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ABSTRACT

PROCESS MODEL FOR SEDIMENTATION
UNDER TUflBULENT FLOW CONDITIONS

BY

LELAND WILBUR YOUNKER

This study involves construction of a
theoretical model of flow/sediment interaction.
Described herein is a mathematical flow/sediment
reaponse model in which deposition occurs analyti-
cally rather than eXperimentally. The model con-
sists of a series of discrete segments which can
be combined into three functional sections:

(1) definition of turbulent state, (2) turbulent
state/sediment interaction, and (3) sedimentary
deposit. Based on current understanding of the
nature of flow and flow/sediment interactions, this
model bears the promise of complementing future

theoretical and empirical studies.

PROCESS MODEL FUR SEDIMENTATION UNDER
TURBULENT FLOW CONDITIONS

By
LELAND WILBUR YOUNKER

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCILKCE

College of Natural Science
Department of Geology

1972

ACKNCWLEDGHLNTS

I extend my appreciation and thanks to the
following peonle: Dr. Robert Ehrlich, Dr. T. A. Vogel,

Dr. B. H. teinberg, and Dr. H.F. Bennett.

ii

ACKNU
LIST
LIST

INTRO

'PA.QAS<JE C<d3hLNTS

{V'LEDGIV’D‘LN T S o o o o o o o o o o 0
OF FIGURLS o o o o o o o o o o 0
0E1 AILJL‘JIJDICbS . o o o o o o o o

DUCTIUI‘. o o o o o o o o o o o o

TURBULLIIT FLOW. o o o o o o o o o o o

GSPEHAL CHARACTERISTICS OF THL MODEL.

TURFULENT STATE . . . . . . . . . . .

Determination of Flow Velocity .
Duration . . . . . . . . . . . .

Shear Stress . . . . . . . . . .

TURBULENCi-SEDIMENT INTERACTION . . .

SLIIILIIIJI

RbeR

APPENDIX I - Flow Chart for Process-Hesconse

Assumptions for the Operation of
theI'."Odelooooooooooo

Size-Frequency Distributions . .

‘cilNTAilY D3130 SIT o o o o o o o 0

Erosion and Deposition . . . .
Accumulation . . . . . . . . . .

*3” O O O O O O O O O O O O O O .

LNChS CITED . . o o o o . o o 0

Model 0 o o o o o o o o

APPLKDIX II - Listing of Computer frogram .

iii

ii
iv

~J \n l4 <

38
AZ

Figure
1.

Figure
2.

LIST OF FIUUHLS

Hydrodynamic reaponse model -
generalized flow diagram . .

Model Velocity Spectrum . .

iv

 

LIST OF APPENDICES

ABPDNDIX I
Flow Chart for Process-Reaponse Model .

APPENDIX II
Listing of Computer Program . . . . . .

INTRODUCTION

It is axiomatic in sedimentology that a
deposited sediment bears the diagnostic imprint of
the depositional environment. A great number of in-
vestigations have studied aspects of sediment such
as size-frequency distributions and minor structure
assemblages in order to deduce depositional environ-
ments. Laboratory studies and observations of recent
sediment indicate that the flow regime is the primary
variable from environment to environment. The complex
motions within the flowing medium produce a sedimen-
tary deposit which in some way mirrors this complexity.
However, it is not known to what degree of fidelity the
deposit mirrors the details of flow.

The fundamental limit for extracting infor-
mation from sediment is a direct function of the faith-
fulness of the sediment in reflecting hydrodynamic
processes. If the transfer of information was perfect,
the fundamental hydrodynamic equations could be com-
pletely deduced from the sediment characteristics. If

no information concerning the fundamental hydrodynamics

equations can be deduced from a sediment, then
study of sediment would be without value for
interpreting depositional environments.

The real situation falls somewhere between
these two extremes. The extent to which the quality
of the record falls short of a perfect imprint of
the hydrodynamic regime marks the absolute limit
of interpretability of the sedimentary record. This
limit plays the same role in sedimentology as does
the uncertainity principle in physics.

The attempt to understand the relationship
between flow regime and sediment has always been one
of the primary goals of sedimentological research.
Much insight into this problem has been provided by

experimental work involving flumes and wave-tanks, as

well as direct observations of natural systems. These

studies as well as most others concerned with evalua-

tion of the flow regime are hampered by the difficulty

of monitoring flow regime without interfering with it.

Therefore, the information such studies have provided
has been mainly qualitative. In general, they have

defined the evolution of a sedimentary accumulation

through time but provide little information concerning

the detailed events that constitute this evolution.

