Vol. 44 . N o . 2 May 1997
SCUIFFSfECHIlK
N u m e r i c a l D e s c r i p t i o n of a P r o p e l l e r B l a d e by Hans Meier B o w F o r m O p t i m i z a t i o n of D i s p l a c e m e n t S h i p s b y M a t h e m a t i c a l P r o g r a m m i n gby Omer G ö r e n , § e b n e m Helvacioglu and M u s t a f a Insel
S t r u c t u r a l a n d A e r o d y n a m i c C a l c u l a t i o n of S a i l s as F l e x i b l e M e m b r a n e s
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A N u m e r i c a l S t u d y of A d d i t i v e B u l b E f f e c t s o n t h e R e s i s t a n c e a n d S e l f - P r o p u l s i o n C h a r a c t e r i s t i c s of a F u l l S h i p F o r m
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S H I P T E C H N O L O G Y R E S E A R C H / S C H I F F S T E C H N I K was f o u n d e d by K . Wendel i n 1952. I t is edited by H . S ö d i n g and V . B e r t r a m i n collaboration w i t i i experts f r o m universities and model basins i n B e r l i n , D u i s b u r g , H a m b u r g and Potsdam, f r o m Germanischer L l o y d and other research organizations i n Germany.
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V o l . 44 • N o . 2 • M a y 1 9 9 7
Hans Meier
N u m e r i c a l D e s c r i p t i o n of a P r o p e l l e r B l a d e Ship Technology Research 44 (1997), 75-79
T h e geometry of a propeller blade can be described by different methods. Surface regions i n the v i c i n i t y o f s t r o n g curvatures and s t r o n g c u r v a t u r e changes show eUiptical characteristics and can therefore be f o r m u l a t e d by series o f t r i g o n o m e t r i c a l f u n c t i o n s . M i x i n g t r i g o n o m e t r i c a l and p o l y n o m i a l f u n c t i o n s p e r m i t s a surface c o n t r o l by a m i n i m u m o f defining parameters. T h e paper offers a concept f o r the propeller geometry t o be defined by a variable number o f characteristic parameters, pending upon the c o m p l e x i t y of the geometry. Generally, the less parameters are introduced, the less smoothness problems occur.
Keywords: propeller, c o m p u t a t i o n a l geometry, surface generation, f a i r i n g
O m er G ö r e n , § e b n e m Helvacioglu, M u s t a f a Insel
B o w F o r m O p t i m i z a t i o n of D i s p l a c e m e n t S h i p s b y M a t h e m a t i c a l P r o g r a m m i n g Ship Technology Research 44 (1997), 80-87
A design procedure optimizes the bow of a displacement ship f o r m i n i m u m t o t a l resistance by m a t h e m a t i c a l p r o g r a m m i n g . T o t a l resistance is assumed t o be the sum of f r i c t i o n a l resistance ( I T T C - 1 9 5 7 ) and wave resistance based on t h i n - s h i p theory. T h e resistance components are reduced t o a q u a d r a t i c objective f u n c t i o n w i t h a set of linear i n e q u a l i t y constraints. T h i s quadratic p r o g r a m m i n g p r o b l e m is then solved by Wolfe's a l g o r i t h m . Results of a case s t u d y are verified by t a n k tests.
Keywords: bow o p t i m i z a t i o n , resistance, quadratic p r o g r a m m i n g , wave resistance, t h i n - s h i p theory
Heinrich Schoop, Michael Hansel
S t r u c t u r a l a n d A e r o d y n a m i c C a l c u l a t i o n of S a i l s as F l e x i b l e M e m b r a n e s Ship Technology Research 44 (1997), 88-97
The three-dimensional steady aerodynamic flow around a sailing boat is s i m u l a t e d . A geo-metrically nonlinear membrane calculation based on finite elements is used f o r the s t r u c t u r a l dynamics c o m p u t a t i o n of the flexible sail. T h e hull is idealized as rigid body, the mast and the boom as elastic beams. T h e aerodynamics combine a v o r t e x l a t t i c e c o m p u t a t i o n of the sail and the hull model w i t h a boundary layer integral m e t h o d . C o m p u t a t i o n a l results f o r a yacht reproduce q u a l i t a t i v e l y w i n d t u n n e l results, b u t q u a n t i t a t i v e agreement of force components is not satisfactory.
Keywords: yacht, sail, air flow, F E m e t h o d , membrane, v o r t e x l a t t i c e , b o u n d a r y layer
George D . Tzabiras
A N u m e r i c a l S t u d y of A d d i t i v e B u l b E f f e c t s o n t h e R e s i s t a n c e a n d S e l f - P r o p u l s i o n C h a r a c t e r i s t i c s of a F u l l S h i p F o r m
Ship Technology Research 44 (1997) 98-108
Systematic numerical tests investigate the influence of flve additive bulbs of varied length and breadth on the resistance and self-propulsion characteristics of a tanker, neglecting the free-surface effects. A finite volume m e t h o d calculates the viscous fiow at model and f u l l scale. T h e propeller is modeled by an actuator disk.
Keywords: Navier-Stokes, bulbous bow, full-scale c o m p u t a t i o n , propulsion, tanker, resistance
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Numerical Description of a Propeller Blade
H a n s M e i e r ^
1. I n t r o d u c t i o n
A numerical procedure f o r generating the exact shape of a propeller blade f r o m a few f u n c t i o n s of section parameters over radius is presented. The method does n o t yet include the hub and the f i l l e t between hub and blade. T h e propeller geometry is defined i n a Cartesian coordinate system XYZ:
X is perpendicular t o the y Z - p l a n e (basically in tangential d i r e c t i o n ) , Y i n the d i r e c t i o n of the propeller axis,
Z perpendicular t o the Y axis to the outside, basically i n r a d i a l d i r e c t i o n .
Upper case symbols indicate physical dimensions (e.g. m ) , whereas lower case symbols indicate nondimensional values.
2. T h e f o i l s e c t i o n
The f o i l (wing) section is defined i n nondimensional f o r m by x,y coordinates between the leading edge x = X / L = 0 and the t r a i l i n g edge x = 1 ( F i g . 1). W i t h i n the i n t e r v a l ü ^ u b < Z < Rmax, the chord length is L = L{Z).
The camber y^a = Vcaix) is defined as a series of basic f u n c t i o n s w h i c h are characterized by VcaiO) = y c a ( l ) = 0 (Meier 1989). T h e basic f u n c t i o n s are:
/o = 2 x - - l h = 2 x . { x - l ) , f2 = f o - f u /3 = / 2 - / l , /4 = / 3 / o , k = h h , /e = / s • / o
-Functions of higher degree may be applied, but are neither required nor recommended. W i t h these basic f u n c t i o n s , the camber is expressed as
yca = [ A / 2 . . . / J - C c a = F ( a ; ) - C , „ .
Cca is a c o l u m n vector w i t h n components. U p to 6 characteristic mean line features are used to determine
Cra'-p o s i t i o n o f m a x . camber 0 = F ' ( a ; „ ) •C q u a n t i t y of max. camber Vca (•''m) = F ( a ; „ ) •C derivative at a; = 0 y'ca (0) = F ' ( 0 ) • c derivative at x = 1 y'ca (1) = F ' ( l ) c c u r v a t u r e at a; = 0 y'L (0) = F " ( 0 ) c c u r v a t u r e at x = 1 y'L (1) = F " ( l ) c
o n l y i f Xm is defined
fo[Xm) l + 3/i(a;,„) 2f2{Xm) i2+5fl{Xm))fliXm) 3/4 (a; m) (3 + 7/i(a;,„))/3(a;,„) " - 1 0
fl{Xm) f2{Xm) fa{xm) fi(Xm) h{Xm) fsiXm) J/ca (^m)
- 2 2 0 0 0 0 yca(o)
2 2 0 0 0 0 yca(i)
4 -12 8 - 8 0 0
4 12 8 8 0 0 y'Ui)
U p t o 5 conditions m a y be dropped by deleting the respective rows o f the equation system and a corresponding number of last columns of the square m a t r i x . A p p l i c a t i o n of c o n d i t i o n no. 2
canilx:r anti üiiu! jiciiü di^iribulioii
re side;
mean lino, Ihickiicss dislribulion
-0.5 O 0.5 X = I • cos(plii)
F i g . 1: Foil section of a propeller blade ( t o p ) , l o g a r i t h m i c a l l y p l o t t e d curvature ( b o t t o m ) , separately f o r suction and pressure side. Zero curvature at the t r a i l i n g edge is, in this case, specified by i n p u t data.
