Sediment Movement Induced by Ships in Restricted Waterways

-~

_26L.21

,

-

r--Sediment Movement Induced by Ships

In Restricted Waterways

Y1-CHUNG LlOU and JOHN B. HERBICH Ocean Engineering Program

TAMU-SG-7~209

COE Report No. 188 August 1976

SEDIMENT MOVEMENT INDUCED BY SHIPS IN RESTRICTED WATERWAYS

by

Yi-Chung Liou and John B. Herbich Ocean Engineering Program

August 1976

TAMU-SG-76-209 COE Report No. 188

Partia11y supported through Institutiona1 Grant 04-5-158-19 to Texas A&M University

by the Nationa1 Oceanic and Atmospheric Administration's Office of Sea Grants

$4.00

Order from:

Department of Marine Resources Information Center for Marine Resources

Texas A&M University

ABSTRACT

A numerical model using the momentum theory of the propeller and Shields' diagram was developed to study sediment movement induced by a ship's propeller in a restricted waterway. The velocity distribution downstream of the propeller was simulated by the Gaussian normal dis-tribution function. The shear velocity and shear stress were obtained using Sternberg's formulas. Once the ship's speed, depth of the water-way, RPM and diameter of the propeller, and draft of the ship are given, the velocity distribution and the grain size of the initial motion

could be obtained from this model. A computer program was developed to solve it. Case studies are presented to show the influence of sig-nificant factors on sediment movement at the channel bottom induced by a ship's propeller.

PREFACE

Research described in this report was conducted as part of the research program in the Coastal, Hydraulic and Ocean Engineering Group at Texas A&M University and was partially supported by the NOAA Sea Grant Program at Texas A&M University.

ACKNOWLEDGEMENT

Computer work was partially supported by the Texas Engineering Ex-periment Station.

The manuscript was edited by Dr. Gisela Mahoney and typed for pub-lication by Ms. Joyce McCabe.

TABLE OF CONTENTS Chapter Page ABSTRACT PREFACE . ACKNOWLEDGEMENT. TABLE OF CONTENTS. LIST OF TABLES . LIST OF FIGURES. LIST OF SYMBOLS. I. INTRODUCTION . . . . 11. LITERATURE REVIEW. i i i

iii

iv

v

vi

viii

3

Sediment Transport. . . .

3

Bed-1oad Transport Theories and Equations . . .

3

Suspended-1oad Transport Theories and Equations

10

Velocity Distribution . . . .

.

. . . .

13

111.

DEVELOPMENT AND DESCRIPTION OF THE NUMERICAL MODEL

16

Problem Statement.

. . . .

16

Momentum Theory of Propeller Action . . . .

18

IV.

THEORETICAL STUDIES OF JET AND SEDIMENT MOVEMENT .

31

Theory of Free Turbu1ence . . . ..

..

31

Theory of the Boundary Layer of a Two-Dimensional

Turbulent Jet of Incompressib1e F1uid. .

34

Theoretica1 Method for Determining the Rate

of Sediment Motion

.

.

.

. . .

.

. .

.

.

41

Simi1arity Consideration on Incipient Motion.

41

The Equation of Sediment Continuity

44

Initial Velocity Downstream

of the Propeller.

Velocity Distribution ...

Initiation of Sediment Movement

Case Studies

.

47 47 47

48

71 82

83

V.

PRESENTATION AND DISCUSSION OF RESULTS

.

VI.

CONCLUSIONS

AND RECOMMENDATIONS.

VII.

REFERENCES

.

LIST OF TABLES

Table Page

2.1 Velocities Required for the Start of Motion ... 4.1 Basic Functions of the Boundary Layer of the Jet. 5.1 Computer Program for Velocity Distribution and

Critical Grain Size of Motion .

13 39

60

5.2 Computer Output for the TEXAS CALIFORNIA. . . . 64 5.3 Summary of Computer Output for the TEXAS CALIFORNIA 70 5.4 Ship Records From Corpus Christi Channe1. . 72 5.5 Maximum Bottom Veloeities . . . 73

LIST OF FIGURES Figure

2.1 2.2

Shields' Diagram for Critical Shear Stress.

..

.

Plot of Einstein Functions

w

*

as a Function

of 4>*.. ••• ••• • •.••

3.1 Definition Sketch of a Ship in a Confined Waterway 3.2 Change in Pressure and Velocity at Propeller Disk,

t~omentum Theory. .. .

3.3 Propeller-Characteristic Curves in Open Water. 3.4 Characteristics of the Normal Probability Curve. 3.5 Definition Sketch of Jet Mixing ..

4.1 Boundary Layer of a Jet ....

4.2 Boundary Velocity Profile of a Submerged Jet 4.3 Force Acting on Grain.. .

4.4 Definition Sketch for Equation of Sediment Continuity. 46 5.1 Longitudinal Velocity Distribution at X

=

D. 49 5.2 Longitudinal Velocity Oistribution at

X

=

2D 50 5.3 Longitudinal Velocity Distribution at

X

= 3D 51 5.4 Longitudinal Velocity Distribution at

X

=

4D 52 5.5 Longitudinal Velocity Distribution at

X

= 5D 53 5.6 Longitudinal Velocity Distribution at

X

=

6D 54 5.7 Longitudinal Velocity Distribution at

X

=

70 55 5.8 Longitudinal Velocity Distribution at

X

=

8D 56 5.9 Longitudinal Velocity Distribution at

X

=

9D 57 5.10 Longitudinal Velocity Distribution at

X

=

100.

vi Page 6 7 17 20 23 26 27 35 40 42 58

lIST OF FIGURES - continued

Figure Page

5.11 Critica1 Grain Size as a Function of Re1ative

Di stance for the OCEAN CHEr~IST .

_·

_{· · · ·}

_·

_{· · · · ·}

74 5.12 Criti ca 1 Grain Size as a Function of Re1ative

Distance for the TEXAS CAlIFORNIA.

_·

_·

_·

_·

_{· ·}

_·

_{· · ·}

75

5.13 Critica1 Grain Size as a Function of Re1ative

Distance for the EAGLE lEADER.

.

_·

_·

_{· · ·}

_{· · · · ·}

_·

76 5.14 Critica1 Grain Size as a Function of Re1ative

Distance for the EXXON NEW ORlEANS

_·

_{· ·}

_{· · · ·}

_·

_·

_·

77

5.15 Critica1 Grain Size as a Function of Re1ative

Distance for the OlONDA.

.

.

. . · · · ·

_·

_·

_·

78 5.16 Critica1 Grain Size as a Function of Re1ative

Distance for the POST CHAllENGER .

_·

_·

_·

_·

_·

_{· · · ·}

_{· · ·}

79 5.17 Critica1 Grain Size as a Function of Re1ative

LIST OF SYMBOLS

The following symbo1s have been used in this paper:

Ao

=

propeller disc area a

=

coefficient Bo

=

slot width b

=

constant C

=

concentration of sediment Cl

=

constant C2

=

constant CT

=

thrust-10ading coefficient

o

=

diameter of propeller

Do

=

diameter of circu1ar orifice Ds = diameter of grain size

d

=

grain size F = force Fg

=

gravity force F.

,

=

inertia force

=

supporting force Ft

=

frictiona1 force Fv = viscous force 9 = gravitational constant h

=

water depth Jt = speed coefficient KT

=

thrust coefficient m

=

mass

n

=

revo1utions per minute P

=

probabi1ity

LIST OF SYMBOLS - continued p

=

pressure

Q

=

discharge

R

=

vertical distance measured from the axis of the propeller R*

=

Reynolds number of the grain

r

=

specific gravity of water Ys

=

specific gravity of grain T

=

thrust

t

=

time

U*

=

shear velocity

u

=

velocity in x-direction

-u

=

average velocity in x-direction Ui

=

fluctuating velocity in x-direction

v

=

longitudinal velocity

Vm

=

maximum longitudinal velocity

V

o

=

initial velocity

v

=

velocity in y-direction

-v

=

average velocity in y-direction Vi

=

fluctuating velocity in y-direction x

=

horizontal coordinate y

=

vertical coordinate P

=

density of water Ps

=

density of grain v = kinematic deviation cr

=

standard deviation n = y/x nl

=

ideal efficiency

CHAPTER I INTRODUCTION

During a ship's passage, the bottom and sides of the canal section are subjected to jet action induced by the ship's engines. In recent years, more serious consideration has b~en given specifically to the

problem of pOllution caused by sediment movement stirred up by the ship's propeller in a restricted waterway. Due to increasing tonnage of crude oil, chemicals, and other commodities over navigation water-ways, the maintenance cost of waterways is also increasing. A large portion of the cost is caused by an increase in dredging operations. It has been observed that much of the material that requires maintenance dredging sterns from sediment movement induced by shipS' propellers. Since sediment movement is a major factor in siltation and subsequently in maintenance dredging, an investigation of the sediment movement in-duced by a ship passing through a restricted waterway is important to minimize dredging costs and to reduce environmental effects.

