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Journal of Science and Medicine in Sport 13 (2010) 444–451

Review

Hydrodynamic glide efficiency in swimming

Roozbeh Naemi a,∗, William J Easson b, Ross H Sanders a

a Department of Physical Education, The University of Edinburgh, Edinburgh, United Kingdomb Institute of Material and Processes, School of Engineering & Electronics, The University of Edinburgh, Edinburgh, United Kingdom

Received 4 October 2008; received in revised form 2 March 2009; accepted 17 April 2009

bstract

The glide is a major part of starts, turns and the stroke cycle in breaststroke. Glide performance, indicated by the average velocity, can bemproved by increasing the glide efficiency, that is, the ability of the body to minimise deceleration. This paper reviews the factors that affectlide efficiency. In the first part of the review the sources of resistive force are reviewed including surface friction (skin drag), pressure (form)rag and resistance due to making waves (wave drag). The effect of body surface characteristics on the skin drag, the effect of the depth of

he swimmer on wave drag, and the effects of posture and alignment, body size and shape on the form drag are reviewed. The effects of theseariables on the added mass, that is, the mass of water entrained with the body are explained. The ‘glide factor’ as a measure of glide efficiencyhat takes into account the combined effect of the resistive force and the added mass is described. In the second part methods of quantifyinghe resistive force are reviewed. Finally, the ‘hydro-kinematic method’ of measuring glide efficiency is evaluated.

2009 Sports Medicine Australia. Published by Elsevier Ltd. All rights reserved.

eywords: Glide performance; Resistive forces; Added mass; Inertia; Glide factor; Hydro-kinematic

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4442. Part 1: factors affecting glide efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

3. Part 2: methods of determining hydrodynamic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

. . . .. . . . .. . . . .

apping, and the feet together and plantar flexed. Maintainingpassive streamlined posture during the underwater period

f starts and turns is beneficial when the body’s velocity isigher than that which can be sustained by kicking.1

∗ Corresponding author.E-mail address: roozbeh.naemi@education.ed.ac.uk (R. Naemi).

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440-2440/$ – see front matter © 2009 Sports Medicine Australia. Published by Eloi:10.1016/j.jsams.2009.04.009

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Average velocity over the period of the glide is an indi-ator of glide performance. It depends on the initial velocity,he magnitude of deceleration, and the glide duration. Initialelocity of the glide is related to the preceding actions ands affected by characteristics of the pre-glide phase includingntry after the start, or the push-off force and body positionuring wall contact in turns. According to Newton’s secondaw of motion, the deceleration during a glide depends on theesistive forces applied to a body and its inertial properties.

The resistive forces act opposite to the direction of travelnd their magnitude is highly related to velocity. Inertia of aliding body is the sum of the body mass plus the mass of

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

The ‘glide’ refers to phases in swimming races duringwhich the swimmer attempts to maintain speed withoutactions to propel the body. Glide phases occur during starts,turns, and within stroke cycles of breaststroke. During theglide phases of starts and turns swimmers typically adopt a‘streamlined’ position characterised by an elongated posturewith arms extended forward with hands pronated and over-

ater entrained with the body. This mass of entrained water,eferred to as ‘added mass’, adds to the inertia and there-ore reduces the rate of slowing of a gliding body. This massf entrained water, referred to as ‘added mass’, adds to the

sevier Ltd. All rights reserved.

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R. Naemi et al. / Journal of Science

nertia and therefore reduces the rate of acceleration of theody during the propulsive phases and can in this respect benegative factor in the preceding actions such as a push from

he wall during a turn. But the added mass reduces the ratef slowing of the body during the glide when only resistiveorces are acting and the body is decelerating. The inertia ofgliding body, being the sum of body mass and the addedass, is termed its ‘virtual mass’.Glide efficiency can be defined as the ability of a gliding

ody to maintain its velocity through time and to minimiseeceleration at each corresponding velocity.2 The lower theesistive force and the higher the virtual mass, the lowerhe deceleration at each corresponding velocity and thushe higher the glide efficiency. Although increasing ini-ial velocity of a glide is one of the ways to optimise thelide performance, improving glide efficiency can be bene-cial because an advantage is gained without increasing theetabolic cost.Factors that affect glide efficiency include posture, align-

ent, and anthropometric and morphological characteristicsf the body. These factors are explored in the first part of thiseview. The methods of measuring glide efficiency and theactors that affect glide efficiency are reviewed in the secondart of the review.

