Ocean Engineering 88 (2014) 472-479
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Ocean Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v l e r . c o m / l o c a t e / o c e a n e n g
A lifting surface optimization method for the design of marine
propeller blades
Kyung-Jun Lee*, Tetsuji Hoshino, Jeung-Hoon Lee
Samsung Ship Model Basin(SSMB), Marine Research Institute, Samsung Heavy Industiies, Science Town, Daejeon 305-380, South Korea
(S>
CrossMaikA R T I C L E I N F O A B S T R A C T
Article history: A lifting s u r f a c e o p t i m i z a t i o n m e t h o d is c o u p l e d w i t h a blade a l i g n m e n t p r o c e d u r e for the design of Received 29 August 2013 p r o p e l l e r i n m a r i n e a p p l i c a t i o n s . T h e m e t h o d is c h a r a c t e r i z e d by the use of a v o r t e x lattice discretization AvaHable online 2°5'ml 2014 ° ^ '"^"^ ^^^'^^ c a m b e r s u r f a c e for the d e t e r m i n a t i o n of the o p t i m u m c i r c u l a t i o n distribution, — 3 ' ^ e on i n e — j u y vvhich is found by n u m e r i c a l l y s o l v i n g a v a r i a t i o n a l p r o b l e m . T h e blade m e a n c a m b e r surface is Keywords: d e f o r m e d to s a d s f y the k i n e m a t i c b o u n d a r y c o n d i t i o n that the n o r m a l c o m p o n e n t of v e l o c i t y becomes Lifting surface z e r o on t h a t surface, in a n iterative w a y w i t h t h e o p t i m i z a t i o n p r o b l e m . T h e s m o o t h blade g e o m e t r y and Variational optimum the robust s o l u t i o n of the a l i g n m e n t are g u a r a n t e e d by the use of a B - s p l i n e s u r f a c e representation. Marine'^ ro'^d'ler ^ v a l i d a t i o n c a l c u l a t i o n is c a r r i e d out for D T N S R D C p r o p e l l e r 4119 o p e r a t i n g in u n i f o r m flow,
anne prope er ^ c o m p a r a t i v e s t u d y w i t h literature for s o m e D T N S R D C p a r a m e t r i c p r o p e l l e r s e r i e s is given to d e m o n s t r a t e the p r o p o s e d a l g o r i t h m for the d e s i g n of propellers that have c o m p l e x geometi-y.
© 2014 E l s e v i e r L t d . A l l rights reserved.
1. Introduction
The design of a marine propeller is a combination of two processes: determination of optimum radial distribution of blade loads and adjustment of blade shape to develop the load distribu-tion. The first step has been mathematically formulated as a variational problem where the thrust is maximized for a given power, or the torque is minimized for a specified thrust. Betz first solved a variational problem of the load optimization for a lifting line propeller model i n open water by an integral approach, Lerbs extended the solution in radially non-unifoim inflow condition (Kerwin, 1994). They used Munk's displacement theorem to obtain a criterion on the hydrodynamic pitch of the trailing vortex sheet. Kerwin et al. (1986) introduced a direct solution method to the variational problem, which avoids the linearization involved in the application of Munk's displacement theorem. A vortex lattice lifting line representation of propulsor blades is used in their numerical variational method. They showed that the obtained distributions of circulation were identical w i t h the Betz optimization for single propellers. The method provided a theoretical background to the recent development of an open source parametric design tool for propellers and turbines (Epps, 2010). Olsen (2001) developed an optimization method using a lifting surface vortex lattice method for propellers w i t h large amount of rake or skew. He solved a variational problem for the radial circulation distribution on a
* Corresponding author. Tel.: +82 42 865 4747; fax: +82 55 630 7671. E-mail address: kj4747lee@samsung.com (K.-J. Lee).
http://dx.doi.Org/10.1016/j.oceaneng.2014.07010 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. All rights reserved.
lattice of quadrilateral panels whose pitch angle is aligned with the hydrodynamic pitch angle at the blade reference line.