Another approach toward the same end involves
construction of theoretical models of flow-sediment
interaction. Described herein is a mathematical
flow/sediment response model in which deposition
occurs analytically rather than eXperimentally. This
model, based on current understanding of the nature
of flow and flow/sediment interactions, bears the
promise of complementing empirical studies.

Explicit and implicit reasons exist for building
such a process model. EXplicitly, similarity between
the product of the model and natural accumulations
can provide information on the workings of an other-
wise unobservable natural system. Implicitly, the lack
of correSpondence between output of the model and
natural deposits represents a measure of the reason-
ableness of the commonly accepted framework of theore-
tical hydrodynamics upon which the model is based.

In order to be reasonable, such a model must
be intimately tied to turbulent flow. An object of
this study, then, is the develonmert of a mathematical
model of small scale sediment variation based on the
concepts of fluctuating hydrodynamic energy regimes,
and further, to demonstrate that the output compares

favorably with empirical observations. with this

established, a certain degree of validity can be

assumed for the model. At this point, deeper analysis
of the output concerning the quality and completeness
of the sedimentary record is justified. Such analysis
can provide a testable framework for future empiricrl

and theoretical studies.

TURBULENT FLOW

In most depositional environments, the flow
regime is turbulent. A wide variety of mathematical
models of turbulent flow have been develoned in hydro-
dynamics and aerodrnrmics. The vast majority of
these, if not all, were developed for non-sedimentological
objectives. A major problem in construction of the
response model involves modification of these models
to make them sedimentologically relevant without de-
struction of their theoretical validity.

Turbulence is that component of flow which is
characterized by random rotational motions. As a
result, the velocity at a fixed point in Space is not
constant but fluctuates irregularly about a mean value
The fluid elements which perform these fluctuations
are macroscooic ”lumps of fluid" (Hinze, 1050). In
a fullf developed turbulent flow, there is a hierarchy
of such fluid packets varying in size from meters to
microns. Each of these packets has its own intrinsic
motion wdich is superimposed on the mean flow. These
packets retain their identity and move as a unit through

the mean flow for a short period of time before being

assimilated by the fluid around them. The length
of time the fluid packet retains its identity is
a function of the packet size and the difference
between the packet velocity and the mean fluid
velocity.

As a consequence, both Eulerian and LaGrangian
models have been deveIOped for turbulent flow. In
the Eulerian models, the reference frame is fixed
and the primary variables of interest are the mag-
nitude and direction of the velocity vector at a point
In the LaGrangian models, a given fluid particle is
followed through the flow field, therebv introducing
the additional variable of location.

In either case, the precise functional relation-
shin between these variables as well as the magnitude
of various constant terms is not known with great
precision owing to the difficulty of obtaining empiri-
cal data from the flow vithout perturbing it. As will
be seen below, turbulent models are a blend of varying
pronortions of theoretical and empirical components.
The model discussed below is develooed for an Lulerian
framework ”hich cancerns velocity fluctuations at a

point.

GENhHAL CHARACTthSTICS OF THE MODEL

The hydrodynamic response model described
below simulates the behavior of turbulent fluid at
one point in snace through time. The fundamental
aspect of the flow is the velocity-shear stress and
the duration of an? pulse.

The competency of the flow is directly re-
lated to the magnitude of the shear stress. The
extent of erosion or deposition is a function of
shear stress magnitude and duration of the pulse.

The model consists of a series of discrete
segments. These individual components can be com-
bined into three functional sections; (1) definition
of turbulent state, (2) turbulent state/sediment
interaction, and (3) sedimentary denosit. The
turbulent state section contains those com onerts
judged necessary for modeling of the turbulent flow,
and defines the duration 'nd intontiti of the h ri-
zontal consonent of f; id velocity. Discussion of
the components of this section.will include the

criteria used for selection or rejection of alter-

native an reaches to turbulent modeling.

7

The turbulent state/sediment interaction
section relates the turbulent state to various size-
frequerct distributions and monitors the nature and
extent of erosion or deposition from velocity
fluctuation to fluctuation.

The sedimentary deposit section represents
the accumulated result of a large number of turbulent
flow/sediment interactions. The time sequence of
erosional and depositional events as well as grain
size analysis of this record will provide an estimate
of the relative preportion of total turbulent events
recorded in the sedimentary record, and the extent
to which the record can be used to deduce the nature
of the turbulent flow regime.

A generalized flow diagram of the model is

depicted in Figure 1.

