Computational fluid dynamics for sport simulation [1 ed.] 3642044654, 9783642044656

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Lecture Notes in Computational Science and Engineering Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

For further volumes: http://www.springer.com/3527

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Martin Peters Editor

Computational Fluid Dynamics for Sport Simulation

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Editor Martin Peters Mathematics, Computational Science and Engineering Springer-Verlag Tiergartenstrasse 17 69121 Heidelberg Germany [email protected]

ISSN 1439-7358 ISBN 978-3-642-04465-6 e-ISBN 978-3-642-04466-3 DOI: 10.1007/978-3-642-04466-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009935691 Mathematics Subject Classification Numbers (2000): 76-XX, 65Kxx, 65Mxx, 65Nxx, 65Yxx © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover illustration: © The Slattery Media Group, Australia, 2009 Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science + Business Media (www.springer.com)

Preface

All over the world, sport plays a prominent role in society: as a leisure activity for many, as an ingredient of culture, as a business and as a matter of national prestige in such major events as the World Cup in soccer or the Olympic Games. Hence, it is not surprising that science has entered the realm of sport, and, in particular, that computer simulation has become highly relevant in recent years. This is explored in this book by choosing five different sports as examples, demonstrating that computational science and engineering (CSE) can make essential contributions to research on sports topics on both the fundamental level and, eventually, by supporting athletes’ performance. Indeed, as diverse as the many kinds of sports are, the basis for the simulation is always to include the relevant laws of physics in the modelling, and due to the complexity of the processes involved, this is a difficult scientific task in itself. Then, in order to obtain results from the computer simulation given this level of complexity, it is necessary to employ the advances in computer power and mathematical methods for the development of algorithms, going to the limit of what is possible. With these results, the scientists proceed to interact with the athletes and coaches to achieve improvements in performance. It is fair to say that without the amazing advances in computer simulation in recent years, it would have been hopeless trying to attack in any meaningful way the scientific challenges posed by the complexity intrinsic to sports. The planning for this volume of Lecture Notes in Computational Science and Engineering started about two years ago, and in the discussions with the invited authors it turned out that all the topics which would be included had computational fluid dynamics (CFD) as their methodological focus. Therefore the book shows how CFD is being used in such diverse sports as sailing, swimming, ski jumping, soccer and Australian football. One of the most spectacular sport events is the America’s Cup, which the Swiss team Alinghi won in the years 2003 and 2007. CSE scientists Danile Detomi, Nicola Parolini and Alfio Quarteroni describe in their survey how cutting-edge computational fluid dynamics made essential contributions to the Alinghi boat’s design and performance. This research and its implementation are an outstanding example of the usefulness of CSE. Swimming is one of the oldest and most popular sports and CFD can be applied here, too. This not only applies to the design of swimwear – which is receiving

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a great deal of attention in the media right now – but also to studying the swimming techniques of the human body itself, aiming to improve performance. The Portuguese-Norwegian team of CSE scientists led by Daniel Marinho examines this very challenging field of research. Moving from water to air, in the third contribution of this book, Helge Nørstrud and Ivar Øye provide an overview of their pioneering work where they applied CFD to ski jumping. This research and the collaboration with the Norwegian national team led to a patented new shape of the ski. As Sarah Barber shows, CFD has become relevant to the most popular sport of all – soccer: studying and computing the behaviour of soccer balls, and consequently their design, represents a very challenging research problem. Her survey includes results which were used for the 2006 World Cup Teamgeist ball. As she points out, this field of research and its industrial implementations will remain very active in the future. Unfortunately, Australian football is not as widely known around the world as this spectacular sport deserves. Hence, even in a scientific book, readers will allow a few general remarks and background information. Australian football is an old sport which celebrated its 150th anniversary in the year 20081. It was originally developed as an off-season fitness exercise for cricket players, and its rules were codified in the year 1859 – for soccer this happened in 1863 and for rugby in 1871. Today it is a nationwide, immensely popular sport in Australia, played at all levels from school children to the elite in the Australian Football League2 . At the professional level, Australian football places extreme demands on the players with regard to all aspects of athleticism, and physical and mental toughness. No doubt its popularity is due to its great speed, precision kicking and competitiveness, as well as to its strong roots in Australian culture. Sports enthusiasts who visit Australia are recommended to visit the Melbourne Cricket Ground or one of the other arenas and watch an AFL match. Due to its oval shape, studying the aerodynamics of Australian footballs is even more difficult than for soccer balls, and Firoz Alam and his team of coauthors present an overview of the results of their CFD studies. As the content of this book shows, sport is an excellent source of challenging problems for CSE, and there are many other relevant methodologies besides CFD which are being used. We plan to cover more of these in future volumes of the series Lecture Notes in Computational Science and Engineering. I would like to thank all the contributors of the book for very interesting discussions and enabling us to introduce sports topics into Lecture Notes in Computational Science and Engineering, a feature which we plan to expand. I would also like to thank Stephen Wright for pointing out Geoffrey Blainey’s book, Svein Linge for giving plenty of background information on computational simulation in sport, 1 Readers who would like to learn more can find plenty of information in The Australian Game of Football since 1858, Geoff Slattery Publishing, 2008, and the scholarly work of the historian Geoffrey Blainey A Game of Our Own, Black Inc., Melbourne, 2003. 2 See afl.com.au .

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and Marco Pilloud for explaining issues with swimming techniques and elite-level swimming. Lastly, my sincere thanks go to Olivia Hudson of The Slattery Media Group and Sarah Davenport of the Australian Football League for granting permission to use the photo on the cover and for guiding us through the application process. Eppelheim, Trondheim, Stockholm, July, 2009

Martin Peters

Contents

Numerical Models and Simulations in Sailing Yacht Design . . . . . . .. . . . . . . . . . . Davide Detomi, Nicola Parolini, and Alfio Quarteroni

1

Swimming Simulation: A New Tool for Swimming Research and Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 33 Daniel A. Marinho, Tiago M. Barbosa, Per L. Kjendlie, Jo˜ao P. Vilas-Boas, Francisco B. Alves, Abel I. Rouboa, and Ant´onio J. Silva On CFD Simulation of Ski Jumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 63 Helge Nørstrud and I. J. Øye Soccer Ball Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 83 Sarah Barber and Matt Carr´e Aerodynamics of an Australian Rules Foot Ball and Rugby Ball . .. . . . . . . . . . .103 Firoz Alam, Aleksandar Subic, Simon Watkins, and Alexander John Smits

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Contributors

Firoz Alam School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia, [email protected] Francisco B. Alves Faculdade de Motricidade Humana, Universidade T´ecnica de Lisboa, Estrada da Costa, 1495–688 Cruz Quebrada, Portugal, [email protected] Sarah Barber Sports Engineering Research Group, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK, [email protected] (now at Laboratory for Energy Conversion, ETH Zurich, Switzerland) Tiago M. Barbosa Instituto Polit´ecnico de Braganc¸a, Departamento de Desporto/CIDESD, Campus de Santa Apol´onia, Apartado 1038, 5301–854 Braganc¸a, Portugal, [email protected] Matt Carr´e Sports Engineering Research Group, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK, [email protected] Davide Detomi CMCS, Institut d’Analyse et Calcul Scientifique, Ecole Polytechnique F´ed´erale de Lausanne, Station 8, 1015 Lausanne, Switzerland, [email protected] Per L. Kjendlie Norwegian School of Sport Sciences, PO Box 4014 Ullev˚al Stadion, 0806 Oslo, Norway, [email protected] Daniel A. Marinho Universidade da Beira Interior. Departamento de Ciˆencias do ´ Desporto/CIDESD, Rua Marquˆes D’Avila e Bolama, 6201–001 Covilh˜a, Portugal, [email protected] Helge Nørstrud Norwegian University of Science and Technology, Department of Energy and Process Engineering, NO-7491 Trondheim, Norway, [email protected] Ivar Øye CFD norway, P.O. Box 1219, Pirsenteret, NO-7462 Trondheim, Norway, [email protected] Nicola Parolini MOX, Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133, Milan, Italy, [email protected]

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Contributors

Alfio Quarteroni CMCS, Institut d’Analyse et Calcul Scientifique, Ecole Polytechnique F´ed´erale de Lausanne, Station 8, 1015 Lausanne, Switzerland, [email protected] and MOX, Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133, Milan, Italy Abel I. Rouboa Departamento de Engenharias, Universidade de Tr´as-os-Montes e Alto Douro, Apartado 1013, 5001–801 Vila Real, Portugal, [email protected] Ant´onio J. Silva Departamento de Desporto, Sa´ude e Exerc´ıcio F´ısico/CIDESD, Universidade de Tr´as-os-Montes e Alto Douro, Apartado 1013, 5001–801 Vila Real, Portugal, [email protected] Alexander John Smits Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey, USA, [email protected] Aleksandar Subic School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia, [email protected] Jo˜ao P. Vilas-Boas Faculdade de Desporto, Universidade do Porto, Rua Dr. Pl´acido Costa, 91, 4200–450 Porto, Portugal, [email protected] Simon Watkins School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia, [email protected]

Numerical Models and Simulations in Sailing Yacht Design Davide Detomi, Nicola Parolini, and Alfio Quarteroni

Abstract In this note, we describe the numerical methodology developed in the framework of the collaboration between the Ecole Polytechnique F´ed´erale de Lausanne (EPFL) and the Alinghi Team, in preparation to the 32nd edition of the America’s Cup which took place in Valencia (Spain) in summer 2007. The mathematical and numerical models adopted to simulate different design aspects (such as appendage design, hull dynamics and sail/wind interaction) are presented and discussed, together with a selection of the numerical results obtained.

