the acceptance of certain assumptions and

nonetheless are forced to accept the inadequacies they offer. While for

a given analysis any of the components of the transport equations may

present evaluation problems, the diffusive terms are invariably a source

of uncertainty and difficulty.

Further discussion of the use of the turbulent advective-diffusion

equation should be preceded by a development and presentation of the

equation itself. The transport of mass, momentum, or heat in a water body

can be described mathematically by considering the fluxes of any of these

quantities into and out of an elemental volume of water. The ensuing

derivation may be found in any of several published works on transport

processes and is offered here for the sake of completeness.



Advective-Diffusion Equation


Consider the elemental volume shown in Figure 2-1. Through each face

of the volume passes a quantity of a substance of concentration, s. If

the effects of molecular diffusion are assumed negligible relative to the

effects of turbulence, the net flux of material into the volume in the

vertical direction, for instance, is
















z (psw)axay
I I (psv)axaz


(psu)ayaz IIy

y [su + ( su)a x]acz
--- [ U


psv+ (psv)ay] xaz [psw+-(psw)a z] AAy

FIGURE 2-1. FLUXES OF SUBSTANCE OF CONCENTRATIONS THROUGH THE FACES
OF AN ELEMENTAL VOLUME











Net flux (z) = (psw)AxAy + [psw -(psw)Az]AxAy

(2-1)
+ rs xAyAz

where p = density of water (mass volume-)

s = concentration (mass substance mass solution-1)

w = instantaneous vertical water velocity
(distance time-1)

x,y,z = orthogonal distances

r = net rate of production of s inside the volume
(mass substance volume-1 time-l)

After simplifying, equation 2-1 becomes


Net flux (z) = -(psw)AV + r AV (2-2)

where AV = element volume = AxAyAz

Given that the velocities in the x and y directions are u and v, respec-

tively, the fluxes in these directions can be similarly expressed so that


Total flux = [- (psw) -(psu) -aypsv)]AV + rs V (2-3)

Now, the total flux must also equal the time rate of change of mass of

substance in the volume, therefore


Total flux -(psAV) (2-4)
at

where t = time

For incompressible fluids changes in density and volume are negligible so

that when equations 2-3 and 2-4 are equated, the result is


a(s) a a a r
a) = (sw) --(su)- (sv) + (2-5)
Dt Bz ax By p











The instantaneous values can be expressed in terms of time-averaged means

plus an instantaneous fluctuations from the mean, as


S s + ', w = w+w', u = + u', v v + v' (2-6)

where the overbar denotes mean values and the prime denotes fluctuation

values. Average values over a time interval T are defined by
t+T
A-f/ tAdt (2-7)
t

The total transport over the time interval is found by averaging equation

2-5 over the interval, which results in the time-average of each term in

that expression. Taking the vertical flux term for example and recalling

the meaning of the overbar


-(s) = -( + s')(w + w') = -(sw + s'w' + s'w + sw') (2-8)
Z as as

Because the coordinate is not a function of time, the averaging can be

performed on the quantities before differentiation. Also, equation 2-7

implies that the averaging may be done on each term separately, and

furthermore, that s'w = sw' = 0 and sw = sw. Therefore, equation 2-8

becomes


--(sw) = -(sw + s'w') = -(sw) + -(s'W') (2-9)

Notice that the term involving the product of two fluctuations does not

vanish. This is because a positive vertical velocity fluctuation must

transport a small amount of substance downward. Depending on the local

gradient of the substance, a small increase or decrease of substance

concentration will occur, and hence, a positive or negative fluctuation

of s will be detected. A reverse, upward velocity fluctuation would cause










a reverse substance fluctuation provided the average gradient were the

same. Therefore, the product of the two fluctuations will always have

the same algebraic sign regardless of the direction of water motion and

unless there is no concentration gradient will always be nonzero. The

flux represented by the term s~' is known as the turbulent diffusive

flux. It is so described because it accounts for the transport due to

the irregular variations from mean values.

Applying a similar averaging process to other terms in equation 2-5

using equation 2-6 yields


( a ) _~ ) -- (< -S-
at az az ax ax
(2-10)

j(s) (s'v')

Substituting the quantities of equation 2-6 into the continuity equation

and time averaging produces the continuity equation for turbulent flow as


a(u) a(v)~a(w) (2-11)
ax + y + S = o (2-11)
3x ay 3as

By substituting this into equation 2-10, the result is

S= -as) -a(s) -a(s) a
-- W- -- V -- ( t
at z a a y az
(2-12)
r
_-- a-Tf-)+

which is the advective-diffusion equation for turbulent flow. It states

that the time rate of change of average concentration over some interval

at a point is due to a flux of material advected by average velocities

and diffused by fluctuations of velocity in each of these directions plus

any production of the substance which occurs during the interval.











Introduction of Diffusion Coefficients


As noted above the terms containing the time average of the product

of fluctuation quantities accounts for transport due to turbulent

diffusion and are called the turbulent fluxes. The usefulness of equation

2-12 is limited because the fluctuations are difficult to evaluate; however,

conventionally the flux terms are replaced by the relationship


s'w' = Ks (s/az)

s'u' = K x(s/ax) (2-13)

s'v' = K ay(s/3y)

where the K values are known as coefficients of turbulent diffusion or

turbulent diffusivities. Generally, Ksz 4 Ksx # Ksy, and they will

only be equal in the special case of isotropic turbulence. Analogous

diffusivities exist for the cases of heat, mass, or momentum transport,

and they are identified accordingly. By relating the turbulent flux to

mean concentration (or velocity or temperature) gradient,the solution of

equation 2-12 is facilitated; yet the theoretical basis for such action is

questionable. This should be restated more specifically: to substitute

equation 2-13 into equation 2-12 does not inherently pose any theoretical

questions because the unknown fluctuation product is merely replaced by a

different unknown K, but at the sqme time no progress is made toward

improving the usefulness of the equation. If on the other hand, the

substitution is made and some specific assumption is made regarding the

variation of diffusivity with coordinate distance (e.g., K = constant),

then the theoretical structure becomes much less rigorous.










The need for more useful forms of equation 2-12 have stimulated and

perpetuated the existence of,the less rigorous use of equation 2-13.

Originally, Boussinesq developed the coefficient of eddy viscosity to

express turbulent momentum transport as an analogy to the Fickian equation

for molecular transport (p. 25, Hinze, 1959); however, the theoretical

basis of the molecular process does not exist for the turbulent process

because the nature of the turbulence is a property of the flow field and

not the transported quantity. A semiempirical basis for the form of

equation 2-13 was presented by Prandtl as his mixing length theory

(p. 277, Hinze, 1959). A review of his salient.points will be presented

at this time.

The one-dimensional transport in the x direction of a substance of

concentration s taking place in the presence of a gradient of average

'concentration s is shown in Figure 2-2. The transport through a y z

plane at x = 0 does not result from the effects of the gradient but from

the effects of turbulence. During a time interval, t, any elemental

volume of fluid may pass through the plane of area BAyz but the average

distance a volume will travel is hi before the initial character of the

volume is lost. The average rate of transport in the positive x direction

is given by


Fs A A t f psdxdzdy s dxdzd (2-14)
Fs El y z l -xj yz o d J 2-]





















PLANE OF AREA 6YAZ
ELEMENTAL
VOLUME





\ ELEMENTAL
VOLUME
I
I I I










-XI 0 XI
x 0 x


FIGURE 2-2. TURBULENT TRANSPORT OF SUBSTANCE THROUGH A X-Y PLANE











Expanding the concentration in a Taylor series about x = 0 gives


s(x) = s(0) + x + 1/2x2 a + ... (2-15)
ax O x2 I
x0-

Prandtl assumed the distance x to be small enough to cause the higher

order terms to be negligible, so after dropping all non-linear terms, and

integrating equation 2-14


Fs = ax O(s) (2-16)


By substituting 2

K = (2-17)
t

into equation 2-16, the resultant direct proportionality between flux

of substance through a plane atx = 0 to the gradient at x = 0 is realized,

and the similarity to equation 2-13 is apparent.

Other semiempirical formulations have been offered. Two of the most

notable of these are by Taylor and von Karman and are discussed by Monin

and Yaglom (1971) who note that attempts to find improved forms of these

relationships were in progress as recently as the time of their writing.

They also indicate the suspect nature of semiempirical attempts at

simplifying transport equations and offer the possibility that radical

new approaches to the question may provide more nearly correct solutions.

On the other hand, the statistical solutions to transport problems are

usually very unwieldly, and the concept of diffusion coefficients continues

to be considered. This.is supported by Okubo (1962) who notes the useful-

ness of the semiempirical approach in relation to the more theoretically

supported turbulence doctrines.











The theoretical concepts of turbulent transport summarized by

equation 2-12 and the semiempirical concepts of turbulent diffusion

given by equation 2-13 have been presented and discussed. These equations

form the basis for the development of methods for predicting the spatial

and temporal distribution of concentrations of substance in water bodies.

Having established this basis, it is now possible to direct attention

toward specific areas of application. In particular, the transport of

heat by turbulent diffusion in the vertical direction in lakes will be

considered.



Heat Transport in Lakes


The focus of this study is on the transport of heat in natural lakes.

The thermal structure of these water bodies may be studied using the

turbulent transport concepts presented in the preceding discussions. As

indicated then, the conveyance of heat is analogous to the conveyance of

substance. To adapt equation 2-12 for use in the study of thermal advec-

tion and diffusion it is only necessary to substitute the scalar quantity,

heat concentration, for the scalar, substance concentration. Also,

analogous expressions to those of equation 2-13 may be written. The

resultant equations will contain the turbulent diffusivities of heat in

each of the three directions. It should be noted that since temperature

is the physical indicator of heat content, it is the quantity most often

used in equations. The substitution is a simple one because the relation-

ship between heat and temperature is given by


H = ce


(2-18)











where H heat concentration (energy mass solution )
-1
c = specific heat of water (energy mass solution -
temperature degree-1)

8 = temperature

The value of c varies only slightly over ambient temperatures and is con-

sidered constant. Its value is one calorie per gram mass per degree

centigrade. Care must be taken to express the production term in the proper

units depending on the transport quantity used.

The use of the equation to describe heat transport in lakes is pre-

sented after a discussion of the factors affecting motion in lakes is

given.



Motion in Lakes


Water movement in natural lakes is nearly always turbulent

(Hutchinson, 1957). The turbulent and advective transport in lakes may

be caused by several factors. In larger lakes the dominant factor is

induced movement from wind. Energy is transferred from the air to water

by the shear at the interface. Surface currents caused by wind action

transport water to the leeward shore where it piles up, and soon a return

current begins to flow due to the tilted water surface. The extent of

the motion varies according to individual situations. In the case of

stratified lakes there may exist additional induced currents at interfaces

between layers of different densities. The shear profile existing at the

surface will increase the levels of turbulence in the water and thereby

contribute to increased vertical transport. Also, the formation of sur-

face waves adds to the turbulent character of the near surface zone. The











degree of water movement caused is directly related to the magnitude of

wind velocity at the water surface and the length of fetch; therefore,

the surface area and surrounding topography are significant factors as

regards wind effects.

Another factor affecting lake motion is convective mixing. Surface

heat exchange at various times may cause the water to cool and become more

dense than water below. The gravitational instability will soon result

in the sinking of the cooler water to a level of equal density. The down-

ward displacement necessarily causes an upward movement of other water

parcels, and if the surface cooling continues, pronounced mixing currents

will form. In most lower latitudes where significant solar heating occurs

during daytime, the convective process will occur nightly to some extent

all year. In other climates this phenomenon may happen on a diurnal

basis during only brief periods, and other lakes may only experience

significant convection currents on a seasonal basis. The converse is not

nearly so common,yet the situation does exist in which bottom waters are

warmed by sediments, especially in shallow regions, causing convectional

mixing.

Seiches are another type of lake motion, however, because seiches

are caused by the relaxation of a wind-tilted water surface or atmospheric

pressure variations, they are phenomena existing in large lakes only.

