:OR PREDICTING THRUST DEDUCTION USING PROPELLER %FACE THEORY Report 77-0087 RESEARCH AND DEVELOPMENT CENTER DAVID W. TAYLOR NAVAL SHIP Bethesda,Md. 20084 A METHOD FOR PREDICTING THRUST DEDUCTION USING PROPELLER LIFTING SURFACE THEORY i | | \ Bruce D. Cox and Allen G. Hansen) 0 U Vii t \ COLLECTION , N\ we APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT November 1977 Report 77-0087 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 00 TECHNICAL DIRECTOR OFFICER-IN-CHARGE CARDEROCK SYSTEMS DEVELOPMENT DEPARTMENT "1 SHIP PERFORMANCE DEPARTMENT Hs STRUCTURES DEPARTMENT as SHIP ACOUSTICS DEPARTMENT ri MATERIALS DEPARTMENT 28 01 OFFICER-IN-CHARGE ANNAPOLIS AVIATION AND SURFACE EFFECTS DEPARTMENT 46 COMPUTATION, MATHEMATICS AND LOGISTICS DEPARTMENT 18 PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT 597 CENTRAL INSTRUMENTATION DEPARTMENT 49 WLU M KONI o032e441 4 Oo 0301 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE Se a ea T. REPORT NUMBER 2. GOVT ACCESSION NO, 3. RECIPIENT'S CATALOG NUMBER DINSRDC 77-0087 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED A METHOD FOR PREDICTING THRUST DEDUCTION USING PROPELLER LIFTING SURFACE THEORY Research and Development 6. PERFORMING ORG. REPORT NUMBER 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(a) B.D. Cox and A.G. Hansen ELEM 9. PERFORMING ORGANIZATION NAME AND ADDRESS M WORK U David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 CONTROLLING OFFICE NAME AND ADDRESS Naval Sea Systems Command Washington, D.C. 20362 ENT, PROJECT, TASK NIT NUMBERS (See reverse side) 12, REPORT DATE November 1977 13. NUMBER OF PAGES 88 15. SECURITY CLASS. (of thie report) UNCLASSIFIED DECL ASSIFICATION/ DOWNGRADING SCHEDULE 11. 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 1Sa. 16. DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Propeller Propeller—Hull Interaction Thrust Deduction Potential Flow Lifting Surface - ABSTRACT (Continue on reverse side if necessary and identify by block number) An analytical method is presented for predicting the added resistance (thrust deduction) arising from propeller—hull interaction. The theory is formulated in terms of the potential flow about the hull and appendages which are represented by surface singularity distributions. The influence of the propeller is derived from lifting-surface theory including the effects of blade number, thickness, skew, rake, and radial and chordwise load dis- tribution. In order to determine the interaction force, propeller-—induced (Continued on reverse side) DD en, 1473 EDITION OF 1 Nov 65 Is OBSOLETE UNCLASSIFIED S/N 0102-LF-014-6601 —— SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 10) IED funding under Element 62766N, Work Unit No. 1552-119, NAVMAT DIRECT Laboratory Funding under Element 62543N, Work Unit Nos. 1520-004 and 1500-200. (Block 20 Continued) velocities and the modified hull pressure distribution are computed with appropriate corrections to the hull singularities. The axial force is then derived by integration of the pressure on the hull and also by appli- cation of the steady-flow Lagally theorem. The usefulness of this technique is illustrated by its application to several stern propeller—body-of-revolution configurations. It is shown that stern appendages contribute up to 25 percent of the total thrust de- duction. The relative contributions of propeller loading and thickness are examined. It is found that the lifting-surface representation predicts 10 to 20 percent lower thrust deduction than the classical lifting-line (sink disk) approximation. Calculations for a series of four raked pro- pellers illustrate the significant (over 50 percent) attenuation of the interaction force as rake is increased. It is concluded that the method is useful for both the analysis of a given design and for parametric in- vestigations of higher efficiency propeller—afterbody configurations. The method may also be extended to treat contrarotating and ducted propellers. UNCLASSIFIED a SECURITY CLASSIFICATION OF“THIS PAGE(When Data Enterod) TABLE OF CONTENTS Page IAESURACT. aE Gere Sue lalsgt Ey Tal Pie ieal CU MESR es! cls ev lies. ee aL il /NONCOESHERAUEIAYS TONTROMIIAIETION Gg 6 Go 6 06b 6 60 0 O60 6600 so) cco i ENTRODUGELON Says, Sacues liu nes) Moat 18. MIMR SU LIAO RS 1 PROPELLER-HULL INTERACTION IN POTENTIAL FLOW .........22-. 5 SINGULARITIES REPRESENTING THE HULL ANDMBAPPENDAGES re An ess Verba cliGoms) co, bebe: oe ce bok Gap i EOL VS ada aa 5 SOLUMEONSEOR THE THRUST DEDUCEEON Ss 2 4 o @oh we gels 20 2 <0 6s PROPELLER PEELD POINE VELOCITIES = . as 2 6 Sw 8 ee SOS S a 10 TALHLSTIN C= TUN FAS hHB ORV aes Oe meie o MAUR ME Lone Moke des ie Slovan Lil IE TRINCSSURFAGE GCORREGEIONS: 5 a «Use cuties cule iu. creek tertiles 17 EXAMPLE CALCULATIONS AND COMPARISON WITH EXPERIMENTS ....... 22 EXAMPLE 1: APPENDED SERIES 58 FORM .........46e6-4+4e88-8 Vi EXAMPLE 2: COMPARISON OF THEORETICAL AND EXPERIMENTAL PRESSURE DISTRIBUTIONS ON THREE BODEES! OF REVOLUDEOND ls) cei ceciienel a enone 32 EXAMPLE 3: BODY OF REVOLUTION WITH STERN APPENDAGE Sic A vice Sed ies 6h feoce te ter sfeesee Ow (Ss Saree esos Ih EXAMPLE 4: APPENDED BODY OF REVOLUTION WITH FOUR RAKED PROPEPLERS PURE. Vise Acde. Re BOSON Aas: 50 CONCLUSIONS AND RECOMMENDATIONS .........s eee -eeee 61 ACKNOWLEDGMENT Samtesacius: lac (a oa ce Hee eis: roles Keane eanel Cohan cc hmen cues 65 RERERENGES:, a «eee nl SarSuces eee ies CUM scchoe te here accn ceca ey, See uM SECU Me 66 APPENDIX A — THE FORCE GENERATED ON A BODY IN POTENTIAL FLOW BY AN ISOLATED SINGULARITY ........... 69 APPENDIX B — SINK DISK REPRESENTATION OF A MODERATELY TOADED sPROPELEERw 3 2 « tkeeaw. Wiliogent. te Sreahwe ts 75 atatat 10 11 12 13 14 iL,5) 16 Ld 18 LIST OF FIGURES Coordinate Systems for Propeller Lifting Line Theory Force and Velocity Diagram at Lifting Line ..... Field Point Velocity Calculations in Propeller Plane Representation of Propeller Blade by Source and Vortex Line Lines (Projected View Looking Forward) Interaction Analysis - Computational Procedure ... Pie@peiiler S3O38. o o o 66 0 0 0 0 oO 6 0 ooo Representation of NSRDC Model 4620 ........ Lifting Line Circulation and Pitch Representing IL@galsiinye OIE Ireepeiliere S38 co 5 6 60 6 0 0 0 0 0 CO Control Points and Propeller Induced Velocities om Nyoeinclad! Series So} Boahy oo bo 6.6 00000606 00 Radial Distribution of Propeller Sink Strength Derived From Lifting Line Calculations ....... Calculated Longitudinal Distribution of Thrust Deduction Force on Model 4620 (Body Only) ....... Representation of DTNSRDC Models 5225-1, 5225-2, and’ S225= 35 \a.cn oe eh ch cee i ey ad edie es oan) Se Measured and Calculated Pressure Distributions on DINSRDC Models 5225-1, 5225-2, and 5225-3 ..... Profile and Quadrilateral Representation of DIUNSIRNG Wiogleil SLA 5 6 56 6 6 oo 6 ol Design Circulation and Hydrodynamic Pitch Dsleerealyoleatoymsy Cue Ireoyoyeulilese USGI 6 56 6 o 0 0 0 0 Oo Control Points and Propeller Induced Velocities on Appended Body of Revolution (DTNSRDC 5224-1) Preopaiilee ASOWN coo 6006 6 6G 6 oO lo Longitudinal Distribution of Thrust Deduction (Cumulative From Stern) on Body of Revolution (DINSRDC Model 5224-1, Propeller 4567A) ...... Stemn) Alliteratiom on) Modell 5224-2) ii 2) ss) eee iv Page 113; 13) 18 23 25) 28 29 30 31 34 35 36 Al 45 46 47 49 Slt IS) = jojo = M3) = 10 = Wil = Lifting Line Circulation and Hydrodynamic Pitch Representing Loading of Propellers 4486, 4487, UWS, alas) HASSE) 5 G6 6 6 6 6 Oo 6 Radial Distributions of Total Rake of Propellers 4486, 4487, 4488, and 4489 ... Longitudinal Distribution of Thrust Deduction (Cumulative From Stern) With Four om Boky S224=2 5 5 696 6 6 6 6 0 Propellers fet Ye iejey! le e ° . ° e Propeller Induced Axial Velocities on Appended Body of Revolution 5224-2... . Contributions to Propeller Induced Axial Velocity on Body 5224-2 ..... Control Volume for Analysis of Force on Body .. LIST OF TABLES Propeller-Body Configurations in Thrust Deduction Cailcullattion's! “sf ssw) ce er es Offsets and Particulars for Series 58 Form, Modedie4G20) Aa ast iar a ar eee ee Computed and Measured Values of Thrust Deduction Fraction for Appended Series 58 Body (DINSRDC Model 4620, Propeller 3638)... Geometry of Propeller 4577 .. Computed and Measured Thrust Deduction Fraction for Appended Body of Revolution (DINSRDC Model 5224-1, Ryeoyaaililere EGY) 6 5 56 9 6 0 4 Principal Characteristics of Four On Wealeil SPAY 5 5 6 6 6 50 6 0 0 Geometry of Propeller 4486 ... Geometry of Propeller 4487 ... Geometry of Propeller 4488 ... Geometry of Propeller 4489 ... Raked Propellers . . e . . ° e . Computed and Measured Values of Thrust Deduction Fraction for Propellers 4486, 4487, 4488, and 4489 on DINSRDC Model 5224-2 . . . . . . ee Page 57 58 61 62 63 69 26 27 33 40 48 S22 53 54 55 56 59 Gar 1j Th mL SL SL BL NOTATION Chord length Coefficients defined by equation (8) Propeller thrust coefficient, Unit vector Camber Fluid force acting on body Advance coefficient, en QR Bessel functions of the first kind Lift/unit radius; body length Momentum Point source strength Unit vector normal to body surface Pressure Pitch Torque Position vector Propeller radius vi Ry Propeller hub radius Rn Body length Reynolds number S Surface Rp Resistance ie Thrust deduction fraction; time; thickness i Thrust Ti (G2,2,0) = (uy ,uy ug) = Induced velocities u(x,r) Circumferential mean induced velocities V Ship speed Vv Fluid velocity w(r) Wake fraction (x,y,z) Cortesian coordinate system (x,r,¥) Cylindrical coordinate system fixed to the propeller Z Number of Blades Bs Hydrodynamic pitch angle Y Vorticity r Bound circulation vii Kronecker delta function