Advancements in free surface RANSE simulations for sailing yacht applications

Ocean Engineering 90 (2014) 11-20

E L S E V I E R

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Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . G o m / l o c a t e / o c e a n e n g

Advancements in free surface RANSE simulations

for sailing yacht applications

Christoph Böhm^''^'* Kai Graf^-'^

'Delft University of Technology, The Netherlands ^ Yacht Research Unit Kiel, Germany

" University of Applied Sciences Kiel, Germany

CO)

CrossMark

A R T I C L E I N F O

Article history:

Received 11 November 2013 Accepted 24 June 2014 Available online 14 July 2014 Keywords:

Computational fluid dynamics RANSE

Volume-of-Fluid Interface smearing Validation and verification

A B S T R A C T

The analysis of yacht hulls performance using RANSE based free surface simulations has become an accepted approach over the last decade. Access to this technology has been eased by the development of user-friendly software and by the increase o f computational power. Results are w i d e l y accepted as superior to previous non-viscous approaches and have to compete w i t h t o w i n g tank results i n terms of accuracy. However, many practical applications suffer f r o m a numerical smearing of the free surface interface between air and water w h i c h can be described as numerical ventilation. This problem occurs when the intersection between bow and calm water surface forms an acute angle and is f u r t h e r pronounced i f the stem is rounded or b l u n t It is therefore especially linked to sailing yacht applications. The problem manifests itself as a non-physical suction of the air-water mixture under the yacht hull, causing a significant under-predicdon of viscous resistance. W h i l e this is the easily observable appearance of the problem, a second issue is its effect on wave resistance. It can be shown that wave damping is significantly increased, causing a prediction o f wave resistance w h i c h is also too low. The paper provides a review of the Volume-of-Fluid method. It discusses the resultant implications for practical applications. A remedy to circumvent the problem is described and its impact o n the accuracy of the result is shown. Simulations on an identical appended hull w i t h and w i t h o u t interface smearing are compared. Effects on free surface visualization and numerical accuracy are shown. The paper finishes w i t h a thorough verification and validation of a f u l l y appended yacht in accordance w i t h ITTC standards.

© 2014 Elsevier Ltd. All rights reserved.

1. I n t r o d u c t i o n

D u r i n g t h e last decade RANSE based viscous fiee surface s i m u l a -tions a r o u n d s h i p hulls have gained a certain degree o f m a t u r i t y . Their capability t o p r o d u c e reliable data w h i c h can c o m p e t e w i t h t o w i n g t a n k e x p e r i m e n t s has been p r o v e d , e.g. b y t h e G o t h e n b u r g 2010 W o r k s h o p o n Ship H y d r o d y n a m i c s (Larsson e t al., 2010). The r a p i d l y d e v e l o p i n g a v a i l a b i l i t y o f c o m p u t a t i o n a l p o w e r has increased t h e p o p u l a r i t y o f t h i s k i n d o f CFD t e c h n o l o g y and the access to i t has been eased by s o f t w a r e packages w h i c h guide t h e user t h r o u g h t h e pre-processing procedure. The once time-consuming procedure o f creating a c o m p u t a t i o n a l g r i d has been i m p r o v e d b y n e w m e s h i n g techniques w h i c h can r e l i a b l y h a n d l e c o m p l e x geometries and a l l o w t a i l o r i n g the m e s h such t h a t i t meets t h e special needs o f s h i p h y d r o d y n a m i c s . These advances i n c o m p u t a t i o n a l p o w e r and n u m e r -ical techniques have changed t h e challenge i n CFD t o w a r d s

* Corresponding author.

E-mail address: christoph_boehm@web.de (C. B ö h m ) . http://dx.doi.Org/10.1016/j.oceaneng.2014.06.038 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. A l l rights reserved.

a c h i e v i n g results t h a t are w i t h i n a n e x p e c t e d u n c e r t a i n t y . As m e n t i o n e d above, v e r i f i c a t i o n s a n d v a l i d a t i o n s f o r s h i p h y d r o d y -namics can be f o u n d i n t h e l i t e r a m r e a n d b e n c h m a r k cases i n c l u d i n g geometries are available. U n f o r t u n a t e l y t h e same does n o t h o l d t r u e f o r yacht h y d r o d y n a m i c s w h e r e v a l i d a t i o n s are rare a n d u s u a l l y n o n - p u b l i c . This m i g h t change i n t h e f u t u r e since results a n d geometries o f t h e D e l f t Systematic Y a c h t H u l l Series (DSYHS) have r e c e n t l y become p u b l i c l y available.

2. M o t i v a t i o n

A n a t t e m p t o f the authors to validate RANSE CFD against t o w i n g tank results o f a/Imerica's Cup Class Version 5 boat (ACCV5) ( B ö h m a n d Graf, 2 0 0 8 ) s h o w e d good results at time o f p u b l i c a t i o n . Resistance i n n o n - l i f t i n g conditions was resolved to - 6 . 2 % o f t h e Experimental Fluid Data (EFD), w h i l e l i f t i n g c o n d i t i o n p r o v e d to be a p r o b l e m w i t h drag and lifl: deltas o f - 2 . 5 % and 19% respectively. W i t h the above-m e n t i o n e d advanceabove-ments i n R/\NSE CFD these s i above-m u l a t i o n s have been repeated i n d u d i n g m o r e recent fi-ee surface m o d e l i n g a n d b o d y

12 C. Bohm, K. Graf/ Ocean Engineering 90 (2014) 11-20 N o m e n c l a t u r e v o l u m e f r a c t i o n o f f l u i d i w i t h i n a cell ( d i m e n s i o n l e s s ) e r r o r e s t i m a t e w i t h s i g n a n d m a g n i t u d e o f /<th p a r a m e t e r ( d i m e n s i o n l e s s ) Sp p a r a m e t e r e r r o r (e.g. i t e r a t i o n n u m b e r /, g r i d size c, t i m e step j) ( d i m e n s i o n l e s s ) SsM s i m u l a t i o n m o d e l i n g e r r o r ( d i m e n s i o n l e s s ) SsN s i m u l a t i o n n u m e r i c a l e r r o r ( d i m e n s i o n l e s s ) Ss s i m u l a t i o n e r r o r ( d i m e n s i o n l e s s ) ^ijk s o l u t i o n change ( d i m e n s i o n l e s s ) X scale f a c t o r ( d i m e n s i o n l e s s )

( l + ' O

f o r m f a c t o r ( d i m e n s i o n l e s s ) n surface n o r m a l v e c t o r ( d i m e n s i o n l e s s )

s

surface v e c t o r ( m ^ ) V v e l o c i t y v e c t o r ( m s"-') Vf, g r i d v e l o c i t y v e c t o r ( m s ~ ' )

~4>c

n o r m a l i z e d value o f c e n t r a l n o d e w . r . t . face ƒ ( d i m e n s i o n l e s s )

m o r t o n techniques a n d a larger and apparently better suited c o m p u t a rtonal g r i d . However, t h e results d i d not reflect the expected i m p r o v e -ments, indeed they w e r e even w o r s e t h a n before w i t h differences b e t w e e n CFD a n d EFD resistance curves o f a p p r o x i m a t e l y - 8%. This o b v i o u s l y led t o t h e q u e s t i o n w h y t h i s behavior occurred. I n general, single phase RANSE s i m u l a t i o n s t e n d t o o v e r - p r e d i c t d r a g values i f g r i d r e s o l u t i o n is n o t s u f f i c i e n t l y s m a l l . This b e h a v i o r is n o t absolutely t r a n s f e r a b l e to f r e e surface ship flows, w h e r e a n i n s u f f i cient r e s o l u t i o n o f t h e w a v e p a t t e r n m i g h t also lead to an u n d e r -p r e d i c t i o n o f drag. Nonetheless, u n d e r - -p r e d i c t i o n o f drag h i n t s t o l o o k a t m o d e l i n g errors. Fig. 1 illustrates t h e volume fraction afwater values o n t h e h u l l . N o r m a l l y one w o u l d expect t h a t these values are zero i n t h e a i r region, one i n t h e submerged area o f t h e h u l l a n d b e t w e e n zero a n d one i n a s m a l l r e g i o n a r o u n d t h e f r e e surface interface. I n t h e v i c i n i t y o f a sharp interface, this r e g i o n s h o u l d n o t s i g n i f i c a n t l y e x t e n d o v e r m o r e t h a n three cells. Fig. 1 clearly shows t h a t t h i s is n o t t h e case. Instead v o l u m e fracrtons are smeared over t h e c o m p l e t e h u l l , expect a r o u n d the appendages a n d i n t h e i r w a k e . This clearly indicates a behavior w h i c h is sometimes r e f e r r e d t o as numerical ventilation b u t can be s h o w n t o be a smeared free surface interface. Due t o t h e n a t u r e o f t h e t r e a t m e n t o f physical properties o f flow phase w i t h i n t h e VOF ( V o l u m e - o f - F l u i d ) m o d e l , t h i s w i l l lead to s m a l l e r resistance values. I t has to be h i g h l i g h t e d t h a t t h e i n t e r f a c e s m e a r i n g as described above has o n l y been e n c o u n t e r e d f o r specific floating bodies. These bodies have i n c o m m o n t h a t t h e y share a r a t h e r b l u n t b o w w h i c h f o r m s a small, acute entrance angle w i t h t h e w a t e r i i n e . For c o n v e n t i o n a l vessel w h i c h n o r m a l l y have sharp b o w w i t h a r i g h t angle at t h e w a t e r line, this p r o b l e m does n o t occur. I t is t h e r e f o r e I d n d o f yacht-specific.

