Marine Propulsors. Design Methodology for Subcavitating. Archief

Transcription

1 TECHNLSCBE UNIVERSITEIT Scheepshydromechanica Archief Mekelweg 2, 2628 CD Delft Te1: /Fax: NATIONAL TECHNICAL UNIVERSITY OF ATHENS DEPARTMENT OF NAVAL ARCHITECTURE AND MARINE ENGINEERING 9, IROON POLITECHNIOU, GR ZOGRAFOU, ATHENS, GREECE LABORATORY FOR SHIP AND MARINE HYDRODYNAMICS (Telephone: , Fax: , Design Methodology for Subcavitating Marine Propulsors By Gerasimos K. Politis Associate Professor Report prepared for the 34th WEGEMT School on 'Developments in the Design of Propulsors and Propulsion Systems', TU Delft, June 2000 Athens, June 2000

2 This document was produced by NTUA, Dept. of Naval Architecture and Marine Engineering for the 34th WEGEMT school on 'Developments in the Design of Propulsors and Propulsion Systems', TU Delft, June Copying and use of the document is restricted within this 'WEGEMT' school. The document may not otherwise be used or disseminated without the consent of NTUA, Dept. of Naval Architecture and Marine Engineering.

3 CONTENTS: Page FOREWORD 4 INTRODUCTION PHENOMENOLOGY OF A SELF-PROPELLED SHIP REDUCTION OF SELF-PROPULSION PROBLEM TO TOWING AND OPEN WATER PROPELLER PROBLEMS 6 DESIGN PARAMETERS, DESIGN CRITERIA AND PRACTICAL CONSIDERATIONS DESIGN PARAMETERS DESIGN REQUIREMENTS CORRELATION OF THE GEOMETRIC DESIGN PARAMETERS WITH THE DESIGN REQUIREMENTS SELECTION OF GEOMETRIC DESIGN PARAMETERS 14 WELL-POSED PROPULSION PROBLEMS 16 DESIGN EXAMPLES USING METHODICAL SERIES PROPELLER DESIGN PROBLEMS FOR CONVENTIONAL SHIPS PROPELLER DESIGN PROBLEMS FOR TUGBOATS AND TRAWLERS MULTI-POINT DESIGN PROBLEMS. 25 ANALYTICAL METHODS IN PROPULSOR DESIGN LIFTING LINE AND LIFTING SURFACE THEORIES PROPULSOR DESIGN USING LIFTING LINE AND LIFTING SURFACE THEORIES THE SURFACE PANEL METHOD FOR THE STEADY AND UNSTEADY PROPELLER ANALYSIS PROBLEMS 33 A FLOWCHART FOR PROPULSOR DESIGN METHODOLOGY 39 REFERENCES 40 3

4 FOREWORD This work intends to give a rational presentation of the modern propulsor-design methodology. The subject shall be presented in the following order: We shall first discuss the phenomenology of a self-propelled ship giving emphasis on the flow of energy in the propulsion system and the terminology connected with this. We shall then present the traditional decoupling process in which a self-propelled ship can be substituted by a combination of a towed ship (i.e. ship w/o propeller) and an open water running propeller (i.e. propeller w/o ship) using the notion of the 'propeller-hull interaction factors'. Assuming that the towing resistance curve of a ship is known (e.g. using systematic series) and that the propeller open water performance (Kt, Kg J) is known as a function of a number of propeller geometric parameters (e.g. B-series) a rational approach shall be presented for preliminary propeller design. Some commonly arising propulsor design problems for conventional ships as well as for trawlers and tugboats shall be elaborated in detail. Finally, we shall present the modern analytical tools used in propulsor design such as lifting line/ surface theories and panel methods, and we shall describe how they can be integrated in the design process in order to increase the freedom of the propeller designer to meet a number of complex design criteria such as optimum hydrodynamic performance, minimum vibratory excitation forces, controlled cavitation development and structural reliability in cooperation with the structural codes. 4

5 1. INTRODUCTION Ideally optimum propulsor design means the selection of a stern arrangement (stern hull lines, propeller geometry, rudder geometry and relative position of the propeller with the rudder and the hull) which propels the ship at maximum hydrodynamic efficiency satisfying other requirements such as minimum vibratory excitation forces, controlled cavitation development and structural reliability. Thus, in the ideal case, the propulsor design problem is a coupled problem of viscous interaction between the propeller, the rudder and the hull flow. This problem reaches the limit of the state of the art in modern CFD and has not yet been attacked in its generality by the researchers. On the other hand, long time ago, model basins and researchers have developed their own heuristic type solutions to the propulsor design problem based on a combination of experimental results (tank tests) and numerical/analytic methods. In the sequel we shall try to present the propulsion design problem in steps of increasing complexity showing that in a modern propulsor design all tools, from systematic series to complex CFD techniques like lifting line/surface theories and panel methods cooperate to provide the required result. We shall start our presentation by making a brief introduction to the self-propulsion phenomenon and the corresponding terminology. We assume that the reader already knows this material from other reference texts, e.g. PNA I* 1.1 PHENOMENOLOGY OF A SELF-PROPELLED SHIP Figure 1 shows the propulsion system of a self-propelled ship together with the various physical quantities connected with it. More specifically: Consider a self-propelled ship traveling at a speed Vs. The main engine provides the required shaft horsepower SHPe at a number of engine revolutions N. The reduction gear changes the engine revolutions to propeller revolutions by a ratio denoted by rg, with a very small amount of energy losses (usually less than 1%). Let SHP denote the shaft power after the reduction gear and n the propeller revolutions (n=ne/rg). SHP is then transmitted through the shaft to the propeller position with an energy loss expressed by the shaft efficiency (due to bearing losses) ris. Let DHP denotes the power delivered to the propeller by the shaft, then is=dhp/shp. Due to the propeller action as an energy transformation device a thrust power is produced denoted by THP=VATB, which propels the ship at a speed Vs. VA denotes the mean effective wake velocity at the propeller disk and TB the thrust produced by the propeller in its operation behind the ship. The ratio of the developed thrust power THP to Superscript numbers indicate references listed at the end of the report 5

