DUN SRC & log! Ocr 168| : DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 DTNSRDC-81/081 LIFTING-SURFACE HYDRODYNAMICS FOR DESIGN OF ROTATING BLADES WHO] DOCUMENT COLLECTION by Terry E. Brockett APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED Presented at PROPELLERS ‘81 SYMPOSIUM The Society of Naval Architects and Marine Engineers Virginia Beach, Virginia, May 26-27, 1981 SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT URFACE HYDRODYNAMICS FOR DESIGN OF ROTATING BLADES October 1981 DTNSRDC-81/081 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 0 TECHNICAL DIRECTOR 01 OFFICER-IN-CHARGE CARDEROCK OFFICER-IN-CHARGE ANNAPOLIS EE <= SYSTEMS DEVELOPMENT DEPARTMENT 11 SHIP PERFORMANCE SNM ENT AVIATION AND SURFACE EFFECTS DEPARTMENT 16 COMPUTATION | Asante AND MATHEMATICS | DEPARTMENT ,, PROPULSION AND SHIP ACOUSTICS DEPARTMENT AUXILIARY SYSTEMS 1 DEPARTMENT 47 were, ———— 1 —— } = | = | —— | MATERIALS CENTRAL =a INSTRUMENTATION a DEPARTMENT =~ | DEPARTMENT a 28 ba =n o=Z fi | S20 | ie a— _ | OS / spb | =n lt —=5 = | E 2 | | NDW-DTNSRDC 3960/43 (Rev. 11-75 GPO 905-679 Dent. pf.) : mf NOV’ 2 1984 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE T. REPORT NUMBER 2. GOVT ACCESSION NO| 3. RECIPIENT'S CATALOG NUMBER DINSRDC-81/081 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED : SHIP PERFORMANCE DEPT. LIFTING-SURFACE HYDRODYNAMICS FOR DESIGN RESEARCH AND DEVELOPMENT OF ROTATING BLADES 6. PERFORMING ORG. REPORT NUMBER . AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(S) Terry E. Brockett . PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK D id W T ily N ile Shi R h AREA & WORK UNIT NUMBERS avid W. Taylor Naval Ship Researc Task Area SF43421001 and Development Center Program Element 62543N Bethesda, Maryland 20084 Work Unit 1500-104 - CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Naval Sea Systems Command (05R) Ship Systems Research and Technology Group Washington, D.C. 20362 Shil . MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) Naval Sea Systems Command (524) Propulsion Line Shafting Equipment Division NCE eo aD Washington, D.C. 20362 1Sa, DECL ASSIFICATION/ DOWNGRADING HEDUL . DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED . DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Marine Propeller Model Experiments Propeller Research Propeller Theory Propulsion - ABSTRACT (Continue on reverse side if necessary and identify by block number) Analysis and numerical results are presented for the design of a system of wide-bladed thin lifting surfaces rotating at constant angular velocity in an axisymmetric onset flow field. Blade sections may be located arbitrarily in space. General chordwise and spanwise loading functions are available as well as a variety of thickness forms. In addition to the final meanline and pitch distribution determined from a chordwise integration of (Continued on reverse side) DD , aontse 1473 EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIED S/N 0102-LF-014-6601 a SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 20 continued) an appropriate combination of geometric variables and induced velocities, additional information not available from other existing techniques includes pressure distributions and surface metrics for an orthogonal streamline coordinate system on the blade surface, as defined in the Appendix. The induced velocity field on the blades is derived from the principal value of a singular integral; the evaluation of this integral is discussed. The predictions are generally supported by experimental data. A new term in the analysis arises from a radial onset flow component and an example illustrates its importance in design. Sufficient information for manufacture is obtained for computer run times from 400 to 1200 seconds on the Burroughs 7700 high-speed computer. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) TABLE OF CONTENTS Page IIS AP Ol WHEUNES>o co 6 6 6 6000 0 0.000 0 0 6,66 0 0-6 5°6)0)'0 00 0 00 4 iii TOUS OF WNSUES co o go o 6 0 6 0000000059600 6066 06.0 66 0 6 0 06 iv /NBGIEBYNGIE Gg 0 G0 080 0 0 8 0 0 00 0 8 oo 615 6 O16 6 G6 6G 0 6 O50) 6 3 0 1 MOMNTION co o 0 6 60600000006 0666 6 56: 6'6 615 0060 6.6 6 0 6 Oo 8 aL TURODUGIION 56 oo 0 6566 6 06 8 Ob oo OO 6 66 OG oO O10 6 O08 2 MATHEMATICAL MODEL - THICK LIFTING BLADE. ....... +2. +. 26 © © © + © © «© 3 NUMERICAL ANALYSTS) PROCEDURE) Ve) 05) ci ve ey ei re) ols) lo) te) @) elo) Gol fel el) fe) (0) 9 COMPUTER CODE - CONVERGENCE AND RUN TIME. ........ 22+ © «© « «© «© » 13 DESICUSSTONT ORS EXAMPLE (COMPUTATIONS 73 Sas 3 te ls) oll ole) el ey ieee) i 14 COMNGIUIDIONG IWIN 5.9.60 0 60.0 0 0105050506066 5 6 6665 610 0 0 10 519 20 ACONORILIRDICIAGISINIES 5 5 56,600 600,010 0 01°86 16 66 6 5 96 5610 06 6 6 66°56 6 6 20 RUIN CS 6 o of oo 0 6 6,0 0910 0 0 6 6 B16 6 0G 66 01G 010 916 0 6 16 6 og 20 APPENDIX - STREAMLINE COORDINATE SYSTEM. . . . « «© «© «© «© «© © © © © © © © © © « 21 LIST OF FIGURES i = lsltesineaGhurtnee CAOMNGEAVo 55 6 6 0 6 6 of 0 6 5 0 6,6 6 6 60 6 6 Oo G6 o 4 2 = i@ac Dnistereidpmiblom 5 oi po 6 460 6060066006006 00.0 65 60 9.0 0 13 3 - Helical Velocity Component, Vv , 2, /V. Pe rt a MAD A oa alten cht 14 4 - Effect of Skew and Rake on Pitch and Camber Values. . ...-.-.-+-.426 - 17 5 - Effect of Chordwise Load and Thickness on Pitch aioveal Gembyere WEULRES oc Go oo. oud) (Oe wo 66 Oo 66 6G Jono Joe oAG® ooo oO. 6 Coll orto nN 6 - Effect of Orientation of Shed Vortex Sheet and Blade Reference Surface on Pitch and Camber WEIGHS PP UE ae rou G0 hGrn ecln Ge WAee De On CHER Om moe pic tuanc nia tracy) PaGunCR ED Mmemattce em Loe ieay Maes iq irc aiomtng 17 7 - Pressure Distribution at Design for Three Blades. . ...-.-+-+-+-+ +e s 17 iii Meanline Shape for Three Blades . Pressure Distribution and Meanline Shape for Variation of Chordwise Load and Thickness . LIST OF TABLES Effect of Parameters on Pitch, Camber and Computer Nia) WALMWSs, GG Seo) oY & 6 6 1 -o 8 67% Definition of Design Example, Sample Data from Computer Code . Thrust Loading and Power Coefficient. Effect of Radial Velocity Component on Pitch and Camber (Geometry Similar to NSRDC Model 4498; Wp = =0,05)) 6 6.6 6 6 66 © iv Page 18 18 14 15 19 19 THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS One World Trade Center, Suite 1369, New York, N.Y. 10048 Paper to be presented al the Propellers 81 Symposium, Virginia Beach, Virginia. May 26-27. 1981 Lifting-Surface Hydrodynamics for Design of Rotating Blades Terry Brockett, David Taylor Naval Ship Research and Development Center, Bethesda, MD ABSTRACT Analysis and numerical results are presented for the design of a system of wide-bladed thin lifting surfaces ;otat- ing at constant angular velocity in an axisymmetric onset flow field. Blade sections may be located arbitrarily in space. General chordwise and spanwise loading functions are avail- able as well as a variety of thickness forms. In addition to the final meanline and pitch distribution determined from a chordwise integration of an appropriate combination of geometric variables and induced velocities, additional infor- mation not available from other existing techniques includes pressure distributions and surface metrics for an orthogonal streamline coordinate system on the blade surface, as defined in the Appendix. The induced velocity field on the blades is derived from the principal value of a singular integral; the evaluation of this integral is discussed. The predictions are generally supported by experimental data. A new term in the analysis arises from a radial onset flow component and an example illustrates its importance in design. Suf- ficient information for manufacture is obtained for com- puter run times from 400 to 1200 seconds on the Burroughs 7700 high-speed computer. NOTATION A,B Cp Cy =P = Poo )/(0V?/2) Cp = 2mQn/(pV?1D?/8) Crp = TeV? nD? /8) C(Xp) D De E(x,, Xp) = ES + Ey E.