Resistance and Propulsion of Ships
Technical University Deift
Course MT512
Prof. Dr. Jr. G.Kuiper1
January 3, 1994
R%k2T --k
'MARIN, Maritime Research Institute Netherlands, Wageningen, Technical
Contents
1 1Hull forms
13 1.1 Displacement Hulls 13 1.1.1 Efficiency. 13 1.1.2 Typical Speeds 14 1.1.3 Hull Forms. 14 1.1.4 Form Parameters. 141.1.5 Considerations for the Stern Form. 16
1.1.6 Considerations for the Bow Form 16
1.1.7 Bulbs. 17
1.2 High Speed Ships. 18
1.2.1 Planing Hulls 19
1.2.2 Hydrofoils. 21
1.3 Air as Carrier. 24
1.3.1 Air Cushion Vehicles. 24
1.3.2 Surface Effect Ships. 25
1.4 Multi Hulls 25 1.4.1 Catamarans. 25 1.4.2 Swath 26 2
Propulsors
29 2.1 Propellers 30 2.1.1 Propeller Arrangements 30 2.1.2 Trusters 312.1.3 Controllable Pitch Propellers. 32
2.1.4 Overlapping Propellers. 34
2.1.5 Contra Rotating Propellers. 34
2.1.6 Surface Piercing Propellers. 34
2.2 Special Types of Propellers. 35
2.2.1 Supercavitating Propellers. 35
2.2.2 Agouti Propellers. 35
2.2.3 Tipplates. 36
2.2.4 Vane Wheels. 37
2 2.3.1 Ringpropellers . 40 2.3.2 Mitsui Duct. 40 2.4 Other Propulsors 41 2.4.1 Voight-Schneider Propellers 41 24.2 Paddle-Wheels. 43 2.4.3 Pump Je1s. 43 2.4.4 Sails 45
2.45 Other Types of Propulsion. 47
3 Intermezzo: Resistance of Simple Bodies
513.1 Non-dimensional Coefficients. 51
32 Drag of a Flat Plate.
523.3 Boundary Layer Flow. 53
3.3.1 Laminar and Turbulent Flows 54
3.3.2 Effects of the Pressure Gradient. 55
3.4 Drag of a Two-dimensional Cylinder. 56
3.5 Drag Components. 59
3.6 Additional References. 59
3.7 Additional Data 60
3.8 Summary. 61
4
Resistance, Wake and Wake Distribution
634.1 Resistance and Wake. 63
4.2 Flow along a Ship Hull 65
4.3 Cross Flow 65
4.4 Separation 67
4.5 The Wake behind Simple Ship like Bodies 69
4.6 Horse-Shoe Vortices 71
4.7 Visualisatjon of the Flow around the Hull 72
4.8 Ship Wake 73
4.8.1 Representation of the wake 75
4.8.2 Relation between hull form and wake distribution 76
4.8.3. Wake Fraction 78
4.9 Design Considerations. 79
5
Wave Resistance
815.1 SurfaEe Waves. 81
5.2 Properties of Surface Waves 82
5.2.1 The Dispersion Relation 82
5.2.2 Energy in a Wave. 82
5.2.3 The Group Velocity. 83
5.3
The Kelvin Wave System...
835.3.2 Resistance due to a Kelvin Wave System. 85
5.3.3 The Wave System of a Ship 86
5.4 Wave Interference. 89
5.4.1 A. Two-dimensional Simplified Hull Form. 90
5.5 EconomicJ Speed.. . 91
5.6
Hull Speed...
. . 935.6.1 High Speed Ships. . 93
5.7 Bulbous Bows. 94
5.8 Shallow Water Depth. 94
6 Intermezzo; Scaling Rules
976.1 Dimension Analysis. 98
62
Physical Meaning of Non-dimensional Parameters 1016.3 Scaling Rules 1:02
6.4 Scale Effects. 102
7 Resistance Prediction using model tests
1037.1 Elements of Ship Resistance 103
7.2 Scaling Laws for Model Tests. 104
7.3 Froudes Hypothesis. 106
7.4 Determination of Resistance Components. 106
7.4.1 Determination of the Frictional Resistance . . . 108 7.4.2 Determination of the Form Resistance. 109
7.4.3 Determination of the Wave Resistance 112
7.5 Extrapolation of Resistance Tests 112
7.5..1 Froude's Extrapolation Method 113
7.6 Effects of Surface Roughness. 113
7.6.1 Equivalent Sand Roughness 115
7.7 Appendage Drag . 115
7.8 Effective Power 116
7.9 Effects of Laminar Flow 116
7.10 Wake Scale Effects 116
7.11 Example of Resistance Extrapolation 118
8
Resistance Prediction using Statistical or Systematic Datal23
8,1 General Considerations for Hull Design. 123
8.2 Systematic Series . 126
8.3 Regression of Available Data. 128
8.4 Design of Curve of Sectional Areas by Lap 131
8.5 The method of Holtrop and Mennen. 132
8.6 Example of Resistance Prediction 135
8.7 Resistance of Small Vessels. 137
4
9 intermezzo:Equations of Motion
1399.1 The Continuity Equation. 140
9.2 The Equations of Motion. 141
9.2.1 Rotation and Deformation. 143
9.2.2 Relatioii between Stresses and Strain. 144
9.2.3 Navier-Stokes Equations 145
9.3 A Simple Example. 146
9.4 The Euler Equations. 148
9.5 The Bernoulli Equation. 148
9.6 Summary. 149
10 Intermezzo:Potentjal flow
15110.1 Singularities in Potential Flow. 152
10.1.1 Uniform Flow. 152
10.1.2 Source. 152
10.1.3 Vortex. 153
10.1.4 Dipole. . . 155
10.2 A Simple Example of Potential Flow. 156
10.3 Forces on a Vortex 158
10.4 Panel Methods. 159
10.4.1 The Lifting Problem 161
10.5 Summary. 163
11 Intermezzo: Boundary Layers
16511.1 The non-dimensional Navier Stokes Equations 166
11.2 The Boundary Layer Equation. 166
11.2.1 Scaling the Thickness of the Boundary Layer. . 168
11.3 Solutions of the Boundary Layer Equations: Blasius. . . 168
11,4 Turbulence. 169
12 Flow Calculations without Waves
17312.1 Potential Flow Calculations
15
12.1.1 Panel Methods without Free Surface 175
12.1.2 Assessment of Various Bulb Designs. 176
12.1.3 Knuckles and Bulge Keels 178
12.1.4 Assessment of the Afterbody. 179
12.2 Navier-Stokes Solutions. 184
13 Flow Calculations with a Free Surface
18713.1 The Linearized Free Surface Condition 187
13.2 Kelvin Sources. 190
13.3 Applications of the Kelvin Sources. 190
5
13.4 Kelvin Sources for Catamaran Hulls, an Example 193
13.5 Dawson's Method. 194
13.5.1 Applications of Dawson's Method. 196
13.6 General Considerations, to Assess Programs. 198
14 Axial Morniitum Theory
20114.1 Axial Momentum Theory. . 201
14.1.1 Effiéiency 204
14.2 Optimum Radial Loading Distribution 205
15 The Propeller G:eometry
20815.1 General Outline 208
15.2 Blade Sections. 209
15.2.1 NACA Definition of Thickness and Camber. 2.12
15.2.2. Root and Tip 213
15.3 Pitch and Pitch Angle 213
15.4 Propeller Plane and Propeller Reference Line. 214
15.5 Rake. 215
15.6 Skew. 216
15.7 Blade Contours and Areas 216
15.8 Warped Propellers 219
15.9 The Propeller Drawing. 219
15.l0Description of a Propeller. 220
15!llControllable Pitch Propellers. 222
16 Systematic Propeller Series
22416.1 Open Water Diagram 224
16.2 The Quality index 22,
16.3 Systematic Propeller Series 227
16.4 Propeller Hull Interaction 229
16.5 Propeller Design Requirements 231
16.6 Choice of Number of Blades and Blade Area Ratio 232
16.7 Propeller Design using B-Series Charts 234
16.8 Elimination of Variables 235
16.8.1 Known Power and Diameter 236
16.8.2 Known Power and Rotation Rate 237
16.8.3 Known Thrust and Diameter 237
168.4 Known Thrust and Rotation Rate 238
16.9 Optimization using the Open: Water Diagrams. 239
16.lOExample 241
16.11 Four Quadrant Measurements. . 243
16.l2Propeller Design using the Optimized Data. . 246
6
17 Profile Characteristics
24717.1 The Pressure Distribution 247
17.2 The Loading Distribution. 249
17.2.1 The Lift Curve 250
17.3 The Zero Lift Angle. 251
17.4 The Leading Edge Suction Peak. 252
17.4.1 The Ideal Angle of Attack 252
17.4.2 Profile Drag. 254
17.5 Profile Series. 256
17.5.1 Thickness Distributions. 257
17.5.2 Camber Distributions. 257
17.5.3 Derivation of the Local Pressure of a Profile 260
17.5.4 Considerations to Choose or Design a Profile. . . 261
18 Cavitation
26518.1 The Cavitation Number 266
18.2 Types of Cavitation. 266