3. V o l u m e - o f - F l u i d m e t h o d

The V o l u m e - o f - F l u i d (VOF) m e t h o d was i n t r o d u c e d by H i r t and Nichols ( 1 9 8 1 ) . I t is a Interface Capturing Methods without

Volume FracHcn of Wolcr

O.OOOOO O.2D0OO 0.40000 0.60O0O O.B0Ü0O 1.0000

Fig. 1. VOF. c c o r r e c t e d e r r o r o r u n c e r t a i n t y ƒ cell face CD drag c o e f f i c i e n t ( d i m e n s i o n l e s s ) Ck c o r r e c t i o n f a c t o r ( d i m e n s i o n l e s s ) CL l i f t c o e f f i c i e n t ( d i m e n s i o n l e s s ) CT t o t a l resistance c o e f f i c i e n t ( d i m e n s i o n l e s s ) CFL C o u r a n t n u m b e r ( d i m e n s i o n l e s s ) E c o m p a r i s o n e r r o r ( d i m e n s i o n l e s s ) Fn Froude n u m b e r ( d i m e n s i o n l e s s ) Pk o r d e r o f accuracy ( d i m e n s i o n l e s s ) Rk convergence r a t i o ( d i m e n s i o n l e s s ) I'k r e f i n e m e n t r a t i o o f p a r a m e t e r k ( d i m e n s i o n l e s s ) Rn Reynolds n u m b e r ( d i m e n s i o n l e s s ) S s i m u l a t i o n results ( d i m e n s i o n l e s s ) T t r u t h ( d i m e n s i o n l e s s ) Up p a r a m e t e r u n c e r t a i n t y (e.g. i t e r a t i o n n u m b e r g r i d size c, time step T) ( d i m e n s i o n l e s s )

UsN n u m e r i c a l u n c e r t a i n t y ( d i m e n s i o n l e s s ) V v o l u m e ( m ^ )

reconstruction a n d thus does n o t treat the free surface as a sharp boundary. Instead the calculation is p e r f o r m e d o n a fixed g r i d , a n d free surface interface o r i e n t a t i o n and shape are calculated as a f u n c t i o n o f the v o l u m e p a r t o f t h e respective fluid w i t h i n a c o n t r o l v o l u m e (CV). The VOF m e t h o d e m p l o y s the concept o f an equivalent fluid. This approach assumes t h a t t h e ( t w o ) fluid phases share the same velocity and pressure fields a l l o w i n g us to solve the same set o f g o v e r n i n g equations describing m o m e n t u m a n d mass t r a n s p o r t as i n a single phase flow. The v o l u m e fi-action a,- describes to w h i c h level the cell is filled w i t h the respective fluid. The free surface is t h e n d e f i n e d as the isosurface at w h i c h t h e v o l u m e fractions take t h e value o f 0.5. As t h e t e r m isosurface i m p l i e s , the location o f t h e free surface is n o t necessarily at a c o n t r o l v o l u m e center. Instead its l o c a t i o n is i n t e r p o l a t e d firom t h e v o l u m e f r a c t i o n values available at the ( 3 / centers. To s i m u l a t e w a v e dynamics, one has to solve an equation f o r t h e filled f r a c t i o n o f each CV a d d i t i o n a l l y to the conservation equations f o r mass a n d m o m e n t u m . A s s u m i n g incompressible flow, the dransport equation o f v o l u m e f r a c t i o n s at is described b y t h e f o l l o w i n g conservation e q u a t i o n :

^JaiöV+ fai(v-Vb)-ndS = 0 (1) otJv Js

The physical properties o f the equivalent fluid w i t h i n a c o n t r o l v o l u m e are t h e n calculated as f u n c t i o n s o f t h e physical properties o f the phases and t h e i r v o l u m e fractions. Sttict conservation o f mass is caicial, b u t this is easily obtained w i t h i n this m e t h o d as l o n g as the s u m o f all volume-fi-actions per cell is 1. The critical issue f o r this k i n d of methods is t h e discretization o f the convective t e r n i . Low-order terms like f o r instance 1st order u p w i n d are l o i o w n t o smear the interface and introduce a n artificial m i x i n g o f the t w o fluids. Therefore higher order schemes are preferred. The goal is to derive schemes w h i c h are able t o keep the interface sharp a n d produce a m o n o t o n e proflle across i L D e v e l o p m e n t o f d i f f e r e n c i n g schemes has been t h e pinnacle o f research i n t h e fields o f VOF m e t h o d s f o r m a n y years. Consequently a large n u m b e r o f schemes are available a n d successfully used i n d i f f e r e n t codes. The vast m a j o r i t y o f these schemes are based o n t h e N o r m a l i z e d V a r i a b l e D i a g r a m ( N V D ) a n d t h e C o n v e c t i o n Boundedness C r i t e r i o n (CBC) i n t r o d u c e d b y L e o n a r d ( 1 9 8 8 ) .

3.1. HRIC scheme

T h e HRIC ( H i g h R e s o l u t i o n I n t e r f a c e C a p t u r i n g ) s c h e m e is o n e o f t h e m o s t p o p u l a r a d v e c t i o n schemes a n d w i d e l y used i n m a n y

C. Böhm, K. Graf/ Ocean Engineering 90 (2014) 11-20 ₁₃

CFD codes, i n c l u d i n g t l i e c o m m e r c i a l codes Comet, S t a r - C C M + a n d Fluent. It has been developed b y Muzaferija and Peric (Muzaferija a n d Peric, 1997, 1999; M u z a f e r i j a et a l , 1999). Like m o s t other schemes, i t is based o n a b l e n d i n g o f bounded u p w i n d and d o w n w i n d schemes. The a i m is to combine the compressive properties o f t h e d o w n w i n d differencing scheme w i t h the stability o f the u p w i n d scheme. Tire bounded d o w n w i n d scheme is f o r m u l a t e d as

H R I C 0 / = 4>c i f 0 c < O 2}c i f O < ^ c < 0 . 5 1 i f 0.5 < 0 c ^ 1 4>c i f l < 0 c (2)

Since the a m o u n t o f one f l u i d convected t h r o u g h a cell face shall be less o r equal t o the a m o u n t available i n d i e donor cell, the calculated value o{ <j>f is corrected w i t h respect to the local Courant n u m b e r (CFL). The CFL is calculated by e m p l o y i n g t h e velocity at the cell face V f , the surface vector S;, the respective cell v o l u m e Vf and the local time step size d t as follows:

CFL: V f S f d t

(3) The correction takes t h e f o r m o f (4) a n d effectively conft-ols tiie b l e n d i n g b e t w e e n HRIC and UD schemes w i t h t w o l i m i t i n g Courant n u m b e r s CL and Cu w h i c h n o r m a l l y takes values o f 0.5 and 1.0 respec-tive 0.3 and 0.7

</'f i f C F L < 0 'PJ = <j <PcHh-~4>c)^^~^ i f C i < C F L < C u

c ifCu<. CK

(4)

E f f e c t i v e l y t h i s c o r r e c t i o n i m p l i e s t h a t t h e HRIC scheme is used f o r a CFL s m a l l e r t h a n t h e l o w e r CFL l i m i t e r a n d t h e U D s c h e m e f o r CFL e q u a l o r greater t h a n t h e u p p e r CFL l i m i t e r . B e t w e e n those v a l u e s a b l e n d i n g o f b o t h schemes is used. T h i s c o r r e c t i o n is a p p l i e d to i m p r o v e robustness a n d s t a b i l i t y w h e n large time v a r i a t i o n o f t h e f r e e surface shape is p r e s e t a n d t h e t i m e step is t o o b i g t o resolve i t . A f t e r t h i s c o r r e c t i o n ^f experiences a final m o d i f i c a t i o n based o n t h e i n t e r f a c e angle, w h i c h is t h e angle 0 b e t w e e n t h e n o r m a l o f t h e f r e e surface i n t e r f a c e n a n d t h e cell s u r f a c e v e c t o r S f . This final m o d i f i c a t i o n reads

= ^ } ( c o s -h^cCl - cos 0)"^' (5) H e r e Cg represents a n angle e x p o n e n t . Its d e f a u l t v a l u e a c c o r d i n g

t o M u z a f e r i j a a n d Peric ( 1 9 9 9 ) is 0.05. T h e final c e l l face v a l u e is c a l c u l a t e d as

^'"' = }f'\4>D-</'u)+4>u

(6) As a consequence o f t h e m o d i f i c a t i o n s d u e t o i n t e r f a c e angle a n d local C o u r a n t number, the I W D can take d i f f e r e n t f o r m s . For t h e t h r e e d i f f e r e n t b l e n d i n g states d e p e n d i n g o n local CFL, Fig. 2 illustrates t h e possible f o r m s o f t h e HRIC scheme w i t h respect t o t h e i n t e r f a c e angle 0. The areas shaded i n r e d represent t h e possible f o r m s t h e scheme can take d e p e n d i n g o n t h e angle f a c t o r f o r the respective local Courant n u m b e r . This k i n d o f b l e n d i n g strategy is m o r e o r less t h e same f o r all i n t e r f a c e c a p t u r i n g schemes, so care has t o be t a k e n w h e n m o d e l i n g free surface flows t o a v o i d u n w a n t e d s w i t c h i n g to a l o w e r r e s o l u t i o n w h i c h is o f t e n a c c o m p a n i e d w i t h i n t e r f a c e s m e a r i n g . 4. T h e o r e t i c a l t e s t case The t h e o r e t i c a l r e v i e w o f t h e HRIC r e v e a l e d t h a t t h e e n c o u n -t e r e d i n -t e r f a c e s m e a r i n g is m o s -t p r o b a b l y r e l a -t e d -t o -t h e use o f h i g h C o u r a n t n u m b e r s . A m o d i f i e r w a s f o u n d w h i c h i m p l i e s t h a t C F L , = Cu, 6 = 9 0 °

