J M a r Sci Technol (2016) 21:501-516
D O I 10.1007/S00773-016-0372-3 ^ H J C r o s s M a r k O R I G I N A L A R T I C L E
( D
Numerical analysis of surface piercing propeller in unsteady
conditions and cupped effect on ventilation pattern of blade
cross-section
E h s a n Y a r i ^ • H a s s a n Ghassemi^
Received: 22 May 2015/Accepted: 11 Februaiy 2016/Published online: 23 February 2016 © J A S N A O E 2016
Abstract The a i m o f this study is to calculate hydrody-namic performance and ventilation flow around wedge, 2D blade and 3D surface piercing propeller (SPP), using computational fluid dynamic based on Reynolds-averaged NavierStokes method. First, numerical analyses f o r t w o -phase fluid flow around the wedge and 2 D blade section (cupped and non-cupped) are presented. F l o w ventilation, pressure distribution and forces are determined and com-pared w i t h experimental data. Then, the method is exten-ded to predict the hydrodynamic performance of propeller SPP-841B. The propeller exhibits a cupped blade. I n the simulated configuration, SPP is one-third submerged (/ = h/D = 0.33) and is w o r k i n g at various loadings w i t h f u l l ventilation occurring at l o w advance coefficient (ƒ). The open water performance, pressure distribution, forces/moments and ventilation pattern on the SPP-841B model are obtained and compared w i t h experimental data. The numerical results are i n good agreement w i t h experi-mental measurements, especially at h i g h advance coefficient.
K e y w o r d s Cupped and non-cupped blades • Surface piercing propeller • Ventilation pattern • Hydrodynamic coefficients
E l Hassan Ghassemi gasemi@aut.ac.ir Ehsan Y a i i
ehsanyari_mechanical@yahoo.com
' Department of Maritime Engineering, A m i r k a b i r University of Technology, Tehran, Iran
1 Introduction
Suiface piercing propellers (SPPs) w o r k i n the air and water during a rotation. It operates about half o f the time i n the air, one-third o f the time is completely submerged and the rest is partly submerged ( i n the entry and exit phases). A l t h o u g h i t w i d e l y used on boats and h i g h speed crafts, some performance estimation techniques o f SPP are still under development and they are mostly based on trial and eiTor. I t is o f t e n recognized as being one of the most e f f i cient propulsion devices i n highspeed vessels. The e f f i -ciency is p r i m a r i l y attributed to the reduction o f appendage drag, since most o f the propeller assembly is elevated above the Water [ 1 , 2 ] . Another advantage is that the SPP effectively eliminates cavitation by replacing i t w i t h ven-tilation phenomenon, i.e. the cyclic blade entry o f the air into the water opens up a ventilated cavity around the propeller w h i c h almost completely prevents the occurrence o f vapor cavitation [ 3 ] .
For the 2 D case, Shiba [4] carried out one o f the best k n o w n experimental studies on the ventilation pattern i n surface piercing propellers. D u r i n g this study, various tests o n ventilation pattern were performed to establish the law o f similarity f o r systematic tests w i t h model propellers and put f o r w a r d a method o f application to actual full-scale propellers. Later, during the 1970s, C o x [5] performed f r e e - f a l l penetration tests f o r a two-dimensional thin, straight wedge at various speeds and wedge incidence angles. W a n g [6] also cairied out water entry and exit of a f u l l y ventilated f o i l . Tsai [7] studied the effects of cupping on a f o i l section w i t h a m a x i m u m thickness ratio o f 3.5 % i n G a w n - B u r T i l l propeller series, both numerically and experimentally.
Regarding ventilation o f the propeller, Koushan pre-sented his experiments on total dynamic loadings o f
ventilated propellers and showed that fluctuations during one ventilation cycle can range f r o m 0 to 100 % o f the average force i n a non-ventilated propeller [ 8 ] . Dynamics of ventilated propeller blade loading on thrusters due to forced sinusoidal heave motion is also carried out by Koushan [ 9 ] . Classification o f different types of propeller ventilations based on analysis conducted on a series o f experiments was investigated by Kozlowska et al. [ 1 0 ] .
