**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Computation of a Nonlinear Rotational Inviscid Flow through a Heavily-Loaded Actuator Disk with a Large Hub." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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Computation of a Nonlinear Rotational Inv~scid Flow through a Heav~ly-Loacled Actuator Disk with a Large Hub B. Yim David Taylor Research Center Bethesda, USA Abstract A heavily-loaded actuator disk with a large hub in an open rotational inviscid flow is considered. With an assumption of axisymmetry for the flow, the governing equation is the well known Helmholtz equation for the stream function. Similar problems have been dealt with by many authors, but with no hub or for an annulus with constant hub and tip radii. The additional freedom in the boundary geometry makes the problem much more difficult but is required to allow extensive applications to ship propulsor hydrodynamics. For the general solution, a nonlinear integral equation for the stream function is formulated by use of the Green function. The boundary conditions require both the stream function and its normal derivative on the boundary. However, although the former is known, the latter has to be derived at each iteration. The solution is obtained by successive iterative approximations substituting the first approximation into the nonlinear integral equation at each mesh point of the flow field. Convergence of the solution has been shown but the computation is quite slow. Therefore the computational method can only be applied using high speed computer of large memory. For the solution at distances far down stream of the propulsor, the Helmholtz equation can be approximated by an ordinary differential equation, and the numerical solution can be obtained without too much difficulty. The velocity components and the pressure are obtained far downstream of an actuator disk with a given set of circulation and shear swirl distributions. The flow near the disk is computed by an iteration method. Any physical quantities can be obtained from the stream functions. As an example of the applications of this program, the effects of shear swirl flow behind a heavily-loaded actuator disk, where the external forces act on the flow, are computed. Introduction Both design and performance prediction procedures for propellers [1] have made continuous progress. Lifting line theory is now only used in preliminary design, and lifting surface theory is used for the final design and performance prediction of propellers. The theory of hull- propeller interaction has also made progress as in the consideration of effective wake in predicting propeller performance. However, present practice still employs rather crude approximations in some areas of propeller theory. In particular, the hub effect on propeller blade design in a shear flow has not been fully considered. The load distribution of a propeller blade produces bound and trailing vortices. Trailing vorticity in a shear flow is different from that in uniform flow, producing additional trailing vorticity which can be called the secondary vorticity. The effect of the secondary vorticity field on blade design and propeller performance is not well understood. In particular the effect of onset shear and swirl flow on the hub vortex has not been fully investigated. For some designs, when the propeller shaft is of small diameter and the inflow is fairly uniform, there is less necessity to handle such things. But in the case of a controllable pitch propeller where the shaft is of relatively large diameter, or a submarine propeller where the hub is part of the stern, hub effects can be very important. Since the boundary layer thickness of a ship at the station where a propeller is located is sometimes the same order of mgnitude as the propeller radius, several types of interactions have received serious attention by previous investigators [2,3]. In the design of wake adapted propellers, the effective wake is used for estimating the thrust and torque of propellers. To estimate the effective wake from the nominal wake velocities measured in the propeller plane in absence of the