F i g . 2: C o m b i n a t i o n of mean line and thick-ness d i s t r i b u t i o n . Off'set points are concent r a concent e d near concenthe greaconcentesconcent curvaconcenture due concent o concent r i -gonometrical p a r a m e t r i z a t i o n .
demands the existence of c o n d i t i o n no. 1, else Xm,ycxi w o u l d not be an extreme p o i n t . Leaving only one (the first) c o n d i t i o n defines a zero mean fine.
The x,y off'set points of the thickness d i s t r i b u t i o n s are parameterized by
X = 1 - cos{(p) and y = [sin((^) sm{2(p) sin(3(/?)...sin(n(/5)] • C f f , = S{(p) • Cth-The f u n c t i o n y is a n t i s y m m e t r i c in. ip and describes the thickness f o r b o t h the pressure side ( - f < < 0) a-nd the suction side (0 < < | ) ( F i g . 3 ) . Similar t o the camber d i s t r i b u t i o n , Cih is an n-component c o l u m n vector. U p to 6 characteristic features are used t o determine
position of max. thickness q u a n t i t y of max. thickness thickness at t r a i l i n g edge x = 1 leading edge radius
slope at t r a i l i n g edge x = 1 curvature at t r a i l i n g edge .t = 1 0=S'{Xm)-Cth ythiXm) = S ( . T m ) -Cth yth{l) = S ( l ) -Cu ^ = S'(0) -Cth y ; , ( l ) = S ' ( l ) - c , < , ( ! ) = S " ( l ) -CtH only i f Xm is defined
COs(v>m) 2cos(2(/>m) 3cos(3vm) 4cos(4v5,„ .) 5cos{5ipm) 6cOs(6(/3m)
Sm(Vrn) sm(2ipm) sin(3v5m) sin(4(/3m] \ sin{5(pm) sin(6(/Jm)
1 0 - 1 0 1 0 1 2 3 4 5 6 0 - 2 0 4 0 - 6 - 1 0 9 0 -25 0 - 1 0 ytha{xm) yth(o) ^/rö y ; . ( i ) ^/rö y ; . ( i ) Cth =
Parameter ipm signifies the m a x i m u m thickness position = 1 - cos(vJm). U p t o 5 rows of the equation system and a corresponding number of last columns of the square m a t r i x may be deleted i f conditions are dropped. The second condition demands the existence of the first condition. Leaving only the first condition defines a zero thickness.
The best fitting ellipse at the leading edge is given by the p r i n c i p a l half axes (a i n x d i r e c t i o n , Ö in y d i r e c t i o n ) :
a =
3S'(0) • Cti,
The leading edge radius is ro = b'^/a = S'(0) • Cth- F i r s t and t h i r d derivatives of 8 ( 9 3 = 0) are S'(0) = [ 1 2 3 ... n] and S'"(0) = - [ 1 2^ 3^ ... n%
Camber and thickness d i s t r i b u t i o n s combine t o a f o i l section, where the thickness yth is added perpendicularly t o the mean line. T h i s operation demands the tangent vector of yca{x):
- 1 / 2 d-Vc 1 dyca/dx dx (l + {dy,a/dxf)' = 2[/o l + 3 / i 4 / 2 (2 + 5 / i ) / i 3 / 4 (3 + 7 A ) / 3 ] - C , The location vector of the f o i l contour is then f o u n d :
P = X X " 0 - 1 '
y — yca{x) J
+
1 0 •T{x)yth{x)Offset points are concentrated near the leading edge, i.e. near the greatest curvature. Defining a f o i l w i t h camber and w i t h the t r a i l i n g edge thickness as the only i n p u t data w o u l d result i n a half-elhpse due t o the t r i g o n o m e t r i c p a r a m e t r i z a t i o n . T h e camber (mean line) d i s t r i b u t i o n is polynomically parametrized as there are no curvature concentrations at the edges.
C o m b i n i n g p o l y n o m i a l and t r i g o n o m e t r i c a l expressions is no obstacle, as there is a clearly defined relation of x = 1 - cos((^) between the t w o f u n c t i o n types which also defines the t w o sides o f the mean line (suction and pressure) pending on the sign of ip, w i t h = 0 indicating the leading edge being assigned t o both sides.
3. T h e b l a d e
T h e blade geometry is also described by a combination of p o l y n o m i a l and t r i g o n o m e t r i c a l f u n c t i o n s . T h e blade outline e.g. f o r m u l a t e d as chord length d i s t r i b u t i o n is f a v o u r a b l y described by a t r i g o n o m e t r i c a l p a r a m e t r i z a t i o n which gives an elliptically approximated blade contour near the t i p radius ü m a x i whereas the pitch d i s t r i b u t i o n w i t h no elUptic terminals demands a p o l y n o m i a l f u n c t i o n . T h e t w o f u n c t i o n types are distinguished by:
- polynomials f [ Z ) = [1 C C ^ - C ] ' C = G(C) • C w i t h C = { Z - i2hub)/(^:max - Rhnh) and 1 < n < 7
- t r i g o n o m e t r i c a l f u n c t i o n s /{tp) = [ s i n ( ^ ) sin(2'^) sm{dip)...sm{mp)] • C = S{^) • C w i t h c o s ( ^ ) = C and 1 < n < 8
A r r a n g i n g predefined f o i l sections at discrete radius stations, followed by a refined inter-polation a l g o r i t h m , is common practice. T h i s proposal first defines characteristic f o i l section parameters as f u n c t i o n s of Z before i t builds up f o i l sections at a r b i t r a r y radius stations.
Up t o 12 f u n c t i o n s f { Z ) and ƒ (C) according to Section 2 are f o r m u l a t e d . T w o f u n c t i o n s are a m i n i m u m i n p u t , one f o r camber and one f o r thickness d i s t r i b u t i o n . Following the f o i l section definition of Section 2, the camber parameter functions are defined as:
Xm{Z) = G(C) • C^rn- chord length Ym{Z) = G ( C ) - C y „ - l e n g t h u n i t
Y^{Z) = G ( C ) - C , „ , Y{iZ) = G ( C ) - C , , '
Y^'iZ) = G ( 0 - C ^ o " / l e n g t h u n i t Y{'{Z) = G(C) • Cy^"/ length u n i t
A l l functions are polynomials G(C) which generally p e r m i t non-zero f u n c t i o n values at i?max-Functions Ym{Z), Yq{Z), and Y{{Z) need t o be forced to G ( l ) = 0 t o avoid surface singularities at the t i p region. S i m i l a r l y the thickness parameter functions are defined as:
position of max. camber q u a n t i t y of max. camber derivative at a; = 0 derivative at a; = 1 curvature at x = 0 curvature at x = 1
position of max. thickness Xm{Z) q u a n t i t y of max. thickness Ym{Z) t r a i l i n g edge thickness Y i {Z)
leading edge radius RoiZ) derivative at a; = 1 (-^) curvature at x = 1
G(C) • C^m- chord length S{tl}) • Cym' length u n i t S{i}) • Cy^- length u n i t S{^) • Cro - length u n i t G ( C ) - C y , ,
G(C) • C y o » / length u n i t Yl'iZ)
T h e coefficient matrices are determined similar t o the examples i n Section 2, by inversion of the i n p u t d a t a matrices. U p t o 5 f u n c t i o n s of the camber and also 5 f u n c t i o n s o f the thickness d i s t r i b u t i o n s may be dropped. T h e m i n i m u m is 2 functions consisting o f one parameter value each, i.e. all f o i l sections can be defined by t w o numbers, e.g. camber slope at the t r a i l i n g edge ^i(-Rhub) = 0-05 and a t r a i l i n g edge thickness y i ( i ? h u b ) = 0-02- length u n i t . A m a x i m u m i n p u t would consist of 12 f u n c t i o n s based on 8 discrete data each, or 96 i n d i v i d u a l parameter values.
