Modeling Hihly Transient Flow,

Mass, and Heat Transport in the

Chattahoochee River near

tlanta, Georgia
Modeling Highly Transient Flow, 

Mass, and Heat Transport in the 

Chattahoochee River near 

Atlanta, Georgia

By HARVEY E. JOBSON and THOMAS N. REEFER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1136

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1979
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Jobson, Harvey E.

Modeling highly transient flow, mass, and heat transport in the

Chattahoochee River near Atlanta, Georgia.

(Geological Survey professional paper ; 1136)

Bibliography: p. 41.

1. Stream measurements-Chattahoochee River. 2. Rivers-GeorgiaAtlanta region-Temperature. I. Keefer, Thomas N., joint author. 

II. Title. HI. Series: United States. Geological Survey. 

Professional paper ; 1136.

GB1225.G4J6 551.4'83'09758 79-607766

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402

Stock number 024-001-03233-3
CONTENTS

Page

Symbols ________„.„..____._„_____„.„„___„.„__.„_____. IV

Conversion table .„„„„„„„„„.„„„„„„„„.„„„„„.„.„„„.„.„„„„„„„ V

Abstract -..........__._.____.___._____.__._._...„_____._____________._._._.__...............__ 1

Introduction --.-.-..........__._._____.__._._........_._______.___.__._._.___.__......_...... 1

Model development „__„„„.„„„_„„„..„„„„„„.„_„„„„_„„......_......... 2

Flow model ———_.-......._._._.____._._._„-_..._...______.___.._____.__.___.______ 2

Mass and heat transport models.............................................. 4

Coupling ................................................................................. 10

Model calibration and verification ................................................ 11

Flow model ........................................................................... 11

Model calibration and verification—continued

Mass transport of a conservative substance .._...__.__. 16 

The temperature model „„„„_„„„„„„._„„„_.„_._„_.„_.._ 21

Discussion of results .„„.„„___.____.____.....__.__.. 26

Flow dynamics ......___...._.—..„...___.__.__.__.. 26

Transport _.___.._„...„...„„.„....„„....„„.....„„..„...„.„.............„.. 32

Temperature __________________________ 35

Summary and conclusions .„„„„...„.„...„.„„.._„„._......._........ 40

References ______________________________ 41

ILLUSTRATIONS

FIGURE 1. Map of Chattahoochee River showing the data/collection points and tributary measurement sites..._.__.......__.__....._._. 2

2. Photograph showing view of Chattahoochee River near Settles Bridge (river km 553.0) at low flow „.„.„„„„„„.„...................._. 2

3. Computation stencil for the linear, implicit finite-difference solution of the flow equation __„....„„„„„..„....._..............._._.-.__. 3

4. Photograph showing Suwanee Creek during a high stage in the Chattahoochee River „„.„„„„—..............._......_.—._......—_... 4

5. Photograph showing aerial view of Chattahoochee River showing shading conditions „„.„„...._..............._........_..„„._._........... 5

6. Schematic of river cross section used to determine the part of the water surface to be shaded by bank vegetation..________. 6

7. Computation stencil for the finite-difference solution of the transport equation ......_...........__..............__.__..._____. 9

8. Graph showing steady flow depth profile for the Chattahoochee River between Buford Dam and Norcross__....._.....————... 14

9. Graph showing calibration of the Chattahoochee River flow model with the March 1976 stage data.............._................————. 15

10. Graph showing comparison of modeled and observed discharge in the Chattahoochee River at the Highway 20 Bridge

during the March 1976 calibration period.........................._....._.............._.——___....._...._.....—.__..___.....___. 16

11. Graph showing comparison of modeled and observed discharge in the Chattahoochee River at the Littles Ferry Bridge

during the March 1976 calibration period .................................................................................——.-„.....-.—..-............—........ 16

12. Graph showing comparison of modeled and observed discharge in the Chattahoochee River at the Highway 120 Bridge

during the March 1976 calibration period ..............................................................._„„„„„„......__........._.....__.._..„.... 17

13. Graph showing comparison of modeled and observed discharge in the Chattahoochee River at the Highway 141 Bridge

during the March 1976 calibration period ..................................................................„....................................„....—............. 17

14. Graph showing verification of the Chattahoochee River flow model using the October 1975 stage data.....__..............——...... 18

15. Graph showing calibration of the Chattahoochee River flow model using the October 1975 stage data.......__..................———.. 19

16. Graphs showing comparison of the modeled and observed discharges in the Chattahoochee River during the October

calibration run..................................................................................................„„„.............................—......—.............—...... 20

17. Photograph of dye injection site for the March 1976 dye study, 200 m downstream of Buford Dam........................—..................... 20

18. Graph showing modeled and observed dye concentrations at Littles Ferry Bridge as computed using the conservative

transport model with unequal grid spacing....................................................................._...._........_......._...........—.— 20

19. Graph showing modeled and observed dye concentrations at Highway 141 as computed using the conservative

transport model with unequal grid spacing _______________________________________________ 20

20. Graph showing modeled and observed dye concentrations at Littles Ferry Bridge as computed using the

conservative transport model with equal grid spacing......._..........................._..........................._.—..„_..........._........... 21

21. Graph showing modeled and observed dye concentrations at Highway 141 as computed using the nonconservative

transport model with equal grid spacing ....................................-..............—..-.......-._........—......—.—.......—..........—.......... 21

22. Photograph of meteorologic instrumentation tower of the R. M. Clayton Sewage Treatment Plant showing the

anemometer and two psychrometers............................................................................—.-.—.--.......-..-............——..————.. 23

23. Photograph closeup of the anemometer showing the Chattahoochee River in the background ....................................................... 23

24. Photograph closeup of the pyranometer and pyrgeometers at the R. M. Clayton Sewage Treatment Plant in Atlanta.................... 23

25. Photograph closeup of ventilated psychrometer at the R. M. Clayton Sewage Treatment Plant in Atlanta ..—.......-.-.............—..— 23

26. Photograph closeup of Top Hat psychrometer at the R. M. Clayton Sewage Treatment Plant in Atlanta.........._..._..........—..... 24

27. Graph showing observed temperature in the Chattahoochee River at the Highway 20 Bridge

during the October 1975 Verification period ................................................................................................................_ 25

28. Graph showing comparison of the observed and modeled temperature at the Littles Ferry Bridge

on the Chattahoochee River during the October 1975 verification period ...........................................................—.—............. 25

III
IV CONTENTS

Page

FIGURE 29. Graph showing comparison of the observed and modeled temperatures at the Highway 120 Bridge

on the Chattahoochee River during the October 1975 verification period —___——._..——_——..„———_...-———...——__.—......— 25

30. Graph showing comparison of the observed and modeled temperatures at the Highway 141 Bridge

on the Chattahoochee River during the October 1975 verification period _-__._____._._________.____.__.____—_—._...___——.——_._——..—— 26

31. Graph showing observed temperature in the Chattahoochee River below Buford Dam during the March 1976 verification period „ 27

32. Graph showing comparison of the observed and modeled temperatures at the Highway 20 Bridge

on the Chattahoochee River during the March 1976 verification period _.....____.___._...-_—.________..__—_—.____.._.———_.-._———..—- 27

33. Graph showing comparison of the observed and modeled temperatures at the Littles Ferry Bridge

on the Chattahoochee River during the March 1976 verification period _—_.________—.——._____._—.„„.—.„„„„„———_„„„.——.... 28

34. Graph showing comparison of the observed and modeled temperatures at the Highway 120 Bridge

on the Chattahoochee River during the March 1976 verification period ..........„.„.............„..............„...............—..........„—— 29

35. Graph showing comparison of the observed and modeled temperatures at the Highway 141 Bridge on

the Chattahoochee River during the March 1976 verification period ..........................................................„—„.....„„——... 30

36. Graph showing loop rating curves for the Chattahoochee River at the Highway 20 Bridge during the March 1976 calibration........ 31

37. Graph showing loop rating curves for the Chattahoochee River at the Littles Ferry Bridge during the March 1976 calibration ...... 32

38. Graph showing loop rating curves for the Chattahoochee River at the Highway 120 Bridge during the March 1976 calibration ..... 33

39. Graph showing loop rating curves for the Chattahoochee River at the Highway 141 Bridge during the March 1976 calibration ..... 34

40. Graph showing excess temperature at the Highway 141 Bridge during the March run due to an excess temperature of 1° at Buford 39

41. Graph showing excess temperature at the Highway 141 Bridge during the October run due to an excess

temperature of 1° at Buford _._________________________._______.___________._____ 39

TABLES

Page

TABLE 1. Internal reach data for the Chattahoochee River between Buford and Norcross ....................................................................... 12

2. Discharge values for the tributaries and withdrawal points on the Chattahoochee River during the October 20-25, 1975,

and the March 21-24, 1976, modeling periods ..................................................................................................................... 13

3. Physical data relevant to the Chattahoochee River between Buford and Norcross .................................................................... 22

4. Daily average values of meteorologic variables at the R. M. Clayton Sewage Treatment Plant in Atlanta .......—...................... 24

SYMBOLS

A = cross-sectional area of the channel;

ALON = longitude of the river (84.2° W);

AZS = azimuth of the sun;

BW = bank width;

C = concentration of dye;

Cjv = Courant number;

Cp = specific heat of water at constant pressure;

CD = heat storage capacity of the bed;

Dx = longitudinal dispersion coefficient;

E = rate of evaporation;

ea = vapor pressure of air;

EBH = effective barrier height;

ELEV = elevation of the sun in degrees;

e0 = saturation vapor pressure of air evaluated at a

	temperature equal to that of the water surface; 

EWi = water-surface elevation at grid point i; 

g = acceleration of gravity; 

HA = hour angle of the sun; 

Hj = sum of last two terms in equation 10 evaluated at grid

	point i and at time jAt; 

HR = time of day, in hours;

H(t) - increase in heat content of the slab between time 0 and t;

I = rainfall rate;

K = kinematic surface exchange coefficient;

L = latent heat of vaporization;

g = latitude of the river (34.0° N);

n = Manning's roughness coefficient;

NH = dimensionless surface exchange number;

P = wetted perimeter of the channel;

Q = discharge;

q = lateral inflow per unit length;

QT = tributary flow rate;

RH = hydraulic radius;

RS = part of incoming solar radiation absorbed by the water;

RSM = part of the incoming solar radiation which would be

	absorbed by the water under shade-free conditons; 

Sf = friction slope;

T = cross-sectional average water temperature;

t = time;

Ta = air temperature;

Te = excess water temperature above ambient;

To = final temperature of the water as it leaves the system;

Ti = initial temperature of the water as it enters the system;

T\ = temperature at grid point i and at time j At;

Tq = temperature of tributary inflow;

Tw = wet-bulb air temperature;

TZM = meridian of the time zone;

U = cross-sectional average velocity;

U{ = velocity at grid point i and at time j A t;

U* = shear velocity;

V = windspeed;

W = top width of the channel;
CONTENTS

x = longitudinal distance along the channel;

XN = normal distance from the tops of the trees to the shade

	point;

y = depth of flow;

2/j = distance above the insulated bottom of the slab;

ymo = measured depth at steady low flow;

Z = thickness of bottom slab;

z = elevation of the bed above some datum;

(3 = acute angle between the azimuth of the sun and the

	azimuth of the river subreach; 

T = psychrometric constant; 

8 = declination of the sun;

AS = change in tributary storage to occur during a time step;

Ai = time step in finite-difference solution;

ATg = temperature rise within the slab;

Ax = distance step in finite-difference solution;

€ = emissivity of water;

770 = Manning's roughness at steady low flow;

771 = rate of change of Manning roughness with stage;

0 = space derivative weighting factor;

K = thermal diffusivity;

p = density of water;

a = Stefan Boltzman constant for blackbody radiation;

T = travel time of a water particle through the system;

3>B = flux of thermal energy from the bed to the water;

<$6 = heat flux caused by longwave radiation emitted by the

	water;

<£6 = heat utilized by evaporation;

Qh = heat conducted from the water as sensible heat;

4%v = net nea-t flux caused by incoming radiation from the

sun and the sky;

<£g = heat flux added to the river by tributary inflow; 

<!>£ = heat added to the water by rain fall directly on the

surface;

<$r = flux of thermal energy from the air to the water; and 

\j/ = empirical wind function.

CONVERSION TABLE

Multiply metric unit

meter (m)

kilometer (km)

millimeters (mm)

meter per second (m/s)

cubic meter per second (m3/s)

pascal (Pa)

watt per square meter

By

3.281

0.6214

0.03937

35.31

0.02832

10.00

0.3172

To obtain inch-pound unit

foot

miles

inch

foot per second

cubic foot per second

millibars

British thermal units per 

square foot per hour

MODELING HIGHLY TRANSIENT FLOW, MASS, AND HEAT TRANSPORT 

IN THE CHATTAHOOCHEE RIVER NEAR ATLANTA, GEORGIA

By HARVEY E. JOBSON and THOMAS N. REEFER

ABSTRACT

A coupled flow-temperature model has been developed and verified 

for a 27.9-km reach of the Chattahoochee River between Buford Dam 

and Norcross, Ga. Flow in this reach of the Chattahoochee is continuous but highly regulated by Buford Dam, a flood-control and 

hydroelectric facility located near Buford, Ga. Calibration and 

verification utilized two sets of data collected under highly unsteady 

discharge conditions. Existing solution techniques, with certain minor 

improvements, were applied to verify the existing technology of flow 

and transport modeling.

The linear, implicit finite-difference flow model was calibrated by 

use of a depth profile obtained at steady low flow and unsteady flow 

data obtained in March 1976. During the calibration period, the model 

was generally able to reproduce observed stages to within 0.15 m and 

discharges at less than 100 m3/s, to within 5 percent. Peak discharges 

of about 200 m3/s were under-estimated by about 20 percent. During 

the verification period, October 1975, the flow model reproduced 

observed stage changes to within about 0.15 m, and its timing and 

over-all performance was considered to be very good.

Dye was added to the upstream end of the river reach at a constant 

rate while the river flow was highly unsteady. The numerical solution 

of either the conservative or nonconservative form of the masstransport equation did an excellent job of simulating the observed 

concentrations of dye in the river.

The temperature model was capable of predicting temperature 

changes through this reach of as large as 5.8°C with a RMS (rootmean-square) error of 0.32°C in October 1975 and 0.20°C in March 

1976.

Hydropulsation has a significant effect on the water temperature 

below Buford Dam. These effects are very complicated because they 

are quite dependent on the timing of the release with respect to both 

the time of day and past releases.

INTRODUCTION

In 1964 the Department of the Interior was designated as the lead agency for coordinating federal activities in water-data acquisition. Responsibility for these 

activities was assigned to the U.S. Geological Survey. 

Federal and nonfederal committees were formed to 

advise the Survey about water-data needs. In the early 

1970's the nonfederal committee recommended that a 

series of interdisciplinary river-quality assessment 

studies be performed to (1) define the kinds and amounts 

of data required to adequately assess various types of 

river-quality problems, and (2) to develop and document 

methods for assessing planning alternatives in terms of 

potential impacts on river quality.

Partly because it is in a developed basin with problems 

at the present time, the Chattahoochee River basin was 

selected as one site for a river-quality assessment study. 

The Chattahoochee is one of the largest rivers in the 

southeastern United States. Its headwaters are in the 

Blue Ridge province of north Georgia. It flows across 

the Piedmont and onto the Coastal Plain near Columbus, Ga. The Chattahoochee River joins the Flint River 

at the Georgia-Florida State line to form the Apalachicola River which drains into the Gulf of Mexico. The 

Chattahoochee's length from headwaters to the confluence with the Flint River is about 710 km, and its 

drainage area is about 23,000 km2 . On April 1, 1975, a 

river quality assessment project began addressing 

problems related to (1) thermal loading and heat 

dissipation, (2) wastes from concentrated urbanindustrial areas, (3) effects on river quality of 

hydropower pulsation, and (4) sediment sources, transport characteristics, and deposition for the Chattahoochee River basin below Buford Dam and above West 

Point Dam.

One element of the Chattahoochee River assessment 

study was to develop and verify coupled flow-temperature models of a 27.9-km reach between Buford and 

Norcross, Ga., with special emphasis on evaluating the 

effects of hydropulsation on the flow and temperature 

regimes. The purpose of this report is to present these 

models with verification and to identify and specify the 

procedures necessary for their successful application.

Buford Dam, located approximately 65 km northeast 

of Atlanta, Ga., creates Lake Sidney Lanier. Its 

hydroelectric units are used for peaking purposes, so the 

flow in the Chattahoochee River below the dam is highly 

unsteady. For example, the U.S. Army Corps of Engineers maintains a minimum flow of 15.4 m3/s through 

the dam but typically releases two pulses of water with 

peak discharges of about 215 m3/s during week days. 

Just downstream of the dam, the total stage rise 

associated with the pulses usually occurs in a timespan 

of 10 to 20 minutes.

