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Twenty-Second Symposium on Naval Hydrodynamics Design System of Marine Propellers with New Blade Sections C.Kawakita, T.Hoshino (Mitsubishi Heavy Industries, Ltd., Japan) Abstract This paper shows the design system of marine propellers with new blade sections based on the lifting-line method and lifting-surface method, i.e. QCM (Quasi-Continuous Method). In order to improve the cavitation performance, the new blade sections with the prescribed three-dimensional pressure distribution over blade surface are designed by the numerical optimization technique, i.e. SUMT (Sequential Unconstrained Minimization Technique) method. The propellers with new blade sections for a pure car carrier and a container ship were designed by this system. The open-water test and cavitation test of the designed propellers were carried out and compared with the conventional propellers with NACA series blade sections. It was found that the designed propellers with new blade sections had higher open-water efficiencies and better cavitation performances than those of the conventional propellers. The present system is a useful tool for designing the high performance marine propellers. Nomenclature a0 Leading edge radius C(r) Chord length CP(Pi) D Propeller diameter eP Efficiency of propeller f(X) Objective function F(X,rk) Modified objective function gic(X) Design constraint J Advance coefficient=VA/(nD) K Number of propeller blades KT Thrust coefficient of propeller =T/ρn2D4 KQ Torque coefficient of propeller =Q/ρn2D5 m Unit outward vector M Number of radial control points n Propeller rotational speed, [rps] n Unit vector normal to blade camber surface Ν Number of chordwise control points Nc The number of design constraints p0 Static pressure at infinity pν Vapor pressure p(Ρi) Pressure on blade Q Propeller torque r Radial coordinate from propeller axis rb Radius of propeller boss ri Radial coordinates of control point rk Perturbed parameter in SUMT method rμ Radial coordinates of loading point R Propeller radius s Chordwise coordinate for blade section t Unit vector tangent to blade camber surface T Propeller Thrust x,y,z Cartesian coordinates ν Velocity VA Speed of advance ν Induced velocity vector VG Induced velocity vector due to vortices VS Induced velocity vector due to sources Vl Velocity vector of relative inflow Wi Weighted coefficients X Design variable ΔΡij Difference between objective and calculated pressure at control point ΔCij Divided chord length θ Angular coordinate from generator line of propeller ρ Fluid density Ω Angular velocity σnc ξ Chordwise station at blade section ξc(ξ,r) Camber distribution ξt(ξ,r) Thickness distribution
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Twenty-Second Symposium on Naval Hydrodynamics 1. Introduction Marine propellers with various blade geometry such as a highly skewed propeller are often fitted to ships in order to improve the cavitation performance, such as reducing the propeller-induced vibration and noise, or to improve the efficiency of the propeller. In the design of such propellers, the design charts based on methodical series tests are to be supplemented by the theoretical calculations of propeller open-water characteristics. The most familiar propeller design method, for given blade contour configuration and a selected blade section, is the use of lifting-surface theory [1,2]. The radial pitch and camber variations are determined by the lifting surface theory in order to match the given radial loading distribution and chordwise loading shape under the condition of circumferential averaged inflow velocity. In practical propeller design procedure which adopts the commonly used NACA66 thickness form and a=0.8 mean line , pitch and camber variation is determined such that the major loading comes from camber in a circumferential averaged inflow. Recently, numerous researchers have adopted the Eppler’s theory  to design new blade sections by prescribing pressure distributions over the blade surface in order to improve the cavitation performance. Since Eppler and Shen [5,6,7] introduced Eppler’s design method of a subsonic wing sections into hydrofoil design, Eppler’s method has been widely accepted by the ship researchers and designers as a very good nonlinear and multi-points design method to explore new sections of hydrofoils and marine propellers. It is theoretically and experimentally verified that the cavitation buckets of the new sections are much wider than any other kind of sections such as NACA series. The application of new blade sections onto propeller blades by Yamaguchi et. al. showed that the cavitation inception was delayed and fluctuating pressure and noise was reduced remarkably. In their first report , a propeller design procedure combining Eppler’s theory and the lifting surface theory were proposed and showed great effect of flat pressure distribution on reducing cavitation volume and fluctuating pressure. In their second report , the influences of designed lift coefficients of new blade sections on the reduction of fluctuating pressure and noise were investigated. The pressure distribution with higher pressure near the leading edge was more effective to increase the open-water efficiency and reduced the noise level. In their third report , the triangular pressure distribution with a negative peak at the leading edge was suggested to stabilize sheet cavitation and to suppress cloud cavity when its generation was possible. Lee et al.  has suggested a different concept of propeller design procedure by developing a new blade section for a key radius of the propeller and using it onto all radii to avoid the difficulty of a different new section for each radius, and the difficulty of surface fairing and smoothing. Dang et al.  have developed a new and different propeller design procedure using new blade sections, which incorporates Eppler’s design code, the steady and unsteady lifting surface prediction codes and concept of equivalent 2-D sections. Realization of the three-dimensional pressure distribution, even though it is known, is difficult as a matter of fact, since the Eppler’s theory treats only a two-dimensional foil section. In addition, blade surface fairing is needed after designing every section at each radius to form smooth blade geometry. This paper shows the design system of marine propellers with new blade sections. The Quasi-Continuous Method (QCM) [2,17], one of the lifting surface theories, is applied to the numerical calculation of the propeller design. The new blade sections with the prescribed three dimensional pressure distributions over blade surface are designed by the numerical optimization technique, i.e. SUMT (Sequential Unconstrained Minimization Technique) method . The new blade section is designed by specifying surface pressure distributions that minimize the cavitation phenomena. For example, surface pressure distributions are prescribed as minimizing the occurrence of local suction peaks at the section leading edge. It is possible to derive the blade sections, which have a superior cavitation performance to those based on the NACA series. Propeller efficiency would be increased by reducing the expanded area ratio while keeping the cavitation performance at the same level as that of the propeller with larger blade area by adapting the propeller with the new blade section which was designed by the present system. The new propellers for a pure car carrier and a container ship were design by the present system. The model test results of the new propellers, such as open-water characteristics, cavitation patterns and fluctuating pressures induced on the hull, are presented. 2. Design Technique of Marine Propellers with New Blade Sections The new propeller design procedure used in the