(X,, Xp) Ey (X,, Xp) (e€),€5,&,) F G (xp) iy (Xp) Point values in linear approxima- tion for distance Blade-section drag coefficient Pressure coefficient Power loading coefficient based on reference speed Thrust loading coefficient based on reference speed Blade-section chord length Rotor diameter Friction drag on blade section Profile shape function Meanline shape function Thickness shape function Unit base vectors in a helical reference system Induction factor Non-dimensional circulation Total rake: axial displacement of blade-section midchord point from x = 0 plane (i, &,, &g) (i,j. k) Jy = V/nD N(x,, Xp) Np (Xo. Xp) N, (X_: Xp) mn n p P(xp) Q GSI cops Goo R=D/2 Th To Sp (Xo. Xp) Sy (Xe, Xp) s Sw, (n, Xp) T t Vv |< =(7, u, 0) Mp No. 20 Unit base vectors in a cylindrical polar reference system Unit base vectors in a Cartesian reference frame Advance coefficient based on reference speed Vector normal to blade surface, pointing into fluid Radial component of N Normal to blade-reference surface (E = O surface) Unit vector normal to blade surface, pointing into fluid Propeller rotational speed, revolu- tions per unit time Pressure Pitch of blade section Torque absorbed by blades Velocity vector Velocity vector far upstream Rotor tip radius Radius of rotor hub Position vector of field point Position vector of field point on blade reference surface Position vector of point on pth blade surface Position vector of point on blade reference surface 6, = 0 Position vector of point on shed vortex sheet Thrust produced by blades Thickness of blade section Reference speed Velocity component due to presence of the blades Average perturbation velocity along blade surface, due to presence of the blades Velocity difference across blade surface Even perturbation velocity com- ponent due to blade loading and shed vortex sheet Wy (Xp) WR (xp) (x, y, Z) Xo ate! BL Zz 6, = 2m(b- 1)/Z 9. (Xp) 9, (xX Xp) A=(0,7) Even perturbation velocity component due to thickness Velocity induced by vortex filament Local wake fraction Radial free stream velocity component, fraction of V Cartesian coordinates Cartesian coordinate for field point on blade surface Fraction of chord, measured from leading edge Fraction of chord for field point on blade surface Hub radius, fraction of tip radius Fraction of radius, measured from axis of rotation Radial coordinate for field point on blade surface Nondimensional thickness offset; maximum Y 7 = 0.5 Number of blades Angular variable in chordwise direction Component of derivative of surface coordinate Advance angle of blade section Circulation distribution Chordwise component of disturb- ance velocity difference across blade section Chordwise velocity difference scaled to give unit magnitude when integrated across the chord Error bound; Increment to pitch angle when radial inflow exists Integration variable along vortex filament Angular coordinate in cylindrical reference frame Angular coordinate of blade- reference line of b'h blade Skew angle; circumferential dis- placement of blade-section mid- chord point from y = 0 plane Angular coordinate of point on blade-reference surface Vorticity vector u(X,, Xp) Normal component of disturbance velocity difference across blade section (source strength due to thickness) (E,,§,1) Helical coordinates on pitch reference surface p Fluid density 0 (x,, Xp) Component of disturbance velocity difference across blade section 6, Surface area i) Potential function for perturbation velocity; polar coordinate for field point P/D dbp (Xp) = tan! (Dy Pitch angle of blade reference ™XR surface; measured on cylinder of radius r o, (xp) Geometric pitch angle vy (x,) Radius of streamline on blade surface w Angular variable in radial direction INTRODUCTION The design of an open marine propulsor is a complex process, involving structural and hydrodynamic considera- tions (1,2). For the hydrodynamic considerations during most of the preliminary design process, approximate models of the lifting surfaces are employed, e.g., the lifting-line model (3, 4) for powering considerations, and two-dimensional flow over equivalent blade sections for cavitation performance. More sophisticated models of the lifting surfaces are used for pre- dicting fluctuating loads (5) and some cavitation predic- tions (6). These approximate models have been acceptable during the preliminary design process and provide a basis for choice of the maximum diameter, advance coefficient and radial variations of chord, skew-angle, rake, thickness, and circulation distribution. The chordwise variation in load has usually been selected during this preliminary stage and is often based on cavitation and propulsion considerations. For the final stage of the design, the meanline distribu- tion and radial pitch variation are determined corresponding to the selections for load and geometry already available. To derive a geometry which accurately produces the specified load distributions, a lifting-surface model of the blades is required. Several procedures already exist for performing lifting-surface calculations for wide-bladed open marine pro- pulsors. In particular, two different approaches to the analysis for blades with arbitrary locations in space have been presented by Kerwin (7) and McMahon (8). Kerwin’s numeri- cal analysis procedure is based on three fundamental assump- tions: (1) that the continuous loading distribution on the nonplanar blade surface can be adequately approximated by a multitude of discrete straight lines of constant-vortex strength and that the source distribution arising from the thickness distribution can be similarly approximated, (2) that the minimum required spacing between lattice elements along the chordline is A@ = 2 degrees, and (3) that the resulting meanline shape for a given chordwise load is similar to the two-dimensional shape for the same chordwise load. The first two assumptions are not acceptable for very narrow blades: for a blade with a 20 degree pitch angle at the 0.9 radius and a chord to diameter ratio of 0.05, the 2 degree spacing equals increments of about 1/3 chord length. The last assumption permits calculations to be performed using only a few points along the chord and the two-dimensional shape is fitted to the data at these points. The resulting computer code is relatively quick running and produces a geometry which, in practice, has an overall speed and power- ing performance generally within a few percent or so of the predicted values, with a general tendency to produce a greater thrust than predicted. The procedure of McMahon employs continuous distributions for the loading and thick- ness functions and calculates the meanline from the induced velocity. Consequently, data at more chordwise points are required to define the pitch and meanline distributions. The resulting computer code is lengthy to run but has shown remarkably different meanline shapes from the two-dimen- sional one at the hub and tip region of the blade where the meanlines can be s-shaped (8). Two models were constructed and experimentally evaluated to provide data on the relative cavitation and propulsion performance of designs having the same input specifications but final geometry according to the Kerwin and McMahon procedures. Some inconsistencies occurred in the experimental measurements but the thrust was closer to the predicted value and the operating point centered in the cavitation bucket for the model designed by the McMahon method. Hence, the determination of specific meanline and pitch distributions, instead of fitting the two- dimensional meanline, is considered to be a superior pro- cedure when the design is based on a narrow range of permis- sible operating conditions and the delay of cavitation is critical. Because the numerical-analysis procedure employed by McMahon results in lengthy computer runs and Kerwin’s procedure is not acceptable for narrow blades, alternative numerical-analysis schemes are investigated in this paper. In addition, a detailed description of the flow field across the blade surface was desired as input into boundary-layer calculations. Two different numerical analysis schemes are described, each involving an expansion of the singular kernel about the singular point. Both approaches employ integra- tion of the specified thickness slope and load distribution over the reference blade in the radial direction first and the remaining chordwise integration then takes the form of the velocity component corresponding to two-dimensional flow modified by the presence of an induction factor in the integral. Regular integration techniques are employed for the other blades and the shed vortex sheet. The induced velocity components are appropriately combined and inte- grated to obtain meanline shapes. The present investigation describes the real-fluid flow about a rotating system of lifting surfaces having both loading and thickness. Several approximations are made. The first of these is the mathematical model for which potential flow equations are employed and the solution to first-order in thickness-to-chord ratio, camber-to-chord ratio and difference in pitch and flow angles derived. Compari- sons with experimental results for other lifting-surface configurations lead to confidence in this linearized approx- imation. In addition to this mathematical model, further approximations occur in the numerical analysis. Confidence in the numerical analysis procedures is justified by compar- ison with analytical solutions or experimental results. That is, results are sought from some discretized numericak analysis procedures involving N by M approximations, which have converged to within some specified tolerance, €, of the Teal or analytical value of the quantity investigated. Math- ematically this may be stated If.y)-tym GylN, M>M, where fy yy = the approximate calculation of a particular ; quantity f S = aregion of the surface of interest N.,M. = minimum numbers of the discrete approx- imations for which the computed results are within € of the values for f For rotating lifting surfaces, neither measured nor analytical solutions exist for details of the flow field on the blade. Hence, comparisons will be made with other procedures. It is assumed that numerical solutions which employ in- creasingly greater pointwise definition of the input variable without change in computed values have converged and that the solution has converged when a smooth curve can be drawn through point values in both the chordwise and radial direc- tions. These assumptions are believed to be necessary but not sufficient for convergence. In the following sections, the mathematical model of the flow field on the blade surface is first reviewed and numerical-analysis techniques for evaluating both regular and singular integrals are described. A FORTRAN computer code is discussed and sample calculations using this code are pre- sented. From example calculations, it is found that greater accuracy in the integral evaluations is required for the deter- mination of smooth pressure distribution curves than for the shape of the meanline and the pitch distributions. The choice of a particular chordwise loading distribution is shown to have an effect on the meanline shape and the pressure dis- tribution. The effects of rake and skew are shown to be important on both pressure distribution and meanline shape. A particular thickness function has hardly any effect on pitch or meanline but a significant effect on pressure distribution. _ MATHEMATICAL MODEL — THICK LIFTING BLADE The mathematical model of a system of rotating lifting surfaces advancing in an unbounded irrotational flow field with an inviscid fluid has been developed on a formal mathe- matical basis by Brockett (9). A reformulation of that analysis in terms of non-dimensional surface coordinates is presented herein for completeness. The propulsor is assumed to be ade- quately represented by the blades alone, i.e., neither the hub nor fillet from the blades to the hub is included in the blade specification. The onset flow is assumed to be directed along the axis of rotation but a new feature included herein is that it may have a small radial component. Overall geometry nota- tion generally follows the definitions given in Reference 10. Coordinate systems are constructed with the same orientation as in Reference 9, and in particular, the helical coordinate system (£), &), r) rotating with the blades is shown in Figure 1. Unit base vectors in a right-handed Cartesian reference frame are the customary (i, j, k) where i is along the x axis and is positive pointing aft, j is along the y axisand k is along the z axis which is generally along the reference blade. Unit vectors along the helical coordinates are e€, =sin dp i + cos dp ep (1) €7 =- cos dp i + sin dp ep (2) le, == sin@j| + cos'@) k (3) r where €9 =-cos6j - sind (4) an Fig. | Lifting-surface geometry The blade surface is given by & = E(é,,1) (5) = E, (Ey, 1) + Ey (Ey, (6) where E. is the meanline shape, and Ey is the thickness shape In the analysis, it is convenient to change the variables of integration to (X,, Xp) instead of (€, r), where Ey = 6s, - 0.5) T D xp/2 (7) c = chordlength at radius r and D = maximum rotor diameter The position vector of a point on the blade surface described by Equation (5) is = it Cc 4 E : 2-{[ + p Xe = 0.5) sin dp - D cos op] D x + = e, «| (8) and a normal, directed out from the blade surface, is (9, 11) ri) i) Nae (9) ax, OXp where the plus sign is used for the suction side of the blade and the negative sign for the pressure side of the blade. After some effort it can be shown that D2 N=+ (5 e _2E/D G + Nee, (10) A ||) 4 Bxe, c 0E/D dc/DdE/D Ne =-2—= +2(x,-0.5) D0 xp C dxp 9x, pple eos + u sin Salar ID. E sal(2\ @. ans: 2222 (5) Be ah aOR |: P/D cos? dp Sin dp cos op dxp ™ Xp XR £ (x. -0.5) cos@ +E sin dé, Direc ‘i 1) 55) P -[Kp — 2 dxp XR AA 0 E/D De ax, cos dp The normal to the blade reference surface, £,=0,0< X. S15 Xp, < Xp < lis Cc N, Busine B [2 *Ne, cr @,) (11) where d P/D cos” ¢p d 6, - Xp —— sin dxp Xp Rd Xp OP Nr , the radial component of the normal, is zero for a constant-pitch blade which is neither raked nor skewed. In Equations (10) and (11): iy = the total rake P = the pitch of the blade bp = the pitch angle, bp = tan7! (P/(r Xp D)) 6 = the angular position of a point on the blade surface, a function of both x, and xp b-1 = 2 m+ O,+ 2 E (x, - 0.5) cos dp Ee: vig SG XR 6, = the skew angle, a function of xp and _ 4 ee! c 6. = Lee arly at Oe (x, - 0.5) cos dp/Xp In the derivation of the expressions for numerical analysis, the reference surface (E = 0) is often employed. Generally no specific mention will be made of differences between variables on the blade surface and on the reference surface. In a coordinate system rotating with the blades, the fluid velocity may be taken to be the sum of the undisturbed velocity and a component due to the disturbance of the blades: q=V(1-w, (xp) fe 2mm r &6 (12) + Vwe (xp) e, + ¥ =q. tv (13) where V = the constant reference speed I-w, = the wake-fraction multiple to obtain the local axisymmetric speed! Wr = the radial component of inflow, fraction of the reference speed n = the rotational speed, revolutions per unit time, and \< | the velocity component due to the presence of the blades If dp is the pitch angle and 6 is the advance angle (B= tan7! [V(1- w,)/(2mn1)]) then , 2 Goo | 5 PR = (1-w,) a J va ; (14) : {cos (dp - B) e| + sin (dp - B) e,}+Wwp ce where the advance coefficient, J,, is given by J, = V/(nD) (15) The boundary condition on the blade is that there be no flow through the surface: q: N= 0 for r on S (16) This condition applies to both the upper and lower surfaces: ai eNen= "0 (17) oN =O (18) The sum of Equations (17) and (18) is: q: (Nt + N7) + vt + Nt +y7 > N7 =0 Now aE,/D i Jeo + (Nt +N7) =-D* ———(q,, * e}) Et abs ax, ea B34 + 0(E, Wp) (19) thus to first order in Wp or E: TG vt 5 Nt + v7 3 N7 =D? v (1-w,)* + J, a (E_/D) cos (pp - B) ) c = Nt + (vy? -v7) (20) 1 As described in Reference 9, the inclusion of a radially variable inflow introduces vorticity into the mathe- matical model. No specific consideration is undertaken to account for this vorticity. The difference of Equations (17) and (18) gives: Vi NG vg TN She deve ONIiaey NG) Be =-g,°|2N¢ - D? D e, 10(Ee, | L xX Cc E pea = D) . + | (cage IN > ax. Goo ©) + O(E, wp) 2 ™XR =-p? v}ya -w,)? +( ) J, f a E,/D c De ie cos (¢p - B) Cc + = wa Ne | + OE, wR) (21) ~ wt + - ING a = wy) or sane P= Iran (op - 8) (22) dx D P Z Nt + (vt + v7)/(D?Vc/D) + WR NR, | Xp 2 cos (p - B) (1 -w,)? + ; Vv This is the fundamental equation to be evaluated to determine the pitch and meanline shape. The slope of the meanline is given as a function of known geometry and inflow quantities plus the normal component of the average induced velocity on the blade surface. Hence to determine the meanline slope, the mean perturbation velocity must be determined on the blade surface. The velocity component due to the disturbance of the blades, v, can be shown to be potential in nature (9). Hence, it can be represented by an equation (9, 1 2) 1 BOS) vo=— >), ff n 4n b=1 r-s, S = T- Sp + (n X v) X——— ] do, (23) In - 559° where n isa unit normal pointing into the fluid from a point s on the surface S (including both the blades and shed vortex sheet), and do, is the area element. In several texts (11, 12), it is shown that n do, = N dx, dxp (24) In Equation (23), it is convenient to let the region of integra- tion be the blade reference surface 3) =0,0< Xo <1, X}, < Xp < |, for which i > XR : eG ey) (25) Hence, Equation (23) can be reduced to an integral over only one side of the blade surfaces and shed vortex sheets: , Z 1 J HO) Sa d x, fe Xp b=1-0 Xh BO 2o. (Nt vt +N °- v7) a og. 2 Po r-s = “Ob | ZL 1 = dx c +—) d dn —I[N* X(vy*-v7 a fox fons DINGS XS (vas ves))] b=1 Xh 0 [=e (26) where: Ss x —w Sn —_ +nsin ¢p iy D D x R eee” (6,. + 2n cos dp/xXp + 4) and As the field point r approaches a point r, (x, , XR) on the surface of the blade, Equation (23) or (26) becomes singular. If a small region about this point is excluded from the surface S and the limit of the integral taken for r>r,, with the excluded area tending to zero, there results (see Reference 9): BG *h iy Sy, » TONt + yt + N7 + v7) ; To So, Li Soh, i) Z | dx ] c 7 = ) dx d 4n il R fo dn Dell a. (27) (cont.) WWeSGR ory) (Gas) 3 where the symbol f means symmetry restrictions occur for the limiting region which excludes the singularity. For example (9), the region may be square, circular, or rectangular centered at r). In the present application, the rectangular region x, -€< x. < 2g. Xp © Xp <1 will be the oO shape of the excluded region. Then this principal value integral is defined 1 1 | £0) fox foe K = lim f& for K 0 Xp mealego xh 1 + fax, fare K (28) Acme Xh The assumption lim [v(1)] = v* (9) (29) + a II (i.e., that the velocity defined in the field does approach the value on the boundary) leads to