A survey of literature revealed a limited amount of published material concerning sediment movement induced by ships in restricted waterways. A numerical model using the momentum theory and Shields' diagram was developed to simulate the prototype. The velocity distri-bution downstream of the propeller is described by the Gaussian normal distribution function.

The parameters in the Shields diagram include the shear stress and shear velocity and are estimated from Sternberg's formulas (28)*.

*

Numbers in parentheses refer to references listed on page 83.

Field data from the Corpus Christi channel were used in the case studies presented.

This research may be considered as an initial step in the sediment transport study in the waterway.

Specific objectives of this study can be summarized as: 1) Literature review

2) Development of a computer program for

(a) calculation of velocity distribution downstream of the propeller,

(b) determination of the critical grain size for the initial motion, and

CHAPTER 11 LITERATURE REVIEW Sediment Transport

Sediment is transported by flowing water as a bed-load, a salta-tion load, or as a suspended load. Each mode of transport may occur singly, or combined with one or both remaining modes. Normally, sedi-ment transport occurs intermittently by all three modes.

The bed-load is composed of larger particles that move on or near the bed. This load travels along the bed by rolling or sliding, and is in substantially continuous contact with the bed.

The saltation load consists of material that bounces along the bed. It is moved directly or indirectly by the impact of the bounc-ing particles. Bagnold visualized the grains in saltation moving like ping-pong balls. It is difficult to distinguish the saltation load from the suspended load.

The suspended load is composed of small particles that are kept in suspension by the upward components of turbulent flow. In regions where clays are eroded, the suspended load is a more important mode of transport compared to the bed-load.

Bed-load Transport Theories and Equations

Although attempts toward a rational approach have been made through the years, the equations to determine bed-load transportation are essen-tiallyempirical. Since the turn of the century bed-load has received much attention; yet, bed-load movement still cannot be accurately pre-dicted.

The basic concept assumed that the loose bed material was sliding in layers under the action of flow. Early investigators reasoned that the top layer of the bed is set in motion by shear between the water and the bed. In flows where energy is dissipated primarily to overcome friction, the shear force, TO' is cal led tractive force and is equal to

T

=

ydS

o e

in which, y is the unit weight of the water, d is the depth of flow, and S is the slope of the energy gradient. If this shear force becomes

e

larger than the force resisting motion of bed particles the bed wil1 move. Rate of transport is therefore a function of the difference be-tween these two forces. The Ou Boys equation for bed-1oad transporta-tion is the first semi-theoretical approach in which rate of transport is re1ated to the flow conditions. Based on the assumption that a cer-tain quantity of sediment is set in motion by an excessive tractive force. Ou Boys (15) conc1uded that the rate of bed-load transport is proportional to the excess of the prevailing tractive force over the critical value required to initiate movement. Thus, Ou Boys proposed the classical bed-load formula:

(2.2)

in which, qs is the rate of transportation, Cs is a sediment coeffi-cient which depends on the character of the sediment, TO and TC are

the prevailing and critical tractive forces, respectively. The Ou Boys equation had been widely used in the past because of its simplicity.

In 1936, Shields (15) developed an equation for bed-load transport. His bed-load equation is dimensional1y homogeneous. By considering the

acting forces to be restricted to shear forces, he deve10ped arelation:

(2.3)

where TO is the critica1 shear stress at the bed. Ys and y are the spe-cific weights of the sediment and f1uid respective1y, d_{. s} is the grain size, v is the kinematic viscosity of the f1uid and U* is the shear velocity. The function f was presented as a shaded area on what has become known as Shie1ds' diagram as shown in Figure 2.1. These data were obtained from f1ume experiments with fully deve10ped turbulent f10ws over artificia11y f1attened sediment beds. The va1ue of the cri-tica1 shear stress was determined by Shields from a graph of observed sediment discharge versus shear stress.

White (32) obtained arelation for f10ws in which motion around the grain was 1aminar,

(2.4)

where 8 is the ang1e of repose of the sediment immersed in the f1uid.

The form of the equation was found by considering the interaction of the drag and weight forces on a grain. The numerical constant was determined experimental1y. For turbulent flow, White found that the critical shear stress was about one half of that for 1aminar flow. He attributed the difference to velocity fluctuations in the turbulent flow which cause f1uctuation in the boundary shear stress and in the forces acting on the grain.

In 1950, Einstein (13) presented a procedure for computation of

1.0 .9 .8 .7 .6 .5

...

..

"a _.3

-

)...

1$1

1

C

.2 0.1 .09 .08 .07 m .06 .05 .0" .03 .02 SYM OESCRIPTION T. g/cm3

•

AMIER 1.06 @ _{LlGNITE 1.27} [J _{GRANITE 2.7} 1!I IARITE _".25

•

•

SA NO (CASEY) _2.65 $ UNO (KRAMER) 2.65 0 SANO (U.S.W.E.S.) 2.65 @ SA NO (GILIERT) 2.65

:s

,Jo.l

(ra

-1)gd

s

1 I\.. 2

..

6 8 10 2

r

~

6 8 100 2

..

6 8 'C.. I I _{L L L L} 1 1 1 1 1 1 1 1 1 1 ~ 1 I1 1 L L I 11 1 1 1 1 "'< I I I1 I .I .L lL II I V I I ..l:.L I~ ./. I V _{lil I I /" I II [Jl}

,ir

-~./ _lLlL~J L_ _{® L_}

v:

_J.l

Jo ~ f..!'"

~.

_'-L h~ _L ,__ IJ •

.

'

.

~ ~

1

1000 .01 0.1 .2 .3.4.5 .6.7.8.91.0 2 3 .. 5 678910 2 3 .. 5 6789100 2 3 .. 5 6 7891000 U.ds

100 ;;...*

...

r- IA

•

1- ~

r..:

~

rt

....

~ I

...

•• ...

_....

it~

L'"

.~

Ut

"~

"r"-I---

+.-...

curve compared with measured points for uniform sediment t-- .d

=

28.65mm[Meyer- Peter et al. (934)] r-- •d =Q.785mm[Gilbert (1914)J

~(r~213d"B.-

Î~I

I

10 -...J lO 0.1 0.0001 0.001 0.01 _{0.1 1.0}

+.

Fig. 2.2- Plot of Einstein Functions

~*

as a Function of

~*.

bed-load discharge from the characteristics of flow and sediment. The Einstein procedure computes the probability of the bed-load movement at the surface as related to intensity of flow. The Einstein bed-load equations may be writte~

J

B*1jJ*-l/no_t2 1 e P

=

1-;.;-B*1jJ*-l/no dt

=

_l-Ä*A*cp_cp*_* (2.5) and (2.6)

where A*, B*, and no are universal constants.

If the bed sediment is not uniform in size, the Einstein procedure requires adjustment of cpand 1jJ.The adjusted values for individua1 size classes of bed sediment are cp*and 1jJ*.The Einstein graph of cp* as a function of 1jJ*is shown in Fig. 2.2.

The graph shows a rapid decrease of bed-load discharge when 1jJ*

(2.7) becomes large. The bed-load discharge of a size fraction is directly proportional to the percentage by weight of that size fraction in the bed sediment for a particular value of CP*.

Kalinske (15) applied fluid turbulence concepts to the critical tractive force theory in 1967. He developed a bed-load transport equation as follows:

in which, U* is the shear velocity, Ds is the grain size, f is a function involving the characteristics of turbulence.

