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Numerical and Experimental Analysis of Propeller Wake by Using a Surface Pane] Method and a 3-Component LDV T. Hoshino (Mitsubishi Heavy Industries, Laid., Japan) ABSTRACT Hydrodynamic modeling of the trailing vor- tex wake of a propeller is one of the most im- portant f actors in developing a propeller theory. A variety of trailing vortex wake models have been proposed hitherto. However, details of geometrical f eatures have not been known clearly . In the present study, f low f ields around propeller are precisely measured in a cavitation tunnel using a 3-component Laser Doppler Velocimeter (LDV). Based on the experimental f inding that the pitch of the tip vortices are smaller than that of the inboard trailing vortex sheets, the surface panel method with a def armed wake model of the trailing vortices is proposed. Then, the pres- sure distributions on the blade and the f low f ields around propeller were calculated by the present surface panel method. A better agree- ment of pressure distributions near the hub is observed when the hub effect is considered in the calculations. It is shown that the calcu- lated f low f ields around propeller are in good agreement with the measured ones. Open-water characteristics of propeller calculated by the present method are also in good agreement with experimental data. NOMENCLATURE 2a(r) Pitch of blade section = P(r) Bi j Influence coefficient due to the j -th source panel on blade and hub surfaces c ( r ) Chord length Ci i Inf luence coef f icient due to the j -th doublet panel on blade and hub surfaces Cp( Pi ) Pressure coefficient = ( P(Pi ~-PO ) /P (VA +( ri2 ) ~ /2 D Propeller diameter e, ,e2 Local coordinates on panel J Advance coefficient = VA /(nD) K Number of propeller blades KT Thrust coef f icient of propeller = T/(pn2D4 ) K? Torque coefficient of propeller = Q./ (p n D ) Number of chordwise wake panels L n n n N 2 NR o P(Pi ) Po P(r) Pw (r) Q r to rh rwh rwT R(P, Q) Rick s SL (r) S Sj t1 ~t2 T V, V] vt Normal coordinate f or blade section Propeller rotational speed, [ rps ~ Unit vector outward normal to surface Total number of blade and hub panels Number of chordwise blade panels Number of radial blade panels Propeller center Pressure Static pressure at inf inity Pitch of blade section Pitch of trailing vortex sheet Propeller torque Radial coordinate f rom propeller axis = ~y2+z2 Propeller radius = D/2 Radius of propeller hub Radius of hub vortex Radius of ultimate wake Distance between field point P and boundary point Q Distance between the i-th control point and the j-th integration point Chordwise coordinate f or blade section Chordwise coordinate of leading edge Boundary surf ace Surface of the j-th panel Tangential coordinates on panel Propeller thrust Cartesian coordinates in the blade-f ixed f rame XR (r) Propeller rake v Velocity Vt Perturbation velocity vector tangent to blade surface Speed of advance Velocity vector of relative inf low Total velocity vector tangent to blade surface Wi j Inf luence coef f icient due to the j -th doublet strip on wake surface SG (r) Pitch angle of blade section Sw (r) Pitch angle of trailing vortex sheet ~ i j Kronecker delta Eli Area of the i-th panel Al Potential j ump across wake surface A¢j Discrete potential j ump in the j-th panel Angular coordinate from generator line of propeller = tank (-y/z) Tetsuj i Hoshino, Nagasaki Experimental Tank, Mitsubishi Heavy Industries, Ltd . 3-48 Bunkyou Machi, Nagasaki 852, Japan 297

p JO Q V Subscripts by TE w + TV Angular coordinate of generator line of k-th blade = 2~(k-1~/K Fluid density Perturbation velocity potential Discrete potential in the j-th panel Angular velocity = 2nn Gradient operator Face and back sides of blade, respectively ~ <=l;face, K=~;b~ck ) B Blade D Drag H Hub i,j Values on panels i,j, respectively k Value on k-th blade P Potential Q Value on boundary point Q r, ~Radial and tangential components, respectively Value on trailing edge Wake x,y,z Components in Cartesian coordinate, respectively Upper surface Lower surface Values on corner points ~,v of panel, respectively 1. INTRODUCTION In recent years, propellers with various blade geometries such as a highly skewed pro- peller have been fitted to ships in order to reduce the propeller induced vibration and noise, or to improve the propulsive perfor- mance of ship. A reliable numerical method is indispensable for the design and analysis of such propellers. A number of propeller design and analysis methods based on lifting surface theories such as Vortex Lattice Method (VLM)[1] and Quasi- Continuous Method (QCM)~2] have been develop- ed. However, the propeller lifting surface methods are essentially based on the thin wing theory. Therefore, they are insufficient to predict the pressure distribution on propeller blade, especially near the hub where the ef- fect of blade thickness and hub would be dominant. On the other hand, surface panel method has been remarkably advanced in the field of aerodynamics for the design and analysis of three-dimensional wing and bodies [3-123. The surface panel method is one of the most ad- vanced methods, because it allows precise rep- resentations of the complicated blade geometry of the propeller such as the highly skewed pro- peller. In the past few years, the surface panel method has been applied to marine propel- lers and also advanced turboprop problems [13- 203. In most of such propeller theories, howev- er, the geometry of the trailing vortex wake of a propeller slipstream has been treated ap- proximately because of the complexity of the slipstream. Since the induced velocities on the blade due to the helical trailing vortex wake of a propeller are much larger than those due to the trailing vortex wake of a wing, hydro- dynamic modeling of the trailing vortex wake behind the propeller becomes important in developing propeller theories. In the past, the trailing vortex wake had been replaced by a prescribed helical surface with a constant pitch obtained from the undisturbed inflow or a constant hydrodynamic pitch calculated from the lifting line theory as described by Hanaoka [212. In the actual propeller, the trailing vortices leave the trailing edge of the propeller blade and flow into the slip- stream with the local velocity at that posi- tion. Therefore, the detailed knowledge of the velocity distributions of the propeller slip- stream would be indispensable to establish the realistic model of the trailing vortex wake. Due to the recent development of measuring technique with Laser Doppler Velocimeter(LDV), the measurements of time dependent flow fields around propeller have been reported by many researchers t22-27~. Based on the results of the flow field measurements, Kerwin and Lee [1] proposed a roll-up wake model which took into consideration the contraction of the slip- stream and the roll-up of the trailing vortex sheets. However, the roll-up model in which the trailing vortices are assumed to be con- centrated into a single hub vortex and a set of tip vortices at a certain distance behind the blade is considered to be too simplified. More realistic geometry of the trailing vortex wake behind propeller has to be taken into consideration. In the present paper, a surface panel method is described for analyzing the flow fields around propeller operating in uniform flow at first. Green's identity is applied to obtain an integral equation with respect to the unknown potential strength over the sur- face of the propeller blades, hub and wake. Such method was firstly developed by Morino for general lifting bodies [6,7J. An improve- ment on Kutta condition is added to the Morino method. That is, the Kutta condition of equal pressure on the upper and lower surfaces at the trailing edge is applied in the present study. Next, flow fields around propeller models operating in uniform flow are precisely measured in a cavitation tunnel using a 3- component LDV. Based on the measured velocity distributions of the propeller slipstream, a deformed wake model is proposed where the con- traction of the slipstream and the variation of the pitch of the inboard helical trailing vortex sheets are taken into account. Then, pressure distributions on the propeller blade and open-water characteristics calculated by the present method are compared with the experimental data. Further, flow fields around the propeller are calculated by the surface panel method and compared with those measured by the LDV. 298