6 the delivered power DHP is the propeller efficiency in the self-propelled condition (behind condition) rib=thp/dhp. Self Propelled Ship: Reduction Gear Mai En. 'ne Note: TB=RB Figure 1. Flow of energy in ship propulsion system. Other interesting quantities connected with the self propulsion phenomenon are: the propeller diameter D and the distance of the propeller axis from the free surface h, the torques Q and QB corresponding to Si-IP and DHP (SHP=27mQ, DHP=27mQB), the resistance of the self propelled ship RB (integral of hull surface normal and tangential hydrodynarnic forces) Obviously at the self propelled condition the horizontal forces are in equilibrium i.e. TB=RB. 1.2 REDUCTION OF SELF-PROPULSION PROBLEM TO TOWING AND OPEN WATER PROPELLER PROBLEMS In the history of science 'cutting in pieces' was always a successful method for solving complex problems. In the same way the self-propulsion problem can be made 'more attractive' if we can 'cut it' or 'decompose it' in simpler problems which can then be attacked separately. In order for the process to be applicable a methodology has also to be provided for the reconstruction of the self-propulsion problem from its pieces. Considering now the self-propulsion problem the decomposition process requires: Definition of 'physically sound' simpler problems Introduction of 'physically sound' assumptions that couple the self-propulsion problem with the simpler ones. 6

7 The physically sound simpler problems for our case are: Towed ship (i.e. ship w/o propeller), figure 2. Towed ship: (Ship w/o propeller) Propeller in open water condition: (propeller w/o ship) 4- (hyd-rw ynamic resistance of towed ship) Figure 2. Towed ship. Free running propeller (i.e. propeller w/o ship), figure 3. 1:209 n DHP,=2Tn Propeller disk flow VA=Vs(1-w), s4) wake velocity peller diameter) Figure 3. Free running propeller (propeller w/o hull). The coupling assumptions are (for details refer to PNAI): Coupling with the hull towing problem: R=RB (1-t)+DF, definition of thrust deduction factor t, Where: R denotes the hull towing resistance (i.e. w/o propulsor) at VS, RB denotes the self-propulsion resistance at Vs (Note that RB=TB, figure 1.), DF is the additional towing-force due to Reynolds number difference between model and full scale. Coupling with the open water propeller problem: KTB =KT0 (thrust equalization method) Where KTB, K10 denote the propeller thrust coefficients at self propulsion and open n2 D4), water conditions respectively (Km-7AP K10=T0/(p n2 D4)). Furthermore: VA=Vs (1-w), definition of effective wake fraction, (20/QB=TIR, definition of relative rotative efficiency tir, Where: To, Qo denotes the thrust and torque of the corresponding open water propeller, TB, QB denotes thrust and torque of the propeller behind ship (self-propulsion), VA is the effective wake velocity arising from thrust equalization. Or KQB =KQ0 (torque equalization method), Where KQB, KQ0 denote the propeller torque coefficients at self propulsion and open n2 D5), K.Q0=Q00 n2 D5)). water conditions respectively (KQB=QB/(p VA=Vs (1-w), definition of effective wake fraction, To/TB=TIR', definition of t The factors t, w and rir are traditionally termed propeller-hull interaction factors. 7

8 Using the previous procedure the self-propelled ship can be 'assembled' from: A towed ship with known resistance curve R(Vs), An open water running propeller with known thrust and torque characteristics as a function of advance coefficient (KT(J), KQ(J), J=VA/(n D)) Knowledge of thrust deduction t, effective wake fraction w and relative rotative efficiency TiR as functions of ship's speed. With the previous process in mind the flow of energy in the propulsion system is shown in figure 4. Note that DHP0 and EHP are 'virtual' quantities with regard to the self-propelled ship. Similarly note that the traditional propulsive coefficient P.0 defined as the ratio of EHP/SHP is also 'virtual' for the self-propelled ship. Its use can be attributed to the fact that towing power EHP represents an ideal minimum (propeller suction is absent). SHP DHP P.C. (propulsive coefficient) DHP0 THP 11Fi EHP Figure 4. Flow of energy in the propulsion system showing the effect of the decomposition of the self-propulsion problem to simpler problems. 8

9 We shall conclude this paragraph with a list of useful formulas (thrust equalization method is assumed): SHP =27c Q n DHP =27c Qg n is =DHP/SHP Assuming thrust equalization method: To =TB =T and TH130= THPB =THP (Note: subsript o is used to denote propeller open water data while subscript g is used to denote corresponding self propulsion data). THP =VAT, DHP0=27c Qo n, no =THP/DHP0 is the open water propeller efficiency, I1R =Qo/Q13= DHPo/DHP is the relative rotative efficiency, Ils =THP/DHP=lo rir is the self propulsion propeller efficiency (behind ship), EHP=R Vs is the towing power (effective power), 11H = THP/EHP=(VA T)/(R Vs)=(1-t)/(1-w) is the hull efficiency, TID = DHP/EHP= 10 TIR TIS is the propeller-hull efficiency, and P.C.=EHP/SHP= TIH 110 1R is is the propulsive coefficient. 9