Fig. 2. NVD o f h i g h resolution capturing scheme (HRIC). (For i n t e r p r e t a t i o n o f the references to color i n this figure caption, the reader is referred to t h e w e b version o f this paper.) t h e HRIC s c h e m e is u s e d f o r a CFL s m a l l e r t h a n t h e l o w e r CFL l i m i t e r a n d t h e U D s c h e m e f o r CFL equal o r greater t h a n t h e u p p e r CFL l i m i t e r . B e t w e e n t h o s e values a b l e n d i n g o f b o t h schemes is used. F r o m a t h e o r e t i c a l p o i n t o f v i e w , t h e sole p u r p o s e o f t h e c o r r e c t i o n o f t h e HRIC s c h e m e f o r local CFL is t o i m p r o v e r o b u s t -ness. I f u n s t e a d y p h e n o m e n a l i k e s l a m m i n g a n d o r s e a k e e p i n g are o f interest, local C o u r a n t N u m b e r s h o u l d b e i n h e r e n t l y l o w e r t h a n 0. 5 a n y w a y . I f r o b u s t n e s s is n o t p r o b l e m a t i c t h e n t h i s s w i t c h s h o u l d be o f n o i n t e r e s t f o r c a l c u l a t i o n s w h i c h seek a s t e a d y state s o l u t i o n . Since s i m u l a t i o n s m i m i c k i n g t o w i n g t a n k p r o c e d u r e s seek s u c h a s t e a d y state s o l u t i o n , t h e HRIC s c h e m e is m o d i f i e d such t h a t t h e s w i t c h is e f f e c t i v e l y r e m o v e d . I f t h i s a s s u m p t i o n is t r u e , t h i s w o u l d r e m o v e t h e necessity t o keep C o u r a n t n u m b e r b e l o w 0.5 f o r e v e n t h e s m a l l e s t cell. The i m p a c t o f t h i s o n p r a c t i c a l a p p l i c a t i o n s is vast because i t has the p o t e n t i a l t o s i g n i f i c a n t l y reduce c o m p u t a t i o n a l e f f o r t b y a l l o w i n g l a r g e r time step sizes. To c o n t r o l t h e v a l i d i t y o f t h i s a s s u m p t i o n a t e s t case has b e e n c o n s t r u c t e d . A i m o f t h e t e s t case is t o p r o d u c e a w o r s t case scenario w h i c h m a k e s i t possible t o j u d g e i f t h e m o d i f i e d d i f f e r -e n c i n g s c h -e m -e can cop-e w i t h t h -e s i t u a t i o n . F r o m a t h -e o r -e t i c a l p o i n t o f v i e w , t h e case w h i c h w o u l d p r o d u c e t h e h i g h e s t a m o u n t o f n u m e r i c a l d i f f u s i o n a n d t h u s t h e h i g h e s t a m o u n t o f i n t e r f a c e s m e a r i n g is a flow t h r o u g h a q u a d r a t i c g r i d cell a t a n a n g l e o f 4 5 ° . T h e r e f o r e a 2 D Cartesian g r i d has b e e n b u i l d w h i c h consists o f 128 X 128 g r i d cells w i t h a n edge l e n g t h o f 0.5 m . T o t a l edge l e n g t h o f t h e d o m a i n is 6 4 m . I n i t i a l v o l u m e f r a c t i o n d i s t r i b u t i o n is such t h a t t h e l i g h t e r fluid ( a i r ) occupies t h e u p p e r l e f t t r i a n g l e o f t h e d o m a i n ( b l u e ) w h i l e t h e h e a v i e r fluid ( w a t e r ) is f o u n d i n t h e l o w e r r i g h t side ( r e d ) . I n f l o w c o n d i t i o n s f o r v o l u m e f r a c t i o n have b e e n set s u c h t h a t t h i s state s h o u l d r e m a i n w i t h i n t h e s i m u l a t i o n . Outiet has been set to N e u m a n n conditions. A sketch o f t h e s e t u p is depicted i n Fig. 3. The test has been conducted using t h e c o m m e r c i a l CFD code SmR-CCM+7.02.008. Depending o n the local Courant n u m b e r , Üie HRIC scheme switches b e t w e e n :

1. A p u r e HRIC s c h e m e i f CFL < 0.5.

2. A l i n e a r b e t w e e n HRIC a n d U D scheme i f 0.5 < CFL ^ 1.0. 3. A p u r e U D s c h e m e i f CFL > 1.0.

The i n f l u e n c e o f t h e s e d i f f e r e n t states o n t h e s h a r p n e s s o f t h e i n t e r f a c e is t e s t e d b y v a r y i n g flow speed a n d t i m e s t e p size s u c h t h a t t h e r e l e v a n t c r i t e r i o n is f u l f i l l e d . First, CFL is set t o 0.3 r e s u l t -i n g -i n a p u r e HRIC s c h e m e (F-ig. 4a). Even t h o u g h t h e f l o w d -i r e c t -i o n w i t h respect t o c e l l faces is u n f a v o r a b l e , t h e HRIC s c h e m e is able t o resolve t h e s h a r p e s t i n t e r f a c e possible w i t h i n t h e VOF m e t h o d (1 cell). N e x t t h e CFL is increased t o 0.75, r e s u l t i n g i n 50% b l e n d b e t w e e n HRIC a n d U D ( F i g . 4 b ) . T h i s b l e n d is also s t i l l s u f f i c i e n t t o r e t a i n t h e s h a r p i n t e r f a c e a n d t h e r e f o r e gives a v a l i d s o l u t i o n . A n e x p l a n a t i o n f o r t h i s b e h a v i o r can be f o u n d i n t h e b l e n d i n g s t r a t e g y

14 C. Böhm, K Graf/ Ocean Engineering 90 (2014) 11-20

d e p e n d i n g o n i n t e r f a c e angle. As d e p i c t e d i n Fig. 2, tire d i f f e r e n c e b e t w e e n t h e p u r e HRIC and t h e b l e n d e d HRIC is reasonably s m a l l f o r a cell f l o w angle o f 4 5 ° w h i c h explains t h e s i m i l a r results. Finally, f l o w speed a n d t i m e step size o f t h e u n s t e a d y s i m u l a t i o n are set t o values such t h a t t h e C o u r a n t n u m b e r i n t h e e n t i r e d o m a i n is 3.0. T h i s leads to s w i t c h i n g t o a p u r e U p w i n d D i f f e r e n -c i n g S-cheme w i t h i n t h e HRIC s-cheme. As a r e s u l t t h e i n t e r f a -c e b e t w e e n air a n d w a t e r becomes severely s m e a r e d a n d is f o r m i n g a c o n e - l i k e shape s t a r t i n g f r o m i n l e t t o w a r d s o u t l e t (Fig. 4 c ) . N o w t h e HRIC scheme is m o d i f i e d b y r e m o v i n g t h e CFL d e p e n d e n c y . This is a c h i e v e d w i t h i n STAR-CCM+ b y c h a n g i n g t h e l i m i t i n g CFL n u m b e r s . The C o u r a n t n u m b e r is k e p t at 3.0 a n d t h e s i m u l a t i o n r e p e a t e d . Fig. 4 d i l l u s t r a t e s t h e r e s u l t w h i c h c l e a r l y s h o w s t h a t t h i s m o d i f i c a t i o n a l l o w s u s i n g h i g h e r CFL n u m b e r s w h i l e a s h a r p i n t e r f a c e is r e t a i n e d . T h i s a l l o w s t h e c o n c l u s i o n t h a t t h e m o d i f i c a -t i o n o f -t h e HRIC s c h e m e is w e l l s u i -t e d -t o s i m u l a -t e f r e e surface f l o w s a t h i g h e r C o u r a n t n u m b e r s , a l l o w i n g us t o c o n v e r g e f a s t e r t o w a r d s a steady state s o l u t i o n . 5. V a l i d a t i o n a n d v e r i f i c a t i o n against t o w i n g tanlc d a t a Outlet

Fig. 3. Slcetcii o f test case setup.

I n m o s t cases v a l i d a t i o n s are c o n d u c t e d b y c o m p a r i n g s i m u l a t i o n results w i t h t r u s t e d t o w i n g t a n k data. D e v i a t i o n s f r o m e x p e r i -m e n t a l data are corrected b y g r i d r e f i n e -m e n t s u n t i l a n acceptable a g r e e m e n t b e t w e e n EFD a n d CFD is f o u n d . H o w e v e r , t h i s a p p r o a c h can lead to false c o n f i d e n c e i n the results i f m o d e l i n g o r g r i d errors are present. T h e r e f o r e , v a l i d a h o n a n d v e r i f i c a t i o n are c o n d u c t e d here w i t h a f o r m a l a p p r o a c h w h i c h a l l o w s d r a w i n g a d d i t i o n a l conclusions w i t h respect to e r r o r types a n d e r r o r sources. First a t all a s h o r t d e f i n i t i o n o f t h e t e i m s v e r i f i c a t i o n a n d v a l i d a t i o n is necessary: • Verification i n c l u d e s t h e assessment o f n u m e r i c a l u n c e r t a i n t y , m a g n i t u d e a n d s i g n o f n u m e r i c a l e r r o r ( i f p o s s i b l e ) a n d u n c e r t a i n t y i n e r r o r e s t i m a t i o n .

0,00

Volume Fraction of Water

0.20 0.40 0.60 0.80 1.0

C, Böhm, K. Graf/ Ocean Engineering 90 (2014) 11-20 15

• Validation is t l i e assessment o f u n c e r t a i n t y o f t f i e s i m u l a t i o n m o d e l b y m e a n s o f e x p e r i m e n t a l data p l u s the assessment o f t h e m o d e l i n g e r r o r itself.

T h e v e r i f i c a t i o n a n d v a l i d a t i o n p r o c e d u r e w i l l be c a r r i e d o u t i n accordance w i t h r e c o m m e n d a t i o n s o f t h e ITTC r e g a r d i n g Uncer-tainty Analysis in CFD ( I T T C 2 0 0 8 ) . For a d e t a i l e d d e s c r i p t i o n see also Stern e t al. ( 2 0 0 6 , 2 0 0 1 ) . The s i m u l a t i o n e r r o r e s is d e f i n e d as t h e d i f f e r e n c e b e t w e e n s i m u l a t i o n r e s u l t S a n d r e a l i t y o r t r u t h T. I t consists o f t h e m o d e l i n g e r r o r SSM a n d t h e n u m e r i c a l e r r o r ÖSN-U n f o r t u n a t e l y SS c a n n e v e r be d e t e r m i n e d exactly since i n s t e a d o f T o n l y e x p e r i m e n t a l results are available w h i c h also c o n t a i n a c e r t a i n l e v e l o f u n c e r t a i n t y .

