Resistance and Propulsion of Ships G Kuiper
Short Description
Ship Resistance and Propulsion Lecture Notes...
Description
Resistance and Propulsion of Ships Technical University Delft Course MT512 Prof. Dr. Ir. G.Kuiperl January 3, 1994 Rev w7. @L%74K
1MARIN, Maritime Research Institute Netherlands, Wageningen, Technical University Delft
Contents 1
Hull forms 1.1
1.2
1.3
1.4
2
13
Displacement Hulls. 1.1.1 Efficiency 1.1.2 Typical Speeds 1.1.3 Hull Forms . 1.1.4 Form Parameters. 1.1.5 .Considerations for the Stern Form 1.1.6 COnsiderations for the Bow Form 1.1.7 Bulbs. High Speed Ships. 1.2.1 Planing Hulls 1.2.2 Hydrofoils. Air as Carrier. 1.3.1 Air Cushion Vehicles. 1.3.2 Surface Effect Ships. Multi Hulls 1.4.1 Catamarans. 1.4.2 Swath
Propulsors 2.1
2.2
2.3
13 13 14 14 14 16 16 17 18 19 21 24 24 25 25 25 26
29
Propellers 2.1.1 Propeller Arrangements 2.1.2 Trusters 2.1.3 Controllable Pitch Propellers. 2.1.4 Overlapping Propellers. 2.1.5 Contra Rotating Propellers. 2.1.6 Surface Piercing Propellers. Special Types of Propellers. 2.2.1 Supercavitating Propellers. 2.2.2 Agouti Propellers. 2.2.3 Tipplates . 2.2.4 Vane Wheels. Ducted Propellers. 1
30 30 31 32 34 34 34 35 35 35 36
37 37
2
Ringpropellers. Mitsui Duct. Other Propulsors 2.4.1 VoightSçhneider Propellers 2.4.2 PaddleWheels. 2.3.1 2.3.2
2.4
2.4.3 2.4.4 2.4.5
40 40 41 41 43
Pump Jets.
43 45 47
Sails
Other Types of Propulsion.
3 Intermezzo: Resistance of Simple Bodies 3.1 3.2 3.3
51
Nondimensional Coefficients.
51 52
Drag of a Flat Plate .
Boundary Layer Flow. 3.3.1 Laminar and Turbulent Flows. 3.3.2 Effects of the Pressure Gradient. 3.4 Drag of a Twodimensional Cylinder.. 3.5 Drag Components. 3.6 Additional References. 3.7 Additional Data 3.8 Summary..
53 54 55 56 59 59 60 61
4 Resistance, Wake and Wake Distribution 4.1
4.2 4.3 4.4 4.5 4.6 4.7 4.8
4.9
63
Resistance and Wake. Flow along a Ship Hull. Cross Flow. Separation. The Wake behind Simple Ship like Bodies HorseShoe Vortices. Visualisation of the Flow around the Hull. Ship Wake. 4.8.1 Representation of the wake 4.8.2 Relation between hull form and wake distribution 4.8.3 Wake Fraction Design Considerations . .
5 Wave Resistance
63 65 65 67 69 71
72 73 75 76 78 79
81
5.1
SurfaCe Waves.
81
5.2
Properties of Surface Waves 5.2.1 The Dispersion Relation 5.2.2 Energy in a Wave. 5.2.3 The Group Velocity. The Kelvin Wave System. 5.3.1 The Froude Number
82 82 82 83 83 85
5.3
3
Resistance due to a Kelvin Wave System The Wave System of a Ship 5.4 Wave Interference. 5.4.1 A Twodimensional Simplified Hull Form. 5.5 Economical Speed. 5.6 Hull Speed 5.6.1 High Speed Ships. 5.7 Bulbous Bows. 5.8 Shallow Water Depth. 5.3.2 5.3.3
6 Intermezzo: Scaling Rules 6.1 6.2
6.3 6.4 7
Dimension Analysis. Physical Meaning of Nondimensional Parameters Scaling Rules Scale Effects.
85 86 89 90 91 93 93 94 94
97 98 101 102 102
Resistance Prediction using model tests
103
Elements of Ship Resistance Scaling Laws for Model Tests. Froudes Hypothesis. Determination of Resistance Components 7.4.1 Determination of the Frictional Resistance . 7.4.2 Determination of the Form Resistance. 7.4.3 Determination of the Wave Resistance 7.5 Extrapolation of Resistance Tests 7.5.1 Froude's Extrapolation Method 7.6 Effects of Surface Roughness. 7.6.1 Equivalent Sand Roughness 7.7 Appendage Drag 7.8 Effective Power 7.9 Effects of Laminar Flow 7.10 Wake Scale Effects 7.11 Example of Resistance Extrapolation
103 104 106 106 108 109 112 112 113 113 115 115 116 116 116 118
7.1 7.2 7.3 7.4
8 Resistance Prediction using Statistical or Systematic Data123 8.1
8.2 8.3 8.4 8.5 8.6 8.7
General Considerations for Hull Design. Systematic Series Regression of Available Data. Design of Curve of Sectional Areas by Lap The method of Holtrop and Mennen. Example of Resistance Prediction. Resistance of Small Vessels.
123 126 128 131 132 135 137
4
9 Intermezzo:Equations of Motion
139
The Continuity Equation. The Equations of Motion. 9.2.1 Rotation and Deformation. 9.2.2 Relation between Stresses and Strain. 9.2.3 NavierStokes Equations 9.3 A Simple Example. 9.4 The Euler Equations. 9.5 The Bernoulli Equation 9.1
140 141 143 144 145 146 148 148 149
9.2
9.6
Summary..
10 Intermezzo:Potential flow
151
10.1 Singularities in Potential Flow. 10.1.1 Uniform Flow. 10.1.2 Source. 10.1.3 Vortex. 10.1.4 Dipole. 10.2 A Simple Example of Potential Flow. 10.3 Forces on a Vortex 10.4 Panel Methods . 10.4.1 The Lifting Problem 10.5 Summary..
152 152 152 153 155 156 158 159 161 163
11 Intermezzo: Boundary Layers 11.1 The nondimensional Navier Stokes Equations 11.2 The Boundary Layer Equation 11.2.1 Scaling the Thickness of the Boundary Layer. 11.3 Solutions of the Boundary Layer Equations: Blasius. 11.4 Turbulence.
12 Flow Calculations without Waves 12.1 Potential Flow Calculations 12.1.1 Panel Methods without Free Surface . 12.1.2 Assessment of Various Bulb Designs. 12.1.3 Knuckles and Bulge Keels 12.1.4 Assessment of the Afterbody. 12.2 NavierStokes Solutions.
165
.
.
.
.
166 166 168 168 169
173 175 175 176 178 179 184
13 Flow Calculations with a Free Surface
187
13.1 The Linearized Free Surface Condition 13.2 Kelvin Sources. 13.3 Applications of the Kelvin Sources. 13.3.1 The Michell Theory.
187 190 190 190
5
13.4 Kelvin Sources for Catamaran Hulls, an Example 13.5 Dawson's Method. 13.5.1 Applications of Dawson's Method. 13.6 General_Considerations to Assess Programs.
14 Axial Momentum Theory 14.1 Axial Mc;Mentum Theory. 14.1.1 Efficiency 14.2 Optimum Radial Loading Distribution
15 The Propeller Geometry 15.1 General Outline. 15.2 Blade Sections. 15.2.1 NACA Definition of Thickness and Camber 15.2.2 Root and Tip. 15.3 Pitch and Pitch Angle 15.4 Propeller Plane and Propeller Reference Line 15.5 Rake. 15.6 Skew.
15.7 Blade Contours and Areas 15.8 Warped Propellers . 15.9 The Propeller Drawing. 15.10Description of a Propeller 15.11 Controllable Pitch Propellers.
16 Systematic Propeller Series 16.1 Open Water Diagram 16.2 The Quality Index . 16.3 Systematic Propeller Series. 16.4 Propeller Hull Interaction 16.5 Propeller Design Requirements. 16.6 Choice of Number of Blades and Blade Area Ratio. 16.7 Propeller Design using BSeries Charts 16.8 Elimination of Variables 16.8.1 Known Power and Diameter. 16.8.2 Known Power and Rotation Rate 16.8.3 Known Thrust and Diameter 16.8.4 Known Thrust and Rotation Rate 16.9 Optimization using the Open Water Diagrams. 16.10Example. 16.11Four Quadrant Measurements 16.12Propeller Design using the Optimized Data. 16.130ther Series.
193 194 196 198
201 201 204 205
208 208 209 212 213 213 214 215 216 216 219 219 220 222
224 224 227 227 229 231 232 234
235 236 237 237 238 239 241 243 246 246
6
17 Profile Characteristics 17.1 The Pressure Distribution 17.2 The Loading Distribution. 17.2.1 The Lift Curve 17.3 The Zero Lift Angle. 17.4 The Leading Edp Suction Peak 17.4.1 The Ideal Angle of Attack 17.4.2 Profile Drag. 17.5 Profile Series. 17.5.1 Thickness Distributions. 17.5.2 Camber Distributions. 17.5.3 Derivation of the Local Pressure of a Profile 17.5.4 Considerations to Choose or Design a Profile.
18 Cavitation 18.1 The Cavitation Number 18.2 Types of Cavitation. 18.2.1 Bubble Cavitation 18.2.2 Sheet Cavitation 18.2.3 Root Cavitation. 18.2.4 Tip Vortex Cavitation 18.2.5 Propeller Hull Vortex Cavitation . 18.2.6 Unsteady Sheet Cavitation.
247 247 249 250 251 252 252 254 256 257 257 260 261
265 266 266 266 267 268 268 269 269
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 18.4 The Cavitation Bucket. 18.5 Cavitation Tests.
19 Lifting Line Propeller Design 19.1 Lifting Line Theory. 19.1.1 Twodimensional Lifting Lines 19.1.2 Lifting Lines in Three Dimensions. 19.1.3 Lifting Line Theory for a Propeller 19.2 Optimum Radial Loading Distribution 19.3 Induction Factors. 19.4 Propeller Design using the Induction Factors. 19.4.1 Determination of the Inflow 19.4.2 Determination of the Blade Sections. 19.4.3 Strength of the Propeller. 19.4.4 Stresses due to Loading.
978 980 284
287 289 289 289 292 993 994 295 295 296 299 299
7
19.4.5 Stresses due to Centrifugal Forces. 19.4.6 Approximate Methods 19.4.7 Lifting Surface Corrections. 19.4.8 Viscous Forces.
300 301 301
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 Equivalent 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 Diagram 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. 21.1 Statistical Prediction of the Model Wake Fraction. 21.2 Statistical Prediction of the Full Scale Wake Fraction 21.3 Statistical Prediction of the Thrust Deduction 21.4 Statistical Prediction of the Relative Rotative Efficiency.
337
.
337 340 340 341
22 TABLES
343
A DICTIONARY
355
B WOORDENLIJST
362
8
G.Kuiper, Resistance and Propulsion, January 3, 1994
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 mathematics 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 computer 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.
9
G.Kuiper, Resistance and Propulsion, January 3, 1994
10
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 dynamics (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 simplifications such as inviscid flow are used in the calculations. 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 Principles of Naval Architecture [38] is more suitable. The basis of the mathematical description of marine hydrodynamics can be found in Newman's book with the same title [33]. Related specialized books are 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 principies 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.
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 most cases 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.
12
G.Kuiper, Resistance and Propulsion, January 3, 1994
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 KoningGans 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. G.Kuiper. January 3, 1994
Chapter 1 Hull forms Objective:
1.1
Introduction of the hydrodynamic features of some ship types.
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 comparison 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 requires 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.
13
G.Kuiper, Resistance and Propulsion, January 3, 1994
14
1.1.2
Typical Speeds.
At higher speeds the influence of waves becomes large and the waves become 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 asgood 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 nondimensional 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 1853.184 meter 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 difference between the waterline length and the length between perpendiculars Lpp will mostly be negligible from a hydrodynamic point of view. ,
January 3, 1994, Hull forms 15
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. The vertical prismatic coefficient Cp, defined as Cp, = 22r A,T, where Au,
16
G.Kuiper, Resistance and Propulsion, January 3, 199.4
is the area at the waterline. It indicates the vertical distribution of the displacement. The longitudinal prismatic coefficient Cp is similarly defined as A.L where A, is the area of the maximum tranverse section, which generally 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 section and is defined as Cm =mABT. It is also indicated by 0.
The longitudinal position of the center of buoyancy is given as a percentage of the ship length L and is indicated as LCB .
The waterline coefficient Cui P  LB in dicates the fullness of the waterline. 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 difficult 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
January 3, 1994, Hull forms 17
Figure 1.3: Fast cargoship with a bulb decreases with increasing speed. Tankers and bulkcarriers have a block coefficient 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
G.Kuiper, Resistance and Propulsion, January 3, 1994
18
41,
A74 AL,

_Lc
.136%7,

;t:o.pu"
orrAr
4s`

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 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.
January 3, 1994, Hull forms 19
07",f3TRAM organ
a
I
AIM
Ia
"'A:1:0m.
Figure 1.5: Various bulbs
1.2.1
Planing Hulls.
In case of planing there is a pressure buildup on the bottom of the ship, such that an upward force is generated. The displacement is thereby re
20
G.Kuiper, Resistance and Propulsion, January 4, 1994
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 bottom and it is displaced downwards. A flat bottom is very sensitive to incoming waves. High loads occur, a phenomenon called slamming. To reduce this sensitivity deadrise is used in the midbody, in combination with a sharp bow. In the stern region the bottom
January 4, 1994, Hull forms 21
is nearly flat (see Fig. 1.7) and ends in a cutoff 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 control 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 resistance (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
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 io operate close to the surface, a condition where the
kf.;
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.
January 4, 1994, Hull forms 23
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 extend 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 Zdrives (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 3Sometimes these propellers are supercavitating propellers, see Chapter 2.
24
G.Kuiper, Resistance and Propulsion, January 4, 1994
canard arrangement, as in the case of airplanes with the stabiliser in front of the wing instead of at the tail.
Air as Carrier.
1.3
Instead of lift also air can be used to create an upward force. This is used in aircushion vehicles.
Air Cushion Vehicles.
1.3.1

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. Typical for an ACV is its amphibious character: it can operate both on land
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.
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 (USNavy). 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
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),
I iL.
 1:'''
,
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 frictional 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
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 sensitive 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 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 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 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 important 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 cavitation . 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 behaviour of large cavities also generate vibrations in the ship structure.
29
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.
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 illustrated 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
G.Kuiper, Resistance and Propulsion, January 4, 1994
32
Figure 2.3: propeller with open stern
r!t,
.anwen
ar......orr
emil
. '.
f.
::',,,
.

i
  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
January 4, 1994, Propulsors 33
Figure 2.5: Thruster configuration
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
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 required to change the direction of the trust. A CP propeller is specifically favourable in case of high_ skew, because a highly skewed fixed pitch propeller 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, although 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 the efficiency
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 cavitation 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 with vapor and on implosion the air will cushion the collaps. As a result the radiated noise of the cavitation 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.
G.Kuiper, Resistance and Propulsion, January 4, 1994
36
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.
January 4, 1994, Propulsors 37
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
38
G.Kuiper, Resistance and Propulsion, January 4, 1994
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 positioning (Fig. 2.13) . These thrusters can freely rotate over the full circle and are therefore also called azymuthing thrusters. The power of these systems 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
January 4, 1994, Propulsors 39
,
'Nara

