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Technical University Delft

Course MT512

Prof. Dr. Ir. G.Kuiperl

January 3, 1994

Re-v

w7.

@L%-74K

1MARIN, Maritime Research Institute Netherlands, Wageningen, Technical

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Contents

1 1

Hull 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. 14

1.1.5 .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 31

2.1.3 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

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2.4 2.3.1 Ringpropellers. 2.3.2 Mitsui Duct. Other Propulsors 40 40 41 2.4.1 Voight-Sçhneider Propellers 41 2.4.2 Paddle-Wheels. 43 2.4.3 Pump Jets. 43 2.4.4 Sails 45

2.4.5 Other Types of Propulsion. 47

3 Intermezzo: Resistance of Simple Bodies

51

3.1 Non-dimensional Coefficients. 51

3.2 Drag of a Flat Plate. 52

3.3 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

63

4.1 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 Visualisation 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

81

5.1 SurfaCe 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. 83

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3

5.3.2 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 Economical Speed. 91

5.6 Hull Speed 93

5.6.1 High Speed Ships. 93

5.7 Bulbous Bows. 94

5.8 Shallow Water Depth. 94

6 Intermezzo: Scaling Rules

97

6.1 Dimension Analysis. 98

6.2 Physical Meaning of Non-dimensional Parameters 101

6.3 Scaling Rules 102

6.4 Scale Effects. 102

7

Resistance Prediction using model tests

103

7.1 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 Data123

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

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9 Intermezzo:Equations of Motion

139

9.1 The Continuity Equation. 140

9.2 The Equations of Motion. 141

9.2.1 Rotation and Deformation. 143

9.2.2 Relation 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:Potential flow

151

10.1 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

165

11.1 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

173

12.1 Potential Flow Calculations 175

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

187

13.1 The Linearized Free Surface Condition 187

13.2 Kelvin Sources. 190

13.3 Applications of the Kelvin Sources. 190

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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 Momentum Theory

201

14.1 Axial Mc;Mentum Theory. 201

14.1.1 Efficiency 204

14.2 Optimum Radial Loading Distribution 205

15 The Propeller Geometry

208

15.1 General Outline. 208

15.2 Blade Sections. 209

15.2.1 NACA Definition of Thickness and Camber 212

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

15.10Description of a Propeller 220

15.11 Controllable Pitch Propellers. 222

16 Systematic Propeller Series

224

16.1 Open Water Diagram 224

16.2 The Quality Index. 227

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

16.8.4 Known Thrust and Rotation Rate 238

16.9 Optimization using the Open Water Diagrams. 239

16.10Example. 241

16.11Four Quadrant Measurements 243

16.12Propeller Design using the Optimized Data. 246

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17 Profile Characteristics

247

17.1 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 Edp 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

265

18.1 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 Cavitation271

18.3 Noise and Erosion. 274

18.3.1 The Implosion of a Single Bubble Cavity 275

18.3.2 Noise Radiation. 976

18.3.3 Thrust Breakdown 978

18.4 The Cavitation Bucket. 980

18.5 Cavitation Tests. 284

19 Lifting Line Propeller Design

287

19.1 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 993

19.3 Induction Factors. 994

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

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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 Propulsion Test

311

20.1 The Additional Towing Force 311

20.1.1 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

20.10Variations on the Extrapolation Method 325

20.11Example of Extrapolation of the Propeller Open Water

Dia-gram 326

20.12Example of the Extrapolation of the Propulsion Test 328

20.12.1 Comparison with Resistance Test 330

20.13Extrapolation of the Example using the Marin Method. . 333

21 Propulsion Calculations.

337

21.1 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 343

A DICTIONARY

355

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Preface

This is an introductory course on ship resistance and propulsion for the Maritime Technology Department of the Delft 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 the 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 able 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.

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The prediction of resistance, propeller and propulsion characteristics each are treated in three ways:

By extrapolation from model test results. By systematic or sfatistical 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 (CFD) 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 having their advantages and diadvantages. 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 simplifica-tions such as inviscid flow are used in the calculasimplifica-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 with the same title [33]. Related specialized booksare Lighthill's book on waves [26]and the books of Knapp [21] and Young [46] on cavitation. An introduction in basic aerodynamics with numerical solutions of potential flow problems can be found in Katz and Plotkin [19]

The emphasis in this course is on the practical application of first princi-pies to the prediction of the behavior of ships and propellers. Insight in these first principles increases understanding of the complex phenomena and forms a basis for intelligent problem solving.

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January 3, 1994, Preface 11

Intermezzo'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 simplificatkins, 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 familiar 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 mostcases users of this course will not yet study literature 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 references are made. This makes the list of references less dependent on the most recent publications.

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Acknowledgements

Many students and colleagues from Marin have given comments, corrections and material 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 furtlier in the future. Therefore the date of printing is present on all pages aiid on the title page. Any comment can be helpful to improve it and will be very welcome.