Internal seiches or waves are similarly stimulated and likewise are

encountered only on large water bodies.

Most lakes exhibit some density change with depth during most or all

of the year. The result is a dampening of vertical motion relative to

horizontal motion. The effects of stratification on the vertical structure











of lakes and other water bodies are usually considered in terms of the

Richardson number


R= - (2-19)



where U is average horizontal velocity and the other terms are as previously

used. The number represents the ratio of the rate of increase in potential

energy due to buoyancy to the rate of turbulence generation by energy trans-

fer from the mean flow. It is a measure of the stability of the lakes.

Higher values of Ri indicate that vertical mixing is being inhibited because

the energy necessary to overcome stratification is not available from the

mean flow. Use of the Richardson number for evaluating turbulent motion in

lakes is difficult because of problems of defining vertical change in

average velocity, and in smaller lakes which are characterized by very low

velocities, only crude evaluations can be made which are not very useful.

Although the Richardson number has been used in some cases of atmos-

pheric turbulence to predict the behavior of vertical diffusivities, the

most workable approach in lakes is to consider the concepts of turbulent

diffusion presented in equations 2-12 and 2-13. From these expressions,

methods may be developed which will yield values for the diffusivities

without the need to know velocity profiles.



Evaluation of Diffusivities


Various methods may be used to define the behavior and magnitudes of

the diffusivities of heat to be used in the transport equations. One such

approach is to assume the diffusion coefficients are constant which permits











an exact solution to the transport expressions. Measured data can then

be used to find the values of the K's that produce the optimum agreement

with data. The insufficiencies of this method are apparent, since obser-

vations clearly indicate that the levels of vertical turbulent mixing vary

with depth. Dutton and Bryson (1962) used this method to describe the

vertical temperature structure of Lake Mendota. When their constant value

of diffusivity failed to produce satisfactory results, they divided the

lake into two layers roughly corresponding to the epilimnion and hypolimnion

and calculated separate diffusion coefficients.

A second and somewhat similar method involves assuming an alternative

manner of variation of the diffusivity with depth (e.g., exponentially

decreasing). This also permits a solution of the transport equation,

either exact or numerical, to be used in conjunction with measured data in

an effort to determine magnitudes of coefficients by fitting predicted

values to measured ones. McEwen's method (p. 468, Hutchinson) is of this

type and has been used extensively to evaluate hypolimnic diffusivities.

A more detailed discussion of this approach will be given in Chapter III.

Another method of diffusivity evaluation entails finding the quotient

of the heat flux and the heat gradient at a particular depth. This may be

accomplished in different ways. For instance, the expression for vertical

heat flux analogous to equation 2-13 is


(w')d = Kd (e/z)d (2-20)

where 6 = water temperature

Kd = diffusivity of heat

d = depth

z = vertical distance


r











Therefore, if the product of the fluctuations can be determined and the

gradient of temperature also determined, then the diffusivity of heat

can be found. This method may be labeled a first principles type of

evaluation because the flux is considered in terms of the turbulent water

motion actually effecting the heat transport.

Another option is to use the one dimensional form of the advective-

diffusion equation for heat transport into which equation 2-20 has been

substituted. Care must be taken to include proper production terms. By

integrating the equation over the vertical distance between depth, d, and

the bottom, an expression results which may be used to evaluate diffusivity

at d over a specific time interval using vertical temperature data taken

over the interval. The principle is to assess the turbulent phenomena in

the water by ascertaining the amount of transport of heat they caused.

This method is often referred to as the flux-gradient method and will be

discussed further in Chapter III.

Finally, it may be noted that diffusivities may be evaluated by

monitoring the variations in content of any of a number of other substances

in the water body and then applying an analysis procedure similar to the

one just discussed. The diffusivities of substance and of heat may then

be related by the fact that they are approximately equal (Hinze, 1959).

Accordingly, the diffusivities for substance transport may be obtained

from studies of thermal transport. That the two diffusivities are nearly

equal can be seen by comparing the turbulent Schmidt and turbulent Prandtl

numbers. The turbulent Schmidt number is the ratio of the diffusivity of

momentum (eddy viscosity) to the diffusivity of mass substance, and the

turbulent Prandtl number is the ratio of eddy viscosity to the diffusivity











of heat. Empirically, the two numbers have been evaluated, and both have

been found to have values of about 0.7. Therefore, the diffusivities of

mass and heat are known to be approximately equal.



Results of Previous Studies


The evaluation of diffusivities has been the goal of many research

efforts because as already discussed the values are necessary to the

application of the advective-diffusion equation for predictions of trans-

port. In most of these studies,heat or some substance, either introduced

into the water or naturally occurring, was used as a traceable quantity.

The volume of these various investigations prevents a comprehensive review

of them here. Most deal with horizontal transport, and many tracer studies

regardless of title are in fact efforts in dispersion measurement. As far

as finding reported values of vertical diffusivities as computed- from con-

siderations of the conservation of either mass or heat, the number of

reports is meager. Bowden (1964) presents results of deep sea studies which

include values of from 1 to 2000 square centimeters per second (9 to 17000

square meters per day) and show much variability with depth. Ryan and.

Harleman (1973) compile results of similar studies in lakes and reservoirs.

They list several cases with values ranging from 0.3 to 13 square feet

per day (0.03 to 1.17 square meters per day) which do not apply to epilimnic

regions. Morris and Thackston (1969) used dye as a tracer in reservoir

studies and found that the vertical diffusion coefficients varied with time

and depth. They obtained values of from 0.5 to 6.3 square meters per day.

In a study of the effects of thermal discharges on receiving waters

Sundaram et al. (1969) used both the conservation of energy and McEwen's












method and found values in the lower lake strata of 18 square feet per day

(1.6 square meters per day). Morris and Thackston (1969) summarize the

results of several studies performed by Orlob and Selna (e.g., Orlob and

Selna, 1970) who used a modification of the conservation of heat approach.

The values reported are grouped according to depth and vary from 0.81 to

9.2 square centimeters per second (7 to 80 square meters per day) at the

surface, 0.02 to 0.17 square centimeters per second (0.2 to 1.4 square

meters per day) at the thermocline, and 1.2 to 1.7 square centimeters per

day (10 to 15 square meters per.day) at selected hypolimnic locations.

The values reported vary greatly. Part of the reason for this variation is.

that the values apply to different lake depths. Epilimnion values are

much higher than values in lower strata. Also, it is interesting to note

that despite the smaller hypolimnic values reported, all of the diffusivities

are at least two orders of magnitude greater than the molecular diffusivity of

heat in water, 0.012 square meters per day.

Regarding investigations of vertical turbulent transport by measuring

velocity fluctuations, the comment by Wiseman (1969, p. 8) is apropos:

In the existent literature one finds not only a lack of
turbulent spectra, but especially a dearth of information
concerning vertical motions in natural bodies of water.

Bowden and Fairbairn (1956) used a custom-made electromagnetic flow meter,

Wiseman (1969) used a custom-made doppler-shift meter, and Grant et al.

(1962) used a hot-film anemometer to measure shear stress in estuaries under

conditions of brisk currents; however, they reported only momentum fluxes

and no diffusivities.











The Problem of Convective Mixing


Earlier a discussion of various phenomena causing turbulent motion in

lakes included mixing due to convective cooling currents. In addition to

wind these currents account for much of the motion in the epilimnion, and

nearly always, this near-surface movement is far greater in magnitude than

the movement occurring below the thermocline in stratified waters. The

lack of knowledge regarding the behavior and vertical variations of

turbulent transport mechanisms has caused much uncertainty in studies of

predicting lake temperatures. Orlob and Selna (1970) developed a methodology

for matching observed and predicted temperatures in reservoirs by using

values of effective vertical diffusivities of heat substituted into a predic-

tive model based on conservation of heat principles. The usefulness of

their model has been questioned because the diffusivities were originally

calculated from the observed data using an inverse solution of the model.

This type of complication prompted Dake and Harleman (1966) and later Huber

and Harleman (1968) to describe the thermal transport in lakes and reser-

voirs alternatively. They have reasoned that the stability conditions

existing in lakes and reservoirs due to density gradients are strong enough

to inhibit virtually all turbulent transport except for the regions influenced

by convective mixing. Their methodology entailed specifying a very small,

constant value of diffusivity throughout the lower strata. In some cases,

the value of molecular diffusivity of heat (0.012 square meters per day)

was used. The upper strata were characterized as completely mixed and

isothermal. Their procedure for defining the single surficial temperature

and the depth to which it extended used thermal energy budgets to predict

a temperature profile. Whenever an unstable near-surface profile was











predicted, a unique average temperature was calculated which- indicated

an amount of energy equivalent to the predicted condition over a

particular depth. The depth was specifically determined by the depth

existing in the stable portion of the predicted profile at which the

unique mixed temperature occurred.

The results obtained by comparing predicted temperatures to observed

values were good in both studies, and the method was deemed satisfactory.

However, Huber's results mainly showed that vertical advection was

dominant in the reservoir, and this condition is not applicable to most

lakes. In the lake study by Dake, much of the success of the model

application depends on the assessment of the vertical distribution of

absorbed solar radiation. This is a highly variable phenomenon as Dake

points out and may affect significantly the predictive performance of the

model when applied to various lakes, especially ones with high or variable

turbidities.

These authors as well as others (Sundaram et al., 1969 and Powell

and Jassby, 1974) have all questioned the ability of the concept of

turbulent diffusivities to adequately describe the process of convective

mixing. Although not stated explicitly in any of these investigations,

much of the problem with describing convective cooling using turbulent

transport equations is as follows. The normal annual thermal cycle of

lakes is reviewed by Hutchinson (1957), among others. When climatic condi-

tions cause more heat to leave the water than enters, the lake cools. This

process is usually a diurnal one wherein most of the cooling occurs at night

and most of the heating is due to daytime solar radiation. If during the

cooling portion of the cycle, the thermal profile of the lake is examined

over a period of days or weeks, the loss of heat is apparent. Yet, in many











cases the shapes of the profiles observed will be very similar, and

temperature structure will be seemingly stable. This is very likely to

happen if, for instance, noontime temperature measurements are made days

or weeks apart. Such behavior cannot be readily explained by conservation

of heat methods because the loss of heat observed is incompatible with the

stable gradients (i.e., a loss of heat indicating surface cooling and

profile slope indicating a downward heat transport). Two possibilities

exist. Either the semiempirical approach to turbulent transport problems

is unsatisfactory or the method of application is in error. The process

of convective mixing seems to have a specific length scale associated

with it. The Dake and Huber models both compute a characteristic depth

over which the mixing extends. Furthermore, the nature of the process

suggests the formation of eddy-type currents. It, therefore, seems

appropriate to describe convective mixing in the semiempirical manner. On

the other hand, there may be some basis for assuming that the usual methods

of applying conservation of heat approaches to conditions of mixing are

in error. This stems from the fact that the mixing is a diurnal process

and cannot be considered properly without giving attention to this fact.

Specifically, the aspect of the length of the time interval used to make

calculations must be considered because there may be marked influences

of this interval on the values of diffusivities calculated. This line

of thought will be developed in the following section.



Averaging


Equation 2-7 indicated the averaging method used to determine mean

quantities; nevertheless, the subject of proper averaging requires some











additional commentary. Although equation 2-7 was presented without

discussion, certain facts are implied by its form. The equation states

that the average value of a quantity, say concentration, in a turbulent

flow is found by time averaging over some interval T. This is true only

as a result of assuming ergodic behavior of the quantity. Ergodicity

implies that the mean value of any individual sample taken over a finite

time interval will be.approximately the value obtained by taking an ensemble

average (i.e., the average of different samples taken at the same time).

Such behavior is in certain cases strictly provable, and in other cases,

safely hypothesized (i.e., the ergodic hypothesis) (Monin and Yaglom, 1971).