Figure 2.11: Ducted Propeller
decelerating
acrelerating
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. Asymmetrical ducts have a different angle over the circumference to make the wake distribution more uniform.
40
G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.13: Dynamic positioning Instead of an asymmetric 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 of a 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
January 4, 1994, Propulsors 41
Figure 2.14: Thrusters for dynamic positioning
also called a "Mitsui Duct".
2.4
Other Propulsors.
2.4.1
VoightSchneider Propellers.
A very special propulsor is the VoightSchneider 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 perpendicular to the direction in which the centerpoint is shifted. The main asset of a VoightSchneider 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
42
G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 2.15: Active rudders
.iwansa C
Figure 2.16: Duct with trailing edge slots (Courtesy v.Gunsteren and Gelling, Delft, the Netherlands)
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
PaddleWheels.
The oldest form of mechanical propulsion after the sails is the paddle . 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. wheel
2.4.3
Pump Jets.
44
G.Kuiper, Resistance and Propulsion, January 4, 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
January 4, 1994, Pro pulsors 45
Genera!plan
Figure 2.20: VoightSchneider Propeller
Figure 2.21: Blade positions of VoightSchneider 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.
The oldest sails were square rigged, using drag as the thrust force,just
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 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
January 4, 1994, Propulsors 47
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 sinusoidal motions perpendicular to the direction of the ship. The construction is very complicated, of course.
A variation on the fish propulsion is WeissFogh propulsion . This is simply a flat plat which moves between two walls in a direction perpendicular to the direction of motion of the ship. The angle of the plate is varying
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 magnetohydrodynamic 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.
January 4, 1994, Pro pulsors 49
Figure 2.26: Modern sailing ship (Wind Spirit)
Figure 2.27: Modern square rigged sailing ship
50
G.Kuiper, Resistance and Propulsion, January 4, 1994
a;a
ro
....., 4



r.
"
Figure 2.28: Flettner rotor propulsion
Chaptei' 3
Intermezzo: Resistance of Simple Bodies ,
Purpose: Brief introduction to nondimensional formulations and hydrodynamic 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
Nondimensional 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
(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 51
52
G.Kuiper, Resistance and Propulsion, January 4, 1994
is a real coefficient, because it is nondimensional. 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
co
.009 .004 .002 .006 .00.3
0.242
e
.004 .003
CF
L3213
/;:r
.
LCIG, F
TURBULENT (SC NOE P44E RR )
I. LAMAR (BLASIUS)
.002
.001 10
107
REYNOLDS NUMBER
to'
UA
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 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 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:
Rn =
VL V
(3.2)
January 4, 1994, Simple bodies 53
The drag coefficient Reynolds number.
Cd
in Fig. 3.1 has therefore been plotted against the
The Reynolds number is again nondimensional, so that in Fig. 3.1 all parameters are expressed nondimensionally. 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 nondimensionally. 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 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 characterized by various quantities. When the boundary layer is replaced by a layer with uniform velocity V outside the boundary layer, with the condition 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 boundary layer is replaced by a layer with velocity V
54
G.Kuiper, Resistance and Propulsion, January 4, 1994
having the same momentum, the momentum thickness 9 is found:
9=
1
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 perpendicular 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 boundary 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 turbulent 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
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
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 pressure gradient reduces the growth of the boundary layer thickness, both for laminar and turbulent boundary layers. It also delays transition to turbulence of a laminar boundary layer. A favorable pressure gradient also increases the velocity gradient at the wall and delays separation, a phenomenon which will be described below. Inversely an adverse pressure gradient 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.
56
G.Kuiper, Resistance and Propulsion, January 4, 1994
causes a strong increase of the boundary layer thickness and stimulates transition and separation.
Drag of a Twodimensional Cylinder.
3.4
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 cylinder is based
... 80 80 40
i li
I
1
Ili
0 k nons!
CD
.
20
0.1
0.3 1.0
....,
10
10
..
Z9
410

4 2
Ifeassfri 8). WieselsOarger
0 WO
300.0  Theoryllvelo lamb

/
.......44.4444....4.00.0.4404.,,,,,
0.3
0.8 Oh
tee
02 0.1
1
10
L.
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 d=
RT
112pV2D(cyl)
The pressure p along the cylinder has been nondimensionalized by the stagnation pressure 1/2pV2. The stagnation pressure occurs where the velocity on the cylinder is zero. The nondimensional 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
January 4, 1994, Simple bodies 57
pattern in this range of Reynolds numbers is given in Fig. 3.5a for subcritical 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 becomes zero, as illustrated 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.
G.Kuiper, Resistance and Propulsion, January
58
PO
p2>pl
4, 1994
p3>p2
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 downstream side of the cylinder remains lower than on the upstream 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 separation 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 pressure, 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 intermediate 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
January 6, 1994, Simple bodies 59
FO" ENTIAL THEORY
locf
i
_
o _
CRITICAL TURB
x7.1
x: __..... ..
ENT
LAMI AR
x
t
.... ....x..\.. \''' ...
i
V\: v
x
%
\
/.
2
ii

1
\
Ix
I
1
\
x
x
3o
30
60
90
120
150
180
210
240
.."
.
x
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.
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].
G.Kuiper, Resistance and Propulsion, January 4, 1994
60
3.7
Additional Data.
Since the flat plate boundary layer is used under many circumstances some data of the laminar and turbulent boundary layer at zero pressure gradient will be given below. Laminar flow: The thickness of the laminar boundary layer 5 is =
(3.4)
So the boundary layer thickness increases with N/X. In nondimensional notation the boundary layer thickness can be written as Rn5 = 5 OR.7z. The displacement thickness of the laminar boundary layer can be written as: 6* =
0.34
. The local friction coefficient is proportional to the slope of the velocity distribution perpendicular to the wall. This local friction coefficient is
Cf =
0.664
(3.5)
The local friction coefficient per unit surface is defined as
Cf =
D9V2
where F is the local friction force on a unit surface.
This translates into a drag coefficient of a flat plate of
=
1.328
(3.6)
where the Reynolds number is based on the length of the plate.
Turbulent flow: An approximation of the velocity distribution in the turbulent boundary layer derived from pipe flow is the socalled 1/7th power /aw. In that case the velocity distribution in the boundary layer is always
T7_ T,
(3.7)
The boundary layer thickness can then be written as
= 0.37xRz4
(3.8)
This means that the boundary layer thickness increases with x.4 instead of with ,N,/ in laminar flow. The turbulent boundary layer will therefore be thicker than the laminar one.
The displacement thickness can easily be derived using the 1/7th power law to be 81 = The local friction coefficient can be written as
C1 = 0.0576(Rn4r+
(3.9)
and the drag coefficient based on a plate of length 1 and unit width as Cd = 0.072(Rn1)1
(3.10)
January 4, 1994, Simple bodies 61
3.8
Summary.
The resistance of a body can be expressed in nondimensional terms as the resistance coefficient. The resistance coefficient or drag coefficient of a submerged body is dep endent on the Reynolds number only. The drag coefficient of a flat plate depefids on the transition from laminar to turbulent boundary layer flow. The loge. ation of transition depends on the local Reynolds number and on the pressure gradient. The drag coefficient e.g. of a cylinder varies with the Reynolds number depending on the type of separation which takes place. Under the subcritical condition laminar separation occurs. In the supercritical condition turbulent separation occurs. The total resistance of a body can be divided in form drag and frictional drag.
Chapter 4 Resistance, Wake and Wake Distribution Objective: A description of the relation between hull form, resistance and wake distribution
The resistance of the ship is caused by the flow around the hull and this flow around the hull is also reflected in the wake of the ship. The wake is the velocity distribution behind the ship hull. The wake is important because its magnitude is related with the ship resistance and the wake distribution is important because it is the inflow distribution of the propeller. When this distribution is very nonuniform the propeller will cavitate more extensively and more violently.
4.1
Resistance and Wake.
The resistance of a body can be related with the wake behind the body. To illustrate this we use the laws of conservation of mass and momentum.
Consider a body,as in Fig. 4.1. The shape of the body is irrelevant. For sake of simplicity the body is assumed to be twodimensional, so the flow is identical in every plane parallel to the drawing. A control volume is defined with plane A (width 2a)upstream of the body, where the velocity is V everywhere. Downstream a plane B (width 2b)is chosen, where the velocity is u(y). The outer boundaries are streamlines, which means that no fluid passes through these boundaries. The outer boundaries are taken at such a distance from the body that the velocity near the boundary of section B is equal to V. This makes it possible to assume a fluid pressure Po everywhere over the control volume. (Note that this is not evident in the 62
January 4, 1994, Wake 63
Streaml me
A
PO
Figure 4.1: Control volume around a body
region where the velocity u is smaller than V. It is an assumption, often made for convienience!) This assumption means that there is no pressure force acting on the control volume. Assuming incompressible fluid the law of conservation of mass requires that
2a V=
çb
b
udy
(4.1)
This relation will be used later.
The drag force RT on the body is related with the loss of momentum over the control volume. Momentum is entering the control volume through
plane A. The volume per unit time entering plane A is 2aV. (Note that this is per unit length perpendicular to the drawing, the dimension of the volume is thus kg Isecm instead of kg Isec) Its momentum is p2aU2. In plane B momentum is leaving the control volume. At an arbitrary position y relative to the centerline a flow volume udy passes plane B. The momentum leaving plane B can therefore be written as Mout
p fb
u2dy
Since the drag force is the only force present and since there is no resultant pressure force the drag is equal to the loss of momentum over the control volume. So: R = p2aV2
p
u2dy
G.Kuiper, Resistance and Propulsion, January 4, 1994
64
Using eq. 4.1 this can be written as
R=
p
u(V
u)dy
(4.2)
b
Since the integrand is zero outside the wake region (because U u = 0) the choice of b is not important. So eq. 4.2 can be used over the wake region only, where u U.1 A further simplification can be obtained when it is assumed that the wake velocity u is not too much different from the incoming velocity U. In that case (V u)2 0 and thus it can be derived that u(V
u) = V(V
u)
(V
u)2
V(V
u)
So eq. 4.2 can be written as
R = pV f!b(V
u)dy
(4.3)
So in this linearized case the resistance is directly related with the velocity deficit behind the body. The velocity deficit is called the wake. Expressed as a fraction of the undisturbed velocity it is the wake fraction .
4.2
Flow along a Ship Hull.
The flow along the ship will remain attached when the hull is well designed. A boundary layer will develop from the bow to the stern. In the bow region
there is a favorable pressure gradient and the boundary layer will remain thin. A typical difference_ with e.g. a flat plate is that the flow, and thus the boundary layer, is not twodimensional. This gives rise to cross flow. In the stern region there is a strong adverse pressure gradient and the boundary layer will become thick. The boundary layer in the stern region will become so thick that its thickness is no longer small compared to the ship length or breadth. Some form of separation may occur there. In the following some threedimensional effects will be described and their effect on the resistance and the wake distribution will be described.
4.3
Cross Flow.
Consider a boundary layer on the ship hull. There is not only a pressure gradient in the flow direction, but also in a direction perpendicular to the 1Zie diktaat SWO I, pag. 25
January 4, 1994, Wake 65 flow. As a result the velocity vectors in the boundary layer will not remain in one plane, but will change direction towards the low pressure region when approaching the wall. This is shown in Fig. 4.2. 2
Figure 4.2: Velocity vectors in a 3D boundary layer
The streamlines outside the boundary layer will therefore have another direction than the streamlines at the wall. This fact is to be remembered when paint is used at the surface of a model hull to find the direction of the flow around the ship, e.g. for the application of fins or stabilizers.
VISCOUS REGION IN EXTERNAL STREAM
SURFACE OF SEPARATION (BUBBLE
LINE OF SEPARATION
IMITING STREAMLINES
SURFACE OF SOLID BODY
Figure 4.3: Transverse separation 2A11 figures in the remainder of this chapter: courtesy of M.Hoekstra
G.Kuiper, Resistance and Propulsion, January 4, 1994
66
4.4
Separation.
Separation in a three dimensional space (3D) occurs in two different manners.
The first is similar to 2D separation , where the flow velocity decreases until zero and becomes negative. This situation is shown in Fig. 4.3. The separation line runs in transverse direction relative to the local flow and downstream of the separation line a "dead water" region occurs. This type of separation is called twodimensional separation or transverse separation. On ship hulls this type of separation has to be avoided, because it increases the drag, just as in 2D flow on a cylinder. Regions where such unwanted separation can occur are specifically the regions in front of or above the working propeller, as shown in Fig. 4.4.
Figure 4.4: Transverse separation on a ships hull In 3D the flow lines can also converge because the body becomes smaller.
In that case the fluid moves away from the surface simply because of the law of continuity (Fig. 4.5). The flow lines in such a region will exhibit
January 4, 1994, wake 67
Figure 4.5: Thickening of stream tube in converging flow
a separation line in streamwise direction, as shown in Fig. 4.6. The flow SURFACE OF SEPARATION
VISCOUS REGION STREAMLINE IN EXTERNAL STREANI
Lai;gab:al:I. lII
LIMITING STREAMLINES SURFACE OF SOLID BODY
Figure 4.6: Streamwise separation
at the separation line has a component both in streamwise and in normal direction. The outgoing flow has the tendency to "rollup" into a vortex. The vorticity thus shed is lost energy and is felt as extra resistance. Such separation lines cannot be avoided on ship hulls and the design of a good ship hull is mainly the control of these separation lines, so that the wake behind the ship remains small. The control of the vortices is also important because it is a means to make the propeller inflow more uniform. Two examples of 3D separation on the bow of a ship are shown in Fig. 4.7
for two different ships. In the first case the separated vortex remains attached to the hull, in the second case the separation line rolls up and forms a vortex, which in this case is a bilge vortex. Sometimes more than one
G.Kuiper, Resistance and Propulsion, January 4, 1994
68
A
SEPARATION LINE ATTACHMENT LINE LIMITING STREAMLINES
SEPARATION LINE
Figure 4.7: Longitudinal separation at bow vortex with different signs are generated at various positions on the hull. Separation at the bow, as shown in Fig. 4.7, is suppressed by proper bow
design. Such vortices are, however, stronger and nearly inevitable in the afterbody region, because the adverse pressure gradient along the afterbody stimulates boundary layer growth and separation. Typical are the vortices originating from the bilge near the end of the parallel middle body, the (bilge vortices).
4.5
The Wake behind Simple Ship like Bodies.
Some simpler forms will be helpful for a good understanding of the wake generation. Consider a simple hull form with Ushaped frames , as in Fig. 4.8. The water will be pressed sideways in this case, causing higher flow velocities at the sides than under the keel. The pressure at the sides of the ship will be lower than at the bottom. The flow will go from bottom to side and a bilge vortex will form which rotates clockwise. This vortex shows up at the aft
January 4, 1994, Wake 69
cwL
5
2
3
4
BODY PLAN
1
CWL
,
,
.., /,
, , / ,
, , , , / / /
i //
/ / /
/ , / / , . / / / 1/4
,,/ It%
1
f bt
/ /
t\ '1/4
N. \
... _
'1/4
TUFT GRID AT A.P.
.... \
"" 1 k
,
\ ./\ 4
1/4 .....
.....

,...
..
I
s
...
/
Figure 4.8: Wake behind a Ushaped hull form perpendicular as shown. Another simplified extreme form is the Pramtype hull shown in Fig. 4.9. The water will be forced down and the lowest pressure occurs here at the bottom, so that a vortex with counterclockwise direction is generated. The vortex rolls up under its own induction and shows up in the near wake as shown.
In these simple cases separation may be suppressed considerably by com
bining Ushaped and Pramtype hull form, as shown in Fig. 4.10. In this case the amount of energy which is left in the wake wil be minimal. No separation occurs and the wake will be completely due to the velocity dis
tribution in the boundary layer along the hull. This body will have the lowest resistance because a minimum of energy has been left in the wake. The main component of the resistance will be frictional resistance. The form resistance will be small.
G.Kuiper, Resistance and Propulsion, January 4, 1994
70
CWL
BODY PLAN
5
CWL S
g
4
4
N.
/
\
\
\
I
i
k
/
I TUF T GRID
AT A. P.
/
/ /
I
/
/
Figure 4.9: Wake behind a pramtype hull form
The effect of 3D separation on the resistance is large. In Fig 4.11 the bilge radius is systematically reduced from model A to D, which means a reduction of the strength of the bilge vortex. The corresponding resistance curves show that the effect on resistance is considerable.
4.6
HorseShoe Vortices.
A different type of separation should be mentioned:horse shoe vortices. These are formed when a strut or fin attaches to a surface while both surfaces have a boundary layer. In this case also 3D separation takes place, but in a special way. A vortex developes around the front of the strut and because of its form this is called a horseshoe vortex. An example of it is given in Fig. 4.12. These vortices are also transported with the flow and, when arriving in the wake, will further complicate the wake structure.
January 4, 1994, Wake 71
CWL
ROOT PLAN
4
///rttt
CWL
///////// /
/
/ / /
/
I
/
I
/
I
1
i1
/ ///1/1 1111,11111 o
/ /
r
/
/
TUFT GRID
AT A.P.
1
Figure 4.10: Wake behind an optimized simple hull
4.7
Visualisation of the Flow around the Hull.
The flow around the hull can be visualised using paint tests. Paint is applied along frames of the hull, approximately perpendicular to the flow direction. During a run at a certain speed the paint forms streaks in the flow direction. Examples of such tests are shown in Figs 4.13 and 4.14 It should be kept
in mind that the direction of the paint streaks is the direction of the flow at the wall. When a strong crossflow is present in the boundary layer the direction of the streaks can be quite different from the flow direction outside
of the boundary layer. Paint tests are useful for the detection of regions of separation, because that occurs in the boundary layer. Paint tests are also used to determine e.g. the proper location of bilge keels, because in those regions the crossflow is generally low. It is very dangerous to determine e.g. the strut orientation in the afterbody with paint tests, because of the crossflow there.
In case of cross flow the use of tufts is better. The tufts are flexible
G.Kuiper, Resistance and Propulsion, January 4, 1994
72
RESISTANCE
A
MODEL A
EN Anil
t Inid0/ r
1"
_AEI
o
MODEL 8
..