In the next stage of a propeller design, the total f l o w including the induced velocities by the optimum distribution of circulation is made to tangentially follow the blade surface. It is a normal practice that the pitch and camber of the surface are adjusted until the normal velocity components are vanished on the control points of the discretized surface. Greely and Kerwin (1982) developed a vortex lattice technique to manipulate the blade shape by integrating the incremental slope expressed in velocity components and surface vectors at each blade control point. Hoshino and Nakamura (1988) applied a similar technique to their quasi-continuous method. A robust technique is developed by Diggs (1991) using B-spline surface defined on a generalized coordinate system. A n over-constrained system is solved for the amount of B-spline vertex movement to find the mean camber surface through which there w i l l be no velocity. These blade design methods did not make a direct use of the lifting surface to get the optimum radial loads, but i t is supposed to be provided from a separate module utilizing such a lifting line technique as mentioned in the previous paragraph. As Olsen (2001) pointed out in his work, inconsistency could be brought about between a lifting line optimum and a lifting surface o p t i m u m due to the complexity of the blade shape.
The current design method extends upon Olsen's work to take a full advantage of the lifting surface optimization procedure. Although Olsen provided the variational approach on the basis of a lifting surface, the blade design problem was not addressed at all. The present study shows that when the variational optimization
K.-J. Lee et al. / Ocean Engineering SS (2014) 472-479 473
procedure is applied to the deforming mean camber surface, contrary to Olsen's formulation on non-cambered hydrodynamic pitch surface, the final design of blade can be successfully achieved in the course of optimization. The simultaneous optimization and design algorithm could be made possible by incorporating a B-spline surface into the representation of a mean camber surface. The blade mean camber surface is deflned via B-spline geometric description for the least square blade alignment problem. In addition, the current method includes trailing wake contraction model (Greely and Kerwin, 1982) w i t h the lifting surface optimiza-tion and blade alignment, rather than a regular helical wake model used by Olsen. An iteratively working algorithm is described to compose of three modules: the solution of variational problem, the blade shape design, and the trailing wake alignment. The method is validated w i t h the experimental results of DTNSRDC 4119 propeller and demonstrated w i t h the optimization of the DTNSRDC parametric propeller series and compared w i t h the literature.
2. Methodology
2.1. A variational optimization metiiod
The numerical model for the propeller blade in the current work is a lattice of concentrated straight line elements replacing the continuous distribution of vortices on the mean camber surface. By defining a cylindrical coordinate system, as shown in Fig, 1, w i t h the angle 6 measured clqclwise f r o m y-axis when viewed in the direction of positive x, which points to downstream and the radial coordinate r^=y'^-i-z^, the coordinates of the endpoints of these elements on the camber surface can be written w i t h the usual definition of skew 9m(r) and rake Xm(r) as
/ i \
Xc = Xm-\-c\^s--^j sin q)-f cos tp
1 \ cos ft) ^sin ffl ^ „ 27r(fc-l)
y, = r cos &c
Zc = r sin 6c
where the subscript c denotes the camber surface. is defined as an indexing angle for /<:-th blade of a Z-bladed propeller, c(r) is the blade section expanded chord length, and cp(r) is the pitch angle.
}'
/
The blade section camber J[s) is defiined w i t h a nondimensional chordwise coordinate s, which is running f r o m leading edge (where i t is set to zero) to trailing edge (where i t is set to unity).
The singularities are arranged i n a uniform spanwise/chordwise manner so that the M + 1 endpoints of the discrete spanwise vortices are located f r o m hub, r^^, to the tip, R, at radii
( i ? - i H ) ( 4 m - 3 )
m = l , 2 , . . . , l W - h l (2) 4M-h2
and the intersections of the spanwise and chordwise singularity elements at each radius are located at points
s „ = ^ , n = l , 2 , . . . , N (3)
Greely and Kerwin's deformed wake model is employed for the discretization of the trailing wake sheet. It is assumed that there are t w o parts i n the propeller wake, a transition wake region up to the roll up point, and an ultimate wake region consisting of helical tip vortex lines together w i t h line hub vortex. The trailing vortices in the transition wake are extensions of the chordwise vortices on the blade.