In addition to this question of theoretical propriety is the practical

matter of what averaging intervals to use. Generally the guidelines

for selecting the proper interval entail using a time spah which is some-

what longer than the largest frequencies of turbulent motion being

studied and short enough so that the mean does not vary during it. Con-

ceptually these requirements present no problems; yet in practice they

may. In studies of large scale, naturally occurring motions, it may be

much harder to establish which are the largest scales of motion existing

than it is to do so in a laboratory study. Large eddies with frequencies

lower than 2 r/T will be excluded from analysis using T as the averaging

time (Okubo, 1962). Furthermore, as Bowden (1964) notes, the method of

getting mean values depends on the scale of movement occurring and also

the particular aspect of that movement being considered. Conversely,

selecting and applying a certain averaging interval to a set of measure-

ments may have an effect on the nature and quality of results obtained,

which is the premise presented at the conclusion of the preceding section.










The emphasis of this discussion so far has related to averaging

procedures for records of fluctuating turbulent quantities versus time;

however, certain parallels may be drawn between computation of diffusivities

by the first principles approach and by integration of the conservation of

heat (flux-gradient) methods. In particular, if the averaging interval

used on a time record of a quantity is long relative to a.specific period

of .fluctuation of the quantity, then the effects of the shorter scale

variations are diminished. If the magnitude of the shorter scale

fluctuations is less than that of the longer scale variation, the

effects may be diminished to the point of not being detected at all. Satis-

factory description of the transport by calculating and using diffusivities

depends greatly on using an averaging interval pertinent to the turbulent

motion. As regards diurnally variable convective mixing, the use of a

one-day averaging time to determine diffusivities by equation 2-13 will

yield the same inappropriate results as would be found using the flux-

gradient method calculated over a twenty-four-hour period. For example,

should the turbulent motion in a lake over the period of one day be

caused exclusively by convective mixing,then the velocities occurring in

the lake would fluctuate (about a zero mean) during the period of cooling.

During the remainder of the day no velocity would exist. .Now the

temperature in the lake at a depth affected by the convective currents would

also fluctuate during the mixing period and coincidently cool. Considering

the daily period, the cooling during the mixing and heating during the

day (regardless of periods of no change should they exist) would produce

a daily mean temperature. If the daily mean of velocity (zero) were used,

even the convective velocities,regardless of size,would be calculated as











instantaneous fluctuations. On the contrary, if the daily mean.of

temperature were used, the entire period of cooling would be repre-

sented by only negative fluctuations. The results of the subsequent

cross-correlating (i;e., w'e') over the averaging period or even a

portion of the day, would not reflect the shorter scale convective

process suitably. Analogously, using a daily time step to calculate

diffusivities in the lake by the flux-gradient approach would not reflect

the shorter scale convective process suitably. Therefore, using a

daily time step to calculate diffusivities in the lake by the flux-

gradient approach would yield poor results due to the insensitivity of

the method to the shorter scale motions.



Goals of This Study


The preceding discussion of averaging requirements tends to support

the suggestion that the inability of the semiempirical theory to satis-

factorily describe convective mixing is due to improper use of the method.

It is a goal of this research to determine whether or not the transport

in a lake can be adequately described by the semiempirical theory of

turbulent transport. Emphasis will be given to the convective mixing

phenomenon. Specifically, the vertical transport of heat will be monitored,

and from it the vertical diffusivities of heat will be determined using the

flux-gradient method. At the same time, the fundamental soundness of the

semiempirical theory will also be tested using the more difficult but more

direct first-principles approach of measuring the turbulent water motion in

the lake. As indicated by equation 2-20 time-averaging the product of

fluctuations of vertical water velocity and temperature at a given depth will










produce the heat flux term needed to find K. The gradient term at the

same depth can be obtained from the flux-gradient study.

Additionally, the computation of the diffusivities by two different

methods could yield some worthwhile insight into the relative merits or

deficiencies of each. Furthermore the study will seek to examine the

behavior of the diffusivities as a function of various vertical positions

in the lake. Not only will diurnal variations of the diffusion coeffi-

cients be evaluated as the convective mixing process is studied, but also

longer term variations which may occur will be documented and analyzed.

A final objective involves the effort to establish to what extent the

vertical diffusivity information obtained can be related to the environ-

mental conditions which exist coincidently. Such relationships might prove

most beneficial to the application of transport theory to practical problems,

especially problems of predicting pollutant transport. The ascertainment

of the diffusivities by separate methods will provide the many values of

diffusivities necessary to accomplish this goal.














CHAPTER III

METHODOLOGY OF DIFFUSIVITY EVALUATION



Introduction


The overall goals of this research were stated in Chapter II.

Specific use is to be made of the flux-gradient method and the correlation

of fluctuations method to evaluate diffusivities of heat in a lake. To

perform the planned analysis,values of diffusivities are needed at various

depths, several times per day, and over a span of several months. This

indicates that a large volume of data must be gathered andsubsequently

analyzed.

This chapter presents the conceptual basis for the flux-gradient

method which is followed by discussions pertaining to the use of the

concepts. Included in the description is a scheme for approximating the

measured temperature profiles with analytical expressions so that the

requisite calculations for the flux-gradient analysis need not be performed

manually. The feasibility of diffusivity computation by this approach is

determined by whether or not such a scheme is workable. Therefore, an

account of a trial of the methodology on Cayuga Lake data is given. The

result of the test is that the procedures developed work well, and that

when applied to data gathered for this study, they should enable many

accurate computations of diffusion coefficients using a digital computer.

Following the discussion of the flux-gradient method is a presentation

of the correlation of fluctuations method for diffusivity evaluation. The











measurement of water velocities-is essential to this approach and the

entire following chapter is devoted to the subject of making such

measurements.



Flux-Gradient Method


The basis for the flux-gradient method is the one-dimensional form

of the advective-diffusion equation for the turbulent transport of heat.

This expression may be obtained by first modifying equation .2-12 to des-

cribe the transport of heat using equation 2-18 to express heat concen-

tration as temperature, then substituting equation 2-20 so that the

turbulent fluxes are represented as the product of diffusion coefficients

and gradients, and finally by amending the three-dimensional expression

to a one-dimensional form. This is done by assuming the lake is

horizontally homogeneous which is valid if-the isotherms in the lake are

horizontal. This condition is often not met in large lakes which may be

affected by seiches; however, for small lakes studies (e.g., Smith and

Bella, 1973) this has been found to be a reasonable assumption. Also the

production term must be included to account for the absorption of inci-

dent solar radiation which varies vertically. The result of these

manipulations is

ad a I6 1 Rd
K ) (3-1)
St z d az pc z (3

where 8 = temperature

t = time











z_ vertical distance

Kd = vertical turbulent diffusivity of heat at depth d
distance2 time-1)

p = density of water (mass volume-)
-1
c = specific heat of water (energy mass -
degree .temperature-1)

Rd = flux of absorbed solar radiation at d
(energy distance-2 time-1)

This equation describes in differential form the vertical transport

of heat in a water column in which no horizontal transport takes place.

Furthermore, the only sources or sinks for energy are boundary fluxes of

heat and solar radiation, and turbulent transport is assumed proportional

to the gradient of the average.heat content.

An explicit expression for the diffusion coefficient is obtained'by

integrating equation 3-1 from depth zl to the bottom'depth z = h,


h Dd h h DR
--dz = -=dz (3-2)
z1 zI z1

which gives


h 1, K (d e h Rd| h h
S dz = K -II (33)
at1 daz Pc
1 zz1 El

Assuming that there is no heat transport or solar flux through the bottom

gives,

h ae-
h -d = K e (3-4)
az zlz pc
z1 1










Solving equation 3-4 for K yields
z1


2 Zl h 8d 1 I-5)

zI 1 (

Because the limits of the integral, z1 and h, are not functions of time

equation 2-5 may also be written


R h
pc Bt d (3-6)

K = zl
1



The first term in the numerator of the right-hand sides of equations 3-5

and 3-6 represents the energy content due to absorbed solar radiation.

The other numerator term represents the rate of change, or flux, of the

total heat stored in the water column below zl, and the denominator is the

gradient of the mean concentration of heat at z1, hence, the nomenclature:

flux-gradient method.



Evaluation of Heat Flux and Gradient Terms


The evaluation of the heat flux term in equation 3-6 may be attempted

by either analytical, numerical, or graphical methods, or combinations of

these. If analytical expressions are known for temperature as a function

of depth at two times, then


h z

z1










may be solved analytically or numerically. Alternatively, plotting 6

versus z and determining the area under the curve between z1 and h can

be done. Analytical or numerical integration is usually easy when

digital computers are used,while plotting points, drawing curves, and

measuring areas are usually time consuming. However, arriving at

satisfactory analytical expressions for 6 = f(z) may not be feasible,

and the graphical approach may be required. Dividing the integral quantity

by the time increment, At, which is determined by the times the two

temperature profiles were measured, yields the flux term.

Another alternative when 6 = f(z) is not known is to evaluate


h
j (a3d/at)dz
z1

as is indicated by equation 3-5. Using temperature versus depth data at

two times, it is possible to approximate 3ad/3t at various depths by

computing Aed/At at depths between zI and h. The flux is computed by

evaluating the integral by analytical, numerical, or graphical means.

Both of these methods will be described in greater detail below as they

are applied to data from Cayuga Lake.

Determining the gradient value a /az is more difficult. Whereas

knowing the amount of stored heat at the end points of a given time span

is sufficient for quantifying heat flux, the gradient of the average

temperature over the same time span may or may not be accurately found

by considering only the end points. If the variation of the gradient

with time is not linear, significant errors may result unless additional

temperatures measured during the original time span are used to improve











the accuracy of the averaging process. Computation of the gradient should

be made using data taken at periods over which the changes in temperatures

are known to be linear, and the likelihood of error should be recognized

when a longer time span is used.

Evaluation of the gradient term entails computing the time average

of the temperatures at various depths, thereby determining an average

profile. If an analytical expression can be found for the average profile,

differentiation gives the gradient at any zj. Otherwise, plotting the

profile and graphically determining the slope at z = zI or adopting some

numerical technique such as straight line interpolation between data points

may be used.



Evaluation of Solar Flux Term


Of the solar radiation striking the water surface, the longer wave

portion is absorbed at the surface while the shorter wave portion is

absorbed within the water body in a manner which decreases exponentially

with depth. The degree and extent of subsurface penetration depends on

wavelength and the physical and chemical quality of the water. Dake and

Harleman (1966) described this twofold behavior with the expression


R = R (1 8)e-1z (3-7)
z 0

where R = solar flux at depth z (energy area- time-1
z
R = solar flux at surface (energy area- time-)
o
B = fraction of R absorbed at the surface
dimensionlesss)

n = extinction coefficient (depth )










They note that values of 8 and n vary greatly depending on the water quality

and suggest that the coefficients be determined for a given lake using

solar absorption data applicable to specific conditions.

The average solar flux over the time span At must be used, and as noted

earlier regarding the suitability of the use of end-point values to compute

averages, the variation of solar flux should be linear during At; otherwise

interim calculations should be made.



McEwen's Method


McEwen (1929) offered an approach for calculating diffusion coeffi-

cients in the hypolianion of lakes which has received much attention.

Hutchinson (1957) presented a description of this scheme which is often

called McEwen's Method, and Powell and Jassby (1974) have restated its

development while including solar radiation terms which are usually

neglected. In many stratified lakes the thermal profile exhibits an expo-

nential form in the region of the lower metalimnion and upper hypolimnion.

The metalimnion is the zone of lake depth wherein the temperature variation

with depth is greatest; it separates the epilimnion and hypolimnion. This

region of exponential temperature decrease is known as the clinolimnion,

and McEwen's Method pertains to diffusivities in it.

Let the depths z1 and z2 indicate the extent of the clinolimnion.

