Calculation of current or manoeuvring forces using a viscous flow solver

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Date June 2009

Author Toxpeus, Serge and Guilherme Vaz

Address

Deift University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2, 2628 CD Deift

TUDeift

Deift University of Technology

Calculation of current or manoeuvring forces

using a viscous-flow solver

by

Serge Toxopeus and Guilherme Vaz

Report No. 1624-P

2009

Proceedings of the ASME 2009 28th International

confe-rence on Ocean, Offshore and Arctic Engineering, OMAE

2009, May 31June 5, Honolulu, Hawaii, USA, ISBN:

978-0-7918-3844-0, OMAE2009-79782)

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WELCOME FROM THE CONFERENCE CHAIRS

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OMAE2009: Welcome from the Conference Chairs

_{Page 1 of2}

R. Cengiz Ertekin H. Ronald Riggs

Conference Co-Chair Conference Co-Chair

OMAE 2009 OMAE 2009

Aloha!

On behalf of the OMAE 2009 Organizing Committee, it is a pleasure to welcome you to Honolulu,

Hawaii for OMAE 2009, the 28th International Conference on Ocean, Offshore and Arctic

Engineering. This is the first conference with the new name, which reflects the expanded focus of the

OOAE Division and the conference.

OMAE 2009 is dedicated to the memory of Prof. Subrata Chakrabarti, an internationally known offshore

engineer, who passed away suddenly in January. Subrata was the Offshore Technology Symposium

coordinator, and he was also the Technical Program Chair for OMAE 2009. He was involved in the

development of the OMAE series of conferences from the beginning, and his absence will be sorely felt.

OMAE 2009 has set a new record for the number of submitted papers (725), despite an extremely

challenging economic environment. The conference showcases the exciting and challenging

developments occurring in the industry. Program highlights include a special symposium honoring the

important accomplishments of Professor Chiang C. Mei in the fields of wave mechanics and

hydrodynamics and a joint forum of 'Offshore Technology', 'Structures, Safety and Reliability' and

'Ocean Engineering' Symposia on Shallow Water Waves and Hydrodynamics. We believe the OMAE

2009 program will be one of the best ever. Coupled with our normal Symposia, we will also have

special symposia on:

Ocean Renewable Energy

Offshore Measurement and Data Interpretation

Offshore Geotechnics

Petroleum Technology

We want to acknowledge and thank our distinguished keynote speakers: Robert Ryan, Vice President

-Global Exploration for Chevron; Hawaii Rep. Cynthia Thielen, an environmental attorney who has a

special passion for ocean renewable energy; and John Murray, Director of Technology Development

with FIoaTEC, LLC.

A conference such as this cannot happen without a group of dedicated individuals giving their time and

talents to the conference. In addition to the regular symposia coordinators, the coordinators of the

special symposia deserve many thanks for their efforts to organize new areas for OMAE. We also want

to express our appreciation to Dan Valentine, who stepped into the Technical Program Chair position

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OMAE2009: Welcome from the Conference Chairs

Page 2 of 2

on very short notice, following Subrata's passing. We also want to thank Ian Holliday and Carolina

Lopez of Sea to Sky Meeting Management, who have done a great job with the organization. Thanks

also go to Angeline Mendez from ASME for the tremendous job she has done handling the on-line

paper submission and review process.

Honolulu is one of the top destinations in the world. We hope that you and your family will be able to

spend some time pie or post conference enjoying the island of Oahu. Whether you're learning to surf in

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Mahalo nui ba,

R. Cengiz Ertekin and H. Ronald Riggs, University of Hawaii

OMAE 2009 Conference Co-Chairmen

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OMAE2009: Message from the Tecirnical Program Chair

Page 1 of 2

MESSAGE FROM THE TECHNICAL PROGRAM CHAIR

Daniel I. Valentine Technical Program Chair

OMAE 2009

Welcome to the 28th International Conference on Ocean, Offshore and Arctic

Engineering (OMAE 2009). This is the 28th conference in the OMAE series

guided by and influenced significantly by our friend and colleague, Subrata K.

Chakrabarti. IL was a shock for me to learn that he had passed away so suddenly;

all involved with this conference express sincere condolence to his family, friends

and colleagues (the sentiments echoed by all of us are eloquently expressed in

the dedication included in this program). It is a great honor for me to have been

asked to continue his work on this conference. I and our community will miss his

leadership and friendship greatly. Although this series of conferences was

formally organized by ASME and the OOAE Division of the International

Petroleum Technology Institute (IPTI), it was Subrata's skill and dedication to this

division of ASME that made this series of conferences the success that it has

been and is today.

The papers published in this CD were presented at OMAE2009 in thirteen

symposia. They are:

SYMP-1: Offshore Technology

SYMP-2: Structures, Safety and Reliability

SYMP-3: Materials Technology

SYMP-4: Pipeline and Riser Technology

SYMP-5: Ocean Space Utilization

SYMP-6: Ocean Engineering

SYMP-7: Polar and Arctic Sciences and Technology

SYMP-8: CFD and VIV

SYMP-9: C.C. Mei Symposium on Wave Mechanics and Hydrodynamics

SYMP-lO: Ocean Renewable Energy

SYMP-1 1: Offshore Measurement and Data Interpretation

SYMP-12: Offshore Geotechnics

SYMP-13: Petroleum Technology

The first eight symposia are the traditional symposia organized by the eight

technical committees of the OOAE Division. The other symposia are specialty

symposia organized and encouraged by members of the technical committees to

focus on topics of current interest. The 9th symposium was organized to

recognize the contributions of Professor C. C. Mei. Symposia 10, 11, 12 and 13

offer papers in the areas of renewable energy, measurements and data

interpretation, geotechnical and petroleum technologies as they relate to ocean,

offshore and polar operations of industry, government and academia.

