HYDRODYNAMIC MODELING OF NETS AND TRAWLS
By
M. KRISHNAMURTHY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
ACKNOWLEDGEMENTS
The author wishes to express his sincere gratitude
to Dr. B. A. Christensen, under whose guidance this work was
carried out, for his advice and invaluable suggestions
throughout the study. This work would not have been
accomplished but for his patient guidance.
The author would like to thank Dr. PopStojavanic
for his discussions during the course of the study. Thanks
are also due to Dr. Huber for serving on the supervisory
committee of the author.
Appreciation is extended to all of his friends in
the hydraulics laboratory, especially to Fred Morris and
Tracy Lenocker for their help during the period of experi
ments. Assistance from the Marine Extension Center of the
University of Georgia is acknowledged. Mr. W. H. Burbank of
Fernandina Beach has provided invaluable suggestions and
assistance in the net designs.
The author appreciates the financial support received
from the National and Oceanic and Atmospheric Administration
Office of Sea Grant.
The typing of this dissertation has been ably per
formed by Mrs. Carolyn Lyons.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ................. ........... ......... ii
LIST OF TABLES ................ ........................ vi
LIST OF FIGURES ...................................... vii
LIST OF SYMBOLS ........ ............................. xi
ABSTRACT .............................................. xvii
Chapter
1 INTRODUCTION .................................. 1
1.1 Statement of Problem ................... 1
1.2 Objectives ............................. 7
2 COMPONENTS OF A GEAR .......................... 9
2.1 Net .............. .. .................. 9
2.2 Ropes ............. .... ............... 28
2.3 Sweeplines ............................. 32
2.4 Doors ........... ... .................. 33
2.5 Towing Warp ................. ........... 36
3 LITERATURE REVIEW .............. .... .......... 41
3.1 Mathematical Modeling of Fish Nets ..... 41
3.2 Physical Modeling of Nets and Trawls ... 44
3.3 Prototype Experiments .................. 70
3.4 Pitfalls in Existing Physical
Model Laws ........................... 71
4 DEVELOPMENT OF MODEL LAWS ..................... 74
4.1 Froude Modeling ........................ 74
4.2 MeshTwine Distortion .................. 74
4.3 Elongation of Twine .................. 80
4.4 Modeling of Ropes ............ ........ 84
4.5 Modeling of Floats and Sinkers ......... 86
4.6 Modeling of Doors ........................ 88
TABLE OF CONTENTSContinued
Chapter Page
5 PROTOTYPE FIELD TESTS .............. .......... 89
5.1 Parameters Measured ..................... 89
5.2 Net and Towing Site ..................... 90
5.3 Instrumentation ........................ 91
5.4 Test Procedure ......................... 107
5.5 Test Results ........................... 109
6 MODEL EXPERIMENTS ............................. 113
6.1 Selection of Scale Ratios .............. 113
6.2 Instrumentation of Model Net ........... 123
6.3 Description of Experimental
Apparatus .......................... 131
6.4 Test Procedure ......................... 140
7 DISCUSSION OF RESULTS .............. .......... 160
7.1 Validity of Model Laws ................. 160
7.2 Effect of the Movable Belts ............ 164
7.3 Comparison Between Physical and
Mathematical Models .................. 166
8 APPLICATION OF MODEL APPROACH IN
DESIGNING A BEAM TRAWL ..................... 171
8.1 Introduction ........................... 171
8.2 Description of the Beam Trawl .......... 172
8.3 Design of the Beam Trawl ............... 175
8.4 Model and Field Experiments ............ 181
8.5 Discussion and Results ................. 185
9 CONCLUSIONS AND RECOMMENDATIONS ............... 191
Appendix
A BIBLIOGRAPHY ON CHARACTERISTICS OF FISHING
TWINES AND THEIR TESTING ................... 195
B BIBILOGRAPHY ON MODEL LAW ..................... 198
C BIBLIOGRAPHY ON MODEL EXPERIMENTS ............. 201
D BIBLIOGRAPHY ON PROTOTYPE EXPERIMENTS ........ 209
E CALCULATION OF FISHING NET DRAG BY THE
NUMERICAL METHOD DEVELOPED BY
KOWALSKI AND GIANNOTTI (1974) ............ 214
iv
TABLE OF CONTENTSContinued
Page
REFERENCES .............................. ............. 220
BIOGRAPHICAL SKETCH .................................... 223
LIST OF TABLES
Table Page
2.1 VALUES OF COEFFICIENT OF FRICTION ............ 32
3.1 BASIC AND DERIVED MODEL SCALES FOR
HYDRAULIC MODELS ........................... 54
5.1 SPECIFICATIONS OF BURBANK FLAT NET ........... 91
5.2 RESULTS OF FIELD TESTS ....................... 110
6.1 SUMMARY OF SCALE RATIO ....................... 127
6.2 MODEL RESULTS WITH MOVABLE BEDS .............. 157
6.3 MODEL RESULTS WITHOUT MOVABLE BEDS ........... 159
7.1 PROJECTED FIELD DATA FROM THE RESULTS
OF MODEL EXPERIMENTS WITH MOVABLE
BEDS ........................ .............. 162
7.2 PROJECTED FIELD DATA FROM THE RESULTS
OF MODEL EXPERIMENTS WITHOUT
MOVABLE BEDS ............................... 165
7.3 COMPARISON BETWEEN PREDICTED VALUE OF
NET DRAG BY NUMERICAL METHOD AND
MEASURED VALUE IN THE FIELD TEST ........... 170
8.1 MODEL TEST RESULTS OF BEAM TRAWL ............. 183
8.2 RESULTS OF FIELD EXPERIMENTS ................. 185
LIST OF FIGURES
Figure Page
1.1 Components of a gear research program ......... 2
1.2 Physical elements of a fishing operation ...... 3
1.3 Iterative procedure in trawl and trawler
design and selection ........................ 4
2.1 Components of a gear .......................... 10
2.2 A netting twine ............................... 11
2.3 Load elongation curves ........................ 13
2.4 Load elongation curves for polyamide
netting twine ............................... 15
2.5 Logarithmic linearization of loadstrain
curve for polyamide ........................ 16
2.6 The mesh ........................ ........... 17
2.7 Net panels ..................... ........... 20
2.8 Net diagram of Burbank flat net ............... 21
2.9 Drag coefficient for nets as a function
of Reynolds number ........................ 25
2.10 Hanging the net on the headrope ............... 29
2.11 Equilibrium condition of door ................. 34
2.12 Forces acting on a towing warp ................ 38
3.1 Force polygon of a fluid flow ................. 46
3.2 Force polygon in the model and full
scale net ................................... 59
LIST OF FIGURESContinued
Figure Page
4.1 Idealized mesh and twine in prototype
and model ....... ............. .......... 76
4.2 Typical stressstrain curve for polyamide
tw ine .................................... 81
4.3 Evaluation of average stress in a twine ...... 83
4.4 Headrope ............................. ........ 85
4.5 Attachments in a rope ........................ 86
5.1 Net diagram of Burbank flat net .............. 92
5.2 Dimensions of door ................ .. ......... 93
5.3 Site of prototype field study ................ 94
5.4 Instrumentation of prototype net ............. 95
5.5 Attachment of bolt to the cable .............. 97
5.6 Bolt load cell, its attachments and the
recorder ......................... ......... 99
5.7 Underwater load cell ................ ........ 100
5.8 Attachment of underwater load cell to
the footrope and headrope .................. 101
5.9 Attachment of underwater load cell to
the door .................. ................ 102
5.10 Twine load cell .............. ........... 103
5.11 Twine load cells in the net .................. 105
5.12 Channel selector switch and strain
indicator for the twine load
cells ............. ........ .......... 106
5.13 Relationship between warp tension and
trawler speed .................. .......... 112
6.1 Tension testing of twines .................... 115
viii
LIST OF FIGURESContinued
Figure Page
6.2 Grip for testing of twines ................... 116
6.3 Elastic characteristics of twines ............ 117
6.4 Model net diagram ................. ........ 119
6.5 Elastic characteristics of ropes ............. 122
6.6 Dimensions of model door ..................... 124
6.7 Model doors ..................... .......... 125
6.8 Model net .......... .......... ........... 126
6.9 Load cell to measure warp tensile force ...... 128
6.10 Load cell to measure the tensile force in
the rope between door and net .............. 129
6.11 Load cell to measure tensile force of the
midsection of headrope ..................... 130
6.12 Gate, main and return flumes ................. 132
6.13 Storage tank, pump and bypass pipe ........... 133
6.14 Relationship between weir head and
flume discharge .......................... 134
6.15 Electric point gauges for measurement
of the head on the weir .................... 135
6.16 Weir and flow straighteners .................. 136
6.17 Sand filter for the flume .................... 138
6.18 Mounting of motor and regulator for
the movable belt ......................... 139
6.19 Movable bed ...................... .......... 141
6.20 Velocity distribution at center
section of right belt .................... 142
6.21 Isovel for run 1 .... ........... ........ 144
LIST OF FIGURESContinued
Figure Page
6.22 Isovel for run 2 ............................. 145
6.23 Isovel for run 3 .................. .......... 146
6.24 Isovel for run 4 ............................. 147
6.25 Isovel for run 5 .................. .......... 148
6.26 Isovel for run 6 ......... ................... 149
6.27 Point gauge and Ott current propeller ........ 150
6.28 Net profile during testing ................... 151
6.29 Plan view of net during testing .............. 152
6.30 Side view of net during testing .............. 153
6.31 Front view of net during testing ............. 154
6.32 Side view of door during testing ............. 156
7.1 Prediction of warp tensile force from
model results ............................ 167
7.2 Prediction of drag force acting on a
net ............. ............. ... ......... 168
8.1 Beam trawl ................... .............. . 173
8.2 Trawl head .......... ......... .......... 174
8.3 Continuous beam .................. ........... 176
8.4 Prototype net for beam trawl ................. 180
8.5 Model net diagram for beam trawl ............. 182
8.6 Ocean testing of beam trawl .................. 184
8.7 Relationship between net spread and
trawler speed for otter door trawler ....... 186
8.8 Relationship between warp tensile force
and trawler speed for otter door trawler ... 187
8.9 Relationship between warp tensile force
and trawler speed for beam trawler ......... 188
LIST OF SYMBOLS
a
4
a
A
b
B
B
B.M.
c
C
a
CD
Cf
C1,C2,C3
Cq,C5,Cg
d
D
D.