2. FORMiLATION OF PROPELLER PROBLEM 2.1 Coordinate Systems and Geometry of Propeller We consider a propeller rotating clockwise with a constant angular velocity Q in an inviscid, incompressible, irrotational flow with a uniform axial speed VA far upstream. The propeller consists of a finite number of axisymmetrically arranged blades of identical shape and a hub. We define a Cartesian coordinate system O-xyz with origin O fixed at the center of the propeller, where x is measured along the down- stream axis of rotation as shown in Fig.1. The z-axis coincides with generator line of the first blade and the y-axis completes right-handed coordinate system. A cylindrical coordinate system O-xrO is also introduced for convenience. Angular coordinate ~ is measured clockwise from the z-axis when viewed in the direction of positive x. Radial coordinate r is measured from the x-axis. Then, the Car- tesian coordinate system O-xyz is transformed into the cylindrical coordinate system O-xrO by the relation x = x, y = -r sine, z = r cosO, (1) where r = ~ , ~ = tan~~(-y/z). Further, we introduce a helical coordinate system (r,s,n) with pitch 2na(r). The e-axis is measured chordwise from leading to trailing edge of the blade section. The e-axis is measured normal to the e-axis from face to back side. Then, the cylindrical coordinate system O-xrO is related to the helical coor- dinate system (r,s,n) by x = ta(r) s-r n]//a(~)2+r2+xR(r), ~ = [sta(r) n/r]//a(r)2+r2, where XR (r) is propeller rake defined by the x coordinate of generating line at radius r. ' Vr Vz V`~: ~_ Vy ~Vx ~Y ~x Fig.1 Coordinate systems of propeller Blade section of propeller is usually defined in a way similar to that of a two- dimensional wing by the ordinates nb<(r,s) of face and back sides along the chord where K = 1 and 2 show the face and back sides, respec- tively. Then the coordinate of a point on the surface of the k-th blade can be expressed as x = xb<(r,s), y = -r sint~b~(r,S)+8k], ~ (3) z = r coStebK(r~s)+Ok where Xb~(r,S) = La(r) S-r nbK(r~s)]t'/a(r)2+r2 +xR(r), ~(r,S) = I S+a(r) nbK(r~s)/r]/?/a(r) +r2, = 2~(k-1~/K, k = 1,2,.--,K, K = number of propeller blades. 2.2 Velocity Potential and Boundary Condition Under the assumption of potential flow, the flow field around a propeller is charac- terized by a perturbation velocity potential¢, which satisfies Laplace's equation (4) and vanishes at infinity. We consider a bound- ary surface S. which is composed of propeller blade surface SB, hub surface SH and wake sur- face Sw, and also unit outward normal vector n to the surface S. Applying Green's identity, the perturbation potential at any field point P(x,y,z) can be written as a distribution of source and doublet over the boundary surface [6,73: (2) where 4nE¢(P) = ~Sl(Q)a3Q(~t Q')dS at(Q) 1 dS S anQ R(P,Q) O for the point P inside S. E ~ 1/2 for the point P on S. 1 for the point P outside S. (5) and R(P,Q) is the distance from the field point P(x,y,z) to the boundary point Q(x',y', z') and a /a nQ is the normal derivative to the boundary surface at the point Q. Kinematic boundary condition is that the velocity normal to the blade surfaces SB and the hub surface SH should be zero. Using rela- tive inflow velocity VI , the boundary condi- tion with respect to a moving frame fixed on the propeller blade can be written as eat = -VI nQ = _(VA+QXE) nQ, on SB and SH, (6) where VA and Q are the advance and angular velocity vectors respectively and r is the position vector of the point P on the boundary surface. 299

We assume that the wake surface Sw is in- finitely thin and there is no flow and no pressure jump across the surface Sw, while the potential jump is allowed. The boundary condi- tion on the wake surface Sw can be written as a++ at = , p+ = p on Sw, (7) and ant where p+ are the pressures on the wake surface Sw , and subscripts + and - denote the upper and lower sides, respectively. For the steady propeller problem the potential jump Al across the wake surface is constant along an ar- bitrary streamline in the wake and can be expressed as ¢ = ¢+ ~ on Sw. Considering Eqs.(6) through (8), the boundary integral equation (5) for the point P on the blade and hub surface reduces to 2~(P)-§S IS l(Q)anQ(R`p Q')dS ~iSW6~(Q')anaQ,( ~[Q'')dS ~SB+SH(VI nQ)R~p Cads on SB and SH ; Here ~ denotes that Cauchy's principal value should be taken and Q' is any point on the wake surface Sw. Eq.(9) is a Fredholm integral equation of second kind for the velocity potential ~ and can be solved uniquely. The resulting surface potential distribution can be differentiated to obtain velocities and pressures, which are integrated to yield the total forces and moments. ~ Upstream View NR=12 NC=12 3. Numerical Procedure 3.1 Discretization of Propeller Blade, Hub and Wake Surface In order to obtain a numerical solution for the boundary integral equation (9), the surface of propeller blades, hub and wake is divided into a number of small elements. In the past application of the panel method to the propeller problem, planar quadrilateral panel has been used to approximate the surface. However, the elements representing the propeller blades must be nonplanar due to the helical blade surface, which results in gaps at the edge of the planar panel and therefore numerical errors. In order to yield a closed surface and avoid such numerical errors, the surface of the blades, hub and wake is approximated by a number of qua- drilateral hyperboloidal panels in the present paper. This paneling is one of the important features of the present method. The discretization of a propeller is divided into three portions, i.e., the gener- ation of blade panels, the generation of hub panels, and the generation of wake panels. We consider the discretization of a propeller blade at first. In the choice of the radial distribution of panel strips for a propeller blade, it should be noted that the better results could be obtained if the finer panel strip was used in the region of rapid varia- tion of sectional properties. Therefore, we will use the cosine spacing which concentrates the panel strips at the hub and tip. If the radial interval from the hub rh to the tip rO is divided into NR small panel strips, the radii of the corner points of each panel strip can be expressed as follows: where rp = 2 (rO+rh)- 2 (rO-rh)cosap, (10) l DIN +~' for ~ = 2 O for ~ = 1, ( Downstream View ) NR=12 NC=12 Fig.2 Panel arrangement for a highly skewed propeller 300 .3,-~.,NR+1. an

In the chordwise spacing of the blade panels, cosine spacing is considered to be the best. Therefore, the chordwise positions of the corner points of each panel are given by Slav = sL(rp)~( 1-cosQv), (11) where sir) = s-coordinate of the leading edge, c(r) = chord length of the blade, OF = vn/Nc, v = 0,1~2,.~,Nc, Nc = number of chordwise division. This concentrates the panels at the leading and trailing edges, where greater resolution is required. A propeller blade surface is thus discretized into NRx2Nc quadrilateral elements per each blade. Propeller hub is considered to be an axi- symmetric body on which the propeller blades are mounted. The blade panels adjacent to the hub surface are shortened or stretched to ob- tain the intersections with the hub surface. Then the axial positions of the hub panels meet with those of the blade panels at the intersections. The hub portion from the lead- ing to the trailing edge is divided into some strips equally spaced in circumferential angle between the root blades. This generates the panels with helical pattern on the hub. The hub portion upstream of the leading edge of the blade is divided into straight panels with equal axial and circumferential spacings. On the other hand, the hub portion downstream of the trailing edge is helically divided with the pitch at the root of the propeller. An ex- ample of the panel arrangement on propeller blade and hub surface for a 5-bladed highly skewed propeller is shown in Fig.2. Trailing vortex leaves the trailing edge of the blade and flows into the slipstream with the local velocity at that position. However, the wake surface is usually ap- proximated by prescribed helical surface in order to avoid time consuming calculation of the slipstream velocities. In the present paper, the surface of the trailing vortex wake is determined based on the measured velocity distributions of the propeller slipstream. The wake surface is divided into NR wake strips, which start from the trailing edges of the blade strips. Then, each wake strip is divided into L wake panels. The axial spacing of the wake panel is finer near the blade and gradually becomes coarser in the downstream. Details of the numerical modeling of the trailing vortex wake is shown in the following chapter. 3.2 Linear Algebraic Equations As mentioned above, the blade and hub sur- face is divided into N small panels Sj and the wake surface is divided into NR X L small panels SQ. The values of the potential and (V~.nQ) are assumed to be constant within each panel and equal to the values hi, A¢j and (V~.nj) at the centroid of the panel, respectively. Then, by satisfying Eq.(9) at the centroid of each panel, one obtained a system of N linear algebraic equations as NR N E(6ij-cij)lj- £~WiiLli = - £ Bij(V~ nj)(12) for i = 1, 2, ...., N. Here Sij is the Kronecker delta and Cij , W and Bij are influence coefficients defined by Cij = 2 n (R. )dSj , k-] So a j i jk K ~ 1 i: a ( )dSQ , (13) k51 Q_~ 2H Stant Rink i j k - ~ 2 HIS j R. i j k i where Rij k and Rick are the distances from the control point of the panel on the k-th blade and hub surfaces to the integration point on S and SQ. The influence coefficients Cij and Bij are evaluated analytically in the near field [7~. On the other hand, they are approximated by a Taylor series expansion in the far field in order to save computation time. The coeffi- cient Wij are also calculated by using the ex- pression for Cij. Then, Eq.(12) can be solved numerically to yield the values of the unknown potential fj. The Kutta condition is applied to obtain the values of the unknown potential jump Alj on the wake surface. An equal pressure Kutta condition is introduced in the present study. A detailed formulation of the numerical Kutta condition is shown in APPENDIX. 3.3 Velocity and Pressure on the Surface The velocity and pressure distributions on the blade and hub surfaces can be evaluated directly by taking the gradient of the in- fluence coefficients for the velocity poten- tial Eq.~13~. However, it takes too much com- putation time because the influence coeffi- cients for the induced velocity must be newly calculated. On the other hand, the velocity and pressure on the surface can be obtained also by differentiating the velocity potential over the body surface which is already known. This method takes much shorter computation time than the former but numerical differen- tiation is used to introduce some numerical errors. In the present paper, the latter method is adopted to calculate the velocity and pressure distributions on the body surface. The numeri- cal differentiation was conducted as follows [12~. The distributions of the velocity poten- tial ~ are approximated by a quadratic equa- tion passing through the potentials at the centroids of three adjacent panels as = at2 + bt + c, (14) where t is the surface distance and a, b, and c are the coefficients of the quadratic equa- tion. Then, the derivatives of the potential 301

