Third International Symposium on Marine smp’13, Launceston, Tasmania, Australia, May 2013

Simulation of Unsteady Interaction Forces on a Ducted with Pre-swirl Stators

Zhi-Qiang Rao, Wei Li, Chen-Jun Yang State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China

ABSTRACT induced by , the discussion on the frequency The frequency characteristics of unsteady forces arising characteristics is inevitable. from the hydrodynamic interaction between the rotor and The research on this kind of propulsor is relatively less stators for a with pre-swirl stators is than on ordinary propulsors owing to the complicated analyzed theoretically. It is shown that the axial force and flow phenomenon around the propulsor. A potential-based moment fluctuate at frequencies equal to nZRN on panel method for the duct coupled with the lifting surface condition that nZR=kZS, where k and n are both positive method for the rotor and stators were presented by Hughes integers, ZR and ZS are the rotor and stator blade numbers & Kinnas (1991), Wang & Yang (1993), and Wang & Liu respectively, and N is the shaft frequency of the rotor. The (2007). In these papers the steady performance was lateral forces and moments also fluctuate at multiples of predicted. The unsteady thrust and torque were presented the rotor's blade frequency, but on condition that by Wang & Liu (2007), however the frequency and nZR=kZS±1. Therefore, the axial and lateral forces do not amplitude characteristics were not analyzed nor discussed. fluctuate at the same frequency, and in many cases the During the past forty years, computational lowest fluctuation frequency of the lateral forces are much (CFD) has been developing rapidly and finding more and lower than that of the axial force. To evaluate the more applications in hydrodynamic research. The amplitudes of unsteady interaction forces, RANS flow field of an axisymmetrical body with a ducted simulations are carried out for a ducted propeller with pre- propeller working behind it was studied by Wang et al swirl stators. The CFD results of time-averaged thrust and (2003) using RANS simulation. The hydrodynamic torque agree fairly well with those experimentally simulation of a torpedo with the pump-jet propulsor was measured. The unsteady force fluctuations are quite conducted by Ivanell (2001). The numerical results show clearly captured at frequencies theoretically determined, that CFD is an effective tool for predicting the and are mostly negligible at other frequencies. The results hydrodynamic characteristics of the ducted propeller with indicate that, for the configuration being simulated, the stators. lateral forces fluctuate at much lower frequencies and higher amplitudes than the axial force. The experimental research on the steady performance and cavitation of an underwater vehicle with the pump-jet Keywords propulsor were conducted by Suryanarayana et al (2010a, propeller, duct, rotor, stator, bearing force, CFD 2010b). The wind tunnel experiment show that most of the 1 INTRODUCTION swirl rectification could be attained by the post-swirl As a multi-component marine propulsor, the ducted stator (Suryanarayana et al 2010a). The cavitation tunnel propeller with pre-swirl stators is mainly used on experiment show that cavitation inception on the rotor of underwater vehicles. The main components of this kind of the pump-jet takes place on the tip face side at higher propulsor are stators, rotor (propeller) and duct. The advance ratios and the stators will be free from cavatation stators are installed on the upstream of the rotor, which over the operating envelop of the vehicle (Suryanarayana can improve the inflow by homogenizing the wake and et al 2010b). yielding pre-swirling inflow, thereby restraining cavitation In this work the frequency characteristics of unsteady and reducing the vibration force and radiation noise. forces arising from the hydrodynamic interaction between Compared with ducted propeller, the performance the rotor and stators for a ducted propeller with pre-swirl evaluation and design for the ducted propeller with stators stators is analyzed theoretically. All the hydrodynamic are much more difficult and the hydrodynamic forces and forces and moments fluctuate at multiples of the rotor’s moments are unsteady even in uniform inflow due to the blade frequency. However, the axial force and moment do interaction between the rotor and stators. Furthermore, for not fluctuate at the same frequencies as the lateral forces the sake of reducing fluctuating force, vibration and noise and moments. RANS simulations are carried out for a ducted propeller with pre-swirl stators. The CFD results