the following expression for the average velocity component on the blade surface es 24 ¥(x.,, RET Ue at ve, +we, 1 x i Ip a 3 far. [or K(x Ry Xe XR)? Vw (31) b=1~0 h where the singular kernel is and the velocity induced by the shed vortex sheet is IN* X (v*-v7)] X (Fo -sy.) a ne [5-809 The first term in Equation (27), the strength of the sources, is known from Equation (20). The strength of the vortex component: Niji Xevaust Ning X vay = IN Xav ava) (32) has yet to be determined both on the blades and along the shed vortex sheet. To find a value for this term, it is appro- priate to look at the condition that the blade section develop a force at a given radial station Fixg)=- fr n dg (33) To first order the local lift is L=F:e,=D— 2 pt = Oy | Mee Ves (34) 0 Bernoulli’s equation can be constructed for a coordinate system rotating with the blade (9) and the pressure dif- ference determined by 2 p--p* wi ffi’ y (9) | (35) = |. “(vt “ep -ot-v9} =p de (Vv -v) (36) Let . N =—[vt-v}=4e,+= = y UNE o Ne X ey +—7 D? V————_ (37) 2 me ING | We (38) TT XR 2 pV\[a-w,)? + ; cos (¢p - B) v AC = + HE (39) 2 TX (1-w,) ( " cos (¢p - B) J, and from Equation (20): ee ea a a 1X =p2 2 u=D Vw) ‘G 2 R - 6) —— _ “4 *) cos ($p - B) ae (40) where: AC, =(p~ - p*) / (eV? /2) (41) It remains to determine a. Now, in general Ng X (vv) =-"D?Voe, +7 (NG Xe,) (42) and the value of o can be determined by setting the diver- gence of the right-hand-side equal to zero(9). However, this results in a differential equation to be solved for o. A more direct procedure is to express the perturbation velocity vector as the gradient of a potential: v=V¢ (43) where r (1) = lim f v- dt (44) Az 2 -Ai Thus T Pee a Reet [eal (oe ae | EOT, “-AL A> 1 T = lim pf» ae | (45) Tiegln -Ai A> oo and Nt x Nt f = x Vf (vt -v7) de 2nD?V 4nD? V le (46) Cc For d& = D—e, dx,, a D c No X Ni enGac —_—__—. = —— xv |D—] vadx,] (47) 2nD?V 4D? V Da It is convenient to define TACcR ran Ce) — 48 D c/D Ce where I (xp) is the bound circulation (¢* - 7 at the trail- ing edge and points beyond), and 7 has unit magnitude when integrated across the chord. Let the nondimensional circulation G be Pa ly mDV (49) Then G (xp) ¥" (x,) = ae 0 y=7V TD (50) Ni x N6 xc A= ————= XV FVDG OR) Manixe 2nD?V— 4nD?-V K (51) No : = A XV G (xp) y* dx, (52) 10} In Reference 9, the gradient of a scalar function is derived for a general helical coordinate system. From this expression one obtains: NG dG (°° N c We oer. Gr" 6, +2(5 y* dx, -aGy* €, se Ae = D dxp 7 D 10} (53) Ke Nt Xe 1 dG £ & =| Be es y* dx. -aGy e, t+—Gy" = 4 D dxp C D2 Cc oO =—_. 72, )d) (54) Hence from Equation (40): xX 7 dG Cc o=— y* dx -aGy (55) D dxp C 0) where £ 5) a @ CE) a= \|— - — (x, - 0.5) dxp D XR i ili d— if XR A dé be 4 D eo) ———11COS sin P dx P XR At the trailing edge and beyond y* (1) =0 (57) 1 * { WT Ce Sl] (58) fo} Hence lGe dG Kee eS 59 i Wi gdm Daad i aur. ore for the shed vortex wake. Thus, the strength of the vortex distribution is ex plic- itly known and the integration to determine the average in- duced velocity at points on the blade surface can be undertaken. Once the induced velocity field on the blade is computed, the meanline slope can be determined and the meanline offset can be found by integrating the slope: i E. (x Xp) E, (x, Xp) (3 aap oO 0x dx, (60) c From Equation (22), the meanline offset for the term with the radial inflow velocity component can be directly com- puted; it consists of an angle of attack term due to gradient of the rake and skew terms and a parabolic arc meanline due to gradients of the pitch. In general, the trailing edge point will not be zero. The trailing-edge offset can be converted into a pitch-angle increment, tan (¢, - dp) =-E, (1, xp)/e (61) E, (1, xp) é = opt tana! ae a ey (62) for which the pitch of the nose-tail ine becomes =1™Xp tan » nt 7Xp tan Op +7Xp (-E, (1, Xp)/c) (63) 1 — tan ¢p (-E. (1, xp)/c) and meanline ordinates measured from the nose-tail line can be determined: oS —_ &- (64) These values of corrected pitch and meanline shapes relative to the nose-tail line are the essential data produced by the lifting-surface analysis. In addition to the meanline and pitch distnbutions, several other quantities of interest can be computed. These are pressure distribution, total forces, and streamline coordi- nates (this coordinate system is described in the Appendix). From Reference 9, the pressure coefficient is P - Poo Ges = (65) oA ico G5 > q y2 An approximation for the non-linear speed on the surface of the blade in the chordwise direction is (13) ee (67) 2 2 Vv 2 q qs a = “a ped a — |S oe -@+—- 68) o a (vs ay ‘) is 2 where @€ is the unit vector in the Ne xe, direction (nearly the radial direction over much of the blade): ri e -Np e SESS et a8 (69) %) INS Xeql Vi + Nr In Equations (67) and (68), the velocity components v + e| and v € are Vv Vv u Y 0 Bp = Cy ols Ss 4 (70) Vio= Vv Vv Vian vN By a Ro T v 0! SUS Win, 2a —-*@=@-(—+ = ————_—_————._ (7l) Viearris IN Each of the components v and has terms due to thickness and terms due to loading. To first order, the pressure coefficient is = TRY /u Y = 39) S: 2 jah ph (REE a) C, 2 Vd wy) + ; \( a a (72) Fs 2 {CORN @ od = =) = < — AG =o = C= 2 WO Cae Ha) a GE) ea TXpY Y*(x,) = 20 \(1- w,)* + F G(Xxp) (74) Y which is independent of rake, skew, or pitch variations (to first order). Once the pressure distribution on the blade surface is known, the thrust and torque in inviscid flow may be cal- culated (9). In terms of a thrust and power coefficient, these inviscid overall performance values are: T; Sunieitas corneas (75) pili y2 Dee pte es ar 1 1 : re) = { ox] dx. AC, Tiga bp + sin Xh oO E _ O55 =, am sin bp (76) 2 : eo ee (77) i 2 1 ov3 De OM wT x 1 1 Aas sal snl dx, ao (5 sin p - cos Y Xh oO E Lay ica bp (78) where AC, =C , Set (at at and C, Gav kr The effects of pitch, skew and rake enter into the determina- tion of pressure and are not explicitly in the integrands for thrust and torque. A correction for viscous drag may be made by assum- ing the drag force acts along the pitch line. For a given drag coefficient Cp, the force will be approximately 2 i's 5) EARN Me DRS (1 - When 2 F D DCp (79) Vv When the effects of this force are included in the thrust and power coefficients, one obtains Crm = Cam, + ACr, (80) Cp = Gp. + AG» (81) where ' , Pez = PW |le AC Thais CHCl wade a j Din op 4XR Vv Xh (82) and 1 c — cos dx iF op 4XR (83) : 2Z FERN ACp =—]| xpCp|(1- w,)? + J Ie X Hence the thrust will be reduced and the torque increased relative to the inviscid values, as expected. h In the Appendix, the formulation for a streamline coordinate system on the blade surface is given. NUMERICAL ANALYSIS PROCEDURE The computation of the meanline slope relative to the blade-reference surface (Equation 22) involves some straightforward computations of geometry gradients, which are handled by spline functions, to determine Ni plus the evaluation of the even velocity component across the blade surface (Equation 30). This even velocity component arises from the odd velocity component (Equation 37) integrated over the blade surfaces and trailing vortex sheets. The mag- nitude of the odd velocity component is known from Equations (40), (50) and (55). A singularity occurs on the reference blade and special procedures must be employed on that blade. Integration over the other blades can be based on conventional integration procedures. Integration over the infinite shed vortex sheet is truncated to a finite extent. Once the slope is known, it can be integrated across the chord to produce the meanline offset (Equation 60). For regular integration over the span or chord of the blade, trigonometric polynomials are employed which pre- cisely fit a set of tabulated values given at equal angular increments in @ or w where 1 x = >(1- cosa), O x, 10} 3 S -aB + 26,4) & fea Vila (3) - 2} This known value at the singular point allows a straightfor- ward analysis procedure to be undertaken using the procedures previously described. (103) Some convergence problems near the leading and trailing edges and over much of the surface for narrow blades (maximum c/D ~ 0.05) have been resolved by computing the linearized form of F (Equation 100) over the entire blade and adding a correction term which is the difference between the actual integrand and this linear approximation. This option has been included in the computer program and Is defined as ‘linear approximation-plus-difference.”” When conventional integration techniques are used everywhere except at the singular point, where Equation (103) is re- quired, the procedure is defined as “‘direct.” For the trailing-vortex sheet, a regular integration can be performed since no singular points occur on the sheet. The strength of the vorticity is given by Equation (59) and the induced velocity field is given by vi ! dG z ——— = Wr or XR» 0 ») dxp (104) Vv 4 dxp 0 =] where s A ele aes D D W ( o> XR» Op) I e, Xx dn (105) 0) ih Wh, D OD For the case when the field point ris given in cylindrical polar coordinates, x ur) Xo ») Xo es R, | a om, 5) a D\D’ Ro D- 2 ep) and 106) SW, x (1, Xp) XR D (, Xp) i te, (0 «n, xp) 9) where x(n, Xp) Xte : D = D + 7 sin dp (107) O(n. Xp) = Fe + 20 COS Hp/xXp Then ina SW, COS pp Byes \ ae D = 5 Partin OE 2S OD t ; x. 5 ALO D D COs Pp sin b >) sin bp 5 (xp, Xp cos (8 + A, - oye. + 2 sy (0 +86 tes ed ‘0S 4 i asi epsces b o) XR + sin op—sin (8 + 6 - )| e€, (d) + cos dp = 5 [an xR, 605 (Are 6,-¢ - n sin er) Cos ¢p sin (4.