along the tangent directions tl and t2 to the 3.4 Field Point Velocity panel surface can be expressed as ttl ~ at = 2alt1+b1, | (15) ~t2 ~ eat = 2a2t2+b2. 2 respectively. Next, we take the e1 axis in the direction of to and the e2 axis in the direction perpen- dicular to t1 in the plane composed of t1 and t2 as shown in Fig.3. Denoting the unit vec- tors in the directions of the el, e2, andt2 axis by el , e2, and t2 , respectively, the derivatives of the potential along the e1 and e2 axes can be expressed as te1 ~ Den (tl, l at (t2-(t2-el)ltl ~ (16) f~2 = ae2 (t2 e2) Then the perturbation velocity tangent to the body surface can be obtained by Vt = filet + fe2e2, (17) Adding the tangential component of the rela- tive inflow velocity VI, we obtain the total velocity tangent to the body surface as Vt = VI_(VI n)n+vt = [(VI el)+¢e1 den + [(VI e2)+le2]e2, (18) where n = elxe2. The pressure on the body surface can be expressed by Bernoulli's equation as P(Pi) = PO+ 2P[ |VI I -|Vt I ], (19) where pa = static pressure at infinity, p = density of water. The pressure is finally expressed in terms of the non-dimensional pressure coefficient Cp(Pi) , which is defined as C ( ) P(Pi) Po (20) where WO = VIVA 2 + ~ rQ) 2 . t2e2 / t2~e2 \ 1 _ e' ' \ - t',e' P2 Fig.3 Local coordinate system on a panel 302 The induced velocities at the field point P outside the closed surface S can be evalu- ated by taking the gradient of the velocity potential ~ as follows: v(P) = Vpt(P) An SB+SH¢(Q) PanQ(R`p Q')dS +4~Sw6~(Q')9pana '(R~P[q',)dS 4~ SB+SH (VI nQ)VP(R(P Q) )dS, (21) Then, Eq.(21) can be approximated by N NR N i j-1 TjVPCij +j=£16ljVPWij-j£1 (VI nj )VPBij, (22) VPCij = Is Vpan (R j )dSj], VpW i i = k Zl Q[~4~§SQVPan'(Ri to )dSt], (23) VpBii = k~lt-4~i7Sj9P(Rijk)dSj]. Here, the influence coefficients VpCij , VpBij and VpWij can be evaluated analytically in a manner similar to the determination of the in- fluence coefficients for the potential t83. 3.5 Thrust and Torque of Propeller The total forces and moments acting on a propeller can be obtained by integrating the pressures over the blade and hub surfaces. Denoting the components of the outward normal vector ni by (nxi' nyi, nzi), the potential components of the thrust Tp and torque Qp of the propeller can be expressed as N Tp = K ~ P(Pi)nXi-6Si' i= N Qp = K ~ P(pi)(nyi zi-nzi yi)6si' i=1 (24) where Nisi = area of panel, (Xi, yi, zi ) = coordinates of the point Pi . With the skin friction coefficient C f(P i) ~ the viscous components of the thrust TD and torque QD of the propeller can be written as TO = 2 pK ~ Cf(Pi)VtXilVtil6Si' 1 N (25) QD = 2 pK. £ Of (Pi )(Vtz i ·Yi-Vtyi ·Zi ) IVt i I AS i 1 = 1 where (Vtxi,vtyi ~ Vtzi) = components of the tangen- tial velocity Vt i at the point P. i

Then we obtain the total thrust and torque of the propeller as T = Tp + TD, Q = QP + QD. (26) Finally, advance, thrust and torque coeffi- cients are defined as VA T where J = D' KT = my, KQ = my. (27) n = propeller rotational speed, D = propeller diameter. 4. MEASUREMENTS OF FLOW FIELD AROUND PROP~.T.RR 4.1 3-component LDV System The LDV system used in the present study is a five beam, two-color, 3-component type with a 3-watt Argon-ion laser as shown in Fig.4. This LDV system allows simultaneous measurement of three components of time depen- dent velocities around a propeller [253. In order to measure the time dependent velocity at the specified field point, one propeller revolution is divided into 256 an- gular positions and the velocity data are com- bined with the present angular positions of the propeller. In the present measurements, total of 5120 data are collected for each velocity component and rearranged according to each angular position. Then, mean values and standard deviations of the velocity are ob- tained at each blade angular position. The water in the tunnel is filtered to 10 ~ m particle size and then seeded with 4 Am metallic coated particle which is best for the present 3-component LDV system because of its high reflection index and adequate size. 4.2 Definition of Field Point Velocity We define the velocity components in the x-, y- and z-axis directions of the flow around a propeller to be vx, vy, vz as shown in Fig.1. Then, the velocity components in the r- and 0-directions vr and vet can be expressed as: LDV optical system | Laser |=- ~ ~ 3-d mensional Photo -multipliers . . traverser green , Display Traverse 1. and controller | printer 1t ~t l ital Mini- computer Memory:512kB HardJ flexible Blade pulses/rev. Disk devices Fig.4 3-component LDV system Table 1 Principal particulars of propeller models Propeller Diameter of Propeller ( mm ) Pitch Ratio at 0.7R Expanded Area Ratio Bass Rado Number of Blades Blade Thickness Fraction Rake Ande (deg.) A 250.0 0.8000 0.6500 0.1800 5 0.0500 8.0 B __50.0 1.000d 0.6500 0.1800 0.0500 8.0 . C 250.0 1.2000 0.6500 0-.1800 5 0.0500 8.0 Table 2 Conditions for flow measurement by LDV Advance Ratio' I 0.4Q, 0.56, 0.72 - 0.581 0.701 0.90 Ohio, 0.84, 1n8 PI . A 8 C . X/rO = -0.?5 O 0.25 0 5 10 20 r/rO= ~-1.1 , 1.0 095 ° 98 0.85 _ 0.7 · Measuring Point Fig.5 Measuring positions around propeller Vr = -vy sinO+vz cos6, ~ (28) vie = -vy cosO-vz sine, If the velocity measurements were conducted in a vertical plane, the y-component vy is iden- tical but opposite in sign with the tangential component vie, and the z-component vz is iden- tical with the radial component v r as follows : vx = vx, Vr = Vz, via = -vy. (29) In a uniform flow' the flow field around propeller is axisymmetrical and oscillating with the blade frequency. If the propeller rotates with a constant angular velocity, the time dependent velocity measurements at a cer- tain radius correspond to the measurements at the same radius for many different angular positions of the measuring points at a fixed propeller position. Hence, the velocity measurements with respect to a certain propel- ler position give the instantaneous velocity distribution of the propeller at a certain time. 4.3 Propeller Models and Measuring Conditions Velocity measurements of the propeller slipstream were conducted in a uniform flow for three propeller models which are five- bladed and different in pitch. Principal par- ticulars of propeller are shown in Table 1. 303