18.2.1 Bubble Cavitation. 266
18.2.2 Sheet Cavitation 267
18.2.3 Root Cavitation. 268
18.2.4 Tip Vortex Cavitation 268
18.2.5 Propeller Hull Vortex Cavitation. 269
18.2.6 Unsteady Sheet Cavitation. 269
18.2.7 The Mechanism of the Development of Cloud Cavitation27l
18.3 Noise and: Erosion. 274
18.3.1 The Implosion of a Single Bubble Cavity 275
18.3.2 Noise Radiation. 276
18.3.3 Thrust Breakdown 278
18.4 The Cavitation Bucket. 280
18.5 Cavitation Tests. 284
19 Lifting Line Propeller Design
28719.1 Lifting Line Theory. 289
19.1.1 Two-dimensional Lifting Lines. 289
19.1.2 Lifting Lines in Three Dimensions. 289
19.1.3 Lifting Line Theory for a Propeller 292
19.2 Optimum Radial Loading Distribution 293
19.3 Induction Factors. 294
19.4 Propeller Design using the Induction Factors. 295
19.4.1 Determination of the Inflow 295
19.4.2 Determination of the Blade Sections. 296
19.4.3 Strength of the Propeller. 299
7
19.4.5 Stresses due to Centrifugal Forces. 300
19.4.6 Approximate Methods 301
19.4.7 Lifting Surface Corrections. 301
19.4.8 Viscous Forces. 308
20 The Propulsioii Test
31120.1 The AdditIonal Towing Force. 311
20.1J Self Propulsion Test with an Additional Towing Force. 312
20.2 Overload Tests 313
20.3 Scaling Laws 313
20.3.1 Scale Effects 314
20.4 Propeller Hull Interaction 314
20.4.1 Thrust Deduction 314
20.4.2 Taylor Wake Fraction 315
20.5 Extrapolation of the Interaction Effects 317
20.6 Extrapolation of the Open Water Characteristics 317
20.6.1 The Equivalent Blade Section 318
20.6.2 Extrapolation of the Drag Coefficient of the
Equiva-lent Blade Section 319
20.7 Extrapolation of the Propulsion Test Results 322
20.8 Trial Condition and Service Condition 323
20.9 Efficiencies 323
20.10 Variations on the Extrapolation Method 325
20.11Example of Extrapolation of the Propeller Open Water
Dia-gram. 326
20.l2Example of the Extrapolation of the Propulsion Test. 328
20.12.1 Comparison with Resistance Test 330
20.l3Extrapolation of the Example using the Mann Method. . 333
21 Propulsion Calculations.
33721.1 Statistical Prediction of the Model Wake Fraction. 337
21.2 Statistical Prediction of the Full Scale Wake Fraction 340
21.3 Statistical Prediction of the Thrust Deduction. 340
21.4 Statistical Prediction of the Relative Rotative Efficiency. 341
22 TABLES
343A DICTIONARY
355Preface
This is an introductory course on ship resistance and propulsion for the
Maritime Technology Department of the Deift University of Technology. The text is written for students who have only basic knowledge of mathe-matics and fluid dynamics. Vector and tensor notation is therefore avoided. The propeller inflow is averaged in time and space to an average uniform inflow, and lhe propeller loading is consequently assumed to be steady. The unsteady conditions will be treated in an advanced course.
The intention of the course is to describe, the models which are used. This means that this course does not contain the complete diagrams, data
and formula's necessary for the actual application of the methods. These
will nowadays often be contained in a computer program. The use of com-puter programs in routine calculations makes it even more necessary that the user understands the model which is used and the restrictions which are
inherent to such a model. For an engineer it is risky to refer only to "a
formula" without knowing the basics behind this formula. It is even more risky to refer to a computer program, which may contain fudge factors, even errors, and which will be changed over time.
It is also essential for an engineer to be ab1 to formulate rapidly a crude approximation of the problem and to grasp the main variables involved. For this and for a proper use of complicated programs understanding the basic approach is more important than the detailed development of a theory. This understanding is the aim of this course.
Structure of the course
The first part of the course is on resistance, the second part on propellers. Only in the last chapters on the propulsion test the interaction between hull and propeller is accounted for. This sequence has been chosen because it allows a gradual introduction of the concepts involved.
10 G..Kuiper, Resistance and Propulsion, January 3, 199.4
The prediction of. resistance, propeller and propulsion characteristics each are treated in three ways:
By extrapolation from model test results.
By systematic or statistical data
By flow calculations
This sequence is natural since model tests form the basis for many
systematic data sets. The development .of computational fluids dynam-ics (C.FD) has been rapid in the last decade, so these methods have become a considerable help in the prediction of the behavior of ships at full scale.
Model tests and computations are often complementary, both haying
their advantages and disad.vantages. Model tests have the disadvantage of possible scale effects, but have the advantage that complex flow phenomena can be simulated..
Calculations have the advantage that the. flow can be calculated in detail and that variations can be made rather easily. However, drastic sim.plifica-tions such as inviscid. flow are used in the calculasim.plifica-tions. An important aim of this course is to explain the complementary role of calculations and model tests.
Textbooks
It is not intended to provide a full inventory of practical methods for ship design or for the prediction of resistance and propulsion. For this the Prin-ciples of Naval Architecture [38] is more suitable. The basis of the math-ematical description of marine hydrodynamics can be found in Newman's book w.ith the. same title [33]. Related specialized books are Lighthiil's book on waves [26]and the books. of Knapp [21] and Yo.ung [46] on cavitation. An
introduction in basic aerodynamics with numerical solutions of potential
flow problems can be found in Katz and Plotkin [19,j
The emphasis in this .course is on the practical application of first
princi-ples to the prediction of the behavior of ships and. propellers. Insight in
these first principles increases understanding of the complex phenomena
Janiary 3, 1994, Preface 11
mt etmezzo's
The basic knowledge on fluid mechanics, required for the understanding
of the introductory course, may not always be available. Therefore some
chapters on the basics of fluid mechanics, such as the equations of motion and its simplificatkns, notably potential flow and boundary layer flow, have been included. These chapters, which are not a part of the introductory
course, are indicated as Intermezzo in the title. The reader who is famiiiai
with these topics can skip these chapters, others can read them without going into great detail. The objective of these chapters is again to show the basic approach, not the details. Some equations are therefore used in
one direction only. This makes it possible to avoid vector analysis, which is nearly unavoidable for the full three dimensional equations.
