**Suggested Citation:**"Computation of Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectagular Blades." 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 Viscous Flow around a Propeller-Shaft Configuration with Infinite-Pitch Rectangular Blades F. Stern and H. T. Kim The University of Iowa Iowa, USA Abstract A viscous-solution method is set forth for calculating marine-propeller flow fields. An overview of the computational method is giv- en, and some example results for both laminar and turbulent flow are presented and dis- cussed with regard to the flow physics, for the idealized geometry of a propeller-shaft configuration with infinite-pitch rectangular blades. It is shown that the flow exhibits many of the distinctive features of interest, including the development and evolution of the shaft and blade boundary layers and wakes, and tip, passage, and hub vortices. Comparisons are made with results from a lifting-surface propeller-performance pro- gram, to aid in evaluating the present method, which show that the present method accurately predicts the blade loading, including viscous effects, and clearly dis- plays the ability to resolve the viscous regions in distinction from the inviscid-flow approach. Introduction - Propeller-type flow fields are encountered in a wide variety of engineering problems, e.g., in the propulsion of marine vehicles, airplanes, and helicopters, in turbomachine- ry, and in inert and reacting swirl-flow sys- tems. The present study concerns the de- velopment of a viscous-solution method for the analysis of incompressible propeller flows. Of particular interest are marine propellers which are unique because they operate in the thick stern boundary layer and wake such that the flow field is interactive, i.e., the propeller-induced flow is dependent on the hull flow which is itself altered by the presence of the propeller. More specifi- cally, here we are primarily concerned with the propeller-induced flow; however, the pre- sent study is an outgrowth of a larger pro- ject concerning propeller-hull interaction and, upon extension, is expected ultimately to handle entire configurations. 553 Presently, only potential-flow methods are available for calculating practical marine- propeller flow fields (for a recent review see Kerwin [13). Lifting-surface methods are available for both steady and unsteady flows (e.g., Kerwin and Lee [2]). Also, surface- panel methods have been developed for steady flow (e.g., Hess and Valarezo [3]). These methods suffer from two major problems: first, they rely on the incorrect assumption that the propeller operates in an infinite ideal fluid, but with a specified spatially varying inflow which represents the hull boundary layer and wake; and second, the results, including the propeller thrust and torque, are very sensitive to the specifica- tion of the geometry of the trailing-vortex wake sheet which requires a viscous-flow analysis for its prediction. Consistent with the first problem, the agreement with exper- imental thrust and torque data for nonuniform inflow has not been satisfactory. Also, the predicted pressure distributions, even for uniform inflow, do not show overall good agreement with experimental data (ITTC [43). A complete evaluation of the theory has been hampered by the lack of knowledge of the effective inflow which is usually assumed to be the nominal wake of the bare hull. Relatively little work has been done con- cerning viscous effects for rotating propel- ler blades. Most of the studies pertain to boundary-layer development and are restricted to laminar flow and idealized geometries (Morris [5]). Only one study has considered practical geometries and flow conditions (Groves and Chang [6]). In general, these methods suffer due to the inaccuracy of the pressure distributions predicted by inviscid- flow methods and are not easily expendable into the wake. Similar difficulties with this approach have been encountered in turbo- machinery applications. Viscous effects have also been studied with regard to the tip- vortex generation process utilizing the par- abolized Navier-Stokes equations (most recently, de Jong et al. [7]).

Most work on propeller-hull interaction assumes that the interaction is inviscid in nature and has focused separately on either propeller influence on hull resistance (thrust deduction) or on hull boundary layer and wake (effective wake). Recently, Stern et al. [8,9] have developed a comprehensive viscous-flow approach to propeller-hull interaction in which a viscous-flow method for calculating ship stern flow (Chen and Patel [10]) is coupled with a propeller- performance program in an interactive and iterative manner to predict the combined flow field; hereafter referred to as the interac- tive approach. A body-force distribution is used to represent the propeller in the vis- cous-flow method. The steady-flow results show good agreement with experimental data and indicate that such an approach can accur- ately simulate the steady part of the com- bined propeller-hull flow field. Although the unsteady-flow results generally follow the trends of available data, these indicate the limitations of this approach for simulat- ing the complex blade-to-blade flow. The work of Stern et al. [8,9] is precursory to the present work. Most of the relevant work from related ap- plications is for high-speed flow in which shock waves have a dominating influence; therefore, the focus of these studies is, in general, quite different from that of the marine-propeller application. The most closely related work is that done to develop energy efficient turboprops and for turbo- machinery applications (see Kim [11] for a more complete discussion, including refer- ences). Although advanced inviscid- and vis- cous-flow methods are under development, in most cases, incompressible-flow calculations are either not possible without major modifi- cations or require the use of the pseudo- compressibility concept. In its usual form, the latter precludes time-accurate unsteady- flow calculations, although some recent stud- ies have shown promising results for such extensions through the use of subiter- ations. Lastly, concerning related applica- tions, the helicopter and swirl-flow calcula- tions are helpful with regard to tip vortices and swirling jets and wakes, respectively; but, here again, involve large differences in both flow conditions and geometry. It is apparent from the foregoing that present methods for calculating marine-pro- peller flow fields are inadequate for analyz- ing the detailed flow structures such as the development and evolution of the unsteady blade boundary layers and wakes, blade-to- blade flow, hub and tip vortices, and overall propeller wake. Furthermore, even the most advanced computational fluid dynamics methods from related applications are either inap- plicable or require major modifications to handle marine propellers. This overall situ- ation motivated the present study. 554 In the following, propeller-flow phenomena are described to aid in understanding the nature of the flow as well as the differences between the present and interactive approaches. Also, the rationale for select- ing the present geometry, i.e., a propeller- shaft configuration with infinite-pitch rec- tangular blades (figure 1) is discussed, including its advantages and shortcomings. Next, an overview of the computational method is provided. Then, some example results are presented and discussed with regard to the flow physics, including the computational grid and conditions and calculations for both laminar and turbulent flow. Subsequently, comparisons are made with results from a lifting-surface propeller-performance pro- gram, to aid in evaluating the present me- thod. Finally, some concluding remarks are made. The details of the computational method and the complete results, including additional calculations to study the influ- ences of a thick-inlet boundary layer, the propeller angular velocity, and the blade number, as well as comparisons with some additional relevant experimental and computa- tional studies are provided by Kim [11]. Propeller-Flow Phenomena Figure 2 displays sketches of both the circumferential-average and blade-to-blade flow for the relatively simple case of a pro- peller-shaft configuration. The circumferen- tial-average flow (figure 2a) clearly dis- plays the expected features based on physical considerations, i. e., axial-velocity U increase (overshoot ) and negative radial- velocity V ( contraction) associated with the propeller thrust, and propeller-induced swirl W. including hub vortex, associated with the propeller torque. Also, there is a jump in pressure p across the propeller plane and a large decrease in pressure along the wake centerline due to the propeller thrust and -induced swi rl, respectively, and a large increase in turbulent kinetic energy k, including two peaks, one near the wake cen- terline and one corresponding to the tip of the propeller blades. As discussed above, the interactive approach is able to predict accurately many details of the circumf eren- tial-average ( steady) f low; however, in order to predict the complex blade-to-blade f low a more detailed representation of the propeller than the body force is required. In comparison with the situation for the ci rcumf Brent ial-average f low, relet ively li t- tle is known concerning the complex blade-to- blade flow due, no doubt, to difficulties in perf arming such experiments and calculations for this type of geometry. Figure 2b dis- plays some of the expected f low structures, including the leading-edge horseshoe, pas- sage, tip, and hub vortices and the blade boundary layers and wakes. It should be emphasized that figure 2b is speculative. It is based on the results f rom the present

