The Behavior of

Large Particles

Falling in Quiescent Liquids

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-C
The Behavior of

Large Particles

Falling in Quiescent Liquids

By G. E. STRINGHAM, D. B. SIMONS, and H. P. GUY

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1969
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price 55 cents (paper cover)
CONTENTS

Abstract_ ________________________________________

Introduction. ______________________________________

Problem and scope_______._____--_--__--_-____

Drag concepts________________________________

Shear drag_________________--_-_-___-_-_-__

Pressure drag-_____________---_--__--_-_-__

Total drag.________________--_--_-_-____-__

Historical development of drag concepts.__________

Mathematical development-_________________

Experimental development.__________________

Acknowledgments___ _ _________________________

Fall behavior parameteis____________________________

Variables affecting particle behavior.______________

Fluid variables._________________-----___-_-

Variables associated with particle motion. _____

Geometric variables_______________________

Dimensional analysis.___________________________

Coefficient of drag versus Reynolds number____

Shape factor._________________-_--------___

Frequency numbers.________________________

Stability numbers_________________________

Force number._______-_____-_--_----_-_____

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Fall behavior parameters Continued

Choice of characteristic diameters and velocities.___ CIO

Particle steadiness______________________________ 11

Particles, liquids, equipment, and procedure__________ 12

Particles___________________________________ 12

Liquids_ _____________________________________ 12

Equipment__ _ _ _______________________________ 13

Procedure__ _ __-_--___-_-_-_-_________________ 15

Experimental results-_______________________________ 16

Description of findings._________________________ 16

Spheres ____________________________________ 16

Disks.---_____________________ 17

Oblate spheroids____________________________ 27

Cylinders__________________________________ 27

Prolate spheroids-__________________________ 31

Comparison of behavior among all particles ________ 31

Interpretation of results_________________________ 32

Parametric relationships.____________________ 32

Patterns of fall___________________________ 32

Shape factors_____________________________ 33

Summary and conclusions._________________________ 35

References cited.----_------_--_----________________ 36

ILLUSTRATIONS

Page 

FIGURE 1. Sketches showing vortex pattern of an oscillating disk.___________________________ C6

2. Sketches of circular fluid flow around a particle._________________-___----__----__ 7

3. Graph showing coefficient of drag as a function of Reynolds number for fixed disks,

spheres, and airship hulls.______________--______--_----^___--------__------- 8

4. Sketch showing forces on a falling disk________________________________________ 11

5. Photographs of particles used in experiments____________________-__--_---------- 14

6. Graph of kinematic viscosity of glycerine-water mixtures as a function of temperature. _ 14

7. Graph of density of glycerine-water mixtures as a function of temperature.--------- 15

8. Schematic diagram of fall column._____________________________-_-------------- 15

9. Sketch showing distortion of light rays caused by fluid refraction and the curved

surface of the fall column.__________________________----__-__--------------- 16

10. Graph of coefficient of drag as a function of Reynolds number and force number for

falling and fixed spheres._____________________________________--_-_--_------ 21

11-15. Sketches showing fall pattern of a disk:

11. Steady....____________________________________________-----_-------- 21

12. Regular oscillation_____________________________________------------- 21

13. Entering glide-tumble____________________________------------------- 22

14. Leaving glide-tumble._______-________________-______-_--------------- 23

15. Tumble.__________________________...______.____--__---------------- 23

16. Graph showing regimes of fall for a free-falling disk__________________------------ 24

17-19. Graphs showing velocity-time relationship of falling disk for 

17. Aluminum in an oscillatory pattern.______________________-_--_--------- 25

18. Aluminum in a glide-tumble pattern_______________________-_----_------ 26

19. Lead in a tumbling pattern.______________________________----__------- 26

in
IV

CONTENTS

Page

FIGURES 20-21. Graphs showing comparison of maximum and nominal particle diameter for coefficient 

of drag as a function of Reynolds number for falling disks, £^,.= 0.1, based on 

20. Path velocity________________--__---__-__ _ ____________ __ _________ 028

21. Vertical velocity __ __________________________________________________ 29

22. Graph showing coefficient of drag as a function of frequency number for falling disks,

S^.= 0.1.___ __ _.___________ __ -- - _ 30

30

31

32

33

34

23-24. Graphs showing coefficient of drag as a function of Reynolds number, SF C =Q.5, for 

23. Falling oblate spheroids-____-__--_________-_-----_-_---------_-_------

24. Falling cylinders ____.-_______-__-__-__-__-_-_-_-____---_-_---___--___

25. Graph showing coefficient of drag as a function of frequency number for falling cylinders, 

/8^.= 0.5__-.- _______________._______________ _ _ -- 

26. Graph showing coefficient of drag as a function of Reynolds number for falling prolate 

spheroids, S-7 <.= 0.71-____- _ _ __ __________________ ___ . __ _ _ . _____

27. Composite graphs of coefficient of drag as a function of Reynolds number for falling 

spheres, disks, oblate spheroids, cylinders, and prolate spheroids based on maximum 

and nominal particle diameter and path velocity.---- _________________________

TABLES

Page 

TABLE 1. Particle properties, measured and computed_--_---_-____------______-__--------_ C12

2-6. Drop data for 

2. Spheres._______-__-_-_-___-_--__-_---_-------__--_---_-_-_--_-._----- 17

3. Disks_-__.____._--____________-__-_______-___-___-__-_______-___----- 18

4. Oblate spheroids_-_________-____-_____--__--__--____-_-_-________----- 19

5. Cylinders___________________________________________________________ 20

6. Prolate spheroids.___________-___--________-_-_____-___-------___-

7. Frequency of oscillation of unstable particles falling in water-glycerine mixtures.

SYMBOLS

22

27

Symbol Definition

A A characteristic area of the particle (cm2). In this paper it is the maximum projected area of the

	particle.

An The area projected on the basis of the nominal diameter (cm2).

Ci... Dimensionless coefficients.

CD Coefficient of drag.

CL Coefficient of lift.

Z> Drag forces (dynes).

F Driving force defined by (PV p/)^g (dynes).

I Moment of inertia of the particle (dyne-cm-sec2).

L Lift force (dynes).

M Virtual mass (dyne-sec2/cm).

Ne Number of corners.

O r Volume of a sphere circumscribing a particle (cm3).

Oe Circumference of a circle of area equal to the plane area of the particle (cm).

Ov Circumference of a particle (cm).

R Reynolds number based on particle size

Rf Resistance due to pressure.

R, Resistance due to shear.

Rt Total resistance.

S Surface area of the particle (cm2).

SF Shape factor.

SFC Corey shape factor (c/Vob) 

SFca Corey shape factor modified by Alger.

SFd Dynamic shape factor («/wn) 2 .

Sr Surface roughness.

¥ Volume (cm3).
SYMBOLS

Symbol Definition

W Bouyant weight of the particle (gm) .

a Maximum diameter of the particle (cm) .

6 Intermediate diameter of the particle (cm) .

c Minimum diameter of the particle (cm) .

d A characteristic diameter of the particle (cm) .

d* Diameter of fluid container.

dA Surface diameter (cm), the diameter of a sphere with surface area equal to that of the particle.

dm Diameter of a circle of area equal to the maximum cross-sectional area of the particle (cm) .

dn Nominal diameter (cm) , the diameter of a sphere of volume equal to that of the particle.

d, Sedimentation diameter (cm), the diameter of a sphere that has the same specific gravity and the

	same terminal uniform settling velocity as the given particle in the same sedimentation fluid. 

d, e Sieve diameter (cm), the length of the side of the smallest square opening of the sieve through which

	a given particle will pass.

/ Frequency of oscillation or tumbling.

g Acceleration of gravity (cm/sec2) .

k Arbitrary constants.

I Distance from center of gravity to the center of pressure.

m Bouyant mass of the particle (dyne-sec2/cm) .

ma Added mass.

m,f Mass of accelerating fluid (dyne-sec2/cm) .

mv Mass of the particle (dyne-sec2/cm) .

rc Radius of curvature of a corner on a particle (cm) .

Ti Radius of the maximum inscribed circle in a particle (cm) .

t Time.

u Horizontal velocity of the particle (cm/sec).

v Verticle velocity of the particle (cm/sec) .

9 Frequency number (fd/ta) .

I Stability number -

64 p, a

X Force number

H Circularity.

T Roundness.

* Density-frequency number (f2pfd*{F).

* Sphericity.

« Angle of attack (radians), the angle between disk trajectory and disk.

18 Angle between vertical and particle trajectory (randians) .

n Dynamic viscosity (dyne-sec/cm2).

v Kinematic viscosity (cm2/sec).

pf Density of the fluid (dyne-sec2/cm4) .

p p Density of the particle (dyne-sec2/cm2) .

T Time interval < t.

u Path fall velocity of the particle (cm/sec) .

<a n Fall velocity of a nominal sphere (cm/sec) .

SEDIMENT TRANSPORT JN ALLUVIAL CHANNELS

THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS

BY G. E. STRINGK&M, IX B. SIMONS, and H. P. GUY

ABSTRACT

The free-fall behavior patterns of specific idealized particles 

falling singly in quiescent liquids were observed with respect 

to fall patterns, fall velocity, and travel path. Particle shapes 

and densities were chosen to represent the limiting conditions 

of natural gravel-sized sediment found in alluvial ehaBnefo. 

The liquids used in the fall column were water, various mixtere» 

of water and glycerine, and pure glycerine.

The particles were dropped in a vertical clear plexiglass fall 

column 3 meters high and 40 cm in diameter. Two 16-mm movie? 

cameras set at right angles to each other and focused on ae 

timer and a 1-meter test section of the column recorded tbe 

particle behavior. The horizontal and vertical coordinates df 

the particles were established by projecting the image of tte 

particles moving through the test section onto a grid.

The spheres usually fell in a straight vertical path, but sometimes the nylon or teflon spheres falling in pure water exhibited; 

small erratic horizontal movements. Disks, on the other hand, 

exhibited four types of fall behavior: (1) steady-flat fall maximum projected area perpendicular to the axis of the fall whea 

the Reynolds number R<100; (2) regular oscillation oscillation 

occurred about a diameter perpendicular to the direction of 

fall, and there was very little horizontal translation; (3) glide1- 

tumble the frequency of oscillation was less than regular 

oscillation, the amplitude increased until the disks fell vertically 

on edge part of the time, and sometimes tumbling occurred at 

the end of each glide; and (4) tumble the axis of the fall path 

was virtually a straight inclined line, the disk continually 

rotated, and the frequency of rotation approached the frequency 

of oscillation in the oscillation behavior pattern.

The steadiness of a falling particle is dependent upon the 

stability of the resultant of the pressure forces in the wake of the 

particle. Shape of the particle and the Reynolds number both 

affect the distribution of the pressure forces in the wake with 

extremes in shape and high R causing the resultant force to be 

the least stable. Thus, the fall pattern of a disk, which is in 

effect a two-dimensional particle, is much less steady than the 

fall pattern of a three-dimensional sphere. The ratio of the path 

length to vertical distance through which the particle falls is 

also related to the steadiness of the particle, the more unsteady 

the particle, the greater the path length.

The effect of particle-fluid density ratio and the thicknessdiameter ratio of the particle on its fall velocity and steadiness 

can be expressed by a dimensionless stability number I. The 

frequency of oscillation can be related to the fall velocity and 

particle diameter by a dimensionless frequency number 6.

INTRODUCTION

The fall velocity of a sediment particle, defined as 

the rate at which the particle settles through a fluid, 

is one of the most important factors affecting the 

credibility and movement of earth material, the time 

that a particle will remain in suspension in streamflow, 

and the nature of deposition and bed roughness formed 

in moving and quiet waters.

The fall velocity for a given particle depends on its 

shape, size, and density as well as on the physical 

properties of the fluid. All these factors, individually 

or in combination, may affect the stability and the 

fall path of the particle. Other properties, such as 

roughness, roundness, or concentration of particles, and 

turbulence in the fluid, may also be important.

PROBLEM AND SCOPE

A particle moving relative to a fluid is acted upon 

by a drag force whose magnitude is a function of the 

relative velocity between fluid and particle, fluid 

density, and particle size and shape. The drag force is 

usually expressed as a function of the particle velocity 

and size and in terms of the fluid properties. The 

relationship is not unique, however, and must be 

correlated by means of a coefficient of drag CD . The 

coefficient of drag is, in turn, related to the Reynolds 

number R. The relationship between CD and R can be 

shown graphically by plotting CD as a function of R 

on logarithmic paper. Such a diagram will be referred 

to as the CD-R diagram.

For some idealized cases, the relationship between 

CD and R, at small values of R, can be determined 

mathematically. Generally, however, it must be 

established experimentally.

The relationship between the coefficient of drag and 

the Reynolds number is often established with the 

particles rigidly mounted in a moving fluid. Different 

and more complicated is the case of freely moving

Cl
C2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

particles in either a quiescent or a moving fluid. The 

resistance forces on a free particle cannot be measured 

directly but must be estimated from limited mathematical theory and from tests on mounted particles. 

Thus, the effects of rotation, density, and inertia in 

the unsteady fall path of a particle cannot be considered if the particle is not free to respond to the 

unbalanced forces which may develop around it. 

This paper is a report of a study made to investigate 

some aspects of free-fall phenomena.

The scope of the study was limited to the examination of the free-fall behavior patterns of specific idealized 

particles falling singly in quiescent liquids. Its purpose 

was to evaluate the steadiness of particles with respect 

to orientation, fall velocity, and path of travel at 

Reynolds numbers greater than 10. The particles 

used were spheres, oblate and prolate spheroids, 

cylinders, and disks. The specific gravity of the spheres 

ranged from 1.14 to 14.95. The specific gravities of the 

other shapes ranged from 2.81 to 10.15. These shapes 

and specific gravities were chosen because they represent 

limiting conditions of natural gravel-sized sediment 

particles commonly found in alluvial channels. In addition, the liquids used were water, various mixtures of 

water and glycerine, and pure glycerine with kinematic 

viscosities ranging from 0.010 to 3.59 sq cm per sec, 

respectively.

DRAG CONCEPTS

The energy imparted to a fluid by a particle falling 

in the fluid is either transmitted to the boundaries of 

the fluid or is dissipated in heat. The internal resistance 

within the fluid to fluid motion is transmitted to the 

particle, in accordance with Newton's third law of 

motion, and is called drag. According to Albertson, 

Barton, and Simons (1960), this drag, or fluid resistance, 

is of two general types: shear drag, the tangential 

component of the resistance, and pressure drag, the 

normal component of resistance.

SHEAR DRAG

A shearing force related to the viscosity of the fluid 

occurs within the fluid as the molecules attracted to 

the particle move past those of the surrounding fluid. 

The magnitude of the resistance due to shear can be 

defined by

(i)

where

£a=a dimensionless coefficient, 

S= surf ace area of particle (cm2), 

p/= density of the fluid (dyne-sec2/cm4), and 

co=path fall velocity of the particle (cm/sec).

Equation 1 is a function of the Reynolds number

(2)

which can be defined as the ratio of the fluid inertia 

forces to the fluid viscous forces and where

d characteristic length of the particle (cm) and 

u=kinematic viscosity (cm2/sec).

At very small R, the shear force is the result of the 

shear stress acting over the entire surface of the particle 

and is distributed throughout the fluid. If the fluid is of 

finite extent, the stress is transferred to its outer boundaries. On the other hand, at large R, separation occurs 

on the lee side of the particle, causing a wake to develop. 

The formation of the wake decreases the surface area 

over which the shear force acts. At the same time, the 

velocity around the particle increases, which tends to 

increase the shear drag. Then, at large R, the most 

concentrated stresses are confined to the boundary 

layer near the surface of the particle and to the wake of 

the particle. Most of the energy imparted to the fluid 

is dissipated in the wake.

PRESSURE DRAG

The resistance to movement of the mass of fluid in 

the path of a particle causes a positive pressure in 

front of, and a negative pressure behind, the particle. 

This pressure differential is the resistance to fall due to 

pressure drag and can be defined by

(3)

where

£72=a dimensionless coefficient and 

A=ihe projected area of the particle (cm2).

At small R, the pressure changes systematically 

around the particle in accordance with the relative 

acceleration and deceleration of the fluid. The pressure 

force is distributed throughout the fluid. On the other 

hand, at large R, a wake develops behind the particle. 

The pressure in the wake is virtually that found in the 

boundary layer at the point of separation. As in shear 

drag at large R, the pressure forces are confined to the 

boundary layer and the wake of the particle.

TOTAL DRAG

The surface area of a particle S, used in equation 1, 

can be expressed as the product of some constant and 

the projected area of the particle A used in equation 3. 