1 Sailing Yacht Design The America’s Cup is the oldest and most prestigious regatta in the sport of sailing. Invited by Lord Wilton, Commodore of the Royal Yacht Squadron, the schooner America sailed away from New York on July 21, 1851, to participate as representative of the New York Yacht Club to the “One Hundred Guinea Cup”. One month later, on August 22, around the Isle of Wight, America won the regatta against the yacht Aurora and other 15 English boats. The prize was represented by 100 guineas and an ornate silver-plated Britannia metal ewer of equal value (made by Robert Garrard) which later took the name of “America’s Cup” in honor of the yacht America. The trophy was donated by the ownership syndicate to the New York Yacht Club in 1857 under a Deed of Gift, originally written in 1852, which stated that the Cup was to be “a perpetual challenge cup for friendly competition between nations”. The current version of the Deed of Gift released in 1887 is the third revision of the original Deed. For 132 years, the America’s Cup was successfully defended by US syndicates, until 1983, when the 12-m Class Yacht Australia II with its innovative winged-keel

N. Parolini (B) MOX, Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133 Milan, Italy e-mail: [email protected] M. Peters (ed.), Computational Fluid Dynamics for Sport Simulation, Lecture Notes in Computational Science and Engineering 72, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-04466-3 1,

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defeated the New York Yacht Club defender Liberty. Since then, the Cup has been conquered three times by the San Diego Yacht Club, twice by Team New Zealand and in the last two editions (2003 and 2007) by the Swiss Team Alinghi. In America’s Cup match racing, two sailing boats race around a course aligned with the wind direction, for this reason America’s Cup yachts are designed to operate optimally in a wide range of sailing conditions. The different components (above and beneath the water surface) of the yacht interact with one another in a very complex way. To describe this system and predict its behavior, the Design Team of each America’s Cup syndicate makes extensive use of advanced experimental and numerical tools, with the final goal of achieving a near to optimal configuration. In upwind and downwind legs different sailing techniques are adopted and the design of the boat should accommodate the conflicting requirements arising from the two regimes. For the sail rig, this problem is overcome through the use of different sets of sails (main and genoa for upwind sailing, main and spinnaker/gennaker for downwind sailing). On the other hand, in the underwater part, the possible changes during the race are restricted to the trimming of rudder and keel trim tab. Yacht appendages (keel, bulb, winglets and rudder) are designed to perform in both downwind sailing, where minimal drag should be attained, and in upwind sailing, where they have to counterbalance the forces and moments generated by the sails. Moreover, this complex optimization problem is constrained by the rules of the International America’s Cup Class (IACC), which was first introduced in 1992 and since then it has continuously evolved from one edition to the next. For the 32nd America’s Cup edition, a new version (Version 5) of the IACC rules has been adopted. The IACC rules impose severe restrictions on a number of design factors, not only on geometrical dimensions (depth, displacement, sail area), but also on flow control devices (e.g. number of underwater moving surfaces) and materials. The main rule that plays a crucial role for the definition of any America’s Cup configuration is known as “the Formula” and is in fact an inequality involving a relation between boat length Lb , sail area As and displacement (that is the boat mass) D: p p Lb C 1:25 As  9:8 3 D  24 m: 0:686

(1)

A longer boat can be realized at the expense of lowering the sail area or increasing the displacement. Further unilateral constraints are dictated for boat length, beam, draft and displacement.

1.1 Performance Evaluation The standard approach employed in the America’s Cup design teams to evaluate whether a design change (and all the other design modifications that this change implies) is globally advantageous, is based on the use of a Velocity Prediction

Numerical Models and Simulations in Sailing Yacht Design

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Aerodynamic Side Force S a Aerodynamic Lift La

Aerodynamic Drag D a

Boat Centerline Course of the boat

Aerodynamic Heeling Moment Mh

Yaw Aerodynamic Angle βy Thrust T a Boat Velocity V b

Hydrodynamic Drag D h

Hydromechanic Righting Moment Apparent Mh Wind Angle βAW

Apparent Wind

True Wind Angle βT W

True Wind

Hydrodynamic Side Force S h

Fig. 1 Forces and moments on the water plane

Program (VPP), which can be used to estimate the boat speed and attitude for any prescribed wind condition and sailing angle ˇT W (the angle between the centerline of the boat and the wind direction). The VPP computes boat speed and attitude modelling the balance between the aerodynamic and hydrodynamic forces acting on the boat. A diagram representing the hydrodynamic and aerodynamic force as well as moment components acting in the water plane is presented in Fig. 1. On the water plane, a steady sailing condition is obtained imposing two force balances in x direction (aligned with the boat velocity) and y direction (normal to x on the water plane) and a heeling moment balance around the centerline of the boat: D h C T a D 0; S h C S a D 0;

(2)

M h C M a D 0; where D h is the hydrodynamic drag (along the course direction), T a is the aerodynamic thrust, S h is the hydrodynamic side force perpendicular to the course, S a is the aerodynamic side force, M h and M a are, respectively, the hydro mechanical righting moment and the aerodynamic heeling moment around the boat mean line. The angle ˇY between the course direction and the boat centerline is called yaw angle. The aerodynamic thrust and side force can be seen as a decomposition in the reference system aligned with the course direction of the aerodynamic lift and drag which are defined on a reference system aligned with the apparent wind direction (Fig. 1). Similar balance equations can be obtained for the other degrees of freedom. For a detailed presentation of Velocity Prediction Programs, we refer to [5, 11]. In a VPP program, all the terms in system (2) are modelled as functions of boat speed, heel angle and yaw angle. The equilibrium condition described by (2) is

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obtained solving a dynamical model for the boat motion in the six degrees of freedom. Suitable correlations between the degrees of freedom of the system and the different force components are obtained either from experimental tests or numerical simulations. Although experimental measurements are still extremely important in some design analysis, their role and relevance w.r.t. numerical investigation has changed in the past few years. Indeed, wind tunnel and towing tank tests are nowadays mainly used in validating and calibrating the numerical tools that are later used to perform extensive investigations on the different design solutions and configurations. The role of advanced Computational Fluid Dynamics (CFD) is to supply accurate estimates of the forces acting on the boat in order to improve the reliability of the prediction of the overall performance associated with a given design configuration. Since 1983, when the keel of Australia II was designed using computational methods [22], the subsequent America’s Cup campaigns have always been characterized by an increasing interest in numerical simulations [3, 4, 6, 7, 10, 15, 16]. Computer simulation was critically important in designing Alinghi below the water line as well as in the air. Mathematical modelling and numerical simulation have been used to reproduce on the computer the complex flow dynamics under a broad variety of sailing conditions. In this note, we will describe the numerical algorithms that have been set up for the different analyses based on CFD and we will report the main results obtained in some of the design solutions that have been subject of investigation. This research has been carried out in the framework of the partnership between the Ecole Politechnique F´ed´erale de Lausanne (EPFL) and the Alinghi Design Team. In addition to the CFD analysis that will be described in this paper, many other research areas have been investigated in the EPFL laboratories during this collaboration. We mention two examples showing how both experimental measures and numerical modelling are also used on board and in real-time to control the boat performances. A new sensor embedding system, based on optical fibers, has been developed to measure the stresses and deformations of different boat components (hull, mast, rigging) [23]. This technology allows the skipper and the other crew members to have an instantaneous monitoring of the boat structure condition, thus helping them to run the boat at its limits. Real-time technology on board has also been developed to exploit all the available meteorological data, which are collected by several meteo boats distributed in the course area and can be communicated to the racing boat till 5 min before the race start. A statistical model of the wind behavior has been developed to support the strategic decision-making on board, based on a real-time stochastic optimization which should propose the sailors the fastest route and the most promising sequence of tacking. Aggregated historical weather information (collected over several months) are processed in order to help the navigator in planning the fastest possible course from any given location. The best starting position is first identified, and then as the race unfolds and the wind changes, regularly updated recommendations will be provided on how the boat’s course should be revised.

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1.2 Yacht Appendages One of the design areas where numerical simulations play a crucial role is the optimization of the appendages. Keel, bulb, winglets and rudder should be shaped and sized (within the degrees of freedom left by the strict IACC rules) in order to guarantee global optimal performances. Full-scale tests are still an invaluable ingredient of the design process: the final step for taking every important design choice is always a test campaign on the real boat. Several days of testing, with the two boats differing by the design detail under investigation, are planned during every America’s Cup campaign by all the syndicates. For instance, the final choice between two keel designs is customary taken on the ground of two-boat testing comparisons and sailors’ preference. However, to select the two final keel shapes to be tested onto the water a full numerical simulation campaign is performed on many different keel candidates. Appendage design parameters that have been considered in our research include bulb length and sections, keel profile and planform, winglet shape and positioning. In addition to this major design characteristics, detailed numerical simulation can also help in refining design details such as, e.g. the intersection regions between keel and bulb where suitable local shaping (strakes and dillets) can improve the hydrodynamic performances.

1.3 Physics of Downwind Sailing Sails are the “engines” of an America’s Cup yacht. Their design can be thought as a sequence of optimization steps which involve high technology and a broad human experience, all focused to ensure the maximum final performance. In particular, modern sails design exploits the information obtained by complex numerical analyses which ideally should provide accurate and reliable representations of the fluid–structure interaction (FSI) between the wind flow and the sails, modelled as deformable membranes. The earlier works in this field were mostly focused on upwind sailing conditions, where the flow stream is mainly attached to the sail surfaces and pressure fields can be computed by simple potential flow approach, thus avoiding the solution of more complicate models based on the Navier–Stokes equations. However, the upwind leg represents only one half of the whole match race, therefore a detailed analysis of the downwind condition is required to provide the designers useful information to optimize the spinnakers/gennakers. A dedicated software, called Virtual Wind Tunnel (VWT), has been developed to solve the coupled FSI problem of racing sails, particularly in downwind conditions. This software merges a commercial finite element structural solver and a finite volume fluid software, also providing a dedicated graphical interface as user front-end. To better understand the reasons suggesting the setup of a complex FSI model interfacing the Navier–Stokes fluid equations with a non-linear sail model, it is

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Fig. 2 Forces on gennaker: the total aerodynamic force Faero can be decomposed according to its driving and leeway or lift and drag components

worthwhile a short introduction about the physical phenomena which characterize downwind sailing. The first distinction to be made is the choice of the kind of sail to be used: a gennaker or a spinnaker. Gennakers are asymmetric sails which are used in light winds and are specifically designed to push the boat by generating lift (a force perpendicular to the wind direction, as for airplane wings). On the other hand, spinnakers are bigger, symmetric sails, used with stronger winds which act somehow like “parachutes”, thus their driving force is mostly based on drag (a force parallel to the wind direction). !  Looking at Fig. 2, we first introduce the difference between true ( V t rue ) and !  apparent ( V app ) wind. The former is the wind as felt from an observer at rest (or positioned on a fixed reference system), while the latter is the wind as felt by an observer standing on the boat (that is positioned on a reference frame moving with the boat speed). Even if the true wind speed is null, the boat may have a nonnull velocity (this can happen, for example, for boats sailing on rivers, carried by the water stream), in this case the sails feel an apparent wind which is equal (and !  !  opposite) to the boat speed ( V app D  V boat ), which can be exploited to further !  increase the boat speed. More generally, if V t rue ¤ 0 the apparent wind felt by the sails can be expressed as follows: !  !  !  V app D V t rue  V boat ;

(3)

where the sum has to be intended in a vectorial sense. In particular, if the boat sails downwind with a true wind parallel and equal (in magnitude) to the boat speed, the apparent wind becomes null and the sails appear completely deflated. On the other

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Faero=Fdrag=Fdriving

A B

Vboat Vtrue Vapp

Fig. 3 Forces on spinnaker: the driving force is drag (for a perfectly symmetric flow)

hand, a true wind coming from ahead generates an apparent wind which magnitude !  is in general larger than V t rue . To better understand the difference between the two1 sails used on the downwind leg, we can compare Figs. 2 and 3. On Fig. 2 a gennaker is sketched (seen from above), and it can be shown that pressures on side A (also called suction side) are usually much lower than the free stream while the opposite happens on side B (pressure side). The contributions of both sides result in an aerodynamic net force for the boat (called Faero in Fig. 4) which can be further split either as driving (Fdri vi ng ) and leeway force (Fleeway ) or lift (Flif t ) and drag (Fdrag ). Trying to understand which is the source of the driving force, from Fig. 4 it is clear that the main contribution is given by lift with Flif t d , but it also contributes to leeway with Flif t l , while drag reduces the driving force (Fdrag d ) and increases leeway (Fdrag l ). From these considerations, it seems clear that the final goal when designing gennakers is the maximization of lift, which in turn becomes reducing separations over the sail surfaces as much as possible. Focusing now on spinnakers, we can observe that the driving force has a completely different source which leads to different design targets and guidelines. Considering Fig. 3, we see that the fluid stream reaches the sail with a completely different angle of attack, with respect to gennakers. This is typical of strong winds, as mentioned above. Now the aerodynamic force is almost completely drag, at least in a simplified case where the spinnaker is perfectly symmetric and the angle of attack of the apparent wind is orthogonal to the spinnaker chord. It is also clear that in this case

1

Actually, several slightly different gennakers and spinnakers are usually available on board, and their choice depends on the wind conditions. However, for simplicity, we will just consider a boat with just one reference gennaker or spinnaker.