D
O
D
w
 Half of wing spread
 Acceleration
 Elastic constant, area
 Half of headline height
 Elastic constant of twine
 Buoyant force
 Bending moment
 Length of net
 Cauchy number
 Drag coefficient
 Friction coefficient of flow
 Constants
 Subscript denoting door
 Twine diameter
 Inside diameter
 Outside diameter
 Depth of water
LIST OF SYMBOLSContinued
E Float diameter
E' Modulus of elasticity
E Euler number
U
f Coefficient of friction, subscript denoting float
F Force
F Froude number
r
g Acceleration due to gravity
H Horizontal dimension
H Rope diameter
I Moment of inertia
kl,k2 Float constants
K Bulk modulus of elasticity
1 Length
L Characteristic length
L Length of towing cable
m Subscript denoting model
m' Number of meshes (depth wise)
m" Mass
M Bar length, moment
M Mach number
a
LIST OF SYMBOLSContinued
n Number of floats or sinkers per unit length of rope
n Frequency of vortices
n' Number of meshes (lengthwise)
ni A factor considering the rate of filling of fish
N Number of meshes
N Total number of floats or sinkers
Nbd Reaction of ocean bottom to door
bd
0 Surface tension
p Subscript denoting prototype
p' A factor denoting twine porosity
p" Pressure
P Power
r Resistant force, subscript denoting rope
R Tensile force in towing cable at door
R Reynolds number
e
s Cable length, subscript denoting solid
S Solidity
S' Circumference
S Strouhal number
n
t Time
xiii
LIST OF SYMBOLSContinued
T Characteristic time
TI Cable tension
T Tensile force
v Velocity of flow
v Velocity at depth y
V Trawler velocity, subscript denoting vertical
dimension
w Weight per unit length or area or volume,
subscript denoting water
W Total apparent weight
W Weber number
n
x Longitudinal axis (along the width of net)
X Total force in the x direction
y Vertical axis
Y Total force in the y direction
z Transverse axis (flow axis)
Z Total force in the z direction
Z Sectional modulus
m
Greek Symbols
a Angle of incidence
w Unit weight of water
LIST OF SYMBOLSContinued
ys Unit weight of twine
y' Unit weight of float or sinker
Yr Unit weight of rope
e Strain
n Mesh number scale
K Force scale
A Length scale
AD Twine diameter scale
AE Float or sinker size ratio
AH Rope size ratio
A Mesh size ratio
24 Total angle between two adjacent bars
p Dynamic viscosity of water
v Kinematic viscosity of water
p Density
o Stress
 Time scale
S Frequency
LIST OF SYMBOLSContinued
Abbreviations
cfs Cubic feet per second
F Fahrenheit
fps Feet per second
ft Feet
gf Gram force
hp Horsepower
in Inch
lb Pounds
OD Outside diameter
rpm Revolutions per minute
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
HYDRODYNAMIC MODELING OF NETS AND TRAWLS
By
M. Krishnamurthy
December, 1975
Chairman: B. A. Christensen
Major Department: Civil Engineering
Design of efficient fish nets by means of smallscale
hydrodynamic models is of relatively recent origin in the
United States. While investigators have developed basic
equations for modeling flow around fish nets by considering
the intertia and gravity forces as primary forces, they assume
that the elongation of twines and ropes in the net is negli
gible. However, the shape of the fish net and, therefore,
the forces acting on it are affected by the stretching of
twines and ropes under load. In order to permit the correct
numerical transfer of model observations to the prototype, it
is necessary to establish special hydrodynamic model laws
which take the stressstrain characteristics of the model and
prototype materials into consideration. Such model laws were
developed in this study.
These developed model laws were verified by fullscale
ocean tests and by laboratory experiments. A movable bed was
xvii
constructed in the laboratory flume in order to obtain uniform
velocity profiles similar to the uniform velocity profiles in
the prototype. Experiments conducted in the flume yielded
results comparable to those of the field tests, thereby,
confirming the validity of the developed model laws.
Using the model approach, a beam trawl was designed
to harvest deepwater shrimp. The results of laboratory
tests and field trials indicated that a substantial decrease
in the overall drag force was achieved with the designed beam
trawl in comparison to a conventional otter door trawl. The
major benefit of the reduction in the drag force would be a
decrease in the fuel consumption of the trawler.
xviii
CHAPTER 1
INTRODUCTION
1.1 Statement of Problem
The manufacture of nets and trawls is an art which
has been handed down from father to son through generations.
Fishing gears of the past were designed by trial and error.
Nowadays an attempt is being made to supplement this empiri
cal approach with analytical theory and with physical tests
in the laboratory and in the oceans.
The optimum performance of a fishtrawling operation
depends on the efficiency of the gear and the behavior of
fish. Therefore, any research program of gear design should
include the biological study of fish reaction. The compo
nents of one such program are shown in Figure 1.1.
The major physical elements of a fishing operation
are the net, the otter doors, the sweeplines, the towing
warps, and the ship or the trawler as indicated schematically
in Figure 1.2. The design of these elements is an iterative
process, an outline of which is given by Dickson (1971) and
is shown in Figure 1.3.
From Figures 1.1 and 1.3, it can be seen that fish
biology should also be considered in the iterative design
I Iprcvemr. of isting Fish Trcl in
I. taoia __S___uof
Engineering Stud. of ; .. ..  c c' 
y .Ic _
Ana iy v ical Mo2 e i I S.mc *s Ele c tri o I
Reserch i !'I
S____ __ ___j
Results 'X :.
SC r_________C_ a _Q
Field Tests Field Tesis
G   c r 
Generalztiion of eas bn Ce nerli:tion of
Gear Design Co c ia ; F ish Eehavior
N ___. _
tiew DK;".g n aind ?ocedL're oa Fishinig
Figure 1.1. Components of a gear research program.
A
0::
w
j
Cu
a
C
0
c
a
L

CO
c
*a
Trawler available
Estimate wind conditions
Estimate current conditions
Estimate weather conditions
Determine towing speed
Calculate propeller thrust
Calculate permissible trawl drag
Estimate trawl size
* Choose mesh sizes
Choose material 
 Choose twine sizes
Determine trawl drag
Consider bottom conditions
Determine rigging to suit bottom conditions
Consider fish species
Consider fish sizes
Consider catch size
Determine any special rigging to handle catch
Determine handling system for trawl and trawler
Consider most vitial environmental conditions,
particularly light and temperature
Consider stage in fish life cycle during the
fishery
Estimate proximity of fish to bottom and
vertical distribution
Consider any special ecological conditions
Decide on trawl shape
Decide on rigging requirements to achieve
shape
Determine trawl size
Figure 1.3. Iterative procedure in trawl and trawler
design and selection.
Source: Dickson, 1971.
5
of a trawl and a trawler. However, these aspects are beyond
the scope of the present investigation. Also, the inter
action of the propulsive system of the trawler with the
performance of the trawl is itself a major study. Therefore,
the current research focuses attention on the engineering
behavior of the gear. This behavior of the gear can be
studied (1) by analytical approaches, (2) by physical tests
in the ocean and (3) by model experiments in the laboratory.
Analytical gear design requires a knowledge of the
hydrodynamic resistance of the net, bottom friction of the
otter doors and the behavior of the rigging under different
towing conditions. The literature on mathematical modeling
indicates that this design procedure involves long and
tedious numerical computations. Further analytical approaches
to these problems are to be made before this approach may be
considered practical.
Ocean tests of fishing gear are expensive and time
consuming. Determination of the shape of the net under
various conditions is cumbersome because underwater observa
tions of the net depends upon the clearness of water. The
outcome of field tests also depends on such factors as
weather conditions, shark problems, etc.
Another method of designing the fish net is by small
scale model experimentation in the laboratory. Hydraulic
models and laws governing their use have been applied suc
cessfully to the problems of dams, spillways, energy
dissipators and ships. The hydraulic model has proved to be
a powerful research tool even in the study of fluvial phe
nomena (Graf 1971), where the interaction between water and
sediment is considerably more complex than the case of nets
and trawls. Therefore, the fishing industry may reap sub
stantial benefits through improved net structure and improved
performance derived from the hydraulic model concept in net
and trawl research.