Additional data
In the text some additional data, such as formula's which are often used,
are printed in smaller print. This text is only given for convenience when the reader is going to use the material for his own purposes. It is not a part of the text and does not add to the understanding of the problems.
Important formula's or statements
Some conclusions,- formula's or definitions are important throughout the text. In order to recognize and retrieve these more easily, a box has been placed around the text involved.
References
Since this course is aimed at an understanding of the basic approach of a
topic, a limited use of references has been made. In most cases users of this course will not yet study literatur.e of the subjects in depth. No efforts have
been made to refer to the most recent literature. For that the references in the Proceedings of the International Towing Tank Conference [16] or
textbooks can be used. When names are linked to formulations or theories,
these names are given with the year, but without the reference. E.g. the
Betz condition for optimum efficiency is mentioned to date back to 1929, but is not referred in the references. Only when full sets of diagrams, data
or formula's can be found in the literature full refercnces are made. This
12 G.Kuiper, Resistance and Propulsion, January 3, 199
Acknowledgements
Many students and colleagues from Mann have given comments, corrections and materiai for this course. The help of Mrs. Raven, Hoekstra, van Gent, van Wijngaarden, Holtrop, de Koning-Gans is gratefully acknowledged. The text will be developed fur tier in the future. Therefore the date of printing is present on all pages aid on the title page. Any comment can be helpful
to improve it and will be very welcome. G.Kuiper. January 3, 1994
Chapter 1
Hull forms
Objective:
Introduction of the hydrodynamic features of some ship types.1.1
Displacement Hulls.
The most common purpose of .a ship is transport of cargo. For such ships the displacement hull. is the most appropriate concept. The weight of the cargo, stores and fuel is called the (deadweight),.. The deadweight and the weight of the empty ship together are equal to the displacement of the ship (Achimedes' law).
1.. 1.1
Efficiency.
The movement of such a displacement ship requires little energy in
com-parison with other means of transport, at least when the. required speed of transport is low . This is because the friction of water is low as long
the generated wave height is small. To give an idea: a ship of 1100 meters lenght, 12.5 meters breadth and 5 meters draft at .a speed of 20, km/hr
re-quires approximately kW. Still the deadweight of such a ship is about
3500 tons. Compared to road transport, where a 20 tons truck requires
some 100 kW to drive at 80 km/hr the amount of fuel, required for water transport is low. This can be expressed 'kWh per tonkm. A characteristic
feature of displacement ships is therefore that a large amount of cargo is moved at low speeds,. The restrictionof a low speed is important, at increas-i.ng speeds. the required power increases very rapidly due to wave generation.
14 G.Kuiper, Resistance and Propusion, January 3, 199.(
1.1.2
Typical Speeds.
At higher speeds the influence of waves becomes large and the waves be-come responsible for most of the resistance. So. the speed is an important
parameter for the type. of ships. For bulk transport (oil, ore, coal, etc)
speeds of up to 16 knots afe common.
1 LN:G is transportedb3 special tankers at a speed of about 20 knots, For large, fast displacement ships such as containerships and reefers, speeds of up to 25 knots are found, with extremes of over to 30 knots for e.g.passenger vessels and up to 40 knots for navy ships.
The displacement hull is by far the most common type of ship and in
this introductory course most attention will be devoted to this type of ship.
1.1.3
Hull Forms.
Since transport of cargo is the basic purpose of most merchant displacement ships the most efficient hull form , both from a viewpoint of cargo stowage
and from the viewpoint of building costs, would be a square box, (Fig, 1.1) However, the consequences of such a simple shape in terms of resistance
are too large. So bow and stern are shaped such that the volume remains
but the resistance is decreased. Efforts have been made to design huluiforms
with chines, preferably with surfaces which could be developed into flat plates (Fig. 1.2). When the chines are accurately in the direction of the
flow such a ship can be as-good as a faired hull. Most of the :hullforms are faired, however.
1.1.4
Form Parameters.
The form coefficients of a ship are given in a non-dimensional form.
An important coefficient is the block coefficient CB , defined asCB = j with V=volume of the displacement, L=length at the waterline 2
B=breadth en T=draft. The block coefficient isan indication fOr the
full-Although SI units should be used the speed of a ship is expressed in knots, which is an international or U S nautical mile of1852 meter per hour A nautical mile is 1
arcminute of a greatcircle. The U.K. nauticaFmile of1853.184meter is also used. Note
that the nautical mile is different from the statute mile of 1609.344 meter, which is in
use for distances over land.
2Hydrodynamically the length is the length of the waterline L, , although the
differ-ence between the waterline length and the length between perpendiculars will mostly
Figure 1.1: Transport of containers
January 3, 1994, Hull forms 15
Figure 1.2: Hull form with chines in the afterbody
ness of the huliform. It is also indicated by 8.
16 G.Kuiper, Resistance and Propulsion, January 3, 1.99.4
is the area at the waterline. It indicates the vertical distribution of the dis-placement. The longitudinal prismatic coefficient C'p is similarly defined as
where Ar is the area of the maximum tranverse section, whichgener-.
ally is the midship section. 'The longitudinal: prismatic coefficient indicates the moment of inertia of the displacement around the midship section. it.
is also indicated by .
The midship section coefficient indicates the fullness of the midship
sec-tion and is defined as Cm = . It is also indicated by fi'
The longitudinal position of the center of buoyancy is given as a
per-centage of the ship' length L and is indicated as LCB.
The waterline coefficient C,,,9 = indicates 'the fullness of the
water-line. is also indicated by c.
These coefficients can be formed similarly for the fore- and afterbody. When a parallel middlebody is present these coefficients can also' be formed for the entrance and run, but because the length of entranceor run is diffi-cult to determine that is not common.
1.1.5
COnsiderations for the Stern Form.
The form of the stern is predominantly determined by the requi'rement of attached flow and proper inflow to the propeller. When the afterbody is too. blunt the flow will detach from the hull, a' 'phenomenon called separation. This drastically increases the resistance. In principle it is simple to avoid this by making a slender ship. But this requiresa larger and more expensive
ship for the same deadweight. So the main topic of ship hull design is
to optimise the conflicting requirements of economics and resistance. In
practice it means that a ship's, hull has to be designed such that the flow
is on the verge of separation. This makes the hydrodynamics of the ship's hull very complicated.
The form of the afterbôdy and the stern is also strongly dependent on the type of the propulsor. Some forms will be shown 'in Chapter 2.
I.1.6
Considerations for the Bow Form.
The shape of the bow is predominantly determined by the generation of waves and: therefore depends on the ship speed. The fullness of the bow
Figure 1.3: Fast cargoship with a bulb
decreases with increasing speed. Tankers and bulkcarriers have a block co-efficient up to 0.85, a slender fast containership has a block of 0.6 or lower. As a consequence, fast ships (Fig. 1.3) have a slender bow, slow ships such as tankers have a very full bow (Fig. 1.4). For very full ships with block coefficients over 0.80 a cylindrical bow has been applied sometimes (See
Fig. 1.5a). Poor ballast performance and high resistance in waves have
made this type of bow obsolete.