MODEL C
SHIP SPEED
MODEL ID
Figure 4.11: Effect of variation of bilge vortex on resistance
wires, mounted on top of a needle perpendicular to the hull. The wire positions are photographed during a run with the model. The direction of the wires indicate the direction of the outer flow. It is even possible to mount wires at more than one position on the needle, so that the crossflow can be
visualised. The needles may disturb the flow by increasing the boundary layer thickness, but that risk cannot be avoided.
4.8
Ship Wake.
The wake behind a ship generally has a complicated structure because it is the result of the retardation of the flow in the ship's boundary layer and of many separated flows around the hull. The wake behind a ship is generally only measured in the propeller plane, which may be only a fraction of the
January 4, 1994, Wake 73
ip 1 SURFACE OF SEPARATION
AIPSEP.:RAT/°N
STREAMLINE
4/// 1A,
UNE OF ATTACHMENT
401,
LINES OF SEPARATION
SURFACE FLOVi PATTERN
Figure 4.12: Horseshoe vortex around a strut
Figure 4.13: Paint test on afterbody of a fast ship
total wake. The wake in the propeller plane without the propeller action is called the nominal wake.
74
G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 4.14: Paint test on the bow
Representation of the wake
4.8.1
The representation of the wake in the propeller plane is done by representing the axial,tangential and radial velocity components separately. An exam
....,
rill .1.039
illKI, ..011
0.672
o°
45
o
90°
135°
18d
POSITION ANGLE
Figure 4.15: Example of axial wake distribution
pie of the axial wake distribution as a simple diagram is given in Fig 4.15. The axial velocities are expressed as a fraction of the ship speed. A rather complicated wake peak is present in this figure in the top position of 180 degrees. The radial and tangential components of the wake can be plotted in a similar way.
Another way to plot both axial and tangential components of the wake
January 4, 1994, Wake 75
170
/so
160
190
200
100
80
o
Figure 4.16: Example of Plotting the Wake is shown in Fig. 4.16. The axial wake distribution is given as contourplots, in which the low speed regions reflect the core of the vortices in the wake. The tangential flow velocities are plotted as a vector diagram, in which the same vortical structures should be visible.
4.8.2
Relation between hull form and wake distribution
The relation between the shape of the hull and the wake structure is complicated. In Fig. 4.17 three variations of the same afterbody are shown. The wake structure in the propeller plane is shown in Fig. 4.18. In general a hull shape with small curvatures will shed few separated vortices and will generate a smoother wake. This is the case with a Vshaped hull form, which shape will have the lowest resistance. However, the nonuniformity of the flow in the propeller plane is large and a large portion of the wake passes outside the propeller plane, which decreases the total efficiency, as will be discussed later. A Ushaped hull form has more longitudinal separation and therefore has a more uniform propeller wake, since the boundary layer from the ship is rolledup into the propeller plane. This Ushaped hull form will have a higher resistance, but the interaction with the propeller may offset this, as will be discussed later. A further increase of the uniformity of the
76
G.Kuiper, Resistance and Propulsion, January 4, 1994 V
STERN
V  STERN SULSOUS STERN
Figure 4.17: Variations of the hull form
(25°7f 030
050
4,rzar. , :LPN
ti,e4C3464 atellga 714.111:1.41korlititi;
uotgo.0 \
V STERN
WOO1/4" BULBOUS STERN
Figure 4.18: Effect of hull form variation on the axial wake
axial wakefield can be obtained using a bulbous stern, where local separation lines from the bulb will rollup an make the wake more uniform.
The determination of the ship's resistance and wake is generally done by model experiments. Model tests do not always show clearly why some forms are better than others and calculation techniques become available nowadays to calculate certain aspects of the flow around the ship's hull.
0.20
January 4, 1994, Wake 77
Both experimental and calculation techniques are necessary to design an optimum hull form.
4.8.3
Wake_ Fraction
As discussed in chapter 3 the velocity deficit behind the ship is a measure for the resistance. It should be kept in mind that this is the case when the wake is measured over a such a region that the velocity at its boundaries is equal to the ship speed. This area is much larger than the area of the propeller plane, in which plane the wake data of a ship are defined. The velocity deficit in the propeller plane (without the propeller present)
can be integrated over the propeller plane. This results in an entrance velocity V, in the wake. Ve
= Ffir
127r fro
urdO clr
This velocity is the average entrance velocity in the propeller plane when the propeller is absent. When the propeller is absent this wake is called the nominal wake. It is made nondimensional with the ship speed V, as Wn =
Vs
(4.4) Vs
This is the definition of the nominal wake fraction and it is the nondimensional form of the velocity deficit V, V, in the propeller plane. The wake fraction determines the relation between the entrance velocity at the propeller and the ship speed by the relation = Vs(1
wri)
(4.5)
In Fig. 4.19 some axial wake distributions are shown for a number of stern shapes. The nominal wake fractions are also given.
The wake distribution is responsible for unsteadiness in the propeller loading during one revolution. In this course only the average wake will be used for the propeller inflow and unsteady effects will be neglected. An effect of the definition of the nominal wake over the propeller plane only instead of over the whole region of the velocity deficit behind the ship is that the nominal wake fraction does not necessarily correspond with the resistance. In case of a pram hull form, with a very flat afterbody, the velocity deficit corresponding with the resistance is distributed over the breadth of the hull and only a small fraction of the velocity deficit is found
78
G.Kuiper, Resistance and Propulsion, January 4, 1994
1/2
o
AoP
AoP
C L_
A
0.80 60 0.40
0,20
\ 010
080 0,10
070 0,50 0.30 0.10.
0,10
wn = 0,399
f.
8L
wn = 0,406 8
Figure 4.19: Examples of wake of various stern shapes
in the propeller plane. In that case a very small nominal wake fraction will be found, although the resistance may be high due to. e.g. strong bilge vortices which pass outside the propeller plane.
4.9
Design Considerations.
The frictional resistance is dominated by the wetted area of the ship. This cannot be influence very much by the shape of the hull. So the ship hull with minimum resistance will be the hull with the lowest form resistance. This can be obtained by avoiding threedimensional separation. In general this means minimum curvature of the frames. Vshaped frames will therefore
wa = 0,371 81..
January 4, 1994, Wake 79 have the lowest resistance. As mentioned the resistance of the hull is reflected in the velocity deficit in
the wake. When the velocity deficit behind the hull can be concentrated in the propeller plane, the propeller will accelerate the fluid again. This reduces the losses due to hull resistance. Uframes and bulb sterns are used to generate vortices in such a way that the velocity deficit behind the ship is rolled up in the propeller plane, although this increases the resistance.
Chapter 5 Wave Resistance Objective: Description of the wave system generated by a surface ship and determination of interference of the various wave systems.
When a ship moves through an undisturbed free surface it generates waves. These waves contain kinetic and potential energy which has to be genereated by the ship propulsion system. In terms of forces the waves result in a drag, which is called the wave resistance. To understand the character of wave resistance some knowledge of basic properties of surface waves is necessary. Unless mentioned otherwise it will be assumed that the water is deep.
5.1
Surface Waves.
IA' "I) riZEC r1C) N
V,,
c:2
C7E) 0 0 O 0 8 8
k l*
o c
Q
C'
l.
Q
CO
Q
o.
O 8 e
9
Figure 5.1: Motions of fluid particles in a wave In a surface wave the fluid particles describe orbital motions. As shown in Fig. 5.1 the radius of the orbital motion decreases with increasing depth. 80
January 4, 1994, Wave Resistance 81
When the wave height is small compared to the length of the wave and the water depth is large compared to the wave length these orbital motions are circles. The circles indicate the path of the fluid particles over time. The particles move in clockwise direction (angular velocity w)and the wave crest moves from left fo right (with a velocity viv). After a time T = 2r/w the particles have comi5leted a full circle and the wave crest has moved over one wavelength A. Inthe crest the 'particles move with the wave direction, in the trough the velocity is backwards. The average position of the particles over time remains unchanged.. The resulting wave height is a sine function and these waves are therefore called sinuoidalwa ves . The wave form is a balance between the gravity force and the centrifugal forces of the orbiting particles and these type of waves is therefore called gravity waves . These gravity waves have some specific properties.
5.2 5.2.1
Properties of Surface Waves. The Dispersion Relation.
At the free surface the combination of gravity forces and centrifugal forces is perpendicular to the surface. This means that a relation between the orbital
velocity w and the wave velocity vu, exists. This leads to the important dispersion relation (5.1)
An arbitrary wave in one direction can be decomposed into various sinusoidal wave components. These wave components have different wavelengths. The various components of the wave will travel with different velocities and after some time the various wave components are therefore found in different locations. This is the dispersive effect of the waves.
5.2.2
Energy in a Wave.
The energy in the wave consists of potential and kinetic energy. In the wave crest the potential energy is highest. Averaged over one period of the wave the energy density per unit area of a plane wave can be calculated as 1
Eu, = pgh,2 8
(5.2)
where h,, is the wave height from crest to trough. Note that the unit of E
is Nlm = NmIm2.
G.Kuiper, Resistance and Propulsion, January 4, 1994
82
5.2.3
The Group Velocity.
Consider a twodimensional wave of one frequency, generated at the end of a deep towing tank by a flaptype wave maker moving at a frequency w. in practice wave generators generate not only the oscillation frequency of the wave maker, but also higher frequencies. The wave velocity of the generated wave is defined as A/T = t. In combination with eq. 5.1 this gives w=
2r g A
and the wavelength follows directly from its frequency.. The wave velocity v is the velocity of the wave crests. However, at t seconds after the wave maker has started the front wave in the tank is not at a distance of t x from the wave maker, but only halfway that distance. This is because a wave front moves with half the velocity of the wave crest. This can ea.sily be observed at a wave front, where it seems as if the waves disappear when reaching the front. The velocity of the wave front is called the group velocity.
It is the velocity with which the wave energy is transported. As will be shown below this property is important for the wave system behind a ship.
5.3
The Kelvin Wave System.
The foregoing considerations can be used to describe the wave system around a ship hull. To simplify matters the ship hull is considered as one point, moving at a speed V. In Fig. 5.2 the ship is at position A at an arbitrary time t. A wave system at an arbitrary direction 9 with the course of the ship moves with the ship when the crest velocity is V/ cos O. After a time At the ship is in position B. The wave crest at an angle O is still emitted. However, the wave
generated at time to at A in not in position C, because it is a wave front which moves with the group velocity.. It is only in D, halfway of AC. This mechanism can be generalized, because the location of C for waves
in an arbitrary direction is a circle through ABC, drawn as circle I in Fig. 5.3. The location of the wave front in arbitrary direction after a time At is a circle with half this radius,drawn as circle II. The wave fronts radiated by the moving pressure point will therefore be restricted to a wedge tangent to this smaller circle. It is a matter of geometry to derive that the top angle of the wedge is 39 degrees. The waves with a crest tangent to the lj
January 4, 1994, Wave Resistance 83
A
Figure 5.2: Single wave generated by a pressure point
Figure 5.3: Boundary of radiated wave system
wedge, in point D, have an wave direction of 55 degrees. This leads to the wave pattern, known as the Kelvin wave pattern2 , as shown in Fig. 5.4. The waves, radiated in the direction of motion are waves with the longest wavelength. They form a transverse wave system behind 2After Lord Kelvin or Sir William Thomson, a British mathematician (1824 1907)
84
G.Kuiper, Resistance and Propulsion, January 4, 1994
TRANSVERSE WAVE CRESTS
19°28'
DIVERGING WAVE CRESTS
Figure 5.4: The Kelvin Wave System
the ship, approximately perpendicular to the direction of motion. At the same time a diverging wave system occurs, originating from waves radiated sideways, which have a lower crest speed (see Fig. 5.2) and thus a shorter wavelength. The crest of a diverging wave is hollow, as shown in Fig. 5.4. The frontal envelope of these diverging waves makes the distinct angle of 19.5 degrees with the path of the ship and the diverging waves near the boundary have a direction of 55 degrees.
5.3.1
The Froude Number.
When a body with length L1 generates a certain wave system, the same pattern can be generated by a geometrically similar body with length L21 when the ratio of the generated waves to the length of the body is the same. Since the radiated wavelength is proportional with V2/g, this means that the wave system of the two bodies is similar when V2 I(gL) is the same. The square of this ratio is written as
Fn =
V
\fir,
(5.3)
and is called the Froude number,, after R.E. Froude who has first used it. Two wave systems are similar when the Froude number is the same. This is important for model testing of ship hulls.
5.3.2
Resistance due to a Kelvin Wave System.
The wave resistance of a pressure point moving at speed V can be found from an integration of the wave energy passing through a control plane at
January 4, 1994, Wave Resistance 85
some distance behind the pressure point perpendicular to its path. This wave energy flux is equal to the resistance R times the velocity V. The resistance can be written in a relatively simple form as
R=
17{pV2 1712 h(9)2 cos' OdO 8
/r/2
(5.4)
From this formulation it can be seen that the waves with 9 close to 90 degrees, which are the shorter waves in the diverging wave system, contribute less to the resistance than the longer waves in the transverse system.
The Kelvin pressure point has no length. A body with a certain length scale however prefers to generate waves with a wavelength of its own length.
This length can be compared with the wavelength of the longest wave = 2T(/2)/g. When the body length L is small relative to the maximum wavelength the waves are primarily radiated in the direction with a large angle O. The ratio L/A is inversely proportional with the square of the Froude nuMber, so this occurs at high Froude numbers. Inversely when L/A is large the Froude number is low and the radiated waves tend to be dominated by the longer waves, of which the crest has a small angle with the path, the transverse waves. So at low Froude numbers the transverse wave system dominates, at high Froude numbers the divergent wave system dominates.
5.3.3
The Wave System of a Ship.
A ship hull can be considered as a system of pressure points, each generating their own Kelvin wave system. When the wave height is small the effects of the various pressure points can be superimposed. The total wave system of a ship has characteristics as in Fig. 5.5. Two major wave systems can be distinguished, one at the bow and one near the stern. The wave system
of the bow generates the wave profile along the hull. The dominance of the transverse wave system at low Froude numbers is illustrated in Fig. 5.6. The dominant diverging wave system at high Froude numbers is illustrated in Fig. 5.7.
From eq. 5.2 it follows that the wave energy per unit surface area is proportional with the square of the wave height h. The energy contained in surface waves over a distance x behind the ship will be proportional with xxbxh,v2, where b is the breadth of the wave system behind the ship. This is the energy necessary to overcome the wave resistance Ru, over the same distance x, so
R.x =
86
G.Kuiper, Resistance and Propulsion, January 4, 1994
Figure 5.5: Wave system of a ship
Figure 5.6: Transverse wave system along the hull
Assuming that b is proportional to the wave length A, the wave resistance is proportional to Alz,,,2 . Since A is proportional with the wave velocity, tit° (eq. 5.1), which is proportional to the ship speedV,, the wave resistance will be proportional with V32 x h. The wave height hu, is related with the pressures, which according to Bernoulli are proportional with V32. So ultimately
January 4, 1994, Wave Resistance 87
7
"":1*.
Figure 5.7: Wave pattern at high speed the wave resistance of the ship will be proportional with V96. This extremely high power is the reason that ships travel at relatively low speedg compared with e.g. cars. At high speeds the wave resistance becomes prohibitive.
The wave energy of the wave systems generated by a ship can be measured by measuring the contours of the waves passing through a control plane aside of and behind the ship. This method is called a wake scan . It requires a complete formulation of the radiated waves, however, to analyse a wake scan and to determine the wave resistance. The analysis is complicated and timeconsuming. The method is therefore not commonly used. The determination of the wave resistance of a ship hull by experiments will be described in chapter 7. The wave resistance is made nondimensional by CW
1/2pV2S
(5.5)
Here S is an arbitrary surface. The area S is taken as the wetted surface of the ship. Since the wave resistance varies approximately with V6 the wave resistance coefficient varies with V4 or, in nondimensional terms, with Frl.
G.Kuiper, Resistance and Propulsion, January 4, 1994
88
5.4
Wave Interference.
As can be seen in Fig. 5.5 the wave system of a ship is composed of different Kelvin wave systems. Grossly simplified the wave system of the ship can be
considered as a wave system at the bow and one at the stern. The transverse waves generated at these two locations will interfere. When the crest of the bow wave coincides with the crest of the stern wave, the two wave systems will reinforce each other. (The two wave systems are in phase). When the two wave systems are exactly out of phase and of equal strength the wave system behind the ship will disappear and the wave resistance will be zero. This illustrates that the interference of the different wave systems generated by the ship hull is important for the wave resistance. Since the wave length depends on the ship speed the wave height behind the ship varies with the speed. Consequently also the wave resistance of the ship as a function of the speed has a wavy character (oscillates), with "hollows" and "humps", as shown in Fig. 5.8. The humps in Fig. 5.8 seem rather 2oone
Hum!) .S
15000 ...
a.
.. ..7. Op
CC
a
10000
t I
Horn
!
Hollow
5000
Hollow o
I
10
15
20
Stn0 soeed ,r. knots
2
30
Figure 5.8: Wave Resistance of a Ship Model shallow because the wave resistance increases rapidly with increasing speed. The wave interference becomes more evident when the wave resistance is plotted nondimensionally as the wave resistance coefficient C versus the Froude number, as in Fig. 5.9.
January 4, 1994, Wave Resistance 89
MODEL DIMENSIONS O 10 ',471.8S' PRISMATIC COEFFICIENT 0.636 MODEL SYMMETRICAL ABOUT AMIDSHIPS
3
0IS
61.1
1{7