Physical model tests in the laboratory offer various
advantages over ocean tests. Costs: Whereas field testing
for one net may require $5,000,laboratory costs may be only
$500, with fewer logistic problems such as weather and ship
time (Hillier 1974). Visual observation: It is possible to
detect easily the deficiencies in net design and to implement
modifications immediately. Time factors: A model test
requires by far less hours than a field test. At least a
week is needed to conduct a particular aspect of a gear test
in the ocean whereas the laboratory tests covering the same
modifications can be done in a day. However, model test
results should be extrapolated with caution because of scale
effects of the models. Therefore, an ideal gear design
would be a method which combines the analytical approach
with model and fullscale experiments.
Summarizing, a simple method is necessary to design
and to test efficient fishing gear. Though not satisfactory,
existing mathematical analyses can be applied to the initial
and preliminary gear design. At the present time, however,
physical scale modeling in the laboratory seems to be the
only reliable alternative tool in the evaluation of gear
performance. Development of such a tool is the general
objective of the present investigation.
1.2 Objectives
The specific objectives of the research are a review
of literature on mathematical and physical models of fishing
gear and their deficiencies, the development of adequate
model laws and the verification of the developed laws and
procedures by field and model tests.
The presentation of the objectives is as follows.
The gear and the behavior of its elements under fluid motion
are introduced in Chapter 2. In Chapter 3, the literature
on mathematical analyses of fishing gear, physical modeling
in the laboratory and fullscale field experiments is
presented. The pitfalls in the existing analyses are
brought out. Development of adequate model laws is pre
sented in Chapter 4. To verify the model laws, prototype
experiments were conducted in the Gulf of Mexico as explained
in Chapter 5. Chapter 6 illustrates the model experiments in
the hydraulic flume. The validity of model laws is given in
Chapter 7. Using the model approach, a special beam trawl
was designed for deepwater shrimping. The design and
testing of this trawl in the laboratory and in the field
are given in Chapter 8. Conclusions on the experimental
8
results in the laboratory and in the field are presented in
Chapter 9. Suggested future research is mentioned also in
the same chapter.
CHAPTER 2
COMPONENTS OF A GEAR
In Chapter 1 the main components of a gear were
briefly mentioned and shown in Figure 1.2. Their physical
structure and behavior, when towed through water, are
described below.
2.1 Net
A net consists of three sections, namely cod end,
belly and wings (Fig. 2.1). Each section may be of different
netting twine and may have different mesh dimension. The
net is usually treated by dipping it in a solution of tar
to give greater resistance to wear, tear and aging due to
storage in the sun.
2.1.1 Netting Twine
Netting twine may be of a natural fiber such as
cotton and coir, or of synthetic polymers like nylon
(polyamide) and polyethylene. Single yarns are'plied
together and twisted to make a netting twine (Fig. 2.2).
The voids in a twine depend on the amount of twist and the
diameter of plied yarns. Nylon is widely used in the United
States in making a net.
11
; z
o Z
IU
0<
WI
O
/ a
ci
Z Q)
z
z
Z 0
W Z
z < 0
9,>
The crosssectional shape of twine and plied yarns
may be approximated by a circle (Fig. 2.2). The solid area
of a twine consisting of three plied yarns may then be
found from the expression
Aolid = 0.508 D2 (2.1)
where
Asolid = Area of solid mass
solid
D = Diameter of the netting twine
Therefore, the ratio of solid area to the total twine area
and porosity in the twine are equal to 0.646 and 0.354,
respectively.
Properties of the twine that are specially important
in the fishing industry are density, tenacity, tensile
strength, knot strength, loop strength, elasticity, tough
ness, stiffness, water absorption, resistance to heat,
sunlight, seawater and mildew. The definition and determi
nation of these properties are described by von Brandt and
Carrothers (1964). Additional information on characteris
tics of fishing twines and their testing may be found in
the bibliography which is given in Appendix A.
The elastic behavior of a netting twine under a load
is described by its stressstrain curve or loadstrain curve.
Figure 2.3 shows the relationship between load and strain of
manmade fibers used in making nets. This relationship for
S0
H \)
D a
"Z o
z
SL 0
L 0 0
S 0 <
u H H \  o "
z <
w o \ \ \
0 0
_02 0 co
z < U 1
w 0 0G nc
>" J w W w :I.c.
J O U)) p
0s o r4
o: t 5 5 S f " \ \ 
Mn W
oD 0 a rL C
S OVO 'V3a iON d7VH
EL EL E L E
w 0
Zw 4: En
OVzl )4OU I0N>
3HiL A0 .LN3OHUd NI OVO9
polyamide netting twines of different linear density is
drawn in Figure 2.4. It can be seen from Figures 2.3 and
2.4 that the fibers do not obey the linear Hooke's law.
However, a nonlinear law is expressed by
o = AEB (2.2)
where
o = Stress
E = Strain
A and B are constants which may be derived from the loglog
plot shown in Figure 2.5. The value of stress is obtained
by dividing the load by the original solid area of the
twine. This definition of stress is used without exception
in this study.
2.1.2 Mesh
The basic structure of a net is the mesh. A mesh
is defined by Kawakami (1964) as a rhombic opening enclosed
by four bars of twine of equal length firmly knotted at the
four corners. Since one bar is common to two meshes and
one knot to four meshes, a mesh consists of two bars and a
knot. The area of a single mesh depends on the angle
between two adjacent bars (Fig. 2.6). This angle may be
changed by different hanging or mounting of the webbing to
the frame line. The surface area of a mesh can be calcu
lated from the geometry of the mesh and is given by the
simple formula
J6bI NI OVOl
,4
.LJ
.
0
o
0
z U 0
O
o M
S, 0
1
z o
nj 0
LZ 0
0 1
43
m 
0 0
A 
4 c0
W p
0i 0
r
s C
*T C~
1 2 5 10 20 50
LOAD kgf
Figure 2.5.
Logarithmic linearization of
loadstrain curve for polya
mide.
Figure 2.6. The mesh.
Amesh = 2M2 sin c cos ( (2.3)
where
A = Surface area of the mesh
mesh
M = Bar length
2I = Total angle between two adjacent bars
The weight of one dry mesh W, in air, is the sum of
the weight of its two bars and the knot.
W = 2([ 1 (M.py.) + [ [MkP ns) (2.4)
where
Ys = Unit dry weight of twine
1p' = Porosity of the twine
Dk = Diameter of the knot
1p' = Porosity of the knot
Mk = Length of twine included in one knot
Since the density of the fiber is unaltered by
tightening of the knots,
iD2 rD2
D p' = P (2.5)
and
Dk = D[P (2.6)
k Pk
From Equations (2.3) to (2.6), the apparent weight,
w, of one mesh in water per unit area is obtained:
sin 2= KJy (2.7)
where
Y = Unit weight of water
2.1.3 Net Panel
A net panel is an array of meshes. It is either
straight or tapered (Fig. 2.7). A net consists of a combi
nation of straight and tapered panels sewed together. The
combination and tapering of panels depends on the judgment
of the net maker who uses a net diagram to show this combi
nation. For example, Figure 2.8 is the net diagram of a
"Burbank Flat Net" which is widely used by Florida and
Texas shrimpers.
The length and depth of a rectangular panel of n'
by m' meshes (m' meshes deep) can be computed from the
geometry and hanging of the net panel
LH = 2n'M cos i (2.8)
and
L = 2m'M sin p (2.9)
where
LH = Length of panel
Lv = Depth of panel
When a net panel is placed in a current with a uni
form velocity, a hydrodynamic force acts on the panel in a
20
I 4
z
c<)
Q)
to
n
l
C')
Al
YX7 v
0 4J
I c o
c c
Cbo
^ O ^41
rC)
direction normal to the panel. The component of this force
to the current is the drag force. The magnitude of drag
force is calculated by
S= C 2 A + C V2 A (2.10)
Db 2 b Dk 2 k
where
F = Drag force
CDb = Drag coefficient of bar
CDk = Drag coefficient of knot
p = Density of water
V = Velocity
Ab = Projected area of bars onaplane normal
to current
Ak = Projected area of knots onaplane normal
to current
The first and second terms in Equation (2.10) represent the
drag to the bars and knots, respectively.
The values of drag coefficient for the bar and the
knot depend on the Reynolds number of the bar and knot,
respectively. The Reynolds number is defined as
R VL (2.11)
e v
where
v = Kinematic viscosity of water
L = Characteristic length
Kowalski and Giannotti (1974) consider the bar length as the
characteristic length. In the turbulent region where the
Reynolds number is greater than 20,000, a value of 2.02 for
CDb is given. Taking the diameter of the knot as the
characteristic length, a value of 0.47 for CDk is shown.
For a net panel placed normal to the current, the
drag on the panel is
F = C n pV A S (2.12)
D panel 2 surface
where
CD panel = Drag coefficient of the panel
D panel
A surface = Total projected area on a plane normal
to the current
S = Solidity, the projected solid area per
unit surface area of the panel.