* 4, - i) cos dp sin bp ) He 2 *Ro XR - Xp cos(6,, + O- 6 + 2n +2n Ks a=s cos “*) XR x oimmate ; + (op Se cos bp cos Gre cos bp sin bp +O,- $+ 2n i + XR in( R COS $p +6,- 6+ 2n e, (9) (108) 5 ie, R 12 A computer code has been developed for a subroutine to compute the approximate value of W: 5S S Boy wiagetiomeen sr] tt Lowsisn Wy l D D 2 DEED Wie e, X- a wD Oe : fy Mb ™ ogeatte DD D (109) Options are included in the subroutine call statement to permit the first integral to be evaluated either by the trapezoidal rule with equal increments in 7 or by Simpson’s rule with equal increments in \/7 . This second integration procedure insures more dense spacing of the integrand near the blade and a more accurate computation. To insure an even number of intervals, the number of points specified in the call statement is doubled when this more accurate pro- cedure is employed. The second integral is evaluated by the trapezoidal rule but is not employed when n < 7 - For a given value of m, and specified number of double intervals, N, the constant increment in /7 Is AVn = 2N for which the increment in 7 between successive points is Any = > 2-1 7 (2i- 1) —— 4n2 il} (110) and the increment in angular variable 9 between successive points is 46; = 2An; cos bp/Xp = (2i- 1) 2 cos dp nj /(xp 4N?) (111) from which ny 2 cos $p ON oe ZING 242 and ANiai) SCY Absn = — n = (4N-1) AQ, 4N2 XR Generally n, => and the equal increments of \/n are used for integration with N = 2 "Nz double intervals. A value of > = 10 has been satisfactory to date. When the distance between points becomes small, special fine point spacing in n is employed to insure convergence. Accuracy of calculations was determined by comparison with analytical results for the tangential velocity component due to a circular arc vortex filament, the axial velocity component at the origin for a general helical filament, all velocity components for a straight line vortex, and induction factors (17) for general helical fila- ments. At individual points of this comparison for filaments, accuracy to the third decimal point was found with the selected parameters and an overall accuracy of the non- dimensional induced velocity component of the sheet to one or two units in the fourth decimal point was found. In order to perform calculations, the form of y* and non-dimensional thickness must also be specified. A general family of loading functions has been selected (18) with the property that they have zero values at the leading and trail- ing edges and resemble conventional NACA loading func- tions(19). The zero values at the ends are necessary for accu- rate fit with a sine-series trigonometric interpolation poly- nomial. For loading distributions which approximate the NACA a series meanlines, the following chordwise form is used 1/K (sin @) O fe Bi | = > fw =| 0 SN ALN at At 4 4 1 oO a 2 3 4 5 6 7 8 9 1.0 x, FRACTION OF CHORD Fig. 2. Load distribution The thickness offset is assumed in the form ais "= (Xp) + Ye (XQ) (114) c where the chordwise distribution, Yq (XQ), remains the same from root to tip, only the maximum value changes with radius. This is true for current propulsor designs. Specific examples of the thickness function included in the computer code are the NACA 4 and 5 digit sections (19), the NACA 16 section (20), an elliptic nose quartic tail section similar to that described in Reference 21, and an approximate NACA 66 (Mod) (22) section. All have been analytically defined. 13 Computer, Godel; Convergenceiand/ Runilitme The complexity of the numerical analysis is such that error estimates are difficult to establish. A few limiting cases exist for which analytic integrations may be performed but comparisons do not usually evaluate the general case. The procedure selected to evaluate convergence was to vary the number of intervals in the radial direction, NR, and the number of intervals in the chordwise direction, NX. In addi- tion, some radii and chordwise points were eliminated from the calculations. Computed values or the pitch and camber at selected radii are shown in Table |, together with a similar variation of data calculated according to the procedure described in Reference 7 for the same propulsor which 1s similar to NSRDC Model 4498. Computer central processing time is for computations at 13 radii between the extreme radii listed and is in seconds for the Burroughs 7700 High- Speed Computer. Current charges are 3 cents per CPU second, resulting in a maximum charge of about $40. All procedures presented produce about equally satisfactory results with about only one percent difference in pitch or camber values, about the same as found for Kerwin’s numerical analysis. The computed pitch, however, is a few percent less than computed by Kerwin’s method. Since unpublished experi- ence at DINSRDC to date has been that Kerwin’s procedure produces designs that are generally slightly overpitched perhaps some improvement in performance may be expected using the present method. Predictions of the pitch and camber by the two procedures developed for computing the induced velocity field on the blade surface which contains the field point, “direct” and “‘approximate-plus-difference,” are shown in Table I to be nearly the same. However, it has been found that overall, the ‘‘approximate-plus-difference” procedure is preferable when dense chordwise spacing is chosen (e.g., NX=19) or narrow blades (maximum c/D ~ 0.05) are involved. In these situations, the “‘direct”’ procedure produces locally erratic values of the induced velocity because of the decreased spacing between adjacent lines of integration with a corresponding lack of accuracy in the numerical integrations for the resulting near-singular integrals. This effect is illus- trated in Figure 3 which shows values of one of the helical components of the average induced velocity at the 0.946 radius of the reference blade. This velocity component is due to only loading on the blade itself; the effects of thickness, the other blades and the shed vortex sheet are not included. All data shown in subsequent figures have been computed by the ‘‘approximate-plus-difference” procedure although only the pressure distribution near the leading edge in the tip region of the blade was significantly different between the two procedures. Overall run time varies with number of points, number of blades, and blade width. Since computer usage charges are so low, the 181 x 19 array size is recommended. For a narrow blade, the linear “‘approximation-plus-difference” procedure 1s recommended,and the run time may increase by a few hun- dred seconds because of special care taken with the shed vortex sheet calculations. Computer execution time for Kerwin’s program is unknown for the Burroughs 7700 high- speed computer but is estimated to be about 150 seconds, for data calculations at 4 chordwise points at 8 radial stations. For the results shown in Table I, data are computed at 13 radial stations with either 8, 11 or 17 chordwise points, depending on input data specification. Further details of the geometry of this example are given in Table II]. Radial variables are titled according to the symbols suggested in Reference 10. Table | Effect of Parameters on Pitch, Camber and Computer Run Time COMPUTATIONAL PROCEDURE 91 x 10 Direct 181 x10 Direct 181 x13 Direct 91 x 19 Direct 181 x 19 XR Direct (0), 5) - = = - - 0.254 1.495 1.495 1.517 LES 377) 1.538 0.4 1.466 1.466 1.477 1.487 1.487 0.6 1.264 1.264 Tee eral 2a W277 0.669 1.184 1.184 1.189 1.194 1.193 0.8 1.038 1,038 1.040 1.043 1.042 0.946 0.883 0.882 0.882 0.884 0.883 0.95 = = = = - 0.25 - - - - 0.254 0.0305 0.0305 0.0315 0.0320 0.0322 0.4 0.0365 0.0365 0.0367 0.0368 0.0368 0.6 0.0291 0:0291 0.0293 0.0295 | 0:0295 0.669 0.0255 0:0255 0:0257 0.0257 0.0258 0.8 0.0189 0.0189 0.0189 0.0188 0.0189 0.946 0.0124 0.0124 0.0122 0.0122 0.0123 0.95 = : ~ = CPU Time, 310 420 600 735 1085 Seconds z Oe Ra aaa 7 iniis i T T 1 Agiaazanal ° f of ween ees JATE + DIFFERENCE a2 eee | 35 hs RP ae Ith a ~ a H Senn ie a eae | is oe | IR ek a toa ao ~ w NB = 18 Xp = 0.946 OF oil Nee 4 1 (ie whee sels th nif ts Sd) X_, FRACTION OF CHORD Fig. 3. Helical velocity component, v > e,/V DISCUSSION OF EXAMPLE COMPUTATIONS In this section, the consequences of choices the designer might make both for overall geometry and for the chordwise variation of the thickness distribution and loading distribution are examined. Some common variations in the location of the blade mid-chord line are investigated to deter- mine the effects of overall geometry on pitch, camber, pressure distribution, and second-order performance coeffi- cients. The variations are unskewed, skewed and warped (23) blades with other input specifications the same. Skewed blades have blade-sections displaced along the pitch helix and warped blades have blade sections displaced circum- ferentially in the plane at x = 0. 