Computational Fluid Dynamics for Sport Simulation (Lecture Notes in Computational Science and Engineering)

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

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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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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

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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 :

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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 

Z 

II

ik

.ıki /dx D

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 , on coupling condition enforces the continuity of the two velocity fields, u and @d @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



: <

Fig. 12 Results of the CFD transition analysis: transition location (right) and wall shear stress (right) on appendages

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Winglet optimization is another active subject of investigation in the America’s Cup design world. Indeed, after more than 25 years since their early adoption on Australia II in 1983, winglets are still placed and sized in very different ways by IACC yachts designer. Traditionally, the fluid-dynamic mechanism underlying the benefits observed when winglets are adopted is the one qualitatively displayed in Fig. 13. Comparing appendage configurations with and without winglets shows that the latter have a strong impact on the vorticity evolution in the appendage wake. Indeed, as in aerospace applications, this vorticity reduction can be related to a reduction of the induced drag on keel and bulb, which is considered to be the most important beneficial effect induced by the presence of winglets. A detailed analysis of the flowfield around different winglet configurations has highlighted a second potential beneficial effect due to the winglets. Indeed, it has been noted that the presence of winglets not only reduces the pressure drag on keel (through the reduction of its induced component) but has also a strong effect on the pressure drag on the bulb. This can be correlated to the reduction of crossflow due to the winglets effect on lift-induced recirculation. In Fig. 14, we present for different winglet configurations, the distribution of local circulation along the bulb,

Circularity

Fig. 13 Cut plane of the vorticity field in the appendage wake: without (left) and with (right) winglets 80 70 60 50 40 30 20 10 0 –10

No Winglets Front Winglets Tapered Front Winglets

Fig. 14 Local circulation along bulb for different winglet configurations

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I

defined as .x/ D

!  i ds; l.x/

where ! is the flow vorticity, i the unit vector in longitudinal (x) direction and l.x/ the local bulb section. The circulation along bulb displays an almost constant value from the nose to the keel leading edge; then the high side force generated on the keel induces a strong increase in the flow vorticity which can be reduced by the winglets. A suitable winglets design taking this effect into account has been adopted in order to maximize the amount of bulb drag reduction associated to this phenomenon. This analysis seems to suggest that a front winglet position may be preferred to an aft position, since the latter could hardly exploit this bulb circulation reduction.

4.2 Free-Surface Simulations The wave component of drag can be quite significant on an America’s Cup hull, as much as the 60% of the total resistance at 10 knots of boat speed. An accurate determination of this component is important when comparing the performances of two hull designs. Local shape modifications require accurate analysis tools to correctly predict the performance differences deriving from these subtle changes. In a typical hull design process, designers explore the performance of a family of hull shapes through a fast free-surface potential solver [20] to determine a set of candidates to be tested in the towing tank. Numerical simulations based on RANS models are integrated into the design process in different ways: on one hand, they can be used to decrease the number of candidate shapes for which models are to be constructed and tested in the towing tank; moreover, they can be used to evaluate the free-surface flow in conditions where codes based on the panel method are unable to resolve critical differences due to viscous effects. The numerical scheme presented here for the prediction of boat dynamics can be a powerful tool in America’s Cup yacht design. Many potential applications are being explored and range from the dynamic response in waves to maneuvering. We expect this kind of numerical investigation could become the golden standard in the coming years. Thus far, in the context of the America’s Cup design, this approach has been used to reproduce towing tank experiments. Two IACC hull shapes, that will be referred to as Hull 1 and Hull 2, have been considered. The two hulls have different bow designs and towing tank experiments have been carried out to estimate drag and sink at different boat speeds. Numerical simulations have been carried out with just the sink degree of freedom activated, since in the towing tank the trim, as well as the other degrees of freedom, were fixed. In Fig. 15 we show the time history of the sink value starting from the initial sink position, correspondent to the hydrostatic

Numerical Models and Simulations in Sailing Yacht Design

0.04

25

Hull 1 Hull 2

0.035

Sink [m]

0.03 0.025 0.02 0.015 0.01 0.005 0 0

2

4

6 8 Time [s]

10

12

14

Fig. 15 Time evolution of sink for the two hulls

Fig. 16 Wave patterns for Hull 1 (left) and Hull 2 (right) with a boat speed of 10 kts

equilibrium, and evolving through a damped oscillation towards the hydrodynamic equilibrium. The wave pattern around the two hulls are displayed in Fig. 16. The numerical results show a good agreement in terms of forces and sink movement. A comparison between the total drag on Hull 1 obtained with the numerical simulations and the towing tank measurements at different boat speed is given in Fig. 17, left. The error is consistently lower than 2% for all boat speeds. A similar comparison for the sink values at different boat speeds is presented in Fig. 17, right. Again, we see a good correlation between numerical and experimental results. In the yacht design context, a numerical tool able to accurately predict forces and attitudes is of utmost importance since it may reduce the need of carrying out expensive experimental sessions in towing tank facilities. In this respect, it is crucial for the numerical results to predict correct trends and variations between different configurations, even more than to give precise estimation of absolute values.

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Sink

Drag

CFD Exp.

3

4

5

6

7 8 BSP [kts]

9

10

11

3

4

5

6

7 8 BSP [kts]

9

10

11

Fig. 17 Comparison between towing tank measurements and numerical predictions for Hull 1: drag vs. boat speed (left) and sink vs. boat speed (right) Table 1 Comparison of the drag deltas [N] between Hull 2 and Hull 1 obtained with towing tank measurements and numerical simulations BSP [kts] 4 7 10

CFD 0:62 0:79 4:15

Exp. 0:32 0:41 2:87

Fig. 18 Free-surface elevation around an appended hull in upwind configuration

For the case at hand, a consistent rating between the two hull designs considered can be obtained from the numerical simulations as well as for the towing tank data, as shown in Table 1 where the drag deltas between Hull 2 and Hull 1 at different boat speeds are given. We have also considered the free-surface flow around appended hulls. These analyses are useful to understand the interaction between the load distribution on appendages and the water surface dynamics. This is particularly relevant in upwind configurations, where the boat is heeled and the suction side of the keel has a strong impact on the local wave shape (see Fig. 18). Although the simulations in calm water are already useful to understand some of the dynamic features of a hull and its natural frequencies with respect to each

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Fig. 19 Contours of free-surface height at different time instant during one wave period

different degree of freedom, the potential benefits coming from the introduction of the boat motion model are fully exploited when the model is adopted for the analysis of the hull response in wavy sea. To this purpose, an incoming wave model based on the fifth order Stokes wave expansion [8] has been implemented and tested on the Series 60 benchmark hull. The wave model imposes a time dependent wave elevation and the correspondent orbital velocity at the inflow boundary of the domain. We have considered a wave amplitude D 0:05L where L is the boat length and different wave frequencies correspondent to incoming wave length of values in the range Œ0:5L; 4L . First a steady state simulation of the flow around the boat with no active degrees of freedom and without incoming wave is performed and the solution is then used as initial condition for a time dependent simulation with incoming waves and the boat free to sink and trim. After a short transient the boat dynamics reaches an asymptotic periodic behavior governed by the incoming wave characteristics. The contours of the free-surface height at different time instants during one period is shown in Fig. 19. We can appreciate the interaction between the incoming waves and the hull-generated wave pattern. The dynamic response of the boat to the periodic forcing generated by the incoming waves is presented in Fig. 20 where the sink amplitude normalized to the incoming wave amplitude,  D z=a, is plotted against the incoming wave frequency !. The figure clearly shows the presence of a resonance peak in correspondence to the natural sinking frequency of the boat. For a detailed description of the implementation of the incoming waves into the flow solver and a complete presentation of the numerical results, we refer to [17].

4.3 Fluid–Structure Interaction for Downwind Sails Finding the optimal trimming of all the cables and sheets governing the sails represents a challenging task and a very useful goal to speed up the design process, and this is true because sails are always designed so that their best performance (in terms of driving force, for instance) can be reached within some given wind and

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Fig. 20 Dynamical response of the boat: sink amplitude as a function of the incoming wave frequency

Fig. 21 Final sail shapes for three different gennaker sheet (GS) trimmings (steady simulations)

trimming ranges. For this goal, an extensive campaign of simulations predicting the sails behavior before they are built can provide the designers a number of extremely useful hints for considerable enhancements, with evident benefits in terms of money and time saving. From the practical point of view, most part of the simulations have been carried on by changing the length of the main and the gennaker sheets, two cables which control the angle of attack of the two sails with respect to the incident wind direction. In Fig. 21 we report a sequence of test cases where the air velocity vector field is plot on a 20 m height plane for three different gennaker configurations. From left to right, the gennaker sheet (GS ) is eased out from GS D 2:0 m to GS D C0:5 m with respect to the design configuration which represents the reference for any adjustment and corresponds to GS D 0:0 m. It can be noticed that the most pulled-in case (GS D 2:0 m) shows a large separated region which extends from the leading to the trailing edge of the gennaker and this region reduces as the GS is eased out. The last case (GS D C0:5 m) shows an attached flow which follows the sail surfaces without any visible recirculation. In Fig. 22 the driving force is plot as a function of the gennaker sheet trimming, and the simulation confirms

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Fx [N]

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Fig. 22 Driving force vs. gennaker trimming (steady simulations)

Fig. 23 Fluid flow and driving force (in top-right corners) at two time instants, T D 6:5 s (left) and T D 18:5 s (right), for an unsteady fluid–structure simulation. The curves plotted on the righttop squares represent the evolution of the forces as functions of time and the red spots highlight the values corresponding to the two time instants

that the design setting (GS D 0:0 m) is the one which gives the larger driving force, hence the best performance. As introduced in Sect. 2.3.2, unsteady simulations allow the investigation of the sail dynamics (the real evolution of the sail shape over time) and the stability properties of the structure. Moreover, the force history on sails, the frequencies of the structural vibrations and any periodic oscillation can also be easily monitored. For all these reasons, when the designers need a deeper investigation of the sail behavior, the unsteady approach can become extremely useful and an important source of information which can not be retrieved by simpler steady runs. In Fig. 23 we compare the same sail simulation at different times, highlighting with a red solid line the left boundary of the big wake behind the gennaker.

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At T D 6:5 s it can be noticed that a small recirculation bubble appears close to the gennaker leading edge and the driving force on sails reaches its maximum value (see plot on the top-right corner). At T D 18:5 s the bubble has clearly enlarged, pushing away the left wake boundary so that the incident flow is now less deflected by the presence of the gennaker. This is clear after comparing the new wake boundary with the dashed line that refers to the previous case and is kept on the second figure as reference. This reduced flow deflection leads to a smaller driving force acting on sails (because of the third Newton’s law). Hence, unsteady simulations allow a truly physical resolution of the fluid flow over the time and highlight the effects of the dynamics of the turbulent wakes on many design parameters, like the evolution of the driving force or the sail stability properties under the action of unsteady loads.

5 Conclusions In this presentation, we have presented some of the most recent results on numerical fluid-dynamic modelling obtained in the framework of the collaboration between the Ecole Polytechnique F´ed´erale de Lausanne and the Alinghi Team in preparation of the 32nd edition of the America’s Cup. We have highlighted the importance that CFD analysis is achieving in the design process of a racing yacht, devoting a particular attention to those modelling techniques that represent a step forward in this field. Among them, we have presented and discussed through several numerical examples the recent advances in transition modelling and its coupling with standard eddy-viscosity turbulence models. We have shown at which extent accurate predictions on transition location can play a key role in the optimization of the yacht appendages. Finally, the coupling of a RANS solver with a 6-DOF dynamic model of the boat has been presented together with recent results of free-surface simulations of boat dynamics in calm and wavy water. Acknowledgements The results presented in this paper have been obtained in collaboration with the Alinghi Team, in the framework of the project “Development of high performance technology applied to Alinghi’s America’s Cup 2007 campaign. Part A: Computational fluid dynamics and flow diagnostics”, supported by the Swiss Confederation innovation promotion agency (CTI) through Project no. 6971.2. The members of the Alinghi Design Team and, more particularly, Grant Simmer, Michael Richelsen and Jim Burgener, are gratefully acknowledged for their invaluable support. The contribution from Geoffrey Cowles and Mark Sawley during the 31st edition of the America’s Cup campaign is also acknowledged.

References 1. R. Azcueta. Computation of Turbulent Free-Surface Flows Around Ships and Floating Bodies. Phd thesis, 2001.

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2. R. Azcueta. RANSE Simulations for Sailing Yachts Including Dynamic Sinkage and Trim and Unsteady Motions in Waves. In High Performance Yacht Design Conference, pages 13–20, Auckland, 2002. 3. C. W. Boppe. Elements of Hull Optimization and Integration for Stars and Stripes. In Proceedings of the Symposium on Hydrodynamic Performance Enhancement for Marine Applications, 1988. 4. M. Caponnetto, A. Castelli, R. Dupont, B. Bonjour, P.-L. Mathey, S. Sanchi, and M. L. Sawley. Sailing Yacht Design Using Advanced Numerical Flow Techniques. In Proceedings of the 14th Chesapeake Sailing Yacht Symposium, Annapolis, USA, 1999. 5. A. Claughton. Developments in the IMS VPP Formulations. In Proceedings of the 14th Chesapeake Sailing Yacht Symposium, Anapolis, USA, 1999. 6. G. W. Cowles, N. Parolini, and M. L. Sawley. Numerical Simulation using RANS-based Tools for America’s Cup design. In Proceedings of the 16th Chesapeake Sailing Yacht Symposium, Annapolis, USA, 2003. 7. F. Jr. DeBord, J. Reichel, B. Rosen, and C. Fassardi. Design Optimization for the International America’s Cup Class. In Transactions of the 2002 SNAME Annual Meeting, Boston, 2002. 8. J. D. Fenton. A Fifth-Order Stokes Theory for Steadywaves. J. Waterway Port Coast. Ocean Eng., 111:216–234, 1985. 9. C. W. Hirt and B. D. Nichols. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comp. Phys., 39:201–225, 1981. 10. P. Jones and R. Korpus. America’s Cup class Yacht Design Using Viscous Flow CFD. In Proceedings of the 16th Chesapeake Sailing Yacht Symposium, Annapolis,USA, 2001. 11. J. E. Kerwin. A Velocity Prediction Program for Ocean Racing Yachts. Technical Report 78– 11, 1978. MIT Pratt Project Report. 12. M. Lombardi. Simulazione Numerica della Dinamica di uno Scafo. Master thesis, Politecnico di Milano, 2006. 13. F. R. Menter. Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA J., 32(8):1598–1605, 1994. 14. F. R. Menter, R. Langtry, S. Volker, and P. G. Huang. Transition Modelling for General Purpose CFD Codes. In ERCOFTAC Int. Symp. Engineering Turbulence Modelling and Measurements, 2005. 15. N. Parolini and A. Quarteroni. Mathematical models and numerical simulations for the America’s Cup. Comput. Methods Appl. Mech. Eng., 194(9–11):1001–1026, 2005. 16. N. Parolini and A. Quarteroni. Modelling and numerical simulation for yacht design. In 26th Symposium on Naval Hydrodynamics, Arlington, 2007. Strategic Analysis Inc. 17. S. Piazza. Simulazioni Numeriche della Dinamica di uno scafo in Mare Ondoso. Master thesis, Politecnico di Milano, 2007. 18. A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations, volume 23 of Springer Series in Computational Mathematics. Springer, Berlin, 1994. 19. P. M. Richelsen. Beregning af vindp˚avirkede b˚adsejl. PhD thesis, Technical University of Denmark, 1987 (in Danish). 20. B. S. Rosen, J. P. Laiosa, W. H. Davis, and D. Stavetski. Splash Free-Surface Code Methodology for Hydrodynamic Design and Analysis of IACC Yachts. In Proceedings of the 11th Chesapeake Sailing Yacht Symposium, Anapolis, USA, 1993. 21. G. E. Schneider and M. J. Raw. Control Volume Finite-Element Method for Heat Transfer and Fluidflow Using Colocated Variables. Num. Heat Trans., 11:363–400, 1987. 22. P. van Oossanen and P.N. Joubert. The Development of the Winged Keel for Twelve-Metre Yachts. J. Fluid Mech., 173:55–71, 1986. 23. D. Yoon, H.-J. Costantini, V. Michaud, H. G. Limberger, J.-A. M˚anson, R. P. Salath´e, C.-G. Kima, and C.-S. Honga. In-situ simultaneous strain and temperature measurement of adaptive composite materials using a fiber Bragg grating based sensor. In Proc. of SPIE, San Diego, USA, 2005.