The first symposium, Symposium 1: Offshore Technology was always Subrata

Chakrabarti's project. It was typically the largest of the symposia at OMAE. His

exemplary work on this symposium provided the experience and guidance for

others to continue to develop the other symposia. Symposium 1 in conjunction

with the OMAE series of conferences is Subrata's legacy. The Executive

Committee has a most difficult yet honorable task of finding a successor to carry

on this important annual symposium in offshore engineering. We are all grateful

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for the inspiration and encouragement provided to all of us by Subrata.

Please enjoy the papers and presentations of OMAE2009.

Daniel T. Valentine, Clarkson University, Potsdam, New York

OMAE2009 Technical Program Chair

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OMAE2009: International Advisory Committee

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INTERNATIONAL ADVISORY COMMITTEE

R.V. Ahilan, Noble Denton, UK

R. Basu, ABS Americas, USA

R. (Bob) F. Beck, University of Michigan, USA

Pierre Besse, Bureau Veritas, France

Richard J. Brown, Consultant, Houston, USA

Gang Chen, Shanghai Jiao Tong University, China

Jen-hwa Chen, Chevron Energy Technology Company, USA

Yoo Sang Choo, National University of Singapore, Singapore

Weicheng C. Cui, CSSRC, Wuxi, China

Jan Inge Dalane, Statoil, Norway

R.G. Dean, University of Florida, USA

Mario Dogliani, Registro Italiano Navale, Italy

R. Eatock-Taylor, Oxford University, UK

George Z. Forristall, Shell Global Solutions, USA

Peter K. Gorf, BP, UK

Boo Cheong (B.C.) Khoo, National University of Singapore, Singapore

Yoshiaki Kodama, National Maritime Research Institute, Japan

Chun Fai (Collin) Leung, National University of Singapore, Singapore

Sehyuk Lee, Samsung Heavy Industries, Japan

Eike Lehmann, TU Hamburg-Harburg, Germany

Henrik 0. Madsen, Det Norske Veritas, Norway

Adi Maimun Technology University of Malaysia, Malaysia

T. Miyazaki, Japan Marine Sci. & Tech Centre, Japan

T. Moan, Norwegian University of Science and Technology, Norway

G. Moe, Norwegian University of Science and Technology, Norway

A.D. Papanikolaou, National Technical University of Athens, Greece

Hans Georg Payer, Germanischer Lloyd, Germany

Preben T. Pedersen, Technical University of Demark, Denmark

George Rodenbusch, Shell lntl, USA

Joachim Schwarz, JS Consulting, Germany

Dennis Seidlitz, ConocoPhillips, USA

Kirsi Tikka, ABS Americas, USA

Chien Ming (CM) Wang, National University of Singapore, Singapore

Jaap-Harm Westhuis, Gusto/SBM Offshore, Netherlands

Ronald W. Yeung, University of California at Berkeley, USA

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OMAE2009: Copyright Information

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Proceedings of the

ASME 2009 28th International Conference

_{on Ocean, Offshore and Arctic}

Engineering (OMAE2009)

May 31

- June 5, 2009 Honolulu, Hawaii, USA

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Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering

OMAE2009

May 31 - June 5, 2009, Honolulu, Hawaii, USA

OMAE2009-79782

CALCULATION OF CURRENT OR MANOEUVRING FORCES USING A

VISCOUS-FLOW SOLVER

ABSTRACT

To investigate the capabilities of using a viscous-flow solver to calculate current or inanoeuvring forces on ship hulls, calcu-lations have been conducted for a test hull under several oblique inflow conditions. All calculations were peiformed wit/I 0 single grid topology to reduce the amount of grid generation time. t'r-ificalion and validation of the predicted loads and Jloiv around the liii!! has been peiforined through grid refinement studies and comparison wit/i experiments and results from previous calcula-tions. Furthermore, the influence of unsteady behaviour on the results for laige inflow angles is shown. Conclusions are drawn regarding the accuracy of the results and recommendations Jbr lmnprovenlents and fur//icr work are given.

INTRODUCTION

ilie application of viscous-flow solvers to calculate current or manoeuvring forces on ship hulls has increased in the past years. Especially for submersihies, viscous-flow calculations arc becoming a useful alternative to conducting complex model tests. One of the main advantages is the absence of struts or stingers to support the model during the force measurements. l-lowevcr, much work is still to be done to demonstrate the capabilities and accuracy of this kind of calculations.

In the present study, the solver F1ESCO has been used to calculate the forces oii a test object. FRESCO is a new unstructured-grid hydrodyiiamic multi-purpose finite-volume URANS solver for calculating viscous-flows around for exam-pIe propellers with cavitation (Vaz et al. [I]), offshore structures (Waals et at. [2]), viscous free-surface liows (Vaz and Jaouen [31) and now ships in manoeuvring conditions. In this paper, the bare-hull DARPA SUBOFF submarine bare-hull form has been selected as test case, because of the availability of extensive validation data for field variables as well as for integral quantities. In litera-ture, several other studies concerning calculations on the hare-hull DARPA SUBOFF can be found, see e.g. Sung et at. 14,5,6], Bull [7], Jonnalagadda et al. [8], Bull and Watson [91 and Yang

and Liihner 11(1.

Calculations have been conducted for straight flight as well as for inflow angles ranging from 0° up to 90°, representing pure beam inflow. The calculations for the different angles were con-ducted using the same grid. The change of inflow angle was modelled by adjusting the boundary conditions. This procedure nilnimises the time spent on grid generation.

Verification of the results has been conducted. By varying the grid density and solver parameters, the changes in the solu-tion as a result of these variasolu-tions have been studied. Valida-tion of the flow field and the forces and moments has been per-formed. Comparison of the calculations with measurements and with calculations conducted previously with the well-established steady ship viscous-flow solver PARNASSOS (see Toxopeus [Ill) demonstrates the accuracy of the calculations. Furthermore, the influence of unsteady behaviour on the results for large inflow angles is shown. Conclusions will be drawn regarding the accu-racy of the results and recommendations for improvements and further work will be given.