The chord length,
Ci{Z) = S(V') • Cci • length u n i t ,
is defined as a t r i g o n o m e t r i c a l f u n c t i o n by up t o 8 discrete d a t a points at iZ^ub < Z < ümax-T h i s leads t o the physical f o i l dimensions of
Ci{Z)
Fig. 4 shows the propeller blade, w i t h o u t p i t c h , skew or rake, w i t h the superimposed f o i l sections, as two-dimensional (plane) figures.
The pitch d i s t r i b u t i o n
F i ( Z ) = G ( C ) - C p < - 2 J ? „ , a x
is defined as a p o l y n o m i a l f u n c t i o n by up t o 8 discrete data points at iï^ub < Z < iïmax- I t describes the helical progress achieved by one revolution, f r o m which the p i t c h angle results as 13 = atan {Pt{Z)/{2-kZ)). T h e nondimensional value of G(C) • C^t signifies the p i t c h / d i a m e t e r ratio. A m i n i m u m i n p u t of one discrete p o i n t defines a constant p i t c h . W i t h k n o w n p i t c h angle /3, the i n d i v i d u a l , still plane f o i l sections are t w i s t e d about the Z axis by t r a n s f o r m i n g the X,Y coordinates of the f o i l contour:
" X ' X Y
L y .
' X ' V ' cos(/?) - s i n ( / ? ) X Y —r L sin(/3) cos(/3) Y
The propeller skew d i s t r i b u t i o n
5fc(Z) = G ( C ) - C , f c - C ' - l e n g t h u n i t ,
i.e. the tangential s h i f t i n g of the f o i l sections {X X + Sk{Z)), is defined as a p o l y n o m i a l f u n c t i o n by up t o 8 discrete data points at iïjiub < Z < i?max- A H f o i l sections, except the one at ü h u b are s h i f t e d by Sk{Z) along the x axis. I n p u t of only one d a t a p o i n t gives a quadratically d i s t r i b u t e d skew.
The propeller rake Rk{Z) = (-RkiRmax), i-e. the s h i f t i n g of the f o i l sections in axial direction (Y Y+Rk{Z)), is linearly d i s t r i b u t e d over the radius and defined by the i n p u t o f the m a x i m u m rake Rk{Rmax) only. U p to here, all f o i l sections are still plane figures prependicular t o the Z axis.
Another t r a n s f o r m a t i o n places the f o i l sections i n t o c y l i n d r i c a l shapes of constant r a d i i r , where a = X/Z:
' X ' sin ( a )
z —y COS(Q;)
Fig. 5: Z,Y ( l e f t ) and Z,X p r o j e c t i o n of the f u l l y developed blade; suction side by solid lines, pressure side by dashed lines
4. C o n c l u s i o n
The propeller blade geometry is expressed by t w o different series o f basic f u n c t i o n s : t r i g o n o -metrical and p o l y n o m i a l . A p p l y i n g one or the other set of f u n c t i o n s depends on the characteristic of the respective curve. I n combination, this method permits a blade d e f i n i t i o n w i t h a m i n i m u m of leading parameters. Radius and cord length dependent sectional curves are n o t segmented and thus continuously differentiable up t o an a r b i t r a r y degree, which reduces not only the error sensitivity b u t also p o t e n t i a l smoothing problems. C o n t r a r y t o other methods, the blade t i p is smooth, f a i r and free of singularities, solely thanks t o the sensibly selected t y p e of basic f u n c t i o n s for the respective parameters.
R e f e r e n c e
MEIER, H . (1989), A useful polynomial representation of multiply continuous functions in the unit domain, in Theory and practice of geometric modeling, Springer, p.3-16
Bow Form Optimization of Displacement Ships
by Mathematical Programming
Ö m e r G ö r e n , § e b n e n i H e l v a c i o g l u , M u s t a f a I n s e l , I s t a n b u l Tech. U n i v . ^
1. I n t r o d u c t i o n
Whereas m a t h e m a t i c a l p r o g r a m m i n g techniques are well-established i n s t r u c t u r a l design problems, studies f o r o p t i m a l hull f o r m s f o r m i n i m u m t o t a l resistance have only recently become a p a r t of the t o t a l design process. T h e problem of hull shape o p t i m i z a t i o n f o r m i n i m u m resis-tance was f i r s t studied by Weinhlum et al. (1957) and later by Webster and Wehausen (1962) using Lagrange multipHers and equality constraints. Hsiung (1981) and Suzuki et al. (1982) applied nonlinear p r o g r a m m i n g methods i n w h i c h the objective f u n c t i o n was taken as the t o t a l resistance a p p r o x i m a t e d by the sum of f r i c t i o n a l resistance and thin-ship wave resistance. Re-cently, Wyatt and Chang (1994) gave an o p t i m i z a t i o n approach using slender b o d y theory. T h i s paper adopts the procedure given by Hsiung and Shenyan (1984) in f o r m u l a t i n g the objective f u n c t i o n and extends Gören and Calisal (1988) t o a design procedure f o r the shape o p t i m i z a t i o n of the f o r e b o d y of vessels w i t h p r o t r u d i n g bulbous bows.
2. O p t i m i z a t i o n P r o c e d u r e 2.1 O b j e c t i v e f u n c t i o n
We chose as objective f u n c t i o n the sum of f r i c t i o n a l resistance and wave resistance. F r i c t i o n a l resistance is calculated using the I T T C - 1 9 5 7 f o r m u l a and wave resistance by Michell's integral. The f o r m u l a t i o n o f the t o t a l resistance in terms of ship's offsets is described b r i e f l y i n the f o l l o w i n g using the same n o t a t i o n as Hsiung and Shenyan (1984) or Gören and Calisal (1988).
_ WL6(DWL) WL5 WL4 WL3 WL2 WL1(B.L) 18 18 19 19 20 20 20 (A.P) F i g . 1: C o o r d i n a t e system
T h e hull surface is defined as ( F i g . 1)
F i g . 2: Meshing of the centerplane (F.P)
^ = i^(e,C)- (1)
C o o r d i n a t e s ^ , rj, and C are nondimensionalized using length L , beam B and d r a f t T:
x = i / L , y = v/{B/2), z = C/T (2) Thus the nondimensional expression f o r the hull surface is
f{x,z) = {2/B)Fi^X)- (3) T h e hull surface is a p p r o x i m a t e d by the t e n t f u n c t i o n s . T h i s requires the centerplane t o be
meshed as i n F i g . 2. G r i d generation i n the forebody must allow a p r o t r u d i n g b u l b in our case. Using the u n i t t e n t f u n c t i o n h^^'^^x,z), one may a p p r o x i m a t e the hull surface as
/ J
fix,z)«h{x,^) = E E y > j h ^ ' ' ' H x , ( 4 )
1=1 i = i
yij is tlie liull offset at p o i n t {xi, Zj).
The f r i c t i o n a l resistance o f a stiip is
1
RF = ^pc'SCf. (5) S is the w e t t e d surface area, p density of water, c ship's speed, and Cj the f r i c t i o n a l coefficient
(according t o I T T C - 1 9 5 7 ) . T h e w e t t e d surface area is calculated by:
S ^ 2 j J [ l + Fl + FlYl^d^d(: (6)
and are the derivatives w i t h respect t o ^ and ( respectively, and So is the p r o j e c t i o n area of the wetted surface o f the ship on the centerplane. I f the i n t e g r a n d i n (6) is expanded i n t o a Taylor series a b o u t = 0 and = 0, the resultant expression can be linearized using the t e n t functions:
Tedious b u t s t r a i g h t f o r w a r d integrations i n (7) yield the explicit expression o f the w e t t e d surface area which is q u a d r a t i c in yij. T h u s , i f the ship's offsets yij are converted i n t o a c o l u m n vector, (7) can be r e w r i t t e n as
S = 2LT + Y , SiV? + E E ""^m = 5o + s, • y + y ^ . A • y . (8) i 1 = 1 j = i
Si and yf^ are numerical i n t e g r a t i o n m u l t i p l i e r s and baseline offsets, respectively, a^j m a t r i x coefficients. S is t h e n used in Rp which is nondimensionalized by SpgB'^T^/(nL):
CF = c^ + cb-y + y'^ -AF -y (9)
T h e wave resistance of a t h i n ship m o v i n g w i t h a constant velocity c can be a p p r o x i m a t e d by Michell's i n t e g r a l :
poo \ 2
T h i s equation is nondimensionalized by SpgB'^T^/{nL) and the s i n g u l a r i t y at A = 1 is removed by the t r a n s f o r m a t i o n A = -|- 1 t o o b t a i n the wave resistance coefficient Cw as:
n ' / l = [ ' U x , z ) h ^ ] (2joxiu' + l))exp[2'ro{j){z-l){u' + l f ] d x d z (116) \Q J JO Jo \ s m J \ ' L
= 1/(2F,^), and F „ is the Froude number. Cw is also q u a d r a t i c in yij and is w r i t t e n f o l l o w i n g the same way i n o b t a i n i n g (8):
IxJ IxJ
= E E dmnVmyn = y ^ • D • y (12) m = l n=l
The coefficients dmn are given in detail i n Hsiung (1981). T h e i m p r o p e r i n t e g r a l in ( U a ) converges quickly t o the l i m i t i n g value f o r the i n t e g r a t i o n interval [ 0 , 2 ] f o r moderate Froude numbers. U l t i m a t e l y , using (9) and (12), the t o t a l ship resistance coefficient can be w r i t t e n as
CT
=
C F
+ CV
= Co + ci, • y + y ^ • C • y (13)C is the s y m m e t r i c coefficient m a t r i x of the quadratic t e r m .