. Part 1: factors affecting glide efficiency

To have a better understanding of the hydrodynamicactors contributing to the glide efficiency, knowledge of dif-erent types of flow and their characteristics is essential. Twoistinct conditions of flow around a body are referred to aslaminar’ and ‘turbulent’ flow. Laminar flow is characterisedy smooth motion in of fluid in ‘layers’. The turbulent flows characterised by the random three-dimensional motion ofhe fluid particles superimposed on the mean motion.3

Whether the flow acting on a body is considered laminar orurbulent can be determined by the Reynolds number whichefines the magnitude of the inertial to the viscous forces onhe flow particles acting on a body. This can be calculated by

e = ρ.v.L

μ(1)

here ρ is the fluid density, v is the body’s velocity; L is theharacteristic length of the object in the direction of the flownd μ represents a constant known as ‘viscosity’.4

For a smooth flat plate with no surface irregularities,he transition from a laminar to a turbulent flow occurs ateynolds numbers of 5 × 105. Assuming that the transitionccurs at the same Reynolds number, if not lower, for theuman body in a streamlined position, at a velocity of about.5 m/s which is common during the glide phase of starts and

urns, only about 20 cm of the body length, that is only theands, remains in a laminar flow. ‘Skin roughness’ whichepends on the height and shape of irregularities on the sur-ace, influences the amount of random motion of the fluid

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dicine in Sport 13 (2010) 444–451 445

articles, causing the transition to occur even earlier forlides under real conditions. Transition occurs at even lowereynolds numbers in decelerating flow, as is the case for glid-

ng bodies, than for bodies moving with a constant velocity.3

hus it can be concluded that for the ranges of Reynoldsumber corresponding to glides of the human body in com-etitive swimming, turbulent flow is dominant along almosthe whole length of the swimmer.

The resistive force, otherwise known as ‘drag’ acts oppo-ite to the direction of motion of the body and is highly relatedo the flow conditions and the body characteristics. The termpassive drag’ refers to the hydrodynamic resistive force act-ng on a body that is not actively changing the orientationf the body segments, that is, on a body during a glide. Theotion of a body travelling through water close to the sur-

ace is affected by ‘friction’; ‘pressure’ and ‘wave making’esistance.5 Alternative terms are ‘skin drag’, ‘form drag’,nd ‘wave drag’ respectively.

Frictional resistance or ‘skin drag’ is the contribution tohe drag that exists due to the presence of a ‘boundary layer’.ccording to the boundary layer theory the flow around aody is divided into two regions, one close to the body surfacend the other covering the volume beyond the region closeo the body surface. The boundary layer is defined as thatart of the flow adjacent to the body in which the effect ofiscosity is important. In this area the flow velocity at theurface of the body is considered to be zero due to no-sliponditions. At increasing distances from the body surfacehe flow velocity increases until it reaches the ‘free-stream’elocity. The free-stream velocity is expressed relative to theoving body. In the case of a body gliding in a swimming

ool in which the water is initially stationary, the free-streamelocity can be regarded as the velocity of the swimmer. Theocation at which this occurs is known as ‘the boundary layerorder’. Beyond this border, the flow is regarded as withoutriction since the velocities of different layers of the flow arehe same.4

Decreasing roughness to create a smoother surfaceecreases the amount of the frictional resistance for a glidingody. Latex swimming caps and body shaving are believed toeduce the friction drag by decreasing the surface roughness.6

uantifying the contribution of the frictional drag to totalrag has been extremely difficult. Recently, using Com-utational Fluid Dynamics analyses (explained later in theanuscript under the same heading) of a human body instreamlined position at constant velocities, Bixler et al.7

ound that the skin friction corresponds to about one fourthf the total drag when the swimmer is submerged to a depthhere wave drag is negligible. The remaining drag is the

esult of pressure drag. In that study surface roughness wasegarded as zero. For a real swimmer with some surfaceoughness the contribution of friction drag to total drag may

ncrease.