10 2. DESIGN PARAMETERS, DESIGN CRITERIA AND PRACTICAL CONSIDERATIONS 2.1 DESIGN PARAMETERS The state of a propulsion device is characterized by a number of geometric, hydrodynamic and material parameters as follows: 1. Propeller geometric parameters: Propeller tip and hub diameters Propeller design pitch and camber distributions Propeller blade outline which can further be considered as a result of blade skew, rake and chord radial distributions Propeller thickness distribution Number of blades 2. Propeller hydrodynamic parameters: Propeller revolutions Ship's speed Propeller-Hull interaction factors (effective wake w (mean value and radial distribution), thrust deduction t and relative rotative efficiency T1R) 3. Propeller material parameters For conventional metallic blades the Young modules and the Yield strength, For modern FRP propellers: Type of construction (monolithic, sandwich, sandwich with internal reinforcing spars) Radial and chord-wise stiffness distribution (mechanical properties of the composite system, number, distribution and orientation of layers, definition of the possible spar trajectories) Note, that in case of specialized propulsion devices, other parameters may also be entered. For example in case of a ducted propeller the duct geometry has to be introduced to the parameter list (duct chord/diameter ratio, duct type with respect to flow (symmetric, asymmetric, accelerating, decelerating, mixed), duct thickness and camber distributions, gap size between duct and propeller tip etc). Designing our propulsor, means to select all the previous parameters in a way such as to satisfy a number of design criteria (or requirements). 10

11 2.2 DESIGN REQUIREMENTS Regarding the design requirements they can be separated into two different categories as follows: 1. Propulsive performance requirements, for example: Minimization of the required power at a given free running ship speed or maximization of ship speed at a given engine SHP. Minimization of the required power for a given towing force (case of a tugboat or trawler) at a usually small ship speed (towing condition) or maximization of towing force at given engine SHP. Compromise efficiency optimisation between two or more conditions, compromise selection of design pitch. 2. Other requirements: Minimization of propulsor unsteady excitations (propeller axis and hull forces) at a range of working ship conditions, minimization of noise. Minimization of cavitation erosion. Adequate strength and fatigue life (structural reliability) with minimization of material and construction/production costs. Other requirements connected with special type propulsors. For example minimization of spindle torque for the working pitch range of a controllable pitch propeller. The previous requirements lead to an ill-posed (design) problem from the mathematical point of view, i.e. there exist usually 'more than one' solutions (in fact there are infinite solutions). The extraction of a single solution is an art in which knowing of the relative significance of the various physical mechanisms entering the game, engineering intuition (experience) and trial and error, are almost always necessary. Fortunately the various design requirements affect to a different extent the propulsor state parameters and thus a serious decoupling of state parameters with design criteria can be made, allowing the invention of a heuristic approach for the design of complex propulsion systems. 11

12 2.3 CORRELATION OF THE GEOMETRIC DESIGN PARAMETERS WITH THE DESIGN REQUIREMENTS A discussion of the relative dependence of the design requirements on the propulsor geometric parameters is as follows: Parameters affecting optimum propulsive performance: From either systematic experimental studies of propeller performance2'3 or from numerical/analytic methods simulating propeller flow (e.g. lifting line theory) we conclude that the propeller geometric parameters that affect to a leading order the propulsive performance are: Propeller (propulsor) diametett Radial distribution of geometric pitch, radial and chord-wise camber distribution! The effect of diameter in propulsive performance can be made clear using the following result of simple momentum theory': 111=2/(1+(cT+ )1 /2) CT=T/(0.5 p Ao VA2) where ill is the ideal propulsor efficiency, Ao is the propeller disk area, T is the propeller thrust and CT is the propeller thrust loading coefficient. Thus for a given propeller thrust increasing the diameter, the ideal efficiency is increased. Furthermore the pitch and camber distributions affect the radial distribution of blade loading and as a side effect the trailing vortex sheet intensity, which is responsible for the 'induced drag' and the corresponding losses in the form of a continuously expanded downwash current. To a less extend selection of blade outline (blade area ratio, chord, skew and rake radial distributions), number of blades and blade thickness distributions affect propulsive efficiency. In the case of ducted propulsor a number of other parameters are entered like the duct geometry (controls duct loading at towing conditions and affects efficiency at different speed ranges), Gap size between duct and propeller tip and geometry of the propeller tip region (propeller tip of Kaplan type or conventional). Parameters affecting 'other requirements' For a given distribution of axial, radial and tangential ship wake velocities, the parameters affecting to a leading order the unsteady excitation forces (hull or duct forces, propeller blade and axis forces) are: number of blades which with propeller revolutions determine the blade frequency, blade skew distribution which determines the way in which the blade leading edge enters the high wake regions, pitch and camber distribution at the tip region which determine the tip loading distribution and consequently the amplitude of the tip induced pressures on nearby boundaries (hull, rudder, shaft brackets) Transient cavitation also induces unsteady phenomena through a virtual 'unsteady cavity' thickness in addition to the blade thickness. Transient cavitation performance is affected by: 12

13 blade outline (expanded area ratio and radial chord distribution) which determines the distribution of pressure difference over the blade surface for a given total thrust, blade thickness distribution which affects the blade pressure distribution and as a result cavity development, pitch and camber distributions which determine the radial and chord-wise blade pressure distribution, blade structural properties which determine the state of deformation of the blade as a function of pressure loading. Structural reliability is affected by: blade thickness and chord distributions structural design Blade spindle torque is affected by: blade skew distribution In the case of a ducted propeller the existence of the duct acts beneficially to both unsteady excitation forces and transient cavitation extend. In particular the duct can be designed to equalize the non-uniformity of the ship wake and thus to reduce the forcing function for the unsteady propeller performance. Furthermore the duct acts as an integrator/cancelator of the unsteady induced pressures produced by the turning propeller blades. Unsteady excitation Optimum forces (hull or duct Transient Structural propulsive forces, propeller cavitation reliability: performance: blade and axis forces) Propeller Radial distribution Blade outline(blade (propulsor) diameter Number of blades of geometric pitch, radial and chord-wise camber distribution area ratio, chord, skew and rake radial distributions) H Blade thickness Structural design (case of FRP blades) Flowchart presenting correlation between propeller geometric parameters and design requirements, only strong correlations are shown. 13