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Twenty-Second Symposium on Naval Hydrodynamics Nagasaki Experimental Tank, MHI. is shown in Fig. 1. Our new approach to propeller design incorporates five steps. Each step is repeated until the desired propeller performance is attained. We mention briefly each step as follows. Step1: Design of standard propeller The particulars of a propeller are determined from the data for resistance of the ship’s hull, self-propulsion factors, which indicate the interference between ship’s hull and the propeller, and model ship correlation factors, representing the scale effect of the hydrodynamic phenomena between model tests and full-scale operations. At this initial stage of the propeller design, propeller design charts obtained from systematic open-water tests of series propellers are used. Cavitation criteria, which gives the minimum expanded area from the viewpoint of cavitation performance of the propellers, and the beam theory to attain the strength of the propeller blades, are utilized. Standard propellers are designed by using Mitsubishi-series (M-series) propeller design charts . Even though Wageningen B-series  Fig. 1 Flow diagram of propeller design system and MAU-series  propeller charts are widely known and used, MHI has developed its own M-series propeller charts and have been making efforts to expand the covering range and to improve their accuracy consistently. Fig. 2 shows the range of M-series propellers in view of the blade area ratio and pitch ratio. By using this chart, the principal dimensions, such as optimum propeller diameter, blade-expanded area and the number of propeller blades are easily decided. Step2: Design of initial propeller with NACA blade section The detailed geometry design follows, using existing propeller lifting-line and lifting-surface theories, to achieve the higher propeller efficiency and better cavitation performance. The lifting-line calculation based on the Lerb’s lifting-line theory  is carried out for the circumferentially averaged ship wake distribution. The hydrodynamic pitch distribution, the induced velocities and the circulation distribution of an optimum wake adapted propeller or non-optimum wake adapted propeller are calculated. In the case of an optimum wake-adapted propeller, the problem is to determine the blade geometry for the distribution of the circulation to obtain the maximum value of the useful power for given quantities of power input, advance coefficient, and ship wake distribution. The lifting-surface corrections to the results of the lifting-line calculation are calculated by QCM . The blade shapes have the NACA a=0.8 camber lines and the modified NACA 66 thickness forms. This step is described in section 3 and 4. Fig. 2 Range of M-series propellers
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Twenty-Second Symposium on Naval Hydrodynamics Step3: Design of new blade section The NACA blade sections for the initial propeller are modified by the numerical optimization technique under the specifying surface pressure distributions for minimizing the cavitation phenomena. For example, the pressure distributions on the back-side of propeller are prescribed as flatter and higher at the section leading edge than those of NACA blade sections. It is possible to derive blade sections, which have a superior cavitation performance to those based on the NACA series. This step is described specifically in section 5. If the performance of the initial propeller is enough to the design purpose, this step is able to skip. Step4: Evaluation of blade strength The blade strength is checked in accordance with the rules of ship classification societies. Then the static stress distribution on the blade surface is analyzed by using Finite Element Method(FEM). The dynamic blade stress is examined by a system in which unsteady QCM(UQCM)  and FEM are combined. In addition, MHI has been accumulating the data of dynamic stress of the propeller blade in transient ship operation mode by model tests. Step5: Estimation of hydrodynamic performance Finally, the propeller open-water characteristics are calculated by the QCM and panel method [19,20]. The unsteady characteristics of the propeller in wake are estimated by the UQCM and unsteady panel method . In propeller design, not only evaluation of propeller open-water characteristics but also evaluation of cavitaion erosion and excitation forces due to cavitating propellers become most important items to be considered. Needless to say, since the cavitation phenomena on the propeller blades are not yet established. Thus, the cavitation performance is usually evaluated by the simple calculation method based on UQCM. In this method, the cavity range on the blade surface is estimated by using the empirical method of equivalent lift, and cavity thickness by open-type cavity models . A performance of the designed propeller is confirmed by the experiments in a towing tank and in a cavitation tunnel with the propeller model, which is precisely manufactured by a 5-axis numerically controlled milling machine. 3. Numerical Solution of Lifting-Surface Problems based on QCM A Quasi-Continuous Method (QCM) was developed by Lan  to improve the conventional Vortex Lattice Method (VLM) through the theoretical considerations so that the wing edge and Cauchy singularities may be properly accounted for. The QCM has both advantages of the Mode Function Method (MFM) and the VLM; the load distribution is assumed to be continuous in the chordwise direction and stepwise constant in the spanwise direction. Simplicity and flexibility of the VLM are also retained. Therefore, the complex blade geometries involving the high skew and extreme changes in pitch of a propeller can be taken into consideration correctly. The QCM was firstly applied to steady propeller problems  and extended to the unsteady propeller problems . 3.1. Coordinate Systems Let us consider now a propeller rotating clockwise with a constant angular velocity Ω in an inviscid and incompressible flow with a uniform velocity VA far upstream. The propeller consists of Κ blades of identical shape axisymmetrically attached to a boss. In representing the geometrical shape of the propeller, we define a Cartesian coordinate system O-xyz fixed on the propeller as shown in Fig. 3. The z-axis coincides with the generating line of the first blade. A cylindrical coordinate system is defined as follows. Angular coordinate θ is measured clockwise from the z-axis when viewed in the direction of positive x. Radial and angular coordinates are given by (1) Then, the Cartesian coordinate system O-xyz is transformed into the cylindrical coordinate system O-xr θ by the relation (2) Fig. 3 Coordinate systems of propeller
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Twenty-Second Symposium on Naval Hydrodynamics 3.2. Numerical Modeling of Propeller and Trailing Vortex Wake Propeller blades are represented by the distributions of vortices and sources on the mean camber surfaces of the blades together with the associated distribution of vortices shed into the wake. The vortex system, i.e. the distribution of horseshoe vortices which consist of bound vortices and free vortices shed from both edges of bound vortices, represents blade loading and wake, and the source distribution represents blade thickness. Continuous distributions of bound vortices and sources are replaced with quasi-continuous ones according to QCM. Thus, the blade surfaces are covered with a number of vortex and source strips. In selecting the radial loading stations, it should be noted that the better results could be obtained if the finer strips are used in the region of rapid variation of the sectional properties. The well-known semicircle method is suitable for this purpose. The radial interval from the boss r=rb to the tip r=R is divided into a suitable number of M strips. The loading stations rμ (radial coordinates of loading points) and the control stations ri (radial coordinates of control points) of the strips are defined as (3) where According to the QCM, the chordwise loading points and the control points at each radius are arranged as follows: (4) where sL(r)=chordwise coordinate of leading edge, c(r)=chord length of blade section, Ν=number of chordwise control points, The arrangement of the loading points and the control points selected for the present method is illustrated in Fig. 4. Fig. 4 Arrangement of loading point and control point The free vortices shed from the bound vortices on the mean camber surfaces are considered to leave the trailing edge of the blade and flow into the slipstream with the local velocity at that position. Usual approach is to approximate the trailing vortex sheet by a prescribed helical surface, in order to avoid the time consuming calculation of the slipstream velocities. In the present method, the roll-up of the slipstream is not accounted for. The trailing vortices are extensions of the chordwise vortices on the blade and leave the trailing edge in the direction tangent to the mean camber surface so as to satisfy the Kutta condition. After that the pitch of the trailing vortices changes linearly with respect to the angular coordinate and reaches an ultimate value. In the downstream, the trailing vortices become the helical lines with the ultimate constant pitch. Then, the trailing vortex sheet consists of M+1 concentrated trailing vortex lines whose trailing edge coordinate match the corresponding values of the vortex strips on the blades.