study to be discussed later as well as on information from other studies both for mar- ine propellers and turboprops and for rele- vant geometries such as turbomachinery and simple tip and juncture flows. Here it is sufficient to point out that there are numer- ous fundamental issues associated with these flow structures which have yet to be expli- cated (see Kim [11] for a partial list). Although some of these issues are in common with other flows and applications, some are unique, and, therefore, must be addressed within the context of the marine-propeller problem. As mentioned earlier, the goal of the pre- sent work is to develop a viscous-flow method for marine propellers, which can analyze the detailed flow structures described above. It is appropriate to initiate such an effort with as simplified a geometry as possible without sacrificing the essential physics of the flow under consideration. The geometry chosen for this purpose is a propeller-shaft configuration with infinite-pitch rectangular blades (figure 1). This geometry has the following important advantages: the grid gen- eration is relatively simple so that the fo- cus of attention can be given to the more basic aspects of the numerics, which is im- portant for the initial development; fine- grid solutions are possible with the avail- able supercomputer resources; the laminar- and turbulent-flow solutions exhibit similar flow patterns such that meaningful compariso- ns can be made between the two flows; and its simplicity facilitates the diagnosis of the important features of the blade-to-blade flow. Also, as will be shown below, the flow field exhibits most of the distinctive fea- tures of interest. However, it should be recognized that this geometry also has short- comings, such as the lack of blade section geometry and, most importantly, thrust. Issues related to these aspects will be ad- dressed in future extensions for practical geometries. Overview of the Computational Method - Consider the viscous flow around a propel- ler-shaft configuration rotating at constant angular velocity ~ in an infinite uniform stream with velocity UO (figure 1). It is assumed that the Mach and cavitation numbers are, respectively, sufficiently small and large such that the fluid is incompressible and noncavitating. Under these conditions, the flow is cyclic in both space and time. Moreover, the flow is steady and spatially cyclic at blade-to-blade intervals in nonin- ertial coordinates, which rotate with the propeller. The situation is similar for propeller-driven axisymmetric bodies; how- ever, for the more general circumstance of propeller-driven three-dimensional bodies the flow is unsteady, even in noninertial coor- dinates. For straight-ahead performance it is cyclic with angular velocity a, whereas for maneuvering,it is noncyclic. 555 As mentioned earlier, the present overall computational method is based on that used previously for calculating propeller-hull interaction (Stern et al. [8,9]) in which a viscous-flow method for calculating ship- stern flow (Chen and Patel [10]) is coupled with a propeller-performance program in an interactive and iterative manner to predict the combined flow field. This is expected to facilitate future extensions for entire con- figurations. In order to extend this approach for the present purpose, a number of major modifica- tions were required, including the following: use of a noninertial coordinate system, which rotates with the propeller, and solution of the corresponding equations; implementation of boundary conditions, including periodic boundary conditions for the blade-to-blade region; adaptation of an ADI scheme at each crossplane; and a complete restructuring of the program for propeller geometries, includ- ing calculations for both laminar and turbu- lent flow. Also, during the time period that the present work was in progress, the basic viscous-flow method of Chen and Patel [10] was upgraded for fully-elliptic calculations of the complete Reynolds-averaged Navier- Stokes equations (Patel et al. [12]). Sim- ilar modifications were made for the present work. Lastly, modifications were required to execute the program efficiently on a super- computer. Below, an overview of the computational method is given. A complete description is p rovi deaf by Kim [ 1 1 ] . Als a, further detai Is of the basic viscous-f low method are provided by Chen and Patel [10] and Patel et al. [ 12 ] . Equations and Coordinate System The Reynolds-averaged Navier-Stokes equa- tions are written in the physical domain (figure 3a) using noninertial cylindrical coordinates (x,r, 6) rotating with constant angular velocity a= (~,0,0) as follows: aU+i a (rV)+{ aW~=0 Dt = ~ aX (P + Uu) + fx ~ ear (uv) _ 1 a (Uw) _ Uv + _ ~7 U Dt r 2~ - ~ r = ~ aa (uv) (2) ar (P + vv) + fr ~ r a ~ ( vw ) - ~ ( vv - ww) + ~ (V2V _ 22 aW~ _ V2 ) (3) r r Dt + r + 2~V = ~ aax (uw) ~ as (vw) ~ r as (P + WW) + f ~3 ~ r (vw)

+ie(V2W+22 aV_ r r with DDt = at + u aaX + v ear + W aid (4) and v2 = a2 + a2 + 1 a ~ a2 t is the time; U. V, W are, respectively, the longitudinal, radial, and circumferential components of mean velocity; p is the pres- sure; uu, uv, etc. are the Reynolds stress- es; f , f , f are, respectively, the longi- tudinal, radical, and circumferential com- ponents of the body force; and Re = U L/v is the Reynolds number defined in terms of a characteristic velocity U and length L, which are used along with the density p to nondimensionalize all variables, and molecu- lar kinematic viscosity v. For laminar flow, equations (1) through (4) reduce to the Navier-Stokes equations by simply deleting the Reynolds-stress terms and interpreting (U,V,W) and p as instantaneous values. Closure of the Reynolds equations is at- tained through the use of the standard k- £ turbulence model. Each Reynolds stress is related to the corresponding mean rate of strain by the isotropic eddy viscosity vim as follows: =vt(ar+ax) 1 au am - uw = It (r as + ax) 1 av aw w - vw = It (r as + ar r) - uu = At (2 aU) _ 3 k - vv = v (2 aV) 2 k - ww = At (r a9 + 2 r) ~ 3 k v is defined in terms of the turbulent kine- t~c energy k and its rate of dissipation £ by 2 v = C k t ~ £ where C is a model constant and k and £ are governed by the modeled transport equations Dk a 1 ak 1 a 1 ak Dt ax (R. aX) + r ar (R. r a k k + 2 as (R. an) + G - r k Dt = ax ( - aX) + r ear ( - r aa£) £ ~ (8) + (aU + aV)2 + (r am + aWx)2 + (l aV + aW _ W)2 The effective Reynolds number R is defined as 1 1 at _ = + . R Re ~ (10) in which ~ = k for the k-equation (7) and ~ = £ for the £-equation (8). The model constants are: C = .09, Cal = 1.44, C ~ = 1~92, MU ~ = ~ = ~ = 1, o~ = 1.3 The governing equations (1) through (10) are transformed into nonorthogonal curvilin- ear coordinates such that the computational domain (figure 3b) forms a simple rectangular parallelepiped with equal grid spacing. The transformation is a partial one since it involves the coordinates only and not the velocity components (U,V,W). The transforma- tion is accomplished through the use of the expression for the divergence and "chain- rule" definitions of the gradient and Lapla- cian operators, which relate the orthogonal curvilinear coordinates x~ = (x,r,8) to ithe nonorthogonal curvilinear coordinates ~ = (t,n, a). In this manner, the governing equations (1) through (10) can be rewritten in the following form of the continuity and convective-transport equations a ~ (blU + b2V + b3W) + a (blU + b2V + b3W) + a: (blU + b2V + b3W) = 0 all ~ + g22 ~ + 33 a: an a: ( 1 1 ) = 2A~ 2~ + 2B~ 2i + 2C~ 2~ + Rib an + So (12) where ~ = (U,V,W,k, £)- (6) Discretization and Velocity-Pressure Coupling 2 a ~ (R a3) + C£1 k G - C 2 k r is the turbulence generation term G = v {2 [(au)2 + (av)2 + (1 aW + V)2] in the form 556 The convective transport equations (12) are reduced to algebraic form through the use of a revised and simplified version of the finite-analytic method (Patel et al. [12]). In this method, equations (12) are linearized in each local rectangular numerical element 65 = An = 6` = 1 , by evaluating the coef- ficients and source functions at the interior node P and transformed again into a normal- ized form by a simple coordinate stretch- ing. An analytic solution is derived by decomposing the normalized equation into one- and two-dimensional partial-differential equations. The solution to the former is readily obtained. The solution to the latter is obtained by the method of separation of variables with specified boundary func- tions. As a result, a twelve-point finite- analytic formula for unsteady, three- dimensional elliptic emanations is obtained