T h e final f o r m of the objective f u n c t i o n is obtained by supposing r variables i n the y vector as unknowns and the remaining s variables as knowns:
r r r
CT = Cc + J2Piyi
+ E E
^ijViVj = Cc + p - y + y ' ^ ' C - y (14) , = 1 ,-=1 j=lCc is the constant t e r m i n which all of the known terms i n (13) are included, and
r + s Pi = 2 J2
ctj-yi-j=r+l y is now an r-component u n k n o w n vector. 2.2 D e s i g n c o n s t r a i n t s
Constraints m o s t l y used in the present o p t i m i z a t i o n s t u d y are: i) A l l the u n k n o w n off'sets are less t h a n or equal t o the half beam:
Vij < 1 (15)
ii) T h e waterhne slope is less t h a n or equal t o t a n ö : 2L
yr+i,j - Vij < {xi+1 - Xi)—t&n6 (16)
iii) T h e block coefficient of the f o r e b o d y volume is fixed:
C'B = ^ E E Vvi^i+i - X i - i ) { z j + i - Z j - i ) (17)
iv) T h e waterplane area coefficient is kept constant:
^ ƒ - ! J - l
i=l j = l
v ) T h e original off'sets of the i n i t i a l ship are taken as the lower b o u n d f o r the u n k n o w n offsets:
mj > yif (19)
Depending on the specific problems of the design procedure, some constraints can be added, and some constraints may be altered t o o b t a i n a realistic h u l l .
2.3 Q u a d r a t i c p r o g r a m m i n g
T h e general f o r m of q u a d r a t i c p r o g r a m m i n g is expressed as minimize P • y + y"^ • C • y
C must be positive semidefinite, provided t l i a t C is a s y m m e t r i c m a t r i x which ensures t h a t any local m i n i m u m calculated w i l l be a global m i n i m u m . T h e C m a t r i x i n (14) satisfies the above condition.
The system (20) can be linearized according t o Kuhn-Tucker (1951) conditions. Wolfe's (1959) a l g o r i t h m t r a n s f o r m s the linearized system i n t o :
A - y + s = B
2 C - y - H • u - v + z = - p (21) y , u , v , z , s > 0
In the system (21): u and v are the vectors of Lagrange multipliers, z is the vector of a r t i f i c i a l variables, s is the vector of slack variables. We extended Wolfe's a l g o r i t h m t o include negative-valued Bi elements (as in (19)) i n the problem such t h a t new a r t i f i c i a l variables, z', are added t o the rows where the corresponding Bi elements are negative. T h e n , according t o Wolfe's a l g o r i t h m , the Simplex m e t h o d solves the system (21), and accordingly s is discarded f r o m the system which is r e w r i t t e n as
A • y + e z' = B
2 C - y - h A ^ - u - v - ^ E - z = - p (22) y , u , v , z > 0
e and E are the coefficients of z' and z, respectively, obtained as a result o f the process discarding s f r o m the system. A n i n i t i a l basis f o r this system can be f o r m e d f r o m the variables z and z'. A t this stage, a recursive step takes place, whereby one change of basis is made i n the Simplex procedure f o r m i n i m i z i n g the f o r m
n+n*
E
(23)under the side conditions: i f yi is i n the basis, u,- is not a d m i t t e d ; i f u,- is i n the basis, yi is not a d m i t t e d . I n (23), n and n* denote the number of positive and negative elements i n B , respectively. I f the f o r m (23) is positive, the recursive step is repeated. T h i s procedure is t e r m i n a t e d when z = z' = 0.
T w o computer programs have been developed which are executed one a f t e r another. T h e first creates the resistance coefficient matrices which are subsequently used i n the o p t i m i z a t i o n p r o g r a m .
Different i n i t i a l hull geometries may lead t o d i f f e r e n t o p t i m i z e d f o r m s which are shaped according t o the prescribed set of constraints defined i n compliance w i t h the i n i t i a l hull geometry. T h i s , in t u r n , is useful i n discarding the undulations on ship surface and i n eUminating the need f o r an experienced designer t o interprete the o p t i m i z a t i o n results. Since the o p t i m a l p o i n t depends closely on b o t h the i n i t i a l hull f o r m and the set of constraints, and since the designer has always some fiexibility t o alter some of the design constraints such as the volume required f o r generating a bulbous bow, i t is n a t u r a l t o look at the effect of design constraints on the o p t i m a l p o i n t . T h e resultant o p t i m a l f o r m generated by the o p t i m i z a t i o n p r o g r a m points o u t at least a general character of the i m p r o v e d hull geometry. T h e designer can t h e n use another set of constraints i f n o t satisfied w i t h the o p t i m u m . T h e f o r e b o d y block coefficient (or volume allowed f o r the o p t i m i z a t i o n region) is the d o m i n a n t parameter i n the search of a satisfactory m i n i m u m . T h e waterline area coefficient is generally kept constant t o conserve the s t a b i l i t y characteristics of the i n i t i a l h u l l .
A comparative analysis of wave m a k i n g characteristics of the i n i t i a l and o p t i m a l hulls are numerically made by a p r o g r a m using Dawson's (1977) a l g o r i t h m . T h e most succesful hull f o r m
computed by Dawson's a l g o r i t h m
is chosen, its Unes f a i r e d by an adaptive surface generator r o u t i n e , and f i n a l l y tested in the t o w i n g t a n k .
3. A C a s e S t u d y a n d M o d e l T e s t s
A f t e r o p e r a t i n g a tanker of 19500t displacement (parent ship) f o r more t h a n 5 years, the owner decided t o b u i l d a sister ship. Due t o complaints t h a t the ship could not a t t a i n the desired service speed and experienced undesirable v i b r a t i o n s , the hull f o r m was analysed numerically and experimentally. A s a consequence, 20 % of the ship's length b o t h i n the bow and the a f t regions were m o d i f i e d based on engineering expertise, and the effective propulsion power was reduced by more t h a n 10 % based on model tests. T h e m o d i f i e d sister ship was taken as the i n i t i a l ship f o r the o p t i m i z a t i o n procedure, Table I .