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Twenty-Second Symposium on Naval Hydrodynamics 3.3. Calculation of Induced Velocities The induced velocity at a control point Pij due to a line vortex segment of the unit strength (see in Fig. 5) can be expressed by the Biot-Savart’s law as (5) where r12=r2−r1. Here we consider a horseshoe vortex of unit strength, which consists of a bound vortex placed at Qμν and Qμ+1ν and two free vortices shed from Qμν and Qμ+1ν in the chordwise direction along loading stations as shown in Fig. 5. The induced velocity at a control point Pij due to this horseshoe vortex is calculated by (6) where the superscripts B, F and W express the induced velocities due to the bound and free vortices and the trailing vortices, respectively. Let the strength of bound vortex on the μ-th vortex strip be γμ(s). Then the induced velocity at Pij due to the closed vortices on the vortex strip and in the wake is expressed by using a characteristic of the QCM as (7) where The induced velocity at a control point Pij due to the whole vortex systems is given by the following equation (8) where the suffix k shows the contribution from the k-th blade. Next, let us consider the induced velocity due to the source distribution. The induced velocity at a control point Pij due to a source segment of unit strength is given in a manner similar to that of the vortex segments by (9) Denoting the strength of the source by σμν, the induced velocity at the control point Pij on the blade due to the whole source distribution is expressed as (10) The total fluid velocity at each control point Pij can be written as a sum of the contributions of the singularity distributions together with the total relative inflow: (11) where i=1, 2,…, M, j=1, 2,…, Ν, k=1, 2,…, Κ, Fig. 5 Vortex and source segments 4. Numerical Method for Initial Propeller Design 4.1. Determination of Vortex and Source Distributions The radial circulation distribution can be obtained from a lifting-line theory such as the Lerbs’ induction factor method  for the circumferentially averaged inflow. The strength of each vortex element is then obtained by distributing the total bound circulation over the chord. The NACA a=0.8 distribution is usually used for the chordwise load distribution. As for the chordwise thickness form, the NACA 16 thickness form or the DTNSRDC modified NACA 66 thickness form is frequently used for propellers. The strength of each source element can be determined by the product of chordwise derivative of the blade thickness and the relative inflow velocity.
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Twenty-Second Symposium on Naval Hydrodynamics 4.2. Calculation of Blade Geometry The components of the inflow velocity can be expressed in the Cartesian coordinate system as (12) where are the cylindrical components of the circumferentially averaged inflow velocity. Then the total fluid velocities at control points are obtained from Eq. (11). The normal and tangential components of the total velocity to the mean camber surface are then obtained as (13) where nij=unit normal vector at the control point to mean camber surface, tij=unit tangent vector at the control point to mean camber surface. The unit normal vector nij is obtained from the cross product of the two diagonal vectors connecting the opposite corners of each quadrilateral element composed of the four loading points surrounding a control point and its positive direction is taken from face to back side. The unit tangent vector tij is obtained from the average of the two chordwise vectors of each quadrilateral element and its positive direction is taken from the leading edge to the trailing edge. Following Greeley and Kerwin , the incremental slope of the mean camber surface are expressed as (14) where ty, tz=y- and z-components of unit tangent vector tij, respectively, my, mz=y- and z-components of unit outward vector mij, respectively. The unit outward vector mij can be obtained as (15) If the incremental slope δ*(r, s) is evaluated at all control points, the new mean camber line can be obtained by integrating the slope from the leading edge to the point considered as (16) The integrated value δ(r, s) is usually divided into the pitch angle correction Δβ(r) and the camber correction Δf(r, s), which are expressed as (17) where sT(r) is the chordwise coordinate of the trailing edge of the blade. If the pitch angle correction and the camber correction are obtained, a new blade surface is formed by correcting the original pitch angle β0(r) and the original camber line f0(r, s) as (18) Then, the new blade surface can be used as the surface on which the source and vortex are to be distributed in the next step. This process is repeated until the correction to the blade surface is sufficiently small. In the present paper, the iteration starts with a trial surface having a pitch distribution corresponding to the hydrodynamic pitch angle βi(r) from the lifting-line theory and a zero camber line. After three iterations, the corrections Δβ(r) and Δf(r, s) become almost zero because of high convergence of the present method. Therefore, the blade surface obtained after three iterations may be considered to be the final one. 5. Design of Blade Sections by Numerical Optimization 5.1. Optimization Problem and Nonlinear Programming The technique, which solves various optimization problems (minimizing or maximizing) by mathematical method, is called the optimization method or the mathematical programming. The optimization method is one of the most powerful methods in the optimization method. We mention briefly the important terms and definitions for optimization problems. Design Variable: The design variables are expressed the object of design, when these variables are given, the design is decided. In the propeller design, it is considered that the design variables are principal dimensions and approximation coefficients that represent the propeller shape. In this paper, a set of variables is written by vector form as X.