5s = S-T = SsM+SsN (7) For s o m e cases m a g n i t u d e a n d sign o f t h e n u m e r i c a l e r r o r can

be e s r i m a t e d , l e a d i n g t o c o r r e c t e d n u m e r i c a l u n c e r t a i n t y US^M-For t h e u n c o r r e c t e d case o n l y t h e n u m e r i c a l u n c e r t a i n t y USN is assessed. T h e r e f o r e t h e n u m e r i c a l e r r o r SSN is d e c o m p o s e d i n t o c o n t r i b u t i o n s f r o m i t e r a t i o n n u m b e r 5,, g r i d size Sc, t i m e step ST a n d o t h e r p a r a m e t e r s Sp. Wth u n c e r t a i n t y U as d e s c r i b e d above t h i s gives t h e f o l l o w i n g e x p r e s s i o n : uj^ = u f + u l + u j + u j ₍₈₎ For v a l i d a t i o n p u r p o s e t h e c o m p a r i s o n e r r o r E b e t w e e n t h e b e n c h m a r k e x p e r i m e n t a l d a t a D a n d t h e s i m u l a t i o n r e s u l t S is d e t e r m i n e d i n o r d e r t o asses m o d e l i n g u n c e r t a i n t y USM-E = D-S = SD-(SSM+SSN) (9) To d e t e r m i n e i f v a l i d a t i o n o f a v a l u e has b e e n a c h i e v e d , c o m p a r i s o n e r r o r E is c o m p a r e d w i t h t h e v a l i d a t i o n u n c e r t a i n t y Uv. Ul = Ul + Uj^ ( 1 0 ) I f | £ | < Uv, t h a n t h e c o m b i n a t i o n o f a l l e r r o r s i n b o t h s i m u l a t i o n a n d e x p e r i m e n t a l d a t a is s m a l l e r t h a n t h e v a l i d a t i o n u n c e r t a i n t y . T h e n v a l i d a t i o n has b e e n a c h i e v e d f o r t h i s v a l i d a t i o n u n c e r t a i n t y l e v e l . I n t h e case t h a t Uy < \E\, t h e m o d e l i n g e r r o r SSM c a n be used t o achieve m o d e l i n g i m p r o v e m e n t s .

5.1. Verification procedure

I n t h e course o f t h e v e r i f i c a t i o n process a g r i d c o n v e r g e n c e s t u d y has t o be c o n d u c t e d . I n o r d e r t o d o t h i s i t is necessary t o use a m i n i m u m o f t h r e e g r i d s w h i c h have b e e n u n i f o r m l y r e f i n e d w i t h a n i n c r e m e n t AXk s u c h t h a t c o n s t a n t r e f i n e m e n t r a t i o r^ exists. (11) ITTC G u i d e l i n e s r e c o m m e n d r e f i n e m e n t r a t i o r^ b e t w e e n V 2 a n d 2. T h r o u g h o u t t h i s w o r k ratios o f 1.5 a n d 2 have b e e n u s e d . N e x t a c o n v e r g e n c e r a t i o Rk is d e f i n e d t o give i n f o r m a t i o n a b o u t c o n v e r -gence respective d i v e r g e n c e o f a s o l u t i o n . I t is d e f i n e d as f o l l o w s : £211 =Sk^-Sk, e32, =Sk,-Sk^ Rk = e2ije32, (12) w i t h Cijk as t h e s o l u t i o n changes f o r t h e i n p u t p a r a m e t e r k b e t w e e n t h r e e s o l u t i o n s r a n g i n g f r o m fine S^, t o coarse S^^. A c c o r d i n g t o t h e ITTC g u i d e l i n e s (ITTC, 2 0 0 8 ) , t h r e e d i f f e r e n t cases are d i s t i n -g u i s h e d : (1) M o n o t o n i c c o n v e r g e n c e : 0 < iJ^ < 1. ( i i ) O s c i l l a t o r y c o n v e r g e n c e : R^ < 0'. ( i i i ) D i v e r g e n c e : > 1. I n t h e case o f ( i ) t h e Generalized R i c h a r d s o n E x t r a p o l a t i o n is u s e d t o assess t h e u n c e r t a i n t y U^ o r t h e e r r o r e s t i m a t e 5^ a n d t h e c o r r e c t e d u n c e r t a i n t y U^^. For o s c i l l a t o r y c o n v e r g e n c e (case ( i i ) ) t h e u n c e r t a i n t y U^ is e s t i m a t e d b y d e t e r m i n i n g t h e e r r o r b e t w e e n m i n i m u m a n d m a x i m u m o f t h e o s c i l l a t i o n . I n t h e case o f d i v e r -gence ( i i i ) i t is n o t possible t o e s t i m a t e e r r o r s o r u n c e r t a i n t i e s .

5.11. Generalized Richardson Exti-apolation

As stated above, i n the case o f m o n o t o n i c convergence generalized RE is used to determine the error 5 j w i t h respect to r e f i n e m e n t ratio r^ and order-of-accuracy P^. Usually 5 j is estimated f o r the finest s o l u t i o n o f t h e i n p u t parameter m = l o n l y W i t h a n u m b e r o f available solutions m = 3 o n l y the leading-order t e r m o f t h e error m a y be evaluated. This gives the f o l l o w i n g equations f o r 5 j and P^.

(14) > ^ - l

Pk In(e32t/e2it)

• In(rfc) (15)

Unless t h e s o l u t i o n is i n t h e a s y m p t o t i c range, Eq. ( 1 5 ) o n l y gives a p o o r e s t i m a t i o n o f t h e rate o f c o n v e r g e n c e . T h e r e f o r e a c o r r e c t i o n f a c t o r C^ is used to i n c l u d e t h e e f f e c t o f h i g h e r - o r d e r t e r m s p r i o r y n e g l e c t e d . Cfc is d e f i n e d as f o l l o w s : r P * _ i " r P t o t - l ( 1 6 ) T h e c o r r e c t e d e r r o r St is d e f i n e d b y c o m b i n i n g Eqs. ( 1 4 ) a n d ( 1 6 ) = CkS^E. = Q ^2h r ^ ' - l ( 1 7 ) D e p e n d i n g o n h o w close t h e c o r r e c t e d e r r o r 5 j is t o t h e a s y m p t o t i c range ( h o w close Ck is t o 1) t h e e x p r e s s i o n t o assess t h e u n c e r t a i n t i e s takes d i f f e r e n t f o r m s . I f Ck is s u f f i c i e n t l y g r e a t e r t h a n o n e a n d l a c k i n g c o n f i d e n c e o n l y Uk is e s t i m a t e d b y t h e f o l l o w i n g f o r m u l a : Uk = \CkS*REj + \0-Ck)5iE,, (18) ( 1 3 ) For Cfc b e i n g s u f f i c i e n t i y s m a l l e r t h a n o n e t h e ITTC r e c o m m e n d s to use e x p r e s s i o n ( 1 9 ) t o assess Uk.

Uk = \StE,J+2iO-Ck)StE,^\ ( 1 9 )

I f Cfc is s u f f i c i e n t i y close to 1, t h e e r r o r i5j c a n be e s t i m a t e d . T h i s a l l o w s us t o d e t e r m i n e a c o r r e c t e d s o l u t i o n Sc a n d t h u s a c o r r e c t e d u n c e r t a i n t y Uk^. Ufc, = | ( l - C f c ) 5 * E ^ J ( 2 0 ) 5.2. Validation procedure As s t a t e d i n Section 5, v a l i d a t i o n is d e f i n e d as a process to t h e m o d e l u n c e r t a i n t y USM a n d , i f possible, sign a n d m a g n i t u d e o f t h e m o d e l i n g e r r o r SSM i t s e l f . This is d o n e b y u s i n g e x p e r i m e n t a l d a t a t o c o m p a r e t h e s i m u l a t i o n results w i t h . T h u s t h e e r r o r i n t h e e x p e r i m e n t a l d a t a has t o be c o n s i d e r e d , m a k i n g i t easier t o v a l i d a t e s i m u l a t i o n s i f t h e e x p e r i m e n t a l e r r o r is large. I t m u s t t h u s be n o t e d t h a t t h e l e v e l o f v a l i d a t i o n is s t r o n g l y d e p e n d e d o n t h e q u a l i t y o f t h e c o m p a r i s o n data. T h e v a l i d a t i o n p r o c e d u r e is based o n t h e r e l a t i o n b e t w e e n v a l i d a t i o n u n c e r t a i n t y Uv, p r e d e fined p r o g r a m m a t i c v a l i d a t i o n r e q u i r e m e n t Ureqd a n d c o m p a r i -s o n e r r o r | £ | . The-se t h r e e variable-s m a y f o r m t h e f o l l o w i n g -six c o m b i n a t i o n s :

\E\<Uv<Ureqd \E\<Uregd<Uv

16 C. Böhm, K. Graf/ Ocean Engineering 90 (2014) 11-20

Uv < \E\ < Ureqd Uv < Urecd < \E\

Ure,d<Uv<\E\ (21) I n cases 1-3 o f ( 2 1 ) t h e results are v a l i d a t e d , V a l i d a t i o n is

a c h i e v e d a t t h e l e v e l o f v a l i d a t i o n u n c e r t a i n t y Uv- T h i s m e a n s t h a t t h e c o m p a r i s o n e r r o r is b e l o w t h e noise l e v e l r e s u l t i n g i n a n i m p o s s i b i l i t y t o e s t i m a t e e r r o r d u e t o m o d e l i n g a s s u m p t i o n Ssm-I n t h e case o f 1, t h e v a l i d a t i o n l e v e l is also b e l o w Ureqd w h i c h m a k e s t h e v a l i d a t i o n successful f r o m a p r o g r a m m a t i c p o i n t o f v i e w . For cases 4 - 6 t h e c o m p a r i s o n e r r o r is above t h e n o i s e l e v e l . Sign a n d m a g n i t u d e o f £ c a n be used t o e s t i m a t e SSMA- I n t h e f o u r t h case t h e v a l i d a t i o n is a c h i e v e d at t h e | £ | l e v e l w i t h r e s p e c t t o t h e used s o f t w a r e .