A linear, implicit finite-difference flow model is 

coupled with two implicit finite-difference transport 

models to describe the flow as well as the transport of
MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

heat and dye in the reach of the Chattahoochee River 

below Burfod Dam. The study reach is illustrated in 

figure 1. It extends from the powerhouse at Buford Dam 

to the Highway 141 Bridge near Norcross. The river 

flows through rolling hills for most of this distance. The 

overall slope is 0.00036. The first 3.2 km are steep and 

rocky with a slope of 0.0011, and supercritical flow is 

present at several places during low flow. A bedrock 

control, below Highway 20, ponds the water in about a 

6.4-km reach at low flow. The channel ranges from 45 to 

65 m wide with a roughly rectangular cross section (see 

fig. 2). A characteristic cross section has large fallen

84

34

ake Sidney Lanier

Littles Ferry BridgeHighway 120 Bridge 

^Highway 141 Bridge, Norcross 

Chattahoochee River

/Atlanta metropolitan area

25

_I_

T 

10 20

50 KILOMETERS 

_I 

T 

30 MILES

FIGURE 1.—Chattahoochee River showing the data-collection points 

and tributary measurement sites (open triangles).

FIGURE 2.—View of Chattahoochee River near Settles Bridge (river 

km 553.0) at low flow.

trees protruding 6 to 15 m into the stream at the banks. 

The trees have fallen in because of bank sloughing which 

results from the large and rapid stage variations. Along 

the tips of the fallen trees, the water is somewhat deeper 

than in the center of the channel. The fallen trees apparently act in muct the same way as jetties or groins. 

The channel bed is primarily coarse sand (1 mm) on bedrock. At low flow, sand dunes, 1.8 to 3 m in length and 

about 0.3 m high, cover the bed. A typical Manning 

roughness coefficient is 0.042.

In the following sections, the flow, mass, and temperature models are described, and the available data, 

as well as the calibration and verification of each model, 

are presented. Two data sets, each containing continuous flow, stream temperature, and meteorologic 

data, were obtained. One set was for the period of October 20 through October 26, 1975, and the other for 

March 21 through March 24, 1976. In addition, a conservative tracer (rhodamine-WT dye) was injected continuously at a constant rate during the March run, and 

samples were collected at 10-minute intervals near the 

center and at the end of the reach. These data are used 

to verify the transport model, and in addition, they 

served as an excellent check for the flow model. After 

the calibration and verification results have been 

presented, each model is discussed with special emphasis being placed on identifying the problems associated 

with analyzing transport in highly unsteady flows and 

the potential accuracy of such an analysis. Finally, the 

effect of the hydropulsation on the flow, transport, and 

temperature regimes is dealt with in some detail.

MODEL DEVELOPMENT

FLOW MODEL

Techniques available for modeling unsteady openchannel flow have advanced rapidly in the past 10 to 15 

years, but almost all models are based on the same basic 

equations. These are continuity equations describing the 

conservation of mass

dA dU ^

dA -5- + -?37 ~ 9 = 

dx dt (1)

and the conservation of momentum

dU

dt dx dx = 0 (2)

in which U= cross-sectional average velocity, A = crosssectional area, x = longitudinal distance, t= time, q = 

lateral inflow per unit length, g = acceleration of gravity,
MODEL DEVELOPMENT

y = depth of flow, z = elevation of the bed above some 

datum, and Sf= friction slope. The friction slope may be 

evaluated from either the Chezy or the Manning equation. The Manning equation

Sf = A2

(3)

will be used in this paper where n = Manning's roughness 

coefficeint, Q = discharge, and RH = hydraulic radius. 

Equation 3 is not dimensionless but is expressed in SI 

units. To convert to the inch-pound system of units, a 

numerical value of 2.22 must be placed in the 

denominator.

Equations 1 and 2 are nonlinear in velocity, and no 

practical analytic solutions are available for unsteady 

flow. Early efforts to develop computer-based numerical 

solutions centered around the method of characteristics 

(Lai, 1967, Yevjevich and Barnes, 1970, Wylie, 1970). 

More recent efforts have centered around direct finitedifference solutions. Explicit techniques, an example of 

which was pioneered by Garrison, Granju, and Price 

(1969), are bounded by rigid stability criteria and tend to 

be expensive. Probably the earliest truly practical solution technique was the nonlinear, implicit finite-difference scheme of Amein and Fang (1970) which is unconditionally stable for any time step and allows an accurate 

and economical solution for most flow problems.

The solution technique chosen here, called linear, implicit, is a subset of the Amein and Fang technique 

which eliminates the need for iteration when advancing 

from time step to time step. In figure 3, which illustrates 

the finite-difference grid, the solid black circles represent points where all variables in equations 1 and 2 are 

known, and the open circles represent points at which 

variables are unknown. The subscript J designates the 

time grid point, and the subscript i designates the space 

grid point. Time and space derivatives are represented 

by the following respective finite-difference approximations:

1 2

3+1 -r • O O

Af

N-I N

O (O Q), O ooo O O • 

S

A U

FIGURE 3.—Computation stencil for the linear, implicit finitedifference solution of the flow equations.

dt

and

+ 1 -

-1-1 (4)

. dx

+ 1 -

(5)

in which A£ = time step, Ax = distance step, and /is the 

variable whose derivative is sought, that is, U, A, or y. 

The approximation of the space derivative at the 

unknown time level (j+l) gives this scheme the name 

"fully forward" implicit. According to Fread (1974), this 

scheme is the most stable of the four-point difference 

techniques. It must, however, be operated with a 

reasonably small grid size to maintain accuracy.

When the difference approximations, equations 4 and 

5, are applied to equations 1 and 2, a system of equations 

of the following form is obtained:

+ +

(6)

and

+ +

(7)

in which B, C, D, and E are coefficients which are functions of Ax, At, U, y, and Manning n at the known time 

level. The friction slope at the new time step was approximate by use of a Taylor series expansion about the 

old time step value. For a given number of grid points, 

N, there are N-2 such equations. Two additional equations are provided by the upstream and downstream 

boundary conditions. For most of the cases reported 

here, the upstream boundary condition was a rating 

curve which related discharge to depth, the relationship 

having been determined by stream gaging techniques. 

This would not normally be good practice under highly 

unsteady conditions. The reasons for its applicability 

here are discussed later. A few runs were made using 

only the measured stage as an upstream boundary condition, and the results were almost identical. The 

downstream boundary condition was specified as y(t), 

where ytf 1 , the depth at the new time step, was approximated by a constant factor times I/N-I- The value of the 

constant factor was determined by experimentation 

with a step-backwater program. This type of boundary 

condition produces some backwater and drawdown ef-
MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

fects but is more stable than a rating curve and more 

realistic than assuming a constant depth. It is 

sometimes referred to as a zero-gradient boundary condition. Measured downstream depth could have been incorporated as the downstream boundary condition but 

was solved at each time step by Von Rosenberg's (1969) 

technique for pentidiagonal matrices.

The lateral inflow term in equation 1, q, was used to 

handle tributary flow and withdrawals along the Chattahoochee. Field observations at low and high flows indicated that while the mean tributary discharge was 

small, several of the tributary valleys have a considerable storage capacity. Figure 4 shows an upstream 

view of Suwanee Creek during a high stage in the Chattahoochee. When the Chattahoochee River was rising, it 

was not unusual to observe reverse flow conditions in 

the tributary channels where they joined the main stem. 

So while the discharge in the tributaries at points well 

away from the Chattahoochee was nearly constant, the 

actual interchange of water between the river and the 

tributaries was quite variable due to constantly changing stage in the main stream. The actual interchange of 

water between the tributaries and the river was simulated by routing the tributary flow through a hypothetical tank, or pond, attached to the main stem. The 

storage in the pond was computed as a function of the 

water-surface elevation in the Chattahoochee, and the 

lateral inflow component to the river, q, was determined 

by the continuity equation

9 = QT ~ 3? (8)

in which QT = tributary flow rate, and A 5=change in 

tributary storage to occur during time step A t. This simple modification produced more realistic recessions on

FIGURE 4.—Suwanee Creek (view upstream) during a high stage in 

the Chattahoochee River.

the hydrographs by releasing additional tributary flow 

during the falling stage of the main stem.

MASS AND HEAT TRANSPORT MODELS

Applying the principle of conservation of thermal 

energy to a one-dimensional open channel, the conservative form of the governing transport equation 

becomes

d (AT) d (UAT) _ d

dt dx dx2 CPP

(9)

in which T= cross-sectional average water temperature, 

Dx - longitudinal dispersion coefficient, 3>T = flux of 

thermal energy from the air to the water, W=top width 

of the channel, Cp = specific heat of water at constant 

pressure, p - density of water, <b E =flux of thermal 

energy from the bed to the water, P= wetted perimeter 

of the channel, and the other symbols are as previously 

defined.

Assuming water to be incompressible and the product 

Dx A to be independent of x, equation 9 can be simplified 

to

dT 

Tt

dT 62T 

~d~x ~~ CpPA CpPA

(10)

which is called the nonconservative form of the 

transport equation. If the velocity field satisfies the continuity of flow equation, the exact solutions of equations 

9 and 10 are identical. When numerical tehcniques are 

used, however, equation 9 will provide a more conservative solution (Roache, 1972).

When the transported substance is dye rather than 

thermal energy,the values of 3> T and $B are zero, and T 

can be replaced by C which represents the concentration 

of dye. Both equations 9 and 10 were solved when 

modeling the movement of dye, but only equation 10 was 

used in the temperature model.

The next to the last term in equations 9 and 10 

represents the rate of change of water temperature due 

to exchange of energy between the atmosphere and 

water. The ratio AJW is the effective water depth, 

sometimes called the hydraulic depth. The rate of exchange of energy between the atmosphere and water 

has been discussed many times, but one of the first and 

most complete analyses of the processes involved has 

been given by Anderson (1954). For the purpose of this 

study, the net exchange was expressed as the sum

(11)
MODEL DEVELOPMENT

in which 3>N = net heat flux caused by incoming radiation 

from the sun and the sky; <S> 6 = heat flux caused by 

longwave radiation emitted by the water; 3> e = heat 

utilized by evaporation; $^ = heat conducted from the 

water as sensible heat; 3>R = heat added by rain falling 

directly on the surface; and 3>9 = heat flux added to the 

river by tributary inflow. The values of <b r <bN- and 3>R 

are positive and the values of 3v 3vand ^are negative 

if the water is gaining thermal energy as a result of the 

respective processes.

The net flux caused by incoming radiation from the 

sun and the sky, $iV -is composed of four components, the 

incoming atmospheric and solar radiation and the 

reflected components of each. The incoming components were measured directly, but the reflected parts 

had to be estimated. The traditional assumption that 3 

percent of the atmospheric radiation is reflected (Anderson, 1954) was made.

The source of all solar radiation is the sun, and if some 

object is between the water surface and the sun, much of 

this radiation will be intercepted. The banks of the 

Chattahoochee River between Buford and Norcross are 

almost completely tree lined (figs. 2 and 5). The incoming solar radiation, however, was measured by use of a 

pyranometer placed in an unobstructed area. The part of 

the measured solar radiation actually absorbed by the 

water is a complex function of the elevation and azimuth 

of the sun as well as the azimuth of the river reach, the 

height and density of the trees, and the width of the 

river.

A two-step procedure was used to estimate the part of 

the measured solar radiation, RS, actually absorbed by 

the water. The first step was to determine the part of 

the available solar radiation which would have been absorbed providing no shading had occurred. Anderson 

(1954) has presented a formula for the computation of 

this shade-free absorption

RSM = 1.0 - 1.18 ELEV-°-77 (12)

FIGURE 5.—Aerial view of Chattahoochee River showing shading 

conditions

in which -RSM=part of the incoming solar radiation 

which would be absorbed under shade-free conditions, 

and ELEV= elevation of the sun in degrees. The sun's 

elevation was computed from

ELEV = sin 8 sin I + cos 8 cos I cos HA (13)

in which 6 = declination of the sun, C = latitude of the 

river (N34.00), and HA=1ciour angle of the sun which 

was computed from

HA = (180 + ALON - TZM) - 15 HR (14)

in which ALON = longitude of the river (84.2°W), 

TZM = meridian of the time zone, and HR =time of day 

in hours. The equation of time was ignored and the 

declination of the sun, obtained from a solar ephemeris, 

was assumed constant during each run.

The second step was to reduce the value of RSM appropriately to account for the shading of the water due 

to trees and other obstructions on the banks. The shaded 

part of the water surface was assumed to absorb solar 

radiation at 20 percent of the measured rate, and the 

clear part of the water surface was assumed to absorb at 

RSM times the measured rate. The value of RS 

therefore, was determined as 0.2 times the part of the 

water surface shaded plus RSM times the part of the 

water surface exposed to the sun. The part of the water 

surface in any subreach to be shaded at any time of day 

was determined from the geometric relation between 

the elevation and azimuth of the sun, the azimuth of the 

river subreach, the effective barrier height, EBH, the 

water-surface width, W, and the bank width, BW. The 

physical relationship between these terms is illustrated 

in figure 6. The river cross section was assumed to be 

symmetric about the centerline, which is a reasonable 

assumption for the study reach. The normal distance 

from the tree tops to the shade point, XN, was determined from the expression

XN =

EBH

(tanELEV) (cos/8) (15)

in which 0 = the acute angle between the azimuth of the 

sun and the azimuth of the river subreach. The azimuth 

of the sun, AZS, was computed from

AZS = arc cos (sin 8 - sin ELEV sin I) -^

(cos ELEV cos I ) (16)

Using the above expressions, the part of the measured 

solar radiation to be absorbed by the water, RS, was
MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

FIGURE 6.—Schematic of river cross section used to determine the 

part of the water surface to be shaded by bank vegetation.

computed once for each subreach and each time step 

during the day.

Analytic expressions must be used to relate all other 

terms in equation 11 to the water temperature and the 

meteorologic variables. The expression used to determine each term will be summarized.

Longwave radiation emitted by the water surface was 

computed using the Stefan-Boltzman law for blackbody 

radiation

= ea-(T + 273.16)4 (17)

in which e=emissivity of water (0.97), a = StefanBoltzman constant for blackbody radiation (5.67E-8 

W/[m2C4]), and 273.16 converts to the Kelvin scale 

when Tis given in degrees Celsius. 

The energy added by rainfall was determined by

- D (18)

in which / = rainfall rate, and Tw = wet-bulb air 

temperature. Equation 18 does not account for the 

water flowing into the river as sheet flow or from small 

ditches during or after a rain. In some cases this nontributary inflow is believed to be a significant, but unaccounted for, item in the energy budget.

It is assumed that the rate of evaporation can be 

estimated by a formula of the Dalton type

E = V(e0 - ea) (19) 

in which E = rate of evaporation in units of length per

time, $ = an empirical coefficient or wind function, 

e0 = saturation vapor pressure of air evaluated at a 

temperature equal to that of the water surface, and 

ea = vapor pressure of the air above the water, which is 

commonly measured at a height equal to the height of 

the measured wind velocity. The wind function was 

estimated from

= 3.01 + 1.13F (20)

in which ^ = wind function that gives the evaporation 

rate in millimeters per day when the vapor pressure 

deficit is expressed in kilopascals, and the windspeed, V, 

is expressed in meters per second. Equation 20 was 

derived from thermal data collected on the San Diego 

Aqueduct in southern California (Jobson, 1977), and, to 

the authors' knowledge, it is the only wind function ever 

derived from an energy balance of an open channel. Empirical wind functions derived from lake or pan data are 

numerous (Ryan and Stolzenbach, 1972; Tennessee 

Valley Authority, 1972; Brutsaert and Yeh, 1970). The 

thermal energy utilized by evaporation was expressed as

(e0 - (21)

in which L = latent heat of vaporization.

Heat exchange by conduction has received relatively 

little attention because its magnitude is usually small in 

comparison to the evaporative heat exchange. Assuming that the eddy diffusivities of heat and mass are identical, which leads directly to the Bowen ratio concept, 

the conduction term can be expressed as

= y (T - ro (22)

in which 7 = psychrometric constant (0.598 based on an 

assumed atmospheric pressure of 98.0 kPa); and Ta = air 

temperature which should be measured at the same 

elevation as the vapor pressure.

The last term in equations 9 and 10 represents the 

thermal flux at the bed of the river. Past attempts to 

model the bed conduction term have estimated the heat 

flux as the product of the thermal conductivity and the 

temperature gradient within the bed. This transient 

temperature gradient was either estimated (Messenger, 

1963) or determined from a few measurements within 

the bed (Brown, 1969; Pluhowski, 1970). Measurements 

of bed temperatures are difficult and seldom available. 

In addition, bed conditions are seldom uniform. Even 

though the bed conduction term has been shown to be 

significant (Brown, 1969, Pluhowski, 1970), at least for 

shallow depths, it is usually ignored because of the above 

difficulties.
MODEL DEVELOPMENT

The earth under a river can be approximated as an infinitely thick conducting medium, the thermal properties of which can be estimated, at least approximately. 

The thermal conductivity of flowing water is much 

greater, because of turbulence, than that of the soil, so 

the surface temperature of the bed can be assumed to 

follow the water temperature very closely.

Mathematical expressions for the temperature 

distribution and heat fluxes within a semi-infinite 

medium which result from an arbitrary temporal variation in surface temperature are relatively simple 

(Carslaw and Jaeger, 1959). Unfortunately, these expressions converge slowly, and their use in a thermal 

model would be expensive. On the other hand, if the 

earth below the river were considered to be a slab, insulated on the bottom and of an arbitrary thickness, Z, 

the equations are still fairly simple but converge much 

faster. If the temporal variations in surface temperature 

are cyclical, the heat fluxes determined by the semiinfinite and finite thickness slab equations become indistinguishable as the slab thickness increases. In 

fact, assuming a diurnal water temperature swing of 

10°C and thermal properties for saturated sand, the 

surface heat fluxes for a slab only 25 cm thick are within 

6 percent of the values for a semi-infinite medium.