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Twenty-Second Symposium on Naval Hydrodynamics Design Constraint: The design constraints are considered the equality and inequality forms that are represented the physical or practical limitations. A function of the design variables are given as follows, gic(X)≧0 (ic=1, 2,…., Nc). Objective Function: The objective function represents the object of optimizing in the problem. In the propeller design, we consider the function that the propeller efficiency or the pressure distribution on the blade is evaluated. A function of the design variables is given as f(X). The definition of the optimization problem is obtaining the design variables X that minimize the objective functions f(X) under the design constraints gic(X)≧0 (ic=1, 2,…., Nc). A nonlinear programming usually solves the nonlinear programming problem that includes a nonlinear function in either the objective functions or the constraints. The nonlinear programming in this paper consists of three stages. Process of constraint We use the SUMT (Sequential Unconstrained Minimization Technique) method  that is one of the penalty function method. The SUMT method changes the constrained optimization problem for the unconstrained optimization problem formally by adding some functions that include the effect of design constraints to the objective function. Minimum value search method for function of several variables After processing of constraint, we use the Zangwill method  for a minimum value direct search method. The Zangwill method is able to search stable in order to have no use for the derivative. Line search method The search method is usually iterative method that consists of repetition by a line search method. This paper uses a quadratic interpolation method without using the derivative. The problem, which applies the nonlinear programming to the propeller design, accompanies with several complicated objective functions and constraints. It is considered that the direct search method is stable for optimization in order to have no use for the derivative and can get the good results for deciding the shape at any calculation time. 5.2. SUMT Method We describe the SUMT method that is used for optimization in this paper briefly. The SUMT method called the interior point method is one of the penalty function method that changes the constrained optimization problem for the unconstrained optimization problem formally by adding some functions (penalty term) that include the effect of design constraints to the objective function. The optimization problem usually uses the following transformation as, (19) This transformation is called the SUMT transformation. We called that rk is the perturbed parameter and F(X,rk) is the modified objective function. Applying the SUMT method, the optimization problem is written as follows, “Obtaining the design variables X that minimize the objective function f(X) without the design constraints, however, rk decreases gradually by k times minimization.” The nonlinear programming that minimized the modified objective function with SUMT transformation by the Zangwill direct search method is simply called the SUMT method in this paper. 5.3. Propeller Design by SUMT Method The new blade sections with the three-dimensional prescribed pressure distribution over blade surface are designed by using the SUMT method. The blade optimization starts from NACA blade sections of the initial propeller. The main purpose of the new blade sections is to improve the cavitation performance, such as fluctuating pressure or cavitation erosion, than NACA blade sections. The pressure distribution (objective pressure distribution) on the back-side of the propeller blade is especially important for the cavitation performance, given as the pressure coefficients . In order to keep the same sectional blade lift coefficients Cl(r), the pressure coefficients on the face-side of the blade are decided automatically as (20) where =pressure coefficients on the back-side of the initial blade sections, =pressure coefficients on the face-side of the initial blade sections.
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Twenty-Second Symposium on Naval Hydrodynamics In this operation, the thrust coefficient KT at design point for the optimized propeller can be kept the same level as the initial propeller. The pressure distributions are prescribed at the several representative radial positions (at the control point radius). The each blade section is modified to satisfy the pressure distribution given at the design condition by the SUMT method. The pressure coefficient distributions on the propeller blade, and are calculated by QCM in a circumferential averaged ship wake. The objective function f(X) is defined as, (21) where ΔCPij: ΔCij: divided chord length, Wi: weighted coefficients. The role of the weighted coefficients takes precedence of optimizing blade section near propeller tip at which occurred the cavitation easily. The pressure distribution on the new blade section is close to the objective pressure distribution by minimizing the objective function. The blade sections at each radial position r can be expressed by the superposition of the thickness and camber distributions in the chordwise direction. Expressing the thickness distribution as ξt(ξ,r) and the camber distribution as ξc(ξ,r), the back-side surface (ξΒ,ηB) and the face-side surface (ξF,ηF) of the blade section are written as, (22) where θc=tan−1(dηc/dξ) Since a coordinate system normalized by the each chord length C(r) of the propeller blade is used in this expression, a range of ξ should be given as, 0≤ξ≤1. The thickness and camber distributions ξt(ξ,r), ξc(ξ,r) are approximated by the polynomial expressions which are used six kinds of functions in the radial direction as follows, ξcmax(r)=maximum camber point in the chordwise direction, θc(0,r)=θc at the leading edge, θc(1,r)=θc at the trailing edge, ξtmax(r)=maximum thickness point in the chordwise direction, θc(1,r)=tan−1(dηt/dξ) at the trailing edge, a0(r)=leading edge radius. These functions in the radial direction are given by 4th or 5th order polynomial. In this paper the polynomial coefficients of these functions are defined as the design variables. The radial distributions of pitch, maximum camber, maximum thickness, skew and rake are kept same as the distributions of the initial propeller. In propeller blade design problem by means of the SUMT method, a set of suitable design constraints can be introduced to get the practical design and available results. In the present problems, the suitable constraints should be introduced in order to get reasonable blade sections. The constraints are given as follows, (23) The system of designing the blade sections by the SUMT method consists of 4 parts as follows. Part 1: Initial set up The NACA blade sections are approximated by the polynomial expressions. The initial propeller with NACA series blade sections is calculated by QCM at design point. Part 2: Set up the objective pressure distribution The objective pressure distributions on the blade sections at the several radial positions are given by the polynomial coefficients. The propeller designer decides the prescribed pressure distribution in order to improve the cavitation performance by comparing with NACA blade section. Part 3: Optimization for blade section by SUMT The polynomial coefficients which represent the
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Twenty-Second Symposium on Naval Hydrodynamics blade section, such as ξtmax(r) and ξcmax(r), set the design variables. They are optimized by the SUMT method as minimizing the objective function. Part 4: Check the optimized blade sections The optimized blade sections are checked whether enough or not, by comparing between the optimized pressure distribution and the objective pressure distribution. If the optimization process is not enough, the objective pressure distribution is changed a little. The optimization process is tried again from STEP3. In order to check the effect of this system, three kinds of typical pressure distributions, (a) triangular pressure distribution, (b) semi-flat pressure distribution, (c) flat pressure distribution with higher pressure at leading edge were given on the back-side of a propeller. The final results (optimum) of the pressure distributions and blades are shown in Fig. 6 respectively, comparing with the initial and objective pressure distributions, and the initial NACA66 a=0.8 blade section. The optimized pressure distributions are almost close to the objective pressure distributions. It is considered that this optimization system is enough for the practical use. Small errors are shown in each case. In this optimization system, the fairness of the blade section in the chordwise and radial direction is prior to the optimization by using some functions for expressing the propeller shape. The fairness is important from the hydrodynamic and blade-strength point of view. It is considered that giving a priority to the fairness causes these small errors. The optimized blade sections for triangular and semi-flat pressure distributions have thin thickness near the trailing edge as shown in Fig 6 (a) and (b). The early pressure recovery on the back-side of blade section tends to be thin thickness near the trailing edge by using the present system. 6. Numerical Examples In order to evaluate the applicability of the present system, the propellers with new blades sections were designed for a pure car carrier and a container ship. The model tests of the designed propellers, such as the propeller open-water test and the cavitation test, were carried out in the towing tank and in the cavitation tunnel at Nagasaki Experimental Tank, MHI. The cavitation test, cavitation observation and fluctuation pressure measurement, were carried out in the wake field simulated by a wire mesh screen. The fluctuating pressure was measured on a flat plate above a propeller. Fig. 6 Comparison of pressure distributions and blade sections at radial position 0.7R (a) Triangular pressure distribution (b) Semi-flat pressure distribution (c) Flat pressure with higher pressure at leading edge