In the LDV-measurements, propeller rota- tional speed was kept constant of n=20 rps and advance speed of the propeller VA was changed to vary the advance coefficient J. Measuring conditions for each propeller are shown in Table 2. Measuring positions of the flow field around the propeller by LDV were taken upstream and downstream of the propeller as shown in Fig.5. 4.4 Results of Measurements of Flow Field around Propeller As an example of the results of the LDV measurements, circumferential variations of three components of the velocities around the propeller B at the advance coefficient of J=0.70 are shown in Figs.6 - 8. It is shown that each velocity component is periodically fluctuating with the blade frequency. Fig.6 shows the velocity fluctuations measured at various radial positions just upstream of the propeller ( x/rO=-0.25 ). The variations of the axial and tangential velocities vx, via are observed at inner radii. Fig.7 shows the velocity fluctuations at axial position of x/rO=0.25 just behind the propeller. The sudden change of the radial velocity component shows the velocity jump across the trailing vortex sheet. The slope of the velocity jump of the radial component at inner radii ~ r/rO < 0.5 ~ is opposite to that at outer radii ( r/rO>0.8 ). This shows that the strength of the trailing vortex changes its sign between the inner and outer radii. Propeller B ( x/rO = 0.25) 4.0 i 3.0 2.0 ~ VX/VI 1.0 =~==Z - QO ma_ ~ - 1.0 VT/VA -20 ' ' -! ~ O 90 180 270 360 O (dim) 4.0 3.0 2.0 1.0 is-0.0 -1.0 -2.0 4.0 3.0 2.0 ~1.0 as-0.0 -1.0 -2.0 4.0 3.0 2.0 1.0 of-0.0 ~ -1.0 -2.Oo Propeller B ( x / rig = -0.25 ) r/r. = 0.90 4.0 _ 3.0 _. 2.0 ~ VX/VA in; 1.0 _ -pi o ~V,/V -20 -- ' ~' O 90 180 270 360 O (de`) 4.0 :# 3.0 _ r/rB = 0.70 ~0 1.0 0.0 ~ - 1.0 _ -2.0 ) . _ . _ Vx/V~ 1 ~- : . - ~ _ -V'/\l~ 1 ~ ~ Vr/V, 1 _ I I I I 90 180 270 360 O (den.) r/rO = 0.30 0 90 4.0 _ - _ :~ 3.0 _ r/r.= 0.50 ~ 2.0 VX/VA - 1.0 ~ on ~ ~ - 1.0 - Vr/VA -20 ~_ I ' O 90 180 270 360 8 (661.) 4.0 _ 3.0 2.0 _ _ 1.0 Jo o.o ~ -1.0 _,n VX /VA _ _ -V/V 180 270 360 O (don) Fig.6 Circumferential variations of veloci- ties upstream of propeller B ~ J=0.70, x/rO=-0.25 ~ _ r/rO = 0.70 r/rO = 0.50 Velocity Defect VT/VA V0I0C;tY JUmP I '. ' ' O 90 1 80 270 360 _ _ _ O (den.) 4.0 O (den.) _ r/rO = 0.30 : ~3.0 - r/rO = 0.90 ~ 1.0~=~, __~ ~;F 00 I ye~10~ _~.~ ~ 90 180 270 360 O (den) 4.0 3.0 - 2.0 1.0 ~ 00 ~ -1.0 -2.0 O - Propeller B ( x/r. = 0.25 ) 0 _ r/r. ~ 1.00 0 vX/v~ 0 ~ O ~0~= O. Vr/V~ O ~ I 1 1 O 90 180 270 360 O (de`) AD 3.0 ;S 12 oO ~ = ;sy~lOo :'' ~ ia -20 n on 180 2~0 360 r/rO = 0.95 Velocity JUmP due to Tip Vortex r/rB = 0.20 ___ _ 90 180 270 360 ~ (de`) v O ~ -Vo/VA V`/VA ~=~C:= AXE l l l l 90 180 ~ (de`) 270 360 4.0 3.O 2.0 1.0 0.0 -1.0 -2.0 90 180 O (deg.) 270 360 Fig.7 Circumferential variations of velocities downstream of propeller B ( J=0.70, x/rO=0.25 ) 304

4.0 3.0 2.0 1.0 ~ 0.0 ;~c -1.0 _ a n Propeller B ( x / r0 = too ) r/r8= 0.70 VX/VA -V'/V~ at_ ~UT/VA -Lou ~1 ~ 0 90 180 270 360 O (de&) 4.0 3.1 ;, 21 1.1 ~ &1 _-~ ~ - 1.0 _ -2.0 1 I 4.0 _ 3.0 ~0 1.D ~ O.C _ - - 1.t 4.1 3.1 2.1 1 1 ~ al - - 1.1 ;b -[1 ~'T;~ 180 270 360 (dew) I _ Propeller B ( x / r0 = 1.00 ) _ r/r.= 0.95 ~ vX/v~ 1 ~ 1 i_ ~Vr/V~ 1 I I I ... I 0 90 180 270 360 O (de`) 4.0 30 _ rare = 0.90 2.0 Vx/VA Velocity lump due to nip Vortex :~_,20 ;a -20 0 90 180 270 ~ (de`) r/rO = 0.30 ~ 4t - r/rO = 0.85 VX/VA- . V8~ by Jump due to Tip V0~8x ~ -~`1.C ~ o.c VT/VA~ -1a -L.U ~~ 90 ~~ 180 270 ~ 360 - 2.a ~ (46~) _ r/r. = 0.20 ~ VX/VA _ Vr/V~ ' ' ' ' 90 180 270 360 O (de&) 4.0 3.0 z0 ~ 1.0 o.o . ~ - 1.0 a _2.0! Fig.8 Circumferential variations of velocities Further, this corresponds to the opposite slope of the radial circulation distribution at each radius. The strong variations in the radial velocity component at r/rO=0.95 seem to be due to strong tip vortices. Therefore, the tip vortices are considered to be located near this radius just behind the propeller. The velocity defects in the axial and tangential velocity components observed at the position where the velocity jump of radial component occurred correspond to the viscous wake of the boundary layer on the propeller blades. Fig.8 shows the velocity fluctuations at axial position of x/rO =1.00 downstream of the propeller. Same tendency is observed on the axial, tangential and radial velocity components. The velocity defects in the axial and tangential components, however, become small. This shows the diffusion of the viscous wake of the blades. The trailing vortex sheets are still observed as the velocity jumps of radial component. The tip vortices are considered to be located between r/rO=0.85 and r/rO=O.90, because the velocity change of axial component is opposite between those two radial positions. Same tendency was observed for the other operating conditions and the other propellers. Velocity distributions around the propel- ler B at the advanced ratio of J=0.70 are shown in Fig.9 as the form of the equi- velocity contour curves of axial component and the velocity vectors of cross components plan parallel to the propeller plane. .. , l . .... .l 270 360 4.0 3.D - . 2.0 1.0 0.0 ~ -1 0 -2.0 90 180 (de`) Van /V 90 180 O (do`) 270 3BO downstream of propeller B ( J=0.70, x/rO=l.OO ) figure is observed from the downstream side of the propeller. Since the propeller is right- handed, the tip vortices are rotating in the anti-clockwise direction as shown in Fig.9. Radial position of the center of the tip vor- tices moves from the blade tip to the inner radii along the downstream direction. This means the contraction of the slipstream. Trailing vortex sheets are also observed in these figures. Angular position of the tip vortex is larger than that of the trailing vortex sheet and the difference of the angular position is increasing along the downstream direction. This means that the pitch of the tip vortex is smaller than that of the trail- ing vortex sheet. 4.5 Hydrodynamic Pitch Using the circumferentially averaged axial and tangential velocities v-x, Van, hydrodynamic pitch angle of the trailing vortex sheets in the propeller slipstream can be calculated by v = tan~1 ( Rev )' (30) ,, .._ _ V-X = mean axial velocity. v- = mean tangential velocity. Radial distributions of the hydrodynamic pitch angle just behind the propeller for three kinds of propellers are shown in Fig.10, In a comparing with the geometrical pitch angles as This follows : 305

Vx/V~ = ~Tip Vortex Trailing Vortex Sheet VX / VA =_~ Trailing //~tl:t Vortex ~ ~1.` Tip Vortex ~ - C _ ~ Propeller B x/r' - 0.25 x/ rO = 1 00 Trailing Tin VnrteY ' ~ . - Trailing . Vortex ? ','t ",;- ~~ Sheet arm= . ·-... 3 . ~. . - Fig.9-a Velocity distributions downstream of propeller B ~ J=0.70 ) 306