The exponential form of the temperature profile for z1 < z < z2 may be

expressed as


(3-8)


8d =.1 + 82e










where 8d = temperature at depth d (degrees)

11 B2 = constants (degrees)
= constant (depth )

The one-dimensional conservation of energy equation (equation 3-1) derived

earlier as the basis for the flux-gradient method also forms the basis for

this approach. Assuming the diffusion coefficient to be constant in the

clinolimnion, this equation becomes


aed 27 1 aRd
-- d p (3-9)


where each term is as previously defined except that since K does not

vary with depth, the subscript denoting its value at a specific depth is

superfluous and is not used. The assumption that K is constant is made

so that some value for it may ultimately be found. The validity of the

assumption is checked by a parallel line criterion that is discussed below.
2 2
Differentiating equation 3-8 to get 3 d /3z and substituting into

equation 3-9 and then rearranging yields


ad 1 BRd 2
at pc az K(8O22e ) (3-10)

Taking the natural logarithms of equation 3-8 and 3-10, respectively,

gives

ln(ed -81) = nB2 z (3-11)

and
ad 1 aRd 2
In(- -- ) =ln(K8 ) =z (3-12)
at pc az 2











McEwen concluded that if plots of In(6d 81) versus z and -

In[3ed/at (l/pc)8Rd/az] versus z produced straight, parallel lines for

a given set of temperature data, then the assumption of constant diffu-

sivity was justified, and the value of diffusivity could be obtained by

finding the slopes and intercepts of the two plots. Powell and Jassby

demonstrated mathematically that this hypothesis was incorrect. They

showed that permitting diffusivity to vary with depth produced a more

complex expression but one which was still resolvable to a form that

gave straight and parallel plots. The result of their analysis was that

the diffusivity could have the form


Kz r1 + r2e= (3-13)

where rl,r2 = constants (distance time-)

The crux of this argument is that assuming K constant defines r2 as zero,

which is not generally justifiable.

Powell and Jassby point out that in some lakes using r2 equal to zero

is appropriate. In other lakes this assumption cannot be made and

applying McEwen's Method leads to erroneous conclusions because the value

of r2 remains to be evaluated. They also use some values of diffusivities

found by the flux-gradient method to lead to values of r2 and show that

about fifty percent of the magnitude of K is due to the r2exp(=z) term.

Powell and Jassby do not offer a new method of finding diffusivities, but

they do show conclusively that in many cases using constant hypolimnic

diffusivities is inappropriate.











Analytical Representation of Temperature Profiles


As already discussed one alternative for evaluating the heat flux

and heat gradient terms in equation 3-6 is to integrate and differentiate,

respectively, an analytical expression which defines lake temperature as

a function of depth. Analytical analysis utilizing digital computers

seems preferable to other methods requiring manual operations, especially

when there are many diffusivities being calculated. The crux of this

approach is to be able to represent the measured temperature profile by

an analytical expression because once such an expression is found, inte-

gration and differentiation may be performed handily. There are at least

two techniques which might be used to approximate measured temperatures

analytically: a polynomial fit and a least squares fit. Both are types

of statistical regression.

Initially, a polynomial fit of the temperature data was tried. This

involved the multiple regression of successive powers of depth, z, on to

temperature, T, thusly,

n
T' = aii-1 (3-14)


where the prime denotes a predicted value and the a's are regression

coefficients. Digital computer programs prepared as part of the Scientific

Package by the International Business Machines Corporation (Anon., 1968)

were used as the basis for the computations. The results of the effort

were unsatisfactory because the fits obtained did not approximate the

measured values well at all depths. Also, the predictedprofiles in some

cases assumed erroneous shapes in regions between data points.











Upon the suggestion of Professor Robert Dean (1974), a new approach

involving the nonlinear least squares fit of a series of terms was tried.

Computer programs included in the UCLA (University of California at

Los Angeles) Biomedical Computer Programs Library (Dixon, 1973) served as

the basis of the analysis,although some custom modifications were made.

A function of the form

n
T' = bicos[(i=l)nz/h] (3-15)


where h is the depth of the .lake bottom, was subjected to least squares

analysis to determine the values of bi giving the best fit to the measured

temperatures. Problems arose because this function yielded zero slopes

(i.e., BT'/az = 0) at both water surface and bottom, so a modification was

made which extended the primary period slightly to permit nonzero slopes

at the water surface. This was accomplished by substituting z h'/h'

for z/h where h' = h + c and c = constant (meters). The results seemed

improved, and further modification was made by making c one of the

regression constants. The effect of this was not helpful because the

regression procedure would not converge to satisfactory values of c

consistently.

It was then decided to revert the cosine expression to its initial

form and amend the entire function to be fit.by adding an exponential term

which was weighted such that it only had an effect near the surface. The

shape of the temperature profiles suggested an exponential term; further-

more, during periods of surface cooling, the profile shape suggested a

skewed function of the general form











T' = zXe-z (3-16)

where X = constant

A more general expression for the upper layer

b2 -b z
T' = bz e (3-17)


was combined with the Fourier-type terms and used as the function to be

fit by the least squares scheme. The coefficient b3 could not be deter-

mined consistently by the scheme and was removed as a regression

coefficient; however, it was kept as a constant. Its primary effect was

to regulate the extent of depth over which the entire term is influential,

and tests made using both integer and noninteger values for it indicated

that best overall performance of the curve fit was realized when the

value six (6) was used. Therefore, the ultimate form of the nonlinear

function to be fit by the least squares scheme became


b n
T' b1z e + = b cos[(i-3)wz/h] (3-18)
1 i=i3

Other slight modifications were tried but none noticably improved the

predictive performance of equation 3-18.

Throughout the process of selecting the optimum function to be used,

the quality or goodness of fit produced by any one function was evaluated

by both analytical and observational criteria. Estimates of error,

residuals, and correlation levels were provided as part of the calculation

of the regression coefficients; however, these parameters only indicated

the relative ability of the particular coefficients to predict temperatures

at the data points used in the analysis. Consequently,each set of











coefficients was used to calculate temperatures at short intervals

throughout the water column. Therefore, erroneous oscillations and

other forms of erratic behavior could be detected. The criteria for

satisfactory temperature profile prediction were that the measured

temperatures should be approximated to within 0.1 degree centigrade,

that no extrinsic behavior exist in the predicted profile, and that

the performance of the regression scheme be repeatable and not erratic.

Temperature versus depth data for three days, one each from October,

November., and December, 1973, were used to test the various prediction

functions. Both daytime and nighttime values were used to insure a

variety of near surface profile shapes. The ultimate choice (as given

by equation 3-18) yielded results which most closely met the criteria

described above. For the October, 1973, profiles the predicted tempera-

tures differed from those observed by as much as 0.2 degrees centigrade

at some of the points. The procedure was deemed a qualified success.



Cayuga Lake


It was desirable to better evaluate the merits and suitability of

using a nonlinear least squares fit of equation 3-18 to analytically

define measured temperature profiles. One possibility was to apply the

regression procedure and subsequent flux and gradient computations to

data from which calculations of diffusivities by the flux gradient

method had already been made by manual means. The recent article by

Powell and Jassby (1974) appeared to provide such an opportunity. In-

cluded in the article is an analysis of data taken at Cayuga Lake,










New York which was originally examined by Sundaram et al. (1969).

Data for periods in 1950 and 1968 were considered.

Although the original data for 1968 were not tabulated by Sundaram

et al., they were presented in graphical form (Figure 49a, p. 277). The

1950 data which were tabulated in another publication were averaged

and perhaps otherwise manipulated by Sundaram before being used by

Sundaram et al. Therefore, only the graphs of lake temperatures for the

weeks of August 14-20 and August 21-27, 1968, were used to tabulate

temperature profiles to which least squares approximations could be

made. It should be noted that the resolution of these plots is poor

due to small size and reproducing effects, and that temperatures read

from them may disagree with the data used to construct the plots

originally. Nevertheless, the value obtained should indicate the

ultimate utility, or lack of same, of the procedures devised for this

study.

Before a comparison of results can be made, it is necessary to

consider in greater detail the alternative methods for calculating

flux-gradient quantities. As discussed earlier in this chapter, the flux

term may be considered in two different ways as given by equations

3-5 and 3-6, respectively. If the form indicated in equation 3-6

is used, temperature data taken at times t1 and t2 are used to

approximate 38/8t by figuring AS/At (At = t2 tl) at various depths.

It is then necessary to integrate this newly created set of data below

the depth being considered, z1. The form of the flux indicated by

equation 3-6 requires that the temperature data taken at t1 be integrated

below z, and the data for t2 be treated likewise. The flux is obtained










by dividing the difference between the two integrals by At. Either

method should yield a proper result; however, one or the other may be

preferable when questions of implementation are considered.

Evaluation of the gradient (at zl) may be made by either one of

two similar methods,assuming the gradient changes linearly during At.

If this is the case, then the average of the slopes of the two temperature

profiles taken at ti and t2 should yield the desired quantity. Alter-

nately, the two sets of temperature data may be averaged-to form a new

set of data the slope of which may be used to provide values of the

gradient.

Sundaram et al. calculated diffusivities of heat by both the flux-

gradient method and McEwen's Method. The emphasis of his work is on the

latter which may explain why the data are presented in a manner which

makes flux-gradient calculations difficult. It should in fact be

noted that their computations of diffusion coefficients by the flux-

gradient method are in error. The text (p. 117) refers the reader to

Figure 54 (p. 284) which shows plots of AG/At, f(AO/At)dz, and K

diffusivityy). The values of K shown (and also discussed in the text)

appear to have been calculated from the quotient of the other two sets of

data in the figure, i.e., I(AO/At)/(AB/At), when in fact K should be given

as the quotient of the quantity /(A8/At)dz and 39e/z (not 38/at).

Fortunately, Powell and Jassby independently compute diffusivities using

Sundaram's data, and their values appear to be more precise..

As part of the presentation of the results of the use-of McEwen's

method, plots are given (Figure 56a, p. 286) by Sundaram et al. which are

easy to discern and which provide data suitable for flux-gradient











calculations. Powell and Jassby tabulate in their Table 3 (p. 196)

these data,which include 6 f8 (cf., equations 3-8 and 3-11) and

A6/At at 1.52 meter (5 foot) intervals to a depth of 27.4 meters. These

data are presented in Table 3-1. Also in Table 3-1 are computed values

for other quantities required to calculate flux-gradient diffusivities.

Powell and Jassby do not elaborate on their methods of obtaining these

other quantities; however, after examining their computed values of

diffusivities, it seems that the gradients are obtained for a specific

depth by taking differences between values on either side of the point

in question. Therefore, gradients calculated in this manner have been

included under the heading A6/Az of Table 3-1. For example, the gradient

Ae/Az, at z = 7.62 meters is given by (13.3 12.2) degrees centigrade

divided by (6.10 9.14) meters = -0.36 degrees centigrade per meter.

While this method may be employed easily, it may not be accurate; there-

fore, a representation of the temperature profile as given by (6 8 )

was plotted and slopes evaluated by visually constructing tangents to the

profile curve at the specified points. This information is listed in

Table 3-1 in the column headed (/z)slope so that comparisons may

be made.

Again,because Powell and Jassby do not describe their methods for

evaluating the flux term, it was assumed they used graphical techniques.

At any rate the data for 36/3t were plotted versus z, and the integrals

were evaluated by measuring areas on the plot. The results of this

procedure are given in Table 3-1 also. Some question remains about the

Powell and Jassby technique because they do not show calculations of

diffusivities (see Table 3-3) at those depths where values of AS/At are










TABLE 3-1. FLUXES AND GRADIENTS IN CAYUGA LAKE 8/14/68 8/27/68
DETERMINED GRAPHICALLY


(1) (2) (3) (4) (5) (6)

za -8 A6/At* A6/Az lh(AO/At)dz (Oe/z)slope
m C *C/day .C/m 'C-m/day *C/m


0.366 --- 6.96

0.293 0.10 6.51

0.27 5.96

0.352 0.17 5.55

--- + 0.10 4.98

0.382 0.36 4.36

0.85 3.73

0.539 0.75 2.96

0.92 2.22

0.311 1.36 1.65

0.259 1.02 1.19

0.173 0.75 0.85

0.138 0.55 0.62

0.075 0.42 0.32

0.065 0.26 0.20

0.045 0.20 0.19

0.025 0.19 0.02

0.0 -- 0.0


- 0.67

- 0.05

- 0.21

- 0.48

+ 0.25

- 0.34

- 0.31

- 0.62

- 0.58

- 0.93

- 0.89

- 0.76

- 0.65

- 0.37

- 0.22

- 0.17

- 0.21

- 0.10


*From Sundaram et al. (1974, p. 196).
**Values at z = 30.5 meters added by extrapolation.