1.1.7
Bulbs.
On many fast ships a bulb is applied, as shown in Fig. 1.3. A bulb is applied to decrease the generation of waves around the ship. Many different shapes have been designed, as shown in Fig. 1.5.
Note that the bulb is mostly designed for one draft. In Fig. 1.3 the
ship is in slightly loaded condition, where the bulb is partly above water and therefore less effective or even counterproductive. For tankers, which operate frequently at ballast draft, a bulb which is effective at various drafts January 3, 199j, Hull forms 17
18 G.Kuiper, Resistance and Propulsion, January 3, 1994
cc-.;--'
*
- -;
'---Figure 1.4: Loaded tanker
is used, as shown in Fig. 1.6. Application of a bulb on a tanker is not very effective because the wave resistance is low (of the order of the air resistance) and the increased frictional resistance of the bulb dominates.
As will be discussed later, a bulb is only effective in a certain speed range. For ships with very high speeds the bulb loses its effect because the wave system changes (see Chapter 5 ) and a sharp bow is applied. (Fig. 1.7)
1.2
High Speed Ships.
As metioned the generation of waves causes a high resistance at high speeds. When these high speeds are required, the speed can also be used to create lift. This reduces the displacement and strongly affects the wave generation and thus the resistance. There are several ways to create lift.
a C e
.1J
"S "I-
4;
Figure 1.5: Various bulbs
1.2.1
Planing Hulls.
In case of planing there is a pressure build-up on the bottom of the ship, such that an upward force is generated. The displacement is thereby
re-January 3, 199, Hull forms 19
r4
b
d
20 G.Kuiper, Resistance and Propulsion, January , 1994
Figure 1.7: High speed displacement ship
duced, but not eliminated entirely. The upward force is obtained by a flat bottom. The water flowing along the hull is mainly flowing along the bot-tom and it is displaced downwards.
A flat bottom is very sensitive to incoming waves. High loads occur, a phe-nomenon called slamming. To reduce this sensitivity deadrise is used in the midbody, in combination with a sharp bow. In the stern region the bottom
'9 20
January 4, 1994, Hull forms 21
is nearly flat (see Fig. 1.7) and ends in a cut-off stern, the transom stern.
This causes the flow to separate smoothly from the hull.
To increase the vertical force of the water in the afterbody and to con-trol the trim a trim wedge can be applied at very high speeds. The bottom
of the trim wedges a continuation of the transom stern. The trim wedges
are also made as adjustable flaps extending from the flat bottom.
Planing may generates excessive spray, which causes additional
resis-tance (see Fig. 1.8). Spray rails are therefore used to reduce the spray.
Figure 1.8: Spray generated by a planing hull
These rails are a kind of longitudinal spoilers on the hull, which deflect the spray downward.
The amount of planing can vary. A small planing force can be generated by using chines. Planing is common for luxury yachts (Fig. 1.9). Extreme planing is used in speedboats and racing boats.
1.2.2
Hydrofoils.
Displacement can be eliminated entirely by using foils to carry the whole ship. Such ships are called hydrofoil ships. These ships are designed
22 G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 1.9: Planing luxury yacht
ensure stability. In such a case these are called surface piercing hydrofoils. (Fig. 1.10).
These hydrofoils have to cpèrate close to the surface, a condition where the
I
__2
--Figure 1.10: Surface Piercing Hydrofoil
lift is reduced. The transition from water to air also causes additional spary resistance . This is avoided by the fully submergedhydrofoil ship (Fig. 1.11).
In that case stability and trim have to be maintained by actively controlled
Figure 1.11: Submerged hydrofoil
Hydrofoils are applied in the speed range up to 40 knots. Especially the fully submerged hydrofoils are rather insensitive to waves, as long as the
hull is not hit by green water.
Hydrofoils are genererally driven by propellers. The shafts, which
ex-tend from the hull into the water, are a source of high resistance. Moreover, the propeller thrust is not exactly in the forward direction due to the rake
of the shafts. This is improved when propellers in front of or behind the
main hydrofoil are used, driven with Z-drives (see chapter 2). In that case the propulsor is integrated in the hydrofoil.
The distribution of the load over the front and rear hydrofoils can differ. When the front foil carries only a very small part of the load this is called a January 4, 1994, Hull forms 23
24 G.Kuiper, Resistance and Propulsion, January , 1991j
canard arrangement, as in the case of airplanes with the stabiliser in front of the wing instead of at the tail.
1.3
Air as Carrier.
Instead of lift also air can be used to create an upward force. This is used in air-cushion vehicles.
1.3.1
Air Cushion Vehicles.
Figure 1.12: Air Cushion Vehicle
Air can carry the weight of the ship when maintained at a high enough
pressure. This pressure is built up in an air cushion, which is maintained below the vessel by skirts around the ship (Fig. 1.12). In such a case the ship is called an air cushion vehicle or ACV. Loss of air will occur in waves or due to forward speed, so the air pressure has to be maintained with air compressors.
An ACV still has a displacement which is equal to the weight of the total ship. The pressure inside the cushion times the area of the cushion has to
be equal to the total weight or displacement. An ACV therefore does not
float above the water, as it does on land. The total resistance of an ACV
is lower than that of a displacement ship due to the lower friction over the bottom and partly because of the more favorable wave forms.
January 4, 1994, hull forms 25 and in the water. They are therefore generally propelled by air propellers. ACV's can also operate over a wide speed range.
1.3.2
Surface Effect Ships.
When the amphibious character is not required the loss of air under the skirts can be reduced by using fixed side walls. These also improve the
behavour in waves (Fig. 1.13). Of course, the side walls have frictional resistance, but the shape of the walls can be better streamlined than skirts and the resistance is therefore lower. Such ships are called Surface Effect Ships or SES. They can be used for very high speeds of up to 60 knots, but a more common speed range is between 25 and 35 knots.
Figure 1.13: Surface Effect Ship
The largest SES vessels nowadays have a length of about 50 meter and
a speed of almost 100 knots (US-Navy). In waves the speed reduction,
however, is larger than e.g. with hydrofoils and it occurs at lowersea states.
1.4
Multi Hulls.
1.4.1
Catamarans.
Long slender ships have a low wave resistance and are therefore good at high speeds. A very slender displacement ship, however, is very narrow and
26 G.Kuiper, Resistance and Propulsion, January 4, 1994
has no deck space nor stability. This can be countered by using two hulls: a
twin hull ship or catamaran.. An example is a passenger ferry (Fig. 1.14),
L
Figure 1.14: Catamaran
where a large deck space is required for a relatively small displacement. The slender hulls have a low wave resuistance, although the wetted area is almost doubled in comparison with a mono hull, which increases the fric-tional resistance. So & catamaran is typically used for higher speeds, where the wave resistance becomes important. Catamarans operate satisfatorily in calm water. Its response to waves is. still a problem. In such conditions
it behaves uncomfortably and there is a risk of hitting the water with the
superstructure.
Efforts' have been made to improve the riding qualities of a catamaran by special bow shapes, such as the wave piercer. The effects still have to be
proven.
1.4.2
Swath.
A variation on a catamaran is a Small Waterline Area Twin Hull or SWATH
ship (Fig; 1.16).
In that case the displacement is brought far below the waterline, thus
reducing the waterline area to a minimum (Fig. 1.17). As a result the vessel will react only sligtly on waves, so it offers a stable platform in waves. A
-Figure 1.15: Wavepiercing catamaran
Figure 1.16: SWATH
disadvantage is of course that its stabvility is very poor, so it is very sen-sitive to changes in loading or even to forward speed. A SWATH therefore must have active fins to control trim and stability. These fins can also be used for further roll reduction.