up 1 + Cp[C + C + R] ~ nb~nb + Cp(Culu+ CD¢D+ ~ up - S)} (13) where the subscript nb denotes neighboring nodes (NE: northeast, NW:northwest, etc.). It is seen that ~ depends on all eight neigh- boring nodal values in the crossplane as well as the values at the upstream and downstream nodes MU and ~D' nanny the values at the pre- vious time step up . For large values of the cell Reynolds number, equation (13) re- duces to the partially-parabolic formulation used previously (Stern et al. [8,9]). Since equations (13) are implicit, both in space and time, at the current crossplane of calcu- lation, their assembly for all elements re- sults in a set of simultaneous algebraic equations. If the pressure field is known, these equations can be solved by the method of lines. However, since the pressure field is unknown, it must be determined such that the continuity equation is also satisfied. The coupling of the velocity and pressure fields is accomplished through the use of a two-step iterative procedure involving the continuity equation based on the SIMPLER algorithm. In the first step, the solution to the momentum equations for a guessed pres- sure field is corrected at each crossplane such that continuity is satisfied. However, in general, the corrected velocities are no longer a consistent solution to the momentum equations for the guessed p. Thus, the pres- sure field must also be corrected. In the second step, the pressure field is updated again through the use of the continuity equa- tion. This is done after a complete solution to the velocity field has been obtained for all crossplanes. Repeated global iterations are thus required in order to obtain a con- verged solution. The procedure is facili- tated through the use of a staggered grid. Both the pressure-correction and pressure equations are derived in a similar manner by substituting equation (13) for (U,V,W) into the the discretized form of the continuity equation (11) and representing the pressure- gradient terms by finite differences. _olution Domain and Boundary Conditions The physical and computational solution domains are shown in figure 3. It is seen that the solution domain is bounded by the inlet plane Si; the shaft surface Ss; the suction and pressure sides of the blade sur- face S. s and Sb , respectively; the exit plane ~e; the periodic symmetry planes S and Spp; the symmetry axis Ls; and the outer boundary SO. The boundary conditions on each of the aforementioned boundaries are as follows: on the inlet plane Si, the initial conditions for ~ are specified from simple flat-plate solutions, initial conditions for p and p' are not required; on the shaft Ss and blade surfaces Sbs and Sb , for laminar flow, the solution is carriedP out up to the actual surface where the no-slip condition is ap- plied, for turbulent flow, a two-point wall- function approach is used; on the exit plane Se, axial diffusion is negligible so that the exit conditions used are ~ ¢/~: = 0, a zero- gradient condition is used for p; on the periodic symmetry planes S and S , an explicit periodicity condition is imPpPosed, i.e., ¢(5,n,() = ¢(g,n,( + ~ ), P(5,N,() = p(5,n,: + ~ ), where ~ corresponds to the blade-to-bladePinterval; in the symmetry axis L , the conditions imposed are V = W = 0 5(U,k,c,p)/an = 0; on the outer boundary SO, the uniform-flow condition is applied, i.e., U = 1, W = arts, p = a(k, c)/an = Be o Grid Generation The computational grid is obtained using the technique of generating body-fitted coor- dinates through the solution of elliptic partial differential equations, i.e., the nonorthogonal coordinates ~ are related to the orthogonal coordinates x by the set of equations V2xi = h h i ( 2 ) where V2 = gi; a + fi a a: aft ail and fi = 1 a (Jai; a: In the present context, fit are called control functions since their specification controls the concentration of coordinate surfaces. For specified boundary conditions and control functions, equations (14) can be solved nu- merically to obtain the coordinates of each grid point in the physical domain. (14) i = 1,2,3 (15) Because of the simplicity of the present propeller geometry (figure 1), it is possible to specify the transverse and longitudinal sections of the computational domain as sur- faces of constant ~ and A, respectively, and moreover, the three-dimensional grid is ob- tained by simply rotating the two-dimensional grid f or the longi tudinal plane. Under these conditions, equations ( 14 ) reduce substanti- ally and can be readily solved once the con- trol functions are specif led. The control functions f1 are determined by the specified grid distributions of axial stations, radial distributions at the inlet and exit, and girthwise distributions at the inlet and on the outer boundary, respectively. These control funcltions 1 were derived under the con3ditions f = f ( i), f2 = f ( i, A), and f = f ( :) only, which are of suff icient gener- ality f or the present application. 557