NUMERICAL PROCEDURES

Coordinate systems and definitions

The same detinilions as used during the experiments were adopted. The origin of the right-handed system of axes used in this study is located at the intersection of the longitudinal axis of symmetry of the hull and the forward perpendicular plane, with x directed aft, y to starboard and z vertically upward. Note that all coordinates given in this paper are made non-dimensional with the overall length Loa of the submarine (Loa = 4.356m). This fcilitates easy comparison with other results published in liter-ature. The velocity field V = (em, v, w) is made non-dimensional with the undisturbed velocity V0. The drift angle is defined by = arctan , with em and i' directed according to the x andy axes

respectively, which means that is positive for flow coming from port side.

All integral forces and moments on the hull are based on

Copyright ® 2009 by ASME

Serge Toxopeus

Guilherme Vaz

MAR IN / Deift University of Technology MAR IN

R&D Department, CFD Projects R&D Department, CFD Projects

P.O. Box 28, 6700 AA Wageningen P.O. Box 28, 6700 AA Wageningen

The Netherlands The Netherlands

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a right-handed system of axes corresponding to positive direc-tions normally applied in manoeuvring studies. This detini-tion was also used during the I)ARPA SUBOFF model tests. This means that the X force is directed forward, Y to star-board and Z vertically down. Similar to the results presented by Roddy [12], all moments are given with respect to the cen-tre of gravity, which is located at O.4Ô2ILOa aft of the nose of the model. Non-dimensionalisation is done with the length between perpendiculars(L = 4.261,n) using

X,Y,Z/pVL1)

and K,M,N/pVL. The pressure coefficient C is defined as = (p - P0) /pV, po being the undisturbed pressure, and the

skin friction coeflicient Gj as C'1 = II/pV.

in some of the results presented in this work, the circum-ferential angle a is used. This angie is defined according to Figure 1, with a = 00 corresponding to the ieeward side and a = 270° to the keel for positive inflow angles.

flow a = 1800

PS SB

a = 270°

Figure 1 Definition of circumferential angle a

Solver

In this study, the flows were calculated with FRESCO, see e.g. Vaz et al. [13] or l-loekstra et al. [141. The code started as a joint development by MARIN, HSVA and TUHH within the EU project VIRTUE in 2005. FRESCO solves the multi-phase un-steady incompressible Reynolds-averaged Navier-Stokes equa-tions together with turbulence models and, if necessary, volume-fraction transport equations for each phase. The equations are discretised using a flexible finite volume face-based discretisa-tion, which permits grids with elements with an arbitrary num-ber of faces (hexahedrals (structured grids), tetrahedrals, prisms, pyramides, etc.). and even h-refinement (hanging-nodes). The code is parallelised using MPI and sub-domain decomposition, and is targeted for Linux workstations, Linux Clusters and Super-Computers. It resembles in global lines the existing commercial CFD packages. However, contrary to the highly general commer-cial codes, FRESCO is intended to deal only with ship-building & offshore related flow problems, and is therefore optirnised for and continuously applied to these types of flows. FRESCO has also some additional attractive points: 1) it is an in-house code, and therefore easy to trouble-shoot, extend, improve and cou-ple with other tools; 2) MAR IN experience on CFD, free-surface niodelling and cavitation iimdclling is incorporated in the code; 3) no licence fees associated, which makes the computations cheaper. FRESCO is however a new code, which still has to ma-ture to the level of accuracy of PARNASSOS and therefore the calculations need to be thoroughly validated.

Solver settings

Menter's SST version of the two-equation k - w turbulence model [15] was selected for all calculations. The governing equa-tions were integrated down to the wall, i.e. no wall-funcequa-tions are used. For the steady calculations, a order time derivative to-gether with a time step size i of 0.07s, was used. For a Reynolds

number of I .2 x I o, this corresponds to about i = tret/20, with

trcf = Loa/V0 = I .575s. This time step size is too large to capture all time dependent fluctuations in the flow, but was adopted to aid the solution process to arrive to a steady solution. A higher-order (3d higher-order in structured grids) QUICK scheme was used to discretise the momentum convection terms. For the turbulence quantities a first order upwind scheme was adopted for the con-vection terms. All diffusive terms were discretized by a 2nd order

scheme.

For the present study, all solutions were run until the max-imum change of the pressure coefficient C1) (the so-called L

norm) between successive iterations had dropped below

Boundary conditions

At the hull surface a no-slip boundary condition is applied. Symmetry conditions are used on the longitudinal plane(s) of symmetry. The exterior domain is formed by a half sphere (sym-metry is assumed between the bottom and top of the hull). The boundary condition is set by calculating the angle between the inflow and the normal vector on a cell face on the boundary. For angles larger than 90°, the boundary condition of the cell face is set to inflow, otherwise to outflow. A Neumann boundary condi-tion stating that the normal gradient is equal to zero is applied on outflow faces for all quantities.

The velocity components in the inflow faces are set to the undisturbed velocity components. The turbulent intensity I = u'/Vo is set to 0.05 and the eddy viscosity p, to lOp. The values applied at the inflow plane are also used as initial condition in the complete flow domain.

GRID GENERATION

Hull form

The submarine form considered is the DARPA SUBOFF hull form, as described in Groves et al. [16]. For this hull, ample comparison material is available. For the current work presented in this paper, only the bare hull, i.e. without sail, planes and pro-peller, is used. This corresponds to Configuration 3 as defined in Roddy [121 and to configuration AFFl* as defined in Liu and Huang [171. The main particulars of the DARPA SUBOFF are sl)ecified in Table I.

Grid

A structured grid has been generated around the hull form using in-house tools. A body-fitted non-orthogonal 0-0 type grid has been used. The hare hull of the submarine is axisym-metric and therefore the grid was made axisymaxisym-metric as well, by first generating the grid in 2D and then rotating the 2D grid around the axis of symmetry. Hexahedral cells are used every-where except at the symmetry line every-where prisms are used instead of degenerated hexahedral cells. The grid was strongly stretched towards the hull to capture the strong gradients in the boundary

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layer. Grid-refinement was also applied at the bow and at the

stern for this reason, In Figure 2 the 2D grid (coarsened for pre-sentation) is shown.