28 G.Kuiper, Resistance and Propulsion, January 4, 1994 DWL AFT SVA II 17 is II 20 21 22 FWD
Figure 1.17: Cross section of SWATH hull
Literature for Fifrther Reading.
Several afterbody forms and their relative merit are given by Vossnack and
Voogd [45]
A very rough estimate of the relative power requirements' of various high speed concepts is given by Dorey [6].
Chapter 2
P ropulsors
Objective: Introduction of various types of pro pulsors and their main
proper- ties
The basic action of a propulsor is to bring water into motion. The force
required for that is the thrust force. The energy of the water behind the
propeller is lost energy.
The amount of concepts for ship propulsion is large. The most impor-tant criterion for a propulsor is its efficiency. The efficiency varies widely
between various types of propulsors, but the screw propeller has not yet
been equalled in most cases.
The propulsor is generally mounted behind the hul1. This is because of efficiency: the water which is set into forward motion by the friction along the ship is reversed by the propeller action. As a result less energy is left
behind in the water.
A risk for every propulsor operating at high rotational velocities is cav-itation . This occurs when the local pressure in the fluid is lower than the
vapor pressure due to local high velocities. Regions with vapor occur e.g on the propeller blades, such as occurs extensively in Fig. 2.8. When these vapor filled (not air filled) cavities arrive in regions with a higher pressure they collapse violently, causing erosion (Fig. 2.1). Strong dynamic behav-iour of large cavities also generate vibrations in the ship structure.
30 G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.1: Example of erosion due to cavitation
2.1
Propellers.
The most common propulsor is the screw propeller. A propeller generates
a force by lift on the blade sections. These blade sections are similar to airfoils, operating at an angle of attack in the flow. The geometry of the propeller blades is quite critical due to the occurrence of cavitation, as
described below. Therefore a separate propeller is generally designed for
each ship to accomodate the specific circumstances behind the ship. The geometry of the propeller blades has to be very accurate too. A propeller
is therefore a delicate piece of equipment. An example of a finished set of Navy propellers is shown in Fig. 2.2. The propeller will be treated in some detail in this course.
2.1.1
Propeller Arrangements.
The propeller is located behind the hull. The traditional afterbody shape is such that the hull ends in a screw aperture in front of the propeller, while the rudder stock forms the after part of the propeller aperture.
4
.'.
re..
Figure 2.2: Newly finished Navy Propellers (courtesy Esscher Wyss)
When the hull is cut away both above and below the propeller shaft the stern is called an open stern , as shown in Fig. 2.3. An arrangement
with the propeller shaft extending under a flat stern under a small angle is typical for Navy ships and twin screw ships. The shaft can be supported by brackets (Fig. 2.4). In large twin screw ships like passenger ships the shafts are covered by bossings
2.1.2
Trusters.
A propeller can be driven from above by a vertical shaft. This makes it
pos-sible to rotate the propeller along the vertical axis and to generate thrust
in all directions. These configurations are called thrusters. . An example is
shown in Fig. 2.5. Thrusters are common for dynamic positioning, as illus-trated in Fig. 2.13. The use of such thrusters for normal propulsion is still limited because the shaft close- to the propeller decreases the efficiency and because of t'he more complicated construction. In fast ships- or in hydrofoi:ls a thruster arrangement can also be used, as shown in Fig. 2.6. In this figure a special arrangement with shafts is also shown. This arrangement reduces
January , 1994, Propulsors 31
32 G.Kuiper, Resistance and Propulsion, January , 199j
Figure .23: propeller with open stern
c-J
/
I.
Figure 2.4; Shafts with brackets the shaft angle as much as possible.
2.1.3
Controllable Pitch Propellers.
In case of a fixed pitch propeller the thrust, and consequently the speed of the ship, is controlled by the propeller revolutions. In case of a controllable
Figure 2.5: Thruster configuration
-A
liii
Il1L
;
January 4, 1994, Propulsors 33
Figure 2.6: Different shaft arrangements (Courtesy Hydromarine, Italy)
pitch propeller or CPP the thrust is controlled by changing the pitch of
the blades. In tha t case the shaft is at a constant rotation rate. This is
often used when the propeller has to operate in more than one condition, e.g. free running and towing. It is also effective when rapid manoeuvring is
34 G.Kuiper, Resistance and Propulsion, January 4, 1994
required. Reversing the thrust occurs by changing the pitch with constant revolutions in the same direction. This decreases significantly the time
re-quired to change the direction of the trust. A CP propeller is specifically
favourable in case of high, skew, because a highly skewed fixed pitch pro-peller will experience extremely large moments on the bIades
The hub of a CPP is of course more complicated and expensive, while the hub diameter is also larger than that of a fixed pitch propeller. This is a disadvantage for hub and blade root cavitation.
2.1.4
Overlapping Propellers.
For large ships with high speeds the thrust is distributed over two or more propellers. Such a twin screw configuration has a lower efficiency because
the propellers operate outside the region of the highest wake. To increase the efficiency the twin propellers can be brought together as close as possible, with one propeller slightly ahead of the other. The blades can than overlap and the twin screw arrangement approaches a single screw arrangement.
The propellers can in principle rotate in opposite direction, so that also a
contra rotating arrangement is approached.
2.1.5
Contra Rotating Propellers.
A rotating propeller also induces a rotating motion in its wake. This is
lost energy. In order to gain this energy two propellers behind each other
are used at the same shaft (Fig. 2.7). These propellers turn in opposite
directions, thus eliminating each others' rotating wake. The diameter of
the front propeller is often slightly larger than that of the behind propeller, to account for the contraction of the propeller wake.
Efficiency gains of over 10 percent are claimed for such configurations, al-though the lost rotational energy in the wake is less.
The construction of the shaft is complicated and costly. Some prototypes have been build.
2.1.6
Surface Piercing Propellers.
When the draft is too small for a normal propeller to operate tunnels are
applied in the hull to lead the water upwards to the propellers. When this is also insufficient the propellers are allowed to operate partly submerged. Such propellers are called surface piercing propellers. The geometry of the blades is generally skewed to soften the impact of the blades on the water surface and the exit from the water. Air suction may decrease t:he efficiency
January , 199, Pro pulsors 35
1-:
: .. c;;Figure 2.7: Contra rotating propeller arrangement
of the submerged blades, but the efficiency drop compared to submerged propellers is not very large. [4]
2.2
Special Types of Propellers.
2.2.1
Supercavitating Propellers.
When cavitati5n occurs extensively, e.g. at very high rotation rates of the propeller, it is advantageous to use blade sections which generatea long
sheet cavity at one side of the blade, about two times the chordlength of
the blades. These propellers are called supercavitating propellers (Fig. 2.8). Because cavitation implodes far behind the blades the danger of erosion is absent, at the cost of a drastically reduced efficiency.
2.2.2
Agouti Propellers.
A special way to control cavitation is to supply air to the cavity. The cavity will then contain air together with vapor and on implosion the air will cush-ion the collaps. As a result the radiated noise of the cavitatcush-ion is lower than without air supply. The amount of air supplied is very critical, because an overdose of air will increase the cavity volume drastically.
36 G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.8: Supercavitating Propeller
The air is supplied through small holes at the leading edge of the blades.
A restricted supply of air will not affect the efficiency of the propeller.
Agouti systems are used only for navy ships.
2.2.3
Tipplates.