The heat exchange between the water and the bed 

was, therefore, estimated by considering the bed to be a 

homogeneous slab, insulated on the lower face and with 

a surface temperature on top equal to that of the overlying water. The heat flux into or out of the bed was then 

determined as a function of the past history of the water 

temperature. Only the thermal diffusivity and heatstorage capacity of the soil needed to be known. A slab 

thickness of 100 cm was assumed to be sufficiently thick 

to give the desired accuracy.

The temperature distribution within a slab, initially at 

constant temperature, for which the surface is subjected 

to a unit increase in temperature at time zero is given by 

(Carslaw and Jaeger, 1959)

cos [2n + 1) >rry b l2Z] (23)

in which A7£ = the temperature rise within the slab, 

K =the thermal diffusivity, Z = thickness of the slab, and 

2/6 = the distance above the insulated bottom of the slab. 

The increase in the heat content of the slab can be 

evaluated at any time by multiplying equation 23 by the 

heat-storage capacity, then integrating over the total 

thickness

H(t) = CVZ

exp [-K(2n + sin [(2n + l)7r/2] (24)

in which H(t) = the increase in heat content of the slab 

between time 0 and t resulting from the unit increase in 

surface temperature at time zero; and Cv =the heatstorage capacity of the slab which is the product of the 

density and specific heat. Of course, this heat must have 

been provided from the overlying water.

Equations 9 and 10 are solved by use of a finitedifference approximation that advances in time by 

discrete steps of duration At. The heat flux to the water 

&H(j) during any time step jAt to (/+ 1)A£ which results 

from a unit increase in temperature at time zero, can be 

computed as

(25)

The AH(j)'s describe the time variation of the response 

of the system to a unit change in water temperature.

Equation 24 is linear with respect to temperature, and 

since water temperature fluctuations can be represented by a series of step changes, the superposition principle is used to determine the heat flux from the bed to the 

water for any temperature history by use of the equation

(26)

k=— s

in which $e(J&t) is the heat flux to the water from the 

bed during the time jAt to (j + l)At; AT(kAt) is the 

change in water temperature which occurred at kAt 

(k<j); AH is given by equation 25; and the water 

temperature is assumed to have been constant for times 

before t = -s&t. Equation 26 is solved for each grid 

point and each time step in the temperature model. The 

river temperature was assumed to have been'constant 

before the model started (A!T = 0 for &<0), and the bed 

conduction term was limited to a 24-hour memory 

(8 = 288-.?).

The heat content of the tributary inflows was modeled 

in equation 10 by treating it as a surface exchange term. 

The equivalent surface exchange was determined from

_

(Tq -

(27)

in which <J»g = heat flux added to the river by the 

tributary inflow; Tq = temperature of the tributary in-
MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

flow; and q = tributary flow rate at the river. The value 

of $q is added to the value of &T for use in the model. 

The value of q represents the actual interchange of 

water between the tributary and the river as computed 

from equation 9. If the flow was from the river to the 

tributary, the value of Tq was set equal to the river 

temperature, otherwise it was set equal to the 

temperature of the water in the tributary storage. The 

temperature of the water in tributary storage was updated each time step by considering the steady tributary 

flow, QT, and storage volume. No surface exchange was 

allowed for the water held in tributary storage.

The dispersion coefficient used when solving equation 

10 was determined from

RHU* = 250 (28)

in which U* = shear velocity, and 250 is the approximate 

average of the indicated ratio for the data summarized 

by Fischer (1973). Roache (1972) gives a method of determining the effective numerical dispersion coefficient 

for a differencing scheme such as that of Stone and 

Brian (1963). Under the combination of grid spacing and 

time steps used for the variable grid model of the Chattahoochee (eq. 9), the effective numerical dispersion 

coefficient often exceeded the value given by equation 

28. Although the model coding allowed for its inclusion, 

the value of Dx was assumed equal to zero when solving 

equation 9.

Equations 9 and 10 are Eulerian equations, meaning 

that they represent a description of the variation of 

temperature with respect to a fixed coordinate system. 

Another description, the Lagrangian, considers the 

variation of the temperature of a given fluid particle or 

fluid lump as the particle moves through the system. In 

the Lagrangian framework, one conceptually follows an 

individual fluid particle while keeping track of the factors which tend to change its temperature. Applying the 

thermal continuity equation to a unit mass of fluid, one 

obtains

d— = D ( —— I ®TW ®sP 

dt ~ * \d^) + AC^p + AC~p

Integrating equation 29 during the traveltime,

(29)

T -T. = T (D^

o * i JQ \ x Qx 2 \

dt (30)

in which T{ = initial temperature of the water particle as 

it enters; T0 = final temperature of the water particle as

it leaves; and T - traveltime of the particle through the 

system. Expanding the right-hand side of equation 30 

yields

- Ti =/„' D* £

W

+ yL(T - dt (31)

Equation 31 cannot be solved before equation 10 

because the value of the temperature as a function of 

distance and time must be known to perform the integration. Equation 31 was solved for each time step in 

the model, however, because it is of great value in the 

process of analyzing the results of the model. Its main 

value lies in evaluating the contribution of each physical 

process to the total temperature change of a particle of 

water passing through the system.

Solution of the one-dimensional transport equation is 

a much less formidable task than solving the flow equations. Most numerical efforts have dealt with ways to 

minimize numerical dispersion of steep concentration 

fronts. To the writers' knowledge, numerical dispersion 

caused by sudden flow changes, such as those that occur 

on the Chattahoochee, have not previously been 

analyzed, however.

In order to find a stable, accurate solution technique, 

which would minimize numerical dispersion with large 

distance steps and widely varying velocities, preliminary 

experiments were conducted with three types of solutions. An explicit scheme was found to be highly dispersive and its stability is strictly limited to values of Ai and 

Ax which satisfy

Ukt < 1 (32)

The quantity on the left is called the transport Courant 

number. A finite-element technique with linear basis 

functions was derived which is more stable and contains 

less numerical dispersion than the explicit technique, 

but it still requires the transport Courant number to be 

less than two. Later research revealed that this 

centered, implicit 6-point scheme is very similar to the 

method presented by Price, Cavendish, and Varga 

(1968).

The technique finally selected for use in this study was 

a slight variation of the implicit scheme of Stone and 

Brian (1963). This centered 6-point scheme considers all
MODEL DEVELOPMENT

points in the time derivative as shown in figure 7. The 

weighting coefficients on the points used to estimate the 

time derivative, 0, are cyclic functions of time, which according to Stone and Brian reduce the numerical dispersion of propagating high frequency harmonics more effectively. With the time derivative weighting factor, 0, 

greater than 0.5, the method is unconditionally stable; 

however, it is still desirable to keep the transport 

Courant number less than 2 when considering high frequency transients.

The weighting coefficients on the known and unknown 

concentration values for the space derivatives derived 

by Stone and Brian (1963) are identical to those for a 

linear basis function finite-element technique. The 

authors modified the values of the centering coefficients, 0 and £, from 0.5 used by Stone and Brian, to 0.6. 

This modification damps the ringing and overshoot of 

the Stone and Brian scheme while maintaining the 

numerical dispersion advantage of the finite-element 

scheme.

Application of the finite-difference forms of equations 

9 or 10 to the stencil shown in figure 7 results in two less 

equations than there are unknowns at the new time 

step. The two additional equations are provided by the 

upstream and downstream boundary conditions. The 

upstream condition was simply a known concentration. 

A zero gradient downstream boundary condition was 

assumed by computing the new concentration at the 

downstream boundary from an explicit upwind differencing scheme without diffusion. The solution for the remaining concentrations involved the inversion of a 

tridiagonal matrix for each time step.

At the beginning of the project, a solution code for 

equations 10 and 31, which had been developed for use 

on the San Diego Aqueduct (Jobson, 1976), was availj+1

i- 1

FIGURE 7.—Computation stencil for the finite-difference solution of 

the transport equation.

able. This model contained three simplifications from 

the modified Stone and Brian technique however. The 

coefficient, 6, was not time variable, the value of ? was 

0.5, and only equally spaced grid points were allowed. 

because of the highly unsteady flow in the Chattahoochee, there was some doubt regarding the accuracy of 

this simplified solution technique to the nonconservative 

form of the transport equation. To address this concern 

the transport of dye was modeled by equation 9, which 

was solved by the modified Stone and Brian technique, 

and equations 10 and 31, which were solved by the 

simplified technique. It was found that the dye concentrations predicted by the two models were essentially 

equal, so the temperature model was not receded. The 

solution to equation 9 will be referred to as the conservative model since it solves the conservative form of the 

transport equation and the simplified solution to equation 10 will be referred to as the nonconservative model. 

The finite-difference formulation for the nonconservative model (eq 10) was

f—

|_ 6A*

D, , J_ Ax 2

i-1 + x T j 

J

(33)

in which 0 = space derivative weighting factor (0.60), 

T\ = temperature at grid point i and time, j&t, U{ = velocity at grid point i and time j&t, and Hi = the sum of the 

last two terms in equation 10 evaluated at grid point i 

and time jA£.

Little has been written about the numerical simulation 

of the source-sink terms, Hi,in equations 9 or 10 but certain precautions are necessary. In equation 33 the surface exchange term is evaluated at the old time step. 

One consideration is the maximum size of the time step 

which can be used with this procedure without seriously 

compromising accuracy. To illustrate the requirements 

consider a Lagrangian excess temperature model with 

no bed conduction or dispersion. The governing differential equation simplifies to
10 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

dTe 

dt

KW Te (34)

in which Te = the excess water temperature, above ambient, and K = the kinematic surface exchange coefficient. For steady uniform flow with constant meteorological conditions, the coefficient on the right of equation 34 is constant and an exact solution is easily obtained. Evaluating the surface exchange at the old time 

step, the finite-difference approximation becomes

(35)

The dimensionless surface exchange number, NH, as 

given by

KWM (36)

governs the accuracy of the numerical solution. A few 

simple calculations will demonstrate that the numerical 

solution is very accurate for values of AT# <0.2 while the 

numerical solution overshoots and becomes oscillatory 

for values ofNH >l. The surface exchange term, Hj, in 

equation 10 is nearly a linear function of temperature 

for temperature differences encountered during any one 

time step in the model. The surface exchange number, 

is, therefore, a meaningful criteria for limiting time step 

size. As shown by Jobson (1973) the value of the 

kinematic surface exchange coefficient will almost never 

exceed 8 m/d while the minimum hydraulic depth in the 

Chattahoochee is always greater than 0.3 m. By use of 

equation 36, it is easily seen that the numerical scheme 

should accurately model the surface exchange for values 

of Ai less than 10 minutes. A 5-minute time step was 

used throughout for both the flow and the transport 

models.

Consideration should also be given to the distribution 

of the surface exchange term between grid points. With 

a steady uniform condition and no surface exchange except at grid point k, the water temperature must remain 

constant both upstream and downstream of grid k and 

increase by the amount HkAx/Uk between grid point 

k - 1 and k. Simplifying equation 33 for steady conditions by observing that the old and new temperatures 

must be the same, one obtains

+

___X I fT<

Aic J i

-^ T, (37)

in which AHi represents the distributed part of the surface exchange to be applied at grid point i. Under the 

assumed conditions, it is easily seen that AHi-0 for 

k-2>i>k + l. that is, in order to get realistic results 

from equation 33, a point source of heat must be 

distributed between two grid points. Applying equation 

37 between grid points k + 2 and k and simplifying, the 

amount to be applied at grid k +1 is obtained

Uk-l (38)

applying the same equation between grid points fc + 1 

and k + 1, the amount to be applied at grid point k is 

obtained

U.k-l D\ Hk -

UL = AHL (39)

Summing equations 38 and 39, it is seen that the sum of 

the distributed surface exchanges is equal to the total 

point source. If a point source is not distributed as indicated in equations 38 and 39, the numerical solution 

will contain errors upstream of the source similar to 

what is sometimes referred to as ringing.

In order to preserve thermal continuity under 

unsteady, nonuniform conditions, the physical surface 

exchange terms were distributed between two grid 

points using

AH, = (40)

This procedure was found to work well in both the conservative and nonconservative models.

COUPLING

The flow and transport models were run independently. At each time step, five items of information for each 

of the 48 grid points in the flow model were stored on 

magnetic disk for use by the transport model. These 

items include the top width, velocity, cross-sectional 

area, tributary inflow (at the river), and the tributary 

flow into the storage volume (beside the river). This arrangement saved a significant amount of computer cost 

because the transport models were not run until the flow 

model was calibrated. After final calibration, the flow 

model was not rerun. Likewise, the transport model 

could be run as many times as necessary without rerunning the flow model.
MODEL CALIBRATION AND VERIFICATION 11

In using the conservative transport model (eq 9), the 

coupling was direct, since this solution allowed for unequal distance steps, and the grid spacings in the flow and 

the transport models were identical.

The nonconservative transport model (eq 10), which 

also contained the solution code for equation 31, required equal grid spacing. In order to make the output of 

the flow model compatible with the solution to equation 

10, the flow data were interpolated to an equal grid 

spacing by use of a processor program. The logic of the 

processor program assumed that the velocity and crosssectional area of the "true" river varied linearly with 

distance between the flow model grid points. The crosssectional area at grid i in the equally spaced model was 

determined by integrating the "true" area from the 

point Xi- Ax/2 to Xi + Ax/2 and dividing by Ax. This procedure assured that the total instantaneous volume of 

water within a subreach was the same for both models. 

The velocity at any grid in the equally spaced model was 

assumed to be equal to the value at the point in the 

"true" river.

Top widths for the temperature model were determined from the cross-sectional areas provided by the 

flow model. For each available cross section, the 

measured relation between width and area was fitted 

with a third degree polynomial of the form

W = ar a2A2 (41)

in which a0, 0-1,0-2, and o3 are fitted coefficients and A is 

the given area.

MODEL CALIBRATION AND VERIFICATION 

FLOW MODEL

The most sophisticated mathematical procedures are 

of little value without adequate data to verify them. 

Data needed for a flow model include internal reach 

data, which describe the physical characteristics of the 

river (geometry, channel elevation, and roughness), and 

boundary condition data such as flow or stage at each 

end of the reach and flow in each tributary.

For modeling purposes the internal characteristics of 

the river are discretized at a number of grid points 

which represent the longitudinal variations of channel 

geometry, elevation, and roughness. The required 

number of grid points is a function of the objectives of 

the study as well as the frequency of temporal variations 

in the boundary conditions. In this case it was desirable 

to model extreme flow changes (discharge varying by a 

factor of 14) which occur in very short time periods 

(about 10 minutes). Assuming these flow changes are

equivalent to a periodic function with a 20-minute 

period, a 10-minute sampling period should be statistically adequate (Bendat and Piersol, 1966). This provides 

two samples per cycle of the highest frequency change. 

A 5-minute time step was used thorughout this study.

The magnitude of the time step generally dictates the 

spacing of the internal grid points. Stability and accuracy are related to the Courant number

(U +

(42)

Explicit models and the method of characteristics 

become unstable at C#>1. Even in highly unsteady flow 

as in the Chattahoochee, implicit models operate 

satisfactorily for values of CN as large as 15, but accuracy decreases as the value of CN departs from unity. 

The average velocity in the Chattahoochee varied from 

0.33 m/s at steady low flow to 0.79 m/s at high flow, 

while the average depth varied from about 1.1 to 2.2 m. 

The distance step required for maximum accuracy 

therefore varied from about 1.1 km at low flow to about 

1.6 km at high flow. The actual spacing of the crosssectional data depended somewhat on field conditions, 

but the average spacing was 0.7 km (table 1).

All hydraulic data were obtained by project personnel 

of the Chattahoochee River Quality Assessment Project, 

and a complete description of the data and methods used 

in its acquisition is in progress (R. E. Faye, U.S. 

Geological Survey, written commun., 1978). Briefly, the 

cross-sectional data were obtained at high flow by use of 

a sonic depth sounder and a boat. Absolute bed elevations were obtained by referencing the water-surface 

elevation, at the time the cross section was taken, to 

references which had previously been set on the bank. 

The field crew also estimated the flow resistance (Manning ri) at the time of the field survey.

The cross-sectional data were processed as follows. 

First the sonic sounder charts were digitized to form 

coordinate pairs which described the shape and elevation of the cross section. A program was then developed 

which produced tabular values of area and top width versus maximum depth and fitted these tabular values with 

an expression of the form

- Tay + (43)

in which ym = maximum water depth in the cross section, 

Ta = bottom width of the channel, and Tb is the shape 

factor which determines the rate of increase of width 

with elevation.

