Numerical analysis of the interference effects on resistance, sinkage and trim of a fast catamaran

O R I G I N A L A R T I C L E

Numerical analysis of the interference effects on resistance,

sinkage and trim of a fast catamaran

Wei He • Teresa Castiglione • M a n i v a n n a n K a n d a s a m y • F r e d e r i c k Stern

Received: 28 A p r i l 2014/Accepted: 24 August 2014/Published online: 16 September 2014 © J A S N A O E 2014

Abstract The purpose o f the present paper is the numerical itivestigation o f the inteiference phenomena between the waves generated by the individual hulls o f a catamaran. The study focuses on the effects o f both Froude tiuiuber and demi-hull separation distance on resistance and on sinkage and triiu. The numerical simulations are cairied out by the U R A N S solver CFDShip-Iowa V . 4 and, to assess the capability f o r prediction o f resistance, sinkage and t r i m of the U R A N S code f o r t w i n - h u l l configuration vessels, a verification and vaUdation study is performed f o r global as well as f o r local variables. A very good agree-luent between the nuiuerical results and the experimental data is obtained, and the validation study demonstrates the high level of accuracy of the cuirent predictions, w h i c h are used to have a better insight into the interference phe-nomena. I n accordance w i t h the experiments, w i t h i n Fr = 0.45-0.65, the catamaran has a significaiidy higher resistance coefficient compared to the mono-hull; further-more, the C T value increases w i t h decreasing the separation distance between the t w i n hulls. On the contrary, at Fr lower than 0.45 and at Fr higher than 0.65, the effects o f h u l l spacing on resistance as w e l l as on sinkage and t r i m

W . He and T. Castiglione contributed equally to the present work. W . He • M . Kandasamy • F. Stern ( E l )

IIHR-Hydroscience and Engineering, The University o f Iowa, Iowa, I A , USA

e-mail: frederick-steni@uiowa.edu W . He

School o f Naval Architecture, Ocean and C i v i l Engineering, Shanghai Jiao Tong University, Shanghai, China

T. CastigUone

Department of Mechanical Energy and Management Engineering, University of Calabria, Arcavacata di Rende, Cosenza, Italy

can be neglected. The flow field characteristics, wave pattern, wave cuts and pressure distributions are analyzed through the C F D analysis. Finally, the effects o f the Rey-nolds number on resistance are also investigated and results show a small decrease i n interference w i t h increasing the Reynolds nuiTiber.

K e y w o r d s U R A N S • Catamaran • Interference • V e r i f i c a t i o n and validation - C F D • Ship motions • Resistance

L i s t of symbols

B Beam of the demi-hull ( - ) Ci Frictional resistance coefficient ( - ) Cp Pressure resistance coefficient ( - ) C- Residuary resistance coefficient ( - ) C T Total resistance coefficient ( - )

C T , cat Catamaran total resistance c o e f f i c i e n t ( - ) ^ T , mono M o n o - h u l l total resistance coefficient ( - )

D Experimental data value

E Comparison eiTor

Fr Froude number ( = U/ gL^^)

8 Acceleration o f gravity (= 9.81 m/s^)

h Water depth (m)

I F Interference coefficient ( - ) k Turbulent kinetic energy (J/kg) Lpp M o d e l length (m)

il N o r m a l direction

P Order o f accuracy, pressure ( N / m ^ ) Pth Estimated order of accuracy

PRE Richardson extrapolation order o f accuracy P _{Estimated order o f accuracy ^ =}

r G r i d refinement ratio

R Ratio between solution changes

J Mai- Sci Teclinol (2015) 20:292-308 293

Re Reynolds number { = pUL/fi) RT Total resistance ( N )

^T, Ciit Catamaran total resistance (N) ^T, mono M o n o - l i u l l total resistance ( N )

S Separadon distance between the hulls [ m ] , wet surface area (m^)

Sj Simulation value w i t h grid j

s Non-dimensional separation distance ( = SI Lpp)

T M o d e l d r a f t (in)

(U, V, W) M o d e l velocity components (m/s) Uo Data uncertainty

f/pB Facility bias uncertainty L^G G r i d uncertainty Ul Iterative uncertainty

UsN Numerical simulations uncertainty t / v Validation uncertainty

(A; }', z) Absolute earth-fixed coordinates {X, Y, Z) Non-dimensional coordinates /? Turbulence model constant ( - )

(5RE Numerical eiTor computed by Richardson extrapolation

E Turbulence dissipation rate (J/kgs) Sjj D i f f e r e n c e between solutions / and j (j) Free surface level-set f u n c t i o n X Wavelength (m)

p Water density {kglxr?) a Non-dimensional sinkage T T r i m (rad)

CniiLx f f i g h e s t wave elevation ( m ) « Specific turbulence dissipation ( l / s )

1 Introduction

M u l t i - h u l l surface ships are attractive f o r high-speed uses due to their favorable resistance, transversal stability, and load characteristics i n comparison to conventional mono-hulls. H u l l types and applications vary considerably: f r o m relatively small foil-assisted high-speed ferries to relatively large S W A T H high-speed sealift vessels designed f o r m i l i t a r y transportation. As a consequence, several studies on m u l t i - h u l l configurations, and in particular on catama-rans, have been carried out both experimentally and numerically over the years.

The experimental tests conducted by Insel and M o l l a n d [1] and M o l l a n d [ 2 ] , on a high-speed catamaran w i t h symmetric demi-hulls, focus on the effects o f h u l l dimen-sions and o f separation distance, on resistance, sinkage and t r i m , over a wide range o f Froude number values (0.2-1.0). They also developed a numerical model, based on the

thin-strip hypothesis, to investigate the catamaran hydrody-namic characteristics. Cheng et al. [3J developed a math-ematical theory f o r the practical design o f an asymmetric d e m i - h u l l , the 5-catamaran, whose h u l l sections were the-oretically determined to eliminate or at least substantially reduce wave resistance at supercritical design speeds i n shallow water. Further experimental and theoretical investigation o f the interference phenomena was also car-ried o u t in M i l l w a r d [ 4 ] and M o l l a n d et al. [ 5 ] . Souto-Igleasias et al. [ 6 ] investigated experimentally the influence o f the distance between catamaran hulls on wave resis-tance. They also measured the wave cuts between the t w o hulls to correlate the shapes o f the inner wave cuts w i t h the interference resistance and f o u n d the possibility o f favor-able interference f o r catamarans.

From a numerical and theoretical point o f v i e w , the approaches used f o r investigation o f wave interference effects were c o m m o n l y based on potential flow theory [ 7 ¬ 10]. Bruzzone and F e i T a n d o [11] used a boundary element method i n order to study hull separation length effects. Despite the mentioned studies being quite useful to esti-mate the wave-making resistance and ship motions, issues remain f o r design improvements, regarding m u l t i - h u l l interference, reduced wake signatures, deep/shallow water maneuvering, potential flow vs, C F D capability, and o p t i -mization. So far, only a f e w studies have introduced C F D methods f o r the analysis o f m u l t i - h u l l vessels. Stern et al. [12, 13] have shown interference-induced longitudinal vortices f o r S W A T H . Kandasamy et al. [14] demonstrated the possibility o f reduced wake signatures f o r foil-assisted catamarans. M i l a n o v et al. [15] discussed about the most relevant issues f o r the improvement o f high-speed cata-maran deep/shallow watercourse stability design and o f system-based method predictions.