An increased loading of the blade tips would be beneficial to efficiency, but the flow around the tips prevent such a heavy loading. In order to prevent such a flow around the tips tipplates have been applied. Because this would also reduce the strength of the tip vortex these propellers have been called
Tip Vortex Free Propellers or TVF Propellers. It is, however, very difficult
to locate the tip plates properly in the wake behind the hull and the tip vortex does not disappear in general.. Moreover, the tip vortex tends to
occur in the corners between the tip plates and the blades.
These propellers have mostly been applied in combination with a duct. A new develop- ment with more sophisticated design techniques is being developed by de Jong [17] He also developed new shapes of the tip plates,
based on numerical calculations (Fig. 2.9).No full scale applications are available yet.
Figure 2.9: Propeller model with tipplates
2.2.4
Vane Wheels.
A large propeller diameter is often benificial for efficiency. When an increase of an existing diameter is benificial or when the diameter of the
main propeler is restricted a vane wheel can be applied. This is a kind of
propeller, which runs freely downstream of the main propeller. (Fig. 2.10).
The inner part of the vane wheel, the impeller part or turbine part, has a
pitch such that the vane wheel is driven by the wake of the main propeller.
The outer part of the blades of the vane wheel, the propeller part, has a different pitch, which causes the vane wheel to generate thrust at these radii. The rotation rate of the vane wheel will be lower than that of the
main propeller. (In German the vane wheel is called after its designer the "Grimmse Leitrad"). The concept is patented.
January 4, 1994, Propulsors 37
2.3
Ducted Propellers.
At high propeller loadings a duct can increase efficiency (Fig. 2.11).
38 G.Kuiper, Resistance and Propulsion, January , 199(
I
Figure 2.10: Vane wheel
propeller. This is the case with an accelerating duct (Fig. 2.12a) , in which
the flow velocity is increased due to the duct. The duct shape can also
cause the flow to be decelerated (Fig 2.12b). This suppresses cavitation,
but decreases the efficiency. A decelerating duct is therefore suitable for navy ships only, and there it is rarely applied.
Ducted propellers are used in a wide range of applications of heavily loaded propellers, such as for tugs and in applications for dynamic
posi-tioning (Fig. 2.13) . These thrusters can freely rotate over the full circle
and are therefore also called azymuthing thrusters. The power of these sys-tems is increasing rapidly with increasing availability of appropriate gears (Fig. 2.14.
A special application is on active rudders (Fig 2.15).
The flow along heavily loaded ducts may separate from the duct, which
decreases their effect and increases their resistance. A method to reduce
this type of separation is the application of slots at the exit of the duct (see Fig. 2.16).
/
The gap between the blade tips and the duct has to be small for a proper interaction of propeller and duct. This makes the construction of the duct
decelerating
January , 199, Propulsors 39
Figure 2.11: Ducted Propeller
accelerating
Figure 2.12: Accelerating and decelerating duct
more difficult, especially the very large ducts on e.g. tankers.
For manufacturing reasons the duct is also generally rotational symmetric: it has the same cross section at every position. A-symmetrical ducts have a different angle over the circumference to make the wake distribution more uniform.
40 G.Kuiper, Resistance and Propulsion, January 4, 1994
Vertical Stinger
Figure 2.13: Dynamic positioning
Instead of an a-symmetric duct other types of finsor ducts can be applied to make the propeller inflow more uniform. These ducts or fins are applied at some distance upstream of the propeller. Generally they accelerate the retarded flow in the upper part of the propeller plane. A patented concept is the "Schneekluth Duct", as shown in Fig. 2.17 . Variations are possib1e
as in Fig. 2.18.
2.3.1
Ringprope.11ers.
A variation on the ducted propeller is the ringpropeller indexringpropeller.
This is a duct similar to the normal duct, but now the duct is connected
to the propeller blades and rotates with it (Fig. 2.19). This eliminates the
gap between blades and duct, but at the cost of a greatly increased viscous resistance. The efficiency of a ringpropeller is therefore relatively low.
2.3.2
'Mitsui Duct.
The position of the propeller of a ducted propeller is generally inside the
duct. The propeller can also be moved towards' the exit of the duct without too much loss of efficiency. Such a position is appropriate when a duct is
used as a retrofit, that is an improvement afterwards. It should be kept in
mind, however, that the application of a duct in front ofan existing propeller will change the propeller loading and may require another propeller design. Mitsui has patented such retrofits with a duct, the combination is therefore
January , 1994, Propulsors 41
I
Figure 2.14: Thrusters for dynamic positioning also called a "Mitsui Duct".
2.4
Other Propulsors.
2.4.1
Voight-Schneider Propellers.
A very special propulsor is the Voight-Schneider Propeller (Fig. 2.20). a number of "knifes" on a rotating plate. These "knifes" can rotate on this
plate and their position is such that they are always perpendicular to the radials from a moving centerpoint P, as shown in Fig. 2.21. When this centerpoint is in the center of the blade circle there is no resulting force
(Fig. 2.21c). When this centerpoint is moved a thrust is generated
perpen-dicular to the direction in which the centerpoint is shifted.
The main asset of a Voight-Schneider Propeller is that in that way the thrust
can be applied in all directions, just by moving the centerpoint. Rudders
and shafts can be omitted. This can be used e.g. for tugs or supply boats, for which manoeuvring is important. Its efficiency,however, is lower than
that of an open propeller due to the fact that the blades generate thrust
42 G.Kuiper, Resistance and Propulsion, January , 1994
Figure 2.15: Active rudders
Figure 2.16: Duct with trailing edge slots (Courtesy v.Gunsteren and
2.4.3
Pump Jets.
F,,
Figure 2.17: Flow improving fins (Schneekluth) the whole revolution.
Voigth Schneider propellers can be mounted under a flat bottom. For protection some cover is sometimes applied (Fig. 2.20).
2.4.2
Paddle-Wheels.
The oldest form of mechanical propulsion after the sails is the paddle wheel. Contrary to the propeller, which uses lift for propulsion, a paddle wheel uses drag, which at higher speeds is less efficient. The blades of a paddle wheel are most effective in the lowest position, in other positions they also generate a vertical force. So a paddle wheel has to be large, with
only a small immersion. In order to improve the entrance and exit of the blades in and from the water, the blades have been made rotating by a
system of rods. This made the wheel very complicated, however.
44 G.Kuiper, Resistance and Propulsion, January4, 1994
Figure 2.18: Flow improving fins
Figure 2.19: Ringpropeller
The basic mechanism of propulsion is acceleration of water. This cannot only be done by e.g. a propeller outside the hull, but also by a pump inside the hull. The water is sucked in from the bottom of the ship, is accelerated inside the ship by a pump and leaves the ship at the stern. This has many advantages when a propeller is too sensitive to damage, or when a propeller is dangerous (e.g. rescue vessels). Also in shallow waters a pumpjet can
Figure 2.20: Voight-S chneider Propeller
The oldest sails were square rigged, using drag as the thrust force,just
January , 199, Propulsors 45
us
a b C
Figure 2.21: Blade positions of Voight-Schneider Propeller
be useful. The inner surface of the pump system is large and the velocities inside are high, so the viscous losses are high too. The efficiency is therefore lower than that of an open propeller.
A special version of a pumpjet is the rotational pumpjet, as shown in Fig. 2.24. The water goes into the pumpjet at the center of the jet and is blow out tangentially. Rotation of this pumpjet along the vertical axis
makes it possible to control the direction of the thrust.