9] Approx + Diff PITCH/DIAMETER 1.539 1.487 1.277 1.193 1.042 0.883 CAMBER/CHORD x 19 181 x19 Approx + Diff Kerwin Computation (7) AO =2°,M=52 Ag =2°,M=70 —= — a 1.580 1.576 = : 0.0266 0.0323 0.0324 0.0268 = 0.0369 0.0369 0.035G 0.0351 0.0295 0.0295 0.0301 0.0294 0.0254 0.0257 0.0257 = 0.0188 0.0188 0.0181 0.0180 0.0123 0.0122 0.0122 = = : 0.0120 0.0113 785 1135 N/A N/A 14 Input quantities and selected output are shown in Table II for a warped blade similar to NSRDC Model 4498 (and one similar to the example of Reference 7). For an unskewed blade, the column labeled TETS, the skew angle 6,, would be zero, and for a skewed blade the column labeled RAKT/D, the total rake i7 /D, would be equal to P-6,/(27D). In Figure 4, the computed pitch and camber ratios are shown for these various overall geometries with all other input the same as in Table II. In Figure S, the effects of the chordwise load distribution and chordwise thickness function on pitch and camber are shown. The effect of rake and skew on pitch and camber follows known trends (7, 24). The effect of thickness distribution on pitch and camber is negligible and the effect of elliptic loading is to reduce the pitch and increase the camber, as would be the case in two dimensional flow at the ideal angle for a given lift coefficient. In Figure 6, the pitch and camber change is shown for another modification of the warped blade. Since a large change occurs in the pitch from the input specifica- tion (Table II) to the computed values (Figure 4), computa- tions were performed with the singularities distributed on the blade reference surface at a pitch taken from Figure 4. This change in pitch places the singularities nearer the final blade surface. To have uniformity in the calculations, the pitch angle of the shed vortex sheet was taken as 8, the advance angle of the shed vortex sheet. (In Figure 4, the shed vortex sheet was taken to be at the input pitch, which is B,, the pitch angle derived from the solution of a straight radial lifting-line representing each blade.) The change in pitch angle of the shed vortex wake from £; to 6 produces a slight increase of pitch near the hub (compare data in Figures 4 and 6). A change in the pitch of the blade reference surface to the values shown by the dashed curve in Figure 4 produces a significant reduction in computed pitch and a compensating increase in camber near the root, with negli- gible change in either pitch or camber from about xp = 0.5 to the tip. Hence the orientation of the free vortex sheet and blade reference surface have significant effects on the pitch and camber values only near the hub. 4498 EXAMPLE Table II DEFINITION OF DESIGN EXAMPLE Sample Data from Computer Code 4496 EXAMPLE LOADING AND THICKNESS» 5 BLADES - SFPT 80 CIRCULATION COEFFICIENTS N GCN) 1 0.029028 2 0.0C2CG10 3 -0.002830 4 70.CC0540 5 0.caocsél 4 0.C00061 DIAMETER = DP = C.3046 ¥ ADVCY = W/C(ND) INPUT DATA XR CH/DP PP/oP RAKT/ OP T 0.2000 0.16500 1.16270 0.00000 0.0 0-25Cuv0 0.19700 1.19530 0.00000 0+0 0.30000 0.22500 1.22790 0.09000 0.1 C.40000 0.2750C 1.75980 Q.000CO 0~3 C-500u0 0.31200 1.25570 0.00000 0.4 0.60000 0.337CC 1.21500 0.000CO 0.6 0-70Cv0 0. 3470C 1.15960 0.00000 0.7 0.800uU0 0.33406 1.09870 0.00000 0.9 C.90C00 0.28C0C 1.°3570 0.00000 1.0 0.95000 0.246C00 1.C0660 0.00000 lel 1.00CvV0 c.00cce 0.57350 0.000 00 122 XR CH/0P (CH/OP)~ PP/OP 0.20000 0.16500 0.€2731 1.16270 0.20606 0.16882 0.€2982 1.16659 0.22412 0.18026 0.€3875 1.17820 0.25539 0.19936 02.€5792 1.19774 0.29338 0.22516 0.€0910 1.22616 0.36288 0.25127 0.46041 1.247 06 0240000 0.27500 0.39203 1.259 40 02663519 0.29924 0. 36619 1.261 43 0253054 0.32119 G.27819 1.24678 0.600u0 0.33700 0.17799 1.216 00 0.66946 0.36573 0.06936 1.17761 0.73681 0.365946 -0.97566 1.13751 0.80000 0.33406 0.85712 0.30705 0.27597 0224651 0.18588 0.99392 0.100038 1.00000 0.00000 0.90662 0.9 4641 0.97588 4496 EXAMPLE xc 0.000000 0.0075 96 02030154 0.060987 0.110978 0.178606 02254000 0.328996 0.415176 0.500000 0.586824 0.671010 0.750006 0.821398 0.885022 0.933013 0-96 7846 0.992404 1.0¢900C "0.23023 1.098 70 "0.59158 1.06273 70.€2509 1.03169 71.18469 1.006 43 “3.11163 0.986 52 "7.05659 0-977 28 99259900 0.97350 LOADING AND THICKNESS» LOADING AND THICKNESS» 5 8l 0.63958 0.64013 0664825 0~6451246 0~ 60024 0~ 33807 012505 -0.08322 -0~ 34858 70651413 -0~58232 0.60499 02623592 -0~63250 70-62462 -0.62063 7062218 70262256 -0~62258 aADES - SEPT 80 CPP/DP)" RAK T/0P 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.0C000 0.00000 0.00000 0.00000 0.00000 0.0C000 0.00000 0.00000 0.00000 0.00000 TETS 0.00000 0.00955 0.03789 0.08418 0.14700 0622444 0.31416 0.61342 0.51921 0.62832 0.73743 0.84322 0.94268 1.03220 1.10964 1.17246 1.21875 1.24709 1.25664 5 BLADES - SEPT 80 ELLIPSE WITH AIRFOIL TAIg ANDO CHCROWISE LCAD ¥T 0.000000 0.086824 0.171010 0.250000 0.321394 0.383022 0.433013 0.469646 0.492404 02500000 0.492202 0.667219 0.421900 0.354824 0.269637 0.176665 0.091629 0.031655 0.010000 TOEAL ANGLE OY/DANG 0.500000 02492604 02469846 0.433013 0.383022 0.321394 0.250000 02171019 0.086824 -0.000000 “0.091115 "0.178511 -0.322508 "02442816 702523365 -0.526605 7024509463 70.2432C6 02143374 22454 DEG STAANG 0.90.000 0.174648 0 234<020 0.501000 0.644748 9.760044 0.860025 0.939695 0.984808 1.004090 0.984808 0.935693 0.860025 0.760064 0.64<768 0.50u000 0.342020 0.173648 9.001000 COSANG 1.000000 0.96 4808 0.939695 0.866025 0.766044 0.642788 0.500000 0-34 2020 C17 3648 - 0.000000 -0.17 3648 -0.342020 -0.590000 =C.642788 007660646 -0.866025 -0.93 9693 =C. 98 4808 =1.000000 15 20 Us¥ 12004224 1.005031 12004562 1~005799 1.005783 1.007620 16008799 16012858 12017103 1.029601 12055582 12095197 1.092884 00998106 0.721556 00229548 -06610699 -1~593591 722014223 = 0.680 NPB ETS TMAX/CH 0000 0.24000 7854 0.19800 5706 0.15610 1416 0-1 8680 7124 0.076890 2832 0.05650 8540 0.04210 4248 0.03140 9956 0.02460 7810 0.02330 5064 0.02460 CTETS)© TMAX/CH 1.57080 0.24C00 1.57080 0.235498 1.57060 0.21999 1.57080 0.19488 1.57080 0.16097 1.57080 0.13002 1.57060 0.10680 1.57060 0.08667 1.57060 0.06970 1.57080 0.05650 1.57080 0.04606 1.57080 0.03775 1.57080 0.03140 1.57080 0.02699 1.57080 0.02432 1.57080 0.02330 1.57080 0.02375 1.57080 0.02437 1.57080 0.02460 GAM @ 0.000000 Co 0.912003 0. 0-992645 0. 1.040899 0. 1.074103 O06. 1.097916 0. 1.114682 0. 12126317 Ce 1.132938 0. 1.135109 0. 1.135109 0. 1.135109 0. 1.103578 0. 02949268 0. 0.712632 0. 0.450324 0. 0.216717 0. 0.056755 O. 0.000000 le CT MAX / OP )- -0.82618 70.82679 -0.83563 -0.87251 “0.77784 0.49729 "0.34634 -0.28746 70221753 -0.16708 -0.13522 “0.11089 0.08871 -0.06529 70204245 =0.00309 0.02963 0.03249 0.C 3802 GAMINT 000000 005193 027568 064678 117962 184685 263966 352310 447614 545946 644652 740063 829166 903254 954948 984272 996675 999765 000000 WXsV 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 2c Yc 0.000000 0.007943 02924846 0.046076 0.069016 0.091134 0.110726 0.126083 0.136207 0.140136 0.137606 0.120200 0-111395 0.086967 0.059375 0.033869 0.014687 0.00345) 0.000090 S*fee°d 99000°0 62100°0 95200°0 68900°O $8900°0 9£800°0 ¥S600°0 £70TO°O 60110°0 291to°od 66110°0 at2to°o d22T0°0 so2t0°O B9TTO°O goTto°o 92010°O 41600°0 32200°0 90990°0 09£00°0 0T200°O 00T00°0 82000°0 00000°0 17996°0 020° 0-S412°0-822T°0-££00°0 4£80°0-0022°0-6550°0-S£00°0 2560°0=£01T2°0-2200°0-9200°0 9011°0-8£61°0-1T220°0 2060°0 BOLT" 0-0S9T°0-S24%0°0 0200°0 OST" 0-£6FT°0=-22%0°0 2400°0 99ST" 0="9TT°0-S270°0 1900°0 28S1°0-800T°0-4990°0 21200°0 699T°0-8080°0-2990°0 2800°0 6T21°0-6T90°0=-"990°0 2019°0 9181" 0-64£0°0-8990°0 12T0°0 2681°0-8ST0°0-02%0°0 af10°0 8002°0-£2T0°0 2012°0-06£0°0 "£22°0-1T980°0 6SS2°0-92TI70 £762°0-2S61°0 un TT3HA 9270°0 09T0°O 9890°0 T8T0°0 6S70°0 9120°0 ¢9£0°O0 S%20°0 "Cr0°O 7Ts0°O IQHN = x0/39350 28626°0 00000°0 6£T00°0 09200°0 £SS00°0 £0800°0 BToTO°O 96T10°O gre£T0°od ££9710°0 sosto°) e7sto°o 99ST0°9 9cct0°0 £2sto°o 299T9°O eZ¢t0°o 2S2T0°0 o1tto°o 92600°0 "0200°O 61%00°9 G7200°9 sTT00°Oo T6000°9 00000°0 27905°0 TE6F°O 9129°0 860S°0O 0£50°0 rT92°O £40%°0 994T°O 0852°0 Of%t°O 9456-0 TeLT°O 9S)T°T Z06T°O 2ETT°T 6B46T°C 9060°T 9T02°O TS40°T £T02°O £T30°T S26T°O Te9Crl 9£6T°O 10.C°T g9et°o Soc’! 