 

DEFIVITIQN CF
TTTRBTTT.:.]‘-IT STATE)

 

 

 

 

 

TURBULth STATL/
SLDIMLKT IRTLHACTIUN

 

 

 

 

SE D IMENTA RY DE 1" x) S IT

 

 

 

Figure l. Hydrodynamic response model -
generalized flow diagram

TURBULENT STATL

A wide varietr of models of turbulence have
been formulated. These models in general contain
two common characteristics:

a. A treatment of t rhulence ranging
from near molecular "otions to
the macrOSCOpic.

b. Incorporation of averwges over
time of such variables as flow
direction and velocity, reflecting
the inherent stochastic nature
of turbulence.

fiith respect to the problem in hand, the
presence of these factors requires modifiCntion of
models to estim tn instantaneous values for the
flow velocitv. In addition, those parts of general
turbulence models involving aspects of turWulence

whose dimensions are too great or too small to in-

fluence sedimentation will not be considered.

Determination of Flow Velocity

Flow velocity is a fundamental parameter for
definition of the turbulent state. In accordance

with general theorv, this velocity can be considered

10

11

to be composed of a mean velocity and a turbulent
component. Setting up orthogonal coordinates,

x, y, and 2 at a fixed point in space, where x is
parallel to the direction of the mean current flow,

we can write:

u =: '5 +' u
v .. 'V + v: (l)
W'Il 'E + 'w

where u, v, and w are the velocities measured parallel
to the x, y, and 2 directions (Schlichting, 1960).

The bar denotes the time averaged component of
velocity. For our purposes we will assume that V
and'w can be set equal to zero. The mean velocity

in the x direction 5 can be set at any value of
interest to the investirator. The prime denotes the
turbulent component of velocity. These comionents

are random variables which may have either positive

or negative sign.

EXperimentally it has been shovn that for the
time averaged conditions described above, the hori-
zontal component of turbulence transverse to the mean
flow Jirectian is negligible (Hinze, 1959). There-
fore equations 1, for our purpose reduce to:

u-‘fi-r u' (p)

I

v==v.

12

Additional experimental evidence has shown
that d is in general greater than v; and the mag-
nitudes of the two components are roughly correlated
(Schlichtina, 1950). As a result, deductions con-
cernine the nature of the horizontal comoonent will
provide substantial information concerning the
vertical component. Therefore, as an additional
simplification the model discussed herein uses as
the primary variable of interest, the magnitude of
the horizontal component of the velocity vector.

For any event based model, the determination
of the probability distribution governing the tur-
bulent components of velocit; is of critical impor-
tance. Empirical investigations of aerodynamics and
hydrodynamics indicate the fluctuations are normally
distributed about the mean velocity (Hinze, 1959;
Schlichting, 1060). Numerous studies have indicated
that the standard deviation of the horizontal com-
ponent of velocity is proportional to the mean
(Kalinsxe, 1947; Slark, 1968; Aozyrenko, 1966).

The degree of turbulence as expressed by
this variation about the mean would be eXpected to
vary depending on the nature of the sediment-fluid

interface , and the heieht above the interface at

13

which the measurement was taken. Recent studies

have indicated that, in general, the magnitude of
the fluctuations decrease with increasing distance
from the interface (Ishihara, Yokosi, Ueno, 1950).

Since this study is concerned with the
movement of sediment as bedload, the nature of the
fluctuations near the fluid-sediment boundary are
critically important. In this region, halinske
has experimentally determined that the velocity
fluctuation is such that 0" is about 0.25 where

c- is the standard deviation of u about 11. This

value of the constant of prouortionality may, in fact,
deviate somewhat from the true value, but the exist-
ence of pronortionalit? of mean and standard devia-
tion in such physical phenomena is generally ob-

served. fimpirical data from both laboratory studies
and natural river studies indicate that near the
boundary, the standard deviation of the velocity is

in general, a small percentage of the mean (Clark, 1968;
Ishihara, :okosi, Ueno, 1969).

An instantaneous horizontal velocity is thus
determined in the model by sampling from a normal
distributiou whose mean equals the mean velocit?
and mhoae standard deviation is set to an empirical

COHStnnto

Duration

Each velocity generated by a random model
such as that described above persists fer a certain
duration. In actuality, the velocit; varies con-
tinuously from moment to moment and duration of
velocity in that sense would cover infinitely small
periods of time.

The complex set of velocities and frequencies
that constitute turbulent flow can be regarded as a
superposition of eddies of various sizes and Spatial
orientations (Hinze, 1959). The behavior of velocity
at any point in space over any interval of time is
simply the sum of the individual eddy velocities.

The largest eddies have dimensions on the
order of the scale of the mean flow. The smallest
fluid packets in natural streams have dimensions in
the millimeter range (Yokosi, 1967). It is obvious
that many of these eddies and their associated
velocities cannot appreciably affect sedizentation
in the sand size range.