propeller, several methods have been used. It was either estimated empirically by multiplying the measured circumferential mean nominal wake by a constant factor or determined theoretically. One of the theoretical effective wake computations for a propeller in an axisymmetric flow field was provided by Huang and Groves [2]. Here, the computed propeller-induced velocities had to be used in solving the vorticity equations. The propeller-induced velocities were computed from potential flow theory using conventional loading and thickness singularities. Also a combination of this program with the conventional lifting surface theory was considered. Experiments showed that the total velocity profiles calculated by Huang & Groves theory immediately upstream of the propeller were in good agreement with measured values. However, Huang and 571

Groves only considered the region upstream of the propeller assuming that the energy was not influenced in that region by the propeller. They do not consider the flow downstream of the leading edge of the propeller, where external forces act on the fluid. Several axisymmetric approximations [4,51 draw attention to the importance of the shear flow effect on the propeller when treated as an actuator disk. Full investigation of the shear swirl flow interaction with the blade-induced velocity field and the secondary flow has not been undertaken so far in propeller theory. In the present paper, the shear swirl flow coming into the heavily- loaded propeller with a large hub is treated as a single.- problem rather than as the combination of separate problems. To make this manageable, we consider it as axisymmetric and the propeller as an actuator disk. In addition the behavior of swirl in the wake of a propeller in shear flow as a function of hub shape is investigated here in an axisymmetric flow analysis. For turbomachine design, similar problems have been considered extensively [6]. Thus the governing equation is well known and even exact solutions for special cases have been found. However, the equation with a general boundary condition has only been dealt with numerically for the purpose of turbomachinery design. For an open propeller without duct, the boundary conditions are different than for turbo-machines. Wu [7] considered a similar formulation of the problem for an actuator disk but no one seems to have considered the wake flow field. A simpler method of computation of the swirl flow is first studied here and computations are performed for an example of a useful general boundary condition which can be used in propeller design, especially for the hub vortex cavity analysis. Then a full axisymmetric nonlinear numerical problem is tackled using an iteration method. Even though this problem is for an axisymmetric actuator disk, this is in the shear swirl flow with a large hub of an arbitrary shape and heavy loading. With an application of the Green theorem and the Green function a full treatment of the boundary conditions is attempted. Both the boundary condition on the hub and the discontinuity of the velocity in the slipstream behind the propeller are properly treated. This analysis is a continuation of a previous investigation [8] with considerable improvements in accuracy and completeness. Shear Swirl Flow Analysis Although there have been many investigations of propeller performance and propeller design, very little is known of the flow in the propeller wake. The swirl that is closely related to hub vortex cavitation should be considered in the wake of a propeller whose hub length is finite. The image vortices inside an infinite hub are well known. These image vortices cannot be placed on the outside surface of the hub, or in the flow. This is because the trailing vortices and the secondary vortices formed near the propeller blades would not disappear behind the propeller unless viscosity were considered, and no other vortex can be considered in addition to them. The problem is how these vortex lines will be rearranged or roll up amongst each other while they are transported downstream where the propeller hub tapers to a point and disappears. In experiments, the vortices around the hub