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1 Propeller Inilow Corrections for Improved Unsteady Force ant! Cavitation Calculations T. S. Mautner Naval Ocean Systems Center San Diego, USA Abstract An existing propeller design method was modified and used to calculate the spatial variation of propeller perfor- mance, velocity components and blade pressures for use in determining unsteady forces and cavitation. The calculations showed only small changes in the magnitude of the velocity components and blade pressures when compared to typical counterrotating propeller design results. Approximate agree- ment was found between the absolute magnitude of both the harmonic coefficients and the total unsteady forces obtained using the calculated axial velocity and the measured wake. However, the unsteady force distributions associated with the calculated axial velocity, which includes propeller effects, resulted in small reductions in the magnitude of the total unsteady forces on the propulsor. The calculated blade pressures and cavitation index also showed only small varia- tions in magnitude with circumferential position. Nomenclature an, bn Fourier coefficients AF Vehicle frontal area c* C CD CL Cp Cp CQ CT I - r D D Propeller diameter DB Vehicle diameter Fx, Fy Unsteady side forces F. (n) /`hBL Hop J k it(k) L LB m M n Nb p Complex Fourier coefficient = anibn Propeller blade chord Vehicle drag coefficient = Drag/~pV2AF Blade section lift coefficient Pressure coefficient Power coefficient = QQ/~hpV3'TR2 Torque coefficient = Q./ ~hpV2xR3 Thrust coefficient = T/~pV2'rR2 Unsteady thrust for the n-th harmonic Change in energy from freestream to local Change in pressure through propeller disk Index taking on values = 1, ,P Advance ratio = or V~/S2 R The reduced frequency = ~Cw/vs Sears' function Lift force on an airfoil or blade section Vehicle length Index taking on values = nNb Moment/Torque on a blade element Order of the propeller force harmonic Number of propeller blades Pressure p PC Q r Ar R RB SL t t/C T T. T. x, Y T. (n) t(k) u vO vi vt rev V Ve TV we wt x,y,z x y (x n p r Subscripts Number of blade elements having width Or Propulsive coefficient = (Thrust (1r) Vs)/(Torque a) Propeller torque Radial coordinate Width of the j-th blade element Propeller radius Vehicle radius Propeller stacking line location Time Blade thickness-to-chord distribution Propeller thrust Unsteady moments Unsteady torque for the n-th harmonic Horlock's function Measured inflow velocity Axial velocity with.propellers present Axial component of the interference velocity Axial inflow velocity with propellers not present Tangential component of interference velocity Change in axial velocity due to propellers Resultant velocity of blade section and fluid Free stream velocity or vehicle speed Overvelocity due to thickness and lift Axial component of self-induced velocity Tangential component of self-induced velocity Rectangular coordinates Radial position = (r-rh)/(R-rh) Non-dimensional chordwise coordinate Angle of attack of a blade element Blade section pitch angle (radians) Propeller efficiency = J CT/7r CQ Frequency Angular velocity of the propeller Potential function Mid-chord skew angle Fluid density Cavitation number Thrust deduction factor = ~ Drag/Thrust Angular coordinate in the direction of propeller rotation Bound circulation Forward propeller A After propeller 701
Introduction One current and important issue in the design of marine propellers is the reduction of propulsor generated noise due to both the transmission of unsteady propeller forces through shafting into the vehicle and propeller noise radiated into the near and far fields. The unsteady blade forces/pressures and cavitation calculations to be discussed in this paper result from the passage of a propeller blade through the spatially varying wake generated by upstream appendages. It is known that when a propeller blade passes behind an append- age, unsteady blade pressures and loadings occur and, with an appropriate velocity and pressure field surrounding the propeller blade, periodic propeller blade cavitation will occur. While there are a wide range of techniques available to calculate unsteady forces and blade pressures, these methods require accurate knowledge of the wake incident upon the propeller. One method of determining the incident flow field required in propeller design is to make wake measurements at the proposed propeller stacking line locations. Typically, these measurements are made without a propulsor present, and only simple corrections, if any, are made to determine an effective wake. For single propellers, one could reasonably assume that the appendage generated wake is incident upon the propeller without further distortion or modification by the propeller induced velocity field. However, for compound propulsors, the wake incident upon the aft blade row now passes through the forward propeller (or stator) and is modi- fied by propeller induced velocities and interactions. In recent years measurements have been made detailing propeller velocity fields. For example, Thompson's [24] fig. 6 shows the