The values of S and CD panel are
(2MD + D2]
S = (2.13)
M2 sin 2)
and
( k AkNk + CDbbAb) (2.14)
D panel AkNk + AbNb
kk k b b
where
Nk = Number of knots
Nb = Number of bars
Fridman (1973, p. 63) considers the diameter of the
bar as the characteristic length and defines the Reynolds
number as
R VD (2.15)
eD v
From the results of the laboratory experiments, he estab
lishes a relation between ReD and CD panel (Fig. 2.9). The
values of CD panel were averaged from the drag coefficients
D
for net panels with M values ranging from 0.0024 to 0.1500.
Carrothers and Baines (1975) present an empirical
equation to compute the value of CD panel for ReD > 500.
CD panel = 16 for S < 0.1 (2.16)
If the axis of the net panel is inclined at an angle
a with the velocity vector, the corresponding drag coeffi
cient of the net panel can be found experimentally.
Carrothers and Baines (1975) analyzed the experimental
results of various investigators and concluded:
C' = C sin2 a (2.17)
D panel D panel
where
CD = Drag coefficient for the panel
D panel for a = 90
CD = Drag coefficient when the panel is
D panel inclined at an angle a with the
velocity vector
For values of a approaching 900, Tauti (1934)
approximates the value of sin2 a and gives an expression
for the drag coefficient.
C' = C sin a
D panel D panel
(2.18)
'41
d
a
0
3
U
,
S,
4.
1o o
c Q)
U a
Q r
U, 0
MP4
a
/ ~ ~ ~ . a 0 i 0 "9 <
/ \ ? (
0 I
Miyake (1927), Revin (1959) and Kanamori (1960) verified this
approximation by laboratory experiments. Giannotti (1973)
has conducted wind tunnel tests on two different net frames
and given results which support this approximation.
However, at low values of a, the component of
resistance normal to the net becomes small. The predomi
nant force will be the component parallel to the frame.
Carrothers and Baines (1975) have found that Equation (2.18)
predicts low values of drag coefficient for low values of a
and, therefore, cannot be applied to compute the drag force.
For this case, Kawakami and Nakasai (1968) determined the
magnitude of the force acting along the length of the panel
by
F = C"D panl 2 [. tan 4 cos a (2.19)
D panel 2 J M
where
C" = Drag coefficient for the net panel
D panel for a =
Equation (2.19) considers the resistance of the knots to the
flow to be small compared with the resistance of the bars.
2.1.4 The Cod End
The cod end is the tail of the net where the catch
is collected. It has a thicker twine and smaller mesh
compared with the other parts of the net. To protect the
cod end from wear and tear, soft strands of twines known
as chaff are attached to each knot.
During a tow, Kowalski and Giannotti (1974) consider
the cod end a hollow cylinder with its longitudinal axis
parallel to the flow. To compute the drag force acting on
the cod end, they suggest:
Fod = (0.82) 22 d od n, + C
= cod 1 2
x [Nb MD + NknD2] 1 ni (2.20)
where
Fod = Drag
dcod = Diameter of the cylindrical cod end
Cf = Friction coefficient
n, = Factor which considers rate of fish filling
in cod end: ni = 0 for t = 0, and nl = 1
for t > 0
Equation (2.20) holds good for Reynolds number,
R =VM greater than 107 and with an aspect ratio greater
eM V
than 2. The aspect ratio is the ratio of the length to the
diameter of the cylindrical cod end.
2.1.5 Belly
The belly refers to the top and the bottom tapered
part of the net extending from wing to wing and from the
hanging lines to the cod end. The shape of the belly
during a tow is complicated. Kowalski and Giannotti (1974)
approximate the belly by a conical net with an elliptical
mouth, where the major axis of the ellipse is given by the
wingspread and the minor axis by the headline height. They
compute the resistance of the belly to the flow by Equation
(2.18).
2.1.6 Wings
The wings are the sides of the net, tapered along
the top seam and straight along the bottom seam. The wings
facilitate the required mouth opening of the net.
2.2 Ropes
2.2.1 Headrope
After the sections of the net are sewed, the top
portion of the net is hung on a headrope (Fig. 2.10). The
headrope is a combination rope of steel and manila. The
floats are attached to the headrope when a larger mouth
opening is required.
Fridman (1973, p. 73) analyzes the shape of the
headrope under towing conditions. He assumes a catenary
shape for the rope and presents a procedure to compute the
magnitude of the tension along the length of the rope.
The resistance of a rope lying at an angle a to the
flow is computed by
F rope = Crope sin3 a] ] H (2.21)
rope D rope 2 r r
where
H = Diameter of the rope
Zr = Length of the rope
r
(J
O
O 0
zw
z U
J
I I
oC
w
0
C = Drag coefficient of the rope when it is
D rope lying perpendicular to the flow
The value of CD rope is given as 1.1 by Kowalski and
Giannotti (1974), whereas, Fridman (1973, p. 56) offers a
value ranging from 1.2 to 1.3. The resistance of the head
rope can be computed by Equation (2.21).
2.2.2 Floats
To achieve a larger fishing height, floats are
attached on the headrope. They may be hollow cast aluminum
or plastic spheres or solid bodies made of plastic foams.
The drag acting on the floats is determined by
F = N C p A (2.22)
float f D float 2 f (2.22)
where
N = Number of floats
CD float = Drag coefficient for spherical float
A = Projected area of float to a plane
normal to flow
For a single spherical float, Fridman (1973, p. 51)
expresses a value of 0.475 for CD float when the float
Reynolds number is greater than (2.0)(105). The float
Reynolds number is defined as
VD
R (2.23)
ef v
where
D = Diameter of the spherical float
However, if the floats are grouped together, there is an
interaction of flow between the floats and, for this case,
Fridman gives a value of 0.93 for CD float
2.2.3 Ground or Footrope
The bottom portion of the net is hung on the foot
rope. Sinkers are attached to the footrope to keep the net
in contact with the bottom. Spacing and weight of the
sinkers vary with the type of the net.
Kawakami and Suzuki (1959) and Suzuki and Kawakami
(1960) studied the shape of the footrope during a tow and
its load distribution. The resistance of the footrope to
the flow are the drag and friction forces. The magnitude
of the drag force may be computed from Equation (2.21) if
the angle between the rope and flow direction is known.
However, the magnitude of the friction force depends on the
bottom condition and the material of the rope. Fridman
(1973) estimates the friction force as
friction = f W (2.24)
friction
where
Ffriction = Friction force
W = Apparent weight of the rope
f = Coefficient of friction between the
soil and the rope
The values of f for two different bottom conditions
and for different materials are given in Table 2.1. The
coefficient of friction is dependent on the angle between the
flow direction and the rope. Fridman (1973) establishes a
graphical relation between a and f.
TABLE 2.1
VALUES OF COEFFICIENT OF FRICTION
Coefficient of Friction
Material
Gravel with
Sand
Fine Sand
Cast iron 0.47 0.61
Wood 0.51 0.73
Stone 0.54 0.70
Lead 0.44 0.53
Sand bags 0.63 0.76
Vegetable ropes 0.70 0.80
hempss)
Source: Fridman (1973, p. 38).
2.3 Sweeplines
To increase the wingspread, it is customary to insert
extra lengths of rope between the door and the wing. These
ropes are known as sweeplines. They may be the same material
as the footrope, the headrope or the towing warp. The length
of the sweeplines depend on the fishing conditions and the
towing speed.
2.4 Doors
The doors in a trawl provide the allimportant fish
ing spread of the net. The doors of the bottom trawls used
in Florida are rectangular in shape. They are wooden boards
reinforced with iron strips and no attempt on streamlining
is made. At the lower edge of the door, a heavy metal shoe
plate is fitted for stability and for protection of doors
against the hard bottom. The towing warp is connected to
the doors by a chain and a shackle arrangement. If the
sweeplines are used in the net, they are connected to the
backstops of the door by shackles.
Figure 2.11 shows the equilibrium condition of the
door. The forces acting on the door are the tensile force
in the towing warp R1; the hydrodynamic force, R2; the
tensile force in the sweepline, R3; the apparent weight of
the door, W; and the friction of the door against the bottom,
F. The governing equations for the equilibrium conditions
are
EX = 0 EM = 0
x
EY = 0 EM = 0
y
ZZ = 0 EM = 0 (2.25)
z
where
ZX, EY and ZZ = Sum of components of all forces
of x, y and z directions,
respectively
x
R3 \R2
R. z
y
n lxy
 z
R3zy
WI
Figure 2,11. Equilibrium condition of door.
Source: Fridman, 1973.
EM EM and EM = Moment of forces about x, y and
x y z z axis, respectively
The equilibrium condition of the doors is detailed by Crewe
(1964) and by Fridman (1973).
The resistance of the door to the flow arises from
the drag of the door and the component of the friction force.
The hydrodynamic drag of the doors is
F C 2V2 A (2.26)
door D door A (2.26)
where
Fdoor = Drag force acting on doors
CD door = Drag coefficient for door
A = Area of door
The drag coefficient for the door depends on the
door Reynolds number which is defined as
Vad
R = d (2.27)
ed v
where
zd = Length of door
It also depends on the angle of incidence of the flow, the
shape of the door and the ratio of the length to height of
the door. For Red > (0.8) (106), Fridman (1973, p. 276)
presents the values of the drag coefficient for oval and
rectangular doors with angles of incidence ranging from 20
to 40. The CD door values vary from 0.5 to 1.0 for a
rectangular door. This range agrees with the trials of
Crewe (1964) and Scharfe (1959).