Experimental works f o r SPP were done by various ventilation parameters and their effects on average thrust and efficiency losses are available i n research works per-f o r m e d by Rose and Kr-uppa [11], Kruppa [12] and Rose and Ki'iippa [13]. A comprehensive work was carried out by Olofsson [14] in his PhD thesis, the forces and flow around the partially submerged propeller at different operating conditions. Dyson [15] conducted experimental tests to determine the dynamic performance o f surface piercing propeUers. The common purpose was to study the time dependent hydrodynamic loading and the stresses induced on the propeller blades, shaft, and the hull struc-ture. Also, hydrodynamic performance, flow visualization o f the ventilated cavities and spray pattern generated by a surface piercing propeller were investigated by Peterson [16], Jeong and Lee [17] A l t a m i r a n o [18].
One o f the numerical conducted studies related to this subject is the time marching boundary element method f o r estimating the ventilated flow around surface piercing h y d r o f o i l w h i c h was earned out by Savineau and Kinnas [19, 20]. I n their research, the non-linear cavity geometry is determined iteratively by applying the kinematic boundary conditions on the exact cavity surface at each time step. Y o u n g and Kinnas developed a 3 D boundary element method w h i c h was extended to model unsteady sheet cavitation on the super-cavitating performance and SPP
[21], V i n a y a n and Kinnas analyzed the flow field around a ventilated 2 D surface piercing h y d r o f o i l and propellers using a robust nonlinear B E M [ 2 2 ] . Prediction o f the hydrodynamic characteristics o f the SPP using an empirical f u n c t i o n f o r the critical advance coefficient i n the transition mode was presented by Ghassemi [ 2 3 ] .
The first application o f Reynoldsaveraged N a v i e r -S t ó k e s ( R A N -S ) methods to -SPP analyses was made by Caponnetto. He showed good correlation o f numerical results w i t h measurements [24]. A n analysis o f different propellers ventilation mechanisms by means o f R A N S s i m i ü a t i o n s v/ere analyzed by Califano and Steen [ 2 5 ] . The commercial R A N S code was used to solve the viscous, incompressible two-phase flow. Recently, a comprehensive research w o r k on series o f four-bladed propellers o f the surface piercing type f o r different operating conditions were carried out by Misra et al. [26]. The effect o f the cupped shape and trailing edge was investigated f o r d i f -ferent immersion depths. A m o n g the blade shapes, the best performance at all immersions was f r o m the propeller w i t h 6 0 ° trailing edge angle.
The m a i n puipose of this paper is to analyze the behavior o f ventilated flow around SPP. First, f o r a better understanding o f the subject, a benchmark wedge and 2-D blade sections (cupped blade and non-cupped) are inves-tigated using Ansys-Fluent 14.5, based on R A N S method. Then, the SPP propeller analysis is carried out on unsteady open water flow under the free surface condition. The propeller is SPP-841B type w h i c h is experimentally investigated by Olofsson [ 1 4 ] . A l l calculations are carried out at zero shaft yaw and inclination angle. The pressure distributions, ventilation pattern and force/moment c o m -ponents o f key blade (cupped and non-cupped) at one cycle are all calculated and discussed.
F i g . 1 Schematic o f the SPP
T a b l e 1 Non-dimensional parameters o f significance f o r ventilated propeller flows
Non-dimensional parameter Advance coefficient Immersion ratio Reynolds number Froude number Weber number Definition
iiD ' D *^ n — j
Va advance speed, v coefficient of kinematic viscosity o f water, rate o f revolution, g gravitational constant, D fliameter of propeller, p mass density o f water, h immersion o f propeller, capillarity constant o f water
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2 Hydrodynamic characteristics of the SPP
SPP is w o r k i n g behind the planning craft that is half i n , half out o f the water. They provide more speed on light fast boat. Figure 1 illustrates a schematic o f the SPP. O w i n g to its easy access to the free suiface, ventilation rather than cavitation occurs at the back o f the propeller. I f the pres-sure i n the ventilated cavities is assumed to be atmospheric and the phenomenon o f cavitation is f u l l y neglected at this point, a dimensional analysis o f the governing fluid dynamic equations reveals five non-dimensional significant parameters f o r ventilated propeller flows (Table I ) .B y studying the behavior of the SPP, i t is f o u n d that both the advance coefficient and the immersion ratio effectively divide the flow encountered by a propeller into three m a j o r flow regimes, as f o l l o w s :
Non-ventilated regime: N o ventilation occurs on the propeller or i n its trailing wake.