change in harmonic content of the axial velocity distribution for a body with and without a propeller. In gen- eral, the results show a reduction in the magnitude of the harmonic components when a propeller is operating. Another example is the work of Blaurock and Lammurs [1] which, for three values of thrust coefficient, illustrates the significant changes in axial, radial and tangential velocity components before and after an operating propeller. Addi- tional examples of propeller flow studies can be found in refs. 7, 10, 11, and 22. The above mentioned experimental results and uncer- tainties in the velocity field used in calculation of unsteady forces, blade pressures and cavitation performance provide the motivation to explore, analytically, the effect of a spa- tially varying wake on propeller forces and cavitation. The continued success of the propeller design method of Nelson [17-20] suggests that if the lifting-line portion of the design method can accurately predict propeller performance using circumferential mean data, the possibility exits of using the same calculation techniques, with some modification, to explore the effect of spatial variations in the wake. The selection of Nelson's lifting-line method was also based upon its availability and ease of modification. Certainly, there are many lifting-surface methods, panel methods and blade pres- sure calculation techniques [2, 5, 8, 9, 23] which might be used for this purpose. The discussion to follow will present a description of the propeller design method and the geometry, velocity fields, harmonic content, unsteady forces (blade-rate) and blade pressures associated with a counterrotating propeller set. Measured Velocity Field In the design of wake-adapted propellers, it is important that the inflow velocity distribution at the propeller stacking lines be properly specified. Even though circumferentially averaged velocity profiles are sufficient for propeller design calculations, the calculation of unsteady forces and pressures requires that both the radial and circumferential distributions of the wake be considered. The velocity data used in this study was obtained from wind tunnel tests [21] where boundary layer measurements were made on a 0.6 scale model. Pitot tubes, oriented approx- imately parallel to the afterbody surface, were used to obtain both the static pressure and the total head over a Reynolds number range of 1.3-2.4x106. To avoid strut interference effects and to utilize body symmetry, measurements were made over the top 90° of the body where the zero degree point coincides with the centerline of the fin trailing edge. The measured wake at the forward and after propeller stack- ing line locations is show in fig. 1. 1.0OT o ,, 9.60 a 0.48 ~ =~ e.20 ~ $~ 0.422 (a) , . . . . . . . 0.00 -50.0 -30.0 -10.0 10.0 30.9 50.0 CICUMFERENTIRL ANGLE (BEG) OF 1.00~ ~ o.ao o _., ,, 0.60 z ~ 0.40 o He 0.20 1 (b) 0.928 - r~ ace ~,~ \-/ 0.498 in- = . ~ I o.~-o5 .0 -30.0 -10.0 10.0 CICUMFERENrIAL ANGLE (BEG) eA . . 30.0 50.0 Fig. 1. Circumferential variation of the measured inflow velocity u/V,, at the a) forward and b) after measurement locations. 702
Propeller Geometry The propeller geometry used in this study is that for a counterrotating propeller set designed using the method developed by Nelson [17-20]. The design utilized the param- eters given in table 1, and the circumferential mean inflow velocity, u/V., and static pressure, Cp, formed from the measured data [21]. Thrust deduction calculations were made using the circumferentially averaged, three-dimensional potential flow velocity profiles calculated using a 3-D body coordinate generator [13] and a 3-D panel method [3, 4]. The details of the input velocity and pressure profiles, the circula- tion distribution and thickness distributions are given in refs. 15-16. Using the inputs described above, a counterrotating pro- peller design was performed. The calculated performance parameters are given in table 1, the radial distribution of chord-to-diameter, local pitch angle and resultant blade sec- tion velocity are listed in table 2 and the geometry is sketched in fig. 2. In addition to calculating the aft propeller circulation distribution required for tangential velocity can- cellation, the design method determines the minimum value of C/D required to meet cavitation (cr=0.75), blade stress (40 ksi maximum) and lift coefficient (CI,)MAX=0-5) require- ments. A distribution, which also satisfies the given hub and tip values of C/D, is fit about this value of C/D and has the shape shown in fig. 2. Vehicle Velocity - V', (knots) 40 Drag Coefficient - CD 0.1 165 Vehicle LB/DB 11.77 Propulsive Coefficient- PC 0.929 Propulsive Efficiency - ,7 1.08 Thrust Deduction - 1-r 0.860 Advance Ratio - J 2.12 Thrust Coefficient - CT 0.212 Torque Coefficient - CQ 0.133 Power Coefficient - Cp 0.196 Torque Ratio QA/QF 1.00 Thrust Ratio TA/TF 1.02 Blade Surface Cavitation - or 0.75 Maximum Stress (ksi) 40 Parameter Forward Propeller After Propeller Blade Number SL/DB R/RB rhub/RB RPM CTI-P/CMAX CHUB/CMAX (C/D)"Ax (t/C)HUB (t/C)T~p (CL)MAX VIIP/V8 6 11.46 0.781 0.3280 