o
c.6 o
Et;
Figure 5.9: Example of Wave Resistance
5.4.1
A Twodimensional Simplified Hull Form.
The location of humps and hollows can be approximated using a simplified hull form, which exaggerates the wave systems generated by a normal ship hull. Such a body was used by Wigley (1930) for calculations of the wave system. The hull is twodimensional and consists of a parallel middle body and two similar wedges, as shown in Fig. 5.10. Four wave systems are present in an extreme form: the bow wave, the shoulder waves at foreand afterbody, and the stern wave. The transverse wave component of each system and its location relative to the hull is also shown in Fig. 5.10 . Note that the bow and stern waves begin with a crest, the shoulder waves begin with a trough. This is understandable when it is realized that a shoulder causes a low pressure region along the hull. The fluid particles tend to move away from the curved wall due to the centrifugal forces, causing a wave trough. On the other hand a bow has a stagnation point, where the pressures are high and a wave crest is formed. The superposition of the wave systems in Fig. 5.10 has a clear resemblance with the measured wave system, showing that the error due to superimposing is small.
G.Kuiper, Resistance and Propulsion, January 4, 1994
90
bow
for ward shoulder
after shoulder
stern
symmetrical surface disturbance
(primary wave Tysteni)
bow wave system
forward shoulder wave system after shoulder wave system
stern wave system X
total wave system N.
(measured)
total wave system(calculated)
'o'
Figure 5.10: Wave systems generated by a Wigley Hull
5.5
Economical Speed.
A merchant ship will be designed in such a way that it operates in a hollow of the resistance curve at the design speed. In practice the design problem is opposite, the desired economical speed is known and the ship has to be designed in such a way that it speed is in a hollow. Such a speed is called an economical speed . This means in the first place a proper choice of the ship length.
January 4, 1994, Wave Resistance 91
In the past the two dominant wave systems discerned were the bow wave and the aft shoulder wave. These two wave systems have an opposite sign because the bow _wave starts with a crest and the aft shoulder wave with a trough. Favorable interference occurs when the distance between both wave systems is equ al to k times the wave length A, where k is an integer. In that case a trough of the bow wave will be at the crest of the stern wave. The wave length of the transverse wave system can be found from eq. 5.1 to be A = 271V2/g. When the distance between both wave systems is called the wave making length L , it is required that
L(economical) =
k2RV2
(5.6)
The problem is, of course, to determine the wave making length, because especially the shoulder waves are not a single wave system but a combination
of many systems distributed over the length of the hull. From Fig. 5.10) the wave making_length can be assumed to be related with 2a + I . The prismatic coefficient Cp of a Wigley hull is VI. So the wave making length can be written as Lw = CpLwi
This relation was used by Baker and Kent in 1913 (!) [2]. They concluded from experiments that the wave making length could be approximated by Lw = CpLw/ +
A
Using the latter wave making length in eq. 5.6, the requirement for minimum wave resistance can be written as: V2
g Li(econornical)
2Cp
7(4k
1)
(5.7)
Note that the left hand of this equation is the square of the Froude number, as defined in eq. 5.3. The assumption that the bow and aft shoulder wave are dominant is true for old hull forms, where a cruiser stern often had a distinct aft shoulder. The modern hull shapes, especially those with a transom stern, have gentler curved waterlines in the afterbody and the bow and forward shoulder wave systems have become more important.
For slender ships the curve of sectional areas can be used as a measure
of the distribution of the wave systems. For full ships the shape of the
92
G.Kuiper, Resistance and Propulsion, January 4, 1994
waterline is more important. The entrance angle of the waterline has been considered as a measure for the strength of the bow wave system. However, at present the interference between the faired bulb and the forward shoulder makes it possible to use a higher entrance angle (4rid consequently a smoother forward shoulder) in combination with a proper bulbous bow design. No rules of thumb are applicable here. The approach from Baker and Kent as described above is more to gain qualitative understanding than a rule for hull design. Optimization is only possible by experiments or by Dawson calculations, as described in chapter 13.
5.6
Hull Speed.
When the wave length of the transverse waves behind the ship is equal to the ship length length a certain limit is reached. There is a wave crest at the bow and at the stern. Frofn eq 5.1 it is found that when A = L and vi = V, the Froude number is equal to 0.4. When the length becomes smaller or the speed higher the ship will trim considerably because it is at the rear side of its bow wave. The resistance increases dramatically in that case and for normal displacement ships the available power will not be enough to attain this condition. The Froude number at which this occurs is about 0.5 and the velocity at which this Froude number is attained is called the hull speed . Such a situation can be approached in case of overpowered yachts or tugs.(see Fig 5.11). The hull speed is the speed with the maximum wave resistance, as is also shown in Fig. 5.9.
5.6.1
High Speed Ships.
High speed ships such as planing vessels will have enough power to overcome the hull speed. For these ships the speed with the maximum wave resistance coefficient is also called the hump speed . At Froude numbers above 0.5 the wave resistance coefficient decreases again, as illustrated in Fig. 5.9. Above the hump speed the diverging waves dominate and these waves have a smaller contribution to the resistance, as follows from eq. 5.4. Also the
extreme trim decreases again. The resistance at the hump speed is often decisive for the power to be installed in high speed ships and for the propeller design. Although at high Froude numbers the diverging waves dominate, the transverse wave system is still there and the angle of the diverging waves still have an angle of 19.5 degrees with the ship, as shown in Fig. 5.7. This is contrary to the situation at restricted depth, as will be discussed later.
January 4, 1994, Wave Resistance 93
atitiNPA "zz
4i"t4t,
4
Figure 5.11: Wave system close to the hull speed
5.7
Bulbous Bows.
The application of a bulbous bow is equivalent to the generation of a separate wave system. If the bulbous bow is at the correct position relative
to the bow it will generate a through at the location of the bow, which partially cancels the bow wave. Of course this is a conceptual simplification because the wave systems are not single systems and the wave height is considerable, so they cannot simply be added. Since a favorable interference of two wave systems occurs only at one speed, the bulb is best for only a limited speed range. At other speeds the interference effects can even be opposite and an increase in wave resistance may occur.
5.8
Shallow Water Depth.
With increasing depth the wave height decreases rapidly. For a sinusoidal wave with small amplitude as described above the decrease is h(y) = hwe21rYIA
(5.8)
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G.Kuiper, Resistance and Propulsion, January .4, 1994
When y/A = 0.3 the reduction factor of the wave height is already 0.15. The energy of the waves at this depth is, according to eq. 5.2, only 2.5 percent of the wave at the surface. For practical purposes there is deep water when the water depth is greater than one third to one half of the wavelength, so h/A 104 and until the boundary layer becomes tur
bulent at R.
105
Turbulence is the occurrence of random motions additional to the mean velocity. So the instantaneous velocity of a fluid particle can be described by the mean velocity and a turbulent component: u = T./ ui with 17 = = i7 + y' with /)7 =
p=
p' with /7 =
To illustrate the effect of turbulence the NavierStokes equation in xdirection (eq. 11.1) is considered:
170
G.Kuiper, Resistance and Propulsion, January 4, 1994
au
au
a2u
ap
82u
tW)
P(ti
Substituting the instantaneous velocity in this equation and taking the time average results in a new formulation, the term p can be written as = P + P' = The latter is true because the timeaverage of the turbulent pressure component is zero. So this pressure term is not changed by turbulence. This is true for all right hand terms, but not for the left hand terms. E.g. the term uti can be written as
uau1 a2u ax
(11.9)
23x2 1 a(ri2 + 2
u'2)
ax
,22
,
2 ax2
2 ax2
The turbulent equation of motion in xdirection becomes in this way:
au ax
52,
or,
au
'17) = ay
ax + '1(ax2 ay2 ) 1 .9%72 au'vi 2.p(8 + ay )
The extra terms require further attention, because they describe the effect of turbulence. The term asat will be small relative to the derivative aui ay vi in y direction, because the variations of the velocities normal to the wall (in ydirection) are much larger than those in flow direction. To assess the effects of turbulence only the second termat4L'''is sufficient. Similarly ay
the termaS 2will be small relative to
ay2
These term can therefore also be
neglected.
When it is assumed that the term 1pu'v' can be written as
it possible to rewrite eq. 11.11 as:
6,,
p(riax+ T)) = ay
ar,
a2t,
+ IL  AT ay2 ax ay2
January 4, 1994, Potential Flow 171
When written in this form is is clear that the turbulence can be taken as an additional viscosity, also called the eddy viscosity. This is of course due to the assumption of eq. 11.11. This assumption is called the turbulence model, which in this case relates the turbulent viscosity in a linear fashion with the velocity gradient in the boundary layer. Other turbulent models have been formulated. No final turbulence model that can describe the phenomena in real flow has been formulated yet. .
The eddy viscosity is generally much greater than the dynamic viscosity,
so A, is greater than ft The determination of A is therefore important. A wellknown model is that of Prandtl, who assumed that A, = p12
where l is the average distance of the turbulent motion in the boundary layer. The model is therefore called the mixing length model. The problem is of course to determine this length. In practice there is not such a single length scale, but this model gives a conceptual physical background to the turbulence modelof eq. 11.11.
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G.Kuiper, Resistance and Propulsion, January 4, 1994
Chapter 12 Flow Calculations without Waves Objective: Review of the use of calculations for resistance and flow. No extensive mathematical formulations are used, the purpose is to be able to use the available programs intelligently.
A complete description of the flow around an arbitrary body is given by the NavierStokes equations, provided that at high Reynolds numbers a solution can be found for the formulation of the eddy viscosity, the ficticious viscosity which is caused by the turbulence in the flow. The flow around a ship hull can be found in priciple by applying the no slip condition at the wall and by applying the appropriate boundary conditions at the free surface. The free surface conditions are twofold. The first is the kinematical free surface condition : the fluid particles at the free surface have to remain at that surface. The second is the dynamical free surface condition : the pressure at the free surface is always equal to the atmospheric pressure. In principle this problem is well formulated and using a volume discretization of the fluid it can be solved.
The solution of this problem is still not really feasible. One reason is that the viscous phenomena require a very fine grid because the scale at which energy is dissipated is very small. On the other hand the scale of the waves at the free surface is large. In principle a large flow region with a very fine volume grid has to be used for the solution and this is still beyond the present computing capacity. Another reason is that the free surface is a part of the solution. It is not known beforehand where the free surface boundary condition has to be applied.
173
174
G.Kuiper, Resistance and Propulsion, January 4, 1994
Programs using either volume discretizations or surface panelling therefore contain essential simplifications. Only when the implications of these simplifications are understood a proper use can be made of Computational Fluid Dynamics (CFD).
The first and basic simplification is that the regions of viscous flow and the regions of potential flow with a free surface are separated. This is an assumption comparable with the Froude hypothesis, which separates viscous and residuary resistance.
By far the most important calculation methods are those which calculate the inviscid outer flow. The presence of a thin boundary layer makes it possible to regard the flow outside the boundary layer as a potential flow and the pressure at the wall is equal to the pressure at the outside of the thin boundary layer. In that case the flow around a ship hull can be regarded as a potential flow, because the viscosity can be neglected and as a result no rotation is generated. The uniform inflow in front of the ship is also irrotational, so the flow aro. und the ship can be described by the Laplace equation. The boundary conditions are the free surface conditions and tangential flow along the outside of the boundary layer. Since the boundary layer is considered thin this can be approximated by tangential flow along the hull. The viscous region can in principle be calculated using boundary layer equations, because the boundary is considered to be thin. Efforts have been made to use twodimensional boundary layer equations along streamThe boundary layer is considered twodimensional when no cross flow perpendicular to the streamline occurs. The streamlines and the pressure distribution along the streamlines have to be found from the potential calculations of the outer flow. The results of such calculations give a distribution of local friction coefficients along the streamline. Integration of the longitudinal component over the hull gives the lines.
resist ance.
These calculation methods have not been very successful. On an average ship hull the effects of crossflow and separation are too important to be ignored, and the presence of a thick boundary layer in the stern region has too much influence on the resistance. crossflow are too important to be ignored. Threedimensional boundary layer calculations are complicated and they blow up in regions of separation, where vortices leave the hull. In those regions the basic assumptions of boundary layer flow are violated. Until now no reliable calculation methods are available to calculate the resistance. Efforts to do this are made using the full NavierStokes solutions, as will be discussed later in this chapter.
G.Kuiper(MT512) January 4, 1994, Flow Calculations I page: 175
The separation of the viscous and the inviscid regions neglects the interaction between the viscous flow region and the potential flow region. The main region where problems occur is the wake, where the flow is highly rotational and viscosity cannot be neglected. Still it is a part of the potential flow region. But also elsewhere on the hull where any type of separation occurs, such as at the bilges in the forebody (see Fig. 4.7) the assumptions are violated. Also when the boundary layer becomes very thick, as in some regions in the afterbody, the solution of the potential theory will be inaccurate. Keeping in mind these restictions the potential flow calculations can be used intelligently to optimize hull forms or to calculate flow patterns which are difficult to measure at model scale.
12.1
Potential Flow Calculations.
12.1.1
Panel Methods without Free Surface.
The simplest category of calculations of the flow around the ships hull are the potential flow calculations without a free surface. The undisturbed free surface is assumed to be unchanged by the flow around the hull. In practice this is realised by mirroring the ship hull at the undisturbed free surface. This socalled double hull is used for flow calculations in an unbounded fluid. Because the undisturbed waterline is a plane of symmetry, the velocities normal to that plane will always be zero. A drawback is, of course, that twice the amount of panels have to be used to cover the whole double body.
When no lift is present the singularity distribution used is a source distribution. The sources are distributed on the discretized hull surface. The elements of the hull surface are called panels. As boundary conditions the condition of tangential flow at the ships hull is used. This condition is applied at the center of the panels, the socalled control points. The simplest panel shape is a quadrilateral flat plane , on which a uniform distribution of sources with constant strength is placed. This implies that the distribution of the sources is a stepfunction instead of a continuous distribution. This approach, first applied by Hess and Smith [11], approximated the shape of a ships double hull with a large number ( N) of such flat panels. On each panel a uniform source/sink distribution is placed with an unknown source strength Q. The potential function (I)(x, y, z) can then be written as the sum of the potential functions of all elements. The velocity at an arbitrary control point can be expressed by influence functions of the panels. The influence function of the own panel is singular, but the
176
G.Kuiper, Resistance and Propulsion, January 4, 1994
singularity can be integrated over the panel.So after some mathematics the velocity in an arbitrary control point can be expressed as a linear function of the N unknown source strengths Q. Next the boundary condition of tangential flow is applied at each control point. This results in N boundary conditions for the N panels. This system can be solved, resulting in the strength Qi at every panel. The velocities at every position at the ships hull and around the ship can then be found from the derivative of the potential function (I)(x, y, z) in the desired direction. Note that the panel size limits the accuracy of the derivative, because the derivative has to be determined from the difference between two panels. The pressures can be derived from the velocities using Bernoulli's law.
It is important to realize the consequences of the simplifications made. First it is potential flow, so the fluid is inviscid. That means that necessarily the resistance is zero (Paradox of d'Alembert) and that flow separation does not occur. Just as in the case of a cylinder the pressure recovery in the afterbody is complete. The water surface is also undisturbed, so the wave resistance is also zero. How can the results of such calculations be used? They can be used to assess qualitatively the relative merit of various alternatives and therefore to optimize the hull before model tests or further calculations are carried out. They can also be used to indicate improvements, because the calculations provide data such as pressure and velocity distributions, which are not measured. Examples are given below.
12.1.2
Assessment of Various Bulb Designs.
In Fig. 12.1 three forebodies are shown of a full bulk carrier. [18] For these designs panel calculations were carried out, The results are shown in Fig. 12.2. The dotted lines in this Figure are lines of equal pressure coefficient. Since the pressure coefficient Cp is defined as 1P/2,2, a low (highly negative) value of Cp means a low pressure on the hull. A bulbous bow is used to minimize wave resistance, but excessive flow separation has to be avoided too, because that increases the residuary resistance again. The risk of flow separation is largest when the pressure gradients are large. In this case the pressure gradients are largest when the minimum pressure is low. From the calculations this minimum pressure occurs at frame 19 near the bottom. The location does not vary significantly between the alternatives. The forebody with the smallest risk of flow separation is the one with the smallest perturbation of the flow, so the one with the highest minimum pressure coefficient. Ship P1 has a minimum pressure
G.Kuiper(MT512) January 4, 1994, Flow Cakulations I page: 177
SHIP
P1
SHIP P2
Figure 12.1: Two bulb configurations for a bulk carrier coefficient of 0.35, which is lower than that of ship P2. This indicates that ship P1 has a potentially higher residuary resistance.
The results from model tests are shown in Tabel 12.3. Ship P1 has indeed a significantly higher power requirement. So although the resistance is not calculated, the relative merit of these designs could be estimated from a comparison of the pressure gradient in the forebody. The assessment of the pressure distribution from potential calculations requires considerable experience and comparison with experimental data. The most important application of this type of calculations is in combination with experimental data, instead of as a replacement of experiments. Suppose that ship P2 was the initial design. After model experiments is would have been found to have a high resistance compared to e.g. statistical data or to other similar ships. The experimental results give no indication of the cause, nor any indication about the remedy. From the pressure calculations
178
G.Kuiper, Resistance and Propulsion, January 4, 1994
SHIP
P1
SHIP P2 Figure 12.2: Pressure distributions on two different forebodies
it can be concluded that the fullness of the lowest waterlines should be reduced, because the pressure gradient there is very steep. The pressure distribution on an alternative design can also be calculated, to see if the change has the desired impact, such as in ship Pl. When this is not the case further improvements can be made. After such an optimisation the new design can be tested again experimentally to determine quantitatively the effects on the resistance.
12.1.3
Knuckles and Bulge Keels.
G.Kuiper(MT512) January 4, 1994, Flow Calculations I page: 179
Speed
Ship P1
Ship P2
PD
PD
in knots 12
100
13
100
14
100
15
100
16
100
17
100
84.6 86.5 88.3 85.8 85.7 90.3
Figure 12.3: Speedpower relations of three forebody configurations
The velocities along the hull can be calculated both in magnitude and in direction. This can be plotted as a vetor diagram. This simulates a "tuft test" in a towing tank, where small wires are observed to find the flow direction and especially to find regions of flow separation. The calculations will never give flow separation, but can only indicate the risk of separation. On the other hand, the calculations give the magnitude of the velocities, and thus the pressure distribution, which is not found from a tuft test.
In Fig. 12.4 an example is given of the calculated flow direction on a hull with a knuckle. The calculations indicate that the flow along the upper knuckle is tangential to the knuckle, but flow separation can be expected at the lower two knuckles. These results can again lead to a change in the design before model tests are carried out. Similarly the calculation of flow lines along the hull makes it possible to make an educated guess about the correct position of bilge keels or stabilisers. It should be kept in mind that these calculations are applicable when the influence of waves is small and when the boundary layer remains thin.
12.1.4
Assessment of the Afterbody.
The boundary layer in the forebody is relatively thin, and the calculated velocities and pressures in the forebody are generally accurate, provided no separation takes place. This is more questionable in the afterbody. In the
G.Kuiper, Resistance and Propulsion, January 4, 1994
180
./1
.
,
// \\
' >,
/
I
co
o o
Figure 12.4: Calculated flow lines along a barge afterbody the neglect of viscosity is more drastic and the results of potential calculations are very much qualitative and should be treated with greater
care. As an example of an assessment of an afterbody configuration the optimisation of a shaft bossing is shown. Fig. 12.5 shows two alternative designs of a bossing. The question is at what angle the bossing should be placed to have minimal interference with the flow. The dotted lines in this Figure are lines of equal pressure coefficients on the hull. The differences are small, so the position of the bossing is not important for the flow over the hull. The lines of equal pressure coefficient on the inside and outside of the bossing, as given in Fig. 12.6, give a better indication to distinguish between the alternatives. The pres
sure coefficient at the inner side of the bossing of ship J is much higher than that on the outer side, indicating that the bossing should be directed outwards. The pressures on both sides of the bossing of ship L are more in equilibrium, and also the pressure gradients are smaller. The calculations indicate that the bossings on ship L are the most favourable. The required power for the three configurations were measured at model scale. The relative merit from the model tests is given in Table 12.7. In this
G.Kuiper(MT512) January 4, 1994, Flow Calculations I page: 181
0=AP. NO INCLINATION
SHIP
.1
0= AP.
SKEG INCLINATION INWARD OVER THE TOP
SHIP
L
Figure 12.5: Two alternative positions of a shaft bossing table the power PE is the power required to tow the ship without propulsor, as discussed in chapter 7.The power PD is the delivered power in the selfpropelled full scale situation, as will be discussed later. Ship L has indeed the lowest power requirement at all speeds. The relative merit as deduced from the panel calculations was therefore confirmed by model tests. The interpretation of such calculation results requires caution and experience, however.
If the bossings are in the proper position can be found from the wake . In Fig 12.8 the measured axial nominal wakes are shown. These results show that the flow at the outer side of the bossing of ship J showed field.
separation or at least a very thick boundary layer. The wake of ship L
182
G.Kuiper, Resistance and Propulsion, January .4, 1994
_ACY21.
C.
0.07?.......z.yVG0 040
0.020
...040
OUTIEI
0 040
0060 00
.060 11020
.020
A.?.
SHIP J
'Loot
A P.
SHIP L Figure 12.6: Pressure distribution on bossings of two containerships
G.Kuiper(MT512) January 4, 1994, Flow Calculations I page: 183
Ship SPeed
in knots
PE PD int./.
PE
PD inw*
20
100
100
92.8
90.6
21
100
100
93.1
91.2
22
100
100
93.2
91.8
23
100
100
93.4
91.3
Figure 12.7: Speedpower relation for two afterbodies of a containership SHIP J ... , ....