5.3. Grid convergence studies on ACCV5 boat for non-lifting cases

V e r i f i c a t i o n a n d v a l i d a t i o n are p e r f o r m e d o n t h e g e o m e t r y o f A m e r i c a s C u p Class V e r s i o n 5 b o a t (ACCV5) f o r w h i c h e x p e r i -m e n t a l t o w i n g t a n k d a t a is available. These boats h a v e a r a t h e r c o m p l e x g e o m e t r y w h i c h besides h u l l , k e e l f i n a n d r u d d e r also i n c l u d e s a t r i m t a b f o r t h e k e e l a n d a b a l l a s t b u l b w i t h w i n g s . Since m o d e l scale A=3, w h i c h is r a t h e r close t o f u l l scale c o m p a r e d w i t h t a n k m o d e l s f o r c o n v e n t i o n a l vessels, i t w a s d e c i d e d t h a t i t is possible t o d o t h e v a l i d a t i o n i n f u l l scale. T h e r e f o r e e x p e r i m e n t a l d a t a have b e e n t r a n s f o r m e d t o f u l l scale b y e m p l o y i n g a m o d i f i e d v e r s i o n o f t h e ITTC p r o c e d u r e s . T h e m o d i f i c a t i o n s a p p l i e d m a i n l y consist o f o w n f r i c t i o n c o e f f i c i e n t s a n d f o r m f a c t o r ( l - l - / < ) values f o r y a c h t appendages. T h e c o n d i t i o n s o f t h e c a l c u l a t i o n s are a F r o u d e n u m b e r Fn o f 0.403 a n d n o r m a l i z e d Reynolds n u m b e r Rn o f 4.75 X 10^. T h e b o a t is a l l o w e d t o s i n k d y n a m i c a l l y , b u t n o t t o p i t c h . The p i t c h a n g l e is p r e s c r i b e d at\fr = 0.46 b o w d o w n t r i m . STARCCMi7.02.008 is u s e d as flow code t o solve t h e R e y n o l d s -A v e r a g e - N a v i e r - S t o k e s e q u a t i o n s f o r t h e flow field a r o u n d t h e y a c h t . T h e s i m u l a t i o n is c o n d u c t e d at f u l l y t u r b u l e n t c o n d i t i o n s a n d t h e k-co based Shear Stress Transport (SST) m o d e l has b e e n u s e d t o m o d e l t u r b u l e n c e . To a l l o w g o o d c o n t r o l o f f r e e s u r f a c e r e s o l u t i o n , u n s t r u c t u r e d t r i m m e d Cartesian g r i d s have b e e n used.

5.3.J. Computational grids

Grid convergence studies have been conducted using t h r e e d i f f e r -e n t c o m b i n a t i o n s o f r -e f i n -e m -e n t param-et-ers t o study t h -e i r i m p a c t o n g r i d densities and c o m p u t a t i o n a l results. The c o m p u t a t i o n a l g r i d has been m o d e l e d such t h a t i t depends o n one base number. This w a y i t can be ensured t h a t a constant g r i d r e f i n e m e n t ratio rk is used. T w o exceptions fi'om this m o d e l i n g p a r a d i g m exist. First tiie p r i s m layer used to resolve t h e b o u n d a r y layer a r o u n d h u l l a n d appendages is excluded fi-om r e f i n e m e n t because this w o u l d lead t o large changes i n dimensionless wall-scale 7 + . M o s t likely this w o u l d lead t o changes i n near-wall t r e a t m e n t like using a low-Reynolds approach f o r one simulations and w a l l fiinctions f o r the other. This w o u l d r e n d e r t h e simulations incomparable. Therefore the t o t a l thickness o f the p r i s m layer, t h e thiclmess o f t h e w a l l nearest node and the n u m b e r o f p r i s m layers are l<ept constant t h r o u g h o u t this v e r i f i c a t i o n and v a l i d a t i o n . The second exception concems the resolution o f the free surface. Since free surface r e s o l u t i o n is v e r y i m p o r t a n t f o r c o n e c t r e s o l u t i o n o f ship drag, i t has been g i v e n its o w n base number. This w a y i t is possible to evaluate t h e i n f l u e n c e o f d i f f e r e n t r e f i n e m e n t ways o n b o t h c o m p u t a -tional g r i d a n d s o l u t i o n . The r e f i n e m e n t ways investigated w i t h i n this w o r k are

1. Global refinement: w h e r e o n l y t h e g l o b a l g r i d base n u m b e r is r e f i n e d .

2. Free surface refinement: w h e r e o n l y f r e e surface p a r a m e t e r s are r e f i n e d b y t h e i r base n u m b e r . Free surface r e f i n e m e n t s c o n s i s t

o f a v e r t i c a l r e f i n e m e n t i n t h e w h o l e d o m a i n at t h e e x p e c t e d l e v e l o f t h e w a v e p a t t e r n a n d a second r e f i n e m e n t i n b o t h l o n g i t u d i n a l a n d t r a v e r s a l d i r e c t i o n s i n t h e v i c i n i t y o f t h e K e l v i n p a t t e r n .

3. Overall refinement: w h e r e b o t h g l o b a l a n d f r e e surface base n u m b e r are m o d i f i e d as a f u n c t i o n o f t h e r e f i n e m e n t r a t i o r^.

For a l l t h r e e cases f o u r g r i d s w i t h c o n s t a n t r e f i n e m e n t r a t i o rfc = 2 have b e e n c o n s t r u c t e d . R e s u l t i n g g r i d sizes v a r i e d f r o m 8.1 X 10^ cells f o r t h e coarsest g r i d t o 1.2 x l O ' f o r t h e finest.

5.3.2. Verification and validation of resistance

T h e v e r i f i c a t i o n o f resistance has b e e n p e r f o r m e d w i t h r e s p e c t t o g r i d c o n v e r g e n c e . I t e r a t i v e c o n v e r g e n c e has b e e n t a k e n i n t o account, b u t since i t w a s i n t h e o r d e r o f 0.05% Cj i t w a s c o n s i d e r e d n e g l i g i b l e . T h e results o f t h e studies have b e e n s u m m a r i z e d i n Tables 1 a n d 2 . Table 1 i l l u s t r a t e s t h e Cj v a l u e s f o r t h e d i f f e r e n t g r i d s as w e l l as t h e s o l u t i o n change e f r o m a coarser t o a finer s o l u t i o n b e t w e e n a d j a c e n t g r i d s . H e r e e is d e f i n e d as

__(Si-Si+i) ^22) S i + i

T h e r e s u l t s s h o w t h a t t h e changes o f Cr b e t w e e n t h e d i f f e r e n t s o l u t i o n s are largest i n t h e case w h e r e f r e e s u r f a c e p a r a m e t e r s v a r i a t i o n s are i n v o l v e d (cases 2 3 ) . V e r i f i c a t i o n r e s u l t s are i l l u -s t r a t e d i n Table 2. H e r e c o n v e r g e n c e r a t i o Rc i n d i c a t e -s m o n o t o n i c g r i d c o n v e r g e n c e o f s o l u t i o n s f o r g r i d s 1-3 f o r a l l t h r e e case ( ] ? c < l ) - For t h e coarser g r i d sequence ( g r i d s 2 - 4 ) o n l y case 1 ( g l o b a l r e f i n e m e n t ) s h o w s m o n o t o n i c c o n v e r g e n c e . For t h e coarser g r i d sequence o f t h e f r e e surface r e f i n e m e n t s t u d y (case 2 ) RG i n d i c a t e s d i v e r g e n c e w h i l e f o r t h e same g r i d sequence o f t h e g l o b a l r e f i n e m e n t s t u d y (case 3 ) t h e s o l u t i o n appears t o b e o f o s c i l l a t o r y n a t u r e . H o w e v e r , t h e l a t e r i n d i c a t o r seems t o b e m i s l e a d i n g , so r e s u l t s f o r case 3 b are also t r e a t e d as d i v e r g e n t . I t is t h e r e f o r e n o t p o s s i b l y t o e s t i m a t e e r r o r o r u n c e r t a i n t y f o r cases 2 b a n d 3 b . W h e r e a p p r o p r i a t e G e n e r a l i z e d R i c h a r d s o n E x t r a p o l a -t i o n is u s e d -t o e s -t i m a -t e s i g n a n d m a g n i -t u d e o f -t h e g r i d e r r o r 5^ a n d a c o r r e c t e d u n c e r t a i n t y Uc, as w e l l as a c o r r e c t e d s o l u r i o n Sc (Eqs. ( 1 4 ) - ( 2 0 ) ) . T h e t h u s g a i n e d c o r r e c t e d s o l u t i o n c a n be Table 1

Grid convergence study f o r total resistance Cj{ x 10 ^) f o r ACCVS.

No. Var Grid number EFD

4 3 2 1 (1) CT 6.46 6.33 6.29 6.28 6.32 (1) e(%) - 2 . 0 - 0 . 6 - 0 . 2 (2) CT 5.87 6.02 6.19 6.28 6.32 (2) 2.6 2.7 1.5 (3) CT 6.06 6.05 6.24 6.28 6.32 (3) e(%) - 0 . 1 3.1 0.6 % S G . Table 2

V e r i f i c a t i o n of total resistance C T ( X 1 0 " ^ ) f o r ACCVS.

No. Grid RG Uc(%) Sc(%)

(1) 1-3 0.34 0.11 0.07 - 0 . 0 7 (1) 2 - 4 0.30 0.26 0.20 0.01 (2) 1-3 0.58 2.06 - 0 . 5 0 0.5 (2) 2 - 4 1.08

-

-(3) 1-3 0.20 0.25 - 0 . 2 0 0.2 (3) 2 - 4 - 4 0 . 3 9

_-

_-

-% S G .

C. Böhm, K. Graf/ Ocean Engineering 90 (2014) 11-20 17

Table 3

V a l i d a t i o n o f total resistance C T ( X 1 0 ~ ^ ) f o r A C C V S .