In 1948, the Meyer-Peter formula (11) was developed at the Zurich Hydraulic Laboratory and has been used quite extensively in Europe. It was first published as fol1ows:

2/3

qb

a +

b

D

s

(2.8)

where qb = VRb - V is the average velocity of the flow and Rb is the hy-draulic radius of the bed.

In 1956, Bagnold (15) developed a theory predicting the relation-ship between transport function and shear function for materials of uni-form grain size. This transport function is composed of (i) a bed-load transport function, ~b' and (ii) a suspended-load transport function,

~s' Assuming the energy distribution is such that the work rate re-presented by ~s is equal to that represented by ~b' Bagnold derived the functions (2.9) and l+B (_s) ~t = ~s Bb

in which ~b' ~s' ~t are the bed-load, suspended load and total trans-(2.10)

port functions, respectively, and Bb is the dimensionless part of ~b'

In 1972, Stern berg (28) suggested the following formula to estimate the bed-load transport.

p -p

(T)gj

s

3

= KpU* (2.11)

where j

=

mass discharge of sediment (gmcm-1sec-1); Ps

=

density of

sediment; p = f1uid density; g = acce1eration due to gravity; K is a proportiona1ity coefficient that expresses the abi1ity of a flow to transport sediment; U* is the friction velocity.

Suspended-load Transport Theories and Equations

If the bed of an alluvia1 channe1 contains some percentage of fine particles, then a certain proportion of the transport wi11 be in sus-pension. Because of the high turbu1ence level generated by the propel-ler, the capacity to transport fine grains is high.

In 1933, O'Brien (19) introduced the basic equation for the distri-bution of suspended material. Assuming the sediment transfer coefficient, ES' is approximate1y equa1 to the momentum diffusion coefficient, Em' he derived the equilibrium equation

dc

Cw

=

ES dy (2.12)

where c is the concentration of sediment at elevation y above the bed,

w is the sett1ing velocity of the sediment and ES is a diffusion coef-ficient for sediment. Integrating the above equation leads to the equation

c ln -

=

ca (2.13)

in which ca is the concentration at an arbitrary reference level y

=

a. According to Vanoni (31), Von Karman, in 1934, introduced Eq. 2.13 to yield the relationship

1

n

s.,

= -pw

J

Y

1(~ )

dy

in which, T

is the shear stress.

In uniform open-channel flow with a

large width to depth ratio the expression for shear at any depth is

(2.15)

in which,

TO

is the shear at the channel bottom and d is the depth of

the flow.

Substituting Equation 2

.

15 into Equation 2.14 yields the

relation-ship

dü

ay

~ d

dy

_(2.16)

Introducing Von Karmanis universal velocity defect law

dü _ 1

Fa

1

dy -

rl ~

y

(2.17)

into Equation 2.16 yields the relationship

(2. 18)

in

which

k

is Von Karmanis constant.

_{Integrating Equation 2.18 gives}

the fol1owing expression

{~_a_ )z y

d-a

(2.19)

where

w

Z=---krçrP

(2.20)

The suspended load distrib

u

tion equation was introd

u

ced by Rouse in

1937.

Braaks (9) assumed that the velocity distribution is logarithmic, and that concentration follows the suspended laad equation:

qs

=

Jd Cu dy Yo

(2.21 )

where

C

=

concentration of suspended sediment, u

=

stream velocity, y

=

distance above the bed, Yo

=

lower limit of integration, and d

=

total depth.

The lower limit might reasonably be selected from one of the fol-lowing equations: (a) 2D _ s na -

cr

Ps-P gDs

'" =

-p- U*2 u

=

5.75 U* log

f-a

(2.22)

-u (b) na = e-k u* -1 u(no) = 0 where n

=

Yo/d o

He also suggested that the most reasonable choice would be the equation giving the largest value of no'

In 1969, Anmar et al. (4), using Einstein's formula and assuming

a logarithmic velocity distribution presented the following:

---

---where

u

=

current velocity at a depth y, U*

=

shear velocity, y

=

o

o

l~ for hydrau1ica11y rough bed, ~

=

Os for hydrau1ica11y smooth bed, and

3ëf

Os

=

bed material grain size.

Ve10cities required to start the motion for particles of specific gravityequa1 to 2.65 are shown in Tab1e 2.1.

Particle Diameter Water Depth Incipient Velocity

D{mm) (m) (cm/sec) 2 33.4 0.2 3 34.4 4 35.3 6 36.4 2 20.2 3 20.6 0.06 4 21.2 6 21.8 2 3.8 3 3.9 0.002 4 3.96 6 4.05

-Tab1e 2.1- Velocities Required for the Start of Motion (from reference 4)

Garre1ts (14), eva1uated the inf1uence of the propeller on the bottom of the cana1. He showed that the distance separating the cir-cle described by the propeller from the bottom and its thrust-load coefficient are important and influential variables.

Velocity Distribution

As shown by Balanin (7), a jet thrust created by a propeller

can be considered, as a first approximation, at any point of this jet relative to the bank (Vs) as being determined according to the follow-ing formula:

~---V =/(Vn2-Vc2)Yo22 _ 2 s 4a

2

x2 4a2x2+Vc2_Vc (2.24) where Vn = Vo + Vc

Vo = speed in the initial section of a jet

(mlsec)

a = coefficient equal to 0.04

Yo = jet radius in initial section (m)

Y = distance from jet axis to the point where the speed is being sought

x = point coordinate where speed is being sought, in the direc-tion of jet axis

v

= ship velocity

(mlsec)

c

He also gave the speed in the jet initial section as follows:

V = Vc (1+/1+ 2Gek )

o 2 S

wh ere

Gek = engine coefficient of load

S = coefficient 1.14

Approximate computation of the influence on the speed field of a jet thrust-back by an engine of limited surface is done by introduc-tion of an addiintroduc-tional, false source located on the opposite side of the limited surface at a distance equal to that of the main source. The unknown value of the speed at any point of jet, allowing for the limited surface, is determined as follows:

(2.25) For cargo ships the effect of this factor can be neglected since the maximum bottom speed is usually at some distance from the stern and the wake effects are small.

Albertson et al. {3} also state that "since the slip-stream of a

propeller differs little from any other type of jet, the velocity

dis-tribution in the wake of aircraft and watercraft should be subjected

to the same method of analysis".

Rouse (21) showed all of the analytic velocity distribution curves

which have been presented for the jet, together with measurements. All

the curves are fitted to the measured distributions at point u/um

=

O.S.

He indicated that two curves fit the major portion of the mean velocity

distribution very well:

the Gaussian distribution ànd that

correspond-ing to the constant mixcorrespond-ing coefficient.

Hilaly (4) gave the following expression for the average return

current resulting from a ship passing through the canal:

u L P { VoL 2 (P)2

-

=

Ko

+

Kl

(B) +

K2

(f) +

K3

Vt

%) +

K4 (

B) +

KS

r

rgn

(2.26)

where

u

=

the average return current, Ko

=

0.70066, Kl

=

-0.07219,

L

K2

=

-0.S699, K3

=

0.000365, K4

=

0.00608, KS

=

0.17125,

B

=

lateral

clearance ratio

=

mean w

i

d

th

of the canal / max ship width, with the

mean width of the canal

=

canal cross

-

sectional area / canal depth,

~ =

vertical clearance

r

at

i

o

=

canal depth/ship draft, V

=

ship

'

s speed,

Vt

=

Schijf's limiting s

p

eed, and h

=

mea

n

canal depth.

CHAPTER III

DEVELOPMENT AND DESCRIPTION OF THE NUMERICAL MODEL Problem Statement

It was the purpose of this study to develop a numerical model for determining the rate of sediment movement induced by a ship's propeller

in a restricted waterway. Figure 3.1 is a definition sketch showing a

ship with a propeller of diameter D traveling in a confined waterway

of depth h. The propeller turns with a speed usually expressed in

re-volutions per minute (RPM). When the propeller turns, a turbulent

ve10-city downstream of the propeller will be generated. Since the propeller

rotates, both axial and transverse velocity components will be induced. In order to simplify the problem only the axial velocity in the

x-direction, (longitudinal velocity) downstream of the propeller was

investigated, and rotational effect of the propeller was neglected.

The literature survey indicated no adequate equation to describe this

velocity. Thus, a numerical model was used to treat the prob1em.