Swimming Simulation: A New Tool for Swimming Research and Practical Applications 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

Abstract This chapter covers topics in swimming simulation from a computational fluid dynamics perspective. This perspective means emphasis on the fluid mechanics and CFD methodology applied in swimming research. We concentrated on numerical simulation results, considering the scientific simulation point-of-view and especially the practical implications with swimmers.

1 Introduction Swimming is one of the major athletic sports and many efforts are being made to establish new records in all events. To swim faster, thrust should be maximized and drag should be minimized. These aims are difficult to achieve because swimmers surge, heave, roll and pitch during every stroke cycle. In addition, measurements of human forces and mechanical power are difficult due to the restrictions of measuring devices and the specificity of aquatic environment. Thus, human swimming evaluation is one of the most complex but outstanding and interesting topics in sport biomechanics. Over the past decades, research in swimming biomechanics has evolved from the study of swimmer’s kinematics to a flow dynamics approach, following the line of research from the experimental biology [1, 2]. Significant efforts have been made to understand swimming mechanics on a deeper basis. In the past, most of the studies involved experimental data, nowadays the numerical solutions can give new insights about swimming science. Computational fluid dynamics (CFD) methodology is one of the different methods that have been applied in swimming research to observe and understand water movements around the human body and its application to improve swimming technique and/or swimming equipments and therefore, swimming performance. One recent example is the D.A. Marinho (B) Universidade da Beira Interior. Departamento de Ciˆencias do Desporto/CIDESD, Rua Marquˆes ´ D’Avila e Bolama, 6201–001 Covilh˜a, Portugal 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 2, 

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cooperation between Speedo swimwear manufacturer and Fluent CFD software provider, in the process of improving the swimwear’s hydrodynamic characteristics. The CFD software was incorporated into Speedo’s design process to evaluate the drag and fluid flow characteristics around the male and female swimmers for various flow conditions. This work allowed Speedo to simulate the flow around the virtual swimmer body, thus making the swimsuits as hydrodynamic as possible [3]. Fastskin FSII and recently the LZR suits are the most visible examples of the application of CFD in swimming research and its influence in the swimming performance. However, other issues related to swimming science, besides the swimwear, were and are also being solved with this methodology. Therefore, the use of CFD can be considered as a new step forward to the understanding of swimming mechanisms and seems to be an interesting approach to the swimming research. In this sense, the main purpose of this book chapter is to present the basis of this methodology and its applications in swimming research. This chapter is divided in seven parts. In the first part, we introduce the issue and the main aims of the paper. In the second, we briefly detail some areas of application of the CFD methodology. In the third part, we show some applications of CFD in biological systems. In the fourth part, some applications of CFD to human beings are presented. In the fifth part, it can be observed the contribution of the different studies in swimming using CFD, where we can analyse some practical application of the CFD technology in swimming research. In the final parts, we present some ideas to future studies in swimming using CFD and the main conclusions.

2 Areas of Application of Computational Fluid Dynamics CFD has a wide field of applications, being used in biomechanical studies applied to several research fields, such as industry, biology, medicine and sports (e.g., Boulding et al. [4–7]). The broad physical modelling capabilities of CFD have been applied to industrial applications ranging from air flow over an aircraft wing to combustion in a furnace, from bubble columns to glass production, from blood flow to semiconductor manufacturing, from clean room design to wastewater treatment plants. The ability of the software to model in-cylinder engines, aeroacoustics, turbomachinery, and multiphase systems has served to broaden its reach. Today, thousands of companies throughout the world benefit from this important engineering design and analysis tool. Its extensive range of multiphysics capabilities makes it an important and interesting tool in engineering studies. Recently, medical applications were also described by this method (e.g., Berthier et al. [8, 9]). Berthier et al. [8] analyzed the blood flow patterns in a coronary vessel digital model whereas Ruiz et al. [9] simulated the complex three-dimensional airflow pattern in the human nasal passageways. In biology, the CFD models started to be used in the middle of the 90s in the flying study of insects as well as in the inquiring of the aerodynamic and hydrodynamic

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forces involved in the propulsion and energy conservation through the generation of organized vortex systems in animals displacing by body undulation [10–12]. In sports scope, the main results suggested that a CFD analysis could provide useful information about performance. Indeed, this methodology has produced significant improvements in equipment design and technique prescription in areas such as sailing performance [13], Formula 1 racing [14] and winter sports [6]. CFD has been applied to swimming in order to understand its relationships with performance mainly by three research groups. One coordinated by Barry Bixler from Honeywell Aerospace (USA), another coordinated by Ant´onio Jos´e Silva from the Research Centre in Sport, Health and Human Development (Portugal), and another coordinated by Bruce Mason from the Australian Institute of Sport (Australia). The numerical techniques have been applied to the analysis of the propulsive forces generated by the propelling segments [15, 16] and to the analysis of the hydrodynamic drag forces resisting forward motion [17, 18].

3 Computational Fluid Dynamics Applied to Biological Systems Fluid dynamic phenomena in animal locomotion are complicated because biological fluid dynamics involves the interaction of elastic or even flexible living issues with surrounding viscous fluid [19]. The biological fluid dynamic phenomena are, in general, characterized by large-scale vortex structure due to the highly unsteady motions and the complex and variable geometry of the object in swimming and flying.

3.1 Computational Fluid Dynamics Applied to Birds/Insects The flight of insects has fascinated physicists and biologists for many years. On one hand, insects owe much of their amazing evolutionary success to flight. One the other, their flight seems improbable using standard aerodynamic theory [20]. The small size, high stroke frequency and peculiar reciprocal flapping motion of insects have combined to prevent simple explanations of flight aerodynamics. Nevertheless, recent developments in high-speed videography and tools for computational and mechanical modelling have allowed researchers to make progresses in the understanding of insect flight. These CFD models, combined with modern flow visualization techniques, have revealed that the fluid dynamic phenomena underlying flapping flight are different from those of non-flapping, two-dimensional wings on which most models are based [20]. In fact, even at high angles of attack, a prominent leading edge vortex remains stably attached on the insect wing and does not shed into an unsteady wake thus enhances the forces generated by the wing, enabling insects to hover and maneuver.

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With recent advances in computational methods, many researchers have begun exploring numerical methods to resolve the insect flight problem [12, 21–26]. Although ultimately these techniques are more rigorous than simplified analytical solutions, they require large computational resources and are not as easily applied to large comparative data sets [20]. Moreover, CFD simulations rely critically on empirical data both for validation and relevant kinematic input. However, several studies have recently emerged that have led to some important CFD models of insect flight. Liu and co-workers [12, 22] using the hawkmoth Manduca as a model, were the first to attempt a full Navier-Stokes simulation by a finite volume approach. In addition to confirm the smoke streak patterns observed on both real and dynamically scaled model insects [21], this study added finer detail to the flow structure and predicted the time course of the aerodynamic forces resulting from these flow patterns. Furthermore, Dickinson et al. [23] used a computational approach to model Drosophila Melanogaster flight for which force records exist based on a dynamically scaled model. Although roughly matching experimental results, these methods have added a wealth of qualitative detail to the empirical measurements [27], and even provided alternative explanations for experimental results [28]. Despite the importance of considering the three-dimensional effects, comparisons of experiments and simulations in two-dimensions have also provided important insight. For instance, the simulations of Hamdani and Sun [25] matched complex features of prior experimental results with two-dimensional airflows at low Reynolds number [29]. In fact, two-dimensional CFD models have also been used to address feasibility issues [20]. Wang [24] reported that the force dynamics of twodimensional wings, although not stabilized by three-dimensional effects, might still be sufficient to explain the enhanced lift coefficient measured in insects. Further, Wang [26], attempting to investigate force generation and energy cost of hovering flight using different combination of lift and drag, studied a family of wing motion parameterized by the inclined angle of the stroke plane. Wang [26] found that the lift-to-drag ratio was no longer a measure of efficiency, except in the case of horizontal stroke plane. In addition, because the flow is highly stalled, lift and drag were of comparable magnitude, and the aerodynamic efficiency was roughly the same up to an inclined angle about 60ı . Based on these data, Wang [26] suggested a strategy for improving efficiency of normal hovering, and a unifying view of different wing motions employed by insects. Interestingly, some swimming researchers suggest a link between swimmers propulsion actions and insects or birds wings actions [30, 31]. Colwin [30] firstly introduced the concept of propulsion through vortex generation in human swimming, based upon the mechanism of flapping wings fly, and Toussaint et al. [31] suggested that, as insects and birds wing’s do, swimmers also use arm rotation that could lead to the establishment of a proximal-distal pressure gradient, which would induce significant axial flow along the arm toward the hand. It was observed that: (a) the flow during insweep and part of the outsweep was highly unsteady; (b) the arm movements were largely rotational; (c) a clear axial flow component, not in the direction of the arm movement, was observed during insweep and outsweep and;

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(d) both the V-shaped “contracting” arrangement of the tufts during the outsweep, and pressure recordings, point to a pressure gradient along the direction of the arm during the outsweep, as predicted on theoretical grounds [31].

3.2 Computational Fluid Dynamics Applied to Fishes Most aquatic animals use the jet-stream propulsion in a form of propagating a transverse wave along the body from head to tail [19]. Physics of fluids around fishes swimming is often of a dynamic vortex structure as their fins usually perform periodically oscillating motions. Liu et al. [10] studied the hydrodynamics and undulating propulsion of tadpoles using a two-dimensional CFD modelling method. The CFD analysis showed that the kinematics of tadpoles is specifically matched to their special shape and produces a jet-stream propulsion with high propulsive efficiency, as high as that achieved by teleost fishes. The authors reported as well that the shapes and kinematics of tadpoles appeared to be specially adapted to the requirement of these organisms to transform into frogs. Liu et al. [11] extended their two-dimensional modelling of tadpole swimming to more realistic three-dimensional situation. Essentially they asked how the threedimensional effects of unsteady undulatory hydrodynamics by swimming tadpoles affected their locomotion performance. Within this study the unsteady flow generated by an undulating vertebrate has been modelled in three dimensions for the first time. This study demonstrated the feasibility of using three-dimensional CFD methods to model the locomotion of undulatory organisms. Tadpoles are unusual among vertebrates in having a globose body with a laterally compressed tail abruptly appended to it. Compared with most teleost fishes, tadpoles swim awkwardly, with waves of relatively high amplitude at both the snout and tail tip. The authors confirmed results from the previous two-dimensional study, which suggested that the characteristic shape of tadpoles was closely matched to their unusual kinematics. Specifically, the three-dimensional results revealed that the shape and kinematics of tadpoles collectively produce a small ‘dead water’ zone between the head-body and tail during swimming precisely where tadpoles can and do grow hind limbs without those limbs obstructing flow. In addition, Liu et al. [11] showed that three-dimensional hydrodynamic effects (cross flows) were largely constrained to a small region along the edge of the tail fin. Although this three-dimensional study confirmed most of the results of the two-dimensional study, it showed that propulsive efficiency for tadpoles was lower than predicted from a two-dimensional analysis. This low efficiency was not, however, a result of the high-amplitude undulations of the tadpole. This was demonstrated by forcing the ‘virtual’ tadpole to swim with fish-like kinematics, i.e. with lower-amplitude propulsive waves. That particular simulation yielded a much lower efficiency, confirming that the large-amplitude lateral oscillations of the tadpole provide positive thrust.

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Fishes CFD data, as reported for the insects and birds condition, can give in a near future some insights or raise questions about propulsive and drag phenomena during non-steady flow with human locomotion in aquatic environment. Especially topics such as the undulatory motion and its relationship with human body undulatory motion in some swim strokes, such as Butterfly stroke, as well as to the kick action in Front Crawl, Backstroke and Butterfly stroke.

4 Computational Fluid Dynamics Applied to Human Beings 4.1 Terrestrial Locomotion In the literature there are not many works that applied CFD to human terrestrial locomotion. However, CFD has been recently used in high-performance sports, such as car racing and motorcycling [32]. One of the reasons for the relatively slow start of the application of CFD in this scope is the enormous complexity of the flow conditions – non-stationary flows, high level of turbulence and complex body shapes, requiring the use of very powerful computational facilities and advanced CFD codes [6]. Dabnichki and Avital [6] focused on the influence of the position of crew members on aerodynamics performance of two-man bobsleigh. The authors studied female crews because they used sleds built for males and thus there is a bigger gap between the crew and the side walls. The position of the brakewoman’s body in terms of upper body inclination and the distance between the cavity and the athlete were studied through computational means. Dabnichki and Avital [6] showed that crew members did influence the drag level significantly and suggested that internal modifications can be introduced to reduce the overall resistance drag. Nevertheless, some experience and background knowledge of human terrestrial locomotion can be useful in a near future for aquatic locomotion. In both environments, powerful computational facilities and advanced CFD codes will be useful for a better understanding of human locomotion for a wide variety of tasks.

