Polidori2006.pdf

ARTICLE IN PRESS

Journal of Biomechanics 39 (2006) 2535–2541

Short communication

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Skin-friction drag analysis from the forced convection modeling in simplified underwater swimming G. Polidoria, R. Taı¨ arb, S. Fohannoa,, T.H. Maia, A. Lodinib a

b

Laboratoire de Thermome´canique, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France Laboratoire d’Analyse des Contraintes Me´caniques, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France Accepted 18 July 2005

Abstract This study deals with skin-friction drag analysis in underwater swimming. Although lower than profile drag, skin-friction drag remains significant and is the second and only other contribution to total drag in the case of underwater swimming. The question arises whether varying the thermal gradient between the underwater swimmer and the pool water may modify the surface shear stress distribution and the resulting skin-friction drag acting on a swimmer’s body. As far as the authors are aware, such a question has not previously been addressed. Therefore, the purpose of this study was to quantify the effect of this thermal gradient by using the integral formalism applied to the forced convection theory. From a simplified model in a range of pool temperatures (20–30 1C) it was demonstrated that, whatever the swimming speeds, a 5.3% reduction in the skin-friction drag would occur with increasing average boundary-layer temperature provided that the flow remained laminar. However, as the majority of the flow is actually turbulent, a turbulent flow analysis leads to the major conclusion that friction drag is a function of underwater speed, leading to a possible 1.5% reduction for fast swimming speeds above 1 m/s. Furthermore, simple correlations between the surface shear stress and resulting skin-friction drag are derived in terms of the boundary-layer temperature, which may be readily used in underwater swimming situations. r 2005 Elsevier Ltd. All rights reserved. Keywords: Skin-friction drag; Forced convection; Underwater swimming

1. Introduction In human swimming under normal circumstances, in which a water/air interface occurs, the total drag (Fd) which is the resistance force on the swimmer’s body, in motion within a viscous medium, is respectively composed of the skin-friction drag (Ff), the profile drag (Fp) and the wave drag (Fw). The contribution of skinfriction drag is generally assumed to be up to 5% of the total drag (Toussaint et al., 2002) whereas the contribution of the wave drag may reach a maximum of 60% of the total drag (Vennell et al., 2005). In the quest for higher levels of performance in swimming, many recent Corresponding author. Tel./fax: +33 326 91 83 10.

E-mail address: [email protected] (S. Fohanno). 0021-9290/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2005.07.013

research strategies have been designed to reduce drag. As well as improved understanding of the fundamental mechanisms of swimming, such as optimisation and coordination of the different movements, technological developments may offer potential benefits. The concept of using specifically designed swimming suits, modeled on shark skin, to achieve drag reduction by controlling the near-wall turbulence and skin-friction forces has received much attention. In idealized laboratory experimental conditions, Bechert et al. (2000) found as much as 7.3% decrease in turbulent shear stress, as compared to a smooth reference plate. Koeltzsch et al. (2002) confirmed this conclusion from fluid dynamic experiments showing that small riblet surfaces induced drag reductions of up to about 10% compared to smooth surfaces. In contrast, Toussaint et al. (2002)

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G. Polidori et al. / Journal of Biomechanics 39 (2006) 2535–2541

showed that a statistically non-significant 2% reduction in drag was found in real pool conditions when wearing fast-skin suits compared to conventional ones. In contrast with previous studies, the present analysis deals especially with the skin-friction drag, which is approached theoretically from non-isothermal fluid dynamic theory. To simplify the analysis, the underwater swimming situation is considered. This swimming condition is of interest because it leads to negligible wave drag (Vennell et al., 2005) and thereby the greatest potential speed increase for elite swimmers, estimated to be up to 40% (Laughlin, 2003). This underwater swimming situation also results in a greater contribution of skin-friction drag to the total drag. The question arises whether varying the thermal gradient between the underwater swimmer and the pool water may modify the surface shear stress distribution and the resulting skin-friction drag acting on a swimmer’s body. As far as the authors are aware, such a question has not yet been addressed. Therefore, the aim of this study was to quantify the effect of this thermal gradient on the surface shear stress distribution and the resulting skin-friction drag acting on a swimmer’s body. For this purpose, the integral formalism was applied to forced convection theory. Furthermore, first-order regression equations for the surface shear stress and deduced skin-friction drag were derived in terms of the boundary-layer temperature for use in underwater swimming situations.