1400 0.4 0.6 0.255 0.18 0.09 0.346 1.733 4 11.67 0.705 0.1823 1400 0.4 0.6 0.380 0.16 0.08 0.435 1.606 Table 1. Summary of Counterrotating Propeller Design Inputs and Calculated Results Using Circumferential Mean Inflow Data RF R.] C/D)F 0.3393 0.1586 0.5567 0.7199 0.3619 0.1698 0.5924 0.7922 0.4072 0.1906 0.6411 0.9300 0.4525 0.2094 0.6797 1.0503 0.4978 0.2259 0.7102 1.1566 0.5431 0.2397 0.7101 1.2567 0.5884 0.2501 0.6894 1.3526 0.6337 0.2548 0.6622 1.4457 0.6790 0.2448 0.6190 1.5378 0.7243 0.2064 0.5657 1.6273 0 7696 0 1286 0.5217 1.7133 OF V/VS)F RA Rim C D)A --- 0.1953 0.2367 0.7779 0.2215 0.2540 0.7593 0.2738 0.2864 0.7562 0.3261 0.3154 0.7708 0.3783 0.3405 0.8100 0.4306 0.3611 0.8260 0.4829 0.3757 0.8074 0.5352 0.3783 0.7724 0.5875 0.3575 0.7117 0.6398 0.2981 0.6218 06920 0.1883 05503 PA V/V0A - 0.6411 0.6924 0.8236 0.9701 1.1155 1.2384 1.3372 1.4177 1.4818 1.5334 1.5919 Table 2. Propeller Geometry and Operating Characteristics Determined Using Circumferential Mean Inflow Data y STACKING LINE Tfly,, ~: ~ R r i DIRECTION OF ADVANCE Fy' Ty //// Art AXIS OF ROTATION / 0, S2 DIRECTION V'')/ ~ LIZ \ I I I ~ l I \'J ~W FX ~ X Fig. 2. Description of a typical propeller and its geometry. Unsteady Force Calculation Method The method used in this paper to calculate unsteady blade forces was developed by Thompson [24] and extended by Mautner [14]. The method divides the propeller blade into strips which are considered two-dimensional airfoils, and two-dimensional unsteady airfoil theory is used to con- sider sinusoidal velocity fluctuations normal and parallel to the inflow velocity. Corrections to the blade lift force due to the presence of adjacent propeller blades, the inclusion of camber and the calculation of the total force and moment on the propeller have been included. The expressions used to calculate the unsteady thrust and torque are 703
PF (n)=Nb ~ Lj(~,Nb~ e-inNb~j (1) and j=1 T(n)=Nb ~ Mj(,lNb) e b j (2) j=1 where the lift and moment are determined using and Lim = L p l jV V C [ ]i {~(klru)coski~ji(knU)sin~j }Arjcos§; (3) Mim = Lim tank rj (4) Examination of the unsteady force equations (1)-(4) and a Fourier analysis of the current four cycle wake where the velocity is represented by vi(r,8) = aO(r) + ~ (an(r)cos(n8) +bn(r) sin(n8)1 (5) V~ _ ~ ~ n=1 show that the harmonic numbers of interest, for a 6x4 pro- peller set, are nNb= 12, 24, 36, for the forward propeller and nNb= 4, 8, 12, 16, for the after propeller. Thus, only the unsteady thrust and torque require calculation. Blade Pressure and Cavitation Calculation Method The starting point in calculating blade pressures is the unsteady Bernoulli equation p = -p ~ - ~pV2 ~ c(t) (6) at applied in a constant total head stream annulus located in an inertial frame of reference. After evaluating the potential function, i, and determining the constant c(t), one obtains the following expression for the pressure coefficient along the blade~s suction surface where ~, p v2 = H~ ~ H2 (7) H~ =1 ~ V2 ~PV2 + ~ V, V,, V,, -2 V. [V,+V,~+ V, +~V,~ (8) ~hBL [ V ~ + C 1 (9) ~pV2 = 2 ~V, ~ [ V, ~ (10) 704 H2=2 ~ ~ ~ ] + ~ ]. 1 (D ~ (11) In the above expression for the blade pressure, ~hB~, represents the change in energy per unit volume from far upstream to where the boundary layer is measured, ~hp represents the change in pressure through the propeller disk and AV accounts for the increase in velocity along an airfoil section due to thickness and lift. In the calculation we varies along thc airfoil while the other induced velocities are con- sidered to be constant. Both w~ (y~ and AV(y~ are functions of the radial distribution of circulation and the chordwise loading. The overvelocity ~V due to thickness is determined from the experimental data for NACA 0010-64 airfoil adjusted for local blade section thickness. If the pressure p on the suction surface is defined as the vapor pressure, a cav- itation number, based upon free stream conditions, can be defined as ~=(poo-pva`>or)/~hpv2 It should be noted that, although not shown here, the pressure equation has been extended to include the axial, radial and tangential com- ponents of the inflow velocity f~eld. Velocity Field Components During the design of wake-adapted propellers, one must account for velocities arising from several sources, and these velocities are shown in the velocity diagram given in fig. 3. The resultant relative velocity (V) between the blade section and the fluid is determined from the tangential velocity due to propeller rotation (Slr), the axial inflow velocity with pro- pellers not present (v0, the axial and tangential components of the self-induced (we, w~) and interference (va, v~) veloci- ties and the change in the axial inflow velocity (Av) due to the presence of the propellers. Development of expressions for the velocity components are given by Nelson [17-20] and have been summarized by Mautner [16]. BrieQy, in determining the radial variation of ~v, one considers an axisymmetric Qow having a radial vari- ation of total head as it passes through a single propeller. As / ~Wa / ~:v! jVj Fig. 3. Relative flow velocity diagram at the lifting-line.