4.2 Aquatic Locomotion Regarding aquatic locomotion, CFD has been applied in swimming attempting to understand deeply the biomechanical basis underlying swimming locomotion. Several studies have been conducted willing to analyze the propulsive forces produced by the propelling segments (e.g., Bixler and Riewald [16, 33]) and the drag force resisting forward motion (e.g., Bixler et al. [18, 34]). To our knowledge, scientific publications concerning the application of CFD methodology in aquatic and nautical activities is almost restricted to swimming. However, CFD for naval applications has

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along standing tradition, not to mention that nowadays the design of competition sail boats often rely on intensive CFD simulations. Thus, it would be interesting to apply this methodology in other fields such sailing, windsurfing, surfing, canoeing and rowing, not only in the analysis of equipment design [13] but also to relate different displacing strategies with performance. In the same way, CFD can also provide new highlights about aquatic activities related to health (e.g., head-out aquatic exercises or water-aerobics) and muscle-skeletal injuries rehabilitation in water (e.g., hydrotherapy). In the following chapter, the application of CFD into competitive swimming will be discussed deeply.

5 Computational Fluid Dynamics Applied to Competitive Swimming 5.1 Experimental vs. Numerical Data CFD analysis in swimming has addressed two main topics of interest: (a) the propulsive force generated by the propelling segments and; (b) the drag forces resisting forward motion, since the interaction between both forces will influence the swimmer’s speed. Some authors attempted to compare the numerical data with experimental data available in previous researches about propulsion and drag. However, not always this is an easy goal because the models included in the CFD simulations are not the same as used in experimental measurements. Bixler et al. [34] tried to overcome this problem and studied the accuracy of CFD analysis of the passive drag of a male swimmer. The aim of this study was to build an accurate computer-based model to study the water flow and drag force characteristics around and acting on the human body while in a submerged streamlined position. Comparisons of total drag force were performed between a real swimmer, a digital CFD model of this same swimmer and a real mannequin based on the digital model. Drag forces were determined for velocities representative of the ones presented in elite competition during the underwater gliding (i.e., between 1.50 and 2:25 m s1 ). Drag force measurements in the real swimmer and in the mannequin were performed using a flume test with a load cell interface. Bixler et al. [34] found drag forces determined from the digital model using the CFD approach to be within 4% of the values assessed experimentally for the mannequin, although the mannequin drag was found to be 18% less than the real swimmer drag (Fig. 1). In fact, the Bixler et al. [34] study has reinforced the idea of the validity and accuracy of CFD in swimming research. This study also showed that the drag of the real swimmer is quite high compared to the model due to little body movements during the gliding position. Another difference between the swimmer and the model is that the swimmer’s skin is flexible while the mannequin’s skin is rigid.

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Fig. 1 Hydrodynamic drag force of the swimmer, the digital Computational fluid dynamics (CFD) model and the mannequin. Adapted from Bixler et al. [34]

Other authors used CFD in swimming research, and compared their results with experimental data available in the literature. Bixler and Riewald [33] and Silva et al. [35] analysed the swimmer’s hand and arm in steady flow conditions using CFD. Drag and lift coefficients computed for the hand and arm fitted well with steady-state coefficients determined experimentally by other researchers [36–39]. For instance, Wood [36] found drag coefficient (CD) values of 0.30 and 1.10 and lift coefficient (CL) values of 0.10 and 0.15; while Silva et al. [35] found CD values of 0.27 and 1.16 and CL values of 0.15 and 0.02 at angles of attack of 0ı and 90ı , respectively. Although the comparison is satisfactory, the differences between experimental and numerical data could be the result of wave and ventilation drag caused by the arm piercing the free water surface in the towing tank experiments [33, 38]. Lyttle and Keys [17] aimed to compare two different dynamic kicking techniques using CFD and needed to validate the model to show the compatibility with actual testing results. Due to the unavailability of empirical testing to accurately measure active drag throughout an underwater kick cycle, the model was validated using steady-state tests. Repeated streamlined glide towing trials showed that the CFD model results were within two standard deviations of the mean empirical passive drag for the subject, thus indicating that CFD predicted results were of sufficient accuracy. Gardano and Dabnichki [40] showed close correspondence between the CFD trends and experimental data measured in a low speed wind tunnel in quasi-static approach using a three-dimensional model of a swimmer arm. Vilas-Boas et al. [41] compared the passive drag values in the two gliding positions assumed during breaststroke starts and turns, calculated through inverse dynamics based upon the velocity to time gliding curve and the swimmers’ inertia,

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and similar results obtained through CFD. Authors found out very similar and coherent results, allowing them to sustain the validity of the CFD approach. Although the emergence of very interesting works applying CFD in human swimming, some limitations still remains. The majority of the digital models have been developed based on approximate analytical representations of the human structures. In most cases, the authors used two-dimensional models [15, 16, 35, 42, 43] and when the authors used three-dimensional models, sometimes these were very simple and reductive representations of the human limbs [40, 44]. Gardano and Dabnichki [40] used standard geometric solids to represent the human arm; while Marinho et al. [44] used a three-dimensional model of the hand and forearm with the fingers slightly flexed. These differences between digital models and the real human segments can lead to some misinterpretation of the biomechanical basis of human swimming propulsion. This fact is one of the causes for the improvement of CFD studies in swimming, developing the models through engineering procedures [17, 34, 45, 46]. Lyttle and Keys [17], Bixler et al. [34] and Lecrivain et al. [45] applied the socalled “reverse engineering process” to build a virtual model geometrically identical to the swimmer body, carrying-out a three-dimensional mapping using a whole body laser scanner. Marinho et al. [46] developed a true three-dimensional model of the human hand and forearm, through the transformation of computer tomography scans into input data to apply CFD methodology. In a general way, the reverse engineering process involves the capture of the point cloud of the real object, editing the point cloud, creating the mesh from the point cloud for viewing and editing, creating smooth surfaces over the mesh, and creating a solid model from the smooth surfaces [45]. These studies have shown the great potential offered by reverse engineering procedures for developing true digital models of the human body to improve the prediction of hydrodynamic forces in swimming.

5.2 Segmental Propulsion 5.2.1 Variation of Drag and Lift According to Angle of Attack As stated by Lyttle and Keys [17] one major advantage of CFD procedures is the possibility to assess how the variance of the inputs affects the resultant flow conditions. Hence, CFD has been used to analyze some concerns arising from empirical data. One of the major themes is related to the relative importance of drag and lift forces to the overall propulsive force production in swimming. Several studies were carried-out using digital models of the human hand and/or forearm and/or upper arms. Bixler and Riewald [33] evaluated the steady flow around a swimmer’s hand and forearm at various angles of attack (Fig. 2) and sweep back angles (Fig. 3). The CFD model was created based upon an adult male’s right forearm and hand with the

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Fig. 2 The angle of attack [37]. The arrow represents the direction of the flow

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Fig. 4 Drag coefficient vs. angle of attack for the digital model of the hand, forearm and hand/forearm (Sweep back angle D 0ı ). Adapted from Bixler and Riewald [33]

forearm fully pronated. Force coefficients measured as a function of angle of attack showed that forearm drag was essentially constant (CD  0.65) and forearm lift was almost zero (Figs. 4 and 5). Moreover, hand drag presented the minimum value near angles of attack of 0ı and 180ı and the maximum value was obtained near 90ı (CD  1.15), when the model is nearly perpendicular to the flow. Hand lift was almost null at 95ı and peaked near 60ı and 150ı (CL  0.60). Axial coefficients were large for the forearm at all angles of attack and for the hand near 90ı . Thus, Bixler and Riewald [33] suggested the employ of three-dimensional lift coefficient incorporating forces acting along the two axis perpendicular to the flow direction.

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Fig. 5 Lift coefficient vs. angle of attack for the digital model of the hand, forearm and hand/forearm (Sweep back angle D 0ı ). Adapted from Bixler and Riewald [33]

5.2.2 Relative Contribution of Drag and Lift to Propulsion The relative contribution of drag and lift forces to overall propulsion is one of the most discussed issues. It was found that more lift force is generated when the little finger leads the motion than when the thumb leads [33, 47]. Silva et al. [47], using a real digital model of a swimmer hand and forearm, confirmed the supremacy of the drag component. They also revealed an important contribution of lift force to the overall propulsive force production by the hand/forearm in swimming phases, when the angle of attack is close to 45ı (Fig. 6). The drag coefficient presented higher values than the lift coefficient for all angles of attack. In fact, the drag coefficient increased with the angle of attack showing the maximum values with an angle of attack of 90ı (CD  90ı ) and the minimum values with an angle of attack of 0ı (CD  0.45). The lift coefficient of the model presented the maximum values with an angle of 45ı (CL  0.50). Silva et al. [47] obtained values of lift coefficient very similar for the angles of attack of 0ı and 90ı , although the minimum values were obtained with an angle of attack of 90ı (CL  0.15). In this study the hand and forearm force coefficients were not analyzed independently but a combined analysis was performed (Fig. 7). Sato and Hito [48] aimed to estimate thrust of a swimmer’s hand and to explore ways to increase it. The computed drag and lift coefficients at each angle of attack showed values of drag coefficient higher than lift coefficient at all angles of attack. From the results of CFD simulations the authors turned out that the resultant force was maximal with an angle of attack of 105ı and the direction of the resultant force in that situation was 13ı . Based on this analysis, the authors suggested stroke backward and with a little-finger-ward, out sweep motion, as the best stroke motion to produce the maximum thrust during underwater path.

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Fig. 6 Drag and lift coefficient of the hand/forearm model for angles of attack of 0ı , 45ı and 90ı (SA: Sweep back angle). Flow velocity D 2:0 m s1 . Adapted from Silva et al. [47]

Fig. 7 The hand and forearm model used by Silva et al. [47] inside the three-dimensional CFD domain

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5.2.3 Studies with Unsteady Flows The studies above mentioned were conducted using steady state CFD analysis. Aiming to approach to more similar real swimming conditions, some authors [16, 33, 49] included the unsteady effects of motion into the numerical simulations. The pioneer study of Bixler and Schloder [15] was conducted both in steady and unsteady state flow conditions. These authors analyzed the flow around a disc with a similar area of a swimmer hand. Different simulations with different initial velocity and acceleration were conducted to model identical real swimming conditions, especially during insweep and upsweep phases of the front crawl stroke. According to the obtained results the authors reported that the hand acceleration can increase the propulsive force by around 24% compared with the steady flow condition. Thus, the drag and lift forces produced by the swimmers’ hand in a determined time are dependent not only on the surface area, the shape and the velocity of the segment but also on the acceleration of the propulsive segment. Sato and Hino [49] showed a numerical method of unsteady CFD simulation to predict swimmer’s propulsive force. The results of the simulations agreed well with the data measured experimentally. The hydrodynamic forces acting on the accelerating hand was much higher than with a steady flow situation and these forces amplifies as acceleration increases. Rouboa et al. [16] analyzed the effect of swimmer’s hand/forearm acceleration on propulsive forces generation using CFD. A two-dimensional model of a right male hand/forearm was studied with angles of attack of 0ı , 90ı , and 180ı. The main results reported that under the steady flow condition the drag coefficient was the one that contributes more for propulsion with a maximum of 1.16, when the orientation of the hand/forearm is plane and the model is perpendicular to the direction of the flow. Under the hand/forearm acceleration condition, the measured values for propulsive forces were approximately 22.5% higher than the forces produced under the steady flow condition (Fig. 8). Analyzing this data, one is tempted to suggest that coaches must advise their swimmers to accelerate their hands during the propulsive movement. However, one should be careful with the practical considerations of this conclusion. There are factors other than instantaneous force to be considered. For instance, Rouboa et al. [16] referred that the gain produced by increase in force magnitude is offset by a decrease in duration of force application. Thus, in the future it could be interesting to calculate the impulse and to compare a lower force applied for longer time to a higher force applied for shorter time. These studies confirm that unsteady mechanisms are present in swimming propulsion. However, both Sato and Hino [49] and Rouboa et al. [16] did not consider direction changes or acceleration in directions other than the hand/forearm velocity direction. Swimmers do not move their hands and arms in a steady velocity or linear direction. The swimmer’s hand/forearm motion is a combination of movements in horizontal (forward-backward), lateral (inward-outward) and vertical (upwarddownward) directions. Therefore, it seems essential to include other aspects of unsteady motions, namely the multi-axis rotations, with the rotation of the mesh

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Fig. 8 Comparison between steady and accelerated drag and lift coefficients for angles of attack of 0ı , 90ı , and 180ı (Sweep back angle D 0ı ). Adapted from Rouboa et al. [16]

relative to the flow. Regarding this issue, Lecrivain et al. [45] used unsteady CFD procedures to analyze the performance of a lower arm amputee swimmer.

5.2.4 Contribution of Arm’s Action to Propulsion Lecrivain et al. [45] used a complex CFD mesh model, representing the swimmer body and its upper arm. The model, including the arm rotation relative to the body and a body roll movement relative to the water, interacted dynamically with the fluid flow. The unsteady evolution of the interaction was achieved through dynamic moving/deforming meshes for the particular body parts which have a relative motion with full computation of the interaction carried out at each successive time step. In further research, the authors intend to analyze the effect of different arm rotations and body roll movements in the arm propulsive force production. Lecrivain et al. [45] were also able to note that the arm provided effective propulsion through most of the stroke, and this must be considered when studying the arm propulsion. In fact, Gardano and Dabnichki [40] underlined the importance of the analysis of the entire arm rather than different parts of it. Thus, the authors concluded that drag profiles differed substantially with the elbow flexion angle, as the maximum value could vary by as much as 40%. In addition, Gardano and Dabnichki [40] stated that maximum drag force was achieved by 160ı of elbow angle. A prolonged plateau between 50ı and 140ı indicated greater momentum generated at 160ı in comparison with the other configurations. This fact suggests a strong possibility for the existence of an optimal elbow angle for the generation of a maximum propulsive

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force. However, these findings are only possible to confirm if an entire model of the swimmer’s arm, its movement relative to the body and the body’s movement relative to the water is computed with CFD. This concern seems an interesting topic to address in further studies.