Drag/lift ratio Fluid density Source strength per unit area Velocity potential Propeller rotational speed viii ABSTRACT An analytical method is presented for predicting the added resistance (thrust deduction) arising from propeller-hull in- teraction. The theory is formulated in terms of the potential flow about the hull and appendages which are represented by surface singularity distributions. The influence of the pro- peller is derived from lifting-surface theory including the effects of blade number, thickness, skew, rake, and radial and chordwise load distribution. In order to determine the inter- action force, propeller-induced velocities and the modified hull pressure distribution are computed with appropriate corrections to the hull singularities. The axial force is then derived by integration of the pressure on the hull and also by application of the steady-flow Lagally theorem. The usefulness of this technique is illustrated by its appli- cation to several stern propeller-body-of-revolution configura- tions. It is shown that stern appendages contribute up to 25 percent of the total thrust deduction. The relative contribu- tions of propeller loading and thickness are examined. It is found that the lifting-surface representation predicts 10 to 20 percent lower thrust deduction than the classical lifting-line (sink disk) approximation. Calculations for a series of four raked propellers illustrate the significant (over 50 percent) attenuation of the interaction force as rake is increased. It is concluded that the method is useful for both the analysis of a given design and for parametric investigations of higher efficiency propeller-afterbody configurations. The method may also be extended to treat contrarotating and ducted propellers. ADMINISTRATIVE INFORMATION This work was performed under the in-house independent research and exploratory development program of the David W. Taylor Naval Ship Research and Development Center (DTNSRDC) (Work Unit No. 1552-119) and the High Speed Submarine Direct Laboratory Funding Program (Work Unit No. 1520-004 and 1500-200). INTRODUCTION The interaction force arising from propellers operating in close proxi- mity to the ship's stern is a familiar concept to naval architects. The propeller accelerates the flow over the hull afterbody. For sufficiently fine ship forms where flow separation effects are minimal, the velocity increase and accompanying reduction in pressure increases the hull pressure drag. The higher velocity also increases the wall shear stress, and hence, the frictional resistance. The net result is that the delivered propeller thrust must be greater than the hull resistance in the absence of the propeller. This increase in resistance due to the propeller-hull interaction is defined in terms of the thrust deduction fraction t, where Ro is the bare hull resistance and T is the propeller thrust. The thrust deduction must be known in advance so that a propeller design will meet the specified propulsion requirements. One approach is to conduct model-scale propulsion tests using a stock propeller with similar principal characteristics. While this technique has proven reasonably satisfactory for many conventional designs, a large number of experiments are required to investigate the effects of different afterbody forms, propeller loca- tions, blade geometries, and loading characteristics. Thus, an analytical prediction technique is desirable both from the standpoint of predicting the interaction force for a given propeller and hull design, and for economi- cally investigating more efficient propeller-hull configurations. Various techniques for the analysis and prediction of thrust deduction have been reported in the last forty years, as shown in the comprehensive bibliography presented by Nowacki and Sihevanas Dielnena was the first investigator to provide a reasonable theoretical analysis of the inter- action force between a hull and propeller. To represent an axisymmetric body, he applied the method of discrete singularities on the body axis together with a single point sink to represent the propeller. With this model, it was possible to relate the thrust deduction to the thrust loading coefficient. During the 1940's, some of Dickman's ideas were extended, as & TNowaeiae H. and §.D. Sharma, "Free Surface Effects in Hull Propeller Interaction,'' The University of Michigan College of Engineering Report 112 (Sep 1971). A complete listing of references is given on pages 66-67. “Dielemaninn, H.E., "The Interaction between Propeller and Ship with Special Consideration to the Influence of Waves," Jahrbuch der Schiffbautechnischen Gesselschaft, Vol. 40 (1939). outlined by Heinen in his survey paper. Ronriacieoukersie also used the method of singularities, but used a constant-strength sink disk as a pro- peller model, and corrected the body sources to account for both the pro- peller induced flow and the boundary layer displacement thickness. More recent developments have logically followed from advances in pro- peller theory and the advent of high-speed digital computing capabilities. Thus, Fevecidiees was the first to apply the Douglas—Neumann three-dimension- al potential flow calculation to represent the body. In a later investiga- tion, Beveridge” introduced a sink disk with radially varying strength derived from propeller lifting-line theory. Based on this method, predic- tions for three different realistic hull forms, including one surface ship, compared reasonably well with experimental data. A similar technique has also been applied to contrarotating seapellilens, 72° In the present work, an extended treatment of the potential flow analy- sis is developed which differs in two fundamental respects from previous ap- proaches. First, a more comprehensive and realistic representation of the propeller is introduced, based on a lifting-surface formulation. This over- comes the limitations of the lifting-line sink-disk approximation by including the additional effects of blade thickness, skew, rake, and chord- wise load distribution. As such, the propeller calculation is comparable in scope and accuracy to currently available lifting-surface design methods. eset im, G., "The Thrust Deduction," American Society of Naval Engin- eers, Vol. 63 (1951). “voicvilmeteeouleoneley. B.V., "Stern Propeller Interaction with a Stream- line Body of Revolution," International Shipbuilding Progress, Vol. 3, No. 17 (1956). ececides- J.L., "Pressure Distribution on Towed and Propelled Stream- line Bodies of Revolution at Deep Submergence," David Taylor Model Basin Report 1665 (Jun 1966). romemadee. J.L., "Analytical Prediction of Thrust Deduction for Sub- mersibles and Surface Ships,'' Journal of Ship Research, Vol. 13, No. 4 (Dec 1969). Ulcer D.M., "Development and Application of a Lifting-Surface Design Method for Counterrotating Propellers," Naval Undersea Center TP 326 (Nov ID)/2)c Sysnecidiee, J.L., "Thrust Deduction in Contrarotating Propellers," Naval Ship Research and Development Center Report 4332 (Nov 1974). In previous investigations, the thrust deduction has been derived by applying the Lagally steady-flow theorem to the propeller sink-disk singu- larities. In the present formulation, it is more convenient to consider the body flow directly. Thus, time-averaged propeller induced velocities and modified hull pressure distributions are computed, including appropriate corrections to the body singularity strengths. The force is then calculated by integrating the pressure and also by applying Lagally's theorem to the body singularities. It is assumed, as before, that for a given representa- tion of the propeller, the added hull resistance arises entirely from the reduction in afterbody pressure and that this pressure distribution may be derived solely from potential flow considerations. Although it is recog- nized that the boundary layer at the stern is relatively thick, the poten- tial flow formulation has been widely accepted on the basis of agreement between predicted and measured thrust deduction. Recently, wind-tunnel experiments were conducted on streamlined bodies of revolution with and without a propeller in opsracielonins The results show that while the theory cannot satisfactorily predict the absolute pressure distributions near the stern, the difference in pressure due to the action of the propeller is predicted remarkably well. It was also found that increases in wall shear stress contribute less than 5 percent of the integrated pressure force. For these reasons, it appears that the Douglas-Neumann potential-flow calcu- lation is a sound approach, at least for nonseparating hull forms. Moreover, the calculation of the detailed pressure distribution will serve as a necessary first step in future treatments of the viscous flow problem. In this report, the theoretical basis and numerical techniques for predicting the thrust deduction are presented. The analysis is restricted to deeply submerged bodies, for which the hull potential flow calculation is only briefly reviewed, being extensively documented in the cited literature. (The theory can, in principle, be extended to surface ship applications; the free-surface would be approximately represented by Eanes, T. et al., 'Propeller/Stern/Boundary-Layer Interaction on Axisymmetric Bodies: Theory and Experiment," DITNSRDC Report 76-0113 (Dec 1976). reflecting the hull and propeller (images) in the waterline plane.) The propeller representation and field point velocity calculations are dis— cussed in some detail. The further analysis considers the determination of the modified hull pressure and solution for the interaction