Propeller B ~x/rn=2.00 z \A f ~ ~ Vortex Sheet Fig.9-b Velocity distributions downstream of propeller B ~ J=0.70 ) Propeller A Propeller B Propeller C x/r'= 0.25 7n tar x/rc = 0.25 blarks I--I-- J = 0.72 J ~6 Pitch Angle _ _o--- 1 = 0.40 . _ 60 _ 50 _ ~50 _ ~- O ~bit ,,'43 \~9w 30 _ '6 ~ 3u _ 20 _ 10 _ \~ x/rD= 0.25 Marks Geometrical Pitch Ang1e I--{}-- J = 0.90 --~--J = 0 70 Hydrodynamic Pitch Angle ---O--- J = 0.50 _ ~ "l to 0.2 0.4 0.6 0.8 1.0 1~2 al 1.0 0.2 0 4 0.6 0.8 1.0 1.2 r/r. on 10 n - 9a r/r. Fig.10 Comparison of radial distributions of hydrodynamic pitch ( Effect of propeller loading ) ~ t -l (P(r) where P(r) = geometrical pitch distribution. In spite of difference of the operating condi- tion of the propeller, the hydrodynamic pitch angles Qw are nearly constant and slightly larger than the geometrical pitch angles FIG. Same tendency is kept for the three propellers which are different in pitch. This is the reason why the geometrical pitch was intro- duced for the pitch of the ultimate trailing vortex wake in QCM [23. The hydrodynamic pitch Pw (r) of the propeller slipstream can be obtained by PW(r) = 2nr tan6W. (32) 7n~ 60 _ 50 _` bit 40 c, 30 20 _ 10 _ O 0.0 Marks Geometrical Pitch Angle I1 = 1.08 ~I = 0 84 HydrodYnarnic ---Lo- J ~ 0.60 b _o 0.2 0.4 0.6 0.8 1.0 1.2 r/r. angle of propeller slipstream An example of radial distributions of the (31) hydrodynamic pitch of the slipstream is shown in Figs.ll - 13 for the propeller B. It is known that the hydrodynamic pitch increases as the distance from the propeller increases. Amount of increase in the hydrodynamic pitch becomes larger as the advance ratio decreases. This shows the tendency opposed to the conven tional wake model, in which the hydrodynamic pitch of the slipstream had been assumed to be proportional to the advance coefficient [21]. Increase in the hydrodynamic pitch would be due to the contraction of the slipstream along the downstream direction. Pitch of the tip vortices is obtained from the axial variations of the angular positions of the center of the tip vortices and also plotted in Figs.ll - 13. Pitch of the tip vortices is considerably smaller than that of the inboard helical trailing vortex sheets. 307

Propeller B t a 0.6 _ 1.0 0.8 _ 1~ Propeller Tip ~^ 0.4 _ 0.2 _ nil Jan 0.50 Hydrodynamic Pitch Do-x/r'- 0.25 - x/r'- 0.50 --to-- x/r'. 1.00 - -v- - x/r.~ 2.00 \~\\` ~ ' 46% ~ __W Fig.ll Variations of hydrodynamic pitch down- Fig.14 stream of propeller B ( J=O.SO ) Propeller B or new . _ 0.6 _ 0.4 _ 0.2 _ 00 ' 0.0 0.5 1.0 Pw/D I, Tip Vortex Hydrodynamic Pitch ,L§L PropelbrTiP ~ x/r'- 0.25 L I`V _ , ---I--- x/r'- 0.50 --o--x/r'= 1.00 ---v- - x/r'. 2.00 Fig.12 Variations of hydrodynamic pitch down stream of propeller B ( J=0.70 ) Propeller B = o.so ; ~Vortex Hydrodynamic Pitch 11 ~x/r=0.25 _~] 0 1.0 n8 0.4 n2 n.o ) - .3 _ .2 _ --Ol 1.0 ~5 Propeller lip Fig.13 Variations of hydrodynamic pitch down stream of propeller B ( J=O.90 ~ ___~--- x/r'= 0.50 __o-- x/r'= 1.00 ~-v--- x/ro" 2.00 2.0 The present LDV-measurement shows that the helical trailing vortex sheets behind propel- ler are not always concentrated into a set of tip vortices and a single hub vortex as shown in Fig.14. The roll-up model of the trailing vortex sheets would be too simplified. In or- der to construct a more realistic model of the trailing vortex sheets, it is necessary to take into account the increase in pitch of the Blade Sex _.~._ Trailing Vortex Sheet ~_,/ Up Vortex A model of vortex pattern of propel- ler ~ from 16th ITTC reportt28] ~ fit Transition Wake Ultimate Wake 1 -' ran ran , ~ - t Fig.15 Model of propeller slipstream inboard helical trailing vortex sheets and the decrease in pitch of the tip vortices near outer edge of the slipstream. 4.6 Numerical Modeling of Trailing Vortex Sheet A linear wake model of the trailing vor- tex sheets which was based on the geometrical pitch of blades and ignored the contraction of slipstream was used in the previous papert203. Based on the measured velocity distributions of the propeller slipstream, a new wake model of the helical trailing vortex sheets is considered. In the present study, the trailing vortex wake is divided into two parts, transi- tion wake region and ultimate wake region as shown in Fig.15. Contraction and variations of pitch of the trailing vortex sheets are con- sidered in the transition wake region. On the other hand, radial positions and pitch of the trailing vortex sheets are kept constant in the ultimate wake region. Contraction of the propeller slipstream is considered at first. If the radial interval from the hub vortex radius r = rw h to the tip vortex radius r = rw T in the ultimate wake is divided into NR small panel strips by the cosine spacing, the radii of each panel strip can be expressed as follows: 308

Propeller B 1.0 _ 0.9 _ 0.8 _ 0.] _ 0.6 _ 0.5 .0 Blade Tip Measured Fitted Cat By_ O J=0.50 , ~____--J - 0.70 = D.90 o I I ~, 0.5 1.0 1.5 2.0 x/r. Fig.16 Comparison of contraction of slipstream rw Table 3 Principal particulars of DTRC propel- ler models Propeller Number Diameter of Propeller Pitch Ratio at 0.7R (mm) Expanded Area Ratio Bass Ratio Number of Blades - Blade Thickness Fraction Skew Angle (deg.) Rake Angle (deg.) Blade Section Design Advanced Coefficient P.4679 610 1.572 0.755 0.300 3 0.099 51 a NACA - 1.077 . P.4718 610 0.888 0.440 0.300 3 0.069 25 O NACA 0.751 T/r 0 = 0.887 - O . 12 5s, ( 36) Linear Wake Model where s = slip ratio = 1 - J/p, p = pitch ratio at 0.7 radius. ~ 1 ~ ~ Z Tip Vortex Lx Fig.17 Comparison between linear and deformed wake models rW~.1 = 2 (rWT-rWh)- 2 (rWT~rWh)cOs(~p' (33) Radial positions of the trailing vortex sheet at the trailing edge of the propeller blade must coincide with those of the panel strips on the blade surface given by Eq.~10). Then variations of the radial positions of the trailing vortex panel strips in the transition wake region can be approximated by a polyno- mial expression as rtp = rp-(rp-rwp) f r ( A) ~ (34) where fry) = ~ +1.0135-1.92052+1.22843-0.32144, X-XTE = ' XF-XTE ' xTE = x-coordinate at the trailing edge of blade. XF = x-coordinate of the point where the ultimate wake region starts. Based on the measured results, the radius of the tip vortices in the ultimate wake region can be expressed as a function of slip ratio s as follows: The radius of the hub vortex and the axial coordinate of the starting point of the ul- timate wake are kept constant as rwh/rO = 0.1, xF/rO = 2.0. (37) Variations of the radial positions of the center of the tip vortices calculated by the above equations are shown in Fig.16, comparing with those obtained from the results of the flow measurements by LDV. It can be said that the contraction of the slipstream is approxi- mated well by using the present formulae. Variations of the pitch distributions in the transition and ultimate wake regions can be also expressed in the similar manner as the contraction of the propeller slipstream. The deformed wake model based on the measurements of the flow field around the propeller are compared with the conventional linear wake model in Fig.17. Large deformation of the tip vortices can be observed in the new wake model, which is similar to the wake model based on the measured wake pitch shown by Jessup [27~. 5.NUMERICAL EXAMPLES 5.1 Pressure Distribution on Blade In order to evaluate the accuracy and the applicability of the present panel method, David Taylor Research Center ~ DTRC ) propel- ler models P.4718 and P.4679 were selected, since very precise measurements of blade sur- face pressure were conducted by Jessupt29,30~. Both propellers are three-bladed and different in skew. Principal particulars of the propel- lers are shown in Table 3. 5.1.1 Effect of Iterative Cotta Condition In the present calculation, the propeller blade surface was divided into 240 hyper- boloidal quadrilateral panels per blade (NR=12, NC=10) and the hub surface was ap- proximated by 155 panels per 1/3 sector for both propellers. Panel arrangements of the propellers are illustrated in Fig.18. 309