The velocity of the propeller's jet can disturb the sediment and

cause sediment movement. In order to maintain channel depth, it

be-comes important to determine the quantity of sediment movement induced

by propeller motion in a restricted waterway. The problem involves

interaction between the velocity distribution induced by the propeller's

jet and the sediment movement caused by this velocity. As the ship passes through the channel, the bottom and sides of the canal section

are subjected to jet action due to the ship's propellers. The velocity

distribution changes from time to time at a given section. A flat

y

$

L,

\

~

-

-

Tx

h

)

R I -...J Channel lottom ~~~~~

development of a numerical model. The sediments considered are cohe -sionless.

Momentum Theory of Propeller Action

Momentum Theory

Propellers derive their propulsive thrust by accelerating the fluid

in which they work. This action is in accordance with Newton's law of

motion, which states that force is required to alter the existing state

of motion of any material body in magnitude or direction, and that the

action of any two bodies upon one another is equal and opposite.

Newton's first law is expressed by the equation:

F

=

m dv/dt

where F

=

force exerted on body, m

=

mass of body, and dv/dt

=

resulting

acceleration of body.

Integrating between 0 and t seconds, we get

I:

F dt

=

mV

2 -

mv,

(3.1)

where Vl and V2 are the velocities at the beginning and end of the time

interval. The expression

I:

F dt (3.la)

is called the impulse of the force in the time interval from zero to

t, and the product of mass and velocity is cal led the momentum. The

equation states that the impulse of the force in a given time

inter-val is equal to the whole change in momentum produced by the force during this interval.

Momentum Theory of Propeller Action

In the ideal concept, the propeller is regarded as a disk or mech-anism capable of imparting a sudden increase of pressure to the fluid passing through it, the method by which it does so being ignored. It is assumed that:

(a) The propeller imparts a uniform acceleration to all the fluid passing through it, so that the thrust thereby generated is uniformly distributed over the disk.

(b) The flow is frictionless.

{c} There is an unlimited inflow of water to the propeller. Consider a propeller disk of area Ao advancing with uniform velo-city VA into an undisturbed fluid. The hydrodynamic force will be un-changed if we replace this system by a stationary disk in a uniform flow of the same velocity VA' as shown in Figure 3.2.

At the cross section 1, some distance well ahead of the disk, the velocity of the flow is VA and the pressure in the fluid is Pl' Well behind the screw, at section 3, the race column, i.e., the fluid which was passed through the screw disk and been acted upon by the pressure or thrust-producing mechanism there, will have some greater sternward velocity, which we may write as VA(l + b}. The fluid must acquire some of this increased velocity before it reaches the disk, and the velocity through it, at section 2, will be greater than VA- We may write this velocity as VA(l + a}, where a is an axial inf10w factor.

The pressure in the race column, which is Pl well ahead of the disk, will be reduced as the fluid approaches the disk, since by Bernoul1i IS law an increase in velocity is accompanied by a decrease

in pressure. At the disk, the pressure is suddenly increased by an

N o VA(I+b) 3 2 1 I IT'

I

----F

-

~peller·C

area Aa ~ncrease in 'pressure I_I1

unspecific mechanism to same value greater than Pl' and then decreases with further acceleration in the race. If section 3 is sa far aft of the disk that the contraction of the race may be assumed to have ceased and, if there is na rotation in the race, the pressure in the race at section 3 wil1 be P1' equal to that in the fluid outside the race.

The quantity of water passing through the disk in unit time wil1 be

(3.2)

Neglecting any effect of rotation which may be imparted to the f1uid the change of momentum in unit time is

and this must be equal to the thrust T on the disk. Hence

T

=

pQVAb

2

= pAo (VA) (l+a)b

_(3.3)

The total work do ne per unit time is equa1 to the increase in kinetic energy of the fluid. Since friction is neglected, and if there is no rotation of the race, the increase in kinetic energy in unit time is given by

~ pQ[(VA

)2(1

+

b)2 -

(V

A)2]

= }

p

Q(VA

)2

b

2

+ 2bVA2)

~

p

Q(VA)2b(1

+ ~)

b

= TVA (1 + 2) (3.4)

This increase in kinetic energy is provided by the work done on the water by the thrust, which is TVA(l + a) in unit time. Hence we have

TVA(l +a)

=

TVA(l + ~)

b

or a

=

'2

(3.5)

or one half of the sternward increase in velocity is acquired by the f1uid before it reaches the disk.

The usefu1 work obtained from the screw, i.e., the work done upon the disk, is TVA• The idea1 efficiency n1 wi11 be

usefu1 work obtained n1

=

work expended

=

TV

A

/TV

A

(l + a)

= 1/(1 + a) (3.6)

If the thrust-loading coefficient is defined as

C

=

T T 1 A (V )2

"2i'

0 A (3.7) then 2 n

=

---:;--;"A 1 1 + (CT + 1)1/2

The thrust coefficient (KT) is given by Comstock (12)

(3.8)

(3.9)

and speed coefficient Jt is given by 101.33V

A

Jt

=

nD

(3.10)

wh ere T = thrust, p = density of water, n = RPM of propeller, D = diameter of propeller, and VA

=

ship speed in knots.

Figure 3.3 shows the prope1ler1s characteristic curve in open water.

N W .8 f::"0

,...

"

c _.6

..

~

...

IU ~

...

~ _.4

-

c

•

-

_"

.

_

...

...

•

•2

°

u

-

..

_::t

_..

..c:

...

•1 .2 .3 .4 .5 _{•6 • 7 • 8 •9 1.0 1.1} 1.2 1.3 1.4 1.5 1.6 Speed Coefficient Jt

momentum theory:

l} Jt is obtained from Equation 3.10, the ship's speed, and diameter and speed of the propeller.

2} KT is obtained from the propeller-characteristic curve. 3} The thrust, T, is evaluated from Equation 3.9.

4} CT is determined from Equation 3.7. 5} nl is obtained from Equation 3.8.

6} Once nl is known, and using Equation 3.6, a may be evaluated. 7} b is obtained from Equation 3.5, and then used to solve for

VA(l + b}.

Once the initial velocity downstream of the propeller is known, a numeri cal model can be developed to simulate the velocity distribu-tion induced by the propeller.

Velocity distribution approach.- The Gaussian normal probability function is used to simulate the velocity distribution downstream of the propeller. It is generally assumed that jets follow the trend of the Gaussian normal probability function {3}. In developing a theory of turbulent jets, Al bertson , et al. (3) assumed that the pressure distribution was hydrostatic throughout the zone of motion, the momen-turn flux was consta nt, and the flow was dynamically similar in every section within the mixing region.

The Gaussian normal probability function can be presented as:

{3.1l}

in which V is the velocity component in the direction of the jet axis, Vm is the axial velocity along the centerline of the jet at any distance

x from the outlet, y is the coordinate norma1 to x, and where crmathe-matica11y is the standard deviation, and physica11y is the distance from the center1ine of the jet to the point of maximum velocity gradient

(Fig. 3.4). There are two zones of flow deve10ped downstream of a jet. As shown in Fig. 3.5, the flow is deve10ped from an initia1 zone (the zone of flow establishment) to a zone of estab1ished flow. The flow establishment zone is just beyond the eff1ux section. The limit of the zone of flow establishment is reached when the mixing region has penetrated to the central portion of the jet which has become turbulent. The flow may then be considered estab1ished. The conditions within the zone of flow establishment and the zone of estab1ished flow were first investigated theoretica11y by To11mien in 1926. Subsequent studies by other investigators (3) introduced either a concept of vorticity trans-port or the assumption of constant eddy viscosity across any section of the diffusion zone. No experimental measurements had been made by To11-mi en (1).

In 1950, A1berston, et al

.

, (3) used volume flux, momentum flux

and energy flux to deve10p a set of equations for these two different

zones.

The formu1as agreed with experimenta1 measurement and are shown

as follows:

Zone of flow establishment

(3.12)

for

Bo

Y >

2" -

Cl

x,

IC 0 >E

'"

~

.

0 11 >)( >)( IC e

.j

.,.... ,.... .,.... ..c t'O ..c o S-c, ,.... ~ S-o z: QJ J:: +-' 4-o Vl U .,.... .,.... LL...