Although the velocity fluctuations with time
at a point are in reality represented by a complex
continuously varying wave form, in terms of sediment

interaction the replacement of this wave form with

15

one much simpler has little appreciable effect on
the sedimentary deposit. This simpler time-av«raae
Spectrum is associated with eddies of a particular
size range. The relevant eddies must penetrate to
the fluid-sediment interface and possess the com-
binstion of duration and intensity necessary to
affect sedimentation in the sand size ranges.

As will be shown below, the magnitude of the
turbulent shear stress is related to the correlation
between the horizontal and vertical components of
turbulence. Numerous experiments have shown that
the smallest eddies may have considerable energy,
but because they do not contribute to the correlation
between u and v, they are of small significance in
generating the turbulent stresses (Ishihara, Yokosi,
Ueno, 1969). In addition, the larger size eddies
would be eXpected to more likely penetrate to the
sediment interface, and persist longer at a point
once they impinged upon the surface.

From this analysis, it is concluded that
larger size eddies with dimensions on the order of
meters possess those attributes which make them
sedimentologically the most important. Experimental
studies have indicated that the dimensions of these

eddies are in Ceneral controlled by the geometry

16

of the channel (Yokosi, 1967).

In the model under discussion, eddy size is
treated as an independent variable fixed by the
investigator at a single value for a depositional
cycle. The range of allowable eddy sizes is deter-
mined from empirical studies of a variety of environ-
ments. Once the eddy size is fixed, the length of
time the eddy will snend over a point is simply the
quotient of eddy size divided by its velocity. Later
modification of this model Will probably include
eddy size as a r'ndom variable rather than as a con-
stant.

The model velocity Spectrum over time
(Figure 2) resembles a "square wave function" with
instantaneous increases and decreases of velocity.
The physical picturt which emerges from the velocity
Spectrum is one in which relatively large vortex
structures, elongate in the streanwise direction,
move past the sediment interface with individual
convection velocities. These structures have widely
varying size and strength, and the velocity fluctua-
tions observed by Kalinske and others can be attributed
to their passage by a point.

It is to be eXpected there aould be a large

VELOCITY

5|

ul

 

l7

 

u3

 

 

 

T2

 

TIME

 

 

Figure 2. Model Velocity Spectrum

 

18

instantaneous value of turbulent shear stress Which
coincides with the passing of each structure (Clark
and Markland, 1971). It is these stresses which are
vitally important in the entrainment of sediment
(Sutherland, 1964), and the follOVin: section
describes a technique for indirectly estimatinr the
fluctuation of turbulent shear stresses associated

with the discantinuous shifts of velocity represented

by Figure 2.

 

 

19

Shear Stress

A function of velocity, shear stress is the
apprOpriate variable for dealing with flow/sediment
interaction. Whereas velocity is concerned with
length per unit time, shear stress is concerned with

force per unit area.
The relationship between velocity and average

shear stress can be expressed as:

:13: ‘ “FE-7' + M A“; (3)
a I)

where x is the longitudinal direction

parallel to the flow
is the vertical axis

du is the velocity gradient in the

dy y direction

P is the density of water

‘4 is the dynamic viscosity of the fluid

u' is the velocity fluctuation in
the x direction

v' is the velocity fluctuation in

_ the y direction

"fly is the force per unit area in the
x direction on a surface normal to
the y axis (Schlichting, 1960).

This relationship expresses the physical fact
that momentum is being transferred by the movement
of the fluid packets, in addition to the normal

transfer of momentum through molecular motions. The

first term of the equation represents the component
due to turbulence and the second term is the laminar
contribution. Under the velocities considered in
this paper, the laminar term is negligible compared
to the magnitude of the turbulent term so that we may

consider the equation to be approximately

Try 7' "Fu'vl . “4-)
Because this function represents time averages,
instantaneous shear stress or shear stress over
short periods of time cannot be deduced from it
directly.

According to Hinze (1959), Shear stress can
be envisioned to arise from the transfer of momentum
across interfaces. Fluid packets carried from one
region to another --— the velocity generally being
different in the two regions --- will either gain or
lose a small amount of momentum. If the packets gain
momentum bv being carried into a region of higher
velocity, they will exert a corresponding retarding
force upon the flow. If the packets lose momentum
conversely, a corresponding force is exerted upon the
flow. The difference between the mean velocity of

the fluid packet and the velocity of the region into

which it moves is recorded as the momentary velocity
fluctuation.