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WAKE ATB _| eT S86, BLADE AT @,. WAKE AT B S PITCH / DIAM PITCH/DIAMETER AND CAMBER/CHORD RATIO o4 SS = 10X CAMBER RATIO 03 02 4 nf | 0 Jee Cone Lee 1 yes 02 03 o4 os 06 07 os o9 10 Xp. FRACTION OF TIP RADIUS Fig. 6 Effect of orientation of shed vortex sheet and blade reference surface on pitch and camber values 17 In Figure 7, blade pressure-coefficient distributions on the warped, skewed and unskewed blades are shown for three radii: one near the hub, one near mid-span and one near the tip. The major difference in pressure distributions occurs near the hub, where the warped blades have greater suction on both sides of the blade, and hence a greater tendency to cavitate when the local pressure reaches the vapor pressure. In Figure 8, the meanline shapes for these blades at the same three radii are shown. The greatest changes in meanline shape occur at the root but are significant all along the radius. In Figure 9, the non-linear pressure distribution (from Equations (66) and (68)) and meanline shape are shown at Xp = 0.669 for the same variations of chordwise load and thickness distributions shown in Figure 5 for a warped = 2 Se a a a nn | N > NV C— GEOMETRY & LOAD FOR MODEL 4408 ay aaten WITH EP THICKNESS So 16F SKEWED —-{}-- St Ve WARPED — -~Y — 4 vale SUCTION SIDE 4 O gqb } — u a 08 (3) ec 06 Ba 9 02 u re oF Fe} is} 02 w 04 c 2 06 we OB8F x a 10 24 =I T = T T T T 1 T wo 22+ UNSKEWED —C)— a qa 20+ SKEWED - {} -- a} x o1eh WARPED — 7 — 4 = 8g a ‘ a " a (3) E 2 a ©) uw re w ° ° w c 2 w c a came ale T aT T T T ea T T > eh —O— UNSKEWED al S - } - skewed 2 16 a 4 = — 7 — warPED =a noe BD-a_LW a \ & a a rs) RS 2 w 9 uw uw 8 w « >) a g x re G a 10 iho. 4 oo 4 ——s = 0 01 02 03 04 0s 06 07 08 09 10 Xe FRACTION OF CHORD Fig. 7 Pressure distribution at design for three blades SKEWED UNSKEWED EC, MEANLINE OFFSET 028 T ale Ts = Verorreal) “If T T T 026 [ it EEA TAS SKEWED ] 3 A RGR ns 024 27° WARPED Sey Ba ot aa te “4 Bear anes Sell y w O 010 y 2 u if UNSKEWEO a 28 / ae oe iS) DE pip K_ 0 669 n at Nl n n 044 T T Toa ol Tr T aT oT T = oa2 | scone | AA ~ oo g K j SKEWED / \, - EVC, MEANLINE OFFSET, FRACTION OF CHORD 0 01 02 03 O46 06 06 07 o8 09 10 Xo FRACTION OF CHORD Fig. 8 Meanline shape for three blades blade similar to NSRDC Model 4498. Both changes in chordwise variables lead to increased suction near mid-chord and reduced suction near the leading edge on the suction side of the blade. An engineering approximation is that cavitation will occur when the minimum pressure on a body equals the vapor pressure of the liquid. Hence at the design point, back bubble cavitation may be degraded by these chordwise variations relative to the conditions of Figure 7 but off-design leading-edge sheet cavitation on the suction side may be improved by these changes in chordwise vari- ables. This conjecture is based on assumed equal incremental changes to the pressure due to off-design operation. Leading edge pressure-side cavitation may be degraded at off-design operation with the elliptic load distribution because of the peak near the leading edge in Figure 9. Meanline shapes are different for these chordwise variations as shown in Figure 9. Thrust loading and power coefficients (computed from Equations 80 and 81 with Cp = 0.0085) are shown in Table ill tor the various overall geometries and chordwise 18 modifications. The performance coefficients computed according to the lifting-line model and first-order linear lifting-surface model (E = 0 in Equations (76) and (78) and ACp from Equation (74)) are nearly identical. The non-linear performance coefficients (E # 0 in Equations (76) and (80) and Cp from Equation (66)) for a blade with only loading (Ey = 0) are increased a few percent relative to the lifting-line values. The addition of thickness and skew or warp changes the pressure distribution and meanline slope, and increases the values by another few percent each, eventually producing values of Cy), and Cp which are more than ten percent greater than predicted by the lifting-line model. In both the pressure distribution and force calculations, non-linear, second-order effects have been included in an intrinsically first-order theory. It is not known if the calculated trends with these second-order effects included are valid or not. Experimental evaluation is required to confirm the predictions. Predictions given in Table III should be interpreted as possible trends in the actual performance. The present lifting-line model employed in propeller design (3, 4) assumes a straight radial line with vanishing chordlength to represent the blade and can hardly be expected to be an acceptable model of wide- bladed lifting surfaces, especially ones with skewed or warped blades. Fortunately, the majority of applications to date have generally performed within a few percent of predictions with this simple model but some significant differences have also occurred. Hence both a curved lifting-line and increased- accuracy lifting-surface performance predictions should be developed and evaluated. MODEL 4498 GEOMETRY __— NACA 66 (MOD) THICKNESS LOAD SIMILAR TO NACA 8-08 — ELLIPTIC CHOROWISE LOAD, EP THICKNESS re, ° e 9 Os=—B---4-__ SUCTION SIDE. ~~ cy p PRESSURE SIDE PRESSURE COEFFICIENT, C_=(p-p y/(% pV?) Ev¢, MEANLINE OFFSET, FRACTION OF CHORD 0 01 02 03 04 06 06 07 os 09 10 Xo: FRACTION OF CHORD Fig. 9 Pressure distribution and meanline shape for variation of chordwise load and thickness Table II Thrust Loading and Power Coefficient Cth Cp 10> CTH/Cp Lifting Line 0.690 1.016 0.679 Ist Order Lifting Surface 0.688 1.013 0.679 Second Order: Unskewed, Loading Only Unskewed Load & EP thickness Skewed Load & EP thickness Warped Load & EP thickness Second Order (warped, load and thickness): 66 thickness 0.782 1.141 Elliptic load, EP thickness 0.782 1.147 Wake at B 0.790 Nol 7S) Ist Order, Blade at eb. Second Order, Blade at , Wake at B 0.794 As shown by Huang, et al (25), some wake fields have mean axisymmetric flow velocities with non-zero radial components. In the example body investigated by Huang, a nearly constant value of Wp = -0.05 was found. In Table IV, the additive effects of this velocity component are shown. When this wp component is isolated from Equation (22) and the incremental addition to the meanline denoted by AE,, one finds AE quae Ne Cc oO = WR ax, 2 TXR ow? + (F) cos (yp ~ B) Vv = €(Xp) + 8(xp) > (x, - 0.5) (115) where WR ee 5) , TXR\" iy + Fi cos (yp - B) v di,/D 46, e 2———C05 fp -Xp Sin Yp (116) dXp XR 2wp c/D R 6 = 2 4 (TRY (1-w,) i ) cos (pp - B) Jy dP/D cos? yp é somes (117) dxp TXR Then AE. AE “ep = dx Cc 0x Cc 10) x, (1 = iis) Ste Xe oO Neer (118) Thus the € term, due to gradients of the rake and skew of the blade, results in an angle-of-attack change, and the 6 term, due to radially variable pitch, results in a parabolic-arc mean- line, with a maximum offset at mid chord of Af = FOr (119) C Table !V EFFECT OF RADIAL VELOCITY COMPONENT ON PITCH AND CAMBER (Geometry Similar to NSRDC Mcdel 4498, Wr -0.05) ©, “Ainitial 0.4 1.449 0.6 1.247 0.8 1.017 *From Table I, lifting-surface design without effect of Wr: Table IV shows the effect of wp = -0.05 on the pitch and camber ratio for a warped blade similar to NSRDC Model 4498. The change in camber ratio is probably not significant for this case but the change in pitch angle is important, resulting in a few percent change in the final pitch values. Hence this effect should be included in designs operating in a wake having a significant average radial inflow component. The present procedure can be improved in several respects: the presence of the hub should be included in the mathematical model, generalized load variations and thick- ness variations should be included, some physically observed phenomena of the shed vortex sheet should be included, consideration of the radially variable inflow (i.e., free-stream vorticity) should be included, steps to improve the accuracy of computations in the tip region should be undertaken, and program execution time should be shortened. Additionally, similar numerical analysis techniques should be applied to both the performance problem and the determination of unsteady blade response in a non-uniform wake. Of particular interest is the physically-observed configuration of the free-vortex sheet. As observed by Nelka (23) and modeled by Cummings (26), for highly-skewed blades, an isolated vortex forms along the leading edge of the blade in the tip region. This vortex will influence the induced velocity field and hence produce changes in the pitch, meanline, pressure distribution and force/moment coefficients relative to the model employed herein. In ad- dition, the trailing vortex will contract and roll up. Present models have a vortex sheet starting at the tip rather than somewhere along the leading edge. Boundary-layer computations on rotating blades performed by Groves (27) utilize output from the computer code described herein. In particular, the pressure distribu- tion, streamline locations, and surface metrics (see Appendix) are input data tor the boundary-layer computations. CONCLUDING REMARKS Procedures have been described for determining meanline and pitch distributions corresponding to a pre- scribed load distribution. Computer run