The coefficients for equation 43 are shown in table 1 

along with the location of the cross section and other
12 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

TABLE I.—Internal reach data for the Chattahoochee River between Buford and Norcross

Section 

number

Comments River 

kilometer

Bottom

elevation

(m)

Coefficients in 

equation 43

Ta (m)

Measured 

hydraulic 

depth at 

low flow 

(m)

Manning's 

roughness

T| 0

1 Buford Gage................................... . 560.20 277.53

2 560.00 277.22

3 Interpolated Section........................... 559.76 276.79

4 Interpolated Section....................... 559.46 276.19

5 559.12 275.47

6 558.56 274.67

7 557.67 274.18

8 556.93 273.85

9 Highway 20........................................ 556.51 273.55

10 Interpolated Section........................... 556.30 273.52

11 James Creek.................................. 556.06 273.00

12 555.26 274.06

13 555.01 274.03

14 554.72 274.00

15 553.87 273.87

16 Settles Bridge................................ 552.97 274.48

17 552.34 274.20

18 551.26 273.98

19 Interpolated Section....................... 550.80 273.55

20 Level Creek................................... 550.41 272.98

21 Interpolated Section........................... 549.86 272.64

22 Dick Creek ........................................ 549.38 272.27

23 548.56 271.72

24 547.47 271.53

25 Littles Ferry...................................... 546.95 270.61

26 546.50 270.74

27 Interpolated Section....................... 546.02 271.39

28 545.54 271.34

29 544.97 271.10

30 Interpolated Section........................ 544.60 271.08

31 Suwanee Creek.................................. 544.26 271.05

32 Gwinnett County Intake.................... 544.04 271.03

33 Interpolated Section....................... 543.81 271.09

34 543.59 271.12

35 543.09 270.79

36 542.24 270.69

37 541.24 269.82

38 540.16 269.41

39 Highway 120.................................. 539.55 268.59

40 538.78 268.53

41 537.92 268.04

42 536.96 267.68

43 536.12 267.46

44 535.61 267.31

45 535.01 267.10

46 534.14 267.10

47 533.35 267.46

48 Highway 141...................................... 532.32 266.98

____Average .-.—-—................................. Length of reach is 27.88 km

interpolated value.

76.2

62.2

54.9

54.9

49.4

36.6

43.9

39.0

64.0

48.8

36.6

46.3

36.6

34.1

41.5

63.4

48.8

41.5

41.5

42.7

46.3

51.2

42.7

43.9

39.6

48.8

47.2

46.0

51.8

48.8

47.2

47.2

50.3

57.9

51.8

45.1

40.2

40.2

51.8

42.7

36.9

56.1

51.8

41.8

53.3

54.9

54.9

64.0

0

0.67

0

0

1.35

.45

.39

.81

.44

.46

.63

.99

.66

.88

.49

.88

.68

3.71

.66

.58

.56

.64

.21

.23

1.02

.97

.98

1.75

.51

.52

.58

.58

.49

.46

.58

.0

.80

1.15

.51

.23

1.01

.0

.0

.93

.59

.33

.73

.91

0.82 

.52

.58 

J .76 

1.05 

1.36 

1.65

u.es

2.19

1.11

1.12

1.14

I.24 

.57 

.74 

.84

I I.04

1.22 1 12Z

1.23

1.34

.80

1.63

1.49 1 .82

.77

.91

.70

.96

.84

1.12

.95

1.01

.87

.94

1.07

1.19

1.29

1.46

1.40

.91

1.25

1.05

0.060

.086

.054

.056

.051

.038

.026

.025

.030

.041

.030

.021

.021

.018

.018

.030

.030

.024

.046

.080

.092

.110

.088

.058

.026

.016

.019

.023

.023

.026

.026

.029

.027

.026

.035

.041

.051

.061

.050

.033

.040

.040

.040

.041

.046

.045

.046

.051

0.042

0.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0

-.003

-.003

-.003 

.0 

.0 

.0 

.0 

.0 

.0

-.007

-.007

-.015

-.012 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0 

.0

pertinent information. The bottom elevation, in table 1, 

is the bed elevation of a cross section with a shape 

defined by equation 43 which was judged to best represent actual measured cross section. In general this was 

the mean elevation of the channel bottom at low flow.

In addition to the above internal reach data, which are 

more or less the standard field data collected for flow 

modeling purposes, the hydraulic depth at each cross 

section was measured on July 17, 1976, under conditions 

of steady low flow. These depths, shown in table 1, were 

obtained by averaging 3- to 10-point measurements obtained at uniform spacing across the river.

Boundary condition information consisted of a continuous record of stage at the upstream end of the reach 

as well as discharge at the four tributaries and at one 

withdrawal point. A stage discharge rating curve was 

also available at the upstream end of the reach. This 

stage-discharge relation was used along with the 

recorded stage to drive the model. Normally this would 

be poor practice but was justified here. Unique rating 

curves apply only to steady flow under constant influence of downstream backwater, if present. Under 

unsteady flow conditions, a different value of stage will 

be obtained for the same discharge depending on
MODEL CALIBRATION AND VERIFICATION 13

whether the discharge is increasing or decreasing and 

how fast. The Chattahoochee River below Buford is 

hydraulically unique. First, the flow is totally governed 

by releases from the Buford powerplant. These releases 

occur in fixed increments. Long periods (3 to 5 hours) of 

steady flow separated by rapid changes (10 to 15 

minutes) are the rule, the reach of river from the dam to 

the Highway 20 bridge is very steep, with supercritical 

flow at several locations. Virtually no upstream reflection of waves is possible and backwater effects are 

nonexistent. Thus, when a flow change is made at 

Buford Dam, the flow at the gage located less than 0.4 

km downstream stabilizes rapidly. During periods of 

unsteady flow, when the hysteresis loop rating should be 

considered, the time period of the loop is shorter than or 

equal to the time steps or the resolution of the model. 

Thus, no great inaccuracy is involved in using the rating 

curve as a boundary condition. Several runs were made 

using the stage directly as a boundary condition with no 

significant change in results.

Field reconnaissance and topographic maps indicated 

that a rock outcrop about 10 km below the Highway 141 

Bridge controlled the depth in the lower end of the 

reach. The downstream boundary conditions were simulated, therefore, by assuming that the water between 

the bridge and this control was ponded to an average 

depth of about 2 m at low flow. The Manning's roughness coefficient over the control was assumed to be 

0.016. Measured stage could have easily been used as a 

downstream boundary condition, but results were considered good enough without this refinement.

In addition to the boundary data, at least partial 

records of stage and discharge were available at the 

Highway 20 Bridge, Littles Ferry Bridge, Highway 120 

Bridge, and the Highway 141 Bridge.

Discharges on tributary streams were virtually constant for both periods, so a constant flow was assumed. 

The observed values are shown in table 2.

The calibration of the flow model centered on the 

March data and involved a two-step process. The first 

step was to calibrate the model at steady low flow, and 

the second step involved additional calibration 

necessary to match the dynamic response of the system.

TABLE 2.—Discharge values for the tributaries and withdrawal 

points on the Chattahoochie River during the October 20-25, 1975, 

and the March 21-24, 1976 modeling periods

Tributary

James Creek —......—„„„„„.„.

Level Creek ..-.._..............______..

Dick Creek .„„„„................„„.

Suwanee Creek .........„...„„„„.

Gwinnett Co. Intake ...............

October 

flow 

(m3/s)

.--„-.........._ 0.5

................... .3

.3

.................... 1.8

................... -.2

March 

flow 

(m3/s)

1.1

.6

.7

4.0 -.2

The steady low flow calibration was accomplished as 

follows. The measured depth was added to the bottom 

elevation (table 1) and an "observed" water-surface profile was plotted. For steady flow the energy equation 

can be integrated between any two cross sections to give

Uf ij +EW>= U2(i +

EWS P2/3 K

(44)

in which EW{ = water-surface elevation at grid point i, 

and Ui = Manning's roughness coefficient applicable to 

the subreach between grid points i and i + 1. Equation 

44 was easily solved for the unknown roughness coefficient, n^ applicable to each subreach since all other 

terms were known. In a few cases errors in the watersurface elevation were detected. These showed up as 

subreaches where the water appeared to run uphill or 

where the computed n value was unrealistically small or 

large (less than 0.005 or greater than 0.1). When this 

situation occurred for a subreach, the value of n was set 

equal to a realistic value, such as the value estimated in 

the field, and a corrected water-surface elevation for the 

grid point was computed. The bed elevation at the grid 

point was then established as the water-surface elevation minus the measured depth.

The roughness coefficients computed by use of equation 44 are applicable to the subreach between grid 

points, and the dynamic model requires roughness coefficients applicable to a subreach centered on the grid 

point. Some judgment was necessary, therefore, in 

averaging the roughness coefficients computed from 

equation 44 to obtain values, which were used in the 

dynamic model. Using the roughness values, tabulated 

in table 1, the dynamic flow model was run to equilibrium at steady low flow and the surface profile computed. 

This computed profile can be compared to the "observed" 

profile in figure 8. The above procedure assured that the 

flow model gave realistic depths, volumes, and surface 

areas, at least at steady low flow.

The ability to match the depths at all cross sections 

under steady low flow conditions does not guarantee 

that the model will reproduce unsteady flow. In order to 

match the dynamic response of the system, it was found 

necessary to vary some of the roughness coefficients 

with depth. These variations were necessary to make 

the modeled and observed rises in stage during the 

hydropulses agree. The roughness was assumed to vary 

with depth as

(45)

in which n(y) = roughness coefficient at depth, y, 

17 o = steady low-flow roughness (table 1), i? ; = rate of
14 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

Flow = 15.6 m 3 /s at Buford

275

270

Highway 120 Duluth 

Highway 141 Norcross

265

560 550 540 530

RIVER KILOMETER

FIGURE 8.—Steady flow depth profile for the Chattahoochee River between Buford Dam and Norcross.

change of roughness with stage, also shown in table 1, 

and ymo = measured hydraulic depth at low flow. The 

grid points at which the roughness was to be varied with 

depth and an approximate value of T^ was determined 

from a sensitivity analysis using a simple backwater program for different flows. Once the model was calibrated 

at steady low flow, dynamic calibration using the March 

data only required seven of the roughness values be 

varied with depth. The variation with depth was not extreme. At section 18 the roughness decreased by 85 percent as the flow increased to its maximum, but variations in the roughness at other sections were less than 

50 percent.

One set of adjustments was made to the flow model as 

a result of observations of the behavior of the transport 

model. The low-flow traveltime in the upper reach was 

increased slightly by arbitrarily increasing the low-flow 

cross-sectional areas, above the values indicated in table 

1 at section numbers 7 through 15 in the pool above the 

control at section 16, by an average of about 36 percent. 

This adjustment was believed to be justified because of 

the rather poor quality of the low-flow depth information in this reach. The adjustment had very little effect

on the modeled stage or discharge values but improved 

the low-flow timing of the transport model at Littles 

Ferry.

The results of the final calibration are illustrated in 

figure 9 in which the observed and modeled stages at 

Buford Dam and the four bridges are plotted for the 3l/z

-day calibration period. The small rise in the observed 

stage at Highway 141 on March 21 was caused by a light 

rain which occurred before 0600 that day. No tributary 

flow measurements were available during this day, so 

constant values were assumed in the model. Visually, 

the stage predictions are good. The cross correlation of 

observed and modeled stages indicates peak correlation 

coefficients of 0.997, 0.997, 0.999, and 0.988 at lags in 

minutes of +20 for Higway 20, -5 for Littles Ferry,

-30 for Highway 120, and -10 for Highway 141, 

respectively. A positive lag indicates the model lagged 

behind the observed. At zero lag the correlation coefficients were 0.989, 0.996, 0.979, and 0.987, respectively. 

During the March run, field crews attempted to continuously measure discharge at the bridges. A detailed 

description of the manner in which these measurements 

were obtained is in progress (R. E. Faye, written com-
MODEL CALIBRATION AND VERIFICATION 15

280

278

L AT\ 7BT 

) O 00 O O 

Buford

- 276

_f

274

5

LU 

O

m

Z 272

O

270

268

DT,

EXPLANATION

O Observed 

Modeled

_ Highway 141 Bridge

21 22 23 

DAY IN MARCH 1976

24

FIGURE 9.—Calibration of the Chattahoochee River flow model with the March 1976 stage data. The 

symbols AT through IT represent the time of arrival of specific water particles at the respective 

locations.

mun., 1978), but briefly it involved the periodic measurement of velocity and depth at particular transverse stations, plotting the data at each station against time, and 

interpolating the data to a particular time in order to 

estimate the instantaneous discharge. A complete traverse of the river required about 1 hour. Figures 10 

through 13 are presented so that a comparison of the 

modeled and measured discharges can be made. A rating curve was also available at the Highway 141 Bridge, 

so that an "observed" discharge could also be determined by use of the table and the observed stage. The 

agreement of the modeled and measured discharges is 

excellent for flows as large as 110 m3/s. The differences 

are less than 5 percent. At higher flows, the model consistently predicts lower than observed discharges. The 

peak modeled discharge on March 23 at Littles Ferry

and Highway 141 was 20 percent lower than the measured value. Because discharge measurements under 

highly unsteady flow conditions are of questionable accuracy, the 20-percent difference in results was not considered serious. Overall, the model results were considered to be very good. The consistency with which the 

transport model reproduced the dye concentrations 

bears out the accuracy of the flow model.

The model was verified by use of the data collected 

during October 1975. A comparison of the predicted 

stage values are within 0.15 m of the observed values in 

most cases, and timing is accurate to within about 30 

minutes. Cross correlation of the measured and modeled 

stage data indicate correlation coefficients of 0.995 and 

0.988 when measured values were lagged by + 25 and 

+ 40 minutes at Georgia Highways 20 and 141 respec-
16 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

250

200

Q

Z

o 

o

IU 

W 

CC 

IU 

Q.

w 150

cciu

100

50

EXPLANATION

O Observed 

————— Modeled

f

21 22 23 

DAY IN MARCH 1976

24

FIGURE 10.—Comparison of modeled and observed discharge in the 

Chattahoochee River at the Highway 20 Bridge during the March 

1976 calibration period.

tively. Zero lag correlation coefficients were 0.971 and 

0.974, respectively. The model does appear to underestimate the stage at Littles Ferry and Highway 20. The 

match could be improved by decreasing the roughness, 

T?O at sections 16 and 17 to 0.021 from 0.030, setting the 

value of r\ 1 to +0.0033 for sections 25, 29, 35, and 37, 

and setting the value of 77^ to +0.0066 and +0.0056 at 

sections 38 and 39, respectively. The predicted and observed stages obtained with these updated resistance 

coefficients are shown in figure 15 which illustrates that 

the visual effect of these corrections is small. Use of the 

updated coefficients with the March data underpredicted the high stage at Littles Ferry by about 0.5 m, 

otherwise, the results were similar.

Only a few discharge measurements were available for 

the October run. These were made on October 23 and 

24 and are illustrated in figure 16. The measured peak 

discharge at Littles Ferry on October 23 was 12 percent 

higher than the modeled value obtained with the updated roughness coefficients. As with the spring run, 

the agreement between the modeled and observed discharge on the rising limb of the hydrograph is good.

It was assumed that a slight change in roughness oc250

200

100

50

EXPLANATION

O Observed 

____ Modeled

21 22 23

DAY IN MARCH 1976

24

FIGURE 11.—Comparison of modeled and observed discharge in the 

Chattahoochee River at the Littles Ferry Bridge during the March 

1976 calibration period.

curred beween October 1975 and March 1976, so the 

roughness coefficients given in table 1 were used when 

generating a flow field for the spring transport model, 

and the updated coefficients, given above, were used in 

generating the flow field for the fall transport model.

MASS TRANSPORT OF A CONSERVATIVE SUBSTANCE

It is generally recognized that neither equation 9 or 10 

accurately represent the longitudinal mixing (dispersion) of a slug injection until considerable mixing has occurred. In fact, the criteria given by Fischer (1973) suggest that the equations will not accurately represent the 

dispersion of a slug injection in the Chattahoochee 

within the first 7.3 km at high flow or within the first 8.6 

km at low flow. On the other hand, it has been shown 

that dispersion plays an almost insignificant role for the 

case of a steady injection rate (Sayre and Chang, 1968). 

In consideration of the above, an independent verification of the transport model was deemed desirable.

To generate data against which the transport model 

could be verified, rhodamine-WT dye was injected into 

the river just below Buford Dam starting at 1100 hours, 

March 21, 1976. The injection rate was held constant for
MODEL CALIBRATION AND VERIFICATION 17

250

200

150

100

50

EXPLANATION

O Observed 

_____ Modeled

O*

21 22

DAY IN MARCH 1976

23 24

FIGURE 12.—Comparison of modeled and observed discharge in the 

Chattahoochee River at the Highway 120 Bridge during the March 

1976 calibration period.

a 3-day period by use of a small positive-displacement 

pump. Figure 17 contains a photograph of the injection 

site. The dye was pumped from the barrels through a 

tub, hung on a cable, to the center of the channel. The 

dye fell about 3 m before striking the water surface. 

Three hours after injection began, sampling began at 

Littles Ferry Bridge, and 6 hours later it began at the 

Highway 141 Bridge. Dip samples were collected in 25 

mL bottles clipped to an angle iron. Samples were taken 

from the thalweg of the river every 10 minutes, and at 

6-hour intervals samples were collected at 6-m intervals 

across the river. During times of steady flow, the maximum variation in concentration across the width of the 

river was about 5 percent. Unfortunately, during 

unsteady conditions, the traverse results were relatively 

meaningless because of the time required to obtain the 

samples.

Dye concentrations at Littles Ferry and Highway 141 

Bridges were then simulated by both the conservative 

(eq. 9) and nonconservative (eq. 10) transport models. 