46 G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.22: Paddle wheel
Figure 2.23 Pumpjet
as the paddle wheels later (Fig. 2.25). Sailing towards the *ind is not
possible with this rigging. Before the steam engine took over longitudinal
sails were also used. When the energy crisis hit, some modern sail designs
were made of both form, either as additional power or as main propulsor.
(Figs. 2.26 and 2.27). The use of computer controlled settings of the sails can highly improve their operation. The development of racing yachts as the 12 meters, used for the America's Cup, can provide more experimental
Impressionof
pumpjet
Figure 2.24: Rotational pumpjet
and theoretical experience with sails. Sails will only become attractive when the fuel price rises again considerably.
2.4.5
Other Types of Propulsion.
When a cylinder rotates in wind a thrust force is generated. This effect resembles sailing. The rotating cylinders are called Flettner Rotors, after
their original designer. Flettner rotors have been applied on experimental
ships only (Fig 2.28). A major problem is the mechanical connection of the rotor to the hull, where the ship motions cause very large forces and
moments.
Even more esoteric types of ship propulsion are ramjets , which are an
analogy of jet engines. In a water jet hot compressed air is injected in a
water stream, and the expanding air accelerates the flow in the engine. Nature has often been an example for technology (although the airplane
only became practical after the bird wing motion was abandoned). Fish
propulsion has also been an example. In this case a flat plate makes sinu-soidal motions perpendicular to the direction of the ship. The construction is very complicated, of course.
A variation on the fish propulsion is Weiss-Fogh propulsion . This is
simply a flat plat which moves between two walls in a direction perpendic-ular to the direction of motion of the ship. The angle of the plate is varying January 4, 1994, Propulsors 47
48 G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.25: Square rigged sail
with its position, so that the water is pressed towards the rear during the motion of the plane. As with fish propulsion this is mainly of theoretical
importance.
The same is true for magneto-h ydrodynamic propulsion . In this case a
strong magnetic field accelerates the flow in a magnetic duct. In principle no moving parts are required for this type of propulsion. The high electrical resistance of sea, water makes the efficiency extremely low, however. It still has to be developed for other fluids with more suitable properties.
Figure 2.26: Modern sailing ship (Wind Spirit)
January 4, 1994, Propulsors 49
':JWMURAL
50 G.Kuiper, Resistance and Propulsion, January 4, 1994
Chapter 3
Intermezzo: Resistance of
Simple Bodies
Purpose:
Brief introduction to non-dimensional formulations and hydro-dynamic concepts relevant for ship resistance.To understand .tie physics of the flow around a ship it is useful first
to look to the flow around a very simple body such as a flat plate in flow direction.
3.1
Non-dimensional Coefficients.
Each flat plate has its own resistance RT. There is a need to compare the resistance of plates of various lengths at various velocities Experiments
show that the resistance of a plate is proportional to the square of the
velocity and:proportional to the area S:[m21 of the plate(heightHxlengthL). When the resistance of the plate is measured in different, fluids it appears that the resistance is proportional with the density p[kg./m3] of the fluid.
The resistance RTEN] of a plate of arbitrary dimensions. at an arbitrary
velocity V[m/sec] can therefore be expressed by a. single number containing these proportionalities. This number is the drag coefficient Cd:
(1 RT
- 1/2pV2'S (3.1)
The resistance RT in this equation is the total resistance. It is called the
total resistance because various components of the total resistance will be distinguished later. The dimension of the drag coefficient is found from the dimensions' of its components to be 1. That means that the drag coefficient
.020 .009 DOS .00? .006 .005 .004 .009 .002 .00I 4 I0
52 G.Kuiper, Resistance and Propulsion, January 4, 1994
is a real coefficient, because it is non-dimensional. Each plate, whatever its size or velocity, has the same drag coefficient in the same circumstances.
An important clause is under the same circumstances. As will be shown later this means that the flow has to be similar in all cases, which is true
when there are no other pirameters for the drag than the size and the speed.
Figure 3.1: Drag coefficients of a flat plate.
3.2
Drag of a Flat Plate.
When the drag coefficient of a flat plate is measured at various velocities the dots in Fig. 3.1 are found. In this diagram the velocity in the abscissa is replaced by the Reynolds number UL/v,. which will be discussed later. It is noted at first glance that the drag coefficient is not a constant! So there
are other parameters involved in the d.rag of a cylinder. Systematic tests
showed that at high velocities the. drag coefficients of plates with the same
product V x L, where V is the velocity of the fluid and L is the length of the plate, are the same, When the temperature is varied the viscOsity of
the fluid is changed and systematic tests showed that the drag coefficient of all cylinders will collapse on one line when plotted on the abscissa V x Liv where v is the kinematic viscosity of the fluid[m2/sec]. This parameter is called the Reynolds number
R- VL (3.2) Ic? 10' REYNOLDS NUMBER V 0242 TURBULCNT ISCN0(Pd.lERlI)
January j, 199ij, Simple bodies 53. The drag coefficient Cd in Fig. 3.1 has therefore been plotted against the Reynolds number.
The Reynolds number is again non-dimensional, so that in Fig. 3.1 all
parameters are expressed non-dimensionally. That means that the drag coefficient is a function of the Reynolds number only and that Fig 3 1 is
valid for all possible flat plates aligned with the flow. This property is the purpose of expressing the parameters non-dimensionally.
In principle the dots should therefore form one single curve, which is not
exactly the case. So there are still other phenomena which influence the
resistance of a flat. plate. The spread of the dots is caused by phenomena in the boundary layer.
3.3
Boundary Layer Flow.
A boundary layer exists because the fluid particles at the wall of the flat
plate stick to the plate (no slip condition). At some distance from the plate
the free stream velocity V occurs. The region where the velocity varies from zero to the outside velocity is called the boundary layer. This is a
region where strong velocity gradients occur. Due to these strong velocity
gradients the viscosity of the fluid has a large influence in the boundary
layer. At Reynolds numbers above 1000 this region is thin con pared to the length of the plate. The pressure from the outside of the boundary layer to. the wall in a thin boundary layer can be considered as constant, so that the pressure in the outer flow, is. equal to the wall pressure. For calculations of the outer flow the thin boundary layer can then be neglected.
In the boundary layer the velocity approaches the free stream velocity
asymptotically. The thickness 6 of the boundary layer is defined as the
distance from the wall where the velocity is 99 percent of V. The velocity gradient at the. wall determines the friction force between the fluid.
The shape of the velocity distribution in the boundary layer can be char-acterized by various quantities. When the boundary layer is replaced by a layer with uniform velocity V outside the boundary layer, with the condi-tion that the same fluid moves through the layer, the displacement thickness
6 is found. This can be expressed by
VS1 = f(V
- v)dy
(3.3)where y is the distance to the wall and v.(y) is the local velocity in the bound-ary layer. When the boundbound-ary layer is replaced by a layer with velocity V
54 G.Kuiper, Resistance and Propulsion, January $, 1994
having the same momentum, the momentum thickness 0 is found:
o=j6v2dy
The ratio between the momentum thickness and the displacement thickness is called the shape facto.rH of the boundary layer.
3.3.1
Laminar and Turbulent Flows.
The boundary layer can have different conditions. In a laminar bound4ry layer the. particles in the boundary layer are gliding smoothly along each
other, so that no motions perpendicular to the flow occur. In a turbulent
boundary layer the smooth motions disappear and violent motions perpen-dicular to the direction of the motion occur. The turbulent motions of the fluid particles cause an exchange of energy between the layers in the
bound-ary layer and as a result the velocity distribution in the boundbound-ary layer is
different, as shown in Fig. 3.2.