10
..
....
. s ...
\
.
1 12
/
r ....
t nt
/
/ /
..'....."'''
V
03
1
... ..../ i
Ct TA
a 56
CLAAtk
r/p  n 14
44 c.k. 00
o
iao
SHIP L
e
,
,
12
10
,
\
0.5
,
%
.
.....,.....;,...
ini
o
360
270
f
.....0
n 7ft Cn.AR14.4_
0 '515.
.4.: C(4
72 a i t
o o
90
160
270
360
Figure 12.8: Measured axial wake distributions of containership
184
G.Kuiper, Resistance and Propulsion, January 4, 1994
confirmed that the bossing was approximately aligned with the flow.
12.2
NavierStokes Solutions.
The most direct solution of the flow problem is of course a full solution of the NavierStokes equations. This is still too difficult and some simplifications have to be made. Hoekstra simplified the numerical scheme for the calculation of the N.S.equations in such a way that he could solve the N.S. equations numerically going from stem to stern. (Parabolisation of the equation). In this way effects near the stern, such as separation, have no effect upstream, which they may have in real flow. This can in principle be solved iteratively in Hoekstra's method, but the time(and thus cost)factor often prohibits this. Hoekstra also neglects the waves and uses a double model for his calculations. The solution of Hoekstra is viscous and in that case the technique of surface distributions is n.o longer applicable. The flow has to be divided into threedimensional elements troughout the flow, which of course increases the calculating costs considerably. These types of calculations are typically suited for supercomputers. The solution of Hoeksta allows flow separation, because the flow is viscous. The rollup of the separated flow can be calculated and this allows the calculation of the nominal wake distribution in the propeller plane. As an example the flow along the ship given in Fig. 12.9 is given at various stations upstream of the propeller plane.(The propeller plane is at 2x/L=1.0, where L is the ships length)
WOE ..WILSONE
L AMA L.N.._11Wil44mail
,1 \6,._'""'% "IilIAMPANYMIN
%_
Ftia6
rlifMill ArAll 11611/11M
.irmiwza) ..=1 /AM, al...
1
Figure 12.9: Hullform for calculation of viscous flow
G.Kuiper(MT512) January 4, 1994, Flow Calculations I page: 185
The interesting region for the calculation is the sharply convex region above the stern bulb. The boundary layer from the forebody becomes very thick in that region, so that a real boundary lauer does no longer exist. The relatively thin viscous region at 2x/L=0.7 is shown in Fig. 12.10. The contours of equal axial velocity indicate the axial velocity as a fraction of the ship speed. The vector diagram indicates the transverse velocity at the crosssection.
1
J
tiffi0
Figure 12.10: Calculated flow pattern at 2x/L=0.7
At 2x/L=0.9 the viscous region is considerably thicker, as shown in Fig. 12.11
The rollup of the separated flow can be seen on the vectordiagram of the transverse flow. This rollup is completed in the propeller plane, as shown in Fig. 12.12. In the contours of axial velocity this results in the strongly curved contours in the upper part of the propeller plane. The effect of the propeller can be incorporated in these calculations when the propeller is represented as an "actuator disk", that is a simple uniformly distributed force over the propeller disk.
G.Kuiper, Resistance and Propulsion, January 4, 1994
186
Figure 12.11: Calculated flow pattern at 2x/L=0.9
_ *
t'::::".
\ \\,,\ %\\
iiI\v\x\O
t
\\\\\N
...
.\\
,""
\
Ill
iiii
,
\ \\\
, \\\\ \\\ \ \\\ \\\ , , /I 1111\ \\ pil ,1110 \\\ \ , \\N
.1,J1
\
: '
\\ \\\
I
%%01\
\
\\
I
'
\\III\ 1
I
i
iitttttit\\\\\\\\ t
Figure 12.12: Calculated nominal wake
Chapter 13 Flow Calculations with a Free Surface Objective: Review of the use of calculations for wave resistance.
When a free surface exists the free surface conditions have to be satified. To illustrate the formulation of such conditions and to illustrate the linearization of the problem the linearized free surface conditions will be derived in twodimensional form.
13.1
The Linearized Free Surface Condition.
Consider a simple twodimensional wave system which is stationary behind a ship hull which moves at a velocity Vs, as shown in Fig. 13.1. Since this is a stationary situation (in a ship mounted system of axes) the kinematic boundary condition is that the fluid velocity at the surface is tangential to the surface:
077,v, vV!,as
A potential flow is assumed. The potential is taken as the potential of the undisturbed inflow V,x and a perturbation potential 0, so that = Vsx +
The velocities vr and vy can then be expressed as
v, =
ao
+ a ;
187
G.Kuiper, Resistance and Propulsion, January 4, 1994
188
Y
Vs
Figure 13.1: Kinematic Free Surface Boundary Condition VY =
ao
The kinematic boundary condition in terms of the potential 0 becomes
017 Vay ax
(1/3
pl)
This condition can be linearized when the perturbation velocities are assumed to be small relative to the undisturbed velocity Vs. That means that the last term can be written as (V8
o (9:c
ao )
= V3(1 F
) = V3(1 F 0( epsilon))
where e is small. Note that also the g is of order e. Neglecting terms of order 62 means that the linearized kinematic free surface boundary condition becomes
1 ao ax = V3 ay
(13.1)
This means that in the linearized kinematic boundary condition the horizontal perturbation motions are neglected.
The dynamic boundary condition at the free surface is the requirement that the pressure at the surface is equal to the atmospheric pressure. This can be formulated using the Bernoulli equation
G.Kuiper(MT512) January 4, 1994, Flow Calculations II page: 189
0(1.2
1
(34:12
Pa = P 75. ) PgY where pa is the atmospheric pressure. At the surface this reduces to
&V
1
/9(
(91,2
Pg71 = °
)