Recently, the D e l f t catamaran [16, 17] has been selected as an international benchmark f o r research and C F D v a l i -dation f o r the O f f i c e o f Naval Research N a v a l International Cooperative Opportunities i n Science and Technology Program ( N I C O P ) and N A T O Advanced Vehicle Tech-nology projects. Several hydrodynamic topics have been assessed about this geometry, by c a i T y i n g out both exper-imental tests and ntimerical studies. They include calm water resistance and seakeeping assessment [ 1 8 - 2 3 ] , interference effects [24, 25], deep/shallow water maneu-vering [15, 2 6 ] , waterjet propelled optimization [27] and stochastic uncertainty quantification and o p t i m i z a t i o n [28¬ 30]. The numerical studies have been earned out b y the use o f R A N S methods; the use o f adaptive g r i d refinements i n Visonneau et al. [26] is w o r t h mentioning.

The present study contributes to this research w i t h the m a i n objective to assess current U R A N S capability f o r a high-speed catamaran, including sinkage and t r i m . I n par-ticular, the effects o f Froude number ( F r ) and d e m i - h u l l

Table 1 Main particulars o f

D e l f t catamaran model

" L is dimensional; other length parameters are

non-dimensionalized by L ^ BSHC uses P M M w i t h 2 contact points on the model " UrB is % M a x (D^,,^ )

M a i n particulars Symbol Unit BSHC T U Delft' I N S E A N CFD

Length between Lpp m 3.63 3.00 3.00 3.00

perpendiculars"

Depth of towing tank h 0.41 0.87 2.17 0.70

Beam overall B 0.31

Beam demi-hull b 0.08

Distance between center o f s 0.23 hulls

Draft T 0.05

Form factor k 0

Longitudinal center o f L C G 0.53 0.53

gravity

Vertical center o f gravity K G 0.04 0.06

T o w point i T b 0.53

0.06

Depth/draft hlT 8.2 17.3 43.3 14.0

Temperature C °C 10.4 20.0 20.1 15.0

Frrange

CT, sinkage and trim Fr 0.17-0.64 0.18-0.75 0.10-0.80 0.30-0.80

Wave cuts 0.3, 0.5, 0.75 0.3, 0.5, 0.75 0.3, 0.5, 0.75 Re (at Fr = 0.5) Re 10" 9.05 8.13 8.12 7.14 Facility Bias C T 3.11 3.07 4.10 C, 3.14 2.63 2.93 Sinkage 12.77 10.13 4.98 T r i m 3.91 2.80 5.59 Springer

J Mar Sci Teclmol (2015) 20:292-308 ₂₉₅

separation distance (s) on resistance, sinkage and t r i m f o r the D e l f t catamaran, are investigated. Simulations are earned out f o r speed values ranging w i t h i n 0 . 3 < Fr < 0 . 8 and include several hull-spacing values varying w i t h i n 0 . 1 7 < 0 . 3 0 . The methodology used i n this work is as f o l l o w s : a quantitative verification and validation ( V & V ) study is performed f o r Fr = 0 . 5 at the design hull sepa-ration distance coiTesponding to ^ = 0 . 2 3 , using available benchmark validation data. Once the code is validated, the numerical results are used to have a better insight i n the flow field characteristics around the t w i n hulls; i n particular wave pattern, wave cuts i n the inner region between hulls and pressure distributions along the h u l l w i l l be analyzed. Finally, a Reynolds number (Re) study is also included to investigate the effects o f viscosity o n interference.

2 Overview of experimental data

The m a i n particulars o f the D e l f t catamaran are provided i n Fig. 1 and Table 1 . F o r the design separation distance, s = 0 . 2 3 , calm water resistance, sinkage and t r i m experi-ments were carried out at three facilides: T U D e l f t [ 1 6 ] , B S H C [ 3 1 ] , and I N S E A N [ 2 4 ] . The experimental uncer-tainty ( L ' D ) is provided by I N S E A N at Fr = 0 . 5 , 0 . 6 5 and 0 . 7 5 . Table 1 also includes the f a c i l i t y bias, U^, w h i c h is

estiiTiated b y c o m p a r i n g the a v a i l a b l e data, a c c o r d i n g to t h e f o l l o w i n g a p p r o a c h : 1 r = A l l / D 1 sr-^M = ^ [ M a x ( | £ / | ) - M i n ( | £ / | ) ; , L . A ' ( 1 ) where M is the number o f facilities where tests are per-f o r m e d , and D / is the observation i n /th per-f a c i l i t y at ;'th Fr. Facility bias is studied i n the Fr range 0 . 2 5 - 0 . 6 f o r w h i c h the data between facilides overiap.

A comparison o f E F D data among the three facilities, at the design separation distance s = 0 . 2 3 , is shown i n F i g . 2 . I t includes total resistance coefficient, C j , w i t h its f r i c t i o n (Cf) and residuary (Cp) components, and non-dimensional sinkage and t r i m as functions o f the Froude number, Fr. The total resistance coefficient is defined as C T = RT/{0.5PU^S), where R-j- is total resistance ( N ) , p is density o f the water, U is the undisturbed velocity and S is the hull-wetted surface at rest; the non-dimensional sinkage is defined as a = sinkage/Lpp, and the t r i m angle is expressed as T, (rad). The measurements have similar trends, but f a i r l y large differences between the facilities occur as indicated b y the large i7pB values, especially f o r

Ü 8 . 0 E - 0 3 6 . 0 E - 0 3 4 . 0 E - 0 3 2 . 0 E - 0 3 - Q-. - B S H C • - V - - TU Delft - G- - INSEAN — « EFD-ave 2 . 5 E - 0 3 O.OE+00 Ö - 2 . 5 E - 0 3 - 5 . 0 E - 0 3 - 7 . 5 E - 0 3 ( a ) 0.2 0.4 0.6 0.8 Fr V

//yo

....Q.... c, B S H C - - - w - - - - C , T U D e l f t . . . . Q . . . . C I N S E A N • C , E F D - a v e - 0 . 0 2 (b) 0.2 0.4 Fr o V O -B S H C TU Delft INSEAN EFD-ave B S H C TU Delft INSEAN EFD-ave O -Ó Ji (d) 0.2 0.4 0.6 Fr

F i g . 2 Comparison o f E F D f r o m different facilities at the design separation distance i = 0.23. a Total resistance coefficient; b residuary

sinkage and t r i m at itigli Fv. Tiiey are due to tlie different iTiodel lengths, depths o f the towing tank, temperature and center of gravity position between facilities. Furthermore, T U D e l f t and I N S E A N use conventional resistance single-point mounts, whereas B S H C uses a Planar M o t i o n Mechanisiu two-point mount. Finally, turbulence simula-tors are used f o r I N S E A N and T U D e l f t models but not f o r B S H C .