Results In the following, first, the computational grid and conditions are described. Then, some example results for laminar flow are discussed to point out the essential features of the solutions. These are followed by a brief presentation of the results for turbu- lent flow to highlight the differences. This order and emphasis of discussion is selected since the former represent solutions to the exact governing equations, whereas the latter are dependent on the choice of turbulence model. Computational Grids and Conditions The geometry of the propeller-shaft con- figuration (figure 1; see Tables 1 and 2 of Kim [11]) was specified based on a config- uration for which calculations had been pre- viously performed, i.e., P4660 (Stern et al. [9] ). Partial views of the grid used in the calculations are shown in figures 4a,b for a longitudinal plane and a typical body tross- plane, respectively. The shaft and blade surface grid is shown in figure 1. Similar grids are used for both the laminar and tur- bulent calculations, but, in the latter case, the near-wall grid lines (y < 30) are deleted in order to implement the two-point wall-function approach. The inlet, exit, and outer boundaries are located at x = (.54,6) and r = .9, respec- tively; for laminar flow, the first grid points off the body and blade surfaces are located at .4 < y < 8 and 1 < x , y , or z < 14, respectively; for turbulent flow, the first grid points off the bo+dy and blade surfaces+ are l+cated at 30 < y < 230 and 40 < x , y , or z < 190, respectively; 62 axial grid points were used, with 18 over the up- stream portion of the shaft up to the blade leading edge, 11 over the blade, 14 over the remainder of the shaft from the blade trail- ing edge to the hub apex, and 19 over the wake; for laminar flow, 40 radial grid points were used with 22 over the blade span and 18 from the tip to the outer boundary; for tur- bulent flow, 36 radial grid points were used with 19 over the blade span and 17 from the tip to the outer boundary; 30 and 26 angular grid points were used for laminar and turbu- lent flow, respectively. In summary, the total number of grid points for the laminar and turbulent calculations are 74,400 and 58,032, respectively. The conditions for the calculations are as follows: characteristic (shaft) length L = 1; characteristic (uniform-stream) velocity U = 1; for la5minar flow, ReL = 2.02 x 10 and iec = 1 x 10 , where ReL and Rec are the shaft- (= U L/v) and chord-length (= U c/v) Reynolds numbers, respectively; for turbid ent flow, ReL = 6.08 x 10 and Rec = 3 x 10 ; the pro- 558 pelter angular velocity ~ = .3n (= 9 rpm) (the blade section angle of attack varies from 1.2 deg at the root to 4 deg at the tip); for laminar flow, on the inlet plane, 6/Rh = .111 (where ~ is the boundary layer thickness and Rh the hub radius) and there is no inviscid-flow overshoot; and for turbulent flow, on the inlet plane, 6/Rh = .489, U = .04, and the inviscid-flow overshoot is 1.01. The ~ values are based on simple flat- plate solutions and the selected Re. For laminar flow, the Re value was selected based on the fact that many investigators have performed two-dimensional flat-plate bound- ary-layer and wake calculations for this same value. For turbulent flow, a reasonable value of Re was selected for which fully tur- bulent flow over the shaft and blades is probable. The propeller angular velocity was taken to be sufficiently low such that no separation occurs over the blades. For the nonrotating condition, the calcu- lations were begun with a zero-pressure ini- tial condition for the pressure field. For the rotating condition, the complete nonro- tating solution was used as the initial con- dition. The values of the time at and pres- sure a underrelaxation factors and total number of global iterations used in obtaining the solutions are .02-.1, .03-.1, and 70-100, respectively. The calculations were per- formed on a CRAY X-MP/48 supercomputer. The central processor unit (CPU) and storage (words) that were required for each of the solutions are about 30min. and 1-1.7M words, respectively. Note that the computer codes were 23% vectorized and optimized to achieve a 65% reduction in CPU, and that the maximum normal system storage limit is 2M words. Laminar Flow The laminar-flow results for both the nonrotating and rotating conditions are shown in figures 5 through 13. Figures 5, 6, and 7 show the variation of some properties in the longitudinal direction, i.e., the shaft and blade surfaces and wake pressure, the wall- shear (magnitude and angle for inertial coor- dinates), and the wake centerline and maximum swirl velocities, respectively. Figures 8 through 11 show the detailed results for some representative axial stations in the form of velocity and pressure profiles (i.e., ~ vs. Y = r/Rp, where Rp is the propeller radius), axial-velocity contours, crossplane-velocity vectors, and axial-vorticity ~ contours, respectively. Lastly, figures 12 and 13 show close-up views of the tip vortex and the tip- vortex trajectory, respectively. Note that, in figure 8, the ordinate is Y such that the distance from the plate is larger near the tip than near the root. Also, the labeling of each of the curves corresponds to the angular grid lines shown in figure 4b. First, consideration is given to the results for the nonrotating condition. The

shaft and blade surfaces and wake pressure variations (figure 5) for the mid-blade plane indicate a minimal influence of the blades and are typical of trailing-edge flow in the presence of a thin boundary layer; however, at this relatively low Re (laminar flow), the adverse axial-pressure gradient associated with the closing of the body is sufficient to cause a small separation region in the vicin- ity of the hub apex, .96 < x < 1.01. Note the rapid rate of recovery of pressure in the radial direction. The pressure variations for the blade plane are similar, but clearly show the effects of the blade leading and trailing edges as well as a small displace- ment effect of the blade boundary layer. The wall-shear velocity magnitude U var- iations (figure 6a) are consistent with those just described for the pressure. For the blade plane, there is a large reduction of U in the juncture region due to the flow retardation and also in conjunction with the relatively large boundary-layer thickness there, and a downstream shift of the region of low U associated with the flow separ- ation asp compared to the mid-blade plane. The latter is consistent with the differences in separation patterns for the blade and mid- blade planes. On the blade, initially U is larger at the mid-span than at the tipT in response to the larger leading-edge pressure peak at mid-span (i.e., more favorable pres- sure gradient), then the trend reverses. The wall-shear velocity vector (figure 6c) is generally aligned with the axial direction except near the blade leading edge where the blade displacement effects are evident and in the separation region where the complex topo- logical nature of three-dimensional separ- ation is displayed. The wake centerline velocity Uc (figure 7a) displays the extent of the separation region and the recovery of the wake. The maximum swirl velocity Nmax is, of course, nearly zero for the nonrotating condition and not shown in figure 7b. The asymptotic forms (figure not shown) display the details of the recovery of the wake. Although the exit plane is 34 diameters downstream of the pro- peller plane (equivalently 5 shaft lengths downstream of the hub apex), the slope of the velocity defect of the shaft wake has not yet reached its asymptotic value. This is con- sistent with our previous turbulent-flow calculations. In contrast, the slope of the blade wake velocity defect is close to the asymptotic value. The exit plane is 103 chord lengths downstream from the blade trailing edge. Lastly, for the nonrotating condition, the detailed results are discussed. The discus- sion to follow is based on the complete results, which include the solution profiles at all the stations designated in figure 4a; however, for brevity of presentation, only the near blade wake station is shown in fig- 559 ure 8. At the near-inlet station, the solu- tion display the characteristics of the inlet conditions, i.e., an axisymmetric, thin, laminar boundary layer. At the leading-edge and mid-chord stations, the solution shows the initiation of the blade boundary layer, including leading-edge (stagnation-point) and displacement effects. Also, the juncture flow indicates a weak leading-edge horseshoe vortex. At the trailing-edge station, the trailing-edge effects of both the blade and the shaft are predominate, including a rever- sal of the juncture flow. At the near blade wake station and hub apex, the solution shows the initial development of the blade wake. Here again, the effects of the shaft trailing edge are quite large. Two corner vortices are apparent near the shaft axis which are an indication of the nature of the flow within the separation region. At the near, interme- diate and far shaft-blade wake stations the , , solution shows the recovery of shaft and blade wakes. The crossplane flow and pres- sure recover more rapidly than the axial velocity component. Next, consideration is given to the results for the rotating condition. Referr- ing to figure 5, in the vicinity of the hub apex and in the near wake there is a decrease in pressure due to the propeller-induced swirl. The lifting effects due to the angle of attack of the blade section are clearly evident. Note that the pressure peak is at the blade leading edge such that just up- stream of the leading edge very large adverse and favorable pressure gradients occur f or the pressure and suction sides of the blade, respectively, whereas just downstream of the leading edge the reverse holds t rue. The wall-shear velocity magnitude U (fig- ure 6b) shows slightly increased valuers over the spinning portion of the shaf t and greater uniformity between the blade and mid-blade planes in the separation region as compared to the nonrotating condition. For the pre- sent conditions, the rotation parameter R = ~ Rh/Uo is quite small, i.e., R = .02, which explains the only slight increase in U as compared to the previous calculations of ~ Stern et al. [9] ~ On the blades, U is smaller on the suction than on the pressure side, in conjunction with the relatively thicker boundary layer on the suction as compared to the pressure side. Consistent with the results for the nonrotating condi- tion, U is larger at the tip than at mid- span except near the leading edge. On the rotating section, the wall-shear velocity vector (figure 6d) shows large effects due to rotation, i.e., the flow is turned towards the direction of rotation. In the blade region, the passage vortex is evident, including its helical nature. In the sep- aration region, the f low is completely turned in the direction of rotation which results in the aforementioned greater uniformity in the separation patterns between the blade and