force by pressure integration and by application of Lagally's theorem. Comparisons between theoretical and experimental thrust deduction are given for deeply submerged stern propeller body-of-revolution configura- tions both with and without stern appendages. These configurations were chosen for initial calculations because the body geometry characteristics afford easier computation. However, it is noted that in application to submarines and torpedoes, the thrust deduction is of great practical im- portance in selecting propeller characteristics (e.g., diameter) for maxi- mum propulsive efficiency. In the examples preSented, it is shown that stern appendages develop as much as 25 percent of the thrust deduction. The relative contributions of propeller loading and thickness are examined and compared with the classical lifting-line sink-disk results. It is found that lifting-surface effects reduce the thrust deduction by 10 to 20 percent. Calculations for a series of four rakes illustrate the significant (over 50 percent) attenuation of the interaction force as rake is increased. Based on these examples, it is concluded that the theory provides a useful technique for both the analysis of a given design and for parametric studies of higher efficiency propeller-hull configurations. PROPELLER-HULL INTERACTION IN POTENTIAL FLOW SINGULARITIES REPRESENTING THE HULL AND APPENDAGES The propeller-hull interaction analysis rests on the computation of the potential flow about the hull in the presence of the propeller. A great deal of effort has been devoted in the past to developing accurate and computationally efficient techniques for calculating the flow about arbitrary three-dimensional bodies. In the method of Hess and Smile, on! the body surface is approximated by planar quadrilateral elements and the solution is derived in terms of simple source distributions. A recently developed computer endle based on this approach is employed in the present work. Briefly, the formulation is as follows. The body is assumed to be deeply submerged and advancing at a constant velocity, V_, in an incompressible, inviscid fluid. By considering only the time-averaged propeller disturbance field, the flow is steady relative to a Cartesian coordinate system fr = (x,y,z) advancing with the body. Outside the propeller blade row and slipstream the flow is irrotational. Thus, a velo- city potential $(r) exists such that V(z) = Vo(t) and 2 BS VY OC) 20 (1) where f is outside the body and the propeller slipstream. The boundary conditions are that the velocity must be tangent to the body surface, ) + Vo(z,) = 0 (2) and approach the free stream value at a large distance from the body- propeller system Vo(r) > Vas as |r| > @ (3) A solution for $(f) which satisfies equations (1) and (3) may be written in . ° ° — terms of a surface source distribution, o(Z,), as o(r_) — —> = =" (@)=-— f—* ds(Z,) +F + Vo +96 () (4) Use ta = | aaa B B Ores, J.L. and A.M.O. Smith, "Calculation of Nonlifting Potential Flow About Arbitrary Three-Dimensional Bodies," Journal of Ship Research (Sep 1964). ese, J.L. and A.M.O. Smith, "Calculation of Potential Flow About Arbi- trary Bodies," Pergamon Press, Progress in Aeronautical Sciences, Vol. 8 (1966). 12 Dawson, C.W. and J.S. Dean, "The XYZ Potential Flow Program," NSRDC Report 3892 (Jun 1972). Here $,() is the potential due to the propeller which satisfies the condition V6, (F) +0 |r] +2 (5) Inserting this expression for $(f) into equation (2) yields an integral equation for the unknown source strengths iL = 1 VA a | 1 EW _ =, lv =y) 5 OQ) - i o(re) n(r,)°Vo | ds (ry) = n(r,) [V+Vo, (FB) (6) B PE 3 It may be observed that the propeller presents a modified onset flow to the body (right-hand side of equation (6)). Moreover, the change in source strength caused by the presence of the propeller depends only on the com- ponent of induced velocity normal to the body surface. Anumerical solution of equation (6) is obtained by representing the body surface using planar quadrilateral elements. It is assumed that the source density is constant over each element and the integral equation is replaced by a set of linear algebraic equations. dy Gis Cia ep (7) J where the coefficients, Ci5? are given by 1 1 ea eles Soe = Sao ds °5 Dl ft is = | j ey j ee a J and Ve is the onset flow evaluated at a selected control point (e.g., cen- troid) of each quadrilateral is is — — Vaz ena! (9) Since the coefficients ee depend only on the body geometry, the inverse matrix om need only be derived once for a given hull form. It is then possible to rapidly compute the source strengths corresponding to a O. ) 5G . Vie (10) al 1 Once the body source strengths are known, it is straightforward to compute velocities and pressures at points on the body surface or in the surrounding field. It is also possible to determine the resultant force acting on the body as described below. SOLUTION FOR THE THRUST DEDUCTION In general, the influence of a stern propeller causes a net increase in the hull resistance. Two methods are available to compute the force exerted on the body. In the first approach, the axial force, Fo is derived by integrating the pressure over the body surface, rose -f 1) = wea) fale) cee) (11) x x B | with without propeller propeller where the pressure is found from Bernoulli's equation Booey ov (12) The velocity is computed at the control points of each quadrilateral as AGS) = 2 1 =) 6s, +H, * HG) j Bioow.! 2 Be ied Oe A distinct advantage of this method is that the effect of body form can be determined by examining the distribution of the pressure integral. Also, a detailed knowledge of the pressure distribution is an important first step in solving the viscous flow over the afterbody. The interaction force may also be derived by application of Lagally's dheeey ee for a body immersed in a steady potential flow. The solution for the force on a body due to an isolated point source is developed in Appendix A as Fe=- of o(Z,) V_(p) ds (r) (13) B where v.@,) is the undisturbed onset velocity generated by the singularity. This equation is also valid for a point doublet singularity. It will be shown subsequently that the propeller disturbance arises from suitable distributions of sources (blade thickness) and doublets or equivalently, line vortices (blade loading). Thus, the axial force arising from the propeller-hull interaction may be written as po =e a0 of o(E,) Vo (Fy) dS(z,) (14) S B or in discrete form Po os @¢ 2 Gi, (E) ey AS, (tr) (15) If only the total force is required, this form is simpler for computation and, in any case, provides a convenient check in the numerical evaluation of equation (11). Once the interaction force is found, the thrust deduction fraction, t, is given by Oe 5 = 7 (16) Se Cameaee W.E., "The Force and Moment on a Body in a Time-Varying Potential Flow," Journal of Ship Research, Vol. 1, No. 1 (Apr 1957). Sala Retin, L.M., "Theoretical Hydrodynamics," The Macmillan Company, New York, N.Y., 2nd edition (1950). where T is the propeller thrust. Up to this point, it has been assumed that the propeller has been represented by an appropriate distribution of singularities external to the body. Within the framework of the potential flow formulation, the solution to the interaction problem is derived com- pletely in terms of propeller disturbance velocities at points on the body surface. It is now appropriate to set forth the theoretical basis and numerical techniques for calculating these velocities. PROPELLER FIELD POINT VELOCITIES In the foregoing analysis, the modified body flow in the presence of a propeller is derived in terms of induced velocities on the body boundary. It is primarily in the treatment of the propeller that the present inter- action analysis differs from earlier investigations. Owing largely to advances in design theory and high-speed computing capabilities, it has been possible to introduce a more realistic analytical representation of the propeller. Previously, the propeller was approximated as a sink disk. In that model, the diameter, axial location, and radial distribution of loading are explicitly represented. In fact, as will be shown shortly, a sink disk of strength generates the circumferential average velocity field of a moderately loaded lifting-line representation of a propeller with bound circulation IT, pitch 27 r tan Bs (r), and Z symmetrically spaced blades. Beverideas has demon- strated that this model satisfactorily predicts the thrust deduction for conventional propeller geometries. However, it is evident that this sim- plification breaks down for raked propellers in which the blade sections are displaced axially. Moreover, the effects of blade thickness and finite chordlength, while perhaps of small consequence to the thrust deduction, may be important to the body pressure distribution and boundary layer characteristics in the immediate vicinity of the propeller. In view of 10 these limitations, a more complete representation of the propeller based on lifting surface theory is required. A number of propeller lifting surface theories have been developed and programmed for computer-aided design calculations. The method of kenein 22 * is representative of the state-of-the-art and was selected for the present application because the theory and numerical techniques have been extended to calculate induced velocities at arbitrary field points. Free space pressure predictions based on this method show excellent agree- ment with experimental nascumenents, Both the theory and numerical analysis are conveniently divided into a lifting-line analysis and lifting- surface corrections