I N S E A N also provides resistance, sinkage and t r i m data for the mono-hull and f o r the catamaran w i t h five separa-tion distances, which vary w i t h i n 0.17 and 0.30, and f o r several speed values {Fr = 0.3-0.8). The experimental data include the inteiference coefficient ( I F ) , w h i c h is defined as the difference between the total resistance coefficients of the catamaran and the mono-hull, C T , cat and CT, mono respectively, divided by the mono-hull total resistance coefficient, according to:

Ci.cat - C T .mono .mono \

1^ = — - 7 ; ' — = — ^ y'^i ^-'T.mono -^-t^T.mono Backqround

% -'^^

Fig. 3 Particulars o f the computation domain

It gives a measure o f the interaction between the t w i n -hull waves systems. It is common, in fact, that f o r a wide range of ship speed values, the resistance experienced by the t w i n - h u l l is higher than twice that o f the mono-hull (^T, cat = 2/?T, mono + ^ I F ) . The additional resistance component is due to the interaction between the waves leaving f r o m each demi-hull, which interact in the inner region. I n some cases, negative I F values can occur; the waves systems tend to cancel each other out and cause, therefore, a decrease o f wave resistance.

For cun-ent U R A N S simulations, the I N S E A N - C N R experimental data [24] w i l l be used f o r validation pur-pose. I t should be noted that the Reynolds number f o r CFD simulations is slightly different f r o m the experi-mental value; this is due to the different water temper-ature values used between the experiments and the luimerical simulations (Table 1 ) . I n fact, during the experimental activity i t is quite complex to maintain the t o w i n g tank water temperature accurate at a prescribed value; consequently, the Reynolds number is usually affected by uncertainty due to changes i n viscosity. It is then conventional to use a Reynolds number correction o f the experimental data to match Re at 15 °C f o r con-sistency, w h i c h is the temperature value used f o r numerical simuladons.

3 Computational and analysis method

3.1 Computational method

The code used f o r the current simulations is CFDShip-Iowa V.4 [32]. I t is based on the solution o f the unsteady Rey-nolds Averaged Navier-Stokes equations f o r the l i q u i d phase of a free surface flow. The free surface is captured using a single-phase level-set method [ 3 3 ] , and the turbu-lence is modeled by a blended k - s/k — co model [34] without w a l l functions. Numerical methods include advanced iterative solvers and second- and higher-order finite difference schemes w i t h conservative formulations.

Table 2 Boundary conditions Description <t> p k 10 U V W

Inlet (X = -0.4) — 7 Bp _ 8)1 ~ = 0 Re 9.0 1 0 0 Exit (X = 3.6) vn 0 _ dlt ' = 0 ¥ = 0 cn a. ~ 0 ts-U _ 8)1- = 0 c-V . dll- = 0 e-ll' dii' " : 0 Far-field (Z = -0.7) M = 3;i 1 Sii ' = 0 1^ = 0 Cll CW 811 ~ 0 1 0 0 Far-field (Z = 0.7) M — _{- 1} - i i = 0 CH do) cn 0 8t/ _ 8/1 0 81' _ & i ~ = 0 811' _ dn 0 Far-field ( 1 ' = 1.3) M = C71 0 = 0 P =_cn o Sco _ bl ~ 0 8Ü „ cn 0 ev _ dn = 0 811' _ dn 0 Symmetry ( 7 = 0 ) M = dn 0 2e = _{= 0} a = o cn dcD 8/1 0 W _ 8/1 0 0 dW _ dn 0

No-slip (ship hull) M = ₀ - 0 60

ISReA\ _I 0 0 0

J Mar Sci Tectinol (2015) 20:292-308 ₂₉₇

Table 3 Grids designed for Dimensions Total v + ( L , - = 0 5

catamaran studies

H u l l in and out Background Refinement = 7.1 X 10") G I R 244 x 93 x 1 3 9 436 x 1 9 6 x 142 6 0 1 x 181 x 181 38.1 M 0.6 G l 244 X 93 x 139 436 x 196 x 142 18.4 M 0.6 G2 172 X 66 X 98 309 x 139 x 101 6.6 M 0.9 G2a 172 x 66 X 98 309 x 139 x 101 6.6 M 0.2 G3 122 X 47 x 70 218 x 98 x 71 2.3 M 1.4 G4 86 X 33 x 49 155 x 70 x 51 0.8 M 2.3

Table 4 Grids and simulation

conditions for the different studies

Study Grid Simulation conditions

Configuration Fr Re (10")

Verification G l , G2, G3, G4 .V = 0.23 0.5 7.14

Interference G2 Mono-hull, J = 0.17, 0.23, 0.30 0.3-0.8 4.29-11.43 Refinement G l , G I R s = 0.17 0.5, 0.75 7.14, 10.72

Re G2a Mono-hull, J = 0.23 0.5 2.53, 7.14, 20.2

Mass conservation is enforced using a PISO algorithm, resulting i n a Poisson equation f o r pressure. CFDShip-Iowa allows the computation o f ship motions ( 6 D 0 F ) by the use o f a dynamic overset-grid approach f o r local grid refine-ment and large-amplitude motions. I n this case, the code S U G G A R [35] is used to obtain the overset domain con-nectivity between the set o f overlapping grids. The simu-lations run on HPC cluster Cray X T 4 (Jade).

3.2 D o m a i n , boundary condition, grids and simulation

conditions-The computational domain includes a background orthog-onal grid and a boundary layer curvilinear grid conforiuing to ship geometry. Some particulars o f the computational domain are depicted i n Fig. 3. Overall, the g r i d consists o f 3 blocks. The background is a Cartesian block, w h i c h is clustered near the free suiface to resolve the wave field. Its boundaries are - 0 . 4 <X< 3.6, 0 < F < 1.3 and — 0.7 < Z < 0.7, where the coordinates X, 7 and Z are non-dimensional w i t h the length between perpendiculars, Lpp. For each demi-hull, two body-fitted " O " type grids are generated around the h u l l geometry. Splash and wave breaking were observed i n the experiment; therefore, an unsteady solution o f the f u l l ship, w h i c h allows asymmetric luodes, may be more accurate. However, o w i n g to the coiuputational demand f o r resolving the cuirent flow, a symmetry boundary condition is applied w i t h respect to F = 0 and only half of the catamaran is simulated. The ship axis is aligned with the .v axis w i t h the b o w at X = 0 and the stern at Z = 1, as shown i n F i g . 1. The free surface at rest lies at Z = 0.

The boundary conditions are summarized i n Table 2. They are specified f o r each face o f the computational domain, and proper values are set f o r each flow variable. A t Inlet (X = - 0 . 4 ) , the free surface level-set f u n c t i o n , w h i c h is zero on the free surface, is given by </> = —z; pressure is zero-gradient; turbulence is set to free stream values and the velocity field is set to the free stream velocity vector {U,V,W). A t Exit {X = 3.6), the boundary is assumed far downstream so that stream-wise viscous effects are zero

( © ^ = 0 = = 0 ^ ; other variables are zero-gradient due to domain truncation. T w o far-field boundary condi-tions are used to l i m i t the domain along Y- and Z-axes. Furthermore, ^ = -1-1 at Z = - 0 . 7 indicates that the sur-face is i n contact w i t h the hquid phase; | ^ = - 1 at Z = -1-0.7 indicates the contact between the top back-ground surface and air. Finally, on the ship h u l l a no-slip boundary condition is imposed, i n w h i c h the value f o r m is suggested b y Menter [34] and is strictiy related to the use o f the blended K - EIK - OJ turbulence model.