mid-blade planes. Over the blade, the wall- shear velocity vector is in the axial direc- tion, except near the tip, where the flow is outward, especially on the pressure side. Figure 7a shows that the recovery of the wake centerline velocity U is slower for the rotating than the nonrotating condition. This is due to the adverse axial-pressure gradient induced by the hub vortex. Also shown is the decay of the maximum swirl ve- locity W in the wake (figure 7b), which is associated with the intensity and decay rate of the hub vortex. Finally, the asymptotic forms (figure not shown) indicate that the shaft wake is unaffected, the blade-wake slope is increased, and the swirl decay is relatively faster than that of the axial- velocity defect. The detailed results vividly display the complexity of the flow for the rotating con- dition. Here again, the discussion to follow is based on the complete results, although only representative stations are displayed in figures 8 through 11. At the near-inlet station, the solution is similar to that for the nonrotating condition, except for the W velocity component which shows a linear in- crease due to the use of noninertial coor- dinates. At the leading edge, the solution shows the initiation of the blade boundary layer, in this case, with significant differ- ences between the suction and pressure sides of the blade due to the influences of the aforementioned abrupt changes in the pressure gradients. Interestingly, the boundary lay- ers on both sides of the blade are thicker for the rotating than the nonrotating condi- tion. The tip-vortex formation initiates with flow around the tip from the pressure to the suction side. The vertical flow is asym- metric such that the tangential velocity component is larger on the suction than the pressure side, whereas the situation is re- versed for the radial velocity component. The passage-vortex formation also initiates and dominates the juncture flow. At the mid- chord station and trailing edge, the effects of the pressure gradient changes are clearly displayed, i.e., on the suction and pressure sides, the flow is decelerated and acceler- ated, respectively. On the suction side, the boundary-layer thickness varies considerably across the span. The tip vortex has lifted off the suction-side surface such that the radial velocity component is positive on both sides. Braiding of the fluid from both the suction and pressure sides is apparent, but particle trajectories were not traced to display this phenomenon. The pressure is surprisingly uniform in view of the cross- plane flow, however, very low values are observed in the tip-vortex core. The passage vortex increases in size and its core moves towards the suction side. The axial-velocity and -vorticity contours are hook shaped near the tip due to the influences of the tip vor- tex. At the near blade wake station and hub 560 apex, the solution shows the development of the blade wakes, which indicate the charac- teristics of the complex mixing of the suc- tion and pressure side three-dimensional boundary layers, including significant ef- fects of the tip, passage, and hub vortices and the hub-induced pressure gradients. The minimum velocity in the wake migrates towards the suction side. There is a rapid recovery of the pressure-side wake such that the ve- locity-defect region is mainly behind the shaft and off the suction side of the blade. The blade wake becomes quite thick as it merges with the wake of the shaft and the tip vortex. There is a region of backward flow near the wake axis associated with the flow separation. The tip vortex reduces in intensity and the passage vortex merges into a large asymmetric hub vortex. Finally, at the near, intermediate, and far shaft-blade wake stations, the nature of the recovery of the wake is displayed. It is clear that the circumferential mixing is faster for the rotating than the nonrotating condition which is also the case for swirling jets. The close-up views of the tip vortex shown in figure 12 clearly display its initiation at the blade leading edge, subsequent migra- tion off the surface along the blade chord, and decay as it is convected and diffuses into the wake. Also, they reveal the mechan- ism of the tip-vortex formation. At the leading edge, nearly all of the fluid forming the tip vortex originates f ram the pressure side, whereas further downstream the suction side f luid is "pumped" into the tip vortex. This indicates a "braiding" process, which is of ten ref erred to as the tip-vortex roll- up. The tip-vortex trajectory is shown in figure 13. Turbulent Flow Some limited turbulent-f low results are shown in f igures 14 and 15. The turbulent- f low results are consistent with and very similar to those for laminar flow. In gen- eral, the differences are as expected based on physical reasoning, i.e., viscous effects are conf ined to narrower regions and the three-dimensionality of the f low is consider- ably reduced f or turbulent as compared to laminar flow. Also, quite apparent for tur- bulent f low is the reduced resolution near solid surf aces and the wake centerplane due to the present wall-function approach. The overall trends described above with regard to the shaf t and blade surf aces and wake pressure, wall-shear velocity, and wake centerline and maximum swirl velocities are quite similar; however, the pressure peak at the hub apex is considerably larger and there are some dif f erences in the wall-shear veloc- ity behavi or due to the absence of separ- ation. The detailed results are also quite s imi far. However, f or the nonrotating condi- tion the juncture effects are minimal and the