According to Kelvin's theorem, the vortex lattice structure can be organized into horseshoe vortices, as shown in Fig. 2a. A horshoe vortex is formed by a spanwise vortex segment and two trailing vortices extending from both ends of the segment follow-ing the chordwise direction on the blade and streamwise direction in the wake. The formulation of variational optimization involves derivatives of force and moment w i t h respect to the circulation. Though the horseshoe structure is appropriate for the analysis problem, i t gives rise to hard-to-arrange terms w i t h the differ-entiation i n the implementation of the variational problem. The more straight-forward evaluation of the expression becomes possible, i f the vortex lattices are rearranged w i t h vortex panels, as shown on the right in Fig. 2b. Each chordwise strip consists of vortex loops on the blade and a trailing horseshoe vortex.
If 7 ^ is the total circulation on the m-th chordwise strip, the strength of the (n, m)th horseshoe vortex, can be assigned through a weight function, K'„, as
z
n ,a = l
or r' = K'„rl (4)
The strength of the (n, m)th vortex panel, Fnm. has the following relation to the total circulation w i t h the sum of the weight function upto the current index:
where Kn= 2 /cj i = l
(5)
An example of the weight function is the well-known NACA a=0.8 loading distribution,
1.0, 5.0(1.0- Sn),
Sn < 0.8
s„ > 0.8 (6)
The total velocity at a point x is the sum of the inflow velocity and induced velocities by the vortex system.
M N
U ( X ) = V „ ( X ) + 2 r l 2 KnKmO^) (7)
It is noted that as the strengths of the trailing horseshoe vortices can be expressed w i t h those of the last t w o panels on the blade by the conservation of circulation and Kutta condition at the trailing edge, the induced velocities from the horseshoes are included in the u* and u* A simple explicit Kutta condition is assumed
Fig. 1. Reference coordinate system.
SO that the last t w o spanwise vortices have strengths proportional to their respective distances away f r o m the trailing edge. The induced velocity at the point x frorri a unit strength panel is found
Fig. 3. B-spline vertex network and surface of a propeller blade: (a) least square fit surface and (b) one of B-spline v é r t e x moved.
by using the law of Biot-Savart,
r3
r . 4 rk
(8)
where 1^ is the length of each side o f a panel. The total forces on the propeller blade are found by adding the contributions from all the panel sides on the blade. They are obtained f r o m the Kutta-Joukowski theorem,
Fside = / ? U ( X ) X r^iie = p r „ d e ( U ( X ) X
1,^,)
(9)The loading distribution of minimizing torque Q for a specified thrust Ty is the solution of a variational problem by forming an auxiliary function w i t h constraint H = Cl+X(J-Ti) (Kerwin et a l , 1986), dH dH dA'' T-T,-d n :0, m = l , 2 , . . . , l W (10)
The evaluation of (10) w i t h the discretized expression of thrust and torque results i n a system of nonlinear equations w i t h M
unknown values of total circulation and an unknown Lagrange multiplier, A. The iterative solution of the nonlinear system can be obtained by the linearization in which the Lagrange multiplier is substituted w i t h the value of the previous iteration for quadratic terms w i t h the circulation
+A' aW)^^tdT(So)_ öQ(V„)
dr,
drl
dTrm = l , 2 . (11)
where f denotes iteration count. The terms o f inflow velocity i n the partial derivatives of Q_ are moved to the right hand side, while those i n the partial derivatives of T becomes the k n o w n coeffi-cients of the Lagrange multiplier Once the solution is converged, the strength of each vortex segment on the blade can be computed using the weight function, and the strength o f trailing wake vortex is constructed w i t h the strengths of the last t w o panels.