The friction force of the door against the bottom
depends on the apparent weight of the doors and the soil
condition. Literature on the evaluation of this friction
force is scarce. As a firstorder approximation, the magni
tude of the friction force may be evaluated from Equation
(2.24) and by using Table 2.1
2.5 Towing Warp
Towing warps are steel wire ropes, the diameter of
which depends on the towing power of the trawler and the
type of trawl. Length of the warps is determined by fishing
conditions, trawl type and drum capacity.
Drag force acting on a towing warp is a function of
its (warp) shape which, in turn, is determined by the system
of forces acting on it. The hydrodynamic drag F of a
straight circular cable inclined at angle a to the flow is
F = CD wr [V2l sin2 a] H( ri) (2.28)
D warp [2 rr
where
CD warp = Drag coefficient for warp
D warp
For steel cables, the value of CD warp is 1.1
(Fridman 1973, p. 57). However, if the frequency of the
vortices forming behind the cable is equal to the natural
frequency of the cable system, the value of C warp will be
greater than 1.1. Literature on the evaluation of this
value is not presently available and research is wanting.
Figure 2.12 shows the forces acting on a towing cable
system. The tangent at any point to the cable makes an
angle a to the incident flow. Therefore, the towing
velocity can be resolved into V sin a and V cos a as the
normal and tangential velocity components, respectively.
Let F be the drag force per unit length of the cable when
the cable lies normal to the flow and F be the drag force
when the cable is tangential to the flow. For the flow
system represented in Figure 2.12, the force acting normal
to the cable is F sin2 a. Similarly, the force tangential
n
to the cable is F cos2 a. But Streeter (1961) approximates
t
the value of F cos2 a to F cos a to fit the experimental
data. The forces acting on an element of length ds are
given in the free body diagram of Figure 2.12. By resolving
these forces in the tangential and normal direction, the
equilibrium condition of the cable system is obtained.
dT
dT + F cos a w sin a = 0
ds t
da
T da F sin2 a + w cos a = 0
f ds n
=d cos a; = sin a (2.29)
S ds
where
38
LO <
o u
+ 0 LL Q_
IL~ 4
0
S>
>
LU M
> r
i
0
4U
Cfb
C
\ 3
\ u > \
c
to
to
\42
\> s V ^C(
\ ^*c
\^ ^ > )
\ s _i *I
\'" ^ Ci^
1^ i
!> ,..
I\
w = Weight of cable per unit length
Tf = Tensile force per unit length
The solution of Equation (2.29), for the values of
z, y, a and Tf as a function of the length of the cable,
yields (Streeter 1961)
T = To Ftz + wy
al
T T w sin a F cos a
=n da
To F sin2 a + w cos a
"0 n
al
f Tf cos a
z = da
F sin2 a + w cos a
a0 n
y = fsin a da (2.30)
F sin2 a + w cos a
a0 n
The magnitude of the tension To in the warp at the
point of attachment to the door can be computed by assuming
that the warp lies in the vertical plane z y (Fig. 2.12).
TO = n + door] + F2 (2.31)
where
Fnet = Drag of the net
net
F = Drag of the door
F = Vertical component of tension in cable
v which is given as
F = W + F N (2.32)
v v door bd
where
W = Apparent weight of door
F = Vertical component of hydrodynamic
v door resistance
Nbd = Reaction of the bottom to door
From Figure 2.12, it is seen that the hydrodynamic
drag of the warps is computed by the following expression:
F = TI cos aI To cos ao (2.33)
warp
The power required to pull one cable is, therefore,
given as
P = T1V cos a (2.34)
where
P = Power
V = Towing velocity
TI = Cable tension
CHAPTER 3
LITERATURE REVIEW
3.1 Mathematical Modeling of Fish Nets
The computation of drag force of a fish net by
analytical approach is known as mathematical modeling of
nets. An estimate of the magnitude of drag at a given speed
determines the capability of a trawler to tow the net at
that speed. Also, if the net is to be physically modeled
and tested, an estimate of drag is necessary to design the
force measuring instruments. However, published research
papers dealing with the calculation of drag on fish net by
mathematical models are scarce. The main features of one
approach found in the literature are described below.
Giannotti (1973) assumes the total resistance of a
fish net to a current to be equal to the sum of the indi
vidual resistances of the components of the net. This
assumption neglects the effect of the interference between
components. The fish net is divided into two major sections,
namely, cod end and belly. The minor components include
foot and headropes and floats. The shape of cod end is
assumed to be cylindrical and the drag acting on the cod
end is computed by Equation (2.20).
Giannotti approximates the belly by a conical net
with an elliptical mouth where the major axis of the ellipse
is given by the wingspread 2a, and the minor axis by the
headline height 2b. To compute the drag force acting on
net panels, he conducted wind tunnel experiments on two net
panels and presented an empirical formula on the form
F = CD p l sin a + 0.005] ( ] (Asurfe
CD panel surface
(3.1)
Applying Equation (3.1) to an elementary surface area of the
elliptical cone and integrating it over the entire surface,
the drag force acting on the belly is obtained.
Belly= (S) (CD panel rab) + 0.005w
x c2ab + a2b2] (3.2)
where
Fbelly = Drag force
belly
c = Length of net
The second term in the radical of Equation (3.2) is
very small compared to the first term. Therefore, this
equation shows that the length of the net does not have any
major effect on the drag force. Giannotti (1973, p. 56)
also infers that "the flow essentially sees only a projected
area as it passes through the cone, and the aspect ratio of
the cone has no effect on the drag force." If the drag on
the belly is independent of the net length and depends only
on the projected area of the net to a plane normal to the
flow, it is to say then, that the shape of the net does not
affect the drag force which is not true. Therefore, the
basic equation (3.1) should not be applied to computing the
drag force on the belly.
The angle of incidence of the flow in the belly
varies from tanI (a/c) to tanI (b/c). For practical cases,
the value lies in the range from 2 to 300. As discussed in
section 2.1.3 for this range, Equation (3.2) will predict
low values of drag force because of the low prediction of
projected solid areas. When a net panel is inclined at a
small angle, a, Equation (3.1) estimates the projected solid
area to be in the order of (SAsurface sin a). But the
actual value of the projected solid area is in the order of
(SAsurfac ) since the projected area of an inclined cylindri
cal bar is in the same order of area of the cylindrical bars.
Therefore, Equation (3.1) is not the proper form of equation
in evaluating the magnitude of the drag force.
The resistance of minor components, footrope, headrope
and floats has been calculated by equations given in section
2.2. The total resistance of the net is, therefore, the sum
of the resistances of the cod end, belly, footrope, headrope
and floats.
As Giannotti predicts very low values of drag force,
his approach is not adequate. Hence, an adequate method of
44
finding the forces on and of testing fish nets is necessary.
At present, physical model testing seems to be the only
alternative and appropriate method of testing fish nets.
3.2 Physical Modeling of Nets and Trawls
Laws relating to the construction of hydraulic models
and the interpretation of the results of model tests are
described below.
3.2.1 Forces in a Fluid Flow
The forces acting on an element of fluid in a flow
can be classified under active and reactive forces. Active
forces include inertia, gravity, viscous, elastic, surface
tension forces and, in rare cases, compressive forces.
Reactive forces are the pressure forces.
Inertia force arises from the acceleration and
deceleration of fluid mass. By Newton's second law of
motion,
F = m" a (3.3)
inertia
where
m" = Mass
4
a = Acceleration
The gravity force if calculated by
F = mr" (3.4)
gravity
where
g = Acceleration due to gravity
Viscous resistance to shear is the property of the
fluid which varies directly with the velocity gradient
normal to the direction of flow. Viscous forces, thus,
depend on the motions in a system at an instant of time
and feedback to modify the motions at succeeding instants
of time.
Surface tension force exists at interfaces between
fluids, whether liquid and liquid,or liquid and gas. The
value of surface tension is the energy required to increase
the surface area one unit.
Elastic forces in a fluid occur by virtue of change
in volume. Reactive forces result from the active forces.
The equilibrium condition is, therefore (Fig. 3.1),
4 * 4 4
F +F + F + F + F
gravity viscous surface elastic pressure
tension
+F. = o (3.5)
+ Finertia
3.2.2 Similitude
To model a hydrodynamic flow system, the prototype
and the model should satisfy the following three types of
similarity.
Geometric similarityTwo objects are said to be
geometrically similar if the ratios of all homologous
Inertia
Felastic
Surface tension
Figure 3.1. Force polygon of a fluid flow.
Gravity
Fviscous
dimensions are equal. Thus, geometric similarity involves
only similarity in form.
Kinematic similarityTwo flow motions are kine
matically similar if the patterns or paths of motion are
geometrically similar, and if the ratios of the velocities
of the various homologous flow particles are equal.
Dynamic similarityThis is the similarity of forces.
Therefore, the force polygon in the model must be geometri
cally similar to the force polygon in the prototype. This
requires that the force scale must be the same for all
forces.