0.0

1.52

3.05

4.57

6.10

7.62

9.14

10.70

12.20

13.70

15.20

16.80

18.30

21.30

22.90

24.40

27.40

30.5**


14.1

13.5

13.8

12.7

13.3

13.0

12.2

10.4

9.9

7.6

5.7

4.5

3.4

2.0

1.5

1.2

0.6

0.0











TABLE 3-2. FLUXES AND GRADIENTS IN CAYUGA LAKE 8/14/68 8/27/68
.FROM LEAST SQUARES FITS


(1) (2) (3) (4) (5) (6) (7)

At tl (8/14-8/20) At t2 (8/21-8/27)

z 6* ae/az fedz 8* a8/az /edz
m oC C/m C-m OC "C/m C-m


23.1

0.30

22.4 0.46

-0.42

21.2 0.38

0.56

19.4 0.96

1.30

15.6 1.23

0.75

13.4 0.25

0.18

12.5 0.56

10.0 0.88

0.46

8.4 0.11

8.1 0.36

7.2 0.0


446.92

411.66

377.04

342.73

310.13

278.46

248.14

219.32

194.17

171.79

151.15

130.46

111.02

76.10

62.02

49.05

23.11

0.0


23.7 108.00

0.32

22.2 0.21

+ 0.04

22.2 + 0.10

0.21

21.5 0.66

0.94

18.7 1.00

1.03

15.6 1.04

1.03

12.5 0.74

11.2 0.34

0.56

9.4 0.74

8.1 0.20

7.1 2.0


*From graphs given by Sundaram et al. (1964, Figure 49a, p. 277).


0.0

1.52

3.05

4.57

6.10

7.62

9.15

10.70

12.20

13.70

15.20

16.80

18.30

21.30

22.90

24.40

27.40

30.50


477.55

442.82

408.83

374.57

341.02

307.32

274.14

241.72

212.20

155.01

160.22

137.19

117.25

81.08

64.74

49.69

23.11

0.0











TABLE 3-3. DIFFUSIVITIES IN CAYUGA LAKE 8/14/68 8/27/68


(1) (2) (3) (4) (5)

z /(de/At)dz K f/(ae/t)dz AflAt
Ae/Az (ae/az) slae/z
m m/day m/day m/day /day
a m2/day m2/day m2/day a2/day


0.0

1.52 (

3.05

4.57

6.10 -

7.62 1

9.14

10.70

12.20

13.70

15.20

16.80

18.30

21.30

22.90

24.40

27.40

30.50

*After Powell


5.0

2.0

33.0

i0.0

L2.0

4.4

3.9

2.4

1.2

1.2

1.1

1.1

0.76

0.76

0.95

1.1


--- 10.4

68.0 130.0

--- 28.0

32.0 12.0

-- 20.0

14.0 13.0

--- 4.6

4.3 4.7

-- 3.8

1.4 1.8

1.5 1.3

1.6 1.1

1.6 0.95

1.6 0.87

1.9 0.90

2.3 1.1

--- 0.09


and Jassby (1974, p.


14.0

13.0

24.0

31.0

11.0

4.5

2.9

2.3

2.1

1.9

1.6

1.4

1.2

0.76

0.21

0.0


196).











missing (see Table 3-1). Since a plot of the data available should permit

evaluation of the integral quantity at any desired point (provided a

smooth curve may be drawn),and gradient values, A6/Az, are easily

determined at every depth, it is hard to understand why some diffusion

coefficient values are missing.

Table 3-3 listsvalues of diffusivities of heat at 1.52 meter

(5 foot) intervals. Column 3 shows the results of Powell and Jassby's

analysis. Column 2 shows the quotient of the pertinent quantities from

Table 3-1,as indicated by the column heading. The close agreement of

these two columns suggeststhat, in fact, Powell and Jassby performed

their analysis identically. Column 4 also is calculated directly from

entries in Table 3-1. A comparison of these values against the first

two sets reveals the effects of more precisely evaluating gradients as the

slopes of a plotted profile. Also, of interest is the negative value of

the diffusion coefficient at 6.10 meters which was among the values

omitted by Powell and Jassby. Such a value should not exist, and dis-

cussion of such occurrences will be presented in a later section.

The final column of Table 3-3 lists diffusivities as computed by

the least squares fit of the temperature data taken from plots in

Sundaram et al. These data and intermediate quantities are shown in

Table 3-2. By comparing items in both Table 3-1 and Table 3-2 it can be

seen that the information is not completely compatible. Values of 6 at

tI and 6 at t2 when averaged do not differ from e L by. some constant

amount (i.e., Bi) and values of 6 at t1 minus e at t2 divided by At equals

7 days do not correspond to Ae/At. These differences may be caused by

either the inability to accurately read the plots from which the data











were taken or an error in the reporting or calculation of e $land

AB/At. Despite the discrepancies,the data were nonetheless subjected

to analysis by the least squares fit method. Each set of temperature

data was approximated by the function given in equation 3-18. The

analytical expression containing the resultant regression coefficient

was then used to evaluate the integral, fedz, at various depths by a

numerical procedure. The mathematical derivative was computed

analytically. Integral and derivative quantities computed at t1 and

t2 are shown in Table 3-2. The difference between the two integrals,

Af, was divided by At = 7 days to obtain the average gradient,

The quotient is presented in Table 3-3 in column 5.

The values tabulated generally agree with those calculated manually

by the alternative approach. The value for the.surface is-not shown.

The entries in Table 3-2 could be used to arrive at a diffusivity;

however, the quantities in Table 3-2 are meaningless because the function

as given in equation 3-18 is not usable at z = 0. The negative values

encountered at 6.10 meters are not predicted by this method. Even

though the slopes at-t2 in that region (see Table 3-2) are positive,

which would yield negative diffusivities, the corresponding slopes at t1

are sufficiently negative to cause the average slope to be negative,

thereby predicting diffusivities that seem satisfactory. As' mentioned

earlier the question of negative diffusivities will be considered below.

The overall agreement between the diffusivities calculated manually

and calculated by analytically representing temperature profiles lends

credibility to the latter approach. Furthermore, the closeness of fit











of the calculated temperatures to curves drawn by eye to the observed

data indicates the satisfactory performance of the nonlinear least

squares approach. The observed and calculated points are shown in

Figure 3-1 for both the week of August 14 and the week of August 20,

1968. The diffusivities shown in column 5 of Table 3-3 were determined

using the analytical expressions and using the digital computer for all

calculations,including finding the analytical expressions. These

results are also plotted in Figure 3-2. The plot indicates a marked

variation of diffusivity over the depth of the lake and although values

below ten meters are much lower than the maximum, there are definite

variations throughout the lower depths.

The preceding discussion indicates the manner in which the flux-

gradient method can yield values of diffusivities of heat at various

depths in a lake using two observed temperature profiles. Different

interpretations of the flux quantities and gradient quantities can

produce different values of diffusivities as was shown. Also, pointed

out was the effort required to make determinations of diffusivities by

graphical methods. The use of a computer-oriented method requiring

significantly less manual effort was demonstrated, and the results com-

pared to those obtained graphically. The generally satisfactory behavior

of the method and apparent suitability of its results suggest the

overall usefulness it can have in applications employing large amounts

of temperature profile data to calculate many diffusivities.






TEMPERATURE DEGREES CENTIGRADE


e7


LEGEND
0 OBSERVED 8/14/68 8/20/68
x LEAST SQUARES FIT 8/14 8/20
r OBSERVED 8/21/68-8/27/68
A LEAST SQUARES FIT 8/21-8/27


S LEAST SQUARES FIT AT DEPTHS OF
OBSERVATION COINCIDENT WITH OBSERVATIONS


I I I I IN CAYU I
FIGURE 3-1. OBSERVED AND CALCULATED TEMPERATURES IN CAYUGA LAKE


A-. I


U)
w
w
I0
I15
x
0.
w
C
20-


I


en. ..


.









VERTICAL DIFFUSIVITY METER2- DAY-'
0 4 8 12 16 20 24 28 32
0- I I I I I I




5-



I0-
10-


I--
tiU


I 1

I


FIGURE 3-2. DIFFUSIVITIES IN CAYUGA LAKE










Correlation of Fluctuations Method


In Chapter II the correlation of fluctuations of velocity and

temperature was shown to be equal to the turbulent heat flux. The

time-averaged term in the fundamental convective-diffusion equation,

equation 2-12, was related to the diffusivity by equation 2-20. By

rearranging equation 2-20, the expression for defining vertical

diffusivity of-heat by the first-principles approach results as



Kd --- (3-19)

d ^d

where the terms are as defined in equation 2-20. Over a given time

interval, T, the individual records of w and 9 are averaged using


t+T t+T
w = t- wdt, = J Odt (3-20)
t t

Then the records are used again to determine w' and 6' by


w' = w w, e' = (3-21)

from which the product w'6' is calculated. The flux is the time-average

of the product over the same integral. Thusly,


t+T
w'' = +T w'9'dt (3-22)
t

The gradient term is determined by the slope of the average temperature

profile at depth d during the same time interval. By computing the

quotient of the two terms the average diffusivity over the time interval

at the depth of the measurements is found.





57




The most important aspect of this first-principles approach is

the determination of the fluctuation terms, especially the vertical

water velocity fluctuations. The successful application of the method

to the problem of evaluating diffusion coefficients depends on the

ability to detect the turbulent water motion. In the following

chapter a detailed survey of turbulence measuring equipment is given

in an effort to determine the manner in which the correlation of

fluctuations method may be used in this study.















CHAPTER IV

SELECTION OF A TURBULENCE MEASURING DEVICE



Introduction


As discussed earlier, the evaluation of turbulent diffusion

coefficients describing the transport of mass or heat may be

accomplished by either a first-principles approach or the flux-

gradient method. The former requires the measurement of turbulence-

caused fluctuations of both a directional water velocity component

and the quantity being studied. The latter requires the measurement

of spatial and temporal concentrations of the quantity being studied.

Depending on the method used the evaluation of turbulent transport.

of heat or mass may or may not require the detection of water velocity;

however, sensing water velocity is an important aspect of most

studies of turbulence, or more specifically, turbulent intensity.

So, although relatively few studies have been made of correlations

between measured fluctuations of velocity components and transportable

quantities, many evaluations have been made of velocimeters from

the standpoint of suitability for use in .studies of turbulent inten-

sity.

The criteria applicable to the selection of a velocimeter used

in a first-principles approach to transport measurement are similar

to those required in a study of intensity. Many of these criteria,











in a general sense, are pertinent to the selection of any device .to

be used in a scientific investigation.

This chapter presents an overview of the factors which must

be considered when selecting turbulence measuring equipment, a

survey of the various types of velocimeters available with comments

about the potential usefulness of each in turbulence measurement

and lastly, a discussion of the specific requirements for velocity

sensing in the study and the ways that the device ultimately chosen

satisfies those requirements.