LAMINAR TURBULENT
Figure 3.2: Velocity distribution in laminar and turbulent boundary layers From the velocity gradient at the wall in this Figure it. follows that a tur-bulent boundary layer results in a higher friction force than a laminar one. The scatter in the dots in Fig. 3.1 is caused by the transition from laminar
to turbulent boundary layer flow. The line at lower Reynolds numbers is
the drag coefficient when the boundary layer is fully laminar. Ata Reynolds
number around iO transition to turbulence occurs. This transition begins
at the downstream edge of the plate. and moves towards the leading edge
of the plate with increasing Reynolds number. At a Reynolds number of
106. transition occurs immediately at the leading edge and the boundary
January 4, 1994, Simple bodies 55 numbers is the drag coefficient for fully turbulent flat plate boundary layers.
in Fig. 3.1 the Reynolds number is expressed based on the length L of the plate. The location of transition depends on a local Reynolds nut ber
R,, based on the distance x from the leading edge of the plate' . Under
ideal conditions transition takes place at
a fixed value of R,. When this
would be the case the dots in Fig. 3.1 would still form a single line. However, transition is very sensitive to disturbances such as vibrations of the plate, turbulence in the incoming flow,, surface irregularities etc. This causes the scatter of the dots in Fig. 3.1.
Because of the higher friction at the wall the boundary layer thickness of a turbulent boundary layer increasesmore rapidly in flow direction than that of a laminar one, as is illustrated in Fig. 3.3
6
TRANSITION
'It should be noted that transition is a very complicated process, which does not occur at one location and in one moment. The description given here -is a very strong simplification of what really happens.
LAMINAR TURBULENT
Figure 3.3: Development of boundary layer thickness
3.3.2
Effects of the Pressure Gradient
In the case of a flat plate there is a constant pressure along the boundary layer. This is not the case when the plate is at an angle to the flow or when the plate has a thickness distribution. The effect of a pressure gradient on the development of a boundary layer is. very strong. A favorable pressure gradient occurs when the pressure decreases in flow direction. Such a pres-sure gradient reduces the growth of the boundary layer thickness, both for
laminar and turbulent boundary layers. It also delays transition to
tur-bulence of a laminar boundary layer.. A favorable pressure gradient also increases the velocity gradient at the wall and delays separation, a phenom-enon which will be described below. Inversely an adverse pressure gradient
56 G.Kuiper, Resistance and Propulsion, January 4, 1994
causes a strong increase of the boundary layer thickness and stimulates
transition and separation.
3.4
Drag of a Twodimensionai Cylinder.
To illustrate some other floj phenomenaa cylinder will now be used instead of a flat plate. The drag coefficient of a cylinder on the basis of Reynolds number is shown in Fig. 3.4 The Reynolds number of the cylinder is based
Figure 3.4 Drag coefficient of. cylinder
on its diameter D(cyl) and the drag coefficient is the drag coefficient per
unit length
RT
d
-1/2pV2D(cyl)
The pressure p along the cylinder has been non-dimensionalized by the
stagnation pressure 1/2pV2. The stagnation pressure occurs where the ve-locity on the cylinder is zero. The non-dimensional pressure coefficient is expressed as the pressure difference between t:he local pressure p and the
undisturbed. pressure
p.
At. low Reynolds numbers the drag coefficient is high and it decreases
gradually to one at R = 1000. At a Reynolds number between 1000 and 2 x i05 the drag coefficient is constant with a value of approx. 1. The flow
887Q6 100 80 80 40 Co 20 10
Ill
I 111.1 o o o 0(mmf 0.05 0.S 03 1.0 7.9 424 aao TherjjI0 MNwrt y Wiesolgbpgg lamb a 8 4 2 I .Jxo 0.8 0.0 0.4 02 0.1 2 4 8 8100 2 4 8 8o 1O2 2 68j 2 4 ' 68ioSUBCRITICAL FLOW
SIJPERCRmcAL FLOW
January 4, 1994, Simple bodies 57 pattern in this range of Reynolds numbers is given in Fig. 3.5a for subcriti-cal flow. The flow separates at a position close to the location of minimum
pressure. At a Reynolds number between 2 x io and 5 x 1O the drag
Figure 3.5: Flow Pattern around a cylinder
coefficient drops drastically to a value of about 03. This is due to the fact that separation is delayed, which causes a much smaller wake, as shown in Fig. 3.5 for supercritical flow.
The wake behind the cylinder is die to separation. Separation occurs
when the velocity gradient perpendicular to the wall becomes zero, as
illus-trated in Fig. 3.6. As a result the friction becomes zero and downstream
of the separation a region with back flow occurs. The streamline along the
wall separates from the wall and becomes the boundary of the separated
flow region.
When the drag coefficient is around 1 the boundary layer on the cylinder is laminar before it separates close to the location of maximum thickness.
58 G.Kuiper, Resistance and Propulsion, January , 1994
-SEPARATION
Figure 3.6: Velocity profiles in the boundary layer around separation
The result of separation on a cylinder is that the pressure at the
down-stream side of the cylinder remains lower than on the updown-stream side, which causes an additional resistance. The pressure distribution over a cylinder at low Reynolds numbers isshown in Fig. 3.7 by the dotted line (laminar). The location of laminar separation is independent of the Reynolds number and consequently the drag coefficient remains constant. This condition is called the subcritical condition.
For comparison the line indicated by potential theory 2 is the pressure distribution without separation in inviscid flow. in that case the pressure at the back of the cylinder recovers to the stagnation pressure and the pressure distribution is fully:symrnetrical. As a result the resist-ance is zero.
At a Reynolds number of about half a million transition of the boundary layer from laminar to -turbulent flow occurs at or upstream of the location of separation-. When the boundary layer becomes turbulent before separa-tion occurs the flow pattern changes also because the turbulent boundary layer separates much later than the laminar one. As illustrated in Fig. 3.5b this strongly decreases the width of the wake and increases the base pres-sure, which decreases the drag. This is reflected in the base pressure on the cylinder, as shown in Fig 3.7 (turbulent).. This- is called the supercritical condition. (Note that there is a critical condition i-n Fig. 3.7 at an interme-diate Reynolds number, where the base pressure- is even- higher than -in the supercritical condition. This complication will be ignored here.)
The reduction of t-he drag coefficient of a, cylinder at a Reynolds number of around 2 x lO is therefore due to -the-transit-ion from laminar separation to
turbulent, separation. -
-2The meaning of potential theory is described in chapter 10
0
2
-3
January 6, 1994, Simple bodies 59
3.6
Additional References.
Drag coefficients of a sphere and of a flat plate perpendicular to the flow
can be found in Schlichting [42J. Experimental drag coefficients of a range of shapes can be found in Hoerner [13].
-ENTIAL THEORY
-_.
___
A
30 60 90 120 150 180 210 240 270 300 330 360
Figure 3.7: Pressure distribution on a cylinder, from Achenbach.
3.5
D'rag Components.
The pressure over the flat plate is constant and the resistance is only due
to frictional forces. The force on the cylinder can be decomposed into
pressure forces perpendicular to the cylinder (the pressure from Fig. 3.7) and friction forces parallel to the cylinder surface. The integration of the
drag component of the pressure forces is the pressure drag or form drag.
The integration of the drag coefficient of the frictionforces is called the
frictional drag. These two drag components are not independent. In the
case of the cylinder they are strongly interdependent, because the frictional resistance determines the location of separation and this location determines the pressure drag. On more streamlined bodies like ship hulls the location of (turbulent) separation is less dependent on the Reynolds number.