Flow Establishment Established Flow N ... ~' " ", .", ....'

TA-;:::-~--~--~'

.:»: la V_o

e ~~::.

_"",-; y_{- I 4'}

"'~-f"~'_"'''

_{~~4c.""CL1"'CI:)LI}cc. ....I

/

.~e

_~~~~~'-~"q ~ QAf'_{:6 C A} ~ _C

o n

~~l~:'CI')

o

C 0 ~ ~~~rv ,~ N . ~~~ ':)

c ')

G

emlne] limits /~~~~~ of Diffusion Re ._910n V <V max 0

(3.12a) for the case of a slot,

and

(3.13)

for

and V

=

1,

V

o (3. 13a)

for the case of a circu1ar orifice with r bein9 the radial coordinate. Given the experimentally determined values of Cl and C2, the above equations can be expressed as

V

y -

1

B

10910

V

o

=

-18.4(0.096 + ~ 0)2

for the slot, and

1

V r -

'2

Do )2

10910

_V

₌

_{-33(0.081 +}

_x

o

for the circular orifice. Zone of established flow

(3.14)

for the slot, and

VlD [1 r2]

V =

2C

__E. exp - -2

2"

for the circular orifice.

Af ter substituting the numerical va1ues of Cl and C2 into the equa-tions, they can be expressed as

Vfi:

10910

V

B

o 0 2 = 0.36 - 18.4 ~

x

(3.14a)

for the slot, and

2 1₀₉₁₀

_V

v

₀

x

0 79 33 ~

= • - x2

o 0

(3.15a) for the circu1ar orifice. The equations for the velocity a10ng the center 1ine in the zone of estab1ished flow, Vm, can be expressed as

V

-m~x = 2.28 Vo Bo

for the slot, and Vm

x

= 6.2 Vo _Do

for the circu1ar orifice, where V = velocity at any

(3.14b)

(3.15b)

point, Vo = initia1 velocity from jet, Do

=

diameter of circu1ar orifice, x

=

point coordinate where the velocity is being sought, in the direction of jet axis, and Cl' C2 = constant.

Once the velocity distribution is found, Shie1ds' diagram is used to estimate the initial motion of a partic1e on the bottom of the channe1. Shie1ds' diagram is se1ected because it is based on ex-perimental data and has been used extensive1y by other investigators. The parameters in Shie1ds' diagram include the shear stress (TO) and

the shear velocity (U*). They can be obtained from Sternberg's (28) formu1as:

-3 -2 '0

=

3

x

10

p

U 100

-2-U*

=

5.47 x 10

U100

(3.16) (3.17)

where

p

=

density of water, and 0100

=

velocity at one meter above the

CHAPTER IV

THEORETICAL STUDIES OF JET AND SEDIMENT MOVEMENT Theory of Free Turbulence

For the two-dimensional flow of an incompressible fluid, the equa-tion of moequa-tion in the x-direcequa-tion of the boundary layer has the follow-ing form:

~ + u ~ + v ~

=

}2U _

1~

at ax ay ay2 p ax (4.1 )

where u and vare the instantaneous values of the velocity components,

p, p, and vare the instantaneous values of density, pressure, and

kine-tic viscosity, respectively.

The equation of continuity for the two-dimensional flow of an

incom-pressible fluid has the form

~+ ~= 0

ax ay

Multiplying by u, the following equation is obtained:

(4.2)

u~+u~=O

a

x

ay

which can also be written in the following form:

a(u2) + auv = u ~ + v ~

ax a y ax ay (4.3)

After substituting equation 4.3 into equation 4.1,

~ + a(u2) + auv

=

v

4 _

1 ~

at ax ay ayL p ax (4.4)

The velocity components of turbulent flow and the pressure can be

separated into a time average and a fluctuating component

u

=

u + UI, V

=

V + VI, P

=

P + pI (4.5a)

where the time averages of the f1uctuating components vanish or

Vi

=

0, Ui

=

0, pi

=

O. (4.5b)

In general , however, this cannot be extended to the square of the f1uctuating va1ues and their products. The instantaneous va1ues in equation 4.4 are rep1aced by the mean and fluctuating va1ues; viscosity and density are assumed constant. Averaging with respect to time, and taking equation 4.5 into consideration we obtain

- - 2 :-:-r2 - :-:-r.:T

~ + ~ + ~ + auv + au v

at ax ax ay ay

a

2

lj 1 ~

= \)

ay2 -

p

ax

The boundary layer of the turbulent flow (Figure 4.1) is assumed

(4.6)

to consist of a very thin, 1aminar sub1ayer bordering directlyon the wa11, whi1e the effect of viscosity in the remaining part of the bound-ary 1ayer is not significant.

A special characteristic of turbulence-free jets is the absence of solid flow boundaries and consequently also of a 1aminar sub1ayer, which

permits the neg1ect of the influence of viscosity in all cases of free

turbu1ence. This a1so exp1ains the se1f-similarity of jets over a

broad range and independent of the Reyno1ds number.

Free jets expanding into an infinite region fi11ed with a quiescent f1uid, and a1so wakes surrounded by an infinite undisturbed stream,

possess such small pressure gradients that in most cases the latter can be neglected.

The equation of motion for the two-dimensiona1 free flow of an incompressible fluid, stationary with respect to mean velocity (;~

=

0),

can be written in the fo11owing form:

-2 -

d

--~ + auv + ~ + aulVi

=

0

or, changing to mean and fluctuating velocities in the equation of con-tinuity [equation 4.2]

(4.8)

It follows from equation 4.5 that with a time average the last two terms of this equation vanish, that is, the conventional farm of the equation of continuity is valid for the mean velocities:

(4.9)

-Multiplying equation

4.9

by u,

-

a

u

-

a

v

u-+u-=O ax ay

(4.9a)

from which follows an equation analpgous to equation 4.3: 2

au + auv

=

u ~ +

v ~

ax ay ax ay

(4.9b)

Substituting this expression into equation 4.7 leads to the equation for the two-dimensional turbulent boundary 1ayer of a free jet:

2

-- au u -_ay + \) au + aul + alJiVT ay ax ay = 0 (4.10) 2 aU'

The term

-ax

can be neg1ected because the velocities and fluctuations of velocity change much more slowly along the flow than in the trans-verse direction; whereas, the magnitudes of Ui and Vi are of the same order. Thus

au + au alJiVT

u

_ax

v-+ =0

ay ay (4.11)

For simplieity, the bars over the mean values of veloeity are disearded, that is, u and v signify the time-averaged ve10eity eomponents.

It is we11-known that the equation of motion for the two-dimensional steady isobaric flow of an incompressib1e f1uid can be represented in the fol1owing form:

~+v~=l~

u ax ay p ay (4.12)

wh ere Txy is the shearing stress aeting in a p1ane perpendicular

tq

the

0Xy plane.

Comparing equation 4.12 with equation 4.11 for pure1y turbulent motion, we arrive at the original re1ation of the Prandt1 theory for the apparent turbulent shearing stress:

(4.13)

Prandt1 used equation 4.2 and assumed t

=

ex to obtain the equation of

two-dimensional motion of PrandtlIS old theory of the free turbu1ence:

au + au 2 2 2 au a2u u

ax

v

a

=

+ C x ay-2

y ay (4.14)

Theory of the Boundary Layer of a Two-Dimensiona1 Turbulent Jet of Incompressib1e Fluid

As shown in Figure 4.1, a uniform initia1 velocity field coincides with the boundary of the initial cross section of the jet. The pro-b1em of a free-p1ane boundary 1ayer is solved in coordinates

><

l'

I

I

I

N 35 35 " -e-" Ol " r-IJ...

the velocity depends only on n, or

(4.16)

In order to eliminate the experimental constant from the equation

of motion, we assume

(4.17)

and we choose the coordinate system

x,

lJ; =..1 =.!l.

ax a (4.18)

where a is an empirical constant characterizing the structure of the flow of a jet. The dimensionless velocity is a function of only one coordinate

u=uF'(lJ!)

o (4.19)

where F'(lJ!)is the derivative of a certain function F(lJ!),which is pro-portional to the stream function

(4.20)

By differentiating equation 4.20 the transverse component of velocity can be found:

v = - ~ = auo(F' - F)

ClX (4.21 )

Substituting equations 4.19 and 4.21 into equation 4.14, and taking into account

3

a = 72c2 ,I, =.1 ~ =

_!