Apart from studying the feasibility of applying FRESCO to the calculation of a submarine in oblique flow, one of the aims of the present study was also to investigate whether the (low at

different drift angles could be calculated using a grid with (lie

same topology for all angles. The idea was to generate one grid and model the inflow angle by adjusting the boundary conditions on the exterior of the grid. To facilitate this, the exterior of the

computational domain was !'ormed by a half sphere. To avoid

interaction between the outer boundary, where undisturbed ve-locities are enforced, and the flow around the hull, the domain size was large. The radius of the spherical exterior was 3.44 Loa. The maximum deviation from orthogonality in the grid was 51°.

In Figure 3, a view of the grid used for this study is

pre-sented. The bow is dii'ected to the lel't of the figure. Table 2

presents an overview of the number of nodes n used in the

FRESCO calculations compared to the grid used previously in

PARNASSOS,using a curvilinear coordinate system notation (see Figure 4).

The number of points given are for the non-zero drift

an-gle calculations. For zero drill, only a quarter of the hull was

modelled and therefore only half the number of grid nodes was

used.

For this study, most calculations were conducted using

grid number 5. Additional calculations were done with other

grid densities, to investigate the grid dependency. Grid number

5 is coarse compared to the

PARNASSOS grids and also

com-°FU5w-.

---a;-

Figure 2. 2D grid around the submarine (coarsened by factor 2)

Figure 3. 3D grid around the submarine (coarsened by (actor 2), fJ 0°

Figure 4. Definition of curvilinear coordinate system (bow to the right)

pared to grids used in similar studies presented in literature when wall-functions are not applied. For grid number 2, the number of nodes on the hull surface is almost the same as for the PARNAS-SOS grid. However, the total number of nodes is less than hull the number of nodes in thePARNASSOScalculations. This shows

that with 0-0 type grids, a large number of cells can be placed

near the hull surface while keeping the total number of cells rel-atively low compared to H-0 type grids. However, with an 0-0 topology, the cell size in the wake of the hull also increases fast when moving away from the wall, and wake resolution is then not optimal.

REVIEW OF THE CALCULATIONS

Calculations for inliow angles ranging l'rom 00 to 90° were conducted for the DARPA SUBOFF bare hull form. In this paper,

the following abbreviations are used to identify the results: NL-TNT PARNASSOS with HO-grid and TNT

tur-bulence model [18,19,11]

N L-Axi l'ARNAssoswith axisymmetric grid and

MNT turbulence model [19, 11]

NL-Axi SST PARNASSOS with axisynimetric grid and SST turbulence model [19,111

N L-fresco-alt FRESCO with axisymmetric 00-grid and

SST turbulence model

The parameter 4' is Ihe largest non-dimensional distance

Table 1. Main particulars of bare hull DARPA SUBOFF submarine

Description Symbol Magnitude Unit

Length overall Loa 4.356 iii

Length between perpendiculars 4.261 Iii

Maximum hull radius Rmax 0.254 flu

Centre of buoyancy (aft of nose) FB 0.4621Loa

Volume of displacement V 0.708 rn3

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Table 2. Comparison of grid densities between FRESCO and PAR-NASSOS (half ship)

be 0 Re= 14x 106,13=00 1000 2000 3000 4000 1000 2000 3000 4000 time steps

Figure 5. Convergence history of local (top) and integral (bottom) quan-tities, f3 = 0°

Discretisation error

In order to determine and demonstrate the accuracy and reli-ability of solutions of viscous how calculations, grid dependency studies are very important. Several methods for uncertainty anal-ysis are available in literature. In the present work, the method proposed by Eça and Hoekstra [20] is followed. The full details and background for the followed procedure for the uncertainty study can be found in their paper. A short summary of the pro-cedure can also be found in Toxopeus [Ill.

For Re = 14 x 106, the discretisation error has been inves-tigated. In Table 3 and Figure 6, the results for 13 = 0° are pre-sented. The characteristic grid size h is calculated for each grid i by h = I /(n - I). The relative step-size h1/h1 indicates the coarseness of the grid with respect to the finest grid /i. For in-stance 115//Il = 2 indicates that grid 5 is two times coarser than grid I. The value of the solution obtained at the finest grid is

indi-cated by t. The solution extrapolated to zero relative step-size

() is an estimate of the solution for zero cell size The graphs

show that scatter exists in the data: the data points are not ex-actly aligned according to the curve. Reasons for this might be

Due to negligible but non-zero contributions of iterative and round-off errors

and the fact that the observed order of convergence may change during

refine-ment, the extrapolated solution may however not be equal to the exact sotution obtained on an inlinitety dense grid.

n x i0

NL-Axi 361 97 49x2 3432 I NL-fresco-alt 24! 105 113 2859 2 NL-fresco-alt 20! 87 93 1626 3 NL-fresco-alt 17! 75 81 1039 4 NL-fresco-alt 138 60 65 538 5 NL-fresco-aIt 12! 53 57 366 6 NL-fresco-alt tO! 3! 47 147 7 NL-fresco-alt 86 38 41 134 8 NL-fresco-alt 61 27 29 48

N L-Axi grid used for PARNASSOS calculations

NL-fresco-alt grid used for FRESCO calculations based on grid I by coarsening with factor 2 based on grid 2 by coarsening with factor 2 based on grid 3 by coarsening with factor 2

ft

_{based on grid I by coarsening with factor 4}

y+ of the first grid node to the wall. In order to capture the full boundary layer and directly apply the no-slip condition at the wall, v should be equal or less than I. Although most of the calculations were conducted with a relatively coarse grid, the y values in these calculations were between 0.7 and 0.9, depending on the drill angle. For the finest grid,y was 0.33 for j3 = 0° and 0.41 for 13 = 18°.

All calculations were conducted for deeply submerged con-dition, so that free surface effects are absent. Based on reference inflow velocities V0 on model scale of 2.7658 and 3.2268 m/s, the Reynolds numbers Re corresponded to respectively 12 x 106 (used for 13 = 0° and 2° only) and 14 x 106 (used for all drift an-gles). In this paper, the Reynolds number is based on the length overall of the model.