5.2.5 Contribution of Leg’s Action and Kicking to Propulsion The majority of the CFD studies regarding swimming propulsion are based on arm analyses, since this is the most relevant segment producing propulsion. Nevertheless, kicking has a lower but also relevant role in overall propulsion. So, it is important to describe a pioneer study about the propulsion generated during underwater dolphin kicking. Based on video images of an elite swimmer, [17] performed a three-dimensional CFD analysis, modelling the swimmer performing two kinds of underwater dolphin kick: (a) high amplitude and low frequency dolphin kick and; (b) low amplitude and high frequency dolphin kick. This model included the addition of user defined functions and re-meshing to provide limb movement. The results demonstrated an advantage of using the large slow kick, over the small fast kick, concerning the velocity range that underwater dolphin kicks are used. In addition, changes were also made into the input kinematics (ankle plantar flexion angle) to demonstrate the practical applicability of the CFD model. While the swimmer was gliding at 2:18 m s1 , a 10ı increase in ankle plantar flexion created greater propulsive force during the kick cycle. These results demonstrated that increasing angle flexibility will increase the stroke efficiency for the subject that was modelled. Even if most of propulsion (85–90%) is generated by the arm’s actions in front crawl [50,51] leg’s propulsion should not be disregarded. In this sense, CFD massive studies about kicking action should also be implemented.

5.2.6 Finger’s Positions Understanding the basis of the propulsive force production can play an important role in the swimmers’ technical training and performance. So, CFD can supply information to coaches on technique prescription, providing answers to some practical issues that remain unclear. The finger’s relative position during the underwater path of the stroke cycle is one of these cases. A large inter-subject range of fingers relative position can be observed during training and competition, regarding thumb position and finger spreading. Concerning thumb position, some swimmers maintain the thumb adducted, others have small thumb abduction, and others have the thumb totally abducted. Concerning finger spreading, some swimmers maintain the fingers close together, others have small distance between fingers, and others present a large distance between fingers. Marinho et al. [52] analyzed the hydrodynamic characteristics of a true model of a swimmer hand with the thumb in different positions using CFD. The authors

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analyzed angles of attack of 0ı , 45ı , and 90ı with a sweep back angle of 0ı (the thumb as the leading edge). These authors showed that the position with the thumb adducted presented slightly higher values of drag coefficient compared with thumb abducted positions. Further, the position with the thumb fully abducted allowed increasing the lift coefficient of the hand at angles of attack of 0ı and 45ı (Figs. 9 and 10). These findings seemed similar to the ones found by Schleihauf [37] with experimental research. In the study of Schleihauf [37] the position with the thumb fully abducted showed a maximum lift coefficient at an acute angle of attack of 15ı , whereas the models with partial thumb abduction showed a maximum value of lift coefficient at higher angles of attack (45ı –60ı ). In these orientations, the position with the thumb partially abducted presented higher values than with the thumb fully abducted. Moreover, Takagi et al. [53] using experimental measurements revealed that the thumb position influenced the lift force. For a sweep back angle of 0ı (as used in the study of [52]) the model with abducted thumb presented higher values of lift force, whereas for a sweep back angle of 180ı (the little finger as the leading edge), the adducted thumb model presented higher values of lift force. In addition, the drag coefficient presented similar values in the two thumb positions for a sweep back angle of 0ı and higher values in the thumb adducted position for a sweep back angle of 180ı. Although some differences in the results of different studies, CFD data seemed to indicate that when the thumb leads the motion (sweep back angle of 0ı ) a hand position with the thumb abducted would be preferable to an adducted thumb position. In addition, when analyzing the resultant force coefficient, Marinho

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et al. [52] found that the position with the thumb abducted presented higher values than the positions with the thumb partially abducted and adducted at angles of attack of 0ı and 45ı . At an angle of attack of 90ı , the position with the thumb adducted presented the highest value of resultant force coefficient. Marinho et al. [54] aimed to study the effect of finger spread on the propulsive force production in swimming using CFD. The authors studied the hand with different finger spreads: fingers close together, fingers with little distance spread (a mean intra finger distance of 0.32 cm, tip to tip), and fingers with large distance spread (0.64 cm, tip to tip), similar to the procedure used by Schleihauf [37]. Marinho et al. [54] found that for attack angles higher than 30ı , the model with little distance between fingers presented higher values of drag coefficient when compared with the models with fingers closed and with large finger spread. For attack angles of 0ı , 15ı , and 30ı , the values of drag coefficient were very similar in the three models of the swimmer’s hand. In addition, the lift coefficient seemed to be independent of the finger spreading, presenting little differences between the three models (Figs. 11 and 12). Nevertheless, Marinho et al. [54] were able to note slightly lower values of lift coefficient for the position with larger distance between fingers. These results suggested that fingers slightly spread can be used by swimmers to create more propulsive force. However, one should be careful transferring these findings to swimming, because the above mentioned studies were conducted only under steady state flow conditions. It is interesting to know if the results would be similar if unsteady conditions were included during the numerical simulations.

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Fig. 11 Drag coefficient for angles of attack of 0ı , 15ı , 30ı , 45ı , 60ı , 75ı , and 90ı for the different finger spread positions (Sweep back angle D 0ı , Flow velocity D 2:0 m s1 ). Adapted from Marinho et al. [54].

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Fig. 12 Lift coefficient for angles of attack of 0ı , 15ı , 30ı , 45ı , 60ı , 75ı , and 90ı for the different finger spread positions (Sweep back angle D 0ı , Flow velocity D 2:0 m s1 ). Adapted from Marinho et al. [54]

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5.3 Drag In addition to the analysis of the propulsive forces generation, CFD methodology can be used to understand the intensity of drag forces resisting forward motion and its effects over swimming performance [17, 18, 34, 43, 55].

5.3.1 Kicking after Start and Turn Lyttle and Keys [17] sought to discriminate between active drag and propulsion produced in underwater dolphin kicking aiming to optimize the underwater phase in swim starts and turns. As mentioned before, using a three-dimensional model of a male swimmer performing two types of dolphin kicking movements (large/slow, small/fast), the authors found that both kick techniques have a similar effect at 2:40 m s1 . It seemed that for velocities higher than 2:40 m s1 there is a trend for the small kick to become more effective whereas for velocities lower than 2:40 m s1 the large kick appeared to be more effective (Fig. 13). Lyttle and Keys [17] compared the dynamic underwater kicking data with the results of experimental studies [56], and suggested that velocities around 2:40 m s1 represent a cross-over point, whereby at higher velocities it seemed more efficient to

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the swimmer to maintain a streamlined position than to initiate underwater kicking. The authors stated that this situation is due to the swimmer creating more active drag than propulsion while kicking compared to remaining in a streamlined position, thus leading to a negative acceleration of the swimmer. Although it appeared that the swimmer would benefit from a smaller kick at higher velocities, it seemed better to maintain a streamlined position.

5.3.2 Gliding Positions Regarding the analysis of the underwater gliding in swimming, Marinho et al. [18] investigated two common gliding positions: a ventral position with the arms placed alongside the trunk, and a ventral position with the arms extended at the front with the shoulders fully flexed (Figs. 14 and 15). A three-dimensional model of a male adult swimmer was used and the simulations were applied to flow velocities between 1.60 and 2:00 m s1 . The gliding position with the arms extended at the front, with the shoulders flexed, presented lower drag coefficient (CD  0.4) values than the position with the arms placed along the trunk (CD  0.7). Regarding the position with the arms extended at the front of the swimmer with the shoulders flexed, the values are very similar to the ones found by Bixler et al. [34], using a CFD approach, as well, and to the ones found by Vilas-Boas et al. [41], through experimental inverse dynamics. Considering the breaststroke turn, Marinho et al. [18] suggested that the first

Fig. 14 The model used by Marinho et al. [18] in a ventral position with the arms alongside the trunk inside the CFD domain

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Fig. 15 The model used by Marinho et al. [18] in a ventral position with the arms extended at the front, with the shoulders fully flexed, inside the CFD domain

gliding, performed with the arms at the front, should be emphasized in relation to the second gliding, performed with the arms along the trunk. Zaidi et al. [43] numerically analyzed the effect of the position of the swimmer’s head on the underwater hydrodynamics performances in swimming. The obtained numerical results revealed that the position of the head had a noticeable effect on the hydrodynamic performances, strongly modifying the wake around the swimmer. The position with the head aligned with the body seemed to allow the swimmer to carry out the best water penetration during the underwater swimming phases, comparing with a lower and a higher head position. The head aligned with the axis of the body induces a decrease in the drag from 17 to 21%, for a range velocity from 2.20 to 3:10 m s1 . For lower velocities (i.e., 1:40 m s1 /, the drag is only slightly affected by the change in the head position. However, it should be kept in mind that Zaidi et al. [43] used a two-dimensional steady flow model to simulate a really unsteady three-dimensional flow.

5.3.3 Drafting Silva et al. [55] aimed to investigate the effect of drafting on the hydrodynamic drag, using a two-dimensional model. The purpose of this study was to determine the effect of drafting distance on the drag coefficient in swimming. Numerical simulations were conducted for various distances between swimmers (0.5–8.0 m) and swimming velocities .1:6–2:0 m s1 / and the drag coefficient was computed for each one of the distances and velocities (Fig. 16). Silva et al. [55] found that the relative drag coefficient of the trailing swimmer was lower (about 56% of the leading swimmer) for the smallest inter-swimmer distance (0.5 m). This value increased progressively until the distance between swimmers reached 6.0 m, where the relative drag coefficient of the trailing swimmer was about 84% of the leading swimmer. The results indicated that the drag coefficient of the trailing swimmer was equal to that of the leading swimmer at distances ranging from 6.45 to 8.90 m. The authors concluded that these distances

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Fig. 16 Two-dimensional model used by Silva et al. [55] to determine the effect of drafting distances on hydrodynamic drag

allow the swimmers to be in the same hydrodynamic conditions during training and competitions. As a suggestion to specific swimming training sets, Silva et al. [55] stated that a swimmer must start swimming at least when the leading swimmer reaches a 10 m distance from the starting wall, rather than the 5 m distance commonly used in training. Nevertheless, concerning open water competitions, the athletes could take important advantages of swimming in a drafting situation. However, these conclusions must be read carefully because this study was conducted using a two-dimensional model of the human body and only the passive drag was computed. Moreover, the simulations were applied with the swimmers under the water and not swimming at the water surface. Therefore, as suggested by the authors, further researches should apply the modelling of bodies on/at the water surface, taking into account the above and underwater body volumes and fluid characteristics.

5.3.4 Relative Contribution of Drag Components to Total Drag In addition to the analysis of the hydrodynamic drag under different body positions, some authors attempted to investigate the contribution of skin-friction drag, pressure drag and wave drag to the total drag [18, 34, 43]. In human swimming, the total drag is composed of the skin-friction drag, pressure drag and wave drag. Skin-friction drag is attributed to the forces tending to slow the water flowing along the body surface of the swimmer. It depends on the velocity of the flow, the surface area of the body and the characteristics of the surface. Pressure drag is caused by the pressure differential between the front and the rear of the swimmer and it is proportional to the square of swimming velocity, the density of water and the cross sectional are of the swimmer. Finally, swimming at the water surface is constrained by the formation of surface waves leading to wave drag. However, the three mentioned studies only considered hydrodynamic drag depending on the

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skin-friction drag and pressure drag since the model was placed underwater. In the study of Bixler et al. [34] the swimmer model was placed at a water depth of 0.75 m. In the study of Zaidi et al. [43] the swimmer was positioned 1.50 m below the water surface while Marinho et al. [18] used a swimmer model placed at a water depth of 0.90 m. This assumption was proven to be correct using experimental tests [57, 57, 58] concluded that there is no significant wave drag when a typical adult swimmer is at least 0.60 m under the water’s surface. More recently, Vennell et al. [58] found that at 0.75 m below the water surface was below the location where “surface effects” begin to influence significantly the drag force. Indeed, the authors showed that to avoid significant wave drag a swimmer must be deeper than 1.8 chest depths and 2.8 chest depths below the water surface for velocities of 0.9 and 2:0 m s1 , respectively. It seems interesting attempting to conduct similar studies with a CFD approach, requiring the simulation of the interface between air and water. Bixler et al. [34] showed that pressure drag represented around 75% of the total hydrodynamic drag. Although pressure drag was dominant, skin-friction drag was by no means insignificant, representing 27% and 25% of total drag for gliding velocities of 1.50 and 2:25 m s1 , respectively. The significantly higher percentage of pressure drag was as well found by Marinho et al. [18] and Zaidi et al. [43]. Zaidi et al. [43] found for the position with the head aligned with the body that pressure drag represented around 80% of the total drag whereas Marinho et al. [18] found a percentage of around 87% and 92% for this drag component in the position with the arms extended at the front with the shoulders flexed, and in the position with the arms along the trunk, respectively (Fig. 17). However, the absolute values of skin-friction drag were about the same in the two gliding positions, being the main differences attributable mainly to the pressure drag component. It

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Fig. 17 Relationship between total drag, skin-friction drag and pressure drag and the gliding velocity for the positions with the arms alongside the trunk (AAT) and with the arms extended at the front with the shoulders flexed (AEF). Adapted from Marinho et al. [18]

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is important to reinforce that these values for the drag components were computed for underwater gliding. If the model were at the water’s surface these percentages would be somewhat different due to the decreasing in wetted area and the generation of wave drag. Moreover, both Bixler et al. [34] and Marinho et al. [18] studies were based on the swimmer model’s surface having a zero roughness. Therefore, the development of roughness parameters for human skin would allow a more accurate CFD model to be built in further studies. It seems possible that if the surface roughness were increased in the models the skin-friction drag would probably be higher, due to increased turbulence around the surface [34]. On the other hand, if the surface roughness were increased the pressure drag could be reduced. Massey [59] stated that the boundary layer, which would be mainly laminar, would change into a turbulent one. When the flow is laminar, separation of the boundary layer at the body surface starts almost as soon as the pressure gradient becomes adverse, and a larger wake forms. However, when the flow is turbulent, separation is delayed and the corresponding wake is smaller, thus decreasing pressure drag [18, 60].