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Flift

Fdrag_d

Flift_d

Fdrag Fdrag_l

Flift_l

Fig. 4 Contribution of lift and drag to leeway and driving force

the drag force is exactly the driving force, while no (or small, if we loose symmetry) leeway is produced. From these considerations, we can conclude that spinnaker design has to be focused toward the maximization of drag force and it can be reasonably compared to a big “parachute”, inflated by the apparent wind. Other examples driven by the same principle are kites, which are kept on air against gravity by drag forces produced by the wind reaching the kite surfaces at very large angles of attack. In general, gennaker design is more critical than spinnaker one, because lift optimization can be influenced by a number of factors (sail curvature and trimming, sail structural stability, wind angle of attack,. . . ), while spinnakers are more a compromise between their wet area (which should be maximized) and their weight (which should be minimized to help the sail flying and capturing higher, faster winds). For this reasons, in the following we will focus our attention on gennakers.

2 Mathematical Model 2.1 Flow Equations Let  denote the three-dimensional computational domain in which we solve O is a volume surrounding the boat B, the computational the flow equations. If  O that is  D nB O domain is the complementary of B w. r. to , (see Fig. 5 for a two-dimensional sketch). The flow around B is governed by the inhomogeneous incompressible Navier–Stokes equations, which read: for all x 2  and 0 < t < T :

Numerical Models and Simulations in Sailing Yacht Design

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Ω

Ω

B

O Fig. 5 A two-dimensional section of the computational domain  D nB

@ C r  .u/ D 0 @t @.u/ C r  .u ˝ u/  r  .u; p/ D g @t r  u D 0;

(4) (5) (6)

where  is the (variable) density, u is the velocity field, p is the pressure, g D .0; 0; g/T is the gravity acceleration, and .u; p/ D .r uCr uT /pI is the stress tensor with  indicating the (variable) viscosity (see [18]). The above equations are complemented with suitable initial and boundary conditions. For the latter we typically consider a given velocity profile at the inflow boundary, with a flat farfield free-surface elevation. In the case we are interested in, the computational domain  is made of two regions, the volume w filled by water and a filled by air. The interface € separating w from a is the (unknown) free-surface, which may be a disconnected two-dimensional manifold if wave breaking is accounted for. The unknown density  actually takes two constant states, w (in w ) and a (in a ). The values of w and a depend on the fluid temperatures, which are considered to be constant in the present model. The fluid viscosities w (in w ) and a (in a ) are constants which depend on w and a , respectively. The set of equations (4)–(6) can therefore be seen as a model for the evolution of a two-phase flow consisting of two immiscible incompressible fluids with constant densities w and a and different viscosity coefficients w and a . In this respect, in view of the numerical simulation, we could regard (4) as the candidate for updating the (unknown) interface location €. This is achieved resorting to the Volume of Fluid (VOF) method [9] in which (4) is replaced by a similar advection equation for a scalar function  (the volume fraction) which is defined to be 1 in w and 0 in a . The interface is then identified by the discontinuity of  and density and viscosity in (5) are given by:  D w C .1  /a ;

 D w C .1  /a :

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Equations (5)–(6) turn out to be equivalent to a coupled system of Navier–Stokes equations in the two sub-domains w and a : @.w uw / C r  .w uw ˝ uw /  r   w .uw ; pw / D w g; @t

(7)

r  uw D 0;

(8)

in w  .0; T /, @.a ua / C r  .a ua ˝ ua /  r   a .ua ; pa / D a g; @t r  ua D 0;

(9) (10)

in a  .0; T /. We have set  w .uw ; pw / D w .r uw C r uw T /  pw I, while  a .ua ; pa / is defined similarly. On the free surface € separating w and a , two interface conditions hold. The kinematic condition (11) ua D uw on €; imposes the continuity of the velocity on the interface, while the equilibrium of forces on the interface is given by the dynamic condition:  a .ua ; pa /  n D  w .uw ; pw /  n C n

on €;

(12)

C Rt1 is the curvature where  is the surface tension coefficient and  D Rt1 1 2 of the free-surface, with Rt1 and Rt2 being the radii of curvature along the coordinates .t1 ; t2 / of the plane tangential to the free-surface (that is orthogonal to n). Based on this formulation, it is possible to account for the surface tension contribution to the free-surface dynamics. In naval hydrodynamic applications, as those considered in the present work, this contribution is often negligible and can be neglected. The flow around an IACC boat in standard race regime exhibits turbulent behavior over the vast majority of the yacht components, both above and beneath the water surface. To account for the turbulent behavior, we have adopted the SST (Shear Stress Transport) model proposed by Menter [13] which combines the standard k  ! model (in the inner boundary layer) with the k  " model (in the outer region of and outside of the boundary layer) and requires the solution of two additional advection–diffusion–reaction partial differential equations. Eddy-viscosity turbulence models, such as the SST model, are nowadays widely adopted for the simulation of turbulent flows in engineering applications. They are able to recover with an acceptable accuracy the global behavior related to the turbulence nature of a flow. In particular, in presence of walls, they supply an accurate description of turbulent boundary layers.

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Unfortunately, they are not satisfactory when the flow displays large portion of laminarity on the surfaces subjected to investigation. A relevant example is the flow around yacht appendages where neglecting the laminarity usually leads to poor estimates of the forces acting on the surfaces. An important improvement has been proposed in [14] where the SST model is coupled with a suitable designed laminar– turbulent transition model. The model requires the solution of two additional partial differential equations. We refer to [16] for a complete description of the model and its application to yacht appendage design. The CFD solver used in this work is Ansys-CFX. The RANS equations, as well as all the partial differential equations required in the turbulence, transition and freesurface models, are solved using a vertex-based finite volume method [21].

2.2 Rigid-Body Fluid/Structure Interaction As the attitude of the boat advancing in calm water or wavy sea is strictly correlated with its performances, it is important for a numerical tool for yacht design to account for the boat motion. This can be done coupling the fluid solver with a code able to compute the structure dynamics. When the motion of the hull is analyzed, the structural deformations can be neglected and only the rigid body motion of the boat in the six degrees of freedom is considered. Following the approach adopted in [1,2], two orthogonal Cartesian reference systems are considered: an inertial reference system .O; X; Y; Z/ which moves forward with the mean boat speed and a body-fixed reference system .G; x; y; z/, whose origin is the boat center of mass G, which translates and rotates with the boat. The X Y plane in the inertial reference system is parallel to the undisturbed water surface and the Z-axis points upward. The body-fixed x-axis is directed from bow to stern, y is positive starboard and z upwards. The dynamics of the boat in the six degrees of freedom is determined by integrating the equations of variation of linear and angular momentum in the inertial reference system, as follows RG DF mX P C   TNN INN TNN 1  D M G ; TNN INN TNN 1 

(13) (14)

R G is the linear acceleration of the center of mass, F where m is the boat mass, X P and  are the angular acceleration and velocity, is the force acting on the boat,  respectively, M G is the moment with respect to G acting on the boat, INN is the tensor of inertia of the boat about the body-fixed reference system axes and TN is the transformation matrix between the body-fixed and the inertial reference system (see [1] for details). The forces and moments acting on the boat are given by F D F Flow C mg C F Ext M G D M Flow C .X Ext  X G /  F Ext ;

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where F Flow and M Flow are the force and moment, respectively, due to the interaction with the flow and F Ext is an external forcing term (which may model, e.g. the wind force on sails) while X Ext is its application point. To integrate in time the equations of motion, the second order ordinary differential equations (13)–(14) are formulated as systems of first order ODE. If we consider, for example, the linear momentum equation (13), it can be rewritten as mYP G D F ;

P G D Y G; X

(15) (16)

where Y G denotes the linear velocity of the center of mass. This system is solved using an explicit two-step Adams–Bashforth scheme for the velocity Y nC1 D Y n C

t .3F n  F n1 /; 2m

and a Crank–Nicolson scheme for the position of the center of mass X nC1 D X n C

t nC1 .Y C Y n /: 2

For a convergence analysis of the scheme (as well as for a detailed description of the integration scheme for the angular momentum equation), we refer to [12], where it is shown that second-order accuracy in time is obtained. Moreover, the schemes feature adequate stability properties. Indeed, the stability restriction on time step are less severe than the time step required to capture the physical time evolution. In the coupling with the flow solver, the 6-DOF dynamic system receives at each time step the value of the forces and moments acting on the boat and returns values of new position as well as linear and angular velocity. In the flow solver, these data are used to update the computational grid (by a mesh motion strategy based on elastic analogy) and the flow equations are solved on the new domain through an Arbitrary Lagrangian Eulerian (ALE) approach.

2.3 Wind–Sails Interaction The sail deformation is due to the fluid motion: the aerodynamic pressure field deforms the sail surfaces and this, in its turn, modifies the flow field around the sails. Obviously, the assumption of rigid body motion does not apply in this case. The mathematical model for the wind/sail fluid/structure interaction is given by the coupling of the incompressible Navier–Stokes equations (9) with constant density and viscosity for air and a second order elastodynamic equation which models the sail deformation as that of a membrane. More specifically, the evolution of

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the considered elastic structure is governed by the classical conservation laws for continuum mechanics.