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Twenty-Second Symposium on Naval Hydrodynamics 6.1. Propellers for a Pure Car Carrier In the first example, we designed new propellers for a pure car carrier (PCC) by the present design system. Main particulars of the PCC are shown in Table 1. Two kinds of propellers (PCC2, PCC2S) were designed for evaluating the new blade sections based on the initial propellers (PCC1, PCC1S). PCC1 and PCC2 are conventional propellers, PCC1S and PCC2S are highly skewed propellers. PCC1 with MAU blade section and PCC1S with NACA66 a=0.8 blade section were initial propellers. PCC2 and PCC2S have new blade sections designed for aiming at low-pressure fluctuations and high efficiency by the present system based on the initial propellers. The design point was selected as . The principal particulars of the designed propellers are shown in Table 2 together with the initial propellers. The full-scale axial wake distribution estimated from the model test result is shown in Fig 7. The pressure distributions at 0.69R of the designed propeller calculated by QCM in a circumferential averaged wake are shown in Fig. 8. The pressure distributions on the back-side of new blade sections were prescribed as the flat with high pressure at the leading edge. Table 1 Principal particulars of PCC Length between perpendiculars 170.0 m Breadth 32.2 m Draft 8.5 m Power of engine (BHP) 14500 PS Propeller revolution 110 rpm Table 2 Principal particulars of designed propeller for PCC PCC1 PCC2 PCC1S PCC2S Diameter 6.1 Pitch ratio 0.889 0.891 0.971 0.962 Expanded area ratio 0.554 0.449 Boss ratio 0.1656 Blade Section MAU NEW NACA NEW Rake angle 0.0° −3.0° Skew angle 13.8° 25.9° Number of blade 5 Fig. 7 Simulated wake distribution of PCC at the propeller plane Fig. 8 Comparison of calculated pressure distributions for designed propellers at radial position 0.69R
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Twenty-Second Symposium on Naval Hydrodynamics r/R=0.949 agree well, with a reasonable match of the double hump in the positive radial velocity values. Figure 13 Tangential wake Velocities The predicted tangential wake velocities (Figure 13) are not as close to the measured values as were the predicted axial wake velocities. This was also the case for the mean tangential velocities and is also reflected in the torque coefficient predictions. The predicted width of the wake is close to the measured values whilst the peak values show some differences. The tangential velocities on the suction side of the wake also show a local minimum that is not present in the model test measurements. This could be due to difficulties with local mesh resolution or turbulence modelling. At the tip vortex radius there are significant differences between the predicted and measured tangential wake velocities. This is most like due to the difference in the actual location of the tip vortex and the numerical position of the tip vortex, the mean tangential values (Figure 7) show significant difference for small changes in the radial location in this region. 4 Scale Effect on a 3 Blade Propeller Flow predictions have been carried out using the wake aligned mesh, shown in Figure 3. The predictions were carried out at one off-design advance coefficient since the propulsion predictions showed the largest scale effect to occur at the lower advance coefficients. Therefore the flow prediction comparison has only been carried out at an advance coefficient of J=0.5. 4.1 Propulsion Predictions The predicted propulsive performance curves for model and full scale are shown in Figure 14. The full scale predicted thrust and torque generally have higher values than those predictions made at model scale. At the design advance coefficient the difference in thrust of the full scale propeller was 2% higher than that for the model scale propeller. The torque coefficient for the full scale propeller was 0.3% higher than the predicted values for the model scale propeller. Figure 14 Propulsion Predictions for DTRC4119 Clearly the scale effect can be seen, from Figure 14, to be more significant at the off-design conditions, particularly at reduced advance coefficient, compared to the design advance coefficient. 4.2 Static Pressure Predictions The blade surface static pressure predictions for the suction side of the propeller blade for the two different scaled propellers are given in Figure 15. The colour contours are of −Cp and the higher the value of
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Twenty-Second Symposium on Naval Hydrodynamics Figure 15 DTRC4119 Suction Face Static Pressure Predictions Figure 16 DTRC4119 Transverse Section X/D=0.1 Static Pressure Predictions Figure 17 DTRC4119 Transverse Section X/D=0.1 Axial Velocity Predictions
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Twenty-Second Symposium on Naval Hydrodynamics −Cp the lower the static pressure. The principal difference in static pressure distributions on the suction side of the blade occur near the leading edge and tip regions of the blade. This region is associated with the leading edge and tip vortex. The remaining area of the suction side are very similar for the two propellers. This leads to the conclusion that the difference in propulsion is associated with the difference in pressure on the suction blade surface associated with vortex flow structure scale effects. A transverse section through the propeller at the axial location X/D=0.1 permit the downstream features of the tip region flow to be shown, Figure 16. At this location the two different scale propellers show significant difference in the static pressure in the tip vortex region. The full scale propeller has much larger values of −Cp and is associated with a much larger and more powerful tip vortex. 4.3 Velocity Predictions Axial velocity contours at X/D=0.1 for the two propellers are given in Figure 17. This Figure shows regions of high axial velocity and regions of negative velocity in the vicinity of the tip radius. This is the result of the strong rotational flow within the tip vortex, leading to negative axial velocity on the outer edge of the tip vortex and higher axial velocities on the inner edge of the vortex. These Figures also show the blade boundary layer developing and the blade wake. The distribution of radial velocity on transverse sections at X/D=0.1 are shown in Figure 18. This figure shows two regions of positive and negative radial velocity which are associated with the tip vortex rotational flow. The full scale propeller has the higher and lower radial velocities compared to the model scale propeller. As the flow developed downstream of the propeller, so the full scale propeller had the higher and lower radial velocities and this is associated with the full scale propeller having the stronger tip vortex flow. 5 Scale Effects on a 5 Blade Propeller The two model propellers used in this section are typical of the modern 5 blade propellers of the controllable pitch propellers (CPP) type fitted to twin shaft escort warships. These skewed propellers were chosen for the scale effect investigation since propellers of this type have had reported performance differences at full scale. Model propeller C659 was designed to have minimal spindle torque whilst model propeller C660 had increased chord length in the tip region and increased blade thickness. 5.1 Propeller Mesh The propeller mesh generation for these propellers did not follow the same procedure as for the 3 blade propeller. A single mesh with no wake alignment was used for two reasons. The first was that only propulsion predictions were used in the propeller design phase and, due to the blade skew, induced skewness within the mesh cell structure led to slow convergency rates. Figure 19 shows the mesh used in both the calculation for the propulsion and flow predictions. The physical boundary and mesh sizes were the same for both the 3 and 5 blade propellers Figure 19 Blade to Blade Mesh 5.2 Propulsion Predictions The predicted propulsive performance curves for model and full scale propellers are shown in Figure 20 for propeller C659 and Figure 21 for propeller C660. At the design advance coefficient, for propeller C659, the full scale predicted thrust was 1.4% higher compared to that of the prediction for the model scale propeller. The torque coefficient was 3.1% higher for the full scale propeller compared with the model scale