roll up and vortex cavities are often formed in the hub wake. To understand this phenomenon, a simple model of axisymmetric swirl flow may be useful. In general, the common dynamic equation of incompressible inviscid steady vortex flow can be [6] represented by qx: = VH (1) where H = _ q2 + P (2) 2 e is the total head in the assumed axisymmetric flow with cylindrical polar coordinates (x,r,8) and the corresponding velocity components q (u,v,w), and vorticity components ~ (`x,cr'`~) Considering the stream function to (x,r), the governing equation for ~ is a2~ ~2~ 1 B~ dH dC + - - r2 _ ~ {~\ Dx2 .,r2 r Br dip din ' ' C(~) = rw (4) is the circulation distribution in the wake. H is constant along a streamline with ~ constant and H = H(Y'). The nominal wake is assumed to be known, i.e. the flow field without propeller is known. Accordingly, H without the propeller is known and designated by Ho. Using an axisymmetric approximation, H can be written as [7] H = Ho + QC~ (5) where Q is the angular speed of the propeller located at x=0 and the circulation Cat is proportional to the blade number times the blade load distribution except for secondary vorticity. If there is no preswirl upstream of the propeller, the total circulation C will be equal to C1. Now both H and C immediately behind the propeller are both known functions of r and A. Thus, IdH dH do dip dr dr (6) dC dC do dip dr dr (7) The boundary conditions at downstream infinity may be written as follows 1 al a~ u = ~ ~ us,,, ~ ~ O as r ~ ~ (8) 572

1 a~ ~ = -a ~ ° as x ~ ~ (9) On the hub, r = h(x), Y' = 0 (10) On the outer edge of the propeller and on the slip stream Y'(r,x) = ~ (1,0) (11) Many problems with similar governing equations have been solved for special cases, mostly for turbomachinery. For a propeller, there are some investigations for a simple case [61. No investigation seems to exist for a propeller wake with shear swirl in the presence of a finite hub. We use non-dimensional quantities r' = r/R, x' = x/R, q' = q/U, C' = C/(UR) hi' = ~/(UR2), H ' = H/U2 and 1 = U/(QR) where U is the speed at upstream infinity, R is the propeller radius. After this substitution the prime will be omitted as understood. Then we obtain the nondimensional governing Equation (3). Although the problem could be solved numerically, it would be very complicated in the present form to obtain an accurate solution. At first, for the purpose of investigating hub vortex cavitation, the solution far downstream of the hub may be very useful. Since Bty '32~ a To, a 2 .0 asx .= may be considered as a function of r only in Equation (3), and the governing equation becomes an ordinary differential equation. For special functional forms of H and C, a closed form solution is possible. With the general functional form of H and C, if ~(a) and B~/8r(a) are known the solution can be obtained as an initial value problem by a method such as the Runge Kutta Technique. When ~ is obtained, u and C will be known and the pressure can be obtained by I' = Ho) - 2 {U2 + r2 (a)} (12) or by 1 dp c2 _ _ = _ Q dr r3 Here we consider C(r) =0 outside the slipstream. Since outside the slipstream at x~= the flow is uniform and the pressure should be continuous, the location of the slipstream and the pressure distribution can be uniquely determined. The solution of the differential Equation (3) with conditions (8) through (12) is obtained by solving two simultaneous ordinary differential equations do 1 dr r F1 + f(~,r) do dr l (13) f(~,r) = r2 dry - C dC with a given boundary condition at the slip stream edge r= rs(x) ~(rs) = Nehru) (14) p = 0 where rsto)ru From Equations (5) and (12) dt s) =rsu=rs[2{HO(ru)~+ A C(ru)-C2(ru)/rs2~]l (15) Because we assumed C(ru) = 0, d~(r5) = rs \/2Ho(ru) However, since rs is not known, these equations may be solved for a range of rs values and the final value selected such that ~(r=0) = 0 (16) either graphically or by an iteration method. Note here that dH/d~, dC/dt4, A, d~/dr at x=0 are all known as functions of r(x=0). Thus, at each step of the solution of the above simultaneous equations, from the known value of ~ at r(x ~c=), r(x=0) can be calculated to determine both dH/d~ and dC/dt4 