Pressure drag is the result of the differences between pres-ure at the leading and trailing edges of the body. Movinglong the body, the fluid particles in the boundary layer are

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46 R. Naemi et al. / Journal of Science

lowed down by the wall shear stress as a result of the skinriction. When the momentum of the fluid in the boundaryayer is insufficient, the flow cannot follow the curve of theody and separates from the surface. Boundary layer separa-ion results in formation of a relatively low-pressure regionehind the body.4 This region, which is deficient in momen-um, is called ‘wake’ although wake is not necessarily theroduct of separation.8 Separation of the flow from the bodyeads to the formation of large and small eddies at the down-tream part of the body resulting in form drag3 because theddies exert less pressure on the body than the water in thepstream sections that has not yet separated from the body.

Form drag is created as a result of the pressure differenceetween the leading and trailing edges of body. The formrag is equal to the amount of this pressure difference timeshe area to which the pressure is applied. Thus in addition tohe effect of flow separation on drag increasing the projectedrea to the flow increases the pressure drag. Numerous studiesevealed that certain actions like having the head above theater, turning the head to breath, lowering the legs, having

egs and arms abducted and body rolling during the stream-ined glide on the surface would increase the total drag forces

ainly due to an increase in the projected area.9–11 Duringhese actions parts of the body protrude beyond the maximumross-sectional area of the chest thereby increasing the pro-ected area and consequently the pressure drag. An increasen the ‘angle of attack’, that is, the angle of the body to theirection of flow, can also result in an increase in the projectedrea. Using CFD analyses Bixler et al.7 found that angles ofttack of 3◦ and −4.5◦ increased the total drag by 2.3% and.4% respectively compared to a zero angle of attack.

Because of the effect of chest cross-sectional area on theressure drag some anthropometric parameters like the chestirth, depth and breadth have been found to be significantlyorrelated with drag values.12–14 In addition to the anthro-ometric parameters, the shape and the contour of body aremportant factors affecting the pressure drag because theyetermine how the flow moves over the body. For exam-le, the bodies of aquatic mammals are contoured so thathe flow particles remain in laminar streams without separat-ng from the body until near the trailing edge of the body.15

n contrast the shape of the human body causes early separa-ion and the flow is turbulent rather than laminar along mostf its length. Consequently the drag of a human in its mosttreamlined position is approximately five times the drag forsubmerged seal with the same weight at a comparable depthnd velocity.16

Counter-intuitively, turbulence may be produced delib-rately to delay separation and reduce drag. Dimples on aolf ball are a particularly well-known example. The dim-les produce turbulence in the layer closest to the ball, thats, the boundary layer. Having a higher momentum, a turbu-

ent boundary layer is more resistant to separation. Recentlyturbulators’ and ‘turbulence amplifiers’ have been designedy some swim suit manufacturers to increase the turbulence inhe boundary layer to delay or minimise separation to reduce

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dicine in Sport 13 (2010) 444–451

rag. Despite these no factual research has been released byhese companies.

Another way of delaying separation is to mix the highpeed free-stream flow with the low-speed flow in theoundary layer. This increases the momentum at the nearall region to reduce flow separation and consequently theressure drag. ‘Vortex generators’ have been installed onwimsuits upstream of the areas where large separations andurbulence tend to occur. Testing of human replicas in a windunnel has indicated that these can decrease resistance.17

owever, independent testing in water in conditions resem-ling actual glides of swimmers is required.

Like frictional resistance, pressure resistance is hard touantify experimentally. CFD analyses revealed that pres-ure resistance corresponds to about 75% of the total drag ofsubmerged swimmer in a streamlined position at velocitiesetween 1.5 and 2.25 m/s and at depths where wave mak-ng resistance was negligible.7 The contribution of pressureesistance to total drag would be less than 75% when waveaking resistance is present.Wave making resistance or wave drag acts on a body when

oving close to the surface. Part of the energy from the mov-ng body is used to lift the water against gravity resulting inhe formation of waves on the surface.18

The wave drag is highly related to a Froude number (Fr)hat determines the ratio of inertial to gravitational forcespplied to fluid particles. This dimensionless ratio can beuantified as:

r = v√L.g

(2)

here v is the velocity of the moving body, L is the length ofhe body in direction of flow and g is the gravitational acceler-tion constant. It is believed that the wave drag increases withhe Froude number. Extending the arms forward increases theody length and thereby reduces the Froude number and con-equently the wave drag compared to a posture in which therms are against the body. For example it was reported thataving arms on side results in 21.5% more drag compared tohe streamlined position.18

The Froude number can be used to indicate a limitingelocity for a swimmer gliding on the surface. At the Froudeumber of 0.45 where the swimmer with an extended heightf 2.5 m reaches the hull speed of 2.23 m/s the wave lengths equal to the extended height of the swimmer.20 As thewimmer is unlikely to hydroplane this can be considereds a limiting factor for a swimmer gliding on the surface.n addition to the excess effect of wave drag on the surface,he limitation in the maximum achievable speed is anotherimiting factor preventing swimmers to glide on the surfacefter starts and turns.