Table 5 Verification f o r s = 0.23 at Fr = 0.5 Param. Triplet R P t / , % UG CT G l , G2, G3 0.71 0.49 0.27 5.66 G2, G3, G4 1.62 M D Sink G l , G2, G3 0.37 1.43 0.43 1.41 G2, G3, G4 - 0 . 3 7 OC 3.91 1.12 T r i m G l , G2, G3 0 7 3 0.45 0.10 20.57 G2, G3, G4 0.753 0.33 0.07 45.82

Six grids are used f o r tlie current studies: the number o f grid points range f r o m 0.8 to 38.1 M . Grids G l , G2, G3 and G4 are generated w i t h refinement ratio \ / 2 . They are used f o r quantitative verification and validation study and enable two grid-triplet studies. G r i d G2 is also used f o r interference study, w h i l e grid G2a, generated f o r the study o f Reynolds number effects are obtained by reducing the near wall distance i n G2. Finally, f o r improved wave ele-vation predicrions, a Cartesian refinement block between the demi-hull and symmetry plane is added to G l to f o r m grid G I R . Tables 3, 4 summarize the grids dimensions and the research studies, respectively.

Table 6 Validation for i = 0.23 at Fr = 0.5 for grid triplets G l , 0 2 ,

G3 Param. I N S E A N EFD"" Param. VD UsN E C T 0.21 5.66 5.66 -4.92 Cf 9.05 Cp - 1 4 . 7 6 Sink 1.31 1.41 1.92 0.85 T r i m 2.91 20.57 20.78 - 1 . 3 9

° UD, USN, (Jy and E are % D

4 Results

This section presents the results o f the numerical simula-tions. Verification and validation study is c a i T i e d out f o r integral variables (resistance coefficients, sinkage a n d t r i m ) as w e l l as f o r local variables (longitudinal wave cuts); the analysis o f the flow field, i n terms o f wave pattern and wave profiles, w i t h focus on demi-hull separation effects, is shown. Finally, the influence of the Reynolds number on interference is investigated.

4.1 Verification and validation f o r resistance, sinkage and t r i m

In order to benchmark current U R A N S capability f o r a high-speed catamaran, including sinkage and t r i m , quanti-tative verification and validation ( V & V ) is required. Ver-ification and validation follows the approach presented i n Stern et al. [ 3 6 ] , while f o r numerical uncertainty, f / s N , the factor o f safety method proposed by X i n g and Stern [37] is adopted. Quantitative V & V is conducted f o r s — 0.23 a n d Fr = 0.5, f o r resistance ( C T ) , sinkage and t r i m .

The verificadon study is carried out f o r two grid triplets: ( G l , G2, G3) and (G2, G3, G4). The results are summa-rized i n Table 5. R is the convergence ratio, defined as:

(a) 8 . 0 E - 0 3 6 . 0 E - 0 3 4 . 0 E - 0 3 0.2 - - - Experimental Data • Numerical Results ' / x-^ 7 I \ ^ V Q 0.4 0.6 Fr 0.8 (b) Ü Q. O 5 . 0 E - 0 3 2 . 5 E - 0 3 O.OE+00 0,2 - - O - - C, ITTC • C , C F D C , E F D • CpCFD ƒ a / 0.4 0.6 Fr (c) 2 . 5 E - 0 3 O.OE+00 \ 13 - 2 . 5 E - 0 3 - 5 . 0 E - 0 3 - 7 . 5 E - 0 3 - O - - Experimental Data — • Numerical Results 0.2 0 - - O 0.4 0.6 0.8 Fr (d) 0.04 0.02 -0.02

- O— Experimental Data - • • Numerical Results

0.2 0.4 0.6

Fr

F i g . 4 CFD simulations (Grid G2) f o r catamaran .s = 0.23 compared w i t h I N S E A N EFD. a Total resistance coefficient; b pressure and friction

coefficients; c non-dimensional sinkage; d non-dimensional trim

J Mar Sci Teclinol (2015) 20:292-308 ₂₉₉

R S2-S, ₍₃₎

where .S',, ^2 and ^3 are the integral or local numerical solutions corresponding to the fine, medium and coarse grid, respectively. When monotonic convergence is achieved (0 < R < 1), the generalized Richardson extrap-olation is used f o r the estimation o f numerical error and uncertainty. Consequently, the Richardson extrapolation numerical eiTor can be estimated as:

5RF. = -£21

where the order o f accuracy, /?RE. is given by:

I n ( e 3 2 / e 2 i ) PRE

Mr)

(4)

(5)

£32 = S3 - S2, £21 = ^2 - ^1 and /• is the grid refine-ment ratio, w h i c h i n our case is \ / 2 . I n the factor o f safety method, a measure f o r the distance f r o m the asymptotic range is given by P, w h i c h is defined as the ratio between the numerical and the theoretical order o f accuracy, P—PKE/PWU vvhen solutions are i n the asymptotic range, P K 1 and the actual order o f accuracy is close to the theoretical value (p,,, = 2). W i t h this method, the

numerical uncertainty, U^n, in the last c o l u m n o f Table 5, is estimated through:

f / s N =

( 2 . 4 5 - 0 . 8 5 F ) | ^ R E | 0 < P < 1

( 1 6 . 4 P - 1 4 . 8 ) | ( 5 R E | P > \ (6)

T h e verification studies f o r Cj, sinkage and t r i m show that, f o r each variable, iterative uncertainty, U\, is accept-ably small compared to g r i d uncertainty, UQ, consequently, the grid quality makes the m a j o r contribution to the numerical uncertainty, (/SN- M o n o t o n i c and oscillatory convergence is achieved f o r all variables (0 < /? < 1) i n the finer g r i d triplets ( G l , G2, G 3 ) ; on the contrary, monotonic divergence is observed i n the coarser grid triplet (G2, G3, G4). Therefore, the finer g r i d triplet ( G l , G2, G3) is used f o r validation study. For this case, Table 5 shows that the order o f convergence f o r C-y is close to f » 0.5, and first-order accuracy is attained; f o r sinkage, P x 7.5, w h i c h indicates a third order accuracy; finally, f o r t r i m a first-order convergence is attained (a value around 7* = 0.5 is f o u n d ) . The numerical uncertainties are very l o w f o r resistance coefficient and f o r sinkage, the highest value estimated is around 20 % f o r t r i m angle.

Table 6 suiumarizes the results o f the validation study. The comparison error is computed as:

F i g . 5 Grid study and validation o f wave profile at Probe 9- The en-or values are normalized with the ma,ximum height o f wave profile C,„^^

£ = ^ % (7)

where D is the experimental data; the validation uncer-tainty, Uy, which takes into account both U^^ and exper-imental uncertainty UD, is defined as:

Ul = Ul^ + Ul (8) W h e n \E\ < Uy, the combination o f all the errors i n

D and S is smaller than Uy, and validation is achieved at the Uy interval. The highest error is observed f o r pressure resistance coefficient and amounts to 14 %. For each var-iable, the comparison error is lower than the corresponding

Uy, indicating thus that validation is achieved at 5, 2 and 20 % D f o r resistance, sinkage and t r i m , respectively. The validation level f o r t r i m is, however, large and reduction i n

Uy requires reduction i n numerical uncertainty.