2j_

=

_1

, 0/ ax' óx x' ay ax

leads to the fundamental equation of the boundary layer of the

two-dimensional flow of a jet of an incompressible fluid, when F'I ~ 0

F'II+F=O (4.22)

This equation was first obtained by Tollmien (1). The characteristic

equation for equation 4.22, 1 + k3

=

0, has three roots:

The complete integral of equation (4.22) equals

(4.23)

The three constants cl' c2' c3 and ordinates 1Ji1 and 1Ji2can be evaluated

using boundary conditions at the inner and outer edge of the jet.

At the inner edge of the boundary layer (1Ji

=

1Jil)

a) the velocity component along the x-axis (u) equals the upper

flow velocity (uo)' that is, according to equation 4.19

F'(1Ji) = 1 (4.24)

b) the velocity component along the y-axis (v) equals zero, that

is, according to equation 4.21 and 4.24

(4.25)

c) the gradient of the horizontal velocity component (~~) equals

zero, that is,

F11(1)i) = 0

1 (4.26)

At the outer edge (1)i=

W2)

d) the velocity component along the x-axis equa1s zero, that is

F'(1)i)=O

2 (4.27)

e) the gradient of the horizontal velocity component equals zero, that is

F"(1)i)=O

2 (4.28)

Using the boundary conditions, we obtain:

1)il= 0.981; 1)i2= -2.040

cl = -0.0176; c2 = 0.1337; c3 = 0.6876

Substituting these numerical va1ues of the constants into equation (4.23), we obtain the fol1owing equation for the desired function:

~

= F(1)i)= -0.0176e-1)i+ O.1337e2 cos (~)

!f.

+ O.6876e2 sin (~) (4.30)

The va1ues of basic and auxiliary functions which a1low calcu1ation of dimension1ess velocities are given in Table 4.1 (1). In Figure 4.2, To11mien's velocity profile [equation 4.30] is compared with the experimenta1 data of Albertson, et al. (3) which are taken from Figure 1.9. The va1ue of the empirical coefficient of the structure of the jet is taken to be a = 0.09 and leads to good agreement between theory and experiment.

Table 4.1

Basic Functions of the Boundary Layer of the Jet (from reference (1))

W 1.0 1jJ F F' F" 1jJ F F' F" 0.98 0.981 1 0 -0.22 -0.021 0.566 0.520 0.93 0.930 0.999 0.048 -0.27 -0.048 0.540 0.519 0.88 0.880 0.995 0.093 -0.32 -0.075 0.514 0.'516I 0.83 0.831 0.990 0.136 -0.37 -0.100 0.489 0.511 0.78 0.781 0.982 0.176 -0.42 -0.124 0.463 0.506 0.73 0.732 0.972 0.214 -0.47 -0.146 0.438 0.499 0.68 0.684 0.961 0.249 -0.52 -0.167 -.413 0.419 0.63 0.636 0.947 0.282 -0.62 -0.206 -.365 0.472 0.58 0.589 0.932 0.313 -0.72 -0.241 0.319 0.450 0.53 0.543 0.916 0.341 -0.82 -0.270 0.275 0.424 0.48 0.498 0.898 0.367 -0.92 -0.296 0.234 0.396 0.43 0.453 0.871 0.391 -1.02 -0.317 O. 196 0.365 0.38 0.410 0.859 0.413 -1•12 -0.335 O. 161 0.333 0.33 0.368 0.838 0.432 -1.22 -0.350 0.128 0.300 0.28 0.326 0.816 0.449 -1.32 -0.361 O.101 0.263 0.23 0.286 0.793 0.465 -1.42 -0.370 0.077 0.226 0.18 0.247 0.769 0.478 -1.52 -0.377 0.056 0.189 0.13 0.209 0.745 0.489 -1.62 -0.381 0.039 O.151 0.08 0.172 0.720 0.499 -1.72 -0.384 0.026 0.113 0.03 0.137 0.695 0.507 -1.82 -0.387 0.017 0.074 -0.02 0.103 0.670 0.513 -1.92 -0.388 0.011 0.036 -0.07 0.070 0.644 0.517 -2.02 -0.389 0.009 0 -0.12 0.038 0.618 0.520 -2.04 -0.389 0 0 -0.17 0.008 0.592 0.521

.75 -+:>0 0 .5 u Uo .25

o

-2 Tollmien's rheery

o X

=

1 AX=2

l:l.

X

=

3 ~Experiments of Albertson, et al.

.X=4

o

0.5 1 V'

-1.5 -1 -0.5

Theoretical Method for Determining the Rate of Sediment Motion

The bed of an alluvial channel contains coarse and fine sediment. Because of the higher level of turbulence generated by the propeller, the capacity to transport sediment is high. It is very difficult to study sediment motion because of different sizes, angularity, and distribution of sediment in the channel bed.

Similarity Consideration on Incipient Motion

The forces acting upon an individual grain as indicated in Figure 4.3 are gravity force Fg' supporting force Fn' frictional force Ft' and inertia force Fi' acting at the points of contact of the grain with the

surrounding grains. Gravity, frictional, and supporting forces resist

the motion of the grain, while the inertia and skin friction forces tend

to move the grain. The relative magnitudes of these forces determine

whether the grain moves or remains stationary. Only the direction of

gravity force is well defined. The frictional and supporting forces

depend upon the orientation of the supporting grains as well as the

shape of the grains. The direction and magnitude of the inertia force

is highly affected by the shape of the grain and the arrangement of the

particles and the Reynolds number. The viscous force can be disregarded

because it is negligible compared to the dominating inertia force. For

these conditions only the net gravity and inertia forces need to be

considered. If their relative magnitudes reach critical values, the

3

grain will start to move.

With Fg proportional to g(ps-

p

) d

and Fi

proportional to

p

u2d2, the equation is as follows:

,.J::.

N

flow

-

Fb

(4.31 )

where u is the local controlling velocity, and it may be selected as the average velocity one grain height above the theoretical boundary. The turbulent-boundary layer theory at a rough wall indicates that these local mean velocities are proportional to the shear velocity, U*

=

{TO/P.

Equation 4.31 can be written as

(4.32)

The derivation of equation 4.32 is based on the assumption that the

viscous force is much smaller than the form drag, i.e., Fv

<

Fi.

If

this assumption is not reasonable, the force ratio Fi/Fv should be

inc1uded.

The viscous shear can be expressed as

(4.33)

The velocity gradient, ~~, is the ratio of shear velocity to grain

size, U*/d.

For the viscous force

(4.34)

and for the ratio of inertia force over viscous force

(4.35)

where R* denotes the Reyno1ds number of the grain. These two dimen-sion1ess parameters are the same as presented by Shie1ds (Figure 2.1).

The Equation of Sediment Continuity

The a1teration in the sediment-carrying capacity and the consequent scour or deposition may be expressed in the form of a differentia1 equa-tion. Consider the flow in an a11uvia1 channe1 of unit width. Let qs be the volume rate of sediment inf1ux across section AA, and q + ~qs the eff1ux across section

BB,

a distance ds further downstream. As aresult of the imba1ance between the inf1ux and efflux of sediment in the section considered, let there be an erosion ~n in a time ~t (Fig.4.4). Then we have

(4.36)

therefore

(4.37)

and hence in the limit

(4.38)

The above equation indicates that the rate of scour in a given section is equa1 to the gradient of the sediment-carrying capacity in the di-rection of flow. A solution of this equation for a particu1ar geometry wi11 therefore afford an approximate description of sediment movement.

The sediment-carrying capacity qs may be expressed in terms of

the bed shear, and if the initial flow pattern is known this can be solved for time t

=

O.

However, the geometry of the bed changes rapidly due to scour and re

-sults in a new flow pattern. If an expression can be found for this new

flow pattern, a procedure describing the motion may be developed up

to the point wh ere there is no further movement of sediment.

•

lIIr <l

+

•

lIIr

f

>, .j...J .,... ::::J C .j...J C 0 u .j...J C QJ E .,....