149 of time-averaged thrust and torque agree fairly well with The total horizontal force of the rotor can be written as those experimentally measured. The unsteady force FyR, fluctuations are quite clearly captured at frequencies  ZR 1  (3) theoretically determined, and are mostly negligible at  Brk sin kZS i  rk sin  i  B tk cos kZS i  tk cos  i  ik01 other frequencies. The radial component in (3) is 2 THEORETICAL ANALYSIS OF FREQUENCIES ZR 1 OF UNSTEADY FORCES  Frisin i As shown in Figure 1, a fixed coordinate system o-xyz is i0 ZR 1  defined with x-, y- and z-axis corresponding to axial,  Brk sin  kZS i  rk  sin i  (4) horizontal and vertical directions respectively. A polar ik01 1  ZR 1              coordinate system (r, θ) is defined in o-yz plane and θ=0 Brk coskZS 1  i rk  cos kZS 1  i rk  when r coincides with z-axis. The coordinate system 2 ki10 o-x1y1z1 rotating synchronously with the rotor coincides And with the fixed coordinate system at initial time. Suppose ZR 1 coskZ  1  the stator and rotor blade numbers are ZS and ZR  S  i rk i0 (5) respectively, where subscript S and R represent the stators ZR 1 21kZS   i and rotor respectively. In o-xyz coordinate system Fx, Fy,   coskZS  1  rk  i0 ZR Fz, Ft, and Fr represent axial, horizontal, vertical, tangential, and radial forces respectively; Mx, My, and Mz Use the following formula, represent torque, horizontal bending moment, and vertical 1 sin (Ry 1) bending moment respectively. R 1 cos(x ry )  cos( x  Ry ) 2 (6)  1 r0 2 sin y 2 (5) can be expressed as

ZR 1   coskZS 1  i rk i0  (ZR  1)( kZS  1) sin(kZS  1) coskZS  1  rk   (7) Z (kZ S  1) R sin ZR 0 (kZ 1 nZ )   SR ZR cos( nZR rk ) ( kZS  1  nZR ) Therefore the radial component in (3) becomes

ZR1  ZR Frisin i  Brk cos nZR  rk  , (kZS  1  nZR ) (8) ik012 Figure 1 Definition of coordinate systems, forces, and moments Similarly the tangential component in (3) can be written as

ZR1  ZR 2.1 Frequencies of Fy and Fz of the Rotor Fticos i  Btk cos nZR  tk  , (kZS  1  nZR ) (9) ik012 Since the periodical angle of rotor is 2π/ZS, the tangential and radial forces of the ith blade are expressed by the The total horizontal force is trigonometric series as follow,  ZR  Fy, R   Br,, t k sin nZR   rk    (10)    2 k1  Fti  Btk cos kZS i tk  k1  in which   (1) F Bsin kZ   ri  rk S i rk  k1  BBcos     arctan rk tk rk tk th   where Btk and Brk are the k -order amplitude of tangential Btksin rk tk  and radial forces respectively; and θi=θ+2πi/ZR is the th angular position of the i blade's reference line, in which θ 22 th BBB cos     B sin     is the reference line angle of the 0 blade. r,, t k  rk tk rk tk   tk rk tk  Then the horizontal force of the ith blade can be expressed The vertical force can be expressed in the same form as as follow, the horizontal force.     FFFyi risin i ti cos i 2.2 Frequencies of Fy and Fz of the Stators and  Duct   Brk sin  kZS i  rk  sin i (2) k1 In the o-x1y1z1 coordinate system, the rotor is stationary  but the stator is rotating in the direction opposite to that of  Btk cos kZS i  tk cos  i k1