DTRC P.4679 ( upstream view ) NR=12 NC-lO DTRC P.4718 ( Upstream View ) NR=12 NC" 10 Fig.18 Panel arrangements for DTRC propeller models ---a--- First Solution -- 0 --NC = 6 -a- Second & Third Iterative Solutions ---a--- NC-10 0 NC = 14 0.3 0.3 DTRC P4718 r/ro = 0.7 DTRC P4718 r/ro = 0.7 & | 3 0 2 Suction Side NR-12 & | 3 0 2 - Suction Side NR = 12 Il' on kit ~I,! of . I I ' ' ~'T -O. 0.3r 0.2 IO 3 I ~^0.1 1 110.0 1 -0.1 _ -02 , . . . 0.0 0.2 0.4 0.6 0.8 1.0 DTRC P4679 r/r' = 0.7 NR=12 ~ Suction Side NC=10 1 ~ \b \\ D /:-- Pressure Side '` Fraction of Chord Fig. 19 Comparison of chordwise pressure distributions at 0.7 radius ~ ef f ect of Kutta condition ~ The chordwise pressure distributions on the blade at 0.7 radius were calculated at several steps of the iterative Kutta condi- tion. Fig. 19 shows comparison between the f irst, second and third solutions . The f irst solution which corresponds to the application of the Morino Kutta condition gives large dis- crepancy of the pressure on the upper and lower sides at the trailing edge, while the second and third solutions give almost equal pressure at the trailing edge by the applica- tion of the iterative Kutta condition. It can be pointed out that the convergence of the present iterative Kutta condition is very fast. -0.1 0.3 0.2 I red O 3 G _ ~0.1 1 11 0. Cal 1 DTRC P4679 r/ro = 0.7 2 - Suction Side NR = 12 O.t ~ -0.1: Pressure Side _ .., 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of Chord Fig. 20 Comparison of chordwise pressure distributions at 0.7 radius ( ef feet of panel size ) 5.1.2 Effect of Panel Size Comparative calculations f or three kinds of discretized models were conducted to inves- tigate the ef feet of panel size . The propeller blade was replaced with 144 (Nc =6), 240 (Nc=lO), and 336 (N c=14) panels per blade respectively and the hub was approximated by 125, 155, and 185 panels per 1/3 sector respectively . The chordwise pressure distribution at 0.7 radius were also calculated and compared as shown in Fig. 20. Results Of NO =6 are fairly different from those of Nc=lO and 14. There 310

-Panel Method with Hub -- --Panel Method O ~ Experiment Without Hub ( Jessup ) 0.4 Suction Side r/ro = 0.5 0.3 I <a 0.2 11 0.1 Cal 0.0 Panel Method with Hub - - --Panel Method ° ~ Experiment without Hub (]essup) 0.4 3 Suction Side ~ r/ ° O. 5 -0.1 ,' , , , , `` 0.3 r/rO = 0~7 Suction Side I0 3:0.2 1 ~ ~1~ 10.1 11 1~ 0.0 ~ -0.1 0.3 r/rO -o.s 2 _ 0 0 to 1 _ NX \\ i. \\ Suction Side I0 3:0.2 I _Ic~ I0.1 If _ I0.0 ~ Pressure Side -01 , . . . 1 1.0 0.2 0.4 0.6 0.8 1.0 '~_ Fraction of Chord Fig.21 Comparison of chordwise pressure distributions with and without a hub for DTRC P4679 is, however, not so large discrepancy of the calculated pressure between the Nc=lO and NC=14, except at the leading edge. 5.1.3 Comparison with Experimental Data The discretized models with 240 blade panels and 155 hub panels were used in the following comparison. The pressure distribu- tions of the blades at 0.5, 0.7, and 0.9 radii were calculated with and without a hub to in- vestigate the hub effect. In the case without the hub, end plate panels are attached to the inner blade panels in order to compose a closed surface. The calculated results were compared with experiments as shown in Figs.21 and 22. They show good agreement between the calculation with the hub and the experiments for the both propellers. On the other hand, the calculation without the hub underestimates the pressure especially at inner radii, where the hub effect is expected to be large. 0.3 I Cod O 3 I - ~ 0.2 1 11 0.1 ~3 1 O.C -0.1 - at? _~ Pressure Side IO 3 0.2 cad H~ 1 0.1 11 cam I 0.1 . -0.1 _ Suction Side r/ro =0.7 ;~-: 1 ~=: /' Pressure Side 'I 0.2 I Suction Side r/ro = 0.9 1~1 r ~ Pressure Side -01 0.0 0.2 0.4 0.6 0.8 ~1.0 Fraction of Chord Fig.22 Comparison of chordwise pressure distributions with and without a hub for DTRC P4718 5~2 Field Point Velocity In order to evaluate the accuracy of the present panel method, flow fields around propeller were calculated for both the linear wake and the deformed wake models and compared with the measurements by LDV. Each of the three propeller models A, B and C is replaced with 240 panels per blade (NR=12, NC=10) and 84 hub panels per 1/5 sector (3 rows). Comparison of the two calculated velocity distributions with the measured ones in the upstream and downstream of the propeller B is shown in Figs.23 and 24 as the form of equi- velocity contour curves and shown in Fig.25 as the velocity vectors in a plane parallel to the propeller plane. Both calculation results of the axial velocity vx agree well with the measurements in the upstream of the propeller. On the other hand, agreement between the cal- culations and the measurements of the veloci- ties is better for the deformed wake model than for the linear wake model in the down- stream of the propeller. 311

x/rO = -0.25 Measured by LDV Calculation with Deformed Wake Calculation with Unear Wake z z z 1 ~1 ~ ~ {''~~t ~ . ~ 1 ~ _1 Fig.23 Comparison of axial components of velocities upstream of propeller B J=0.70, x/r =-0.25 x/rO = 0.50 Measured by LDV Calculation with Deformed Wake Calculation with Linear Wake z z z ~2 ~2 am/= W1~: Fig.24 Comparison of axial components of velocities downstream of propeller B ~ J=0.70, x/r =0.50 ) Measured by LDV In' " x/rO = 0.50 Calculation with Deformed Wake Calculation with Linear Wake z z I'. it, _y Fig.25 Comparison of cross components of velocities downstream of propeller B ( J=0.70, x/r =0.50 ) 312 v,

~ Panel Method with Deformed W. Propeller B (~=0.70) · Panel Method with Linear Wake 0 Measured by LDV ~-~~~ Expenment · Calculation Ynth 0.5 0.0 1.0 2.0 V'/Y ~Defonned Wake Propeller A 0.0 1.0 2.0 Unear Wake 0 4 ~ ~ 0.0 1.0 2.0 0.0 1.0 I., mu- . . ,~: =~0:2L ~ KT ~ x/r. 0.1 _ I, O ~1;02;000 1.0 2.0 VT/VA O 12 ~AFT_ 00 10 20 00 10 20 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ~ 06L ~(T (T Advanced Coe cient. - 04 ~o c ~ , Fig.27 Comparison of open-water characteris o' ~ __] ~tics of propeller A _ _ -025 0.25 0.50 1.00 x/rO i i 0.O 10 2.0 -Vo/ 2 00 0~5 1.00 x/r Fig.26 Comparison of circumferentially ~ aged velocities for propeller B ~= aver Radial distributions of the circumferen- tially averaged velocities calculated by the present method are compared with the measure- ments for the propeller model B. Calculated results using the linear wake model are also shown in Fig.26. The agreement between the calculations and the measurements is generally good. There is, however, disagreements in the axial velocities at outer radii for the cal- culations with the linear wake model. It can be said that close agreement between the cal- culations and the measurements is due to the consideration of both the contraction of the propeller slipstream and the variation of pitch of the trailing vortex sheets in the present calculation. 5.3 Open-Water Characteristics The open-water characteristics of the three kinds of propellers calculated by the present panel method with the deformed wake model are shown in Figs.27 - 29, comparing with the experiments and the calculations by the previous panel method with the linear wake model [203. The present panel method gives slightly lower values of thrust and torque than the previous panel method. This dif- ference would be due to the effect of the con- traction and the pitch variations of the trailing vortex sheets considered in the present panel method. It can be said that the open-water characteristics of propeller calcu- lated by the present panel method with the deformed wake model are also in good agreement with the experimental data. 313 Panel Method with Deformed Wake -- · Panel Method with Linear Wake ----- Experiment 0.5: 0.4~ by to - 0.3 _ 0.2 _ 0.1 _ :~ Propeller B i. 00 ~ ~ ~ ~ 1 1 0 3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 Advanced Coefficient, J Fig.28 Comparison of open-water characteris- tics of propeller B Panel Method with Deformed Wake Panel Method with Unear Wake ~Experiment 0.6 "is, 0.5 If 0.4 _ ~ _ lo\ I 0.3 0.2 _ 0.1 _ Propeller C :10Ke O.g 4 0.5 0.6 0.7 ~ 0.8 0.9 1.0 1.1 Advanced Coefficient, J Fig.29 Comparison of open-water characteris- tics of propeller C