14 2.4 SELECTION OF GEOMETRIC DESIGN PARAMETERS We can now propose a selection methodology for the design parameters as follows: Propeller diameter the propeller revolutions are selected freely the propeller diameter is always selected asi, he maximum permitted taking into account the hull lines at the stem region and thej Lf ecessary propeller-rudder-hull clearances prescribed by the Classification Societies. In case of given propeller revolutions (e.g. case of preselected engine and reduction gear) the optimum propeller diameter is selected according to the methodology discussed in sections 4.1 and 4.2. Blade pitch and camber distributions Are selected in order to: )rovide the propulsor with a maximum hydrodynamic efficiency, Control the radial distribution of propeller loading that affects both excitation forces and transient cavitation. For example unloading the blade tip region acts beneficially on both excitation forces and transient cavitation w/o serious efficiency reduction. Number of blades Blade number Z is selected to avoid resonance between blade frequency (=Z n) and various ship structural components. Blade number is also selected to minimize shafting system unsteady stress distribution (torsional vibration study). Blade outline Expanded-blade area-to-disk-area-ratio is selected: To reduce the cavitation extension (increasing the blade area the cavitation extend decreases). A useful formula for estimating a starting value for AE/AO is Keller's formula, PNA, Vol II, p1831. To increase the propulsion efficiency (increasing expanded area ratio propulsion efficiency is slightly decreased) Note that the previous two requirements are contradicting! Radial distribution of chord lengths is selected: To control (reduce) cavitation area (radial distribution of chords should follow the radial loading distribution) In combination with thickness to control the cavitation free bucket and the propeller strength. Radial distribution of blade skew is selected to: Minimize the amplitude of unsteady excitation forces on hull and shafting system and to minimize the transient cavitation Minimize the spindle torque at various pitch settings (case of controllable pitch propeller) Radial distribution of blade rake is selected to: 14

15 Evenly arrange the blade at the stern aperture minimizing thus the unsteady interaction between propeller, hull and rudder. Note that by increasing blade rake the centrifugal stresses are increased and a thicker blade root is required. Blade thickness distribution Is selected in order to: Result in better 'lift' to 'form drag' ratio (decreasing thickness acts beneficially in efficiency) Result in satisfactory blade cavitation free bucket (increasing thickness acts beneficially) Satisfy structural reliability (together with chord distribution) at a minimum propeller weight (cost minimization). Note that the first and second of the previous requirements are contradicting! From the previous discussion it has been made clear that efficiency optimisation for a propulsor is mainly achieved by a proper selection of: Propulsor diameter. Blade pitch and camber distributions. All other parameters are selected with 'other criteria' in mind and to a less extend with regard to optimisation of its hydrodynamic performance. 15

16 3. WELL-POSED PROPULSION PROBLEMS The subject of a propulsion problem (for a given ship) is either: To propose a new propulsion installation satisfying certain requirements or criteria To analyze the performance of an existing propulsion installation In the first case the hull is known while the propulsion installation (i.e. the propeller geometry, the engine and the reduction gear) has to be determined. In the second case the hull and the propeller geometry are known while the propeller law SHP-Vs-N has to be determined. We shall term the previous problems the design problem and the performance problem respectively. Inspecting the various physical quantities appearing in the energy flow of a propulsion installation (figure 1) we observe that there are two types of relations between them: Linear relations (e.g. definitions: rir =Q0/QB, DHP =27r QB n, etc) Nonlinear relations: Expressing the dependence of propeller thrust on the ship resistance R(Vs) Expressing the open water propeller thrust and torque coefficients as functions of advance coefficient J and a non-dimensional propeller geometry G: KT=KT(J,G), 1(Q=K,Q(J,G). By applying the linear and nonlinear relations to all quantities entering the problem the remaining independent variables are three in number (by inspection). Thus determination of the solution of a propulsion problem requires three additional constraints that can be either simple equality constraints: for example: SHP=given, D=given, N=given or one of them be a variational (or extremum) constraint: for example: SHP=minimum, D=given, Vs=given With the previous considerations in mind a well-posed propulsion problem requires three additional constraints. It is now possible to visualize the totality of hydrodynamic propulsion solutions of a given ship by introducing a coordinate system with reference axis the quantities SHP, D and N, figure 5. Obviously each point in figure 5 defines a well-posed propulsion problem that can be solved providing all other important quantities (e.g. propeller geometry G or hull speed Vs), which can then be appeared in figure 5 parametrically. 16

17 SHP Re sentation of solutions or given ship Figure 5. Representation of the totality of solutions of a propulsion problem for a given ship. In practical applications we represent the totality of solutions of figure 5 in a 2-D form by introducing a number of 'cut planes' either at constant D (i.e. parallel to SHP-N plane) or at constant N (i.e. parallel to SHP-D plane). Finally note that in the case of propulsion problems for tugboats and trawlers towing at a given speed Vpull, the propeller thrust is no longer a function of ship's speed (which is assumed given) but it is a function of the towing force F which replaces the ship's speed in the previous discussion. 17