150 the rotor. Then each stator blade has a periodical angle of The frequencies of interaction forces were derived by 2π/ZR, and the tangential and radial force can be written as Strasberg & Breslin (1975) for contra-rotating   (CRPs). The alternating frequency of thrust is F Bcos kZ  t1, i  t1, k R i t1, k    k 1  fkm  kYM mZN (18)   (11) F Bsin kZ   r1, i  r1, k R i r1, k  where Y and Z are the forward and aft propeller blade k1  numbers respectively; M and N are the rotating speed of where subscript 1 denotes variables in the rotating the forward and aft propellers respectively. When viewing th coordinate system. The reference line angle of the i the ducted propeller with stators in a coordinate system stator blade θi equals to -θ+2πi/ZS, where θ is the which rotates in the same direction as and at a half speed th reference line angle of the 0 stator blade, which is equal (N/2) of the rotor's rotation, the stators (and duct) rotate in th to that of the 0 rotor blade. opposite direction against the rotor. Then (17) and (18) The horizontal and vertical forces can be obtained by yield the same results. projecting tangential and radial forces to the y- and z-axis 2.4 Frequencies of Mx, My and Mz respectively, The moment is the result of multiplying force and force FFFcos sin  y1,, i t 1 i i r1, i i  arm. Assuming that the change of force arm with the  (12) FFFcos sin rotor's angular position is negligible, the frequencies of z1,, i r 1 i i t1, i i  Mx, My, and Mz and the conditions for them to occur The horizontal force on one stator blade in the fixed would be the same as those of Fx, Fy, and Fz. coordinate system is expressed as 3 RANS SIMULATION OF UNSTEADY FORCES FFF cos   sin (13) yi y11,, i z i 3.1 Computational Model The total horizontal force of the stators is The hydrodynamic performance of a ducted propeller with pre-swirl stators (Hughes & Kinnas 1991) is analyzed by ZS 1 22ii        the RANS method. The rotor is four-bladed with a FFy, S  t11, i cos Fr, i sin   (14) i0 ZZSS    diameter D of 254mm. The stators are nine-bladed, with the NACA63-018 section, a chord length of 50.8mm, and (14) is in similar form to (3). In the same way as for the a stator setting angle of 3 degrees (a pitch angle of 87 rotor, the total horizontal force on the stators can be degrees). The length of the duct is 192.405mm. Geometry derived as information of rotor and duct is listed in (Hughes & Z  F S B sin kZ       (kZ  nZ  1) Kinnas 1991). The geometrical model of the propulsor is y, S  r1,, t 1 k  R t1, k   RS (15) 2 k1 shown in Figure 2. where BBcos   t1,, k r 1 k r1,, k t 1 k     arctan B sin  r1, k r1,, k t 1 k 

22     BBB  cos    B sin   stator blade: NACA 63-018 r1,, t 1 k  t1,, k r 1 k r1,, k t 1 k   r1, k r1,, k t 1 k  The vertical force on the stators can be expressed in the same form as the horizontal force. The duct is also stationary and can be divided into ZS parts of identical geometry. Then the unsteady forces acting on the duct are duct in the same form as on the stators. From (10) and (15) the total horizontal force of the propulsor can be expressed in the same form as Figure 2 Geometry of the computational model 

FyR B sin nZ     (16) 3.2 Simulation Approach  k 1 The computational domain is a cylinder. The inlet and where nZR=kZS±1. Let θ=0 when t=0, since θ=2πNt (N is outlet boundaries are located at 2.5D and 5D upstream the shaft frequency), the frequency of the total horizontal and downstream of the rotor's disk plane respectively. The force is f=nZRN. domain is divided into a rotating part enclosing the rotor and its hub with about 1.34 million cells, and into a 2.3 Frequencies of Fx Employing the same analysis method, the frequency of stationary part with about 1.5 million cells. Boundary propulsor thrust can be derived as layer grids are attached to the rotor, stator, and duct surfaces. The first layer height is about 0.1mm. The f kZ nZ N /2 x, kn S R  (17) volume and surface grids are shown in Figures 3 and 4 respectively. where kZS=nZR.

151 The fluctuation frequencies are accurately predicted for single blade, which has a base frequency of 9N.

0.082

0.081

0.080

0.079

of key blade 0.078 T K 0.077

0.076 0 40 80 120 160 200 240 280 320 360 Key blade angular position of rotor 0.30 0.25 0.20 0.15 0.10 Amplitude /% 0.05 0.00 0 18 36 54 72 90 108 126 144 162 180 198 Figure 3 Computational domain grids Multiples of shaft frequency Figure 5 Unsteady thrust on one rotor blade and its amplitude spectrum According to the theoretical analysis, the frequencies of unsteady forces and moments less than 180N are listed in Table 2. The base frequency of thrust Fx and torque Mx is 36N, which is much higher than the lowest frequency (8N) of horizontal and vertical forces and moments.