at Ax =0) and therefore, at r(x All. Such solutions are exact at x em, under the assumed conditions. However, the solution with (or = h) = 0 can also be considered to be an approximate solution at any appropriate strip of a slender hub, i.e., the hub slope is small with respect to x. In the present study, a sample calculation of ~ and pressure are rgade with the following form of the boundary conditions. At x=0, for Ho u = (r - h)~/n/D at r < rO D (rOhen u= 1 atr~rO v = 0 in rH < r < 1 (17) and C = an + air + a2r2 with constant coefficients an, al, a2 and n. Using the above representations, the incoming velocity profile is similar to the flow in a turbulent boundary layer, and the circulation distribution has a radially parabolic shape which can take on zero values at the blade hub and tip. Then, at x = 0 573

Jr 1 (r - h)l/n + 2 h(r - h)l/n + l r (or) = u(r)rdr = D 1 /n + 2 1 /n + 1 r = rH ' s = J rH at rH < r ~ rO (or) = intro) + 2 - 2 at r ~ rO R d = D {(r-h)l/n+l + h(r-h)l/n} at rH < r ~ rO d = D {(rO-h)l/nr} at r > rO or Also assumed at x=0, from Equation (12) dr 2 dr {U2 + r2 C2(~)) + 13 C2 dH ~ {(r - h)l/n ~ (r - h)l/n - l + an + air + a2r2 (a + 2a r) at rH < r < rO (19) aO+a~r+a2r2 = (al + 2a2r) at r ~ rO The pressure distribution induced by the particular shear swirl flow considered here through Equation (17) is given in Figure 1. Even if the circulation near the hub is zero, the pressure rapidly decreases near the axis at x loo. If the pressure is lower than the vapor pressure Pv, a cavity may be formed and will behave like a solid boundary. If a streamline has ~=0 and P= Pv, this will be a cavity boundary. In Figure 1, the axial and the tangential velocity components, u, w and the pressure p/g are given at each radial position for three assumed slip stream radii at x em, rS=0.98, rs=0.95 and rs=0.915. ~=0 at r=0 is achieved only in one case rs=0.915. In the other two cases, ~=0 for r>0 when the hub is infinitely long. Even when the radius of an infinitely long hub decreases, the quantities u, w and - p/g increase near the hub. These quantities downstream of the actuator disk near r = 0 increase very rapidly to oo. The boundary values of u and w at x=0 are shown as broken lines. Full Axisymmetric Solution Now we try to solve Equation (3) in its full axisymmetric form. We consider to =r2 + rt4 in Equation (3). Then it changes to V214 - ~ = - g (21 ) 574 1.0 one 0.95 0.915 0.8 . 0.6 0.4 O. at x=o )at~ if/ / \<oo\tx~oo \~x~oo (18) 0.0` ).0 0.2 0.4 0.6 0.8 1.0 1.2 ~ , w, u Fig. 1. Shear swirl flow field at U/(QR) = 0.44. where dH C dC 2 '3214 1 an at -g = r dip -d,4 ~ V it' = aX2 + r fir ~r2 On the hub ~ is constant, say 0. Then to = - 2 on the hub. At the upstream station x=xL, ~ is assumed to be known. At x=xN at far downstream ~ will be obtained as a solution but '4x will be safely considered to be zero. At r ~c=, ~ is considered to be the stream function for uniform flow and ~ and ton are known. At r = 0 behind the hub ~ and In are both zero. Then this is mixed boundary value problem for a nonlinear elliptic equation. V 24' = f(x,r,~) (22) The existence and the uniqueness of the Dirichlet problem for this equation has been considered by Courant [9]. However, he contends that the uniqueness is guaranteed only in a sufficiently small domain or if ~f/~y>0. In the present boundary value problem ~ is known on x=x and at x em. Ax is known. Thus the application of Green's theorem may be most appropriate. Therefore, at x= xL ~ and ton cannot be given arbitrarily. Besides, 14 ~ dH C dC f =" - ~ = + r - r2 ~ r2 do r dip' where H and C are both given functions at the boundary and df 1 d2H d / dC = 2 + r2 don - dip (C d`4 ~ O (23) may be another condition to guarantee a unique solution. (20) Application of the Green Function We consider the Green Function G(r,x;g,4) that is related to two points P(r,x) and Q(Q,4) and that has the following properties: V2G - G/r2 = - d(r - Q) (X - 4)/r (24)