NUMERICAL RESULTS

Iterative error

In Figure 5, the iterative convergence histories for the straight-ahead sailing condition and for 13 = 18° are presented. All results are given in non-dimensional values. It is seen Ihat after some initial transients, the L norms of the pressure and velocity components (top graph) converge smoothly to below l0 for = 00. The convergence history of the integral vari-ables shows convergence to well below 5 . 10-8. The conver-gence for the calculations with non-zero drift angles is similar, but for increasing drift angles, some stagnation in the conver-gence is found. Further examination of the different iterations shows that the stagnation is caused by minor changes in the

ve-locities, probably caused by instationary eFfects. The changes are

within 0.01% of the final values.

Based on the aforementioned observations, iterative conver-gence errors in the calculations are assumed to be negligible with respect to discretisation or modelling errors.

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e.g. the non-evenly spaced ccli nodes, the use of numerical lim-iters or lack of perfect geometrical similarity between the grids. For this high Reynolds number, i.e. when convection dominates,

and when using an unstructured-grid QUICK scheme for

con-vective fluxes, it is expected that FRESCO will be second order accurate. Based on the verification study, it is found Ihat the ob-served order of convergence p is just below 2 indicating that the convergence with grid refinement follows the expected order of accuracy of FRESCO. The uncertainty U in X is 2.2% which is judged to be small.

Compared to the uncertainty study conducted using

PAR-NASSOS calculations, see [lIj, it is found that uncertainty val-ues in the same order of magnitude are li.und: for IARNASSOS 11x = 2.1% when using the TNT turbulence model. Interestingly, in FRESCO the absolute value of the friction component Xj in-creases with increasing grid density, while in PARNASSOS the

opposite trend is found. As expected, the pressure coniponent

X reduces in absolute value for both solvers when a finer grid is

used.

Table3. Uncertainty analysis, X,/3

=

00

In Tables 4 through 6 and Figures 7 and 8, a selection of the results for/3= 180 are presented. In this case, the observed order of convergence p ranges from 0.46 for X1 to 3.91 for X11. This may indicate that a liner grid needs to be used to obtain a reliable solution.

For the transverse force Y and yaw moment N the observed orders of convergence are between 0.8 and I .7, which may in-dicate that for the transverse force and yawing moment the grid

density is acceptable. The uncertainty in X is found to be

rel-atively large. The large value is caused by the fact that for the

overall force X rnonotonic convergence was not obtained. How-ever, the value is acceptable from an engineering viewpoint. The uncertainty in the overall transverse force or yawing moment is judged to be small.

The verification study for FRESCO has been conducted for

/3= 18° while for PARNASSOS the results for fJ = 100 were used.

Therefore, the results cannot be compared directly.

However,

qualitatively, it is found that for FRESCO the uncertainty in Y is considerably smaller than for PARNASSOS due to monotonic convergence of the values in FRESCO and monotonic divergence in PARNASSOS. The uncertainties in X and N are of similar

mag-ii tudes.

It is also concluded from this study, that using grid 5 (or

relative step size equal to 2) the numerical solutions for drift an-gles ranging up to 18° are within a plausible uncertainty region,

where the discretizalion error is under control. For the rest of

the results presented in the paper this grid will then be used with confidence on the reliability of the nunierical solution.

-7.5

-8.5

-9,5

relative step üie

0.5 I 5 2 2.5 3 3.5 4

,.Jath step 4I

Figure 6. Discretisalion uncerlainly analysis, /3 = 0°

Table4. Uncertainly analysis, X, /3 = 18° -8.22x l0

2.41 x

-I.06x l0

Oscillatory convergence

Unsteady flow

In all calculations, the flow converged to a steady flow,

al-though time stepping was used. The main reasons for the

ab-sence of unsteady effects are the poor grid resolution away from the hull surface and the coarse time step. Furthermore, unsteady

p 6.3% 24.7% 3.91 5.0% 0.46 5

Copyright © 2009 by ASME

Item U p X

-l.05xl03

-1.05x103 2.2% 1.75 X -8.78x io -9.91 x io 18.5% 1.92 X1 -9.60xIO -9.52xI04 2.2% 1.97 05 I5 2 2.5 3.5 4 -10 -10.5

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5.8 5.6 5.4 5.2 0 0.5 x l0 6.6 6.4 6.2 5

Table 6. Uncertainty analysis, N, J3 = I 8°

2 Monotonic divergence 3.65 3.6 3.55 3.5 3.45

z

3.4 3.35 3.3 3.2S 3.2 0 0.5 2 2,5

reIath step size

Figure 8. Discretisation uncertainty analysis, N, /3 = 18°

Full domain (also top half sphere modelled), same time iter-ation settings as before. This calculiter-ation was conducted to investigate the influence of the symmetry assumption. Full domain (also top half sphere modelled), improved time iteration settings. 'fhis calculation was conducted to investi-gate whether with improved time resolution different results would be obtained,

The forces and moments for the first full domain calculation

show an oscillating behaviour, as presented in Figure 9.

How-ever, the differences in the forces and moments predicted using

the half domain and lull domain, see Table 7, are judged to be

relatively small, compared to the forces and moments found for other drift angles. When looking at the transverse velocity field around the hull, pronounced vortex shedding is not clearly

ob-served.

The calculation with improved time resolution was made

us-ing a 2d order implicit three-time-level integration scheme in

combination with a time step & of 0.OlSs or & = trcf/9O. Ad-ditionally, 50 outer iterations were conducted for each time step, to sufficiently reduce the residuals between the time steps (about 4 orders of magnitude, instead of 2).