Table 2 Theoretical frequencies of hydrodynamic forces and moments (ZS=9, ZR=4)

Figure 4 Grids on rotor, stator, and duct surfaces k n Fx, Mx Fy,z, My,z k n Fx, Mx Fy,z, My,z The sliding mesh model and SST k-ω turbulence model 1 2 - 8N 11 25 - 100N are applied to simulate the unsteady flow around the 3 7 - 28N 12 27 108N - propulsor. The SIMPLEC scheme is applied to solve the 4 9 36N - 13 29 - 116N coupled pressure and velocity fields. 5 11 - 44N 15 34 - 136N The unsteady time step size is Δt=5 × 10-5s, which corresponds to 0.36 degrees rotating angle per step. The 7 16 - 64N 16 36 144N - rotating speed of rotor is 1200r/min, and the advance 8 18 72N - 17 38 - 152N coefficient J=0.6. 9 20 - 80N 19 43 - 172N 3.3 Results First of all, the sliding mesh model is validated in terms of The total thrust acting on the rotor and its amplitude the steady performance. As shown in Table 1, the time spectrum are shown in Figure 6, and those for the stators averaged thrust and torque are both higher than and duct in Figure 7. It is seen from the figures that the experimental results (Hughes & Kinnas 1991), though the base frequency of thrust is 36N for the rotor, stators, and simulated open water efficiency agrees well with that duct, which coincides with theoretical analysis results. measured. The highest amplitude is merely about 1 ‰ of the averaged thrust. Table 1 Comparison of calculated and measured

open water performances 0.330 Experiment Present CFD 0.325 0.320 0.315 KT 0.3512 0.3712 of rotor T

K 0.310

10KQ 0.6032 0.6380 0.305

0.300 KTS+KTN 0.0756 0.0555 0 40 80 120 160 200 240 280 320 360 Angular position of rotor η0 0.5560 0.5556 0.30 0.25 0.20 The unsteady thrust of one rotor blade during one 0.15 0.10 revolution and its amplitude spectrum are shown in Figure Amplitude /% 0.05 5, where the amplitude is shown in percentage of the time- 0.00 0 36 72 108 144 180 216 252 288 324 360 396 averaged thrust. The bold bars in the lower figure indicate Multiples of shaft frequency the frequencies that match with the theoretical analysis. Figure 6 Unsteady thrust of the rotor and its amplitude spectrum

152 0.060 and eleven-bladed stators to replace the nine-bladed ones,

0.058 while using the same rotor and duct. Numerical results of

0.056 the propulsor with seven-bladed stators are shown in 0.054 Figures 10 through 14. Again, the frequency results in this of stator and duct

T 0.052 case agree with the theoretical analysis. K 0.050 0 40 80 120 160 200 240 280 320 360 Angular position of rotor Simulation results of the propulsor with eleven stator 0.15 blades are shown in Figures 15 through 19. CFD results 0.12 agree well with the theoretical analysis. The base 0.09 frequency of thrust is 44N, which is much higher than the 0.06

Amplitude /% lowest frequency of lateral forces (12N). The largest 0.03 amplitude of lateral force is higher than that of thrust. It 0.00 0 36 72 108 144 180 216 252 288 324 360 396 Multiples of shaft frequency indicates that lateral force should be given more attention. Figure 7 Unsteady thrust of the stators and duct The results are similar for the 4-bladed rotor combined and its amplitude spectrum with 7- and 9-bladed stators. It is showed that the more stator blades, the higher amplitude of rotor lateral forces. The horizontal force acting on the rotor and its amplitude So the numbers of stator and rotor blades not only have a spectrum are shown in Figure 8, and those for stators and direct relationship with the fluctuation frequencies, but duct in Figure 9. It is seen from the figures that the lowest also influence the amplitude of rotor forces which may be frequency of rotor, stator and duct is 8N, which agrees important for propulsor vibration and noise. with theoretical analysis results. But the amplitude of lowest frequency of rotor is larger than those of higher 0.0800 frequencies. 0.0790