Partially-ventilated regime: Pai1 o f the trailing wake shows streaks or sheets o f ventilated cavities, w h i l e the propeller may either be f u l l y wetted or partly covered w i t h cavities that vent to the atmosphere.
Fully-ventilated regime: A single ventilated cavity existing on each blade o f the propeller starting close to the leading edge on the suction side and at the trailing
Table 2 Hydrodynamic characteristics o f the propeller
Thrust coefficient Torque coefficient Efficiency
edge o f the pressure side, thus f o r m i n g a sheet cavity w h i c h extends into the helical wake o f t h e propeller. The flow is normally quite stable i n this regime, covering l o w advance coefficients and partially submerged immersion ratios.
A s concluded by Shiba [ 5 ] , the effect o f Weber number
(Wu) characteristics which is closely related to suiface
tension vanishes when W„ > 180. Below this critical value, the Weber number e f f e c t i v e l y determines the transition advance coefficient through w h i c h the flow becomes f u l l y ventilated. The transition advance coefficient asymptoti-cally approaches a certain fixed value w h i c h is character-istic o f the propeller as the Weber number is increased. I n all cases o f this paper, the Weber number is greater than 180. The hydrodynamic characteristics o f the propeller are defined in the usual non-dimensional f o r m , i.e. thrust and torque coefficients and efficiency (KT,KQ,II)- f h e s e defi-nitions are shown i n Table 2.
KT • r Kn . _ e _ 'I KQ • 2iz
Table 3 Particulars o f propeller model o f the SPP-841B
Parameter Symbol Value
Diameter (mm) D 250
Hub diameter (mm) d 85
Pitch at 0.7 radius (mm) P 310
Hub-diameter ratio d/D 0.34
Pitch-diameter ratio at 0.7 radius P/D 1.24
Expanded Area ratio AE/AO 0.58
Number of blades Z 4
Rotation R . H .
F i g . 2 SPP-841B propeller and
definitions o f yaw angle W, immersion ratio / = h/D, shaft inclination angle y and rate o f revolution /; n y A -1 D
h
e = o feta=9 0 = 9 0 (b)F i g . 3 a 3D modeling and actual of the SPP-841B, b l<ey blade f r o m entry to exit
3 Study cases
3.1 S P P - 8 4 1 B propeller
I n order to validate the surface piercing propeller R A N S analysis, numerical predictions f o r 841-B model cupped propeller are compared w i t h experimental measurements collected by Olofsson [14]. For non-cupped SPP model, geometry o f propeller 841-B model is m o d i f i e d and its cup
is removed. A l l calculations are carried out at / = /?/ D = 0.33 and zero shaft yaw and inclination angle. As shown i n Fig. 2, h is the blade tip immersion and D is the propeller diameter. A photograph of the suiface piercing propeller and the corresponding R A N S model are shown i n Fig. 3a. The key blade f r o m entry to exit is also presented i n F i g . 3b and the geometrical characteristics o f the pro-peller are shown i n Table 3.
Fig. 4 The geometric data of
the cupped and non-cupped blades Back (Upper) Face (Lower) L = 0 . 1 9 m Non-cupped blade L = 0.047 m Back (Upper) L = 0.047 m ^ X ,
Face (Lower) Trailing E d g e j
; L=0.02m Cup L = 0.19m Cupped blade ->< • L = 0.08 m Outlet
/
Tank wallFig. 5 Grid generated and computational domain around the blade section (fine condition)
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3.2 Cupped and non-cupped blade sections
SPP has a sharp blade at leading edge and cusped (or blunt) at the trailing edge like super-cavitating propeller (SCP).