18 4. DESIGN EXAMPLES USING METHODICAL SERIES We can now give a number of propulsor design examples using methodical series2'3. In this case the freedom of selecting the propulsor geometry G is limited to very few geometric parameters i.e. the pitch ratio P/D, the number of blades Z and the expanded blade area ratio AE/AO. The main advantage of using series is the simplicity and the speed of the corresponding calculations, which allow the designer to search for optimum over the independent geometric parameters in a minimum time and cost. The main disadvantage is the lack of freedom to optimise the details of propeller geometry (wake adapted propulsors) and to customize them according to other design criteria as discussed in section 2.2. These disadvantages can be relaxed using lifting line and lifting surface theories (next chapter). 4.1 PROPELLER DESIGN PROBLEMS FOR CONVENTIONAL SHIPS. The following four optimum design problems are very likely to occur at the preliminary design stage of a new build conventional ship: Minimization of required shaft power_shp at given ship speed Vs Type 1: Given propeller diameter Type 2: Given propeller revolutions Maximization of the ship's speed Vs at riven engine shaft power SHP Type 3: Given propeller diameter Type 4: Given propeller revolutions All the previous problems are well posed optimum design problems according to the definition of chapter 3. Their solution can be 'extracted' from the totality of solutions of a propulsion problem as presented in chapter 3. For example assume that we have a twin screw ship with a resistance curve R-Vs given in table 1. Then the totality of solutions of figure 5 can be represented in a 2-D form using a number of 'cut planes' either at constant D (i.e. parallel to SHP-N plane) or at constant N (i.e. parallel to SHP-D plane). Figures 5, 6 show cut planes at D=l 83m and D=2 Om while figures 7 and 8 show cut planes at N=300 rpm and N=400 rpm for a B5-71 propeller. For simplicity the propeller-hull interaction factors have been chosen to be t=w=0, 1liz=1 while shaft efficiency has been taken is= 1. It is now obvious that all possible questions regarding propulsive performance of our ship can be answered using plots similar to that of figures 5 to 8. We can now solve the design problems introduced in the beginning of this paragraph as follows: 18

19 Vs (m/s) Resistance (Kp) Table 1. Resistance curve for an example hull. Assume that we have selected the maximum permitted propeller diameter D=1.83m, the number of blades Z=5 and the expanded blade area ratio AE/A0=0.71 (B- series assumed regarding geometric details). Then: The solution of a 'type 1' problem with Vs=17 knots can be directly found from figure 5 to be SHP,,,,,,=-1800 PS (draw a horizontal line tangent to the iso-vs line at Vs=17 knots) with corresponding optimum pitch P/D=1.1 and corresponding propeller revolutions N=375 rpm. The solution of a 'type 3' problem with SHP=1800 PS can be directly found from figure 5 to be V5ni.=17 knots (draw a horizontal line at SHP=1800 PS and find the iso-vs line at which the horizontal line is tangent) with corresponding optimum pitch P/D=1.1 and corresponding propeller revolutions N=375 rpm. Similarly assume that we have selected the propeller revolutions N=400 rpm (i.e. engine revolutions and the reduction gear are known), the number of blades Z=5 and the expanded blade area ratio AE/A0=0.71 (B- series assumed regarding geometric details). Then: The solution of a 'type 2' problem with Vs=17 knots can be directly found from figure 8 to be SHP,r,,=1800 PS (draw a horizontal line tangent to the iso-vs line at V5=17 knots) with corresponding optimum pitch 13/D=0.96 and corresponding propeller diameter D=1.89 m. The solution of a 'type 4' problem with SHP=1800 PS can be directly found from figure 8 to be V5.=17 knots (draw a horizontal line at SHP=1800 PS and find the iso-vs line at which the horizontal line is tangent) with corresponding optimum pitch P/D=0.96 and corresponding propeller diameter D=1.89 m. Except optimum design problems already discussed, other well-posed design problems with equality constraints only are very likely to occur. For example: Assume that we have selected the maximum permitted propeller diameter D=1.83m, the number of blades Z=5, the expanded blade area ratio AE/A0=0.71 (B- series assumed regarding geometric details) and the propulsion installation i.e. the main engine with SHP=1800 PS at 1800 rpm and the reduction gear with rg= Then the corresponding propeller revolutions should be N=1800/3.273=550 rpm. 19

20 This problem is a well-posed design problem (not optimum design but design with three equality constraints: SHP=given, N=given, D=given) according to the definition of the previous paragraph. The required solution can be found from figure 5 as the intersection of a vertical line at N=550 rpm with a horizontal line at SHP=1 800 PS. In this point the propeller pitch ratio is P/D=0.63 while the attainable ship's speed should be 16.5 knots. WAGENIGEN 8 series D= 1.83 m Z=5 AE/A0= Propeller revolutions RPM Figure 5. Cut plane at a constant diameter D=1.83 m (iso-p/d, iso-vs grid) Closing this section we have to mention the existence in the literature4 of specialized diagrams capable of providing the optimum solution for problems of type 1,2,3 and 4 as well as problems of type 5,6,7 and 8, which we shall discuss in the next paragraph. They are based on the observation that the values of KT, KQ, J at which rio is maximized are fiuictions of the following non-dimensional parameters: KT/J2 for problem of type 1,5 KT/J4 for problem of type 2,6 1(Q/J3 for problem of type 3,7 KQ/J5 for problem of type 4,8 We believe that knowing the optimum solution w/o a filling of the 'sharpness' of the optimum point is a disadvantage for the propeller designer. For this reason we have omit to present in more detail the corresponding methodologies. 20