Figure 4 shows the comparison between nuiuerical results and experimental data over the whole Fr range, f o r the catamaran configuration corresponding to ^ = 0.23. Overall, a good agreement between the numerical solution and the experiments is achieved. The average comparison errors over all Fr amount to E = 5, 9 and 4.7 % D f o r C T , sinkage and t r i m , respectively, where the highest error

computed f o r sinkage is m a i n l y due to the differences at larger Fr. The differences between Cf and the I T T C model correlation line, and between Cp and C- equally contribute to the difference between computed and experimental

CT-4.2 Verification and validation f o r wave elevation

A further validation test o f the numerical results is pro-vided by the comparison o f the numerical wave cuts w i t h the measurements obtained by I N S E A N . Quantitative V & V is conducted f o r s = 0.23 and Fr = 0.5.

The verification and validation procedure f o r point variables f o l l o w s the approach presented i n W i l s o n et al. [38]. The convergence ratio, R, and the order o f accuracy,

p, f o r point variables, are defined through the separate L 2

norms o f £21 and £32 according to the f o l l o w i n g formulations:

W - l | £ 2 l | | 2 / l | £ 3 2 | | 2 (9)

^j,^ J^ihÉiIMA (10)

I n ( ; G )

where () and ||||2ai'e used to denote a profile-averaged quantity and L 2 n o r m , respectively. Numerical errors and

F i g . 6 Grid study and ( a ) 0.03 validation of wave profile at

Probe 11. The eiror values are normalized with the maximum height of wave profile

0.3

-1 i I

0 0.5 1 1.5 2 x/Lpp

J Mar Sci Teclmol (2015) 20:292-308 _{30 J}

Table 7 Verification and validation f o r wave profiles (s Fy = 0,5)

= 0.23,

Location Verification Validation

P UG UO Uy E

Probe 9 0.67 0.58 15.01 7.45 1676 11.56 Probe 11 0.66 0.58 10.74 4.72 1173 11.58 " Uo, UG, Uy and E are % C,„ax

Table 8 C F D simulations with different grids

i Fr Grid C T X 10^ Sink T r i m S 2 i 2 % 5 £ „ % Sl SJ S 6 , 2 % Sl 0,23 0.50 G l 7.322 1.17 - 1 1 . 7 9 0.27 -1.14 3.71 G2 _{7.236 1.66 - 1 1 . 7 6 0.73 - 1 . 6 7} 5.24 G3 _{7.116 - 1 1 . 6 8 - 1 . 5 8} 8/2 % S, = (5, - S2) % Sl, where is the result of finer grid, and ^2 is the coarse grid

grid uncertainties distributions are estimated throitgh Eqs. 4 and 6, respectively, where p^^ is given b y 10. The L 2 norms o f these distributions are, then, used to assess verificadon levels. I n order to judge whether validation is

globally achieved, the L 2 norm o f E and Uy distributions, obtained through Eqs. 7 and 8, respectively, is finally evaluated.

Figures 5a and 6a show the longitudinal wave cuts at distances Y/b = 0.[25 and Y/b = Q.4\66 f r o i u the h u l l inner side, respecdvely (Probes 9 and 11). I t can be observed that there is a good agreement among numerical and experimental wave profiles. The verification and validation results are sutnmarized i n Table 7. The values are normalized w i t h the m a x i m u m value o f wave profile,

^ n i a x - I n both cases, monotonic convergence is achieved;

furthetmore, results are validated at a level o f 16.7 and 11.7 % f o r Probe 9 and Probe 11, respectively. The V & V procedure shows that, overall, to reduce the validation uncertainty, numerical improvements related to g r i d quality are needed. Nonetheless, the numerical results can be advantageously used f o r the analysis o f the flow physics i n v o l v e d i n the interference phenomena. Distri-butions o f E and Uy as a function o f j / L p p are plotted i n Figs. 5b and 6b.

4.3 Resistance, sinkage and t r i m

The results presented i n this section are obtained by the use o f grid G2, w h i c h is the best compromise between grid quality, computational effort and solution accuracy. The

Fig. 7 CFD siinulations for mono-hull and catamaran (Grid G2). a Total resistance coefficient; b pressure and friction resistance coefficients; c non-dimensional sinkage; d non-dimensional trim

( a ) 0.5 0.3 0.1 -0.1 0.2 s = 0 . 1 7 , C F D A s = 0 . 1 7 , E F D s = 0 . 2 3 , C F D O s = 0 . 2 3 , E F D s = 0 . 3 0 , C F D 3=0.30, E F D I ) Q'' O - . 0 O 0.4 -1 i — _0.6 0.8 ( b ) 1.5 O a o 0.5 0 0.2 0.4 F i g . 0.6 Fr Fr

a Interference factor: numerical values compared with I N S E A N E F D ; b numerical Cp/Cf

0.8 Moiioliiill

m

m. .\

m m m

Fig. 9 Wave patterns and surface pressure distribution on the hull surface at three Fr f o r mono-hull and for catamaran at three separation

distances

Study, s u m m a r i z e d i n Table 8 , s h o w s t h a t t h e s o l u t i o n c h a n g e s f o r the finest g r i d s are less t h a n 1.2 % Si f o r r e s i s t a n c e , less than 0 . 3 % Si f o r s i n k a g e , a n d less t h a n

about 4 % Sl f o r t r i m , o w i n g to the d i f f e r e n t order o f convergence o f each variable. In conclusion, even though the solutions are far f r o m the asymptotic range, the

J Mar Sci Teciinol (2015) 20:292-308 303

differences between the coarser and the finer grids are relatively small, and j u s t i f y , therefore, the use o f the results obtained w i t h grid G2 f o r the discussion o f the numerical fitidings.

Numerical results both f o r the mono-hull and the cata-maran, at several separation distances, are plotted i n F i g . 7, including C T , Cp, Cf, sinkage and t r i m . A b o u t the resistance coefficient (Fig. 7a), similarly to E F D data, at l o w and high speed, i.e., Fr < 0.35 and Fr > 0.75, the differences between the mono-hull and the catamaran are small; also, at Fr < 0.45 and Fr > 0.65, the h u l l configuration seetns to have littie influence on C T . On the contrary, a clear dependency o f C T on the separation length, s, is observed near the huirip {Fr « 0.5): as the separation distance decreases, the m a x i i u u m value o f C T increases and occurs at higher Fr values. Figure 7b shows that the augmented resistance experienced by the catamaran is mostiy due to an increase i n the wave resistance, Cp, and consequentiy to the interference phenomena between the two hulls. Another hump around Fr = 0.3 is f o u n d i n the experiments, as reported i n F i g . 2 [ 2 4 ] . However, the hump is not distinct in C F D simulations. A local m i n i m u m is, finally, observed

i n the experiments at Fr = 0.35 ( F i g . 2), where C T decreases as the distance between the hull increases. Negligible or slightly favorable inteiference is obtained around this point. This local m i n i m u m is not predicted i n C F D simulations.