1000 / T o t a l n u m b e r o f e l e m e n t s 32000 58000 81000 92000 -1000 30 60 90 120 150
Angular position of cupped blade section (deg)
T o t a l n u m b e r o f e l e m e n t s - = - 32000 58000 81000 92000 1B0 » \ \ N. \ 30 60 90 150 Angular position of cupped blade section (deg)
Fig. 6 Verification of FH_total and FV_total versus angular position
of cupped blade section in various total numbers o f elements
Such a blade generates high lift-drag ratio ( L D R ) . A cup at trailing edge o f the blade section is indirectly made to increase the camber. So, it physically grips the water and generates higher l i f t . 2 D cupped and non-cupped blade sections are shown i n F i g . 4. Both blades have the same chord and thickness. The overall chord length and the cup height are 0.27 and 0.02 m respectively.
4 Governing equations
The equation f o r mass conservation, or continuity equation, can be written as f o l l o w :
6p
0/ + V • (pu) (1)
Equation (1) is the general f o r m o f the mass conserva-tion equaconserva-tion. The source is the mass added to the continuous phase f r o m the second dispersed phase and any user-defined sources.
Conservation o f momentum i n an inertial reference frame is described by
8
8/ (pu) + V • {puli) = - V p + V •{pz)+ pg +F (2) where p is the static pressure, f is the stress tensor, and pg and F are the gravitational body force and external body forces, respectively. F also contains other model-depen-dent source terms such as porous-media and user defined sources.
The standard k - B model is a model based on model transport equations f o r the turbulence kinetic energy {k) and its dissipation rate (e) [ 2 7 ] .
The volume o f f l u i d ( V O F ) model can model tvvo or more immiscible fluids by solving a single set of momen-tum equations and tracking the volume f r a c t i o n of each o f the fluids throughout the domain. The tracking of the interface between the phases is accomplished by solving a continuity equation f o r the volume f r a c t i o n of one (or more) o f the phases. For the phase, this equation has the f o l l o w i n g f o r m :
1
(3) where ihqp is the mass transfer f r o m q phase t o p h a s e and ihpq is the mass transfer f r o m p phase to q phase which represents the entry o f air into the water fluid at low pressure areas near the propeller surface and vice versa.
4.1 Solver
The commercial R A N S code (Ansys_Fluent 14.5) is used to solve the viscous, incompressible, two-phase (air and water) flow. The conservation equations f o r mass and momentum are solved i n integral f o r m using a finite v o l -ume method ( F V M ) . The integrals are approximated using thé m i d p o i n t rule and simple algorithm w h i c h couples pressures and velocities. The R A N S method is modeled using the standard k-B turbulence model. T i m e is dis-cretized using an i m p l i c i t Euler scheme. A t the inlet, a fixed velocity and a zero normal gradient condition f o r the pressure, turbulent energy and its dissipation rate are pre-scribed. A t the outlet, a zero normal gradient condition f o r the pressure is enforced. On the propeller surface, a no-slip condition is applied using a w a l l f u n c t i o n . The interface between water and air is also determined i n a surface
0.06 0.04 0.02 0 i " -0.02 -0.04 -0.06 -0.08 -0.1
* •
1 . 1 1 1 Present study EXP -0.04 -0.02 0 0.02 0.04 0.06 0.08 X ( m )F i g . 8 Comparison between numerical free surface elevation and experimental data [5] due to the wedge entry, at a = 6°, 8° and 10° e n ü y angle
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capturing method. A scalar function is defined between zero and one value which describes the volume percentage o f water i n each cell. One is defined f o r cells filled com-pletely w i t h water, and zero f o r cells filled comcom-pletely w i t h air. This scalar f u n c t i o n allows modeling the two phases i n ventilated flow as one effective fluid w i t h locaUy weighted material properties. It should be noted that at the inlet
boundary condition, the velocity values are given, and at the outlet the pressure value is given.
5 Numerical results and discussion
5.1 Wedge and blade sectionÜ 1.4 1.2 1 0.8 0.6 0.4 0.2 -0.1 -0.05 Exp Present study y (m) 0.05
Fig. 9 Pressure coefficient along the wetted part o f the wedge
cross-section, at a = 10° entry angle compared w i t h expenmental mea-surements [5]
I n the 2 D numerical simulation, two-phase R A N S / V O F flow model is applied and dynamic g r i d is used f o r the analysis. User defined f u n c t i o n ( U D F ) is prepared i n the Ansys_Fluent f o r the model movement m o t i o n i n a circular path. The wedge is selected to validate the 2 D numerical calculation.