Table I : M a i n particulars of i n i t i a l ship
LwL 137.28 m B 22.80 m T ( m e a n ) 7.60 m
CB 0.822 Cp 0.824 CM 0.998 Cwp 0.873 LCB 3.02 m (fore)
T h e f o r e b o d y volume of the i n i t i a l hull t o be o p t i m i z e d was chosen as the volume between bow profile and the 18th s t a t i o n . T h e half-breadths on stations 20, 19^/^, 19 and 18 each being described by 6 waterlines were the unknowns i n the o p t i m i z a t i o n study. T h e resistance coefRcient matrices were c o m p u t e d only f o r the loaded c o n d i t i o n ( T = 7.60m) and at the speed of 15 knots which corresponds t o a Froude number of F „ = 0.21. T h e f o r e b o d y volume was increased g r a d u a l l y f r o m an acceptable m i n i m u m value, and i n the end the f o l l o w i n g set gave a satisfactory m i n i m u m :
i) y^3 < 1
ii) T h e waterline slopes were taken t o be less t h a n or equal t o t a n 25°
iii) T h e f o r e b o d y volume (the volume between bow and the 18th s t a t i o n ) was increased by 2% as compared t o the i n i t i a l hull
iv) CwL was kept constant
Fig. 5: C o m p a r a t i v e wave resistance ciiaracteristics of t l i e parent ship, i n i t i a l ship and o p t i m a l f o r m 100 c o 60 40 20
• Poreril ship (Experiment Residual resistance) O Parent ship (CofTiputation)
• Initial ship (Computation) • Optimal form (Compulation)
12 13 14 15 15 17 V (kn)
F i g . 6: Wave {Cw) and wave p a t t e r n {Cwp) resis-tance of the i n i t i a l and op-t i m i z e d hull f o r m s a (J 0.002( J 0 ~ C i — < T! r g^ " l . I 12 13 Speed Cknots) yiis > OMylf f o r s t a t i o n 19 Vij > 0.97ylf f o r s t a t i o n 18 where y\°^ are the o r i g i n a l offsets.
F i g . 3 compares the transverse cross-sections of the resultant o p t i m a l hull (Hues faired) t o those of the i n i t i a l h u l l . T h e i n i t i a l bow profile was kept t o be the same f o r the o p t i m a l h u l l . T h e computed t o t a l resistance was reduced by more t h a n 12%.
T h e o p t i m a l hull and the i n i t i a l hull were also compared w i t h respect t o t h e i r wave deforma-tions along the ship and wave resistances by a p r o g r a m based on Dawson's a l g o r i t h m . F i g . 4. Theoretical wave resistance results f o r the o p t i m i z e d hull and i n i t i a l h u l l show a nearly 20% reduction of wave resistance. F i g . 5.
M o d e l tests (scale is 1/35) validated our c o m p u t a t i o n s (Figs. 6, 7 ) . T h e experimental wave resistance coefficient is defined as Cw = CT - 1.28CF; the effective power p r e d i c t i o n refers t o f u l l scale. T h e gains at the o p t i m i z a t i o n speed of 15 knots are a p p r o x i m a t e l y 20% i n the wave resistance and 7% in the effective propulsion power.
To ascertain the c o n t r i b u t i o n s of resistance components by direct measurements, wave pat-t e r n analysis is conducpat-ted f o r b o pat-t h models. T h e wave p a pat-t pat-t e r n is analysed in m u l pat-t i p l e l o n g i pat-t u d i n a l cuts, Insel (1990), by a f u l l y a u t o m a t e d acquisition-analysis system using f o u r wave probes. T h e wave elevation is expressed as a sum of a series of discrete waves w i t h angle of 6m and wave
L 1
sooa-n sooa-n
re t7 F i g . 7: Effective power f o r the i n i t i a l and op-timized hull f o r m
F i g . 8: Wave p a t t e r n resistance d i s t r i b u t i o n [DCwp) across t l i e wave angle f o r the i n i t i a l and o p t i m i z e d hull f o r m s
TRACE LENGTH [m)
F i g . 9: Measured wave elevations f o r the i n i t i a l and o p t i m i z e d hull f o r m s , {y coordinates are given as measured f r o m the centerline of the model)
number of Km '•
^ 2rmTy C = E cos(A'm cos dmx) + Tim s i n ( / t „ cos Omx)] cos( —p ^ ) (24)
m = 0
Km and 6m are solutions of Km — KQ sec^ 6m t a n h ( 7 l m ^ ) — 0 and Km sin 6m — 2nnr/W. ]'V is the t a n k w i d t h and H the t a n k d e p t h . Wave resistance is t h e n
/!,VP = - P [ ( Ö + . , 5 ) ( 1 - - | ^ )
+ (25,
m=l
T h e reduction i n wave p a t t e r n resistance is about 30% at 15 Itnots which c o n f i r m s the t o t a l resistance tests. A review of Cw — Cwp of b o t h models suggests t h a t the wave breaking effects are pronounced above 12 knots f o r the i n i t i a l and 13 knots f o r the o p t i m i z e d h u l l . So the optimized f o r m benefits b o t h f r o m wave p a t t e r n and wave breaking resistance gains.
F i g . 8 gives wave p a t t e r n resistance d i s t r i b u t i o n across the wave angle s p e c t r u m f o r 15 knots. The reduction i n transverse waves is effective up to 45° which make up a b o u t 90% of the wave p a t t e r n resistance. F i g . 9 gives measured wave traces f o r b o t h the i n i t i a l and the optimized hulls at 15 knots. The reduction on wave disturbance w i t h the o p t i m i z e d f o r m is clear.
4. C o n c l u s i o n
M a t h e m a t i c a l p r o g r a m m i n g w i t h a proper set of constraints can s u b s t a n t i a l l y reduce the t o t a l resistance of commercial ships by o p t i m i z i n g p r o t r u d i n g bulbs. T h e choice o f i n i t i a l hull f o r m and the set of constraints are i m p o r t a n t f o r the success of the o p t i m i z a t i o n procedure. The procedure removes guess w o r k on hull f o r m improvements and can find o p t i m u m solutions having commercial significance.
5. A c k n o w l e d g e m e n t
T h e s t u d y was p a r t i a l l y f u n d e d by the Undersecretariat f o r Defence I n d u s t r y . T h e authors are t h a n k f u l t o P r o f . A . Y . O d a b a § i whose efforts made this s t u d y possible. T h a n k s are also due to M r . K . T i i m e r f o r his k i n d help d u r i n g the experimental studies.
R e f e r e n c e s
DAWSON, C W . (1977), A practical computer method for solving ship-wave problems, 2nd Int. Conf. Num. Ship Hydrodyn., Berkeley, 30-38
G Ö R E N , Ö.; C A L I S A L , S.M. (1988), Optimal hull forms for fishing vessels, 13th STAR Symp., SNAME, Pittsburgh, 41-51
HSIUNG, C.C. (1981), Optimal ship forms for minimum, wave resistance, J. Ship Res. 25, 95-116 HSIUNG, C . C ; SHENYAN, D. (1984), Optimal ship forms for minimum, total resistance, J. Ship Res. 28, 163-172
INSEL, M . (1990), An investigation into the resistance components of high speed catamarans, Ph.D. thesis, Univ. of Southampton
K U H N , H . M . ; T U C K E R , A . W . (1951), Nonlinear programming, 2nd Berkeley Symp. on Math. Statistics and Probability, 481-492
SUZUKI, K.; HIGUSHI, M . ; M A R U O , H . (1982), Hull form design based on Michell's theory by means of nonlinear optim.ization. B u l l , of the Faculty of Eng. Yokomaha, Natl. Univ. 31, 112-120
WEBSTER, W . C ; WEHAUSEN, J.V. (1962), Schiffe geringsten Wellenwiderstandes mit vorgegebenem Hinterschiff, Schiffstechnik 9, 62-68
W E I N B L U M , G.; W U S T R A U , D.; VOSSERS, G. (1957), Schiffe geringsten Widerstandes, J. Schiffbaut. Ges. 51, 175-204
W O L F E , P. (1959), The Simplex method for quadratic programming, Econometrica 27, 382-398
W Y A T T , D . C ; C H A N G , P.A. (1994), Development and assessment of a total resistance optimized bow for the AE 36, Mar. Techn. 31, 149-160
structural and Aerodynamic Calculation of Sails
as Flexible Membranes^
H e i n r i c h S c h o o p , M i c h a e l H a n s e l , T U Berlin-^
1. I n t r o d u c t i o n
Tlie aerodynamics of sails is similar t o those of airfoils of low-speed aeroplanes, b u t sails are flexible structures subject to a complicated interaction between aerodynamics and s t r u c t u r a l mechanics. The flow around the sail depends upon the sail shape. The flow pressure i n t u r n deforms the mast, the b o o m , the sail cloth, and - due t o geometric restrictions - the sail shape.
Numerical analysis o f sails as flexible membranes started w i t h 2-d theories. Voelz (1950), Thwaites (1961), Nielsen (1967), Jackson (1983), and Grennhalgh et al. (1984) discuss different sail profiles and give basic explanations how sails w o r k . B u t f r o m the s t r u c t u r a l p o i n t of view, f o r the 2-d case a membrane degenerates t o a rope which can carry loads only when i t is fixed at both ends. The leading edge of a sail is indeed fixed at the mast or the forestay, b u t the t r a i l i n g edge is free. T h u s a 2-d sail model cannot describe real sail characteristics.