3.2.3 Model Scales
The term model scale refers to a quantity in the
prototype divided by the corresponding quantity in the model.
Greek letters are used for the model scales while the sub
scripts p and m refer to the prototype and model, respec
tively.
The fundamental model scales are:
L
Length scale A = L (3.6)
m
T
Time scale T = Tp (3.7)
F
Force scale K = (3.8)
m
in which L, T and F stand for a characteristic length time
and force. All other model scales may be derived from the
fundamental scales.
Velocity scale:
L /T
LmTm
(3.9)
Pressure scale:
F /L 2
F P/L
F /L 2
m m
K
X2
(3.10)
Mass scale:
m" F F
m" L /T 2 L /T 2
p force scale:
Inertia force scale:
K inertia
inertia
p L3 L /T2
P p p p
p L3 L /T
mm m m
Gravity force scale:
K = p ___
gravity 3 p
m mgm m
Viscous force scale:
pITL L p2
Kp TR J p
viscous L
m 1 L2
ST L m
mm
KT2
A
P p A
2
P_ T
(3.11)
(3.12)
(3.13)
= p 2
(3.14)
Surface tension scale:
0 L 0
K = f P (3.15)
surface 0 L 0
m m m
tension
Elastic force scale:
K
Pi K A (3.16)
elastic K
m
where
u = Dynamic viscosity of fluid
0 = Surface tension
K = Bulk modulus of elasticity
Dynamic similarity condition requires that
K= =K K K
inertia gravity viscous surface elastic
tension
(3.17)
From Equations (3.12) to (3.17),
p 4 p p 2 k
Pm T2 m m m
The length scale, A, can be selected at will but all
other unknown scales are to be solved as a function of A.
Equation (3.18) is actually three equations. They cannot
be simultaneously solved for the unknown time scale, T,
unless a suitable fluid satisfying these equations is found
by chance. Therefore, the usual procedure is to neglect
minor forces in the flow under consideration.
3.2.4 Model Laws
The ratio of the gravity force per unit mass to the
inertia force per unit mass is Lg/V2. The inverse of this
ratio is called the Froude number. Therefore,
F = V = Inertia force .
r / [Gravity forceJ (3.19)
The Froude number is an important parameter whenever gravity
is a factor which influences fluid motion. In such a case,
the Froude number in the model and prototype must be the
same.
The inverse of the ratio,
Viscous force/mass V/pL2 _(
Inertia force/mass V2/L pVL (3.20)
is the Reynolds number. It is important when viscous forces
influence fluid motion. To model such fluid motion, the
Reynolds number should be kept the same in the model and
prototype.
The relationship between the inertia and pressure
force defines Euler number E
u
Inertia force pL3 L/T2 pV2
Pressure force L2pt pr
Therefore,
E V2
u P'
(3.21)
If the predominant forces are inertia and surface
tension, the Weber number should be kept constant both in
the model and the prototype. The Weber number W is defined
as:
W Inertia force V2/L
n Capillary force 0/pL2
W p2L (3.22)
If similitude with respect to elastic force is to be
insured, the Cauchy number or Mach number should be kept the
same in the prototype and model. This is defined as:
Inertia force pL3 L/T2 (L/T)2
Elastic force L2E' r L/L E'/p
2
C 2 pV (3.23)
a a EF
where
C = Cauchy number
M = Mach number
a
E' = Modulus of elasticity
In short, the predominant force scale ratios and the model
laws are:
K K. = = K =K K
inertia gravity viscous surface elastic
tension (3.24)
Froude Model
 Reynolds Model
Weber Model
Cauchy Model
3.2.5 Derived Model Scales
In all the cases mentioned above, if the length
scale, A, is chosen, depending on Lhe predominant forces,
the time scale can be found from the model law. Then, the
model scale ratio for any derived quantity can be solved.
For example, in the case of Froude's model law
which considers the gravity and inertia forces as predomi
nant forces, the inertia and gravity force scale ratio
should be equal. From Equations (3.12) and (3.13), the tii,
scale, T, is:
T = / (3.25)
When g = g the corresponding velocity scale is:
V
P = (3.26)
V T
m
Similarly, the force scale ratio is obtained as:
K = _3 L (3.27)
P T P
and so on.
Scales for derived quantities for different model laws
can be obtained in a similar fashion. Table 3.1 summarizes
the basic and derived scales for the model laws considered
above (Christensen 1975a).
3.2.6 Limitations
In model studies, the following limitations are
usually encountered.
Reproduction limitAs the size of the model decreases,
the magnitude of the predominant forces in the model decreases
significantly, such as,by the cube of the length scale in the
case of Froude modeling. If the model is small, a surface
tension force may become large enough to be of the same
order of magnitude as that of the active forces since sur
face tension force is proportional to the first power of the
length scale. In this case, extrapolation of model results'
to prototype conditions, where surface tension forces are not
predominant, will be incorrect. However, this difficulty can
be overcome by using the Weber model law in deriving the
model scales, in which case,the Froude model law,of course,
will be violated.
Cavitation limitWhen atmospheric pressure is not
reproduced in the model, cavitation phenomena in the proto
type are not reproduced correctly in the model.
Lo
C
C
0
l
> L
r=
41
cu
C N
0) 0
C cU
C) C') t'f Q. 0 '
a 42 0
'la
> 0 7
4J
01 4' NM __
C) C)
3 C. NN 
0 I 0 < I o'
p! U M m <<  *
>^ I r< r<
& >ElS .D*01a6
43
Ic
0)
K
: 6 :
r
0
o a
a) N C)
Q *il4' 4L.l 4' C) .fl
g U/)  44 '4 '44 c) 
H p~
)0 U.
a0
n].I 'a
a< 4 < ^ ^ " V
C)0OU
0 0
U.
CO
o *H
0 *H
c) a
a)
WO
C.O
H
0
43
a
C)
C)
__ 1 I
4)
t 4
0 0
U )
0 0
a a
So
() II
o.I
a CE
a C: a
QH N
P4
l 6
Of
I O B
oo 1 L4
Ld
01 0
a ap
a 6
0C e
.41U
201 i
ol B
;> 0.1
al a
N
1 2E
N
U
4..
IA
IN N
a a
da E
cli
H
N
OJ
'" ,
tm to
o 4
*r( *Hl N ( U C N
U) C U U (1) () (
0 U) cIu [0 U) U) U
Sl to ) Q )
rE U) UN
*H pi ,o 4 .U 4 4 1
4 .0
fU4
;(U (U Ci .
4p *H0 U) U
40 0 U ,C *H (V
W U1 U 00 U l
S H ) V H *H U
0 0
N
Ln
r<
a
M
7
'3
U
44
0)
U
C
0
U
II
r<
cfi  Q
C
0<
43
04 0
U 4 0
Q)
,4 <1
" C __l 
do o c 0
01 0 
ol 0 37oJ 3'
s 2j 0440 0MK 'I7' 04j1 s
u
V] i CN I i l
[fl N I
13 O C >^ CM
i C^ .alM  >,, <
o o  o j
C e jS ?1? :S Q
oiSec "t ~ a v e &)la e e &1E
43
'3
O
44
) '3
Ili r
1 0
0 U
(4
(4 II
0*<
rob
04 ^
b04 &
0<
a&i 6
a3~ I34
0M
e
o1 Q
I 3 I
r3 4'
43 U) Nr'
o r s ,n tu I m
c) 0 ) y i
Q h '1
C LO
43 U 4
u 0
00 1 0
'34
M O PO O Or
33 04
Wave limitIf waves are to be reproduced correctly
in the model, wave velocities in the model should be greater
than 0.755 ft/sec corresponding to the minimum capillary
wave velocity.
Laminarturbulent limitThe model should not be so
small that the flow in the model becomes laminar while the
corresponding prototype flow is turbulent. By obeying
Reynolds' model law, this limit can be avoided.
Subcritical and supercritical limitSometimes it
is necessary to use a different scale for the vertical
dimension than for that of the horizontal dimension. This
is known as vertical distortion. Where subcritical and
supercritical flows of the prototype are to be reproduced
properly, the Froude model law should be used.
Sediment transport limitIf a sediment transport
phenomenon is to be modeled, the sediments cannot usually
be scaled down to the length or depth scale. E.g., if
normal sand is the prototype bed material and if it is to
be scaled down to a realistic scale, say, x = 50, then clay
or silt should be used in the model. But clay and silt
behave differently than sand due to their cohesive proper
ties. Therefore, special model laws are usually needed for
these types of models (Christensen 1975b).
Roughness limitSmall wall and bed roughness in the
prototype cannot be reproduced correctly in a small model.
Christensen and Snyder (1975) present a procedure to model
the roughness elements by distortion.
3.2.7 Model Laws Applied to Fish Net
Tauti (1934) has applied the hydraulic model laws to
fishing nets by assuming: (1) that the elongation of the
netting twine is negligible, (2) that the netting twines are
flexible, (3) that Newton's law of hydrodynamic resist
ance is valid for every portion of the net irrespective of
its Reynolds number and (4) that any change in the form of
the net occurs so slowly that the external forces acting on
each element of the net can be considered to be in quasi
equilibrium. Considering an element of the net with area, A,
and circumference,S'(Fig. 3.2), he examines the forces acting
on it under equilibrium conditions. From assumptions (1)
and (4), the forces are gravity, drag and webbing tensile
forces.