Because of the mixing process in turbulent
motion, momentum is constantly being transferred back
and forth across any imaginary plans. As a result,
the instantaneous shear stress fluctuates about the
mean value expressed by equation A. These fluctua-
tions are closely tied to the velocity fluctuations.

Ideally, shear stress estimates should arise
from calculations involving incremental velocity
differences over time as the turbulent velocity
changes. However, as discussed above, the velocity
duration model used herein resembles a square wave
function involving constant velocity for a certain
duration, succeeded instantaneously by another
velocity level fixed for a given duration. In the
absence of a continuous velocity function, shear
stress for the model was estim ted in an indirect
fashion.

The average shear stress was calculated for
a particular mean.velocity and standard deviation,
using equation A. The equation contains u' and v',
horizontal and vertical fluctuations of velocity

respectively. It has been shown that v' is correlated

92

with u' with a constant of prOportionality equal

to k. Therefore ‘37‘= k . [57] and equation k
may be rewritten as :3“! = P IE'I ' K-I'CN (5)
Experimental evidence indicates k is roughly equal
to the value 0.2 (Schlichtina, 1960).

Experimentally it has been shown that shear
stress varies as the square of the velocity (Schlichting,
1955; Kalinske, 19kg). Using the above expression
for average shear stress and this emairicrl relation-
ship between shear stress and velocity, we are now
in a pnsition to determine changes in bottom shear
stress caused by velocity fluctuations. ie can
express the empirical relationship between shear
stress and velocity as fiiy=' c . u2 (6)
where c is the constant of prOportionalit;. Assuming
”c" can be evaluated, shear stress o‘n be estimated
directly from the horizontal component of velocity.
This constant was determined in the following manner.

For each mean velocity used in the study, a
series of horizontal fluctuations were generated
in the manner described above in the section concern-
ing determination of instantaneous flow velocity.
These fluctuations more used in equation 5 to deter-

mine the average level of the shear stress. At the

 

23

same time, the instantaneous mannitude of the velocity
associated with each pulse was recorded. A series
of instantaneous values for bottom shear stress were
generated according to equation 6. This process was
allowed to continue for a set number of velocity
fluctuations.

An alternative expression for average shear

stress can now be written:

 

__. 1 1 . 1
TXY: CU|T|+ Cut-1.2+ ' ' ° CUATR
.T.

where 137 is the average shear stress
0 is constant of prOportionality
ui is magnitude of the 1th velocity
squared
T1 time 1th pulse persisted at a point
T total amount of time represented

by n velocity fluctuations.

The constant of prooortionality can be cal-

culated according to the folloving relationship:
—' n
c: Txy'T/ i u'slTi
Ln
where n is the number of iterations.
Using this heuristically derived constant,
the shear stress can be calculated and used to deter-

mine the competency of the system e.g., the largest

grain size the system can move.

TURBULENCE-SEDIMENT INThRACTIUN

With the fixing of the horizontal component
of velocity, the results of the preceeding section
define the shear stress and time duration associated
with that shear stress. If the relationship between
shear stress and grain size can be established, it,
along with the duration will allow us to translate
the time series of turbulent velocities into a sequence
of depositional or erosional events, involving Specific
grain sizes.

The relationShip between flow velocity and
critical grain size has been apuroached from many
directions. A wide variet; of experiments have shown
that as velocity of flow over a bed of sediment is
increased, there is a critical point at vhich the
sedimentarv grains start to move.

Earliest studies of the relationship between
the flow and the movement of sedimentarv crains
centered around the velocity. Brahms (1753) suggested
the folloving relationship between velocit: and grain

V
size: v == k- W'b , where v was the critical
or or
pick-up velocity, k a coefficient, and w the submerged

weight of the particles.

24

Realizing it was physically more correct to
use force per unit area rather than velocity as the
critical variable for defining the competency of a
stream, duaovs (1879), white (1940). Shield (1936),
and other investigators develOped a wide variety of
relationships between shear stress and grain move-
ment. These studies indicate that movement will not
take place until a critical shear stress that is a
function or gr'in size is exceeded.

The dujovs equation relates the tractive force
to the product of the depth of water and gradient of
the stream,‘1'= d D S , where A is the Specific
weight of water, D is the depth of water, and S is the
gradient of the stream. Rubey (1938) showed in labora-
tory eXperiments that at low velocities a grain of a
given size moves only if the product of depth and
gradient exceeds some critical value.

Shields (1936), using experimental data, pro-

posed a dimensionless form of the stress:

¢ 1’ 1'ch
err

where D is the grain size, 0" and p are the Specific

 

weights of the particles and fluid reSpectively, and
¢¢fl{ is a constant for a particulr level of

turbulence.