Equipment Requirements


The criteria which must be satisfied when selecting a device

for measuring turbulent water velocity fluctuations are similar

to those criteria pertinent to the selection of most devices used

in scientific investigations. Consideration must be given to

instrument performance in the areas of range, sensitivity, stability,

ruggedness, power requirements, noise generation, calibration

requirements, and frequency response, among others. Turbulent

velocity fluctuations may be of a small magnitude when compared

to the average velocity, and velocimeters used in such studies

must have sufficient range to measure the mean and be sensitive

enough to detect the fluctuations. The frequency of the fluctuations

may vary from near zero hertz to several thousand hertz. Frequency

response limits indicate the maximum frequency at which the device

can accurately detect the existing velocity. Stability is a parameter











which may be better considered in three categories: lack of drift,

retention of calibration, and the ability tn be unaffected by

changes in the water other than velocities. Drift, short term

deviation, must not occur to a significant degree during the length

of time of continuous measurements. The same may be said for a

permanent alteration of the device calibration, and if water quality

changes during measuremen; the device must not respond to the change.

The nature of the turbulence will dictate the length of time stability

must persist.



Hot-wire, Hot-film Anemometers


Hot-wire anemometers have been used for many years as turbulence

measuring tools. A short wire of platinum or similar metal is

heated above ambient temperature by an electrical current, and subse-

quently cooled by fluid flowing across it. The amount of cooling

can be related to the velocity. The hot-wire probe is more useful

in gas flows than in liquid flows because the greater electrical

conductivities usually experienced in liquids interfere with normal

operation. This problem stimulated the development of hot-film

probes for use in liquids. Hot-films are essentially hot-wires

coated with an electrically insulating substance such as quartz or,

more recently, Teflon. The application of the film is done in such

a manner as to preserve as much of the thermal response of the probe

as possible. Also, increased structural strength resulting from the

film coating makes them preferable for use in water where more impact











force is exerted on the probe.

Hot-wire and hot-film probes are used in two basic electrical

models: constant current and constant temperature. The constant

current method maintains a uniform current through the probe, and

changes in velocity cause changes in wire temperature which cause

changes in wire resistance. Therefore, the voltage drop through

the probe is an indication of the velocity across the probe. An

electronic feedback network helps constant temperature anemometers

to maintain the probe at a uniform temperature. Hence, the amount

of current through the wire is indicative of level of velocity

passing it. Problems of electronic stability within the feedback

circuits inhibited the early usefulness of constant temperature

anemometers; however, now sufficient electronic capability exists,

and the constant temperature method is prevalent among commercial

anemometer manufacturers.

Because turbulence measurements in water are nearly always

made using hot-film probes, the remainder of this discussion will

focus on anemometry using them. Hot-film probes are available in

various geometric shapes and configurations. The shape of the probe

may make it more or less suitable for a particular application.

Furthermore, probes and associated electronics are now available

which sense water velocity in the three component directions. Thermal

response and electronic response of modern anemometer systems are

such that very high frequency turbulence can be accurately sensed.

Also, electronic instruments are available which perform correlation

and averaging operations as the velocity measurements are being made.











Although, hot-films have been used successfully in many studies

of turbulent flow, they do have some drawbacks. The physical size

of the probes is very small; however, the somewhat larger size of

the probe supporting apparatus restricts the use of them in some

instances. The use of hot-films in natural water causes probe

deterioration due to accumulations of dirt, biological growth, gas

bubbles, and scale. The result of this degradation is, among other

things, loss of calibration and thermal response (see Morrow and

Kline, 1971). Another problem arises from the heating of the prohe.

The quality of output signal is proportional to the amount over

ambient the probe is heated, and in some cases this heating may -

in fact inject significant amounts of heat into the water being

studied.



Laser-Doppler Velocimeters


Laser-Doppler anemometry is one of the newest velocity measuring

techniques and has only recently been applied to studies of turbulence.

A beam of light is focused on a small volume of fluid in which

particles reflect and scatter the beam. If the particles are moving,

then they reflect the light with an apparent difference in wavelength,

the Doppler effect. An indication of the'particle velocity is

obtained by determining the frequency shift. This is done by hetero-

dyning (i.e., producing a new frequency by adding or subtracting,

the shifted and the unshifted signal). In some cases,two signals

shifted in a controlled manner are heterodyned. Lasers provide






63


illumination sources with very narrow bandwidths, which permit

measurement of even relatively low velocities. If the positions

of the laser and the scattered beam sensor are fixed, the: .

velocity detected represents a specific direction of particle motion.

Arranging three mutually perpendicular sensors to detect the.scattered

light enables a determination-of the three velocity components of

the particle (Fridman, Huffaker, and Kinnard, 1968).

The most significant advantage of-laser-Doppler anemometry-is

realized in studies of fluid flow through transparent ducts or. --

pipes. In these cases the-velocity'sensing equipment need not-be

inserted into the flow,"and no alteration or other disturbance of

the flow should occur.- This is especially useful for investigating

fluid motion very near conduit walls. Investigations of flow in

opaque conduits or natural waters would require that much of the

laser-Doppler equipment be enclosed .in a submersible package which

would make satisfactory measurements unlikely.

Many problems encountered in the formative stages of laser-

Doppler velocity measurement have been resolved or are being resolved.

Developments in electronic capability, beam splitting techniques,

and heterodyning procedures are providing better equipment perfor-

mance than in the past. Indications are that this method of anemometry

will permit much progress in laboratory studies of turbulence.



Rotating Element Velocity Meters


Velocity meters which use rotating elements such as propellers,











turbines, and cup assemblies are very common. Propellers and turbines

are similar to each other; however, propellers usually have fewer

blades and propeller blades usually have more curvature. Turbines

are used to measure high velocities in pipes, and lower pipe velocities

and open channel (free-surface) currents are measured by propeller-

type meters. Cup-type meters are used for free surface flow. Propeller

and cup-type velocimeters sense water motion in a plane parallel to

the propeller axis or pendicular to the cup axis of rotation. However,

the response of each to the direction of-velocity within the plane

being considered is different. The propeller is affected only by

flow components parallel to its axis while the cup is affected byall

velocity components in the plane of its rotation. For these reasons

the cup type meter is often preferred for use in studies of total

planar velocity, and propeller type meters are required when it is

desirable to distinguish various directional components of velocity.

There are several techniques used to indicate the rate of revo-

lutions of the propeller or cup element. The most common is pulse

generation by means of electrical contact closure. The pulses are

counted as clicks heard through ear phones or as digital signals

which trigger electronic counters. Other meters operate by sensing

changes in electrical resistance or direct current voltages generated.

at different rotational speeds. The design.of these meters has been

refined to the point that they are highly reliable and rugged, and

they are used extensively in both field and laboratory studies. They

have been used mostly in studies of flow or current measurement and

similar applications requiring detection of average velocities.











Attention has been given to the effects of turbulent water velocities

on propeller meters, and besides improving the reliability of average

flow data obtained using them, the better understanding of propeller

response to turbulence has permitted them to be used in studies of

turbulent flow structure (Plate and Bennett, 1969).

Propellers do not foul or degrade or otherwise lose calibration

when used in natural waters. Furthermore, they resist damage due

to impact from foreign matter. Using two meters positioned at right

angles enables resolution of separate velocity components from

simultaneous measurements. Advanced designs using lightweight materials

and low friction pulse generation techniques allow quick, accurate

response to even turbulent velocity fluctuations. However, the limit

of this response due to inertia effects of even improved designs

precludes the use of propellers in investigations of very low velocities.

The lowest threshold velocity of meters currently being used is in

the range of one-half centimeter per second. Slower water motions

are not detected. The physical size of the propeller also limits the

minimum eddy size which it can detect.



Acoustic or Ultrasonic Devices


Methods have been developed and improved which detect fluid

motion using acoustic signals whereby the behavior of ultrasonic

waves emitted into the flow is monitored and processed into velocity

information. Acoustic devices function according to either one of

two basic principles: time of travel or Doppler shift. Measuring






66


water velocity by measuring the Doppler shift of high-frequency

signals reflected from particles in the flow is the same principle

of operation used by laser velocimeters. The chief difference is

the wavelength of the emitting source. Time of travel devices

sense the incremental propagation of the emitted signal caused by

the fluid motion. In other words, the ultimate velocity of the

acoustic signal is the vector sum of its generated velocity plus

the spatially integrated velocity of the fluid in the region

through which the signal passes. Indication of water velocity is

obtained by discerning that portion of the received signal altered

by the water motion.

Various techniques may be employed to detect the alteration of

emitted waves. The simplest in concept is to produce a pulse or

wave form and transmit it toward a receiver unit which senses it.

By accurately measuring the time of travel of the pulse and sub-

tracting it from the expected time of travel in still water, calcu-

lated from a knowledge of the speed of sound for ambient water

conditions and sender-receiver separation distance, the water velocity

along the sender-receiver path may be determined. However, the

necessity of knowing the ambient speed of sound inhibits the usefulness

of the technique, and it may be averted by adopting a dual path

procedure. If the times of travel of two signals traveling the

same path but in opposite directions are measured, then the difference

between the two should indicate twice the water velocity only, due

to the canceling out of the speed of sound component common to each.

A modification of this elementary concept is the sing-around circuit











which operates by emitting an initial signal that is received by

a secondary unit and upon reception triggers the emission of a

return signal which,when detected by the primary unit,triggers yet

another signal. The process continues for a selected time or number

of rounds. It uses the dual path concept and is essentially a

summation of individual measurements. This design yields improved

resolution or sensitivity while indicating an average velocity over

the repetition period. Another modification of the time of travel

method is the differential time circuit which also uses a dual path

instrument arrangement. Two signals are originated at each end of

the path coincidentally. The reception of one signal at the other

end triggers the start of a timing circuit which is stopped when

triggered by the reception of the slower signal. Knowing the

difference in time and which signal arrived first permits computation

of water speed and direction along the instrument path.

Acoustic methods are desirable because they may be non-intrusive;

hence they do not disturb the water at the point of measurement. They

do not foul or degrade rapidly when used in "dirty" waters, and they

are rugged. They have given very satisfactory results when used

as flow quantity meters because they intrinsically average flow

between sensors. Stream flows have been measured using time-of-travel

acoustic systems installed on opposite banks. Doppler shift systems

may be the more satisfactory in studies of turbulence which examine

local flows. It is possible to construct a Doppler shift device

using properly oriented sensors which will detect water velocity in

three directional components. This meter used in a study of estuarine











turbulence was calibrated over a wide velocity range of from about

one-half centimeter per second to thirty meters per second (Wiseman,

1969). Problems with acoustic velocimeters are caused by electronic

noise, sensing of spurious waves, inadequate alignment of emitter

and receiver units, and interference from foreign matter in the

water.



Lagrangian Methods


An alternate method to measuring water velocity past a fixed

sensor is to mark water parcels and follow the movement of the

markers. Such studies of the Lagrangian nature of turbulence are

carried out by a wide range of methods among which are flow visual-

ization techniques, tracer tests, and drogue studies. Flow visual-

ization methods are often used in laboratory investigations of

turbulence. They involve making a succession of photographs, which

record the positions of fluid markers at the time of exposure.

Markers must be detectable by the photographic equipment and possess

physical properties which permit them to behave as suitable indicators

of fluid behavior. Types of markers include small spheres, hydrogen

bubbles, and small volumes of dyes, among others. Merritt and

Rudinger (1973) studied flume turbulence by using a solution of

water and pH sensitive dye. A pulse of a current through a thin

metallic wire positioned in the flow created a surplus of hydrogen

ions at the wire surface which changed the pH and consequently the

color of a thin line of water particles. The fate of these particles











was then recorded photographically and subsequently analyzed.

The travel of water parcels may also be monitored by tracer

tests. Tracers have been used in both laboratory and field studies

of turbulence. They may occasionally be detected visually, but more

often, the nature of the tracer is such that it may be sensed in

invisible concentrations by equipment designed for such a purpose.

Fluorescent dyes, radioactive substances, and salt solutions,

among others,have been used extensively in investigations.

Field studies of larger.scale motions have been performed

using drogues or similar devices which are transported by water

currents. Various schemes have been employed to reduce unwanted

interference, for example' those of wind. The primary problem

in such studies is that drogues which can be located and monitored

successfully are often affected by wind or buoyancy or inertia and

do not accurately indicate water motion.