0.0780 of key blade

T 0.0770 K

0.0760 0 40 80 120 160 200 240 280 320 360 Key blade angular position of rotor 0.6

0.5

0.4

0.3

0.2 Amplitude /% 0.1

0 0 14 28 42 56 70 84 98 112 126 140 154 168 182 196 Multiples of shaft frequency Figure 10 Unsteady thrust on one rotor blade and

its amplitude spectrum for ZR=4, ZS=7

0.340 0.335 Figure 8 Unsteady horizontal force of rotors 0.330 and its amplitude spectrum 0.325 0.320 0.010 of rotor

T 0.315

0.008 K 0.006 0.310 0.004 0.305 0.002 0.300 0.000 0 40 80 120 160 200 240 280 320 360 -0.002 Angular position of rotor -stator & duct

T -0.004 0.25 K -0.006 -0.008 0.2 -0.010 0 40 80 120 160 200 240 280 320 360 Rotor angular position 0.15 0.25

0.20 0.1 Amplitude /% 0.15 0.05

0.10 0

Amplitude /% 28 56 84 112 140 168 196 224 252 280 308 336 364 392 0.05 Multiples of shaft frequency 0.00 8 28 44 64 80 100 116 136 152 172 188 208 224 244 260 280 296 316 332 352 368 388 Figure 11 Unsteady thrust of rotor and its amplitude Frequencies spectrum for Z =4, Z =7 Figure 9 Unsteady horizontal force of stators and duct R S and its amplitude spectrum

3.4 Investigation of Blade Number Combination on Unsteady Force Characteristics In order to further validate the present CFD approach in terms of frequency prediction accuracy, and investigate the effect of blade number combination on unsteady force characteristics, simulations are conducted using seven-

153 0.35 0.060 0.34

0.059 0.33

0.32

0.058 of rotor

T 0.31 K

of0.057 stator and duct T 0.3 K

0.056 0.29 0 40 80 120 160 200 240 280 320 360 0 40 80 120 160 200 240 280 320 360 Angular position of rotor Angular position of rotor 0.16 0.06 0.12 0.05 0.08 0.04

Amplitude /% 0.04 0.03 0.00 0 28 56 84 112 140 168 196 224 252 280 308 336 364 392 0.02 Multiples of shaft frequency Amplitude /% 0.01 Figure 12 Unsteady thrust of stator and duct and 0 its amplitude spectrum for Z =4, Z =7 44 88 132 176 220 264 308 352 396 440 484 R S Multiples of shaft frequency 0.002 Figure 16 Unsteady thrust of rotor and its frequency

0.001 spectrum for ZR=4, ZS=11 rotor

of 0.000 y

K 0.056 -0.001 0.054 0.052 -0.002 0 40 80 120 160 200 240 280 320 360 0.050 Angular position of rotor 0.25 0.048 of stator and duct

T 0.046 0.20 K 0.044 0.15 0 40 80 120 160 200 240 280 320 360 Angular position of rotor 0.15 0.10

mltd /% Amplitude 0.12 0.05 0.09 0.00 8 20 36 48 64 76 92 104 120 132 148 160 176 188 204 216 232 244 260 272 288 300 316 328 344 356 372 0.06 Multiples of shaft frequencies

Amplitude /% 0.03

Figure 13 Unsteady horizontal force of rotors and 0.00 0 44 88 132 176 220 264 308 352 396 440 484 Multiples of shaft frequency its amplitude spectrum for ZR=4, ZS=7 0.012 Figure 17 Unsteady thrust of stators and duct and 0.008 its amplitude spectrum for ZR=4, ZS=11 0.004 and 0.000 0.002 stator-0.004 duct 0.001 of y -0.008 rotor K

-0.012 of 0.000 0 40 80 120 160 200 240 280 320 360 y Angular position of rotor K 0.40 -0.001 0.35 0.30 -0.002 0 40 80 120 160 200 240 280 320 360 0.25 Angular position of rotor 0.20 0.15 0.20 mltd /% Amplitude 0.10 0.05 0.15 0.00 8 20 36 48 64 76 92 104 120 132 148 160 176 188 204 216 232 244 260 272 288 300 316 328 344 356 372 384 0.10 Multiples of shaft frequency