26

White (19h0), by equating the fluid forces
that just cause movement of the grain to the force
holding the grain in place, arrived at the following
relationship to define the critical threshold stress:

Tc: ca’sal t“ ¢

where c is a constant that depends on the grain
packing, (é is the angle of repose of the grains, and
F” is the difference between the particle density
and fluid density. This expression holds for the case
in which the grain is mainly enclosed in the turbulent
flow.
Kalinske (19A?) extended this relationship for

sand mixtures, arriving at the following eXpression:

Tc. ‘—' (15) pp'edi fond:

where p is the portion of the bed area taxing the
shear, ’0' is the difference between the particle density
and fluid density, g is the force of gravity, 4) is the
angle of repose of the grains, and d1 is the grain

diameter. The size, d1 , thus determined represents
the diameter of a particle, presumably quartz, Which
Just moves from a position of cohesionless contact

with fellow grains.

 

 

27

Kalinske's relationship between shear stress
and grain size is the one utilized in the present
model. This relationship is more thoroughly grounded
in theoretical concepts than the more empirical
relationships of others (Shield, 1936; Leliavsky, 1959).
The empirically-based estimates were derived from a
simple inepection of the result of increasing flume
velocity over a substrate of known grain size. Such
results, by their very nature, represent a sort of
time average of the turbulent flow-sediment interaction.
Because the individual events that constitute this
average are the principal elements of this model, it

is felt that using such estimates would be inapprooriate.

Assumptions for the Operation of the Model

The relationships discussed above are enough
to construct a process model for deposition in
turbulent flow at a point in space. As can be seen,
it is a very simple model, indeed probably one of the
simplest that could be constructed. In order to view
it in a realistic fashion, a number of assumptions

which underly the simple model should be mentioned:

28

(l) The total volume of sediment in transport
with respect to the volume of the trans-
porting medium is small so that complex
grain to grain interactions are at a
minimum.

(2) The assumption of a critical diameter
carries with it an implicit assumption
that the grains have uniform diape.

(3) The model takes no account of small
grain entrapment between larger grains.

Size-Frequency Distributions

In any localized portion of a dynamic sediment
situation, the size-frequency distribution available
for interaction with the turbulent flow is not uniform.
That is, not all grain sizes are present in equal
prOportions . Obviously the shape of the size-frequency
distribution and the location of the most abundant
grain size classes with resnect to mean velocity will
have a tremendous bearing on the deposited sediment.

We are, in this investigation, concerned with
erosion and deposition occuring at a point in Space.
The real worLi is composed of an infinite number of such

points which can be envisioned as selectively feeding

m“,

 

 

29

sediment from one to another. The more distant a
point, the less influence it has on the grain size-
frequency distribution occuring at the selected point
of interest at a given time.

Later parts of this discussion ”ill concern
the nature of the deposit at a soecific point. The
sediment being supplied to that point can be envisioned
as emanating from a great number of points nearby. The
grain sizes departing from this neighboring cloud of
points are all those sizes less than some critical
size. Lith this in mind, the size-frequency distri-
bution fed into the point of interest was simulated
by providinr the neighboring cloud of points with a
uniform size-frequency distribution and assuming a
fixed level of mean velocitfi passing a fraction of
that distribution.

Each size-frequency distribution generated
from each member of the cloud differs from each of
the others only in a random fashion. The sediment
that reaches the central point can be considered the
sum total of these events and tends to conform to

some sort of limiting distribution.

 

SEDIMENTARY DEPOSIT

Erosion and Deposition

Although both deposition and erosion occur
during model iterations, the dominant process is
that of deposition. This is due primarily to a
"lag effect", with the coarser portions of previous
deposits protecting the underlying strata from
erosion by any but extreme turbulent events. Even
when such an event occurs as discussed above, the
higher velocities tend to Operate with least dura-
tion and the length of time for scouring is very
limited.

In the real world, such upbuilding is probably
limited. Indeed, it is very likely this upbuilding
effects evolution of the bottom configuration which
fundamentally changes the hydrodynamics of the system.
This change could conceivably result in a shifting of
the average velocity, making scouring of the existirg
deposit likely.

‘Each velocity pulse which results in deposition
is assumed to deposit a layer whose grain size is a
function of velocity, and thickness is a function of
duration. Thickness of each layer was determined

from an analysis of the rate of sediment movement into

30

 

31

the area of our depositional site. The unit of
area of concern around our point is assumed to be
one square centimeter. This area was judged to be
great enough to completely capture the result of a
depositional event occuring at its upstream edge.
The sediment entering this area is assumed to spread
uniformly through the area. Then, assuming a constant
mineralogy, and that total mass of sediment delivered
to the area is known, thickness can be easily cal-
culated.