The results are presented in figures 18, 19, 20, and 21 

along with the measured concentration values. The 

model results in figures 18 and 19 were obtained by use

250

200

100

50

EXPLANATION

O Observed (current meter)

D Observed (rating table) 

— Modeled

diS>

21 22 23 

DAY IN MARCH 1976

24

FIGURE 13.—Comparison of modeled and observed discharge in the 

Chattahoochee River at the Highway 141 Bridge during the March 

1976 calibration period.

of the conservative model (solution to eq. 9 with unequally spaced grid points), and the results in figures 20 

and 21 were obtained by use of the nonconservative 

model (solution to eq. 10 with equally spaced grid 

points).

The model results presented in figures 18 and 19 are 

considered to be excellent. Cross correlation of the 

measured and modeled concentrations indicated correlation coefficients of 0.992 and 0.952 when the measured 

values were lagged by 15 and 0 minutes, respectively, at 

the Littles Ferry and Highway 141 Bridges. At Littles 

Ferry the zero-lag correlation coefficient was 0.986.

The results presented in figures 20 and 21 are also 

considered to be excellent. Comparison of figures 18 and 

20 shows that the conservative model maintains slightly 

better timing of results at Littles Ferry but that its solution is slightly more dispersive even though it was run 

with the dispersion coefficient set equal to zero. The conservative model also provides a better simulation of the 

anomalous rise in concentration beginning about 1000 

hours on March 22. On the other hand a comparison of 

figures 19 and 21 indicates that the non-conservative 

model gave slightly better timing results at Highway
2B2

280

278

UJ

w

Z 274

<

UJ

O

CO

<

Z 272

O

H

270

268

266

i i r i r 

-CT

EXPLANA

• • • • • 

—————— 

• • • 

Buford

••t

• •

* •

• 1

•*!

.

.

.

.

*•

\

*

.

•,

i*

^

. 

\

. 

.

j

«-^AT BT— ~ 

. «

CT

•". BT~-

: : $ \ ••

•: •

•* •

v • •

^

* (

• .

;

•

.

y\

•7 \

•

*• , .

• * *

• * *

A

— Highway 141 Bridge

•AT

20 21 22 23

DAY IN OCTOBER 1975

24 25 

FIGURE 14.—Verification of the Chattahoochee River flow model using the October 1975 stage data. The symbols represent the time of arrival of spe

ticles at the respective locations.
MODEL CALIBRATION AND VERIFICATION 19

C3 CO

T3 C

•

I B 

6|

Sd3±31AI Nl 'T3A3T V3S NV31AI 3AO8V NOI-LVA3H3
20 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

250

200

1-50 

Q

Z

o

CJ 100

50

I Littles I Ferry ' 

October 23. 1975 

EXPLANATION 

• •••• Observed

Modeled

cj

" 250 

CJ

Z

iu 20°

o

(C

I 150 

CJ

100

50

6 12 18 

TIME, IN HOURS

24

Highway 120 Bridge 

October 23, 1975 October 24, 1975

EXPLANATION

• • • • • Observed

————— Modeled

12 18 0 6 

TIME. IN HOURS

12 18 24

FIGURE 16.—Comparison of the modeled and observed discharges in 

the Chattahoochee River during the October calibration run.

FIGURE 17.—Dye injection site for the March 1976 dye study, 200 m 

downstream of Buford Dam.

141. On the whole, however, the results of both models 

appear excellent, and their differences appear to be 

trivial. Because the nonconservative model appeared to 

satisfactorily represent the transport under the highly

15

12

EXPLANATION 

• • • • Modeled 

Observed

21 22 23 

DAY IN MARCH 1976

24

FIGURE 18.—Modeled and observed dye concentrations at Littles 

Ferry Bridge as computed using the conservative transport model 

with unequal grid spacing.

15

12

(C

O 9

O

OC

CJ

EXPLANATION

• ••••• Observed

————— Modeled

21 22 23 

DAY IN MARCH 1976

24

FIGURE 19.—Modeled and observed dye concentrations at Highway 

141 as computed using the conservative transport model with unequal grid spacing.

unsteady conditions in the Chattahoochee and because 

the existing conservative model did not allow for surface 

exchange or the solution to equation 31, the temperature model was not recoded to solve the conservative 

form of the transport equation.
MODEL CALIBRATION AND VERIFICATION 21

22 23 

DAY IN MARCH 1976

24

FIGURE 20.—Modeled and observed dye concentrations at Littles 

Ferry Bridge as computed using the nonconservative transport 

model with equal grid spacing.

15

12

i i i 

EXPLANATION 

• • • • • Observed 

Modeled

21 22 23 

DAY IN MARCH 1976

24

FIGURE 21.—Modeled and observed dye concentrations at Highway 

141 as computed using the nonconservative transport model with 

equal grid spacing.

THE TEMPERATURE MODEL

Four types of data are needed for use with a temperature model. These are: hydraulic data which describe 

the flow; physical data which define the geometric relation between the river and its surroundings; meteorologic data which help define the heat flux at the airwater interface; and temperature data at the upstream 

boundary and at all tributaries.

Hydraulic data input to the temperature model included flow velocity and area as well as the tributary 

flow (both the steady and variable component) at each 

grid point and time step. These data were passed to the 

temperature model directly from the flow model.

Physical data included the azimuth, effective barrier 

height, bank width, and the relation between top width 

and flow area for each grid point. Top widths were 

determined for each cross section by use of equation 41. 

Spot checks indicated the relations defined by equation 

41 were accurate to within 1.5 percent most of the time 

for within-bank flow, which is all that is of concern here. 

The azimuth of each subreach, the effective barrier 

height, and the coefficients for use in equation 41 are 

shown in table 3. The azimuth values, centered at the 

grid point, were determined by use of a 1:24,000 scale 

topographic map, and the effective barrier heights, 

figure 6, were estimated in the field at the time the 

longitudinal depth profile was measured. The banks of 

the Chattahoochee are very steep and the trees often 

lean out over the river as shown in figure 5. A constant 

bank width of 0.3 m was assumed. In so far as the 

shading computations were concerned, the river top 

width was assumed to have a value representative of 

steady low flow at each grid point.

The thermal properties of the river bed might also be 

classified as physical data. The bed of the Chattahoochee 

River is mostly covered with small sand dunes so the 

thermal diffusivity and heat storage capacity were 

assumed to be 0.77 mm2/s and 0.68 cal/cm°C, respectively. Braslavskii and Vikulina (1963) suggest these 

values are applicable for saturated sand.

Stream temperatures in the Chattahoochee River and 

its tributaries were monitored by personnel of the Chattahoochee River Quality Assessment Project. These 

data and the details of their collection will be reported 

elsewhere (R. E. Faye, written commum., 1978). The 

locations of the data collection points, however, are 

shown in figure 1.

During the October run the temperature recorder at 

the upstream end of the reach malfunctioned, so the 

upstream boundary of the temperature model was set at 

the Highway 20 Bridge, which is 3.7 km downstream of 

the boundary of the flow model. Water temperatures at 

Highway 20 were available on 15-minute intervals, starting at 1300 hours on October 21, 1975, and continuing 

for the duration of the run. Before 1300 hours on the 

21st, the upstream temperatures were estimated. Hourly temperatures were available for all tributaries and at 

the Highway 141 Bridge for the entire Fall run. Inter-
22 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

TABLE 3.— Physical data relevant to the Chattahoochee River between Buford and Norcross

Temperature 

model 

grid

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Coefficients in equation 41*

<»o

61.10

60.29

17.15

35.34

36.28

5.29

12.15

22.23

20.19

40.48

56.86

17.19

40.26

16.29

26.75

36.81

11.39

17.12

35.26

20.45

34.32

40.13

50.82

22.02

10.43

16.54

20.31

14.40

34.30

10.66

25.37

34.72

20.90

18.73

25.77

«i

39.79

-13.68

80.59

62.55

30.05

68.89

85.74

60.64

72.47

45.86

7.68

137.49

41.25

134.83

63.42

52.39

74.16

93.75

61.05

75.95

42.99

41.08

16.91

78.01

190.32

96.45

85.66

45.43

31.69

150.71

71.79

65.76

102.24

112.03

48.90

a 2

-19.459

13.130

-47.530

-48.568

-8.855

-12.038

-36.101

-35.677

-34.403

-14.771

-1.768

-138.739

-26.820

-89.536

-36.898

-35.384

-36.025

-68.102

-43.448

-41.749

-23.634

-27.701

-7.528

-49.816

-281.511

-60.425

-80.764

-22.654

-16.888

-157.014

-61.229

-51.248

-114.297

-77.910

-15.380

«3

2.921

-2.027

8.878

11.658

.948

-5.080

4.886

6.918

5.035

1.754

.323

42.648

5.800

17.956

6.639

7.307

5.532

15.606

9.683

7.144

4.442

6.164

1.880

10.118

145.283

11.483

23.757

5.028

3.272

48.941

16.904

12.714

41.513

16.383

1.729

Azimuth

227.3

214.4

185.7

212.7

194.4

209.9

208.4

180.2

208.7

212.4

192.1

191.3

186.2

198.9

130.8

154.8

113.7

201.5

241.0

217.2

237.3

258.7

254.2

270.6

281.7

246.5

224.3

199.5

195.9

186.3

207.6

234.3

259.9

339.0

359.5

Effective 

barrier 

height 

(m)

12

12

18

12

12

11

18

18

18

18

18

18

18

18

18

18

20

15

18

11

6

12

12

14

18

15

15

12

18

18

18

15

18

18

18

"Top widths are given in meters when the area is given in hundreds of square meters.

mittent temperature data were available at the Littles 

Ferry and Highway 120 Bridges.

Much more complete temperature data were collected 

during the March run. Beginning at 0000 hours on 

March 21 and continuing to 0700 hours on March 24, 

1976, 5-minute data were recorded at the upstream 

boundary as well as at Highway 20, Littles Ferry, 

Highway 120, and Highway 141. Becasue of a recorder 

malfunction, approximately 24 hours of data were missing at Highway 120, however. Five-minute data were 

also recorded on Suwanee Creek from 0000 hours on 

March 21 until 1800 hours on March 23,1976. Only a few 

temperature measurements, obtained at random times, 

were available for the other tributaries. Using available 

temperatures, regression equations were derived for 

each tributary which predicted the instantaneous 

tributary temperature from the Suwanee Creek temperature. After 1800 hours on March 23, all tributary 

temperatures, including the Suwanee Creek temperature, were estimated from the air temperature using a 

regression expression which had been derived from the 

random data. Before 1800 hours on March 23, the predicted temperatures in Level, James, and Dick Creeks 

were probably accurate to within +2°C and after this 

time about +3°C.

All temperature data were first placed on cards and 

then plotted to check for keypunching or instrument errors. Once the data were verified in this manner, the 

hourly or 15-minute data were expanded to a 5-minute 

time base by straight-line interpolation. The expanded 

data sets were stored on magnetic disk for use with the 

temperature model.

The initial temperature distribution in the river was 

assumed to vary linearly with distance between points of 

observation.

Meteorologic data needed to drive the temperature 

model include windspeed, incoming solar radiation, incoming atmospheric radiation, air temperature, wetbulb air temperature, and rainfall intensity. All 

meteorologic data were obtained at the R. M. Clayton 

Sewage Treatment Plant in Atlanta, Ga. This site was 

selected primarily because of its security, its nearness to 

the study headquarters, its proximity to the river, and 

because it offered good exposure to the sun and wind.
MODEL CALIBRATION AND VERIFICATION 23

The R. M. Clayton plant is about 55 km southwest of 

Buford Dam and about 35 km southwest of the Highway 

141 Bridge.

A propeller-type anemometer was used to sense the 

windspeed. The starting speed of the propeller was 

about 0.45 m/s with full tracking at about 1.4 m/s. The 

wind direction was also recorded but not used in the 

model. The general exposure of the anemometer is 

shown in figure 22. A closeup of the anemometer with 

the Chattahoochee River in the background in shown in 

figure 23.

The total incoming solar radiation was determined by 

use of an Eppley1 precision spectral pyranometer. The 

instrument is sensitive to radiation with a wavelength 

between 0.3 and 3 /urn. The pyranometer is the instrument on the left in figure 24. The incoming atmospheric 

radiation was determined by use of two Eppley 

pyrgeometers which are sensitive to radiation in the 

range of 4 to 50 /^m. These instruments are shown to the 

right of figure 24.

Two psychrometers were also used to determine the 

wet- and dry-bulb air temperatures. These can be seen 

projecting from the tower in figure 22.

A closer view of the ventilated psychrometer is shown 

in figure 25. The temperatures in this instrument were

FIGURE 22.—View of meteorologic instrumentation tower of the R. 

M. Clayton Sewage Treatment Plant showing the anemometer and 

two psychrometers.

sensed by use of platinum resistance temperature devices, and the wet-bulb probe was covered by a wick

FIGURE 23.—Closeup of the anemometer showing the Chattahoochee 

River in the background.

FIGURE 24.—Closeup of pyranometer and pyrgeometers at the R. M. 

Clayton Sewage Treatment Plant in Atlanta.

FIGURE 25.—Closeup of ventilated psychrometer at the R. M. 

Clayton Sewage Treatment Plant in Atlanta.
24 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

which was continually wetted by distilled water. The 

probes projected across a plastic tube which was 

shielded from radiation by a curved aluminum sheet and 

through which air was drawn at a speed of 4.5 m/s by a 

vane axial fan. A detailed description of this psychrometer has been given by Jobson and Sturrock (1976).

A closer view of the nonventilated psychrometer is 

shown in figure 26. This instrument was originally designed for use during the Lake Hefner studies and has 

been described in detail by Anderson and others (1950). 

This instrument is generally called a Top Hat psychrometer because of the shape of the radiation shield. Temperatures are sensed by copper-constant thermocouples 

which are housed above a distilled water reservoir and 

within a housing that facilitates natural ventilation. 

Rainfall was measured by a tipping bucket rain gage.

An Esterline Angus D2020 recorder was housed in the 

tower and used to record all meteorologic data. At a 

specified time interval, the time as well as the millivolt 

values of the 10 parameters were printed on a paper 

tape. No averaging of the readings was possible, so the 

recorded value represented only an instantaneous 

reading. A sampling of the 10 channels required about 5 

seconds. The recording was at hourly intervals in October and at 5-minute intervals in March.

FIGURE 26.—Closeup of Top Hat psychrometer at the R. M. Clayton 

Sewage Treatment Plant in Atlanta.

No special processing of the windspeed and solar 

radiation data was required for the October period 

because the records were complete. October 20, 22, and 

24 were essentially clear days and the radiation values 

indicated only a little cloudiness around noon on October 

21 and 23. On October 25, it was partly cloudy. Daily 

average values of all meteorologic data are presented in 

table 4. The output of the two pyrgeometers agreed to 

within 3 percent, so their results were averaged for use 

in the model. Previous experience has indicated that the 

pyrgeometers indicate too much diurnal variation in the 

incoming atmospheric radiation, so daily average, 

rather than instantaneous values, were used in the 

model. No rainfall occurred during the October run.

Both psychrometers worked satisfactorily during the 

October run. It was believed, however, that the ventilated psychrometer was a little more accurate, so its 

readings were used throughout. Air temperature contained a typical diurnal swing of about 19°C and a 

gradual warming trend (table 4) occurred during the run.

Data coverage during the March period was not as 

complete. No meteorologic data were available before 

1140 hours on March 21. It was assumed that meteorologic conditions between midnight and 1140 hours on 

March 21 were identical to those which occurred between midnight and 1140 hours on March 23. Although 

rain is known to have occurred during this period, no 

record was available, so it was assumed to have been 

zero. After 1140 hours on the 21st, complete data were 

available for the windspeed, solar radiation, and rainfall 

(which was zero). These values were used directly. The 

days of March 23 and 24, as well as the afternoon of 

March 21, were almost free of clouds. It was quite 

cloudy until about 1500 hours on March 22. The outputs 

of the two pyrgeometers differed by about 19 percent 

during the March run. The outputs of the two sensors 

were averaged, however, to give the values used in the 

model (table 4). The temperatures from the ventilated 

psychrometer were again used in the model except for 

an 8-hour period on March 24. During this period the

TABLE 4.—Daily average values of meterologic variables at the R. M. 

Clayton Sewage Treatment Plant in A tlanta

Date Windspeed Solar 

(m/s) radiation 

(Watts/m2)

Atmospheric Air Vapor

radiation temperature pressure

(Watts/m2) (°C) (kPa)

October 20, 1975

October 21, 1975

October 22, 1975

October 23, 1975

October 24, 1975

October 25, 1975

March 21, 1976

March 22, 1976

March 23, 1976

March 24, 1976

1.22

.84

.44

.51

.92

.43

1.92

1.29

.86

2.64

203

189

192

158

201

122

264

171

261

263

315

323

335

350

353

368

346

336

348

373

10.4

12.6

13.6

15.2

16.6

16.5

11.0

10.7

10.5

13.6

0.78

.86

.98

1.25

1.26

1.52

.64

.61

.67

.95
MODEL CALIBRATION AND VERIFICATION 25

values from the Top Hat psychrometer were used 

because the wick of the ventilated psychrometer was 

partly dry. During the March run the diurnal air 

temperature swing was about 15°C.

Like the temperature data, all meteorologic data were 

placed on punch cards, plotted, and edited. The hourly 

values, for the fall run, were then expanded to a 5- 

minute time base by straight-line interpolation. The 

vapor pressures were then computed and the data sets 

stored on magnetic disk for use by the model.