the boundary layer flow moves through the propeller, the induced velocity field of the propeller rearranges the vorti- city in the boundary layer and consequently alters the boun- dary layer profile. Using a potential flow model, an approxi- mate calculation is made to determine ~v. The interference velocities are calculated using an exten- sion of the work of Hough and Ordway [6]. Using classical vortex system representation, Hough and Ordway developed expressions, in terms of Fourier coefficients, for the induced velocities of a finite bladed propeller with arbitrary circula- tion distribution. The zeroth harmonic, or steady com- ponent, of their expressions have been extended by Nelson to the case of a moderately loaded, wake-adapted propeller with non-zero circulation at the hub. The self-induced velocities are calculated using an extended version of Lerbs' [ 12] induction factor method. Lerbs' original method was restricted to circulation distribu- tions which go to zero at the hub. For wake-adapted, coun- terrotating propellers this restriction is not desirable because circulation distributions having non-zero circulation at the hub are more efficient (more work done on slower moving fluid near the hub) and the after propeller may be used to remove the tangential velocity from the forward propeller so that the hub vortex can be avoided. Thus, Lerbs' method was extended by Nelson to include non-zero circulation at the hub. The blade pitch and the relative flow angle are related by r tank = V~/Q and this expression has been extented to the moderately loaded, wake-adapted propeller case by replacing V~/Q with its equivalent r tank. The result is ~ t -I v + va + wa (12) From this expression it can be seen that the calculated velo- cities and propeller geometry are not independent but must be determined in an iterative fashion to account for the effects of both the forward and after propellers. Propeller Calculations As stated before, one problem inherent in the calculation of unsteady forces involves the use of wake data obtained without a propeller present. While the propeller design method of Nelson calculates changes in the circumferential mean inflow velocity field due to the presence of a propulsor the uncorrected, spatially varying inflow has been used to determine the unsteady forces acting on the propeller blades. It is known that the presence of a propulsor will cause changes in streamlines due to acceleration of the flow that there may be additional unsteadiness due to the relative motion of the blade rows and that the propeller will change the amplitude and phase of the incident flow distortions. From these few facts it is apparent that the measured wake should be "corrected~to reflect propulsor induced velocity field and then be used in unsteady force, pressure and cavi- tation calculations. Calculation Procedure Before velocity or unsteady force calculations could begin, complete specification of the input velocity and static pressure profiles was required. Due to the fact that the design method requires the velocity and static pressure at the hub surface (numerical requirements specify a slip condition at the wall), the measured radial distributions of u/V,, and Cp were extrapolated to the wall for -45°<~<+45°. Since the 3-D potential flow distributions, for the body with append- ages, show only minor deviations from the circumferential mean, (max =0.3% of V8), the circumferential mean profiles will be used for all cases. The measured wake data was specified in Look up. table format. In an attempt to account for, at least first order, propeller effects in the calculation of unsteady forces and pressures, the lifting-line portion of Nelson's design method was modi- fied to provide the calculation of the circumferential varia- tion in velocity components required to form the resultant velocity V/V (fig. 3) at each blade section. The technique used fixed the velocity profile, u/V',, at a particular forward propeller angle, OF, and then calculations were performed as the after propeller angle PA was varied from45° to +45°. In the region of large inflow velocity changes, -10° < ~ < +10°, single degree increments were used, and outside this region calculations were made at increments of 4-6 degrees. Velocity Components and Performance Parameters For comparison purposes, the counterrotating propeller design results obtained using circumferential mean inflow data are presented in fig. 4 and tables 1 and 2. Referring to fig. 4, it is seen that, for the forward propeller, the constant magnitude of va is small compared to wa over the central portion of the blade, while at the after propeller these veloci- ties are similar in magnitude. Furthermore, it is found that the maximum value