21 WAG EN IGEN B series D m Z=5 AE/A0= ri 2000 :, 1800 wt S k /. ";-"i"- -` ` ".:1 ',-- -7 i 2', A / ' , , ;/- / A 2-7, :..---.:./: /--7F--7 : A"1, / 117 \. \c,t,c)( Propeller revolutions RPM Figure 6. Cut plane at a constant diameter D=2. m (iso-p/d, iso-vs grid) WAGENIGEN B series N= 300. rpm Z=5 AE/A0= Propeller diameter (m) Figure 7. Cut plane at a constant revolutions N=300 rpm (iso-p/d, iso-vs grid) 21

22 WAGENIQEN B series N rpm Z=5 AE/A Propeller diameter (m) Figure 8. Cut plane at a constant revolutions N=400 rpm (iso-p/d, iso-vs grid) 4.2 PROPELLER DESIGN PROBLEMS FOR TUGBOATS AND TRAWLERS. The following four optimum design problems are very likely to occur at the preliminary design stage of a new build tugboat or trawler ship: Minimization of required shaft power SHP at given ship speed Vpull and towing force F Type 5: Given propeller diameter Type 6: Given propeller revolutions Maximization o the towin orce Fat a iven shi 's s eed V,11 and shaft power SHP Type 7: Given propeller diameter Type 8: Given propeller revolutions The totality of solutions of a towing problem can be represented in similar plots like those of figures 5 to 8 with the towing force as parameter instead of ship's speed which is now given (towing speed Vii). Figure 9 presents results for a cut plane at a constant diameter D=1.83 m and pull speed Vpu11=1 knots (iso-p/d, iso-f grid) for a B5-71 propulsor. Figure 10 present results similar to figure 9 but with a Ka 4-70 with 19a duct3 propulsor. Figure 11 present results for a cut plane at constant propeller revolutions N=300 rpm and pull speed Vp11=1 knots (iso-p/d, iso-f grid) for a B5-71 propulsor. Figure 12 presents results similar to figure 11 but with a Ka 4-70 with 19a duct propulsor. 22

23 ' ' ' I WAGENIGEN B series D= 1,83 m Z=5 AE/A0= ri I L / 1,1 / Q 7- / I / y.: N:'4',..,... / : I / i /. i. 1 i / / : :...7, 77.7,5L, "N.".--. / 1,..._ /,,. i. i. / / ', ,_ 2/ - : e / /....'--1., / : I : : //.4 : 0,:li 'N.. /,._-.,z'':; N. z., z //' / I '-''.4.'r-'.1.:-':--::: / / V., 1 /. ' / / / -,,... -,,x '..I, I. N Propeller revolutions RPM l ' I 1 I',, Figure 9. Cut plane at a constant diameter and pull speed V1,u11=1. knots (iso-p/d, iso-f grid) for a B series propulsor KA4-70, DUCT 19a D= 1.83 m Propeller revolutions RPM Figure 10. Cut plane at a constant diameter and pull speed grid) for a Ka4-70, Duct 19a propulsor. knots (iso-p/d, iso-f 23

24 WAGENIGEN B series N= 300. rpm Z=5 AE/A0= q=. r :,., "1' Propeller diameter (m) Figure 11. Cut plane at a constant revolutions and pull speed Vpu11=1. knots (iso-p/d, iso-f grid) for a Ka4-70, Duct 19a propulsor. ns KA4-70, DUCT 19a N= 300. rpm Propeller diameter (m) Figure 12. Cut plane at constant revolutions and pull speed V1,u11=1. knots (iso-p/d, iso-f grid) for a Ka4-70, Duct 19a propulsor. 24

25 We can now 'extract' the solution of the optimum design problems for tugs and trawlers, as defined in the beginning of this paragraph, from the totality of solutions as presented in figures 9 to 12. For example assume that we have selected the maximum permitted propeller diameter D=1.83m, the pull speed Vpu11=1 knots, the number of blades Z=5 and the expanded blade area ratio AE/A0=0.71 and a B series propeller. Then: The solution of a 'type 5' problem with towing force F=30 tons can be directly found from figure 9 to be SHP,,1n=1690 PS (draw a horizontal line tangent to the iso-f line at F=30 tons) with corresponding optimum pitch P/D=0.64 and corresponding propeller revolutions N=420 rpm. The solution of a 'type 7' problem with SHP=1690 PS can be directly found from figure 9 to be F=30 tons (draw a horizontal line at SHP=1690 PS and find the iso-f line at which the horizontal line is tangent) with corresponding optimum pitch P/D=0.64 and corresponding propeller revolutions N=420 rpm. In case of a ducted propulsor, which is usual for tugboats and trawlers (e.g. Ka 4-70 with 19a duct), the solution of the previous 'type 5' and 'type 7' problems can be found in a similar way from figure 10. Comparing results of figures 9 and 10 we note that a ducted propulsor has much better hydrodynamic performance characteristics at towing condition than a conventional propulsor. In our example selecting a ducted propulsor result in a reduction of the required power by =560 PS (i.e. 33%) for the same towing force of 30 tns. Finally problems of 'type 6' and 'type 8' can be solved similarly using figures 11 or MULTI-POINT DESIGN PROBLEMS. In many practical cases a propulsor is required that operates in a nearly optimal way at more than one point. In this case a compromise optimal design is required taking into account the operational requirements at the different propulsive conditions. Some examples follow: Trawlers (or Tugboats): Maximum ship's speed when going to (or return from) the fishing place Maximum towing force at a smaller speed during fishing Destroyer ship: Maximum service speed Maximum battle speed Tank landing craft: Maximum service speed Maximum reverse thrust at zero speed (in order to leave the shore after tank loading) It is obvious that for multi-point design problems selection of a controllable pitch (C.P.) propeller can result in absorption of the full engine power at all conditions which is what is ideally required. On the other hand it should be noted that a C.P. propeller has a hydrodynamically optimum geometry only at the design pitch. At all other pitch settings 25