The gravity force is the apparent weight of the
elemental area of the net and is expressed as:
w = f() ,(, ,j D + c, 2 (3.28)
where
fl() = T = Function depending on degree of
2 sin slackness of netting
Mk
cl = /p'p' = Constant which depends on type
k of knot and its tightness
w = Apparent weight of net per unit
surface area
/
PROTOTYPE
Cir
/ \
\ / Circ
2"
MODEL
Figure 3.2.
Force polygon in the model and
fullscale net.
The resistance r acting on a unit area of the webbing
is expressed by
r = f2( ) + C2 V2 (3.29)
where
f2W() = Function which depends on incident angle
of the current and half of the angle
between two adjacent bars
Geometric similarity requires that a an ( should be
the same in the model and the prototype and dynamic simi
larity yields:
T S'
wA rA f p F
P P P P P P (3 30)
w A r A TS' F.
mm mm fm m
where
Tf = Tension per unit length
F = Total force acting on net
Tauti sets the twine diameter scale to be equal to
the bar length scale.
D M
D > A P (3.31)
m m
He gives the scales of the area and the circumference to be
equal to the square of the length scale and the length
scale, respectively. From Equations (3.28) to (3.31),
Tauti gets:
Ps Tf F
SD pE p2 = (3.32)
2A{ o23 =V FA
s w m f m
m m m
Equating the first and second terms in Equation
(3.32), the scale ratio for velocity is obtained as:
V Ps Pw
P AD (3.33)
V p p D
m s s
m
m m
From Equation (3.33), it is seen that the velocity scale
depends on the mesh scale AD' rather than the length scale
ratio A.
From the third and second terms, the scale for the
tension per unit length is:
Sf V 2
T = X P2 (3.34)
f m
and, the second and fourth terms give:
F V
P = 2 RP (3.35)
m m
Summary points of Tauti's model laws for fishing
nets: The linear length scale A is chosen arbitrarily; the
mesh length scale is distorted and chosen arbitrarily; the
velocity scale depends on the mesh scale rather than the
linear scale and the force scale is given by Equation (3.35)
which is in agreement with Froude's law.
Using the dynamic similarity concept, Dickson (1959)
argues that the velocity scale ratio should be dependent on
the linear scale A instead of the mesh length ratio. Neg
lecting the small density difference between the prototype
and the model fluid, he presents:
V
P = T (3.36)
V
m
Fridman (1973) considers the diameter of the bar as
the characteristic linear dimension and reasons that the
drag coefficient for the net is independent of the linear
length of the net. Therefore, he supports the model laws
developed by Tauti.
Kawakami (1959) extends the results of Tauti to
model ropes, floats and sinkers. He considers two cases of
modeling of ropes. In the first case, the rope is used as
the main part of the net, the reduction ratio is equal to
that of the main body of the net, i.e.,
r
= (3.37)
m
In the second case, the rope length is independent of the
size of the main body of the net, as in the case of the
towing warp of a trawl.
In both cases, Kawakami (1959) considers the three
forces acting on a rope: The hydrodynamic force, the
apparent weight of the rope, and the tension in the rope.
He also assumes the elongation of the rope to be negligible.
He expresses the hydrodynamic resistance rr acting per unit
length of the rope as:
r = r sin2 a H V2 (3.38)
r r 0 r
where
r = A constant
r 0
H = Diameter of rope
a = Angle between rope and current
The apparent weight of the rope per unit length wr is given
by
nH2
Wr = r ( Yw) (3.39)
in which
Yr = Specific weight of rope
The force ratios must be equal in order to satisfy
the dynamic similarity. From Equation (3.35),
w 'r T 2
r r r r r V
P P= P = T = X2 k (3.40)
zw i r T V
r r r r r m
mm m m m
where
r = Length of rope
T = Tension in rope
For the first case, where the rope is used as a main
body of the net,
r
9.
Substitution of Equations (3.38) and (3.39) in (3.40),
yields:
H
r
P=
H
r
Yr Y
P P
Yr wY
m m
(V /V) 2
(3.41)
For the second case where the rope length is independent of
the size of the net,
r
9. r
r
m
Kawakami gives:
p A
H
r r
m
(3.42)
Tr i w V 2
P p = P r (3.43)
L 2
Yr w V x
m m
Therefore, the diameter and density of the rope in
the model should be chosen to satisfy Equations (3.41) or
Equations (3.43).
To model floats and sinkers, Kawakami (1959) analyzes
the forces acting on them which are the apparent weight and
the hydrodynamic resistance. Denoting the number of floats
or sinkers per unit length of rope by n, the apparent weight
w, per unit length, is calculated by
w = k, E3 [' y] n (3.44)
where
kI = A constant depending on the shape of float
or sinker
E = Diameter
y' = Specific weight of material
S
The hydrodynamic resistance is expressed in the form,
r = k2 E2 V2 n (3.45)
where
r = Hydrodynamic resistance per unit length
a of rope
k2 = A constant
From the dynamic similarity condition and from
Equations (3.35), (3.44) and (3.45),
s w V n
p p = Vp p P IP 1
Y' y V n A
s w m m
m m
and
E = n ,
P = A (3.46)
m p
Therefore, the floats and sinkers of the model should satisfy
Equation (3.46). In cases where Equation (3.46) cannot be
satisfied, an approximation should be made in considering
the forces acting on floats or sinkers. Neglecting the
hydrodynamic resistance force compared to the apparent
weight, the similarity condition yields:
E 3 ^s w V 2 n
{E] {YRJ= [j2 A n^ (3.47)
E v Y V n P
Im s w m p
m m
A bibliography on development of model laws is given
in Appendix B.
3.2.8 Dimensional Analysis
The variables that can influence the profile of a
net characterized by a linear dimension S'are: (1) series
of linear dimensions defining the boundaries L, M, D
(2) kinematic and dynamic quantities such as mean velocity
V and force F and (3) the physical properties of the fluid
and of the net such as density p, viscosity j, specific
weight of twine ys and elastic properties. These variables
and the functional relation can be expressed in the form:
S'= f[p, L, V, p, t, M, D, F, e, e, ui, u2, Y (3.48)
Using Buckingham's 7 theorem, Fridman (1973) derives
the following dimensional numbers:
S'_ r fD M Vt pV2 pVL F
L L' L' L w L pV, 1, ul, u2, 2 (3.49)
where
5 = Elastic displacement
ul = Hanging coefficients
u2
E = Unit elongation of the twine
Therefore, for the complete similarity condition, the follow
ing terms should be the same for the model and for the
prototype:
1. M/L and
2. D/L or these terms can be grouped into single
one term, D/M
3. Vt/L which is the Strouhal number
4. V a modified Froude number
y*gL'
5. VL/v, the Reynolds number
6.
7. 5, ul, u2, E (geometry of the net)
Fridman (1973) analyzes each term in detail and
determines whether it is possible to keep the term constant
both in model and prototype. He derives the model laws and
scales considering only the major forces acting on the net,
which vary from case to case.
3.2.9 Model Experiments
Model experiments are conducted in hydraulic flumes
generally by two different methods. In the first method,
the water is at rest and the net is towed by a carriage
moving on rails. In this case, the length of the flume
must be sufficient to allow enough time to conduct the tests
while not accelerating or decelerating. If the behavior of
the net is to be visually observed and photographed, the
sides of the flume should be transparent. Few such tanks
are available in the U.S.A. for testing of nets.
In the second method, the net is stationary and the
water flows through it. This facilitates visual observa
tion of a conveniently stationary net. In this case, the
water velocity varies from zero, at the bottom of the flume,
to a particular value at the top of the water surface. The
mean velocity of water in the flume is found by the Froude
model law. However, the variation of velocity with depth
follows a logarithmic law (Nikuradse 1933):
S= 8.48 + 2.5 In ] (3.50)
Vf
where
V = Friction velocity
y = Depth from bottom
k = Equivalent sand roughness
v = Velocity at distance y from the bed
Literature on model experiments (Appendix C) indicates that
most of the tests on fixed nets are conducted in flume where
the velocity distribution of water follows Equation (3.50).
But in the prototype, the net is pulled through stagnant
water which means that the profile of water velocity with
respect to the trawl is uniform. Therefore, the relative
velocity has the same value at all distances from the bed.
To satisfy similarity conditions, the same velocity distri
bution should be obtained also in the model. The uniform
velocity distribution can be achieved by constructing a
movable bed which can be moved at a velocity probably equal
to the spatial mean velocity.
The Chamber of Commerce at BoulognesurMer, France,
owns and operates a flume in which such a movable bed is
constructed. The details of the flumeare reported by World
Fishing (1972). However, this flume was used as a simple
observation tank rather than a testing tank. Therefore, in
the literature, quantitative test results of model experi
ments of fish net, using a uniform velocity distribution
from the bed to water level, are not found, so far.