No. Grid Error Type £(%) Uv{%) UD C%) UsN (%}

(1) 1-3 E 0.6 2.0 2.0 0.11 Ec 0.7 3.2 2.0 0.04 2 - 4 E 0.4 2.0 2.0 0.26 Ec 0.6 3.2 2.0 0.05 (2) 1-3 E 0.6 2.9 2.0 2.04 Ec 0.1 4,1 2.0 1.55 2 - 4 B 2.1

-

2.0 -Ec

_-

_-

2.0

-(3) 1-3 E 0.6 2.0 2.0 0.25 Ec 0.4 3.2 2.0 0.05 2 - 4 E Ec 1.2

:

2.0 2.0 %Sc. c o m p a r e d t o t h e s o l u t i o n Sc. This gives an e s t i m a t i o n o f t h e l e v e l o f v e r i f i c a t i o n o f the s i m u l a t i o n . I n a l l cases w h e r e a n e s t i m a t i o n o f t h e n u m e r i c a l u n c e r t a i n t i e s was possible, t h e c o r r e c t e d s o l u t i o n does n o t d i f f e r m u c h f r o m t h e o r i g i n a l l y c a l c u l a t e d w i t h d i f f e r -ences i n t h e range o f - 0 . 0 7 t o 0.5%SG- h can t h u s be c o n c l u d e d t h a t i n a l l t h o s e cases t h e l e v e l o f v e r i f i c a r i o n is r a t h e r g o o d a n d t h e r e s u l t s can be c o n s i d e r e d v e r i f i e d . V a l i d a t i o n o f t h e s i m u l a t i o n r e s u l t s is p e r f o r m e d w i t h respect t o t h e results o f t h e t o w i n g t a n k tests. T h e r e f o r e t h e c o m p a r i s o n e r r o r is c a l c u l a t e d a c c o r d i n g t o Eq. ( 9 ) t a k i n g i n t o a c c o u n t t h e s i m u l a t i o n r e s u l t S a n d t h e e x p e r i m e n t a l d a t a D. I n o r d e r to c o n d u c t t h e v a l i d a t i o n as d e f i n e d i n ( 2 1 ) , t h e v a l i d a t i o n u n c e r t a i n t y Uy has t o be c a l c u l a t e d ( 1 0 ) . T h e c o r r e c t e d c o m p a r i s o n e r r o r Ec is d e f i n e d as i n ( 9 ) b u t u s i n g Sc i n s t e a d o f S. Table 3 s u m m a r i z e s c o m p a r i s o n e r r o r E, v a l i d a t i o n u n c e r t a i n t y Uy, e x p e r i m e n t a l u n c e r t a i n t y Uoand s i m u l a t i o n uncer-t a i n uncer-t y UsN as p e r c e n uncer-t a g e o f D f o r b o uncer-t h c o r r e c uncer-t e d a n d u n c o r r e c uncer-t e d a p p r o a c h e s . I t has t o be n o t e d t h a t d a t a u n c e r t a i n t y UD has n o t b e e n s p e c i f i e d i n t h e e x p e r i m e n t a l t o w i n g t a n k d a t a . Details r e g a r d i n g e x p e r i m e n t a l u n c e r t a i n t i e s o f large t o w i n g t a n k f a c i l i t i e s are r a r e l y f o u n d i n t h e l i t e r a t u r e . L o n g o a n d Stern ( 2 0 0 5 ) give values b e t w e e n 0.6% a n d 1.5% f o r a s y s t e m a t i c i n v e s t i g a t i o n o f t h e s u r f a c e c o m b a t a n t D T M B 5415 m o d e l w i t h r e s p e c t t o e x p e r i m e n -t a l e r r o r s w h i l e Yan e-t a l . ( 2 0 0 8 ) g i v e values o f 2.8% f o r -t h e same s h i p . S i m i l a r data f o r y a c h t i n v e s t i g a t i o n have n o t b e e n available. The o n l y source f o u n d f o r u n c e r t a i n t i e s o f y a c h t i n v e s t i g a t i o n has b e e n a p r e s e n t a t i o n g i v e n b y Frank D e B o r d at Stevens I n s t i t u t e ( D e B o r d , 2 0 0 6 ) . T h e d a t a g i v e n i n t h i s p r e s e n t a t i o n s h o w t h e l o n g t e r m r e p e a t a b i l i t y o f t o w i n g t a n k tests t o be a p p r o x i m a t e l y 3%. A l s o t h i s o v e r v i e w o f t o w i n g t a n k u n c e r t a i n t i e s is b y n o m e a n s c o m p l e t e , i t can be c o n c l u d e d t h a t t h e d a t a u n c e r t a i n t y n o r m a l l y s h o u l d n o t exceed 3%. I t w a s t h e r e f o r e d e c i d e d t h a t i t is feasible t o t a k e i n t o a c c o u n t a n e x p e r i m e n t a l u n c e r t a i n t y Up o f 2% f o r v a l i d a t i o n p u r p o s e . By c o m p a r i n g E a n d Uy o f Table 3 o n e c a n easily see t h a t f o r a l l cases i n w h i c h t h e c o m p a r i s o n e r r o r c o u l d be c a l c u l a t e d , E<Uv is t r u e . T h e r e f o r e results have b e e n v a l i d a t e d f o r a l l cases except case b ( g r i d s 2 - 4 ) o f b o t h f r e e s u r f a c e a n d o v e r a l l r e f i n e m e n t studies. T h i s c o i n c i d e s w i t h t h e findings o f t h e v e r -i f -i c a t -i o n s t u d y a n d a l l o w s t h e c o n c l u s -i o n t h a t b o t h v e r -i f -i c a t -i o n a n d v a l i d a t i o n have b e e n a c h i e v e d f o r a l l r e f i n e m e n t s t u d i e s e x c e p t t h e t w o cases stated above. T h e f o r m a l v a l i d a t i o n a n d v e r i f i c a t i o n p r o c e d u r e as c o n d u c t e d above o n l y a l l o w s d r a w i n g c o n c l u s i o n r e g a r d i n g t h e finest g r i d i n t h e study, i n t h i s case g r i d 1 r e s p e c t i v e g r i d 2. W h i l e n o t g i v i n g t h e same l e v e l o f c e r t a i n t y a p l o t o f results d e l t a s o v e r g r i d cells is a feasible a p p r o a c h t o j u d g e t h e s e n s i t i v i t y o f t h e s o l u t i o n t o g r i d changes. Fig. 5 i l l u s t r a t e s resistance c o e f f i c i e n t ACT o v e r g r i d p o i n t s . I t is i n t e r e s t i n g t o n o t e t h a t w i t h o n g o i n g r e f i n e m e n t cases i n c l u d i n g f r e e surface g r i d p a r a m e t e r s s h o w a n i n c r e a s i n g d r a g w h i l e f o r t h e g e n e r a l r e f i n e m e n t case t h e +4.00% +2.00% +0.00% ^ -2.00% C -4.00% i -6.00% -8.00% X Finest Grid s G e n e r a l Refinement •e-Free Surface Refinement •. Overall Refinement

0 2 4 6 8 Grid Points [x 10»6]

Fig. 5. ACr over g r i p points w.r.t. experimental data.

10 12

PositionCZ) (mm)

-300.00 -113.00 70.000 250.00 440.00 625.00

Fig. 6. Wave contours f r o m iniüai studies ( t o p ) and f r o m Grid Convergence studies ( b o t t o m , g r i d 1 - finest grid).

o p p o s i t e h o l d s t r u e . T h e l a t e r one is a t y p i c a l r e s u l t f o r single phase RANSE s i m u l a t i o n s . I n c r e a s i n g g r i d r e f i n e m e n t gives a b e t t e r r e s o l u t i o n o f p r e s s u r e peaks w h i c h u s u a l l y r e s u l t s i n s m a l l e r f o r c e s values u n t i l g r i d i n v a r i a n c e o f r e s u l t s is r e a c h e d . This i n v e s t i g a r i o n suggests t h a t w h i l e t h i s c e r t a i n l y h o l d s t r u e f o r single phase i n v e s t i g a t i o n o f d e e p l y s u b m e r g e d b o d i e s , i t is n o t a p p l i c a b l e t o f r e e surface flows a r o u n d floating b o d i e s . T h e r a t i o n a l e b e h i n d t h i s b e h a v i o r p r o b a b l y is t h a t a t o o coarse r e s o l u t i o n o f f r e e s u r f a c e leads t o increased w a v e d a m p i n g t h u s a l t e r i n g t h e p r e s s u r e fluctuations o n t h e h u l l s u c h t h a t a l o w e r w a v e resistance is p r e d i c t e d . H o w e v e r , t o be s u r e t h i s t h e o r e m w o u l d have t o be p r o v e d . T h e d i s t r i b u t i o n o f r e s u l t s also i l l u s t r a t e s t h e h i g h i m p a c t o f f r e e s u r f a c e r e f i n e m e n t p a r a m e t e r s o n o v e r a l l g r i d d e n s i t y a n d r e s u l t accuracy. I t c a n be c o n c l u d e d t h a t special a t t e n t i o n has t o b e d e v o t e d t o t h e s e p a r a m e t e r s i n o r d e r t o achieve r e l i a b l e r e s u l t s . Since t h e c o r r e c t d e t e r m i n a t i o n o f w a v e r e s i s t a n c e is c r u c i a l f o r r e l i a b l e results o n t o t a l resistance o f ships, a r e f i n e m e n t s t u d y f o r f r e e surface flows also has t o t a k e i n t o a c c o u n t i t s i n f l u e n c e o n g e n e r a t e d w a v e p a t t e r n s . Fig. 6 c o m p a r e s w a v e r e s o l u t i o n f r o m i n i t i a l s t u d i e s ( t o p ) w i t h r e s u l t s g a i n e d w i t h t h e m o d i f i e d HRIC scheme. T h e t o p p i c t u r e s h o w s t h a t t h e c o m p u t a t i o n a l d o m a i n is t o o s h o r t a n d t h e w a v e p a t t e r n is d i f f u s e a n d d a m p e d . E s p e c i a l l y t h e l a t e r suggests a n i n s u f f i c i e n t r e s o l u t i o n o f t h e f r e e surface. T h e b o t t o m o f Fig. 6 s h o w s t h e finest g r i d o f t h e i n v e s t i g a t i o n .

18 C. Bdhm. K. Graf/ Ocean Engineering 90 (2014) 11-20

Volume Fraction of Water D.40IX}0 0.60000

VoiumeFrOQtion cfWater

O.OOOOO 020000 D.AOOOO 0.60000 O.eOOGO t.0000

w t z : _ n

Fig. 7. Numerical v e n t i l a t i o n w i t h Courant number dependency.

Vdume Ffaclion of Water

O.moO 0.60000

O.OOOOO 0.200OO

~7SumeTiacHon of Wats! 0.40000 0.60000

Fig. 8. Numerical v e n t i l a t i o n without Courant n u m b e r dependency. (For i n t e r p r e -t a -t i o n o f -t h e references -to color i n -this figure cap-tion, -the reader is referred -to -the w e b version o f this paper.)