Studies of turbulence by Lagrangian means require constant

monitoring throughout the course of the experiment. Observations

of the entire flow field must be made often enough to detect the

effects of the smallest eddies being studied and over sufficient

total times to satisfy statistical requirements. Laboratory studies

of small scale eddies may last only several seconds while field

studies of larger scale motions may take days or weeks. Improvements

in photographic techniques, electronic drogue tracking capabilities,

and tracer detection equipment have made the performance of Lagrangian

studies easier; however, they are still difficult and laborious to

carry out because they are not adaptable to high levels of automation.










Electromagnetic Flowmeters


The movement of charged particles through a magnetic field

induces currents. This was discovered by Farraday in 1831 with

ordinary solid electrical conductors. The same principle can be

used to measure fluid flows, and since all particles have at least

atomic level electrical charges, they are affected by magnetic

fields regardless of whether or not the fluid is an electrically

conductive medium. Electromagnetic flow meters have been used

primarily to measure pipe flow (Grossman, et al., 1958), and blood

flow in humans and animals, but instruments for measuring local

velocity have also been developed (Bowden and Fairbairn, 1956).

There are two methods of electromagnetic flow meter operation.

The first senses the induced voltage generated by the flow of charged

particles through a magnetic field. The second senses an induced

magnetic field generated by a conducting fluid flowing through an

established magnetic field. The second method is often used to

measure flow in electrically conductive media because it need not

be in electrical contact with the media. The induced voltage method

requires electrical contact between the sensing probes and the flow.

Cushing (1958) showed by theoretical arguments and experimental

results that the sensitivity of induction flow meters is not affected

by variations in water conductivity as long as the conductivity is

above a threshold value of about 10-5 mho's per meter. The vector

nature of the induced voltage field enables the determination of

velocity direction, and instruments exist which can measure at least











two velocity components simultaneously.

In theory, velocities down to zero can be detected; however,

electronic noise in the signal processing portion of electromagnetic

meters prevents detection of motion below some threshold level.

Also, some error in measurement is caused by a "transformer effect"

voltage which is due to the use of alternating current to create the

magnetic field. Electronic circuitry is capable of diminishing the

resultant variations in zero baseline due to this "transformer

effect", but some drift still occurs. Boundary-layer effects at

the probe-water surface cause minor errors in the response of the

instrument; however, the probe is affected by water motion over a

distance equal to two or three times the probe radius and the boun-

dary-layer is small in thickness relative to the overall distance

of influence. The device has no moving parts, is rugged, and does

not lose calibration quickly because of fouling.



Other Devices


Several other types of devices have been developed for measuring

turbulent flow. Some are altogether different from those already

discussed, and some are similar to or use similar principles to those

mentioned previously. Several kinds of pressure transducers have

been devised to gage instantaneous changes in water velocity. The

dynamic pressure at the probe is often.sensed by electronic means;

for instance, Ippen and Raichlen (1957) used a diaphram which moved

slightly due to pressure change. The diaphram also served as a











variable capacitance in an electrical circuit, and the response

of the circuit indicated flow velocity. Force transducers have

also been used to measure water velocities. The force of the fluid

impinging on a small sensor surface is converted to an electrical

signal which is calibrated to velocity (Earle et al., 1970; Siddon,

1971). Pressure and force transducers cannot measure very low water

velocities, and the use of them is limited to velocities above 1

centimeter per second. -

Electro-kinetic transducers have also been used in studies of

turbulence (Binder, 1967). This type of sensor is unusual because

it is not energized. Fluctuations in water velocity cause a thin,

small wire probe to emit detectable electrical impulses. However, the

wire does not respond to laminar flow and can only be calibrated

dynamically, which makes it useful only in studies of the relative

intensities of turbulence. Other studies have been performed using

thermal methods which impart small quantities of heat to the flow

and then attempt to sense the heat nearby. Similarly a device known

as the Deep Water Isotropic Current Analyzer (DWICA) has been

developed for use in investigations of reservoir transport and

turbulence. It omits a small quantity of radioactive material near

the center of a ring of sensors. Flow direction and magnitude are

determined by which sensor detects the radioactivity and the time

lag between emission and detection.











Price Comparisons


No information on prices of the various types of devices has

been given, although this is an important consideration when

selecting suitable instrumentation. Some of the reasons for not

stating prices follow. Many devices reported in the literature

are custom-made, and often cost information is not given. Even

when costs are noted,they may only reflect a fraction of the

total resources necessary to produce the equipment. Furthermore,

many research systems are integration of commercially available

devices and custom-made ones. Also, quoting prices for commercially

available equipment is of itself difficult because optional or

auxiliary equipment is usually available. The desirability and,

more importantly, the necessity of any or all of these supplemental

devices varies greatly according to the specific application.



Requirements of the Lake Mize Study


As discussed in Chapter VI, Lake Mize,near the University of

Florida campus,was selected as the site for the field study of

vertical turbulent transport. The nature of the lake together with

the overall objectives of the study imposed a set of rather rigorous

constraints on the selection of a velocity measuring device. The

primary requirement-was for the instrument to be able to measure

water velocities in two or three directional components at a point

in the lake. It had to be suited to field use, although battery

operation was not necessary because of the 110 vac supply at the lake.












The scheme of the field study was to take measurements over a long

period of time, so stability of calibration was essential. The

time, scales of turbulence were suspected to be such that stability

over at least a period of one day was necessary while the maximum

frequency response needed would be 1 hertz. Furthermore, an

acceptable flow meter had to be unaffected by changes in chemical

and physical water quality such as ion concentration, temperature,

and concentrations of dissolved substances,since significant variations

occurred with depth in the lake and with time; Probably the most

stringent criterimwas that of sensitivity. Although definite values

of water velocities were not known, it was known that the rate of

movement would be very low. A target value of 0.3 cm/sec (0.01

ft/sec) was established. This value represented a compromise between

the desire to sense large scale water motion and the awareness that

ultra-low velocity measurement was not feasible at the time. For the

approximate lake depth of 25 meters,a 0.3 cm/sec velocity was

indicative of vertical time scales on the order of 1 day. Another

requirement involved instrument output. Because the entire field

study would be automated,the device had to indicate water velocity

in analog form. Manual operation was unacceptable. It was felt

that the design and fabrication of an adequate velocimeter was beyond

the capabilities of the personnel and physical resources available on

hand. Funds for the purchase of a suitable instrument were limited

to well under ten thousand dollars,which was the total amount allotted

for all equipment expenses of this study.











Selection of the Electromagnetic Flow Meter


Of all the types of devices available for use in turbulence

measurement only one, the electromagnetic flow meter, seemed capable

of satisfying the criteria established above. Laser-Doppler meters

were unacceptable for field work. No Lagrangian technique could

be used for a continuous, highly automated study. Pressure and

force sensors and propellers could not respond to low enough water

velocities. Hot-film probes would foul too quickly. The acoustic

Doppler meters seemed to exhibit promising characteristics, but

they were not available commercially, and similarly dual-path

acoustic devices were only supplied in massive units for measuring

streamflow. The electromagnetic meters were previously available

only for pipe flow measurements; however, it was learned that a

meter for..local velocity measurements in natural waters was newly

available. The meter measured two perpendicular flow components and

seemed to meet all the other criteria established. The lower

limit of its sensitivity was 0.3 cm/sec; however, drift and noise

limits were on the same order. Despite these possible shortcomings,

it was decided that the electromagnetic meter was acceptable and

possessed superior overall capabilities to any other device being

considered. Its cost of less than five thousand dollars was also

acceptable.















CHAPTER V

DESIGN OF THE DATA ACQUISITION SYSTEM



Introduction


Because the ultimate objectives of this study involved the

collection of large amounts of data over a period of at least one

year, there existed a need for acquiring this information in an

automated fashion. This chapter describes the projected requirements

for a data acquisition system capable of satisfying these needs.

The design of such a system was made by incorporating the data needs

for each phase of analysis planned. The result was a systemcapable

of performing a combination of many types of in situ measurements

and recording them in a compatible fashion on a single punched paper

tape recorder. Brief descriptions of the principal components

of the system are given in this chapter, and a discussion of the

physical installation of these components at the field research

site is presented in the following chapter. Also included in this

chapter is a description of the calibration procedures used on each

of the sensors in the data acquisition system.



Data Requirements for Flux-Gradient Method


Evaluation of heat flux and gradient quantities is done by using

temperature profile data taken at two separate times. Temperature at












various lake depths must be known, and the times at which the

temperatures occur must also be known.

The number of locations within the water column at which

temperatures are recorded and the frequency at which the obser-

vations are made which will provide sufficient information for

flux-gradient analysis depend: on the individual situation. For

instance, the lake temperature may be nearly constant over several

meters of depth in certain regions (e.g., the hypolimnion) and

demonstrate large spatial gradients at other levels.(e.g., the

epilimnion).

Furthermore, the zones of highest gradient may change with

time. Similarly, changes in temperature at a point may occur

regularly over a long time span or may become irregular relatively

quickly. The adequacy of using a given vertical spacing or time

interval is indicated by whether or not actual temperatures occurring

between measurements may be inferred from the data collected.

Since the field experiment also involved measuring water

velocities, the use of a single temperature sensor mounted on a

.moving support was not feasible. Therefore, a survey of water

column temperature required multiple sensors at fixed positions.

Consideration of temperature profiles taken by Nordlie (1972)

at Lake Mize indicated vertical spacing of less than one meter near

the surface and about one or more meters in the lower region would

be advisable. Diurnal changes in temperature were foreseen so the

sampling frequency was specified as at least six times per day.

The required precision of temperature measurements was established











as 0.1 degree centigrade. While resolution to minute fractions of

a degree would have been desirable, preliminary study indicated a

practical limit of resolution of commercially available temperature

sensors of about 0.1 degree centigrade.



Data Requirements for Correlation of Fluctuations Method


Evaluation of heat flux by the correlation of fluctuations

method entails the measurement of the vertical component of water

velocity (i.e., magnitude and direction) and coincident water

temperature at the place of velocity detection. The gradient

evaluation requires knowledge of the change in temperature with

depth at the depth of the velocity measurement. Although'discussed

in detail in the preceding chapter an additional comment regarding

velocity measurement is in order here. Of particular import is the

ability to sense water velocity in only one direction and at low

levels. A magnitude of water velocity was not known a prior

however, the small size and sheltered nature of Lake Mize suggested

that very low levels of water speeds might be'expected. By estimating

distance scales of motion in the lake to be from less than one to

about ten meters and time scales of from several seconds to several

minutes, a crude estimation of a minimum velocity of 0.003 meters

per second (0.01 ft/sec) was made.

The requirement for temperature measurement was established as

0.1 degree centigrade due to prior knowledge of equipment limitations.

Since evaluation of the gradient was to be accomplished as part of











the flux-gradient method analysis, it was felt that sufficient

gradient data would exist from those measurements, and no new data

would need be collected.

As noted above, the time scales of the turbulent motion were

estimated to be as low as several seconds. Therefore, a maximum

sampling frequency of one second was specified for the velocity-

temperature correlation data.

It was also desired to evaluate diffusivities by the correlation

of fluctuations method at various depths throughout the water column.

To do this the velocity probe and its companion temperature sensor

had to be capable of being positioned at any depth. Of course, once

at a specified depth the apparatus had to be rigid so that even

slight water movement might be detected.



Other Data Requirements


Other data needs were those related to environmental-meteorological

conditions. Wind speed measurements at three different heights were

planned, and wind direction was also specified. Incident short-wave

solar radiation, relative humidity, and air temperature complete

the list of required parameters. Since these parameters all change -

daily and most exhibit marked diurnal fluctuation, a sampling frequency

of several times per day was deemed desirable. After the study

began,it became apparent that instantaneous measurements of these

variables were subject to great error. A discrete observation of wind

speed, for instance, could not accurately be applied to a lengthy time











interval as.the average value over the interval. Therefore,

during the field study the requirements for wind speed and solar

radiation were changed such that summed or integrated values .of each

were needed. Instead of wind speed in meters per second, wind in

total meters past the sensor during the interval since being last

recorded was specified. Similarly, total langleys of solar radiation

were called for. The variations in air temperature, relative

humidity, and wind direction were such that readings taken half-hourly

(as will be discussed later) sufficed.