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Twenty-Second Symposium on Naval Hydrodynamics Figure 18 DTRC4119 Transverse Section X/D=0.1 Radial Velocity Predictions Figure 22 C659 Suction Face Static Pressure Predictions Figure 23 C660 Suction Face Static Pressure Predictions
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Twenty-Second Symposium on Naval Hydrodynamics propeller, with a consequent reduction in efficiency. Figure 20 Propulsion Prediction for C659 For the second skewed propeller C660, at the same design coefficient, the full scale propeller produced 1% less thrust than the model scale propeller. This is attributed to the flow separation that is predicted for this propeller. The torque predictions for the full scale propeller were 0.3% higher that those of the model scale propeller. Figure 21 Propulsion Prediction for C660 In the study, at off-design conditions, the effect of propeller scale for the 5 blade propeller is significantly less than that of the 3 blade propeller. It is hypothesised that the scale effect has its maximum effect for the vortex flow structure. In that case the three blade propeller by having a much stronger tip vortex, due to the number of blades and the radial blade loading, will exhibit a greater scale effect for the propulsion predictions. 5.3 Static Pressure Predictions Blade surface static pressure predictions have been made at one advance coefficient of J=0.8. It should be noted that at this off-design advance coefficient neither of the two skewed propellers showed any significant scale effect on the propulsion predictions, in contrast with the 3 blade propeller. The most notable feature of the predicted static pressure distributions on propeller C659 for both model and full scale is the region of high −Cp in the region of the leading edge and the hub (Figure 22). There are not significant scale effect on the suction blade surface static pressure distributions. For the second 5 blade propeller (C660) there are significant scale effect on the blade suction static pressure. The region of high and low static pressure near the trailing edge in the tip region are associated with a flow separation. The skewed propeller has a boundary layer separation near the trailing edge of the propeller. The location of the separation is best estimated from the hook shape in the static pressure contours near the trailing edge of Figure 23. As the flow separates so the boundary layer thickens as the axial velocity is reduced and the static pressure increases. At the same time the centrifugal effect within the boundary layer starts to dominate the flow and the resulting flow is predominately radially outwards. 6 Conclusions 6.1 3 Blade Propeller Validation The propulsive predictions produced by the RANS code compare well with the model test results. The degree of mesh alignment with the wake field had a significant effect on the predictions of the wake field. The upstream circumferentially averaged velocities show some differences between the predicted and measured values and this is considered to be associated with the modelling of the upstream fairing cone by a long constant radius cylinder.
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Twenty-Second Symposium on Naval Hydrodynamics The downstream circumferentially predicted velocities agree well with the measured ones and this is reflected in the closeness of the propulsive predictions. The blade-to-blade and wake predictions, generally, agree well with the measured data. 6.2 3 Blade Propeller Scale Effect The propulsive predictions for the full scale propeller show the scale effect when compared with the predictions for the model scale propeller. The scale effect is largest at off-design condition when the advance coefficient is less than the design value. The full three dimensional nature of the flow around the propeller blades has been investigated. The flow results show the complex nature of the flow around a propeller blade including the boundary layer, blade wake region and the tip vortex flow in the tip region. The scale effect is most noticeable in the region of the tip vortex flow for both static pressure and flow velocities. These scale effects account for the difference in propulsive performances between the model and full scale sized propellers. 6.3 5 Blade Propeller Scale Effect The propulsive predictions produced by the RANS showed that scale effect was present in the propulsion prediction. This was particularly true for propeller C660 with a reduced thrust at the design condition. The scale effects were most noticeable, for the skewed propeller C660, on the blade surface static pressure predictions. Also predicted was the formation of a boundary layer separation near the trailing edge of the full scale propeller that was not present at the model scale. These scale effects on blade surface static pressure predictions account for the difference in propulsive performances for the skewed propeller between the model and full scale. Acknowledgements The Author would like to thank the Staff of DERA Haslar for their help in compiling this paper, especially Dr. C.Jenkins. (C) British Crown Copyright 1998/DERA Published with the permission of the Controller of Her Britannic Majesty’s Stationery Office. References Denny S.B., 1968, “Cavitation and Open-Water Performance Tests of a Series of Propellers Designed by Lifting-Surface methods”, David W.Taylor Naval Ship Research and Development Center Report 2878, September. Jessup S.D., 1989, “An Experimental Investigation of Viscous Aspects of Propeller Blade Flow”, Doctor of Philosophy Dissertation, The Catholic University of America, Washington, D.C. Stanier M.J., 1992, “Design and Evaluation of New Propeller Blade Sections”, International STG Symposium on Propulsors and Cavitation, Hamburg, June. Stanier M.J., 1994, “Propeller Blade Sections with Improved Cavitation Performance”, Proceedings NAV’94 Conference, October 5–7, National Central Library, Rome. Notation CP C0.7 Propeller Chord at r/R=0.7 (m) D Propeller Diameter (m) Re
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Twenty-Second Symposium on Naval Hydrodynamics J KT KQ n Propeller Rotation Speed (rps) P Local Pressure (N/m2) P∞ Free Stream Pressure (N/m2) Pv Vapour Pressure (N/m2) Q Propeller Torque (Nm) R Propeller Radius (m) r Local Propeller Radius (m) r0.7 Propeller Radius at r/R=0.7 (m) T Propeller Thrust (N) V∞ Free Stream Velocity (m/s) Vx Axial Velocity Component (m/s) Vr Radial Velocity Component (m/s) Vt Tangential Velocity Component (m/s) Vrel Blade Section Relative Velocity Vrel=√[(V∞)2+(2πnr0.7)2] ρ Density of Water (Kg/m3) υ Kinematic Viscosity (m2/s) DISCUSSION S.Brizzolara Universita di Genova, Italy I appreciate very much the very well presented and design oriented work. I was impressed by the good prediction of the increasing of negative pressure in the tip vortex region of the DTRC 4119. This fact is experienced also in reality, and each cavitation tunnel has its own empirical formula to extrapolate tip vortex cavitation from model scale to full scale (MacCormick or other). Do you think it will be possible to define new correlation formulas based on CFD calculations in the near future? AUTHOR’S REPLY The current use of the ‘RANS’ code at DERA is to predict the full scale performance directly and it is used in conjunction with model test and model to full scale correlation procedures. The use of full scale predictions in the design stage hopefully will lead to better propeller designs. With regard to the formulation of correlation procedures, based on CFD calculation, more correction between full scale propeller performance and predictions is needed to assure the full scale accuracy of the method. Since full scale trials of this type are both difficult and costly, such correction may have to be undertaken based on model scale flow validation only and an act of faith for the full scale predictions. DISCUSSION S.Jessup Naval Surface Warfare Center, Carderock Division, USA The author’s demonstration of the application of RANS solutions to practical propeller problems is greatly appreciated. The time has come to use advanced viscous codes for real design problems. Also, the inclusion of validation with data is completely appropriate and today, required. When comparing tip vortex flows to propeller 4119 results, the adjustment of the vortex location, I believe, is proper. Overall wake contraction seems to be difficult to capture accurately enough to expect spatially accurate