O b v i o u s l y t h e r e are l a r g e d i f f e r e n c e s b e t w e e n t h e t w o s i m u l a t i o n s , t h e l a t e r o n e s h o w i n g a sharp r e s o l u t i o n o f p r i m a r y a n d secondary w a v e t r a i n s . H e r e w a v e d a m p i n g seems t o be l a r g e l y r e d u c e d . One o f t h e goals o f t h i s i n v e s t i g a t i o n w a s t o r e d u c e n u m e r i c a l v e n t i l a t i o n caused b y t h e s m e a r i n g o f t h e f r e e s u r f a c e i n t e r f a c e . Fig. 7 s h o w s t h e v o l u m e f r a c t i o n s o f w a t e r at t h e y a c h t surface f o r t h e o l d a p p r o a c h w i t h C o u r a n t n u m b e r d e p e n d e n c y w h i l e Fig. 8 i l l u s t r a t e s t h e same f o r t h e n e w a p p r o a c h w i t h o u t . C o m p a r i n g t h e t w o cases o n e c a n c l e a r l y see f r o m t h e p r o f i l e v i e w t h a t t h e n e w a p p r o a c h gives a m u c h s h a r p e r i n t e r f a c e b e t w e e n a i r ( b l u e ) a n d w a t e r ( r e d ) . T h e d i f f e r e n c e s are m o s t d i s t i n c t i v e at t h e b o w w a v e w h i c h takes a n e n t i r e l y d i f f e r e n t shape. T h e b o w w a v e o f t h e o l d a p p r o a c h (Fig. 7 ) has a large r e g i o n o v e r w h i c h t h e i n t e r f a c e is s m e a r e d a n d t h i s s m e a r i n g is t r a n s p o r t e d s i g n i f i c a n t l y d o w n -s t r e a m . For t h e n e w a p p r o a c h (Fig. 8 ) t h e b o w w a v e i-s m u c h m o r e d i s r i n c t i v e a n d t h e f r e e surface i n t e r f a c e is u s u a l l y c a p t u r e d o v e r 3 - 4 cells. T h i s c l e a r l y s h o w s a n advantage o f m o d i f i e d a p p r o a c h o v e r t h e o l d . H o w e v e r , p l a n v i e w reveals t h a t t h e v o l u m e f r a c t i o n a c h i e v e d w i t h t h e n e w a p p r o a c h s t i l l is n o t p e r f e c t . While t h e i m p r o v e m e n t s b e t w e e n o l d a p p r o a c h a n d n e w a p p r o a c h are o b v i o u s a n d pleasant, p l a n v i e w s r i l l reveals s o m e r e m a i n i n g i n t e r f a c e s m e a r i n g . Still t h e i m p r o v e m e n t is large since t h e v o l u m e f r a c t i o n f o r t h e o l d a p p r o a c h ranges b e t w e e n 0.4 a n d 1.0, w h i l e f o r t h e n e w approach t h e range is b e t w e e n 0.85 a n d 1.0. It seems t h a t w i t h i n the VOF m e t h o d achieving perfect results w i t h o u t smeared interfaces f o r this rather b l u n t b o w is still v e r y h a r d i f n o t impossible. Nonetheless f r o m an engineering p o i n t o f v i e w t h e s i m u l a t i o n is absolutely applicable since w i t h respect to t h e v e r i f i c a t i o n and v a l i d a t i o n results the e n o r i n t o t a l resistance is small.

5.4. Grid convergence studies including lift

A f t e r t h e successful v e r i f i c a t i o n a n d v a l i d a t i o n o f t h e p o i n t v a r i a b l e Cj f o r t h e s a i l i n g y a c h t i n u p r i g h t c o n d i t i o n s r e p o r t e d i n

Section 5.3, a f u r t h e r s t u d y has b e e n c o n d u c t e d i n o r d e r t o p r o v e t h e f e a s i b i l i t y o f t h e a p p r o a c h f o r h e e l e d c o n d i r i o n s o f t h e y a c h t . H e e l e d c o n d i t i o n s i n c l u d e t h e g e n e r a t i o n o f h y d r o d y n a m i c l i f t b y the y a c h t a n d its appendages. T h e r e f o r e a v a l i d a t i o n a n d v e r i f i c a -tion f o r these c o n d i t i o n s c a n n o t be r e s t r i c t e d t o t h e e v a l u a t i o n o f t o t a l resistance Cj. I n s t e a d i t has t o i n c l u d e t h e l i f t i n g c o m p o n e n t to consider t h e c o m p l e t e state o f t h e y a c h t . T h e r e f o r e t h e t w o p o i n t values t o t a l d r a g c o e f f i c i e n t C D a n d t o t a l l i f t c o e f f i c i e n t Cj, are evaluated t o g e t h e r . The c o r r e c t e v a l u a t i o n o f t h e s e f o r c e s w i t h i n t o w i n g t a n k e x p e r i m e n t s o r CFD s i m u l a t i o n s r e q u i r e s t h e m o d e l -i n g o f a e r o d y n a m -i c forces -i n w h -i c h a s a -i l -i n g y a c h t e n c o u n t e r s . I n o r d e r t o c o r r e c t l y s i m u l a t e t h e i n f l u e n c e o f t h e a e r o d y n a m i c f o r c e generated b y t h e sails, one has t o i n t r o d u c e a n a d d i t i o n a l d y n a m i c sail t r i m m i n g m o m e n t a r o u n d t h e y - a x i s o f t h e y a c h t w h i c h is equal to h y d r o d y n a m i c d r a g D t i m e s t h e v e r t i c a l c e n t e r o f e f f o r t s o f t h e sails VCEaero-MY,,,==D.VCEaero ( 2 3 ) A d d i t i o n a l l y , t h e g e n e r a t i o n o f l i f t b y t h e y a c h t h u l l a n d a p p e n -dages i n t r o d u c e s a v e r t i c a l f o r c e p o i n t i n g u p . S i m i l a r t o t h e t r i m m i n g m o m e n t e x p l i c a t e d above, t h i s f o r c e has t o be c o u n t e r e d b y a c o l l i n e a r a e r o d y n a m i c v e c t o r o f e q u a l l e n g t h a n d d i f f e r e n t sign. This sail f o r c e has to be m o d e l e d d u r i n g t e s r i n g as a n a d d i t i o n a l d y n a m i c s i n k f o r c e Fz^^. I t is m o d e l e d as h e e l i n g f o r c e FH t i m e s t h e sine o f t h e h e e l i n g angle ^ .

FZ,^„=FH sin </> ( 2 4 )

C o n t r a r y t o t h e u p r i g h t resistance g r i d c o n v e r g e n c e study, t h i s s t u d y has b e e n c o n d u c t e d i n m o d e l scale. This a p p r o a c h n o t o n l y a l l o w s easier c o m p a r i s o n b e t w e e n r e s u l t s b u t also m a k e s t h e a p p l i a n c e o f t h e v a r i o u s a d d i t i o n a l i n p u t p a r a m e t e r s easier. W h i l e f o r t h e n o n - l i f t i n g t e s t cases v a l i d a t e d i n S e c t i o n 5.3 t r i m w a s k e p t fixed a n d o n l y sinkage w a s d y n a m i c a l l y c a l c u l a t e d , t h e p r e s e n t case sets b o t h state v a r i a b l e s f r e e . This is a m a j o r c h a n g e since i t m a k e s i t necessary t o a c c o u n t f o r s i m i l a r t r i m a n d s i n k a g e f o r c e s i n o r d e r t o c o m p a r e s i m u l a t i o n a n d e x p e r i m e n t . For t h e t o w i n g t a n k e x p e r i m e n t p r e s c r i b e d t r i m m o m e n t s a n d v e r t i c a l forces e x i s t as i n p u t values. These values have b e e n u s e d as i n p u t data f o r t h e CFD s i m u l a t i o n i n s t e a d o f d y n a m i c c a l c u l a t i o n o f t h e s e values, w h i c h w o u l d also have b e e n possible.

5.4.1. Computational grids T h e g r i d c o n v e r g e n c e s t u d y has also b e e n c o n d u c t e d a c c o r d i n g t o ITTC s t a n d a r d s as e x p l i c a t e d i n Section 5.3. T h e p r i n c i p a l d e s i g n o f t h e g r i d s is i d e n r i c a l t o t h e o n e used i n S e c t i o n 5.3. I t i n c l u d e s r e f i n e m e n t o f t h e f r e e surface i n t h e v e r t i c a l d i r e c t i o n a n d a d d i t i o n a l l y i n h o r i z o n t a l d i m e n s i o n s i n t h e v i c i n i t y o f t h e k e l v i n angle a r o u n d t h e boat. T h e results o f t h e n o n - l i f t i n g v e r i f i c a t i o n a n d v a l i d a t i o n s t u d y c l e a r l y s h o w e d t h a t t h e m a j o r f a c t o r t o w a r d s a g r i d i n d e p e n d e n t s o l u t i o n is t h e r e f i n e m e n t o f t h e f r e e surface. Fig. 5 i l l u s t r a t e s t h a t surface g r i d r e f i n e m e n t is a l r e a d y s u f f i c i e n t . T h e r e f o r e o n l y f r e e surface r e f i n e m e n t has b e e n v a r i e d f o r t h e p r e s e n t g r i d c o n v e r g e n c e study. G r i d p a r a m e t e r s have b e e n s y s t e m a t i c a l l y v a r i e d a c c o r d i n g t o Table 4 . I n c o n t r a s t t o t h e g r i d c o n v e r g e n c e s t u d y f o r t h e n o n - l i f t i n g case i n S e c t i o n 5.3 t h e Table 4

Grid parameter f o r g r i d invariance study.

Ref. Interface spacing Grid size (dimensionless)

Factor (dimensionless) dz ( m m ) dx & dy ( m )

1.0 10.0 0.0625 1.25 X 1 0 '

1.5 15.0 0.0938 7.07 X 10^

C. Böhm, K. Graf/ Ocean Engineering 90 (2014) 11-20 19 2.5% 1.5% £ 0.5% H

I

-0.5% -1.5% -2.5% 1 1 l l l l 1 1 1 1 1 J •••Delta cD "D-Delta cL 4 6 8 10 12 14 Grid Points x l 0 « 6 [-]

Fig. 9. A Q over g r i d points.