The effect of viscous drag can be simply included in the optimization procedure f r o m stripwise application of a two-dimensional sectional drag force Fv(r) = lp(u*(r))^CD(r)c(7-)Ar in the flow direction (parallel to total velocity u*) at each radius. The sectional drag coefficient C d may be either determined experi-mentally, or theoretically calculated. It can be considered, w i t h the
K.-]. Lee et al. / Ocean Engineering 88 (2014) 472-479 475
inclusion of corresponding components i n the second constraint equation of (10), that the circulation should be developed enough to compensate the viscous drag for the desired total thrust.
2.2. Blade alignment
The surface used for finding the optimum circulation distribu-tion does not necessarily agree w i t h the one on which the kinematic boundary condition is satisfied. The design is accom-plished by aligning the blade shape tangent to the total flow. In order to do the alignment in a robust way, a B-spline surface is selected as a mean of blade shape representation, because i t has a significant number of advantages over other blade shape descrip-tions. By using B-splines, i t is possible to uniquely define all points on the surface without interpolation process. The blade may be deflned w i t h a relatively few number of vertex points, whose
manipulation has a local effect on the surface as shown i n Fig. 3b. The blade surface is evaluated using a B-spline w i t h 8 x 10 grid of vertex points.
The flrst step of the blade shape design is to define B-spline surface f r o m the vortex lattice grids. A B-spline surface approx-imation technique is used for this purpose. The B-spline vertices are determined so that the sum of all squared error distances to the approximated surface is minimized in a least-square sense. The points on the leading edge and trailing edge are added as the boundaries of the surface. Next, the change i n the normal velocity \A(u„)j is computed at each center of the vortex panel w h e n each
vertex i n the B-spline vertex net is perturbed by a small amount, Asj. If the amount of perturbation is sufficientiy small, the ratio of the t w o can be thought of as influence coefficient. A system of linear equations can be constructed for the amount to move the B-spline vertices such that the normal velocity is cancelled on the
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surface as A(u„)
K.-], Lee et al. / Ocean Engineering 83 (2014) 472-479
<5(U„)
ds S + (U„)base = 0 (12)
where (u.Oiase's the normal velocity before the B-spline vertices are perturbed. As the number of B-spline vertices are usually less than the number of vortex panels, the system becomes over-constrained, namely there are more equations than unknowns, for which the singular value decomposition is the adequate algorithm to get the converged solution. The vertex is perturbed i n the direction normal to the chordal line on the cylindrical surface at corresponding radii, since this direction is natural to describe the section camber i n the blade shape design.
Ov = cos (pi- sin tp sin 6 j + sin cp cos 9k (13)
The leading edge is not perturbed for the unique solution, while the trailing edge is included i n the perturbation for the adjustment of the pitch of the blade.
2.3. Algorithm
Fig. 4 shows the current methodology. The design parameters, such as the advance coefficient, thrust loading coefficient, and the i n f l o w information ^re specified together w i t h a trial blade surface for the initial lattice generation. A stock propeller can be a best choice for the trial surface. The pitch and camber of this surface is modified during the optimization, but other geome-trical features such as chord length remain unchanged through-out the procedure. The trailing vortex geometry is initially set up f r o m the rotational speed and the undisturbed inflow. It is the circumferentially averaged effective inflow, f o r the propeller blade is assumed to operate i n steady flow in an optimization problem.
The system of equations (11) for the numerical variational problem is constructed w i t h the initial assumptions that the induced velocities are zero and the Lagrange multiplier is - 1 . The regular helical surface is used for the trailing wake i n the veiy first iteration. In the iterative solution of the system, the induced velocities are updated each time the new circulation distribution is obtained. Since the trailing wake geometry is flxed in this stage of the algorithm, the velocity influence function u*„,(x) needs to be computed once and saved at the beginning of the module.