Wave drag is also dependent on the depth at which the

ody travels.19 At a depth of three times the body thicknesshe wave drag becomes negligible, and has its maximum valuehen it is submerged just beneath the surface. Recently, Lyttle

t al.13 and Vennell et al.20 established that the wave drag is

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R. Naemi et al. / Journal of Science

egligible at a depth of about 0.6 m underwater. It was foundhat at a velocity of 2.5 m/s on the surface the wave dragontributes to at least 40% of the total resistance, while atm/s and at a depth of 0.4 m the wave drag corresponds tonly 15% of the total drag.20 Therefore, in the glide phasesf starts and turns, swimmers should glide at sufficient deptho minimise the effect of the wave drag.

The sum of different types of resistance, ‘total drag’, cane estimated by the following formula:

d = 1

2cd.ρ.A.v2 (3)

d represents the total hydrodynamic resistance, ρ is theater density, v is the velocity of the body and A is the refer-

nce area. In the case of humans, because of the dominancef pressure drag compared to other types of drag for a sub-erged body at an adequate depth, the projected area replaces

he reference area in the total drag formulae.21 When the bodys well aligned to the flow with zero angle of attack the pro-ected area has its minimum value equal to the maximumross-sectional area of the body.

The dimensionless drag coefficient (cd) indicates the levelf ‘streamlining’. It is an empirical constant dependent onhe body shape, angle of attack, surface roughness and theow characteristics. cd has been reported to be between 0.65nd 0.75 for a swimmer in their most streamlined positiont the surface,21 while the drag coefficient of a submergeduman body at an adequate depth was found to be about.30.7 Recently Vennell et al.20 found that the drag coefficientaries with velocity, a fact that is not traditionally consideredn swimming research.

It is also highly popular in the swimming literature toeport a resistive factor as the ratio of total drag force (Dd) tohe velocity squared. A resistive factor (CR) in kg/m incorpo-ates the maximum cross-sectional area (A), the fluid densityρ), and the dimensionless drag coefficient (cd) according tohe following formula:

R = 1

2cd.ρ.A (4)

The resistance to change in motion of a body is its ‘iner-ia’. Thus, a gliding body tends to keep moving withoutny changes in the velocity while the resistance is actingo slow it down. In a vacuum the only source of inertia is the

ass of the body, while in the presence of a fluid a quan-ity of the fluid around the body, not only does the massf body need accelerating (or decelerating) so does somef the adjacent fluid. Thus the apparent (virtual) mass thatesists acceleration is larger than the physical (body) mass.uring a glide, the inertia is the sum of the mass of the swim-er and the added mass of water. This sum is called ‘virtualass’.

The added mass represents the particles of fluid adjacent to

he body that move with the body to varying degrees, depend-ng on their position relative to the body. In principle everyuid particle would accelerate to some extent as the body

nd

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dicine in Sport 13 (2010) 444–451 447

oves, and the added mass is the weighted integration of thisntire mass.22 It can be argued that the added mass is a resultf the same phenomena that contribute to the resistive forces,hese being the boundary layer, the flow separation and theresence of waves at the surface.

The mass of fluid in the boundary layer presents a sourcef added mass. When the fluid is stationary and the bodys moving due to no slip conditions at the body surface (asas explained under skin friction), the fluid particles at the

urface of the body move with the same velocity as theody, while outside the boundary layer the fluid particlestay still. In the boundary layer, shear stress causes the dif-erent layers of fluid to move with speeds that decrease withncreasing distance from the body surface. The moving waterepresents a source of added mass in the areas of attachedow. The amount of the added mass then depends on the

hickness of the boundary layer and the relative velocity dis-ribution of the flow in the boundary layer. A simplifyingssumption is that a fraction of the boundary layer movesith the same speed as the body and the remaining part

tays still. The thickness of that fraction depends on the vis-osity of the fluid.23 The velocity of flow and thickness ofhe boundary layer can be determined using methods thatllow flow visualisation such as CFD.7 However, there is aaucity of data regarding the added mass of gliding humanodies.