of i\v (change in v;) is of comparable magnitude to the maximum value of we. This infers that Av plays an equal role with wa in determining the radial distri- bution of blade pitch. The data also shows a small, constant value of vie on the forward propeller while the after propeller vie and both the forward and after propeller values of we have comparable magnitudes. The radial variation of a indicates that the region on the blade most prone to blade surface cavi- tation occurs at x-~0.75, and the CL. profile (not shown) reflects the change from large loading at the blade root to the unloading of the blade tip region. It should be noted that while these results are for a specific design, they are typical of counterrotating propeller designs obtained using Nelson's design method. Next, the results obtained by variation of the forward and after propeller inflow will be presented. During the cal- culation procedure, 49 values of x were used to determine the radial variation of parameters. An example of the calcu- lated variation of propeller parameters with both OF and PA iS the variation in the change in the axial inflow velocity (rev). The results, given in fig. 5, show that the most significant variation in rev occurs in the region of ~F=O, ~A=0 and near the hub surface (x=0). As one moves away from the hub surface the region of large parameter change narrows from d:20° at x=0 to only a few degrees at x=1.0. Although not shown here, similar variations are found for v and ,B while the spatial distribution of the induced and interference velo- cities, lift and drag coefficients have a nearly constant magni- tude except for very small changes in a narrow region cen- tered about ~F=0 and ~A=O- The spatial distribution of propeller performance param- eters (ref.. 16) indicates that, with the design constraint of constant Cat the performance parameters PC, 1-~, ,7 and CQ show only small variations in the region of ~F=0 and ~A=O- It should be remembered that the circumferential variation of potential flow was not included. Thus, due to the fact that 705
~ / o o . . !l / ~ ~. / X . . . 1' ,. .. , , , , , i -0.2 o 0.2 0.4 0.6 0.8 V/VS OELV r~ o o- _ -o . . . o.s 1 1.5 CflP V/VS Q o- CD ry o ~: X ~ o ~_ ~';~ 1,- .../~`r; . ;/~, . i,, : ~ .! I O_ _ o CD . G X . _ o- C~ o C-- 0.4 - a' o~ . (D _ ~ o - G X . _ o- C~ o~ 1 C- / 2 0.4 0.6 .'.~W 1:'.^ \,)' ~"' , _- ,--- ·, , o o.oo 0.14 -0. VR/VS WR/VS o.o 0.1 VT/VS WT/VS 1 / / BETR "~ ) 0.8 1 ,-- 0.2 o o.s SIGMR Fig. 4. Velocity components and geometric parameters for the forward and after propellers calculated using circumferential mean inflow data v~s , forward; ~ -, after. /~v/V~,: . , forward; · · · · ·, after. V/V,,: , forward, -------, after. v'~/V',: , forward; -------, after. Wa/VB:· , forward; · · · · · ·, after. v`/V~: , forward; -------, after. wt/V,,: - · , forward; · ~ , after. ,8: , forward; -------, after. a: , forward; -------, after. ~, NS:' X BRR - 0.0000 X BRR - 0.2500 X BRR - 0 .5000 ~°°'~0 WO 1 o~ ~ o(G) X BRR - 0.7500 (st o Fig. 5. Circumferential variation of the calculated change in axial inflow veIocity, Av/V,,, for the forward propeller at various x . ~F- FWD (DEG). 6A - AFT (DEG). 706 ~- :~,45U Pkb °~ [orG) X BRR - 1 .0000
:: l.. i 1 1 -I -0.2 o ~ o G X A_ Cat I of o~ G . Xo._ i; l, A o __ .. o ,,; , I, , , 0.2 o.4 0.6 0.8 V/VS DELV .. It'd ): ,'.! . _ . /~? of ,.,, (.'' -o. 14 o.oo VR/VS WR/VS :/ o ~ ~ G ~ ~ Xo ~ Xo_ o ,' / o_ ,,' o- ., , , o- o.s 1 1.5 - CRP V/VS ~ o 0.14 -0.1 G · ~ \ , ·. =0- ;~~~~ O- {it, to l .\ ·--. o.o 0.1 VT/VS WT/VS 2 0.2 o ~ / '" 1 o.4 0.6 0.8 BETH _ %%,%~ CD . to °- G X,= 0.5 SIGMR Fig. 6. Velocity components and geometric parameters for the forward and after propellers. ~F=0 and ~A=14°. v/V8: - , forward; -------, after. Av/Vs:. , forward; · · · · · ·, after. V/V,,: ~ , forward, -------, after. va/V8 , forward; -------, after. walVa ·~ forward; · · · , after. vt/V8: , forward; -------, after. wt/V8: ·, forward; · - , after. ,6: - , forward; -------, after. cr: , forward; -------, after. PC is a function of the thrust deduction factor, one can con- clude that there would be additional small changes in PC with ~F,A if the circumferential variation of the potential now were included; however, calculations made for the forward propeller indicate that changes in PC would be on the order of 0.5%. To further illustrate the results obtained by varying OF and ~A, the radial distribution of parameters for the case when ~F=0° and bA=14° is given in fig. 6. Comparison of this data set with the circumferential mean data presented in fig. 4 indicate that the magnitude and radial distribution of the profiles are very similar. However, the complete set of calcu- lated data shows the sensitivity of the results to