Wind velocity up to about five meters per second was judged

suitable. The capability to measure air temperature from zero to

forty degrees centigrade and relative humidity from twenty to one

hundred percent was specified. 1.2 to 1.5 langleys per minute was

estimated as the maximum levels of solar radiation it would be

necessary to measure.



Aggregate Requirments


Before considering other factors associated with the plan of

the data acquisition system, a recapitulation is in order. Two

groupings of data needs existed due primarily to different requirements

in frequency of collection. One was velocity-temperature correlation

data sampled as often as once per second. As the study developed,

this group became known as Group A. The other, a larger group of

data signals, monitored the environmental conditions at a rate as

slow as several times per day. Thislatter group, eventually labeled











Group B, included water temperatures at a series of depths, air

temperature, solar radiation, wind speed at a series of heights

above the water surface, wind direction, and relative humidity.

Finally, although not stated earlier, a precise knowledge of the

time of sampling of each parameter had to be available.



Data Recording Requirements


The amount of data to be collected was forecast to be very large

because not only was the overall length of the field study projected

at one year but also the number of separate data sources was great

and the frequency of sampling high. The need to employ special

procedures to accommodate this volume of information was evident.

Certainly manual data collection was not feasible. A high level

of automation was required. Due to the number of inputs (parameters),

some type of multiplexing or switching was indicated, because a

different recording device for each signal would necessitate the

use of well over ten units. The satisfactory recorder must then be

able to accept data from several sources either in parallel or

serially and also be able to operate accurately at the rate of at

least one measurement per second. Finally, the form in whichthe

data were recorded should permit easy access for subsequent analysis;

*in other words,converting the data to a form which could be read

into digital computers should be a process requiring as little effort

as possible.











Additional Data Acquisition Needs


The major requirements of the data gathering equipment have

already been stipulated or alluded to; however, some factors should

be discussed further.

The speed of operation of the system had to be at least one

data item per second to accommodate the velocity-temperature correlation

information. Yet, since the actual time scales of motion in the lake

were not known and could perhaps be very much slower, say, on the

order of minutes, the ability to select from a range of sample rates

was desired. This sample rate flexibility could permit optimum

system operation by enabling the selection of the slowest sample

rate providing adequate data and thereby reducing the volume of

data collected.

The data collecting equipment had to be suitable for use in

the field. Furthermore, the ability to function on, in, or very

near the water was requisite of most of the system components.

There were few requirements on the various sensors to be used

other than that they be capable of measuring the levels of the

respective parameter of each which were specified earlier. It was

necessary, however, that each of the types of sensors be compatible

with the overall data gathering effort.

The budget for the procurement of field equipment was ten

thousand dollars. This figure was less than requested and considered

to be a significant constraint on the ultimate comprehensiveness of

the data acquisition system.












Commercial Equipment


As a general rule,equipment needs can most efficiently be met

by purchasing commercially available equipment whenever possible.

The necessity to use custom-made devices arises whenever there is

no satisfactory equipment on the market or when there is not enough

money to afford equipment which is otherwise available. A survey

of manufactured data acquisition equipment was made. This included

a study of sensors capable of measuring each of the physical parameters

required and also the available alternatives for gathering and

recording these measurements. In summary, as could well be expected,

no single device capable of performing all of the stipulated tasks

was found.

As described in detail in the proceeding chapter, the search

for a suitable velocity meter culminated in the purchase of an

instrument which detected velocities in two perpendicular directions

and the output of which was an analog (voltage) signal at a level

of one volt out per foot per second of water speed. Other instruments

were found which could measure each of the other parameters; however,

several drawbacks were encountered. Among' these was the fact that

the total cost of all of these exceeded available funds. Additionally,

the aggregate of instruments would occupy much space if they all were

operated at the same location and, besides, there was much duplication

of duty because most were designed to function independently.

The data assembling and recording phases of the acquisition

system could also be performed by commercially available equipment.











Disadvantages encountered included very high costs, incompatibilities

with certain types of sensors, and unworkable requirements of power

and operation. No system was found which offered satisfactory

performance yet was affordable. It should be noted that the high

costs usually were associated with instruments which provided

greater sampling speeds or capacities than needed for this project.

As could well be expected,considering the range of performance

requirements associated with this field study,very little of the

necessary equipment could'be purchased already assembled. Most of":

the sensors were available commercially, but even they had to be

acquired in a form which would permit the incorporation of them into

a mostly custom-fabricated instrument package.



Design of Data Acquisition System


The preceding actionss have discussed in detail requirements

of the data acquisition system. What follows is a description of

the system as it was ultimately fabricated within the constraints

of available resources (especially money).

To stay within the budget of ten thousand dollars it was

necessary to amend the goals of the project by limiting its scope

or modifying technique,or find alternate sources of supply of

equipment,or make better use of material already on hand. The

result was an optimization procedure which combined all of these

alternatives to various extents. While every step in the decision

making process cannot be explained, suffice it to say that many












characteristics of the completed system were dictated by the

availability of instruments at low or no costs, even when preferences

would have been otherwise. However, in most cases,the requirements

outlined earlier were met or exceeded.

The system consisted of several stages. One stage was the

sensors which sensed the level of a physical parameter (e.g.,

temperature, wind speed) and reflected that level as an electrical

signal (e.g., resistance, current). Another stage of the system

received the sensor outputs and conditioned them, individually, to

yield compatible voltage signals. The voltages from the various

conditioners were multiplexed or switched one at a time into the

recording device. The multiplexing and recording stages were supported

and controlled by a stage which contained all the digital logic

circuitry. This digital stage was also the source of the system

timing.

As a rule the sensors were manufactured devices and the signals

from each were conditioned by custom-made circuits. An exception

was the water velocity meter which sensed and conditioned its own

signal so that the output was a voltage directly related to water

speed and direction. The only other system component not custom-

made was the paper tape punch which included an analog-to-digital

converter.

The requirements for a suitable recording device outlined earlier

indicated the use of telemetry, magnetic tape recording, or punched

paper tape recording. Other alternatives such as photographic methods

and manual recording seemed infeasible, and even strip-chart recording












was thought to be acceptable only as a last resort. The degree

of complexity and the high cost of installing and operating a

telemetry system eliminated that option. Magnetic tape recording

satisfied all of the stipulations; however, its cost was high

and could only be justified for data collection at much higher

speeds (thousands of items per second). Also, magnetic tape was

thought to be susceptible to extraneous electronic noise, especially

when used at slow rates. The capabilities of punched paper tape

seemed matched to the needs of this study. Data could be recorded

sequentially at rates up to tens of items per second, and the stored

information could be read directly into digital computers for sub-

sequent use. No cumbersome data handling was required.

It was determined that the local computing facility had the

capability of reading the punched paper tape, so the decision to

procure such a device was made providing costs were within budgetary

limitations. Fortuitously it was learned that an apparently suitable

piece of equipment might be obtained gratis from a nearby govern-

mental agency which no longer used it. Ultimately this tape punch

was acquired and after testing proved most satisfactory; therefore,

the remainder of the data acquisition system was planned such that

optimum use could be made of it.

The recorder was manufactured by Towson Laboratories (Model

DR-100P-35) and contained an analog-to-digital converter as part

of the unit. Zero to ten volt signals of either polarity were stored

in digital form by punches in a one-inch wide paper tape. The device











punched whenever a pulse command from the digital control stage was

received.

The nerve center of the system was the digital control stage.

This stage used integrated circuits of transistor-transistor logic

(TTL) type to generate pulses which controlled all the system

functions. Precise timing was achieved by using a sixty cycle per

second signal from the alternating current power line as a source

for a digital timing chain. The chain was composed of several

serial stages each dividing the frequency into it to produce a lower

frequency. Any one of several time chain outputs could be selected

as the time control for the velocity-temperature correlation data,

Group A. By actuation from shore,any of a range of sample frequencies

from one sample per second to one per three minutes, could be selected.

A one pulse per minute signal was used to control and activate

the Group B sensors. Ultimately, there were twenty thermistor

probes which sensed water and air temperatures, a pyranometer for

measuring solar radiation, three cup anemometers detecting wind

speed, a wind direction vane, a polymer-type sensor for relative

humidity, a voltage signal indicating depth of the Group A probes,

another voltage signal indicating the rate of Group A sampling, and

two signals indicating power supply levels in the system. These

thirty parameters were sampled at the rate of one:per minute making

a complete scan every half-hour.

The timing chain controlled the switching and recording of Group

A and Group B so that they were sampled in an integrated, coincident

manner. The design of the chain was such that the Group B signal






88


could be interspaced between Group A recordings even at the fastest

Group A sampling rate.

The switching in the system was done by solid state FET (Field-

effect transistor) switches. The analog signals were amplified

using integrated-circuit operational amplifiers. The amplifiers

were of a common style (741) but specially selected for low levels

of signal drift due to ambient temperature changes.

Figure 5-1 gives a conceptual diagram of the system in its

ultimate form. The various analog and digital control phases are

as indicated in the preceding discussion. No further description

of the electronic control circuitry will be presented, but the

sensors used to detect parameter values will be discussed in greater

detail. Brief descriptions and identifications of each of the sensor

types used and the circuits used to condition the sensor outputs

are given. Following that is a summary of calibration procedures.



Velocity Measurement

The velocity meter purchased was made by the Engineering-Physics

Company, Rockville, Maryland. The unit selected was Model EMCM-3BX

(Serial number 620). The meter has been discussed earlier and is

not described in detail here. It produces a voltage output of one

volt per foot per second.water velocity in either of two perpendicular

directions. The algebraic sign of the output voltage indicated the

orientation of the movement in either direction.















Island bridge) "...... o..ga-


.9 r .
26_^ ------------ --'------'---->* p
ir--



Thermistor A/D
S String Temperature g Final Convertor
S. Multiplexer Bridge 6opp Multiplexer
19 -
20

r-------^------_--_____^..

,o _ KEY

Wind DATA SIGNAL
2 .Speed
Counter -- -- AMPLIFICATION
S--, Group B
fEnvironmental ._
Sensors
Wind Direction Multiplexer DATA COLLECTION

Solar Radi.on nearer AND

Others (inl. rel. hum.) ACQUISITION SYSTEM

FIGURE 5-1. CONCEPTUAL DIAGRAM OF THE DATA ACQUISITION SYSTEM USED AT LAKE MIZE











Temperature Measurement


The choice for temperature detection was between using thermistors

or thermocouples. Thermistors were picked because the voltage

outputs of thermocouples was lower than that which could be tolerated

due to electronic noise associated with boosting.the signal. Originally

thermistors manufactured by the Yellow Springs Instrument Company

(Part No. 44018) were selected because they offered linear change

in resistance with change in temperature. Each thermistor was

connected to a multiconductor cable, and the joints sealed with potting

resin. One by one the joints leaked, and new probe making techniques

were attempted. Before a satisfactory technique was developed nearly

all of the thermistors deteriorated to the point of uselessness. New

thermistors were bought from Fenwal Electronics, Inc. (Part No.

GA51J1). They were conventional non-linear devices but were ordered

because of lower cost. The revised probes were constructed by

connecting the thermistor to two-conductor, vinyl-covered wires

(thereby using one wire per probe) and covering the thermistor and

joint with a short (2-3-inch) piece of copper tubing which had been

crimped and soldered at one end. The region of juncture between the

tubing and wire covering was sealed by wrapping the area with over-

lapping turns of plastic electrical tape and then repeating with

another wrapping of tape. These probes performed satisfactorily even

though with time some failed for various reasons.

The probes were initially spaced at intervals of two feet

(0.610 meters) from lake surface to bottom, but when the first cable




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