2.3.1 Modelling the Sail Structure O s is the reference 2D domain occupied Considering a Lagrangian framework, if  by the sails, the governing equation can be written as follows: s

@2 d D r  s .d/ C fs @t 2

O s  .0; T ; in 

(17)

where s is the material density, the displacement d is a function of the space coordiO s and of the time t 2 Œ0I T , s are the internal stresses while f s are the nates x 2  external loads acting on the sails. (These are indeed the normal stresses .u; p/  n O s represents a wider (bounded on the sail surface exerted by the flowfield.) In fact,  and disconnected) domain which includes also the mast and the yarns as parts of the O s and Œ0I T   RC is the time O s is denoted by @ structural model. The boundary of  interval of our analysis. For suitable initial and boundary conditions, and assigned an appropriate constitutive equation for the considered materials (that provides the stress s as a function of the strain d), the displacement field d is computed by solving (17) in its weak form: Z

Os 

@2 di s 2 .ıdi /dx C @t

Z

Os 



II

ik

.ıki /dx D

Z

Os 

fs i .ıdi /dx;

(18)

where  II is the second Piola–Kirchoff stress tensor,  is the Green–Lagrange strain tensor and ıd are the test functions expressing the virtual deformation. The second coupling condition enforces the continuity of the two velocity fields, u and @d , on @t the sail surface. The structural model can be partitioned into three different elements: the sails, the yarns and the mast (see Fig. 6). Each of these parts has different materials, different stress conditions and, then, different constitutive equations. The sails are modelled as thin elastic isotropic membranes, with large displacement and small strains. This model implies a linear description for the material, but also includes a non-linear term due to the geometry in the strain expression. The expression of the Green–Lagrange strain tensor is non-linear in terms of the displacements, but there is a linear relation between the strain tensor  and the second Piola–Kirchoff tensor  II . The mast is thought as an elastic beam under large displacements and small strains. Once again the Green–Lagrange tensor is nonlinear, whereas the relation between the stress and the strain tensor is linear. Finally, the mast rig is modelled as a set of elastic cables under large displacements and large strains. Both the Green–Lagrange strain tensor and the stress–strain relation are therefore non-linear. Once the materials properties are known, both the stress and strain tensors can be written as non-linear functions of the displacement d,

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Fig. 6 Components considered in the structural model

respectively  II .d/ and  .d/. These expressions close problem (18), thus allowing the determination of the displacements field d. As previously noticed, large displacements are allowed for all the components of the structural model. In this view it is convenient to formulate the problem in a Lagrangian framework always referred to the last deformed configuration and not to the original (design) one. A difficulty arises when dealing with sails: the latter are described as membranes with zero flexural stiffness and, thus, the idealized membrane cannot sustain any compressive stress. When compressive stresses are about to appear in the membrane, they are immediately released by out-of-plane deformations, i.e. the membrane wrinkles. This phenomenon has to be included in the model. The stress field after wrinkling is modelled as a uniaxial tension field, in which one principal stress is tensile and the other one is zero, moreover the wrinkle direction is assumed to be identical to that of tension lines. In addition, the out-of-plane deformation caused by wrinkling is replaced by the in-plane contraction, whose direction is perpendicular to that of wrinkle. Such in-plane contraction is the zero-energy deformation because its direction coincides with the principal axis corresponding to zero stress. In this context the wrinkled membrane can be treated as a planar problem, and local buckling caused by wrinkling is simplified to the in-plane contraction. For a thorough analysis about the wrinkle model used the reader can refer to [19]. Problem (18) has been solved numerically by a classical finite element formulation, even if the different nature of sails, mast and yarns require the implementation of specific techniques. The sails have been discretized using three-points triangular elements with linear displacements and constant stresses and strains. Numerical integration on the elements makes use of a one point Gaussian quadrature scheme.

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The yarns have been discretized using one-dimensional two points linear elements while the mast has been modelled by using one-dimensional cubic Hermite elements. The resulting formulation of problem (18), after the introduction of the finite elements discretization, reads M vR .t/ C K .v.t// D F .t/;

(19)

where v are the finite elements nodal variables, M is the mass matrix, K .v.t// contains the non-linear elastic nodal forces and F are the nodal external loads. In the model adopted the damping forces are assumed to be null.

2.3.2 Fluid–Structure Coupling Algorithm As previously introduced, the coupled problem is solved iteratively. At each step, the fluid solver receives sail velocities and position: the former provide Dirichlet boundary conditions, the latter identifies the new flow domain in which the flow computation is carried out, then the structural solver receives the pressure field on sails and the procedure loops until the structure undergoes no more deformations because a perfect balance of forces and moments is reached. When dealing with transient simulations, this must be true for each time step and the sail geometry evolves over time as a sequence of converged states. On the other hand, a steady simulation can be thought as a transient one with an infinite time step, such that “steady” could be regarded as an average of the true (unsteady) solution over time. More formally, we can define two operators called Fluid and Struct which represent the fluid and structural solvers, respectively. In particular, Fluid can be any procedure which can solve the incompressible Navier–Stokes equations while Struct should solve a membrane-like problem, possibly embedding suitable non-linear models to take into account complex phenomena as, for example, the structural reactions due to a fabric wrinkle. The fixed-point problem can be reformulated with the new operators as follows: Fluid .Struct.p// D p:

(20)

In this formulation we can clearly recognize the structure of a fixed-point problem. A resolving algorithm can be devised as follows. At a given iteration the pressure field on sails p is passed to the structural solver (Struct) which returns the new sail geometries and the new sail velocity fields. Afterwards, these quantities are passed to the fluid solver (Fluid) which returns the same pressure field p on sails. Clearly, the “equal” sign holds only at convergence. The resulting fixed-point iteration can be rewritten more explicitly as follows: Given a pressure field on sails pk , do: .GkC1 ; U kC1 / D Struct.pk / pNkC1 D Fluid.GkC1 ; U kC1 / pkC1 D .1  k /pk C k pNkC1 ;

(21)

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where GkC1 and U kC1 are the sail geometry and the sail velocity field at step k C 1, respectively, while k is a suitable acceleration parameter. If an initial pressure field pk is not available, it is also possible to start from the second of (21) providing a design geometry GkC1 and null initial velocities U kC1 D 0. Even though the final goal is to run an unsteady simulation, the fluid–structure procedure has to run some preliminary steady couplings to provide a suitable initial condition. The steady run iterates until a converged sail shape and flow field are obtained, where converged means that it does exist a value of kc such that (20) is satisfied 8 k > kc (within given tolerances on forces and/or displacements). When running steady simulations the velocity of the sails is imposed to be null at each coupling, thus somehow enforcing the convergence condition (which prescribes null velocities at convergence). This explains why convergence is slightly faster when running steady simulations with respect to transient ones (clearly only when such a solution reflects a steady state physical solution).

2.3.3 Pressure Relaxation The experience made on a number of numerical experiments has shown that a good convergence rate is ensured even without any pressure relaxation (like that used in the (21)). Hence, most part of the computed solution have been obtained by setting k D 1, which corresponds to directly applying the last computed pressure fields to the sails without any memory of the past ones. However, in some cases a pressure relaxation step is needed, in particular when the mesh motion algorithm fails and when the leading edge starts curling. Mesh motion is very useful to skip a new domain discretization by moving the grid vertices according to the sail displacements and relaxing the internal mesh vertices as nodes of a fictitious elastic net. Moreover, mesh motion represents a natural way to map the previously computed fluid solutions onto the newly available mesh (which fits the deformed sail shapes) to be used as reasonable guess (for steady runs) or initial condition (for transient ones) for a new fluid solution. Unfortunately, mesh motion algorithms are usually prone to failures, mainly when the required displacements largely exceed the local mesh size or geometries are particularly complex. On the other hand, edge curling is a typical issue to be faced when the gennaker is too eased out, in this case the incident wind produces regions of high pressure near the suction side of the leading edge which can trigger a local structural wrapping. Both these issues can be fixed if the previous FSI coupling (corresponding to the pressure field pk on sails) is assumed to be “acceptable”, that is a valid mesh is available and no edges are curled. In this case, even if the newly calculated pressure field (pNkC1 ) generates one of the above problems, it is possible to compute a “safe” pressure field as follows: pkC1 D .1  k /pk C k pNkC1 :

(22)

From the numerical viewpoint this procedure is quite difficult because the two pressure fields pk and pNkC1 do not refer to the same geometry (there is a deformation

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step in-between so pk refers to the geometry Gk while pNkC1 refers to GkC1 ) and, sometimes, even the mesh topology may have changed (for example if we allow the generation of a new domain discretization when mesh motion fails and pressure relaxation is only used to fix edge curling, in this case we have to deal with two different meshes: Mk and MkC1 ). In order to perform the pressure relaxation we have to execute the following steps (jGk ! jGkC1 indicates a node displacement from the sail geometry Gk to GkC1 while jMk ! jMkC1 denotes interpolation from the mesh Mk to MkC1 ): 1. Move the sail pressure field pk jMk ; Gk to the new sail geometry GkC1 , that is pk jMk ; Gk ! pk jMk ; GkC1 . 2. Interpolate the moved pressure field pk j Mk ; GkC1 (hat has been computed on the mesh Mk ) onto the new mesh MkC1 , that is pk j Mk ; GkC1 ! pk jMkC1 ; GkC1 . The first step is simple because the displacements are provided by the structural solver, while the second step is quite demanding in terms of computational effort and introduces numerical errors. However, once pk jMkC1 ; GkC1 is available, the relaxation (22) is straightforward and by suitably tuning the parameter k it is possible to apply to the sails any intermediate pressure field between pk and pNkC1 . In particular, it is always possible to choose k D 0 which corresponds to the old, “safe” pressure field pk .

3 Mesh Generation and Mesh Motion In the solution of any discrete problem, the accuracy and efficiency of the numerical scheme is strictly related (and depends on) the choice of the computational grid. The generation of the grid should account for a precise definition of the complex shapes characterizing the domain boundary. Moreover, the grid should be able to comply with the expected behavior of the fluid-dynamics solution: in regions where high solution gradients occur, such as boundary layer and moving interface (e.g. the water–air free-surface), the grid should be fine enough to capture these flow features. Further requirements on mesh generation arise when dynamic problem, such as hull motion and sail deformation, are solved. Hereafter, we give a short description of the different mesh generation procedures adopted for hydrodynamic and aerodynamic simulations.

3.1 Block-Structured Mesh for Hydrodynamic Studies For the hydrodynamic analysis of hull and appendages, we have adopted blockstructured grids. The computational domain is first subdivided into hexahedral subdomains each filled with a structured grid. The mesh is conformal through the subdomain interfaces and, where required, the blocks can be collapsed from

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Fig. 7 Block structured grid around the boat: global blocking structure (left) and section of the blocking around the appendages (right)

Fig. 8 Detail of the volume grid: cut plane in the keel/bulb (left) and winglet/bulb (right) intersection regions

haxahedral to prismatic shapes. Other mesh generation strategies would be possible. For example, unstructured tetrahedral grid with prismatic boundary layer mesh is an alternative approach which ensures a higher level of automatization in the grid generation process. However, a block-structured approach usually guarantees mesh of better quality with beneficial effects on the accuracy of the numerical solution. Moreover, in block-structured grid the most time consuming phase (which may require several weeks for a complex geometry such as a fully appended yacht) is the generation of the blocking subdivision (see Fig. 7). When several configurations with the same topology have to be analyzed (and this is usually the case in parametric yacht design), the blocking generation is done once for a reference geometry and any other configuration can be obtained just projecting the blocking onto the new geometric entities. The mesh around an appended hull consists in a H-type grid (extending to the external domain boundaries) which contains O-type grids around hull, keel, bulb, winglets and rudder. The blocking generation complexity comes from the need of designing these local blocking such that the generated grid is valid, of good quality and with local spacing adequate to capture the relevant flow features. In Fig. 8, details of the grid around the appendages are shown.

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Due to the grid requirements imposed by the transition and turbulence models, these simulations entail very large size computational grids (up to 20 millions elements). The simulations were run on an EPFL cluster (450 AMD Opteron processors connected by a Myrinet network). The CPU time required for each simulation to reach convergence was about 30 h on 32 processors.