the relative angular position of the propellers and to the lack of total symmetry in the measured wake. Unsteady Forces The axial velocity component of the measured wake and the calculated axial inflow velocity were used in both Fourier analysis and unsteady forces calculations. The data was supplemented with the geometric parameters found in tables 1 and 2. First, Fourier analysis and unsteady force calculations were made using u/V8, which due to the meas- urement procedure, contains both axial and radial com- ponents. Figure 7 presents the radial distribution of Fourier coefficient magnitude and unsteady thrust and torque for the forward and after propellers. For the forward propeller, the results show the dominance of the 12th harmonic (an), espe- ,,; By) l cially in the region of r/RF<0.6, and the rapid approach toward zero of the 24th and higher harmonics. The har- monic content distribution for the after propeller shows the dominance of the 4th harmonic and, when compared to the forward propeller, a slower decrease in magnitude of the 8th, 12th and higher harmonics. As in the forward propeller case, the large magnitude of an is located in the region of r/RF<0.6. For both propellers, the magnitude of bn is nearly zero for all harmonic numbers, and the magnitude of an becomes nearly constant for r/RF> =0.6. Results for higher harmonics can be found in ref. 14, and, in general, harmonic component magnitude changes are in agreement with previ- ous work [24]. The calculated radial distributions of Fit and Tz associ- ated with the forward propeller's 12th harmonic (fig. 7) show that regions of large forces occur in both the inner and outer portions of the propeller blade. Also, there is a distinct minimum force region, located at r/RF ~ 0.7, which coin- cides with the minimum velocity defect/excess region of r/RB~ 0.45 shown in fig. la. For the after propeller, the unsteady forces for the 4th harmonic show a minimum point at r/RF ~ 0.38 which coincides with the region of greatest velocity excess at r/RB ~ 0.3 (fig. lb). Even though the 4th harmonic has a small and nearly constant magnitude over the outer region of blade, the force and moment distributions have large magnitudes in both the inner and outer regions of the blade. With increasing harmonic number, nNb=8 and 12, the minimum force point moves outward along the after pro- 707
petter blade's span to r/RF~ 0.52 and coincides with the region of minimum velocity excess/defect, r/RB ~ 0.48 (fig. lb). Finally, for nNb=8 and 12, the shape and magnitude of the force and moment distributions become more like that found for the forward propeller. Next, data will be presented to illustrate the unsteady forces obtained by varying the propulsor inflow. Figures 8 and 9 present the radial distribution of harmonic coefficient magnitude and unsteady forces for the calculated axial velo- city fields obtained by specifying (a) bA=0° and (b) ~F=0° while the other propeller ((a) forward, (b) after) used wake data over the range45°<8A<~45°. Fourier coefficients and unsteady forces were calculated using the measured axial velocity vi and the calculated axial velocity, v=vi+^v. For ~A=0° (fig. 8), the harmonic coefficient distributions show total removal of the small magnitude of bn previously obtained in the Fourier analysis of u/Vs . Also, the sign of an has been reversed while its (absolute) magnitude remains nearly the same. The radial distribution of Fz and Tz, obtained using vi are nearly equal to that calculated using u/V8. For the calculated axial velocity distribution, v/V',, the magnitude of an for the 12th harmonic has been reduced in the region of r/RF<0.5 where the peak magnitude has been reduced by =25%. The unsteady forces associated with v show a reduction in magnitude over the inner radii, an increase in magnitude over the outer radii and movement of the minimum force point to a smaller radii, r/RF ~ 0.65. When ~F=0 and HA iS varied, the measured axial velocity field results in a sign change for the 4th and 12th harmonics while the magnitude of an remains approximately the same. As in the previous case, the small magnitude of bn has been removed. The characteristics of the unsteady force distribu- tions are nearly the same as those obtained for u/V~; how- ever, they show a lower magnitude of Fit and To at the other radii. The harmonic content obtained from the Fourier analysis of v/V8 reveals a reduction in an, for all harmonic numbers, over the after propeller blade's inner radii. Also, the minimum force point has been shifted to a smaller radii, r/RF ~ 0.35 for the 4th harmonic and to r/RF ~ 0.57 for 8th and 12th harmonics. Only small deviations from the results given above were found for other combinations of OF and ~A. In addition to