Next, the converged circulation distribution is used for the determination of the blade shape. The grid points of the vortex lattice are passed on to the surface approximation to obtain the B-spline vertices for the manipulation of the blade. The induced velocities are calculated at the center of each panel and the right-hand side of the overdetermined system of equations (12) is established from the normal components of the total velocity. The variations of the normal components w i t h small perturbation of each B-spline vertex are computed i n order to get the elements of the influence coefflcients matrix. Due to the direction of B-spline vertex manipulation, which is made normal to the pitch helix, it is not likely to keep chord length of the blade between iterations. Thus, each time a new B-spline surface is defined f r o m the vertex manipulation, the pitch and camber are extracted f r o m the surface and the lattice grid is updated w i t h the original chord length and extracted features. From the cylindrical coordinates of the leading (subscript '(') and trailing (subscript 't') edges, the extraction is a simple matter of conversion using Eq. (1),
tan (p(r) =
fO\s)-.
r(dtir)-0,(r))
(r)-Xc(r,s) + c(r)(s(r,s)--i/2) sin (p(r)
cos (p(r)
The iteration loop of blade surface alignment is terminated i f the total sum of the normal velocities is examined to have within tolerance, It is noted that the trailing wake geometry is fixed during the process, but the starting points of the horseshoes are perturbed w i t h the trailing edge. It is simply joined to the rest of the trailing wake vortices.
At the final module of the algorithm, the determination of the wake geometiy is conducted with the optimum circulation distribu-tion and the designed blade shape. A partially numerical formuladistribu-tion of Greely and Kerwin (1982) is adopted for its fast evaluation time while preserving reasonable accuracy. They have the evaluation of induced velocities along the vortex done two dimensionally by prescribing the conti-action of the transition wake with user-specified parameters. The induced velocities are computed only at some designated points and interpolated at the desired points using smoothly varying functions. The pitch angle of the wake vortex is aligned by the iterative calculation of induced velocity and the geometry of vortex lines. The next iteration of the numerical varia-tional problem starts again w i t h the aligned ti-ailing wake geometry and the whole procedure is repeated until the nomalized difference of circulation and Lagrange multiplier between iterations falls within a prescribed tolerance. 0.08 0.06 11 O 0.04 0.02 0.00 0.2 ƒ 0.4 0.6 Rndius (r/R) 0.8
Fig. 5. Optimum distributions of circulation . for varying values of advance coefflcients. 5.0 4.0 3.0
I
2.0 1.0 0.0 -Expeiiuientiil Diitii(Jessup) Cun-ent Method -• / m^^^ ^: . /
• //
\
\
1
1\
\
0.2 0.4 0.6 0.8 RiltJiiis (i/R)Fig. 6. Comparison of circulation distributions.
K.-J. Lee et al. / Ocean Engineering S8 (2014) 472-479 477
1
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0IVJ), cunvnl nicOjotl F / C , cunvnl incUuitl I ' / i ) , D T U C l ' 4 U y F / C , D T R C P 4 n ! ) 0.2 Fig. I 0.8 0.4 0.6 Rildhis (r/R)
7. Pitch and camber distribudons, P4119-design. 1.0
Table 1
Propeller 4119 forces from the current computation and experiment.
Method KT I O I C q
Present computation 0.150 0.273
Experiment 0.146 • 0.280
Difference (%) + 2.7 - 2 . 5
This blade design procedure can be readily extended to the design of the contra rotating propeller, for each propeller of a contra rotating set can be designed as single if the induced f l o w by the other propeller is treated as the inflow to the current propeller The t w o propellers are designed alternately by solving each propeller in the flow field including the induced velocity by the other propeller until simultaneous convergences are obtained. An external script controlling the alternate runs should be prepared for this purpose. Buchoux (1995) developed a technique for the calculation of circumferential mean velocities induced by a three-dimensional vortex segment. This technique is utilized i n the current work for the interaction velocity calculation. At the end of one propeller design, the program computes the circumferen-tially averaged induced velocity at the other propeller position and outputs in an interface file to read in the successive design.