Written in terms of the perturbation potential 0 this is 80 2 802 2P{(u 1+ P9.17 = ° Both a. and ay are of the order 6. Neglecting terms of order 62 gives the linearized dynamical boundary condition 1
1
,
g
2 's
(13.2)
ax Elimination of the wave height i from both the kinematic and the dynamical boundarycondition makes it possible to formulate the free boundary condition without the (unknown) waveheight 77. Differentiation of eq. 13.2 to x gives a77
1
=
a2 q5
ax g 3ôx2 In combination with eq. 13.3 this results in vs2 520
50
g ax2
ay The variables can be made nondimensional by a length L. So
Y=L ,
x= V' =
x
L V
L
Omitting the primes gives the nondimensional free linearized free surface condition vs2 520
50
gL ax2+ Oy = °
(13.3)
The coefficient gL 11 is the square of the Froude number. As expected the Froude number enters the free boundary conditions.
190
G.Kuiper, Resistance and Propulsion, January 4, 1994
The linearized free surface condition eq. 13.3, which is also called the Kelvin condition 1 does not contain the wave height 77. It can therefore be applied before the wave height is known. This is a consequence of the linearization. However, it still has to be applied at the unknown wave surface. It can be shown, however, by expanding the wave height into a Taylor series in 77 that the boundary condition can also be applied at y = 0. The error is than of 0(62). This makes it possible to eliminate the wave height completely from the formulation of the problem.
13.2
Kelvin Sources.
The sources mentioned until now were solutions of the Laplace equation, with the property that their influence disappeared at infinity. These type of sources are also called "Rankine sources" , because it is possible to formulate other types of solutions of the Laplace equations. For example a basic solution exists which satisfies both the Laplace equation and the linearized free surface condition. The properties of this solution are similar to a regular source: fluid is issuing from the source in all directions; but the formulation is much more complex. It will not be given here. These sources are called "Kelvin sources" of "Havelock sources". Using a distribution of these sources instead of regular sources the free surface problem can be solved by applying the hull boundary condition. By using Kelvin sources the free surface condition is than satisfied implicitly. When the potential has been calculated, the wave height can be found from the linearized dynamical boundary condition eq. 13.2.
13.3
Applications of the Kelvin Sources.
13.3.1
The Michell Theory.
A classical application of Kelvin sources was introduced by Michell (1898). The undisturbed flow is the uniform inflow with the undisturbed free surface. The disturbances due to the presences of the ship are described by Kelvin sources. The basic assumption is the assumption that the breadth
of the ship is small relative to the length and to the draft. The approach is therefore called the thin ship theory. The sources can consequently be positioned at the ship centerline. The power of this approach is that the problem can be solved analytically, 1Lord Kelvin or Sir William Thomson (18241907) was a British mathematical and physicist who formulated the properties of vortices in potential flow
G.Kuiper(MT512) January 4, 1994, Flow Calculations H page: 191 which was a prerequisite in the precomputer era. The solution of the problem had the form of eq. 13.4
AM= fi(0)
ab(x'Y) f2(x,y,O)dxdy
(13.4)
A is the wave amplitUde. The angle O is the wave direction. (For coordinates
see chapter 5). b(x, y) is the half beam of the hull, the x,y plane being the centerplane of the hull. This integral makes it possible to calculate the radiated wave height in all directions. Since the radiated waves moves
with the ship the wave velocity is found from V, = Vslcos(0) and the corresponding wave length is found from the dispersion relation eq. 5.1. So
using eq. 13.4 the wave energy radiated in an arbitrary direction can be found (see eq. 5.2. The integral of the radiated energy over the full circle is the wave resistance of the ship. The result is Michell's integral for the wave resistance [33] 2
sec 0 [.1 D = 2L4 rvs2 f2ir 0
ax
exp( "g± sec' 0)(y Vs2
ix cos 0) dxdy] de (13.5)
This integral is not easy to solve analytically. A numerical approach is better suited nowadays. In that case panels with a constant Kelvin source strength are placed at the ship centerline. The boundary condition of tangential flow can be reduced for a thin ship to
ax ab=
vY
V,
where y is in the transverse direction and x is in the longitudinal direction of the ship. b is half the breadth of the local waterline. In the case of a thin ship the vertical velocities along the hull are thus neglected. Application of the boundary condition in the control points of the panels results in a set of equations. The solution gives the strength of the Kelvin sources at each panel. When the strength of the Kelvin sources is known, the potential at each location can be calculated.Differentiation with respect to x and y gives the local velocities. Since it is a potential flow the Bernoulli equation can be used to translate the velocities into pressures. Integration of the longitudinal components of these pressures over the hull gives the wave resistance. At the undisturbed surface z = 0 (with z as the vertical axis) the linearized dynamical free surface condition eq. 13.2 can again be used to determine the wave height. In practice only the wave height at the centerplane is calculated, which gives the wave contour along the hull.
G.Kuiper, Resistance and Propulsion, January 4, 1994
192
This calculation is carried out with the model in a fixed immersion, the position without speed. Integration of the pressures in vertical direction gives a force distribution over the hull in vertical direction. This force distribution can be used to calculate trim and rise. This method is very approximate. Further iterations, with a corrected wave resistance calculation in a trimmed position, did not increase the accuracy of the predictions, however. An example of a calculation of the wave resistance of a hull with LI B = 12 and BIT = 2.5 is given in Fig. 13.2. A special element is the effect of a transom stern, which is quite common. In that case the hull is modelled as an open stern. The breadth at the stern has a finite value. The pressure over the transom stern is taken as the atmospheric pressure. The resistance"due to this transom stern is responsible for about one third of the residual resistance coefficient in Fig. 13.2. 3.00 
8 o ,tp
2.50 e
..
e
2.00 
o
...,
. . . MIL
0
vi 1.50 ti) CC
1.00 
Measured
471