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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 100 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 restriction of a low speed is important,at increas-ing speeds the required power increases very rapidly due to wave generation.

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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 ade common.

1 LNG is transported* 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 hullforms 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 C2 , defined asCB =

7 with v=volume of the displacement, L=length at the waterline 2 LBT

B=breadth en T=draft. The block coefficient is an 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 of 1852 meter per hour. A nautical mile is 1

arcminute of a greatcircle. The U.K. nautical mile of 1853.184meter 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 L1 ,although the

differ-ence between the waterline length and the length between perpendiculars Lpp will mostly be negligible from a hydrodynamic point of view.

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_

Figure 1.1: Transport of containers

Figure 1.2: Hull form with chines in the afterbody

ness of the hullform. It is also indicated by S.

January 3, 1994, Hull forms 15

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is the area at the waterline. It indicates the vertical distribution of the dis-placement. The longitudinal prismatic coefficient Cp is similarly defined as where A, is the area of the maximum tranverse section, which

gener-A.L

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 O.

The midship section coefficient indicates the fullness of the midship sec-tion and is defined as Cm =---m-ABT. It is also indicated by 0.

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 CuiP - LB in dicates the fullness of the water-line. Cwp is also indicated by a.

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 entrance or 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 requirement 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 requires a 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 afterbody and the stern is also strongly dependent on the type of the propulsor. Some forms will be shown in Chapter 2.

1.1.6

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

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Figure 1.3: Fast cargoship with a bulb

decreases with inc-reasing 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

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_Lc

-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 fesistance 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 thewave 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 thewave generation and thus the resistance. There are several ways to create lift.

-.136%-7, --;t:o.pu" --orr-Ar 4s` -41, A--74 AL,

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I I a

"'A:1:0-m.

AIM

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, 1994, Hull forms 19

07-",f3TRAM

-organ

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Figure 1.6: Bulbcontour for various drafts

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

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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 wedge is 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 specif-ically for high speeds. The foils can pierce through the water surface to

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Figure 1.9: Planing luxury yacht

ensure stability. In such a case these are called surface piercing hydrofoils. (Fig. 1.10).

These hydrofoils have io operate close to the surface, a condition where the

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

fins.

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ft

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. 3 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

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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.

-a

is

tinzintvp=e----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.

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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 lower sea 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 January 4, 1994, Hull forrns 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.

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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),

--1:''' I iL. , ;,..,-44,-4_ -1.74111 *sin

7--

iressisoirsiarir

1, N -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 a catamaran is typically used for higher speeds, where the wave resistance becom- es 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

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January 4, 1994, Hull forms 27

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.

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DWL STA 111 17 le 11 20 21 22 AFT FWD

Figure 1.17: Cross section of SWATH hull

Literature for Further Reading.

Several afterbody forms and their relative meritare given by Vossnack and

Voogd [45]

A very rough estimate of the relativepower requirements of various high speed concepts is given by Dorey [6].

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Chapter 2

Propulsors

Ob jective: 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 hull. 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.

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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.

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January 4, 1994, Propulsors 31

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 the more complicated construction. In fast ships or in hydrofoils 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

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.

Figure 2.3: propeller with open stern

emil

f.

'. ::',,, . -- - 4..:..:1-= ,---!

--

,

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

i

r!t, .anwen

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Figure 2.5: Thruster configuration January 4, 1994, Propulsors 33 NUPE! -'11; A111111111 111111.110111111116110,10,111MINIMO

171:3 tinn5111.15._Aii

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

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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 blades.

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 suctionmay decrease the efficiency

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'

..`"" .

-January 4, 1994, Propulsors 35

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 cavitatio-n occurs extensively, e.g. at very high rotation rates of the propeller, it is advantageous to use blade sections which generate a 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 withvapor 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.

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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.

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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.

2.3

Ducted Propellers.

At high propeller loadings a duct can increase efficiency (Fig. 2.11).

A duct generates part of the total thrust due to its interaction with the

January 4, 1994, Propulsors 37

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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

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decelerating

January 4, 1994, Propulsors 39

, 'Nara

-Figure 2.11: Ducted Propeller

acr-elerating

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.

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Figure 2.13: Dynamic positioning

Instead of an a-symmetric duct other types of fins or 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 possible,

as in Fig. 2.18.

2.3.1

Ringpropellers.

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 ofa duct in front of an 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

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January 4, 1994, Propulsors 41

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 over part of the revolution only, while the viscous resistance is present over

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Figure 2.15: Active rudders

.iwansa

C

Figure 2.16: Duct with trailing edge slots (Courtesy v.Gunsteren and

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-2.4.3

Pump Jets.

,

January 4, 1994, Propulsors 43

'

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.