5.3.5 Swimsuits and Training Equipments The study of the effects of different swimsuits on the hydrodynamic drag was one of the first applications of CFD in swimming [3]. As stated in the introduction section, the cooperation between Speedo and Fluent allowed developing some well-known swimsuits, as FastSkin and LZR suits. However, Speedo is not the only manufacture using numerical solutions to enhance the swimsuits. Arena in straight cooperation with the Mox Institute at Politecnico de Milano (Milan, Italy) developed mathematical models and simulations to measure the water flow around the swimmer using the PowerSkin new generation swimsuits. With CFD methodology it is possible to analyze the velocity and the direction of the water flow around the body, thus allowing evaluating the different paths due to different suit tissues and body compressive effects. However, the major research conducted in this field is performed with great secrets. To our knowledge, there is no study published about this issue. Therefore, new lines of research concerning the effects of different swimsuits on performance should be attempted in the future. For instance, it seems important to evaluate the use of different suit tissues, different ways to sew the tissue pieces, different suit types and sizes, and the effect of swim suits upon wobbling body masses, and full body (and body parts) compression during different swimming phases. Based on these assumptions, it seems CFD can also be an interesting tool to help developing training equipments. For example, different paddles, fins, kickboards, pull-buoys, cups, swim goggles and training aids used by the swimmers can be evaluated using numerical simulation techniques. The effect of different lane lines in the swimmer’s performance can also be analysed with a CFD approach.

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6 Computational Fluid Dynamics Methods Contribution for Near Future Development of Swimming Science As one can note, CFD can be a good approach to study swimming hydrodynamics and can contribute to the development of swimming science. However, despite the important steps forward in the application of CFD in swimming, there are several aspects that can be improved. The concern of Gardano and Dabnichki [40] and Lecrivain et al. [45] of taking into account the entire arm when studying the arm propulsion should be considered. In addition, the effect of whole body movements on the arm propulsive force production must also be attempted in the future. Moreover, the analysis of hydrodynamic forces must be conducted with the body at the water surface, taking into account the interface between air and water. This fact will require the simulation of two different fluids around the swimmer body, allowing including wave drag in the evaluations. The modelling of whole body swimming movements seems to be the next step in swimming research applying CFD methodology. Furthermore, the development of roughness parameters for human skin would allow a more accurate CFD model to be built in future studies, to more accurately understand the relative contribution of skin-friction drag to the total hydrodynamic drag. As stated by Bixler et al. [34], as CFD methods continue to develop, it will be possible to evaluate the effects of different techniques, body positions, and swimwear on performance, thus optimizing swimmers’ performance. Therefore, with these assumptions we can state some ideas and some purposes for other studies following and complementing the ones we have presented during this chapter: 1. Propulsive forces studies: (a) The computation of the ideal shape for a swimmers hand, arm, foot, or other body segment; (b) The computation of the effects of acceleration (positive and negative), and multi-axis rotations on lift and drag; (c) The computation of the added mass of water as an inertial effect to the body displacement during the stroke cycle; (d) The computation of the effect of different stroke patterns on propulsion in front crawl, backstroke, butterfly and breaststroke. 2. Drag forces studies: (a) The computation of total drag force on a swimmer moving through the water, and the relative contribution of pressure drag, skin-friction drag and wave production drag for the total drag; (b) The effect of different forms of streamlining on the hydrodynamic drag; (c) The computation of the effect of underwater turbulence and waves on a swimmers motion;

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(d) The effect on hydrodynamic drag of “dragging” off a swimmer, either in an adjacent lane and/or behind; (e) The evaluation of the effects of different swimming suits and other equipments on hydrodynamic drag; (f) The computations of the ideal body shape and size to minimise drag; (g) Eventually, to calculate active drag, using moving meshes would be an important task.

7 Conclusions In summary we can state that the recent evidence strongly suggests that CFD technique can be considered as an interesting new approach for evaluation of swimming hydrodynamic forces. In the near future, as in the present, CFD will provide valorous arguments for defining new swimming techniques or equipments. Therefore, within this chapter we attempted to present the already applied CFD techniques and to propose new procedures that may be used by the research community in further studies under similar research topics in order to improve swimming performance. On the other hand, we tried to make some contribution to the dissemination of the main results, not only stimulating young researchers in the fulfilment of the existent gap between the sports sciences and other sciences (hydrodynamics in the present case) but also to the spreading of the use of this recent technique (in sports context) by the ones that are really interested in the development of new concepts and applications. We also aimed to contribute to the application of the knowledge gathered into practical situations, trying to introduce some new insights in the designing of new propulsive techniques in swimming, new ways of streamlining the body during the displacement or even the development of new materials (suits and others) helping the swimmer moving faster. Acknowledgements Our research has been supported by the Portuguese Government by a grant of the Science and Technology Foundation (SFRH/BD/25241/2005; PTDC/DES/098532/2008).

References 1. Dickinson MH (2000) How animals move: an integrative view. Science 288: 100–106. 2. Arellano R, Nicoli-Terr´es JM, Redondo JM (2006) Fundamental hydrodynamics of swimming propulsion. Port J Sport Sci 6(Suppl.2): 15–20. 3. Fluent (2004) Speedo goes for gold with CFD. Fluent News 13(1): 4–6. 4. Boulding N, Yim SS, Keshavarz-Moore E, Ayazi Shamlou P, Berry M (2002) Ultra scaledown to predict filtering centrifugation of secreted antibody fragments from fungal broth. Biotechnol Bioeng 79(4): 381–388.

Swimming Simulation: A New Tool for Swimming Research and Practical Applications

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5. Marshall I, Zhao S, Papathanasopoulou P, Hoskins P, Xui Y (2004) MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models. J Biomech 37: 679–687. 6. Dabnichki P, Avital E (2006) Influence of the position of crew members on aerodynamics performance of two-man bobsleigh. J Biomech 39(15): 2733–2742. 7. Guerra D, Ricciardi L, Laborde JC, Domenech S (2007) Predicting gaseous pollutant dispersion around a workplace. J Occup Environ Hyg 4(8): 619–633. 8. Berthier B, Bouzebar R, Legallais L (2002) Blood flow patterns in an automatically realistic coronary vessel influence of three different reconstruction models. J Biomech 35(10): 1347–1356. 9. Ruiz P, Ruiz F, L´opez A, Espa˜nol C (2005) Computational fluid dynamics simulations of the airflow in the human nasal cavity. Acta Otorrinolaringol Esp 56(9): 403–410. 10. Liu H, Wassersug R, Kawachi K (1996) A computational fluid dynamics study of tadpole swimming. J Exp Biol 199(6): 1245–1260. 11. Liu H, Wassersug R, Kawachi K (1997) The three-dimensional hydrodynamics of tadpole locomotion. J Exp Biol 200(22): 2807–2819. 12. Liu H, Ellington C, Kawachi K (1998) A computational fluid dynamic study of hawkmoth hovering. J Exp Biol 201(4): 461–477. 13. Pallis JM, Banks DW, Okamoto KK (2000) 3D computational fluid dynamics in competitive sail, yatch and windsurfer design. In: Subic F, Haake M (eds) The engineering of sport: research, development and innovation: 75–79. Blackwell Science, Oxford. 14. Kellar WP, Pearse SRG, Savill AM (1999) Formula 1 car wheel aerodynamics. Sports Eng 2(4): 203–212. 15. Bixler BS, Schloder M (1996) Computational fluid dynamics: an analytical tool for the 21st century swimming scientist. J Swim Res 11: 4–22. 16. Rouboa A, Silva A, Leal L, Rocha J, Alves F (2006) The effect of swimmer’s hand/forearm acceleration on propulsive forces generation using computational fluid dynamics. J Biomech 39(7): 1239–1248. 17. Lyttle A, Keys M (2006) The application of computational fluid dynamics for technique prescription in underwater kicking. Port J Sport Sci 6(Suppl. 2): 233–235. 18. Marinho DA, Reis VM, Alves FB, Vilas-Boas JP, Machado L, Silva AJ, Rouboa AI (2009a) The hydrodynamic drag during gliding in swimming. J Appl Biomech 25(3): 253-257. 19. Liu H (2002) Computational biological fluid dynamics: digitizing and visualizing animal swimming and flying. Integr Comp Biol 42: 1050–1059. 20. Sane S (2003) The aerodynamics of insect flight. J Exp Biol 206: 4191–4208. 21. Ellington CP, Van den Berg C, Willmott AP, Thomas AL (1996) Leading-edge vortices in insect flight. Nature 384: 626–630. 22. Liu H, Kawachi K (1998) A numerical study of insect flight. J Comput Phys 146: 124–156. 23. Dickinson MH, Lehmann FO, Sane SP (1999) Wing rotation and the aerodynamic basis of insect flight. Science 284: 1954–1960. 24. Wang ZJ (2000) Two dimensional mechanism for insect hovering. Phys Rev Lett 85: 2216– 2219. 25. Hamdani H, Sun M (2001) A study on the mechanism of high-lift generation by an airfoil in unsteady motion at low Reynolds number. Acta Mech Sin 17: 97–114. 26. Wang ZJ (2004) The role of drag in insect hovering. J Exp Biol 207: 4147–4145. 27. Ramamurti R, Sandberg WC (2002) A three-dimensional computational study of the aerodynamic mechanisms of insect flight. J Exp Biol 205: 1507–1518. 28. Sun M, Tang J (2002) Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J Exp Biol 205: 55–70. 29. Dickinson MH, Gotz KG (1993) Unsteady aerodynamic performance of model wings at low Reynolds numbers. J Exp Biol 174: 45–64. 30. Colwin C (1984) Fluid dynamics: vortex circulation in swimming propulsion. In: Welsh T.F (ed) American swimming coaches association world clinic yearbook 1984: 38–46. American Swimming Coaches Association, Fort Lauderdale. 31. Toussaint HM, Van den Berg C, Beek WJ (2002) “Pumped-up propulsion” during front crawl swimming. Med Sci Sports Exerc 34(2): 314–319.

60

D.A. Marinho et al.

32. Hannah RK (2002) Can CFD make a performance difference in Sport? In: Ujihashi S (ed) The engineering of sport: 17–30. Blackwell Science, Oxford. 33. Bixler BS, Riewald S (2002) Analysis of swimmer’s hand and arm in steady flow conditions using computational fluid dynamics. J Biomech 35: 713–717. 34. Bixler B, Pease D, Fairhurst F (2007) The accuracy of computational fluid dynamics analysis of the passive drag of a male swimmer. Sports Biomech 6(1): 81–98. 35. Silva AJ, Rouboa A, Leal L, Rocha J, Alves F, Moreira A, Reis VM, Vilas-Boas JP (2005) Measurement of swimmer’s hand/forearm propulsive forces generation using computational fluid dynamics. Port J Sport Sci 5(3): 288–297. 36. Wood TC (1977) A fluid dynamic analysis of the propulsive potential of the hand and forearm in swimming. Master of Science Thesis. Dalhouise University Press, Halifax. 37. Schleihauf RE (1979) A hydrodynamic analysis of swimming propulsion. In: Terauds J, Bedingfield EW (eds.) Swimming III: 70–109. University Park Press, Baltimore. 38. Berger MA, de Groot G, Hollander AP (1995) Hydrodynamic drag and lift forces on human hand arm models. J Biomech 28(2): 125–133. 39. Sanders RH (1999) Hydrodynamic characteristics of a swimmer’s hand. J Appl Biomech 15: 3–26. 40. Gardano P, Dabnichki P (2006) On hydrodynamics of drag and lift of the human arm. J Biomech 39: 2767–2773. 41. Vilas-Boas JP, Costa L, Fernandes R, Ribeiro J, Figueiredo P, Marinho D, Silva A, Rouboa A, Machado L (2008) Determinac¸a˜ o do arrasto hidrodinˆamico em duas posic¸o˜ es de deslize, por dinˆamica inversa e por simulac¸a˜ o computacional (CFD). In II Congreso Internacional de Biomec´anica de Venezuela. Minist´erio del Poder Popular para el Deporte de Venezuela, Isla de Margarita, Venezuela. 42. Moreira A, Rouboa A, Silva AJ, Sousa L, Marinho D, Alves F, Reis V, Vilas-Boas JP, Carneiro A, Machado L (2006) Computational analysis of the turbulent flow around a cylinder. Port J Sport Sci 6(Suppl. 1): 105. 43. Zaidi H, Taiar R, Fohanno S, Polidori G (2008) Analysis of the effect of swimmer’s head position on swimming performance using computational fluid dynamics. J Biomech 41: 1350–1358. 44. Marinho DA, Reis VM, Alves FB, Vilas-Boas JP, Machado L, Rouboa AI, Silva AJ (2009b) The use of Computational Fluid Dynamics in swimming research. Int J Comput Vis Biomech (in press). 45. Lecrivain G, Slaouti A, Payton C, Kennedy I (2008) Using reverse engineering and computational fluid dynamics to investigate a lower arm amputee swimmer’s performance. J Biomech 41: 2855–2859. 46. Marinho DA, Reis VM, Vilas-Boas JP, Alves FB, Machado L, Rouboa AI, Silva AJ (2009c) Design of a three-dimensional hand/forearm model to apply Computational Fluid Dynamics. Braz Arch Biol Technol (in press). 47. Silva AJ, Marinho DA, Reis VM, Alves F, Vilas-Boas JP, Machado L, Rouboa AI (2008a) Study of the propulsive potential of the hand and forearm in swimming. Med Sci Sports Exerc 40(5 Suppl): S212. 48. Sato Y, Hino T (2002) Estimation of thrust of swimmer’s hand using CFD. In: Proceedings of 8th symposium on nonlinear and free-surface flows: 71–75. Hiroshima. 49. Sato Y, Hino T (2003) Estimation of thrust of swimmer’s hand using CFD. In: Proceedings of second international symposium on aqua bio-mechanisms: 81–86. Honolulu-USA. 50. Hollander A, de Groot G, Schenau G, Kahman R, Toussaint H (1988) Contribution of the legs to propulsion in front crawl swimming. In: Ungerechts B, Wilke K, Reischle K (eds) Swimming science V: 39–43. Human Kinetics Books. Champaign, Illinois. 51. Deschodt V (1999) Relative contribution of arms and legs in human to propulsion in 25 m sprint front crawl swimming. Eur J Appl Physiol 80: 192–199. 52. Marinho DA, Rouboa AI, Alves FB, Vilas-Boas JP, Machado L, Reis VM, Silva AJ (2009d) Hydrodynamic analysis of different thumb positions in swimming. J Sports Sci and Med 8(1): 58–66.