arising from blade thickness, blade location (skew and rake), and chordwise variation in loading. Therefore, it is possible to examine the separate contributions of various propeller characteristics to the thrust deduction. LIFTING-LINE THEORY The basis of analytical propeller design methods is the moderately loaded lifting line dheomys 22 The analysis considers the flow field associated with the steady loading on a propeller with symmetrically spaced blades. In accordance with circulation theory, the pressure loading on the blades arising from camber and incidence can be represented by distributions of bound and free vorticity. In the lifting line approxima- tion, each blade is replaced by a single concentrated vortex line with 1 evant, J.E. and R. Leopold, "A Design Theory for Subcavitating Propellers," Transactions SNAME, Vol. 72 (1964). “nema. J.E., "Computer Technique for Propeller Blade Section Design," International Shipbuilding Progress, Vol. 20, No. 227 (Jul 1973). Tease. S.B., "Comparisons of Experimentally Determined and Theoreti- cally Predicted Pressures in the Vicinity of a Marine Propeller," Naval Ship Research and Development Center Report 2349 (May 1967). iS rage. H.W., "Moderately Loaded Propellers with Finite Numbers of Blades and an Arbitrary Distribution of Circulation," Transactions SNAME, Vol. 60 (1952). Monee W.B. and J.W. Wrench, "Some Computational Aspects of Propeller Design," Methods in Computational Physics, Vol. 4, Academic Press Inc., New York, N.Y. (1965). iil bound circulation, [(r). Conservation of vorticity requires that a free- vortex sheet of strength - a) is shed from each lifting line. These sheets are assumed to lie on the surfaces x(r,#) = r tan B, (x) le - eal R, Sr sin 7”) oL I = - i = sin ¢¥ a (x, r cos ¥ - ( cos ve r sin ¥- 0 1 20 Wu, T.Y., "Some Recent Developments in Propeller Theory," Schiffs- technik, Vol. 9, No. 47 (1962). 12 BOUND VORTEX LINE FREE VORTEX SHEET Figure 1 - Coordinate Systems for Propeller Lifting Line Theory edL V(r) =V [1-w(r)] Figure 2 - Force and Velocity Diagram at Lifting Line 13 Similarly, the velocity induced by the free-vortex sheets, ur» is given by —\ — Z R Be 72 3B. * DW ay 1 ar [2 uz(x,r,¢) = Do -- | = zero? [ += da do (22) =] 1 Ry p 1 = IP where ae 1 ; e = (A.(p), - © cos a, Pp sin a) Ss 5 7 i dr + , 0) +e D. = (x - d, () a, © cos - p cos a, r sin ¥ - p sin a) d, (©) = tan B. (9) If these expressions are evaluated at the lifting line (0, r, ?) and inserted into equation (20), an integral equation is obtained relating T(r) and 8, (x). In design applications where a prescribed thrust, T, is to be developed, a second relationship can be derived by applying the Kutta-Joukowski law (with an empirical correction for profile drag - (see Figure 2) yielding R od dT T=Z ae R oz f T(r) [ar - ug(0,r,4 ) J fh = e@)tan B, (r) J dr Ry (23) The solution is normally found by an iterative procedure, starting with an estimate of tan B- T(r) is then computed from equations (20) and (22) and used to calculate a thrust according to equation (23). This procedure is repeated until the desired value of thrust is obtained. Highly efficient numerical techniques based on asymptotic series expansions have been developed to perform these computations. In the present work, it is assumed 14 that both I(r) and B, (x) have been determined, either from design calcula- , Soke PA 5 AP tions or from a separate performance prediction. The induced velocity varies with angular position, Y, corresponding to a time dependence in a reference frame fixed to the hull, i.e., oe ae By virtue of the propeller symmetry, the induced velocity may be resolved into a time-averaged or steady velocity U(x,r) and harmonics in blade passage frequency, Z2. The steady component or zero harmonic which gives rise to the thrust deduction, is given by 1 27 u(x,r) = a= ‘ a’ (x,r,¢) dy (24) 27 0 The separate contributions from the bound and trailing vorticity can be determined by introducing expressions of the type co ©o eee len =0 Bay a ye € cos n(¥ - # ) f J (kr) J (kp)e Sue 2 = n k n n [Do n=0 0 and performing the integration in equation (24). The following results are obtained: A eninge, D.E., "Numerical Prediction of Propeller Characteristics," Journal of Ship Research, Vol. 17, No. 1 (Mar 1973). SVE ey S.S., "Documentation of Programs for the Analysis of Performance and Spindle Torque of Controllable Pitch Propellers," Mass. Inst. of Tech- nology, Dept of Ocean Engineering, Report No. 75-8 (May 1975). 15 ~ ba R 0 kx eet Z dar @) e 2 up (tr) = - fe. || dk Jj(kr) J, (ke) dk do x £0 x 4 Ry i 0) 2- -kx e u,, (x,r) —— at 2 dk J, (kr) J, (kp) as |x| dk dp 1 At 0 dr. (p) 1 1 Te Ry i 0) me fr ae [Pe k u, (x,r) = -— — dk J. (ko) J, (kr) e as xO T At p 0 1 v7 Ru 0 g-e KX The total tangential velocity component is given by ZL KN dr < 0 = = ——— < ug (xr) Oa + On | x i) Jo (ke) J, (kr) i} dk do x $0 or _ Ae) 27r uy(x, r) 0 elsewhere It follows that the steady velocity induced by a moderately loaded lifting line propeller at points outside the slipstream is given by ( pera ee he fe) ao) oe lan < oO aie am a do 2, (p) Qe ee sai ae aaa > Ry aL 0 6x, j oe & enaligito He eG te) a Al pee (26) Ch acre aie Ry fe) d, (0) 0 1 Sane k= e ug(x,r) = 0 16 and arises solely from the trailing-vortex sheets. As shown in Appendix B, this is also the velocity field due to a sink disk of strength R o(r) = = i] a= dp = -2u,(0,r) (27) 1 i The method of Kemnay = originally developed to compute all harmonics of the induced velocity, is based on a direct numerical integration of equations (21) and (22). The sonuilamemsnoeeee sheet is divided into a set of discrete helical vortex lines of constant strength. For computational efficiency, the integration interval is divided into segments of increasing size with distance from the field point and evaluated using a five-point Gauss integration formula. The velocity is evaluated at a selected number of angular positions between two blades and resolved into blade frequency harmonics by Fourier analysis. Only the zero harmonic is used in the present application. As an example of the numerical accuracy of this method, a comparison between the computed and exact solutions of the axial velocity at the propeller disk plane for two selected loading distributions are shown in Figure 3. The velocity computed at the lifting line is also shown for comparison with the circumferential average. At distances greater than about one radius, the propeller disturbance velocity is generated essentially by the lifting-line sink disk. In order to more accurately derive the near-field influence of the propeller, the lifting-surface representation must be applied. LIFTING-SURFACE CORRECTIONS Propeller lifting-surface analyses have been developed as a logical. extension of thin planar hydrofoil theory. Thus, blade loading and thick- ness are represented in terms of vorticity and source-sink distributions located on a reference surface approximating the actual blade. This is normally taken to be the pitch surface derived from lifting-line calcula- tions. Within the blade outline a distribution of bound and free vorticity is established to represent both the radial and chordwise load variation. A free-vortex sheet, previously assumed to be shed from the lifting line, now originates from the blade trailing edge. 17 SLO (4d3Tq JUeISUOD) 4EZ7°O = (a) *g ueq = 89g uOTANTOS JoOeXy YIM Suotjegndwog AIFOOTEA JUTOG PTeTA JO uostaeduog - e¢ @An3Tq ANW1d 43173d0Ud LV S3ILIDOTSA IVIXV A pas a NOLLWINDYIO TWNOISNAWIG NON n n OL'0 g0'0 000 z00- GLO 0L0'0 00'0 Z'0 v0 A D SNOILVINO1VO ANIT D ann z ONILIIT WOUS NOILVINOID z= ONILIIT LV ALIOOTSA 90 = = (op) (op) D D NOILVINDIWO Ad4 e ; (a)!guer A we A =e 80 veto = ‘guely/s IZ n NOILN1OS LOVX4 ALIDO1SA JOVYAAV TVILNAYSSWNOYID ize'0 = “to OL eueTd AeTTedoag UE suoTqeTNoTeD AITIOTEA JUTOd PTetdA - € oIn3Tq 18 (YOIEd PTGPEAPA) O°T = (a)*g ue] AOF WOTAINTOS 4oexy YIIM SuoTze3NndwWoD AITOOTSA JUTOg PTeT| Jo uostzedwoyg - qE san3Ty N INV 1d OSIG Y43114Sd0Ud LV ALIDOTSA 1VIXV a n 0L0 S0'0 00°0 S0'0- OL'0- SLO- NOILV1IND1V9 Ads O ul eee Se = 60190 + 5-1 NOILN10S LOVX4 OP WA oN uz (ania a) OOL = (4)'g ued (penutjuoy) ¢ ean3T¥q 19 It is convenient to consider the induced velocities as the sum of the lifting-line result and separate lifting-surface corrections. The induced velocity due to the bound and free vorticity, @ Vp (e>9) and @_ ¥ (659), may be written as a Ot (he A Ik e, x D, uo (x,r,) = > Sam eee Yn (p,a) Saadp da 2a) B =a Gn 1 e ( B \D | 3 H ne s and (29) Tile (Gxaren@) S I Mr el — 2) = He po} 2 | 4 m | el g ae Saw OD 2 a (1) =| x tH n where on oT, (r) are the blade leading and trailing edge positions. Continu- k ity of vorticity requires that 7.) vee S Peete 2 2 y (59%) Feta Y_(r>o) Ni (e)+p” da G3) . (30) roa = Yo e k and hence U’, can be rewritten as #.(0) Ds 1 R TIE 7 k ES Xx D, Shr uc(x,r,”) = D> = >; (0)+0 (Oe) = T At i Ss = ,3 k=1 #, (o | k (31) 20 where yk (r5¢) ¥,(r>9) ¢ (32) ¥,(F>) , ar) gy>”¢ The induced velocity due to a distribution of sources over the blade surfaces is given by ¢., (0) b: ai fia egal i Ea GaeN?)a= = b= f Wrz +0 o(p,0) VY —— dp do = (33) k=l (0) [D.| SE “ee Ss where o(r,¢%) is the source strength density. It follows that the lifting surface induced velocity may be considered as the sum of four terms $y (0) & 2, 5 kk é, x D U(x,r, 9) =u; (x,r, 9) + » Lf A. (p) +0 y*(p,a) PRE aT — An \ i s = k=1 Ry g (0) [D.