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Twenty-Second Symposium on Naval Hydrodynamics near-tip results. Therefore, radial movement of the vortex solution is the only way to correlate the details of the result. I would comment that the differences between measured and predicted results are the largest anywhere shown, in Figure 13, the tangential velocity at 0.94R, even well away from the vortex, in the blade to blade passage. It should be noted that the tangential velocity is also related to the blade circulation, and propeller thrust. It is not clear that the discrepancies in circumferential average tangential velocity are consistent with the comparison of thrust coefficient. Further investi-gation of this component may point to why the tangential velocity, in general, does not match the experimental result. I would suggest to the author that full 3-d comparisons of the vortex structure may assist in diagnosing the discrepancies in this region of the flow. The investigation of scale propeller effects is certainly one of the most important applications of the RANS code. I would like the author to explain in more detail how the grid was modified to perform the high Reynolds number calculations. One would expect the blade boundary layer to become much thinner, requiring the migration of the mesh closer to the wall. The author, again, has demonstrated the use of RANS for design problems. We are also beginning to use the codes for design purposes, particularly in the calculation of tip vortex flows. A question we have is whether the near wake must be gridded to a sufficiently refined level to fully capture the downstream vortex structure. Our calculations have shown Cpmin to occur very close to or at the blade trailing edge. It would be most desirable to detail grid around the blade surface only, and not worry about extreme grid refinement in the downstream vortex, which requires iterative calculations and regridding. Can the author comment on this? The full scale solution for propeller C660 indicating flow separation near the trailing edge around 0.85R is curious. The model scale solution is what would be expected. Unfortunately, full scale surface flow visualization is not common or possibly has never been performed, and is therefore out of our experience base. I would recommend further inspection of the solution to understand its cause. A milestone in the use of CFD for design problems will be reached when the user has sufficient confidence in the calculations to make critical decisions based on those solutions, even when results may not pass all tests of intuition. Can the author reflect further on his experience using RANS in propeller design, and for instance, were your conclusions on scale effects on thrust used to impact changes to full scale propulsion hardware? Concluding, I would like to thank the author for his contribution. The use of RANS in design should be promoted fully. The practical problems of gridding, computation times, and geometry handling must be addressed along with education on what analysis parameters are important. In this way CFD can be integrated effectively into the unfortunately, time limited design process. AUTHOR’S REPLY The author would like to thank Dr. Jessup for his kind comments. With regard to the meshing strategies I would refer to the reply to Dr. Uto discussion. With regard to meshing the wake downstream of the propeller, previous mesh sensitivity studies have shown that the wake meshing strategy has little effect on the flow predictions through the propeller blade row provided that the propeller blade row is suitably meshed in detail. Therefore if the region of interest is close to the blade then there is no need to carry out wake meshing. It should be noted that the wake alignment strategy does not increase the cost of performing these calculations. The flow separation predicted for the full scale propeller C660 is not unique and a similar separation prediction can be found, Stanier 1998. Further work is being currently undertaken to investigate this effect further and would appear at the present to be a geometry related phenomenon, of pitch, rake, skew and blade thickness. In recent propeller designs, the RANS code calculations have been used to make propeller parameter selections, both at model and full scale and now form a standard part of the propeller design procedures at DERA, for advanced propeller design.
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Twenty-Second Symposium on Naval Hydrodynamics Reference Stanier M.J., 1998, “Investigation into propeller skew using a ‘RANS’ code—Part 2: Scale effects,” International Shipbuilding progress, Vol. 45, No. 443, September 1998. DISCUSSION J.English Consultant, United Kingdom Over the last few years I have been fortunate to follow the progress of Mr Stanier’s work, using, I believe, some of the most powerful computers in the UK’s MoD. The most vivid recollections of our conversations concerned the problems of suitable mesh generation and the large differences found between the propeller thrusts and torques from panel methods compared with the RANS solutions. My questions to him were mostly concerned with how long it would be until further significant stages were reached in the quest for answers to straightforward practical questions that could make our dependence on physical modelling less than it currently is. It is pleasing to see, therefore, that some aspects of the fundamental questions concerning viscous scale effects have been answered, albeit in the idealistic conditions of near uniform inflow and the absence of cavitation. On the other hand it is disconcerting that it has taken so long to reach this position considering the gulf that remains between what has been attained and what we need to know. Does the author consider that these advanced calculation techniques will eventually make physical testing unnecessary or are they likely to remain complementary tools in support of the latter? Some of the author’s results are most impressive, especially the pressures on the suction sides of the blades, the conditions in the wake close to the propeller and the comparison of the calculated and measured propulsion results, although there must be some concern about the apparent lack of resolution of the flow in the close vicinity of the tip vortices. How sensitive are the results in the downstream propeller wake to the mesh particulars and the size of the control volume and has the author checked on this by varying them? In this context do limitations arise from the capacity of the computer facilities at his disposal and perhaps the personnel working on the project with him? The author’s References suggests that he is working virtually alone internationally at this level of detail. Is this correct as far as he is aware? I notice the author refers to the “method of artificial compressibility” in calculating pressure fields. Is this because he is using a computer code developed for use in aerodynamics rather than liquids that are virtually incompressible? Does he eventually expect to include cavitation and the change of state from water to vapour in his work? A curious finding is that whereas the 3 bladed straight propeller behaves as expected re scale effect, the 5 bladed skewed propeller does not, as the predicted efficiency of the full scale propeller is slightly less than that of the model. The author refers to the radial flow strengthening from the centrifugal effect when flow separation occurs in the vicinity of the trailing edge. This flow feature can be removed of course by reshaping the blades there; however, of much more importance in my opinion is the beneficial flow separation that occurs at the leading edge, the presence of which is fundamental to the production of the cavitating leading-edge vortex and the prevention of rotational, cavitating break-off passing over the blades, with the attendant reduction in vibration excitation from this source. I suggest it would have been most instructive if the author had devoted time to studying the details of this flow, especially in the blade planeform view and in planes normal to the leading-edge. I would be very interested to know, for example, the magnitude of the axial velocity in the core of a propeller leading edge vortex and how it varies with sweep back and radial load distribution. Presumably it is this effect which sweeps partially vaporised rotational flow into the slipstreams of propellers with high leading-edge sweepback, through the vortices along the leading-edge and via the tip vortices. AUTHOR’S REPLY The author would like to thank Dr. English for his support over the years and for his contribution to this discussion. The author is of the opinion that physical testing will always be required, since numerical modelling by definition is a simplification of a very complex, real world. The use of numerical methods can give an insight into the flow that can only be achieved by complex and costly experiments. The author uses this particular method in the design of high performance propellers, to allow the checking of detailed flow in uniform inflow before committing the design to model manufacture and tests in ship wake fields.