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53. Takagi H, Shimizu Y, Kurashima A, Sanders R (2001) Effect of thumb abduction and adduction on hydrodynamic characteristics of a model of the human hand. In: Blackwell J, Sanders R (eds) Proceedings of swim sessions of the XIX international symposium on biomechanics in sports: 122–126. University of San Francisco, San Francisco. 54. Marinho DA, Barbosa TM, Reis VM, Kjendlie PL, Alves FB, Vilas-Boas JP, Machado L, Rouboa AI, Silva AJ (2009e) Swimming propulsion forces are enhanced by a small finger spread. J Appl Biomech (in press). 55. Silva AJ, Rouboa A, Moreira A, Reis VM, Alves F, Vilas-Boas JP, Marinho DA (2008b) Analysis of drafting effects in swimming using computational fluid dynamics. J Sports Sci Med 7(1): 60–66. 56. Lyttle A, Blanksby B, Elliott B, Lloyd D (2000) Net forces during tethered simulation of underwater streamlined gliding and kicking technique of the freestyle turn. J Sports Sci 18: 801–807. 57. Lyttle A, Blanksby B, Elliott B, Lloyd D (1999) Optimal depth for streamlined gliding. In: Keskinen KL, Komi PV, Hollander AP (eds) Biomechanics and medicine in swimming VIII: 165–170. Gummerus Printing, Jyvaskyla. 58. Vennell R, Pease DL, Wilson BD (2006) Wave drag on human swimmers. J Biomech 31: 664–671. 59. Massey BS (1989) Mechanics of fluids. Chapman & Hall, London. 60. Polidori G, Taiar R, Fohanno S, Mai TH, Lodini A (2006) Skin-friction drag analysis from the forced convection modeling in simplified underwater swimming. J Biomech 39(13): 2535–2541.

On CFD Simulation of Ski Jumping Helge Nørstrud and I. J. Øye

Abstract Numerical analyses have been performed on a generic ski jumper in flight based on Computational Fluid Dynamics (CFD) flow codes and a geometric simulation of the jumper with skis. Various configurations have been simulated and results are presented for several body/ski positions and flight angles. A theoretical analysis of the performance of the ski jumper is also briefly presented. In addition, a proposal for new jumping skis design is described as an outcome from the present numerical analysis.

1 Introduction It is believed that the art of ski jumping has evolved from children’s play in snow where natural and self-made jumps where a most exiting sport to prove the individuals ability to fly in the air on a pair of skies, see Fig. 1. Note that the first world record in ski jumping (official accepted by the International Ski Federation, FIS) was made in Eidsberg, Norway by Olaf Rye on November 22, 1808 with the length 9.5 meter. The present world record is held by Bjørn Einar Romøren (NOR) at the length of 239 m in Planica, Slovenia in 2005. The flying style was obviously a matter of trying and failing until Jan Bokl¨ov of Sweden introduced the aerodynamic V-style of jumping in 1987. This configuration of spreading the skis out from the body in a V-shape is the current professional way, and possible the ultimate, to fly down from a ski jump in order to earn good points for the style and the jumping length, see Fig. 2. The aerodynamic interaction between a ski jumper in flight and the surrounding air can be described through the dimensionless pressure coefficient Cp Œ , i.e. Cp;i D

pi  p1 1  V2 2 1 1

(1)

Helge Nørstrud (B) Norwegian University of Science and Technology, Department of Energy and Process Engineering, NO-7491 Trondheim, Norway, 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 3, 

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Fig. 1 A group of children trying to adopt the Bokl¨ov jumping style (Photo: Høyeggen skole, Melhus)

Fig. 2 Numerical simulation of a V-style ski jumper (Simulation: CFD Norway)

where the lower index i refers to a general point i on the surface of the jumper and on the skis. The static pressure pi [Pa] is in (1) referred to the atmospheric pressure p1 [Pa] in the free stream (identified by the lower index 1). Furthermore, (1) is nondimensionalized with respect to the dynamic pressure 1=2 ¡1 V1 2 where the air density is denoted by ¡1 Œkg m3 and the free stream velocity is V1 Œm s1 . Hence, the pressure coefficient distribution around the jumper with skis as shown in Fig. 2 gives overpressure (related to the atmospheric pressure) as Cp;i > 0. and

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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 Π, 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

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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

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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]

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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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Fig. 15 Distribution of lift- and drag coefficients along the longitudinal ski axis for ’ D 20ı and various V-angles .20ı  “  40ı /. Note that the aft part of the ski start at the abscissa 0 and that the tip is at 248, i.e., the length of the ski is 248 cm. Note the rapid drag increase from the tip of the ski and this drawback will be more discussed in Sect. 5

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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0.7

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0.5

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0.1

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20 30 Angle of Attack, alfa [deg]

40

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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.

On CFD Simulation of Ski Jumping

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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On CFD Simulation of Ski Jumping

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

Alpha = 20 deg. Beta = 40 deg.

Pressure coeff [–] 1.00 0.80 0.60 0.40 0.20 Top view

0.00 –0.20 –0.40

Bottom view

–0.60 –0.80 –1.00 –1.20 –1.40 –1.60 –1.80 –2.00 –2.20 –2.40 Side view

PHIZ=

20.0

PHIX=

–20.0 0.

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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13

15 12 14

Fig.2 13 12 17

11

Fig.3

Fig.1

14

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Fig.4B Fig.4C Fig. 21 Norwegian patent for the proposed ski design with asymmetric skis

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LIFT / DRAG RATIO, CL / CD [-]

3

Beta = 30° 2,5

= 25° 2 = 20°

1,5 14

16

18

20

22

24

26

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ANGLE OF ATTACK, Alfa [deg]

Fig. 22 Experimental values for the aerodynamic coefficients for the proposed new ski design [7]

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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Normalised Position (y/ymax)

1

Laminar

Turbulent 0 0

1 Normalised Velocity (v/vmax)

Fig. 1 Laminar and turbulent boundary layer velocity profiles [3] um(max) y

um

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‘Edge’ of boundary layer

δ

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Separation streamline

B A ∂u >0 ∂y 0

Reverse flow causing eddy Ideal X

Actual P

Pmin

Fig. 2 Mechanism of boundary layer separation over a curved surface [3]

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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0.6 Subcritical Drag coefficient, CD

0.5 Supercritical

Transcritical

0.4 0.3 0.2 0.1 Critical Reynolds number 0.0 1.0×104

1.0×105

1.0×106

1.0×107

Reynolds number, Re

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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AIR FLOW

WAK 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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Magnus Coefficient, CM

0.4

0.2

0.0

0.2

–0.4 0.0

Re = 2.1e5 Re = 1.7e5 Re = 1.1e5 Re = 0.9e5 0.2

0.4

0.6

Spin Parameter, Sp

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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Magnus Coefficient, CM

0.3 0.25 0.2 0.15 0.1 Scale model ball, Re=2.1e5 (Came et al., 2005) BallR_32s BallR_26s BallR_20s BallR_14b

0.05 0 0

0.2

0.4

0.6

0.8

1

Spin parameter, Sp

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

10

20

30

40

50

60

70

80

90

100

–0.1 –0.15

CFD: Ball 1

–0.2

CFD: Ball 3

–0.25 Angle (q), °

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 (<1 rev/s) are often seen to “move in the air” during their flight and behave erratically, as opposed to spinning balls that swerve in the air due to the Magnus Effect in a predictable manner. An example is shown in Fig. 17, an approximate plot of the trajectory of a free-kick scored by Cristiano Ronaldo for Manchester United against Fulham in the English Premiership in 2006, taken from television footage. The analysis in the previous section suggests that this motion could be due to the variation in CS or CL with orientation. The variations in CL and CS were entered into a trajectory simulation programme for a kick similar to Cristiano Ronaldo’s goal. The kick was taken at 33 m from goal, slightly to the right of centre, launched at approximately 36 m s1 with 0.25 rev/s of sidespin. The effects of a change in initial orientation and ball type on the resulting trajectory were investigated. At an initial orientation of 0ı , Ball 1 showed large amounts of bend two or three times during the flight, whereas Ball 2 did not appear to bend a great deal, as shown in Fig. 18. An indication of general behaviour was given by the final positions of the balls with initial orientations ranging from 0ı to 180ı,

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Fig. 18 Trajectories of Balls 1 and 2 at an initial orientation of 0ı

Fig. 19 Final positions for initial orientations ranging from 0–180ı for Balls 1 and 2

shown in Fig. 19. The ranges were 1.41 m for Ball 1 and 1.18 m for Ball 2 (a reduction of 16%). Thus it can be concluded that balls that swerve more do not necessarily have a larger range of final positions (because they are more likely to self-correct); the frequency of swerve is important as well as the magnitude of the force coefficient. This analysis did not take into account all possible combinations of initial orientation and rotation axis and therefore did not cover the full range of possible trajectories. This means that the results can only be used to give a general idea of how a ball might behave, which are discussed below.

5.3.5 Factors Affecting the Trajectory The analysis of the results brought up the following factors that appear to affect the trajectory of a ball launched with low amounts of spin:

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 Number of seams: more seams (especially ones that are perpendicular to the flow

near separation) increase the maximum force coefficient, which would increase the amount of swerve, which may or may not lead to a less consistent range (depending on the frequency of force coefficient variation, spin rate and initial orientation).  Depth of seams: deep seams increase the maximum force coefficient, which would increase the amount of swerve.  Symmetry of pattern in a given direction: the less symmetry about an axis, the larger the force coefficient is perpendicular to that axis and the flow direction, increasing the amount of swerve; if the pattern is symmetrical, the force coefficient would be zero perpendicular to the axes of symmetry.  Frequency of repeating pattern in flow direction: a lower frequency pattern (i.e. larger panels in the flow direction) reduces the frequency of the force variation, which would reduce the number of swerves during flight and could make the flight less consistent (depending on the spin rate and the initial orientation). The most consistent balls are the ones that have the optimum combination of maximum force coefficient and frequency relative to the amount of applied spin.

5.4 Conclusions A CFD analysis tool was applied to three different soccer ball geometries and the predicted force coefficients entered into a trajectory prediction programme. Highvelocity, low-spin kick types, such as those used for direct free-kicks and volleys, were examined. The following conclusions relating to ball flight were drawn:  The drag coefficient of a soccer ball is only significantly affected by the addition









of a row of large seams at approximately 90ı from the front of the ball (Ball 3), which induces early separation and increases the drag coefficient by approximately 50%. In comparison to a standard 32-panel ball, this ball was predicted to end up approximately 0.5 m lower in the goal, which it reaches 0.1 s later and moving approximately 5 m s1 slower. Both the lift and side force coefficients were found to vary significantly with orientation and with ball type due to asymmetrical geometry causing variation in separation around the seams. Erratic trajectories were observed, in which the balls could move suddenly up and down or side-to-side in the air. These trajectories altered with the initial placement of the ball. The nature of the trajectories were dependent on ball geometry, and the standard 32-panel ball and the 14-panel 2006 World Cup ball were found to behave quite differently – the World Cup ball behaved more consistently and had a range of final positions in the goal when the initial orientation was varied that was 16% lower than the standard ball.

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Ball consistency is a ball property noticed by players and spectators alike. Interesting reactions from the players go hand-in-hand with the introduction of new balls and new technologies – this was especially apparent with the introduction of the adidas Teamgeist for the 2006 World Cup. The work discussed here shows that these effects can certainly by real, physical phenomena. The most consistent balls are the ones that have the optimum combination of amplitude and frequency of the varying force coefficient relative to the amount of applied spin. With the introduction of new ball manufacturing techniques, it should be possible to tailor ball surface patterns to give some interesting ball flights. It should be equally possible to optimise a ball’s consistency. In any case, the future of soccer ball design promises to be an exciting and controversial one!

References 1. I. Newton. New theory of light and colours. Philosophical Transactions of the Royal Society, London, 80:3075–3087, 1672. ¨ 2. L. Prandtl. Uber Fl¨ussigkeitsbewegung bei sehr kleiner Reibung. In 3rd International Mathematical Congress, Heidelberg. 3. B. Massey. Mechanics of Fluids. Seventh Edition. Cheltenham, UK, 1998. 4. E. Achenbach. Experiments on the flow past spheres at very high reynolds numbers. Journal of Fluid Mechanics, 54:565–575, 1972. 5. M. van Dyke. An album of fluid motion. Parabolic Press, Stanford, 1982. 6. S. Taneda. Visual observations on the flow past spheres at reynolds numbers between 10,000 and 1,000,000. Journal of Fluid Mechanics, 85:187–192, 1978. 7. M.J. Carr´e, T. Asai, T. Akatsuka, and S.J. Haake. The curve kick of a football 2: flight through the air. Sports Engineering, 5:183–192, 2002. 8. S. Barber, S.B. Chin, and M.J. Carr´e. Sports ball aerodynamics: a numerical study of the erratic motion of soccer balls. Computers and Fluids, 38(6):1091–1100, 2009. 9. S. Barber. The aerodynamics of association footballs. PhD thesis, University of Sheffield, 2007. 10. J.M.T. Penrose, D.R. Hose, and E.A. Trowbridge. Cricket ball swing: a preliminary analysis using cfd. The Engineering of Sport, 1996. 11. K. Aoki, M. Nonaka, T. Goto, M. Miyamoto, and M. Sugiura. Effect of the dimple structure on the flying characteristics and flow patterns of a golf ball. In M. Hubbard, R.D. Mehta, and J.M. Pallis, editors, The Engineering of Sport 5, Davis, California, 2004. 12. H.C. Kim, K. Nakahashi, and H.J. Kim. Three-dimensional flow analysis around a cylinder with dimples. In Asia-Pacific Congress on Sports Technology, Tokyo, Japan, 2005. Australasian Sports Technology Alliance. 13. S. Barber, S. Seo, T. Asai, and M.J. Carr´e. Investigating the effects of orientation on the flight of a non-spinning soccer ball. In Asia-Pacific Congress on Sports Technology, Tokyo, Japan, 2007. Australasian Sports Technology Alliance. 14. T. Asai, K. Seo, O. Kobayashi, M. Ajiki, and S. Shiozawa. A fundamental study on aerodynamics of soccer ball. In 83rd Japan Society of Mechanical Engineering Conference (Fluid engineering division), 2005. 15. S. Barber, S.J. Haake, and M.J. Carr´e. Using cfd to understand the effect of seams on soccer ball aerodynamics. In E. Moritz and S.J. Haake, editors, The Engineering of Sport 6, Munich, Germany, 2006.