/% Amplitude 0.05 Figure 14 Unsteady horizontal force of stators and duct 0.00 12 32 56 76 100 120 144 164 188 208 232 252 276 296 320 340 364 384 and its amplitude spectrum for ZR=4, ZS=7 Multiples of shaft frequency Figure 18 Unsteady horizontal force of rotors and its 0.084 amplitude spectrum for ZR=4, ZS=11 0.082

0.080

of key0.078 blade T K 0.076 0 40 80 120 160 200 240 280 320 360 Key blade angular position of rotor 0.6 0.5 0.4 0.3 0.2 Amplitude /% 0.1

0 0 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 Multiples of shaft frequency Figure 15 Unsteady thrust on one rotor blade and

its amplitude spectrum for ZR=4, ZS=11

154 0.006

0.004 number of rotor blades, the more the stator blades the duct 0.002 higher amplitude of the rotor lateral forces. and 0.000

stator

of -0.002 y K -0.004 REFERENCES -0.006 0 40 80 120 160 200 240 280 320 360 Hughes, M. J., Kinnas, S. A. (1991). An analysis method Angular position of rotor 0.25 for a ducted propeller with pre-swirl stator blades. 0.20 Propellers/Shafting ’91 Symposium, Virginia, USA. 0.15 Ivanell, S. (2001). Hydrodynamic simulation of a torpedo 0.10

mltd /% Amplitude with pumpjet propulsion system. Stockholm: Royal 0.05

0.00 Institute of Technology. 12 32 56 76 100 120 144 164 188 208 232 252 276 296 320 340 364 384 Multiples of shaft frequency Strasberg, M., Breslin, J. P. (1975). Frequencies of the Figure 19 Unsteady horizontal force of stators and duct Alternating forces due to interactions of contrarotating and its amplitude spectrum for ZR=4, ZS=11 propellers. Journal of Hydronautics, 10(2), pp.62-64. Suryanarayana, Ch., Satyanarayana, B., Ramji, K., et al. 4 CONCLUSIONS (2010a). Experimental evaluation of pumpjet In the present work, the frequency characteristics of the propulsor for an axisymmetric body in wind tunnel. ducted propeller with pre-swirl stators are analyzed by International Journal of Naval Architecture and Ocean theoretical and CFD methods. The following conclusions Engineering, 2(1): 24-33. are drawn, Suryanarayana, Ch., Satyanarayana, B., Ramji, K., et al. (1) The frequencies of interaction forces and moments (2010b). Cavitation studies on axisymmetric acting on the ducted propeller with pre-swirl stators are underwater body with pumpjet propulsor in cavitation determined from theoretical analysis. It is shown that the tunnel. International Journal of Naval Architecture and axial force and moment fluctuate at frequencies equal to Ocean Engineering, 2(4): 185-194. nZRN on condition that nZR=kZS. The lateral forces and moments also fluctuate at multiples of the rotor's blade Wang, G.-Q., Liu, X.-L. (2007). A potential based panel frequency, but on condition that nZR=kZS±1. The axial method for prediction of steady and unsteady and lateral forces do not fluctuate at the same frequency, performance of ducted propeller with stators. Journal and in many cases the lowest fluctuation frequency of the of Ship Mechanics, 11(3), pp.333-340. lateral forces are much lower than that of the axial force. Wang, G.-Q, Yang, C.-J. (1999). Hydrodynamic (2) The comparison of CFD simulation for three blade performance prediction of ducted propeller with number combinations with theoretical analysis results stators. Journal of Ship Mechanics, 3(3), pp.1-7. indicates that the present RANS approach is capable of Wang, T., Zhou, L.-D., Zhang, X. (2003). Numerical predicting accurately the unsteady force characteristics of simulation of 3-D integrative viscous complicated flow the ducted propulsor with pre-swirl stators. CFD field around axisymmetric body with ducted pro- simulations also show that the largest amplitudes of the pulsion. Journal of Ship Mechanics, 7(2), pp.21-32. lateral forces are larger than that of thrust. With the same

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