2. Background of the problem Swimmer’s body geometry can be modeled as a linkage of complex convex, concave, truncated, ellipsoidal non-rigid surfaces (Yanai, 2001). Both swimmer’s body shape and the specific swimming propelling motion induce very complex separated shear layers combining turbulent fluctuations and large-scale vortices. Such a physical problem seems theoretically too complex to be analysed and a simplified approach is required. In order to reduce profile drag underwater swimmer should adopt a streamlined shape, as flat as possible, to prevent the boundary layer from separating. This is the reason why, using a rough assumption, the physical model used in the present analysis and presented in Fig. 1 consists of a flat surface at the human body surface temperature Tw flowing at a steady constant velocity UN in a quiescent water pool whose temperature TN varies in the range 20 1CpTNp30 1C. It should be noted that this model is also applicable when the swimmer is wearing a full-body competitive swimsuit. In such a case, Tw is defined as the external (waterside) swimsuit surface temperature. Even if suits’ materials are good thermal insulators (l
0:05

Y, V U∞

LAMINAR T∞

TURBULENT TRANSITION δ DYNAMICAL LAYER TS

δΤ THERMAL LAYER

x, U, T

ISOTHERMAL BODY SURFACE

Fig. 1. Summary of the simplified boundary-layer forced convection model.

0:15 W m1 K1), the competitive suits are so thin (100 mm) that the swimsuit surface temperature will remain close to that of the real body surface. The simplified model assumes a uniform temperature over the whole body surface corresponding to the average body surface temperature. This choice is difficult due to the real non-uniformity of the skin temperature in swimming (higher temperature where muscles are active, lower elsewhere) and the lack of measurements in the literature. For example, Brandt and Pichowsky (1995) measured a 33 1C local skin temperature over the deltoid muscle after strenuous exercise. More recently Jansky et al. (2003) investigated skin temperature changes induced by local cooling. For example, they measured temperatures between 32.5 and 34.5 1C at the start of their experiments over various body areas (trunk, forehead, thigh, forearm, fingers, and palms). These temperatures correspond closely to those of swimmers at the start of swimming races. Therefore, an estimated average skin temperature Tw ¼ 33 1C was chosen for this analysis. The flat surface is supposed to model a 2D-surface placed parallel to the mean uniform stream and acting like the smooth skin surface of a rigid body. Because a thermal gradient occurs between the flat surface and the ambient fluid, both dynamic and thermal boundary layers develop at the body surface. A common approach in forced convection consists of evaluating the fluid properties at an average boundarylayer temperature, called the film temperature, and defined as T¯ ¼ ðT w þ T 1 Þ=2.

(1)

Table 1 summarizes the various physical properties of water in the realistic pool temperature range 20 1CpTNp30 1C and used for further analysis.

3. Theoretical modeling Initially the boundary-layer development is laminar but at some critical distance xc from the leading edge small disturbances in the flow begin to be amplified,

ARTICLE IN PRESS G. Polidori et al. / Journal of Biomechanics 39 (2006) 2535–2541 Table 1 Thermo physical properties of water at different film temperatures Swimmer Tw Water TN (1C) (1C)

Density r Film temperature (kg/m3) ¯  CÞ Tð

Kinematic viscosity n (m2/s)  106

33 33 33 33 33 33

26.5 27.5 28.5 29.5 30.5 31.5

0.865 0.846 0.829 0.811 0.794 0.777

20 22 24 26 28 30

996.68 996.41 996.12 995.83 995.53 995.21

Fig. 2. Anthropometric parameters of subject Yann in a prone position with the arms extended above the head and the head facing the bottom of the pool.

characterizing a transition process towards turbulence (Fig. 1). In submerged swimming, the governing parameters reduce to the Reynolds number ReL ¼ U 1 L=n (UN is the swimmer’s speed, L is the swimmer’s body length with stretched arms defined in Fig. 2 and v is the water kinematic viscosity). It is usually assumed that the swimmer’s body acts like a rigid streamlined body and the critical location xc of the transition is estimated to correspond to a Reynolds number Rexc ¼ 5  105 . For example, for a swimmer’s speed U 1 ¼ 2 m=s and for L ¼ 2 m, the transition from laminar to turbulent occurs for xc
0:2 m. About 10% of the skin surface is subjected to a laminar flow regime so that the major part of the body surface will experience a turbulent boundary layer. Because both dynamic and thermal viewpoints are of interest, the simplified model is based on external boundary-layer theory applied to steady forced convection flow conditions. For this purpose the Karman-Pohlhausen integral approach (Kakac- and Yener, 1995; Padet, 1997; Polidori et al., 2003) has been used and developed in the range of swimming Reynolds numbers inducing both laminar and turbulent effects.