Another source of added mass is a result of the wakeormation. In the areas of flow separation a wake formsn which the velocity of the fluid particles relative to theody is zero. During deceleration this bulk of fluid is decel-rated at the same rate as the body and acts as an addednertia.

Added mass increases when a swimmer is gliding closeo the surface. The waves created by a swimmer move at theame velocity as the body and act as a source of added mass.hese include the wave formed at the leading edge of thewimmer, the ‘bow wave’, as well as the waves formed at therailing edge, ‘stern wave’.

The total added mass is the sum of the added massesccording to the contributions of all sources. When a dimen-ionless added mass coefficient (ca) is defined as the ratiof the accelerated mass to displaced mass of the fluid by theody,22 then the added mass can be written according to theollowing formula:

a = ca.ρ.V (5)

here ma is the added mass, ρ is the density of water, andis the body volume. Determining added mass coefficient

equires details of the geometry and physics of the flow, andas determined for certain geometries in a frictionless fluid.2

ecause of the irregular shape of a human body there is

o information available on the amount of added mass foreceleration during a glide.

Like the drag coefficient the added mass coefficientecreases with improvement in streamlining. For a por-

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48 R. Naemi et al. / Journal of Science

oise the added mass coefficient is about 0.045.24 For auman body it can be speculated that in a fixed position andepth the added mass coefficient may vary across velocities.his can be attributed to the fact that changes in velocityould result in changes in the boundary layer thickness

nd variations in the wake volume. Since the wake andoundary layer are the sources of added mass at adequateepth, depending on the extent at which the changes inelocity affect the amount of added mass from these twoources, the added mass coefficient may vary across veloci-ies.

The more streamlined a body the less the added mass.y adopting a streamlined position and gliding at an ade-uate depth, the swimmer decreases the size of the wake andhe amount of wave entrained. This results in a decrease inhe added mass. Although it may seem that the reduction inhe added mass may be detrimental for glide efficiency, theverall effect is beneficial. As the added mass makes a smallroportion of the virtual mass, an equal decrease in the dragnd added mass coefficient will result in an increase in thelide efficiency. This can be seen in Eq. (7) in the followingection.

The hydrodynamic parameters of a body including theesistive and inertial characteristics change with modifica-ions in the body position during a glide. Thus in determininghe effect of different factors on the glide efficiency, theesistive and inertial parameters should be considered in con-unction with each other. Contrary to the resistive force thatan be determined by the conventional methods (which areoing to be explored in the second part of the review) thedded mass cannot be quantified easily. Recently Naemind Sanders2 introduced a ‘glide factor’ as a holisticeasure of glide efficiency that takes into account the

ombined effect of both the resistive and inertial parame-ers.

Naemi and Sanders2 argued that based on the equation ofotion of a representative body during a glide (Eq. (6)), at

ach corresponding velocity, the higher the virtual mass andhe lower the resistive factor the less the body decelerates,eading to a higher glide efficiency.

.dvx

dt= −CR.v2

x (6)

Using Eq. (6) Naemi and Sanders2 proposed that the ratiof velocity squared to deceleration can be used as a measuref glide efficiency. This ratio is called glide factor by Naemind Sanders,2 denoting the ratio of virtual mass to the resistiveactor from a hydrodynamic point of view (according to Eq.6)).

Thus a glide factor was introduced by Naemi and Sanders2

s the ratio of virtual mass (M) to the resistive factor (CR)Eq. (7)) that indicates the ability of a body to minimise

eceleration at each corresponding velocity.2

G = v2x

−ax

= M

CR

= m + ca.ρ.V

1/2(A.ρ.cd)(7)

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dicine in Sport 13 (2010) 444–451

The glide factor (CG) in meter can be quantified with aydro-kinematic method (reviewed at the end of the secondart) without the need to know the resistive factor or the addedass separately.Naemi and Sanders2 showed that the effects of size and

hape on the glide efficiency can be distinguished by con-idering the glide factor as the product of a size-relatedlide constant and a shape-related glide coefficient. Thisas achieved by replacing the virtual mass as the productf dimensionless virtual mass coefficient (cm); affected byhe body shape; and the body mass (m) and rewriting Eq. (7)s follows:

G = m

1/2(A.ρ).cm

cd

(8)

Naemi and Sanders2 defined the first term on the right sidef Eq. (8) that incorporates the known constant parametersncluding the body mass (m), the maximum cross-sectionalrea (A) and the water density (ρ), as the glide constant (λ)n meter:

= m

1/2(A.ρ)(9)

The ratio of the virtual mass coefficient (cm) to the dragoefficient (cd) (second term in the right side of Eq. (8))as defined by Naemi and Sanders2 as a dimensionless glide

oefficient (cg).

g = cm

cd

(10)

The glide coefficient was used by Naemi and Sanders2 toetermine the shape-related glide efficiency for each swim-er independent of body size. According to this the glide

actor is the product of the glide constant and the glide coef-cient, while the former is influenced by the body size, the

ater is affected by the shape characteristics of a body in atreamlined position.

In the second part of the review the methods of determininghe resistive force and the glide efficiency are explored.

. Part 2: methods of determining hydrodynamicarameters

The total drag can be quantified either by directly measur-ng the force or by calculating drag based on the kinematicsf a gliding body.

The force can be measured directly while a body is beingowed in still water or exposed to a flow at a constant veloc-ty. The total drag force is equal to the towing or holdingorce.

Forces can be measured during towing using either sta-

ionary or moving apparatus. In the stationary apparatus aable on a rotating winch on the waterside pulls the bodyhrough still water. The towing force is equivalent to the dragorce and is measured at different constant velocities and the

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orce would be measured directly by a strain gauge or loadell.9,12,25

In the moving apparatus the swimmer is attached to thearriage via a telescopic rod to which a force transducer isttached.26 This type of apparatus was used in a number ofifferent studies in 70 s.27–29 In order to collect force dataver a longer period of time, the moving apparatus can besed continuously over a circular path.30

Drag measurement in a fixed position requires expos-ng the body to fluid moving in a flume and measuring theorce required to hold it in the flow.10,11,31 To increase theccuracy of the drag force measurements the free-stream tur-ulence and the wave created by the flume current shoulde controlled.32 Further, attachments to the swimmer mayffect the posture and comfort of the swimmer. To over-ome these problems, Vennell et al.20 used model replicasf swimmers fixed with a specifically designed towing rodn a sophisticated flume enable of producing even current atredetermined velocity across the whole testing area. Thisnabled testing the model at a consistent posture at differentelocities and depths. Bixler et al. found that the results showbout 18% more drag for a live subject than for a model.7 Thisay result from the non-compliant nature of the mannequin’s

ody surface.‘Numerical simulations’ involving solving the

avier–Stokes differential equations that govern thequation of motion of the fluid particles is a relatively newpproach that provides the advantages like the ability toisualise the entire flow domain and can distinguish betweenhe different sources of resistance including the pressure andriction drag. Bixler et al.7 used such a technique to calculatehe drag forces on a virtual body model based on a scan ofhuman body. Conducting steady state analyses at differentelocities the resistive forces were found to be within 4%f the drag values of the mannequin replica measured in aume. The discrepancies between the CFD results and therag forces on a live subject might be related to the possibleifferences in the level of skin roughness and upstreamurbulence in the model and in reality.33 The disadvantagef CFD can be seen as the inability to model the complianturface of a body using software designed to model flowver solid bodies.

The total drag force applied to a body during a glide isqual to the deceleration multiplied by its inertia. Klaucknd Daniel34 solved the differential equations of motion of aliding body to model the velocity as a hyperbolic functionf time. The ‘resistive factor’ was determined by fitting ayperbolic velocity function to the measured instantaneouselocity–time data. The added mass was not considered andhe body mass was assumed to be equal to the virtual masseading to underestimation of the drag forces compared tohose obtained during the towing experiments at the constant

elocities.34 This re-emphasises the need for considering thedded mass as an important part of the inertia.35

The added mass of aquatic animals can be approximatedy assuming that they resemble bodies of revolution.36 Due

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dicine in Sport 13 (2010) 444–451 449

o the irregularity of the human morphology the added massf a human body cannot be approximated by these methods.