3.2 Mesh Handling in Wind/Sails Interaction Fluid–structure interaction analysis for sail design has slightly different targets with respect to hull and appendages optimization. While the latter mainly focuses on drag reduction by minimizing the skin friction on the wetted surfaces (or the induced drag produced by lifting surfaces), the former essentially concerns about an accurate detection of the flow separation. Skin friction evaluation requires extremely refined boundary layer meshes (the typical thickness of the first layer may be less than 0:1 mm), which are critical in order to properly predict the laminar-to-turbulent transition over the boat surfaces. On the other hand, the fluid flow over sails is almost everywhere turbulent (from small scale vortices to large turbulent wakes, which frequently occur behind gennakers and always behind spinnakers), and frictional drag effects are negligible with respect to other physical phenomena. For these reasons, sails simulation generally requires isotropic meshes which are suitable for turbulent, chaotic structures, and even if boundary layer anisotropic meshes may be useful for a more accurate detection of the stream separation, they usually do not require the huge amount of elements which is typical of skin drag evaluation. Moreover, sail simulations are often characterized by large deformations which require huge displacements of the mesh elements during mesh motion, this is usually impossible to be achieved if very thin and stretched elements have been built within the boundary layer. Focusing on mesh refinement, experiments have shown that the surface mesh over the main sail should not be coarser than the one over the gennaker (or spinnaker), even if the main sail structure is much stiffer. In fact, suitable mesh refinements on both sails ensure a good refinement also within the sails channel, which has proven to be critical for a good prediction of the sails performance. Different meshing techniques and refinements have been compared, and a mesh sensitivity analysis has been carried out to evaluate the optimal mesh refinement which might ensure nearly asymptotic behaviors of the computed solution (see Fig. 9). To run the tests, sails have been enclosed within a cylinder, and the inner volume has been meshed by means of tetrahedra (either with Octree and TGrid techniques). A far field bounding box encloses the cylinder and this second region has been discretized with an hexahedral structured mesh. The connection between the two grids over the cylinder surface has been realized by means of pyramids. In Fig. 10 two sections are shown corresponding to the two different meshing approaches.

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Fig. 9 Aerodynamic forces on sails as function of mesh refinement and mesh generation technique (Octree and TGrid): (a)–(d) forces and regions of reversed flow behind gennakers for two different trimmings of the gennaker sheet, (e)-(f) streamline and wake comparisons for Octree and TGrid meshes of about 20 millions of elements. The purple line shows almost identical regions of reversed flow

4 Numerical Results The numerical techniques described in the previous sections represent a relevant contribution for the improvement of the CFD analysis adopted for IACC yacht design. In this section, we present an overview of the numerical results obtained on different design solutions during the preparation of the 2007 edition of the America’s Cup, in collaboration with the Alinghi Team, defender of the Cup.

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Fig. 10 Mesh sections around sails: Octree (left) and TGrid (right) unstructured grids. It can be noticed how Octree tends to generate finer meshes close to the sail surfaces while TGrid are more graded

4.1 Advances in Appendage Design The design analyses that can be carried out by CFD simulations cover all the possible design variables that define a set of appendages. The great advantage of the numerical approach, when compared to experimental tests, relies on the possibility to test many different configurations and to obtain a complete picture of the flow behavior at every time instant. Information about local distribution of flow quantities (such as, e.g. pressure, vorticity and turbulence intensity) can be very useful to improve the hydrodynamic performances. These information can be hardly obtained during a full-scale test and even in a fully equipped experimental facility (wind tunnel or towing tank) each of these data requires the setup of suitable measurement equipments. On the other hand, numerical simulations supply as outcome a complete database of relevant quantities about the considered flow problem. A complete reconstruction of the flow around the appendages can help understanding the formation of the main flow features and their interaction with the boat components. A large collection of simulation results on this subject can be found in [6, 10, 15, 16]. Hereafter, we report some examples of recent improvements in appendage analysis. One is the adoption of laminar–turbulent transition model in modern CFD tools which represents a big step forward in naval hydrodynamics. Indeed, in appendage design, the optimization process is often governed by trade-off analyses where pressure and viscous drag play one vs. the other. A typical example is provided by the comparison between a slender bulb and a shorter one (for constant weight/volume). If the former usually guarantees a lower pressure drag, this advantage is counteracted by a larger viscous drag due to the larger wetted surface. For this comparison, an accurate estimation of the transition location is required to predict the viscous component of the drag with an acceptable precision. Indeed, bulb and keel are

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often designed to work in a transitional regime where slight differences in shape can induce a significant change on the location where laminar–turbulent transition occurs. The transition model has been calibrated through a large experimental campaign in wind tunnel. Force components on each appendage element, transition locations on keel and bulb, sensitivity analysis with respect to freestream turbulence intensity have been used to compare numerical and experimental data. After the calibration phase, the new model has been used in all the simulations carried on to design the final appendage configurations. The set up of design details, such as, e.g. the strakes and dillets adopted in the design of keel-bulb junction, have benefitted of the more accurate flow prediction that the new model guarantees. The laminar and turbulent regions on three keel planforms characterized by different strake design are displayed in Fig. 11. As mentioned above, the adoption of the transition model, which gives an automatic way to estimate the transition location, is essential to compare different shapes with different levels of laminarity. In Fig. 12, we display the extension of the laminar region on the appendages for a low value of free-stream turbulence intensity, together with the corresponding wall-shear stress distribution. This picture gives an idea of the importance of accounting for transition in order to get accurate prediction forces (in particular their viscous component).

Fig. 11 Transition location on suction and pressure keel sides for three different keel planform: laminar (blue) and turbulent (red) regions

;

Y

2.000

0 1.000

(m)

 O



X Z



:
0. and

On CFD Simulation of Ski Jumping

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under pressure (suction) as Cp;i < 0. At the stagnation point we have Cp;i D 1 and the atmospheric pressure is at Cp;i D 0 .pi D p1 /. As Fig. 2 shows, the suction (green and blue colors) on the sides of the jumper is felt by the jumper as relative cooler areas on their body and a stagnation point (maximum overpressure) is visible on the forehead of the jumper (white color). An integration of the pressure around a closed body will yield the force acting on the body and this resultant force can be divided into the two components D [N] and L [N] acting respectively in the free stream direction and normal to it, see Fig. 3 for a ski jumper in two dimensions. The aerodynamic drag force D is conveniently expressed through the dimensionless drag coefficient CD Œ, i.e. 1 2 D D CD A 1 V1 2

(2)

where A Œm2  is a specific reference area. Similarly, the lift force coefficient CL Œ is defined in the equation 1 2 L D CL A 1 V1 (3) 2 Finally, the gravity force G [N], i.e. G D mg

(4)

represents the product of the gravity constant g Œm s2   9:81 and the mass m [kg] of the jumper including the weight of the equipment, see again Fig. 3.

Lift, L

Drag, D

α

VELOCITY, V∞

Gravity, G

Fig. 3 The force balance on a ski jumper in steady flight. The angle of attack is designated as ’ [deg]

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2 Geometric Simulation of a Ski Jumper The first step to undertake in order to characterize our jumper is to define an average man (or woman) and to create a simplified model of that body, see Fig. 4. The next task to consider is to select parts of the simplified human body and generate a surface grid around that geometry. The computational grids around the jumper with skis have been constructed with the use of Computational Fluid Dynamics (CFD) Norway’s in housemesh generation packages TwoMesh and ThreeMesh. These are of an algebraic type which applies transfinite interpolation technique [2, 3] to mapping between user specified boundaries. For improved grid generation resolution control, integral control curves or surfaces have been specified based on macro-block concept. In addition, both algebraic and elliptic type smoothening algorithms have been utilized for the smoothing of discontinuities. Since the simulated ski jumper shall be able to obtain various jumping positions, the link between the selected geometric elements must be flexible as illustrated e.g., in Fig. 5 for the knee joint [4]. All main joints are made as singular points which the surfaces can be rotated around. A membranous surface that is smoothly defined between surfaces that intersect at the common joints is generated. This membranous surface is generated by cubic-spline curves using the point information of surfaces that intersect at this common joint. An automatically generated smooth surface is then produced and the surface grid is represented by the same topology.

1

6

2

7

3

8

4

9

10

11

12

13

14

15

5

1 – Head 2 – Upper trunk 3 – Lower trunk 4,5 – Hand 6,7 – Upperarm 8,9 – Forearm 10,11 – Thight 12,13 – Shank 14,15 – Foot

Fig. 4 Measurements of an average man of height H D 176 cm and a mathematical model of the human body [1]

On CFD Simulation of Ski Jumping

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Fig. 5 Samples of jumper surface grid including grid covering knee joint [4]

Fig. 6 Surface grid for the complete jumper with skis. (“ D 30ı ; • D 5ı , see Figs. 9 and 10 for the definition of the angles)

Adding a pair of skis (of length 248 cm and width 11 cm) to the jumper completes the surface grid for the athlete which includes 212,000 grid points for half of the ski jumper., see Fig. 6. After the surface grid for the jumper with skis has been defined a computational space grid must overlap the surface grid in order to pursue a flow analysis around

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Fig. 7 Space grid (or computational space) over the simulated ski jumper (in blue in the middle of the figure, see also the figure below)

Fig. 8 The simplified geometric ski jumper with flow visualization of streamlines in the computational space (’ D 20ı ; “ D 20ı )

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Fig. 9 The V-angle of the skis identified by “ D 0ı ; 10ı ; 20ı ; 30ı and 40ı (from upper left). The distance between the aft part of the skis is twice the skiwidth

Fig. 10 The •-angle between the jumper and the skis identified as • D 0ı ; 5ı and 10ı (from above) for “ D 25ı

the geometry of the ski jumper. This is illustrated in Fig. 7 where 620,000 mesh points are distributed among 16 blocks Fig. 8. The jumper with skis has been analyzed for various geometric reference angles, see Figs. 9 and 10, and example results are given in the next chapter.

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3 Numerical Results 3.1 The Flow Solver The CFD Norway’s flow solver used the present study is a general purpose code for solving the time-dependent two- and three-dimensional Euler or Navier-Stokes equations on multiblock type meshes. The in-house developed code is a second order accurate explicit 3-stage Runge-Kutta time integration scheme based on an upwind-biased 3rd order accurate finite-volume flow evaluation. Including is a local time stepping for convergence acceleration to steady-state and various turbulence models (algebraic, k-©, k-¨, non-linear k-¨) are available.