3.1. Shear stress modeling in the laminar case This section deals with the laminar boundary-layer theory of the forced convection developed in the range Rexc o5  105 .

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The continuity and momentum integral equations are expressed as
Z q d qU

ðU 1  UÞU dy ¼ n
, (2) qx 0 qy y¼0 Using the integral method inevitably leads to a choice of a polynomial order characterizing both velocity and temperature profiles. From mathematical criteria a fourth order shape has been advocated by Polidori et al. (1999). Thus the velocity profile across the dynamical boundary-layer is expressed as a fourth-order polynomial using the Pohlhausen method (Polidori and Padet, 2002)  3  4 ¯ Uðx; y; TÞ y y y ¼2 þ . 2 ¯ ¯ ¯ U1 dðx; TÞ dðx; TÞ dðx; TÞ (3) Resolution of momentum integral Eq. (2) leads to the velocity boundary layer thickness rffiffiffiffiffi 35 x pffiffiffiffiffiffiffiffiffiffiffiffiffi . dðxÞ ¼ 6 (4) 37 ReðTÞ ¯ In laminar forced convection, the local shear stress at the surface tw is defined as   ¯ qUðTÞ ¯ ¯ tw ¼ rðTÞnðTÞ . (5) qy y¼0 Incorporating Eqs. (3) and (4) in the shear stress expression (5) leads to the analytical surface shear stress expression sffiffiffiffiffiffiffiffiffi rffiffiffiffiffi ¯ 1 37 ¯ nðTÞ tw jLAM ¼ ½U 1 3=2 . (6) rðTÞ 3 35 x 3.2. Shear stress modeling in the turbulent case The transitional region is very complex and not yet understood, so that the turbulent regime is usually considered as idealized and the corresponding turbulent theory is applied beyond the critical Reynolds number value. In turbulent flow the governing parameters are made up of both mean and time-fluctuating components, so that it is convenient to consider their timeaveraged expressions to solve corresponding problems. It has been shown (Kakac- and Yener, 1995) that in the fully turbulent region (outside the laminar sub-layer) the average velocity increases approximately as a oneseventh-power law resulting in the corresponding profile   U y 1=7 ¼ . (7) ¯ U1 dðTÞ Incorporating this velocity profile in the integral momentum Eq. (2) gives the following analytical form

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of the surface shear stress ¯ 7 ¯ 21 qdðTÞ . rðTÞU (8) 72 qx A common way to express the surface shear stress developing at the surface of a flat wall is as an empirical form from measurements of turbulent flows in circular ducts (Kakac- and Yener, 1995)  1=4 ¯ 45 nðTÞ ¯ 21 tw ¼  103 rðTÞU (9) ¯ 2 U 1 dðTÞ

tw ¼

Thus combining Eqs. (8) and (9) leads to the expression of the dynamical boundary-layer thickness ¯ dðTÞ 0:371 ¼ . ¯ 1=5 x Rex ðTÞ

(10)

Incorporating (10) in Eq. (9) provides an expression for the turbulent surface shear stress ¯ 21 tw jTURB ¼ 2:88  102 rðTÞU

1 . ¯ 1=5 Rex ðTÞ

(11)

First-order regression equations for both laminar and turbulent regimes have been deduced from the previous parametric analyses in the complete range of considered parameters and can be expressed in terms of film temperature and swimming speed. It should be noted that these correlations are only suitable in the range of ¯  CÞp31:5 corresponding to film temperature 26:5pTð realistic indoor pool temperatures rffiffiffiffiffiffiffiffi U1 3 ¯ tw jLAM ¼ ð3:40  10 T þ 0:408ÞU 1 ; x Rex o5  105 ,  4 1=5 U1 3 ¯ tw jTURB ¼ ð7:95  10 T þ 1:970ÞU1 ; x Rex 45  105 .