5.7.7 Grid generation
I n 2 D blade section case, the domain grid is composed o f two rectangular zones. A fixed external rectangular zone and a smaller zone containing the 2 D blade section are both set inside the external zone. The numerical solution domain is considered as a real test tank to p e r f o r m the numerical simulation. On its lateral walls (tank waU) a slip condition is imposed. A t each time step the internal zone along w i t h the 2 D blade section is rotated by a small angle. The grid inside each zone is produced by the combination o f structured and non-structured grid. T o increase the accu-racy o f calculations, boundary layer grid o n 2 D blade
N o n - c u p p e d
9 = 50 deg 9 = 1 0 0 deg 9 = 150 deg Cupped
Fig. 10 Volume fraction of the cupped and non-cupped blade f r o m water entry to exit
9 = 180 deg
600 400 LL -200 400
K
T I \ i - A Non-Cup o Cup\
- ^ A ^ A A A •%
• 1 1 1 1 1 1 1 1 1 1 > 1 I 1 1\
, , 1 1 1 1 1 1 1 r 1 1 l \ 30 60 90 120 150 Angular position of blade cross-section (deg)300 CO D - - Non-Cup -o Cup o / a P 0 30 60 90 120 150 180 Angular position of blade cross-section (deg)
Fig. 11 Total horizontal and vei-fical forces versus angular position on 2 D blade (cupped and non-cupped) f r o m water entry to exit
F i g . 12 Grid generation around the SPP-841B propeUer and computational domain, a upstream shaft, structured grid on the free surface and
outline o f domain, b computational domain and boundary conditions
section is generated. Tiie overall computational grid size viscosity ratio is set to 1 % . The no-slip condition is set o n needs approximately 92000 cells to satisfy the g r i d inde- the 2 D blade section walls. A pressure boundary condition pendency condition. The boundary condition o f rotational is set on the outlet and a slip boundary condition is set on velocity is set to 2100 R P M . The turbulence intensity and the lateral walls (tank walls). The g i i d generated around the
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^ 4
o
? 2
1/
Total number of elements 3.6 million 3.4 million — — — - 2.6 million • — 1.7 million 1 I . . . I
\
45 90 135 180 225 270 315Angular position of l<ey blade (deg)
360 1.5 O O 0.5 h , / .' / /" .'/•
Total number of elements 3.6 million 3.4 million - 2.6 million •• 1.7 million / 45 180 225 270 315
Angular position of l<ey blade (deg)
Fig. 13 Verification o f K F x and K M x versus angular position o f
SPP-841B key blade at various number o f elements, / = 0.8
^ 2 O O 1.0E-4 8.0E-5 4.0E-5 8.0E-e V /
•A
60 120 180 240Angular position of l<ey blade
300 360
F i g . 14 Sensitivity o f K F x versus angular position o f SPP-841B key
blade in various time steps, i = 1.2
blade section and computational domain is presented i n F i g . 5.
5.1.2 Verification of grid study
A convergence analysis is carried out i n f o u r element numbers. Near the free surface and near the wedge are collocated fine elements and coarse elements are defined i n
I SSe+OO 1 . 5 l e » G 1 2 . e 7 e t 0 1 4 2 2 e » 0 1 5 5 7 e»( J l 6.93etOI 8 2 8 e + 0 1 9 1 8 e + 0 1
Fig. 15 Contour o f Y*- on SPP-841B key blade
Table 4 Flow conditions in
various advance coefficients J Fn V ( n i / s ) c 0.4 2 3.13 20.5
0.6 4 6.26 5.1
0.8 4 6.26 5,1
1.0 6 9.39 2.3 1.2 6 9.39 2.3
the far field. Figure 6 depicts the influence o f elements number on the cupped blade cross-section horizontal (FH_total) and vertical (FV_total) total forces, w h i c h converged quickly w i t h an increase i n number o f total cells. W h e n the element increases into 92000 cells, the forces reach to constant value and are converged.