Aerodynamics of an Australian Rules Foot Ball and Rugby Ball Firoz Alam, Aleksandar Subic, Simon Watkins, and Alexander John Smits

Abstract The aerodynamic behavior of a Rugby ball and an Australian Rules foot ball is complex and significantly differs from spherical sports balls due to their complex ellipsoidal shapes. Although prior aerodynamic studies have been conducted on soccer, tennis, cricket and golf balls, scant information about the Australian Rules and Rugby balls is available in the public domain. In order to understand the aerodynamic properties of Rugby and Australian Rules foot balls, experimental and computation studies have been undertaken for a range of speeds and yaw angles. The airflow around Rugby and Australian Rules foot balls was visualized and the average drag coefficients for both balls were determined and compared. Minor Reynolds number sensitivity at zero yaw angle was found in the experimental studies for both balls. However, significant Reynolds number variations were noted at yaw angles between 75ı and 85ı . In contrast, no major Reynolds number dependency was found in the Computational Fluid Dynamics (CFD) results for both balls.

Nomenclature D S  V A NPL AFL CFD

Drag force (N) Side force (N) Air density .kg=m3 / Wind velocity .m=s/ Projected frontal area .m2 / National Physical Laboratory Australian Football League Computational Fluid Dynamics

F. Alam (B) School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia 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 5, 

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1 Introduction Airflow around a sport ball plays a significant role in performance as it influences the speed, motion, trajectory and ultimately place of landing of the ball. Despite the popularity of games such as Rugby1 and Australian Rules foot ball2 , there appears to be scant research on ball aerodynamics apart from some fundamental studies of the effect of Reynolds numbers around circular cylinders and spheres and some work on axisymmetric flows around spheroids, Chang et al. [1]. In the recent World Rugby Cup it was clearly evident that distance kicking plays an increasingly significant role in the outcome of the game. In Australia, a game known as “Australian Rules” uses a somewhat similar shaped ball (hereafter denoted the Australian Football League (AFL) ball – details about the balls will be given later). Some exciting moments during the AFL and Rugby games are shown in Fig. 1. Although AFL and Rugby games are very popular and widely played around the world, little knowledge on the aerodynamic properties of AFL and Rugby balls are available in the open literature. Prior aerodynamic studies on spherical sport balls include Alam et al. [4], Mehta et al. [5], Mehta and Pallis [6] and Smits and Ogg [7]. However, no knowledge about the aerodynamic forces of more “ellipsoidal” balls is available in the public domain except a limited study conducted by Alam et al. [8,9] and Seo et al. [10]. The airflow around a Rugby ball and AFL ball is believed to be very complex and three dimensional, especially because these balls can experience both lateral and longitudinal rotational motion during flight. The primary objective of this study, therefore is to investigate the steady-state aerodynamic properties,

a) An exciting moment in AFL game [10]

b) An exciting moment in Rugby game [11]

Fig. 1 AFL and Rugby games. (a) An exciting moment in AFL game [2]. (b) An exciting moment in Rugby game [3]

1

Rugby is played in over 120 countries across four continents. Australian Rules Foot ball is played over 60 countries and has great influence on Australian national culture and the use of leisure time. In 2000, AFL football attracted 6.3 million spectators in Australia alone. 2

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such as drag and side force (or lift) acting on a Rugby ball and an AFL ball. Ultimately the work will be extended to understanding the complexities of spinning balls but here it is restricted to non-spinning flight. The steady aerodynamic properties were measured experimentally for a range of wind speeds and yaw angles to simulate the crosswinds effects. Additionally, Computational Fluid Dynamics (CFD) was employed to study the aerodynamic properties of a simplified Rugby ball and an AFL ball and the simulated results were compared with the experimental findings.

2 Experimental Procedure and Equipment 2.1 Description of Balls The new Summit Rugby ball and the AFL Sherrin ball (made by Heritage Sports Australia and Sherrin Australia, respectively) were selected for this work as they are officially used in tournaments around the world including Australia. The dimensions of the Rugby ball used for this work are 280 mm in length and 184 mm in diameter. The ball is made of four synthetic rubber segments as shown in Fig. 4. External dimensions of a typical Rugby ball are shown in Fig. 5. The dimensions of the AFL ball are approximately 276 mm in length and 172 mm in diameter when inflated to between 62–76 kPa. The ball is made of four leather segments (see Fig. 2). The ball’s external dimensions are shown in Fig. 3. Although the AFL ball was closely circular in cross-section, the Rugby ball departed from a circular section by up to about 10 mm. A sting mount was used to hold each ball, and the experimental set up in the wind tunnel test section is shown in Fig. 7. The aerodynamic effect of sting on the ball was measured and found to be negligible. The distance between the bottom edge of the ball and the tunnel floor was 420 mm, which is well above the tunnel boundary layer and considered to be out of significant ground effect.

a) Side view (Longitudinal)

b) Side view (Lateral)

Fig. 2 Australian Rules Foot ball (AFL ball). (a) Side view (Longitudinal). (b) Side view (Lateral)

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Fig. 3 External dimension of an AFL ball. (a) Side view (Longitudinal). (b) Side view (Lateral)

AFL Ball

AFL Ball

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172 mm

a) Side view (Longitudinal)

b) Side view (Lateral)

a) Side view (Longitudinal)

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Fig. 4 Rugby ball. (a) Side view (Longitudinal). (b) Side view (Lateral)

Rugby Ball

Rugby Ball

280 mm

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b) Side view (Lateral)

Fig. 5 External dimension of a Rugby ball. (a) Side view (Longitudinal). (b) Side view (Lateral)

2.2 Experimental Set Up and Equipment The RMIT Industrial Wind Tunnel was used to measure the aerodynamic properties of an AFL ball and a Rugby ball. The tunnel is a closed return circuit wind tunnel with a maximum speed of approximately 150 km h1 . The rectangular test section

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Fan Anechoic Turning Vanes Diffuser

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Car Entrance Retractable Turning Vanes

Anechoic Turning Vanes Turning Vanes

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Heat Bench System

Control Panel

Fig. 6 A plan view of RMIT University Industrial Wind Tunnel, Alam et al. [11]

a) Experimental setup in wind tunnel (Rugby ball)

b) Experimental setup in wind tunnel (AFL ball)

Fig. 7 Experimental setup in RMIT University Industrial Wind Tunnel. (a) Experimental setup in wind tunnel (Rugby ball). (b) Experimental setup in wind tunnel (AFL ball)

dimension is 3 m (wide) 2 m (high) 9 m (long), and is equipped with a turntable to yaw the model. A plan view of the tunnel is shown in Fig. 6. The tunnel was calibrated before conducting the experiments and tunnel’s air speeds were measured via a modified National Physical Laboratory (NPL) ellipsoidal head Pitot-static tube (located at the entry of the test section) connected to a MKS Baratron pressure sensor. The mounting stud (sting) holding the ball was mounted on a six component force sensor (type JR-3), and purpose made computer software was used to digitize and record all 3 forces (drag, side and lift forces) and 3 moments (yaw, pitch and roll moments) simultaneously. More details about the tunnel can be found in Alam et al. [11].

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3 Computational Fluid Dynamics Models and Computational Procedure A simplified (e.g., smoother oval shape compared to real balls) Rugby ball and AFL foot ball with four segments was developed by using Solid Works . The models are shown in Fig. 8 and Fig. 11. A smooth oval shape was achieved by joining all four equal size segments. The joint between the two segments possesses a seam to approximate the real ball. The Rugby model was of length 280 mm, width (diameter) 184 mm and seam radius 2 mm. The AFL ball model was of length 275 mm, diameter 170 mm and seam radius 5 mm. The simplified models with seams were imported into GAMBIT, a pre-processor of the CFD software used here, FLUENT 6.1. A wind tunnel with reduced dimension (in order to reduce computational time) was also created using GAMBIT with a length of 2,500 mm, width 2,000 mm and height 2,000 mm. The size function was used to mesh the volume, and a tetrahedron grid with mid-edged nodes was used to mesh the ball. A total of 900,000 cells were required to mesh the model effectively. The finished meshed models are shown in Figs. 9 and 10. The inlet boundary condition was a constant velocity condition, and the outlet was a constant pressure condition. The other boundaries were specified as walls including the Rugby ball and the AFL ball, which is also considered as wall. The boundary parameters are shown in Table 1. The accuracy of the CFD solution is primarily governed by the number of cells in a grid, a larger number of cells equates to a more accurate solution. However, an optimal solution was achieved by using fine mesh at locations where the flow experienced large gradients in velocity and relatively coarse mesh where airflow was more uniform. The 2nd order upwind interpolation scheme was used for all computations.

Fig. 8 Simplified CFD model of Rugby ball

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Fig. 9 Surface meshing of the Rugby ball in CFD

Fig. 10 Wind tunnel meshing in CFD

Figure 9 shows a computer generated model of the Rugby ball with the tetrahedron mesh. Generally, a structured (rectangular) mesh is preferable to tetrahedron mesh as it tends to give more accurate results but there are difficulties in using structured meshes in complex geometry. Seven models were constructed to simulate the yawed wind conditions (crosswinds effects): one for each yawed condition. A verification of grid independency was performed and the 900,000 cells used here gave

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Fig. 11 Simplified CFD model of AFL Foot ball Table 1 CFD modeling boundary parameters 1 2 3 4 5

Description

Boundary Conditions

Inlet Outlet Rugby ball AFL ball Control volume

Velocity Inlet Pressure Outlet Wall Wall Wall

grid independent results. In order to accelerate the computations, a parallel solver was used. Each run took around 8 h (wall clock time). The Segregated (Implicit) solver was used for the computation as it is faster and produced results close to experimental findings. The k-epsilon model with enhanced wall treatment was used for the turbulence modeling. The non-equilibrium wall function was used as the flow is complex involving separation, re-attachment and impingement. Other models such as the k-omega model were also used but no significant differences were observed. The velocity and turbulence were specified at the flow inlet. The flow inlet velocity was varied from 60 to 140 km h1 (typical speeds achieved in AFL and Rugby games) in an increment of 20 km h1 , and the flow angles were varied between ˙90ı yaw angles in increments to compare the CFD modeling results with experimental findings. The direction of airflow was normal to the inlet and the reference frame was set as absolute for the velocity. The convergence criterion for continuity equations was set to be 1  105 (0.001%).

4 Results and Discussion 4.1 Experimental Results The AFL ball and the Rugby ball were tested at 60, 80, 100, 120, and 140 km h1 speeds under ˙90ı yaw angles with an increment of 10ı . The balls were yawed relative to the force sensor (which was fixed with its resolving axis along the mean flow direction) thus the wind axis system was employed. Flow visualizations by

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wool tufts at 0ı and 90ı yaw angles for the AFL ball and the Rugby ball are shown in Figs. 12 and 13 respectively. The direction of wind with respect to the direction of drag force .D/ and side force .S / is shown in Fig. 14. The aerodynamic forces were converted to non-dimensional parameters (drag coefficient, CD and side force coefficient, CS ) and tare forces were removed by measuring the forces on the sting in isolation and removing them from the force of the ball and sting. The influence of the sting on the balls was checked and found to be negligible. Only drag force and side force coefficients are presented in this work and they are plotted against yaw angles. The CD and CS were defined as CD D 1 D 2 and CS D 1 S 2 , where A is =2 V A

2

=2 V A

. The repeatability of the the projected frontal area of the ball defined as A D  D 4 measured forces was within ˙0:1 N and the wind velocity was less than 0:5 km h1 . The drag coefficient .CD / variation with velocity and yaw angle for the AFL ball and the Rugby ball are shown in Figs. 15 and 16 respectively, and the corresponding side force coefficients .CS / are shown in Figs. 17 and 18. At zero yaw angle, the Rugby ball had the higher drag coefficient . 0:18/ than the AFL ball . 0:10/ probably caused by the lower aspect ratio of the Rugby ball. With increasing yaw angle, the drag coefficient increases due to a large and very complex flow separation. An average increase of CD at 90ı yaw angle for the AFL ball and the Rugby ball is approximately 0.65 and 0.55 respectively. No significant Reynolds number (varied by tunnel wind speeds) dependence was found at zero

Wind Direction Wind Direction

a) 0° yaw angle

b) 90° yaw angle

Fig. 12 Airflow around an AFL ball at 100 km h1 (a) 0ı yaw angle (b) 90ı yaw angle

Wind Direction Wind Direction

a) 0° yaw angle

b) 90° yaw angle

Fig. 13 Airflow around a Rugby ball at 100 km h1 (a) 0ı yaw angle (b) 90ı yaw angle

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D Wind Direction

S Fig. 14 Direction of drag (D) and side force (S) relative to wind direction (top view) 1.00

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–80

–60

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0.00 –20 0 20 Yaw Angle in degree

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Fig. 15 Experimentally measured drag coefficient .CD / as a function of yaw angles and speeds for AFL ball

yaw angle for the AFL ball except at the lowest speed tested and at yaw angles between C80ı and C90ı . At these high yaw angles (between C80ı and C90ı ) for the AFL ball, a variation in the drag coefficients with speeds was noted. However, the variation was negligible at high speeds (e.g., 120 km h1 and over). No significant Reynolds number variation was seen for the Rugby ball except at the lowest speed .60 km h1 / as shown in Fig. 16. The side force coefficients .CS / for both balls displayed a similar behavior (see Figs 17 and 18). They display a minor off-set from 0ı yaw angle for the Rugby ball which is believed to be due to a small mounting error. A Reynolds number dependence in CS for the AFL ball is clearly evident in between 50ı and 90ı yaw

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–60

–40

–20

0

20

40

60

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Yaw Angle in degree

Fig. 16 Experimentally measured drag coefficient .CD / as a function of yaw angles and speeds for Rugby ball

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0.80

120 k m /h 140 k m /h

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–80

–60

–40

–20

0

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–0.20 –0.40 Yaw Angles in degree

Fig. 17 Experimentally measured side force coefficient .CS / as a function of yaw angles and speeds for AFL ball

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–80

–60

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–20

0

20

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Fig. 18 Experimentally measured side force coefficient .CS / as a function of yaw angles and wind speeds for Rugby ball

angles, and a minor dependence was also noted for the Rugby ball at the lowest Reynolds number .60 km h1 /. The flow visualizations with wool tuft around the AFL ball and Rugby ball shown in Figs. 12 and 13 indicate that the flow is complex and three dimensional. Broadly speaking, flow separation starts at approximately three-quarter length from the front edge at zero yaw angle for both types of ball, but the surface flow patterns in the separated zones at 90ı yaw angle are complicated due to three dimensional flow separations. A comparison of drag coefficients at all speeds and yaw angles for the Rugby ball and AFL ball indicates that there is a slight lack of symmetry in the results (Figs. 15 and 16). Whilst some errors are associated with airflow and force sensor asymmetry (due to small misalignment of sensor axis and ball axis) and some minor deflections of the ball under wind loading (mainly from the bending of the support strut), the errors are primarily due to asymmetries in the ball themselves.