Sinkage difference among all the configurations is noticeable i n the whole Fr range ( F i g . 7c), even f o r l o w or high Froude number values, where the resistance difference among the configurations is very small. Sinkage reaches a m i n i m u m value f o r Fr = 0.5 i n mono-hull, and f o r Fr — 0.45 i n catamaran configurations. Sinkage i n catam-arans is more sensitive to Fr change than mono-hull. Finally, the decrease o f separation distance increases the sinkage magnitude.

T r i m is negligible up to Fr = 0.35 f o r all the configu-rations (Fig. 7d). Similarly to Insel et al. [1] and M o l l a n d et al. [2] results, t r i m difference becomes clearer when Fr > 0.4 and catamarans display significant higher t r i m than mono-hull. W h e n the separation distance increases, t r i m angle approaches the monohull value. Large d i f f e r -ence between mono-hull and catamaran can be seen i n the Fr range 0.5-0.7, where t r i m reaches a m a x i m u m . A t

0.02 0.01 0 -0.01 -0.02 -0.03 • mono 5=0.-17 • s=0.23

/ ^

Fig. 10 Numerical wave profiles on the hull surface at three Fr, f o r mono-hull and catamaran, at thi'ee separation distances. Inner side on hull (left), outer side on hull (right)

higher speed values (Fr > 0.7), die effects o f demi-hull s e p a r a ü o n distance on t r i m are negligible except f o r the i- = 0.17 case, w h i c h shows a slight difference w i t h respect to the other configurations.

4.4 Interference

Figure 8 shows the comparison between predicted and experimental IF and C / C f values vs. Fr, f o r a l l the cata-maran configurations. The numerical values used i n this section refer to grid G2. The agreement between experi-ments and computed results is satisfactory. The average errors over the whole speed range amount to 6.4, 6.2 and 8.4 % f o r s = 0.17, 023 and 0.30, respectively.

A t l o w speed ( F i g . 8a), i.e., Fr < 0.35, and at super-critical speed {Fr > 0.75), the I F value is close to zero. I n this case, i n fact, the C T curves (Fig. 7a), f o r all the s val-ues, collapse on the mono-hull line and the separation distance does not have any influence on catamaran resis-tance, w h i c h is consistent w i t h previous studies [ 6 ] .

However, d i f f e r e n t l y f r o m Souto-Iglesias et al. [6] and f r o m Broglia et al. [24], i n our computations negative I F values are not observed at Fr values near 0.35. As Fr increases, 0.35 < Fr < 0.45, interference becomes large, o w i n g to the larger differences between mono-hull and catamaran C T values, but w i t h siuall influence o f the sep-aration distance. The inteiference coefficient reaches a peak around Fr = 0.5, similariy to the resistance c o e f f i -cient behavior, and it is higher f o r narrower than f o r wider separation distance. For higher speed {Fr > 0.5), I F redu-ces substantially.

Figure 8b shows C / C f versus Fr. The C / C f ratio directly correlates w i t h the interference factor I F , so that f o r the D e l f t catamaran the interference is largely due to the increased pressure component Cp, w h i c h increases significantly as the separation distance decreases.

The behavior o f resistance, interference factor, sinkage and t r i m as a f u n c t i o n o f the Froude number w i l l be related to the flow field and wave patterns obtained by the numerical simulations i n the f o l l o w i n g sections.

0.04 -0.02 Numerical Experimenlal I 0.04 • 0.02 Numerical Experimenlal I -0.02 -0.04 0-04 0-02 -0.02 -0.04 0.04 0.02 -0.02 -0.04 0.5 1.5 Fr=0.5 / / . . ' G f i d G 2 G r i d G l 3 G r i d G I R ^ A ^ r' ^/ -0.02 -0.04 • 0.04 0.02 0.5 1 X rvJ O - - ^ ' -0.02 • -0.04 0.04 0.02 N 0 -0.02 -0.04 Fr=0.75 _{& G r l d G !} G r i d G 2 T G f l d G I R 1.5

Fig. 11 Longitudinal wave cuts at Fr = 0.3, Fr = 0.5 and Fr = 0.75 f o r the catamaran configuration s = 0.17. Comparison between numerical

results obtained using grids G2, G l and G I R , and I N S E A N EFD. Left: Location 3 {outboard); right: Location 11 {inboard)

J Mar Sci Teclinol (2015) 20:292-308 ₃₀₅

4.5 Wave pattern and suiface pressure

Figure 9 presents the computed v/ave pattern and h u l l suiface pressure distribution f o r the mono-hull and f o r all the three catamaran configurations, at Fr = 0.3, 0.5, 0.75.

Overall, the mono-hull displays a typical K e l v i n wave pattern o f diverging and transverse waves w i t h crest at the bow. A t Fr = 0.5, //Lpp » 2 and the first trough is at the stern; as a consequence, the pressure drag is m a x i m u m . The wave patterns f o r the catamarans are more complex than those f o r the mono-hull. Strong inteiference phe-nomena are observed i n the inner region, where the inter-acting wave systems between the hulls result i n larger wave amplitudes (higher wave crests and deeper wave troughs). These are more prominent as the separation distance decreases. On the contrary, the external wave pattern is slightly affected by the presence o f the t w i n h u l l .

The variation o f the resistance coefficient as w e l l as the interference factor as functions o f speed and separation distance (Figs. 7, 8) is strictly related to the position and magnitude of the waves crests and troughs which develop i n the inner region between demi-hulls. For a fixed sepa-ration distance, at l o w Fr (Fr = 0.3), the crests and troughs

are relatively small and l o w Cy values occur; the whole wave pattern is quite similar to the mono-hull and, as the interaction between the waves systems i n the inner region is weak, the effects of separation distance on I F are neg-ligible ( I F X 0). As Fr increases, (Fr = 0.5), the wave trough is deeper and moves downstream closer to the stern; therefore, the resistance reaches its m a x i m u m ; furthermore, due to the higher amplitude o f the interfering waves w i t h respect to the mono-hull, I F increases significantly. A t FJ- — 0.75, the wave trough moves behind the stern and a reduction o f Cj- occurs; the interaction occurs downstream and a consequent reduction of I F is observed.

I n the range o f speed values where significant depen-dency of Cf on the separation length is observed (Fr X 0.5), as the gap between the hulls increases, the wave trough moves behind the stern, w i t h lower depres-sion; consequently, a reduction o f resistance coefficient is obtained. This also implies that, at small separation length, the wave trough reaches the stern at higher speeds than the larger separation gaps, and consequently the C j peak is shifted to higher Fr.

Sinkage and t r i m are strongly affected by the pressure distributions on the h u l l suifaces. F r o m F i g . 9 it is clear

(b) 0.04 0 . 0 2 0 N - 0 . 0 2 -0.04 — s=0.17 • 8=0.23 • s=0.30 yy~~~^'^^:r~~~._^ 0.5

F i g . 12 Numerical longitudinal wave cuts at Fr = 0.3. a outboard

locadon; b inboard location I I ; c center line

Fig. 13 Numerical longitudinal wave cuts at Fr = 0.5. a outboard

that the wave trough, which arrives at the stern w i t h the highest intensity (Fr = 0.5, s = 0.17), causes a large depression on hull surface. Under these conditions, sinkage and trim reach their absolute maxima (Fig. 7c and d ) . Figure 10 shows the wave profile on the hull surface both f o r mono-hull and catamaran configurations. A t Fr — 0.5, the wave elevation has a crest at the bow and a trough at the stern; it creates large pressure resistance as w e l l as large sinkage and t r i m values, w h i c h vary w i t h the separation length according to F i g . 7. A t l o w speed, Fr = 0.3, the wave height is l o w and only a small area o f the h u l l suiface

Fig. 14 Numerical longitudinal wave cuts at Fr = 0.75. a outboard

location; b inboard location 11; c center line

is affected by the wave trough depression. Finally, at high speed, Fr = 0.75, the wavelength is large enough to shift the wave trough downstream of the stern; the inner wave profile on the catamaran hulls is similar to the mono-hull one-; therefore, the sinkage and the trim approach the mono-hull values.