5.1.3 Comparison with experimental results
I n the 2-D analysis, a wedge shape is selected [ 5 ] . The chord o f the wedge is 0.04 m w h i c h rotates w i t h 2100 R P M . The inlet velocity is defined by the radius o f the section m u l t i -plied by the angular velocity. The numerical solution domain was considered as a real test tank to p e r f o r m the numerical simulation. T o validate the 2 D numerical calculation, experimental observation available on the wedge object is used. Figure 7 shows the numerically obtained contour o f volume f r a c t i o n and experimental observations on water level rising on wedge model. A c c o r d i n g to the results, the increase i n water level is also an important factor w h i c h is modeled correctly b y R A N S method.
There is a good agreement between the numerical con-tour o f v o l u m e f r a c t i o n and experimental observations. The eiTor between the numerical and experimental results is due to lack o f surface tension parameter simulation and the ehmination o f the Weber number effect. It can be seen that, the present scheme is able to predict the ventilated pattern over the entire submerged cycle o f rotation.
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 J (advance coefficient) 0.025 O Exp (Cup)^ - - Num (Cup) — N u m (non-Cup) 0,4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 J (advance coefficient) O Exp (Cup) Num (Cup) ^ — Num (non-Cup) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 J (advance coefficient)
Fig. 16 Comparison o f hydrodynamic coefficients (KT, IOKQ, i/q) o f
the SPP-841B
Comparison between numerical free suiface elevation and experimental data [5] due to the wedge entry, at attack angle of a = 6°, 8° and 10° entry angle is presented i n F i g . 8. Based on the observations, an overall agreement between the predicted and the experimental ventilated pattern is observed. Figure 9 shows a comparison o f the pressure coefficients on the wetted side (back side) o f the wedge, w i t h good agreement between numerical results and experimental data.
Next, we deal w i t h the SPP blade section w i t h and without cup. The volume f r a c t i o n contour f o r cupped and non-cupped blades during a cycle is shown i n F i g . 10. Based on the numerical results, the cupped blade profile plays significant roles i n the stability o f ventilation area and i n preventing its dissipation. The ventilation thickness of cupped blade is about 1.5-3 times compared to that o f non-cupped blade. The difference is less i n the water entry zone o f the blade section w h i c h increases w i t h blade secfion
movement into the f u l l y ventilated zone. The ventilation zone thickness is one o f the most important and effective parameters i n the design and analysis o f a suiface piercing propeller. Thus, it can be seen that a lot o f changes have been created i n the ventilation zone due to the cup profile between the trailing edge and the lower side.
Another important flow parameter is the amount o f the fluid thrown into the air w i t h the cross-section. This is clearly shown i n position 9 — 180°.
O w i n g to the collision between the blade section and the water surface i n the water entry zone, the symmetrical rising o f water level i n lower and upper side regions can be observed. I n this case, the ventilation thickness i n com-parison w i t h the rotation radius is greater than other zones. I n the transition region, the ventilafion pattern and its thickness are changed and decreased. I n the stable region, the ventilation pattern and its thickness become stable and are not subject to significant changes. Most of the rotation time is allocated to the stable area. A t the exit zone, the blade section carries and throws some water out w i t h i t , so that i n the region, the ventilation pattern dissipates at the lower side region due to the fluid adhesion forces; however it s t i l l continues at the upper side o f the blade section.
Figure 11 presents the total horizontal force and vertical force on the blade f o r both cupped and non-cupped blades during entry to exit. As shown i n this figure, cupped blade has significant effect o f the pressure distribution, w h i l e non-cupped blade is almost constant i n w h i c h no changes occurs.
5.2 Surface piercing propeller
5.2.1 Grid desciiption and grid independency
I n this approach, the domain grid is composed o f t w o zones (cylinder- rectangular). A fixed external rectangular zone simulates the cavitation tunnel waUs; while a slip condition is imposed on its lateral surface and on the f o r e and a f t surfaces to w h i c h the inlet and outlet condition are applied respectively. A smaller zone (cylinder) containing the propeller is set inside the external zone. A t each time step, the internal zone is rotated by a small angle and compu-tational variables are interpolated at the sliding interface o f the internal and external zones. The rotation axis o f the internal zone can be oriented arbitrarily to represent the exact propeller shaft inclination. The geometry o f the propeller has been developed by home code. The g r i d inside each zone is produced by the combination o f structured and non-structured grid. T o increase the calcu-lations accuracy, boundary layer grid on the propeller w a l l surfaces, hub, boss, and shaft is generated. G r i d generation around the SPP-841B propeller and computational domain are shown i n F i g . 12.