times are as great as 20 minutes on tne DINSRDC Burroughs 7700 high-speed computer. Computed meanline and pitch distributions are dependent upon blade mid-chord location and chordwise variation of load distribution, with negligible effect due to chordwise variation of thickness distribution. A radial inflow component was found to have a significant effect on the design pitch of the blades. Calculated pressure distributions are dependent upon blade rake and skew and both chordwise load distribution and chordwise thickness distribution. Overall performance coefficients calculated from second-order effects show significant differences compared to lifting-line predictions for highly-skewed and warped blades. Experimental con- firmation of the predictions is required to evaluate these second-order modifications of a first-order theory. Suggestions have been given for improving the pro- cedures for calculating design geometry. Some of these suggestions involve an improved mathematical model of the flow field and some are improvements to the numerical analysis techniques. ACKNOWLEDGEMENTS The development of the numerical analysis pro- cedures for lifting-surface design was initiated while the author was an Exchange Scientist at the Defence Research Establishment Atlantic, Dartmouth, Nova Scotia and completed at DINSRDC under work unit 1500-104, Task Area SF43421001, Program element 62543N. The help of Ms. K. Tymchuk and Mr. R. Brian of the DREA staff and Ms. J. Libby of the DINSRDC staff has been essential to completion of this task. REFERENCES 1. R.A. Cumming and Wm. B. Morgan, “Propeller Design Aspects of Large High-Speed Ships,” Symposium on High Powered Propulsion of Large Ships, Nether- lands Ship Model Basin Publication 490, December, 1974. N 9) .G. Cox and Wm. B. Morgan, “The Use of Theory in Propeller Design,’ Marine Technology, Vol 9, No. 4, October, 1972. . B. Caster, et al, “A Lifting-Line Computer Program for Preliminary Design of Propellers,’ DTNSRDC Report SPD-591-01, November, 1975. _ E. Brockett, “A Lifting-Line Computer Program Based on Circulation for Preliminary Hydrodynamic Design of Propellers,’ Defence Research Establish- ment Atlantic, Technical Memorandum 79/E, December, 1979. 5. Tsakonas, S., et al, “An ‘Exact’ Linear Lifting-Surface Theory for a Marine Propeller in a Non-uniform Flow Field,” Journal of Ship Research, Vol 17, No. 4, December, 1973. 20 21. i) te 23. Re A. van Oossanen, ‘“‘Calculation of Performance and Cavitation Characteristics of Propellers Including Effects of Non-uniform Flow and Viscosity,” Netherlands Ship Model Basin Publication No. 457, 1974. _ E. Kerwin, “Computer Techniques for Propeller Blade Section Design,’’ International Shipbuilding Progress, Vol 20, No. 227, July 1973; also MIT, Dept. of Ocean Engineering, October, 1975. . F. McMahon, ““LFTSUR — A Computer Program for Determining Hydrodynamic Pitch, Meanline Shape, and Pressure Distribution for Marine Propellers,” DTNSRDC/SPD-731-01, 1976 (unpublished). try Brockett, ‘‘Propeller Perturbation Problems,” NSRDC Report 3880, October, 1972. . Lackenby, editor, ‘International Towing Tank Conference Standard Symbols 1976,” British Ship Research Association Technical Memorandum No. 500, May, 1976. P. Wills, Vector Analysis with an Introduction to Tensor Analysis, Dover Publications, Inc., 1931. _B. Phillips, Vector Analysis, Wiley, 1933. . Weber, ‘‘The Calculation of the Pressure Distribution on the Surface of Cambered Wings and the Design of Wings with Given Pressure Distribution,” Aero- nautical Research Council R&M 3026, 1955. I. S. Sokolnikoff and R. M. Redheffer, Mathematics of H. Physics and Modern Engineering, McGraw-Hill, 1958. Glauert, The Elements of Aerofoil and Airscrew Theory, Cambridge, 1926. Terry Brockett, ‘“‘A Subroutine for Evaluation of One- Dimensional Singular Integrals (SINGINT),” Defence Research Establishment Atlantic, Technical Memorandum 79/F, December, 1979. W. B. Morgan, ‘‘Propeller Induction Factors,” DTMB Report 1183, November, 1957. Terry Brockett, “The Design of Two-Dimensional L. Profiles from a Specified Surface Speed Distribution, Part | — Meanline at Ideal Angle of Attack,” Defence Research Establishment Atlantic, Tech- nical Memorandum 79/G, December, 1979. H. Abbott, and A. E. von Doenhoff, Theory of Wing Sections, Dover, 1959. W. F. Lindsey, et al, ““Aerodynamic Characteristics of 24 NACA-16 Senes Airfoils at Mach Numbers between 0.3 and 0.8,”” NACA TN 1546, September, 1948. H. P. Rader, “‘Cavitation of Propeller Blade Sections,” Admiralty Experimental Works, Hasler, Report 22/54, March, 1954. Terry Brockett, ““Minimum Pressure Envelopes for J. Modified NACA-66 Sections with NACA a=0.8 Camber and BuShips Type I and Type II Sections,” DTMB Report 1780, February, 1966. J. Nelka, ‘“‘Experimental Evaluations of a Series of Skewed Propellers with Forward Rake,” NSRDC Report 4113, July, 1974. 24. R.A. Cumming, et al, “Highly Skewed Propellers,” Trans. SNAME, Vol 80, 1972. 25. J.T. Huang, et al, “Stern Boundary-Layer Flow on Axisymmetric Bodies,”’ Twelfth Symposium on Naval Hydrodynamics, National Academy of Sciences, Wash. D.C., 1978. 26. Damon E. Cummings, ‘‘Numerica] Prediction of Propeller Characteristics,” Journal of Ship Research, Vol 17, No. 1, March, 1973. 27. Nancy Groves, “An Integral Prediction Method for Three-Dimensional Turbulent Boundary Layers on Rotating Blades,” Paper presented at ‘‘Propellers *81 Symposium,” SNAME, May, 1981. APPENDIX — STREAMLINE COORDINATE SYSTEM It is often convenient to have an orthogonal coordi- nate system on the surface of the blade. In particular, for performing boundary-layer computations, an orthogonal coordinate system with one variable along the streamlines reduces the number of terms in the governing equations. To determine the differential equation of the streamline path, let KR = V(x) (120) be the radius of the streamlines as a function of the chord- wise coordinate x. Then e = S*(x,) = S(x_, W (x,) (121) is the position vector of the streamlines on the blade surface. Hence a tangent to the streamline is ds* as as dy Cy =e = + —— . dx, ax, xp=¥ OXR piv dxp + c No X£1\ dy DE Oy | |@Cy Se || D— Pe D5 dx I For this tangent vector to be parallel to the velocity vector on the surface, the vector cross product, ty xX q, must be zero. Hence, for the velocity on the blade surface given by Eee Vv Vv Vv U Ww. “We ape (123) 1S dv Ws = Dae (124) Oc oin U W ; Vie es ets 2 DAN Re, 29 For lines along the surface which are normal to the streamlines, let X= K (Xp) (125) be the chordwise position as a function of radius. Then a vector on the blade surface tangent to this line is ds (kK (Xp), Xp) dxp =) dk (=. ar (126) Ox, dxXp OXpR XM XO TK c dk I = ds fsa geten tome | The condition to be satisfied is that t, be perpendicular to the velocity vector, or = 6 (127) Ufc dk ‘4 nul ——— W py ogc ma V\D dxp BN Re ay, (128) Thus the slope of lines cn the surface which are normal to the streamline is: 1 W U — a= & — ye NRO Y Oy = 5 as (12) dk dxp Ge UE DV One now has differential equations to determine an orthog- onal network over the blade surface. The differential arc length along the streamlines is ds =4(—— +| — dx. Ox, Xp =¥ OXp Kyiv dx. € Since ds = |ds| = hy dx, Gish) PAL then or hy c dy¥ 1+Npo dy\2)% — =; (—+a-—]+ D D dx, 4 \dx Similarly the differential arc length along the orthogonal surface coordinate is (132) — dk (=. | ds = + dx R Ox, Xoo k dxp OXp x= K (133) = hj dxp (134) where i Vie ks z 1 +Np? 4 Sal ee) (135) D D dxp 4 22 Copies 38 ARMY CHIEF OF RES & DIV ARMY ENGR R&D LAB CHONR 1 1 NRL Code 438 LIB ONR BOSTON ONR CHICAGO ONR LONDON, ENGLAND USNA 1 all LIB Johnson NAVPGSCOL LIB NROTC & NAVADMINU, MIT NADC NOSC PRPeERPR UBWRFPWPWRPEPHEEPE SLAG als} 6005 13101 LEB 2501/Hoyt Nelson SEA 032 SEA 0321 SEA 03D SEA 052 SEA 052P SEA 0521 SEA 0522 SEA 0524 SEA 0525 SEA 05D SEA 05H SEA O5R INITIAL DISTRIBUTION 23 Copies PMS 378 PMS 380 PMS 381 PMS 383 PMS 389 PMS 391 PMS 392 PMS 393 PMS 397 PMS 399 PMS 400 SEA Tech Rep Bath, England DET Norfolk (Sec 6660) NP RPP RPRPRPRPREPRPRPR RB FAC 032C MILITARY SEALIFT COMMAND (M-4EX) NAVSHIPYD/PTSMH NAVSHIPYD/PHILA NAVSHIPYD/NORVA NAVSHIPYD/CHASN NAVSHIPYD/LBEACH NAVSHIPYD/MARE NAVSHIPYD/ PUGET NAVSHIPYD/ PEARL DTIC HQS COGARD U.S. COAST GUARD (G-ENE-4A) LC/SCI & TECH DIV MARAD DIV SHIP DES COORD RES Schubert Falls Dashnaw PRP R Hammer Lasky Siebold LIB MARITIME RES CEN STIF 2 DIR RES NSF ENGR DIV LIB DOT LIB U BRIDGEPORT/URAM U CAL BERKELY/DEPT NAME 1 NAME LIB 1 Webster U CAL SAN DIEGO/Ellis UC SCRIPPS 1 Pollack 1 Silverman U MARYLAND/GLEN MARTIN INST (GILL 1 AERO LIB 1 Acosta 1 Plesset 1 Wu CATHOLIC U COLORADO STATE U/Albertson U CONNECTICUT/Scottron CORNELL U 1 AERO DEPT 1 THEOR APPL MECH DEPT FLORIDA ATLANTIC U OE LIB 24 Copies i FLORIDA STATE/OCEAN ENGR JL HARVARD U 1 MCKAY LIB 1 Birkoff 1 Carrier 2 U HAWAII/Bretschneider iL U HOUSTON/Dalton al U ILLINOIS/Robertson yD. 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