3.3 Prototype Experiments
Any model law should, of course, be verified both by
model and field (prototype) experiments. Reports on field
tests of fishing gear detail the parameters measured and the
instrumentation used in the tests. A substantial list of
such reports is found in Appendix D. The parameters measured
in the field tests can be grouped under the following head
ings: (1) Linear dimensions, such as net spread and height;
(2) Angles, such as angle or attack, tilt, etc.; (3) Forces,
warp tensile force and (4) Velocity, such as towing speed
and current velocity.
The instruments to measure these parameters can be
grouped under two divisions: (1) Decktype instruments and
(2) Underwater instruments. Decktype instruments that are
used on the ship deck include warp tensile load cells, warp
declination and divergence meters, and ship's speed log.
Underwater instruments are generally batterypowered and
selfrecording. Some of the underwater instruments are
load cells that measure door spread, tensile force on head
and footropes, net height, etc.
de Boer (1959) used a differential manometer to
measure the net opening. Motte et al. (1973) describes an
accoustical transducer to measure the linear dimension of
door spread and net height.
The forces in the footrope and headrope can be
measured by a selfrecording load cell. One such load cell
is described by Nicholls (1964). He also gives a method of
finding the loads acting on netting twines. The principle
of this method is the same as the Brinnell Hardness test, a
hardened steel used to give an indentation in a relatively
soft metal plate.
The angles measured in a fullscale field test are
the angle of attack, the pitch, and the heel of the doors.
If a rod is attached to a door and suspended to move in
horizontal and vertical directions, the free end sliding
over the ground will adopt the towing direction of the door.
This is the principle underlying the angle of attack meter
used by de Boer (1959) and Nicholls (1964). The warp
divergence and dip angles can be measured by a protractor
and indicator arrangement as described by Motte et al.
(1973).
The towing speeds of the trawler, with respect to
the water and with respect to the bottom, can be obtained
by a conventional ship's taffrail log and navigational
instrumentation.
3.4 Pitfalls in Existing Physical Model Laws
The existing model laws described in section 3.2.7
do not consider the effect of elongation of the netting
twine under load. If the geometric similitude is to be
satisfied, the strain of the twine in the model and in the
prototype should be the same. However, if the same netting
twine material is used, both in the model and in the proto
type, strain cannot be the same because of the high change
in the order of magnitude of forces in the two cases. There
fore, the geometric similarity will not be satisfied in this
case and predictions of prototype linear dimension from the
model results will be erroneous. Thus, adequate model laws
that take the elongation of twine into consideration should
be developed.
These model laws should be verified both by field
and model data. While conducting the model experiments, it
is emphasized that the velocity of water should have the
same distribution from top to bottom as in the field experi
ments, i.e., in most cases,be constant. It will not be
adequate to set the mean velocity of water in the flume as
the desired velocity. Because the mean velocity occurs at
a point whose distance is 0.368 times the depth of flow from
the bed if a logarithmic velocity distribution is assumed.
This means that the water velocity below that point will be
less than the required velocity. This distance is crucial
for model tests of bottom trawl since the major portion of
the model lies in that region. Therefore, the model drag
forces will probably be less than the actual forces.
Consequently, the projection of model test results to
prototype conditions may not be correct.
In short, the model laws which take the elongation
of netting twine into consideration should be verified by
73
tests in the flume where the velocity distribution is uniform
from top to bottom as in the case of field experiments.
Such tests are conducted and will be explained in subsequent
chapters.
CHAPTER 4
DEVELOPMENT OF MODEL LAWS
4.1 Froude Modeling
To model a flow through and around a trawl, the
predominant forces acting on it must be considered. The
predominant forces are due to gravity, inertia and the
viscosity of the water. The viscous forces will have the
least influence on the model if its twine is not thin.
Therefore, it is assumed that this type of force has
negligible effect on the trawl and that the predominant
forces are due only to gravity and inertia. As discussed
in section 3.2.4 to model such a flow field, the Froude
model law is required.
Equating the inertia and gravity force scales, the
time scale ratio is obtained as given by Equation (3.25).
The model scale ratios for other derived quantities can
then be expressed in terms of the fundamental model scales
(Table 3.1).
4.2 MeshTwine Distortion
While the principal dimensions of the prototype
trawl (length and width of the net) are reduced by the
length scale X, it is not advisable to reduce the twine size
by the same length scale. Such a reduction will, in most
cases, result in such low values of the model Reynolds
number, which is based on twine diameter, that viscous
forces may predominate in the model but not in the proto
type. This may happen in spite of the fact that the length
scale used in net and trawl models is usually approximately
one order of magnitude smaller than the length scales
usually used in models of rigid hydraulic structures. The
problem may be overcome by using a larger twine size and
compensating for the twine distortion by also distorting
the mesh size in such a way that the hydrodynamic drag
forces acting on any section of the prototype trawl and the
corresponding section of the model have the correct force
scale.
Figure 4.1 shows a prototype section of a net and
its corresponding model section. The drag force acting on
a section due to the flow velocity V, relative to the
trawl, is computed by calculating the drag force acting on
the bars and on the knots. The drag force acting on a
twine may be written:
F = C pV2 MD (4.1)
bar Db b
Similarly, the drag force acting on a knot is given by
PV2 k4
F knot k =C I7 (4.2)
knot DkZ
PROTOTYPE
FORCE Fp 7 FORCE
Y vm
Lm
Fm
Idealized mesh and
and model.
twine in prototype
Figure 4.1.
where
Dk = Diameter of knot
It is assumed that the diameter of the knot is proportional
to the diameter of the twine. The total solid areas of the
bars and the knots on which the drag force is acting may be
written:
Ab = (]LD) (4.3)
and
Ak =[ L2 D] (4.4)
From Equations (4.1) to (4.4), the total drag force
F, acting on the prototype section, is computed from
fL V L 2
S c LD p V2 + c2 D2 V2 (4.5)
P P P P P M ppp
where
cl and c2 = Constants of twine and knot,
respectively
If the Reynolds number based on the twine diameter is not
small, both in the model and in the prototype, then the
drag coefficients will have the same value in the model
and in the prototype. Therefore, the constants cl and c2
also remain the same for the model. The drag force in the
model is expressed as:
L 1 2 L 2
F = c LD p V2 + C2 D2 V2 (4.6)
m 1 M m m m mM m
m m
Therefore, the drag force scale ratio is:
clL2D D2
P + c L2 Pp 2
F M 2 p M2 p
P p p (4.7)
F D D2 p
m 2m c2 mD m
m m
But, the drag force and gravity force scales must be equal
to satisfy the dynamic similarity condition. Therefore,
from Equations (3.26), (3.27) and (4.7),
D c + p
M M
P P
=1 (4.8)
D D
M c + Dm
m m
where
c = cl/c2 = A constant
The mesh scale, AX, and twine scale, AD, are defined
as:
M
m= p (4.9)
M M
m
and
D
AD (4.10)
Substitution of Equations (4.9) and (4.10) into (4.8), yields:
D
SD c D p
P P
1M c + M c + M
P P
The solution of is:
AM
D
M
AD
hM
(4.11)
(4.12)
(4.13)
D
P_
M
P
D
P +p
D
P
Equation (4.13) gives a negative value for which is not
AM
possible. Therefore, the only solution of Equation (4.11)
is given by Equation (4.12), i.e.,
AD = xM
(4.14)
which is independent of the constant c.
Tauti (1934) has applied Equation (4.14) in deriving
the model laws, but he has chosen the result as an arbitrary
value instead of a necessary condition. Equation (4.14) is
also consistent with the findings of Fridman (1973) who
derives the condition from dimensional analysis. A special
case of Equation (4.14) occurs where both scales are equal to
one indicating that the same twine and mesh size may be used
in both model and prototype. However, the model twine size
is determined from the elastic properties of the prototype
twine material.
4.3 Elongation of Twine
As discussed in section 2.2.1, the stressstrain
relationship of netting twines is expressed in the form:
o = AcB (4.15)
in which
a = Stress corresponding to the strain E
A and B = Constants which are properties of the considered
twine
The stressscale is, therefore, obtained:
B
o A P
= P _ (4.16)
o A B
m m m
E
m
Geometric similarity requires that the strain in the model,
Em, is equal to the corresponding strain in the prototype,
C '
Equation (4.15) is valid within the strain ranges
normally encountered in nets and trawls, and the dimension
less exponent B is nearly the same in most net and trawl
materials (Fig. 4.2). A representative value is B = 0.9.
90/
09
 =99000 &
I I I I I I I _LLO
0.01 0.1 1.0
STRAIN
Figure 4.2.
Typical stressstrain curve for
polyamide twine.
105
(I)
c3
104
a
di
10"
T
bg
The dimensional constant A varies substantially from material
to material. Therefore, Equation (4.16) reduces to:
o A
= P (4.17)
o A
m m
The average stress on a twine oriented in the
direction of flow may be evaluated by considering the number
of twines existing in a prototype length L (Fig. 4.3). The
total area on which the total drag force F acts is calcu
lated to be M c3D2, where c3 is a constant. Therefore, the
average stress is obtained as:
S= F (4.18)
Sc3D2
and, hence, the stress scale is:
F
_
F
o m
P = (4.19)
m L D P
L M c3 D
m p 3 m
M m
Because of the requirement of geometric similitude c3 and
m
c3 must have the same value. Therefore, Equation (4.19)
P
becomes:
F
P
o F
= m (4.20)
0 2
AM
NX