6. CON(~UDING REMARKS A surface panel method to analyze the hydrodynamic properties of a propeller operat- ing in a unif orm f low was presented . Further, f low f ields around three propeller models were investigated precisely by using a 3-component LDV. Based on the measured velocity distribu- tions of the slipstream, a def armed wake model of the helical trailing vortex sheets behind propeller was proposed. Comparison of the cal- culations by the present panel method with the measurements led the following conclusions: ( 1 ) Convergence of the iterative Kutta condi- tion used in the present method is remarka- bly fast. ( 2 ) The present panel method can predict blade pressure distributions precisely, especially at the inner radii where the thickness and hub ef f ects are large . (3) Three-dimensional f low f ield around a propeller can be predicted by the present panel method with reasonable accuracy. (4) Open-water characteristics of propeller calculated by the present panel method with the def armed wake model are also in good agreement with experimental data. ACIDIOWLE~I~TS The author would like to express his sin- cere gratitude to Prof essor Emeritus R . Yamazaki of Kyushu University f or his ini- tial motivation of the present work. The present work could not have been ac- complished without the cooperation of stat f of the Nagasaki Experimental Tank of the Nagasaki Research and Development Center, Mitsubishi Heavy Industries, Ltd. The author feels very grater ul to Dr . E . Baba, Manager of the Ship & Ocean Engineering Laboratory of the Nagasaki R & D Center, for his valuable guidance in preparing this paper. REF=DCES 1. Kerwin, J.E. and Lee, C.S., "Prediction of Steady and Unsteady Maine Propeller Perf or- mance by Numerical Lifting Surface Theory, " Transactions SNAME, Vol . 86, 1978, pp . 218-253. 2. Hoshino, T. and Nakamura, N., "Propeller Design and Analysis Based on Numerical Lifting-Surface Calculations, " Marine and Offshore Computer Applications, Springer Verlag, Berlin, 1988, pp. 549-574. 3. Rubbert , P. E ., Saaris , G . R ., et al ., "A General Method f or Determining the Aero- dynamic Characteristics of Fan-in-Wing Conf figurations. Volume 1 Theory and Application, "Technical Report No . 67-61A, Dec. 1967, USAAVLABS, Fort Eustis, Va. 4. Woodward, F.A., "Analysis and Design of Wing-Body Combinations at Subsonic and Su- personic Speeds, " Journal of Aircraf t, Vol.5, No. 6, Nov.-Dec . 1968, pp. 528-534. 5. Hess, J.L ., "Calculation of Potential Flow About Arbitrary Three-Dimensional Lif tiny Bodies. Final Technical Report, " Report MDC J5679-01, Oct. 1972, McDonnell Douglas Co ., Long Beach, Calif . 6. Morino, L ., "A General Theory of Unsteady Compressible Potential Aerodynamics, " CR- 2464, Dec . 1974, NASA. 7 . Morino , L ., Chen , L . -T . and Suciu , E. O ., "Steady and Oscillatory Subsonic and Super- sonic Aerodynamics around Complex Conf igura- tions," AIAA Journal, Vol. 13, No. 3, Mar . 1975, pp. 368-374. 8. Suciu , E. O . and Morino, L ., "A Nonlinear Finite-Element Analysis of Wings in Steady Incompressible Flows With Wake Roll-Up, " AIAA Paper 76-64, 1 4th Aerospace Sciences Meeting , Washington, D. C ., Jan. 1976. 9 . Johnson , F . T ., "A General Panel Method f or the Analysis and Design of Arbitrary Con- f igurations in Incompressible Flows, " CR- 3079, May 1980, NASA. 10. Suzuki , S . and Washizu, K., "Calculation of Wing-Body Pressures in Incompressible Flow Using Green' s Function Method, " Journal of Aircraft, Vol .17, No. 5, May 1980, pp . 326-331 . 11 . Maskew, B . "Prediction of Subsonic Aerody- namic Characteristics: A Case for Low-Order Panel Methods, " Journal of Aircraft, Vol.19, No. 2, Feb. 1982, pp. 157-163. 1 2 . Yanagizawa , M., "Calculations f or Aerody- namic Characteristics on a 3-D Lif tiny Body in Subsonic Flow Using Boundary Element Method," TR-835, Sept. 1984, National Aerospace Laboratory, Mitaka, Japan. 1 3 . Ling , Z ., Sasaki , Y . and Takahashi , M ., "Analysis of Three-Dimensional Flow Around Marine Propeller by Direct Formulation of Boundary Element Method ( 1st Report: in Unif orm Flow), " Journal of the Society of Naval Architects of Japan Vol. 157 June , , 1985, pp. 85-97. 14. Hess, J.L. and Valarezo, W.O.,"Calculation of Steady Flow About Propellers Using a Sur- face Panel Method " Journal of Propulsion and Power, Vol.1, No.6, Nov.-Dec. 1985, pp.470-476. 15. Feng, J. and Dong, S., "A Panel Method for the Prediction of Unsteady Hydrodynamic Per- formance of the Ducted Propeller with a Finite Number of Blades," Proceedings of the International Symposium on Propeller and Cavitation Wuxi Apr. 1986 pp.126-135. , , , 16. Koyama, K., Kakugawa, A. and Okamoto, M., "Experimental Investigation of Flow Around a Marine Propeller and Application of Panel Method to the Propeller Theory," Proceedings of the 16th Symposium on Naval Hydrodynamics, Berkeley, Calif., July 1986, pp.289-311. 17. Chen, S.H. and Williams, M.H., "A Panel Method for Counter Rotating Propfans," AIAA-87-1890, AIAA/ASME/SAE/ASEE 23rd Joint Propulsion Conference, San Diego, Calif., June-July 1987. 18. Yang, C.-I. and Jessup, S.D., "Marine Propeller Analysis with Panel Method," AIAA-87-2063, AIAA/ASME/SAE/ASEE 23rd Joint Propulsion Conference, San Diego, Calif., June-July 1987. 19. Kerwin, J.E., Kinnas, S.A., Lee, J.T. and Shih, W.Z., "A Surface Panel Method for Hydrodynamic Analysis of Ducted Propellers," Transactions SNAME Vol .95 1987 pp. 93-122 . , , , 20. Hoshino, T., "Hydrodynamic Analysis of Propeller in Steady Flow Using a Surface 314