3. Applications
Fig. 5 shows the optimum distribution of circulation for a constant thrust loading of C 7 = 0.662 w i t h varying advance coeffi-cients in open water operation. The outiine of the figure is similar to the results presented by other authors, for example Olsen (2001). The two notable behaviors are clearly seen that circula-tions fall off to zero at the root and tip of the blade (zero value is forced at both hub and tip in the figure) and the maximum load is moved toward the tip of the blade for increasing advance coeffi-cient. This proves that the current method captures well the underlying characteristics of the lifting surface in marine propeller application.
478 K.-]. Lee et al. / Ocean Engineering SS (20t4) 472-479
The developed optimization roudne is tested on the design of DTRC propeller 4119. The propeller has three blades w i t h no skew and rake. The propeller was designed to deliver /f7-=0.150 at 7s=0.833 for uniform flow. Jessup (1989) obtained experimentally
for this propeller the radial distribution of circulation and propel-ler forces. The experimental circulation distribution is reproduced from Taylor (1996) in Fig. 6 and compared w i t h the optimized distribution by the current method. Some discrepancy in the circulation is noticeable at inner radii and it is attributed to the distribution that the blade alignment of the present method produces different camber at these radii, which is compared in Fig. 7. The main reason is thought that the present method ignores the existence of hub and the effect of thickness. The existence of hub is known to increase the circulation due to the image effect. The blade should be thickened at the root for the strength, so the thickness has more effect in the inner radii. Otherwise, the method is considered to agree well w i t h the experimental results. Also, there is a reasonable agreement between the optimized and measured forces, as shown in Table 1. It is noted that the calculation is performed w i t h a coefficient of viscous drag 03=0.0085.
In order to demonstrate the lifting surface optimization for the propeller w i t h complex geometry, the five DTNSRDC propellers are tested for an advance coefficient J=1.0 w i t h desired thrust /f7-=0.2055. The outiines of the propellers are compared in Fig, 8. DC4381 is the reference propeller, which has no skew and rake. The other propellers have 36° skew (DC4382 and DC4497), or 72° skew (DC4383 and DC4498), but only DC4382 and DC4383 have skew-induced rake. The propellers are originally designed for the same radial distribution of circulation w i t h the same expanded area ratio (AE/AO = 0.725) and thickness distribution. The viscous drag force is ignored ( C d = 0 . 0 ) in the optimization.
Fig, 9 shows the distributions of circulation for the reference propeller DC4381, and the two propellers with both skew and skew-induced rake, DC4382 and DC4383. The maximum value of the circulation around r/R=0.7 is decreasing with increasing skew. This obsei-vation is also given in Olsen (2001) and, though small, the skewed outline can be said to have an influence on the optimum dishlbution of circulation. Fig. 10 gives comparisons of the results with the option whether the skew-induced rake is included in the shape of the propeller or not. Tlie circulations of 36° skew propellers are compared in the upper figure and 72° skew propellers in the lower Tlie results for the reference propeller are also included in the figures. The distributions of the circulation for the propellers w i t h
0,04 li O 0.03 0.02 0.01 0" .skew (DC4381) 30° skew (DC4382) 72° ske\i' (I)C4383)
- /
/\
• / - //
1 I 0.2 0.4 0.6 RadiiLs ( r / R ) O.S 1.0and without skew-induced rake are seen to be almost identical regardless of the amount of the skew.