4.6 Longitudinal wave cuts

I N S E A N experimental wave cuts were measured at out-board locations ( 1 - 4 ) , inout-board locations ( 9 - 1 2 ) and at centerline. Figure 11 shows a comparison between the computational and experimental wave cuts, f o r the inboard location 11 (on the right) and the outboard location 3 (on the l e f t ) . I n F i g . 11, the wave cuts are obtained f o r i = 0.17 using grids G2, G l and G I R , to evaluate the effects o f grid refinement on wave elevation predictions. This study is carried out f o r Fr — 0.5 and Fr = 0.75. Overall, the simulations show f a i r l y good agreement f o r all the cases. The finer grid G l improves the wave predictions w i t h respect to G2, and further improvements are obtained w i t h the finest g r i d G I R . However, the improvements i n solution accuracy do not j u s t i f y the higher computational e f f o r t due to the finer grids. I n fact, f o r Fr = 0.75, the average error computed using grid G 2 amounts to 8.02 %, it reduces to 7.6 % using grid G l ; the eiTor amounts to 7.31 % using grid G I R . Similar values are obtained f o r Fr = 0.5,

Figures 12, 13, 14 show the numerical longitudinal waves elevations w i t h varying the separation distance, i n the outer side, i n the inner side and at the center plane, f o r several speed values. I t can be observed that the external wave cuts are weakly affected by the demi-hull separation distance; i n particular, along the h u l l (0 < X < 1) the d i f -ferences between the wave cuts are negligible, w h i l e , o w i n g to the waves interactions at and after the stern, some differences can be observed f o r X> L However, the most relevant effects o f inteiference occur i n the inner region, where large differences occur between the inner wave profiles, w i t h varying the separation length. The figures put

Table 9 Effects of Reynolds number on resistance, sinkage and trim f o r catamaran confi guration i = 0.23 and mono-hull. at Fr = 0.5 s L ( m ) Re{\<f)

s L ( m ) Re{\<f)

Cf X 10' Cp X 10' C T X 10' Sinkage (mm) T r i m (deg) C T , cai — C T , nionc , IFf IFp

0.23 1.5 2.53 18.5 0.4 9.5 - 0 . 5 - 0 . 4 1.0 - 1 . 7 0.7 3.0 7.14 6.0 20.2 - 1 5 . 4 0.02 - 7 . 8 0.2 0.5 - 0 . 3 5 3.0 - 0 . 6 Mono 1.5 2.53 19.2 O.I 11.5 0.3 - 0 . 1 3.0 7.14 6.0 20.2 - 1 5 . 9 0.23 - 9 . 5 - 0 . 1 0.1 (S - S") % S\ 5" is the result of Lpp = 3.0 m Ö Springer

J Mar Sci Teclinol (2015) 20:292-308 ₃₀₇

in evidence tlie position and the magnitude o f tlie wave trough and c o n f i r m what is already observed: the wave trough becomes deeper and moves downstream w i t h increasing ship speed, while, at fixed Fv, it is deeper w i d i reducing the separation length.

4.7 Study o f Reynolds number effects

The effects o f Reynolds number {Re) on resistance, sink-age, t r i m and on the interference factor are studied f o r

5 = 0.23, Fr = 0.5 and Re = 2.53 x 10^ 7.14 x 10^ and 20.2 X 10^ con-esponding to Lpp = 1.5, 3 and 6, respec-dvely. Simulations are conducted using grid G2a, i n which the near w a l l distance is reduced w i t h respect to grid 0 2 , so that 3'+ ranges f r o m 0.1 to 0.6.

Results are summarized in Table 9 f o r both s ^ 0.23 and the mono-hull. They show that, overall, the total resistance decreases by about 20 % over the whole Re range, both f o r s = 0.23 and f o r the mono-hull, mainly o w i n g to the reduction o f frictional resistance component, Cf, rather than the pressure component, Cp (it decreases by less than 1 % over the Re range).

A reducdon i n sinkage and t r i m is also registered over the whole Re range. Both f o r ^ = 0.23 and the mono-hull, it amounts to less than 1 % and can then be neglected.

Finally, to evaluate the effects o f Re on the interference coefficient, the fricrional and pressure I F are computed w i t h i n the Re range and the results display an increase amounting to 4.7 % f o r f r i c t i o n a l I F and a reduction by 1.3 % f o r the pressure I F . Overall, the global I F reduces f o r increasing Re. However, the Re effects are small and can be, therefore, neglected.

5 Conclusions

A numerical study has been presented w i t h focus on the effects o f Froude number and o f the separation distance o f the hulls on the resistance, sinkage and t r i m o f a m u l d - h u l l vessel. The aim o f this w o r k was to assess the predictive cun-ent U R A N S capability f o r a high-speed catamaran, including sinkage and t r i m . Consequently, a verification and validation study was carried out both f o r global and local variables.

The f o l l o w i n g conclusions can be drawn:

• Ship motions are predicted w i t h reasonable accuracy f o r most o f the cases under investigation, the m a x i m u m average error, over the whole Fr range, amounting to 8.9 % D for sinkage.

• The verification and validation study proves that the major source o f errors is due to grid quality. Numerical errors are, however, acceptably small. Overall, the

numerical model is validated and the numerical analysis constitutes a useful tool to gain a deep insight into the flow physics i n v o l v e d i n the interference phenomena, e The effects o f the Reynolds number, and therefore o f

viscosity, on inteiference are small compared to waves interaction and can be therefore neglected.

Acknowledgments This work v\'as supported by Office o f Naval

Research (ONR), grant o f N000141010017, under the administration of Dr. Patrick Purtell. The authors appreciate I N S E A N who provides the EFD data f o r this study.