26 (except the design) the efficiency of the C.P. propeller is reduced (sometimes significantly). This observation together with the high cost for purchasing and maintaining of a C.P. propeller make interesting to search for a conventional propeller that solves the multi-point design problem. Lets give an example. Consider a trawler with the requirement of F=30 tons (towing force) at Vpun=4 knots and maximum free running speed. Consider also that other propulsor characteristics have already been selected (propeller diameter D=1.83m, Ka 4-70 with 19a duct propulsor). Then the totality of solutions for free running and towing cases can be found in figures 13 and 14 respectively. Observe now that the optimum regions of both figures overlap at a range of revolutions N=300 to 370 rpm. Thus selecting N=320 rpm a towing force of 30 tons can be suceded optimally with a P/D=1.2 and corresponding absorbed engine power SHP=1690 PS, figure 14. At the same propeller revolutions and pitch ratio, the free running speed can be found from figure 13 to be V knots with corresponding absorbed power SHP=1100 PS. Note that in case of a controllable pitch propeller the absorption of full power at free running can increase the corresponding speed by nearly 1.2 knot (estimated from figure 13). KA4-70, DUCT 19a 1J 1.83 rn Propeller revolutions RPM Figure 13. Cut plane at constant diameter (iso-p/d, iso-vs grid) for a Ka4-70, Duct 1.9a propulsor, free running case. 26

27 KA4-70, DUCT 190 D= 1.83 rn 2200? F Propeller revolutions RPM Figure 14. Cut plane at constant diameter (iso-p/d, iso-f grid) for a Ka4-70, Duct 19a propulsor, towing at Vpu11=4. knots. 27

28 5. ANALYTICAL METHODS IN PROPULSOR DESIGN 5.1 LIFTING LINE AND LIFTING SURFACE THEORIES Lifting line and lifting surface theories are mathematical models simulating the flow around a marine propulsor without any limitations regarding incoming flow conditions and selection of blade geometric parameters. As a result the designer can select the blade outline (skew, rake and chord radial distributions) with other design requirements in mind (section 2.2) and calculate then the radial pitch distribution and the radial and chord-wise camber distributions that minimize the induced drag in a radial varying effective wake field. Propulsors designed in this way are usually termed 'wake adapted' in contradiction to the propulsors designed from methodical series where only a 'reference mean pitch' is adapted to the mean value of effective wake over the propeller disk area. We shall now give a brief description of what lifting line and lifting surface theories are. The interested reader can find more details on the subject using the references5'6'7'8 at the end of this paper. Lifting line and lifting surface theories are mathematical models simulating the flow around a marine propulsor using a number of simplifying assumptions (linearising assumptions) as follows. LLT (lifting line theory) assumes that the blade chord is small relative to the blade span. Thus the blade collapses to a 'lifting line'. Radial distribution of blade loading is modeled using bound vorticity with a radial varying strength F(r). For the satisfaction of the first Helmholtz law (continuity of vorticity) a trailing vortex sheet emanates from the trailing edge of each blade. For a heuristic satisfaction of the other two (vorticity dynamics) Helmholtz laws, a predetermined roll up shape is usually used for the trailing vortex sheet geometry. By introducing a delineation of flow with the trailing vortex sheet at points on the lifting line (and satisfying thus exactly the dynamical laws of Helmholtz at trailing edge) a nonlinear version of lifting line theory can be obtained which usually gives impressively good predictions of propeller hydrodynamic performance. Figure 15 shows a lifting line modeling of a two-component propulsor6. LST (lifting surface theory) assumes that blade thickness is small relative to blade chord. Thus the blade collapses to its mean camber surface'. By applying a rational linearization process the blade thickness effect can be introduced by means of a suitably selected distribution of point sources a(r,$). Radial and chord-wise distribution of blade loading is modeled using bound vorticity with chord-wise and span-wise varying strength y(r,$). For the satisfaction of the first Helmholtz law a trailing vortex sheet emanates frrn the trailing edge of each blade. A predetermined roll up shape is usually used for the trailing vortex sheet geometry using information from similar calculations at a lifting line leve114. Figure 16 shows a lifting surface modeling of a propeller. Figure 17 28

29 shows the surface paneling used in numerical lifting surface calculations together with the total velocity on control points. VA (r) Trailing vortex sheets Propeller Grimw heel Figure 15. Lifting line flow model for a two component propulsor (propeller and Grim wheel). a osed vortex for bl ade bound vartidty modeling 13 mintrd Horsmhoevorte< Figure 16. Lifting surface flow model for a propeller. 29

30 Figure 17. Surface paneling used in lifting surface calculations together with the total velocity on control points. 5.2 PROPULSOR DESIGN USING LIFTING LINE AND LIFTING SURFACE THEORIES Before we give an example of propulsor design using LLT/LST we shall try to explain the philosophy under their usage. In this respect it is necessary to retrieve some theoretical results from lifting line theory of 3-D wings9. For the case of a 3-D wing in a uniform flow field the application of lifting line theory can give some very interesting 'closed form' relations regarding flow hydrodynamics. Thus the lift of the wing is given by: L = 2n-pV2 s2 and the lift coefficient (L =CL 1 pv2 S) given by: 2 where: CL = icalar 30