Kalinske (19h7), on the basis of theory, held
that for any single grain size, the rate of sediment
tranSport can be determined from the following re-

lationship:

6‘ “ii Pixfi U3

where G is the rate of sediment movement per unit
width in grams / sec-centimeter
C is a constant
d1 is the grain size
p1 is the prOportion of the bed occupied by
grain size d
A, is Speciiic gravity of the particles
u is the difference between the flow velocity
agd the velocity required to start movement.

If a size-frequency distribution is partitioned

into a number of segments each covering a small grain

 

32

size range, Kalinske's relationship may be applied
to the mean size of each segment without too great
a sacrifice in precision.

A given velocity pulse will cause the sediment
in each segment of the distribution to move with a
characteristic velocity. The distribution in question
is that portion of the size-frequency distribution
supplied by the upstream cloud of points coarser than
that defined by the critical shear stress at our
location.

The rate of accumulation of each size fraction
within our area of deoosition can thus be determined.
Because the duration of the turbulent pulse is known,
the total weight of each size fraction is simply
determined. The relative proportions in the size

classes constitute the size frequency distribution

1

that has resulted from the depositional event. Tne

thickness of the deposit is calculated by the relation-
shim Th: (m). w
_____,_.

2435
where Th is the thickness, X1 is the mass of sediment

delivered for size class i, and assuming a density of
quartz of 2.65 gm/cm3 and porosity of ho».

The size intervals used in this model are

33

.01 millimeters wide. For most distributions in the
sand range, this would result in 50-103 divisions
of the size-frequency distribution.

whereas the conditions for deposition Can be
defined from the above discussion, the conditions
for erosion require knowledge of the previously
deposited grain size-frequency distribution. In the
case of the present model, a homogeneous grain size of
1 mm. was assumed to exist at the model site. This
grain size ras too coarse for the vast majority of
random fluctuations to move.

After a few pulses, sediment deposits
accumulated on this coarse substrate. This material
is potentielly erodable under the turbulent regime
of the model. Erosion was judged to occur when the
turbulent shear stress exceeded the critical stress
of the mean grain size of the surface packet of sedi-
ment. This was judged apprOpriate, rather than size
fraction by size fraction erosion, because empirical
studies have shown that initiation of erosion in-
volves bulk movement of the body rather than grain
by grain sortine (Bagnold, 1955).

The thickness removed is calculated in a

similar manner as described above for deposition.

 

31;

For a given level of turbulent shear, erosion
will proceed until a sediment packet is un-
covered which has a mean grain size vhich is too
coarse to be moved, or until the fluid packet
associated vitn the shear stress passes by the

depositional site.

Accumulation

A series of turbulent pulses are generated
by the model and each is translated into either an
erosional or depositional event. The primary out-
put of the model is then a complete listing of
events with associated grain sizes and thicknesses.

From this list which includes every hydro-
dynamic event vhich occured over the time range, a
second shorter sequence was compiled, containing
only those sediment packets still remaining after
erosion. This second list constitutes the simulated
sedimentarr record. The characteristics of this
simulated deposit can be compared to those of the
turbulent medium. In this way, the objectives for

construction of the model can be realized.

 

SUMMARY

The flow diagram for the model whose
components are discussed above is illustrated
in Appendix I.

Two parameters are under control by the
investigator. The nature of the incoming size-
frequency distribution which is controlled by
the Specific mean velocity passing through the
upstream cloud of points, and the mean velocity
at the model observation point.

The model is incorporated into a computer
program, Appendix II. This program was constructed
in such a manner that modification of any or all
parts of it can be easily accomplished. That is,
the fundamental logical structure of the model should
always be suitable. The functional relationships
within each logical element can be modified or re-
placed according to the used to which the model
is put, and contemporary understanding of the nature

of water/sediment interaction.

35

 

REFERENCES C ITED

 

REFERENCES CITED

Bagnold, R. A., "The Flow of Cohesionless Grains
in Fluids", Royal Society of London, PhilOSOphical
Transactions, Vol. 2A9, Ser. A, pgs. 235-297, 1956

Brahms, A., Anfangsgrundeder Deich-u tasserbaukunst,
1753

Clark, J. A., "A Study of Incom ressible Turbulent
Boundary Layers in Channel Flow', Transactions,
American Society of Mechanical Engineers, Journal

of Basic Engineering, Vol. 90(D), pgs. uSS-h68, 1968

Clark, J. A., and E. Markland, "Flow Visualization

in Turbulent Boundary Layers", Journal of the Hydraulics
Division, ASCE, Vol. 97, # HY 10, pgs. 1653-1643,
October, 1971

DuBoys, P. F. D., "LeRhone et le rivier a lit
affouillable", Annales des Ponts et Chaussees,
Vol. 18, Serial 5, pgs. 1&1-195, 1879

Hinze, J. 0., Turbulence, McGraa Hill Book Co., 1960

 

Ishihara, Y., Yokosi, S., and T. Ueno, "Reynolds
Stresses in a River Current", Annals Disaster
Prevention Research Institute, Kyoto University,
KYOtO: # 1239 p830 503-514: 1969

Kalinske, A., "The Movement of Sediment as Bed Load
inuRivers", Transactions AGU, Vol. 28, pgs. 615-620,
19 7

Kozyrenko, L. 3., "Problems of Large-Scale Turbulencefi
Izvestiza Vysskikh Uchebnykh Zavedenu Energetika,
Moscow, # 10, pgs. 78-83, 1966

Leliavsky, 8., an Introduction tg_Fluvial fiydraulics,
Constable and Co. Ltd, London, 1955

36

37

Rubey, W. J., "The Force Required to Move Particles
on A Stream Bed", U. 3. Geological Survey, Prof.
Paper 189-E, pgs. 121-141, 1938

Schlichting, H., Boundary Layer Theorv, MoGraw Hill
Book Co., 1959

 

Shield, A., "Applications of Similarity Principles
and Turbulence Research to Bed Load Movements",
Translation by w. P. Ott and J. C. van Ucheler,

U. S. Dept. of Agriculture, Soil Conservation
Service Coop Lab., California Inst. Tech., 1936

Sutherland, A. J., "Entrainment of Fine Sediments
by Turbulent Flows", W. M. Keck Laboratory of
Hydraulics and Water Resources, California Institute
of Technology, Report # KH-R-13, 1966

White, C. M., "The Equilibrium of Grains in the
Bed of a Stream", Proceedings Royal Society A,

Vol. 171». pas. BELL-338. who

Yokosi, 8., "Large-Scale Turbulence in a River",
Annals Disaster Prevention Research Institute,
Kyoto University, Kyoto. # 108, pgs. 199-206, 1967

 

APPENDICES

 

APPENDIX I

FLOW CHART FUR PROCESS-
RESPONSB MODEL

 

FLOW CHART FOR PROCESS-RESPONSE MODEL

 

READ
Mean velocities
at upstream and 1
depositional sites

 

 

 

READ
Size of eddies
Nature of bottom 2
sediment initially
present

 

 

 

 

READ
Size range of ‘
available size- 3
frequency dis-
tribution

 

 

 

DBTBRMILB average
level of shear a
stress at upstream
and depositional
sites

 

 

 

 

DETERMINE constant
of proportionality 5
between velocity and
shear stress

 

 

 

38

NO EROSION

39

 

 

DETLRMINE nature

of size-frequency
distribution emanat-
ing from upstream
cloud of points

 

 

l

 

 

SELECT velocity
from probability
distribution

 

 

 

 

 

COMPUTE instan-
taneous shear
stress

 

 

 

 

COMPUTE duration
of velocity pulse

 

 

j

 

 

CALCULATB critical
grain size that
shear stress will
just move

 

 

  
    

 

  
 
 

CONPARE
with bottom grain
size

11

 

 

lO

. EROSION

 

 

no

i

 

 

DLTDHUINE amount
of underlying 12
sediment removed

 

 

 

 

to 11

 

If all of under-
lying laminae is 13
removed and time
remains,

return

 

 

 

 

 

DETBRMINB portion
of distribution
generated in 6
which is not passed
through the site

 

 

l

 

 

DETFPWINE relative
rates of movement
of each size class

 

 

l

 

 

DETLRMIKB amount of
sediment in each size
class which accumu-
lates in depositional
area

 

 

| 17-

DBTBREINB thickness
of unit deposited '

 

 

15

l6

 

 

Output parameters
of each pulse-
velocity duration
and nature of
deposit

 

 

 

 

 

Return to 7
for desired number
of iterations

 

 

 

 

 

Output nature of
final deposit and
associated flow
parameters

 

 

 

 

 

Analyze nature of
deposited sediment

 

 

 

 

Output aggregate
size-frequency dis-
tribution of the
deposit

 

 

 

STOP

18

19

20

21

APPENDIX II

LISTING OF COMPUTER
PROGRAM

 

LISTING OF COMPUTER PROGRAM

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