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Furthermore, Fig. 13 shows the verification analysis o f the grid generated over SPP (for 3D condition) by deter-m i n i n g the sensitivity o f thrust ( K F x ) and torque ( K M x ) coefficients i n open water condition at J = 0.8. It is shown that w i t h an increase i n number of elements up to 3.6 m i l l i o n , the results are w e l l converged. T i m e step has very Iittie impact on the result. A diagram o f K F x versus angular position o f key blade i n various time steps is shown i n Fig. 14.
The total number o f cells around the SPP-84IB pro-peller is about 3.8 m i l l i o n . On this refined mesh the M is less than 90 on the propeller key blade, as presented in Fig. 15.
5.2.2 Hydrodynamic coefficients
The undisturbed free-surface elevation is assigned i n both inlet and outiet boundaries. I n order to validate the treat-ment of surface piercing propellers, numerical predictions
of the SPP-841B propeller model are compared w i t h experimental measurements collected b y Olofsson [14]. A photograph o f the SPP ventilation and the corresponding R A N S model results are also compared. A l l numerical results in this study were extracted after several rotations. To compare numerical results w i t h experimental mea-surements by Olofsson [14], the flow conditions i n each advance coefficient were set as shown i n Table 4.
Hydrodynamic coefficients (Kj, IOKQ, and i]o) are cal-culated by averaging force and moment distributions around X axis (Fig. 16). The cupped propeller results have a good agreement w i t h the experimental data. A t l o w advance coefficients, differences between numerical results and experimental data are greater where the m a x i m u m relative error is about 20 %. But at high advance coefficients, the en'or is less than 2.5 %. The numerical results obtained f r o m non-cupped propeller are far f r o m that o f the cupped pro-peller. Here, it is shown that f o r cupped propeller, m a x i m u m thrust may be as twice as non-cupped propeller.
0.05 h Q . O -0.05 -0.1 Face(Cup) Face(Non-Cup) Back(Cup) Back(Non-Cup) 0.2 0.4 0.6 x/c A n g u l a r position, 8 = 30 deg 0.8 a. O 0 Face(Cup) Facê(Non-Cup) Back(Cup) Back(Non-Cup) 0.2 0.4 0.6 0.8 X/C A n g u l a r position, d = 80 deg -4
\
Face(Cup) • • Face(Non-Cup) Back(Cup) Back(Non-Cup) 0.2 0.4 0.6 X/G 0.8 Q -o -2 Face(Cup) Faoe(Non-Cup) Back(Cup) Back(Non-Cup) A n g u l a r position, d = 130 degFig. 17 Pressure coefficient on the key blade at different angular positions, r/R — 0.65, 7 = 0
0.2 0.4 0.6 0.8
X/C
A n g u l a r position, 0= 180 deg
5.2.3 Pressure, forces/moments and ventilation pattern
Figure 17 shows the pressure coefficient (Cp) on the key blade (cupped and non-cupped, back and face) at ;•/ /? = 0.65, y = 0.8 and different positions o f propeller rotation. Cupped blade generates high pressure on the face side. Here, 0 = 0 means blade enters into the water and means blade leaves the water at Ö = 180. The cupped blade gives a better grip on the water; and hence generates plenty o f thrust.
Comparison between calculated ( w i t h and non-cupped) and experimental data o f six-component force/moment coefficients at three advance coefficients ( / = 0.8, 1.0 and 1.2) are shown i n Figs. 18, 19 and 20. The figures show the calculation o f a l l six-components o f the force/moment o f the key blade (cupped and non-cupped) compared w i t h the experimental measurements. As can be seen, both numerical and experimental curves have the same trends. H o w -ever i n some cases, numerical results are much larger than experimental data. The error is due to the weakness o f
Exp (with cup) Num (with cup) Num (without cup)
O
CO
M i
100 200 ^ 300
Angular position of key blade (teta)
0.8
X 0.6
0.2
f
- - ' s. VExp (wilh cup) Num (with cup) Num (without cup)
9
100 200 300
Angular position of key blade (teta)
400
Angular position of key blade (teta) Angular position of key blade (teta)
Fig. 18 Comparison between the calculated and measured rotational fluctuation o f six-component force/moment versus angular position o f key
blade at J = 0.8
J Mai- Sci Technol (2016) 21:501-516 513 1.8 1.6 1.4 1 . 2 ^ 1 -0 . 6 . 0.4 0.2 -O r :
> — Exp (with cup) Num (with cup) Num (without cup)
v . '
C.