"

QJ I ti)

,

4-,

0 I c I 0 I .j...J ttl ~ ::::J 0-W

,

'"

S-I _<J 0

4-,

..c I u .j...J QJ .::L. Q) ti)

'"

D s::

-0

...

;) .j...J

'"

s::

...

'r-Q)

4

-Q)

i

Cl <:::t-

_.

<:::t-en

,

LL.

•

lIIr

CHAPTER V

PRESENTATION AND DISCUSSION OF RESULTS Initial Velocity Downstream of the Propeller

As described in Chapter 111, the momentum theory of the propeller is used to develop the initial velocity just downstream of the propel-ler. The procedures are summarized as follows:

1) J

t i

s obtained

f

rom Equation 3.10, t

h

e s

hip·s speed, and

dia-m

eter and speed of the propeller.

2)

KT is obtained from the propeller - characteristic

curve.

3)

T

he thrust, T, is evaluated from Equation 3.9

.

4)

CT is determined from Equation 3.7.

5)

n

l is obtained from Equation 3.8.

6

)

On

c

e

n

l is known, and using Equation 3

.

6

, a may b

e

evaluated

.

7)

b

is obtained from Equation 3.5, and then

us

e

d

t

o

s

olve for

Velocity Distribution

T

he velocity distribution downstream of the propeller a

t different

p

o

s

it

i

on

s

is simulated by using the formulas fo

r

t

h

e o

rifice

s

(Chapter

I I1) .

Zone of f

low establishment

u

=

(3.13)

(3.13a)

Zone of established flow

(3.15)

Velocity distributions based on these formu1as are shown in Fig-ures 5.1 through 5.10, which show the variation of velocity at different locations downstream of the propeller. In Figures 5.1 through 5.10, x is the di stance from the propeller,

D

is the diameter of the propeller, V is the velocity at any point, Vo is the initia1 velocity we11 behind the propeller, and

R

is the distance from the axis of the propeller to the point where the velocity is being sought as shown in Figure 3.1. C2 is an experimenta1 constant. The va1ue C2

=

0.081 was used in the study.

Initiation of Sediment Movement

Shie1ds' diagram was used to estimate the size of grain which wi11 move. Wh en the shear Reynolds number (U*d/v) is greater than 30, a 1inear function was found between dimension1ess critica1-shear stress and shear Reyno1ds number.

(5.1)

1.0' • • • • 0.7 V 0.6

-

_Vo ~ _O.S 1.0 0.4 0.3 0.2 0.1 0,9 0.8

o

• 1 ,2 .3 .4 .5 .6 ,7 R

-

_o

.8 .9 1.0 1.1 1.2 1.3

0.7 V 0.6

-

_Vo ln _0.5 0 0.4 0.3 0.2 0.1 1.0· • 0.9 0.8

o

.1 .2

.

...

.3

"

.

\ \ "\ \ .4 _{.6 .7} R

o

.8 .9 _{1.0 1.1 1.2 1.3}

0.7 V 0.6 Vo en _0.5 0.4 0.3 0.2 0,1 ~ 1.0' • .._ 0,9 0.8

o

.1 .2 .3 .5 .6 .7 R

o

.8 .9 1.0 1.1 1.2 1.3

CO)

·

.-C"'f • .-Cl o::t 11 .-• X .-~ ttl 0 s::

·

0 .- .,... ~ ::J x: 0- .,..._s,

·

~ Vl .,... Cl Cl) _e-, ~ .,... U 0 ,...

"

QJ ::::-~IQ ,... ttl s:: >0 .,... -0 ::J ~ .,... t))

'"

s::0 _J o::t ~

.

LO t)) .,... L!... CO) C"'f 0-

.

o Cl) o

"

o -0

.

o lil

.

o ~

.

o M o C"'f

.

o .- o o

0.7 V 0.6

-Vo (.J1 _0.5 w 0.4 0.3 0,2 0.1 .\ 1.0· • --0.9 0.8

o

• 1 .2 .3 .4 .5 .6 .7 R

-

_o

.8 .9 1.0 1.1 1.2 1.3

11"'" 0.7 V 0.6

-

_Vo en _O,S ~ 0.4 0.3 0.2 0,1 1.0-0,9 0.8

o

.1 .2 .3 .4 .5 .6 .7 _{1.0 1.1 1.2 1.3} R

-

o

.8 .9

o r-, o co o ..0

.

o _ov

.

lil

.

o ('oe

.

o M

.

...

o o 55 CO) •

...

('oe •

-

_{,..._}Cl 11

...

_·

-

>< .jJ rtS 0

.

_t::

-

_.

_....

0 .jJ :::3 ..c 0-

...

_s, .jJ VI

.

....

Cl co _~ .jJ

.

....

u 0

"

...

QJ ::> ~IQ

_...

rtS -0

.

....

t:: e _"0 :::3 .jJ

...

Ol

'"

t:: 0 _J ,..._ 'V_• LO Ol

...

CO) u, N

...

o

o cc o ..0

.

o "I:t

.

o lil o M

.

o N o

....

o ("')

·

....

N

...

Cl co 11

....

·

x

....

+-'

"'

0 I::

·

0

...

'r-+-' ::::::I .0 0- 'r-s; +-' Vl 'r-Cl co _>,

·

+-' 'r-U 0

"

r-<IJ ::> aclC <ti I:: -0 ' r-"'0 ::::::I +-' 'r-0'>

'"

I:: 0 ....J co ~

.

LO 0'> 'r-("') L1.. N

....

o

o ... o co o -0

.

o 'O:t_o

.

lI)

.

o c...

.

o

...

.

o o 57 C") •

...

c...

...

_Cl Ol 11

...

_•

...

>< +-> rtl 0

.

_C

...

0 "r-+-> :::J ..a 0- " r-s, +l VI "r-Cl co _c-, +-> "r-U 0

...

r-Q) >-ac:IC r-rtl -0 "r-c -0 :::J +l "r-Ol lI) C 0 _J Ol "lilt

.

l.() Ol -e--(") LL.. c...

...

o

0.7 V 0.6 Vo CJ1 _0.5 ex> 0.4 0.3 0.2 0.1 1.0 0.9 0.8

o

.1 .2 .3 .4 .5 .6 .7 R

-

_o

.8 .9 1.0 1.1 1..2 1.3

=

'0

= 0.06

(rs - r)

(5.2)

In equations 5.1 and 5.2 shear velocity (U*) and shear stress

(

'0

)

can be estimated by uSing the equations given by Steinberg (28) as shown in Chapter 111.

-3

-2

'0

=

3

x

10 p U100

(3.16)

-2-U*

=

5.47 x 10 U100

_(3.17)

All the steps mentioned previously in this chapter were programmed as shown in Table 5.1 for computer solution. Once the input of the ship's speed, the speed and diameter of the propeller, water depth and draft of the ship are read in this program, the velocity distribution and critical grain size of motion at each location could be obtained. A sample of output for the Tanker TEXAS CALIFORNIA is shown in Table 5.2. Bottom vélocities and critical grain sizes from the output are summarized in Table 5.3.

C C C C C C C C C C c c c en 0 _C C C C

,

TABLE 5.1 COMPUTER PROGRAM FOR VELOCITY DISTRIBUTION AND CRITICAL GRAIN SIZE OF MOTION

THIS PROGR'M COMPUTES VELOCITY OISTRIBUTION AND INITIATION OF SEOIM:NT MOVEMENT INOUCEO BY SHIP'S PR~PELLER I~ A RESTRICTEO WATERWAY

M= NUMBER OF CASE TO BE STUDIEO VA= SHIP VELOCITY IN KNOTS

N= RPM OF PROPéLLER

0= OIA~ETER OF P~OPELLER IN FEET Y= OEPTH OF WATERWAY

SD= ORAFT OF SHIP Cl= OENSITY OF WATER

VO= AVERAGE VELOCITY AFTER PROPELLER GAMS= SPECIFIC GRAVITY OF SAND

GAM = SPFCIFIC GRAVITY OF WATER

VI = VELOCITY lMETER FROM BOTTOM OF THE CHANNEL U = SHEAR VELOCITY U*

T = SHEAR STRESS

050 = CRITICAL GRAIN SIZE BEGIN TC MOVE

AEGIN TO CALCULATE THE AVERAGE VELOCITY AFTER PROPELLFR REAO.M Cl= 1.99 C2= 0.081 GAMS= 2.65 GAM= 1 CU= 0.0000121 00 400 K=l.M READ. VA.N,O.Y.SO CJ= 101.JJ* VA/(N*O) CKT = 0.45 - 0.41 *CJ T= CKT * Cl*N*N*CO.*4)/3600 OT= T/(.5*Cl*3.1416/4*O**2*VA*.2) CT=OT/(1.689··2) l = 1+S0RT(CT+l) ETA=2/Z B=2/ETA-2

