VETP - SHORT COURSE ON NEW TECHNIQUES FOR ASSESSING A N D Q U A N T I F Y I N G VESSEL STABILITY A N D SEAKEEPING Q U A L I T I E S M A R I N T E K , Trondheim 8 - 1 1 March 1993
Participants: Company
Abeking & Rasmussen, Germany Brodosplit, Jugoslavia
Danyard A/S, Denmark
Gusto Engineering bv, Holland Insean, Italy
Karlskronavarvet AB, Sweden
Kvaerner Masa-Yards, Finland Ministry of Defence
Royal Netherlands Navy, The Netherlands Netherlands Coast Guard, The Netherlands Norwegian Marine Directorate
Sedco-Forex, France
SSPA Maritime Consulting AB, Sweden STN Systemtechnic Nord, Germany
Name:
Erwin Karabinski, Dipl.Ing. Damir Bezinovic, Naval Architect Christian Schack, Naval Architect Carlo van dee Stoep, Senior Eng. Spec. Dr. R. Penna
Jan Bergholtz, Naval Architect Mats Olsson, Naval Architect Ismi Lindstr0m
J.L. Perluka, R. Brouwer
A. Schaap, Ing./Naval Architect Dag Liseth, Div. Eng.
Georges Barreau, Naval engineer Jan Lundgren
W A V E I N D U C E D M O T I O N S A N D L O A D S O N S H I P S . T H E O R Y A N D N U M E R I C A L M E T H O D S O. Faltinsen Department of M a r i n e Hydrodynamics Norwegian I n s t i t u t e of Technology N-7034 T r o n d h e i m , N o r w a y
To be presented at U E T P Course on New Techniques for Assessing and q u a n t i f y i n g vessel stability and seakeeping quahties, T r o n d h e i m , M a r c h 1993.
I N T R O D U C T I O N
N o r m a l l y ship design is based on s t i l l water performance. However, i t is possible to incorporate seakeeping considerations f r o m the b e g i n n i n g of the design either by experimental or numerical methods. Im.portant seakeeping variables can be local motions, accelerations, added resistance, s l a m m i n g , w a t e r on deck, l i q u i d sloshing i n tanks and wave bending moments a n d shear forces. We w i l l focus our a t t e n t i o n on heave and p i t c h motions.
We w i l l discuss the state of the a r t of computer aided ship m o t i o n predictions b o t h for conventional and high-speed vessels. For conventional ships t h i s includes linear theories l i k e s t r i p theories, u n i f i e d theory and complete three-dimensional theories. A h i g h speed theory t h a t accounts for the divergent wave systems, are presented. Nonlinear theories are also discussed. The importance and possibility to predict the influence of flow separation on the vertical motions of conventional ships are studied. N u m e r i c a l methods t h a t accurately describe s l a m m i n g on h u l l -sections are discussed.
We w i l l discuss i n details s t r i p theory calculations of heave and p i t c h of a ship at moderate f o r w a r d speed i n head sea. I t is not common to present errors i n ship motion calculations. However we feel t h a t i t is i m p o r t a n t to ensure t h a t the errors are smaller t h a n the m a x i m u m variations of the heave a n d p i t c h of a realistic f a m i l y of h u l l forms. E r r o r s can be divided i n t o h u m a n errors, numerical errors and physical errors. H i m i a n errors mean for instance "bugs" i n computer programs, w r o n g i n t e r p r e t a t i o n of i n p u t and o u t p u t . T h i s error source can be m i n i m i z e d by documentation of proper v e r i f i c a t i o n procedures f o r the computer programs and by standards for q u a l i t y control of use of computer programs. We w i l l present a procedure to estimate n u m e r i c a l and physical errors i n ship motion calculations. T h i s is d i f f i c u l t because we s t i l l do n o t understand properly a l l physical phenomena associated w i t h ship motions. However, we w i l l make a n a t t e m p t and present examples on error estimates.
S H I P M O T I O N T H E O R I E S
Generally speaking s t r i p theories are still the most successful theories for wave induced motions of ships at moderate forward speed. However f r o m a theoretical p o i n t of view one can question strip theories. A s t r i p theory is based on l i n e a r i t y . T h i s means for instance t h a t the ship motions are small relative to the cross-sectional dimensions of the ship. I n practice one "forgets" the l i n e a r i t y assumptions and applies s t r i p theory programs w h e n parts of the ship go out and i n of the water or i n predicting green water on deck. Due to the l i n e a r i t y assumption there are only hydrodynamic effects of the h u l l below the mean free surface level. A s t r i p theory program w i l l not d i s t i n g u i s h between a l t e m a t i v e above-water h u l l forms.
A s t r i p theory is based on potential flow theory. T h i s means for instance viscous effects are neglected. The most severe consequence of this is i n the prediction of r o l l at resonance. I n practice viscous roll d a m p i n g effects are accounted for by empirical formulas.
The way t h a t the forced motion problems are solved i n s t r i p theory, the method cannot be j u s t i f i e d w h e n the frequency of encounter is low l i k e i t may be i n f o l l o w i n g and q u a r t e r i n g seas.The Seakeeping Committee of the 16th I T T C reports for instance substantial disagreement between calculated results and experimental investigations of v e r t i c a l wave loads i n following waves.
S t r i p theories account for the interaction w i t h the f o r w a r d speed i n a simplistic way. The effect of the steady wave system around the ship is neglected. The free surface conditions are s i m p l i f i e d so t h a t the unsteady waves generated by the ship are propagating i n directions perpendicular to the centre plane. I n r e a l i t y the wave systems may be f a r more complex. For instance for h i g h Froude numbers unsteady "divergent" wave systems become i m p o r t a n t . T h i s effect is neglected i n strip theories.
S t r i p theory is also questionable to apply for ships w i t h low l e n g t h to beam ratios. The reason is t h a t s t r i p theory is a slender body theory. O n the other h a n d the Seakeeping Committee of the 18th I T T C concludes t h a t s t r i p theory appears to r e m a r k a b l y effective for predicting the motions of ships w i t h l e n g t h to beam ratios as low as 2.5. There exists d i f f e r e n t types of striptheories. One commonly used method is the STF-method (Salvesen et al. (1970)).
There have been d i f f e r e n t attempts to improve s t r i p theories by f o l l o w i n g a r a t i o n a l approach. One example is Ogilvie & Tuck's theory (1969) for l i n e a r forced heave and p i t c h m o t i o n of a ship. They make a h i g h frequency assumption and show t h a t s t r i p theory is consistent i n the near-field of a slender ship at zero f o r w a r d speed. The story is d i f f e r e n t for f o r w a r d speed. Ogilvie & T u c k include interactions w i t h the local steady flow. This effect occurs both i n the body boundary conditions, i n the free surface conditions and how the pressure forces are integrated. I n the body boundary conditions they include the socalled nL-terms w i t h second derivatives of the steady motion potential. These are d i f f i c u l t to compute i n areas on the body surface w i t h h i g h curvature. Special care has to be
shown at sharp corners, where the terms are singular. Ogilvie & Tuck avoided the problem by using an integral theorem to compute the effect of the nij-terms. The waves t h a t Ogilvie & Tuck predict i n the near-field of a ship at f o r w a r d speed are propagating t n directions perpendicular to the centre plane of the ship. However, t h e i r f o r m are d i f f e r e n t fi-om w h a t s t r i p theory predicts. I n solving the steady f o r w a r d m o t i o n p a r t of the problem Ogilvie & Tuck account only for the transverse wave system. T h i s is appropriate for moderate f o r w a r d speed. A t h i g h f o r w a r d speed the divergent wave system is i m p o r t a n t . There is no clear borderline between h i g h and moderate f o r w a r d speed. One existing d e f i n i t i o n is t h a t h i g h f o r w a r d speed means Froude numbers larger t h a n 0.4.
N e w m a n (1978) has presented a u n i f i e d theory f o r the forced m o t i o n problem of a ship. N u m e r i c a l results for both zero and f o r w a r d speed have been presented by N e w m a n & Sclavounos (1980). B y vmified theory is meant t h a t b o t h h i g h and low frequencies are covered. The theory is based on linear p o t e n t i a l flow and is l i m i t e d to moderate f o r w a r d speed. T h e i r nimierical results f o r added mass and d a m p i n g at zero speed show quite s i m i l a r results as strip theory f o r laJJLIg)^'^ > ~2 (cOg = circular frequency of oscillation of the ship, L = ship l e n g t h , g = acceleration of g r a v i t y ) . For lower frequencies at zero f o r w a r d speed the i m i f i e d theory shows a correct behaviour relative to complete linear 3-D solutions w h i l e s t r i p theory gives unsatisfactory resiolts f o r added mass and damping i n the l o w frequency range. The f o r w a r d speed effect is d i f f e r e n t i n strip theory and u n i f i e d theory. T h e comparative studies w i t h experimental data for added mass and d a m p i n g coefficients i n a moderate f o r w a r d speed range are not conclusive about w h a t theory shows the most correct f o r w a r d speed effect. For zero-speed problems there is no need to base the analysis on slender body theories l i k e s t r i p theory, Ogilvie & Tuck's theory a n d u n i f i e d theory. There are several l i n e a r 3-D n u m e r i c a l methods and commercial computer codes available for calculation of wave induced motions and loads on stationary ships. The theoretical basis of the computer codes are the same. Also for very low f o r w a r d speed or combined current-wave effect there are practical 3-D methods available (see Zhao & F a l t i n s e n (1989) f o r instance). B y l o w f o r w a r d speed is meant (üeU/g < - 0 . 1 5 . ( U = f o r w a r d speed of the ship).
For analysis at moderate or h i g h f o r w a r d speed the s i t u a t i o n is more d i f f i c u l t . Several research groups have developed linear 3-D methods based on the classical linear unsteady fi-ee surface condition w i t h f o r w a r d speed. T h i s free surface condition is appropriate i n the f a r - f i e l d of a ship and accounts properly f o r a l l frequency a n d Froude number effects. However, one can question i f t h i s is the correct free-surface condition to use i n the near-field of a ship irrespective of Froude number and bluntness of the h u l l . Inglis & Price (1981) have developed a l i n e a r 3-D method based on the classical linear imsteady free-surface condition w i t h f o r w a r d speed. Froude numbers ( F n = U/(Lg)^^) 0 and 0.25 were studied. The difference w i t h s t r i p theory was most significant i n the low-fi-equency range. Nakos & Sclavounos (1990) presented a linear three-dimensional fi-equency domain solution, where they modified the classical free surface condition close to the ship to account for local effects. T h e i r results are p r o m i s i n g . The method is l i m i t e d to WgU/g > 0.25. A complete 3-D theory hke IngHs & Price (1981) or Nakos & Sclavounos (1990) requires significant computational t i m e w h i c h makes i t
presently questionable for routine calculations of motions and wave loads on ships. Chapman (1975) has presented a simplified high-speed theory for a vertical surface-piercing flat plate i n unsteady yaw and sway motion. Chapman's method shows good agreement w i t h experimental results, while s t r i p theory is not able to satisfactorily predict added mass and damping values for h i g h Froude numbers. Faltinsen & Zhao (1991a&b) have generalized Chapman's method to any type of slender high-speed ships i n waves. This is described i n the next chapter.
H I G H - S P E E D T H E O R Y
Details about the theoretical and numerical method used to analyze the steady and l i n e a r unsteady flow about high-speed non-planing m u l t i h u l l s i n c a l m water and waves are described by Faltinsen & Zhao (1991a&b). I t is asstmied t h a t the hulls are h y d r o d y n a m i c a l l y independent of each other. T h i s is a reasonable assumption at h i g h speed as long as the hulls are not to close. One can u n d e r s t a n d this b y a n a l y z i n g the steady and unsteady wave systems generated b y a h i g h -speed monohull. The problem is f o r m u l a t e d i n terms of l i n e a r p o t e n t i a l flow theory. A n u m e r i c a l solution for the flow around one h u l l is f o u n d by s t a r t i n g at the bow. The fi*ee surface conditions are used to step the solutions of the free-surface elevation and the velocity potential on the mean free free-surface i n the l o n g i t u d i n a l direction of the h u l l . The velocity potential for each cross-section is f o u n d by a two-dimensional analysis. Transom stern effects are accounted f o r by assuming t h a t the flow leaves the transom stern t a n g e n t i a l l y i n the downstream direction so t h a t there is atmospheric pressure at the t r a n s o m stern.
The wave resistance, the steady vertical forces and pitch moments are f o u n d f r o m the steady flow analysis. The l a t t e r can be used to calculate the vertical position a n d t r i m . The transom stern has an i m p o r t a n t effect on the steady l o n g i t u d i n a l force on the ship. A reason to t h i s can be seen by i n t e g r a t i n g the hydrostatic pressure force over the body surface below the mean free surface level. Since there is atmospheric pressure at the transom s t e m , the hydrostatic pressure force causes a l o n g i t u d i n a l force on the vessel.
The wave excitation forces i n regular waves and the frequency-dependent added mass and d a m p i n g coefficients are f o t m d f r o m the unsteady flow analysis. B y combining t h i s vrith i n f o r m a t i o n about mass d i s t r i b u t i o n and hydrostatic considerations the equations of motion i n six degrees of fi-eedom can be solved. F i g . 1 shows the steady wave elevation according to l i n e a r theory a r o u n d a parabohc s t m t w i t h l e n g t h 1 m , b r e a d t h 0.1 m and d r a u g h t 0.25 m . The Froude n u m b e r was 1. A comparison is made w i t h t h i n ship theory. The agreement between the two methods is reasonable. Since the method by F a l t i n s e n & Zhao (1991 a) neglects the transverse wave system (see O h k u s u & F a l t i n s e n (1990)), i t indicates t h a t the transverse wave system is not i m p o r t a n t for high-speed ships. Roughly speaking the high-speed theory is v a l i d for Froude numbers higher t h a n 0.4. F i g 2 shows a comparison between experimental and n u m e r i c a l values for the wave resistance of a high-speed m o n o h i i l l . The agreement is reasonably good.
Fig. 3 shows computed results of unsteady wave generated by a high-speed ship. The high-speed theory is compared w i t h a t h i n ship theory. A s t r u t w i t h parabohc waterplane area and wedge-shaped cross-sectional area i n unsteady heave motion w i t h f o r w a r d speed is studied. The length of the s t r u t is 1 m , the breadth 0.05 m and the d r a u g h t is 0.2 m . The draught is constant along the whole l e n g t h of the s t r u t . The Froude number is 1.0 and the circular frequency of oscillation co = 8 rad/s. The t h i n ship theory calculations by H o f f (1990) are based on d i s t r i b u t i n g three-dimensional sources over the center plane of the ship. The sources satisfy the classical free surface condition w i t h f o r w a r d speed. The agreement w i t h the high-speed theory is reasonable, b u t not as good as for the steady f l o w problem. I t should be noted t h a t the t h i n ship theory is also an approximate theory. W h a t the comparison indicates is t h a t the neglection of the transverse wave system is reasonable a t h i g h Froude nimiber.
The high-speed theory accounts for the transom s t e m effects only i n a n approximate way. The reason is t h a t the numerical method only accounts for u p s t r e a m effects, i.e. the method has no knowledge t h a t the pressure should be atmospheric at the t r a n s o m stern. The predicted pressure d i s t r i b u t i o n w i l l therefore be i n error i n a close v i c i n i t y of the t r a n s o m stern.
The i n a b i l i t y to properly describe the t r a n s o m s t e m flow w i l l have an influence on the predictions of the v e r t i c a l motions. T h i s is i l l u s t r a t e d by F i g . 4. There are two types of theoretical results. I n one case there are included t r a n s o m s t e m effects. T h i s was done by using the n o r m a l approach up to the s t a t i o n next to the t r a n s o m s t e m . A t the t r a n s o m stern i t is used t h a t the pressure m u s t be atmospheric. T h i s value was used f o r the whole last station. There is no theoretical j u s t i f i c a t i o n f o r doing this for the whole station. The m a i n purpose is to i l l u s t r a t e a possible effect from the t r a n s o m stern on the ship motions a n d accelerations. No special t r e a t m e n t of the local transom stern flow was made i n the other case. The ship model is the same as presented i n F i g . 5. The p i t c h radius of g y r a t i o n is 25% of the ship l e n g t h .
The experimental values presented i n F i g . 4 were given by Blok & B e u k e l m a n (1984). We note t h a t the theory is i n good agreement w i t h experimental values. The description of the local flow around the transom s t e m has smaU effect on the heave m o t i o n , while there are some effect on the p i t c h a n d the v e r t i c a l accelerations i n the bow. I n c l u d i n g "transom s t e m effects" i n the n u m e r i c a l predictions improve the agreement w i t h experimental results.
The high-speed theory presented above can easily be used for catamarans since no hydrodynamic i n t e r a c t i o n between the h u l l s is assumed. O h k u s u & F a l t i n s e n (1990) showed reasonable agreement w i t h experimental values f o r heave and p i t c h added mass and d a m p i n g coefficients of a catamaran. N e s t e g è r d (1990) has shown how the effect of the a i r cushion can be i n c l u d e d a n d Falch (1991) has shown how foilcatmarans can be dealt w i t h .
N O N L I N E A R S H I P M O T I O N S
A l l unsteady theories mentioned above are linear theories. Committee 1.2 of the 10th ISSC has reviewed the state of the a r t i n prediction of strong non-linear wave loads on ships. Presently there are no r a t i o n a l methods available. E x p e r i m e n t a l results of wave bending moments, shear forces and torsional moments i n steep waves can show strong influence of non-linearities. The same is t r u e i n predicting large relative v e r t i c a l motions between the ship and waves, bow flare forces, s l a m m i n g loads and effect of green water on deck. Committee 1.2 of the 10th ISSC recommends practical methods to calculate non-linear wave loads. I m p o r t a n t parts of the methods are exact calculations of F r o u d e - K r i l o f f and hydrostatic pressure forces on the wetted h u l l surface. The incident wave field is described by linear theory. The non-linear hydrodynamics forces due to ship motions and the d i f f r a c t i o n of the incident waves corresponds closely to conventional s t r i p theories i n the l i m i t of small ship and wave motions. The method is therefore not applicable to h i g h f o r w a r d speed and to low frequency of encounter. The non-linear p a r t of the hydrod5mamic forces are reasonable f o r m u l a t i o n s i n the case of water i m p a c t l o a d i n g on the ship due to large vertical relative motions, b u t questionable i n the general case.
I n order to develop physically based numerical tools for nonlinear ship motions, m a n y f i m d a m e n t a l physical problems have to be better understood. We w i l l concentrate on two aspects. Those are the effects of flow separation and the water e n t r y (or slamming) problem. I n a d d i t i o n we may m e n t i o n t h a t the water exit problem a n d the modelling of steep (including breaking) i r r e g u l a r waves need to be addressed.
T h e e f f e c t of flow s e p a r a t i o n o n the motions of c o n v e n t i o n a l s h i p s
I t is w e l l accepted t h a t flow separation matters i n describing r o l l of conventional ships a r o u n d resonance, b u t i t is common practice to neglect flow separation i n the prediction of heave and pitch motions of a ship. However, B e u k e l m a n (1980, 1983) presented experimental results t h a t suggest flow separation can have a n influence on v e r t i c a l ship motions. T h i s was evident i n his studies of a ship model w i t h rectangular cross-sections (see F i g . 6) i n regular head sea waves. The Froude nmnbers were 0.16 and 0.26. As a p a r t of his studies he presented experimental heave d a m p i n g coefficients as a f u n c t i o n of Froude number, Fn = VI^Lg , non-dimensional circular fi-equency of oscillations iHg^Llg a n d a m p l i t u d e of forced heave oscillations 1)3^. Here L is the ship l e n g t h and g is the acceleration of g r a v i t y . P a r t of the damping coefficient is due to l i n e a r wave r a d i a t i o n damping. The nonlinear effects can be i n t e r p r e t e d i n terms of a drag coefficient Cp. This means one w r i t e the vertical force due to flow separation on the ship as
F^ V3 (1)
where p = mass density of the water, = waterplane area, drig/dt = heave velocity. B y equivalent linearization i t follows t h a t
n dt
The drag coefficient depends on the geometrical f o r m , the free surface, the Reynolds m m i b e r and the Keulegan-Carpenter number K C = Ta]^^/D. Here D is the draught. The free surface effect is a f u n c t i o n of (£>g^L/g and the Froude number. For cross-sections l i k e a rectangular cross-section where separation occurs f r o m sharp comers, i t is not expected any i m p o r t a n t Reynolds n u m b e r dependence as long as viscous shear forces do not matter. The l a t t e r may be t m e f o r s m a l l Reynolds a n d Keulegan-Carpenter numbers and can m a t t e r f o r small models and l a m i n a r boundary layer flow.
F i g . 7 shows Beiikelman's experimental results w h e n the nonlinear p a r t of the heave d a m p i n g coefficient is i n t e r p r e t e d i n terms of a drag coefficient. There is a clear frequency and Froude number effect. The experiments were done f o r r\^/D = 1/15, 2/15, 3/15. The data d i d not show any i m p o r t a n t K C - n u m b e r dependence. The Reynolds number dependence is not k n o w n . F i g . 7 shows also n i m i e r i c a l value of C^ obtained by the two-dimensional vortex t r a c k i n g m e t h o d presented by Faltinsen & Pettersen (1987) and B r a a t h e n & F a l t i n s e n (1988). No effects of Reynolds number and viscous shear forces are included. The m i d s h i p cross-section was used i n the calculations and the Froude n u m b e r was zero. The two-dimensional vertical drag force was non-dimensionalized by the beam w h e n the drag coefficient was calculated. I n practice three-dimensional end effects should have been accounted for. T h i s w i l l result i n lower Cj)-values. The n u m e r i c a l results show a s i m i l a r frequency dependency as the experimental values. The frequency dependency implies t h a t the free surface waves influence the vortex shedding. There are no experimental results for zero Froude number, b u t the n u m e r i c a l values for F n = 0 are reasonable relative to the experimental values f o r F n = 0.16.
The vortex t r a c k i n g method can be described by means of Fig. 8. The p r o b l e m is solved as an i n i t i a l value problem. The v o r t i c i t y is concentrated i n t h i n boundary layers a n d free shear layers (S^). The separation points are assumed k n o w n . Outside the v o r t i c i t y domain a p o t e n t i a l flow problem is solved at each t i m e i n s t a n t by means of Green's second i d e n t i t y . On the instantaneous position of the body boundary i t is required t h a t there are no flow t h r o u g h the body surface. O n the free surface Sp inside | y | = b(t) (see F i g . 8) the exact dynamic and k i n e m a t i c free surface conditions according to p o t e n t i a l theory is used. As long as the body surface is nearly vertical at the w a t e r l i n e , there are no numerical d i f f i c u l t i e s i n describing the flow at the intersection between the free surface and the body surface. For y > b(t) where b(t) is a large m m i b e r dependent on time, the flow is approximated by a v e r t i c a l dipole i n i n f i n i t e fluid w i t h s i n g u l a r i t y at y = 0, z = 0. T h i s impHes t h a t a l l waves are inside | y | = b(t). Faltinsen (1977) has shown i n details how the fi-ee surface problem can be handled.
The viscous forces i n the numerical model are due to pressure forces and can be related to the v o r t i c i t y d i s t r i b u t i o n i n the free shear layers and the motions of the free shear layers. B o t h the v o r t i c i t y d i s t r i b u t i o n and the motions of the free shear layers depend on the presence of free surface waves. This is w h y the C^-values presented i n F i g . 7 are dependent on (njLlg . However the presence of the free shear layers do not have an i m p o r t a n t influence on the free surface waves w h e n the KC-number is small. This implies t h a t the linear wave r a d i a t i o n d a m p i n g is not influenced by flow separation.
The method described above has a clear advantage i n analysing separation from sharp corners at small KC-numbers. The effect of the free surface can be included i n an easy way. A n y modes of m o t i o n can be studied. However the method has disadvantages i n long t i m e simulations and i n describing f l o w separation f r o m continuously curved surfaces. I n the l a t t e r case i t is better to use a v o r t e x - i n cell method or a Navier-Stokes solver.
Fig. 12 shows the heave d a m p i n g coefficient for d i f f e r e n t amplitudes of oscillations for the m i d s h i p cross-section of the ship presented i n F i g . 6. The d a m p i n g force was w r i t t e n as the sum of a linear t e r m and a quadratic drag t e r m (see Eq. 1). The linear potential flow d a m p i n g due to wave r a d i a t i o n was f o u n d to agree w i t h a frequency domain solution based on F r a n k Closefit method. The results for Tig^/D = 1/15 were used to derive the Cp-values presented i n F i g . 7. However i t should be noted t h a t the numerical results presented i n F i g . 9 show K C - m u n b e r dependence. The figure also illustrates t h a t the effect of flow separation can be large relative to wave r a d i a t i o n damping, i n p a r t i c u l a r for h i g h frequencies. I n a practical context i t is frequencies a r o i m d a n a t u r a l frequency t h a t is of p r i m e interest. For the two-dimensional body analysed i n Fig. 9 the non-dimensionalized n a t u r a l circular frequency of heave oscillations ö)j^(D/g)'^^ according to the n u m e r i c a l method. The r a t i o between viscous drag damping at y\^JD = 1/15 and linear potential flow damping is 0.09 and 0.29 for respectively oXD/g)"^^ = 0.588 and 0.835. This indicates t h a t this ratio can be large w h e n rjg^ is the order of magnitude of D . However w h e n r\^^ becomes t h a t large, nonlinear p o t e n t i a l flow effects may also matter. The ratio between linear potential flow d a m p i n g and the critical d a m p i n g is 0.13 and 0.066 f o r respectively oXD/g)^^ = 0.588 and 0.835. This suggests t h a t the effect of flow separation m a t t e r i n predicting heave (and pitch) motions of the ship model presented i n F i g . 6 when the frequency is close to resonance. However i n analysing the vertical motions of the vessel i n head sea one should also account for the effect of flow separation due to p i t c h m o t i o n and the incident waves. B e u k e l m a n (1980, 1983) showed experimentally t h a t viscous effects also mattered for p i t c h damping, heave and pitch motion. A n example on p i t c h results for F n = 0.16 are presented i n F i g . 10. The influence of viscous effects is largest around resonance. For instance f r o m the results i n F i g . 10 we find t h a t
1.09 - 0 . 3 4 ^ for co, 'Ig = 2.85
(3)
^5a
1.15 - 1 . 3 5 ^ for CO,
For ships w i t h cross-sections w i t h o u t sharp corners the effect of f l o w separation w i l l be less i m p o r t a n t , while the presence of bilgekeels can make the effect of flow separation more i m p o r t a n t . W e i n b l u m & St. Denis (1950) presented experimental results f o r vertical motions, t h a t showed influence of bilgekeels.
I n order to n u m e r i c a l l y describe the influence of viscous effects on heave and p i t c h motion i t is necessary to generalize the method presented above to include three-dimensional and f o r w a r d speed effects.
S l a m m i n g l o a d s
W h e n calculating slamming on h u l l sections a two-dimensional flow i n the cross-sectional plane is n o r m a l l y assumed. For a general presentation of s l a m m i n g sea F a l t i n s e n (1990).
Zhao & F a l t i n s e n (1993) have presented a boundary element method apphcable for water e n t r y of a broad class of two-dimensional bodies and relative velocity and o r i e n t a t i o n between the body and the water. They have been able to satisfy the exact non-linear free surface conditions w i t h o u t g r a v i t y and at the same t i m e properly describe the local flow at the intersection between the free surface and the body surface. I n c l u d i n g g r a v i t y effects does not represent a problem. The intersection problem represented the biggest challenge. I t was concentrated on the i m p a c t between an i n i t i a l l y calm free surface and a two-dimensional r i g i d body of a r b i t r a r y cross-section. The effect of flow separation from knuckles or from other parts of the body was not incorporated. I t was assmned t h a t the a i r flow h a d no influence on the water flow. The l a t t e r means t h a t a body w i t h h o r i z o n t a l flat bottom or a small deadrise angle (< ~ 2-3°) is excluded. A t the intersection between the free surface and the body surface a j e t flow is created. As a first a p p r o x i m a t i o n the pressure is constant a n d equal to atmospheric pressure t h r o u g h the j e t . This enables one to s i m p h f y the problem. The problem is solved as an i n i t i a l value problem. The method predicts pressure as a f u n c t i o n o f t i m e and
Zhao & F a l t i n s e n (1993) compared t h e i r method w i t h the s i m i l a r i t y solution developed by Dobrovol'skaya (1969) f o r water e n t r y of wedges t h a t are forced w i t h a constant downward velocity. There is no effect of g r a v i t y i n the s i m i l a r i t y solution. Since no numerical s i m i l a r i t y solution results existed f o r deadrise angles a lower t h a n 30°, Zhao & Faltinsen (1993) presented numerical results f o r a d o w n to 4°. The numerical method was based on the analytical f o r m u l a t i o n by Dobrovol'skaya (1969), b u t a d i f f e r e n t n u m e r i c a l solution technique was used i n order to handle smafler deadrise angles accurately. The agreement between the space.
boundary element method and the s i m i l a r i t y solution was good.
A n asymptotic method was used by Zhao & Faltinsen (1993) for bodies w i t h small deadrise angles. "The method is based on matched asymptotic expansions i n a s i m i l a r way as outlined by Cointe (1991). I n the jet-flow region Wagner's (1932) solution was used. A simple composite solution for the pressure d i s t r i b u t i o n was presented. The body has to be two-dimensional and symmetric about a vertical line. The assumptions are otherwise similar to the boundary element method by Zhao & Faltinsen (1993). The m a x i m u m pressure is the same as Wagner's f o r m u l a , b u t the pressure d i s t r i b u t i o n along the body is generally d i f f e r e n t from w h a t Wagner predicted. The asymptotic method presented by Zhao & Faltinsen (1993) seems to converge towards the results predicted by the s i m i l a r i t y solution w h e n a —> 0. The m a x i m u m pressure is well predicted by the asymptotic method (Wagner's f o r m u l a ) even for larger deadrise angles < - 40°. For instance Cpj^^^^ is only 7% larger by Wagner's f o r m u l a t h a n by the s i m i l a r i t y solution w h e n a = 30°. Wagner's f o r m u l a is often used i n practical calculations of m a x i m u m s l a m m i n g pressure f o r any value of a. However i t has no r a t i o n a l basis for very large a-values where i t clearly underpredicts the m a x i m u m pressure. For instance at a = 8 1 ° Wagner's form.ula shows Cpj^^^ = 0.08, while the s i m i l a r i t y solution gives 1.16 (see F i g . 11). I t should be noted t h a t the m a x i m u m pressure occurs at the apex of the wedge w h e n a > - 4 5 ° , while at smaller values of a i t occurs at the spray root of the j e t flow. T h i s is demonstrated i n F i g . 12, w h i c h shows s i m i l a r i t y results f o r wedges w i t h deadrise angles f r o m 20° to 81°. W h e n a is very large the pressure shows a r a p i d change around the apex. Also w h e n a is small, there is a r a p i d change i n the pressure around the m a x i m u m pressure.
One should be careful i n a p p l y i n g results f o r wedges to other cross-sections. The local deadrise angle is not the only i m p o r t a n t body parameter. For instance the local curvature does also matter. F u r t h e r the assumption of constant body velocity, does not account for t h a t acceleration may have importance, i n p a r t i c u l a r for drop test experiments. We w i l l discuss t h i s f u r t h e r by s t u d y i n g s l a m m i n g loads on a bow flare section, w h i c h have been experimentally examined by Yamamoto et al. (1985). The bow flare section was inclined a constant angle d u r i n g the drop tests to account for i n an approximate way the r o l l i n g of the coiresponding vessel. Fig. 13 presents comparisons between experimental and n i m i e r i c a l values of the pressure for the pressure gauges P-2, P-3 and P-4 as a f u n c t i o n of the t i m e . The vertical velocity d u r i n g the experiments and i n the n u m e r i c a l simulations are shown i n F i g . 13. The difference i n the numerical and experimental velocity i n the first p a r t of the time record is of no importance. The large r e t a r d a t i o n (about 3 g) t h a t occurs later on, is of importance. The small discontinuity i n the numerical pressure calculated is due to added mass effects connected w i t h the sudden r e t a r d a t i o n 3g. The numerical simulations are l i m i t e d i n t i m e relative to the experiments. I n the last t i m e i n s t a n t the spray root is at the k n u c k l e . Since flow separation f r o m the knuckle w i l l m a t t e r later on and this feature is not incorporated i n the numerical method, the n u m e r i c a l computation had to stop. We note f r o m F i g . 13 t h a t the n u m e r i c a l method predicts well w h e n the pressure starts to deviate f r o m atmospheric pressure at P-2, P-3 a n d P-4. The magnitude
of the pressure is well predicted for P-2 and P-3, w h i l e the numerical predictions are too large for P-4. A r a i & Matsunaga (1989) have also made n u m e r i c a l comparisons w i t h Yamamoto et a l . (1985) drop test experiments. T h e i r results are also presented i n F i g . 13 and show good agreement w i t h the results by Zhao & F a l t i n s e n . The effect of g r a v i t y as well as separation f r o m the k n u c k l e was incorporated. The finite difference method based on the Volume of F l u i d method by Nichols et al. (1981) was used to solve the t i m e dependent E u l e r equations.
U N C E R T A I N T Y A N A L Y S I S O F S T R I P T H E O R Y C A L C U L A T I O N S
I t is not common practice to make error estimates of ship motion calculations. However, s t r i p theory programs are often used to find h u l l forms w i t h o p t i m u m seakeeping qualities. One w o u l d feel more confident i n o p t i m a l i z a t i o n studies like t h i s i f one could ensure t h a t the errors i n predictions are smaller t h a n the variations of the ship motions and accelerations of the f a m i l y of h u l l forms t h a t is studied.
I t is d i f f i c u l t to make error estimates of s t r i p theory calculations. A reason is t h a t we s t i l l do not understand properly a l l physical phenomena associated w i t h ship motions. However, we w i l l make an a t t e m p t and realize t h a t our estimates can be easily criticized.
A n u n c e r t a i n t y analysis w i l l include three steps 1. L i s t of errors
2. S e n s i t i v i t y of final results to each error source 3. Combination of errors i n final results
E r r o r s can be classified as 1. N u m e r i c a l errors 2. Physical errors 3. H i m i a n errors
N u m e r i c a l errors are errors measured relative to the theoretical basis of the computer program. Physical errors are errors measured relative to physical r e a l i t y , b u t do not contain numerical errors. H u m a n errors are due to misuse of computer programs i n terms of specifying w r o n g i n p u t or i n wrongly i n t e r p r e t i n g o u t p u t f r o m the computer program. I t can also be due to "bugs" i n the computer program. Comparative computer program calculations performed recently show t h a t h u m a n errors should not be disregarded. However, we w i l l exclude h u m a n error sources i n the f o l l o w i n g discussion. One way to m i n i m i z e the possibility of h u m a n error sources is to establish standards f o r verification procedures of computer programs and for q u a l i t y control of use of computer programs.
We w i l l consider s t r i p theory predictions of heave and p i t c h of a ship i n head sea waves. F r o m the heave and p i t c h motions we can obtain the relative v e r t i c a l motions and v e r t i c a l accelerations along the ship. The relative v e r t i c a l motions are i m p o r t a n t i n calculating wave impact loads, green water on deck and added
resistance due to waves. There are two classes of strip theories. One is named O S M and the other one is named S T F M (Salvesen, Tuck, Faltinsen (1970)) or N S M . We w i l l refer to results based on S T F M . We w i l l assume the two-dimensional added mass and damping coefficient are calculated by F r a n k Closefit method, w h i c h is based on a d i s t r i b u t i o n of wave sources over the mean wetted body surface of a cross-section. Lewis-form technique is also sometimes used. T h i s is a conformal m a p p i n g technique t h a t assumes the two-dimensional hydrodynamic properties of a section is adequately described by the cross-sectional beam, draught and cross-sectional area. This approximate representation of the cross-sectional f o r m causes additional error sources t h a t w i l l not be f u r t h e r discussed.
There can be n i m i e r i c a l errors associated w i t h source f u n c t i o n calculation, m a t r i x i n v e r s i o n and the presence of i r r e g u l a r frequencies. I n the f o l l o w i n g text we w i l l concentrate on numerical errors due to number and placement of strips and segments used to describe each cross-section (or strip). T h i s w i l l result i n errors i n added mass and damping coefficients, volume calculations, hydrostatic restoring coefficients, and F r o u d e - K r i l o f f and d i f f r a c t i o n forces and moments. The error w i l l also depend on w h a t numerical integration procedures are used to sum up the effects f r o m each segment and strip. Presently the only w a y to obtain t h i s error estimate is by convergence studies. T h a t means by systematically increasing n u m b e r of segments and strips. The rate of convergence is dependent on the frequency, wave heading, Froude number, response variable, shape of the body, n u m e r i c a l i n t e g r a t i o n procedure, assumed v a r i a t i o n of source density and velocity p o t e n t i a l over each segment, and specified choice of segments and strips.
There are d i f f e r e n t ways to present the errors. One way is to present the error of a response variable i n regular waves as
where f j ^ is the computed value w i t h totally N strips a n d segments and f ^ is the value one would obtain i f there were an i n f i n i t e number of segments and strips. A n estimate of f^ could be obtained by p l o t t i n g f j ^ as a f u n c t i o n of 1/N and e x t r a p o l a t i n g to 1/N = 0. This procedure has the drawback t h a t i t provides us w i t h a large sum of Ej^-values t h a t are functions of the frequency of the regular waves. I t does no directly t e l l us w h a t the error is i n prediction of ship motions i n i r r e g u l a r sea. We have therefore decided to use the f o l l o w i n g measure of the error i n heave and p i t c h predictions
where j = 3 and 5 correspond to heave and pitch, oj is the calculated standard deviation of heave or pitch i n longcrested i r r e g u l a r sea w i t h a given set of strips and segments on each s t r i p . is the extrapolated value w i t h i n f i n i t e number of
strips and segments, is a f u n c t i o n of the wave heading, the Froude ntmiber and a non-dimensionahzed mean wave period . We have presented an example on calculations of e^g and E^^ i n Table 1. A modified Pierson-Moskowitz spectrum was used to represent the wave spectrum. This is uniquely defined by the two parameters H^yg and T2, where H ^ g is the significant wave height a n d T2 is the mean wave period defined by the second moment of the wave spectrum, e^^ is independent of the s i g n i f i c a n t wave height. I n the example presented i n Table 1 n u m b e r of strips are either 20 or 25. I n the case of 25 strips we selected a d d i t i o n a l strips i n the bow and s t e m region of the ship relative to the case w i t h 20 strips. The distance between each s t r i p is never larger t h a n the distance between the stations of the ship. ( I t is assumed t h a t the ship is divided i n t o 20 stations). N u m b e r of offset points are either 7, 10 or 19. T h i s means t h a t t o t a l number of segments on each s t r i p was 12, 18 or 36. The results w i t h 25 strips and 19 offset points were used as Oj^. S t r i c t l y speaking this is incorrect. However we do n o t t h i n k t h i s choice influences our conclusions.
We note t h a t the error is most sensitive to number of strips. The m a x i m u m error is 0.13 for heave and 0.11 f o r pitch. T h i s occurs f o r r 2 ( ^ / L ) ^ 2 ^ 5 ghip lengths L = 100 m , 200 m , 300 m this means respectively T2 = 4.8 s, 6.8 s, 8.3 s. T h i s example provides a w a r n i n g t h a t we should be aware of possible n u m e r i c a l errors due to placements of strips and offset points. However, we believe t h a t i t is possible to keep the error due this error source on a sufficient low level by u s i n g enough strips and offset points.
P h y s i c a l e r r o r s
Physical errors i n s t r i p theory calculations w i l l be categorized as errors due to l i n e a r potential flow effects, viscous effects and non-linear potential flow effects.
h L i n e a r potential flow effects
I n a previous section we discussed t h a t strip theory is a n approximate l i n e a r theory. For instance i t does n o t properly account for a l l oscillatory wave systems generated by the body, the i n t e r a c t i o n w i t h the local steady flow around the ship a n d the three-dimensionahty of the flow. I t was stated t h a t s t r i p theory is questionable f o r h i g h speed problems and for low frequency problems. O u r discussion concentrated on the calculation of added mass and d a m p i n g . We w i l l now discuss the error i n heave and p i t c h predictions. Since there is no generally accepted exact linear theory f o r seakeeping predictions of ship at f o r w a r d speed, we have to r e l y on experimental results to estimate w h a t " t m e " l i n e a r values are. E x p e r i m e n t a l values have also errors, b u t we believe t h a t c a r e f u l l y conducted experiments give less errors t h a n numerical predictions.
G e r r i t s m a et al. (1974) presented comparisons between s t r i p theory and experiments for heave and p i t c h of a series of ships i n regular head sea waves of small amplitudes. The models were derived f r o m the standard Sixty series h u l l f o r m w i t h L / B = 7 and C 3 = 0.7. The w i d t h was m u l t i p l i e d by a constant so t h a t L/B varied between 4, 5.5, 7, 10 and 20. One of the s t r i p theories they used i n t h e i r numerical prediction is quite s i m i l a r to the Salvesen-Tuck-Faltinsen method.
We w i l l refer to t h i s method w h e n we t a l k about s t r i p theory. N u m e r i c a l errors are assumed negligible. I n Table 2 is presented the relative error i n s t r i p theory prediction at the wavelength where m a x i m u m non-dimensionalized experimental heave or p i t c h a m p l i t u d e occurs. T h i s wavelength w i l l be a f u n c t i o n of L / B , F n and type of response. The mean of all relative errors i n Table 2 is 0.12. The m a x i m u m and m i n i m u m relative errors are 0.27 and 0.02. The relative error is frequency dependent and w i l l generally be smaller for wavelengths outside the domain where m a x i m i m i heave and p i t c h amplitudes occur. For small frequencies (large wavelengths) the relative error is negligible. The reason is t h a t the heave and p i t c h motions are very m u c h determined by hydrostatic and F r o u d e - K r i l o f f forces. I t does not m a t t e r t h a t strip theory predicts wrong added mass and d a m p i n g coefficients. For very h i g h frequencies the heave and p i t c h are small and i t is i r r e l e v a n t w h a t the relative errors are. The relative error of heave and p i t c h predictions i n i r r e g u l a r sea w i l l obviously be largest i f the peak period of the wave spectrum is i n the v i c i n i t y of a period where m a x i m u m heave and p i t c h occur. The relative errors of heave and p i t c h i n a short t e r m sea state w i l l be analyzed by s t u d y i n g the standard deviations of heave and pitch. We w i l l l i m i t ourselves to longcrested sea. The errors i n heave and p i t c h are denoted Egg and Egg and are presented i n Table 3. The largest calculated value f o r b o t h Egg and Egg are 0.05. T h i s occurs a t Tc/^IL)^'^ = 2.2. For ship lengths 100 m , 200 m , 300 m t h i s means respectively Tg = 7s, 9.9 s, 12.1 s.
2^ Viscous effect
We have a c t u a l l y already given examples on the errors due to viscous effects w h e n we were discussing Beukelman's (1980, 1983) experiments (see equation (3)).
3^ Non-linear p o t e n t i a l f l o w effects
A s t r i p theory p r o g r a m assimaes linearized free surface and body boundary conditions and use linearized force expressions. We w i l l base our estimates of errors due to non-linear p o t e n t i a l flow effects by analyzing experimental results i n regular waves.
I f we assume the wave slope (2 Q/i^) of the incident waves is s m a l l , a correction to oscillatory forces oscillating w i t h the f u n d a m e n t a l frequency 0)^ of the i n c i d e n t waves can be f o u n d f r o m a t h i r d order approximation. For instance the p i t c h responsethatoscillates w i t h frequency Gag can be w r i t t e n as rig = tig^cosCwgt + 65), where
^ ^ A y ^ A ^ i ^ f (5)
A-^ and A2 are functions of non-dimensionalized frequency, Froude number, wave heading a n d the shape of the ship. A2 is a f u n c t i o n also of the above-water h u l l f o r m . A n i m p o r t a n t factor i n f l u e n c i n g A2 is the shape of the h u l l surface t h a t is p a r t of the t i m e i n the water and p a r t of the t i m e out of the water. I f this p a r t of
the h u l l surface is nearly vertical, i t is expected t h a t A2 is relatively small. For a frequency where the relative vertical m o t i o n is large and the ship has a pronoimced bow flare A2 may be significant. Another case i n f l u e n c i n g A2 may be water on deck or i f the ship has cross^sections t h a t move out and i n to the water. I n the latter cases we cannot outrule t h a t even stronger non-linear effects exist. O'Dea & Walden (1985) studied experimentally the non-linear behaviour of a ship i n regular head sea waves. The Froude n u m b e r was 0.3 and A/L = 1.2. D i f f e r e n t bow forms w i t h v a r y i n g f l a r e were investigated. This is one source of non-linear behaviour. A n o t h e r reason may be t h a t the ship had shallow d r a u g h t at the a f t stations. By f i t t i n g t h e i r experimental data for the parent f o r m to equation (5) we f o u n d t h a t
= 1.1 - 200 ( ^ ) 2 (6)
The v a r i a t i o n w i t h bow f o r m was not very strong. T h i s indicates t h a t large relative vertical motions at the shallow d r a u g h t a f t sections may be an i m p o r t a n t source of non-linearities. The r e s i ü t s for the heave transfer fimctions d i d not show a s i m i l a r strong dependence on (2(^3/^.) as the p i t c h . The results f o r the d i f f e r e n t bow
shapes were more scattered t h a n the p i t c h results. E q u a t i o n (6) shows t h a t non-l i n e a r effects can be i m p o r t a n t . O n the other h a n d i f we w a n t e d to study the occurrence of deck wetness, t h i s occurred a t 2^^/^ = 0.02 i n the experiments t h a t equation (6) is based on. The relative error i n s t r i p theory calculations caused by non-linear potential flow effects is t h e n 0.08.
O'Dea & Troesch (1986) studied the non-linear behaviour of the S7-175 h u l l i n regular head sea waves at F n = 0.2 and (ü(L/g)^^ = 2.4. The non-linear effect was stronger i n heave amplitude t h a n i n the p i t c h a m p l i t u d e . I t is d i f f i c u l t to conclude f r o m these experiments t h a t the data fitted to equation (5). T h i s is p a r t l y due to scatter i n the experiments. The experimental values f r o m D T N S R D C showed approximately 15% lower values for rig^/^g at 2^^/^ = 0.02 t h a n for 2C,/k, = 0.01.
S T R I P T H E O R Y A N D O P T I M I Z A T I O N O F S E A K E E P I N G P E R F O R M A N C E
The errors i n s t r i p theory predictions t h a t we have discussed i n the previous text should be related to the sensitivity of heave and p i t c h to h u l l f o r m and to how m u c h freedom one has to change h u l l parameters i n practical design. We w i l l i l l u s t r a t e t h i s by the example presented by T a k a k i (1989). He studied systematically a f a m i l y of container ships. Bales procedure was used to determine w h a t ship has the best seakeeping qualities. T a k a k i presented t r a n s f e r functions of heave and p i t c h f o r the prototype ship and the new ship. The differences i n t r a n s f e r functions are largest i n the v i c i n i t y of heave a n d p i t c h resonance frequencies. For smaller and larger frequencies the differences i n transfer functions are u n i m p o r t a n t . We have presented i n Table 4 the differences i n heave and p i t c h transfer functions for the periods where the largest transfer functions of the prototype ship occur. These numbers should be compared w i t h the n i m i e r i c a l and physical errors i n s t r i p theory calculations. We w i l l assume the
errors t h a t we have hsted earher are independent of each other and estimate the total error as the square root of the sum of the squares of each error. A first step i n the comparison would be to assume t h a t motion responses are so small t h a t hnear theory is applicable. This has relevance when we w a n t to compare h u l l forms i n moderate sea conditions. We should then combine numerical errors and physical errors due to hnear potential flow effect. We have no way to say accurately w h a t the l i n e a r physical errors are for the ship presented by T a k a k i . We w i l l base our discussion on the data i n Table 2. Generafly speaking we may neglect n u m e r i c a l errors relative to physical errors. A general t r e n d is t h a t there are larger differences i n the heave and pitch transfer fimctions for the new ship and the prototype ship t h a n the errors i n strip theory calculation due to Unear potential flow effects. (Compare Table 4 w i t h Table 2). We can conclude s i m i l a r l y i f we study Takaki's results i n i r r e g u l a r sea and compare t h e m w i t h the results i n Table 2 and possible numerical errors. A next step i n the comparison w o u l d be to consider non-hnear effects. We then have a problem i n how to use the results for regular waves i n i r r e g u l a r sea predictions. There exists no theory f o r doing that. One approximate way is to use a design wave approach. I f we compare the possible errors due to viscous effects (see equations (3) and possible errors due to non-linear p o t e n t i a l effects (see equation (6)) w i t h the results i n Table 4, we see t h a t the error i n s t r i p theory predictions may become larger t h a n the predicted difference i n heave and p i t c h of the alternative h u l l f o r m s .
C O N C L U S I O N S
A n overview over seakeeping theories for ships is given. Generally speaking s t r i p theories are s t i f l the most successful and practical theories for calculations of wave induced motions of conventional ships. The Hmitations of seakeeping theories are discussed. S t r i p theories cannot be j u s t i f i e d for h i g h f o r w a r d speed, for l o w frequency of encounter between the ship and the waves and f o r large relative vertical motions between the ship and the waves.
For high-speed monohulls and catamarans i t is pointed out t h a t the most i m p o r t a n t ship generated waves are due to "divergent" wave systems. A high-speed theory t h a t accoimts for the divergent wave systems, are presented. I n order to develop physicafly based numerical tools for nonhnear ship motions, many f u n d a m e n t a l physical problem have to be better imderstood. I t is concentrated on the effect of flow separation and the water e n t r y (slamming) problem.
I t is indicated t h a t the effect of flow separation can m a t t e r i n the description of vertical motions of conventional ships, i n p a r t i c u l a r f o r h u l l forms w i t h sharp corners l i k e bilgekeels. A n u m e r i c a l two-dimensional method t h a t accounts for the i n t e r a c t i o n between free surface waves and flow separation is presented. I t gives reasonable predictions for F n = 0.16, b u t not for F n = 0.26. The effect of flow separation are both frequency and Froude number dependent.
S l a m m i n g on huU cross-sections are discussed. V e r i f i e d results by an asymptotic method f o r locally small deadrise angles, a s i m i l a r i t y solution and a nonhnear
boundary element method w i t h j e t flow approximation are presented. The intersection problem between the free surface and the body surface requires h i g h accuracy. The nonlinear boundary element method is a useful tool for a broad class of h u l l forms. The local deadrise angle is not the only h u l l parameter i n f l u e n c i n g the s l a m m i n g pressures.
A n u n c e r t a i n t y analysis of s t r i p theory calculations of heave and p i t c h of ships a t moderate f o r w a r d speed i n head sea is given. The most i m p o r t a n t numerical error source is placement of strips. However, the error can be m i n i m i z e d and neglected. This requires convergence studies w i t h increasing number of strips. Physical error sources are divided i n t o errors due to l i n e a r potential flow effects, viscous effects and non-hnear potential flow effects. These error sources cannot always be neglected. However, i f s t r i p theory program are used to find h u l l forms w i t h o p t i m u m seakeeping qualities i n moderate sea conditions, the physical errors i n heave and pitch predictions are expected to be smaller t h a n the m a x i m u m variations i n heave and p i t c h of a realistic f a m i l y of hull forms. I t is d i f f i c u l t to conclude s i m i l a r l y for extreme sea conditions.
T A B L E 1
Example on relative numerical errors due to strips and offset points i n i r r e g u l a r long crested head sea waves on a ship w i t h Cg = 0.66, beam-draught ratio = 2.4, F n = 0.21. (Og^ and Og^ = "true" values of standard deviations of heave and p i t c h a m p l i t u d e , £^3 and = relative error i n prediction of standard deviation of heave, and p i t c h respectively (see Equation (1)), To = mean wave period, L = ship length).
Number of strips Number of offset points «^3--^173 Heave error: Pitch error: 1.0 20 25 25 10 7 10 0.01 0.06 0.02 0.00 0.03 0.06 0.00 0.08 1.5 20 25 25 10 7 10 0.08 0.50 0.13 0.02 0.00 0.11 0.01 0.01 2 . 0 20 25 25 10 7 10 0.15 0.74 0.09 0.00 0.01 0.08 0.00 0.00 2.5 20 25 25 10 7 10 0.20 0.74 0.06 0.01 0.01 0.05 0.00 0.00 TABLE 2.
Example on relative errors i n strip theory calculations of heave and pitch i n head sea due to linear potential flow effects. (Cheave ~ relative error of heave amplitude at the frequency where maximum value offy^Jr]^ occurs i n the experiments by Gerritsma et al. (1974), e j^^y^ = relative error of pitch amplitude at the frequency where maximum value of ^^JO^ig) occurs i n the experiments by Gerritsma et al. (1974), rig^ = heave amplitude, rjg^ - pitch amplitude, C = incident wave amplitude, k = wave number of incident waves).
L/B Fn p heave ^pitch 4 0.2 0.05 0.08 5.5 0.2 0.05 0.14 7 0.2 0.11 0.17 10 0.2 0.27 0.05 4 0.3 0.04 0.05 5.5 0.3 0.19 0.02 7 0.3 0.24 0.15 10 0.3 0.17 0.17
TABLE 3
Examples on relative errors in strip theory calculation of heave and pitch in irregular head sea due to linear potential flow effects. (Tg = mean wave period, L = ship length, H^^g = significant wave height, a (j = 3,5) = "true" values of standard deviations of heave and pitch amphtudes based on linear potential flow, (j = 3,5) = relative errors i n strip theory prediction of standard deviations of heave and pitch amplitudes
^1/3 ^3 - s 3 ••s5 1.4 2.2 3.0 4.0 6.0 150 19 16 15 16 5.8 1.5 I . 9 3.6 I I . 0 0.002 0.05 0.03 0.01 0.002 0.001 0.05 0.05 0.03 0.02 TABLE 4
Example on difference i n transfer functions for heave and pitch for prototype ship and new ship i n Takaki's (1989) optimalization study of seakeeping performance (tigap/Ca = Maximum value of heave transfer function of prototype ship, tlsayCa = Heave transfer function of new ship at the same frequency as rjg^p is calculated, r\^Zj{M,g) = Maximum value of pitch transfer function of prototype ship, rig^^jAk^^) = pitch transfer function of new ship at the same frequency as rjg^ is calculated.
Fn
'n3ap~'n3a7i ^5<ip ^5a 7i
^5an
0.2 0.11 0.4
0.25 0.25 0.62
REFERENCES
Arai, M . , Matsunaga, K., 1989a, A numerical study of water entry of two-dimensional ship-shaped bodies. Proceedings, PRADS'89, Varna, Bulgaria, Vol. 2, pp. 75-1 to 75-7.
Arai, M., Matsunaga, K, 1989b, A numerical and experimental study of bow flare slamming, (in Japanese), J. of Soc. N.A. of Japan, Vol. 166, Dec. pp. 343-353.
Beukelman, W., 1980, Added resistance and vertical hydrodynamic coefficients of oscillating cylinders at speed. Report No. 510, Ship Hydromechanics Laboratory, Delft University of Technology.
Beukelman, W., 1983, Vertical motions and added resistance of a rectangular and triangular cylinder in waves. Report No. 594, Ship Hydromechanics Laboratory, Delft University of Technology.
Blok, J.J., Beukelman, W., 1984, The high speed displacement ship systematic series hull forms, SNAME Transaction, Vol. 92, pp. 125-150.
Braathen, A., Faltinsen, O., 1988, Application of a vonex tracking method to roll damping, Intemational Conference on Technology Common to Aero and Marine Engineering, London. Chapman, R.B., 1975, Free surface effects for hydrodynamic forces on a surface-piercing plate oscillating in yaw and sway, Proc. 1st. Int. Symp. Numer. Hydrodyn., pp. 333-350, David W. Taylor Naval Ship Center, Bethesda, Maryland.
Cointe, R. 1991, Free surface flow close to a surface-piercing body, "Mathematical approaches in hydrodynamics". Editor: T. Miloh, SIAM, pp. 319-333.
Dobrovol'skaya, Z.N., 1969, On some problems of fluid with a free surface, J. Fluid Mech., Vol. 36, part 4, pp. 805-829.
Falch, S., 1991, Seakeeping of foilcatamarans. Proceedings FAST'91, Trondheim, Norway, Tapir publishers. Vol. 1, pp. 209-221.
Faltinsen, O., 1977, Numerical solutions of transient nonlinear free-surface motion outside or inside moving bodies. Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, California, USA.
Faltinsen, O., Pettersen, B., 1987, Application of a vortex tracking method to separated flow around marine structures, Joumal of Fluids and Structures, 1, 217-237.
Faltinsen, O., 1990, Sea loads on ships and offshore structures, Cambridge University Press, Cambridge, England.
Faltinsen, O., Zhao, R., 1991a, Numerical predictions of ship motions at high forward speed. Philosophical Transactions of Royal society. Series A.
Faltinsen, O., Zhao, R., 1991b, Flow prediction around high-speed ships in waves, "Mathematical approaches in hydrodynamics", Editor: T. Miloh, SIAM.
Gerritsma, J., Beukelman, W., Glansdorp, C.C., 1974, The effects of beam on the hydrodynamics characteristics of ship hulls. In Proc. Tenth Symp. on Naval Hydrodynamics, Eds. R.D: Cooper and S.W. Doroff, pp. 334, Ariington, Va.; Office of Naval Research -Department of the Navy.
Hoff, J.R., 1990, Three-dimensional Green function of a vessel with forward speed in waves, Dr.ing.Thesis 1990-25, Division of Marine Hydrodynamics, Norwegian Insdtute of Technology, Trondheim, MTA Report 1990:71.
Inglis, R.B., Price, W.G., 1981, The influence of speed dependent boundary conditions in three dimensional ship motion problems, ISP, 28(318).
Keuning, J.A., 1988, Distribution of added mass and damping along the length of a ship model at high forward speed. Report No. 817-P, Ship Hydrodynamics Laboratory Delft University of Technology.
Nakos, D.E., Sclavounos, P.D., 1990, Ship motions by a three-dimensional Rankine Panel method. Proceedings 18th Symposium on Naval Hydrodynamics, Univ. of Mich., Ann Arbor, National Academy Press Washington D . C , pp. 21-40.
NestegSrd, A., 1990, Motions of surface effect ships, A.S. Veritas Research, Report No. 90¬ 2011.
Newman, J.N., 1978, The theory of ship motions. Adv. Appl. Mech., 18, 221-83.
Newman, J.N., Sclavounos, P., 1980, The unified theory of ship motions. In Proc. Thirteenth Symp. on Naval Hydrodynamics, ed. T. Inui, pp. 373-98, Tokyo: The Shipbuilding Research Association of Japan.
Nichols, B.D., Hirt, C.W., 1981, Volume of fluid method (VOF) for dynamic firee boundaries, J. of Computational Physics, No. 39.
O'Dea, J.F., Walden, D.A., 1985, The effect of bow flare and non-linearities on the prediction of large amplitude motions and deck wetness. In Proc. Fifteenth Symp. on Naval Hydrodynamics, pp. 163-76, Washington D . C ; National Academy Press.
O'Dea, J.F., Troesch, A.W., 1986, Comparative seakeeping model experiments, 21st ATTC, Washington.
Ogilvie, T.F., Tuck, E.O., 1969, A rational strip theory for ship motions. Part 1, Report No. 013, Dept. Nav. Arcit. Mar. Eng., University of Michigan, Ann Arbor.
Ohkusu, M . , Faltinsen, O., 1990, Prediction of radiation forces on a catamaran at high Froude number, Proc. of 18th Symp. on Naval Hydrodynamics, Ann Arbor, Michigan.
Salvesen, N., Tuck, E.O., Faltinsen, O.M., 1970, Ship motions and sea loads, Trans, SNAME, 78 : 250-87.
Takaki, M., 1989, Effect of hull forms oh ship motions and optimalization of hull forms for seakeeping performance, Joumal of the Society of Naval Architects of Japan, 166 : 239-49. Wagner, H., 1932, Über stoss- und gleitvorgange an der oberflache von fliissigkeiten, Zeitschr. f. angew. Math un Mech., Band 12, Heft 4, pp. 194-235.
Weinblum, G., St. Denis, M., 1950, On the motions of ships at sea, SNAME Transactions. Yamamoto, Y., lida, K., Fukasawa, T., Murakami, T., Arai, M . , Ando, M . , 1985, Structural damage analysis of a fast ship due to bow flare slamming. International Shipbuilding Progress, Vol. 32, No. 369, pp. 124-136.
Zhao, R. Faltinsen, O.M., 1989, Interaction between current, waves and marine structures, Intemational Conference on Numerical Hydrodynamics, Hiroshima.
Zhao, R. Faltinsen, O., 1993, Water entry of two-dimensional bodies, J. Fluid Mech., 246, pp. 593-612.
Fig. 1 The wave elevation around a vertical strut with parabolic water plane area in steady forward motions. Fn = 1.0. Length = 1 m. The results by Hoff (1990) based on thin hip theory are shown in the upper half, and the results in the lower half based on the linear high-speed theory by Faltinsen & Zhao (1991a).
Fig. 2 Wave resistance for a high-speed hull. Comparison between the high-speed theory by Faltinsen & Zhao (1991a) and experiments.
Fig. 3 The amplitude of the wave elevanon around a vertical strut with parabolic water plane in unsteady heave motion with forward speed. Fn = 1.0, CO = 8 rad/s. Unit heave amplitude. Strut length 1 m. The results from thin ship calculations by Hoff (1990) in the upper half and the results by the high-speed theory by Faltinsen & Zhao (1991a) using the classical free surface conditions with forward speed in the lower half.
A A Heave 00 Pitch 1.5000 -I 1.1250 J 0 . 7 5 0 0 H 0 . 3 7 5 0 H 0 . 0 0 0 0 - I — 0 . 0 0 0
4 Heave, pitch and vertical acceleration amplitudes for the model presented in Fig. 5 in head sea regular waves. Fn = 1.14. Trim 1.62°. Experiments by Blok & Beukelman (1984). C,^ = wave amplitude of the incident waves, k = 2k/X wave number of the incident waves, = vertical acceleration amplitude, L = ship length.
A A A Experiments
theory (the flow at the transom stem gives no difference in results)
Experiments
theory without transom stem effects theory with transom stem effects
o o o Experiments
theory without transom stem effects theory with transom stem effects Heave:
Ca
Pitch:
Acceleration at station 19 (in the bow)
Fig. 5 Body plan of the model used in the experiments by Keuning (1988). Length of test wateriine 2.00 m
Beam of test wateriine 0.25 m Draught 0.0624 m Block coefficient 0.0396 2.50 m 0,2 Ö
>
0.25 , -4—! 2.00 m 0,25 , 2.50 m ^ F' 0,25 0.25 , 2.00 m 0.25 4 r I i 0.25 iFig. 6 Form and dimensions of ship model with rectangular cross-sections (Beukelman (1980, 1983)).
C VORTEX METHOD Fn = 0, n 3 3 / D = 1/15
3.0 H
2 . 0 i
1.0-a ' A EXPERIMENTS (BEUKELMAN (1983)) Fn = 0.16 • EXPERIMENTS . . . . . . . ( B E U K E L M A N (1983)) Fn = 0.26 A A A A A • • Q0.0
1-0 2.0 3.0 4.0 5.0 6.0
e\J g' S r e a f e l ? . ' ' ^ T '''''' '^^^^ ^^^^ °f ^he ship model
with rectangular cross-sections shown in Fig. 6. The data are presented as a funcdonFig. 8 Flow situation around a two-dimensional cross-secrion performing forced heave motion with effect of free surface waves and vortex shedding.
LINEAR POTENTIAL FLOW
DAMPING
DAMPING AT ^
DAMPING AT ^ =
15
15
• CDe\ i lFig. 9 Numerically calculated two-dimensional heave damping coefficients S^f*^ for the midship cross-section of the ship model presented in Fig. 6. The data are presented as a function of non-dimensionahzed frequency of oscillation (ü,^/D/^ for different oscillation amplitudes T]^^. Fn = 0. B = beam, D = draught.
I5a
1.0
0.5
0
Fn =
0
.16
0.02 m
0.03 m
0.04 m
2Fig. 10 Pitch amplitudes in regular head sea waves of the ship model presented in Fig. 6. Modeltests by Beukelman (1983). L' = 2.333m. = incident wave amplitude, k = wave number of the incident waves.
lil a'
20
4-10
Wagner jet flow solution
Similarity solution
10° 20° 30° 40° 50° 60° 70° 80°
Fig. 11 Prediction of maximum pressure coefficient Cp^^ during water entry of a wedge with constant vertical velocity V by means of similarity solution and Wagner's jet flow solution, a = deadrise angle.
Fig. 12 Predicdons of pressure (p) distribution during water entry of a 2-D wedge with constant vertical velocity V by means of a similarity solution (Zhao & Faltinsen (1993)). PQ = atmospheric pressure, p = mass density of the water.
3. 5 ;
Fig. 13 Comparisons between numerical and experimental pressure measurements on bow flare secdon. The experiments are drop test results by Yamamoto et al. (1985).
!["Nadużycie prawa w świetle rzymskiego prawa prywatnego", Franciszek Longchamps de Bérier, Wrocław 2004 : [recenzja]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)