crossplane flow and pressure variations are reduced, whereas, for the rotating condition, the tip and passage vortices are larger and persist longer, the latter merges into a larger hub vortex, lower pressures are observed in the tip-vortex core, and the recovery of the wake is considerably faster. The turbulent kinetic-energy pro- files show two peaks, one near the wake cen- terline and one corresponding to the tips of the blades. Comparison With Results from a Lifting- Surface Propeller-Performance Program Unfortunately, no experimental information is available for the present geometry. Therefore, to aid in evaluating the present work, comparisons have been made with some relevant experimental and computational stud- ies, including the following topics: juncture flow which is related to the present flow in the blade-hub juncture region for the nonro- tating condition; tip flow which is related to the present flow in the tip region for the rotating condition; turbomachinery flow which is related to the present blade boundary- layer and wake development and blade-to-blade flow; and propeller flow which is, of course, the topic and goal of the present study. Although in most cases, the comparisons are only qualitative due to the large differences between the topic and present geometries, they support the present results in that the predicted flow structures are similar and consistent with the results from these stud- ies. The complete comparisons are lengthy and beyond the scope of the present paper (see Kim [11]). Herein, only the direct comparisons between the present turbulent- flow results and those from a lifting-surface propeller-performance program, i.e., PUF-2 (Kerwin and Lee [2]) will be presented. Special modifications of PUF-2 for the present idealized geometry were not deemed necessary, and, therefore, not done. A con- stant pitch ratio PlDp = 105 was used to represent the infinite pitch of the present geometry. All other geometry input data was given the same values as those for the pre- sent turbulent-flow calculations. Also, the open-water condition value was used for the advance coefficient J = 44.44, i.e., the effective wake due to the interaction between the propeller and the shaft boundary layer was neglected. For the wake-model param- eters, the standard values for the wake pitch and zero contraction were used. A value of .005 was used for the section-drag coeffici- ent which is based on the present calcula- tions. Figures 16a,b show a comparison of the chordwise and spanwise distributions of the blade loading in terms of the pressure jump (figure lea) and section-lift coefficient (figure 16b), respectively. A large differ- ence in the pressure jump is observed near 561 the leading edge. Differences are also seen in the section-lift coefficient. The viscous results show considerably larger values near the root and the tip, but smaller values for the mid-span region. The higher root loading for the viscous flow is, no doubt, a result of the increased effective angle of attack due to the oncoming shaft boundary layer. However, a part of the difference may be due to the lack of hub effects in PUF-2. The lower mid-span loading is consistent with the aforementioned differences in chordwise load- ing near the leading edge. The higher tip loading may be due to the reduced pressure on the suction side due to the tip vortex. Interestingly, in spite of these differences in the loading distributions, the total for- ces and moments show remarkably close agree- ment. Figures 16c,d show a comparison of the propeller-induced velocities just upstream and downstream of the propeller at the mid- span radius. For the viscous-flow solution, the propeller-induced velocity (u,v,w) is defined as the total velocity, (U,V,W) minus the freestream (UO,O,O) value. Results are shown using the blade angle coordinate ~ = at as the abscissa for the entire blade- to-blade region from the suction (0 = 0 deg) to the pressure side (D = 90 de"). The velocity components just upstream of the propeller (figure 16c) clearly show the effects of the leading-edge stagnation point. The u velocity components show sim- ilar trends, i.e., the point of the minimum velocity shifts to the pressure side which suggests that the stagnation point also shifts to the pressure side. The increased magnitude f or the viscous solution may be due to the prescribed overshoot for the oncoming shaf t boundary layer. The v velocity compon- ent is nearly zero f or both results. The w velocity components also show similar trends; however, the inviscid solution indicates a stronger local effect of the leading-edge stagnation point than the viscous solution such that the circumf erential-average is zero for the inviscid but not the viscous solu- tion, i.e., the viscous solution indicates small negative preswirl. The velocity components just downstream of the propeller (figure led) highlight the differences between the viscous and inviscid solutions. The inviscid u velocity component shows very small positive values f ram the suction to the pressure side, whereas the viscous u velocity component shows a large change f ram the suction to the pressure side, i. e., the vi scous blade wake appears as a sharp drop on both the suction and pressure s ides and the ef f ects of the retarded suc- tion- and accelerated pressure-side boundary layers are clearly evident. The v velocity components show similar trends, but with somewhat larger variations f or the viscous solution. The w velocity components also

show similar trends, but with larger swirl for the viscous solution in spite of the 2. smaller loading. Concludi ng Rema rks The present work was motivated by the limitations of the interactive approach for s imulating the complex blade-to-blade f low. This has certainly been accomplished by the present viscous-solution method, albeit for an idealized geometry. In fact, the present work provides, for the first time, a very detailed documentation of the viscous f low around a propeller for both laminar and tur- bulent f low. It is concluded that the pre- sent approach is capable of simulating marine-propeller f low f ields, including both the propeller loading and the complex blade- to-blade f low, and should be extended f or practical geometries. It is also concluded, based on the comparison of the laminar and turbulent results, that, although most aspects of the f low are governed by pressure- gradient ef f ects, improvements in turbulence- modeling procedures, especially near-wall treatment, are important to resolve certain flow features, including transition, separ- ation, and small-scale vertical structures such as leading-edge horseshoe and secondary vortices. Of course, much more work needs to be done to extend the method to realistic propeller and body geometries. Some of the issues that need to be addressed are as f allows. Optimum coordinates, including investigations of inertial and helical systems. Optimum grid- generation techniques f or complex, three- dimensional, propeller-driven bodies, includ- ing investigations of moving, adaptive, and multi-block grids. As already mentioned, improved turbulence-modeling procedures are essential and possibly a pacesetting issue. Also, further development of solution algor- ithms is a necessity in order to perform the required large-scale computations even on the most advanced available supercomputers. It should be recognized, that none of these issues are trivial, on the contrary, all require substantial effort so that it is expected that the present problem will remain a challenge f or many years to come. Acknowledgements This research was sponsored by the Of f ice of Naval Research, Accelerated Research Ini- tiative Program in Propulsor-Body Hydrody- namic Interactions, under Contract N00014-85- K-0347. The Graduate College of The Univer- sity of Iowa and the National Center for Supercomputing Applications Academic Affil- iates Program provided a large share of the computer funds. Ref erences 1. Kerwin, J.E., (1986), "Marine Propel- lers, " Ann. Rev. Fluid Mechanics, Vol. 18, pp. 367-403. 562 3. 4. Kerwin, J.E. and Lee, C.S., (1978), "Pre- diction of Steady and Unsteady Marine Propeller Performance by Numerical Lift- ing-Surface Theory," Trans. SNAME, Vol. 86, pp. 218-253. Hess, J.L. and Valarezo, W.O., (1985), "Calculation of Steady Flow about Propel- lers using Surface Panel Method, " J. Propulsion, Vol. 1, pp. 470-476. ITTC, (1984), "Report of the Propeller Committee," Proc. 17th Int. Towing Tank Conf ., pp. 139-194. 5. Morris, P.J., (1981) , "The Three-Dimen- sional Boundary Layer on a Rotating Heli- cal Blade," J. of Fluid Mech., Vol. 112, pp. 283-296. 6. Groves, N.C. and Chang, M., (1984), "A Differential Prediction Method for Three- Dimensional Laminar and Turbulent Bound- ary Layers of Rotating Propeller Blades, " Proc. 15th ONR Symp. on Naval Hydro., pp. 429-444. 7. deJ ong, F. J. ., Govi ndan, T. R., Levy, R. and Shamroth, S.J., (1988), "Validation of a Forward Marching Procedure to Com- pute the Tip Vortex Generation Process for Ship Propeller Blades, " Proc. 17th ONR Symp. on Naval Hydro., Hague, The Netherlands. 8. Stern, F., Kim, H.T., Patel, V.C. and Chen, H.C., (1988), "A Viscous-Flow Ap- proach to the Computation of Propeller- Hull Interaction," J. Ship Research, Vol. 32, No. 4, pp.246-262. 9. Stern, F., Kim, H.T., Patel, V.C. and Chen, H.C., (1988), "Computation of Vis- cous Flow Around Propeller-Shaft Config- urations, " J. Ship Research, Vol. 32, No. 4, pp. 263-284. 10. Chen, H.C. and Patel, V.C., (1985), "Cal- culation of Trailing-Edge, Stern and Wake Flows by a Time-Marching Solution of the Partially-Parabolic Equations, " Iowa Ins t i tute of Hydraul i c Resear~ch, The University of Iowa, IIHR Report No. 285. 11. Kim, H.T., (1989), Computation of Viscous Flow Around a Propeller-Shaf t Conf igur- ation with Infinite-Pitch Rectangular Blades, " Ph. D. Thesis, The University of lowa, Iowa City, IA. - 12. Patel, V.C., Chen, H.C. and Ju, S., (1988), "Ship Stern and Wake Flows: Solu- tions of the Fully-Elliptic Reynolds- Averaged Navier-Stokes Equations and Comparisons with Experiments, " Iowa Institute of Hydraulic Research, The University of Iowa, IIHR Report No. 323.