Panel Method, " Journal of the Society of Naval Architects of Japan, Vol. 165, June 1989, pp. 55-70. 21. Hanaoka, T., "Fundamental Theory of a Screw Propeller ( Especially on Munk's Theorem and Lif ting-Line Theory ), " Report of Ship Research Institute, Vol . 5, No . 6, 1968, pp. 1-41. 22. Min. K. S ., "Numerical and Experimental Methods f or the Prediction of Field Point Velocities around Propeller B] aces, " Report 78-12, June 1978, Dept. of Ocean Engineer- ing, MIT, Cambridge, Mass. 23. Kobayashi, S., "Propeller Wake Survey by Laser-Doppler Velocimeter, " Proceedings of the International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, 1982. 24. Jessup, S.D., Schott, C., Jeffers, M. and Kobayashi, S., "Local Propeller Blade Flows in Unif orm and Sheared Onset Flow Using LDV Techniques, " Proceedings of the 15th Sym- posium on Naval Hydrodynamics, Hamburg, Germany, Sept . 1984, pp. 221-237. 25. Hoshino, T. and Oshima, A., "Measurement of Flow Field around Propeller by Using a 3- Component Laser Doppler Velocimeter (LDV)," Mitsubishi Technical Review Vol. 24 No.1 , , , Feb. 1987, pp. 46-53. 26. Blaurock, J. and Lammers, G ., "The In- f luence of Propeller Skew on the Velocity Field and Tip Vortex Shape in the Slipstream of Propeller," Proceedings of the S NAME Propellers ' 88 Symposium, Virginia Beach, Va., Sept. 1988. 27 . Jessup, S. D., "An Experimental Investiga- tion of Viscous Aspects of Propeller Blade Flow, " PhD Thesis, The School of Engineering and Architecture, The Catholic University of America , Washington, D . C ., 1 989 . 28. Report of Propeller Committee, Proceedings of the 16th ITTC, Leningrad, Sept. 1981, pp. 61-128. 29. Jessup , S . D., "Measurement of the Pressure Distribution on Two Model Propellers, " DTNSRDC-82/035, July 1982, David Taylor Re- search Center, Bethesda, Md. 30. Jessup , S . D ., "Further Measurements of Model Propeller Pressure Distributions Using a Novel Technique, " DTNSRDC-86/011, May 1986, David Taylor Research Center, Bethesda, Md. APPENDI]1 Equal Pressure lLutta Condition The so-called Kutta condition is a physi cal condition that the velocity at the trail ing edge of the blade should be f inite . This physical Kutta condition cannot be applied directly to a general numerical procedure. Therefore, three forms of numerical Kutta con dition f or three-dimensional lif tiny bodies were proposed by Hess as follows (Ref .5 , p. 36): (a) A stream surface of the flow leaves the trailing edge with a direction that is known, or at least can be approximated . (b) As the trailing edge is approached, the surface pressures ( velocity magnitudes ) on the upper and lower surf aces have a common limi t . (c) The source density at the trailing edge is zero. Morino introduced an approximate Kutta condition that the strength of the doublet in the wake surf ace is equal to the dif f erence in the value of the doublet strength of the upper and lower panels adj acent to the trailing edge[6, 7 ]. This can be expressed as AT = A¢T E (38) However, this form of the Kutta condition was f ound to contain a f undamental error when the free stream from the trailing edge had a cross f low component as pointed out by Kerwin et al . t19] and the present author [203. Therefore, in the present study the Kutta condition (b) is employed, where the pressure is same at the two control points of the upper and lower panels adj acent to the trailing edge. The equal pressure Kutta condition becomes non- linear f unction with respect to the unknown potential ~ and cannot be solved directly. Ling et al . ~ 13 ~ introduced Simplex method to satisf y the equal pressure Kutta condition iteratively. However, that method takes many iterative calculations f or obtaining a conver- gent solution in general. A Newton-Raphson iterative procedure is, theref ore, adopted in the present method for faster convergence. The equal pressure Kutta condition will be applied to determine the unknown Aq) j of the doublet strength in the wake surface. In the numerical calculation, the Kutta condition that the pressure dif f erence at the control points on the upper and lower blade panels ad- j acent to the trailing edge should be zero can be expressed as APi = P+T~ (Pi )-P-TE (Pi ) ~ O (39) for i = 1,2, ,NR, where subscript TE indicates the value at the control point of each panel adj acent to the trailing edge . If we def ine the derivative 3(6Pi) A = lJ a(~¢j )' Eq. (39) can be solved iteratively by solving the f ollowing sets of linear equations . Namely, using the values of ~<p . ( m ) and ~pj ( m ) at the m-th step, the equation for the next step solution li~j ( m+1 ) can be derived from Eq. (40) as N Ai; [~] +1 ) -~(P( m) ] = ,L (m ) (41) for i = 1,2,3, ,N-R (40) The above process is repeated until the value of the pressure difference Api(m) becomes zero within the desired accuracy. The initial guess of the solution A¢j ( 1 ) was obtained by applying the Morino Kutta condition given by Eq. (38). 315

DISCUSSION Jinzhang Feng Pennsylvania State University, USA (China) 1. Numerical prediction of a propeller wake is very difficult because it demands an adequate representation of the deformed wake vortex. With the experience the author has, would he comment on the feasibility of using an empirical model with a few open parameters, such as the one proposed in Eq. (33) to predict the flow field behind the propeller? 2. In both Figures 21 and 22, the hub effect on pressure distribution at sections r/rO > 0.5 is larger than expected. Usually one would assume the hub effect is very much confined to r/rO < 0.5 region and to a less extent. Is this because in his computation the author used a conical cap proceeding the propeller disk to close the hub? AUTHORS' REPLY (1) The present empirical wake model was based on the LDV- measurements of slipstream for a few propellers and therefore applicability of the wake model would be limited. Further measurements and/or calculations of propeller slipstream for various propellers would be indispensable to construct more general wake model. (2) A conical cap proceeding the propeller was not only used in the calculations but also in the measurements of pressure distributions and the measurements were conducted under the same condition. The reason why the hub effect on the pressure distribution is larger than expected would be large hub to diameter ratio for the controllable pitch propeller. DISCUSSION Gun iL Choi Hyundai Heavy Industries Co., Ltd. Korea According to your expression the contraction of the propeller slipstream in the ultimate wake region is obtained as follows: for the case of neglecting slip ratio rWT/rO = 0.887 for the case of J = 0 rostra = 0.762 The above calculations indicate that the contraction of slipstream in the lightly loaded propeller will be more than 10% which is difficult to understand. AUTHORS' REPLY In reply to the question by Mr. Choi, according to the momentum theory, the radius of propeller slipstream is expressed as follows: rat = 1.0 for the case of slip ratio = 0.0 rat = 0.707 for the case of slip ratio = 1.0 The above values are different from those obtained by Eq. (36) because of extreme off-design conditions of the propeller, however, the tip radius given by Eq. (36) is close to the measured one as shown in Fig. 16. DISCUSSION William B. Morgan David Taylor Research Center, USA The author draws the conclusion that his panel method with his deformed wake model is in good agreement with experimental data. However, the data given in Figs. 27 through 29 would indicate that there would be little improvement with the deformed wake model. Does the author have any comment? Also, the deformed wake model is apparently based on measurements. Did the author try to base the wake model on calculated deformed wake? It would appear that even a lifting line model could be used to calculate the deformed wake. Also, in Figure 16, how much did the contraction of the slipstream deviate from a momentum theory analysis? AUTHORS' REPLY (1) The effect of wake model on the integrated values of pressure distribution over the surface such as thrust and torque of a propeller would be small but the flow fields around propeller especially in slipstream were greatly affected by wake model as shown in Figs. 23- 25. Therefore, wake model would become important in solving the interaction problem between two propulsors such as counter-rotating propellers. (2) Amount of contraction or hydrodynamic pitch of slipstream in the ultimate wake region would be calculated by a momentum theory or a lifting-line theory. However, variations of the radial positions and the pitch distributions of the trailing vortices in the transition wake region would be too difficult to be estimated by numerical calculations only. Therefore, a deformed wake model based on LDV-measurement was proposed in the present paper. DISCUSSION Wolfgang Falter Sulzer-Escher Wyss GambH Ravensburg, Germany On Fig. 24 of your paper you show calculated velocities in the tip vortex regions. Does your method provide pressure values in these regions also and/or have you compared these pressures to tip vortex cavitation data obtained from your experiments? AUTHORS' REPLY I have never compared the calculated pressure with the results of observation of tip vortex cavitation because the present panel method could not calculate the pressure inside the tip vortex. Viscous effect would be dominant for the inception of tip vortex cavitation and therefore must be considered in the calculation method to estimate the tip vortex cavitation. 316

DISCUSSION Kuniharu Nakatake Kyushu University, Japan I admire your elaborate experimental work and calculations. I have two questions: (1) Do you have any explanation about the smaller pitch of tip vortices? (2) Even your modified wake model does not express the hub vortex. Do you have any idea? AUTHORS' REPLY (1) The smaller pitch of tip vortices would be due to the strong induced velocities by the strong tip vortices themselves. (2) A wake model including the hub vortex were considered, in which the hub vortex started from the end of hub surface as shown in Fig. A. However, propeller characteristics were not so affected by consideration of the hub vortex. Precise measurement of flow fields around a hub vortex would be indispensable to construct a more realistic hub vortex model. DISCUSSION Shi-Tang Dong China Ship Scientific Research Center, China Congratulate on author's excellent work. I would like to ask two questions: (1) You calculate the total forces acting on the propeller by integrating the pressure distribution on the blade surface. How do you ensure the accuracy in calculating the force acting on the nose region of the leading edge of the blade section especially at off design condition since the direction of surface normal changes rapidly and the shape is rather thin and sharp at the leading edge, and there exists pressure peak there at off-design condition? (2) Have you any special consideration in arranging the panels at the tip region for highly skewed propeller? Thank you. r -- r AUTHORS' REPLY First of all, I would like to thank all discussers for their interest and their stimulating comments on my paper. In reply to the questions by Prof. Dong: (1) As Prof. Dong pointed out, the direction of surface normal and the surface pressure change rapidly at the leading edge. Therefore, the cosine spacing was adopted for the chordwise spacing of the blade panels, which concentrated the panels at the leading edge. Then the number of chordwise divisions Nc= 10 or 14 was enough for calculating the force acting on the blade as shown in Fig. 22. (2) Twisting of the panel becomes large at the tip region especially for tip unloaded and highly skewed propeller. Therefore, the cosine spacing was also used for the radial spacing to concentrate the panels at the tip region. ll ~ Fig.A New modeling of trailing vortex wake including hub vortex 317 z