The temperature model contains several physical constants and empirical coefficients. The usual modeling 

procedure is to select a calibration period during which 

one or several of the empirical coefficients are adjusted 

until a "good" fit occurs. The calibrated model should 

then be run with an independent set of data to verify 

that the coefficients, determined during the calibrating 

period, are "universally" applicable to the particular 

river reach.

In this case all model coefficients were assumed a priori on the basis of assumptions already given. Because 

no coefficients were adjusted to fit the measured temperatures, both the March and October data should be 

considered as verifications of the temperature model.

The measured and predicted water temperatures for 

the October verification are shown in figures 27, 28, 29, 

and 30. The observed temperatures at the Highway 20 

Bridge, shown in figure 27, were used as the upstream 

boundary condition of the temperature model. The 

water temperature at the Highway 20 Bridge for times 

before 1130 hours on October 21 had to be assumed 

because no measured values were available. The assumption of these values of course invalidates any 

verification of the model during these first few hours, 

but it does allow the starting times of the flow and 

temperature models to be the same. Very few measured 

water temperatures were available at Littles Ferry 

Bridge (fig. 28) or Highway 120 (fig. 29). At Littles 

Ferry the data on October 22 consisted of three spot 

measurements which would appear to have been too 

high by 0.5 to 1°C. It is possible that these spot 

measurements were taken too near the bank or that 

poor procedures were used. The accuracy of periodic 

temperature-measuring techniques used by the U.S. 

Geological Survey has been estimated to be ±0.8°C 

(Moore, 1969; Rawson, 1970; and Blodgett, 1971). On 

October 23 the spot field measurements are within 

0.5°C of the computed values, and the general shape of 

the computed and measured curves are alike. Except for 

the two points on October 23, the general shape and 

magnitude of the measured temperature distribution at 

the Highway 120 Bridge (fig. 29) agree closely with the 

predicted values. A complete temperature record was 

available at the Highway 141 Bridge, figure 30. The

§15

13

12

EC 11

3 

H 

<

EC10

UJ

Water temperature at Highway 20

20 21 22 23 24 

DAY IN OCTOBER 1975

25

FIGURE 27.—Observed temperature in the Chattahoochee River at 

the Highway 20 Bridge during the October 1975 verification 

period. The symbols represent the time of arrival of specific water 

particles.

15

5 10

EXPLANATION

O Observed 

Modeled

20 21 22 23 24 

DAY IN OCTOBER 1975

25

FIGURE 28.—Comparison of the observed and modeled temperatures 

at the Littles Ferry Bridge on the Chattahoochee River during the 

October 1975 verification period.

« 15

3 S 14

13

12

11

£ 10

\ i i l I i 

EXPLANATION

20 21 22 23 24 

DAY IN OCTOBER 1975

25

FIGURE 29.—Comparison of the observed and modeled temperatures 

at the Highway 120 Bridge on the Chattahoochee River during the 

October 1975 verification period.
26 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

15

14

13

12

10

I I I I I I n i i

EXPLANATION

Water temperature at Highway 141

20 21 22 23 

DAY IN OCTOBER 1975

24 25

FIGURE 30.—Comparison of the observed and modeled temperatures at the Highway 141 Bridge on the Chattahoochee River 

during the October 1975 verification period. The symbols represent the time of arrival of specific water particles.

agreement between the computed and measured temperatures is very good. Ignoring the first 24 hours of 

record, the RMS (root-mean-square) difference between 

the measured and computed values is 0.32°C, and the 

mean difference is 0.21 °C. The near-perfect fit before 

noon on October 21 is of course the result of judicious 

estimates of the upstream temperature and should be 

disregarded in evaluating the verification of the model.

The measured and predicted temperatures for the 

spring verification are shown in figures 31, 32, 33, 34, 

and 35. It can be seen from figure 31 that the 113 m3/s 

discharge pulse on March 23 increased the temperature 

by almost 2°. The shapes of the modeled and observed 

temperature curves at Highway 20 (fig. 32) are very 

similar; however, the modeled temperatures appear to 

be about 1.5°C higher than the observed values. It is difficult to believe the model could be off this much since 

very little surface exchange can take place during the 

short traveltime between Buford Dam and Highway 20. 

This was particularly true during the high flow pulses 

around noon on March 22 and 23. At Littles Ferry 

Bridge the agreement between the modeled and observed temperatures is very good (fig. 23). The poorest 

agreement occurs between noon and 2100 hours on 

March 21. No meteorologic data were available before 

1100 hours on this day, and a light rain had occurred 

during the night. In order to account for the heat input

of the nontributary inflow caused by the rain, 150 

Watts/m2 was added to the incoming radiation term for 

the first 9 hours of March 21. At Highway 120 (fig. 34), 

the poorest fit occurred between 1500 and midnight on 

March 21, or about 3 hours later than at Littles Ferry. 

The errors at Highway 120 are also believed to be the 

result of the rainfall and lack of meteorologic data during the first part of March 21. Finally at Highway 141 

(fig. 35) the region of poorest fit occurred between 1800 

hours on March 21 and 0300 hours on March 22, again 

about 3 hours later than at Highway 120. It appears that 

the combination of rainfall and lack of data created an 

area of poor fit which shows up at each measurement 

point as the water is convected through the system. The 

RMS difference between the observed and computed 

temperatures at the Highway 141 Bridge was 0.20°C, 

and the mean difference was + 0.09°C. These results are 

considered to be very good. Temperature's occurring 

before 2315 hours on March 21 were not considered in 

these statistics becasue the time required for a particle 

to traverse the system was 23.24 hours on this day.

DISCUSSION OF RESULTS

FLOW DYNAMICS

The results of the flow model may be summarized as 

follows. A linear, implicit finite-difference flow model
DISCUSSION OF RESULTS 27

10

Water temperature at Buford

I i I i ' i i i i

21 22 

DAY IN MARCH 1976

23 24

FIGURE 31.—Observed temperature in the Chattahoochee River below Buford Dam during the March 1976 verification period. The symbols

represent the time of arrival of specific water particles.

21

DAY IN MARCH 1976

FIGURE 32.—Comparison of the observed and modeled temperatures at the Highway 20 Bridge on the Chattahoochee River during the March

1976 verification period.

was used to simulate the depth, velocity, and discharge 

at 48 points in a 27.9-km reach of the Chattahoochee 

River under highly unsteady flow conditions. The reach 

is fairly uniform in cross section but varies considerably 

in local slope (fig. 8). Simulated results can be compared

at all points for a single, steady low-flow condition and 

at four points during unsteady conditons. Generally the 

results of the flow model are considered to be very good. 

Visually the comparison of the measured and modeled 

stages (figs. 9 and 15) are excellent. In all cases the error
28 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

14

EXPLANATION

• • • • • Observed

————— Modeled

12

10

21 22 

DAY IN MARCH 1976

23 24

FIGURE 33.—Comparison of the observed and modeled temperatures at the Littles Ferry Bridge on the Chattahoochee March 1976

verification period.

in the modeled stage for the March calibration run was 

less than 0.16 m, while for flows of less than 100 m3/s the 

error remained less than 0.1 m.

The comparison of the modeled and observed discharge values (figs. 10-13), while not as good as the 

stage comparison, was considered good. Two types of 

differences between the observed and modeled discharges were apparent. These were differences in phasing and in peak discharge values. Differences in the peak 

discharge for the small pulse of March 22 were less than 

5 percent, an excellent agreement, but differences as 

large as 20 percent occurred for the peak discharge of 

the large pulse of March 23. The percentage error 

generally seemed to increase in the downstream direction. Although the accuracy of the observed peak 

discharges, obtained under such highly unsteady conditions, may not be very good, they are consistently higher 

than the peak discharge computed by the model. Thus, it 

is apparent that the model underestimated the peak 

discharge.

The second type of difference appears in the timing, or

phasing, of the discharge curves. The phasing difference 

appears to increase systematically in the downstream 

direction, and the worst case appears to occur at the 

Highway 141 Bridge (fig. 13). Here the modeled and 

observed discharges agree very well on the rising limb of 

the hydrograph. The discharge computed by use of the 

rating table appears to be inadequate on the rising limb 

of the hydrograph. During a rising stage the hydraulic 

gradient in a river will be larger than under the steady 

state conditions upon which a rating curve is based, 

thus, the velocity and discharge will be larger than the 

rating table value. On the recession the reverse effect 

should occur although perhaps to a lesser extent. The 

observed and rating table values agree very closely on 

the recessions of figure 13, but the modeled values are 

much lower. The general phenomenon discussed here is 

commonly known as a looped rating curve. The reasons 

this was not significant at the upstream end of the study 

reach have been discussed earlier under boundary conditions. The discharge results for the March run are plotted in figures 36 thorugh 39 for the bridges at Highways
DISCUSSION OF RESULTS 29

14 i I I i i r n i i r i i -r -|- 

EXPLANATION

Observed 

Modeled

12

10

21 22

DAY IN MARCH 1976

23 24

FIGURE 34.—Comparison of the observed and modeled temperatues at the Highway 120 Bridge on the Chattahoochee River during the

March 1976 verification period.

20, Littles Ferry, 120, and 141, respectively, in a form 

which demonstrates the looped ratings. Except at Highway 120, the model consistently predicts a more pronounced loop than was observed. Both the model and the 

observations indicate that the stage-discharge loop is 

affected by the magnitude of the pulse. The results 

shown in figures 36 through 39 probably present the 

model results in the worst possible light because for the 

modeled and observed curves to match, the model must 

simultaneously reproduce the correct stage, discharge, 

and rate of change of discharge with stage. The results 

at Littles Ferry and Highway 120 Bridges are considered very good. The model does appear to have some 

systematic bias at Highway 141 which may be caused by 

some bias in the assumed channel characteristics 

downstream of the bridge, that is, the self-setting 

downstream boundary condition. The channel characteristics in the vicinity of the Highway 20 Bridge were 

believed to be less accurate than the values for other 

parts of the reach. 

The major contribution of this study was to apply existing solution techniques, with certain minor improvements, to a very comprehensive data set in order 

to verify the existing technology of flow modeling. 

There are two questions that should be asked at this 

point. What can be done to improve the mathematical 

model, and what additions to the data base or calibration 

procedures could be incorporated to improve the 

results?

The first question is difficult to answer. In general, the 

linear-implicit technique is an entirely satisfactory 

method for routing flow. It is not perfect, however, and 

an uninitiated user will have problems. These problems 

will probably be related to boundary or initial conditions 

at first and later to schematization problems.

Contrary to the literature, the linear-implicit scheme 

is not unconditionally stable. Very good initial conditions are required, at least for models configured with 

nonuniform grid spacing and nonprismatic cross sections. For the Chattahoochee, a compatible backwater 

program which computed the initial velocities and 

depths was required. The use of the term compatible is
30 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

14 n i i I I r I T I

Water temperature at Highway 141

12

10

EXPLANATION 

• • • • Observed CTModeled

ET

t • • •/ • • ••

21 22 

DAY IN MARCH 1976

23 24

FIGURE 35.—Comparison of the observed and modeled temperatures at the Highway 141 Bridge on the Chattahoochee River during 

the March 1976 verification period. The symbols represent the time of arrival of specific water particles.

intentional. The friction slope terms in the backwater 

program must be averaged in the same manner as in the 

dynamic model. To reduce programming difficulties in 

the dynamic model, a linear average was used in both 

the dynamic and backwater programs.

Instabilities can also result from boundary conditions. 

An unrealistic rating curve or a self-setting boundary 

condition based on discharge only can produce oscillations or waves at the upstream end. Each stream will 

have its own peculiarities, and experience is the only 

way to discover them.

Schematization problems not related to boundary conditions usually take the form of an anomalous grid point 

depth. For instance, the flow may proceed smoothly 

down a uniform slope through uniform sections and suddenly double in depth or produce some peculiar watersurface irregularity. The problem is not an instability, it 

just doesn't appear realistic. In the Chattahoochee 

model, two such places were encountered. The first occurred at river-kilometer 551.26 (fig. 8), and the other 

was at river-kilometer 541.24. These problems can

usually be alleviated by adding an interpolated section to 

shorten the distance step. Apparently the friction and 

bed slopes are too different, and extrapolating these 

slopes between grids gives unrealistic conditions.

A compatible backwater program is very valuable for 

spotting and correcting problem areas. Experience with 

the Chattahoochee model indicates that any change in 

stage at a given steady discharge, accomplished by 

changing the roughness coefficients or other parameters in the backwater program, will produce an equal 

change in stage under unsteady discharge in the flow 

model. The backwater calculations can be performed 

quickly on a desk-top calculator, so many trials can be 

evaluated quickly. Efforts which have gone into improving backwater programs, such as expansion loss coefficients, should now be incorporated into unsteady flow 

models.

Another area deserving further study is the method 

used to represent the cross-sectional shape. Aesthetic 

appeal is certainly important in some instances. The 

question can be raised as to whether equation 43 is an
DISCUSSION OF RESULTS 31

279

EXPLANATION

Observed for large pulse 

Observed for small pulse 

Modeled for large pulse 

Modeled for small pulse

CC 278

LU

H

UJ

HI

w

Z 277

276

275

50 100 150 

DISCHARGE, IN CUBIC METERS PER SECOND

200 250

FIGURE 36.—Loop rating curves for the Chattahoochee River at the Highway 20 Bridge during the March 1976 calibration.

adequate representation of the cross-sectional shape, or 

should the actual measured shape be used for each section. Little research has been done on the question, and 

this study did not address the problem. It is the writers' 

opinion that computed results are not very sensitive to 

the actual cross-sectional shape as long as the correct 

areas are given as a function of stage.

In summary, three areas should be investigated in 

order to improve the mathematical model. First, more 

needs to be known about the best way to handle boundary and initial conditions. Second, much of the existing 

knowledge used in sophisticated backwater programs 

should be incorporated into dynamic models. Finally, the 

question of how to best represent the cross-sectional 

shape should be studied.

The collection of necessary and sufficient field data as 

well as the intelligent use of these data appears to be 

more critical in the development of an accurate flow 

model than the selection of the particular solution

technique. It was found that adequate cross-sectional 

information could be obtained quickly at high flow by 

use of a sonic depth-sounder and a boat. Top widths at 

the time of the traverse were obtained by stadia. Vertical control, between the bridges, while desirable, was 

not critical provided a depth profile at one steady flow 

conditions was available. The depth profile obtained at 

steady low flow was perhaps the most useful set of data 

available in calibrating the flow model. Furthermore 

these data were fairly easy to obtain. It would have been 

helpful to check the top widths when this profile was 

taken, however, so that cross-sectional areas could have 

been checked more closely.

Optimization is currently popular in the model field. 

Optimization in this case was obtained on a trial and 

error basis wherein the modeler selected each new trial 

on the basis of the results of his previous trials and 

engineering judgment. It would appear that a formalized optimization procedure which incorporates all
32 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

276 —————————I———————— 

EXPLANATION

Observed for large pulse 

Observed for small pulse

—— Modeled for large pulse

•—— Modeled for small pulse

275

Z 274

Z 

O

< 

>

"J 273

272

50 100 150 

DISCHARGE, IN CUBIC METERS PER SECOND

200 250

FIGURE 37.—Loop rating curves for the Chattahoochee River at the Littles Ferry Bridge during the March 1976 calibration.

factors that were considered in calibrating the Chattahoochee flow model would be extremely difficult to 

design. Furthermore, if all these factors are not considered, there exists a very real possibility that the 

resulting model would be far from realistic.

TRANSPORT

Many dye studies have been conducted in rivers of the 

United States. Virtually all of these, however, were performed as nearly as possible under conditions of steady 

flow and using an instantaneous slug injection. From 

these studies the steady flow traveltime and dispersion 

coefficients are inferred. Using the continuous injection 

procedure, the traveltime and dispersion characteristics 

can also be inferred from the timing and shape of the 

first rise in concentration under steady flow conditions. 

In addition, the response of the system to unsteady flow 

can also be determined. The continuous injection focuses 

the concentration changes in the regions of most interest, that is, in the regions where the flow is unsteady. 

The results in figures 18 and 19 or 20 and 21 indicate 

that the transport response of the system is being 

reproduced very well by either of the transport models.

Any differences in the modeled results in figures 18 

and 20 or figures 19 and 21 are due to differences in the 

solution routines for the conservative and nonconservative forms of the transport equation. Although several 

significant differences existed in the solution schemes, 

there appears to be little difference in the results. As 

with the flow model, it appears that the key to successful 

modeling results is good data and calibration techniques 

and not the use of highly sophisticated solution schemes.