In this case, as seen in Table 7 and Figure 10, the X force re-duces considerably, while the yawing moment N increases. In the out-of-plane forces, see the bottom graphs in Figures 9 and 10, a large change is seen compared to the calculation with less ac-curate time resolution: high-frequency oscillations of especially the vertical force are observed, caused by vortex shedding. Close inspection of the solution shows that the pitch moment is in

Table 5. _{Uncertainty analysis, Y, JJ =} 18° xl0'

Item U,, p 3.7 3.6 V 5.52x103 5.66x103 3.1% 1.17 3.5

),

5.20x103 5.36x103 3.7% 1.15 3.4 0 '1 3.18x104

3.0lxl0

5.5% 0.85 Z 3.3 3.2 _cxp 3.1

0

cfd p - 1.7 3 U= 2.9 0 0.5 IS 2 2.5 3,5 4 relalhe ,tep size

1.5 2 2.5 3.5

reIati step l,e

0 0.5 I 5 2 2.5 3.5

relailve Iep size

Figure 7. Discretisation uncertainty analysis, Y, j3 = 18°

behaviour is suppressed by the assumption that the flow is sym-metric for the top and lower halves of the spherical domain. This effectively prohibits vortex shedding in vertical direction around the submarine hull form. However, for large inflow angles, it is expected that vortex shedding will occur and that this might in-fluence the forces and moments acting on the hull. Although a complete investigation into the influence of unsteady flow on the forces on the submarine is outside the scope of the present study, two additional calculations for J3 = 90° were conducted:

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site phase compared to the vertical force. This is caused by oscil-lating pressure fields in the bow area, due to the vortex shedding. This vortex shedding can also be seen in the transverse velocity field. In the present calculation, however, the grid density in the wake is quite poor due to strong stretching of the cells towards the hull surface to capture the boundary layer adjacent to the hull. To obtain even more accurate vortex shedding, the grid should be reliiied ri the wake region as well.

Re = 14 x 10',J3

= 90° (full domain) 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0.003 0.002 0.00! 0 -0.001 -0.002 -0.003 0 1000 2000 3000 4000 5000

K

M

0 1000 2000 3000 4000 5000 lime steps

Figure 9. Convergence history of forces and moments, /3 = 90°, full domain, original settings

COMPARISON WITH EXPERIMENTS,_/3

=0°

Integral values

txperiinental force measurement results arc available for the straight-ahead condition and were published by Roddy [121. The experiments were conducted in the towing basin of the David Taylor Research Center. During the tests, the model was sup-ported by two struts. The speed used during the experiments resulted in a Reynolds number of 14 x 106. For this condition the experimental value of the longitudinal force was found to be:

X = average(Xtestj,X1e512) = average(l.061,l.OSl) x

l0-= l.056x

io-Re = 14 x 106,/i = 90° (full domain, time) 0.003 0.0025 0.002 0.0015 0.001 0.0005 0,003 0.002 0.00 I 0 -0.00 -0.002 -0.003 0 A

x

0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 tine steps

Figure 10. Convergence history of forces and moments, /3 = 90°, full

A detailed uncertainty analysis of the force measurements was not conducted. However, to obtain an estimate of the uncertainty in the experimental data, the uncertainty U1) in the experimental data is estimated using a factor of safety of 1.25 by

U = 1.25 x abs (X111 Xtest2) = 1.25 X l0 = 1.2% xX

Table 8 presents the longitudinal Force components obtained from the calculations. As can be expected for submarine hull

domain, time-accurate

Table 7. Forces and moments (X 1000), /3 = 90°

Domain Half Full Full

Time Coarse Coarse Accurate

X 1.766 l.560 0.820 Y 44.653 44.029 44.489 Z - 0.954 0.729 K - 0.000 0.000 M - -0.024 0.080 N 0.53 I 0.345 1.579

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forms, the largest part (about 90%) of the total resistance is

caused by friction resistance.

The difference Lx between the

FRESCO prediction of X and the measurement is about 0,5%,

which is judged to be very good for practical applications when also the uncertainty in the experimental data is taken into con-sideration, Furthermore, it is found that the total resistance

pre-dicted by FRESCO is close to the value prepre-dicted by PARNASSOS

(NL-Axi): the skin friction coefficient is almost the same and the difference is mainly caused by a slightly higher pressure coeffi-cient in FRESCO,

The influence of the Reynolds number can be observed in

the calculations. As expected, a slightly higher Reynolds

num-ber results in a slightly lower resistance coefficient, due to the

decrease in frictional resistance coefficient.

Table 8. Longitudinal force X, Re = 1.4 x l0, /3 = 00

Local quantities

'the experimental values are obtained from low field and

pressure measurements in a wind tunnel conducted by Huang et at. 121]. These experiments were conducted at a Reynolds

num-ber of 12 x 106. Figure II shows comparisons of the pressure

and skin friction coefficients along the hull, It should be noted that the results for NL-Axi were obtained for a Reynolds number

of 14x 106.

These graphs show that the differences in pressure

coefli-cient between the FRESCO and PARNASSOS results arc negligi-ble, which corresponds to a small difference in the longitudinal pressure coefficients X,,.

For the skin friction coefficient, it is

seen that the FRESCO results with the SST model are slightly

closer to the experimental data than the PARNASSOS results us-ing the TNT model, except for some points in the aft ship.

The predicted distribution of the pressure coefficient is close

to the experiments. The trends in the predicted distribution of

the friction coefficient correspond well to the trends found in the experiments. Although some discrepancies at the how and stern

area are found, it is concluded that the prediction of the

pres-sure and skin friction coefficients is good. It is noted that the

1.5 0.5 .0,5

I

IrntT_Csd)I E.W NL.TNT NL'A\i NL-Iresco-alt

Figure 11. Pressure (fop) and skin friction (bottom) coefficients along the hull, /3 = 00

discrepancies at the bow and stern were also present in

calcu-lation results published by Bull [7] and Yang and Löhner ItO]

and in results submitted for a collaborative CFD study within

the Submarine Hydrodynamics Working Group (SHWG, see

www.shwg.org) [22].

Figure 12 presents the streamwise and radial velocities at

specific longitudinal locations in the aft part of the hull.