Table 5

Grid convergence of drag and l i f t f o r ACCVS.

Variable Grid 3 Grid 2 Grid 1 EF data

CD 8.94 9.00 9.01 9.05 c(%) - 0.7 0.1 CL 1.89 1.88 1.87 1.86

e{%)

- - 0 . 7 - 0 . 4 CL/CD 2.12 2.09 2.08 2.05 - - 1 . 4 - 0 . 5 %Sc. Table 6

Verification o f drag and l i f t f o r ACCVS.

Variable

_Rc

_{UGC%) Sim Ucc (%)}

CD 0.21 0.19 - 0 . 1 2 0.08

Cl

0.54 0.44 0.30 0.14

CL/CD 0.37 0.52 0.41 0.11 VcSa.

Table 7

Validation o f drag and l i f t f o r ACCV5.

Variable Error Type E(%) Uv(%)

CD £ 0.4 2.0 2.0 0.19 Ec 0.3 2.0 2.0 0.08 CL E - 0 . 9 2.0 2.0 0.44 Ec - 0 . 6 2.0 2.0 0.15 CL/CD E - 1 . 3 2.1 2.0 0.53 Ec - 0 . 9 2.0 2.0 0.11 %Sc. c o n s t a n t g r i d r e f i n e m e n t f a c t o r has b e e n decreased f r o m 2 t o 1.5. T h i s has b e e n d o n e t o get a m o r e u n i f o r m r e f i n e m e n t i n t e r m s o f c e l l sizes w h i c h e n h a n c e s t h e c o m p a r a b i l i t y o f t h e results. T h e d i f f e r e n c e s o f l i f t a n d d r a g c o e f f i c i e n t t o t h e e x p e r i m e n t a l d a t a d e r i v e d f r o m t h e g r i d c o n v e r g e n c e s t u d y are s h o w n i n Fig. 9. T h e f i g u r e i l l u s t r a t e s t h a t t h e d r a g c o e f f i c i e n t Q j is a l w a y s u n d e r -e s t i m a t -e d , w h i l -e f o r t h -e l i f t c o -e f f i c i -e n t Q t h -e c o n t r a r y h o l d s t r u -e . H o w e v e r , d i f f e r e n c e s t o EFD are r a t h e r l o w f o r b o t h c o e f f i c i e n t s a n d i n t h e same o r d e r o f m a g n i t u d e . G e n e r a l l y b o t h c o e f f i c i e n t s c o n v e r g e q u i t e s a t i s f a c t o r i l y , g i v i n g t h e first i n d i c a t i o n o f a h i g h q u a l i t y s o l u t i o n . Table 5 gives t h e n u m e r i c a l values o f t h e c o n v e r g e n c e o f drag, l i f t a n d l i f t / d r a g - r a t i o . T h e s o l u t i o n change f r o m a coarser t o a finer s o l u t i o n e, as d e f i n e d i n ( 2 2 ) , decreases

c o n t i n u o u s l y . The results o f t h e v e r i f i c a t i o n p r o c e d u r e (Table 6 ) s h o w t h a t t h e convergence r a t i o i? G< l is t r u e f o r a l l cases, a l l o w i n g t h e c o n c l u s i o n t h a t t h e decrease is m o n o t o n i c f o r a l l values. T h e biggest u n c e r t a i n t y o f t h e c o m p u t a t i o n a l g r i d Uc is 0.52% f o r t h e l i f t - t o - d r a g r a t i o CL/CQ w h i c h is a l r e a d y v e r y l o w . Since t h e convergence is m o n o t o n i c , i t is p o s s i b l e t o use General-ized Richardson Extrapolation i n o r d e r t o a p p l y a c o r r e c t i o n f o r n u m e r i c a l error. I n p a r t i c u l a r , i t is possible t o c a l c u l a t e a c o r r e c t g r i d u n c e r t a i n t y Uc^ a n d a c o r r e c t e d s o l u t i o n So W i t h a m a x i m u m d e r i v a t i o n o f 0.14%, these c o r r e c t e d values are e v e n closer t o t h e e x p e r i m e n t a l values. I t can be g e n e r a l l y said t h a t f r o m a n u m e r i c a l p o i n t o f v i e w t h e results o f t h e g r i d c o n v e r g e n c e s t u d y s h o w a docile b e h a v i o r a n d s t e a d i l y converge t o w a r d s t h e e x p e r i m e n t a l values w i t h i n c r e a s i n g r e f i n e m e n t . This a l l o w s t h e c o n c l u s i o n t h a t t h e s i m u l a t i o n is v e r i f i e d . Table 7 gives a n o v e r v i e w o f t h e values necessary f o r t h e v a l i d a t i o n p r o c e d u r e . Data u n c e r t a i n t y UD a n d n u m e r i c a l s i m u l a t i o n u n c e r t a i n t y USN are c o m b i n e d t o t h e v a l i d a -t i o n u n c e r -t a i n -t y Uy. Uy is -t h e n c o m p a r e d -t o -t h e c o m p a r i s o n e r r o r Ec w h i c h is d e f i n e d as data D m i n u s s i m u l a t i o n r e s u l t S as p e r Eq. (9). The table lists a l l values b o t h f o r t h e u n c o r r e c t e d s o l u t i o n and t h e s o l u t i o n corrected b y means o f Generalized Richardson E x t r a p o l a t i o n . Per d e f i n i t i o n , a s i m u l a t i o n is v a l i d a t e d i f t h e c o m p a r -ison e r r o r is less or equal t h e v a l i d a t i o n u n c e r t a i n t y . T h i s clearly is t h e case f o r a l l six c o m p a r i s o n cases. The s i m u l a t i o n can t h e r e f o r e be considered v a l i d a t e d a t t h e v a l i d a t i o n u n c e r t a i n t y level.

I t can be s u m m a r i z e d t h a t v e r i f i c a t i o n a n d v a l i d a t i o n f o r l i f r i n g c o n d i t i o n s w e r e h i g h l y successful. A c h i e v e d r e s u l t s are n o t o n l y c o n s i d e r a b l y b e l o w v a l i d a t i o n u n c e r t a i n t y l e v e l b u t also v e r y close to e x p e r i m e n t a l data. A l t h o u g h t h i s f o r m a l l y does n o t decrease t h e u n c e r t a i n t y o f t h e results, i t s t i l l increases t h e c o n f i d e n c e i n t h e a p p l i e d m e t h o d s . I t also s h o w s again t h a t t h e a s s u m p t i o n s r e g a r d i n g f r e e surface i n t e r f a c e s m e a r i n g m a d e i n t h e p r e v i o u s sections are c o r r e c t .

6. S u m m a r y

The m o t i v a t i o n f o r t h i s i n v e s t i g a t i o n has b e e n a f a i l e d first a t t e m p t t o c o r r e c t l y d e t e r m i n e t o t a l resistance o f f r e e s u r f a c e flow a r o u n d a n ACCVS h u l l . A r e v i e w o f t h e first s i m u l a t i o n s l e d t o t h e a s s u m p t i o n t h a t t h e p r o b l e m c o u l d be t r a c e d b a c k t o t h e occur-rence o f e x t e n s i v e i n t e r f a c e s m e a r i n g at t h e y a c h t h u l l . T h i s l e d t o a t h o r o u g h r e v i e w o f t h e t h e o r y b e h i n d t h e i n t e r f a c e c a p t u r i n g m o d e l i n S e c t i o n 3. This r e v i e w s h o w e d t h a t t h e p r o b l e m s e n c o u n t e r e d m o s t l i k e l y w e r e s i t u a t e d i n t h e use o f C o u r a n t n u m b e r s e x c e e d i n g 0.5, t h u s c a u s i n g t h e s w i t c h t o a 1st o r d e r u p w i n d d i f f e r e n c i n g scheme. Since r e d u c i n g t h e o v e r a l l t i m e step size s u c h t h a t i t w o u l d a l l o w t h e m a x i m u m C o u r a n t n u m b e r t o be l o w e r t h a n 0.5 w o u l d lead t o u n d e s i r a b l e l o n g s i m u l a t i o n times a n a l t e r n a t i v e a p p r o a c h w a s s o u g h t t o a l l o w t h e use o f h i g h e r o r d e r schemes e.g. t h e HRIC s c h e m e w i t h i n acceptable time s t e p size. I t w a s c o n c l u d e d t h a t i t m i g h t be possible t o m o d i f y t h e V O F m o d e l s u c h t h a t i t does n o t s w i t c h t o u p w i n d d i f f e r e n c i n g e v e n i f t h e local C o u r a n t n u m b e r w o u l d be larger t h a n 0.5. T h i s a p p r o a c h seems feasible as l o n g as o n l y a steady state s o l u t i o n is s o u g h t -after. S e c t i o n 4 s h o w s a n u m e r i c a l t e s t case w h i c h a l l o w s t h e c o n c l u s i o n t h a t t h i s a p p r o a c h is feasible. T h e r e f o r e , t h e m o d i f i e d scheme w a s a p p l i e d t o t h e s i m u l a t i o n o f t h e t o t a l r e s i s t a n c e o f t h e ACCVS y a c h t . V e r i f i c a r i o n a n d V a l i d a t i o n a c c o r d i n g t o t h e ITTC g u i d e l i n e s w e r e t h e n c o n d u c t e d against e x p e r i m e n t a l d a t a f o r l i f t i n g a n d n o n - l i f t i n g t e s t cases, Extensive g r i d s t u d i e s h a v e b e e n c a r r i e d o u t , t h u s also a l l o w i n g j u d g i n g t h e s e n s i t i v i t y o f t h e r e s u l t s to t h e c h a n g e o f v a r i o u s g r i d p a r a m e t e r s . T h e r e s u l t s s h o w e d a m u c h s h a r p e r c a p t u r i n g o f t h e f r e e surface i n t e r f a c e w i t h t h e n e w a p p r o a c h . I t w a s also s h o w n t h a t t h e i n i t i a l d i f f e r e n c e s i n o v e r a l l resistance w e r e m a i n l y caused b y t h e p o o r f r e e s u r f a c e r e s o l u t i o n