cc 0.50
Calculated
0.00
0.15
0.2
0.3
0.35
0.4
0.45
0.5
0.6
0.7
0.8
Fraude number
Figure 13.2: Wave Resistance of a Slender Hull Calculated with Thin Ship Theory
This picture shows a general trend in the results of thin ship theory: overestimation of the interference effects of the waves. The humps and hollows are more
pronounced than found experimentally. The position of the hump is properly predicted. The result can also be very sensitive to the number of panels and their distribution over the centerplane. The calculation results deteriorate rapidly with increasing BIT ratio. The calculated trim and rise shows considerable discrepancies with experiments.
0.9
1
G.Kuiper(MT512) January 4, 1994, Flow Calculations II page: 193
13.4
Kelvin Sources for Catamaran Hulls, an Example.
An application in which the thin ship approach is more suitable is in case of catamarans, where the breadth of each hull is indeed small relative to the length. ( The ipproach is used in Marin's program CATRES.) In that case the Kelvin sources are positioned at the centerplanes of each hull, as shown in Fig. 13.3.
Figure 13.3: Position of Sources on a Catamaran
The fact that only sources are used is a simplification, because it assumes that the flow around each hull is symmetrical. However, the induced velocity of one hull at the location of the other induces also an asymmetrical flow around the other hull, which can only be described by dipoles
or vortices on the centerplanes. The result is a sideforce on the hull and a corresponding induced resistance due to trailing vorticity. This effect is neglected.
The strength of the Kelvin sources at the centerplane is calculated using the boundary condition at both hulls. This implies that hull interference is
194
G.Kuiper, Resistance and Propulsion, January 4, 1994
taken into account, because the velocities induced by both hulls are taken into account in the calculation of the velocity in the control points. The resistance is calculated by integration of the pressures in longitudinal direction. Since the problem is symmetrical integration over one hull is sufficient. The wave resistance is found by doubling the wave resistance of one hull.
An example of the calculated residual resistance using this approach is given in Fig. 13.4 The LIB ratio of each hull was 8.64, the BIT ratio was 12.00 co, 10.00
Q
8.00 a)
8 4.00 
Measured
:0 cc
2.00 
 Calculated
0.00 0.4
0.45
0.5
0.6
0.7
0.8
0.9
Froude Number
Figure 13.4: Residual Resistance of a Catamaran
1.85. The hull spacing was 32 percent of the length. A good indication of the residual resistance coefficient can be found in this way, especially below Fri = 0.5. Above that speed the effects of sprayrails and trim wedges can cause serious errors in the prediction.
13.5
Dawson's Method.
The "Thin Ship Theory" considers the ship hull as a perturbation of the undisturbed water. The linearized free surface condition does the same with the waves, it is assumed that the waves are small disturbances to the smooth water surface. These are severe restrictions for practical applications. Dawson (1977) designed a practical method to overcome these restrictions. He used the "Hess and Smith" solution of the double body without free surface
1
G.Kuiper(MT512) January 4, 1994, Flow Calculations II page: 195
as the base flow. The source strength of the panels of the double body are thus used as a base potential. The difference between this double body potential and the potential around the hull including a free surface is then assumed to be small. This difference is thus used as a perturbation of the double body potential. This makes it possible to redefine the free surface conditions. When the base potential as found from the double hull is used for the half body (the ship hull until the waterline without its image) the vertical velocities at the undisturbed free surface are no longer zero, because they are no longer canceled by the velocities induced by the image hull. Using the dynamical boundary condition eq. 13.2 these velocities can be translated into a wave height. This is the "double body waveheight", on which the perturbations are superimposed. In principle the free surface boundary conditions should now be applied at the wavy surface generated from the base potential. Dawson formulated a free surface condition in terms of the double body potential and the perturbation potential, which could be applied at the undisturbed free surface. 2. Having defined the free surface condition, a panel distribution over an area around the ship at the free surface can be formed. At these free surface panels a uniform distribution of regular Rankine sources is placed. Care should be taken of the radiation of wave energy at the boundaries of this area. In combination with the panels of the hull the total region at which boundary conditions should be applied is covered. At the free surface the
panels have a source strength equal to the perturbation potential, at the hull the panels have a source strength equal to the perturbation potential plus the (known) double hull potential. At the free surface the newly formulated boundary condition is applied, at the hull the condition of tangential flow is applied. This set of equations can be solved.
This technique circumvents the restrictions of the thin ship theory. It can also be applied to full hullforms, as long as the viscosity does not play a significant role. It still has its limitations in the waveheight, because the disturbance potential is assumed to be small. Application to moderately high waves appears to be possible however.
2The transition from the double hull waves to the undisturbed surface has some complications, as discussed by Raven [39]
196
13.5.1
G.Kuiper, Resistance and Propulsion, January 4, 1994
Applications of Dawson's Method.
An illustration of a hull form with a high wave resistance is shown in Figs. 13.5. This wave pattern has been calculated with the Marin program "Dawson" . It shows a very strong interaction of all wave systems involved. After optimization using the Dawson program the wave resistance could be drastically reduced, as shown in Fig. 13.6. Especially the interaction be
. . ..........  4414116,
j='''
10Sile.
'
................................... .. WAVE HEIGHT MAGNIFICATION FACTOR
5.0
Figure 13.5: Dawson Calculation of Wave Pattern before Improvement of the Hull tween the bow wave system and the forward shoulder wave system has been improved. This optimization meant a complete redesign of the hull shape, as the drawn frames in Fig. 13.7 show. Actually only the main dimensions
and the displacement were maintained. The redesigned hull form has a softer forward shoulder with a more slender forebody, moving the center of buoyancy backwards. The bulb is more pronounced . The midship coefficient has been increased, resulting in a lower prismatic coefficient . The afterbody has become very full afterbody, which may increase the frictional resistance. This should be verified using other calculation methods or by experiments. Although the threedimensional graphs of Figs. 13.5 and 13.6 are nice to see, it is often sufficient to study the wave pattern along the hull, as shown in Fig. 13.8 for the same case. Here the drawn line is from the improved
G.Kuiper(MT512) January 4, 1994, Flow Calculations II page: 197
Figure 13.6: Dawson Calculation of Wave Pattern after Improvement of the Hull
hull form. Note that the waves contours at the stern are not too much different. These contours are somewhat distorted by the (violation of) the linearization of the free surface condition. Viscosity will also reduce the height of these waves. The most important gain for the wave resistance is in the forebody. In practice it is never allowed to redesign a ship hull so drastically as in the foregoing example. For reasons outside hydrodynamics it is often necessary to maintain the length of the parallel middlebody (containerships), to maintain the position of the center of buoyancy (trim) and to maintain the deck area and midship section. An example of a more realistic improvement in the wave pattern is shown in Fig. 13.9. The original design is the dotted line. In this case it is important to reduce the interference between bow wave and bow shoulder wave. The wavelength of the shoulder wave is indicated as Ao in this Figure. The bow wave should be at a position where the shoulder wave has a minimum. This can be obtained by moving the bow wave forward, resulting in the drawn wave profile. This profile has a lower wave amplitude along the hull. Although these realistic improvements seem small, the decrease of the wave resistance is considerable.
198
G.Kuiper, Resistance and Propulsion, January 4, 1994
,..4rAT 2o
Figure 13.7: Hull Forms before (dotted) and after (drawn) Optimization of the Wave Resistance
These examples show that the bulb cannot be considered as a separate part of the hull. A Dawson type calculation is not done to design a bulb, but to improve the shape of the hull including the bow form.
13.6
General Considerations to Assess Programs.
Many calculation methods are made available to customers commercially. The calculations are usually done by specialized institutions, because the generation of the grid for the calculations and the performance of calculations require some measure of expertise. Simpler programs such as the "Michell" programs, are generally run by
G.Kuiper(MT512) January 4, 1994, Flow Calculations II page: 199
77.7.7..7Mfel,...1211Sr.
/
Figure 13.8: Wave Contours along the Hull before (drawn) and after (dotted) Optimization of the Wave Resistance
STERN
WAVE HEIGHT MAGNIFICA1 ION FACTOR 120.
Figure 13.9: Wave Contours along the Hull before (dotted) and after (drawn) Optimization of the Wave Resistance
the user himself. The user is confronted with programs which are "black boxes" for him. To judge the possibilities of a calculation method it is important to realize the simplifications involved. This means that it is important to know: which equations of motion are used.
where are the boundary conditions applied which quantities have been assumed small (linearized)
200
G.Kuiper, Resistance and Propulsion, January 4, 1994 which quantities have been assumed twodimensional
which quantities have been neglected (physical parameters) which corrections have been applied to overcome previous simplifications 
which empirical coefficients or data are used
The simplifications are often hidden, because corrections are applied. As an example a simple twodimensional boundary layer calculation can be added to the "Hess and Smith" calculation to calculate the viscous resistance or even the wake pattern. These elements should be mentioned in the documentation of the program. Examples of the documentation of some previously mentioned calculation methods are attached to this chapter.
Chapter 14 Axial Momentum Theory Objective:A description of the actuator disk model for the calculation of the induced velocity and the ideal efficiency. Determination of optimum loading distribution and effects of rotation.
The propeller induces velocities in the flow around it. The determination of the induced velocities in the propeller sections is a main problem in propeller design theory. The simplest model to estimate the induced velocities is the axial momentum theory, in which viscosity is neglected, the number of blades is assumed to be infinite and the rotation induced by the propeller is also neglected. The only action of the propeller is to excert a uniformly distributed axial force on the fluid.
14.1
Axial Momentum Theory.
The fluid is considered to be inviscid and incompressible. The propeller is described as an actuator disk, which is a disk with diameter D which causes a pressure jump Ap over the propeller disk. Because the thrust is distributed uniformly the pressure jump at the propeller disk will be the same at every position of the disk. The flow pattern is axially symmetric and a cross section is shown in Fig. 14.1.
Nomenclature: The inflow has a velocity Ve at a pressure po, which is the pressure in the undisturbed flow. The diameter of the streamtube far upstream is Do and the area Ao is therefore 0.25rDg.The streamtube contracts downstream of the propeller in the slipstream until the diameter remains constant. This diameter is called D2 and the area A2. The veloc201
G.Kuiper, Resistance and Propulsion, January 4, 1994
202
Ve
Po NI
PROPELLER DIAMETER
Vs +2Va
Vs + va
Po
SECTION A2
SECTION A, STREAMTUDE
SECTION Ao
Figure 14.1: The axial actuator disk model
ity is axial there and the pressure is Po again, with an increased velocity vaa. (Note that in Figure 14.1 the velocity increase is 2va, or twice the increase at the propeller. This still has to be proven.) At the propeller disk the pressure rises from p upstream to p Ap downstream of the actuator disk. The thrust is then ApAi, where A1 = 0.257rD2. The velocity at the actuator disk is continuous because the water is incompressible and has a value Ve va. It is assumed that the flow in the slipstream has no rotation, so only axial induced velocities occur. A relation between the axial induced velocities va and vea with the propeller thrust T can now be formulated using the conservation laws for mass and momentum. The first relation to be used is the conservation of mass or the continuity equation: VeA0 = (Ve + va)Ai = (Ve
va.)A2
From these two relations the diameters Ao and A2 can be written in terms of the propeller diameter D:
January 4, 1994, Momentum Theory 203
D02 .
(14.1)
[ve v+ Va 1,02
D22 = [
]D2
Ve
Ve + Vaa
The law of conservation of momentum equates the force excerted on the fluid with the net outflow of momentum . The control volume is the streamtube from Ao to A2. The mass per unit time through Ao is pV,A0 and the momentum inflow is pl/e2A0 Similarly the momentum outflow through A2 can be written and the conservation of momentum requires that 1: pVe2A0
T=
vaa)2A2
p(Ve
Using eqs. 14.1 this can be written in terms of the propeller diameter D as: n2
T
4
/ T7
pk,
\,
(14.2)
r va)vaa
The thrust can also be written as T ApS and the pressures and velocities at the propeller disk are related with those upstream and downstream by Bernoulli's law, which can be applied over those regions where no force is applied on the fluid, that is upstream and downstream of the propeller separately.
11
Upstream of the propeller:
1,
1
2PVe` = P + 2P( Ve + v. )2
Po
Downstream of the propeller:
Po + 2P(Ve + Vaa)2 = P
AP
2P(Ve
va)2
Subtracting these two equations gives: 1
Ap = P(2VeVaa
Vaa2)
2
(14.3)
and a second formulation for the propeller thrust is: ]n
==
4
1)2p(Ve
lvaa)vaa 2
(14.4)
'This assumes that no net pressure force is present on the outside of the control volume. It can be shown that this is true when the fluid outside the control volume is large. A proof is given e.g. in [7].
G.Kuiper, Resistance and Propulsion, January 4, 1994
204
From eq. 14.2 and eq. 14.4 it is found that vaa = 2va.
The axial induced velocity at the propeller is half the axial induced velocity the slipstream The relation between the propeller thrust and the axial induced velocity is now:
T = D2 p(Ve 4
va)2va
(14.5)
The propeller thrust is made nondimensional with the propeller area and the inflow velocity V, CT =
'D21pV2 2 e
4
where CT is a thrust coefficient indicating the propeller loading. 2 Eq. 14.5 becomes CT =
)
V
Va
Ve
Ve
or inversely
14.1.1
Va_
1
_1
V:
2
2V
(14.6)
Efficiency.
The induced velocity in the slipstream represents energy supplied to the flow behind the propeller. This is due to the fact that the fluid "gives way" when a thrust is excerted to it. The loss of energy is reflected in an efficiency
which is lower than one. To formulate the efficiency we use a reference system in the undisturbed water instead of connected to the moving disk. In that reference system the propeller disk moves with a velocity Ve and excerts a force T. The power delivered by the propeller is therefore TVE. In the slipstream an velocity 2va is present. With the mass flow expressed as the mass flowing throught the propeller disk, which is equal to that flowing through the slipstream, this represents an energy of: 2This thrust coefficient CT is different from the thrust coefficient KT based on rpm. To distinguish between both CT will be called the loading coefficient and KT the thrust coefficient
January 4, 1994, Momentum Theory 205
(14.7)
E108t = p(Ve + va)(7iD2)(2va)2
The efficiency of the propeller can be written as =
TV, TV, + Elost
Using eq.. 14.5 this can be rewritten as qo
or:
=
VeP(Ve
va)f irDi22v.
Vep(Ve + va)14,0122va + 2va2piDi2(Ve + va)
170(14.8) 1
1 F tige.
The induced veloctiy va can be written in terms of the propeller loading using eq. 14.6 as:
2 710 =
1 + 1/1 + CT
(14.9)
This represents the maximum efficiency which is theoretically possible in an inviscid flow with a propeller which does not introduce any rotation in the slipstream. It is therefore called the ideal efficiency. Its value is a function of the propeller loading and this relation is graphically shown in Fig. 14.2. When an efficiency is claimed which is larger than the ideal efficiency, basic laws of nature have been violated or a simple error has been made.
In this actuator disk model the thrust Ap is considered uniformly distributed over the propeller disk. This is an assumption. Below it is shown that this situation of uniform propeller loading gives the maximum efficiency, so the actuator disk calculations really give the ideal efficiency in uniform inflow.
14.2
Optimum Radial Loading Distribution.
In the preceding section we have assumed a constant radial loading distribution Ap on the actuator disk. The momentum theory can also give an indication of the optimum radial loading distribution when an annular disk
G.Kuiper, Resistance and Propulsion, January 4, 1994
206 100
090
0130
070
ni 060
150
040
a30
025
a5
2
16
CT
Figure 14.2: Ideal efficiency as a function of propeller loading
with radius r to r + dr is considered. The energy lost in the slipstream is similar as in eq. 14.7, but now for an annular ring with area 2irrdr:
dE(r)
1 p(Ve 2
va(r))(2va(r))2dS
(14.10)
Similarly the thrust of an annular element is: dT(r) = p(Ve
va(r))2v3(r)dS
(14.11)
Note that here the total pressure force on an annular element is again considered to be zero, thus neglecting any possible interaction between the elements.
From these two equations for an annular disk the loss of energy in the slipstream can thus be written as:
dE = vadT
(14.12)
The aim is now to minimise the energy loss for a given propeller thrust. An increase of the trust AT at an arbitrary radius ra gives an increase of the lost energy AEa = va(a)AT. To keep the total thrust constant a similar decrease of the thrust AT has to be applied at another radius rb. This gives a decrease of the lost energy AEbva(b)AT. When va(a) is smaller than va(b) the total efficiency is increased. An optimum is therefore obtained when va is constant over the radius. In that case the radial distribution of the thrust at the propeller disk is also constant, so this is the actuator disk model as
January 4, 1994, Momentum Theory 207 used above.
The optimum radial loading distribution gives a uniform axial velocity distribution in the slipstream _
This property will be used later when the hydrodynamic pitch of the vortices in the wake is chosen.
Chapter 15
The Propeller Geometry Objective: Description of the propeller geometry and the names and definitions used to describe it. Once the geometry is understood, the figures and the definitions in the text should be sufficient for further use.
15.1
General Outline.
A sketch of a propeller is given in Fig. 15.1. The propeller blades are attached to the hub, which is fitted at the end of the propeller shaft.
The propeller rotates about the shaft center line. The direction of rotation is as viewed from behind, that is towards the shaft. In normal forward operation a right handed propeller rotates in clockwise direction when viewed from behind . The propeller in Fig. 15.1 is righthanded. The front edge of the blade is called the leading edge. The other edge of the blade is called the trailing edge. The outermost position, where leading and trailing
edges meet, is called the blade tip. The radius of the tip is the propeller radius. The propeller diameter is, of course, twice the radius. The surface of the blade which is at the side of the shaft is called the propeller back. The other side is the face of the propeller. ( When the ship moves forward the propeller inflow is at its back.) Because in forward speed the back side has a low average pressure and the face side has a high average pressure (this pressure difference generates the thrust), the face is also called the pressure side and the back the suction side.
208
January 5, 1994, The Propeller Geometry 209
Di recti on of rotati on
Trailing
edge
7Leading edge
Fi 1 1 et area
Hub
Shaft
Back Face
Figure 15.1: Sketch of a propeller The propeller hub is of course rotationally symmetrical because it should
not disturb the flow. The attachment of the propeller blade to the hub is gradual, which is done in the fillet area or blade root. A streamlined cap is generally fitted to the hub.
15.2
Blade Sections.
Consider an arbitrary propeller, as drawn in Fig. 15.2. The intersection of a cylinder with radius r and a propeller blade, the blade section, has the shape of an airfoil. Such a shape is also called just a foil or a profile. Some characteristic parameters of a foil will be defined first. A general shape of a profile is shown in Fig. 15.3. The side which meets the flow is the leading edge of the profile. The trailing edge is generally sharp. A sharp trailing facilitates the definition of a coordinate system in which the profile coordinates are defined. The leading edge is found as the point on the contour with the largest distance from the trailing edge. Other names for leading and trailing edge are nose and tail.
G.Kuiper, Resistance and Propulsion, January 5, 1994
210
P/4
Direction of rotation Figure 15.2: Cylindrical cross section of a propeller blade
Suction sid
Camber
Tail
Y
Thickness
Nose
Pressure side
Figure 15.3: Geometry of a propeller blade section
January 5, 1994, The Propeller Geometry 211
The straight line between the leading and the trailing edge of the profile is the chordline of the profile and the distance between nose and tail is the chord length c. The chord line is also called the nosetail line.
The trailing edge is not always sharp, however. In that case the chordline is defined as the direction of the maximum distance between two points
on the contour. This direction has to be found iteratively in such a case. (A different definition of leading and trailing edge will be given below)
Generally, the origin of the local coordinate system of a profile is taken
at the leading edge. The xdirection is towards the tail, the y direction upwards, perpendicular to the chord. The angle between the nosetail line and the undisturbed flow is the angle of attack a. Its positive direction is given in Fig. 15.3. At a positive angle of attack the pressure at the upper side of the profile is lower than the pressure in the undisturbed flow and this side is therefore called the suction side. The pressure at the lower part is higher than the pressure in undisturbed flow over most of the chord and is therefore called the pressure side. These names match with the names of the corresponding blade surfaces. The distance between the suction side and the pressure side, measured perpendicular to the chord, is the thickness t(x) of the profile (see Fig. 15.3). The line through the middle of the thickness over the chord is the camber
line of a profile. The vertical distance between the camber line and the nosetail line is the camber f (x). The camber and thickness distributions are often made nondimensional with their maximum values, so that the camber and thickness distributions are given in values between 0 and 1, or as percentages of the chordlength . When the same camber and/or thickness distribution is used for the blade sections at all radii, as is often the case, the blade sections can simply be described by this distribution and the radial distribution of maximum thickness and maximum camber. The maximum thickness and maximum camber are often given as percentages of the chord length. Variations in a given camber and thickness distributions are often made by varying the chordwise position of the maximum. These positions are generally expressed in percentages of the chordlength, measured from the leading edge. (A section is described e.g. as having 2% maximum thickness and 1% maximum camber, with the position of maximum camber and thickness at 35% from the leading edge. The distributions of camber and thickness are then assumed to be known).
212
G.Kuiper, Resistance and Propulsion, January 5, 1994
15.2.1 NACA Definition of Thickness and Camber. An alternative definition of thickness and camber has been given by the NACA organisation in the United States (now NASA). This organisation has carried out a massive amount of experiments on profile series [1] since the thirties of this century. In the NACA definition the thickness is measured perpendicular to the camber line. In the NACAmethod the nose of a profile is no longer the extreme of the contour, but the intersection of the camber line with the contour. Since the camber line is connected with the thickness distribution, the camber line in the NACA method always has to be determined iteratively.
Figure 15.4: Two ways of combining camber and thickness
For a given profile geometry the Naca definition results into a different camber line than the definition using the maximum length. As a result the nose point will also be different (as will be the tail location in case of a blunt tail). Inversely, the construction of the geometry of a profile from a given camber line and thickness distribution results in a different geometry. The difference in profile contour when both definitions of the camber are used is especially apparent near the nose. This is illustrated in Fig. 15.4. For thin profiles the differences between both definitions of camber and thickness are small. Since propeller blade sections are generally thin it is generally not mentioned which method is used to define camber and thickness.
January 5, 1994, The Propeller Geometry 213
15.2.2
Root and Tip.
Two important points on the propeller can now be defined more precisely. The first is the tip of the propeller blade. This is the location where the chord length of a cylindrical section becomes zero. This occurs at radius r, which is the outer radius R of the propeller. The second location is the root section. This is more difficult to define. The root section of a propeller is the intersection of the blade with the hub. However, the hub is often not a circular cylinder, but has a conical or even more complex shape. Therefore, a (design) hub diameter is defined as some average hub diameter in the root region. The blade section in the cylinder
at the design hub radius is the blade root section. This section is found when the blade is extended virtually into the hub. In this definition of the blade root section the fillets are neglected.
The plane perpendicular to the shaft through the mischord of the root section is the propeller plane. It is used as a pine in the coordinate system in which the propeller geometry is defined.
15.3
Pitch and Pitch Angle.
The cylindrical cross section of a propeller blade, as shown in Fig. 15.2, is now developed into a plane. (Fig. 15.5). In this figure the xaxis is the projection of the center of the propeller shaft. In this developed plane a number of parameters can be defined. The chordline or nosetail line of the blade section changes from a helix
on the cylinder into a straight line, and its extension is called the pitch line. 1. The propeller pitch P is defined as the increase in axial direction of the pitch line over one full revolution 27rr. The dimension of the pitch is a length. The pitch angle (1) is the angle between the pitch line and a plane perpendicular to the propeller shaft.
The pitch distribution is given in a pitch diagram, which is simply a graph of the pitch at every radius. The pitch diagram is given in the propeller drawing (Fig. 15.9), as will be discussed later. A significant radius, which is often used as representative for the propeller, is the radius at r R = 0.7. If a pitch value is given in the case of a variable pitch distribution it is usually the pitch at 0.71t. 1When the blade sections have a flat face (pressure side), the pitch line is sometimes defined as the line trough the section face instead of the nosetail line.
G.Kuiper, Resistance and Propulsion, January 5, 1994
214
\\
Propel 1 er r
erence 1 in
Pro eller plane Rake
Skew
Generator 1 ine
i nduced
rake
Blade reference line
Figure 15.5: Expanded cylindrical cross section of a propeller
15.4
Propeller Plane and Propeller Reference Line.
The pitch and pitch angle could be defined without the definition of a coordinate system. A coordinate system in which the geometry of a propeller is expressed is always chosen as a cylindrical coordinate system (x, r, 0), fixed to the propeller, with the positive xaxis as the shaft center line in the direction of the propeller back. (see Fig. 15.1).The origin of the coordinate system is chosen on the shaft center line in such a way that the plane through the origin and perpendicular to the xaxis, called the propeller plane. , goes through the middle of the chord of the root section. The coordinate O = 0 in the propeller plane is the radial through the shaft center and the midchord of the root. This line is called the propeller reference line.
In Fig. 15.5 the intersection of the propeller plane with the expanded cylinder at an arbitrary radius is given. The x/axis is the intersection of the plane x, r, O = 0 with the expanded cylinder. The intersection of the cylinder with the propeller plane gives another line, which is perpendicular to the x'axis. Both form the coordinate system in the plane of the expanded cylinder.
January 5, 1994, The Propeller Geometry 215
15.5
Rake.
Having defined the coordinate system some other parameters can be defined in Fig. 15.5. The x/axis intersects the pitch line at a point on the generator line and the distance between the generator line at a certain radius and the propeller plane is called the rake. Rake therefore has the dimension of a
length. When the rake is away from the ship hull (in the direction of the negative xaxis), thus increasing the tip clearance, it is called positive rake or also backward rake. This direction is the common direction for propellers. When there is no rake the propeller reference line coincides with the generator line.
Only in case of a linear rake distribution from root to tip the generator line is a straight line in the plane z, r, 8 = O. In that case the angle between the generator line and the propeller reference line is called the rake angle. The rake angle is positive in case of backward rake. An example of linear backward rake (without skew, see later) is shown in Fig. 15.6.
Propeller plane
Rake
h ft centreline
Hub
Figure 15.6: Longitudinal cross section of a propeller with rake
The axial displacement of the blade sections has little effect on the propeller performance. It increases the wetted surface of the blades somewhat
G.Kuiper, Resistance and Propulsion, January 5, 1994
216
and thus decreases the efficiency slightly. Because the blade thickness is measured in axial direction rake decreases the thickness of the blades when measured perpendicular to the blade surface. This may become important in cases of extreme rake. Backward rake is used to increase the tip clearance, the distance between a propeller tip in top position and the hull. When this is the only purpose of the rake the rake distribution is mostly linear. Rake may also be used in the casting process to prevent gas inclusions.
15.6
Skew.
The midchord of the blade section in Fig. 15.5 does not coincide with the generator line. The section is shifted along the pitch line. The location of the midchord of the propeller section is now called the blade reference point and its position is indicated in Fig. 15.5and the distance between blade reference point and the generator line is called the skew. . When skew is in the negative direction of O it is called backward skew.
Since skew moves the blade reference point along the pitch line, the blade reference point also moves in axial direction when skew is changed. The axial displacement of the blade reference point due to skew is called skew induced rake . A propeller without skew has a generator line which coincides with the blade reference line. Unlike the rake, the skew distribution is never linear. Because the skew
is defined along the pitch line the skew distributions used can be better shown in other projections than Fig. 15.5, as will be discussed in the next section.
15.7
Blade Contours and Areas.
The projection of the blade contour on the propeller plane gives the projected blade contour. An example is given in Fig. 15.7a as the drawn contour.
The propeller reference line and the generator line coincide in this projection as the vertical axis. Consequently rake is not visible in the projected blade contour. The blade sections in this projection are segments of a circle. Apart from the projected blade contour the developed blade contour can be defined. The blade sections in the cylinder of Fig. 15.2 are rotated
218
G.Kuiper, Resistance and Propulsion, January 5, 1994
blades and the area of the propeller plane Ao (Ao = 0.2571D2, where D is the propeller diameter). Two blade area ratios are used: the projected blade area ratio Ap I Ao and the expanded blade area ratio Ae/A0. The latter ratio is physically most significant and when no further indication is given this blade area ratio is meant.
Because the skew is measured along the pitch line, the skew distribution can be plotted directly in the expanded blade contour, as shown in Fig. 15.8. The skew varies over the radius. To indicate the amount of skew as a property af the whole propeller the skew angle is used. It is the angle between the blade reference line and the line from the shaft center to the tip. This angle can be defined in the expanded contour. In the ITTC nomenclature, however, the skew angle is rather inconsistently defined in the plane of rotation, and the skew angle has to be drawn in the projected contour, as is done in Fig. 15.8. Skew angle extent