The anomalous rise in concentration on the leading 

edge of the power wave, shown in figures 18 and 20, is 

probably the most interesting result to be obtained from 

the dye study. The plateau concentration of 11 //g/L at 

Littles Ferry, which occurred between about 0600 and 

1000 hours on March 22, is the value which would be
DISCUSSION OF RESULTS 33

273

272

271

270

EXPLANATION

Observed for large pulse 

Observed for smaH pulse 

Modeled for large pulse 

Modeled for small pulse

269

50 100 150 200

DISCHARGE, IN CUBIC METERS PER SECOND 

FIGURE 38.—Loop rating curves for the Chattahoochee River at the Highway 120 Bridge during the March 1976 calibration.

computed by dividing the flow rate of dye by the river 

discharge at Littles Ferry. The observed peak concentration which occurred about 1100 hours, on the other 

hand, is larger than the value which would be predicted 

by dividing the dye flow rate by the low flow discharge 

from Buford Dam. The cause of this anomaly served as 

the subject of many discussions, and full agreement has 

not been reached. A plausible cause, however, appears 

to be the interaction of tributary inflow with the river 

stage. Stage variations in the Chattahoochee River are 

large and rapid, while considerable storage volume exists in the tributary channels (fig. 4). In fact, backflow 

up the tributaries was observed in the field during times 

of rapidly increasing stage. The reduced dilution which 

occurs as the water wave passes a tributary and reduces 

or stops the tributary inflow would certainly appear to 

be at least a partial explanation of the anomaly. Assuming complete mixing in the cross section, however, this 

phenomenon should not raise the peak above 12.7 jug/L. 

Some of the anomaly may be due to inhibited transverse 

mixing which perhaps occurs on the leading edge of the 

power wave.
34 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

270

269

EXPLANATION

Rating curve 

Observed for large pulse 

Observed for small pulse 

Modeled for large pulse

—— — —— Modeled for smell pulse

268

50 100 150 200

DISCHARGE, IN CUBIC METERS PER SECOND

FIGURE 39.—Loop rating curves for the Chattahoochee River at the Highway 141 Bridge during the March 1976 calibration.
DISCUSSION OF RESULTS 35

It can be noted that no negative anomaly occurs when 

the effect of the flow reduction passes Littles Ferry 

bridge at about 0800 and 2200 hours on March 23. 

However, the effect of the tributaries at high flow would 

be expected to be much less. Furthermore, the water 

stored in the tributaries before the flow reduction 

probably contains about as much dye as the river water.

The flow and transport models were run for one 

hypothetical case in which the tributary inflow was set 

equal to zero, but the discharge at Buford Dam was the 

same as used in figures 18 through 21. An interesting, 

perhaps disturbing, result was that the anomaly, 

although reduced in size, remained.

TEMPERATURE

Both the October and the March runs are considered 

to be verifications of the temperature model because no 

calibration was involved. All physical parameters and 

coefficients were determined by deduction on the basis 

of other studies. In order to assess the sensitivity of the 

results to these determinations, however, the parameters were arbitrarily changed one at a time and the effect on the predicted temperatures determined. In the 

interest of economy and simplicity, the sensitivity 

analyses were restricted to the March data. The sensitivity to bed conduction, bank shading, and the two 

coefficients in the wind function was checked. In addition, the sensitivity of the computed temperatures to the 

measured values of top width and atmospheric radiation 

was determined. These two quantities were considered 

to be of somewhat questionable accuracy.

Rerunning the model assuming no heat transfer at the 

bed (no bed conduction) decreased the mean error in the 

computed temperature by 0.006°C, yet increased the 

RMS error by 7 percent. Visual changes in the predicted 

temperatures were subtle, but generally bed conduction 

acts as a slight damper to the computed temperatures. 

The October data were also rerun with no bed conduction and again the RMS error increased even though 

the mean error decreased. Clearly, inclusion of the bed 

conduction term improved the model results.

The effect of shading by the banks and trees was 

thought to be a weakness in the temperature model. 

Estimates of the effective barrier heights were rather 

crude, so two cases were rerun. The first case eliminated shading entirely, and the second case reduced the 

effective barrier height by 20 percent. Completely 

eliminating the shading caused more damping of temperature swings at times and less at other times. It 

increased the mean error by 0.269°C and the RMS error 

by 86 percent. In general the changes were hard to 

characterize except that the computed temperatures

with shading looked more like the measured values than 

did the temperatures computed without shading. Assuming that there was a tendency to overestimate the 

tree height in the field, the effective barrier heights 

were reduced 20 percent. This change increased the 

mean error by 0.073°C and the RMS error by 11 percent. The visual effect of this change was small.

Wind function (eq. 20) was developed from a thermal 

balance of the San Diego aqueduct which is in the arid 

region of southern California. Because of the sheltering 

due to trees and the humid climate in northern Georgia, 

the applicability of this wind function to the Chattahoochee River could be questioned. The constant term, 3.01, 

and the mass transfer coefficient, 1.13, were varied independently. Reducing the mass transfer coefficient by 

25 percent increased the mean error by 0.053°C and the 

RMS error by 4 percent. Reducing the constant term by 

50 percent increased the mean error by 0.118°C and the 

RMS error by 29 percent. Neither of these changes affected the visual fit significantly.

The cross-sectional area and flow velocity were 

believed to be fairly accurate because of the accuracy in 

timing of the transport model. The top width measurements, however, were subject to some doubt because of 

the dense brush and fallen trees along the banks (see fig. 

2). Without changing the flow fields in terms of velocity 

and area, the top widths were increased by 10 percent 

and the model rerun. This change, which of course 

decreased the hydraulic depth by 10 percent, increased 

the mean error in the computed temperature by 0.055°C 

and the RMS error by 8 percent.

Reducing the atmospheric radiation by 10 percent 

lowered the mean temperature at the Highway 141 

Bridge by 0.24°C and increased the RMS error by 42 

percent.

In summary, adjusting each coefficient or questionable measurement by an amount judged to be roughly 

equal to its maximum probable error always resulted in 

a poorer fit (in terms of the RMS) of the measured 

temperatures. Nevertheless, the fit could probably have 

been improved by adjusting more than one of the parameters simultaneously. For example, the mean error 

could have been forced to zero in any of several ways, by 

reducing the atmospheric radiation by 3 percent, by 

decreasing the top width by 13 percent, by increasing 

the effective barrier height by 20 percent, or by some 

combination of these. Some combination of adjustments 

could undoubtedly have been found which would have 

reduced the RMS error significantly. Furthermore, 

none of the adjustments would be large in comparison 

with the confidence limits of the original estimate or 

measurement. For example, a 3-percent error in the at-
36 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

mospheric radiation term seems to be reasonable when 

it is remembered that the measured atmospheric term 

represents the average output of two pyrgeometers 

which differed by 19 percent in March and were located 

about 30 km from the study reach. No juggling of the 

terms to improve the fit was attempted, however.

Within the limitations discussed above, the temperature model provides a powerful tool for determining the 

effect of hydropulsations on the temperature regime of 

the Chattahoochee. The effect on the temperature at the 

upstream end of the reach will be discussed first, and 

then the cause of some specific anomalies at the downstream end of the reach will be investigated. These 

investigations will also provide some insight into the 

effects of hydropulsations on other river-quality 

parameters.

Because of the simpler flow release pattern in March, 

these data will be discussed first. The data presented in 

figure 31 indicate that the Buford outlet temperature, 

during times of low flow, was constant at about 7°C. 

The small temperature rise, of about 0.25°C, occurring 

between 0300 and 0800 hours on March 21 is probably 

due to heating from inflow of the rainfall which occurred 

during the night. The solar heating, which occurred between 0900 and 1800 hours is obvious. It is perhaps surprising that for low flow near noon almost a 1°C 

temperature rise occurred over the short distance (0.48 

km) between the dam and the recorder.

At 0700 hours on March 22 (point A in fig. 31), the turbines were started to provide the first flow pulse of 110 

m3/s. The river temperature increased by 0.6°C almost 

immediately. At 2200 hours on March 22 the flow was 

quickly reduced to 15.4 m3/s (point B in fig. 31) and held 

constant until the start of the second pulse of 220 m3/s 

which occurred at 0700 hours on March 23 (point C). At 

1200 hours on March 23, point D, the flow was quickly 

returned to 15.4 m3/s and remained constant at this 

value for the remainder of the study. The small amount 

of heating after 1200 hours on March 23 is probably due 

to solar radiation. A gradual decay in temperature occurs at the completion of each flow pulse. Considerable 

water is ponded between the dam and the temperature 

gage, and this gradual decay is probably the result of 

the rather slow flushing of this ponded water at low 

flow.

At high flow the traveltime and surface exchange between the dam and the temperature gage is negligible so 

the immediate temperature rises of 0.6°C and 1.8°C, 

respectively, for the small and large pulses indicate that 

the turbines withdraw mostly hypolimnion water at low 

flow but at least a mixture of hypolimnion and epilimnion water at high flow. The higher the flow the greater 

proportion of epilimnion water withdrawn. If any waterquality parameter has different values in the hypolimnion and epilimnion waters of the lake, the temporal 

variation of the parameter in the river can be easily 

predicted from the data in figure 31. At steady low flow, 

the river parameter will be close to the hypolimnion 

value, but with the starting of the turbines, the value 

will immediately shift toward the epilimnion value. The 

magnitude of the shift will be a function of the 

discharge. After the turbines are shut down, the river 

value will gradually shift back to the hypolimnion value.

During the October verification period, the upstream 

temperature distribution as well as the inflow hydrograph is much more complex. In addition, the first 

temperature measurements were at the Highway 20 

Bridge, 4.2 km downstream of the dam. However, the 

steep gradient in this short reach (fig. 8) precludes much 

surface exchange, except at steady low flow. Consider 

October 22 which was a rather typical day. More 

specifically consider a water particle, which is labeled A 

in figure 27 and which passed the Highway 20 Bridge at 

1000 hours. Since the traveltime for this water (from the 

upstream end of the model to Highway 20) was 2.75 

hours, it had been released from the dam just before 

0715 hours, and steady low flow had occurred during its 

entire passage (fig. 15). The model indicates that this 

particle cooled by only about 0.25°C during the passage, 

so the water released from the dam must have had a 

temperature of about 9.8°C during steady low flow.

The particle labeled B in figures 15 and 27 was 

released from the dam a little before 1240 hours and arrived at Highway 20 at 1520 hours, just after the turbines were started (fig. 15). The entry temperature of 

this particle would have been 9.8°C, therefore, and the 

2.8°C temperature rise between points A and B in figure 

27 resulted from surface exchange.

Between 1445 and 1500 hours on October 22, the 

power release at Buford began. Just before this power 

release started, the particle represented by point C in 

figures 15 and 27 entered the system with a temperature of about 9.8°C. It did not experience as much 

surface exchange as particle B, however, because the 

power pulse, which followed, flushed it through the 

system more rapidly then the particles which preceded 

or followed it. Its traveltime from the upstream end of 

the model was only 0.74 hours as compared to 2.75 hours 

for particles A and B or 1.01 hours for particle D. The 

reduction in temperature between points B and C, 

therefore, is the result of the flushing action of the 

power pulse.

The water particle presented by point D in Figures 15 

and 27 entered the system a little before 1700 hours 

which is about the time of the peak discharge in figure 

15. Assuming that surface exchange was negligible for 

this particle because of the large depth, short traveltime, and the low solar radiation after 1700 hours, its
DISCUSSION OF RESULTS 37

entry temperature would have been 12.8°C. As in March 

the turbines appear to be withdrawing much more epilimnion water during high flow. A sudden temperatue 

increase of 3°C (from 9.8 to 12.8°C) probably occurred 

at the upstream boundary when the turbines were 

started. Likewise, a sudden shift in values of other 

water-quality properties, which differ between the 

warm surface water and the cooler deep waters, probably occurs each time the turbines are started. Gerald 

Wiley, U.S. Army Corps of Engineers (oral commun., 

1977), has observed sudden shifts in the dissolved oxygen content of the discharge waters upon starting or 

stopping of the turbines at Buford.

Finally, the particle of water labeled E left the dam a 

little before 2000 hours, which corresponds to the second peak in discharge in figure 15. This water traversed 

the 3.7-km reach in 1.07 hours. The minimum temperature between points D and E in figure 27 undoubtedly is 

the result of a reduced release temperature during the 

reduced flow occurring at 1900 hours. The gradual flushing of the system after the flow shutdown is apparent in 

the gradual decay in temperature after point E just as in 

the March run.

In light of the above discussion, it is interesting to 

compare the observed temperatures at Highway 20 on 

October 24 to those of October 22. Meteorologic conditions were very similar on these two days so any temperature differences are primarily dependent on differences in flow scheduling. On Friday, October 24, the 

first power pulse began 3 hours earlier than on the 22d, 

figure 15. Moving the pulse ahead to 1200 hours increased the flow in the river before the solar heating had 

time to influence the river temperature. This eliminated 

the local minimum in temperature which occurred about 

1600 hours (point C) on the other days.

In summary, hydropulsation had a significant effect 

on the water temperature below Buford Dam during 

both the March and October verification periods. At low 

flow the water temperature remained constant and low, 

but during the power releases, it almost immediately 

rose by between 0.6°C and 1.8°C in March and by as 

much as 3°C in October. Furthermore, the flushing action, illustrated by the temperature changes between 

points B and D in figure 27, helps accentuate these 

temperature changes and carry them downstream.

At the lower end of the reach, thp effects of hydropulsation are still present although somewhat damped. 

As before, consider the March data first. Before 0700 

hours on March 22 (point AT in fig. 35) a steady low flow 

of 15.4 m3/s existed throughout the reach. The diurnal 

swing occurring before this time should, therefore, be 

fairly typical of steady low flow conditions. A light rain 

occurred during the early morning hours of March 21, 

and although its exact influence is not known, it is

believed to have increased the river temperature by as 

much as 1.2°C at 0900 hours on March 21 and to have increased the peak temperature that day by as much as 

0.6°C. The influence of this warming is not completely 

dissipated until 0700 hours on March 22. Nevertheless, 

water particles released from the dam just after midnight on March 21 with a temperature of 7.0°C gained 

5.8°C during their 23.25-hour traveltime and arrived at 

Highway 141 at 2315 hours with a temperature of 

12.8°C.

The thermal effect of the first power wave is first 

observed at Highway 141 about 1200 hours on March 22, 

just 5 hours after the turbines were started. The 

traveltime of the particles passing the 141 Bridge at 

1200 hours on March 22 (labeled BT in figs. 9, 31, and 

35) was reduced to 23.16 hours meaning that the 

flushing had just begun.

For reference, the times at which particles left Buford 

Dam and arrived at Highway 141 are also indicated in 

figure 9 and figure 31.

Stream temperatures decreased rapidly after 1200 

hours on March 22 due to the flushing action and 

reached minimum values at 1630 hours (labeled CT in 

figs. 9, 31, and 35) on March 22. This water entered the 

reach only 11.5 hours earlier, at 0500 hours on March 

22, 2 hours before the turbines were started (figs. 9 and 

31). Even though this water was in the river during the 

entire heating part of the day (0500-1630 hours), the 

short traveltime and relatively large depth (1.7 m) during transit prevented much warming from surface exchange. The water-temperature increase during transit 

was 1.6°C, and surface exchange accounted for only 

1.0°C of this change. The 0.6°C occurred due to mixing 

with the warm water which was both upstream and 

downstream of the slug. Flushing action decreased the 

water temperature at Highway 141 by 1.9°C in 4.5 

hours.

The water particles that traversed the reach most 

rapidly (9.75 hours) passed the Highway 141 Bridge at 

1730 hours (DT in figs. 9, 31, and 35) on March 22. This 

water entered the system at 0745 hours, 45 minutes 

after the turbines were started, with a temperature of 

7.4°C. The warming from CT to DT in figure 35 is the 

result of the arrival of the warm epilimnion water 

withdrawn by the turbines at high flow. The epilimnion 

water warmed only 1.2°C during transit, because of the 

shorter traveltime and higher initial temperature, compared to a warming of 1.6°C for the hypolimnion water 

represented by CT.

Some warming occurred even at night, indicating the 

natural river temperature is higher than the release 

temperature, even at high flow. Consider the water 

passing Highway 141 at 0600 hours on March 23 (labeled 

ET in figs. 9, 31, and 35). This water entered the system
38 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

at 1845 hours on March 22 with a temperature of 7.3°C 

(fig. 31). Even though it traversed the system during the 

cool part of the night, it warmed by 0.3°C during the 

transit.

The warming of the river between points ET and FT 

in figure 35 at Highway 141 is caused by natural heating 

due to surface exchange and longer transit times due to 

low flow. The water particles reprsented by point FT 

had the longest traveltime of any water to traverse the 

system between the two flow pulses (15.75 hours). This 

water entered the system, during the high flow, on 

March 22 at 2130 hours (figs. 9 and 31) with nearly the 

same temperature as the water represented by ET, but 

it absorbed the morning solar radiation.

The drop in temperature from point FT to GT in figure 

35 is partly due to flushing action, but it is mainly due to 

the reduction in the upstream temperature which occurred during the low flow between power pulses. The 

flushing action on March 23 had little effect on the temperature because the river was already cold, whereas, it 

had been warm on March 22 when the flushing action 

began. The traveltime of the water labeled GT was 14.0 

hours, indicating it entered the system at midnight with 

a temperature of 7.0°C, figure 31.

The water particles with shortest traveltime on March 

23 (labeled HT) rode the rising stage like the particles 

labeled DT on March 22. This water entered the system 

at 0730 hours with a temperature of 8.0°C.

Finally, the water leaving the system at 1800 hours on 

March 23 (IT) traversed the reach at approximately a 

high steady flow of 225 m3/s. This water had a traveltime of 8.9 hours, a depth of 2.75 m, and temperature 

rise of 1.2°C. Most of the temperature rise was due to 

absorbed solar radiation.