To quantify glide efficiency the distance travelled duringlide intervals of identical initial velocities have been mea-ured by calculating the area below the velocity–time curveuring the glides after maximum push-offs from the wall.37

ince the initial velocities were calculated based on the posi-ion data in two frames, the initial velocity values may notepresent a reliable method of selecting two glide intervalsor comparison. Sharp and Costill38 used regression analysiso determine an exponential decay rate constant of the veloc-ty when the velocity drops from 2 to 1 m/s. Using regressionnalysis38 provides more accurate results by fitting a curveo a number of data points over the whole period of glideompared to the method used by Starling et al.37

Despite this, the exponential velocity decay rate does notrovide an insight into the resistive or inertial parameters ofody. Naemi and Sanders in 2004 (as cited in Ref. [2]) estab-ished that a hyperbolic function which assumes that drags proportional to velocity squared (Eq. (6)) provides a bet-er fit to the raw velocity data than an exponential functionhat assumes drag to be proportional to velocity. Unlike thexponential fit, when a hyperbolic parametric curve is fittedo the velocity data of a glide, a parameter can be deducedrom the curve that has an identity in terms of the hydrody-amic function.2 Despite the fact that a measure of the glidefficiency can be extracted by fitting a hyperbolic functionas a parametric fit) to the velocity data, Naemi and Sanders2

eported fluctuations in velocity data that could affect theccuracy of the obtained values.

To relate the hydrodynamic characteristics of a humanody in a streamlined position including resistive and iner-ial properties to the kinematics of glide, Naemi and Sanders2

roposed a method of quantifying glide efficiency using para-etric curve fitting. In the method dubbed ‘hydro-kinematicethod’ the displacement over time is fitted by a parametric

quation (Eq. (11)) that is obtained by solving the differentialquation of motion of a representative body during glide (Eq.6)).2

= CG.Ln

[Vx

CG

.t + 1

](11)

here Vx0 is the initial velocity and CG is the glide fac-or. The parameters of the curve that provide the best fit tohe displacement data gathered from video analyses deter-

ine the glide factor and the initial velocity. Naemi andanders2 provided an example of using a parametric curvetting technique to fit the displacement function (Eq. (11))

o the displacement data of a body during a relatively shortlide interval of about 0.4 s. The glide factor of 4.62 m and thenitial velocity was 2.34 m/s, were the parameters of a curve

hat provided the best fit to the data with an R-squared valuef 0.99. The glide factor is a measure of glide efficiency ofhe glide interval and corresponds to the average velocity oflide. For example in this case an average velocity of 2.14 m/s

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50 R. Naemi et al. / Journal of Science

an be calculated based on the distance travelled after 0.4 sEq. (11)). Repeating the same procedure for a number oflide intervals selected form underwater glide trials follow-ng static push-offs from the wall for each swimmer, Naemind Sanders2 found a profile of glide factor versus averageelocity for each swimmer, and reported a linear regressionine with a distinctive intercept and gradient for each individ-al, that can be used to compare the glide efficiency betweenndividuals across different velocities.

The advantage of the method is that fitting the displace-ent function to the displacement data, rather than fittingvelocity function to the derived velocity data, increases

he accuracy of the calculation as the errors in the displace-ent data do not get amplified when deriving velocity. Theethod enables glide efficiency to be determined in realistic

lide conditions and without the need to know the resistivend inertial parameters (virtual mass) as separate parameters.ith this method, the effect of body surface characteristics,

osture, body alignment and depth on the glide efficiency cane determined accurately and reliably.

Because the glide factor takes into account the inertialffect that includes both added mass and body mass it is aore valid and realistic measure of glide efficiency; defined

s the ability of a body to maintain its velocity and minimiseeceleration; than measures of the resistive force alone. Forxample, as larger and heavier swimmers experience largealues of passive drag, it may be interpreted incorrectly thathis type of swimmer decelerates faster during glides and

ay have a disadvantage compared to slim swimmers whoncounter less drag. This interpretation ignores the fact thathe heavier swimmers have higher inertia, which togetherith the resistive force affects their deceleration during alide.2

Naemi and Sanders2 found that the method was able toistinguish between swimmers in their glide efficiency, andeported the glide factor to vary between 2.7 and 4.8 m (atverage velocity of 2 m/s) for the individuals tested at theirost consistent streamlined position. While the variations

f the glide factor within individuals was much less than theariations between individuals, the former was related it to theifferences in the body shape and size.2 Naemi and Sanders2

lso found that the glide factor increases as the swimmereaches the lower velocities during a glide. This is in lineith the findings of Vennell et al.20 that indicate a lower drag

oefficient, so a lower resistive factor, at the lower towingelocities.