Milgram (1968) presented a 3-d aerodynamic theory of sails t h a t resulted i n some guidelines f o r the design of sails. B u t whether the aerodynamic pressure is in equiUbrium w i t h the i n t e r n a l forces i n the membrane was n o t answered. Jackson (1985) published a complete 3-d theory for sails. Schoop (1984) and Chatzikonstantinou (1987) presented a theory f o r sails t h a t starts f r o m the sail p a t t e r n used by the sailmaker. Similar t o Jackson (1985), Schoop (1984) and
Chatzikonstantinou (1987), our theoretical description is based on the s t r u c t u r a l dynamics of membranes and on a 3-d aerodynamic theory f o r sails. The membrane is elastic and i t w r i n k l e s in negative stress regions. The flow around the sail is assumed inviscid, except in the boundary layers on the sail surfaces. T h e inviscid flow is modelled w i t h a v o r t e x lattice method ( V L M ) . T h e method of Stock (1978) is applied f o r the boundary layers. The e q u i l i b r i u m condition couples external aerodynamic and i n t e r n a l s t r u c t u r a l dynamic forces.
F i g . l : V o r t e x g r i d ( l e f t ) and F E s t r u c t u r a l g r i d ( r i g h t )
The vortex g r i d of the aerodynamic model consists of q u a d r i l a t e r a l and t r i a n g u l a r and horse-shoe vortex elements. T h i s allows t o model the flow around the h u l l , the j i b , and the mainsail considering the reflection on the water surface. T h e s t r u c t u r a l g r i d consists of q u a d r i l a t e r a l
'This study was supported by Deutsche Forschungsgemeinschaft ^Inst. für Mechanik, Jebenstr. 1, D-10623 Berlin, Germany
and t r i a n g u l a r membrane f i n i t e elements ( F E ) , beam F E and rope F E . F i g . 1 shows a t y p i c a l f o r m a t i o n w i t h t w o sails, several ropes, the mast, and the b o o m . T h e hull is assumed r i g i d .
2. S t r u c t u r e d y n a m i c s of the sail as a d e f o r m a b l e m e m b r a n e
The sail is assumed t o be a t h i n membrane w i t h o u t bending stiffness. T r i a n g u l a r F E , Oden and Sato (1967), and q u a d r i l a t e r a l F E , Haug and Powell (1972), are applied. Four integration points are required f o r the q u a d r i l a t e r a l elements. A s the geometrical conditions of the t r i a n g u l a r element lead t o constant strains, one integration point is sufficient f o r the t r i a n g u l a r element. The membrane strains are defined by c o m p a r i n g the " a c t u a l c o n f i g u r a t i o n " w i t h the undeformed and stressless "reference c o n f i g u r a t i o n " . T h i s leads t o the Green-Lagrangian s t r a i n tensor which is calculated w i t h the d e f o r m a t i o n gradient Fai i n each integration p o i n t as
£a/3^^{FaiFp,-6ap). (1) Following Schoop (1984,1990), the sail p a t t e r n is chosen as reference c o n f i g u r a t i o n . I t consists
of several 2-d parts ("panels"). Each panel of the pattern is divided i n t o several FEs. The actual configuration connects these panels to a 3-d membrane. The material properties of the sail cloth under consideration are similar to those of foils. Thus in the stretching d o m a i n , the constitutive equation is approximated as isotropic elastic
aa/3 = '^G{eal3 + ^ _ ^e-y-ySap)-^ (2) Due t o the effect of w r i n k l i n g in negative stress domains, a f i c t i t i o u s , much smaller stiffness
( " d u m m y stiffness") is assumed. T h e v i r t u a l i n t e r n a l work at the i n t e g r a t i o n p o i n t is the product of the 2nd P i o l a - K i r c h h o f f stress tensor aajS and the v a r i a t i o n o f the s t r a i n tensor Sap
SW = AptaaphanFpiSxin = ApTapEapinSXin. (3) Ap is the subarea corresponding t o the i n t e g r a t i o n p o i n t , the geometry m a t r i x , t the
mem-brane thickness, Tap the memmem-brane force tensor, Eapin the derivatives of the strains, and 5xin the v i r t u a l node displacement. T h e expression i n f r o n t of Sxin i n (3) denotes the i n t e r n a l force P/„ which is assigned t o the integration p o i n t .
W i t h the derivates T^p^s of membrane forces w i t h respect t o the strains
Kp^s = j f f = 2Gt{5a^6ps + Y^S^ysSap), (4)
the stiffness m a t r i x Sinkm for the integration p o i n t is obtained as dPj^
Sinkm = -E—~ = Ap{EapinE^SkmTap.yS + Taphanhpm^ik) • (5) tJXkm
The sum of the nodal forces f r o m the adjacent elements must be i n e q u i l i b r i u m w i t h the external loading P/^ for ^ach node
For given loading, the e q u i l i b r i u m equations (6) are a nonlinear algebraic system f o r Xin- Start-ing f r o m a f i r s t estimate x^^, (6) is solved by the Newton-Raphson m e t h o d . T h e constructive elements bearing the sail - the mast, the b o o m , and the shrouds - are considered by 3-d elastic beams and ropes.
3. A e r o d y n a m i c s of the s a i l a n d the h u l l
The inviscid üow is described here. Chapter 7 describes the boundary layer influence. T h e V L M describes the steady inviscid flow around the hull and the sails, s i m i l a r to Schoop (1990), Chatzikonstantinou (1987). F i n i t e and semi-infinite straight v o r t e x Unes generate q u a d r i l a t e r a l , triangular, and horseshoe v o r t e x elements on the hull and on the sails.
The vortex lattice fiow Vik is superimposed by the homogeneous flow field of the w i n d , rep-resented by the u n d i s t u r b e d w i n d velocity U i o o - The K u t t a condition is satisfied by a p p l y i n g horseshoe vortices at the leech edges of the sail. The water surface is represented by refiection of the vortex lattice ( s y m m e t r y plane X2 = 0) to compensate f o r the n o r m a l components of the velocities. Thus the velocities are tangential t o the water surface.
The non-penetration c o n d i t i o n demands a tangential flow i n the c o n t r o l points of all v o r t e x elements on the hull and the sails, F i g . 1. The center of the v o r t e x element is used as c o n t r o l point. I n each c o n t r o l p o i n t , the non-penetration condition is described by
UiVik + niVioo = 0. (7) Ui is the n o r m a l vector of the vortex element. The induced velocity Vik i n a c o n t r o l p o i n t Xik
is computed by Biot-Savart's law. I t results as the sum of the induced velocities of all vortex elements. T h e induced velocity of a v o r t e x element consists of the sum of the velocities of the legs
" e l / ria,finite ^ « 3 , 0 0 -j^ ^ Vik = J2^' J2 -^^ijpi^32 - Xji){Xpk- Xpi)J + -^^ijpej^iXpk- Xpi)J . (8)
; = 1 \ m / , „ , t e = l m o o = l / I
In the r.h.s. of ( 8 ) , the influence of finite vortex hnes w i t h the actual coordinates xn and Xi2 and the infiuence o f semi-infinite v o r t e x lines w i t h the wake u n i t vector e,oo are summed up. J and J are the integral kernels of Biot-Savart's law. I n t r o d u c i n g the infiuence m a t r i x
ns,finite ^ n s , o o .j^
Aiki= E ^£ijp{xj2 - Xji){xpk - Xpi)J + —eijpCjooixpk- Xpi)J , (9)
\mfinite = l m c o = l / ;
Vik = AikiTi. (10) (8) reads
Considering (7) yields then
UiAikiTi -{niVioo)k- (11) The vortex strength o f each element is computed by solving the equation system (11). These
algebraic equations describing the aerodynamic problem are linear, because the induced velocity depends linearly on the v o r t e x strength F . T h e force vector Ki acting on a s t r a i g h t line v o r t e x f r o m node xa t o node Xi2 is determined by K u t t a - J o u k o w s k i ' s law
Ki = l } t e i j p V j { x p 2 - X p i ) . (12)
p is the air density, f the correspondent vortex strength, Vj the resultant fiuid velocity, usually taken at the m i d p o i n t of the line. For this velocity, the influence of all v o r t e x lines according to (10) is taken i n t o account, excluding the singular influence of the v o r t e x fine f r o m node 1 t o node 2 itself. T h e non-penetration condition is f u l f i l l e d i n the c o n t r o l points only. T h e velocities in the line m i d p o i n t s thus w i l l not be tangential.