31 Aspect Ratio = (2s)2 = AR, S denotes the wing surface, s the wing span, AI is the first coefficient in a sin series expansion of the span-wise circulation distribution F(y) (Re) = 4s VEA sin n9, y = -s cos 19) and V is the undisturbed velocity at infinity (incoming flow). 1 The induced drag coefficient of the wing ( D = CD - pi/2 S) is given by: 2 2A2 3A2 A 4A2 where 8 is a positive number given by: 8 = , + Let now consider the consequences of the previous relations in wing design. Assuming constant CL, there exists an absolute minimum for the induced drag CD! which can be achieved if 8=0 i.e. A2=A3=A4=...=0. It can be proved that if the wing is planar with similar thickness forms along span, the validity of 8=0 implies an elliptical wing outline and vice versa9. In case now that some of the wing geometric parameters have been selected in advance, a number of coefficients in the circulation distribution F(y) have been constrained to non-zero values. By selecting the remaining coefficients in the 8 expansion equal to zero, a 'local minimum' for the induced drag can be obtained. Thus by constraining the geometry of the wing further and further the minimum induced drag is trapped to greater values. Systematic propeller series introduce strong constraints in propulsor geometry. Regardless of this, by leaving free the pitch-ratio P/D that has a leading effect to induced drag, they can be used in a preliminary design stage to obtain indications regarding optimum design point. By freeing more geometric parameters (e.g. radial distribution of pitch and radial and chordwise distribution of camber) a better minimum can be obtained for the induced drag. This is exactly what the LLT/LST tool is doing. AL /11 Az CDI = + 8) rrar In practical applications the LLT/LST tool is usually used to calculate the radial distribution of pitch and radial and chord-wise distribution of camber, which maximize the wake adapted efficiency under the following constraints: Given chord-wise distribution of loading (e.g. NACA a=0.8). Given both radial and chord-wise distribution of loading. Blade outline is assumed given (selected to satisfy other criteria, section 2.4). In the first case selecting an evenly distributed chord-wise loading results in minimization of cavitation danger. In the second case by properly prescribing the radial load distribution at the tip region a 'tip unloaded' propellers can be designed with beneficial geometric characteristics regarding induced vibration and noise. After the previous decisions have been made the theory of chapter 3 regarding well-posed propulsion problems can be applied and the totality of solutions can be obtained in the form of figure 5. We can also use 'cut planes' in order to represent the totality of solutions in a 2- D form. 31

32 Lets give an example. Assume, that we have a twin-screw ship similar to that taken in section 4.1 with resistance curve according to table 1 and propeller-hull interaction factors t----w(r)=0 (i.e. uniform speed of advance at propeller disk), 1R=1 and shaft efficiency 1is=1. Assume further that we have select a NACA a=0.8 chord-wise loading distribution. Finally, assume that the blade outline has been selected according to table 2. r (m) Chord (m) Mid-chord line Mid-chord Maximum Skew angle rake (m) thickness (m) (deg) , , , , Table 2. Propeller blade outline and maximum thickness distributions The totality of solutions as obtained by LLT/LST is shown in figure 18 (blue grid). In the same figure the totality of solutions using B-series is also shown (red grid). Comparing the two grids we observe that both predict nearly the same optimum revolutions with a small shift of LLT/LST prediction to the left. In this point please note that the results of B-series contained in figure 18 refer to model scale while the LLT/LST results are for the full-scale propeller. Taking into account the beneficial effect of scaling in model efficiency, the real difference between LLT/LST and B-series at optimum revolutions is estimated to be 2-3% (and not 5% which is shown in figure 18). At 'of optimum' values of propeller revolutions the differences in efficiency between LLT/LST and series prediction is increased. Furthermore note that the differences observed in the pitch ratio between the two methods are due to the fact that a LLT/LST propeller has always a variable pitch along radius and a different camber distribution at each pitch ratio while in order to produce figure 18 we have to: Omit the effect of camber distribution, Make an assumption to extract a 'reference pitch' from the pitch distribution. It is usual to consider as a reference pitch of a variable pitch blade the pitch at 75%R (pitch at equivalent radius")) and this has been used for the extraction of the reference pitches shown in figure 18 (blue grid). 32

33 c,. 41 c n WAIVILVINIAAWAF MIBMIN111011/ Pills\*Si Jr/1VMM "WAWA! FINIMMI/1_47,,,m 1f ,,,,... r ariv IgArlOr. WAN rilrapil rarji 0p ITIPfrAir ri FAp awra...1. tadi 5, kn rpm r I I I I I Figure 18. Comparisons of B-series with LLT/LST. Red grid denotes B-series calculations while blue grid denotes LLT/LST calculations. ue to the qualitative similarities in the predictions between series and LLT/LST alculations it is usual in practice to use series initially to investigate for the optimum pointl estimation of optimum revolutions, pitch ratio etc) and then use LLT/LST for 'fine tuning' lof propeller geometric details and design a 'wake adapted' propulsor.,' 5.3 THE SURFACE PANEL METHOD FOR THE STEADY AND UNSTEADY PROPELLER ANALYSIS PROBLEMS The surface panel method (or boundary element method) is a great innovation in propeller steady and unsteady performance calculations. Although LLT/LST and panel methods can solve similar problems, the panel method uses the exact propeller-duct-hub geometry without any linearising assumptions. It can thus predict the blade (and duct) pressure distribution with much greater accuracy than LST. Thus Panel codes can be used in the final stage of propulsor design to predict the steady (and unsteady) blade pressure distributions (especially at the leading edge region) and permit the designer to forecast and correct possible cavitation inception problems. The method described in the sequel has been developed at NTUA (National Technical University of Athens) and is based on a modification of the Hess and Valarezo 1 formulation. Detailed results of this method have been presented at an ITTC workshop on Panel and RANS methods in Grenoble12. A panel method for the unsteady performance 33