When the total resistance Rt of a freely falling particle 

becomes equal to the buoyant weight or total driving
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C3

force F of the particle, equations 1 and 3 may be 

written in the form

(4)

where

<7.D=(7i(constant) + Oz= coefficient of drag,

F=¥g (pp pf) in dynes,

g= acceleration of gravity in cm/sec2,

¥= particle volume in cm3 , 

pf=fluid density in dyne-sec2/cm4, and 

pp=particle density in dyne-sec2/cm4.

This is the general form of the law of resistance to 

motion of solids in a fluid as expressed first by Newton 

(Prandtl and Tietjens, 1934).

At small R (less than 0.5), Stokes' solution (Lamb, 

1932) for resistance to flow around spheres shows that 

two-thirds of the total resistance is due to shear, and 

one-third is due to pressure. The drag at small R is 

sometimes called deformation drag because the shear 

and the pressure forces are distributed throughout the 

fluid and thus "deform" the fluid.

At large R, the shear drag becomes insignificant, and 

the total drag becomes predominantly pressure drag.

HISTORICAL DEVELOPMENT OF DRAG CONCEPTS

The present knowledge of resistance to motion 

experienced by a solid particle falling through a fluid 

has been developed largely within the past 125 years. 

Both mathematical and experimental studies have 

contributed to this fund of knowledge.

MATHEMATICAL DEVELOPMENT

From the Navier-Stokes equations, rather concise 

equations have been developed that define the nature 

of flow around particles for Reynolds number R less 

than about 0.5. However, solutions to the NavierStokes equations have not been found for flow conditions where R»0.5 as in this study; hence, the 

Navier-Stokes equations are not applicable herein.

For values of R greater than the Stokes range, 

concepts will be based on the Lagrangian description 

of flow phenomenon, or from the point of view of an 

observer traveling with the particle through a stationary fluid. For unsteady motion of a particle in a 

fluid, Stelson and Mavis (1957) show that the force 

required to cause acceleration is greater than the force 

required to accelerate the mass of the particle mv, and 

that the mass of the displaced fluid mf must be considered. The mass of the fluid that must be considered 

is related to the shape as well as the size of the particle. 

The total mass, or virtual mass M is, therefore, the 

sum of the particle mass and an appropriate fluid mass.

The character of the resistance of an accelerating 

particle is defined by Brush (1964) on the basis of 

Basset's equations. When the nonlinearity of the 

resistance term, for motion beyond the Stokes range, 

is accounted for, the equation becomes:

M ( 

dt

where

g= acceleration of gravity (cm/sec2),

d=& characteristic diameter of the particle (cm),

M=dynamic viscosity (dyne-sec/cm2)

£=time (sec),

T=time interval <t, 

M virtual mass (dyne-sec2/cm), 

mj,=mass of the particle (dyne-sec2/cm), and 

wi,/=mass of the accelerating fluid (dyne-sec2/cm).

For steady motion, equation 5 reduces to equation 4. 

A solution to equation 5 was presented by Brush in the 

form of series expansions, which is suitable for computer 

analysis.

Odar and Hamilton (1964) also worked with Basset's 

equation. They expanded its range of applicability into 

the nonlinear zone by accounting for the ratio of the 

convective to local acceleration.

EXPERIMENTAL DEVELOPMENT

The flow characteristics around a particle become 

so complex when R>1 that resistance to motion and 

the behavior of the falling particle must be determined 

experimentally. Thus, Newton's law of resistance, 

equation 4, has been used as the basis of experimental 

development. The experimental development has been 

achieved by determining CD from measurements of the 

fluid and particle properties as needed in equation 4 

and from measurement of the particle velocity relative 

to that of the fluid.

Pernolet (Wadell, 1934), a French engineer, made 

some of the first experiments on fall velocity in 1851. 

He discovered that the fall velocity of lead particles 

of similar weight varied with shape. Among the first 

contributors to use a systematic approach to determine the effect of shape on fall velocity of a particle 

included Rubey (1933) Zergrzda (Schulz and others, 

1954) and Wadell (1932). Included in Wadell's (1932,

307-965 O 69-
C4 SEDIMENT TRANSPORT IN AUJJVIAL CHANNELS

1933, 1934) work were the following particle descriptive 

parameters:

1. The nominal diameter dn, the diameter of a sphere 

having a volume equal to the volume of the particle -F,

64* 1/3

(6)

2. The degree of sphericity ¥, the cube root of the 

ratio of the volume of the particle to the volume 

of the circumscribing sphere Ov ,

1/3

(7)

3. The degree of roundness T, in a plane with respect 

to the particle, the ratio of the sum of the ratio 

of the radius of curvature of the corner r c to the 

radius of the maximum inscribed circle rt to the 

number of corners Nc ,

(8)

4. The degree of circularity E, the ratio of the length 

of the circumference of a circle of area equal to 

the plane area of the particle O c to the length of 

the circumference of the particle 0P,

(9)

Of these four shape parameters, the nominal diameter 

has greatest application because of the relative ease 

of making the necessary measurements. It is used 

extensively as the characteristic length in computations of the Reynolds number.

In addition to Wadell's nominal diameter, two other 

particle diameter concepts are commonly used and 

defined by B. C. Colby (in U.S. Inter-Agency Comm. 

Water Resources, 1957).

1. The sedimentation diameter d s is the diameter of a 

sphere that has the same specific gravity and same 

terminal uniform settling velocity as the given 

particle in the same sedimentation fluid. The 

sedimentation diameter varies with the viscosity 

of the fluid. Chatuthasry (1961) studied the sedimentation diameter as a function of the nominal 

diameter and found that the ratio d s/dn tended to 

decrease when R increased beyond 100.

2. The sieve diameter dse is the length of the side of the 

smallest square opening of the sieve through which 

the given particle will pass. Serr (1948) proposed

that the ratio dse/ds be used as a measure of particle shape.

Shape description was advanced independently by 

Corey (1949) and McNown and Malaika (1950) with 

an expression of the ratio of the three mutually perpendicular axis lengths,

(10)

where

a=maximum diameter of the particle (cm),

b=intermediate diameter of the particle (cm), and

c=minimum diameter of the particle (cm).

It was soon learned that this relationship did not adequately relate the fall velocity of all sizes and shapes of 

particles to that of the fall velocity of a sphere. Alger 

(1964) used the ratio of the diameter of a sphere whose 

surface area is equal to that of the particle to the nominal diameter of the particle, dA /dn, to modify equation 

10.

In a study of the various shape parameters affecting 

fall velocity, Wilde (1952) found that the maximum 

projected area of a particle could be closely approximated by the product of the lengths of the major and 

intermediate axes. He also found that the fall velocity 

increased greatly with the roundness of a particle. 

Roundness has not been used extensively because it is 

difficult to measure.

A dynamic shape factor SFd has been derived by 

Briggs, McCullock, and Moser (1962) by solving equation 4 for w 2, dividing both sides by the square of the 

fall velocity of a nominal sphere W B and combining 

terms to obtain

-§_?L

T pfV

1 (11)

At large R, the fall pattern of particles is usually 

irregular. Wilmarth, Hawk, and Harvey (1964) studied 

the fall pattern of disks in various fluids and found that 

the dimensionless mass moment of intertia I of disks 

correlated with particle stability. The inertia term 

"I," referred to in this paper as the "stability number," 

is defined by:

I if PP C .

p/a6 64 pf a,

(12)

where 7=the mass moment of inertia of the particle 

(dyne-cm-sec 2). Experimentally, I determines a point 

on the CD-R diagram at which a particle would no 

longer be steady in orientation. Disks, for example, 

were found to be stable with the maximum area perpendicular to the fall line when R is less than 100. At
THE BEHAVIOR OF LARGE PARTICLES FALLESTG IN QUIESCENT HQUIDS C5

R>100 the orientation of the disk and its path of 

travel were no longer constant.

Once oscillation of the particle began, Wilmarth, 

Hawk, and Harvey found that I could be correlated 

with a dimensionless "frequency number" defined by

(13)

where/ is the frequency of oscillation or tumbling. The 

relation between I and 6 is linear for a given R.

It is evident from this summary of previous work on 

particle behavior that parameters controlling stability 

of the free falling particle have received too little 

attention, and therefore, should receive further attention in this paper.

ACKNOWLEDGMENTS

This report is a modification of a Ph. D. dissertation 

presented to Colorado State University by the senior 

writer (Stringham, 1965) while employed by the U.S. 

Geological Survey. Notable among those giving counsel 

and guidance during the study were Professors M. L. 

Albertson, R. S. Creely, and E. J. Plate, of Colorado 

State University, and E. V. Richardson, of the U.S. 

Geological Survey. Ashok Kumar, Anwiya Andrews, 

Reinhard Weiss, and Darell Zimbelman assisted with 

data collection and the reduction of voluminous raw 

data to a usable form.

Appreciation is extended to P. C. Benedict, R. W. 

Carter, B. C. Colby, B. R. Colby, J. V. Skinner, G. P. 

Williams, and F. Witzigman in connection with review 

of this paper.

FALL BEHAVIOR PARAMETERS

In accordance with the scope of this paper, it is 

necessary to understand how the various parameters 

involved in fall velocity phenomenon at R > 10 

operate. A discussion of the major fall velocity parameters as deduced from dimensional analysis is presented. 

The general form of the resistance equation is that of 

equation 4.

VARIABLES AFFECTING PARTICLE BEHAVIOR

The three major kinds of variables which affect the 

behavior of a freely falling particle are fluid properties, 

particle motion characteristics, and particle characteristics. Acceleration or deceleration, as expressed in 

equation 5, occurs if the drag forces do not balance the 

buoyant weight. When drag and buoyant weight 

balance, the motion is steady and is governed by 

equation 4. It should, therefore, be possible to define 

all types of falling particle behavior, including fall 

velocity, acceleration, and pattern of fall, in terms of 

the force balance between buoyant weight and drag.

FLUID VARIABLES

Fluid variables of concern are density and viscosity. 

The density ratio between particle and fluid pp/pf, is a 

measure of the driving force and hence the fall velocity 

of the particle. The fall velocity increases with the 

density ratio. The coefficient of drag CD, the Reynolds 

number R, and the stability number I are also functions 

of fluid density.

Both shear and pressure forces are dependent upon 

the viscosity of the fluid. The magnitude of the shear 

force is a direct function of viscosity. Viscosity controls 

the pressure force to the extent that it affects the 

nature of the boundary layer and the point at which 

the boundary layer separates from the particle. As 

discussed earlier, the point of separation affects the 

pressure within the wake behind the particle, and hence 

affects the pressure drag.

It has been noted, as will be discussed later, that 

patterns of fall for some particles are dependent upon 

their fall velocity. It is probable that changes in the 

fall velocity, because of viscous changes in the fluid 

(brought about usually by temperature changes), can 

significantly affect the fall pattern of the particle.

As mentioned in the introduction, the fall velocity 

of sediment particles found in the bed of an alluvial 

channel is one of the factors which control the bed 

form of the channel. Fahnestock and Haddock (1964) 

note that the bed form of the Rio Grande near El 

Paso, Tex., changes with the annual fluctuation of the 

water temperature and the corresponding change in 

viscosity.

VARIABLES ASSOCIATED WITH PARTICLE MOTION

Velocity, vortex formation and shedding, and circulation are the variables considered which affect fall 

pattern. In accordance with equation 5, resistance to 

motion is a function of the square of the velocity. 

Both shear and pressure drags increase with increasing 

velocity.

Vortex formation and. sliedcllng

When separation occurs behind an object moving 

in a fluid, vorticies form in the wake. The vorticies remain bound to the particle under certain conditions, 

and under other conditions they are shed. Prandtl and 

Tie tj ens (1934) have shown that hi a two-dimensional 

flow around a cylinder, at a certain value of R, the 

vorticies are alternately shed, first from one side of 

the cylinder, and then the other, at regular periodic 

intervals. At greater R, the periodicity disappears, but 

the shedding continues. This process of vortex shedding, 

radically changes the pressure distribution in the wake, 

which, in turn, may cause oscillation of the cylinder.

Wilmarth, Hawk, and Harvey (1964) painted some 

of their disks with a water-soluble dye. As the disk
C6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

dropped through the water, the dye in the wake remained giving a visible trace of the wake and its associated vorticies, which they photographed. Pictorial 

conception of those photographs is given in figure 1. 

At small R where the disk orientation was stable, the 

wake formed a straight streak behind the particle. 

When oscillation first began, (for R = 100), and motion 

was close to vertical with very little lateral translation, 

a complete horseshoe-shaped vortex ring was shed at 

the end of each swing (fig. L4). At greater R, when

lateral translation was associated with the oscillation, 

vortex trails formed on the edges of the disk on the ends 

of a diameter perpendicular to the direction of motion 

much the same as occur on wing tips of aircraft (fig. 

IB). At the apex of each swing, a vortex was shed from 

each end of the perpendicular diameter.

Particles of other shapes will have different vortex 

patterns. The vortex pattern and the associated pressure 

distribution is one of the major factors controlling 

particle orientation and path of travel.

A B

FIGURE 1. Vortex pattern of an oscillating disk. After Wilmarth, Hawk, and Harvey, 1964. A, Oscillation accompained by very 

little horizontal translation, path of travel almost vertical. B, Oscillation accompanied by horizontal translation, path of 

travel almost parallel to disk face.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C7

Circulation

Circulation can be considered as a circular fluid 

flow around a submerged particle (fig. 2A). If circular 

flow is superimposed on the steamline flow (fig. 2B), 

the streamlines will be distorted (fig. 2C). This distortion of the flow pattern causes a nonsymmetrical 

pressure distribution around the particle and gives 

rise to lateral thrust and a lift force, which combines 

with the drag force to resist motion.

Wilmarth, Hawk, and Harvey (1964) also show that 

circulation exists around the disk when lateral translation occurs. The combination of circulation of the 

fluid and translation of the particle is the cause of the 

vortex trails mentioned in the preceding section. The 

circulation produces a lift force on the particle similar 

to that associated with the aerodynamic effects upon 

which atmospheric flight depends. Thus, through at 

least part of the oscillation of the disk, its motion is 

controlled by a force system similar to the system which 

controls flight. Circulation may have some small effect 

on the shear drag, but this effect would be small in 

comparison to its effect on pressure drag.

GEOMETRIC VARIABLES

The geometric variables that affect particle behavior 

are shape, roundness, surface roughness, and orientation with respect to the direction of motion. The 

density of the particle affects the buoyant weight as 

indicated in the discussion of fluid density.

The relative lengths of the three mutually perpendicular axes of a particle are used as an index of 

the particle shape. The shape of the particle controls 

the geometry of fluid flow. The fluid flow geometry 

affects the shear and pressure drag by controlling the 

contact area between the fluid and particle, the pressure 

distribution around it, and the point of separation 

of the wake.

If a particle is not symmetrical, the shape, with 

reference to the direction of motion, will vary with 

variation in the orientation of the maximum crosssectional area. Resistance will be controlled by the 

geometry of flow as discussed in the preceding section.

The roundness of a particle modifies the effect of 

shape on the point of separation. Sharp corners cause 

separation at smaller Reynolds numbers than rounded 

ones, and a wake develops at lower velocities. The 

development of the wake increases the pressure drag. 

Rounded particles, therefore, will have less resistance 

to motion; hence greater free-fall velocities than sharpcornered ones with similar characteristics such as size.

Surface roughness of a particle may increase or 

decrease the resistance to motion. Shear drag will be 

increased by roughness if the roughness protrudes 

beyond the laminar boundary layer. Prandtl and 

Tietjens (1934) show that separation around a sphere 

is controlled by flow conditions in the boundary layer. 

When the flow in the boundary layer changes from 

laminar to turbulent, the point of separation moves 

from the upstream to the downstream side of the sphere. 

This reduces both the size of the wake and the pressure 

drag. The total drag will be affected to the extent that 

the balance between shear and pressure drag is affected.

DIMENSIONAL ANALYSIS

Measurable variables pertaining to the resistance to 

motion of a single particle falling at a uniform velocity 

in a quiescent fluid include:

w=fall velocity (cm/sec), 

Pf=fluid density (dyne-sec2/cm4), 

M=dynamic viscosity of the fluid (dyne-sec/cm2), 

pp=particle density (dyne-sec2/cm4), 

a=major axis of the particle (cm), 

b=intermediate axis of the particle (cm), 

c= minor axis of the particle (cm), 

F= driving force (dynes), 

/= frequency of oscillation or tumbling (cycles/

sec),

ST=surf ace roughness (cm), and 

S= surf ace area (cm2).

ABC 

FIGUKE 2. Circular fluid flow around a particle. A, Pure circulation. B, Pure translation. C, Circulation and translation.
C8 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

These variables can be expressed in the functional 

equation:

*i (w, Pf, M, p,, a, b, c, F,j, Sr, £)=0. (14)

Selecting co, a, and pf as the repeating variables, the 

following groupings can be established:

PfWCb

u

F

a29 p/co 29

Pp C > > ) Pf a

b

a

s-,

a

S a*' fa

CO

£2,±,X,i^,^,£? =o. (15)

Because surface roughness will not be considered now,

nr

 - can be neglected. 