There are several possibilities t o apply (12) to get the loading of the s a ü membrane: The direct force method applies the K u t t a - J o u k o w s k i law i n version (12) w i t h Vj as the velocity at the line m i d p o i n t . T h e force vector on the vortex fine is d i s t r i b u t e d i n equal parts t o the
adjacent nodes as loading (see ( 6 ) ) . From the v i e w p o i n t of the potential flow theory this seems to be a logical discrete a p p r o x i m a t i o n of the t r u e c o n t i n u u m p r o b l e m . B u t i n connection w i t h flexible membranes, this method has some spurious effects: T h e exact c o n t i n u u m solution of the 2-d rigid plate or arc problem has a singular tangential suction force at the leading edge
(inviscid flow w i t h K u t t a c o n d i t i o n ) . The n o r m a l components of the v o r t e x m i d p o i n t velocities (violating the non-penetration condition) are not j u s t numerical errors. These n o r m a l velocity components produce v i a (12) tangential forces w i t h a resultant corresponding to the leading edge suction force. Thus the force vectors of the V L M have the (nearly) correct resultant, b u t the d i s t r i b u t i o n of the tangential components is w r o n g . For a r i g i d s t r u c t u r e , this substantial d i s t r i b u t i o n error is not interesting. For a highly flexible membrane, the w r o n g d i s t r i b u t i o n leads to a wrong shape, and this influences the aerodynamics and the force resultant. Lan (1974), Stark (1971), and Guermond (1989) recommend a m o d i f i c a t i o n of the direct force method. Due to the exact potential f l o w theory solution, the resultant of the tangential forces is s h i f t e d t o the leading edge.
The air pressure metiiod, Cfiatzikonstantinou (1987), sums the forces according t o (12) up f o r each membrane element. The component norm al to the membrane is averaged over the element (average element air pressure). T h e tangential components of the force are neglected as inviscid flow cannot induce shear stresses on the membrane. We apply a modified air pressure metiiod. Instead of the velocity Vj in the v o r t e x Une m i d p o i n t , we use the velocity i n the c o n t r o l p o i n t of each element. T h i s velocity has no n o r m a l component (see (7)) and thus the force vector according t o (12) has no tangential components. T h e numerical differences between b o t h the air pressure methods are small ( F i g . 2 ) , but the m o d i f i e d air pressure method requires less computing e f f o r t . There is no leading edge suction force when air pressure is the only loading. Thus the air pressure method violates the pure potential flow theory i n a global manner. The tangential force components o f the direct method are local violations of the p o t e n t i a l theory that essentially influence the sail shape: Consider a 2-d profile w i t h i n f i n i t e s i m a l thickness i n a parallel fiow. T h e p o t e n t i a l flow theory solution has a singular suction force, except f o r the ideal angle of the parallel flow. I n a real flow, separation w i l l occur. F i g . 3 shows three possible cases. The case in the m i d d l e corresponds t o 'fore w i n d ' or a 'reach course' and cannot be modelled by a potential flow. We have the other cases in m i n d i n this paper. B o t h correspond t o 'close to the wind sailing' (small angles t o the w i n d ) . The small separation bubble at the leading edge in the upper case has the effect t h a t there w i l l be no singular suction force at the real sail. T h i s bubble acts as an a r t i c i f l c a l profile nose. For real sails, there w i l l be a mast or a forestay at the leading edge t h a t w i l l have a similar effect like the bubble.
20 22 24 26 28 am' 20 22 24 26 28 a in"
Fig. 2: Coefficients of propulsion (c^) and drag (CD) over angle of attack a f o r the j i b
Fig. 4 shows a realistic case t h a t corresponds t o measurements of Marcliaj (1982). B y c u t t i n g off the suction force i n the air pressure method, we violate the pure p o t e n t i a l theory, b u t hopefully are closer t o a realistic flow around the sail. The sum of all element forces over the whole sail is the resulting w i n d force Km. F i g . 5 shows the components parallel t o the water surface. T h e tangential velocity j u m p lS.Vik (luff side - lee side) at each c o n t r o l p o i n t follows by c o m p a r i n g the line vortices of an element w i t h an adequate panel v o r t i c i t y t o
K j , - lift force
K q - drag force K y - propulsive force K j , - heeling force
K r - resulting total force v „ - undisturbed wind
velocity D P - pressure point
a. - angle of attack P - position angle
Fig. 5: D e f i n i t i o n of sail forces
Sk are the side vectors of the element, F the corresponding v o r t e x strength. T h e l u f f and leeward velocities at the c o n t r o l p o i n t f o l l o w t o
Vi,l^ff ^ v f f . -^Avik, Ui.iee = •yf^ + ^Ai;,-fc. (14)
vfl. is the m e d i u m velocity at the c o n t r o l p o i n t . Bernoulli's equation relates then
Pk = ^{vL - <n) (15) the velocity j u m p t o the air pressure j u m p pf. at the c o n t r o l p o i n t .
T h e v o r t e x g r i d on the hull is a t o o l to model the displacement effect of the air i n the v i c i n i t y of the water surface as an i m p o r t a n t boundary effect f o r the flow around the sail. Because of separation effects, this model of the hull is rather rough. Hansel (1995) describes the details of the hull g r i d i n c l u d i n g the wake.
4. W a k e c o r r e c t i o n
O n l y the v o r t e x lines on the sail c o n t r i b u t e t o the sail force i n the c o m p u t a t i o n . T h e semi-i n f semi-i n semi-i t e v o r t e x lsemi-ines of the horseshoe vortsemi-ices at the leech edge representsemi-ing the wake are n o t considered. Nevertheless, the wake has an indirect effect on the velocity field of the sail. I f the direction of the wake v o r t e x lines are assumed to be the same as the direction of the u n d i s t u r b e d w i n d , forces on parts of the wake result according to (12). Therefore, the wake should be corrected t o m i n i m i z e these spurious forces. Consequently, the cross p r o d u c t i n (12) should be a zero vector, i.e. the vector of the v o r t e x segment should be parallel t o the velocity vector (streamline). T h e semi-infinite curved v o r t e x line is modelled by a number o f s t r a i g h t f i n i t e
segments and one s t r a i g h t semi-infinite segment, F i g . 6, f o l l o w i n g Kandil (1974). A t the s t a r t of the i t e r a t i o n , all v o r t e x lines of the wake coincide w i t h the direction of the u n d i s t u r b e d w i n d . T h e directions o f all finite v o r t e x segments are corrected according to the velocities i n the leading node points. T h e correction procedure is repeated u n t i l all angles between the velocity vectors and the segment vectors are smaller than 2 ° . T h i s method is called nonlinear v o r t e x method. I f the wake correction is only applied to the leeches of the j i b and the mainsail, the influence on the resulting force is insignificant (less than 1%). B u t substituting the closed vortices at the lower leech ( " f o o t " of the sail) by horseshoe vortices, the influence of the correction increases up t o 10%. F i g . 7 explains: T h e v o r t e x lines at the leech show a small change t o the direction of the undisturbed w i n d , except f o r the domain by the f o o t of the sail. The free vortices at the f o o t are characterized by a roll-up eff"ect because the pressure j u m p is larger. T h e strong changes i n directions m o d i f y the velocity field and the resulting force. Using more finite vortices i n the wake does n o t essentially change the shape of the v o r t e x hnes or the resulting force.
F i g . 6: Wake correction F i g . 7: J i b and mainsail w i t h v o r t e x lines of the wake
5. I n t e r a c t i o n b e t w e e n j i b a n d m a i n s a i l
The combination o f the j i b and the mainsail is investigated by simultaneously solving the equation systems of b o t h single sails. Hansel (1995). T h e angle of attack a is l i m i t e d t o small angles because the inviscid fluid assumption combined w i t h the K u t t a c o n d i t i o n w o u l d n o t be admissible otherwise. T h e angle of attack is related to the boat axis. T h e angle between the boat axis and the chord of the sail is cahed position angle. There is another angle between the sail chord and the sail tangential plane. Thus f o r a < 3 0 ° , the eff'ective angle of attack relative to the sail cloth is small.
Table I shows results f o r a = 2 4 ° . T h e sum of the propulsion forces o f the t w o single sails is smaller than the combination due t o the gap between the leech of the j i b and the f r o n t edge of the mainsail. C o m b i n i n g the sails, the leech of the j i b is placed i n a high velocity domain of the mainsail lee flow. Accordingly, the velocity at the j i b leech is higher t h a n f o r the single j i b leech and this increases the air pressure j u m p . F i g . 8. The j i b causes a displacement of the mainsail stagnation p o i n t i n direction to the mast. There is a pressure j u m p increase also i n the top domain of the mainsail under the influence of the j i b . F i g . 8. T h i s agrees q u a l i t a t i v e l y w i t h measurements of Marchaj (1982). The numerical s i m u l a t i o n c o n f i r m s the outstanding propulsion properties of a j i b . W i t h only 47% of the area of the r i g , the j i b produces 72% of the propulsion force.
Hansel (1995) investigates the g r i d influence f o r a single j i b . A possible g r i d refinement is characterized by the constraints: T h e edges of the sail panels are edges o f FEs. T h e F E nodes are also nodes o f the v o r t e x g r i d . F i g . 9 shows a possible g r i d refinement i n the m a i n stream direction and the corresponding computed propulsion coefRcient Cy. T h e sparse g r i d m = 2 is too inaccurate. T h e convergence should be better. For the aerodynamic convergence, a quasi-continuous method ( Q C M ) is best, Lan (1974), Stark (1971), and Guermond (1989). B u t a
Table I : Results f o r single sails and f o r the combination of t w o sails result, force vector heeling moment position angle of sails
[N] [Nm] [degree]
j i b - 3 . 7 0.8 10.8 4.4 9.4
mainsail - 3 . 6 - 0 . 5 10.9 6.0 6.4
j i b and mainsail - 8 . 2 0.4 33.1 17.4 9.4/6.4
s i n g l e s a i l s c o m b i n a t i o n
F i g . 8: Pressure d i s t r i b u t i o n i n the j i b and the main-sail as singles and as combination; isoline no. x 40 gives pressure j u m p i n N / m ^ 0.18 0 , 1 7 0.16 0.15 O . H 0.13 0.12 0,11¬ 0.10
applied model grid
F i g . 9: Propulsion coefficient c„ of the j i b depending on g r i d refinement
Q C M g r i d does n o t f i t i n t o the panel structure and f u r t h e r m o r e w o u l d lead t o poor convergence in the s t r u c t u r a l F E c o m p u t a t i o n . For better convergence, different grids f o r aerodynamics and structure dynamics should be applied. A n o t h e r problem is that the simple t r i a n g u l a r and quad-rilateral FEs employed approximate the n o r m a l vector of the sail surface rather roughly, i.e. n o t smoothly at the element boundaries. T h i s influences the aerodynamics substantially. I m p r o v e -ments of the convergence is subject t o current reserach.
6. C o m p a r i s o n of n u m e r i c a l a n d e x p e r i m e n t a l r e s u l t s
We computed results f o r j i b and mainsail (footlength 0.33m) of a yacht model o f about 1 m height. O u r results do n o t agree well w i t h w i n d tunnel measurements, Conradi and Neubelt (1989), F i g . 10. T h e measured propulsion decreases f o r a > 2 8 ° , probably due t o flow separation which is not modelled in our c o m p u t a t i o n . T h e differences f o r small a are s u r p r i s i n g , b u t the exact test conditions (positions of j i b clew and boom) are u n k n o w n and the calculations show that the propulsion depends strongly on these conditions.
7. C o u p l i n g of the 3 - d V L M w i t h the 3 - d b o u n d a r y l a y e r i n t e g r a l m e t h o d
A n integral method is applied f o r the calculation of the 3-d t u r b u l e n t boundary layer. Stock (1978), Stock et al. (1988). T h e x and y m o m e n t u m integral equations and the e n t r a i n m e n t equation are used. T h e two-parameter Coles (1956) profile is assumed t o describe the velocity d i s t r i b u t i o n inside the boundary layer i n m a i n direction and the Mayer (1952) profile and John-ston (1960) profile are used f o r the n o r m a l d i r e c t i o n . The V L M is coupled t o the integral m e t h o d after convergence of the fiow c o m p u t a t i o n . From this e q u i l i b r i u m s i t u a t i o n , the boundary layer calculation is started. The shape of the membrane is the i n p u t f o r the i n t e g r a l method in the discrete f o r m of the Cartesian coordinates .x'i,iufi, a'i.iee above and below the c o n t r o l points Xik w i t h the velocities Vi^y^n and Uj-.iee according t o E q . ( 1 4 ) , F i g . 11 ( l e f t ) .
O
r . ^ . .
20 24 28 32 36 40 d i n "F i g . 10: Propulsion coefRcient Cy over angle of attack a
F i g . 1 1 : I n p u t parameter ( l e f t ) and o u t p u t parameter ( r i g h t ) of the boundary layer cal-culation
edge of the sail. Lacking experimental results f o r sails, we assume a m o m e n t u m thickness of Oil = 0.000177m and a displacement thickness of 5* = 0.000248m f o r the model. These values correspond to the boundary layer f o r a plate i n parallel flow at the f i r s t d o w n s t r e a m c o n t r o l p o i n t . The boundary layer calculation evaluates the development of the boundary layer thickness, the displacement thickness, and the m o m e n t u m thickness, and the skin f r i c t i o n coefRcient. T h e boundary layer influences the n o r m a l velocities according t o the displacement effect. T h i s is taken i n t o account according t o the o u t f l o w concept. T h e n o r m a l t r a n s p i r a t i o n velocities on the contour o f the sail and the skin f r i c t i o n forces Kij are fed back t o the p o t e n t i a l method, F i g . 11 ( r i g h t ) . T h e kinematic flow c o n d i t i o n of the p o t e n t i a l theory (7) is therefore extended t o
riiVik + niVi^ = v_i_. (16) Simultaneously, the external node forces (at the membrane nodes) are altered by the a d d i t i o n a l
skin f r i c t i o n forces K i f . Consequently, the membrane is not i n e q u i h b r i u m . W i t h these f r i c t i o n forces, a new e q u i l i b r i u m i t e r a t i o n is started, v a r y i n g the membrane node coordinates and the vortex strengths according t o the inviscid calculation. T h e n a next boundary layer calculation is started, and so o n . T h e coupling of inviscid and boundary layer c o m p u t a t i o n has converged when the change of the membrane coordinates are under a small l i m i t . I n most cases, less than six coupling iterations are required. Hansel (1995).
T h e consideration of the boundary layer f o r a j i b increases the drag by approximately 23% and reduces the propulsion by 3%. Furthermore, the separation behaviour is interesting f o r sailors. T h e separation c r i t e r i o n is the detection of a negative boundary layer thickness 5 (defined as k/a • H6ii/[Il + 1); Stock 1978) or a value > 2.4 of the velocity profile shape parameter H. F i g . 12 shows separation domains depending on angle of attack a and sheet length f o r an effective w i n d velocity v^o = 10.3m/s. A l u f f separation i n the t o p region is detected f o r a small angle of attack a = 16° and a large sheet length = 0.018m, again f o r the model indicated i n chapter 6. The sail is t o o f a r opened and tends t o flutter. T h i s flutter tendency of the sail f o r t h a t "close t o w i n d " course can be prevented by a shorter sheet length.
The flow f o r = 0.018m under angles of attack f r o m a = 17° t o 33° lies f u l l y on the sail surface. T h e region of attached flow is reduced to a = 26° when the sheet length is shortened. Outside these regions, the flow separates lee sided. For a short sheet length = 0.014m, t h a t is possible f o r a > 2 6 ° . The sail is held too close. These results are c o n f i r m e d by practice of sailing. B u t the corresponding curve of the propulsion coefRcient Cy contradicts the sailing reality. T h e curve should decline when separation occurs. B u t Stock's integral boundary layer method cannot evaluate separation effects correctly. I t s v a l i d i t y is l i m i t e d t o attached flows, Hansel (1995).