Propeller reference line Generator line
Blade reference line
Hub radius
Root
Projected contour
Expanded contour
Figure 15.8: Projected and expanded contour of a propeller with skew The skew at inner radii is generally forward skew, at outer radii backward skew is applied, as shown in Fig. 15.8. Such a skew distribution is
January 5, 1994, The Propeller Geometry 219
called balanced skew. This is done to reduce blade spindle torque and to avoid excessive stresses in the blade root, which would occur due to centrifugal forces if the skew was not balanced. As an indication of the shape of the propeller blade the skew angle alone can be misleading. In that case the skew angle extent is defined in the projected contour. A drawback of the
skew angle extent is that minor changes in the skew distribution at inner radii, which have little impact on the propeller performance, can greatly influence the skew angle extent.
15.8
Warped Propellers.
Because of its definition skew also generates skew induced rake, which causes the blade sections to move out of the propeller plane. For the same struc
tural reasons as with balanced skew a balanced rake can also be required. This is done using, negative rake together with positive skew, such that the negative rake compensates the skew induced rake. In that way the blade reference line can be kept in the propeller plane. Such a propeller is called a warped propeller.
15.9
The Propeller Drawing.
A propeller drawing contains the elements discussed before. An example is given in Fig. 15.9.
At the left the longitudinal projection on the plane x = 0, r, O = 0 is given. It gives the contour of the hub and a projection of the blade with the generator line in the plane of the drawing. The propeller in Fig. 15.9 has no skew, so the tip of the contour is at the outermost radius. With skew this is no longer so. In this drawing also the the envelope of the blade over a revolution ( the clearance curve or sweep) is sometimes given. The tip contour is given separately in detail. This tip contour is not a projection but an axial cross section at the location of the blade reference line.
Next to the longitudinal cross section the pitch distribution is given graphically. The distance to the generator line of the projected contour is the pitch. The middle figure gives the projected blade contour. It is characterized by the circle segments of the blade sections. Above this contour are the details of the leading and trailing edges of a propeller section. The trailing edge is not smoothly rounded, but has a knuckle at some distance from the
220
G.Kuiper, Resistance and Propulsion, January 5, 1994 Anti singing 2g9e
IINErr:"smiumnomvAmairLW
=EMS.
111MW, =1MM M1%1W =1
Figure 15.9: Example of a propeller drawing
trailing edge. This is the antisinging edge , which serves to fix the separation of the flow from the blades. If there is no antisinging edge vibrations of the blade (singing) may occur. The vibrations are caused by excitation of vortices leaving the trailing edge periodically, often at an audible frequency. The figure at the right hand side of Fig. 15.9 is the expanded blade contour. The nosetail line is a straight line here and the shape of the blade sections is given relative to this line. In this drawing the location of maximum thickness is also drawn. In Fig. 15.9 this location differs from the midchord line at inner radii only.
To illustrate the effects of camber and skew on the propeller drawing these parameters are varied in Fig. 15.10. The interpretation of these diagrams should be clear now.
15.10
Description of a Propeller.
The propeller geometry is generally defined by the following data: 1. Number of blades.
January 5, 1994, The Propeller Geometry 221
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222
G.Kuiper, Resistance and Propulsion, January 5, 1994 Diameter.
Radial distribution of rake.
Radial distribution of pitch. Radial distribution oi skew.
Radial distribution of chord length.
Type of camber and thickness distribution. (locating also the positions of maximum camber and thickness) Radial distribution of maximum thickness. Radial distribution of maximum camber. details of antisinging edge.
hub shape root fillets.
Instead of the nondimensional camber and thickness distributions and the radial distributions of the maximum camber and thickness the section contours (pressure and suction side coordinates) may also be given. In that case the definition of the nose region requires much attention and at least 50 points over the chord are required for a proper definition. The precise definition of the root fillets is still lacking in practice. The shape is often indicated by a radial cross sections of the blade. More precise definitions are required, especially when the blades are being milled numerically.
15.11
Controllable Pitch Propellers.
A special type of propeller is the controllable pitch propeller,, often abbreviated as CPP or CPPropeller. The blades of such a propeller can rotate about an axis perpendicular to the shaft centre line, the spindle axis. The spindle axis is the propeller reference line of Fig. 15.5. In exceptional cases the spindle axis does not pass through the centre line of the shaft.
Because the hub is a complex mechanism in this case, the blades are manufactured separately and mounted to the hub. The blades end in a circular disk, which is called the palm. This disk is bolted to another circular disk in the hub, the carrier, which can be rotated mechanically or
January 5, 1994, The Propeller Geometry 223 hydraulically.
The variation of the propeller pitch poses additional constraints on the hub geometry, because the blades should fair smoothly into the hub at all blade positions. This requires that the palm has the shape of a sphere. This is often not so and then there is a clearance between blade and hub. In case of CPpropellers the propeller reference line is not always chosen through the midchord of the root section, but at another position which is
favorable from a manufacturing point of view. The spindle axis can then be used as the propeller reference line, but it should be kept in mind that such a shift of then reference line can strongly change the values of skew and rake while the actual blade geometry is unchanged.
Chapter 16 Systematic Propeller Series Objective: Propeller design in uniform flow using systematic test results The performance of a propeller is characterized by its open water performance as represented in the Kt Kg J diagram. This diagram will be explained first.
16.1
Open Water Diagram.
As shown in the description of the propeller geometry the blade sections have a certain pitch. When the propeller moves forward over that distance during one revolution the chordline of the blade sections is in line with the flow along the blades. When the forward displacement over one revolution is smaller the propeller will develop more thrust. An important parameter for the thrust of the propeller is therefore the axial displacement per revolution,
or the ratio V/n. This can be expressed in nondimensional terms as the advance ratio J T
nD
where D is the propeller diameter and n is the rotation rate (sec') and V, is the undisturbed velocity upstream of the propeller. When the axial distance covered per revolution is smaller than the pitch the difference is called the slip of the propeller. In non dimensional terms the slip is expressed as PIDJ. The ratio PID is the pitch ratio. The slip
can also be expressed as a percentage of the pitch ratio. The slip is used in older literature, but has been replaced by the advance ratio in modern 224
January 5, 1994, Systematic Propeller Series 225
literature.
In the foregoing the advance velocity has been made nondimensional with the velocity nD (proportional with the tip speed). The thrust and torque can be made nondimensional with the same speed and with the propeller diameterD. The result is the thrust coefficient IfT KT = pn2D4
and for the torque the torque coefficient KQ AQ
pn2D5
where T is thethrust in Newton, Q is the torque in Nm. These are the nondimensional parameters in which the propeller performance is expressed. These nondimensional parameters can also be derived from dimensional analysis. Two additional parameters are introduced now: the propeller diameter D and the rotation rate of the propeller n. As has been mentioned in chapter 6 the introduction of a propeller with diameter D introduces a nondimensional parameter DIL, which means that in model tests the propeller should be geometrically scaled. The introduction of the propeller revolutions n introduces the advance ratio J.
The propeller performance in uniform flow has the characteristics as given in Fig. 16.1 1. In addition to the thrust and torque coefficients the propulsive efficiency of the propeller is shown. The efficiency is the ratio between the delivered power by the torque and the effective power of the thrust. The power delivered to the propeller is the delivered power PD: PD
27rQn
The effective power PE is the power delivered by the propeller thrust
PE = TV, 'Note that the torque coefficient is multiplied by 10 to separate it from the thrust
coefficient
226
G.Kuiper, Resistance and Propulsion, January 5, 1994 0.7
ETA
0.1
0.3
0.2
0.1
0.0
0.2
0.1
0.6
AOVANCE COEFF IC I ENT
0.8
0.0 1.0
J
Figure 16.1: Open water diagram of a propeller The velocity V, is the entrance velocity. It is the velocity of the propeller relative to the undisturbed water. With propeller mounted coordinates this means that the entrance velocity is the undisturbed velocity far upstream of the propeller. The velocity in the propeller disk will be higher due to the induced velocities, both in axial and tangential direction. The axial induced velocities were calculated in the actuator disk model of chapter 14. The efficiency of the propeller is the ratio between delivered and effective power:
770 =
PE PD
January 5, 1994, Systematic Propeller Series 227
This efficiency is defined without interference between propeller and hull and with uniform inflow. These conditions are met when a propeller is mounted on the front of a sting. It is therefore called the open water efficiency.. 770 can be written in terms of thrust and torque coefficients as: qo =
KtJ
(16.1)
27rit,q
The efficiency is commonly plotted in the open water diagram as shown in Fig. 16.1. Note that at a certain advance ratio the thrust coefficient becomes zero. Then by definition the efficiency is also zero. This condition will be close to the zero slip condition and thus depends on the propeller pitch ratio.
16.2
The Quality Index.
At zero advanCe ratio the efficiency is also zero. This condition is called the bollard condition . The thrust coefficient is a measure of the thrust delivered
at zero speed. This can be an important design parameter (e.g. for tugs) but the efficiency loses its meaning in that condition. In that condition the quality index is sometimes used. The efficiency indicates the quality of the energy conversion. It does not say very much about the quality of the propeller itself, because a heavily loaded propeller will always have a lower efficiency than a lightly loaded one.
A measure for the quality of a propeller is therefore the ratio between the ideal axial efficiency, as derived in chapter 14 in eq. 14.9, and the efficiency as defined above. This ratio can be written as 110
TVe(0.5
0.5V1 + CT) 27rmn
KT 47rKQ
j + J2 + 8 KT) V
7r
This "quality index" does not go to zero in the bollard condition, when the advance ratio J is zero. Instead it is
Quality Index (J=0) =
16.3
K1.5 T VT71KQ
Systematic Propeller Series.
The open water diagram gives a characteristic of the powering performance of a propeller. Systematic series of propeller models have been tested to form a basis for propeller design. The starting point of a series is its parent form. The extent and applicability of the series depends on the parameters
228
G.Kuiper, Resistance and Propulsion, January 5, 1994
which are varied and on the range of the variations. There are several series, but one of the most extensive and widely used series is the Wageningen
Bseries. The basic form of the Bseries is simple and it has a good performance. The extent of the series is large: some 210 propellers have been tested. The basic characteristics of the Bseries are shown in Fig. 16.2.
Pitch distribution 1.0R
100%
B440 0.045
4
' B455
' 8470
.B4100'
i B485
_
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1.0R
.....,..._
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m v. ,
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B5105/
0.040
Figure 16.2: General plan of the B4.40, B4.55 and B4.70 propellers As shown the Bseries propellers have
a constant radial pitch distribution at outer radii very little skew
15 degrees backward rake angle (linear rake distribution)
a blade contour with fairly wide blade tips circular back blade sections of NSMBdesign 2NSMB:
Netherlands Ship Model Basin, nowadays MARIN: Maritime Research Institute
Netherlands

January 2, 1994, .Systenta,tie Propeller Series 229 The followiag parameters have been varied
the expanded blade atea, ratio AE/Ao from 0.30 to 1.2
the number of blades Z from 2 to 7
the pitch ratio
D from 0.5 to 1.4
The propellers are indicated by their blade munber and blade area ratio_ Propeller B4_85 e.g.. Has four blades ;qTfd a bia.de area ratio of 0.85. From each propelle,r an open water diagram NVaS Irte.as:urea. . Until now 210 propeller models have beert tested_ The re.suits are given in. open water diagrams per series of one blade number and area ratio. An example is given in Figs. 16.3
The open water tests of the Bseries were done at various rpm, so at a variety of model Reynolds numbers, The Bseries diagrams ha.ve been corrected to a Reynolds number3 of 2 x 106 along the lines of the IT'TC57 method., as will be discussed in chapter 20. The correction is only
16.4
Propeller Hull Interaction.
The propeller works behind the ship hull. Before a propeller can be designed from open water diagrams it is necessary to estimate the interaction between hull and. propeller. In this preliminary design stage this vyill be done in a very simple way using the wake fraction a_nci the thrust decluction factor. The velocity V, as used in the open water diagrams? is the velocity far
upstrearn of the propeller. Behind the hull without propeller the velocity at the propeller disk is called the nominal wake_ This wake is Used as the velocity V. This assumes first that the nominal wake fraction is the same in the propeller plane (w.here it is measured or calculated) and in a plane several propeller diameters upstream of the propeller plane. It assume.s next that the wake, is uniform over the propeller disk, because the inflow is used in open water diagrams. where the irlow is uniform. ft finally a.ssnmes
that the nominal wake distribution in this upstream plane is not afreeted by the propeller aciion. rFor the propeller inflow the nominal wake fraction is used: Ve
Vr,;(1
tu,)
Tb.e propeller has an effect on. the hull, however The propeller increases
the resistance of the ship bull by increasin!4 the velocity along the hull 3A Reynolds number based on the chord len,gth and inflow velocity at 076R.
230
G.Kuiper, Resistance and Propulsion, January 5, 199.4
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Figure 16.3: Open Water Diagrams of B4.70 Propeller Series
(generally a small effect) and by decreasing the pressure around the stern. The thrust to be developed by the propeller should thus be greater than the resistance without propeller at the design speed, because the thrust has to be equal to the increased resistance. The increase of the resistance due to the propeller action is expressed as the thrust deduction factor t:
January 5, 1994, Systematic Propeller Series 231
t=
TR
T is the thrust to maintain a certain design speed and R is the resistance without proReller at that speed, as found e.g. from the resistance test.
The assumption that the nominal wake fraction can be used to determine the propeller inflow is not consistent with the increase of the resistance
due to the propeller, as expressed by the thrust deduction factor. To be more accurate the effective wake fraction should be used, but this refinement is left for later. In the preliminary design stage some estimates have to be made of the thrust deduction factor and of the nominal wake fraction. These estimates can be made using statistical data or even comparable ships.
16.5
Propeller Design Requirements.
The charts of the Bseries can be used for a preliminary design of a propeller. To be able to design a propeller the requirements should be defined. The most open situation is when a ship is designed for a given design speed. Using the wake fraction the propeller entrance velocity is known. Also the resistance of the hull without propeller should be determined, e.g. from statistical calculations or from a resistance test. Using the estimated thrust deduction factor t the required thrust can be found from
T=
(1
t)
The thrust deduction can be estimated using statistical formula's. A common procedure is to measure it at model scale using a stock propeller with an approximate diameter and with the required loading at the design speed.
The propeller designer now has the freedom to choose the number of blades, the blade area ratio, the rotation rate, the diameter etc. The criterium is optimum efficiency and the result is a propeller and the required shaft power for the design speed.
The situation is not always as open as in the foregoing case. A limited number of engines is available and especially the rotation rate of these
232
G.Kuiper, Resistance and Propulsion, January 5, 1994
engines is prescribed. The engine has a maximum power at which it can operate continuously ( the Maximum Continuous Rate or MCR) and the design condition of the engine is often chosen at 80 percent MCR. At that operating condition the engine delivers its power at a certain rate of rotation. That means that the _propeller designer has to design the propeller to absorb that power at that required rotation rate. Another restriction which occurs frequently is a restriction on the allowable diameter. To maintain sufficient tip clearance while staying at or above the baseline of the hull with the propeller often requires a limit to the allowable diameter. In practice this maximum diameter is often smaller than the optimum one, so the designer has to design a propeller with restricted diameter.
Another possibility is that the ship exists and the engine has already been chosen or even installed. Maximum propeller efficiency in this case means maximum ship speed. In such a case the available power and the rotation rate is prescribed. Several combinations of such restrictions and prescriptions are possible. The open water diagrams can be used to meet these requirements, but first some other considerations are necessary.
16.6
Choice of Number of Blades and Blade
Area Ratio. Before the open water diagrams of the systematic series can be used to fulfill the design requirements the parameters which are chosen on other considerations than efficiency have to be defined. These parameters are the number of blades Z and the expanded blade area ratio AE/Ao The number of blades is chosen in relation to possible vibrations. An 8 cylinder engine and a four bladed propeller may suffer from resonance frequencies because the blade frequency and the engine frequencies have common harmonics . In that case the vibrations will become excessive, resulting in damage. The structure of the wake is also important for the choice of the number of blades. When the wake has strong second and fourth harmonics, which oc4Harmonics are distinct frequencies in a periodical signal. The frequency content of a periodical signal can be found by a Fourrier transformation, which describes the signal as a sum of sine wave with distinct frequencies
January 5, 1994, Systematic Propeller Series 233 curs when there is a wake peak both in top position and in bottom position of the blades, an even bladed propeller is at a disadvantage with respect to shaft vibrations. For Navy ships the number of blades is often chosen as high as possible to reduce the danger of tip vortex cavitation. The blade area ratio is chosen such that cavitation is avoided as much as possible. Empirical formulas have been developed to choose the arearatio. An old and very simple formula is that of Taylor AE/Ao = 1.067
0.229P/D
An empirical chart which is still frequently used is the chart of Burrill (1943),as given in Fig. 16.4. The line based on the experience of MARIN is also given in that chart. (Note that in this chart e = p, and AEIA° = SI Fp)
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