ð12Þ

It may be recalled that there is no satisfactory theory dealing with the transitional region between laminar and turbulent regimes but the analysis suggests that the transitional region corresponds to the beginning of the turbulent regime. Table 2 shows the various transition critical locations for the chosen parameter range. 3.3. Overall skin-friction drag The skin-friction drag force in the present analysis reduces to the skin friction Ff exerted by the external ambient water on the body surface area of the swimmer in motion. To avoid complex difficulties in the body geometry simulation the simplified theoretical model considers the body swimmer to act like a fictitious parallelepipedal shape made of planar surfaces. To suggest anthropometric dimensions both the average ¯ H and the swimmer’s height body hydraulic perimeter P

Table 2 Transition critical location xc between the laminar and turbulent regimes Underwater swimming speed

T¯ ¼ 26:5  C T¯ ¼ 31:5  C

1 m/s 0.43 m 0.39 m

1.5 m/s 0.29 m 0.26 m

2 m/s 0.22 m 0.19 m

2.8 m/s 0.16 m 0.14 m

with outstretched arms L are of interest in the modeling, as shown in Fig. 2. The global skin-friction drag may be obtained by integrating the local shear stress over the simplified anthropometric dimensions of the swimmer with respect to the flow regime conditions.  Z xc ¯ ¯ U 1 Þ dx tw jLAM ðx; T; F f ¼ PH 0  Z L ¯ þ tw jTURB ðx; T; U 1 Þ dx . ð13Þ xc

Incorporating Eq. (12) into expression (13) leads to the skin-friction drag acting on the swimmer’s body model expressed as a function of the average thermal boundary-layer temperature: pffiffiffiffiffi ¯ H fð6:80  103 T¯ þ 0:816ÞU3=2 Ff ¼ P xc 1 3 ¯ 9=5 4=5 þ ð9:94  10 T þ 2:462ÞU1 ½L  x4=5 c g. ð14Þ 4. Results and discussion The question that has motivated the present analysis was to theoretically quantify the potential influence of varying the pool temperature on the laminar and turbulent surface shear stress distributions and the deduced skin-friction drag calculations. To make that quantification easier a drag performance percentage was also established from a reference thermal situation. 4.1. Shear stresses analysis First, the influence of the temperature gradient on the surface shear stresses was evaluated at given streamwise body abscissae for both laminar and turbulent regimes. For elite swimmers, underwater turn techniques lead competitors to have forward speed of gliding after turn and push up to 2.8 m/s. In Fig. 3 the surface shear stress is plotted against the streamwise body abscissa for the range of standardized and real maximum average underwater swimming speeds (1.5–2.8 m/s). It is shown that the surface shear stresses are strongly dependent on the location of the laminar-turbulent transition, i.e. the flow regime. The laminar to turbulent mathematical transition results in a drastic increase in the surface shear stress

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SURFACE SHEAR STRESSτw (Pa)

G. Polidori et al. / Journal of Biomechanics 39 (2006) 2535–2541

xc (T, U∞) 17

For example, consider the increase of the average boundary layer temperature from T¯ ref ¼ 26:5  C to T¯ ¼ 31:5  C. Applying the expression (15) leads to an estimated 5.3% local shear stress enhancement in the laminar regime and a 1.5% in the turbulent regime.

T = 26.5°C T = 31.5°C

LAMINAR MODELING 13

4.2. Deduced skin-friction drag

2.8 m/s 9

TURBULENT MODELING

2 m/s 5 1.5 m/s 1 0.01

0.10 1.00 STREAMWISE BODY ABSCISSA (m)

10.00

Fig. 3. Surface shear stress evolution versus the streamwise body abscissa.

level. To minimize the friction drag it would be ideal to keep a laminar boundary layer along the whole body surface. However, it seems difficult to maintain a laminar boundary layer with such complex flow behavior where adverse pressure fields and inherent layer separations occur. Now, especially focusing on the turbulent regime, which is the state of the majority of the flow around the swimmer, one can observe that increasing the streamwise body abscissa leads to a decrease in the local surface shear stress acting on the swimmer’s body model. Moreover, whatever the swimming speed may be, the increase of the film temperature results in a decrease of the local surface shear stress. This conclusion is important. Indeed, the lack of maintenance of a laminar regime over the whole body surface may result, a priori, in the inconvenience of an earlier transition to the turbulent regime when increasing the thermal gradient. However, this is rapidly balanced by the reduction of the local surface shear stress due to the increase of the film temperature. To quantify the surface shear stress reduction for a ¯ by comparison with that at a given temperature, T, reference temperature, T¯ ref , and at a given swimming speed, the following formula is deduced from Eq. (12) and written as     ¯ tw ðT¯ ref Þ  tw ðTÞ T¯  T¯ ref E t ð%Þ ¼ 100  ¼ 100  b  T¯ ref tw ðT¯ ref Þ (15) ( with