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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

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Genera!plan

Figure 2.20: Voight-Schneider Propeller

January 4, 1994, Pro pulsors 45

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.

2.4.4

Sails.

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Figure 2.23: Pumpjet

as the paddle wheels later (Fig. 2.25). Sailing towards the wind 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 Figure 2.22: Paddle wheel

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Impression of 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

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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-hydrodynamic 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.

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January 4, 1994, Pro pulsors 49

Figure 2.26: Modern sailing ship (Wind Spirit)

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a;a

r o ..., 4 r.

- -- "

(52)

,

Chaptei' 3

Intermezzo: Resistance of

Simple Bodies

Purpose:

Brief introduction to non-dimensional formulations and hydro-dynamic concepts relevant for ship resistance.

To understand the 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 platés 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[m2] of the plate(heightH x lengthL). When the resistance of the plate is measured in different fluids it appears that the resistance is proportional with the density p[kg Im3] of the fluid. The resistance RT[N] of a plate of arbitrary dimensions at an arbitrary velocity V[mIsec] can therefore be expressed by a single number containing these proportionalities. This number is the drag coefficient Cd:

Cd 1/2pv2s

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

51

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co

is a real coefficient, because it is non-dimensional. Each plate, whatever its size or velocity, has the same drag coefficient in the sarne 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 pa- rameters for the drag than the size and the speed.

.020 .010 .009 .004 .002 .006 .00.3 .004 .003 .002 .001 10 CF L3213 I. LAMAR (BLASIUS) e . 107 UA REYNOLDS NUMBER

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 theabscissa is replaced by the Reynolds number ULlv, 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 drag 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 dragcoefficient 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:

V L Rn = V 0.242 LCIG, /;:r F

TURBULENT (SC NOE P44E RR )

(3.2)

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January 4, 1994, 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 ex-pressed non-dimensionally. That means that the drag coefficient is a fundion 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 lay-er 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 compared 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 81 is found. This can be expressed by

V(51 = f ( V v)dy (3.3)

o

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

(55)

having the same momentum, the momentum thickness 9 is found:

1

9 = 15

-V2 o v2dy

The ratio between the momentum thickness and the displacement thickness is called the shape factor H of the boundary layer.

3.3.1

Laminar and Turbulent Flows.

The boundary layer can have different conditions. In a laminar boundary 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. At a Reynolds number around 105 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 layer on the plate is fully turbulent. The line in Fig. 3.1 at higher Reynolds

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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 number R, based on the distance x from the leading edge of the plate 1 . 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 increases more rapidly in flow direction than that of a laminar one, as is illustrated in Fig. 3.3

8

TRANSITION

1It 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.

LAM I NAR T URBULENT

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

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causes a strong increase of the boundary layer thickness and stimulates transition and separation.

3.4

Drag of a Two-dimensional Cylinder.

To illustrate some other floW phenomena a 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 cylinderis based

CD ... 80 80 i li I 1 Ili 40 0 k nons! . 0.1 1.0 Ifeassfri 20 0.3 10 ...., .. 10 8). 0 WO 300.0 410 Z9 WieselsOarger 4 -2 - -- Theoryllvelo lamb 0.3 0.8 Oh 02 0.1 / ...44.4444-....4.00.0.4404.,,,,,

tee

1 L. 10- 10 10i 102 103 104 4 6 8105 2 8 8We 113,2,

Figure 3.4: Drag coefficient of a cylinder

on its diameter D(cyl) and the drag coefficient is the drag coefficient per unit length

RT

d =

112pV2D(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 the local pressure p and the undisturbed pressure pco.

At low Reynolds numbers the drag coefficient is high and it decreases gradually to one at ./77, = 1000. At a Reynolds number between 1000 and 2 x 105 the drag coefficient is constant with a value of approx. 1. The flow

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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 105 and 5 x 105 the drag

SUBCRMCAL FLOW

SUPERCRMCAL FLOW

Figure 3.5: Flow Pattern around a cylinder

coefficient drops drastically to a value of about 0.3. 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 becomeszero, 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.

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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 numbérs is shown 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 symmetrical. As a result the resistance 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 in 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 the drag coefficient of a cylinder at a Reynolds number of around 2 x 105 is therefore due to the transition from laminar separation to

turbulent separation.

2The meaning of potential theory is described in chapter 10

(60)

3

o

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 [42]. Experimental drag coefficients of a range of shapes can be found in Hoerner [13].

_

FO" ENTIAL THEORY

locf

_ x7.1 CRITICAL ENT x t

-i

ii

TURB x---: __... V\: v

i

/.

. . LAMI AR

.... ....

%

\

x..\ .. \''' x 1 Ix

\

I 1 ... .." x

\

x . x 30 60 90 120 150 180 210 240 270 300 330 360

Figure 3.7: Pressure distribution on a cylinder, from Achenbach.

3.5

Drag 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 friction forces 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. o

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