Then, combining:

 and - to obtain and - 

pf a pf a

results in:

- and - to obtain s and - 

a a a2 a

T'1 pP c cb c S fa ' -9 2' ~' ~T' -' ~2'

a Pfw Of a cb cb cb co

(16)

If the repeating variables used are a, p/, and F, a 

functional equation similar to equation 16 results and 

and is expressed as:

FPf F p c cb S fPfat

 if"' "' 9 9 9" " 9 

a2p/co 2 p/ a as a? Si F

=0. (17)

Because -j' -> and 2 are functions of the geometry

Cb Cb Cb

of the particle, they can be combined into a term called 

the shape factor SF. 

Assuming terminal fall velocity:

oc R Keynolds number

D coefficient of drag& ap/co

 - oc I Wilmarth's stability number

Pfd

 = force number

 oc 6 frequency number

D

oc $ density-frequency number;

anxl equations 16 and 17 reduce to:

, I, 5^,0=0, (18)

and

, GD, (19)

From equations 18 and 19, CD can be expressed as:

and

I, SF, 0), (20)

(21)

Equations 20 and 21 indicate that R, X, I, SF, 6, and 

$ are parameters of major importance governing the 

resistance of a particle falling at a constant velocity in 

a quiescent fluid. Although they are all used for 

clarification of certain concepts, these parameters are 

not independent of each other because X can be shown 

to be a function of R, and $ is a function of 8.

COEFFICIENT OF DRAG VERSUS REYNOLDS NUMBER

The coefficient of drag and the Keynolds number 

have been defined in previous discussions. Experimental 

evidence and theoretical solutions of the equations of 

motion for small Reynolds numbers clearly indicate 

that the coefficient of drag is a function of the Reynolds 

number. It has, therefore, become a "universal" 

procedure to express the relationship between GD and 

R by means of a logarithmic plot (fig. 3) that is further 

described as a (7D-R diagram. The curves show how 

CD varies with R when a disk, sphere, and an airship 

hull are held stationary in the flow of a wind or water 

tunnel. When a disk is mounted with its maximum 

cross-sectional area perpendicular to the direction of 

flow, the coefficient of drag becomes nearly independent 

of Reynolds numbers at R> 1,000 (fig. 3, upper curve). 

The curve for fixed disks represents an extreme condition 

in which the drag force is almost entirely pressure drag; 

it occurs when the size and shape of the wake are fixed 

by separation of the fluid from the boundary of the

102 1Q3 104

REYNOLDS NUMBER (R)

105

FIGURE 3. Coefficient of drag as a function of Reynolds number 

for fixed disks, spheres, and airship hulls. After Rouse, 1946.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C9

particle at a fixed place around the particle periphery. 

Because shear drag is a function of velocity, as is the 

Reynolds number, the fact that CD does not increase 

with R indicates the minor role played by the shear in 

the total drag on a disk at high Reynolds numbers.

The relationship of CD to R for a smooth sphere (fig. 

3, lower curve) is typical for flow conditions around a 

particle where a combination of both shear and pressure 

forces are significant for greater Reynolds numbers than 

for conditions previously discussed. Here, the place of 

separation is affected by flow conditions in the boundary 

layer as well as by shape.

The relationship of CD to R for an airship hull (fig. 3, 

lower right corner) mounted with the long axis parallel 

to the flow may be considered a case where the resistance 

to flow is due almost entirely to shear drag on the surface 

of the hull for all ranges of R. The hull is a streamline 

shape, and separation does not occur except at the 

extreme downstream end.

SHAPE FACTOR

The shape factor combines the lenghts of the three 

mutually perpendicular axis into one parameter. The 

particular form that will be used in this study is the 

Corey shape factor as defined previously (equation 10).

FREQUENCY NUMBERS

Two numbers which are functions of frequency of 

oscillation or rotation are deduced from the dimensional 

analysis. The frequency number 6 has been used 

by Wilmarth, Hawk, and Harvey (1964) and others in 

studies of resistance to motion. This number is a significant factor only when cyclic oscillations or complete 

rotation occurs.

The density-frequency number $ reflects the density 

ratio between the fluid and the particle. Logically, 

density should affect the frequency of oscillation through 

the inertial forces; this number, therefore, may be 

significant.

STABILITY NUMBERS

The steadiness of a particle does not influence resistance to motion in studies on rigidly mounted particles, nor on particles which maintain a stable 

orientation during free fall. For freely falling disks, 

the stability number I can be used in conjunction with 

the Reynolds number to define a region of stable fall, 

and hence is termed a "stability number" (Wilmarth 

and others, 1964). Since particles with shapes other than 

disks are used in this study, the number will be specifically defined for each shape on the basis of the 

definition for disks as a pattern.

The stability number for disks, as defined by Wilmarth, Hawk, and Harvey, is the ratio of the mass 

moment of inertia of a thin disk about its diameter to

a quantity proportional to the mass moment of inertia 

of a rigid sphere of fluid of the same diameter as that 

of the disk about its diameter. Or, in equation form:

pfa

where

then

I==_T pp C

64 pf a

(22)

(23)

(24)

Thus, it can be seen that I is a constant multiplied by 

the density ratio of the disk and the fluid in which 

the particle is submerged and by the thickness-diameter 

ratio of the disk.

The stability number for oblate spheroids is defined 

in the same way as for disks. Therefore, with:

(25)

(26)

20

the stability number becomes:

-P a2 + C2 pp

a5 20 pf

The stability number for cylinders is defined as the 

ratio of the mass moment of inertia of a cylinder 

about an axis perpendicular to its major axis to a 

quantity proportional to the mass moment of inertia of 

a rigid disk of fluid whose diameter is equal to the 

length of the cylinder about an axis perpendicular to 

the diameter of the fluid disk. 

Therefore, with:

_ _ 

16+12

the stability number is:

ft n* ^__i_^_ 

16^12

(27)

(28)

The stability number for prolate spheroids is defined 

in a manner similar to the stability number for cylinders. The moment of inertia is:

(29)

(30)

20

and the stability number becomes:

a5 20 pf
CIO SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

The stability number for spheres, if defined in terms 

of the moment of inertia of the sphere and the moment 

of inertia of the associated fluid sphere of the same 

diameter, is the density ratio of the sphere and the 

fluid.

FORCE NUMBER

The force number, known as a size coefficient by other 

investigators, has been used to estimate fall velocity of 

a nominal sphere in fall-velocity studies. Not only does 

it come from the dimensional analysis, but it can be 

derived from Newton's drag equation. By proper 

manipulation of equation 4, and assuming the area 

term A can be written as some constant k times a 

characteristic diameter, the drag equation can be 

written as:

(31)

or

JL

pfv2=kCD R2 (32)

where k is ki/2.

Since F is the buoyant weight of the particle, for 

any discrete particle falling in a fluid of constant 

density and viscosity, the force number X is unique. 

This indicates that the product ODW is constant, and 

that a line of constant X on the OD-R diagram should 

have a slope of 2.

The unique nature of the relationship among X, CD, 

and R indicates to some extent the influence that 

shape, diameter, and velocity have on the position 

on the (To-R diagram for any discrete particle. If for 

any given value of X equation 32 is rewritten as:

k=Cn R2 (33)

CD is replaced by its definition as derived from equation 

4, and the Reynolds number is written in accordance 

with its definition, then k is:

(34)

or by letting A=kid?, then 

, 2F I

kipf (cod) 2 v2

(35)

In equation 35, note that u>d is common to both the 

Reynolds number and the coefficient of drag. Therefore, any factor which affects either to or d affects both 

CD and R, changing the individual terms, but not the

constant-product relationship. If by some means the 

shape of a particle could be changed while maintaining 

the same buoyant weight, the relationship of CD to R 

for the particle would shift on the CD-R diagram to 

a new position along the line of constant X. Any other 

parameter which might affect velocity, such as roughness or roundness, would also simply change its position 

along the line of constant X.

In the literature, the question as to what diameter 

should be used in computing CD and R for irregularly 

shaped particles is discussed. Equation 35 indicates 

that the only effect of picking one particular diameter 

over another would be to shift position on the CD-R 

diagram on a line of constant X. To be specific, because 

the area associated with the nominal diameter of a 

nonspherical particle is less than the maximum projected area of the particle, its characteristic diameter 

is the smaller for the same particle. Hence, the point 

based on the nominal diameter plots at a greater value 

of CD and a lesser value of R than the point based on 

the characteristic diameter associated with the maximum projected area, but both points are on the same 

Xline.

CHOICE OF CHARACTERISTIC DIAMETERS AND 

VELOCITIES

The computations for CD and R require the use of 

a characteristic length and velocity. Two characteristic 

diameters and two characteristic velocities are used in 

this report.

The characteristic lengths are the maximum diameter 

dm and the nominal diameter dn . The diameter dm is 

the diameter of a circle which has an area equal to the 

maximum projected area of the particle. When 10<CR<C 

100, the particle falls along a straight vertical path, 

and does not oscillate or rotate; the maximum area of 

the particle is perpendicular to the direction of fall. 

Under these circumstances, according to Newton's law 

of resistance, the resistance is a function of the maximum projected area; hence dm is used. To maintain 

consistency throughout the entire range of Reynolds 

numbers, dm is used even when the particle does 

oscillate and rotate and when the path of fall is not 

vertical. The nominal diameter dn is used as a basis of 

comparison with the work of others because fall velocity 

is usually expressed in terms of dn in the literature.

In a following section, the data for each diameter 

are plotted separately on the (To-R diagram. However, 

if the data for both diameters, dn and dm, were plotted 

on the same diagram, points could be moved from one 

curve to the other simply by moving them along lines 

of constant force number X.

The characteristic velocities are the path velocity, 

or the time-rate of change in distance along the path
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS Cll

of the particle, and the vertical velocity or the timerate of change of distance in a vertical direction. 

When the difference between path and vertical velocities is not significant, only the relationship of CD to 

R using path velocity will be given.

PARTICLE STEADINESS

The forces affecting the steadiness of a particle need 

to be discussed because the steadiness has a large effect 

on its fall pattern. The forces acting on a disk when 

R >100 will be used as an example (fig. 4).

When the plane of the disk face is neither perpendicular nor parallel to the direction of motion, the resultant pressure force opposing motion acting normal 

to the face of the disk has both horizontal and vertical

FIGURE 4. Forces on a falling disk. Component L is the lift 

force normal to the drag, and component D is the drag force 

parallel to the direction of travel. See text for description of 

symbols.

307-965 O 69 3

components. The vertical component resists the vertical 

fall of the disk, whereas the horizontal component 

causes its horizontal movement. If, as a result of circulation or the nonsymmetrical formation of the trailing 

vortex, the center of pressure does not act through the 

center of gravity of the disk, a torque will be exerted 

on the disk. The magnitude of the torque will be equal 

to the product of the magnitude of the resultant pressure 

force and the distance along the disk between the center 

of pressure and the center of gravity (fig. 4). The plane 

of the disk face must continually change because there 

is no movement to counterbalance the torque. This 

change in the angle of inclination is the reason that an 

oscillating disk swings through curvilinear arcs. 

With reference to figure 4:

mg=buoyant weight of the particle, 

Z)=drag force, 

L=lift force, 

1=distance between the center of gravity and

center of pressure,

u=horizontal velocity of the particle (cm/sec), 

v=vertical velocity of the particle (cm/sec), 

a=angle of attack or the angle between the plane

of the disk face and direction of travel, and 

/8=angle between direction of travel and the

vertical.

By appropriate substitutions, the differential equations 

of motion for a freely falling disk can be written:

TO

dv mg+D cos @-\-L sin /3=0, (36)

du TO j?+D sin p L cos 0=0, and (37) dt

D sin2 a+CL cos2 a) -7^=0. (38)

At the instant an oscillating disk stops at the apex 

on one of the arcs, the plane of the face is inclined with 

the horizontal, and the disk weight is the only force 

acting on it. Once motion begins, the horizontal component of the resultant pressure changes the horizontal 

position, the vertical component opposes the motion, 

and the torque changes the angle of inclination, turning 

the disk towards the horizontal. This process continues 

until the apex at the opposite end of the arc is reached. 

This apex is reached rather abruptly once the angle of 

inclination is upward, because the resultant force and 

the weight of the particle combine to reduce the velocity 

of the disk and cause it to stall It is then in a position 

to begin its next arc of motion.

The net result of all the forces acting on a disk is a 

dynamic situation involving the momentum of the
C12 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

disk, and the horizontal, vertical, and torque components of force acting on it. All these variables continually 

change with time.

PARTICLES, LIQUIDS, EQUIPMENT, AND PROCEDURE 

PARTICLES

The particles studied included spheres, cylinders, 

disks, and oblate and prolate spheroids. Oblate spheroids 

are solids of revolution generated when an ellipse is 

rotated about its minor axis. The maximum projected 

area of an oblate spheroid is circular. Prolate spheroids 

are solids of revolution generated when an ellipse is 

rotated about its major axis. The maximum projected 

area of a prolate spheroid is elliptical. Nylon spheres, 

teflon spheres, and steel spheres, of three sizes and one 

tungsten carbide sphere were used (fig. 5). The diameters 

of the nylon, teflon, and steel spheres were 1.91, 2.54, 

and 3.81 cm, respectively. The tungsten carbide sphere

was 2.54 cm in diameter. In all, there were 10 spheres, 

no two being alike in both material and size. The other 

dimensions and properties of the spheres are given 

in table 1.

Three sizes of cylinders, disks, oblate and prolate 

spheroids were machined from lead and aluminum, 

making a total of 24 nonspherical particles, no two 

being the same size, shape, and density. The lengths 

of the major axes were 1.91, 2.54, and 3.81 cm. The 

thickness to diameter ratio was 0.1 for the disks and 

0.5 for the oblate spheroids. The length to diameter 

ratio was 4.0 for the cylinders and 2.0 for the prolate 

spheroids. Additional properties of these particles are 

also summarised in table 1.

LIQUIDS

Seven different liquids, pure glycerine, water, and 

five different mixtures of glycerine and water were used 

in the expert?ent. Percentages of glycerine used to

TABLE 1. Particle properties, measured and computed

Measured

Particle No. Material and Specific gravity Weight 

shape i (g)

Length

Maximum 

a

(cm;

Minimum 

c

(cm)

Ratio 

c/o

Maximum 

projected 

area 

A

(cm 2)

Maximum 

projected 

diameter 

dm

(cm)

Volume 

V

(cm 3)

Surface area 

S

(cm 2)

9

10

11

12

13

14

15

16

17

18

19

20

N-S 

N-S 

N-S 

T-S 

T-S

T-S

st-s st-s

St-S 

TC-S

L-OS 

L-OS 

L-OS 

L-PS 

L-PS

L-PS

L-C

L-C

L-C

L-D

1. 14

1. 14

1. 14

2. 15

2. 15

2. 15

7.68

7. 68

7.90

14.95

10. 15

10. 15

10. 15

10. 15

10. 15

10. 15

10. 15

10. 15

10. 15

10. 15

4. 17

9.87

33. 22

7.82

18.60

62. 92

27.76

65.68

229. 56

128. 34

21.38

45.30

156. 85

9.94

23. 26

76. 52

3.88

8. 12

27.49

5. 50

1.91

2. 54

3. 81

1.91

2. 55

3. 82

1.91

2. 54

3. 80

2. 54

1.91

2.49

3.85

1.91

2. 52

3.85

1.91

2. 54

3.81

1.91

1.91

2. 54

3. 81

1.91

2. 55

3.82

1. 91

2. 54

3.80

2.54

1.04

1. 33

1. 89

.96

1.28

1.92 

.48 

.63 

. 95 

. 19

1.00

1.00

1.00

1. 00

1.00

1. 00

1.00

1.00

1.00

1. 00

.54 

.53 

.49 

.50 

. 51

. 50 

.25 

.25 

.25 

. 10

2.86

5.08

11. 40

2. 86

5.09

11.43

2.85

5. 07

11. 36

5. 06

2.88

4.87

11. 62

1.44

2. 54

5.81

.91

1. 61

3.63

2.87

1.91

2. 54

3.81

1.91

2. 55

3.82

1. 91

2. 54

3.80

2. 54

1.91

2.49

3.85

1. 35

1. 80

2. 72

1.08

1.43

2. 15

1.91

3. 64

8.61

28. 96

3.63

8.64

29. 07

3. 62

8.58

28. 84

8.58

2. 11

464

15.45

.98

2.29

7.54

.33

.80

2.71

. 54

11.46

20. 25

45. 53

11.44

20.41

45.78

11.40

20. 25

45.28

20.25

8.20

13.86

31.89

4. 79

8. 71

19. 9

3. 24

5.49

12.79

6.28

21

22

23

24

25

26

27

28

29

30

31

32

33

34

L-D

L-D

A-OS

A-OS

A-OS

A-PS

A-PS

A-PS

A-C

A-C

A-C

A-D

A-D

A-D

10. 15

10. 15

2. 81

2.81

2. 81

2. 81

2.81

2.81

2. 81

2.81

2. 81

2. 81

2. 81

2. 81

13.07

44 20

5.60

12.66

45.05

2.48

5.98

20.00

.90

2. 15

7.62

1. 54

3. 62

12.27

2.54

3. 81

1.91

2.50

3. 78

1.83

2. 50

3.78

1.91

2. 54

3.81

1. 91

2. 54

3.81

.26

.38

.99

1.31

1.96

.95

1. 27

1.88

.47

.64

.96

. 19

.26

.38

. 10

. 10

.52

.52

. 52

. 52

. 51

. 50

.25

.25

.25

. 10

. 10

. 10

5.08

11.42

2.85

490

11. 21

1.37

2. 49

5.57

.94

1. 62

3.64

2.87

5. 08

11. 39

2. 54

3.81

1.91

2.50

3.78

1. 32

1. 78

2.67

1. 09

1.44

2. 15

1.91

2. 54

3. 81

1. 29

4 35

1.99

4 51

15. 99

.88

2. 13

7. 11

. 32

.76

2. 71

. 55

1.29

4 37

12.20

26. 13

8.02

15.63

31.70

4.72

8. 52

19. 06

3. 17

5. 76

12. 74

6.87

12. 17

27. 39

1 N, nylon; T, teflon; S<, steel; Tc , tungsten carbide; L, lead; A, aluminum; S, sphere; OS, oblate spheroid; PS, prolate spheroid; C, cylinder; D, disk.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS 13

TABLE 1. Particle properties, measured and computed Continued

Computed

Particle No.