In these graphs, the difference between the FRESCO and

PARNASSOS results is considered to be small. Comparing the

computed results with the experiments, it is observed that the

sends in the development of the boundary layer are very well

predicted by both solvers, hut quantitative discrepancies are seen. Especially the magnitudes of the radial velocities are different. It is seen that in the experiments the radial velocity (positive cor-responds to a radial velocity away from the wail) changes sign between (r - R0 )/R100 = 2 and (r - R0 ) /Rinax = 0.8, suggesting outward radial flow in the far field. This may be caused by the use of an open-Jet wind tunnel.

Reynolds shear stresses

In this study, also the correlation between the measured and the predicted Reynolds shear stresses is investigated by com-parison of Following the eddy-viscosity assumption, the Reynolds stresses are defined by

= 2 v, 'S,

v, being the eddy viscosity, and the strain rate tensor Sj being

calculated by

i

a

Solver Grid Integral values x

X X1 X1, Ex (%) Exp(DTRC) - -1.061 - - -Exp (l)TRC) - -1.051 - - -Mean Pexp -1.056 - - -I'fl'C-57 - -0.936 - -Schoenherr - -0.918 -Katsul - -0.905 - -Grigson - -0.932 - -NL-Axi - -1.027 -0.958 -0.069 -2.7 NL-fresco-alt I -1.051 -0.952 -0.099 -0.5 NL-fresco-aR 2 -1.040 -0.943 -0.097 -1.5 NL-fresco-alt 3 -1.032 -0.935 -0.097 -2.3 NL-fresco-alt 4 -1.022 -0.920 -0.102 -3.2 NL-fresco-alt 5 -1.054 -0.926 -0.128 -0.2 NL-fresco-alt 6 -1.017 -0.893 -0.124 -3.7 NL-fresco-alt 7 -1.035 -0.883 -0.152 -2.0 NL-fresco-alt 8 -1.144 -0.844 -0.300 8.3 0 0.2 0.4 0.6 0.8 CL ,uI,offCai,4)2

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'.5 U.S 0.2 oI,oflC,i,e03ft.90.4 * Exp NL-TNT NL-A\i NL-ftesco-aIt

X0.927Loa

E.'p NL-TNT - NL-freso-aII I; = 0.956Loa x = 0.978Loa 0 ((.2 1)4 0.6 05 Vek,ily/V0

Figure12. Axial (lines in right-handed parts of each graph) and radial (lines in left-handed parts of each graph) velocities, 13 = 0°

For a = 180° and a = 270° (see Figure 1), the

CompoflentS

are given respectively by,

faa

dw'\

fda

dv\

-VV,=-v,.( -+-- 1

and

\dz

dx)

In Figure 13 and Figure 14, the Reynolds shear stresses for four longitudinal stations are presented.

It is observed that curves representing the FRESCO results correspond well with the measurements, except for the decay of

the Reynolds stresses further away from the hull.

Tile results

are however better than forPARNASSOSwith TNT, which under-predicts the peak of the distribution and shows even slower decay

'.5 U 0 I,,,fl'(o4i70.9(I4 Ep NL-TNT NL-A' - NL-frcsco-aI x = 0.904L00 Exp NL-TNT NL-k'o NLtrcsco-aIL x = 0.956Loa Eop NL-TNT NL-Axi NL-freco-aII x = 0.978Loa

COMPARISON WITH EXPERIMENTS, p 0°

Integral values

Figure 15 gives an impression of the results of the

calcula-tions for the bare hull l)ARI'A SUBOFF for J3 = l8.

In this

picture a pair of vortices on starboard is seen. These vortices

modify tile pressure distribution on the submarine and also the wake field of the submarine completely.

Figure 13. Reynolds shear stress,₁₃

= 0, X/Loa = 0.904,0.927

of the stresses. ffC.'09II.956 0.1 0.2 0.3 ((4 0.5 0.6 0.7 O.K (1.9 (rk))(fR,, ,uI,off_C,,cIOj(.57S U.S ((.4 VIolIyA'11 ,uIrnffC*.05lL956 -02 2-(5 0 .02 Or, 0.8 ((.2 11,4 Veh,chOV0 ,uI,off_CuiO6_0.975 U Ep NL-TNT NL-k'i NL-fre5co-aIl Figure14. 0.3 0.4 0.5 0.6 (r-R0)/R

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Figure 15. Impression of calculation results for bare hull DARPA SUB OFF. Axial velocity contours, pressure contours on the hull, /3 = I 8°

Experimental results for oblique angles were published by Roddy [12]. The experiments were conducted at a Reynolds number Re of 14 x 106. Figure 16 presents the force and mo-inent components obtained from the calculations and the values from the experiments for oblique inflow.

In Tables 9 through II the results for /3 = 18° are shown. The comparison shows that FRESCO predicts a longitudinal force X of the same order as niagnilude as the value found with l'ARNAssOs using the SST turbulence model. Compared to the PARNASSOS MNT results, a considerable improvement is found (see Toxopeus [II] for a discussion on the clilTerences between the MNT and SST calculations with I'ARNAssos). The deviation ex from the measurement is about 8%, which is within the un-certainty band of the measurements. The trends in the transverse force Y and yaw moment N and the de-stabilising arm N/Y are predicted reasonably well, but slightly less good than the PAR-NASSOS SST results. This may be caused by the coarse grid away from the wall used itt FRESCO.

Although experimental validation data are not available, cal-culations for large inflow angles were also conducted to inves-tigate the convergence behaviour and consistency of the forces and moments calculated by FRESCO. It appeared that without any modification of the input files, reasonable results could be obtained. As seen in Figure 16, the trends in the forces and mo-ments are consistent and appear to be realistic. However, as dis-cussed previously, unsteady behaviour of the flow influences the flow and loads on the ship at the large drift angles and therefore the results should only be interpreted qualitatively.