The effects of hydropulsation during the October 

verification can be illustrated in the same manner. Consider the temperature variations observed to occur at 

Highway 141 between 1900 hours on October 23 and 

2400 hours on October 24. The flow release schedule and 

meteorologic conditions were similar during most other 

days. As before, the times for which the temperatures 

are discussed are labeled in figures 15, 27, and 30. Consider first the water particles that passed the Highway 

141 Bridge at 1900 hours on October 23. This water, 

labeled AT, passed the Highway 20 Bridge at midnight 

with a relatively high temperature of 11.6°C. It 

traversed the reach during relatively low and receding 

flow, figure 15, and was in the river during all of the 

daylight hours. The water temperature increased by 

about 1.8°C during its transit. Radiant exchange should 

have increased its temperature by 1.72°C, so 

evaporative cooling and other exchange processes nearly balanced one another.

Next consider the minimum water temperature on October 23, which occurs at Highway 141 at about midnight, point BT. This water that left the dam just before 

the turbines were started was from the hypolimnion and 

had a low temperature of about 9.8°C. Furthermore, it 

was flushed through the system very quickly, and it 

entered the stream about sunset so surface exchange 

was also minimal. The model indicates that surface exchange accounted for only 0.1 °C of the 1°C temperature 

rise actually measured. The remaining 0.9°C 

temperature rise was the result of dispersive mixing 

with the warm water ahead and behind it in the river. In 

summary, three factors combined to give the local 

temperature minimum at point BT in figure 30. These 

were flushing, low initial temperature, and small surface 

exchange because of the nighttime passage.

The water passing Highway 141 at 0200 hours on October 24 is labeled CT in the figures. The traveltime of 

this water was only a little longer than that represented 

by BT (8.9 versus 8.0 hours), but it entered the system 

after the turbines were started (fig. 15) and, therefore, 

at a higher temperature (fig. 27). The increase in temperature from BT to CT in figure 30 represents the 

arrival of the warmer epilimnion water being withdrawn 

at high flow. The net surface exchange for this water 

was less than 0.1 °C, and in fact the total temperature 

change of the water as it traversed the reach was less 

thanO.PC.

The minimum water temperature at Highway 141 on 

October 24 occurred at 0715 hours and is labeled DT in 

figures 15, 27, and 30. This water entered the reach at 

2000 hours on March 23 between the high flow pulses 

(fig. 15) with a low temperature of 11.4°C (fig. 27). 

Because of nighttime passage and relatively rapid transit speed, little surface exchange occurred. Most of the 

0.5°C temperature rise of the water was the result of 

dispersive mixing with the warm water ahead of and immediately behind it in the river.

Meteorologic conditions on October 24 were similar to 

those of the other days, however, the flow release was 

started about 3 hours earlier. This change in the flow 

schedule had a significant effect on the temperature of 

the river. Consider the maximum temperature at Highway 141 on October 24, point ET. The water represented by this point was released from Buford Dam about 

2100 hours on October 23 (fig. 15) passing Highway 20 

about midnight (fig. 27). This water started out warm 

(fig. 27), slowly traversed the reach under low-flow conditions (fig 15), and was in the river during the entire 

warm part of the day. As a result it absorbed enough 

radiation to warm it by 2.10°C while evaporative cooling 

lowered its temperature by 0.3°C. It is interesting that 

the temperature minimum which occurs at Highway 20 

between points AT and BT as well as the temperature 

maximum which occurs between DT and ET have been
DISCUSSION OF RESULTS 39

completely dissipated by the time the water reaches the 

Highway 141 Bridge.

By tracing the route of a few individual water particles through the river reach, the effects of hydropulsation on the thermal regime of the river have been 

illustrated. These effects are complicated, however, in 

that they depend on several factors such as the timing of 

the release in relation to the time of day and of earlier 

releases. Rapid temperatures rises of as much as 3°C 

are common at the upper end of the reach. At the lower 

end of the reach, these effects are still significant 

although they are somewhat damped. Rapid temperature decreases caused by flushing are more likely at the 

lower end of the reach. By studying the travel times and 

flow release schedules, it is possible to predict the origin 

of the river water (epilimnetic or hypolimnetic) for any 

particular time. This could be important if water-quality 

parameters vary significantly between hypolimnetic and 

epilimnetic waters.

One way to view the effect of the reservoir on the river 

temperature is to consider an excess temperature, 

either positive or negative, as the difference between 

the observed water temperature and the temperature 

which would have occurred had the reservoir not been 

present. The question can then be asked: How fast does 

this excess temperature decay in the downstream direction?

To illustrate the rate of decay of a temperature perturbation downstream of the reservoir, the thermal 

model was rerun with all conditions identical to those in 

figures 30 and 35 except that the upstream temperature 

was increased by 1°C. The difference between the two 

predicted temperatures is then a direct measure of the 

percentage of the excess temperature (or effect of the 

reservoir) which has been dissipated within the reach. 

Figure 40 is a plot of the excess temperature remaining 

at Highway 141 for the March run. To facilitate analysis 

the letters AT through IT shown in the previous figures 

are also included. Under steady low flow conditions, 

before point BT, at least 65 percent of any thermal effect caused by Buford Dam should be dissipated by the 

time the water reaches the Highway 141 Bridge. At a 

steady flow of about 100 m3/s, points DT to ET, only a 

little more than 10 percent of the perturbation is 

dissipated, and at a steady flow of about 200 m3/s, points 

HT to IT, less than 10 percent of any excess heat would 

be dissipated by the time the water reaches Highway 

141. The percentage of the excess heat to be dissipated 

within the reach is very dependent on the travel time.

The excess temperatures at the Highway 141 Bridge 

during the October run are illustrated in figure 41 and 

again specific points are labeled. In general it appears 

that a smaller percentage of the excess heat is dissipated 

during the October run. At high flow, points BT to CT,

1.0

0.8

UJ

o 

w

UJ 

UJ

cc 

CD 

g 0-6

UJ

cc

D

H

< 0.4

CC

UJ

0.^

UJ

UJ 0.2

O

X

UJ

1 I I I I I I

DT

-BT

21 22 23 

DAY IN MARCH 1976

20

40

60

80

100

FIGURE 40.—Excess temperature at the Highway 141 Bridge during 

the March run due to an excess temperature of 1° at Buford.

UJ

1.0

0.8

cc

3

cc "J 0.6

UJ

S UJ 

UJ "J •-£0.4

8§uj O

o z

X ± 

UJ 0.2

20

40

60

80

Q. <

O w

. w

20 21 22 23 

DAY IN OCTOBER 1975

24

100

FIGURE 41.—Excess temperature at the Highway 141 Bridge during 

the October run due to an excess temperature of 1° at Buford.

only about 5 percent is dissipated, and at low flow, AT, 

only slightly over 30 percent is dissipated. There are a 

number of reasons for the apparent decrease in the rate 

of return to natural temperature for the October run. 

First, the traveltime at low flow in October, point AT, is 

only 19 hours compared to 23.2 hours in March. The 

frequent power releases in October never allowed the 

river to reach steady low flow. The release schedule was 

also different between the two runs. In October the low 

flow water, AT, spent most of its time in the river at 

night when meteorologic conditions are not optimal for
40 MODELING FLOW, MASS, AND HEAT TRANSPORT, CHATTAHOOCHEE RIVER, GEORGIA

rapid decay.

At high flow the March and October results are in better agreement. The minimum traveltimes in October 

and March were 8.8 and 7.7 hours, respectively. In 

March the fast moving water, DT, ET, traversed the 

reach during the daylight hours, while in October the 

fast moving water, points BT, CT, traversed the reach 

at night.

It thus appears that the results of the two runs are 

fairly consistent. At high flow as little as 5 to 10 percent 

of the excess temperature is dissipated before reaching 

Highway 141. Unless low flow is maintained for more 

than 18 hours, seldom would more than 40 percent of the 

excess temperature be dissipated before reaching 

Highway 141.

SUMMARY AND CONCLUSIONS

A coupled flow-temperature model has been developed 

and verified with data collected on a 27.9-km reach of 

the Chattahoochee River between Buford and Norcross, 

Ga. A major contribution of this study has been to apply 

existing solution techniques, with minor improvements, 

to a very comprehensive data set in order to verify the 

existing technology of modeling flow and transport 

under highly unsteady flow conditions. The modeling 

analysis has identified transport phenomena unique to 

unsteady flow.

A linear, implicit finite-difference flow model was 

coupled with implicit finite-difference transport and 

temperature models. Both the conservative and nonconservative forms of the transport equation were 

solved, and the differences in the predicted concentrations of dye were found to be insignificant. The 

temperature model, therefore, was based on the simpler 

nonconservative form of the transport equation.

Two extensive data sets were available for calibration 

and verification of the models under conditions of extremely unsteady flow. Continuous flow, temperature, 

and meteorologic data were available for the periods of 

October 20-26, 1975, and March 21-24, 1976. In addition, rhodamine-WT dye was injected to the flow at the 

upstream end of the reach at a constant rate during the 

March run, and frequent samples were obtained near 

the center and at the downstream end of the reach. The 

dye concentrations were used to verify the transport 

models.

The flow model was calibrated using the depth profile 

obtained at steady low flow and dynamic stage data collected in March 1976. A comparison of the modeled and 

observed stages at four points in the reach indicated 

that timing errors were generally less than 15 minutes, 

and errors in absolute stage were generally less than 

0.15 m. Agreement of the modeled and observed discharges for flows less than 100 m3/s was excellent, the

difference being less than 5 percent. At higher flows the 

model generally predicted a peak discharge which was 

about 20 percent less than the observed value.

The flow model was verified by use of the data collected during October 1975. The modeled values of the 

stage were generally within 0.15 m of the observed 

values and timing errors were generally less than 30 

minutes. The verification results were considered to be 

very good.

Numerical solution of either the conservative or nonconservative form of the transport equation did an excellent job of simulating the observed concentrations of 

dye in the river under highly unsteady flow conditions. 

Timing errors generally appeared to be less than 15 

minutes. The agreement between the observed and 

modeled dye concentrations was considered to be an excellent verification of the transport models.

Both the October and March runs were considered as 

verifications of the temperature model because no 

calibration was involved. Results were considered to be 

very good. Temperature change as large as 5.8°C occurred as the water traversed the reach. The model was 

able to predict the downstream temperature with a RMS 

error of 0.32°C in October and 0.20°C in March.

Hydropulsation has a significant effect on the water 

temperature below Buford Dam. At low flow the release 

temperature is low and constant. At the beginning of a 

power release, the water temperature increases by as 

much as 3°C almost instantaneously because the turbines withdraw warm epilimnetic water at high flow. 

Any water-quality property which differs markedly between the epilimnetic and metilimnetic waters of the 

reservoir is likely to experience similar sudden changes 

in the river. The flushing action of the power release 

often causes a rapid temperature reduction to occur 

prior to the arrival of the warm epilimnetic water. The 

effects of hydropulsation are somewhat damped at the 

lower end of the reach. The thermal effects of hydropulsation are complicated because they are dependent on 

the timing of the releases with respect to both the time 

of day and past releases.

At least during the March run, the temperature of the 

water released from the dam was lower than the natural 

river temperature, even at high flow. Unless the river is 

held at steady low flow for longer than 18 hours, at least 

60 percent of the thermal effect caused by Buford Dam 

will remain at the Georgia Highway 141 Bridge.

By use of both the Lagrangian and Eulerian forms of 

the transport equation, it was usually possible to predict 

the origin of the water in the river, whether from the 

epilimnion or hypolimnion. This ability could be very 

helpful in predicting water quality, if the parameter of 

interest differs significantly between the epilimnetic 

and hypolimnetic waters of the reservoir.
REFERENCES 41

REFERENCES

Amein, M. M., and Fang, C. S., 1970, Implicit flood routing in natural 

channels: American Society of Civil Engineers Proceedings, 

Journal of the Hydraulics Division, v. 96, no. HY12, p. 2481-2500.

Anderson, E. R., Anderson, L. J., and Marciano, J. J., 1950, A review 

of evaporation theory and development of instrumentation, Lake 

Mead water loss investigation: Navy Electronics Laboratory, Interim Report 159, February 1.

Anderson, L. F., 1954, Instrumentation for mass-transfer and energy 

budget studies, in Water-loss investigation: Lake Hefner studies, 

technical report: U.S. Geological Survey Professional Paper 269, 

p. 35-45.

Bendat, Julius S., and Piersol, Allan G., 1966, Measurement and 

analysis of random data: John Wiley and Sons, Inc., New York, 

387 p.

Blodgett, J. C., 1971, Water temperatures of California streams, 

Sacramento Basin Subregion: U.S. Geological Survey open-file 

report, 29 p.

Braslavskii, A. P., and Vikulina, Z. A., 1963, Evaporation norms from 

water reservoirs: Translated from Russian by Israel Program for 

Scientific Translations, Jerusalem, p 85.

Brown, G. W., 1969, Predicting temperatures of small streams: Water 

Resources Research, v. 5, no. 1, p. 68-75.

Brutsaert, W., and Yeh, Gour-Tsyh, 1970, Implication of a type of empirical evaporation formula for lakes and pans: Water Resources 

Research, v. 6, no. 4, p. 1202-1208.

Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in solids [2d 

ed.]: Oxford University Press, New York, p. 68-75.

Fischer, Hugo B., 1973, Longitudinal dispersion and turbulent mixing 

in open channel flow: Annual Review of Fluid Mechanics, 

p. 57-98.

Fread, D. L., 1974, Numerical properties of implicit four-point finite 

difference equations of unsteady flow: U.S. Department of Commerce, National Oceanic and Atmospheric Administration, 

March, Technical Memorandum NWS HYDRO-18, 38 p.

Garrison, J. M., Granju, J. P., and Price, J. T., 1969, Unsteady flow 

simulation in rivers and reservoirs: American Society of Civil 

Engineers Proceedings, Journal of the Hydraulics Division, v. 95, 

no. HY5, p. 1559-1576.

Jobson, Harvey E., 1973, The dissipation of excess heat from water 

systems: American Society of Civil Engineers Proceedings, Journal of the Power Division, v. 99, no. P01, p. 89-103.

___1976, Thermal modeling and its relation to canal evaporation: 

National Conference on Complete Water Reuse Proceedings, 3d, 

American Institute of Chemical Engineers, Cincinnati, 

p. 370-379.

___1977, Thermal model for evaporation from open channels: Congress of the International Association for Hydraulic Research 

Proceedings, 17th, Baden Baden, Germany, v. 2, p. 95-102.

Jobson, Harvey E., and Sturrock, Alex M., Jr., 1979, Comprehensive 

monitoring of meteorology, hydraulics and thermal regime of the 

San Diego aqueduct, California: U.S. Geological Survey Professional Paper 1137,00 p.

Lai, C., 1967, Computation of transient flows in rivers and estuaries by 

the multiple reach method of characteristics, in Geological Survey 

research 1967: U.S. Geological Survey Professional Paper 

575-D, p. D273-D280.

Messenger, Harry, 1963, Dissipation of heat from a thermally loaded 

system, in Geological Survey research 1963: U.S. Geological 

Survey Professional Paper 475-C, p. C175-C178.

Moore, A. M., 1969, Correlation and analysis of water-temperature 

data for Oregon streams: U.S. Geological Survey Water-Supply 

Paper 1819-K, p. 8-14.

Pluhowski, E. J., 1970, Urbanization and its effects on the 

temperature of the streams on Long Island, New York: U.S. 

Geological Survey Professional Paper 627-D, p 110.

Price, H. S., Cavendish, J. C., and Varga, R. S., 1968, Numerical 

methods of higher-order accuracy for diffusion-convection equations: Society of Petroleum Engineers Journal, American Institute of Mechanical Engineers, v. 8, September, p. 298-303.

Rawson, J., 1970, Reconnaissance of water temperature of selected 

streams in southeastern Texas: U.S. Geological Survey in 

cooperation with the Texas Water Development Board, p. 2-4.

Roache, Patrick J., 1972, Computational fluid dynamics: Albuquerque, 

Hermosa Publishers, 434 p.

Ryan, P. J., and Stolzenbach, K. D., 1972, Chapter I of engineering 

aspects of heat disposal from power generation: Environmental 

Heat Transfer, MIT Summer Session, June 26-30, Ralph M. Parsons Laboratory for Water Resources and Hydrodynamics, Cambridge, Mass., 75 p.

Sayre, W. W., and Chang, F. M., 1968, A laboratory investigation of 

open channel dispersion for dissolved suspended, and floating 

dispersants: U.S. Geological Survey Professional Paper 433-E, 

71 p.

Stone, H. L., and Brian, P. L. T., 1963, Numerical solution of convective transport problems: American Institute of Chemical 

Engineers Journal, v. 9, no. 5, p. 681-688.

Tennessee Valley Authority, 1972, Heat and mass transfer between a 

water surface and the atmosphere: Water Resources Research 

Laboratory Report No. 14, Norris, Tenn., April, p. 4.20.

Von Rosenberg, D. W., 1969, Methods for the numerical solution of 

partial differential equations: New York, Elsevier, 128 p.

Wylie, E. Benjamin, 1970, Unsteady free-surface flow computations: 

American Society of Civil Engineers Proceedings, Journal of the 

Hydraulics Division, v. 96, no. HY11, p. 2241-2251.

Yevjevich, V., and Barnes, H. H., 1970, Flood routing through storm 

drains, Part 1, Solution of problems of unsteady free surface flow 

in storm drains: Fort Collins, Colorado State University 

Hydrology Paper No. 43,108 p.

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