3.2 Euler Calculation The jumper with skis has been first analyzed for various ski jumper reference angles (’; “) and some few results are given in Table 1. The results are obtained from Euler calculations, i.e., the viscosity in the flow is neglected. This means that flow separation from sharp corners (like the sides on the skis) is obtained with realistic values for the aerodynamic coefficients, see Figs. 11–14. From Table 1 we can conclude with the following statements: (1) The lift on the jumper alone increases up to a maximum between the limits 30ı < “ < 40ı . (2) The lift on the skis shows similar trend and, hence, the optimum angle for the V-style ski jumper lies in the same limits. This is also indicated by the quality ratio CL;t =CD;t if we neglect the suspect value CD;j at “ D 20ı . We will at this point remind the reader that the presented results were obtained by an Euler code and, hence, that viscosity is neglected. It is, however, believed that

Table 1 Aerodynamic dimensionless coefficients vs. “–angle from Euler calculations were the lower index j, s and t refers respectively to jumper, skis and total. The distinct difference between Figs. 11 and 12 is the vortical flow over the oblique ski relative to the oncoming flow and the sideways spreading of the streamlines. This is more pronounced in Figs. 13–14 “ [deg] CL;j CD;j CL;s CD;s CL;t CD;t CL;t =CD;t CL;j =CL;s 0 10 20 30 40

0.428 0.475 0.492 0.625 0.551

0.245 0.258 0.147 0.300 0.325

0.183 0.262 0.361 0.375 0.289

0.092 0.123 0.155 0.156 0.127

0.611 0.737 0.853 1.000 0.840

0.337 0.381 0.302 0.456 0.452

1.813 1.934 2.825 2.193 1.858

2.339 1.813 1.363 1.667 1.907

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Fig. 11 Pressure distribution and flow visualization around a ski jumper with parallel skis (’ D 20ı ; “ D 0ı )

Fig. 12 Pressure distribution and flow visualization around a ski jumper with V-style (’ D 20ı ; “ D 10ı )

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Fig. 13 Pressure distribution and flow visualization around a ski jumper with V-style (’ D 20ı ; “ D 20ı )

Fig. 14 Pressure distribution and flow visualization around a ski jumper (’ D 20ı ; “ D 30ı )

Table 2 Aerodynamic coefficients for a single ski at ’ D 20ı and various “-angles “ (deg) CL CD CL =CD 20 0.400 0.165 2.42 25 0.440 0.175 2.51 30 0.425 0.170 2.50 35 0.385 0.150 2.57 40 0.345 0.130 2.65

this restriction is not so severe for the flow calculations on the skis and the values given in Table 2 are representative. The values in Table 2 are extracted from Fig. 15 which gives the distribution of the aerodynamic characteristics along the ski axis.

On CFD Simulation of Ski Jumping

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3.3 Navier-Stokes Calculation It has been indicated that an Euler calculation will underestimate the aerodynamic drag of the jumper alone and overestimate the lift on the body. Hence, a viscous correction must be applied and for this a viscous Navier-Stokes code has been introduced. The results are shown in Fig. 16 [5].

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20 30 Angle of Attack, alfa [deg]

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Fig. 16 Comparison of lift and drag for the jumper without skis .“ D ” D 0ı / based on viscous (Navier-Stokes) and non-viscous (Euler) calculations

As can be seen from the above figure, the lift on the jumper’s body is highly overestimated for all angles of attack. The reason for this is that the flow over the jumpers rounded back is in reality partly separated.

4 A Ski Jumper in Flight A theoretical analysis of a ski jumper in flight from a ski jump has been performed describing the ski jumper as a point mass and using the aerodynamic results from the previous chapters, Fig. 17. After applying Newton’s second law for the forces in the x, y directions we will obtain the relations Fx D L sin ˛  D cos ˛ D max

Fy D L cos ˛ C D sin ˛  mg D may

where the symbol a [m s2 ] denotes acceleration. This will yield a set of four ordinary differential equations, i.e.

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75 y Lift, L Drag, D Vx α

Vy

x V

Gravity, mg

Fig. 17 Force balance on a ski jumper in flight

dx dt dy dt dVx dt dVy dt

D Vx D Vy  A  CL Vy  CD Vx V 2m  A  D C L Vx  C D Vy V  g 2m D

The above equations are integrated with respect to the time t with a fourth order Runge-Kutta method. The instant values of the variables must be specified in the x,y – coordinates at the jump table and also the velocity at that point. Furthermore, the aerodynamic CL and CD – values must be determined. These are obtained by a polynomial representation of the second order with respect to the angle of attack ’ and where the lift and drag data are described from the preceding analysis. The reference ski jumper has a total mass of m D 80 kg, the “-angle is 25ı and the ”-angle is 5ı . With an exit velocity of 25:8 m s1 .92:9 km h1 / the jumper will achieve a jumping length of ` D 105 m. By changing these reference values we can summarize the following benefits: – – – – – –

For ” D 0ı the jumper flies to ` D 111 m .` D 5:7%/ For “ D 30ı and ” D 0ı the jumping length will be 125.5 m (` D 19:5%) For m D 70 kg; “ D 30ı and ” D 5ı we will obtain ` D 122 m .` D 27%/ For an increase of the exit velocity of 0:1 m s1 the ` will give 4 m. A wind speed of 1 m s1 against the jump will produce approximately ` D 13 m. The best angle of attack is 12–15ı above the horizontal plane.

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5 A Proposal for New Ski Design The numerical analysis over the skis using an Euler code has given a valuable insight into the flow field in particular at the tip of a ski. This is illustrated in Fig. 18 and a geometric modification of the ski tip is at hand since the vortical flow leaving from the outer part of the tip will produce drag, see also Fig. 2. The right picture in Fig. 18 shows the computed skin friction lines emerging out from an attachment line (or positive bifurcation line) and the shaded area indicates the footprint of the separation bubble behind the curved ski tip. Hence we can observe a three-dimensional steady flow separation which will create an additional drag force on the ski. This is clearly seen in Fig. 19 which also shows the large suction area on the upper side of the ski for better lift performance. The flow visualization in Fig. 20 illustrates the earlier discussed vortical flow development on the ski tip for a lower angle of attack as compared to Fig. 19. And the drag producing vortex leaving from the outwards side of the tip is clearly seen. A geometric modification of the ski has been patented (1992) and Fig. 21 illustrates this ski with a bulb. The Austrian ski producer Atomic has produced two pairs of the asymmetric skis and one pair has been tested by several members of the Norwegian jumping team in 1993. Furthermore, wind tunnel tests were undertaken in 1992 at the Norwegian Institute of Technology, Trondheim and some results are shown in Fig. 22.

Fig. 18 Detailed flow visualization of the right-footed ski with skin friction lines

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Fig. 19 Pressure distribution and vortical flow visualization on the right-footed ski at ’ D 20ı and “ D 30ı

The above given data for “ D 30ı have been compared with similar experimental data for a standard ski and results shows that the lift/drag ratio is about 8% higher for the modified ski as for the standard ski for angle of attack in the range 15ı < ’ < 30ı . The main reason for this is that the standard ski yields higher drag values at high angles of attack.

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CFD norway as

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Fig. 20 Pressure distribution and flow vortical flow visualization on the right-footed ski at ’ D 20ı and “ D 25ı .

6 Concluding Remarks The present report summarizes a numerical analysis for the flow over a ski jumper in various positions and has given several results from non-viscous (Euler) and viscous (Navier-Stokes) computations. It is concluded that the V-style represents an improved aerodynamic art of jumping and that the skis plays a major part in producing favorable lift, see Fig. 23. The work is over 15 years old and it does not give a state-of-the art overview. Further simulations, however, can be found in the recent publication [5]. Two factors can be mentioned in describing the favorable V-style jumping aerodynamics. Firstly, it is known from aeronautics that sharp leading edges on oblique wings will generate additional lift due to the vortical flow on the upper part of the wing surface. An illustrative example is a supersonic airliner similar to the historical Concorde shown in Fig. 24. Secondly, the jumper and the skis represent in a simplified form three lifting bodies and each part has a similar wake system, Fig. 25. Such a vortical flow system can also be observed behind a moving car on a misty day or visualized from the

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Fig. 23 Final view of a ski jumper in flight showing the important vortex generation on the upper side of the skis

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Fig. 24 Vortical flow visualization over the wing of a supersonic airplane (Simulation: CFD Norway)

Fig. 25 Illustration of the trailing vortices behind three lifting plates (Black arrows represent downwind whereas white arrows represent upwind)

exhaust pipes. Since the mentioned vortex systems are connected to up- and downwash as illustrated in the figure, it is necessary to separate the bodies apart. The goose (Branta bernicla) in formation flying have learned this trick for power saving. And the V-style ski jumper automatically spreads the skis out from the body. Acknowledgements The presented work was performed for Olympiatoppen and the Norwegian Ski Federation prior to the Olympic Winter Games in Lillehammer, Norway in 1994 under the

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auspices of jumping Coach Trond Jøran Pedersen. Hence, it represents an early task to analyze a ski jumper in flight and not a review of similar simulations at that time and the authors would like to acknowledge the fundamental contributions from Erland Ørbekk and Ernst Meese.

References 1. Hanavan, Jr., E.P., A personalized mathematical model of the human body, AIAA Paper No. 65–498, 1965. 2. Eriksson, L-E., Transfinite Mesh Generation and Computer-Aided Analysis of Mesh Effects, Ph.D. thesis, Uppsala University, 1984. 3. Eriksson, L-E., Practical Three-Dimensional Mesh Generation Using Transfinite Interpolation, SIAM J. Sci. Comput., Vol. 6, No. 3, 1985. 4. Ørbekk, E., Algebraic and Elliptic Grid Generation for CFD Applications, Ph.D. thesis, Norwegian Institute of Technology, Trondheim, 1994. 5. Øye, I.J., On the Aero thermodynamic Effects on Space Vehicles, Ph.D. thesis, Norwegian University of Science and Technology, Trondheim, 1996. 6. Søyseth, O., Aerodynamisk analyse av ulike hoppski (in Norwegian), M.Sc. thesis, Norwegian Institute of Technology, Trondheim, 1992. 7. Nørstrud, H. (Ed.), Sport Aerodynamics, CISM Courses and Lectures, vol. 506, Springer, NewYork, 2008, pp. 183–216 and pp. 217–228.

Soccer Ball Aerodynamics Sarah Barber and Matt Carr´e

Abstract This chapter describes an interesting new application of computational science to sports engineering. The flight of sports balls (and in particular soccer balls) through the air is often a key part of the sport. In this work the physics behind the flight of soccer balls is introduced and discussed. This includes basic concepts such as boundary layer separation and the Magnus Effect. Computational Fluid Dynamics (CFD) and trajectory simulations are then combined to assess the erratic nature of different soccer ball designs, including the 2006 World Cup ball. It is found that both the lift and side force coefficients on a low- or non-spinning soccer ball vary significantly with orientation, which can result in varying erratic trajectories. These trajectories can also vary strongly with ball design and with the initial orientation of the ball. Ball consistency is one property that is often commented on by professional players. It is found that the most consistent balls are the ones with the optimum combination of amplitude and frequency of the varying force coefficients relative to the amount of spin. With the recent introduction of new manufacturing techniques, it should be possible to tailor ball surface patterns to give some interesting ball flights or to optimise consistency.