| k @ @ (34) + [up (xr, 9) J + U-(x,r, ¢) lifting line ©) ® where @) = velocity due to a distribution of bound vorticity on the blade surface (equation (28)) @) = velocity due to a distribution of free vorticity on the blade surface G) = velocity due to the lifting line free-vortex sheet (equation (12)) ZL @ = velocity due to a distribution of sources In the method of eee = (available as a FORTRAN computer program FPV), the numerical evaluation is again based on discrete singularities. On the blade surface, a grid of radial and helical lines is constructed to form a lattice of source and vortex elements as illustrated in Figure 4. The elemental singularity strengths are determined as follows: 1. Radial vortex elements are required to produce the desired chord- wise load distribution and the known bound circulation I(r) at each radius. 2. Helical vortex elements must satisfy conservation of vorticity. 3. Source and sink elements are required to generate the same chord- wise velocity distributions as the section thickness form would produce in two-dimensional flow. Also the sum of sources and sinks must equal zero. As in the lifting-line analysis, the velocity is computed at a set of angu- lar. positions for each field point (x,r) and resolved into blade frequency harmonics. It may be noted that radial vortex lines (bound circulation) do not contribute to the steady axial and radial velocities and need only be computed to determine the strength of the helical (free) vortices on the blade. It is also straightforward to show that the steady velocity is g +9 rf iy as 2 On the other hand, blade rake, the axial displacement of blade sections aft independent of the skew angle, ss as would be expected physically. of the lifting line plane, is of marked importance. This is manifested in "corrects" for the the free vorticity term (2) in equation (34) which starting position of the trailing-vortex sheets. These and other features of the propeller calculations are best illustrated by considering specific examples. EXAMPLE CALCULATIONS AND COMPARISON WITH EXPERIMENTS The equations derived in the foregoing theoretical analyses have been programmed for computer-aided numerical solution. The calculation is per- formed by interfacing three separate programs: (1) the hull potential flow solution (PF), (2) propeller field point velocities (FPV), and (3) the 22 ROTATION TRAILING VORTEX SHEET BLADE SOURCE AND VORTEX SHEET y SOURCE AND VORTEX LINE MODEL OF PROPELLER BLADE Figure 4 - Representation of Propeller Blade By Source and Vortex Line Lines (Projected View Looking Forward) 23 interaction analysis (CALCTD). A block diagram illustrating this procedure is given in Figure 5. It should be noted that the first, and most time- consuming task, is to assemble the necessary hull offsets and propeller geometry and loading data in a suitable form. Example calculations have been conducted for several propeller body- of-revolution configurations, with and without cruciform stern appendages. These examples were chosen to illustrate important features of the analy- tical method and to provide experimental verification of the theory. A summary of the results is presented in Table 1 showing pertinent character- istics of each propeller-hull configuration and a comparison of predicted and measured thrust deduction fractions. Detailed numerical results and discussions of each example are given in the following sections. EXAMPLE 1: APPENDED SERIES 58 FORM As a first check on the computational procedure, a Series 58 Form originally calculated by Renecildee® was selected for analysis. The hull is a streamlined body of revolution, DINSRDC Model 4620, fitted with cruciform stern appendages. The propeller, DINSRDC 3638, is a 5-bladed wake-adapted design located 98 percent of the hull length from the bow. Offsets and particulars of the hull are listed in Table 2. A drawing of the propeller is given in Figure 6. The quadrilateral representation of the hull and appendages is illus- trated in Figure 7 (identical to that used by Beveridge). Note that for ease of computation, the horizontal control surfaces are also used to represent the upper and lower rudders and the forebody is approximated by reflecting the afterbody about the hull midlength (this latter simplifica- tion is shown to be valid in later examples). The propeller circulation and hydrodynamic pitch distributions, I(r) and 2mr tan B.(r), were obtained from lifting-line calculations and are shown in Figure 8. The axial and radial propeller field point velocities induced at control points on the body boundary are presented in Figure 9. Since the propeller has no rake, the lifting-surface corrections arise solely from blade thickness and chord- wise load distribution. These effects are seen to decay more rapidly than the lifting-line disturbance field, contributing less than 10 percent at distances beyond one propeller radius. 24 Assemble body data in potential flow program (PF) format Check data and correct as necessary using computer graphics Execute first section of PF program (PF1) to organize data and make preliminary calculations Check results, make corrections and rerun PF1 as necessary Execute second section of potential flow program (PF2) to calculate the influence coefficients Execute program RITPNT which defines field points for propeller calculation Assemble propeller geometry and loading data Execute FPV program to determine the propeller-induced velocities at the ship hull Execute program COMPVN to calculate the normal component of the propeller-induced velocity at each quadrilateral centroid Execute the third and fourth sections of the potential flow program (PF3 and PF4) to determine the panel source strengths and make velocity and pressure calculations Execute program CALCTD to compute the interaction force between propeller and hull If the interaction force due to another propeller design is to be computed, repeat the above five steps Figure 5 - Interaction Analysis - Computational Procedure 725) WLP t aaa 00S °T €7L°0 O1Z°0 | 064°T (pepueddy) 7-727S °Y ¢40°0 | 160°0 G4¥S"0 786°0 oL€°0 | oSz°T VLOG (pepueddy) 1-47z¢ er €-G2¢S (sa8epueddy c-S¢2S S7S°0 €86°0 ON) 1-S7@7S O61°0 | SEt‘o 40%"0 086°0 0g%"0 | 726°0 SE9E (pepueddy) ozo i *dxq | Aroauy Si : “ON sntpey Apog on A, | CON D@ESNEG) | (tapoy oqusnaa) | etdwexa ips 3e eo a Ssntpey JeTTedorg | uotjwe.07 Terxy Apog SNOILVINOTVO NOILONGHG LSNYHL NI SNOILVYNSIANOD AGOG-aATTdOdd - T ATAVL 26 TABLE 2 - OFFSETS AND PARTICULARS FOR SERIES 58 FORM, MODEL 46200773 FPoodcooaooqocoooo0o0o°ceo°o°0o0oeoeo0ca°ecoocd od DFOANNHDOFWANAHAOFANAAHAOHFANADOFLAN DAD DFONNHDOFANAHDOFANAHAODOLANAOCHLANA OO DOrRFNWFUUUDANNWOWWM WOO 0. 0. QO. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. QO. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. MOSS SGOOOOOOOO OOO OOOO S2 Model 4620 Wetted Surf. Coeff. Serial 40050060-73 LCB, x = 0.4456 Formula: L/D = 7.339 2 2 3 4 5 6 : a y =a, xta)x tax ta,x tax Haex Model Particulars: where: - 000000 Length, ft Diameter, ft o ESIESS Nose radius, ft - 774885 Tail radius, ft 9 Wetted Surf., ft ae 2e8 Volume, ft3 - 792534 LGB, £t 645977 .0000' (OA (Beso 53) aim) 51392 CloS7/O atin) i} Wow tl Sedeners L. and M. Gertler, "Mathematical Formulation of Bodies of Revolution,'’ DIMB Report 719 (Sept 1950). 27 IN’S. PITCH | pitcy | MEAN MODEL | MODEL | pazi9| WIDTH RATIO | BLADES IN’S. OR | 1485 a tot TL = LITTZZ: \} 7S DearS PROJEC TION : SL a a +. GO / fro, odeas e\ — 003 6 MMMM Mises era OO MMMM Mii ooee Se A TITLE LLLL db biase® 9 7544 \\ MM ek is @ Le ATIT ALT? oH Figure 6 —- Propeller 3638 28 HORIZONTAL RUDDERS STABILIZERS Figure 7a - Stern Appendage Configuration - NSRDC Model 4620 NOTE: ALL FOUR APPENDAGES REPRESENTED BY HORIZONTAL STABILIZERS IN CALCULATIONS Figure 7b — Quadrilateral Representation of NSRDC Model 4620 Figure 7 - Representation of NSRDC Model 4620 29 CIRCULATION 2nRV 0.02 0.01 0.2 0.4 0.6 0.8 1.0 RADIUS — R Figure 8 - Lifting Line Circulation and Pitch Representing Loading of Propeller 3638 30 te | SS 9] ue} ji Th OILVY HOLId DINWWNAGOYGAH CONTROL POINTS 25 APPENDAGE POINTS RADIUS 29 15 BODY POINTS i aa MAXIMUM R ae BODY RADIUS 0.0 0.0 { i 6.0 8.0 10.0 32% OF AXIAL DISTANCE FORWARD OF PROPELLER x/R TOTAL BODY LENGTH 0.08 FROM TAIL LIFTING SURFACE (LOADING AND THICKNESS OVER FINITE CHORD- LENGTH — —— LIFTING LINE (LOADING ONLY, 0.06 CONCENTRATED AT RADIAL LINE) PROPELLER VELOCITIES 0.04 s|> xo} {= oO AS 0.02 AXIAL AND RADIAL INDUCED VELOCITIES ate 10.0 -0.02 Figure 9 - Control Points and Propeller Induced Velocities on Appended Series 58 Body 31 The computed values of thrust deduction and the measured result ob- tained from model resistance and self-propulsion experiments are shown in Table 3. Since no correction for increased frictional drag has been includ- ed, the potential flow prediction of (1-t) should be higher than the measured value*. The agreement between the two calculated values (pressure integration and Lagally theorem) indicates good numerical accuracy. Also note that the correction to body singularity strengths changes the thrust deduction fraction by less than 3 percent. The comparison between lifting- line and lifting-surface predictions shows that the influence of blade thickness and chordwise loading reduces the thrust deduction fraction by only 4 percent. The agreement between theory and experiment is quite en- couraging and is considered to be within experimental accuracy. It is believed that the discrepancy between Beveridge's result and the current work lies in the derivation of the propeller sink-disk strength. Beveridge used the induced axial velocity at a lifting line. The present method (equation 27) properly averages the velocity field at the disk plane, resulting in a lower sink strength, as shown in Figure 10. It is also of interest to examine the distribution of the interaction force over the afterbody. This is a function of both the body sectional- area distribution and the breadth and intensity of the propeller disturb- ance field. The longitudinal distribution of the thrust deduction, (dt (x)/dx), acting on the body only, is shown in Figure 1l. The signifi- cant contribution is over the last 25 percent of the body length with 50 percent concentrated in the last 6 percent of the length. EXAMPLE 2: COMPARISON OF THEORETICAL AND EXPERIMENTAL PRESSURE DISTRIBUTIONS ON THREE BODIES OF REVOLUTION Wind-tunnel experiments were recently conducted to determine the flow characteristics of three streamlined bodies of eerrolwelens” Measurements of the afterbody pressure distribution with and without a stern propeller in operation were obtained for comparison with analytical predictions. The profile of the parent body (DINSRDC Model 5225-1) is shown in Figure 12 *The added frictional resistance is expected to increase the value of t by no more than 5 percent, based on Reference 9. 