References

1. Insel M , Molland A F (1992) A n investigadon into the resistance components o f high speed displacement catamarans. Royal Inst Naval Archit 134:1-20

2. Molland A F , Welhcome JF, Couser PR (1996) Resistance experiments on a systematic series of high speed displacement catamarans forms: variation o f length-displacement ratio and breadth-draught ratio. Trans Royal Inst Naval Archit 138:59-71 3. Cheng X N , Sharma SD, Stiintz N (2003) Wave reduction by

S-catamaran at supercritical speeds. J Ship Res 47:145-154 4. M i l l w a r d A (1992) The effect of hull separation and restricted

water depth on catamaran resistance. Trans Royal Inst Naval Archit 134:341-349

5. Molland A , Wilson P, Taunton D , Chandraprabha S, Ghani P (2004) Resistance and wash measurements on a series o f high speed displacement monohuU and catamaran forms i n shallow water. Int J Marit Eng 146:19-3^

6. Souto-Iglesias A R , Zamora Rodriguez R, F c r n a n d e z - G u t i é r r e z D, P é r e z - R o j a s L (2007) Analysis o f the wave system o f a catamai an for CFD validadon. Exp Fluids 42:321-332

7. Aubault A , Yeung R (2009) M u l t i - h u l l interference wave-resis-tance in finite depth waters. I n : Proceedings o f the 24th interna-tional workshop on water waves and floating bodies, Zelenogorsk 8. Moraes H B , Vasconcellos J M , Laton-e R G (2004) Wave

resis-tance f o r high-speed catamaran. Ocean Eng 31:2253-2282 9. Tarafder M S , Suzuki K (2007) Computation o f wave-making

resistance o f a catamaran in deep v^'ater using a potential-based panel method. Ocean Eng 34:1892-1900

10. Yeung R W , Wan H (2008) M u l t i h u l l and surface-effect ship configuration design: a framework f o r powering minimization. J Offshore Mech Arct Eng 130:1-9

11. Bruzzone D, Fenando M (1995) Numerical evaluation o f the steady free surface flow for catamaran hull forms. I n : Proceedings of the I I I symposium on high speed marine vehicles, Naples, A p r i l 1995, pp 187-198

12. Stern F, Carrica P, Kandasamy M , Gorski J, ODea J, Hughes M , M i l l e r R, Hendrix D , Ka-ing D , Milewski W , H o f f m a n R, Cai-y C (2006a) Computational hydrodynamic tools f o r high-speed transports. Trans S N A M E , V o l . 114

13. Stern F, Carrica P M , Kandasamy M et al. (2008) Computational hydrodynamic tools f o r high-speed sealift: phase I I final report. I I H R Technical Report N o . 465

14. Kandasamy M , Peri D , Ooi SK, Carrica P, Stem F, Campana EF, Osborne F, Cote J, Macdonald N , Waal N D (2011) M u l t i - f i d e l i t y optimization of a high-speed foil-assisted semi-planing catamaian for l o w wake. J M a r Sci Technol 16:143-156

15. M i l a n o v E, Zlatev Z (2011) Experimental and computational studies of high-speed catamaran calm water maneuvering. R T O / A V T - 1 6 1 Final Report, Chapter 24

16. Van't Veer R (1998a) Experimental results o f motions, liydro-dynamic coefficients and wave loads on the 372 catamaran model. T U Delft Report, No. 1129

17. Van'tVeer R (1998b) Experimental results of motions and structural loads on the 372 catamaran model in head and oblique waves. T U D e l f t Report, N . l 130

18. Bouscasse B , Broglia R, Stern F (2013) Experimental investiga-tion of a fast catamaran in head waves. Ocean Eng 72:318-330 19. Broglia R, Bouscasse B , Jacob B , Zaghi S, Olivieri A , Stern F

(201 la) Calm water and seakeeping investigation for a fast cat-amaran. In: Proceedings of the 11th international conference on fast sea transportation (FAST2011), Honolulu

20. Broglia R, Zaghi S, Mascio A D (2011b) Numerical simulation o f interference effects f o r a high-speed catamaran. J Mar Sci Technol 16:254-269

21. Broglia R, Aloisio G, Falchi M , Grizzi S, Zaghi S, Felli M , Miozzi M , Pereira F, D i Felice F, Stern F (2012) Measurements of the velocity field around the D E L F T 372 catamaran in steady drift. In: Proceedings o f the 29th symposium on naval hydrody-namics, Gothenburg

22. Castiglione T, Stern F, Kandasamy M , Bova S (2011) Numerical investigation of the seakeeping behavior of a catamaran advancing in regular head waves. Ocean Eng 38:1806-1822 23. Zaghi S, Broglia R, D i Mascio A (2011) Analysis o f the

inter-ference effects for high-speed catamarans by model tests and numerical simulations. Ocean Eng 38(17-18):2110-2122 24. Broglia R, Jacob B , Zaghi S, Stern F, Olivieri A (2014)

Experi-mental investigation of interference effects f o r high-speed cata-maran. Ocean Eng 76:75-85

25. Castighone T , He W , Stern F, Bova S (2013) U R A N S simulation o f catamaran interference in shallow water. J Mar Sci Technol. doi: 10.1007/S00773-013-0230-5

26. Visonneau M , Deng G B , Queutey P, Wakers J, M a l l o l B (2012) Anisotropic grid adaption f o r RANS simulation o f a fast manoeuvring catarnaran. I n : 4th high performance yacht design conference, Auckland, pp 172-180

27. Kandasamy M , He W , Takai T, Tahara Y , Peri D , Campana E, Wilson W , Stern F (2011b) Optimization o f waterjet propelled high speed ships - JHSS and D e l f t catamaran. I n : Proceedings o f the 11th international conference on fast sea transportation (FAST201I), Honolulu

28. Chen X , Diez M , Kandasamy M , Campana EF, Stern F (2013) Design optimization of the waterjet-piopelled D E L F T catamaran in calm water using U R A N S , design of expeiiments, metamodels and swarm intelligence. I n : Proceedings o f the 12th international conference on fast sea transportation (FAST2013), Amsterdam 29. Diez M , He W , Campana E, Stern F (2013a) Uncertainty quan-tification of D E L F T catamaran resistance, sinkage and trim f o r variable Froude number and geometry using metamodels, quad-rature and Karhunen-Loeve expansion. J Mar Sci Technol (in press)

30. Diez M , Chen X , Campana EF, Stern F(2013b) Reliability-based robust design optimization for ships in real ocean environment. In: Proceedings o f the 12th international conference on fast sea transportation (FAST2013), Amsterdam

31. Zlatev Z , M i l a n o v E, Chotukova V , Sakamoto N , Stern F (2009) Combined model-scale EFD-CFD investigation o f the maneu-vering characteristics o f a high speed catamaran. I n : 10th inter-national conference on fast sea transportation F A S T 2009, Athens 32. Carrica P, Hosseini H , Stern F (2012) CFD analysis o f broaching f o r a model suiface combatant with explicit simulation o f moving rudders and rotating propellers. Comput Fluids 53:117-132 33. Carrica P, Wilson R V , Noack R W , Stern F (2007) Ship motions

using single-phase level set with dynamic overset grids. Comput Fluids 36:1415-1433

34. Menter FR (1994) Two-equation eddy viscosity turbulence models f o r engineering applications. A I A A J 32:1598-1605 35. Noack R (2005) SUGGAR: a general capability for moving body

overset grid assembly. I n : A I A A paper 2005-5117, 17th A I A A computational fluid dynamics conference, Ontario

36. Stern F, Wilson R, Shao J (2006) Quantitative approach to V & V of CFD simulations and certification o f CFD Codes. Int J Numer Methods Fluids 50:1335-1355

37. X i n g T, Stern F (2010) Factors o f safety f o r Richardson extrap-olation. J Fluids Eng 132:1-13

38. Wilson R, Stern F, Coleman H W , Paterson E G (2001) compre-hensive approach to verification and validation o f c f d simula-tions- part 2: application f o r R A N S simulation o f a cargo/ container ship. J Fluids Eng 123:803-810