The results of torque calculation from the optimization are summarized in Table 2, together with those presented in Olsen (2001). The tendency is quite the same for both methods that the torque is decreasing, hence the efficiency is increasing, with increasing skew, and the rake has a minor effect on the results, as implied by the comparisons of the circulation distributions. However, it is interesting that the torque of 72° skew propeller is decreased by no more Üian
ft
0.04 0.03 0.02 0.01 • OVskcw, n o m k e ( I ) C 4 3 8 1 ) 36° skc^^^ r.ike (DC4382) 36" skew, no rake (DC44!>7) -y \ y \\
/
/
• /\
Y
/
0 0.2 0.4 0.6 0.8 Radius (r/R) 1.0 1=; II 0.04 0.03 0.02 0.01 -0" skew, no m k e ( D C 4 3 8 1 ) 72" ske^v, rake (DC4383) 72" skew, no nikc ( 0 0 4 4 9 8 ) - / ^ X \\
• \
• /
: /
\
\
, , , 0.2 0.4 0.6 Radius ( r / R ) 0.8 1.0Fig. 10. Comparison of circulation distributions with the option of rake (J=1.0): (a) comparison between the reference propeller and 3 6 ° skew propellers and (b) comparison between the reference propeller and 72° skew propellers.
Table 2
Comparison of torque coefflcients with different propellers. () indicates the ratio of the torque to the reference propeller.
Fig. 9. Comparison of circuladon distributions between the reference propeller and the propellers with skew and skew-induced rake (/=1.0).
Propeller 1 0 / f a
ID Skew (°) Rake (skew-ind.) Present Olsen
DC4381 0 No 0,5099 0.5371 DC4382 36 Yes 0.5095 (0.999) 0.5330 (0.992) DC4383 72 Yes 0.5075 (0.995) 0.5279 (0.983) DC4497 36 No 0.5096 (0.999) 0.5324 (0.991) DC4498 72 No 0.5070 (0.994) 0.5263 (0.980) •
K.-J. Lee et at. / Ocean Engineering 88 (2014) 472-479 479
0.6% i n tlie present study, when compared with the reference propeller, while it is decreased by as much as 2% in Olsen's result.
He analyzed, fi-om the terms involved in (l—yF^-zFy using the expression in (9), that the largest contribution to the torque is from the panel side on the U-ailing edge where the onset flow contriburion does not vanish and the induced velocities are fully included. As the induced velocity is most influenced by the trailing vortex sheet and the development of the sheet near the trailing edge can be affected by the blade pitch, the forces can be differently estimated depending on these informations even with the same circulation distributions. It should be noted that Olsen estimated the forces on a blade lattice aligned with hydrodynamic pitch, not on the final design, with the trailing wake modeled in a simpler configuration of regular helical surface. On the other hand, the current optimization method estimates the performance on the designed blade, which is fully aligned with the total flow calculated via the wake modeling with contraction for accurate induced velocity. The dependence of the circulation distribu-tion on the.skew angle in Fig. 9 is thought to be compensated by the designed pitch and camber in the current method. It is standard to apply the skew along the pitch helix, so the design pitch has an effect on the position and direction of the ti-ailing edge lattice relative to the inflow. The current optimization is considered as compatible w i t h synchronous blade design, while Olsen's one is not likely to represent the flnal performance.
4. Conclusion
In this work, a coupled optimization and blade shape design method is developed for marine propellers. The optimization method solves a variational problem, for radial circulation dis-tribution, that is formulated on the deformed mean camber surface whenever it is aligned w i t h the local flow induced by the circulation distribution. The deformation of the surface includes pitch and camber to give the final design of the propeller blade, so that the iterative algorithm converges to the blade shape compa-tible w i t h the optimum circulation. The success of the algorithm is
achieved f r o m the robust representation of blade shape through B-spline surface and the inclusion of the trailing wake modeling for the accurate calculation of induced velocities. It is shown by the coupled method that a more reliable performance estimation, conforming to the optimized circulation, is also made possible through the simultaneous convergence of the blade shape.
The state of the art marine propeller analysis is coupled w i t h a Reynolds-Averaged Navier-Stokes solver for the estimation of effective inflow. The current method is believed promising for the wake adapted design of marine propeller, when combined w i t h such a RANS simulation technique, due to the feature of the flnal blade shape production. The research on this item w i l l be started soon by the authors.
References
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