1 Introduction The flight of a ball through the air is a key part of many popular sports, including soccer, golf, baseball, cricket, tennis and volleyball. Sports balls have been studied aerodynamically since 1672 when Newton commented on the deviation of a tennis ball [1]. The study of sports ball aerodynamics requires the consideration of a number of fundamental fluid mechanics phenomena, including boundary layer flow, transition and separation, turbulence, flow over rough surfaces, the Magnus Effect and both steady and unsteady wake behaviour. Gaining an understanding of the aerodynamics of sports balls, particularly the behaviour of the boundary layer and its separation, can help equipment designers, players, coaches and game regulators Sarah Barber (B) Sports Engineering Research Group, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK, e-mail: [email protected] (now at Laboratory for Energy Conversion, ETH Zurich, Switzerland) M. Peters (ed.), Computational Fluid Dynamics for Sport Simulation, Lecture Notes in Computational Science and Engineering 72, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-04466-3 4,

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and can also make the game more interesting for the fans. The recognition of this required understanding, along with recent advances in computational power, has pushed computational modelling to the forefront of sports ball aerodynamics studies. The aerodynamics of sports balls is interesting from a fundamental point of view due to the domination of boundary layer separation and its variation due to surface details and roughness. Correctly computing the flow of air around a smooth sphere is challenging enough itself, without the added complexity of detailed surface geometry. However, Computational Fluid Dynamics (CFD) has been used successfully to help understand and compare the air flow around different sports balls, and the results have the potential to significantly affect the relevant sport. This chapter introduces some basic concepts that are relevant to sports ball aerodynamics, discusses methods of measurement and analysis and gives some real-life examples. A recent soccer ball CFD study that discusses the erratic nature of kicks and new design technologies is then revealed.

2 Basics of Sports Ball Aerodynamics 2.1 Basic Definitions The starting point for the aerodynamic analysis of sports balls is to consider the forces acting on them. The forces acting on a ball, mass m, moving through the air with velocity v are given by mg (weight) and the aerodynamic forces FD (drag force), FL (lift force) and FS (side force). FD is a combination of skin friction drag, caused by friction between the air and the ball’s surface, and pressure drag, caused by separation of the air from the ball’s surface and the subsequent formation of a low-pressure wake. For a smooth ball, FL and FS only act when the ball is spinning, as a result of the Magnus Effect (see Sect. 2.4). Non-dimensional parameters are often used in aerodynamic analysis for comparison purposes. The non-dimensional aerodynamic force coefficients assigned to a ball are given by, CX D

FX ; 0:5v2 A

(1)

where  D air density.kg m3 /; v D ball velocity.m s1 /; A D ball cross-sectional area .m2 / and FX D force in a given direction (N). The non-dimensional velocity-related parameter is given by the Reynolds number, Re D

vd ; 

(2)

where d D ball diameter (m) and  D air dynamic viscosity (Pa s). The non-dimensional spin-related parameter is given by the spin parameter, r! Sp D ; (3) v where r D ball radius (m) and ! D ball angular velocity (rad s1 ).

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2.2 Boundary Layers The viscosity of a fluid is defined as its resistance to deformation under shear stress, and if plane shear flow is assumed it is given by,  D

du ; dx

(4)

du where  D shear stress exerted by the fluid (Pa) and dx D velocity gradient perpen1 dicular to shear direction (s ). The boundary layer, first defined by Prandtl [2], is the area of a flow next to a surface where viscosity dominates. Away from the boundary layer the viscosity can be considered negligible without significant effects on the flow. The limit of the boundary layer is usually defined to be where the flow is moving at 99% of the free-stream velocity. At the surface the no-slip condition occurs and the local flow velocity is zero. A boundary layer can be laminar, turbulent or in transition between the two. For flow at low velocities, the boundary layer over a surface is laminar, but when the flow travels at a high enough velocity the boundary layer transitions to turbulence. The Reynolds number (Re) at which this occurs depends upon the shape and roughness of the surface. For a flat plate, Re at a given position is usually calculated using the distance from the leading edge, but for a ball it is calculated using the diameter. Laminar flow is very smooth and the fluid travels in parallel layers with no disruption, whereas turbulent flow is “chaotic” and mixing. A turbulent boundary layer has more energy than a laminar one because it mixes with the faster-moving flow outside the boundary layer, and thus has a steeper velocity profile close to the surface and is thicker, as illustrated in Fig. 1 [3]. This means that a turbulent boundary layer will exert a higher skin friction drag than a laminar boundary layer [according to (4)]. It will also separate less readily over a curved surface, the consequences of which are discussed in the next section. Additionally, a turbulent boundary layer has a clear distinctive edge which is strongly fluctuating, and hence usually averaged with time in analysis and computations.

2.3 Boundary Layer Separation and Drag Separation occurs when the adverse pressure gradient caused by the curved surface on the rear side of the ball pushes the flow back and slows it down, until it eventually comes to a standstill and even moves backwards, forming a region of recirculating flow behind the ball, called the wake. This mechanism is illustrated in Fig. 2 [3]. A graph of CD vs. Re for a smooth sphere is shown in Fig. 3 for the Re range relevant for most sports balls. CD drops suddenly when the boundary layer transitions from laminar to turbulent flow at the critical Re (Recri t ) of approximately 3:85105 [4]. The turbulent boundary layer has more energy close to the surface and hence separates much later, resulting in a smaller wake and hence a lower pressure

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drag. This is illustrated in the flow visualisation in Fig. 4. Increasing Re beyond this causes CD to rise slightly because the boundary layer transitions to turbulence earlier on the sphere and thickens, increasing the skin friction drag. Once the boundary layer is fully turbulent CD is just less than 0.2 and independent of Re. The four regions marked on Fig. 3 are key to sports ball aerodynamics due to the particular range of Re they experience; boundary layer transition and thickening are of particular importance.

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Fig. 3 CD vs. Re for a smooth sphere, game-relevant Re [4]

Fig. 4 Smoke flow (top) [5] and oil flow visualization (bottom) [6] for a smooth sphere, flow from right to left

2.4 The Magnus Effect In match situations, sports balls are frequently launched through the air with spin. In general, sidespin causes a ball to swerve to one side, topspin causes it to dip, and backspin pushes it upwards. When a sphere is spinning, CL and CS must be considered in addition to CD , due to the Magnus Effect, which is illustrated for a ball moving from right to left with sidespin (top view) in Fig. 5.

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E

B

Fig. 5 The Magnus Effect for a ball moving from right to left with sidespin (top view)

If the ball were not spinning then the streamlines would flow symmetrically around the ball. However, the spinning ball pulls the air round at A, increasing its velocity, and pushes it back at B, reducing its velocity, producing late separation on the top and early separation on the bottom. This causes an asymmetrical wake. Due to the velocity differences, the pressure is lower at A than at B, and this pressure difference causes a Magnus Force, FM , to act sideways on the ball.

3 Soccer Ball Aerodynamics The aerodynamics of soccer balls is especially interesting, not only because the game involves a large variety of kick types and trajectories, but also due to the introduction of new manufacturing techniques. This has recently enabled more freedom in the surface design and structure. Soccer balls are either kicked with spin, where the Magnus Effect dominates and causes the ball to swerve, or without spin, where another, more erratic effect occurs. The general asymmetry of the surface geometry of a ball in a random orientation relative to its flight direction causes asymmetric separation of the air and an asymmetric wake. Consequently, the ball experiences a side or lift force. The force continues to alter in a seemingly erratic manner throughout the flight and can result in a trajectory that curves in several different directions. As shown in Sect. 5, this trajectory depends on the surface geometry and roundness of the ball as well as on its initial launch orientation. New manufacturing techniques that enable new surface pattern designs could hence be used either to suppress or enhance this effect as desired. This erratic effect only occurs when the ball is launched very fast with little or no spin, that is, when it is kicked very centrally. If the ball is kicked offcentre it will be launched with spin but with a lower velocity, because less energy is transferred to it from the foot. In general, for spin-type kicks, soccer balls are launched with a combination of spin about the horizontal and vertical axes perpendicular to the flight direction. They can be made to swerve, dip and rise through the air. It has been shown in wind tunnel tests of a scale model soccer ball that the seams on a soccer ball actually

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Fig. 6 CM vs. Sp for a scale model soccer ball [7]

stabilise its behaviour when spinning. Additionally the Magnus Force Coefficient (CM ) perpendicular to the spin axis increases with the spin parameter of the ball, as shown in Fig. 6 [7]. Trajectory measurements of a range of different soccer balls have shown that increasing the number of seams from 14 to 36 increases the CM for a given Sp as shown in Fig. 7, which can alter the final position of the ball by up to 1 m [9] when kicked from approximately 30 m from goal.

4 Measurement and Analysis Methods The measurement and analysis techniques for the study of sports ball aerodynamics can be basically split up into three different parts: CFD, wind tunnel testing and trajectory methods. CFD analysis is especially effective for computing the fully turbulent drag coefficient (CD ) and the quasi-steady lift (CL ) and side force (CS ) coefficients acting on a ball as well as for steady-state flow visualisation and the analysis of altered and rotated geometry (see Fig. 8a). However, the technique is difficult to set up and validate, and a study is largely limited by the available software and the computer’s processing speed, disk space and memory. Phenomena such as vortex shedding, the effects of spin and boundary layer transition have not presently been studied in great detail. Wind tunnel tests are especially effective for measuring CD at a wide range of Re, unsteady CL and CS , for identifying steady-state flow regimes and for the analysis of unsteady wake behaviour (e.g. using flow visualisation – see Fig. 8b). However, full-size wind tunnel tests can be very time consuming, partly due to

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Fig. 7 CM vs. Sp for a range of soccer balls [8]

considerations such as a suitable mounting technique and the suppression of unwanted vibrations of the ball. Trajectory analysis can be especially effective for measuring CL at a range of Sp and for the comparison between different balls in a controlled environment (see Fig. 8c). However, there are a number of approximation and measurement errors that can create a fairly large scatter in the results. Player testing can be especially effective for measuring typical launch conditions and for comparison of applied spin between balls. Problems with player testing include the difficulty of controlling the environment and the consistency of the kickers. Additionally, trajectory prediction models can be used to predict and compare trajectories with measured force coefficients, and can be especially effective for the evaluation of the effects of quasi-steady and unsteady CL and CS , the comparison of trajectory shapes and comparisons of ball consistency. As for the majority of engineering problems, CFD studies are most effective when used in conjunction with the experimental tests and field testing.

5 CFD of Soccer Balls: The Nature of Erratic Flight Two computational techniques are used to assess the aerodynamic behaviour of three different soccer balls, including the 2006 World Cup ball (adidas Teamgeist). It is found that ball construction and design can have a significant effect on a ball’s performance, in terms of both the flight and the consistency.

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Fig. 8 (a) Example of CFD flow visualisation, (b) example of wind tunnel tests, (c) example of trajectory measurements

A validated 3D CFD analysis technique for balls that have been scanned with a 3D laser scanner or drawn in CAD, using the commercial code FLUENT, is described. This utilises a surface wrapping meshing method and the ReynoldsAveraged-Navier–Stokes approach with the realizable k- turbulence model. The effects of three different ball geometries on their flight is examined using a trajectory simulation programme. The force coefficients of the balls are compared, and CD is only significantly affected when a row of deep seams is introduced around a circumference of the ball perpendicular to the flow, which causes early separation and increased CD . CL and CS are found to be significantly affected by the orientation of the ball relative to its direction of travel. This means that balls kicked with low amounts of spin could experience quasi-steady lift and side forces and move erratically from side-to-side or up and down through the air. The variation of CD , CL and CS with orientation for the balls is approximated and entered into a modified trajectory simulation program. It is found that certain kick types could cause the ball to move erratically from side-to-side through the air. The erratic nature of this type of kick is found to vary with details of the surface geometry including seam size, panel symmetry, number, frequency and pattern, as well as the velocity and spin applied to the ball by the player. Exploitation of this phenomenon by players and ball designers could have a significant impact on the game.