The fact that the glide factor values, as the ratio of virtualass to the resistive factor, are more than unit indicates that

he virtual mass represents a larger value than the resistiveactor. During the glide the only force that is applied to theody in horizontal direction is the drag force, at each instanturing a glide the product of deceleration and the virtual mass

s equal to the resistive factor times velocity squared (Eq.6)). Thus the ratio of the virtual mass to the resistive factors equal to the ratio of velocity squared to deceleration. Thisndicates that the higher the glide factor the higher the ratio

•

dicine in Sport 13 (2010) 444–451

f velocity squared to the deceleration, or a less decelerationt each velocity denoting higher glide efficiency.

By measuring the maximum cross-sectional area (A) usingphotogrammetric method and knowing mass of the swim-er Naemi and Sanders,2 calculated the glide constant (λ)

or each swimmers and reported it as 1.80 ± 0.14 m for theemales and 1.88 ± 0.08 m for the males. The glide constantas used to determine the body’s size suitability to glide

fficiently. For example heavier swimmers with a lower max-mum cross-sectional areas (i.e. tall and slim body types) thatosses a higher glide constant have a better potential to gliden terms of their size compared to the swimmers with a lowerody mass and a higher maximum cross-sectional area (i.e.hort swimmers with broad chest) who posses a lower glideactor.2

Naemi and Sanders (2008) reported a dimensionless glideoefficient (cg) that was obtained by dividing the glide factorCG) by the glide constant (λ). The reported glide coefficientas 2.14 ± 0.35, indicating a dimensionless coefficient that

eveals the potential of the body shape to glide more effi-iently. It was also indicated that the shapes that are able tontrain a large quantity of water while minimising the dragan glide more efficiently.2 The glide coefficient as a dimen-ionless ratio provides a way of comparing swimmers withifferent body size in terms of their body shape includingtreamlining posture and alignment. Also the glide coeffi-ient has practical applications in testing the swimming suitsesigned to improve performance it considers the ability toeduce the drag force in conjunction with the effect on thebility of a body to entrain the added masses of water.2

When the curve fitting technique cannot be used, Naemind Sanders2 provided a formula (Eq. (12)) to estimate theverage glide factor when the initial velocity (Vx0), the finalelocity (Vx) and the duration (T) of a glide interval arenown.

G = T

(1/Vx) − (1/Vx0)(12)

Based on this Eq. (12) Naemi and Sanders2 reported anverage glide factor of 4.19 m based on the reported averagenitial velocity (2.86 m/s), average glide duration (1 s) andhe average final velocity (1.7 m/s) of five male participantseported by Lyttle.13 As the concept of glide efficiency and theay to measure it are new,2 there has not been any previous

iterature to compare the results with what was reported byaemi and Sanders.2

. Conclusion

Inertial and resistive characteristics of a body in a stream-lined position affect the glide efficiency.

Most if not all the swimming research studies on the humanbody in streamlined position have focused on the resis-tive characteristics only. The effect of inertial factors, inparticular, added mass, has received little attention.

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R. Naemi et al. / Journal of Science

The conventional methods of the direct resistive force mea-surements may not represent the drag forces in realisticglide conditions, as the natural posture and flow character-istics around a swimmer’s body would be affected.Calculating drag based on the kinematics of the glide over-comes the deficiencies of the conventional direct forcemeasurements by allowing the drag to be measured in anon-invasive way during more realistic gliding conditions.However, the results do not take into account the effect ofadded mass and the effect cannot be predicted.The hydro-kinematic method2 is a new method that canquantify the glide efficiency without the requirementof knowing the resistive and inertial parameters inde-pendently and promises huge benefits for the study ofhydrodynamic characteristics of a human body duringglide in future.

cknowledgements

We thank Stelios Psycharakis for reviewing the draft ofhe manuscript. No financial assistance is declared.

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