TABLE 5.1 ( CONTINUEO)

0'\

VO=VA*(1+B)·1.689 YO=V-SO+0/2 R 1= VO-3.3

C END OF CALCULATING AVERAGE VEL.OCITY AFTER PROPELLER X0=0 / ( 2 ..C2 ) PRINT lS0.K 150 FORMATC·l·.T5.· ••• CASE'. 13) DO 20 1=1.10 x=,.O PRI NT 100,1 I F (X. GT. XO) G 0 TO 1 0 00 30 .J=1.21 R=( .J-l) .0/10 RD=R/O IF(R.GT.YO) GO TO 25 S=)/2-::2*X IF(R.LT.S) GO TO 55 C3=(R+C2*X-O/2) •• 2 C4=-:: 3/ (2.( C 2. X)•• 2) IFC C4 .L T .-20) GO TO 30 VXI =VO.E XPCC4) GO Ta 45 55 VXI =VO 45 VRI=VXI/VO PRINT 200.R.RO.VX1,VRI 30 CONTI NUE 25 PRINT 80

80 FORMATC'Ot,t***** VELOCITV IN REGION I ••••••• ) IF( RI.LT.O) GO TO 20

C7= (RI+C2.X-0/2) •• 2 CB: -C7/(2*,C2*X) •• Z)

IF(CS.LT.-20) GO TO 20 VI: Vo. EXP( CS)

TABLE 5.1 ( CONTINUEO)

0\ N GO TO 50 C VELOCITV IN REGICN 11 10 00 40 J=l,31 R=(J-l) .0/10 RD=R/O IFCR,GT.VO) GO TO 35 C5= -1/(2.C2 •• 2).CR/X, •• 2 C6=C1/C2.C2'*O/K).EXPCC5) VKII= VO*C6

VRI I=VK II/VO

PRINT 200,R,RO,VKIl,VRII 40 CONT INUE

35 PRI NT 50

60 FO~~ATC .0 •••••••• VELO:ITY IN REGION 11 •••• ··, IFC RI.LT.O) GO TO 20

C9= -l.lC2*C2.*2'.CRI.lX'''.2 Cl 0= C1.I (2.C 2'*0.1 X ) .E XPCC9) VI=

va.

CIO

50 u= 5.47.0.01.VI T=3.0.001*Cl*(Vl.*2) 61= T.I(GAMS- GA~) 62= (U/CU, ·.0.183 BJ= 1/82 64=a1.33.S0 BS=ALOG(B4) 66=8S/1 .183 050= EXP( aei RE=U. 05 O/CU IF(R~.GT.400) GO TO S1 GO Ta 52 510S0=T.I(0.06*(GAMS-GAM') 52 PRINT 250

TABLE 5.1 ( CONTINUEO)

250 FORMAT('O'.T3.'BOTTOM ~ELOCITY'.T2~. 'SHEAR STRESS'.T40.'SH:AR V:LO I CI TY' •T60 • ' GRAINS I ZE' )

PRINT 300.VI,T,U.050

300 FORMAT('0' .TS.EII .3.T21.EI0 .3.T42.EI0 .3,T62,EIO.3'

100 FOR'I4AT('O'.T25,' X/O=' ,I3,3( /) ,TIO,' R' ,T20,' R/O' .T3) ,'LONGITUOINAL 2 VELOCITY',T60,'V/VO" 200 FORMo\T(' ',T7,F5.I,T20,F5.1,T35,FIO.S,T60,F5.3' 20 CONTINUE 400 CONTI NUE STOP END en w

TABLE 5.2 C(J<1PUTEROUTPUT FOR THE "TEXAS CALIFORNIA"

0'1

.j::o.

XI'D= 1

R RI'D LONG ITIJDINAL VELOCITY (ft/sec) VI'VO

0.0 0.0 42.09247 1.000 ?4 0.1 42.09247 1.000 4.8 0.2 42.09247 1.000 7.':' 0.3 42.09247 _1.000 9.7 0.4- 42.09247 1.000 12.1 0.5 25.53036 0.607 14.5 0.6 3.46677 _0.082 16.9 0.7 0.10253 0.002 1°.4 0.8 _{0.00066 0.000} .**$* VELOCITY IN REGION I **.***

BOTTO~ VELOCITY SHEAR STRESS _{SHEAR VELOCITY (ft/sec)}

_{GRAIN SIZE}

_(ft/sec)

0.218E 00 0.283E-03 _{0.119E-OI 0.617E-02}

Xl'o= 2

P RI'O LONG nUL) rNAL VELOC ITY VI'VO

0.0 0.0 42.09247 _1.000 ?4 0.1 42.09247 _I.COO 4.8 0.2 42.09247 _1.000 7•.3 0.3 42.09247 _1.000 CJ.7 0.4 _{39.11996 0.929} 12. 1 0.5 25.53043 _0.f>07 14.5 0.6 11.38241 _0.270 16.9 0.7 _{3.46679 0.082} 1CJ. 4 0.8 _{0.72133 0.017}

TABLE 5.2 ( CONTINUED )

••••• VELOeJTY IN REGION I •••••• ROTTOM VELOelTY SHEAR STRESS

SHE"ARVELOelTY _{GRAIN SJZE} 0.443E Ol 0.117E 00

0.242E 00 _0.118E

Ol X/D= 3

R _R/D

LONGITUDINAL _{VELOeITY v/va}

0.0 _0.0 42.09247 _1.000 2.4 _-0.1 42.09247 _1.000 4.8 _0.2 42.09247 _1.000 7.3 _0.3 41.43857 _0.9R4 0'1 _9.7 0.4 _35.40015 0.841 (1'J 12.1 _0.5 25.53038 _0.607 14.5 _0.6 15.54395 _0.369 16.<;1 _0.7 7.98950 _0.190 19.4

_o.a

3.46679 _0.082

•••••

VF.LOeJ_{TY IN PEGION I}

_{••••••}

BOTTO~ VELOCITY _{SHFAR STRESS}

SHEAR VELOCITV _{GRAtN SIZE} 0.913E Ol _{0.49BE 00} 0.499E 00 _{0.503E Ol} X/D= 4 R 0.0 ?4 4.R R/O 0.0 0.1 0.2 LONGlTUOINAL VELOelTY 42.09247 42.09247 41.97714 v/va 1.000 1.000 0.997 ...--~---.__. -..-._....-....-.-"----

-TABLE 5.2 (cONTINUEO

)

1.3 0.3 39.11996 0.929 9.1 0.4 33.14467 0.181 12.1 0.5 _{25.53041 0.601} 14.5 0.6 11.81849 _0.425 16.9 0.1 _{11.38240 0.210} 19.4 0.8 6.58818 0.151 ••••• vELnCITV IN RF.GION( ••••••

BOTTOM VFLOCITY SHEAR STRESS _{SHEAR VELOCITY GRAIN SIZE} 0.124E 02 0.924E 00 _{0.681E 00 0.934E Ol}

~

~ X/D= 5

R RI'O LONG ITUDINAL VELOC ITV _v/va

0.0 0.0 42.09247 _1.000 2.4 _{O. I 42.08925} 1.01l0 4.8 _{0.2 40.10134 0.961} 1.3 0.3 31.03130 _0.8RO 9.7 0.4 _{31.69946 0.153} 12.1 _{0.5 25.53038} 0.601 14.5 0.6 19.34515 _0.460 16.9 0.1 13.79230 _0.328 19.4 _{0.8 9.25148} 0.220

•••••

VFLOCITV IN REGION I

••••••

BOTTO~ VELOCITY SHEAR STRESS _{SHEAR VELOCITY GRAIN SIZE'}

O.141E 02 _{oel30E Ol 0.806E}