the radial distribution of unsteady forces given above, the total forces on the forward and after pro- pellers were calculated. The results for the measured and calculated wakes are given in table 3. It can be seen that removal of the radial component from the measured inflow, u/V~, results in the introduction of unsteady forces associ- ated with the 6th and 18th harmonic components, on the for- ward propeller, which were zero for the measured inflow case. The magnitudes of these forces are substantially lower (3.4-4.9 times) than those obtained for the 12th and 24th har- monics and are probably an artifact of the computation pro- cedure. The data in table 3 also shows a general reduction in unsteady force magnitude for vi/V,, data on both the forward and after propellers when compared to the u/V`, results. While the unsteady forces obtained using v/V8 show a reduc- tion in magnitude for most harmonic numbers, small increases in magnitude, over the u and vi data, are found for T~)F at the 12th harmonic and TEA at the 8th harmonic. Cavitation and Blade Pressures An integral part of Nelson's propeller design method is the ability to determine the blade geometry (C/D) required to satisfy a given blade surface cavitation requirement. The counterrotating propeller geometry detailed in tables 1 and 2 reflect the the chord-to-diameter ratio required to meet the cavitation requirement of a=0.75. In all subsequent cavita- tion and blade pressure calculations, the blade geometry was held constant, and, since cavitation is the parameter of interest, only suction surface pressures will be calculated. The variation of a with both forward, IF, and after, ~A, pro- peller angles is presented in figs. 10 and 1 1. The results show that a has the same type of variation from the circumferential mean data as did the various velocity components, blade pitch and performance factors (see fig. 5 and ref.. 16). The variation of a is greatest in the region of -20° < ~F,A < 20°. As shown by equations (6)-(11), a and the blade section Cp distribution depend on all the velocity components compris- ing the local blade section velocity diagram (fig. 3). However, as mentioned before, both wb and TV, are functions of the chordwise trapezoidal loading distribution and the radial dis- tribution of circulation. Thus even with corrections for thickness, lift and circumferential variation of velocity com- For Use Fz for Nb = PA Vel 6 12 18 24 u/Vs)F O 32.3 O 12.4 0-45 vi/V~)F 3.9 25.5 3.7 12.1 0 45 v/V.)F 4.8 23.3 3.2 10.8 For Use Fz for Nb = F Vel 4 8 12 16 u/V8)A 79.1 24.4 20.7 8.1 32.9 7.3 7.5 0-45 vi/V8)A 60.4 24.7 17.5 7.4 26.7 7.4 6.0 0-45 v/V8)A 63.3 20.8 14.8 7.2 26.6 7.7 5.6 = For OF or HA = 0-40, the values of Fz (lb)and Tz (ft-lb) are representative OF or HA was fixed while the other propeller angle was varied Tz for Nb = 6 12 18 o 1.5 1.3 24 4.9 4.9 4.4 _ 0 10.8 2.0 8.0 22 9.8 4 Tz for Nb = 8 12 16 3.0 2.8 2.9 Table 3. Calculated Total Unsteady Forces for the Forward and After Propellers 708
~o o o ~ I NB ~o o o G o ,, o~ ~:~ C~ ~ o- NB - 12 · NB - 12 1 1 O- 1 0.4 0.6 0.8 1 0.4 0.6 0.8 R/RF R/RF a, o \ o \ ~U, \ - - C o \ ~ _ . , ~ L~ ~ ~ o - o ~ o- C: o NB - 4 o_ ~, o 0.2 0.4 0.6 0.8 R/RF ~ ,~ ~% 0.2 0.4 0.6 0.8 R/RF Fig. 7. Variation of Fourier coefI;cient magnitude (an, ten) and unsteady thrust (Fz) and torque (Tz) with radial position (r/RF) for the forward and after propellers using the measured inflow u/V.. a) forward propeller, nNb=12. b) after propeller, nNb=4. ~0 o co R/RF o CD ~ ~1 .= · ~o ,`K ~ o / `~] ~ mC_ r. Ln I . . 0.4 0.6 0.8 . o o \~ . / , ~ \\ {/ , ~ ~, . . . . 0.4 0.6 0.8 R/RF ,\ ,.'''' I NB - 12 o.4 0.6 0.8 i o.4 0.6 R/RF R/RF Fig. 8. Variation of Fourier coefficient magnitude (an, ten) and unsteady thrust (F~) and torque (Tz) with radial position (r/RF) for the forward propeller. ~A=O- nNb=12. a) vi~s; b) v~. - F. --------- T. ~ Z. ~ b 709
a, o- o G o . o rid L" ° a' o_ a' lo o G ° . o ~1 lo C) lo o _ ' T 0.2 0.4 0.6 R/RE / / 0.8 1 0.2 0.4 0.6 R/RF _ ~ ~ U. , . _ ~ U) o o NB - 4 LO o_ 0.2 0.4 0.6 0.8 R/RF me l 0.8 1 _~ ~ , l 0.2 0.4 0.6 0.8 R/RF Fig. 9. Variation of Fourier coefficient magnitude (an, ten) and unsteady thrust (Flu) and torque (To) with radial position (r/RF) for the after propeller. 