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b ¼ 120 in the laminar region

Rex o5  105 ;

b ¼ 247 in the turbulent region

Rex 45  105 :

One will notice that b is a parameter that is deduced from the present calculation and depends only on the flow regime. In Eq. (15), b, T¯ and T¯ ref are given in degrees celsius (1C).

In the considered film temperature range, the kinematic viscosity of water decreases as its temperature increases (Table 1). Therefore, the mathematical transition to turbulence occurs earlier for warmer water. This leads to a strong increase in the local surface shear stress (Fig. 3) for the warmer water that will only occur a little further downstream for the colder water. As a consequence, the benefits of a higher temperature gradient are locally overcome by the disadvantages of the turbulent regime, compared to the laminar regime, with respect to the local surface shear stress. When the transition to the turbulent regime has occurred for both film temperatures, the benefits of the increase of the film temperature are recovered for a sufficiently large streamwise body abscissa. To clarify and quantify the previous conclusion the skin-friction drag performance has been expressed as a function of the swimmer’s body length with stretched arms. For this purpose, the low film temperature T¯ ref ¼ 26:5  C was taken as the reference case. The skin-friction drag performance was defined as:   ¯ F f ðT¯ ref Þ  F f ðTÞ Performanceð%Þ ¼ 100  , F f ðT¯ ref Þ ( drag reduction; Performance40 ) ð16Þ drag increase: Performanceo0 The skin-friction drag performance parameter, Performance (%), is plotted against the streamwise body abscissa in Fig. 4 for different swimming speeds varying from 1 to 2.8 m/s. It can be seen that the magnitude of the skin-friction drag performance depends on the regime of the flow. Indeed it appears that according to the flow regime, the skin-friction drag performance is either swimmer speed dependent in the case of the turbulent flow regime, or not in the laminar case. Increasing the average boundary layer temperature in the laminar flow regime results in a better situation for skin-friction drag enhancement up to 5.3% whatever speed the swimmer has. However, the majority of the flow regime around the swimmer is actually turbulent. Therefore, turbulent regime analysis is of great importance. By contrast to the laminar case, the turbulent skin-friction drag performance is highly Reynolds number or swimmer’s speed dependent. The major conclusion was that at the beginning of the turbulent regime, one observes a degradation in the skin-friction drag performance due to the sudden and

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FRICTION DRAG PERFORMANCE (%)

10 ENHANCEMENT (FRICTION DRAG REDUCTION)

5

skin-friction

drag

evaluation

for

U 1 ¼2.8 m/s;

2.8 m/s

Water temperature (1C)

Film temperature (1C)

Friction drag (N) swimmer female

Friction drag (N) swimmer male

1 m/s

20 30

26.5 31.5

19.7 19.4

22.0 21.6

0 -5

Table 3 Example of P¯ H ¼0.81 m

DEGRADATION (FRICTION DRAG INCREASE)

-10 1.5 m/s -15 2 m/s -20 0

0.5 1 1.5 2 STREAMWISE BODY ABSCISSA (m)

2.5

Fig. 4. Friction drag performance when increasing the average thermal boundary-layer from T¯ ref ¼ 26:5  C to T¯ ¼ 31:5  C.

sharp increase in the surface shear stress. Then, from a given streamwise body abscissa the trend is reversed and the performance is observed to reach 1.5% for fast underwater speeds. This enhancement (i.e. reduction) in the skin-friction drag occurs earlier as the speed is increased. Comparisons with experimental results are essential but difficult, since solutions for solving such problems have not yet been developed. 4.3. Tentative dimensional analysis A tentative evaluation is made from the male and female anthropometric parameters given in the USA swimming data (see for example the web site www. usswim.org). The difference between male and female swimmers is statistically significant. The average height and weight for male swimmers are respectively 187.177.2 cm and 80.477.0 kg while the average height and weight for female swimmers are 173.075.5 cm and 65.676.6 kg. From the following commonly used formula (Mosteller, 1987) defining the body surface area: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Height ðcmÞ  WeightðkgÞ 2 Body Surface Area ðm Þ ¼ 3600 (17) corresponding body surface areas are respectively found to be 2.04 m2 for males and 1.78 m2 for females. To get some idea of the theoretical skin-friction drag, ¯ H ) is the average swimmer hydraulic perimeter (P defined as the ratio between the body surface area and the L swimmer length with outstretched arms. The average swimmer’s body length with stretched arms is L ¼ 2.50 m for male swimmers and L ¼ 2.20 m for female swimmers. The values for the average hydraulic perimeter were found to be comparable for both female