Local quantities at leeward symmetry plane

Figuic 17 presents comparisons of the piessure coefficient along the hull, the axial V, tangential V0 and radial Vr veloci-ties and Reynolds shear stress respectively, given for the leeward symmetry plane (a = 00, see Figure I) located at x = O.978LOa. The experimental values of the pressure distribution and the flow field and Reynolds stresses were obtained using an inflow an-gle of J3 = 20 and for a Reynolds number of 12 x 106. All cal-culations were therefore done for this Reynolds number, except for the NL-Axi calculations which were obtained for a Reynolds

e.es 004 0.03

'

0.02 em .0.0! ff.lI_KI.-k7_i) Ii uhoff_fJRI.1.7_aII

o

Figure 16. Force and moment coefficients against drift angle

number of 14x 106.

These graphs show that the distribution of the pressure co-efficient along the length of the ship and the velocity distribution at the stern is quite well represented. The difference between the

FRESCO and I'ARNASSOS results is considered to be small. The

distribution of the Reynolds shear stress predicted by FRESCO shows good correspondence with the measurements.

CONCLUSIONS

Calculations on the unappended hull-form of the DARIA SUBOFF sailing straight ahead and at oblique motion were con-ducted in order to verify the accuracy of the predictions using the new viscous-flow solver FRESCO. Based on this study, it is concluded that FRESCO is well capable of predicting the

JO 20 1)) 4))

Io_3 ,ui,off_l)_K

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Table 9. _{Longitudinalforce X, Re = 1.4 x io, fi = 18°}

Table 10. Transverse force Y, Re = 1.4 x J3 = 18°

Tablell. Yawing momentN,Re= 1.4 x iO,j3 = 18°

flow around a bare-hull submarine. Comparison of file predicted resistance with experimental data gives very good agreement. The difference between the calculations and the measurements is -0.5% for the straight-ahead sailing condition. This differ-ence is well within the estimated uncertainty band around the experimental value and the numerical uncertainty.

1.5 0.5

Iii

Ep NL-Axi NL-Iresco-aii

and radial velocities (middle) and Reynolds shear stress (bottom). J = 2°

For oblique motion, encouraging results are found for tile predicted trends in the forces and yawing moment as a func-tion of the oblique inflow angle. Comparisons between the re-sults from the new code FRESCo and the well established solver I'ARNASSOS show that only small differences exist between the results of the two solvers.

The approach adopted to model the oblique flow by chang-ing the boundary conditions while uschang-ing one common 0-0 type grid has proven to be feasible for this hull form and therefore large amounts of grid generation time could be saved. However, for common applications, the wake is probably not sufficiently resolved and therefore it is recommended to study the feasibility of this approach for other ships as well.

FRESCO can be used to calculate the flow around the hull

for large in flow angles, both using a steady or unsteady

computa-tion approach. For these large inflow angles, it was seen that un-steady vortex shedding influences the flow field around the hull and the forces and moments acting on the submarine. However, to obtain fully accurate unsteady results, adequately small time steps and line grid resolution on all possible shear-layers is re-quired. This has to be further investigated.

Solver Grid Integral values x

X

Xj

X,, eX (%) Exp (DTRC) - -0.670 - - -Exp (l)TRC) - -0.852 - - -Mean -0.761 - - -NL-Axi - -1.071 -1.103 0.032 40.8 NL-Axi SST - -0.8 13 -1.093 0.279 6.8 NL-fresco-alt I -0.822 -1.063 0.24! 8.0 NL-fresco-alt 2 -0.814 -1.053 0.239 7.0 NL-fresco-alt 3 -0.810 -1.044 0.234 6.4 NL-fresco-alt 4 -0.817 -1.029 0.212 7.3 NL-fresco-alt 5 -0.827 -1.020 0.193 8.7 NL-fresco-alt 6 -0.959 -1.022 0.063 26.0 NL-fresco-alt 7 -1.056 -1.017 -0.039 38.7 NL-fresco-alt 8 -1.326 -0.999 -0.327 74.3

Solver Grid integral values x IO3

Y _'1 Y, Cy(%) Exp (DTRC) - 7.355 - - -Exp (DTRC) - 7.438 - - -Mean Pexp 7.397 - - -NL-Axi - 6.241 0.320 5.921 -15.6 NL-Axi SST - 6.309 0.306 6.003 -14.7 NL-fresco-alt I 5.661 0.30! 5.359 -23.5 NL-fresco-att 2 5.692 0.299 5.394 -23.0 NL-fresco-alt 3 5.722 0.296 5.426 -22.6 NL-fresco-alt 4 5.792 0.29! 5.501 -21.7 NL-fresco-alt 5 5.828 0.288 5.540 -21.2 NL-fresco-alt 6 5.793 0.286 5.508 -21.7 NL-fresco-alt 7 5.856 0.282 5.574 -20.8 NL-fresco-alt 8 6.296 0.277 6.019 -14.9

Solver Grid Integral values x l0

N N1 N,, eN (%) Exp (DTRC) - 2.986 - - -Exp (DTRC) - 2.939 - - -Mean p1, 2.962 - - -NL-Axi - 3.407 -0.041 3.448 15.0 NL-Axi SST - 3.296 -0.041 3.337 11.3 NL-fresco-alt I 3.414 0.018 3.397 15.2 NL-fresco-alt 2 3.406 0.017 3.389 15.0 NL-fresco-alt 3 3.399 0.017 3.382 14.7 NL-fresco-alt 4 3.384 0.017 3.366 14.2 NL-fresco-alt 5 3.371 0.017 3.354 13.8 NL-fresco-alt 6 3.417 0.017 3.399 15.3 NL-fresco-alt 7 3.452 0.017 3.435 16.5 NL-fresco-alt 8 3.517 0.017 3.500 18.7 0.2 0.4 0.6 i/L ,ol,oITC,nel 20.978 0.8 I ii 0 K 0 9 0 0.2 ((.3 04 0.5 (.6 (r.R)f K

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ACKNOWLEDGMENT

Part of the work presented here was funded through TNO

Defence, Security and Safety within the framework of

Pro-gramma V705 carried out for DM0 of the Royal Netherlands

Navy. Their support is greatly acknowledged.

Another part of the work conducted for this article was

funded by the Commission of the European Communities

through the Integrated Project VIRTUE under grant 516201 in the sixth Research and Technological Development Framework Programme (Surface Transport Call).

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