8~0. nNb=4. a) vi/V~; b) v/V8. , Fz, ---------, T., portents, only small changes in a are noted. The variation of a in figs. 10 and 11 also represent the maximum (negative) pressure on the blade's suction surface, and, if the blade geometry were allowed to vary as required by the calculation at each 8, one might obtain larger changes in a. Figures 12 and 13 are representative of the variation in pressure along the suction surface of the blade at various radial locations. First, fig 12 gives the values obtained for the counterrotating propeller set using circumferential mean inflow data The data shows that the shape and magnitude of suction surface pressure coefI;cient are typical of that found on various airfoils and are in general agreement with the results of other researchers (for example refs. 2, 5 and 9). It can also be noted that the magnitude of Cp is nearly the same for both the forward and after propellers. Figure 13 gives the results for the case when ~F=0° and FA=14° where the Cp calculation uses the data presented in fig 6. Comparison of figs. 12 and 13 indicate very small differences in CF This result is consistent with the a data presented in figs. 10 and 11 and is typical for the range of ~ used in the study. When the equation for the suction surface pressure is derived from the Euler equations (see for ref. 9) in a rotating frame of reference, an additional term equal to Hip (eqn. 10) is obtained. The blade pressure results for this formulation reject small changes in magnitude due the additional Ahp however, the magnitude and shape of the suction surface pressure remain in general agreement with previous work [2,5,9]. Conclusion An existing propeller design method was modified and used to calculate the spatial variation of propeller perfor- mance, blade pressures and velocity components for use in determining changes in blade-rate forces and cavitation per- formance. The calculations showed only small changes in the magnitude of the various velocity components, forces and blade pressures when compared to the counterrotating pro- peller design results. In general, there was approximate agreement between the (absolute) magnitude of both the har- monic coefficients and unsteady forces obtained using the axial component of the inflow velocity and the measured wake data. However, the unsteady force distributions associ- ated with the calculated axial velocity, which includes pro- peller effects, showed an increase in magnitude at the inner radii with minimal change in its general shape. Overall, there was a small reduction in the magnitude of the total unsteady forces on the propulsor. The circumferential variation of both the cavitation index and suction surface pressure distribu- tions showed only small variations with blade position. Also, the difference between the forward and after propeller blade pressure distributions were small. While the simple approach used in this paper did not reveal large changes in the inflow velocity field, cavitation or unsteady forces, it did show that the design method has the tendency to modify the measured inflow velocity field in such a way that both the forward and after propeller "see. nearly identical velocity fields. This result indicates a "smoothing. of the incident velocity field. From a design standpoint this is desirable since one would not want pro- peller performance to be highly sensitive to small perturba- tions in the inflow. However, these results can only be con- sidered first order and require extension of the calculation procedure to account for additional effects due to the unsteady flow field such as blade interaction terms and flow acceleration due to the moving blade rows. Acknowledgments This work was supported by the Naval Ocean Systems Center's Independent Research and Independent Explora- tory Development Programs. 710
k~ (0 tore) rWO ~ torG) ~~ it' X BRR - 0.0000 X BRR - 0.2500 X BRR - 0.5000 : ' :35 W torG X BRR - 0.7500 Fig. 10. Circumferential variation of the cavitation index a, for the forward propeller at various x locations. OF- FWD (DEG). PA - AFT (DEG). X BRR - 1 .0000 ~n X BRR - 0.0000 X BRR - 0.2500 X BRR - 0.5000 WO 'o ~or§] X BRR - 0.7500 Fig. 11. Circumferential variation of the cavitation index ~, for the after propeller at various x locations. ~F- FWD (DEG). ~A - AFT (DEG). 711 WO 1 torG} X B8R - 1 .0000
~ o- - o BRR - 0 .0000 0.5 Y BRR _ ~ 1 ~ o~ " _ ~ - e X 8RR - 0.2500 , 1 o 0.5 Y BRR ~_ - - -- ~_ X BRR - 0.5000 l '':_ - 1 ~ 0 0.5 1 Y BRR o- ~ X BRR - I .0000 X BRR - 0.7500 , , _ u 0.5 1 0 0.5 1 Y BRR Y BRR Fig. 12. Variation of suction surface pressure coefficient, Cp = (pOO-p)/~ p V2, calculated using circumferential mean data for the forward and after propellers at various x locations. . , forward propeller; o , after propeller. BRR - 0.0000 ,_ ~ , 0 0.5 Y BRR ~ o - 1~ _ X BRR - 0 .7500 1 - `~ - `~ , , . - 0 0.5 Y BRR ~, X BRR - 0.5000 - -~o , _- o 0.5 Y BRR I X BRR - 1 .0000 _ 1 _ 1 0.5 1 ' ~ 0.5 i Y BRR Y BRR Fig. 13. Variation of suction surface pressure coefficient Cp = (pOO-p)/ ~ p V2, calculated for the forward and after propellers at various x locations with ~F=0° and ~A=14°. · , forward propeller; 0 , after propeller. 712
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