Figure 1. Propeller-shaft configuration with inf inite-pitch rectangular blades. (b) blade-to-blade f low Figure 2. Propeller-f low phenomena. L \ ~~mm~:lry Antis ,~ \ ~ SOD (a) physical domain - 1 (a) circumferential-average flow ~//J ~ so Exit Plans 1~ ~ _ ~ . ~ ~~~ Figure 3. Solution domain. 563 (b) computational domain

C,~ Q7 T 0.8,~ T ~ ~ r - 1.1 12 1 0.7619 o e?o2 Q9~2. 1.0S NEAR ,410- NEAR NEAR INL£T I CHORO eLAD£ SHAFT- Qe455 C18. ,',0 wAKE ,5 '0 9LAO£ LEADING rRAlLlNG Hue EDGE EDCE APEX X/L (a) longituc inal plane NONROTATINC NONROTArlNG 8LAOE PLANE ~tO-BLAOE PLANE ROOT ROOT ROT AT I N C P (PRESSURE ~ ROTAT I NC ROOT 5 [SUCTION) ~er~_ R~ AnF ROTATING 1'410- SPAN Figure 5. Shaf t and blade surfaces and wake pressure: laminar flow. . oa ~ N<>NRoTAT~NG ur . 06 ur . 04 .06 t t34 O2 Mld-Span ~\ Tip Med -Span_ Root (Shcft) ,~ ~1 Root ( Bl Cde) '. C. ' .S .7 .a .9 X/L (a) magnitude: nonrotating . 3a ROTO.TING . P {PRESSURE S {~[J~TInM, .02 1. Mid-SpOn~~ Roo~ t13lOde)~~ :Shat ~) . . 5 . 7 . ~ . 9 .. X/L (b) magnitude: rotating ~ 3.61 1.532 fAR INTER~EOlATE SHAfT- SHAFT-eLAOE 8LAO£ WAKE waKE Figure 4. Computat tonal ,,rid, ~ c (OEG) -45 t DEG ~ O. .04 .08 .12 Y/L (b) body crossplane ~ 1 0.8 o.ss o.s n.s~ Y'1 (d) vector: rotating Figure 6. Wa '1-shear velocity: laminar flow 564 1 0

Qel 0.6 . Uc, . 0.4 0.2 O- _Q? NONROTATI~==~ ~< ROTATING 1 =~ W max 0.28~ 0.24( 0.2011 0.16 t 0.12 0.08 0041 _.2 1 ._ 0.9 1.3 1.7 2.1 2.5 2.9 3.3 1.0 X/L (a) centerline Figure 7 . Wake velocit ies: laminar f low . w,o Y .n .6 .4 .2 _ 1. .e Y .6 .4 o. -.2 ~ ROTATING I I I I I I ~ I I I I I 1 1 1 1 1 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 X/L (b) maximum swirl . Is -.os w/o .< a w~n 1.4 1 . - .6 .~ a 1 ~ y .1 . 1 .2 .05 .15 W/O . q424 - '5 - 05 V J6 o C os p ~-16 · 1 .2 .5 .15 .25 S . 1 . 2 . . .05 .15 S ~ . ~ , . . . . . W ·05 .15 .2 -.25 - 15 - 05 C ·05 15 .25 ·5 D (b) rotating: suet ion s ide Figure S. Velocity and pressure profiles: laminar flow. 565

p . ~ 1. .e ~. X . q424 . oe . 07 . 06 .05 . 04 Y/L 5: - L -. O I ~ . 072 ~ . 0 1 . 03 . 05 . 07 -. 001 Z/L -. 01 . 1~, _ . 1 oa os Y/L 04 02 O. 02 - .08 .07 . 06 . 05 . 04 .03 . 02 . 0 1 O. . O 1 4oo, ,.200 . 077 ~- 100 X/L-. 8702 ,so .07B so 075 . ~~so \~ 100 -. 01 n' p X- R4~d y 1.2 1. _ ac . 077 X/L-. 8742 . 07~ .07S Y'L . D73 .025 - __ ~ \ .02 :' \ \ Y/; t.". `, ~ `, - ~\ \ \ .01 ~ ~ .005 X/L-. 8702 SCALE, UO-. 08 X/L-. 8702 SCALE 1 UO-. 025 .02 .06 .1 0. .01 .02 .03 Z /L Z/L (c) rotating: pressure side Figu re 8 . ( cont inued ) . 07 O9S X/L-. e742 0. .001 .002 .003 .004 -.03 .01 .03 Z/L Z'L Figure 9. Axial-velocity contours: laminar :,,~,: :,~.. 2 -. 02 -. 02 . 02 . oe . 1 -. 02 . 02 Z'L05 . I . 1 toe . os YlL . 04 . 02 O. -. 02 30 ~1F X/L1. nD4 .08 .07 . 0( . O . O Y/L . O' . O; . 0 1 O. - -. 01 ~e 32 //~ 09S 31 W. J l ~ ~ .05 .07 -.01 .01 .03 .OS .07 ZIL f low, rotas ing. `, .12 ~ X,L-I. Rsl .08 .06 Y'L . 04 . D2 1 o. _ n~ Figure 10. Crossplane-velocity vectors: laminar flow, rotating. X/L-. 97O2 oOO: so 50 ,00 073 - - .072 . _. .03 .05 Z/L n7` . 08 . 07 .06 . 05 .n4 Y/L . 03 . 02 . 0 1 O. n. ~>lo X/L- 1. .07 -. 001 0. .001 .002 .003 . nn4 -. 01 .01 Z/L -. 01 .01 . 03 Figure 11. Axial-vorticity contours: laminar flow, rotating. 566 X/L - ~ . S32 . .05 . 07

.085 ~ .085 . .083 _ _ .083 . ' . 031 ~ ~ . 081 . . 079 ; _ _ _ ~ . 079 , Y/L,07s ~ '~ ~ ~ ~ - ~ ~ ~ Y/~.075 1 ! i. ~ ,` - , i ~ i . 069 _ ~ ~ _ _ _ . 069 - ~ ~ ~~ _ , .067 _ X/L~. 8455 SCAL£: UO-. 016 .067 ~ X/L-. 942B - ~ ~~ ~uD~. 016 .065 1 - . 06S . ' ~ -. 02 ~. 01 0. .01 .02 -. 02 -. 01 0. .01 .02 Z/L Z/L .085 _ _ _ _ .08 . , ... ~. .083 ~ ~ ~ .078 . .081 _ _ ~~_ _ _ .076 , , .... _ . 079 ; _ ~ ~ . 074 . . ' 'C77g 5 ~ - i ~ ~ 06SI~ ' ~ ~~~~~ . 069 ~ r te. ~ ~ ~ ~ ~ . 064 ~ _ . 067 ~ X/L-. 8702 SCALE: UD-. 016 ~ .062 ~ X/L 1. _~_ UO~. 016 .065 .06 -. 02 -. 01 0. .01 .02 -. 02 -. 01 0. .01 .02 Z/L Z/L .085 _ _ _ _ _ .08 . .083 ~ ~ .078 . .081 _ _ ~~ . .076 , ,,,_ . . . 079 ; ~1_ . ~ , 074 . Y/L, O75~ ~ ~ ~ ~ ~ 05B , , ~ ~_~- _ ~ ~ _ _ . 069 ~ ~ ~ r ~ ~ ~t~ , , ~ . 064 ~ = = . 067 ~ _ X~'L-. 895 SCALE: UO~. 016 ~ .062 - X/L-;. 05 SCA.LE : UO-. 016 ~ ~ .065 .06 -. 02 -. 01 0. .01 .02 -. 02 -. 01 0. .01 .02 Z/L Z/L Figure 12. Close-up view of the tip vortex: laminar flow. 2.0 , , , , , 1.0 . -10 , · · · _~ Helical Trace of T.E. = -1 0~ ~ ~ -° 10~ ~ .-_~ _3 0t Turbulent(Large`) ~ ~ _ ~ Helicol T~cce of T.E. ~ ~ ,_ -40L + Laminar(Z=g) ~g 1 40~ . -5.0 50 0 1 2 3 4 5 0 ~ 2 3 4 5 ( X XLE)/ C LE ) Figure 13. Tip-vortex traj ectory. toe .077 toe .oa w-.: L~w'-~- Y~2~ :'"~ -.01 .01 .03 .0~ .07 . -.002 -.001 0. .001 .002 .003 -.01 .01 .03 .05 .07 -.01 .01 .03 .OS .07 Z/L 2'L Z'L Z'L Figure 14. Axial-velocity contours: turbulent flow, rotating . 567

1.4 I .4 1.4 1.2 X..8702 1.2 X-1. t.2 X-1.532 r ~ := _: _, ~ ~ '; ~ 0. 0. 0. 0 4 ~ 12 0 4 8 12 0 4 8 12 2 6 10 2 6 10 2 6 10 K ~ 1000 K ~ 1000 K ~ 1000 (a) nonrotating 1.4 1.4 ~ .4 1.2 X-.8702 1.2 X-1. 1.2 X-1.532 Y ~ ~ ~ ~ Y '0 0. 0. 0. 0 4 8 12 0 4 8 12 0 4 8 12 2 6 10 2 6 10 2 6 10 K ~ 1000 K ~ 1000 K ~ t000 (b) rotating: suction side ~ .4 ~ .4 1.4 1.2 X..8702 30 1.2 Xa1. 1.2 X-~.532 2 ~ ~ ~ ~[ Y ~ ~ ~ ~ 16 0. 0. 0. 0 4 8 12 0 4 8 12 0 4 8 12 2 6 10 2 6 10 2 6 10 K 41000 K 61000 K c 1000 (c) rotating: pressure side Figure 15. Turbulent kinetic energy profiles. 0.5, . , . . . . . . . . 0.06 r ~ 2 :~==~1- ~ ~ Viseous (Turbulent Flow) -0.06 . ~ . . . . . . . . 0 10 20 30 40 50 60 70 80 90 ~ (D.gr~o) (c) upstream propeller-induced velocity _ 0.4 n 0.2 0.1 Inviscid (PUF2) r/R. - 0.65 -. Viscous (Turbulent Flow) (laidSpan) 1't\ _ _~\ . ~__ 0.0 _ 0.2 0.4 0.6 0.8 1.0 X/C (a ) chordwise load ing 0.9 ~ Inviocid (PUF2) ~~ . Cr ~ 0.~70. C4 - 0.116, '` \ 0.8 - r/Rp 0.680 '` 0 7 ~ Vlacous (Turbulont Flow) ,' r ~ 0 162. Cux ~ 0 109. , / 0.6 r/RF ~ 0.674 ,' / . ," / 0.5 _ ~ 0.4 - . ~ ,, 03 ~, . ~ 0.00 0.02 0.04 0.06 0.08 a. 0 Cl (b) spanwise loading Figure 16. Comparison of turbulent-flow and lifting-surface propeller-performance program results. 568 on~ 0.02 ~1 ° _n n, 0.04 . , . X/R~ - 0.3606 r/R, - 0.648 0041- 0.02 ,.1~- o _nn' ° ::: r.' v.v~ ~ 0 10 20 30 40 SO 60 70 80 90 ~ (Dogreo) (d) downstream propeller-induced velocity

DISCUSSION by K. Mori Although an explicit description about a systematic accuracy analysis is requested by the paper committee, no descriptions are found in the paper. Because the accuracy analysis is primarily important for the computational fluid dynamics, it should have been mentioned, although the procedures are not definite yet. DISCUSSION by S. Kinnas I would like to congratulate the authors for their interesting paper. I have however two questions to raise. 1) Concerning the circulation distribution that they show in Fig.16(b) as predicted by the presented method: is it a convergent result with respect the chordwise and spanwise grid discretization on the propeller blade? 2) In the case of a realistic propeller, with blade thickness included, what would they expect to be a reasonable grid on the propeller in order to capture the detailed flow at the propeller leading edge and tip? Author's Reply We thank both the oral and written discussers of our paper for their pertinent remarks. With regard to Prof. Mori's comments, we apologize for not including an explicit statement of accuracy in the paper, and, at this time, offer the following. As stated in the paper, the present overall computational method is based on that used previously for calculating propeller-hull interaction [8,9] in which a viscous-flow method for calculating ship-stern flow [10,12] is coupled with a propeller-performance program in an interactive and iterative manner to predict the combined flow field. References [8,9] and [10,12] provide numerous applications for propeller-hull interaction and bare bodies, respectively, including validation studies through grid-dependency and convergence check as well as comparisons with experimental data and other analytic and numerical solutions. Some limited grid dependency and convergence check were also done for the present application to test the extensions and modifications for calculating marine-propeller flow fields. That is, some preliminary turbulent-flow calculations were performed using a coarse grid, i.e. 36x22x16 (16,672). The coarse-grid solutions converged more rapidly (i.e. in about 40 global iterations) than the fine-grid solutions. Qualitatively the coarse-grid solutions were very similar to the fine-grid solutions, but with considerably reduced resolution. Also, as stated in the paper, unfortunately, no experimental information is available for the present geometry; therefore, to aid in evaluating the present work, comparisons were made with some relevant experimental and computational studies, including the direct comparisons between the present turbulent-flow results and those from a lifting-surface propeller- performance program which are provided in Fig.16 (see [11] for the complete comparisons). With regard to Dr. Kinnas's comments, the solutions presented are fully converged for the present grid. As discussed in the Concluding Remarks, grid-generation for complex geometries is an important issue which must be considered in extending the present method to realistic propeller and body geometries. Presently, calculations are in progress for the SR-7 turboprop using a single-block, H-grid of somewhat higher density than the present one (i.e. 64x46x36=105,984),but with x1=x1( ~,n,: ) in order to have the grid conform to the three- dimensional curved boundaries of the skewed and twisted blades and the nacelle. The results are very encouraging: however, it is anticipated that in order to completely resolve all the details of the flow field, especially for marine propellers, multi-block grids will be necessary, including H-, C, and O-types. 569