where d is the delta function and G(r,x; Q,4) = G(e.~; r,x) G(r,x; Q,4) ~ O as PQ Ge(r,x; Q.~) ~ O as PQ Such a function has been obtained by Wu [7] and is represented by the 2nd order Legendre function which has a logarithmic singularity at PQ .0. From Equations (21) and (24) GV24,-~V2G = - Gg + (or - Q)(X - t)/r Integrating this equation in the space bounded by the Besides, at ~ = x hub boundary and planes at x= x~ and XN we obtain Ji (GV2~-~V2G) pdedt = - J:GgQdQd4+~(r,x) V D where Using the Green theorem From Equation (21) am 1 am 1 or On r an - 2 an on r = rH (27) From Equations (26) and (27) am 1 altar ax ar I ax 1 ar an = r ar ~ an - an at / at -a (28) where a / a is the slope of the hub with respect to x and do= dx, do= -dr (29) an an ~ = (l in_ e ~ Q 2 t=xL in = ( ~ ~~) tptr,x) = - JIG an ~ an A') pdg + JiGgtp,4)pdedk When we insert Equations (28~-~30) into Equation (25) Q D we obtain -Jo is GV aa9' pdg where Q is along the contour at x =xL and XN and the hub boundary, n is the inward normal to the fluid, Qs is along the slip stream where at4/8n is discontinuous as much as ~ a ala n while ~ and a G./ a n are continuous. Along the contour Q. the boundary conditions are given as where ~ = 0 on the hub r = rH or ~ = _ H The normal velocity on the hub is zero, or am = 0 at where t is the unit vector along the tangent to the hub. That is, am am ar am ax _=_ + _= 0 (26) at ar at ax at However, (30) (25) JiG ( 6) d do _JG am {1 ( de )2t do D Q - JGA a { 1 + ( do ) ~ do - iG~QdQ + f(r,x) (31 ) Is XN f(r,x) =J( 2 - ap 2 ) Q do +l am 2 do do Q=rH Q=rH +l am (a - 2 ) do -; G am do (32) t=XL t=XL al' am ar am ax = + an ar an ax an Since 14~ .0 as Amp, the line integral at ~ loo disappears. Note the effect of slipstream is first included here among solutions solved by the similar technique. Although ~ is continuous throughout the flow including the slipstream, we know that the velocity or the derivatives of ~ may not be continuous across the slipstream especially when the circulation distribution is not continuous. This may cause the line integral along the slipstream to be non- negligible. Numerically this gives a considerable complexity because neither the shape of the slipstream nor the velocity discontinuity is known a priori. However, in each iteration this could be known approximately. Equation (31) is a complicated nonlinear integro-differential equation for ~ because g is a complicated nonlinear function of A, and ~g is not known on the boundary. Wu formulated the problem for 575

a simple case of no shear and no hub without slipstream, only with the heavily-loaded actuator disk. He obtained the Green function and suggested solving the nonlinear integral equation by an iteration technique, showing as an example the case with a given load distribution on the propeller disk but did not obtain numerical results. Later Greenberg [10] performed the numerical computations for the same problem in a somewhat different manner with a uniform loading and a nonuniform loading, and demonstrated the convergence of the iteration with the same Green function obtained by Wu [7]. The present problem includes a large hub and incoming shear flow with pre-swirl in addition to the heavily-loaded propeller disk. Because of this complexity, there exists a line integral term, in addition to the area integral that was handled by Wu and Greenberg et al. The line integral includes the normal derivative of the unknown which changes the integral equation to an integro-differential equation. When an approximate solution ~ is assumed everywhere, Bt4/8n on the boundary will be known accordingly, and H. C and their derivatives along the streamlines can also be determined. Therefore when these quanitities are inserted in Equations (31) and (32) the iterated solution can be obtained. If the continuation of this process converges to a solution, it will be the desired one if the solution exists. Computation of tY The first order solution may be considered with a straight slipstream through the blade tip (r= 1). As a first approximation the streamlines are assumed to be constant along r =(rT - rH(x)} m= 1, 2, ·~. mO (33) The stream functions at x= XL is given as Fir) from which we obtain the shear distribution 1 al r or -, or vice versa. At x = A, g can be calculated from the given A, H and C at x = A. Then along each streamline A, H. C, dH/d~ and dC/d~ are constant. Therefore when ~ is known at any field point H. C, dH/d`Y, and dC/d~ will be known automatically. Thus g will be known and we can evaluate the area integral of Equation (31). The mesh is created with lines (33) and the vertical lines x = + Ax (34) with lix intervals. At each mesh point, the value of G is precalculated by a good approximation [10]. Then Jru AH GgQdg (35) is first calculated at each vertical line by Simpson's rule. The logarithmic singularity of G in the numerical integration of Equation (35) and in the slipstream line integral is treated as follows: when x=t, and Q ~r(/O) G ~ - 1 1 log (r-e)2 4n r r2 Though the integration of the log function does not produce any singular behavior, the numerical treatment requires care. J GgQde =J { GgQ + 2g° log ~r-e~ } do rH rH + 2 r [(rHr){log~rHrig1 }(rur)(log~ruret1~] where gO is the value of go at Q = r, the singular point. Then the values of Equation (35) at ~ = + Ax are integrated by Simpson's rule to produce the area integral. Since ~ on the boundary is given from the boundary condition, the function f(r,x) in Equation (32) is determined and it does not change by iteration. However 3~/8g on the hub r=rH, a~/at on x=x and ~ at x = XN are not known a priori. The line integral terms have to be iterated by calculating 3~/ap and 3~/~t from the first approximated ~ distribution. When the mesh (33) is set from the beginning, the iterated solution ~ will change at the given mesh points at each iteration and will converge to a solution if the solution exists. There, H. C and the derivatives which are only functions of ~ have to be interpolated numerically at each point from the given values of Hi), C(~), etc. at x= xL. Since the values of to on the boundary are already known from the boundary conditions, they do not have to be calculated on the boundary. This fact is very convenient because the boundary integral/the line integral is more singular than the area integral. However the values of the line integral at the points other than at the boundary have to be calculated and need special care. Iteration and Convergence Iteration techniques for nonlinear equations are familiar to those who use high speed computers. However, since the present problem may be considered to involve simultaneous equations with an extremely large number of variables, it is rather impractical to use the conventional Newton Raphson method. Because it might be rather simpler and more economic to consider a naive first order iteration without modification of values, this approach was tested first. However in this case sometimes the solution seemed to oscillate after a certain course of convergence, so that a simple modified quasi Newton Raphson method was considered [8]. The convergence using the simple modified quasi Newton Raphson method was found to be very sensitive 576

to the computational error. Whenever any error exists in the computation the solution does not converge. This has been studied in detail in a previous paper [8]. However, even though it just converges it is not necessarily the right solution. Besides the question of uniqueness, the solution has to satisfy all of the boundary conditions. In fact it converged without line integrals along the lines x=x and the slipstream, giving a wrong solution. It also converged to an unreasonable solution when unreasonable values were given as the boundary condition. Therefore not only the convergence but also satisfaction of boundary conditions must be checked. Propeller Characteristics Now the axisymmetric stream function is obtained everywhere, all the physical quantities can be computed. These are used for the characteristics of an infinite bladed propeller. Well known formulae for propeller characteristics are given as follows: The thrust coefficient T neQZR4 = 4A J w1(r,° + ) { A - w2(r,0 + ) } rdr rH where 2w~r equals the circulation distribution F set of data obtained in a water tunnel experiment. The first example has already been used for an approximate analysis as shown in Figure 1. The full axisymmetric analysis with the boundary conditions given in Equation (17) are now tested for the hub geometry shown in Figure 2. The efficiency changes corresponding to the inflow velocity distributions shown in Figure 3 are shown in Figure 4. These changes are compared with the results calculated by simple momentum theory without considering the thrust deduction. It is obvious from the CQ equation that CQ will decrease when u deceases, thus increasing the efficiency. However, the reduction in u may be as the result of increment in the friction and/or form drag of the ship. The second example corresponds to one of the sets of test data of Huang and Groves as shown in Table 1 with the hub geometry shown in Figure 5. The velocity distribution is taken from the measured longitudinal velocity component at the station x=xL= -0.482 as shown in Figure 6. XjX~ corresponding to the propeller blade circulation ~ with a Z bladed propeller such that km ZE = Foo/K = 2wlr where K is the Goldstein factor and w2 is the ~ component of the velocity at the actuator disk. The torque coefficient CQ = QR5 = 4A2 J u(r,0) w~(r,0) r2dr rH The efficiency The pitch angle 71 = A CT/CQ = cot- ~ (rr7C(r)/A) where rlC(r) is the radial distribution of the propeller efficiency r1c(r) = { 1 - r W2(r,0 + )}/u(r,0) The other conventional thrust coefficients CTS= T/(e/2 U2nR2) has the relation with CT, CT=O.SA2CTS. Numerical Examples and Discussions For checking the numerical accuracy and solution behavior for parameter changes, two examples were numerically tested. One example uses the boundary conditions given in Equation (17) and the other uses a ~ 1.0 1 1 1 1 ~ -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Fig. 2. Hub geometry 1. 0.8 0.6 0.4 0.2 The head distribution from Equations (2) and (5) is approximated from the nominal wake values because not enough information is available for the head. The blade circulation given in the Table is converted by the relation zr ~ r00 = 2wlr taking the Goldstein factor K= 1 because for the 7-bladed propeller k is very close to 1. With this approximation, the thrust coefficient CTS, using the r values of Table 1, is very close to the measured value and the value from the lifting surface computation. The velocity components at the propeller plane (x =0) and a little downstream of the plane (x=0.192) are shown in Figure 6. These values are taken at the iteration where the average relative error is less than 0.005~Vo with 20 iterations. A simple approximation assuming that u = nominal wake + F<~,/(2rtan+) from actuator disk theory is also plotted in Figure 6. The three calculated 577

1 x=x1 1.0 ~ 0.9 L 0.8: 0.7 g 0.6: 0.5 g 0.4 0.3 1 1 1 1 1 0.4 0.6 0.8 1.0 u Fig. 3. Shear distribution at x= -0.8. 1 n _ _ PRESENT THEORY 0 8 _=~/ n=2 0.8 MOMENTUM 0.6 - THEORY n=4 0.4 /// /// n=2 /3 /4/ 1 1 1 ~ 1 1 1 1 1 =\ / // - 0.5 - 0.3 - 0.1 0 0.1 0.3 0.5 0.7 Fig. 5. Hub geometry 11. 1 .2 _ 1.0 _ 0.6 _ 1.0 0.8 0.6 0.4 0.2 0.0 // MEASURED AT / /L X=0 WITH PROPELLER 7/ AWL X0. 192 0 2 NOM NAL WAKE ~ 0 I // \ + ?CO/~2rtan l) 0O0 2 4 6 8 10 CT10-2 0 Z: ~ ~ ~ ~ 2 0 2 0.4 0.6 0.8 1.0 Fig. 4. Efficiency-thrust coefficient relation. Table 1. Tested propeller data. CTS = 0.356, HA = 1.268 r/R 0.211 0.250 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.950 1.000 1.050 1.100 1.200 Nominal Wake 0.387 0.417 0.454 0.520 0.579 0.631 0.677 0.720 0.763 0.785 0.806 0.826 0.845 0.880 Fig. 6. Axial velocity components near the propeller disk. r curves are very close to each other. The efficiency computed with the boundary condition given by the measured longitudinal velocity at x= -0.480 IS 1.15 which is close to the value 1.17 from actuator disk theory. 0 0023 If uniform inflow is assumed, the propeller efficiency 0.0051 with the same value of F`,0 in open water is 0.82 this 0.0107 means the approximate corresponding wake fraction Wf is 0.0150 0.287 from 1.15 (1-wf)=0.82. It is well known that the 0.0172 overall propulsive efficiency also depends upon the thrust O. 0140 deduction which reduces the efficiency. The numerical 0 0089 results show that when F00 =0 at the blade tip, the effect 0.0055 of the slipstream line integral is negligible. This is 0.0000 reasonable because it has been analytically proven [11] 0 0000 that both u and w are proportional to F0O on the 0.0000 propeller plane when hub Is absent. 578

Since the governing equation (31) holds in a shear swirl flow and no explicit vortex distribution is used for the solution, it is not clear here how the conventional bound and trailing vortices plus secondary vortices [12] are involved. However, this could be interpreted using the vortex ring method [13]. The numerical program with iterations adopted in the present paper is mainly for the purpose of checking the feasibility of the application of the iteration method to a nonlinear problem with the reasonable convergent results. The convergence is quite stable with the change of mesh size and the degree of interpolation (when a Lagrangian interpolation is used). Because the present problem neglects viscosity, the effect of viscosity on the shear flow cannot be computed. The effect of shear on the propeller can be computed from the measured velocities or the computed velocities by the method of reference [2]. Therefore the present program could be most effectively used to find the propeller shear interference at or behind the propeller plane with the measured upstream velocities or the effective wake computed by the Huang and Groves method. Since experiments show that the propeller does not affect the flow beyond about two propeller diameters upstream [2] of the propeller the nominal wake at a station two propeller diameters upstream of the propeller plane may be effectively used to compute total flow using Equations (2), (5) and (3). If there is no shear or swirl upstream of the propeller, the problem becomes much simpler because the area integral outside the domain bounded by the slipstream, the propeller plane, and the hub disappears. However, the line integrals along the hub and the slipstream (when FOO (tip) ¢0) are needed in addition to the area integral. The present program may be further extended to evaluate ducted propellers and multistage propellers such as contra-rotating propellers in the shear swirl flow. In fact for the problem of contra-rotating propellers in shear swirl flow, the present computer program can be applied without too much change. Acknowledgments The investigation reported herein was supported by the David Taylor Research Center's Independent Research Program. The author would like to thank Dr. T.T. Huang, Mr. Justin McCarthy, Dr. William B. Morgan, and Dr. Milton Martin for their continuous encouragement. References 1. Kerwin, J.E. and C.S. Lee, "Prediction of Steady and Unsteady Marine Propeller Performance by Numerical Lifting Surface Theory," Trans. SNAME, Vol. 86, 1978. 2. Huang, T.T. and N.C. Groves, "Effective Wake Theory and Experiment," Proceedings of 13th Symposium of Naval Hydrodynamics, ONR, 1980. 3. Nagamatsu, T. and T. Sasajima, "Effect of Propeller Suction on Wake," Jour. of SNJE, Vol. 137, pp. 58-63. c 4. Dyne, Gilbert, "A Note on the Design of Wake- Adapted Propellers," J.S.R., Vol. 24, No. 4, Dec. 1980, pp. 227-231. Goodman, Theodore R., "Momentum Theory of a Propeller in a Shear Flow," J.S.R., Vol. 23, No. 4, Dec. 1979, pp. 242-252. 6. Horlock, J.H., "Actuator Disk Theory, Discontinuities in Thermo-Fluid Dynamics," McGraw-Hill, Inc., 1978. 7. Wu, T.Y., "Flow Through a Heavily Loaded Actuator Disk," Schiffstechnik, Bd. 9, Heft 47, 1962, pp. 134-138. 8. Yim, B., "An Iteration Method for a Nonlinear Shear Flow Through a Heavily-Loaded Actuator Disk With a Large Hub," Proceedings of the International Conference on Numerical Methods in Engineering; Theory and Application, Edited by G.N. Pande & J. Middleton, Martinus Nijhoff Publishers, 1987. 9. Courant, R., "Method of Mathematical Physics," Vol. II, Interscience Pub. New York, 1862. 10. Greenberg, M.D., "Nonlinear Actuator Disk Theory," Zeitschr, Flugwissensch, Bd. 20, Heft 3, 1972. 11. Morgan, W.B. and Wrench, J.W. Jr., "Some Computational Aspects of Propeller Design," Method in Mathematical Physics, Vol. 4, Academic Press Inc., New York, pp. 301-331, 1965. 12. Yim, B., "A Note With Secondary Vortex Caused by a Thin Foil in a Nonuniform Flow," J.S.R., Vol. 32, Nov. 1988. 13. Cox, B.D. "Vortex Ring Solutions of Axisymmetric Propeller Flow Problems," MIT, Dept. of Naval Arch. & Marine Eng., Rept 68-13, June 1968. 579