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Twenty-Second Symposium on Naval Hydrodynamics The predictions for the downstream velocity are sensitive to mesh orientation to the flow as shown in Figure 5. The blade passage flow is less sensitive in this region since the mesh is body fitted. Extensive mesh sensitivity studies have shown that provided the number of cells within the blade passage is sufficiently large (approximately half the current mesh nodes), the thrust and torque results are insensitive to mesh. To increase the resolution of the detailed flow downstream of the propeller would require increased mesh nodes and hence reduce the cell volumes. The volume of the region of the tip vortex is a small fraction of the domain volume. Increased computing can lead to a simple increase in the number of nodes used, but more effective use can be made by the use of more complex mesh structure such as unstructured mesh solver schemes. It is regrettable that the author did not include a short review of this type of numerical method within this paper and would have indicated the small number of other groups active in this area. The continuity equation for incompressible flows does not allow the pressure term to be directly obtained. Numerical schemes are required to both satisfy the continuity and to allow the evaluation of the pressure term. The method of artificial compressibility is one way of obtaining this requirement for steady converged flows. There are a number of cavitation models that could be used with this type of numerical methods and have been applied to a number of nonpropeller geometries, including pumps. Since the code is steady state this would be akin to conducting uniform inflow cavitation tests. Since cavitation problems of interest usually involve unsteady propeller operating conditions, an unsteady (time accurate) solver would first be required. The flow around propellers is complex and for skewed propellers the flow appears even more complex. Dr. English’s description of leading edge flow are an important characteristic of skewed propellers and should rightly be investigated further. The results for the skewed propeller with the separation show that significant unexpected scale effect can change the performance of skewed propellers; the propeller designer must be aware of this to prevent poor full scale performance. DISCUSSION S. Uto Ship Research Institute, Japan I would like to thank Dr. Stanier for his successful work. First, I would like to confirm the simulation condition. Did the author use the same grid or not for the different Reynolds number flow? Does the turbulent flow develop from the leading edge of the propeller or not? My second question is how accurately the blade boundary layer flow and the tip vortex are simulated at the model Reynolds number, because that gives the basis for the scale effect study. Would the author compare the intensity of tip vortex and the blade boundary layer characteristics (displacement thickness distribution at 0.7R location, for example) with the experiment data? My third question is about the scale effect of the blade boundary layer characteristics. The 3-bladed propeller has so simple a geometry that the boundary layer flow around the mid-span location can be treated approximately as the 2-dimensional flow. Would the author compare the differences of the displacement thickness and the local skin friction coefficient between the two Reynolds numbers with those by the simple analytical solution (for the boundary layer flow over the flat plate, for example)? In modeling the scale effects of propulsion coefficients, changes of friction and pressure components are treated independently in the theoretical framework. It gives us useful information if the author would present the contribution of these two components in the scale effects of propulsion coefficients for three propellers. This is my last comment. Finally I congratulate the author again for his excellent achievement. AUTHOR’S REPLY The author would like to thank Dr. Uto for his contribution to the discussion. The grid generation is based on geometric conditions and does not take Reynolds number directly into account. There are small differences in the grids due to grid space weighting. For the full scale propellers the turbulent flow develops from the leading edge
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Twenty-Second Symposium on Naval Hydrodynamics cell, for the model scale transition is specified at a fixed percentage of the blade chord. The accuracy of boundary layer flow can in part be obtained from the comparison of predicted and measured wake flow variables. The comparisons suggested would be useful but at the present time these have not been undertaken. Although the 3-bladed propeller has a simple geometry the flow over the blade surface is far from simple and the author questions the validity of a comparison with 2-dimensional flow as suggested by Dr. Uto. The propulsion coefficients are made up from two components, one pressure, the other wall shear stress. The pressure term also includes the influence of viscosity due to the boundary layer thickness resulting in blade to blade blockage, one of the effects of the viscous boundary layer. For completeness a comparison with Euler and potential calculations may give a better indication of the effect of viscosity. The following tables give a breakdown of the two components and have been non-dimensionalized using the resultant component obtained from the RANS code. Pressure and Blade Surface Viscous Forces Propeller J δΚΤp/ΚT δΚTτ/ΚT δΚQp/KQ δΚQτ/KQ Size dtrc4119 0.84 1.015 −0.015 0.897 0.103 model dtrc4119 0.50 1.006 −0.006 0.956 0.044 model dtrc4119 1.13 1.148 −0.148 0.692 0.308 model dtrc4119 0.84 1.013 −0.0.13 0.906 0.094 full dtrc4119 0.5 1.004 −0.004 0.961 0.039 full dtrc4119 1.12 1.088 −0.088 0.722 0.278 full C660 — 1.012 −0.012 0.911 0.089 model C660 — 1.012 −0.012 0.910 0.090 full RANS and Euler Predictions Propeller J (KT)EULER/(KT)RANS (KQ)EULER/(KQ)RANS Size dtrc4119 0.84 1.066 0.925 model dtrc4119 0.50 1.029 0.967 model dtrc4119 1.13 1.644 0.761 model dtrc4119 0.84 1.056 0.935 full dtrc4119 0.5 1.001 0.948 full dtrc4119 1.12 1.259 0.792 full C660 — 1.170 1.032 model C660 — 1.145 1.016 full
Representative terms from entire chapter: