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A New Propeller Design Method for the POD Propulsion System Ching-Yeh Hsin1 Shean-Kwang Chou2 Wei-Chung Chen2 ([Department of System Engineering and Naval Architecture National Taiwan Ocean University Keeling Taiwan ~ ~ 7 ~ 2United Ship Design and Development Center Keelung' Taiwan) ABSTRACT This paper presents a new design method for propellers used on the POD propulsion system. Two major features of this method are first the propeller geometry is designed on a conical coordinate system instead of the traditionally used cylindrical coordinate system. Secondly, the inflow of the propeller is simulated by a coupled viscous/potential flow calculation including both the POD body and the propeller for the purpose of designing a propeller in a more realistic condition. In order to verify the designs, both the potential flow method and the coupled viscous/potential flow method are developed to calculate the flow around a POD propulsor. The numerical calculations for the flow around a POD propulsor are presented in the paper, and the computational results are compared to the experimental data. A design case is also included in the paper, and the designed geometries are shown to be different from the traditional method. INTRODUCTION Recently, the POD propulsion systems have drawn more attention than ever, and they are not only used on cruise ships, but also on semi-submersible heavy lift vessels, chemical tankers and other types of ships. The major advantage of a POD propulsion system is that the propeller inflow is more uniform than that of a conventional propulsion system, and this can often improve the propeller blade cavitations and powering performance. Since the hydrodynamic characteristics of the POD propulsion system and the traditional propulsion system are different, it is questionable that the traditional propeller design method is appropriate for the POD propulsion system. The object of this paper is to develop a new propeller design method that can design propellers used on a POD propulsion system. The researches in marine propeller designs are mainly in two topics, one is to find the blade geometry 1 that produces the desired loading, and the other one is to find a new sectional geometry that can reduce the cavitaion. Recently, researchers also begin to use various optimization methods to find out better loading distributions (Cho, 1998) (Jang, 2001~. In the presented method, we would like to emphasize on the calculation of the inflow, and a different definition of the blade geometry. The reason to study the inflow is to understand the performance of the propeller on a POD propulsor with yaw angles. This is similar to study the effective inflow problem of a conventional propulsor. A coupled viscous/potential flow method is thus used for the computations. The propeller of a POD propulsor is often installed on two ends of the nacelle, and it is equivalent to having a conical hub. The traditionally used cylindrical coordinate system may no longer be appropriate for this kind of configuration. Therefore, we will investigate the use of a conical coordinate system to define the blade geometry. In order to understand the performance of POD propulsors, computational methods are needed to calculate the flow field. Ghassemi (Ghassemi, 1999) has developed a potential flow method to calculate the flow around POD propulsors, and Sanchez-Caja (Sanchez-Caja, 1999) has applied a viscous flow code to the calculation of flow field around POD propulsors. In this paper, we will present two different computational methods for the computations of flow around POD propulsors. BLADE GEOMETRY DEFINITION Traditionally, the propeller geometry is defined on the cylindrical coordinate system. The propeller blade geometry can be considered as a group of two-dimensional sections at different propeller radii connected by a centerline, and this centerline is defined by the rake and skew distributions. When designing a propeller, an optimized radial loading distribution is first obtained, and the blade sectional geometry at each
conical section - ----------------- cylindrical section Figure 1: Blade sections defined on the cylindrical coordinate system and on the conical coordinate system 3.5 3.0 2.5 2.0 1 .5 1 .0 0.5 ~ ~ ~ _ - era At/ 0 ~ \ \ \ ~ \ At\ At\ Hi\ cylindrical section ~~\ conical sect ion gt 00 ~ 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r/R Figure 2: The calculated circulation distributions for blade sections defined on the cylindrical coordinate system and on the conical coordinate system with the same blade outline. radius producing the corresponding loading will then be found. For example, the widely used blade geometry design program MIT-PBD-10 (Greeley, 1982) is a lifting surface method for determining the pitch and camber distributions by giving a prescribed load distribution. In most design programs like MIT-PBD-10, the blade sections are aligned with constant-radius lines, and each section is assumed to be on the same stream surface. Therefore, the hydrodynamic characteristics of the two-dimensional section are assumed to be valid. Since the stream surfaces are assumed aligned with the cylindrical surfaces, the thickness and the camber of a section are defined on the cylindrical surface with the same radius. The pitch angle (angle of attack) is defined in the same way in the cylindrical coordinate system. However, for propellers used on POD propulsion systems, they are often installed at two ends of nacelles, and the geometry of that part is often like a cone. The inflow of the propeller thus aligns with the cone shape geometry. This is also true for propellers used on any axis-symmetric bodies such as submarines or torpedoes. That is, if we still define blade geometries on the cylindrical coordinate system, the assumption that each section on the same stream surface can no longer be true. It has been observed in the NTOU (National Taiwan Ocean University) cavitation tunnel that the root cavitation happened for a well-designed propeller used on an axis-symmetrical body. A possible reason for the root cavitation is that the propeller blade geometry is designed on the cylindrical coordinate system, and the manufacturer extrapolated the blade sections near the root to fit the conical body. This may suggest that a propeller design on the cylindrical coordinate system may not be a good design for conical bodies. Therefore, in order to remedy this, and to more accurately design blade geometry for propellers on conical bodies, the propeller blade geometry defined on the conical coordinate system is used in the presented design method. When the blade geometry is defined on the conical coordinate system, the blade sections are aligned with the conical surfaces, not the cylindrical surfaces. Therefore, blade sections can still be assumed to be on the same stream surface. The propeller geometry defined on a conical coordinate system has been discussed and used for several years, for example, the design method described by Kerwin et. al. (Kerwin, 19941. To demonstrate the difference between blade geometries defined on a cylindrical coordinate system and a conical coordinate system, we first calculate the circulation distribution of a propeller with a conical hub. With the same blade outline, Figure 1 shows the blade sections defined on two different coordinate systems. Figure 2 shows that the calculated circulation distributions are different when the blade sections defined on a cylindrical coordinate system and on a conical coordinate system with the same blade outline, and the difference is not negligible. Notice that when the blade geometry defined on the conical coordinate system, the centerline is vertical to the conical surface; therefore, there is a negative rake distribution for the above case (Figure 3~. Another convenience that the geometry defined on the conical coordinate system has is that the same blade geometry produces the same loading distribution even the cone angles are different. 2
zero rake center line ~ ~ i~ ~ ~~\ 3.5 3.0 2.5 l 2.0 1.5 1.0 ~ 0.5 Figure 3: In the conical coordinate system, the centerline is vertical to the conical surface, therefore, there is a negative rake distribution for a propeller with zero rake in the cylindrical coordinate system. cone angles. Figure 4 shows that the calculated circulation distributions for propellers with the same blade geometry but different conical hubs (different cone angles). The computational results show almost no difference between these propellers. EFFECTIVE INFLOW CALCULATION Currently, the inflow of a propeller is assumed to be the circumferential mean of ship wake velocities in propeller design procedures. The propeller-hull interaction is ignored, or considered with an empirical correction. In the presented method, the effective inflow is used as the propeller inflow, and it is calculated by a coupled viscous/potential flow scheme (Hsin, 2000~. This method follows the method proposed by Kerwin (Kerwin, 1994), and the "equivalent body force" concept is adopted. In this coupled viscous/potential flow calculation, the propeller flow field is solved by using either the propeller steady flow analysis program MIT-PSF-2, or the propeller blade design program MIT-PBD-10 depending on solving the analysis problem or the design problem. Both programs are lifting surface vortex lattice method developed at MIT. The viscous flow around a ship is solved by a viscous flow RANS code "UVW" developed at United Ship Design and Development Center, and a brief description of this code is shown in Appendix A. In the following descriptions, we will use "U" to represent the propeller flow solved by potential flow calculations, and "V" ship flow solved by viscous flow calculations. , v 'it. cone angle= 0 ~ ~ cone angle=5 At, -e cone angle=10 | 00 ,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I 0.4 0.5 0.6 0.7 0.8 o.s 1.0 OR Figure 4: Calculated circulation distributions for propellers with the same blade geometry but different Considering the propeller inflow, U. it can be U=Ue+Up (la) Up = Ui + Ui (lb) where Ue is the effective inflow, Upis the propeller induced velocity, and can be divided into Ui, the circumferential mean of the propeller induced velocity, and U., the fluctuating part of the propeller induced velocity. We then consider the ship flow, it also consists of two parts, which are the effective inflow and the propeller induced velocity: V = Ve + Up (2) Kerwin (Kerwin, 1994) has suggested that the coupling of viscous and potential flow calculations can be carried out consistently only under the assumption that both the ship flow and the propeller flow at the propeller plane are axis-symmetric. This assumption is reasonable since that the induced velocity upstream of a propeller is usually dominated by the circumferential mean value. Under this assumption, the propeller induced velocity component in ship flow is also assumed to be the mean value. The total propeller inflow in ship flow can then be written as: V = V = Ve + Up = Ve + Ui ~ ~ 3
where Ve is the circumferential mean value of the effective inflow in ship flow. Similarly, the total propeller inflow becomes: U =Ue +UP =Ue +Ui (4) It should be remembered that it is not possible to separate the effective inflow and the propeller induced velocity in the ship flow, and therefore a method to identify each one has to be developed. First, to couple the solutions from two different flow solvers, the flow field at the propeller plane has to be the same Therefore, V=U (5) From equation (3) and (4), the following equation can be then derived: (2) U =Ue +Ui =Ve +Ui V (5) and the effective inflow can be calculated as Ve = Ue = V Ui In equation (7), the effective inflow, Ve, is defined as the difference between the ship flow and the circumferential mean of the propeller induced velocity. It thus explains how the effective inflow can be calculated. In the solution of the ship flow, the body-force terms in the Navier-Stokes equation simulate the propeller effect, and the body forces are calculated by the potential flow. In propeller analysis program MIT-PSF-2, the Kutta-Joukowski's law and the Lagally's theorem are used to calculate concentrated forces on each vortex element. For example, forces on each element,f, resulted from the vortex lattice are calculated as f = puny (8) where u is the local velocity, and ~ is the vortex strength. However, according to the above description, the local velocity has to be taken as the circumferential mean. The formula can then be modified as: f=paxy (9) The total force on the propeller is the summation of forces on each element. These forces are then transferred into the body-force terms in the Navier-Stokes equation for the ship flow calculations. The procedure to solve the propeller-hull interaction is described as follows: (1) The viscous flow around a ship hull without the propeller in operation is first solved by the RANS code UVW; The circumferential mean value of the flow velocity at the propeller plane is then extracted from the solution of UVW, and then used as the inflow of propeller flow calculations; For solving the analysis problem, MIT-PSF-2 is used to calculate the circumferential mean induced velocities and propeller forces. These forces are then transferred to RANS code UVW; (4) The flow around the ship with body forces is calculated by UVW, and the flow at the propeller plane is extracted and taken circumferential mean again, and this is the total velocity V; The circumferential mean propeller induced velocity calculated in last iteration will be used as Ui in equation (7), and it is deducted from the total velocity calculated in step (4) to become the effective inflow. (6) Repeating steps (3) to (5) until the convergence of the effective inflow is reached. In the case of simulating a self-propulsion test, step (3) should be modified. MIT-PSF-2 is still used to calculate the circumferential mean of induced velocities and the propeller forces. However, the rotational speed of the propeller is adjusted to obtain the self-propulsion point. For solving the design problem, MIT-PBD-10 is used instead of MIT-PSF-2. The same procedure can be used to calculate the interaction between a POD body and a propeller. Recent computational results have shown that this coupled viscous/potential flow calculation has successfully simulated the hull-propeller interaction. Figure 5 and Figure 6 show the comparison of the computational results and the experimental data for a velocity cut at the propeller plane without and with the propeller in operation. This is the case that the operating condition of the propeller is known (analysis problem), and a good agreement can be seen from these figures. As described above, with some modifications, this coupled viscous/potential flow method can also used to simulate a self-propulsion test. Table 1 shows the comparison of the computational results and the experimental data of a self-propulsion test. The simulation of a self-propulsion test is relatively difficult, 4
and the computational errors of this case are satisfactory. Please refer to Chou et. al. (Chou, 2001) (Chou, 2002) for the detailed configurations of ships and propellers used in this computation. 1 }0.51 - ~ O ~0 - ~ ~ O ~ ~ O O\ / °q~°° ~;~ -05 1 1 1 1111 1 1111 1 1111 1 1111 1 1111 1 11 0.03-0.02-0.01 0 0.01 0.02 0.03 yL Figure 5: The computational results (lines) and the experimental data (symbols) of the velocities at the propeller plane of a bare hull. 1 0.5 t ~~ O ~ - w/U~USDDC-UVW ~ ·~1 ~ -0 5 0.03-0.02-0.01 0 0.01 0.02 0.03 y/L Figure 6: The computational results (lines) and the experimental data (symbols) of the velocities at the propeller plane when propeller in operation. Table 1: The measured and the calculated forces on the ship hull and on the propeller of a self-propulsion test J Measured 0.852 Calculated 0.863 Error 1.20% I KT 0.175 0.183 4.39% KQ 0.0261 0.0281 7.66% Figure 7: The computer depiction of a POD propulsor and its calculated pressure distribution. NUMERICAL COMPUTATIONS In this section, we will first demonstrate the flow around a POD propulsor calculated by a potential flow boundary element method, and then the same flow calculated by a coupled viscous/potential flow code. Results from both computational methods will be compared to the experimental data. Finally, a design case is shown. A POD with a pulling propeller is used for the numerical validation. The experimental data is obtained from an experiment conducted by Szantyr (Szantyr, 2002), and the computer depiction of this POD propulsor is shown in Figure 7. The length of this POD propulsor model is 16.22 inches (0.412m), and the maximum diameter of the nacelle is 2.91 inches (0.074m). The vertical length of the strut (from top to the centerline of the nacelle) is 5.51 inches (0.14m), and it has a constant chord length 4.29 inches (0.109m). The strut has an elliptical section, with a maximum thickness 1.85 inches. The geometric parameters of the propeller are listed as follows: Propeller type: Diameter: Hub diameter: Number of blades: Area ratio: Pitch ratio, P/D: Gawn-Burrill series 7.166 in. (0.182m) 2.52 in. (0.064m) 3 0.8 0.8 . CT The propeller used on this POD propulsor is a . 3 28*10-3 Gawn-Burrill series propeller, and we first validate the . 3.32*10-3 computational results of MIT-PSF-2. Figure 8 shows 1.32% the calculated KT and KQ comparing to the experimental data (Gawn, 1957~. s
0.35 o 0.30 7 0.25 0.20 0.15 0.10 0.05 0.50 _ 0~ 0.45 _ 0.40 A' _~` 1 -13 - - experiment KT - - 0- - experiment KQ*10 calculated KT calculated KQ*10 ., . . · ., -~.40 0.50 0.60 0.70 0.80 o.go 1.00 J Figure 8: Calculated KT and KQ of a Gawn-Burrill series propeller comparing to the experimental data. A boundary element method for analyzing the flow around a multi-component propulsor (such as ducted propellers, a stator/propeller combination, contra-rotating propellers, and a pumpjet, etc.) is developed for the potential flow computations. A brief description of this boundary element method can be seen in Appendix B. Figure 7 shows the pressure distribution on this POD propulsor at 15 degrees yaw angle and J=0.8. Figure 9 shows the computed and measured axial forces on the POD propulsor at different yaw angles, and the advanced coefficient, J. is 0.8. One can see that the measured forces are asymmetrical with respect to the yaw angles; however, the computational forces are almost symmetrical. The asymmetrical effect apparently results from the interaction between the propeller and the POD body. This asymmetrical effect is not seen in the potential flow computations due to the interaction between the propeller and the POD body is taken a circumferential mean value (Appendix B). Considering a rectangular strut behind a propeller, and the propeller drifts with the strut. If the vertical length of the strut is the same as the propeller diameter, and the chord length and the thickness form of this strut are constant. Then, the circumferential mean values of the induced potentials between the strut and the propeller in the potential flow computations are the same for the same drift angles but opposite directions. Although the strut and the propeller of the POD propulsor used for the computations are not exactly as described above, the geometries and relative position are almost the same as above. Therefore, it is reasonable to see that the axial force computed is almost symmetrical with respect to the yaw angles. Figure 10 to 12 show the comparisons between the computational forces and the measured forces at different yaw angles. The computational forces are almost linear with the advanced coefficients, and the measured axial force has a sudden jump at J=0.6. One can see that the difference between the computational results and the measured data is larger for J smaller than 0.6 due to the sudden jump. For all the J's, the difference between the computed results and the measured data is relatively large for-15-degree yawing angle. The negative yaw angle means that the direction of the yawing and the direction of the propeller rotation are opposite, and the measured data show that the interaction between the propeller and the POD body is larger at this case. We will then turned to the viscous flow calculation to see if we can reach the same conclusion as the experiment. We then use the viscous/potential flow code to calculate the same POD propulsor. Figure 13 shows the calculated streamline in the flow field for J=0.8 and 15 degrees of yaw angle. Figure 14 shows the effective axial inflow calculated at J=0.7 and J=0.8 for different yaw angles. From Figure 14, one can see that the axial velocities are almost the same for yaw angles 15 degrees and -15 degrees, and the axial inflow is larger for zero yaw angles. Figure 15 to Figure 17 show the velocity vectors at the propeller plane for different yaw angles at J=0.7, and it is clear that the flow filed is asymmetrical. Figure 18 shows the calculated forces comparing to the experimental data for J=0.8, and the potential flow calculations are also included. From Figure 18, we can see that the forces on the POD propulsor calculated by the viscous flow code are indeed asymmetrical with respect to the yaw angles. After studying the computational results, it is found that the over-prediction of frictional forces (negative axial force) is the reason why the forces are under-predicted for yaw angles other than zero degree. It is still under investigation to understand the cause of this over-prediction. Finally, the numerical computation we demonstrate is a propeller design case. As described earlier, in this design method, we first get the desired propeller loading distribution, and then obtained the effective inflow thru the coupled viscous/potential flow computation. A modified MIT-PBD-10 using the conical coordinate system to define the blade geometry is then used for the blade geometry design. Here, we will demonstrate the different designs in a cylindrical coordinate and in a conical coordinate. Figure 19 shows the pitch (P/D) distributions and the camber distributions designed by two different geometry definitions, and the difference is obvious. 6
0.25: ~ 20 ~ _ Q X Ox 0.15 - ~ 0.10 - x cr 0.05 _ Jut ~ 1 1 / _. . ~.00 -10.00 0.00 10.00 20.00 yaw Figure 9: The calculated and measured axial force on the POD propulsor at different yaw angles (J=0.8~. 0.30 0.25 Q 0.2C x x 0.1~ Cal 0.1 C x 0.0c O.OC YAW=0.0 O Rx(cal) _ _ - _ - ~ Rx(exP) I" ma; ` - ~ . . . . . . . . . . . . ).50 0.60 0.70 0.80 0.90 yaw Figure 10: The calculated and measured axial force on the POD propulsor at different J's (yaw angle=0.0~. 0.3C 0.25 Q 0.2C x 0.15 to O. 1 a - x t 0.05 0.00 0 RX(cal) _ _ - - - · Rx(exP) _ _ _ ~ YAW=15.0 -v.v~ '0 0.60 0.70 0.80 0.90 yaw Figure 11: The calculated and measured axial force on the POD propulsor at different J's (yaw angle=15.0~. 0.30 ~ _ RX(Cal) nest _ _ - - - · Rx(exP) . _ `2. 0.20 x 0.1 5 - ~ 0.10 - ~ 0.05 0.00 . ~_. - 0 Rx(cal) - - - - - Rx~exp) -0.08 so 0.60 0.70 0.80 O.gO yaw Figure 12: The calculated and measured axial force on the POD propulsor at different J's (yaw angle=-15.0~. \ Figure 13: The calculated streamline in the flow field of a POD propulsors at J=0.8 and 15 degrees of yaw angle. 1.2t 1 .1 1 .0 0.9 n R 1.2 ~1 .1 c) 1.0 0.9 0.8 t J=0.7, Yaw=O.O J=0.7, Yaw=15.0 0 J=0.7, Yaw=-15.0 J=0.8, Yaw=O.O ~ J=0.8, Yaw=15.0 0 J=0.8, Yaw=-15.0 ......... .............. ............... . 0.3 0.4 0.5 0.6 r°/lk 0.8 0-9 1.0 1.1 Figure 14: Calculated effective axial inflow at J=C and J=0.8 for different yaw angles. 7 1_7
Figure 15: The velocity vectors at the propeller plane for J=0.7, yaw angle=0.0. Figure 16: The velocity vectors at the propeller plane for J=0.7, yaw angle=15.0. 1 ~ w Ye._15 ,J - .7 __ - ~5J If~~arm,_ Figure 17: The velocity vectors at the propeller plane for J=0.7, yaw angle=-15.0. 0.25 In 8 .u>, 0.20 Q 0.15 x x 'A 0.10 - - ~ 0.05 - x _ JO -,,,,1,,,,1,,,,1,,,,1 .0~ .00 -10.00 0.00 10.00 20.00 yaw Rx(potential) _ _ - - - · Rx(exP) O Rx(viscous) JO Figure 18: calculated forces comparing to the experimental data for J=0.8. 1.40 ~ 1.35 .., 1.30 : 125 _ e..20 . 1.15 _ 1.10 _ ~ no : ............ cylindrical coord. - Mob_ conical coord. . . on ' ~.2 0.4 0.6 0.8 ·.0 r/R 0.030 n non (' 0.020 = 0.015 _ 0.01\ 2 ............ cylindrical coord. ~ conicalcoord. ~~ 0.4 0.6 0.8 1.0 r/R Figure 19: With the same given loading distribution, designed pitch and camber distributions are different when blade sections defined on different coordinate systems. CONCLUSIONS In this paper, the blade geometry definition and the propeller inflow are studied for a new, integrated propeller design method applied to a POD propulsor. The presented design method can also be used for propellers installed on any axis-symmetrical bodies, or for the design of propellers on a traditional propulsion system. The computational results show that a given blade geometry defined in different coordinate systems results in noticeably different loading distributions. 8
For a given loading distribution, different blade geometries are obtained when geometries defined in different coordinate systems. Therefore, it is concluded that the blade geometry defined in the conical coordinate system is more appropriate for analyzing and designing a propeller on POD propulsors. The flow around a POD propulsor is calculated by both the boundary element method and the coupled viscous/potential flow code. The results from the boundary element method show better agreements with the experimental data; however, they do not show the asymmetric effect when a POD propulsor in yaw angles. The results from viscous calculations show the asymmetric effect; however, the force predictions are less satisfactory. Results from two computational methods conclude that the viscous effect cannot be ignored when calculating the interaction of a POD body and a propeller especially when they are in a yaw angle. Therefore, when designing a propeller on a POD propulsor, the inflow calculation should include the viscous computation. For the further research, the boundary element method used in the POD propulsor calculation should include the unsteady effect, not just the circumferential mean values. The wake of a propeller should be aligned to the inflow when POD in a yaw angle. The over-predicted frictional forces should be carefully examined in the viscous flow calculations. REFERENCES Baldwin, B.S. and Lomax, H., "Thin-layer Approximation and Algebraic Model for Separated Turbulent Flows," AIAA paper, No. 78-257, 1978. Benek, J.A., Steger J.L., Dougherty, F.C., and P.G Buning P.G., "Chimera: A Grid Embedding Technique," AEDC-TR-85-64, 1986. Cho, J. and Lee, S.-C., "Propeller Blade Shape Optimization for Efficiency Improvement," Computer & Fluids, Vol. 27, No. 3, 1998, pp. 407-419. Chorin, A.J., "A Numerical Method for Solving Incompressible Viscous Flow," Journal of Computational Physics, Vol.2, 1967, pp.12-26. Chou, S.-K., Hsin, C.-Y., Chen, W.-C. and Chau, S.-W., "Simulating the Self-propulsion Test by a Coupled Viscous/Potential Flow Computation," Proceedings of PRADS 2001, Shanghai, China, Sep. 2001. Chou, S.-K., Chen, W.-C., Hsin, C.-Y., and Chau, S.-W., "Computations of Ship Flow Around Commercial Hull Forms with Free Surface or Propeller Effect." Journal of the Chinese Society of Naval Architecture and Marine Engineering, Vol.21, No.l, 2002. Gawn, R.W.L. and Burrill, L.C., "Effect of Cavitation on the Performance of a Series of 16 in. Model Propellers," Transactions INA, Vol. 99, 1957, pp. 690-728. Ghassemi, H., and Allievi, A., "A Computational Methods for the Analysis of Fluid Flow and Hydrodynamic Performance of Conventional and Podded Propulsion Systems," Oceanic Engineering International, Vol. 3, No. 1, 1999, pp. 101-115. Greeley, D.S. and J.E. Kerwin, "Numerical Methods for Propeller Design and Analysis in Steady Flow," SNAME Transactions, Vol.90, 1982, pp.415-453. Hsin, C.-Y., "Development and Analysis of Panel Methods for Propellers in Unsteady Flow," PhD thesis, Department of Ocean Engineering, M.I.T., 1990. Hsin, C.-Y., Tzeng, I.-W. and Chang, C.-Y., "Propeller Analysis and Design Using a Coupled Viscous/Potential Flow Method," Proceeding of the 4th International Conference on HydrodYnamics, Yokohama, Japan, Sep. 2000, pp.145-150. Jameson, W. S. and Turkel, E., "Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes," AIAA paper, No.81- 1259, 1981. Jang, T.S., Kinoshita, T. and Yamaguchi, H., "A New Functional Optimization Method Applied to the Pitch Distribution of a Marine Propeller," Marine Science and Technolocy, Vol. 6, No. 1, 2001, pp. 23-30. Kerwin, J.E., Keenan, D.P. and Black S.D.K., "A coupled viscous/potential flow design method for wake-adapted, multi-stage, ducted propulsors using generalized geometry," SNAME Transactions, Vol. 102, 1994, pp.23-56. Lee, J.-T., "A Potential Based Panel Method for the Analysis of Marine Propellers in Steady Flow", PhD thesis, Department of Ocean Engineering, M.I.T., 1987. Sanchez-Caja, A., Rautaheimo, P., and Siikonen, T., "Computation of the Incompressible Viscous Flow Around a Tractor Thruster Using a Sliding-Mesh Technique," Proceedings of the 7th International Conference on Numerical Ship Hydrodynamics June , , 19999 France. Szantyr, J.A., "Hydrodynamic Model Experiments with POD Propulsors," Oceanic Engineering International, Vol. 5, No. 2, 2002, pp. 95-103. APPENDIX A: THEORY AND NUMERICAL SCHEME OF THE RANS CODE "UVW" In this appendix, we will briefly describe the theory and numerical scheme of the viscous flow RANS code, UVW. The 3-D RANS equations with pseudo- compressibility (Chorin, 1967) are used by UVW, that iS, 9
aq +aF + Do + aH = 1 (&FV + OGv + aHV ) (10) At ax ay az Re ax By Oz where u v w . IF= Fv=L pU u2 +v UV UW o T" T T~ (u,v,w) denote the velocity components directions, respectively. fir is the dynamic pressure, Re is the Reynolds number and ,B is a positive constant related to artificial compressibility. On the hull surface, the zero normal gradient for the pressure is imposed and the velocity components are all set to zero. The propagation properties of characteristic variables in the corresponding Euler Equation ~ pv law vu , H = wu vv + ~ we vw ww +v O Tyx T' T' TV = aq+3F+a6+3H=0 (12) At ax ay an are used to construct the far field (inflow, outflow and outer) boundary conditions. The application of Finite Volume Method to the governing equations (10) in a curvilinear coordinate space leads to the transformed equation in computational domain. The convective terms are evaluated by the central difference approximation and a 4th order background dissipation term is added for stabilization of scheme and elimination of the non-physical oscillations (Jameson, 1981~. The multi-stage Runge-Kutta explicit scheme is adopted for the time-integration. Local time step and residual smoothing technique are also employed for the acceleration of convergence. The algebraic eddy-viscosity model of Baldwin-Lomax (Baldwin, 1978) is used to evaluate the turbulent viscosity. A composite grid approach is used in this code, and it mainly follows the method proposed by Benek et al. (Benek, 1986~. Two types of grids are employed in the composite grid system: foreground and background grids, and both grids are structured grids. The background grids are applied to cover the whole computational domain, possibly without considering local geometry for the sake of easy grid generation. The foreground grids are designed to take care of the local regions, where complex geometry or high gradient of physical variable exists. After the foreground and background grids are created in the computational domain, cells of background grids resided inside foreground grids will be removed from the solution procedure. The removed cells in background grids are known as hole regions. In order to have enough coupling between two overlapping grids, sufficient overlapping region must be assured. Besides, the cells (11 ) falling inside the body should be also excluded from the solution process. The hole regions define new inner boundaries (hole boundaries) of background grids. The field variables on hole boundaries are interpolated from the neighboring cells (stencils) in foreground grids. In in x,y,z the same way, the ones on outer boundaries of foreground grids are interpolated from the neighboring cells in background grids. Hence the flow solution is obtained by alternatively solving background and foreground grids until the numerical solution converged. APPENDIX B: A BOUNDARY ELEMENT METHOD FOR THE MULTI-COMPONENT PROPULSOR In this appendix, we will introduce a boundary element method for the analysis of flow around a multi- component propulsor. The detail of the theory and numerical algorithm of the propeller boundary element method can be referred to Lee (Lee, 1987) and Hsin (Hsin, 1990~. We will begin with the governing equation of a perturbation potential based boundary element method: 2~(p)=||[~(q)C,; R( )-~(q) R( )]dS (13) where ~ is the strength of perturbation potentials, or equivalent to the dipole strength, an is the source an strength, ha ~ is the potential induced by a unit strength dipole, and ~ is the potential induced by a unit strength source. The discretized form of the equation (13) is 10
N N Thai Alp = >,bi jaj i = 1,N (14) j=l j= where hi and of represent the discrete forms of ~ and an, and al j, bi j represents the discrete forms of an a ~ and ~ . an R R calculated. We now divide this body into two parts, and the panel number of these two parts are Nl and N2 separately. Equation (14) thus can be rewritten as N is the total number of panels N. N2 N. N. i,ai Alp + Thai Ale = >,bi j(;j + 2,bi jaj (14) j=l j=l j=l j=l We can solve equation (14) directly to get unknown potentials, hi, or solve it by an iterative way. That is, j=, N N. N. N. N. Thai Alp = ~,bi jaj + ~bi joj - Ma j=l j=l j=1 . ~ N. Nl N. ,ai Alp = 2,bi jaj + ~,bi jaj - Thai Ale =l j=l j=l j=l .j~j (15) In equations (15), two equations are solved separately, and an iterative procedure is needed to obtain a convergent solution. To validate the numerical method, a wing is used for the computation. We first solve equation (14) to calculate the loading distribution on this wing, then divide this wing into two half-wings, and solve equation (15) to obtain loading distributions on two half-wings. Figure 20 shows the circulation distribution obtained from these two solutions, and they are exactly the same. If we consider each component of a two-component propulsor as one part of equation (15), then the flow around this two-component propulsor can be calculated. Similarly, we can extend equation (15) to more than two components for multi-component propulsors. For a multi-component propulsor with both rotating and static components, such as a POD propulsor (the propeller is rotating, and the nacelle is not), a circumferential mean of the component-to-component induced potential is taken to simplify the solution procedure. Equation (15) thus becomes: 2.5 r 2.0 1.5 1.0 nst ~ --a I- ~ " - - - direct solution ~- iter 0. -~- iter 1. -e- ~ iter 2. _ -a- - iter 3. iter 4. o n I , I , I I , I I ~ I I I I I I , I ~ , , I , ~ I I , , I , , I I ~ I I I , I I I I I I I I , , I '~0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 OR Figure 20: The circulation distribution calculated by using equation (14) (direct solution) and equation (15) (iterative solution).. N. Nl N. _ N2 I,ai j0j = ~bi,j~j +~bjtjaj -Iai,jf j=1 j=1 j=1 j=1 N2 N. N _ N I,ai.jfj = ~bi,j~j +~b,,j~j -Zai,jf j=l j=1 j=1 j=1 (16) Equation (16) is the equation used to analyze the flow around a POD propulsor in this paper. 11
DISCUSSION V.I. Krasilnikov MARINTEK, The Netherlands I have a number of questions on this extremely interesting paper. My first question is addressed to the Figure 19 in the paper where two different design geometries are shown as obtained using the cylindrical and conical definitions. They are different. The question is: which geometry does meet the required integral performance better? Did you perform the analysis at both geometries? The conical geometry definition applied in the design method for podded propellers allows the improvement in terms of the blade root geometry. I wonder if the conical definition works equally good near the blade tip where the flow may be far from being conical. DISCUSSION Y.L. Young University of Texas at Austin, USA 1. Wake alignment should be applied, especially at high yaw angles. 2. Why is the comparison for yaw=0 degrees better than yaw=15 degrees considering that wake alignment was not applied? DISCUSSION G. Kuiper Maritime Research Institute, The Netherlands I am surprised about the large effect of the conical coordinate system. When we have an oblique flow the mean thrust does not change (while there is a radial inflow added periodically at the same axial velocity). But now the thrust distribution changes when we change from oblique to conical inflow, so when a radial velocity is added everywhere. Did you take the lift of the (conical) blade sections in the (conical) axial correction or in the cylindrical axial correction? What is the physical explanation for this difference? REPLY The authors would like to thank to discussers' valuable comments. First, we will answer Mr. Krasilnikov's questions. In Figure 19, both geometries satisfy the design requirements; however, the two geometries are different since they are defined in different coordinate systems. It's not clear if the propeller blade tip flow can be beneficial from the conic definition. The main purpose of the conic definition is to make the section aligned with the streamline; however, if this can reduce tip cavitation has to be investigated. Dr. Young asked about the wake alignment of the propeller of a pod propulsion system. In the computations of pod propulsors in yaw angles, we take the mean velocities as the inflow of the propeller; therefore, the wake alignment is not critical. For propellers in a yaw angle, if we take the inclined flow as the propeller inflow, then it is an unsteady flow problem, and the wake geometry has to be changed at each time step. We didn't make any conclusion from Figure 9, and we won't say that the comparison for yaw=15 degrees better than yaw=0 degrees. More comparisons between the computed results and the experimental data have to be made. Finally, we will try to answer Dr. Kuiper's question. Figure 2 shows the circulation distributions of two propellers in an axial inflow (parallel to the cylindrical coordinate system), and Figure R1 shows the mean sections of two propellers. Both propellers have a conic hub, and one propeller has sections aligned with the cylindrical coordinate system except the root section, and one propeller has sections aligned with the conical coordinate system. We have carefully recomputed the case, and proved that the computed circulation distributions have no mistakes. The thrust and torque coefficients of two propellers are actually not very different: KT KQ Cylindrical 0.3367 0.06157 Conical 0.3207 0.05930 Difference 4.75% 3.69% Notice that the circulation distributions for two propellers are at different "cuts", namely, cylindrical sections and conical sections. We believe that the different circulation distributions are due to the radial cross inflow to blade
sections, and for propellers with an inflow aligned with the conic hub (no radial cross inflow to blade sections), Figure 4 then shows that the circulations distributions are the same. · _ cylindrical ,~ Figure R1. The mean sections of two propellers computed.