1

2

3

4

5

6

7

8

0 fy

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

Nominal 

diameter 

dn

(cm)

1.91

2.54

3.81

1.91

2.55

3. 82

1.91

2.54

3.80

2.54

1.59

2.04

3.09

1.23

1.64

2.43

.86

1.15

1.73

1.01

1.35

2.03

1.56

2.05

3. 12

1. 19

1.60

2.39

.84

1. 13

1.73

1. 02

1.35

2.02

Nominal 

area

An

(cm 2)

2.86

5.08

11. 40

2.86

5.09

11. 00

2.85

5.07

11.36

5.06

1.99

3.27

7.46

1. 19

2. 12

463

.58

1. 04

2.36

.80

1.42

3.22

1.90

3.30

7.63

1. 12

2.00

4.47

.56

1. 00

2.34

.82

1.43

3.22

Surface 

diameter 

dA

(cm)

1.91

2.54

3.81

1.91

2.54

3.82

1.91

2.54

3.80

2.54

1.62

2. 10

3. 17

1.24

1.67

2.52

1. 02

1.32

2.02

1.41

1.97

2.88

1.60

2.23

3. 16

1.23

1.65

2. 46

1.00

1.36

2.03

1.48

1.97

2.95

Ratio 

dA/dn

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1. 00

1.00

1.00

1.02

1.03

1. 03

1. 01

1. 02

1.04

1. 18

1. 15

1. 17

1.40

1.46

1.42

1.03

1.09

1.01

1.03

1.03

1.03

1. 19

1.20

1. 17

1.45

1.46

1. 46

Shape 

factor 

SF

1.00

1.00

1.00

1.00

1.00

1.00

1. 00

1.00

1.00

1.00

.54

.53

.49

. 71

.71

. 71

.50

.50

.50

. 10

. 10

. 10

.50

.50

.50

.72

.71

.70

.49

.50

.50

. 10

. 10

. 10

Sphericity 

*

1.00

1.00

1.00

1.00

1.00

1 j.. on\j\j

1.00

1. 00

1.00

1.00

.83

.82

.80

.64

.60

.63

.43

.44

.43

.53

.53

.53

.83

.82

.82

.65

. 64

.63

.43

.43

.40

.53

.53

.53

0.0

16.8

16.8

16.8

8.38

8.38

8.38

438

4.39

4 39

4.98

4.98

4.98

465

465

4.65

2.32

2.32

2. 32

1.22

1.22

1.22

1.38

1.38

1.38

Stability number 1X102

70.0

13.9

13.9

13.9

6.90

6.90

6.90

3.65

3.65

3.65

4.21

4.21

4.21

3.88

3.88

3.88

1.95

1.95

1.95

1. 02

1. 02

1.02

1. 16

1. 16

1. 16

Percent glycerine 

83.3 90.1 05.2

13.6

13.6

13.6

6.80

6.80

6.80

3.56

3.56

3.56

4 10

4 10

4 10

3.80

3. 80

3.80

1.90

1.90

1.90

1.00

1.00

1.00

1. 13

1. 13

1. 13

13.4

13.4

13.4

6.70

6.70

6.70

3.50

3.50

3.50

4 02

402

402

3. 74

3.74

3.74

1.87

1.87

1.87

.98

.98

.98

1. 11

1. 11

1. 11

13. 3

13.3

13.3

6.64

6.64

6.64

3.47

3.47

3.47

3.99

3.99

3.99

3.71

3.71

3. 71

1.86

1.86

1.86

.97

.97

.97

1. 10

1. 10

1. 10

08.0

13.3

13.3

13.3

6.64

6.64

6.64

3.47

3.47

3.47

3.97

3.97

3.97

3.71

3.71

3.71

1.86

1.86

1.86

.97

.97

.97

1. 10

1. 10

1. 10

100.0

13.2

13.2

13.2

6.60

6.60

6.60

3.45

3.45

3.45

3.96

3.96

3. 96

3.68

3.68

3.68

1.84

1.84

1.84

.96

.96

.96

1.09

1.09

1.09

define the specific mixtures were determined by weight. 

The glycerine and water were mixed with a large paint 

mixer for 5-10 minutes in two large interconnected 

barrels. During the mixing process the fluid was circulated from one barrel to the other to insure a homogeneous fluid. When mixing was completed, the fluid 

was pumped into the fall column. The viscosities of the 

liquids (fig. 6) were measured with calibrated CannonFenske viscometers. The densities of the liquid mixtures 

were taken from standard density tables (fig. 7).

Temperatures were measured at the top, center, and 

bottom of the test section at the beginning of each day 

and periodically throughout the day. Rarely did the 

temperature vary more than 0.5° C through the test 

section or more than 0.2° C throughout the day.

EQUIPMENT

The fall column was a vertical clear plexiglass 

cylindrical column 3 meters high and 40 cm in diameter

(fig. 8). The wall thickness was 0.625 cm. A 1-meter 

test section was located near the bottom of the column 

to allow the particles to fall through the liquid for 

about 1.5 meters to ensure that they reached terminal 

fall velocity before they entered the test section. The 

test section was circled with black plastic tape at 10 

cm intervals. The clear plexiglass column and the fluid 

made it possible to see the tape on the back of the 

column and permitted a close approximation of the 

vertical position of the particle. After the experiments 

were started, a mirror was added at the bottom of 

the column to reflect the top light. Then, both the top 

and the bottom of the particle were illuminated.

A wire basket was suspended below the test section 

to retrieve the particles. Periodically it was withdrawn 

from the column and the particles removed. A minimum of 10-15 minutes was allowed for the fluid to
C14 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

!\

IS

- L . I _

FIGURE 5. Particles used in experiments. Particles were marked 

with black stripes to make rotational behavior observable. 

A, Nylon, teflon, steel, and tungsten carbide spheres in order 

from top to bottom. B, Aluminum disks, oblate spheroids, 

prolate spheroids, and cylinders in order from left to right.

return to equilibrium after the particles had been 

retrieved.

Photographs of each drop were made with two 16-mm 

reflex movie cameras equipped with 10-mm wide-angle 

lenses and set 90° to one another with respect to the 

column (fig. 8). The speed of each camera could be 

varied from 8 to 64 frames per second. All drops were 

made at 16, 32, or 64 frames per second, depending on 

the rate of particle fall and detail required to define 

particle behavior.

A 10-second sweep timer was used to obtain direct 

readings to 0.1 second and estimated readings to 0.01 

second. The timer was mounted at the side of the 

column and set level with the center of the test section. 

It was illuminated by two 100-watt incandescent light 

globes shielded to protect the film from direct exposure 

by this source. One camera photographed the timer 

directly, the other photographed an image of the timer 

reflected to it by a mirror. Thus, a reference time 

image was recorded on the pictures obtained with both 

cameras.

0.005

25 30 35 40 

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 6. Kinematic viscosity of glycerine-water mixtures in 

terms of percent glycerine as a function of temperature. Fluid 

1 was 100 percent water.

Two 500-watt photo floodlights and a 300-watt incandescent lamp were suspended at the top of the 

column to direct light down through the column. The 

lights were turned on only during the run to prevent 

excess absorption of heat in the top layers of the fluid.

The developed pictures were projected onto a grid 

to make the necessary particle location measurements. 

The projector was equipped with a single or multiple 

framing switch and a reverse switch that enabled the 

operator to change the pictures one frame at a time, 

watch a series of pictures as a slow-motion movie, or 

to reverse the film thus permitting "re-runs" of short 

sections. These switches were mounted in a remote 

control box to enable the operator to control the projector from a position near the grid.

The grid was a sheet of 10- by 10-to-the-inch graph 

paper. The projector was positioned to give a projection
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C15

1.27

1.25

1.23

1.21

1.19

1.17

10 15 20 25 30

TEMPERATURE, IN DEGREES CELSIUS

35

FIGUKE 7. Density of glycerine-water mixtures in terms of 

percent glycerine as a function of temperature.

scale of 1 inch on the grid for 10 cm of the fall column. 

This arrangement permitted reasonably accurate estimates of the particle position at any time.

PROCEDURE

A permanent record was made of each drop of each 

particle with the two 16-mm movie cameras. The 

motion pictures thus made permitted a more accurate 

visualization of particle behavior and fall pattern than 

could be obtained with still pictures. The cameras were 

adjusted so that the camera lens were level with the 

center of the test section. The distance from the 

camera to the fall column was 7.5 feet for the first 

series of tests and 6 feet for the last series. The distance 

was changed to get better lighting characteristics.

The cameras were manually operated throughout 

the experiment, and the particles were dropped by 

hand.

Clock

_v

Camera

Mirror

Camera 

PLAN VIEW

I Lights

Clock

Tube

.Grid

Camera

Screen

Mirror

SIDE VIEW

FIGURE 8. Schematic diagram of fall column.

Determination of path and vertical drop lengths 

required measuring the distances between the positions 

of the particles in successive pictures as they were 

projected onto the grid.

Inherent in this procedure is the assumption that the 

particle fell in a plane through the center line of the
C16 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

column perpendicular to the camera. Generally, this 

was not the case. However, when the angle from the 

camera to the top or bottom of the measured section 

and the angle of refraction of the light in the fluid were 

considered, a maximum error of ±6.5 to ±8.25 percent 

in water and ±5.95 to ±7.45 percent in glycerine can 

result. No attempt was made to compensate for the 

error in defining particle behavior.

The distortion of the curved surface of the column 

was neglected. The error thus caused was affected by 

the position, parallel to the line of sight, of the particle 

in the column (fig. 9).

The three-dimensional length of the fall path of the 

particle was determined by taking the x and z, or the 

horizontal and vertical, coordinates from one camera 

and the y coordinate, from the other camera. The 

square root of the sum of the squares of the differences 

in position of successive pictures, summed for the 

entire measurable length, determined the length of the 

fall path. The vertical path was measured by taking 

the difference in the z coordinates of the particle at the 

beginning and end of the run. Except for disks, the fall 

and vertical paths were usually identical.

The path and vertical velocities of the particles were 

determined by dividing the path and vertical distance 

by the elapsed time of fall.

Some of the drops for disks required special analysis 

to determine the fall velocities. For example, during the 

drop through the test section, a disk would often 

tumble from one side of the column to the other, strike 

the wall, bounce along it for a short distance, then 

move away and begin a tumbling descent through the 

remainder of a test section. In these cases, only those 

parts of the drop where the disk was away from the 

wall were used in the computations. To arrive at a

Wall of fall column

Light ray

10 20 CENTIMETERS

FIGURE 9. Distortion of light rays caused by* fluid refraction 

and the curved surface of the fall column.

single value of the velocity for that run, the distances 

which the particle fell in the center of the column were 

added and the sum divided by the elapsed time for 

falling that distance. For example, if a particle fell 

through the first 18 cm in 2 seconds and then struck 

the wall, fell down the wall for 10 cm, and then fell 

freely for 52 cm in 5 seconds before striking the opposite 

wall, the length of the fall path was treated as being 

70 cm, elapsed time as 7 seconds, and the velocity as 

10 cm/sec. If the same disk was dropped in the same 

fluid more than once, then the aggregate free-fall 

distance of all drops was divided by the accumulated 

free-fall time to determine the single velocity used in 

computations of CD and R for that disk in that fluid. 

The procedure outlined above gave the greatest 

weight in the averaging process to the longest free-fall 

distance.

EXPERIMENTAL RESULTS

In the presentation of results, a few symbols have 

been changed from those common to current literature. 

Fall velocities and resistance to fall are expressed in 

terms of the coefficient of drag and the Reynolds 

number diagrams. The basic drop data and some parameters computed therefrom are given in tables 2-6 for 

spheres, disks, oblate spheroids, cylinders, and prolate 

spheroids. The results for each differently shaped 

particle are plotted separately to avoid obscuring any 

significant relationships in a maze of points. The curves 

thus developed are later plotted together for comparison. The curves of the relation of CD to R for fixed 

spheres and for disks, and a curve computed from 

Stokes' equation, are plotted on all diagrams for reference. The particular curves for fixed disks and spheres 

were taken from Rouse (1946).

DESCRIPTION OF FINDINGS

Terms relating to orientation, velocity, and path of 

travel of a particle need to be clearly defined. When a 

particle falls with its maximum projected area perpendicular to the direction of fall, in a straight vertical 

path at a constant velocity, the particle is said to be 

steady, and the condition of the fall is stable. When the 

particle orientation, direction of path, and velocity 

change during fall, the particle is said to be unsteady, 

and the condition of fall may be either stable or unstable. Reference to fall patterns of particles and their 

effect on the appropriate descriptive parameters will 

be made in these terms.

SPHERES

Generally, the spheres fell in a straight, vertical 

path. Exceptions were small erratic nonrepeating 

horizontal movements exhibited by two or three of the 

nylon and teflon spheres as they dropped in water. The
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C17

TABLE 2. Drop data for spheres 1

Run No. 2

Travel

distance

(cm)

Mean 

velocity 

(cm/sec)

CD and R values for 

dm and dn

CD R X 10-2

1-1-10

3-1-

100.0

103.4

102.7

102.0

100. 6

6-1-6--------------- 104.6

7-1-6--------------- 101.6

8-l-5-__--_____-_--- 102.6

15-l-l__---_-_-__-_-- 100.1

____.-__-_-__-- 102.5

17-l-3-_---_-_-_-.__- 102.4

18-l-2___________ 101.5

57-2-5___--___ _. 100.0

58-2-6__-_-_------__ 100.0

_____ _ _______ 100.0

____________ 99.0

61-2-9___-___-_ __ 100.0

62-2-9___-_-_--____-_ 102.0

_-___-____-- 98.0

._-___-_____- 100.0

90-3-5--------------- 100.0

91-3-6--------------- 100.0

-_-___-_____- 100.0

_-___-___-___ 101.0

--_--_-_-____ 98.1

98. 1

98. 0

100.0

100. 0

100.0

99. 1

98. 1

69.7

97.8

91.3

100. 0

___.---._-___ 99.0

185-5-6----_----_---__ 100.3

190-5-9_-___--_- -___ 100.8

194-5-7_-___-__-__-__- 100.1

195-5-10.--.---------- 77.0

240-6-4_____.--.__.._. 100.5

241-6-5___-___-_______ 61.2

243-6-6----.______ 100.9

247-6-7___-__----.___ 101.5

_----___-__ 97.4

98.2

102.7

101.3

101.8

101. 1

249-6-9.

345-7 6_________ _____ 95. 6

346-7-7.___. __________ 97.1

347-7-8__--_-_._____- 100.2

348-7-9___----.______ 96.7

349-7-10.-_-__._______ 100. 3

303.0

258. 6

209.6

185.4

192.8

94. 7

98.4

87.5

29. 9

80.0

39.0

34.4

24. 9

44. 5

83.9

114.0

180.0

151.0

206. 0

22. 0

31.4

49.6

91.0

122. 0

184. 5

190.0

202.0

25. 0

35. 0

56.0

101. 0

134.0

201. 0

231. 0

197.0

37.0

48.8

70.2

252.0

129.8

256.5

45.7

55.6

80. 8

145. 2

198.2

233. 0

267.0

158. 3

67. 1

77.4

101.0

162.5

208.0

263.0

305. 0

0.51 

.51 

.49 

.51

.42

. 64 

.60 

.51 

.42 

.46

.39

.43

3.81

1. 79

1.80

1.29

.82

.96

1.07

4.05

2.42

1.47

1.54

1. 14

.78

.74

.88

2.90

1.98

1. 16

1. 26

.94 

. 66 

.87 

.69 

1.34

1. 03 

.76 

.42 

.76 

.56

.91 

. 83 

. 59 

.62

.44

.51 

.53 

.53 

.45 

.40

. 41 

.50 

.40 

.41 

.40

927

1, 180

641

426

461

435

453

268

69

184

179

105

18

.48

.45

.80

1.85

1.55

1.46

. 15

.29 

.64 

.62

1. 11

2. 51

2.58

1.84

.21

.39

.93

.84

1.48

3.33

2.55

3. 25

.59

1.03

2.22

7.75

2. 05

5. 39

1.40

2.27

4.95

4.45

8.05

14. 1

10.8

4.85

7.64

11.7

23

18.5

31.4

58.5

46

1 The vertical and path data for travel distance, velocity, CD and R are identical 

(or spheres.

2 The run number is a combination of drop number, fluid number (fig. 6), and 

particle number.

horizontal movements increased the fall-path length a 

maximum of 3 percent. Average increase in length was 

1 percent. Occasionally, a steel sphere exhibited a 

horizontal translation as it fell through the test section. 

This translation also increased path length to a maximum of 3 percent. Specific characteristics of this unstable condition could not be repeated with other 

drops.

Only one sphere exhibited any perceptible rotation. 

The 3.81-cm teflon sphere falling in water rotated 

approximately 45° during the last 50 cm of fall.

The CD-R curve for the falling spheres was virtually 

the same as for spheres rigidly mounted in a moving 

fluid (fig. 10). The scatter in the data can be ascribed 

to experimental error.

DISKS

Several broad systematic patterns of fall existed for 

a freely falling disk.

Steady-flat fall

When the Reynolds number for a falling disk was 

less than 100, the disk fell at a uniform velocity with 

the maximum projected area perpendicular to the direction of fall and had no tendency to oscillate. This indicated a very stable condition because oscillations 

caused by some outside influence damped out with time 

and distance. The stable fall pattern of a falling disk 

is typified in figure 11. Drawings from successive pictures of the movies are used to represent the steady-flat 

fall pattern (fig. 11).

Regular oscillation

At Reynolds numbers slightly greater than 100, the 

disk oscillated about a diameter perpendicular to the 

direction of fall. Very little horizontal translation was 

associated with the oscillation (fig. 1). As R increased, 

the amplitude of the oscillation and the horizontal 

translation increased. The path of travel was almost 

parallel to the face of the disk, and the drag was mainly 

shear drag (figs. 1, 12).

In theory, the amplitude of the arc through which the 

disk oscillates is a function of the balance between the 

weight of the disk and the lift and drag forces acting 

on it. The greater the lift force with respect to the 

disk weight and the drag force, the greater will be the 

amplitude of the arc through which the disk oscillates. 

Wilmarth, Hawk, and Harvey (1964) found that the 

amplitude of the arc was inversely proportional to the 

stability number I of the disk. Because I is the product 

of (a) the density ratio of the disk and the fluid and 

(b) the thickness-diameter ratio of the disk, decreasing 

I merely decreases the ratio of the weight to area of 

the disk. Therefore, assuming a constant velocity, 

decreasing I will increase the lift-weight ratio of the
CIS SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS 

TABLE 3. Drop data for disks

19-1-34. _ . 

20-1-33. ._ 

21-1-32-.- 

23-1-33--. 

24-1-33.-.

25-1-32. __ 

42-1-21--. 

43-1-22--. 

73-2-20. _ . 

74-2-21---

75-2-22. _ _

87-2-34. __

106-3-20--.

107-3-21. ._ 

108-3-22.--

119-3-33--. 

120-3-34-.- 

121-3-34. . . 

143-4-33--.

144-4-34- _ .

145-4-20- . - 

146-4-21--. 

147-4-22--. 

148-4-22-.- 

156-5-32.--

157-5-33-.- 

158-5-34- __ 

159-5-20--. 

160-5-21. ._ 

161-5-22--.

162-5-22. . . 

169-5-22. _ _ 

170-5-22-.- 

186-5-22... 

187-5-22-.-

188-5-22--. 

189-5-22-.. 

192-5-22 .. . 

193-5-22.-. 

196-1-32. _ _

197-1-32- . _ 

198-1-33-.- 

199-1-33-.- 

200-1-34. _ _ 

201-1-34..-

202-1-20- ._ 

203-1-20--. 

204-1-21... 

206-1-22. ._ 

207-1-22. . .

224-6-32...

225-6-33-.- 

226-6-34..-

227-6-20--. 

228-6-21--.

229-6-34-.. 

230-6-20--. 

231-6-21- . _ 

232-6-22. __ 

233-6-22. __

Travel distance

Path 

(cm)

121.8 

61.8 

95.0 

30.4 

129.0

121.2 

51.7 

102.9 

100.0 

100.0

100.0 

100.0 

100.0 

80.0 

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

101.0 

100.0 

99.0 

99.0 .

100.0 

100.0 .

100.0 .

100.0 .

100.0 .

78.0

91.6 

75.2 

83.4 

119.5 

121.0

118.4 

68.2 

103.2 

104.4 

93.3

58.0 

114.5 

58.9 

147.6 

55.5

76.8 

74.0 

64.0 

72.9 

98.6

101.0 _

100.0 .

99.5 _

115.8 

112.4

80.0 .

111.9 

89.0 

101.2 

96.4

Vertical 

(cm)

70.5 

31.0 

46.5 

21.0 

91.5

69.5 

33.5 

67.5 

100.0

61.0

72.5 

60.0 

70.0 

100.0 

96.5

78.5 

52.0 

74.5 

80.5 

56.5

26.5 

63.5 

39.0 

74.5 

30.0

43.0 

42.0 

45.5 

59.0 

66.5

100.0 

76.5

100.5 

66.5 

68.0 

66.5

Velocity

Path 

(cm/sec)

86.5 

74.3 

58.0 

73.5 

58.5

70.5 

76.2 

102.5 

26.2 

36.2 _

53.9 .

17.4 .

29.4 _

42.0 .

61.0 _

12.2 _

19.6 _

20.2 .

13.6 _

20.4 .

33.8 .

46.1 _

67.9 _

66.0 _

14.2 .

18.7 _

27.4 _

45.0 _

56.9 _

95.1

92.3 

94.0 

100.5

87.5 

97.0

101.8 

89.1 

102.0 

99.2 

57.5

56.9 

75.0 

93.7

78.5 

99.7

51.4 

78.8 

66.8 

109.2 

91.5

17.1 -

22.2 _

29.3 .

53.5 

78.6

30.0 .

50.9 

79.1 

79.5 

94.5

Vertical 

(cm/sec)

50.2 

37.1 

28.4 

50.9 

41.6

40.5 

49.8 

67.5 

26.2

74.5

72.5 

75.0 

84.4 

73.2

77.4

67.7 

67.8 

73.5 

76.4 

34.8

26.0 

41.5 

62.1 

39.8 

54.1

28.8 

44.8 

47.3 

88.5 

61.8

46.7 

53.5

44.6 

59.1 

53.4 

65.4

CD and R values for dm

Path 

CD RX10-2

0.28 

.16 

.20 

.17 

.26

.14 

.79 

.65 

3.82 

2.68

1.82 

3.05 

3.04 

2.00 

1.42

4.16 

2.44 

2.28 

3.38 

2.26

2.31 

1.67 

1.16 

1.22 

2.33

1.80 

1.28 

1.30 

1.11 

.60

.64 

.61 

.54 

.70

.57

.52 

.68 

.52 

.55 

.20

.21 

.16 

.10 

.22 

.14

1.29 

.54 

1.02 

.62

.82

1.56 

1.32 

1.15 

.94 

.59

1.09 

1.05 

.58 

.87 

.61

324 

226 

133 

224 

179

163 

234 

472 

.14 _

.26 .

.57 _

.19 _

.20 _

.38 _

.83 .

.11 _

.27 .

.28 _

.15 _

.34 .

.28 _

.51 _

1.12 .

1.09 .

.22 _

.39 .

.79 .

.70 _

1.20 _

3.00

2.90 

2.96 

3.17 

2.76 

3.06

3.20 

2.80 

3.21 

3.12 

108

107 

187 

234 

294 

373

97.6 

148 

168 

409 

343

.52 _

.90 _

1.79 .

1.64 

3.20

1.83 _

1.51 

3.22 

4.84 

5.76

Vertical 

CD RX10-2

0.54 

.66 

.85 

.35 

.48

.42 

1.84 

1.52

.97

.99 

.96 

.76 

1.01 

.90

1.17 

1.17 

1.00 

.92 

.56

1.00 

.52 

.23 

.86 

.47

4.09 

1.69 

2.02 

.88 

1.80

1.24 

1.27

1.43 

1.04 

1.93 

1.28

231 

114 

65 

155 

134

93.4 

1.53 

310

2.35

2.28 

2.36 

2.68 

2.30 

2.44

2.13 

2.14 

2.31 

2.40 

65.6

49 

104 

155 

149 

203

54.1 

84.2 

119 

332 

232

1.43 

2.18

1.37 

2.40 

3.24 

3.98

CD and R values for dn

Path 

CD RX10-2

0.98 

.59

.72 

.60 

.94

.48 

2.80 

2.33 

13.6 

9.56

6.45 

10.8 

10.9 

7.15 

5.05

14.7 

8.60 

8.05 

11.8 

7.97

8.30 

5.96 

4.11 

4.36 

8.23

6.40 

4.53 

4.63 

3.94 

2.10

2.24 

2.16 

1.89 

2.50 

2.03

1.84 

2.42 

1.84 

1.94 

.72

.73 

.57 

.36

.78 

.48

4.66 

1.95 

3.64 

2.23 

2.90

5.46 

4.69 

4.06 

3.40 

2.10

3.86 

3.76 

2.08 

3.08 

2.18

172 

121 

70.5 

120 

95.4

86.6 

124 

250 

.07 

.14

.30 .

.10 

.11 

.20 

.44

.06 .

.14 .

.15 .

.08 

.18

.15 

.27 .

.60 

.58 .

.12 .

.21 .

.42 .

.38 .

.64 .

1.60

1.55 

1.58 

1.69 

1.47 

1.63

1.71 

1.49 

1.71 

1.67

57.8

57.1 

99.5 

124 

156 

198

51.2 

78.6 

88.9 

222 

183

.27 _

.48 .

.94 .

.86 

1.70

07

.82 

1.71 

2.58 

3.07

Oi

Vertical 

CD RX10-2

1.93 

2.34 

3.02 

1.24 

1.86

1.47 

6.55 

5.40

3.44

3.48 

3. 38 

2.68 

3.57 

3.20

4.14 

4.16 

3.52 

3.26 

1.95

3.50 

1.86 

.83 

3.05 

1.65

14.65 

6.04 

7.25 

3.11 

6.36

4.46 

4.55

5.14 

3.72 

6.85 

4.56

123 

60.4 

34.6 

83.0 

67.9

49.7 

81.0 

164

1.25

1.22 

1.26 

1.42 

1.22 

1.30

1.14 

1.14 

1.23 

1.28 

35.0

26.1 

55.1 

82.4 

79.0 

108

28.6 

44.7 

63.0 

177 

124

.75 

1.16

.72 

1.28 

1.73 

2.12

ability *

GT 

GT 

GT 

GT 

GT

GT 

T 

T

S 

S

S

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s

GT

GT 

GT 

GT 

GT 

GT

GT 

GT 

GT 

GT 

GT

GT 

GT 

GT 

GT 

GT

T 

T 

T 

GT 

GT

S 

S 

S

o

T

S 

0 

T 

GT 

GT

See footnotes at end of table.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C19 

TABLE 3. Drop data for disks Continued

Travel distance Velocity CD and R values for da CD and R values for d n

244-6-34. . _

245-6-20.--

246-6-2 1__ _

356-7-32. _.

357-7-32...

358-7-20--.

359-7-20. . _

366-7-33..-

367-7-21. _.

375-7-34. _.

376-7-22...

Path 

(cm)

100.0

111.2

102.1

146.8

120.2

57.1

82.5

132.8

93.4

186.1

108.9

Vertical 

(cm)

100.0

69.0

100.0

101.0

35.5

57.5

100.5

50.5

100.0

60.0

Path 

(cm/sec)

30.6

51.2

87.5

26.3

22.8

66.9

57.9

28.8

89.0

47.1

107.9

Vertical 

(cm/sec)

46.8

59.1

18.1

19.8

41.4

40.4

21.8

48.0

25.1

59.5

Path 

CD RX10-4

1.05

1.04

.48

.74

.99

.63

.83

.87

.48

.46

.49

1.86 .

1.56

3.55

3.03

2.62

7.70

6.65

4.46

13.8

10.9

25.0

Vertical 

CD RX10-J!

1.24

1.04

1.57

1.31

1.63

1.72

1.52

1.63

1.63

1.60

1.43

2.40

2.08

2.28

4.76

4.65

3.38

7.45

5.83

13.8

Path 

CD

3.71

3.71

1.71

2.66

3.55

2.26

3.01

3.08

1.70

1.64

1.74

RX10-J!

0.99 _

.83

1.89

1.61

1.40

4.06

3.51

2.39

7.31

5.82

13.4

Vertical 

CD RX10-2

4.44

3.72

5.50

4.69

5.88

6.18

5.37

5.81

5.75

5.70

.76

1.28

1.11

1.21

2.52

2.45

1.81

3.94

3.10

7.37

S

O

T

OR

O

T

T

0

T

O

T

i The ran number is a combination of drop number, fluid number (fig. 6), and particle number. 2 Q, glide fall; T, tumbling fall; S, stable orientation and path; O, regular 

oscillation; R, rotation about vertical axis.

TABLE 4. Drop data for oblate spheroids

Travel CD and R values CD and R values

Run No.2 distance Velocity for dm for dn Sta- (cm) (cm/sec) - bility 3

CD .RXIO-2 CD

26-1-25......

28-1-23......

34-1-25. --.-.

45-1-12. .... .

46-1-11......

65-2-12---..-

66-2-13......

77-2-24......

78-2-25..-..-

97-3-11--..--

98-3-12----.

99-3-13......

109-3-23. .....

110-3-24......

111-3-25......

130-4-23.-...-

131-1-24......

132-4-25 ......

133-4-11-....-

134-4-12......

135-4-13. .....

163-5-23......

164-5-24......

165-5-25......

166-5-11. .... .

167-5-12 ......

168-5-13......

234-6-23......

235-6-24. .....

236-6-25......

237-6-11--..-.

238-6-12......

239-6-13...--.

354-7-23. .....

355-7-11--..-.

1R4-7- 94

368-7-12 ..... .

374-7-13......

377-7-25......

29.0

108.4

102.1

103.4

101.2

102.4

100.0

100.0

100.0

100.0

100.0

100.0

98.0

100.0

100.0

100.0

100.0

100.0

QQ ft

100.0

100.0

101.1

99.0

99.0

99.0

102.1

99.0

100.0

100.0

101.5

99.6

QQ ft

97.6

101.6

100.0

81.5

100.5

99.6

96.5

97.5

88.1

65.6

78.6

89.6

168.9

148.5

91.5

135.5

9fi 7

41.6

74.0

OK A

137.7

22.4

31.0

48.8

25.3

34.5

51.0

90.0

110.0

146.5

36.2

45.4

61.9

113.0

128.9

166.5

42.7

53.4

70.8

123.8

139.2

177.5

54.1

139.0

64.7

151.8

177.8

81.2

0.66

.58

.83

.41

.58

.60

1.52

.99

3.12

1.98

1.86

1.39

.97

3.39

2.32

1.47

2.75

1.88

1.35

1.27

1.06

.87

1.32

1.10

.94

.81

.78

.58

.98

.83

.73

.69

.68

.63

.64

.56

.63

.59

.63

.59

400

750

358

270

506

342

.64

1.4510

.44

.50

.85

1.89

.15

.28

.66

.21

.38

.83

.75

1.19

2.45

.57

04

1 04

1.79

2.65

5.30

1.31

2.13

4.25

3.79

5.56

10.6

6.22

16

9.85

23

41.5

18.8

0.98

.87

1.22

.61

.86

.87

2.26

1.56

4.61

2.91

2.67

2.06

1.51

5.06

3.44

2.16

4.13

2.80

1 OS

1 84

1.57

1.35

1.99

1.65

1.37

1.18

1.16

1.06

1.47

1.23

1.07

.99

1.01

.98

.99

.80

.93

.88

.97

.86

331

124

296

221

415

285

.52

1.17

.15

.36

.43

.70

1.52

.13

.23

.55

.17

.31

fiQ

.62

.98

1.96

.47

.77

1.59

1.48

2.18

4.25

1.07

1.75

3.54

3.14

4.55

8.56

5.08

13.3

8.08

18.9

33.4

15.5

OT

OT

O

T

S

S

S

S

S

S

S

8

S

S

S

S

S

S

S

S

S

S

S

8

S

8

S

8

S

S

8

S

S

S

8

S

S

S

8

S

1 The vertical and path data for travel distance, velocity, CD, and R are identical 

for oblate spheroids.

2 The run number is a combination of drop number, fluid number (fig. 6), and 

particle number.

3 O, regular oscillation; T, tumbling fall; S, stable orientation and path.

disk because the lift is directly proportional to the 

area. Hence, the amplitude of oscillation should increase with I, as observed.

GIMe -tumble

The third general fall pattern which the disks exhibited as R continued to increase was a combination of

oscillation, gliding, and tumbling (figs. 13, 14). In the 

glide-tumble pattern, the amplitude of the disk oscillation increased until the plane of the disk face was 

almost vertical at the end of the arc. At this point, the 

disk fell vertically, on edge. The torque caused by 

circulation then changed the angle of inclination from 

vertical to horizontal and to vertical again. Associated 

with the change in angle of inclination was a change 

in path direction from vertically downward to horizontal to vertically upward. When the drag and gravity 

forces stopped the upward motion of the disk, the 

cycle was repeated. Whether or not the particle continued in the same horizontal direction on the next arc 

or returned on its path depended on the position of the 

disk at the end of the arc. If, for example, the disk 

approached the end of the arc from the right and 

stopped with the angle of inclination slightly to the 

right of the vertical, it would move back to the right 

on the next arc (see upper half, fig. 13). If the disk 

stopped with the angle of inclination to the left of the 

vertical, it would move on to the left in the succeeding 

arc (see lower half, fig. 13). The oscillation frequency of 

the disk was much less during the glide-tumble fall 

pattern than during the regular oscillation fall pattern.

Tumble

With further increases in R, the disks exhibited a 

fourth fall pattern, the tumble pattern. In the tumble 

fall pattern, the disk rotated through 360° at a nearly 

constant angular velocity, and the path of travel was 

nearly a straight line inclined at an angle with the horizontal (fig. 15). The tumble fall pattern is a stable one 

where the frequency of angular rotation approaches the 

frequency of oscillation in the regular oscillation 

pattern.
C20 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS 

TABLE 5. Drop data for cylinders

Travel distance

Run No. »

32-1-31.-- 

33-1-30. _ _ 

38-1-29--. 

50-1-19.-- 

51-1-18--.

7 1-2-18 _ .

72-2-19.-.

84-2-31.--

103-3-17.--

104-3-18.--

105-3-19. . _

117-3-31.--

149-4-19.--

150-4-18. _ _

151-4-31..-

154-4-17. . .

171-5-29.--

1 72-5-30. . _

173-5-31. _ .

174-5-17. . .

175-5-18.--

176-5-19.--

208-1-18.-- 

209-1-19.-- 

210-1-31---

212-6-29.--

213-6-30---

214-6-31. _ _

215-6-17.--

2 16-6-18. _ _

217-6-19.--

350-7-29. _ .

351-7-17. _ . 

360-7-30--.

361-7-18. _ .

369-7-31. .. 

370-7-19-.-

Path 

(cm)

106.1 

104.1 

103.9 

103.1 

89.1

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

99.0 .

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

100.0 .

99.0 .

100.0 .

100.0 .

33.1 

106.4 

111.4

100.2 .

100.0 .

100.0 .

101.1 .

99.0 .

100.0 .

100.0 .

102.8 

100.0 .

51.1

100.7 

54.41

Velocity

Vertical Path 

(cm) (cm/sec)

100.0 60 

100.0 47 

100.0 39 

100.0 164 

85.0 137

._____-. 42

._-.____ 67

.-______ 18

._-.__-- 33

.---.-_- 46

.-_.__-- 65

..___-__ 21

.-____-. 77

..___--- 52

.----._- 23

.---._-. 38

.-_-._-- 14

1Q

.---__.- 31

._---._- 50

.____._. 66

.-_-_.-- 102

31.0 132 

100.0 152 

98.5 66

._._-_-- 18

.__.._-- 24

.-____-- 39

._-._-_. 60

.___-.-- 82

._--___- Ill

..--_-_- 29

99.5 90 

.-_-..-- 37

50.0 105

100.0 50 

52.5 146

.5 

.3 

.0 

.0 

.0

6

.8 

.8 

.1 

.1

.4 

.3 

.4 

.1 

.6

.8 

.4

7

.9

.7

.7 

0

.4 

.2 

.5

.6 

.9

,7

.6 

6

2

,6

.0 

?,

.6

.3 

.2

Vertics 

(cm/se<

57. 

45. 

37. 

157. 

130.

124. 

143.

58.

87.

103.

50. 

141.

d i-\

1 

5 

6 

0 

5

0 

0

8

2

0

0 

2

CD and R values for dm

Path

CD

0.73 

.76 

.80 

.50

.48

3.78 

2.32 

5.06 

4.64 

3.27

2.44 

4.00 

1.73 

2.56 

3.27

3.41 

4.11 

3.02 

1.82 

2.02

1.58 

1.01 

.51

.58 

.60

2.51 

1.96 

1.22 

1.44 

1.05

.87 

1.04 

.67 

.92 

.66

.79 

.52

RX10-2

157 

82 

51.5 

425 

236

.17 

.40 

.12 

.13 

.24

.50 

.16 

.72 

.32 

.22

.18 

.13 

.23 

.57 

.45

.79 

1.82 

187 

322 

141

.32 

.57 

1.37 

1.05 

1.89

3.82 

1.94 

5.85 

3.27 .

9.19

6. SO 

19.2

Vertical

CD and R values for <J»

Path

CD R X 10-2 CD

0.81 41.5 1. 

.83 79. 1. 

.86 49.6 1. 

.55 407 

.52 225

._---_-----.---- 5.

.__---_---__---_ 3.

--_-__-_--_-.-- 7.

.---_.------_--. 7.

..__......_...-- 5.

.--_--.-_._-____ 3.

.---_---.-___--- 6.

. _ _ __ _ _ 2.

.--____--_.....- 3.

.__.___-_.__-.-- 5.

.---_-----.____- 5.

.--_--_______--. 6.

._.__-__--___.-- 4.

2.

...--.__._.----- 3.

.__---___-___--_ 2.

._____------__-- 1.

.58 175 

.65 303 

.76 125

...._____-_-__-- 4.

3.

. ___ _. _ _ _ 1.

.. __ ______ 2.

_______________ 1.

.__-__--_-___--- 1.

.---_--.-__.___- 1.

.72 5.66 1. 

1.

.69 8.96 1.

.79 6.55 1. 

.56 18.5

14 

22 

33

77 

74

85 

58 

85 

31 

02

74 

16 

63 

96 

09

33

86 

86 

80 

17

44 

58 

79 

89 

92

22 

17 

89 

26 

62

33 

75 

07 

49 

02

22 

79

RXHH

126 

64.6 

40 

341 

191

.14 .

.32 

.09 

.10 .

.19

.40 

.13

.58 .

.26 

.18 .

.15 .

.10 .

.18 .

.46 .

.36 .

.64 .

1.46 .

150 

260 

113

.25 

.45 .

1.10 .

.83 .

1.52 .

3.08 .

1.50 .

4.65 

2.56 .

7.40

5.30 

15.4

Vertical Stability 2

CD R X 10-2

1.27 118 OR 

1.32 62.1 OR 

1.43 38.6 O 

.85 326 O 

.81 182 O

._------_.._-.__ S

.-_-----_.-__.__ S

._-----_-._._._- S

.-_.___._.---.-_ S

.-----_--.---_._ S

.__-_._-_-__-_-- S

.______-__._-.-- S

._--__---._.___. S

.-__-___---_-__- S

..-.__._.___._.- S

.__..___--.--__- S

.--_-__--.__..-_ S

_-__--__-_--__- S

__-_______-__-- S

.-___ __ S

.-----_------.-_ S

._-_--____-_..-- S

.90 140 OR 

1.01 244 OR 

1.18 100 OR

..-_-.-..-.---__ S

.-____----____-- S

.-_--_--_.-__.-_ S

._-.__-._.-.-_-- S

.--_-__--_.-_-_- S

.-_---___----__- S

. .--_.-_-__--- S

1.14 4.51 O 

.-_.-...-..--.-- S

1.07 7.21 O

1.24 5.26 O 

.85 14.9 O

1 The run number is a combination of drop number, fluid number (fig. 6), and particle number. 2 O, regular oscillation; R, rotation about vertical axis, S, stable orientation 

and path.
10

103

THE BEHAVIOR OF LARGE. PARTICLES FALLING IN QUIESCENT LIQUIDS

FORCE NUMBER (X)

104 1Q5 106 1Q7 1Q8 1Q9

C21

101°

\ \

0.1 \

10 102 103 104

REYNOLDS NUMBER (R) 

FIGURE 10. Coefficient of drag as a function at Reynolds number and force number for falling and fixed spheres.

105

FIGURE 11. Steady fall pattern of a steady disk. The example 

is from drop 107 consisting of a lead disk with dm=2.54 cm 

in a fluid of 98 percent glycerine and showing a time interval 

of 0.075 sec. Path parameters are w=42 cm/sec, 11=38, 

CD=2.00. Shaded surface is bottom of disk.

FIGURE 12. Regular oscillation fall pattern of a disk. The 

example is from drop 375 consisting of an aluminum disk 

with dm =3.81 cm in a fluid of 70 percent glycerine and 

showing a time inverval of 0.055 sec. Path parameters are 

w=47.1 cm/sec, R= 1,092, C/>=0.46. Vertical parameters 

are «=25.1 cm/sec, R=583, CD =1.63. Shaded surface is 

bottom of disk.
C22 SEDIMENT TRANSPORT IN ALLtJVIiAL CHANNELS

TABLE 6. Drop data for prolate spheroids l

Run No.2

29-1-28......

30-1-26......

31-1-27---..-

36-1-28......

37-1-26......

48-1-15......

49-1-16......

53-1-15......

67-2-14.... . .

68-2-15......

69-2-16--.-.-

81-2-28.--.--

100-3-14......

101-3-15......

102-3-16... ...

113-3-27......

114-3-28......

136-4-16-....-

137-4-15. .... .

138-4-14.-. _.-

140-4-27......

141-4-28......

177-5-26......

178-5-27.... ..

179-5-28. .....

180-5-14... ...

181-5-15......

182-5-16.. ... .

218-6-26.... . .

219-6-27......

220-6-28......

221-6-14......

222-6-15......

223-6-16......

352-7-26......

353-7-14......

362-7-27......

363-7-15--..--

371-7-28......

372-7-16......

Travel 

distance ^ 

(cm) (c

104.1

103.1

100.4

103.7

104.3

104.4

52.2

105.2

100.0

100.0

100.0

100.0

100.0

100.0

100. 0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

100.0

99.0

97.2

100.0

101.0

100.0

100.0

102.2

99.6

100.5

92.5

100.0

72.0

101.5

90.0

CD and R values 

Velocity for dm

85.5 

65.7 

75.2 

89.5 

65.2

174.0 

196.0 

178.0 

57.2 

81.3

114.2 

33.6 

62.1

85.5 

128.0

24.3 

42.5 

141.0 

96.5 

66.7

26.6 

42.0 

27.6 

37.2 

55.6

94.4 

122.0 

165.8 

35.7 

47.8

65.0 

108.5 

132.8 

175.3 

50.2

138.0 

61.5 

164.5 

81.9 

210.0

CD

0.63 

.53 

.54 

.57 

.54

.54 

.61 

.52 

2.88 

1.89

1.38 

2.71 

2.46 

1.71 

1.10

3.50 

1.72 

.91 

1.35 

2.13

2.82 

1.79 

2.10 

1.55 

1.01

1.09 

.86 

.67 

1.31 

.96

.79 

.83 

.73 

.61 

.69

.53 

.63 

.50 

51 

.43

RX10-2

275 

104 

159 

288 

104

378 

644 

386 

.21 

.41

.87 

.25 

.30 

.55 

1.24

.54 

.40 

1.66 

.76 

.55

.21 

.49 

.30 

.55 

1.23

1.06 

1.82 

3.73 

.75 

1.36

2.77 

2.34 

3.82 

7.63 

4.00

11.2 

6.67 

18.0 

13.3 

34.8

CD and R values 

ford.

CD

0.78 

.66 

.67 

.71 

.68

.65 

.76 

.67 

3.49 

2.26

1.72 

3.38 

2.99 

2.08 

1.38

4.39 

2.14 

1.14 

1.62 

2.59

3.50 

2.20 

2.62 

1.92 

1.26

1.32 

1.03 

.84 

1.60 

1.19

.96 

1.00 

.88 

.76 

.48

.64 

.80 

.59 

.64 

.54

RX10-2

245 

940 

145 

258 

93.3

344 

572 

352 

.20 

.37

.78 

.22 

.27 

.50 

1.11

.14 

.36 

1.49 

.68 

.36

.19 

.44 

.27 

.49 

1.11

.96 

1.65 

3.34 

.68 

1.22

2.49 

2.14 

3.48 

5.82 

3.40

10.2 

6.0 

16.4 

12.0 

31.2

Stability a

UO 

UO

s 

uo 

uo

R 

R 

R

S

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

s 

s 

s 

s 

s

1 The vertical and path data for travel distance, velocity, CD and R are identical 

for prolate spheroids.

2 The run number is a combination of drop number, fluid number (fig. 6), and 

particle number.

3 S, stable orientation and path; O, regular oscillation; U, unstable path; R, rotation about vertical axis.

In summary, the four general fall patterns for disks 

are:

1. Steady-flat

2. Regular oscillation

3. Glide-tumble, and

4. Tumble.

Except for the boundary between steady-flat fall and 

regular oscillation, no definitive boundaries between 

each pattern exists. Instead, there was a progressive 

transition from one pattern to another.

All the disks did not exhibit all the patterns described 

as a result of the step-changes in the viscosity of the 

fluids and the subsequent step-changes in the Reynolds 

number. The lead disks exhibited the glide-tumble fall 

pattern at much smaller Reynolds numbers than did 

the aluminum disks, indicating that the stability number I was a significant parameter in defining the fall patterns of disks. Since the thickness-diameter ratio of all 

the disks was the same, the apparent affect of I must be 

due solely to the density ratio of the disk and the fluid.

FIGURE 13. Fall of a disk just after entering the glide-tumble 

pattern. The example is from drop 24 consisting of an aluminum disk with dm =2.54 cm in a fluid of water and showing 

a time interval of 0.051 sec. Path parameters are co=58.5 

cm/sec, R= 17,900, C/>=0.26. Vertical parameters are co=41.6 

cm/sec, R= 13,400, CD =QA8. Shaded surface is bottom of disk.

Changes in I as a result of changes in the thicknessdiameter ratio of the disk would produce similar changes 

in fall patterns if the density ratio of the disk and fluid 

were kept constant. Further evidence of the importance 

of the density ratio will be discussed in connection with 

the <7D-R diagrams in a following section.

Regimes of fall

As the Reynolds number for a falling disk was further 

increased, the fall of the disk changed from a stable 

pattern of steady-flat fall, through a period of transition 

until the disk reached a second stable condition of constant angular rotation or tumble. Thus, the patterns of 

fall for a disk can be divided into three regimes: a 

steady regime, where a stable pattern of steady-flat 

fall exists; the transition regime which includes the 

oscillation and glide-tumble fall patterns; and the tumble regime where a second stable condition exists as a 

uniform tumble and constant angular rotation.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C23

J

FIGURE 14. Fall of a disk just before leaving the glide-tumble 

fall pattern. The example is from drop 161 consisting of a lead 

disk with dm =3.81 cm in a fluid of 90.1 percent glycerine and 

showing a time interval of 0.029 sec. Path parameters are 

w = 95.1 cm/sec, R=300, (7/3=0.60. Vertical parameters are 

w=74.5 cm/sec, R=235, (7/3=0.97. Shaded surface is bottom 

of disk.

The regime classification is dependent, not only upon 

the coefficient of drag CD and the Reynolds number R, 

but also upon the stability number I and the frequency 

number 6. Thus, the regimes of fall can be related to the 

CD-Hi diagram (fig. 16).

Ojj-R curves and fall regimes

The steady regime exists when 10 <R<100. The 

limits of the steady regime are fixed by the Reynolds 

number; they are not dependent upon particle density.

The lower limit of the transition regime is coincident 

with the upper limit of the steady regime. Limits between the transition and tumble regimes are not 

so clearly defined. It is often difficult to tell whether 

a disk is in the transition or tumble regime. In this 

particular study, the transition regime includes the 

part of the CD-R curve when 100<R<2,000, and the 

lower branch of the CD-R curve in figure 16 when R 

>2,000. The tumble regime consists of the upper branch

FIGURE 15. Tumble fall pattern of a disk. The example is from 

drop 204 consisting of a lead disk with dm =2.54 cm in a fluid 

of water and showing a time interval of 0.03 sec. Path parameters are w=66.8 cm/sec, R= 16,800, (7D =1.02. Vertical parameters are w=47.3 cm/sec, R= 11,900, (7/3=2.02.

of the CD-R curve of figure 16 where the angular 

velocity is nearly constant.

Unsteady conditions

The preceding discussion on the fall pattern and 

regimes of disks makes it clear that the velocity of a 

disk is not constant once oscillation begins. The disk, 

thereafter, continually accelerates or decelerates in a 

vertical direction, angularly, or both.

The method of data collection permitted both vertical 

and path velocities to be determined for short periods 

of time, that is, periods down to less than 0.015 second. 

Graphs of the time-variation of velocity were obtained 

plotting the incremental velocities against time for 

several of the drops (figs. 17-19).

The three horizontal lines on the graphs represent 

the average velocity of fall through the entire test 

section. The long-dashed line represents the vertical 

fall velocity of the disk, the short-dashed line represents 

the path velocity, and the large solid dots represent the 

three-point moving average of the path velocities. The 

three-point moving average was determined by plotting 

the average of three consecutive path velocities at the 

midpoint of the three. When the data were taken from 

the film, movements of less than one-half centimeter 

were difficult to measure. Because of the high speed of 

the camera and slow speed of some of the particles, 

movement of that magnitude was not uncommon. It 

should also be noted that a gradual change over two or 

three consecutive intervals was recorded in one interval 

because the minimum length recorded was one-half 

centimeter. When the lengths were converted to velocities, a sudden change seemed to have occurred at
C24 S1DIMEPST TRANSPORT IN ALLUVIAL CHANNELS

In i.o I 

Steady-flat 

Transition |||lt|l|||||||

Q.I 

10 102 103 

REYNOLDS NUMBER (R)

FIGTJKE 16. Regimes of fall for a free-falling disk.

such intervals. Therefore, the three-point moving 

averages should better represent path velocity than the 

singly determined values shown by the short-dashed 

line.

When a disk falling in a glide-tumble pattern (fig. 18) 

is accelerating vertically (positive slope on the velocitytime curve), the vertical and path velocities are identical. When the path of travel is not vertical, the 

velocities are no longer identical and the vertical 

velocity decreases earlier in time than does the path 

velocity. After a disk begins a regular tumble (fig. 19), 

it moves laterally across the column throughout the 

entire fall. Therefore, the path velocity is always 

greater than the vertical velocity.

The cyclic nature of the velocity-time relationship is 

typical for all disks in all fall patterns except the 

steady-flat fall pattern. Several times, particularly in 

the glide-tumble pattern, negative vertical velocities 

were recorded. By definition, the path velocity is 

always positive except at the apecies of the arcs where 

it goes to zero.

Average velocities

Current methods of expressing resistance to motion 

do not account for accelerating motion because average 

velocities are used. The apparent cyclic nature of the 

fall of disks, indicating that the disk does not continually

accelerate or decelerate, tends to justify this approximation.

In all tests in the transition and tumble regimes, the 

average path velocity was greater than the average 

vertical velocity (see heavy horizontal lines, figs. 17-18) . 

In one test the average path velocity was 81 percent 

greater than the average vertical velocity.

The path velocity of a falling disk will generally be 

greater than the velocity of a fluid, of the same density 

and viscosity as the one in which the given disk is 

falling, flowing past a rigidly mounted disk, of the same 

cross-sectional area as the falling disk, which would 

exert a force on the fixed disk equal to the weight of the 

falling disk. Such a velocity can be computed from the 

CD-R diagram for fixed disks and from the force number 

of the falling disk (see heavy broken lines, fig. 17-19).

Frequency of oscillation

Two variables which affect the frequency of oscillation 

or rotation are disk size and density. If the densityfrequency number $ previously derived by dimensionless analysis is written as :

c 9, 

pf a a

it is evident that for a constant thickness-diameter 

ratio of the disk, the square of the frequency is inversely
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C25

180

160

120

80

40

o

o o

u

>

-40

DROP NUMBER 366

oj CD R 

Path 28.8 0.87 446 

Vertical 121.8 1.52 338 

Fixed 20.0 1.82 308 

-- Path

- Vertical

Three-point 

moving average

Path average

\ V v \/ \ /

Fixed disk Vertical averageST

v

0.5 1.0 1.5 

TIME, IN SECONDS 

A

2.0 2.5

180

160

120

80

40

o 

°, o

-40

DROP NUMBER 366

Path average

Path

Vertical

Three-point 

moving average

Vertical average

I____I____I____I____I

0 0.5 1.0 1.5 2.0 2.5

TIME, IN SECONDS

B

FIGURE 17. Velocity-time relationship for a 2.54-cm aluminum disk falling in an oscillatory pattern. The time interval between

successive position of the disk is five times longer in (B) than in (A).

proportional to the diameter of the disk and directly

proportional to 

Pf For example, the 3.81 cm (diameter) aluminum and lead disks falling in water at 

average vertical velocities of 50.2 and 67.5 cm/sec, had 

rotational frequencies of 0.6 and 2.9 revolutions per 

second, respectively. Table 7 summarizes the frequency 

of oscillation for each experimental drop.

CD-R curves and fall velocity

Four diagrams (figs. 20, 21) present the CD-R 

relationships for disks. Particle identity is maintained 

by use of different symbols as indicated in the legend. 

The stability number I and the frequency number 6 

appear beside each point in figures 20A and 21A 

where the disks were unsteady.
SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

1i- oo 0

>- en0

n i | i r 

DROP NUMBER 24

-i i r

i-- i\>

0

£ 80

-p» 0

O

9 o

-40

co C0 R

Path 58.5 0.26 17,900

Vertical 41.6 0.48 13,400

Fixed 26.6 1.10 8,560

 Path

- Vertical

Three-point 

moving average

0.5 1.0 1.5 

TIME, IN SECONDS

2.5 2.7

FIGURE 18. Velocity-time relationship for a 2.54-cm aluminum disk falling in a glide-tumble pattern.

180

160

120

80

40

-40

co Co R

Path 102.9 0.65 47,200

Vertical 67.5 1.52 31,000

Fixed 76.0 1.18 35,000

Path 

Vertical

Three-point 

moving average

0.2 0.4 0.6 

TIME, IN SECONDS

0.8 1.0 1.2 1.3

FIGURE 19. Velocity-time relationship for a 3.81-cm lead disk falling in a tumbling pattern. (Break in curve occurs where the disk

fell down the wall of the column.)

When 10 < R < 2,000, the curve for the relationship of CD-R for falling disks is approximately the same 

as the CD-R relationship for fixed spheres (fig. 20A). 

For 10 < R < 120, the relationship is well established 

because the disks fell vertically, at a constant velocity 

without oscillation and with their maximum projected 

area perpendicular to the direction of fall. The relationship is not so well defined in the range between 

120 < R < 2,000, where the disks were unsteady.

Here, the data are more sparse, the points scatter more, 

and the stability aud frequency number are beginning 

to influence the CD-R relationship.

At R equal to about 10,000 for the given disks of 

this experiment, the CD-R relationship is no longer 

unique, but has two values. The only difference between the disks used to develop the nonunique relationship is the density of the disks. The lead disks 

have a greater CD value for a given value of R than do
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C27

TABLE 7. Frequency of oscillation of unsteady particles falling in

water-glycerine mixtures, in cycles per second

[Particles falling in mixtures of 94.24 percent or more of glycerine were always stable! 

A, aluminum; O, cylinder; D, disk; GT, glide tumble; L, lead; O, oscillation; OS, 

oblate spheriod; PS, prolate spheroid; R, rotation; S, steady fall; T, tumble; U, 

undetermined.

Particle material, shape, Percent glycerine

L-D-1.91__--_-_

L-D-2.54_______

L-D-3.81___-__-

A-D-1.91-_---__

A-D-2.54_ _____

A-D-3.81_--_--_

L-OS-1.91______

L-OS-2.49______

L-OS-3.85______

A-OS-1.91______

A-OS-2.50______

A-OS-3.78 ___

L-PS-1.91______

L-PS-2.52______

L-PS-3.85------

A-PS-1.83--___-

A-PS-2.50______

A-PS-3.85-_____

L-C-1.91_--____

L-C-2.54_---___

L-C-3.81_______

A-C-1.91.______

A-C-2.54_______

A-C-3.81. ______

90.1

_____ S

S

_____ 2.3-GT

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

_____ S

S

83.33

4.0-O

2.9-GT

2.6-GT

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

70.00

4.2-GT

3.6-GT

2.9-GT

2.8-O

2.2-O

1.6-O

S

S

S

S

S

U

S

S

S

S

S

S

3.2-O

2.4-O

2.0-O

S

S

1.6-O

0.00

4.2-T

3.9-T

2.9-T

1.4-GT

1.2-GT

.6-GT

S

S

U

4.6-T

U

2.66-O

U

4.81-OR

4.0-OR

3.5-O

S

2.69-O

U

2.0-OR

2.0-OR

2.3-O

2.0-OR

1.7-OR

the aluminum ones. Thus, somewhere between a 

Reynolds number of 2,000 and 10,000, the resistance 

to motion for a disk becomes a function of the density 

ratio between it and the fluid as well as Reynolds 

number and shape factor. The division of the CD-R. 

relationship into two branches on the basis of the 

stability number I further emphasizes the importance 

of the density ratio between the disk and the fluid on the 

behavior of disks at certain values of R.

The shape of the curve representing the (To-R 

diagram is not affected by the choice of characteristic 

diameter used in the computations. The position of the 

curve CD-R diagram, however, is the result of using 

different characteristic diameters (fig. 20).

Associated with the effect of I on resistance to 

motion is the effect of the frequency number 6. CD 

increases with 0 independently of R (fig. 22). Thus, it is 

evident that the stability number I and the frequency 

number 6 are important parameters in describing the 

behavior of freely falling disks.

The solid lines in figures 22^1 and 22B represent 

the relationship for the glide-tumble and tumble 

patterns of fall. Insufficient data were available for the 

oscillating pattern of fall to establish a definite trend 

in the relation of CD to 6; therefore, the broken lines, 

which represent that fall pattern, were arbitrarily drawn 

parallel to the solid ones. The position of the lines 

indicate, however, that the drag for a given frequency

number is greater when the disk tumbles or glides 

than when it oscillates.

OBLATE SPHEROIDS

The fall pattern of the oblate spheroids was steady 

throughout most of the range of Reynolds numbers 

covered in the experiment. The only exceptions 

occurred for the aluminum and the largest lead 

spheroids falling in water. Of the aluminum spheroids, 

only the smallest one tumbled, whereas the largest 

one oscillated in a vertical plane about a horizontal 

diameter. The two smallest lead spheroids were perfectly steady while falling in the water. The middlesized aluminum particle and the largest lead one, for 

which the data were insufficient to make computations 

of CD and R, had a tumbling pattern. The fact that 

the lead spheroids were steady whereas the aluminum 

ones were not, at least for the drops having valid data, 

is again evidence that the density ratio of the particle 

and fluid is an important parameter in particle 

behavior.

As expected, the oblate spheroids were steady over 

a larger range of Reynolds number than disks were.

When 10<R< 150, the (7fl-R relationship for oblate 

spheroids is almost identical to that for spheres (fig. 

23). The divergence of the C^-R relationships for 

spheres and oblate spheroids when 150<R<5,000 

probably occurs because the pressure drag affects the 

fall of the oblate spheroid at smaller Reynolds numbers 

than it does the sphere. The divergence is not due to 

an unstable fall condition because the spheroids remain 

steady throughout their fall, but it was probably due 

to the earlier development of the pressure drag behind 

the spheroid. The development of the pressure drag 

occurs more rapidly for the spheroid than for a sphere 

because its shape induces separation and full development of the wake at smaller Reynolds numbers. When 

5,000<R<50,000, the drag is independent of the 

Reynolds number.

* - -a CYLINDERS

All cylinders fell with their maximum projected 

area perpendicular to the direction of fall for Reynolds 

numbers less than 400. When R>400, the cylinders 

oscillated in a vertical plane about a horizontal axis 

normal to the major axis of the cylinder. At R> 8,000, 

another oscillation was superimposed on the first, an 

oscillation in a horizontal plane about a vertical axis. 

The frequencies of oscillation for cylinders are summarized in table 7.

The curve representing the relationship of CD to R 

for cylinders is higher on the CD-R diagram than for 

spheres (fig. 24). The two curves slowly diverge to 

R«400, after which the increased rate of divergence 

is coincidental with the increased oscillation in the
C28 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

103 104 105

FORCE NUMBER (X)

106 107 108 109 1010

Alumi- Major

num axis Lead

o 1.91 

'0.050

0/48 0050

if^^^

0.1

102

10

103 104 105

103

REYNOLDS NUMBER (R) 

A

FORCE NUMBER (X)

106 107

104 105

108 109 1010

U. 1 

Ld 

O 

O

CO 

CO

Ld

_l

z 

g

CO

10

\ \

Alumi- Major

num axis Lead

o 1.91 

a 2.54 

* 3.81 A

Fixed disk

'\

--- ^nere

\

\

A I_______l_l________i I \ I i

10 102 1Q3 1Q4 105

REYNOLDS NUMBER (R)

B

FIGURE 20. Relationship of coefficient of drag and Reynolds number for falling disks having SFC 0.1. A, Computations are 

based on dm and path velocity. I and 6 are given for points where disks were unsteady. B, Computations are based on dn and 

path velocity.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C29

10

103 104 105

FORCE NUMBER (X)

106 107 108 109 1010

Iz

LJ 1 

O

o

-.

v \\\ \ \ \\\ \ N\\\ \ \\\\ \ N \\\ \ \\\\ \ K\\\ \ \ \\\

Alumi- Major

num axis Lead

o 1.91 

D 2.54 

10 102

103 104 105

103

REYNOLDS NUMBER (R) 

A

FORCE NUMBER (\)

106 107

104 105

103 109 1010

\ K \ \\ \ \ \ \\ \

Alumi- Major

num axis Lead

o 1.91 

n 2.54 

* 3.81 A

102 103

REYNOLDS NUMBER (R) 

B

104 105

FIGURE 21. Relationship of coefficient of drag and Reynolds number for falling disks having SF C 0.1. A, Computations are 

based on dm and vertical velocity. I and 0 are given for points where disks were unsteady. B, Computations are based on dn 

and vertical velocity.
G30 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Alumi- Major

num axis Lead

o fi 1.91 

a p1 2.54 

* X 3.81 *

0.04 0.08 0.12 0.16 

FREQUENCY NUMBER (6) 

A

0.20 0.24

2.4

2.0

1.2

0.8

0.4

Alumi- Major

num axis Lead

o & 1.91 

o X 2.54

3.81

0.08 0.16 0.24 0.32

FREQUENCY NUMBER (6)

B

0.40

FIGURE 22. Coefficient of drag as a function of frequency number for falling disks having SF C 0.1. A, Computations are based on 

dm and path velocity. B, Computations are based on dm and vertical velocity.

10

103 104 105

FORCE NUMBER (X)

106 1Q7 108 109

0.1

\\\ \ \ \\\ \1V\\\ \ \ \\\ \ l\ \\\ \ \\\\ \ N\\\ \ \ \\\

i, Alumi- Major

num axis Lead

o 1.91 

D 2.54 

A 3.81 *

Fixed disk

0.046

0 0465 0. 130 _.A 0. 145 _^- -^ '0.046 

0 128

Path velocity

- - ____fixed_sphere

\ 

-\

10 102 103

REYNOLDS NUMBER (R)

104 105

FIGURE 23. Coefficient of drag as a function at Reynolds number for falling oblate spheroids, SFC =0.5. Computations are 

based on dm and path velocity. I and 9 are given for points where the spheroids were unsteady.
10

o 

<

K 

Q

LL. 

O

ILlJ

O

t 1 LJ 

O

O

THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS

FORCE NUMBER (X)

103 1Q4 1Q5 106 1Q7 108 1Q9 1Q10

C31

0.1

\

\\\ \ \ \\\ \ |\ \\\ \ \ \\\ \ |\ \\\ \ \\\\

' i.^2 sphere^

0.044

0044 00/2 

0027 0.0/2* \ 00/0 AaU/ ^

0.022 1<^__=r^*5.044 

0.0/2

__

0.04~4 

0009

10 102 103 

REYNOLDS NUMBER (R)

104 105

FIGURE 24. Coefficient of drag as a function at Reynolds number for falling cylinders, SFC =0.5. Computations are based 

on dm and path velocity. I and & are given for points where cylinders were unsteady.

cylinder fall pattern. At R« 12,000, the drag coefficient 

suddenly decreased.

In all instances where the cylinders were unsteady 

during fall, the increase in path length due to horizontal 

motion was small. The resulting increase in the path 

velocity over the vertical velocity was a maximum of 

11 percent and a minimum of 1.6 percent. Average 

increase was 5.6 percent. As for disks, a strong relationship exists between CD and 6, further evidence that 

resistance to motion increases as the degree of particle 

instability increases (fig. 25).

PROLATE SPHEROIDS

The prolate spheroids, a shape between the sphere 

and the cylinder, exhibited much the same patterns of 

fall as the cylinders. The first indication of instability 

was an oscillation in a vertical plane followed by an 

oscillation in both the horizontal and vertical planes as 

the Reynolds number increased. One particle, the 2.54 

cm aluminum spheroid, exhibited a very slow rotation 

in the horizontal plane and no oscillation in the vertical 

plane. All particles were steady for Reynolds number 

ranging from 10 to 8,000.

For R<200, the CD-R, relationship for prolate 

spheroids is almost identical to that for spheres (fig. 

26). Differences in path and vertical velocity were 

small. The path velocity was greater than the vertical 

velocity by a maximum of 5 percent and a minimum of 

2.1 percent. The average increase was 3.3 percent.

COMPARISON OF BEHAVIOR AMONG ALL 

PARTICLES

When the CD-R relationships for all particles computed by using dm and path velocity as characteristic 

parameters are compared, the relationships for spheres, 

oblate and prolate spheroids, and disks are almost 

identical in the range where 10<R<400 (fig. 27^1). 

This consistancy in the relation of CD to R for spheres, 

oblate spheroids, and prolate spheroids indicates that 

the shape factor has only small effect on the fall velocity 

of a particle when R<400. The coefficient of drag for 

cylinders is consistently higher when 10<CR<C400 than 

for the other four particles. The higher drag for cylinders 

is probably due to some characteristic not included in 

the shape factor. Perhaps the characteristic is the sharp 

corners which the cylinders have, but which the spheres 

and spheroids do not have. The sharp corners are part of 

the particle shape, but they are not included in the 

shape factor.

The effect of shape is more noticeable when R>400. 

The extreme shape of particles with small values of the 

Corey shape factor SFC causes separation of the flow and 

subsequent development of the pressure drag at smaller 

Reynolds numbers than for particles more uniformly 

shaped. The more rapid development of pressure drag 

of particles with small SFC causes the resistance to become independent of the Reynolds number at smaller 

values or R and larger values of CD .
C32 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Alumi- Major

num axis

° 1.91

D

Lead

0.01 0.02 0.03

FREQUENCY NUMBER (6)

0.04 0.05

FIGURE 25. Coefficient of drag as a function of frequency 

number for falling cylinders, SFC=Q.5. Computations are 

based on dm and path velocity.

When R> 10,000, the <VR relationships for disks and 

cylinders, is not uniquely fixed by particle shape. It is 

a function of I, 0, and $ as well.

When R<2,000, the curves representing the relationship between CD and R are separated according to 

SFe, except for cylinders and oblate spheroids, on the 

CD-R diagram (fig. 275). The cylinders and oblate 

spheroids have the same shape factor, but the relationships do not coincide. More will be said about the effect 

of particle shape in the section below.

INTERPRETATION OP RESULTS

The foregoing discussion presented experimental 

findings; the following interprets these findings.

PARAMETRIC RELATIONSHIPS

The ratio between the maximum diameter dm and 

minimum diameter dn obviously is constant for all 

particles of one shape. Thus, when dm and dn are used 

to compute CD and R, the constancy of the ratio of 

dm and dn preserves the "shape" of the curves representing the relationship of CD to R regardless of which

characteristic diameter is used. Therefore, the shift in 

location of the respective curves on the CD-R diagram 

is due to particle geometry, and the choice of diameter 

can be arbitrary as long as it is used consistently.

The difference found between the path and vertical 

velocities is a direct function of the effect of particle 

steadiness on the length of the fall path. The greater 

the degree of oscillation or rotation, the greater the 

difference between the velocities. Therefore, the C^-R 

curves using the respective characteristic velocities in 

the computations for CD and R, are the same as long 

as the particles are steady; they are different when the 

particles are unstable. Unlike the shift in the position 

of the (Tp-R curve because of the change of characteristic diameter, the shift due to difference in velocity 

changes the shape of the curve. Probably much of the 

scatter in the data of other investigators is due to the 

change in vertical velocity brought about by the unstable condition of the particles during fall. All the 

given CD-R diagrams that were based on the path 

velocity, except for the curve for disks when R> 10,000, 

show a smooth curve, even when the particles oscillate 

and rotate. Hence, the fall velocity predicted from 

these (To-R diagrams should be a close approximation 

of path velocity of the particle.

PATTERNS OF FALL

When the magnitude of the pressure drag on a particle 

approaches the weight of the particle, its pattern of 

fall is greatly influenced by small changes in the distribution of the pressure forces around it. Thus, the 

pattern of fall is a function of the steadiness of the 

center of pressure on the particle. Inasmuch as the 

pressure drag is dependent upon the degree of wake 

development, particle shape affects the pattern of fall 

to the degree that it affects the development and 

stability of the wake downstream from the particle.

If the wake stability is a function of the symmetry of 

the particle, then particles with a circular cross section 

should be more stable than ones with an elliptical cross 

section. Except for the disks, this was found to be true 

The oblate spheroids were more stable than either the 

prolate spheroids or the cylinders.

The fact that the disks were the most unstable indicates that extremeness in shape also affects the fall 

pattern. That is, extreme shape causes separation and 

wake development, and hence full development of pressure drag at smaller Reynolds numbers than for particles of more streamlined shapes, such as the oblate 

spheroids.

Apparently, the fall pattern of a particle becomes 

more steady (less oscillation and rotation and more 

vertical) as the Corey shape factor increases from numbers less than one to one. Also, the particle becomes
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C33

10

103 104 105

FORCE NUMBER (X)

106 1Q7 108 109 1010

V

0.1

\ \ \ \ 1\ \ \ \ \ \

___________ \\\ \ \ \\\ \ N x Alumi- Major

num axis

o 1.91

\\\\

Lead

\

\

\

Falling prolate spheroid-c/m

- J 0023 _ 

 ____A^ixed_spJTere.____ g^g

\

\

\

\

\

\ V \ \ \ I \ I

10 102 103

REYNOLDS NUMBER (R)

104 105

FIGURE 26. Coefficient of drag as a function at Reynolds number for falling prolate spheroids, SFc=Q.7l. Computations are 

based on dm and path velocity. I and 9 are given for points where spheroids were unsteady.

more stable as SFC approaches one because of crosssectional changes from elliptical to circular. The effect 

of shape on the fall pattern of a particle is, therefore, a 

combination of the effects of the shape of the cross section and the extremeness of the overall shape or Corey 

shape factor SFC .

The steadiness of the fall pattern of a particle has a 

marked effect on the ratio of path distance to vertical 

distance that the particle travels. The ratio increases as 

the degree of steadiness decreases. The degree of unsteadiness has a greater effect on the length of the path 

of nonspherical particles having a circular cross section 

than on those having elliptical cross sections. The path 

of travel for both disks and oblate spheroids was increased more by an unsteady fall pattern than it was 

for cylinders and prolate spheroids.

The separation of the curve into two parts in the 

C/>-R relationship was found only for disks. It is not 

known whether particle shapes other than disks will 

become sufficiently unsteady as a result of density differences for a similar division in the C^-R relationship.

Two characteristics of an unsteady particle may 

reduce its fall velocity in turbulent natural flow from 

that obtained in a quiescent fluid. As previously noted, 

resistance to fall increases with the frequency of oscillation or rotation. If a particle, which has one fall pattern 

when falling in a quiescent fluid, were falling in a turbulent fluid, the turbulence may increase its oscillation, 

and hence its resistance to fall. The velocity of the particle would decrease a corresponding amount. The second 

characteristic is the increased length of the fall path as 

the particle becomes more unsteady. As the path of 

travel increases, the particle may be exposed to more 

turbulent eddies, which, in turn, would increase its 

unsteadiness. The net result would be a slower fall 

velocity than the same particle would experience in a 

quiescent fluid.

The turbulenece of a moving fluid may affect the behavior of a particle in another manner. Since resistance 

to motion is a function of w2 , squaring the mean velocity 

plus the velocity fluctuations will give a greater number 

to be used in equation 4 than the square of the mean 

velocity. The result will be a greater mean resistance to 

fall and a smaller mean fall velocity.

SHAPE FACTORS

Of the three shape factors presented in the data 

(fig. 27B), the Corey shape factor SFC fits the 0/,-R 

relationships in the most logical manner. It is still an 

inadequate measure because two curves having the 

same shape factor (cylinders and oblate spheroids) plot in 

quite different positions on the diagram. Alger's shape 

factor SFca, which modified SFC by accounting for 

surface area, fits the C/>-R curves (in fig. 27) except
C34 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

103 104 105

FORCE NUMBER (X)

106 1Q7 108 109 1Q10

PARTICLE 

 Fixed disk

Fixed sphere 

Falling disk 

\ Falling cylinder

Falling prolate spheroid 

Falling oblate spheroid

0.10 0.15 0.53

0.50 0.59 0.43

0.71 0.73 0.63

 0.50 0.53 

1.00 1.00

102

10

103 104 105

103

REYNOLDS NUMBER (R)

A 

FORCE NUMBER (X)

106 1Q7

104 105

108 109 1010

o.i

\ \ \\

PARTICLE 

Fixed disk

Fixed sphere 

Falling disk 

Falling cylinder 

Falling prolate spheroid 

Falling oblate spheroid 

Falling sphere

\ I______I \ i

0.10 0.15 0.53 

0.50 0.59 0.43 

0.71 0.73 0.63 

0.50 0.53 0.82 

1.00 1.00 1.00 

 I_______L_

\

10 102 103 

REYNOLDS NUMBER (R)

B

104 105

FIGURE 27. Composite graphs of coefficient of drag as a function of Reynolds number for falling spheres, disks, oblate spheroids, 

cylinders, and prolate spheroids. A, Computations are based on dm and path velocity. B, Computations are based on dn and 

path velocity.
THE BEHAVIOR OF LARGE PARTICLES FALLING IN QUIESCENT LIQUIDS C35

for the cylinders and oblate spheroids. The larger value 

should be with the oblate spheroids rather than the 

cylinders to fit properly. Thus, the modification by 

Alger may overcompensate for some deficiencies of SFe 

for some particles. Wadell's sphericity parameter seems 

to be the least significant of the three shape factors.

Apparently, rough surfaces and sharp corners measurable increase the drag of falling particles. Comparisons of the curves representing the CD-R relationship 

for crushed and naturally worn gravel particles have 

been made by Wilde (1952) and Colby (U.S. InterAgency Comm. Water Resources, 1957) for naturally 

worn sand and gravel. The curves for particles studied 

in the Wilde experiments indicate that roughness and 

roundness have as great an effect on particle fall 

velocity, if not greater, than the effects of shape.

Alger (1964) shows that the modification of CD and 

SFe by a factor proportional to the surface area of the 

particle is significant. By appropriate modification of 

CD and SFC with the ratio dAfdn, Alger was able to 

predict the fall velocity of large concrete and steel 

cylinders from small aluminum models with an accuracy 

of about 2 percent. Alger assumed that CD was independent of R when R >2,000. The relationship of CD 

to R for cylinders found in this study (fig. 275) is not 

independent of R when R >2,000. This appears to 

contradict Alger's findings.

SUMMARY AND CONCLUSIONS

The fall velocities of disks, cylinders, oblate and 

prolate spheroids, and spheres falling in quiescent 

fluids has been studied for the range of Reynolds 

numbers between 10 and 100,000. Curves that present 

the relationships between the coefficient of drag and 

the Reynolds number based on the path velocity, 

vertical velocity, maximum diameter, and nominal 

diameter as characteristic variables have been developed.

These characteristic variables are used to determine 

the coefficient of drag and the Reynolds number which 

are plotted on the CD-R diagram. Individual particles, 

however, plot on lines of constant force number regardless of which parameters are used. Curves from 

points based on the nominal diameter and the maximum 

diameter of the particles are identical in shape when 

the path velocity is used in the Oz>-R computations. 

The curves are also identical when the vertical velocity 

is used, but they are different from those for the path 

velocity. Curves using the path and vertical velocities 

are the same as long as the particles are steady, but 

deviate, depending on the degree of difference in 

velocity, when the particles become unsteady.

The falling spheres exhibited very little unsteadinessThe experimental curve based upon the data herein 

presented coincides with the curve for fixed spheres.

No rotation was apparent in any of the drops made 

with spheres except one that rotated through 45° 

during its fall in the test section.

The disks were the least steady of all the particles 

tested. Unsteadiness first occurred in the form of oscillations in a vertical plane about a horizontal diameter 

at Reynolds numbers of approximately 100. As the 

Reynolds number increases over its entire range, the 

results of the drops show a series of fall patterns ranging 

from a stable pattern of steady-flat fall where the disks 

fell vertically with their maximum projected area 

perpendicular to the direction of fall, to one of quasistable pattern of constant rotation. The patterns have 

been categorized in four general classifications which 

are, in turn, combined into three fall regimes as follows:

Fall patterns Regimes 

Stable____________________ Stable

Oscillation 1 Glide-tumble/ --------------

Tumble_________________ Tumble

Transition

Boundaries of the suggested regimes are defined by 

CD, R, and I.

The density ratio between the particles and the fluid 

affects fall velocity by virtue of its relation to the 

stability number I and to the frequency of oscillation. 

When R>10,000, two branches of the CD-R curve 

were defined for two different stability numbers of the 

disks. The stability number is greater for lead than 

aluminum disks, and the points for the lead disks plot 

higher on the CD-R diagram than do the points for 

the aluminum ones. Since all variables of I are the same 

for all disks, including the thickness-diameter ratio, 

except the density ratio, the division in the curve must 

be due to particle density.

The coefficient of drag is a function of the frequency 

of oscillation as reflected by the linear relationship 

between CD and 6. Thus, it can be concluded that both 

I and 6 are important parameters in fall behavior 

studies.

The studies indicated that once oscillation began, 

disk unsteadiness was characterized by a rather uniform 

cyclic frequency. The oscillations were not random with 

respect to time.

The oblate spheroids were the most stable of the 

nonspherical particles. Reynolds numbers as great as 

50,000 were obtained without indication of unsteadiness. 

When unsteadiness did develop, the patterns of fall 

were oscillation and tumble.

Prolate spheroids and cylinders had very similar 

patterns of fall, except that instability began at smaller 

Reynolds numbers for cylinders than for prolate spheroids. The general pattern of unsteady fall was that of 

oscillation in a vertical plane about a horizontal axis
C36 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

followed by the imposition of a horizontal oscillation 

about a vertical axis as Reynolds numbers increased.

When all curves based on the "path velocity" and 

the "maximum diameter" as characteristic parameters 

were plotted together, it was apparent that the shape 

of the particle had only a small effect on fall velocity 

for Reynolds numbers less than 400. For R>400, the 

particle shape had a more pronounced effect on the 

location of the 6^-R curves. Curves for particles with 

the smallest values of SFC became horizontal, or independent of R, at lower Reynolds numbers than the 

curves for particles with larger SFC values.

When the fall patterns of all the particles were compared on the basis of shape, it was evident that the 

more extreme the shape, the greater the degree of instability. When extremes were not considered, it was 

apparent that particles with circular cross sections 

were more stable than particles with elliptical cross 

sections of the same SFC .

This study indicates that more research is needed to 

investigate the various problems related to the behavior 

of large particles falling in quiescent fluids. The following studies would be helpful:

1. A systematic study of the effect of roughness and 

roundness on particle behavior. A comparison of 

these results with those of other investigations 

show that neglect of roughness and roundness may 

introduce errors of as much as 100 percent in 

estimating fall velocities.

2. A study to enlarge the concept of fall regimes to 

particles of shapes other than disks.

3. A more systematic study of the effect of the shape 

of the cross-sectional area needs to be made. The 

study herein reported appears to indicate that the 

OD-R curve for particles with nearly circular cross 

section is not the same as the CD-R curve for 

particles with elliptical or rectangular cross section 

even though the Corey shape factor is the same 

for both particles. Therefore, a study of particle 

behavior as the particle shape changes through 

progressive stages from an oblate spheroid to a 

sphere as compared to the particle behavior as 

the shape changes from a prolate spheroid to a 

sphere would be helpful.

4. The steadiness of a particle is important in its behavior. Therefore, a more detailed analysis of the 

equations of motion of a particle should be made.

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