¯ H ¼0.81 m. Examples of and male swimmers, namelyP friction drag values in steady flowing state are given in Table 3 for a 2.8 m/s speed. Average friction drags of 20 and 22 N were found respectively for female and male swimmers. Increase in pool temperature from 20 to 30 1C resulted in a decrease of the friction drag of about 0.4 N. Because the theoretical model used is a simplified one, the present intention is only to provide the order of magnitude of the skin-friction drag. 4.4. Comparison with Toussaint’s results A thorough comparison with previous results cannot be made because, to the authors’ knowledge, no numerical solution or experimental data concerning friction drag in underwater swimming are available in literature. Nevertheless a rough comparison can be attempted with the measurements of Toussaint et al. (2004) in the case of swimming at the air/water interface. In such a case, they established the following relationship between the total drag, Fd, and the swimming velocity, UN, for velocities less than 2 m/s. F d ¼ 21:33ðU 1 Þ2:34 .

(18)

According to Toussaint et al. (2002), the friction drag contribution is less than 5% of the total drag. At a speed of 2 m/s, this gives: F f jToussaint
5:4 N. Now let consider the present analysis and try to arrive as close as possible to Toussaint’s experimental conditions. Under the assumption that the friction drag mainly occurs at the water/skin interface and that half of ¯ H ¼ ð0:81=2Þ m, the the body is underwater so that P present modeling gives, respectively F f jMale ¼ 5:8 N and F f jFemale ¼ 5:2 N for male and female swimmers corresponding to about 4–5% of the above mentioned Toussaint’s total drag, which is in good agreement with Toussaint’s results.

5. Conclusion The main objective of this study was to evaluate and theoretically quantify the effect of the temperature

ARTICLE IN PRESS G. Polidori et al. / Journal of Biomechanics 39 (2006) 2535–2541

gradient between the swimmer’s body and the pool ambient water on the surface shear stress distribution and deduced skin-friction drags, in a pool temperature range varying from 20 to 301C. This was achieved using the integral formalism modeling extended to laminar and turbulent forced convection theory under the assumption of a streamlined swimmer. The results of this study were limited to steady flow. For different swimming speeds, the surface shear stresses and the skin-friction drags were investigated and presented in both analytical and graphical forms. In order to check the performance degree in friction drag with varying the pool temperature both laminar and turbulent regimes have been considered. The laminar flow regime provides a better situation for friction drag enhancement up to 5.3% whatever speed the swimmer attains. This conclusion is in close agreement with the common boundary layer theory. By contrast, the turbulent skin-friction drag performance is highly Reynolds number or swimmer’s speed dependent. In turbulent and fast swimming speed increasing the average boundary layer temperature in the considered range gives a 1.5% friction drag reduction for sufficient body lengths. Additional parametric dimensional analysis gives average skin-drag values of about 20–22 N. Increasing the pool temperature from 20 to 30 1C results in a decrease of the friction drag of about 0.4 N. It is not our intention and it seems difficult to say if these skin-friction drag performances are significant or not. Indeed relative weighting of both skin-friction drag and profile drag is unknown and varying the thermal field inevitably leads to altered boundary-layer separation points along the body and consequently the turbulent wake size and the profile drag. A compromise has to be taken between favoring either the laminar or the turbulent regimes although a laminar boundary layer is normally considered as ideal. When the flow regime is laminar, separation at the body surface starts almost as soon as the pressure gradient becomes adverse and a larger wake forms while when the flow regime is turbulent, separation is delayed and the corresponding wake is smaller. The importance of keeping the boundary layer attached to the swimmer body surface

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is so important that swimwear manufacturers sometimes purposely cause the boundary layer to become turbulent. The authors hope that the work presented here be used as an starting point for further discussion.

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