32 QoUeISTSSY TPUOCTIOTAY peseestDUT TOF suoT}IeATIOD ANoYATMs (9 e3ptreAeq) sy73uei9g 804n0g Apog 03 suOoT}DeA1I0D ANoYATM pourweW YSTA-YUTS teTTedorzg 03 pet{ddy werzosyy ATT eseT (istd-4Uts) ((€1) Uotzenby) werzo0sys, ATT eseT SIRE Ghat ste (sya3uet97Sg 901n0Sg Apog oF “suoT}09I10D JNOYITM) wesrosyy, ATTeseT poursy ((€LT) uotzenby) wer0sy, ATT eseT aoezans ((I1) Uwotjenbg) uotjeAzZeqUy sAansserzg Apog Saye peanduoy (Juowtiedxy uots[ndoig-jJ[es pue voUeISTSey) peanseow GOHLAW (8€9€ UATIddOUd ‘0797 THGOW OGNSNLG) AGOT 8S SHLYAS CAANaaav WYOA NOILOVAA NOILONGIAG LSNYHL AO SAN IVA GAYNASVAW GNV xC@lNaWod - € ATAVL 3)3) SINK STRENGTH 0.6 BEVERIDGE® 0.4 PRESENT METHOD (EQUATION 27) Sis OA 0.0 0.2 ; 0.6 r/R RADIUS -0.2 Figure 10 - Radial Distribution of Propeller Sink Strength Derived From Lifting Line Calculations 34 (ATuO Apog) 0794 TepoK Uo vdA0q UoTIONpeg Jsniy, JO UOTINGTIASTG TeUTpNITSsuoT peyeTNoTe) - TT eansTy 0297 TAGOW NOILINGIG LSNYHL 40 NOILNGIYLSIG IVNIGNLISNOT GALVINIIWO OL S60 06°0 - §8°0 08°0 SZ°0 f) ¥a113d0ud 35 SOTpoqieqsy OTAWouMASTXY JO SeTTJorg - e7T o1insTy Vx OL 60 80 LO 90 c0'0 €0'0 yoo s0'0 OL 60 80 £0 90 G0 v0 c0 c0 LO 0 d'V dq ¢/6'0l = a/7 ae 1-SzzS 1300W €-S77G pue *%-GZZS ‘T-SZZS STPPOW OGUSNLG JO uoTeJUesSeadey - ZT 2aANsTy 36 Figure 12 (Continued) 5225-1 — AFTERBODY Figure 12b - Quadrilateral Representation of Bodies of Revolution, DINSRDC Models 5225-1, 5225-2, and 5225-3 37 Figure 12 (Continued) > a e) a oc uw Kk LL < | oie ite) N nN ike} 5225-3 — AFTERBODY Figure 12b (Continued) 38 along with the quadrilateral representation used in the calculations. The other two bodies, DINSRDC Models 5225-2 and 5225-3, differ from the parent in afterbody shape only and are also illustrated in Figure 12. One 7-bladed propeller model, DINSRDC Model 4577, was used for the three body experiments. The principal characteristics and geometry of the propeller are summarized in Table 4. Pressure measurements were obtained at two advance ratios, J = 1.25 and J = 1.07. Since the propeller was not operating in its design wake, it was necessary to predict the load distribu- tion using an inverse performance calculation. This was an iterative process, starting with the measured nominal wake, computing a load distribu- tion, recomputing the wake, and so on. Convergence was achieved after two iterations. The computed and measured differences in afterbody pressure distribu- tion are shown in Figures 13a, 13b, and 13c. The agreement is excellent for body 5225-1 and very good for body 5225-2. There is a marked dis- crepancy for 5225-3. This body was intentionally designed to have boundary layer separation in the absence of the propeller with the hope that the propeller influence would reattach the flow. However, flow separation occurred at x/L = 0.92 which is too far upstream. The thrust deduction fraction, Saye was derived by integrating the measured pressure difference and dividing by the calculated propeller thrust (no self-propulsion experi- ments were conducted). The results are compared in Figure 13 with the theoretically computed values, t The agreement for body 5225-3 is considered fortuitous in view ieee difference in pressure distribution. The thrust deduction for body 5225-2 is larger than for 5225-1 which is expected on physical grounds since the fuller afterbody places the frontal area closer to the propeller. Body 5225-3, which has the fullest after- body, is actually finer in the immediate vicinity of the propeller; hence, the calculated thrust deduction is also lower than for body 5225-2. 39 TABLE 4 — GEOMETRY OF PROPELLER 4577 Number of Blades 7 Expanded Area Ratio 0.584 Section Meanline NACA a = 0.8 Section Thickness Distribution BUSHIPS Type Tara Rake Angle, deg 6.964 Skew, deg 30 0. 0. 0. 0. 0. 0. O. 0. l. So 2 28o 2 2 2 2 © So eo eC 2S 282 2 S&S © PErpPrmm pao Oo © © So oO So 2 oOo Ee 2S OS ©& pproceres T., "Minimum Pressure Envelopes for Modified NACA-66 Sections with NACA a = .8 Camber and BUSHIPS Type I and II Sections," DTNSRDC Report 1780 (Feb 1966). 40 Figure 13 - Measured and Calculated Pressure Distributions on DTNSRDC Models 5225-1, 5225-2, and 5225-3 0.15 8.8 5 : 3 0.10 POTENTIAL FLOW re ° ea 5 N ~o| > ao Q|N fe {ot £0 sa ror I 0.05 fom oO | 0.0 0.6 x/L Figure 13a - Body 5225-1 41 AC, 0.10 0.05 0.0 0.6 Figure 13 (Continued) EXPERIMENTS R,,x 10° Jyeies dyaion a 5.9 > 8.8 POTENTIAL FLOW Cy, = 0.420 0.637 = 0.143 0.140 = 0.129, and 0.126 Figure 13b — Body 5225-2 42 Figure 13 (Continued) 0.15 EXPERIMENTS Rx 10° Jy = 1.25 Jy = 1.07 5.9 8.8 0.10 POTENTIAL FLOW 0.428 0.646 AC, 0.109 0.103 = 0.106 0.103 0.05 0.0 x/L Figure 13c — Body 5225-3 43 EXAMPLE 3: BODY OF REVOLUTION WITH STERN APPENDAGES This example provides a further comparison of computed and measured thrust deduction and exhibits the separate contribution of stern appendages. The analysis is applied to an appended body of revolution, represented by DINSRDC Model 5224-1. This model, with appendages removed, is a geosym of the wind-tunnel model, DINSRDC 5225-1, described previously in Example 2. The afterbody profile and appendages are illustrated in Figure 14, together with the quadrilateral representation used in the calculations. Note that each control surface is represented properly (in contrast to Example 1), but the forebody is again approximated by the afterbody image. The propeller, DINSRDC Model 4567A, (a geosym of the wind-tunnel model 4577), was designed specifically for the appended body. All calculations are therefore based on the design loading characteristics. The circulation and hydrodynamic pitch distributions are shown in Figure 15. Calculated propeller induced velocities on the body are given in Figure 16 showing both the lifting-line sink disk and lifting-surface contributions. The results derived from a simple point sink located at the shaft centerline are also presented. At distances beyond one propeller diameter the veloci- ties derived from the point sink, lifting-line, and lifting-surface propeller representation agree to within 10 percent. Closer to the propeller, the lifting-line velocities are about 25 percent too large, while the point-sink velocities are as much as 50 percent too large. Calculated and measured values of the thrust deduction are presented in Table 5. The higher lifting-line velocities result in a prediction of t which is 21 percent larger than that derived from the lifting surface calcu- lation. The predicted value of (1-t), while lower than the measured value, is within experimental accuracy. The contribution of the stern appendages, as shown in Table 5, is 24 percent of the total thrust deduction. (Although not discussed in Example 1, a comparable value of 25 percent is found in that case). Finally, the distribution of interaction force on the afterbody is displayed in Figure 17, shown as the integrated force as a function of distance from the stern. The significant contribution is over the last 30 percent of the body length with 50 percent of the force concentrated in the last 5 percent (i.e., within a distance of one propeller diameter). 44 = aw > Ww a iL (e) oc a. 7 + N nN Ys) 5224-1 PLAN VIEW 45 Figure 14 - Profile and Quadrilateral Representation of DINSRDC Model 5224-1 CIRCULATION 2nRV r wR tan Bi(r) r/R Figure 15 - Design Circulation and Hydrodynamic Pitch Distributions of Propeller 4567A 46 in 4 (4)!g ued OlLVY HOLId DINVNAGOY GAH AXIAL AND RADIAL INDUCED VELOCITIES CONTROL POINTS 25 APPENDAGE POINTS Ce ©) 6) Or 26 BODY POINTS 2.0 S$ © 2 wo 6 © fF eoooe @ & as eee @ MAXIMUM = 1.0, OO O 6d, @0 @ BODY | Baeg® Q RADIUS 1.833 0.0 2.0 4.0 6.0 8.0 10.0 t 25% OF AXIAL DISTANCE FORWARD OF PROPELLER x/R BODY LENGTH 0.08 PROPELLER VELOCITIES 0.06 — LIFTING SURFACE ——— LIFTING LINE ONLY @ = POINT SINK OF STRENGTH m: ra m=mR2V [1-1 + Cy] , Crp, = 0.370 +|> mo} Cc oO => 0.02 0.00 -0.02 Figure 16 - Control Points and Propeller Induced Velocities on Appended Body of Revolution (DINSRDC 5224-1) Propeller 4567A 47 TABLE 5 — COMPUTED* AND MEASURED THRUST DEDUCTION FRACTION FOR APPENDED BODY OF REVOLUTION (DINSRDC MODEL 5224-1, PROPELLER 4567A) METHOD Measured (Resistance and Self Propulsion Experiment) Computed* Pressure Integration (Equation (11)) = 0.070 Body Only ny endages _ D024 Lifting PP 8 Surface Lagally Theorem (Equation (13)) Lifting : Pressure Integration Line *Without Correetions for Increased Frictional Resistance 48 0.08 ar ak F ) dx 0.06 Pwithout propeller i) -3 0.04 = 0 = = ee x —a Ale W 0.02 zs 0.00 PROPELLER PLANE DISTANCE FORWARD OF AP. _ x F BODY LENGTH L —= 0.016 Figure 17 - Longitudinal Distribution of Thrust Deduction (Cumulative From Stern) on Body of Revolution (DINSRDC Model 5224-1, Propeller 4567A) 49 EXAMPLE 4: APPENDED BODY OF REVOLUTION WITH FOUR RAKED PROPELLERS In this example the thrust deduction analysis is applied to compare four different propellers fitted to the same hull. The hull is an appended body of revolution, represented by DINSRDC Model 5224-2. This model is identical to DINSRDC 5224-1 described in Example 3 except for a slight modi- fication to accommodate a large propeller hub (see Figure 18). The four propellers are wake-adapted designs with equal diameters and thrust loading coefficients. A comparison of principal characteristics is given in Table 6 and propeller geometry is presented in Tables 7 through 10. The radial distributions of bound circulation and hydrodynamic pitch shown in Figure 19 were derived from lifting-line calculations using terse? criterion for optimum loading. The propellers are distinguished by varying amounts of skew with corresponding ''skew back" along the respective geo- metric pitch helices. As a result, the blade rake differs considerably among the four propellers, as illustrated in Figure 20. Physically, it was expected that these differences in blade axial positions would strongly influence the thrust deduction. Calculated and measured values of thrust deduction are compared in Table 11. Both the theoretical and experimental results exhibit a pro- nounced decrease in the interaction force with increasing propeller rake. However, the computed values of 1-t are substantially lower than measured in each case. This discrepancy is very likely due to difficulties in the experiment owing to the weight of the propeller models.* Comparison with the result of Example 3 provides evidence that the measured values of t are too high. The calculated contributions of the stern appendages to the thrust deduction are given in Table 11. These vary from 19 to 24 percent of the total, which is comparable to the result of example 3. In Figure 20 the rake distribution of propeller 4567A is shown as a dashed line and may be compared with propeller 4487. Since propeller 4487 is further aft on the body and has a larger diameter, the thrust deduction fraction should be lower. This is, in fact, predicted from theory (0.059 versus 0.09), while the experiments yield a higher result for 4487 (0.109 versus 0.09). *These models were approximately 5 times heavier than typical models due to material (bronze instead of aluminum) and size. 50 Z-"7ZG T@POW UO UOTIeTeITY UIAIS - gT saNn3Ty ZZ) NOILVYSLT1V @NH ¥3114d0ud 51 TABLE 6 — PRINCIPAL CHARACTERISTICS OF FOUR RAKED PROPELLERS ON MODEL 5224-2 Propeller Number of Blades Propeller Diameter Hull Diameter Expanded Area Ratio Blade Thickness Fraction P/. at 0.7R D X Axial Position, =* Skew, Deg *Here SS is defined as the longitudinal distance from the bow to a propeller reference plane passing through the midchord of the blade root section. Dy TABLE 7 — GEOMETRY OF PROPELLER 4486 Number of Blades 5 Expanded Area Ratio 0./07 Section Meanline NACA a = 0.8 Section Thickness Distribution NACA 66 (DINSRDC modified nose and tail) Skew, Deg 43 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SIO C2 28 2 © & Oo & Jo C2 Oo Co L2 S&S OS S&S ooo Oo oo © C&C S&S © 53 TABLE 8 — GEOMETRY OF PROPELLER 4487 Number of Blades 7 Expanded Area Ratio 0./07 Section Meanline NACA a = 0.8 Section Thickness Distribution NACA 66 (DINSRDC modified nose and tail) Skew, Deg 43 0.2 i 0.3 1 0.4 1 0.5 1 0.6 1 0.7 1 0.8 1 0.9 1 1.0 1 54 TABLE 9 - GEOMETRY OF PROPELLER 4488 Number of Blades 3 Expanded Area Ratio 0.707 Section Meanline NACA a = 0.8 Section Thickness Distribution NACA 66 (DINSRDC modified nose and tail) Skew, Deg 72 oo © oo & &| & oOo eo 2 2 & © 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 So 2 2S 2S 2 Oo Oo & © ee oe eo 55 TABLE 10 —- GEOMETRY OF PROPELLER 4489 Number of Blades 3 Expanded Area Ratio 0.709 Section Meanline NACA a = 0.8 Section Thickness Distribution NACA 66 (DINSRDC modified nose and tail) Skew, Deg 120 oO Eo So oC Lf CO EC CO CO oOo 2S S| 2 2 2 OS OC Ce ce 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 56 CIRCULATION 2nRV DESIGN Cy, = 0.210 4488 AND 4489 a wy 4 Ig ue} 0.2 0.4 0.6 0.8 1.0 r RADIUS R Figure 19 - Lifting Line Circulation and Hydrodynamic Pitch Representing Loading of Propellers 4486, 4487, 4488, and 4489 /| OILVY HOLId DINWNAGOYGAH 6877 pue “egy “/8hh ‘OR8Hr sieTTedorig jo ayeYy TeIO] Jo suoT§AnqTAIsTG TeTpey - OZ eAnsTy SNIGWY 4¥37114ad0u"d GHOHOCIW OL SJONVLSIG 1VIXV 4903 SNIGVaT SNH Z'0 1 HLONAT AGO 4O NOILOVYS dy J SsNiavY 58 TABLE 11 — COMPUTED* AND MEASURED VALUES OF THRUST DEDUCTION FRACTION FOR PROPELLERS 4486, 4487, 4488, AND 4489 ON DINSRDC MODEL 5224-2 Gee) Experiment appendages Propeller *Without corrections for increased frictional resistance 59 By considering the longitudinal distribution of thrust deduction (body only) shown in Figure 21, it may be concluded that virtually all of the interaction force develops on the last 20 percent of the body. Within this region the effect of rake is manifested as a change in propeller induced velocities. The total axial induced velocity on the afterbody is shown in Figure 22. The separate lifting line and lifting surface contributions are displayed in Figure 23, showing corrections for skew and rake to the essen- tial difference. The effect of blade thickness is to decelerate the flow ahead of the propeller and hence, reduce the interaction force. In the present example, this effect is more pronounced even for the least raked propeller (4487) because of the relative volume of each blade compared to the loading. For propellers of equal blade volume, the influence of thickness would diminish with increasing rake. Such is not the case here because, for increased rake, more material is required for equal strength. Thus propeller 4489, while raked the most, also has the largest thickness contribution because of blade volume (for example, it is approximately twice the volume of propeller 4487). CONCLUSIONS AND RECOMMENDATIONS An improved theory and computer-aided numerical analysis have been developed for predicting the added resistance (thrust deduction) arising from the propeller-hull interaction. The theory is formulated in terms of the diffracted potential flow about the hull in the presence of the propeller-induced velocity field. The propeller representation is derived from lifting-surface theory which includes the effects of blade thickness, skew, rake, and chordwise load distribution. It is shown that these effects may be regarded as corrections to the moderately-loaded lifting-line (sink disk) approximation. Important features of the analysis and comparisons with experimental results are illustrated by application to several stern propeller-appended body-of-revolution configurations. Based on these sample calculations, the following conclusions are drawn: 60 0.05 PROPELLER 4487 0.04 4486 s | xX nod io) SS 23 Oo ® = = 0.08 a? 4488 s s a 4489 265 0.02 Q _—— x (o) &|F il} x 0.01 [ps x 0.0 0.0 0.1 0.2 0.3 0. DISTANCE FORWARD OF ALP. BODY LENGTH exe L Figure 21 - Longitudinal Distribution of Thrust Deduction (Cumulative From Stern) With Four Propellers on Body 5224-2 61 4 Ux V AXIAL VELOCITY 0.2 PROPELLER 4487 0.01 0.0 0.1 DISTANCE FORWARD OF A.P._ x BODY LENGTH L Figure 22 - Propeller Induced Axial Velocities on Appended Body of Revolution 5224-2 62 0.2 Ux V AXIAL VELOCITY 0.04 0.03 4488 & 4489 LIFTING LINE TRAILERS 0.01 0.0 02 AXIAL DISTANCE x -0.01 THICKNESS -0.02 AXIAL DISTANCE = 0.1 iL 0.2 0.0 CORRECTION FOR SKEW AND RAKE -0.01 Figure 23 - Contributions to Propeller Induced Axial Velocity on Body 5224-2 63 1. For nonseparating hull forms, potential flow theory predicts the change in afterbody pressure distribution and total thrust deduction within experimental accuracy. 2. Calculations of the thrust deduction fraction based on pressure integration and Lagally's theorem agree to within 4 percent. 3. The influence of the propeller rapidly decays with distance. In all cases, the thrust deduction develops within the last 30 percent of the hull length. 4. Corrections to hull source strengths (interference or diffraction effect) changes the thrust deduction fraction by less than 3 percent. 5. Lifting-surface corrections reduce the calculated thrust deduction fraction by as much as 20 percent for conventional propeller geometries (or equivalently, a 2 to 3 percent reduction in required thrust). This correc- tion will increase for more highly raked propellers. 6. Conventional cruciform stern appendages contribute up to 25 percent of the thrust deduction fraction. 7. The use of an afterbody image to represent the forebody reduces the computational effort without loss of accuracy. In view of these findings, the following extensions and applications are recommended: 1. Further comparisons with available experimental data should be conducted to aid in refining the analysis. 2. For purposes of preliminary design calculations, the thrust deduc- tion analysis based on the lifting-line model, as described in the report, should be incorporated into existing propeller design computer programs. In order to properly account for rake, it will be necessary to develop a curved lifting-line representation of the propeller. 3. A parametric study should be undertaken to systematically examine the role of hull form and propeller characteristics (e.g., diameter, axial location, radial distribution of loading, and rake). 4. The method should be extended to include other propeller configura- tions -- contrarotating, tandem, ducted, and twin-screw. 5. The method should be extended to apply to surface-ship configura- tions. 64 ACKNOWLEDGMENTS The authors are indebted to a number of people for their assistance in the course of this work. Mr. T.A. LaFone, Mr. J. Diskin, and Ms. R.B. Hurwitz aided in the development and execution of the various computer pro- grams. Mr. C. Dawson and Ms. J. Dean provided invaluable guidance in the use of the XYZ potential-flow computer program. Finally, the authors wish to thank Messrs. J. McCarthy, T. Huang, and H. Wang for their continued interest in the analytical work and support in obtaining experimental data presented in this report. 65 REFERENCES 1. Nowacki, H. and S.D. Sharma, "Free Surface Effects in Hull Propeller Interaction," The University of Michigan College of Engineering Report 112 (Sep 1971). 2. Dickmann, H.E., "The Interaction between Propeller and Ship with Special Consideration to the Influence of Waves," Jahrbuch der Schiff- bautechnischen Gesselschaft, Vol. 40 (1939). 3. Weinblum, G., "The Thrust Deduction," American Society of Naval Engineers, Vol. 63 (1951). 4. Korvin-Kroukovsky, B.V., "Stern Propeller Interaction with a Stream- line Body of Revolution," International Shipbuilding Progress, Vol. 3, No. 17 (1956). 5. Beveridge, J.L., "Pressure Distribution on Towed and Propelled Streamline Bodies of Revolution at Deep Submergence," David Taylor Model Basin Report 1665 (Jun 1966). 6. Beveridge, J.L., “Analytical Prediction of Thrust Deduction for Submersibles and Surface Ships," Journal of Ship Research, Vol. 13, No. 4 (Dec 1969). 7. Nelson, D.M., "Development and Application of a Lifting-Surface Design Method for Counterrotating Propellers,'' Naval Undersea Center TP 326 (Nov 1972). 8. Beveridge, J.L., 'Thrust Deduction in Contrarotating Propellers," Naval Ship Research and Development Center Report 4332 (Nov 1974). 9. Huang, T. et al., "Propeller/Stern/Boundary-Layer Interaction on Axisymmetric Bodies: Theory and Experiment,'’ DINSRDC Report 76-0113 (Dec 1976). 10. Hess, J.L. and A.M.O. Smith, "Calculation of Nonlifting Potential Flow About Arbitrary Three-Dimensional Bodies," Journal of Ship Research (Sep 1964). 11. Hess, J.L. and A.M.O. Smith, "Calculation of Potential Flow About Arbitrary Bodies,'' Pergamon Press, Progress in Aeronautical Sciences, Vol. 8 (1966). 12. Dawson, C.W. and J.S. Dean, 'The XYZ Potential Flow Program, " NSRDC Report 3892 (Jun 1972). 13. Cummins, W.E., "The Force and Moment on a Body in a Time-Varying Potential Flow,'' Journal of Ship Research, Vol. 1, No. 1 (Apr 1957). 14. Milne-Thomson, L.M., "Theoretical Hydrodynamics,'' The Macmillan Company, New York, N.Y., 2nd edition (1950). 15. Kerwin, J.E. and R. Leopold, "A Design Theory for Subcavitating Propellers," Transactions SNAME, Vol. 72 (1964). 16. Kerwin, J.E., "Computer Technique for Propeller Blade Section Design," International Shipbuilding Progress, Vol. 20, No. 227 (Jul 1973). 66 17. Denny, S.B., "Comparisons of Experimentally Determined and Theo- retically Predicted Pressures in the Vicinity of a Marine Propeller," Naval Ship Research and Development Center Report 2349 (May 1967). 18. Lerbs, H.W., "Moderately Loaded Propellers with Finite Numbers of Blades and an Arbitrary Distribution of Circulation," Transactions SNAME, Vol. 60 (1952). 19. Morgan, W.B. and J.W. Wrench, "Some Computational Aspects of Propel- ler Design," Methods in Computational Physics, Vol. 4, Academic Press IMCs g New York, N.Y. (1965). 20. Wu, T.Y., "Some Recent Developments in Propeller Theory,'' Schiffs-— technik, Vol. 9, No. 47 (1962). 21. Cummings, D.E., "Numerical Prediction of Propeller Characteristics," Journal of Ship Research, Vol. 17, No. 1 (Mar 1973). 22. Tsao, S.S., "Documentation of Programs for the Analysis of Perform- ance and Spindle Torque of Controllable Pitch Propellers," Mass. Inst. of Technology, Dept of Ocean Engineering, Report No. 75-8 (May 1975). 23. Landweber, L. and M. Gertler, "Mathematical Formulation of Bodies of Revolution," DTMB Report 719 (Sept 1950). 24. Brockett, T., "Minimum Pressure Envelopes for Modified NACA-66 Sections with NACA a = .8 Camber and BUSHIPS Type I and II Sections," DTNSRDC Report 1780 (Feb 1966). 67 eheavzt Pett oot et dred cleghcd 'gane® cea Cae et ; lonaesietl eeT earei srg! styeo asiekir! aehy efute’ gactiseace:! “ «48¢esec2$> 40 enteedisteht Views =) Ss <4] ra? " ,t3lsegoet eters *3, Sinks 7). eels ef ei sft oe] ty* . ia’ Ye oo34/ omett. ai Gadend + 5. rere Sod (Jen teiivscs — wile? Ld patie > 267 19%-Se eater tanks re te ¢@ : fa) * 2 ; bye 9 See - Himejee@® od: “ alt. 7. a thet ence Ot 59 dS 7 2 LE ope A sveqal sae wean ee ee oe oe om aie EPI EY * ave : fcmh, VG) 7}. a2 ae @ teiheger? aj i tesequdiines® Stee. i= mae “a Ltepe ct iES Aseaoe 5h st eS eet ao et ie es ners . = , be icteaetbes 6e¢neen Fant ig? vio 2ECEE ee Ls | Doe PRR Aevic hake ree : ret catt? 2 cm (Tl A) iiogeedy tae “st sar poPT Ths sep . Wyo hear Mined emery) « ores GT cane 4 Veta + Lag. : Ce ert Oe APPENDIX A THE FORCE GENERATED ON A BODY IN POTENTIAL FLOW BY AN ISOLATED SINGULARITY Consider a body immersed in a uniform strean, Wee with an isolated point singularity located external to the body aie Te S 3 The flow is Q° assumed to be steady, irrotational, and incompressible. If the body is represented by a surface distribution of source singularities o(t,)> then the force exerted on the body is given by (Equation (13) in the text) F326 f o(r,) Vg (Fp) ds (F,) S B where AG) is the velocity induced by the singularity at the body surface. This relation may be derived by considering the control volume illustrated in Figure A-l. CONTROL VOLUME BOUNDARY Figure A-1 — Control Volume for Analysis of Force on Body 69 The pressure forces acting on the control volume surface are equal to the — time rate of change of linear momentum, M, of the fluid within the control volume. For steady incompressible flow, one obtains z mpl i f (pds = = M dv Sea OmGho Control volume C.V. () B Q Se ence Gavic (A-1) — —\ = 0 VeVV dw GoWo = 0 Vv (Ven) ds +§ + S.. So S. On the body surface, it is required that V-n=0. The force acting on = — F = I pn ds SB the body is or, from Equation A-l, ha i [pn + ov(Ven) Jas S The pressure is related to the velocity by means of the Bernoulli theorem as — —\ » So oO Wow 2 and so the force may be expressed entirely in terms of the velocity on the boundaries S. and So? as Q = anGion)]) as (A-2) It is now necessary to develop expressions for the velocity at large distances (|z| +o) and in the immediate vicinity of the singularity (rr, +0). For convenience, the velocity may be regarded as the sum of three terms, vT@®) = TS he) = 1e® a bea * Free Body Singularity Stream Induced Induced Velocity Velocity The body velocity, V,(#), is given by i iss —J _\ ah shes, —\\ Seo ks We) = ae 4, a(t) zee ISG) B B As |z| OS —\ 3 7-\ i —" Sire (Gu, Pe) B 1 B 1 VOR Reamerst 8) ae Ore. 2 4n|r eal |r| where i —\ Up = f o(r,) t ds (z,) B As t+ Xo: => .\ = ay = = Sd 2 ~ -_— e + V, (©) V, (ZQ) en o VE ) @x(ew) where en=r,-ft Q 71 If the singularity is a potnt source of strength m, then see OS re Saers [F-29 Q Then, as eal =e Dp uA = 3 Y(t .°r) nr l Ne ae eo ancl ee a a ele Fl and as r Ti aN an m T me ag a If these expressions are inserted into Equation A-2, the following results are obtained by carefully evaluating the limits io Elee * — F —\\ _—\ — my = War fou [Ve + V3 (FQ) = pm V, (9) To 2 7 _ om eh SOM EB. aay eae f o(r,) = eeea e ds(r,,) B CO LB —A\ —A\ van! —\ ) m nee Q —\ aia i o(rp) tt een : ey B 3 “O 72 or Teo@ f of.) a) ds (Z,) which is the desired result. The solution for the case of a doublet singu- larity may also be written in this form, although the derivation is con- siderably more involved. 13 Tr cr arity 3¢@ 4.5 vox inileh POLE i eR iy siti ee Gq. ti ti m ‘al | | ie Ha Ne 7 F i ~ ms A> o exgng cr PS j4aO8 i “ie erin ot’ ok wok gubag ont stuns bering: rus a! nvlsavriad mid rguodsis etter Mh obs * ears vé' “tal Vee ry aS € | 7 ott ey ae ay ari OTR ‘ in va awe Vivi) APPENDIX B SINK DISK REPRESENTATION OF A MODERATELY LOADED PROPELLER The circumferential mean axial and radial velocities induced by a moderately loaded propeller (outside the slipstream) can be related to the flow generated by a sink disk. Consider the velocity potential associated with a distribution of sources, o(r,¥), located at the propeller plane = W5 © < we S Ip Git.) oe ey GE Wh) we aie On sty ee an 0 0 Vx teeter ore’ cos (Y-¢" ) If the source strengths are independent of angular position, g, the integral may be rewritten as -k |x| R eo) o(x,r) = -4 || o(r~) il dk J (kx) Jo (kr “de dk dr~ 0 0 Integrating by parts yields R L © Jase) i (se) ncaa il 2 8) f sie Ola pclae ce: 2 0 dr 0 k where it is required that o(R) = 0. The axial and radial velocities are then given by R 5 oo 20, r) = oe ao Ss) J,(kr) J,.(kr7) e iS [>| dk dr7 5 2 O > we 2 0 ie 0 0 1 | -_-o_ ~ a ——s il} R ne PES Li a7 ES) i. Ge) 5. Ca) eo [| dk dr all x Af r eae 1 75 These results are equivalent to the induced velocities of the propeller (Equation (26) in text) provided that do (x) dr = 0 es Ry lige | A che) 1 dr 20 dr rx tan 8, (r) apse Ss It follows that R D , a(x) 2 { dO) dee ‘ 21 Gke™ r° tan B, (r*) where the constant of integration is set so that o(R) = 0. For typical propeller loading distributions, o(r) is negative over most of the disk, corresponding to a sink distribution. 76 Copies I 1 OF FF FF FE Fe Soy WES U.S. ARMY TRAS R&D Marine Trans Div CHONR/438 Cooper NRL 1 Code 2027 1 Code 2629 ONR/Boston ONR/ Chicago ONR/New York ONR/ Pasadena ONR/San Francisco NORDA USNA 1 Tech Lib 1 Nav Sys Eng Dept 1 B. Johnson NAVPGSCOL 1 Library 1 T. Sarpkaya 1 J. Miller NADC NOSC, San Diego 1 Library 1 T. Lang 1 J.W. Hoyt 1 D.M. Nelson NCSL/712 D. 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