100 200 300
Angular position of key blade (teta) Angular position of key blade (teta)
Fig. 19 Comparison between the calculated and measured rotational fluctuation o f six-component force/moment versus angular position o f key
blade at / = 1.0
R A N S method i n modeling and analyzing the ventilation around the SPP. I n other words, (due to the instability i n the transition region) particularly i n the l o w advance coefficient, the numerical modeling o f the flow does not have the appropriate ability to show fluctuations o f the cross-flow and ventilation.
Figure 21 shows the ventilation pattern on the SPP key blade i n three different positions i n one cycle. The results are obtained f o r / = 1.2 and as i t can be seen, there are f a i r l y good c o n f o r m i t y between experimental observation
and numerical contours. This adaptation is reduced i n l o w advance coefficient due to the propeller operation i n heavy and unstable conditions.
6 Conclusions
A computational method is presented to predict the unsteady hydrodynamic forces acting on SPP and the behavior o f ventilation flow around 2 D section and 3 D
o r . 0.5 N o L L L t
r
o ° -0.5 1-Exp (with cup) Num (with cup) Num (without cup)
Exp (with cup) Num (wilh cup) Num (without cup)
100 200 300
Angular position of key blade (teta)
Exp (with cup) Num (with cup) Num (without cup)
-100
':-7
200 300
Angular position of key blade (teta)
100 200 300
Angular position of key blade (teta)
0.6 •
Exp (with cup) Num (with cup) Num (v/ithout cup)
X 0.4 o o 0.2 • 100 ^ 200 300
Angular position of key blade (teta)
1.6 1.4 1.2 1 2 0.8 X 0.6 O ° 0.4 0.2 0 - , " -0.2
Exp (with cup) Num (with cup) Num (without cup)
O
^u
u^
r.
. . .. . .
100 2O0 300
Angular position of key blade (teta)
0.21- - o — Bcp (with cup) Num (with cup) . , « , « • . Num (without cup)
N O O -0.2 -0.4 100 200 300
Angular position of key blade (teta)
Fig. 20 Comparison between the calculated and measured rotational fluctuation o f six-component force/moment versus angular position o f key
blade at 7 = 1.2
SPP-841B o f the cupped and non-cupped blades. R A N S simulation using the V O F method is also employed. Var-ious numerical results are presented and compared w i t h experimental data. As a result of the present w o r k , the f o l l o w i n g conclusions can be drawn:
• For the wedge, ventilation pattern and pressure distri-bution are validated and a good agreement is reached. • The cup on the blade has significant e f f e c t on the
ventilation pattern, pressure and forces.
• Numerical results o f the hydrodynamic coefficients o f the SPP (cupped and non-cupped) are compared w i t h the cupped SPP-841B o f the experimental data. No-cupped SPP has produced lower thrust and lower efficiency. • Numerical results regarding the six-components o f the
force/moment are i n relatively good agreements w i t h experimental measurements, especially at higher advance coefficient. This adaptation is reduced i n l o w advance coefficients.
J Mar Sci Teclinol (2016) 21:501-516 515 (9) = 120 deg (9) = 150 deg (a) E x p (9) = 1 8 0 deg Water (9) = 120 deg (9) = 1 5 0 deg (b) Contour o f V O F
F i g . 21 Comparison o f the experimental observed (Olofsson [14]) and simulated ventilation patterns ( i = 1.2) at angular positions o f SPP-841B
key blade
• Greater understanding o f the flow around the SPP and the resulting nonlinear free surface is required. This is a research area that the authors intend to pursue i n more detail i n the future.
Acknowledgments The work presented i n this paper has been
supported by the H i g h Performance Computing Research Center (HPCRC) at Amirkabir University ofTechnology ( A U T ) . The authors gratefully like to thank the reviewers f o r their comments that helped us to improve the manuscript.
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