5.1 Introduction New ball manufacture and design technologies are leading to the production of balls with seam and panel patterns that differ from the conventional stitched ball with 32 hexagonal and pentagonal panels. The aerodynamic effects of such radical changes are not yet known in detail. Relatively few CFD studies have been undertaken on sports balls due to the required computational power, difficulty of meshing and perhaps a previous lack of demand for a highly detailed understanding (e.g. [10–12]).

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There is no consensus yet as to the most appropriate CFD technique for modelling sports ball aerodynamics. This work presents a new validated 3D technique for the CFD analysis of soccer balls using FLUENT [8] and applies it to three different soccer ball designs. The technique is used to help understand the aerodynamics of soccer balls and the effects of surface geometry on their trajectories. The type of kick studied is a low-spin, high-velocity kick that is generally launched at a velocity of 30–35 m s1 , and used commonly for direct freekicks and volleys.

5.2 Establishing the Method This study was limited to the use of the commercial code FLUENT Version 6.2. The computational resources available for this study were those of the University of Sheffields high performance computing server. The system has 2.4 GHz AMD Opteron processors and runs 64-bit Scientific Linux, which is Redhat based. There are 160 processors available for use, 80 of which are in 4-way nodes with 16 GB of main memory coupled by a double speed low latency Myrinet network, and 80 of which are in 2-way nodes with 4 GB of main memory. The validation procedure was central to this study, and more details can be found in Barber [9] and Barber et al. [8]. In this procedure, the geometry was made progressively more complex, and solutions were continuously compared to experimental results for validation purposes. The studies included a 2D smooth sphere, a 3D smooth sphere, the CAD geometry of a scale model soccer ball and a scanned soccer ball. The study culminated in the production of a preferred CFD methodology for the 3D comparative analysis of soccer ball aerodynamics. The technique uses the realizable k- turbulence model, a surface wrapping meshing technique and a hybrid mesh mainly consisting of structured hexahedral elements with a near-wall cell size of 0.01 mm. The velocity discretisation scheme was second order upwind and the PRESTO! scheme was used for pressure discretisation. PISO pressure-velocity coupling was employed. The solving methods were chosen following considerations such as accuracy of results, speed of convergence, whether convergence could be achieved at all and required grid size and computer memory. The chosen general mesh structure is shown in Fig. 9; the ball was placed 5d from the inlet, 20d from the outlet and 5d from the domain sides. The mesh varied slightly between balls, but in general had about 9 million cells with prism cells growing from the surface and a section of unstructured tetrahedral cells joining them to the outer structured mesh. A refined region in the wake was found to bring no benefit to the results. For each case, the outlet was specified as an outflow, and the inlet defined by the fluid velocity. The surface of the geometry was defined as a no-slip wall, and the far field boundaries of the solution domain were defined as a slip-wall, i.e. a wall with zero shear stress. The CFD technique was found to be capable of predicting average wake structures that compare well with wind tunnel flow visualisation for

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Fig. 9 Mesh structure most suitable for soccer ball analysis

soccer balls. It can be used as an effective tool for comparison between different ball designs. As discussed in more detail by Barber et al. [8], the limitations in computational power did not allow the prediction of laminar–turbulent boundary layer transition. However, previous studies found the flow for the kick-type studied to be fully turbulent. Hence it was deemed valid for this study to assume that the flow was always fully turbulent. The computational limitations also required the use of the steady-state Reynolds-Averages Navier–Stokes (RANS) approach. It was thought that over the range of Re experienced by the flow around a football (106 ), unsteady effects would be of secondary importance, and hence the aerodynamic coefficients would be largely unaffected by this and a steady state flow analysis would provide useful information at relatively low computational cost. Additionally, an error analysis showed that the inherent error in the calculation of CL and CS in the CFD process due to the mesh was 0.01. A 2–3 mm discrepancy in the

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Fig. 10 Balls 1–3 at 0ı

diameter was found for the balls due to inconsistencies in manufacturing, scanning and surfacing processes.

5.3 Soccer Ball Analysis 5.3.1 Set-up Scanned geometries of three different balls were entered into the solver, which was set up at various orientations about the vertical y-axis (rotated at 10ı intervals from 0–90ı) at Re D 1:0  106 . Different orientations were studied in order to analyse the behaviour of balls launched with very low amounts of spin. For the orientations of 0ı , the study was repeated for Re D 1:6  105 and 6:0  105 . The resolution of the mesh was such that each seam contained at least 10 grid points. The balls studied were as follows (defined for each ball at 0ı in Fig. 10):  Ball 1: 32 panels, 20 pentagonal and 12 hexagonal, hand-stitched together in the

traditional manner (e.g. adidas Fevernova).  Ball 2: 14 pre-curved panels, thermally bonded together (e.g. adidas Teamgeist,

World Cup Ball 2006).  Ball 3: altered geometry: central row of seams blanked out, taken from CAD

geometry of a one-third scale model soccer ball.

5.3.2 Drag Results The CD results for Balls 1–3 are shown in Fig. 11 and compared to known data [4, 13]. CD is higher than for the smooth sphere, as expected due to the seams. For

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Fig. 11 CD vs. Re for (a) Balls 1 and 2, (b) Ball 3, compared to known results [4, 13]

Ball 2, CD was under-predicted compared to the wind tunnel results (by about 3– 5% in the critical regime and about 25% in the trans-critical regime). The reasons for this were because the flow was modelled as fully turbulent for the entire Re range, and the effects of the flow interaction between the ball and its supporting device were not deducted from the wind tunnel CD values. The CD values were split into skin friction and pressure drag components, and the skin friction drag was approximately 0.030–0.035 for the lowest Re and 0.020–0.025 for the highest Re (depending on the ball). CD for Ball 2 was approximately 6% smaller than for Ball 1 due to a decrease in both pressure and skin friction drag with the smaller and fewer seams. These trends match the wind tunnel results around the super-critical regime [13]. Flow visualisation (velocity contours taken at a central plane, side view and total pressure contours on the surface) for the Ball 1, Ball 2 and a smooth sphere at Re D 1:0  106 is shown in Fig. 12. The wakes were all similar in size, explaining the fairly similar CD values seen, and compared well with experimental flow visualisation [14]. For the consideration of Ball 3, results from a previous study [15] of the unaltered CAD geometry of a one-third scale model soccer ball were used for comparison purposes. This ball has 32 standard hexagonal and pentagonal panels, with larger,

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Fig. 12 Velocity contours taken at a central plane (m s1 , side view) and total pressure contours (Pa) for Ball 1, Ball 2 and a smooth sphere

Fig. 13 Velocity vectors close to the surface for Ball 3 and the original scale model ball geometry (side view, air flow from left to right)

sharper and more exaggerated seams than a standard ball such as Ball 1. For this ball,the CD values at Re >1:6  105 agreed well with experiment (under-predicted by about 4%, Fig. 11). The CD of Ball 3 was found to be approximately 50% less than the unaltered geometry at Re D 1:0  106 , from 0.15 for Ball 3 to 0.22 its original geometry. The velocity vectors at the first cell on the ball’s surface for Ball 3 and its unaltered, 32-panel version are shown in Fig. 13. The velocity vectors show that the presence of the vertical row of seams induces early separation. For a high-velocity, low-spin kick from approximately 30 m from goal, such an increase was found to result in the ball ending up approximately 0.5 m lower in the goal, which it reaches 0.1 s later and moving approximately 5 m s1 slower.

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0.25 Side force coefficient, Cs

0.2 0.15 0.1 0.05 0 –0.05 0

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CFD: Ball 1

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Fig. 14 Variation of CS with orientation for Balls 1 and 2, Re D 1:0  106

5.3.3 Variation with Orientation Side and lift forces were examined for Ball 1 and Ball 2. They showed significantly different CL and CS behaviour, and this behaviour varied with ball orientation due to the asymmetry of the scanned geometry. CD did not vary with orientation. The predicted variations of CS for Balls 1 and 2 are shown in Fig. 14. These variations can be explained by considering how the position of separation relates to the surface geometry of the balls. AS an example, Fig. 15 shows the oil flow visualisation (from four different views) around Ball 1 at 0ı for Re D 1:0106. This demonstrates the pathlines of particles released from the surface and shows clearly the separation position towards the rear of the ball. Each particle had its own “particle ID” and the particles were coloured according to their ID. For each ball, images were then built up that indicated the position of separation at all points around the ball, and the influence of the seams were seen. Figure 16 shows shear stress contours (Pa) on the rear surface of Ball 1 and Ball 2, along with approximate separation points obtained from the oil flow visualisation, for x deg and x deg as examples. Black lines indicate approximate separation and grey lines indicate separation where the flow is particularly affected by the presence of a seam. The results suggest that seams that are perpendicular, or nearly perpendicular, to the flow had an effect on the position of separation, due to the sudden change in curvature of the surface. The less perpendicular to the flow a seam was (i.e. the more aligned with the flow) the more likely the flow was to continue in its original direction and not be affected largely by the seam. In general, the seams of Ball 1 that were perpendicular (or nearly perpendicular) to the flow seem to have been more likely to hold back the flow and alter its position of separation than the seams of Ball 2. This was probably because they were larger and had more influence on the flow. This meant that the CS varied more often with angle for Ball 1. The seams of Ball 2 only rarely held back the flow and influenced separation, but was sometimes

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Fig. 15 Oil flow visualisation for Ball 1 at 0ı at Re D 1:0  106 (coloured by particle ID)

Fig. 16 Shear stress contours (Pa) on the rear of Balls 1 and 2, with separation points marked in black (indicating separation) and red (indicating separation at a seam)

swayed heavily by the presence of long, perpendicular seams, e.g. at 40ı and 50ı (Fig. 16). The magnitude of CS was correspondingly lower for Ball 2. The significant “stepped” pattern in CS occurred when particular seams held back the flow in a certain position, and continued to do so as the ball was rotated. Eventually it reached a point where the seam suddenly lost its influence because it was too far away from the natural position of separation (i.e. where separation would have occurred without the presence of the seams), and the separation point then retreated to the previous perpendicular seam. This was particularly evident between 50ı and 80ı for Ball 1. The pattern occurred less frequently for Ball 2 because a seam was less likely to hold back the flow near separation and remain holding it back as the ball rotated. The variation in CS was less for Ball 2 in general; however the occasional sudden peak was seen. The peak at 40ı was explained by the strong

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Fig. 17 Approximate trajectory of Cristiano Ronaldo’s erratic free-kick

bias of the flow to separate near the long, vertical seam on one side. At 50ı the separation region jumped back to the previous vertical seam. The results suggested that the exact positioning of the balls in the mesh had as much effect on the flow as their seam arrangement and hence the trajectories were thought to be very sensitive to orientation.

5.3.4 Effects on Trajectory Balls launched with high velocity and low spin (