4.2 Computational (CFD) Results 4.2.1 Rugby Ball The CFD simulation was conducted at a range of speeds (60–120 km h1 with an increment of 20 km h1 /. The velocity vectors and static pressure distributions around the ball at 0ı and 90ı yaw angles for the 100 km h1 speed are shown in

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Fig. 19 Velocity vectors (in m s1 ) around the simplified Rugby ball at 0ı yaw angle

Figs. 19 to 22 and the drag coefficients .CD / for 60–140 km h1 at ˙90ı yaw angles are shown in Fig. 23. The side force coefficient .CS / distributions as a function of yaw angle and wind speed are plotted in Fig. 24. A significant variation in velocity between 0ı and 90ı yaw angles is evident (see Figs. 19 and 20). At zero yaw angle, the flow separated zone is smaller than in the case of the 90ı yaw angle as expected. No significant Reynolds number variation was found in the CFD results. The CFD computed minimum drag coefficient at 0ı yaw angle was approximately 0.14 compared to 0.18 found experimentally. The drag coefficient increases with yaw angle (see Fig. 23), and the maximum drag coefficient was found at ˙90ı yaw angles ( 0:50 for the CFD compared to 0.55 found experimentally). As expected, no asymmetry of drag coefficients between the positive and negative yaw angles was noted in the CFD study. The side force coefficient demonstrates the highest magnitudes (0.25) at approximately ˙50ı yaw angles. As expected, a zero side force coefficient was found at zero yaw angle.

4.2.2 Australian Football League Football The CFD simulation for the AFL ball was conducted at speeds of 60 to 140 km h1 with an increment of 20 km h1 . The velocity vectors and static pressure distributions around the ball at 0ı and 90ı yaw angles for 140 km h1 speed are shown in Figs. 25 to 30. The computed drag coefficients .CD / at ˙90ı yaw angles for 60–140 km h1 are shown in Fig. 31. The computed side force coefficient .CS / distribution as a function of yaw angle and wind speed is presented in Fig. 32.

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Fig. 20 Velocity vectors (in m s1 ) around the simplified Rugby ball at 90ı yaw angle

Fig. 21 Computed static pressure (in Pa) distribution around the simplified Rugby ball at 0ı yaw angle

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Fig. 22 Computed static pressure (in Pa) distribution around the simplified Rugby ball at 90ı yaw angle

1.00

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Drag Coefficient, CD

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 –100

–80

–60

–40

–20

0

20

40

60

80

100

Yaw Angle in degree

Fig. 23 Computed drag coefficient .CD / as a function of yaw angles and wind speeds for simplified Rugby ball

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–80

–60

–40

–20

–0.10

0

20

40

60

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–0.20 –0.30 –0.40 Yaw angle in degree

Fig. 24 Computed side force coefficient .Cs / as a function of yaw angles and wind speeds for simplified Rugby ball

Fig. 25 Flow around a simplified AFL ball at 0ı yaw angle

Aerodynamics of an Australian Rules Foot Ball and Rugby Ball

Fig. 26 Flow around a simplified AFL ball at 90ı yaw angle

Fig. 27 Static pressure (in Pa) distribution around the simplified AFL ball at 0ı yaw angle

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Fig. 28 Static pressure (in Pa) distribution around the simplified AFL ball at 90ı yaw angle

Fig. 29 Velocity vectors (in m s1 ) around the simplified AFL ball at 0ı yaw angle

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Fig. 30 Velocity vectors (in m s1 ) around the simplified AFL ball at 90ı yaw angle

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1.00

80 km/h

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100 km/h

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120 km/h 140 km/h

Drag Coefficient, CS

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–100

–80

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–40

0.00 –20 0 20 Yaw Angle in degree

40

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Fig. 31 Drag coefficient .CD / as a function of yaw angles and wind speeds for simplified AFL ball

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Side Force Coefficient, Cs

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–80

–60

–40

–20

0

20

40

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–0.20 –0.40 –0.60 –0.80 –1.00 Yaw Angle in degree

Fig. 32 Side force coefficient .Cs / as a function of yaw angles and wind speeds for simplified AFL ball

Between 0ı and 90ı yaw angle, the flow field changes significantly (see Figs. 25–26, 29–30). The general flow features are similar to that seen for the Rugby ball in all respects, except the behavior of the separated flow regions in CFD simulation of AFL ball as the laminar flow and boundary layer transition were not modeled. However, study is underway to take into account these effects and the findings will be published as soon as data is available. No Reynolds number variation for the AFL ball was found in the CFD results. The computed minimum drag coefficient at zero yaw angle was approximately 0.14 compared to 0.11 found experimentally, and at ˙90ı yaw angles, it was approximately 0.57 compared to 0.65 found experimentally. It is believed that the higher drag coefficient is due to not modeling turbulent boundary layer. As mentioned earlier, an effort is undertaken to study this effect. In CFD study, as expected, no asymmetry of drag coefficients between the positive and negative yaw angles was noted in contrast to the experimental findings where significant variation in drag coefficients between the highest positive and negative yaw angles was found (see Fig. 15 for experimental results). The side force coefficient demonstrates the highest magnitudes (0.47) at approximately ˙60ı yaw angles. As expected, zero side force coefficient was found at zero yaw angle. The velocity vectors clearly indicates the separated flow region at 90ı yaw angles (Fig. 26), however no apparent flow separation is noted at zero yaw angle (see Fig. 25). Further CFD study is undertaken for the AFL ball to clarify it by modeling laminar flow and turbulent boundary layer transition.

Aerodynamics of an Australian Rules Foot Ball and Rugby Ball

123

5 General Discussion The flow visualization with wool tuft (and smoke not shown here) around the Rugby ball and the AFL ball indicated the presence of complex, three-dimensional separation zones on the leeward side of the balls with a larger separated region present at 90ı yaw angle compared with 0ı yaw angle. The experimentally determined drag coefficient for the Rugby ball (0.18 at 0ı yaw angle and 0.60 at 90ı yaw angle) is higher than the computed drag coefficient (0.14 at 0ı yaw angle and 0.50 at 90ı yaw angle). For the AFL ball, the experimentally measured drag coefficient is 0.10 at zero yaw angle compared to the CFD value of 0.14 and it is 0.70 at 90ı yaw angle compared to the computed CD of 0.48. A comparison of measured and computed CD and CS values for 100 km h1 speed is shown in Figs. 33 to 36 for the Rugby and AFL balls respectively. Further study is currently being undertaken to clarify the computational results for AFL ball at zero yaw angle by modeling laminar flow and turbulent boundary later. At 90ı yaw angle, the experiment revealed a small Reynolds number dependence for both balls, while only a small dependence at lower Reynolds numbers was noted in the computation for the Rugby ball, and no Reynolds number variation was found in the CFD results for the AFL ball. However, further computational study is required to understand why there was no Reynolds number variation in AFL ball results. The Rugby ball and the AFL ball were found to be slightly asymmetrical along the longitudinal axis. The Rugby ball surface was also rough and it was not perfectly oval in shape as it was made of four segments. In contrast, the surface of the AFL ball was smooth although it was made of four segments as well. For the CFD analysis, the Rugby ball was assumed

1.0

100 km/h, EFD

0.9

100 km/h, CFD

Drag Coefficient, CD

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 –100

–80

–60

–40

0.0 –20 0 20 Yaw Angle in degree

40

60

80

100

Fig. 33 Experimental (EFD) and Computed (CFD) drag coefficients .CD / as a function of yaw angles for Rugby ball

124

F. Alam et al. 1.0 100 km/h, EFD 0.9

100 km/h, CFD

0.8 Drag Coefficient, CD

0.7 0.6 0.5 0.4 0.3 0.2 0.1

–100

–80

–60

–40

0.0 –20 0 20 Yaw Angle in degree

40

60

80

100

Fig. 34 Experimental (EFD) and Computed (CFD) drag coefficients .CD / as a function of yaw angles for AFL ball

100 km/h, EFD

1.0

100 km/h, CFD

Side Force Coefficient, CS

0.8 0.6 0.4 0.2 0.0 –100

–80

–60

–40

–20

0

20

40

60

80

100

–0.2 –0.4 Yaw Angle in degree

Fig. 35 Experimental (EFD) and Computed (CFD) side force coefficients .Cs / as a function of yaw angles for Rugby ball

to be completely symmetrical along the longitudinal and lateral axes with a smooth surface. The cross sectional area was approximately circular compared to the real Rugby ball which had a cross sectional geometry that was slightly larger compared to a circular geometry modeled in CFD.

Aerodynamics of an Australian Rules Foot Ball and Rugby Ball

125 100 km/h, CFD

1.0

100 km/h, EFD

Side Force Coefficient, CS

0.8 0.6 0.4 0.2 0.0 –100

–80

–60

–40

–20

0

20

40

60

80

100

–0.2 –0.4 –0.6 Yaw Angle in degree

Fig. 36 Experimental (EFD) and Computed (CFD) side force coefficients .Cs / as a function of yaw angles for AFL ball

The geometries of a real Rugby ball and the AFL ball vary from the designed shape due to the restrictions of the manufacturing process and they are often asymmetrical. In the CFD, an idealized shape was assumed. Nevertheless, CFD and experimental studies show broadly similar results. The error in force coefficient was estimated to be less than 1% in the experimental studies, which is considered to be within acceptable limits. We expect, therefore, that the experimental approach is more accurate than CFD as it incorporates more realistic airflow and the real ball. It is worthwhile to mention that it is impossible to replicate all turbulence scales in the wind tunnel environment that generally experienced by the ball in the open air stadium. An average turbulence effect is usually generated in wind tunnel testing. The turbulence intensity of the wind tunnel air in experimental measurements was approximately 1.8%. More details about the RMIT Industrial Wind Tunnel and its air turbulence can be found in Alam et al. [11]. Spherical balls, such as those used for soccer, have better aerodynamic properties and balance than non-spherical balls as the design of soccer balls has undergone a series of technological changes over the years, in which the ball has been made more spherical and aerodynamically efficient. On the other hand, the AFL ball and the Rugby ball had not significantly changed since its inception. There is scope to improve the aerodynamic efficiency and predictable flight trajectory of the Rugby and AFL balls by improving design and making more ellipsoidal shape.

126

F. Alam et al.

6 Concluding Remarks The following conclusions are made from the work presented here:  The aerodynamics resulting from the flight of irregular shaped sporting balls is

extremely complex even when the ball is not spinning.  The average drag coefficients for the AFL ball and Rugby ball at zero yaw angle

are 0.10 and 0.18 respectively.

 The maximum drag coefficients for the Rugby ball and the AFL ball at 90ı yaw

angle are approximately 0.68 and 0.80 respectively.  The average drag coefficient found by CFD for the Rugby ball at zero yaw angle

is lower than the experimentally determined results.  The average drag coefficient found by CFD for the AFL ball at zero yaw angle is

higher than the experimentally determined results which is believed to be due to not modeling turbulent boundary layer.  The average drag coefficient at 90ı yaw angle determined by CFD is significantly lower compared to experimentally found results for both AFL and Rugby balls.  The CFD findings are indicative only. The experimental study is more reliable due to the complexity of 3D oval shapes of AFL ball and Rugby ball and the inherent limitations of CFD incorporating two-equation turbulence models.  The primary goal of this work as part of a larger study is to understand the aerodynamic behavior of the current Rugby and AFL balls in order to develop new generation balls with improved aerodynamic performance.

7 Recommendations for Further Work We are in the process of finalizing a rig to experimentally spin balls about the longitudinal and lateral axes, and also propose to model this in CFD. Additionally, we plan to enhance the accuracy of the CFD by refining the mesh in the areas of the seams and to conduct further studies on the influence of the turbulence models, especially modeling the laminar flow and turbulent boundary layer transition for the AFL ball. As mentioned previously, the ultimate objective of the future work is to develop new generation Rugby and AFL balls with improved aerodynamic performance by designing more ellipsoidal shape. Acknowledgements The authors express their sincere thanks to Mr. Wisconsin Tio, Mr. Pek Chee We and Mr. Lachlan Orr, School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Melbourne, Australia for their invaluable assistance with the CFD modeling of the Rugby ball and AFL ball. The authors are also highly indebted to Mr. Greg McClure, Heritage Sports Australia for his strong encouragement and support.

Aerodynamics of an Australian Rules Foot Ball and Rugby Ball

127

References 1. Chang C-C, Liou B-H and Chern R-L (1992), “An Analytical and Numerical Study of Axisymmetric Flow around Spheroids”, Journal of Fluid Mechanics 234, 219–246. 2. http://www.australianrules.com.au (accessed on 20 March, 2008) 3. http://uk.sports.yahoo.com (accessed on 22 March 2008) 4. Alam, F., Watkins, S. and Subic, A. (2004), “The Aerodynamic Forces on a Series of Tennis Balls”, Proceedings of the 15th Australasian Fluid Mechanics Conference, University of Sydney, 13–17 December, Sydney, Australia. 5. Mehta, R. D., Alam, F. and Subic, A. (2008), “Aerodynamics of tennis balls- a review”, Sports Technology, 1(1), 1–10. 6. Mehta, R. D. and Pallis, J. M. (2001), “The aerodynamics of a tennis ball”, Sports Engineering, 4(4), 1–13. 7. Smits, A. J. and Ogg, S. (2004), “Golf ball aerodynamics”, The Engineering of Sport 5, ISBN 0–9547861–0–6, 3–12. 8. Alam, F., Subic, A., Watkins, S., Naser, J. and Rasul, M. G. (2008), “An Experimental and Computational Study of Aerodynamic Properties of Rugby Balls”, WSEAS Transactions on Fluid Mechanics, 3(3), 279–286. 9. Alam, F., Subic, A. and Watkins, S. (2005), “Measurements of aerodynamic drag forces of a Rugby ball and Australian Rules football” in The Impact of Technology on Sport 1 (edited by A. Subic and S. Ujihashi), ISBN 0–646–45025–5, Tokyo, Japan. 10. Seo, K., Kobayashi, O. and Murakami, M. (2004), “Regular and irregular motion of a rugby football during flight”, The Engineering of Sport 5, ISBN 0–9547861–0–6, 567–573. 11. Alam, F., Zimmer, G. and Watkins, S. (2003). Mean and time-varying flow measurements on the surface of a family of idealized road vehicles, Journal of Experimental Thermal and Fluid Sciences, 27 (5), 639–654.

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