MEASUREMENT OF TURBULENT SKIN FRICTION DRAG
COEFFICIENTS PRODUCED BY DISTRIBUTED SURFACEROUGHNESS OF PRISTINE MARINE COATINGS

Frederik Zafiryadis, Knud Erik Meyer
Department of Mechanical Engineering
Technical University of Denmark
DK-2800 Kgs. Lyngby
kem@mek.dtu.dk

F. Gökhan Ergin
Dantec Dynamics
DK-2740 Skovlunde
gen@dantecdynamics.com

ABSTRACT

count for up to 90% of the total drag experienced by a ship
in motion, even if the skin is free of biofouling (Schultz
et al., 2010). The accumulation of micro- and macroorganisms on the wetted surface of a ship hull causes an increasein surface roughness, frictional resistance, and fuel consumption as shown by Vorburger et. al. (1982). To avoid
settlement on ship hulls, coatings with antifouling properties, whether through biocides toxic to marine organisms orother technologies, are applied as a top coat on the wetted
hull surface. The frictional resistance of the pristine coating is important as long as fouling with slime or barnacles
remain absent, a period which increases as the coatings become more efficient. Since the ban of tributyl tin (TBT)
in biocides in 2003 (Claire et al., 2009) efforts to develop
new and improved antifouling coatings within the legislation have been made. The state-of-the-art foul release coatings (FR) have shown most promising results by use of nontoxic silicone elastomers forming a self cleansing non-sticksurface when subjected to fluid shear forces. Compared to
traditional toxic based antifoulings, FR coatings require asmooth surface finish. This has spurred an increase in frictional resistance prediction of antifouling coatings and effects of surface roughness on frictional resistance. These include studies by Grigson (1992), Schultz (2004), Yigit et al.
(2014) and Benson (2005), presenting numerical as well
as experimental ways of evaluating skin friction of roughplates. Candries et al. (2001) and Schultz (2004) further
carried out experimental studies of the effect of fouling on
drag using towing tank experiments to determine the resistance increase caused by the application of different paintsystems. Despite the fairly large body of research that has
been conducted on the subject, less attention has been given
to acquiring accurate skin friction coefficients for different
coatings across as span of hydraulically smooth and roughsurfaces.

Skin friction drag coefficients are determined for marine
antifouling coatings in pristine condition by use of Constant Temperature Anemometry (CTA) with uni-directional
hot-wires. Mean flow behaviour for varying surface roughness is analysed in zero pressure gradient, flat plate, turbulent boundary layers for Reynolds numbers from Rex =1.91 × 105 to Rex = 9.54 × 105 . The measurements wereconducted at the Technical University of Denmark in aclosed-loop wind tunnel redesigned for investigations as
this. Ensemble averages of the boundary layer velocity
profiles allowed for determination of skin friction drag coefficients as well as roughness Reynolds numbers for thevarious marine coatings across the range of Rex by fittingof the van Driest profile. The results demonstrate soundagreement with the present ITTC method for determining
skin friction coefficients for practically smooth surfaces at
low Reynolds numbers compared to normal operation mode
for the antifouling coatings. Thus, better estimates for skinfriction of rough hulls can be realised using the proposed
method to optimise preliminary vessel design.

INTRODUCTION
Drag on marine vessels comprise of three fundamental
parts: skin friction drag, pressure drag and residual drag.
Each part contributes to the overall drag of the ship (Bixler& Bhushan, 2013). The size of the pressure drag is directly
linked to the shape of the underwater hull. The skin friction
drag, on the other hand, is caused by the streamwise shear
stress on the hull surface exerted by the flow of the water,thus dependent on the morphology of the surface (Vorburger
et. al., 1982). Lastly, residual drag accounts for the amountof energy responsible for the formation of bow- and stern
waves. Chambers et al. (1978) proposed a method for determining each of these components, an empirical method
which is still highly used today. Extensive effort has gone
into developing ways of accurately determining and minimizing skin friction drag on marine vessels, since it can ac-

Based on the work of Ergin et al. (2016) a measurement method is developed to allow for evaluation of equivalent sand grain roughness ks of a given antifouling coatedsurface, relating the drag increment of the surface to an1

vessel in operation (e.g. ReL = 108 − 109 ) to achieve fulldynamic similarity, was initially overcome by attempting toreach a suitably high Reynolds number for the skin frictions
to become practically constant. In total, four identical aluminium plates were constructed with a surface roughnessof Ra ≈ 0.2 µm in order to limit the roughness of the plate
itself, thus contaminating the results. To minimize flow disturbance leading edges of the plates were rounded.

equivalent surface composed of uniform sand grains, aswell as predicting skin friction coefficients to be applied in
the ITTC-performance prediction method (Chambers et al.,1978). This will allow for better skin friction estimates than
traditional empirical relations do, and will provide the necessary quantification of novel antifoulings hydrodynamic
performance compared to traditional ones, needed for predictions in full-scale conditions.
The objective of the study is to develop a method
for accurately determining skin friction drag coefficients ofmarine antifouling coatings to supersede the empirical relations used for smooth and rough surfaces in the ITTC
method, thus allowing for improvements of modern ship
design to be made. Numerical simulations to attain thisgoal are highly dependent on input parameters and boundary conditions in reference to the surface in question, without which the solution is inaccurate. Furthermore, when
dealing with near wall flows numerical approaches are often simplified and inadequate. In this case proper experimental investigations allow for enlightenment in the matter. Using a hot-wire Constant Temperature Anemometry(CTA) measurement setup, it is feasible to evaluate the hydrodynamic performance of antifouling coatings used forship hulls. By exposing a rough surface to an increasing
range of flow speeds up to a point where the skin friction
coefficient C f becomes independent of the Reynolds number, Re = U0 x/ν, it is possible not only to make accurate
predictions of the non-dimensional equivalent sand grainroughness ks+ and C f , but also allow for extrapolation to
full-scale applications. For general purposes the dimensionless group ks+ = ks uτ /ν is used for non-dimensionalisingp
the rough and smooth flow conditions, with uτ = τw /ρ
being the friction velocity, where τw is the wall shear stressand ρ is the density of the fluid. From knowledge of theroughness Reynolds number, ks+ , distinctions
between ahydraulically smooth flowregime ks+ ≤ 5 , a transitional
regime 5 ≤ ks+ ≤30 − 70 , and a fully rough flow regime
for ks+ ≥ 30 − 70 can be made (Sumer, 2013).

Coatings
In order to experimentally investigate the boundary
layer response to changes in surface roughness due to various types of ship hull coatings, three of the four plates were
coated with different antifouling coatings. Coatings wereapplied to the plates under ideal conditions and in agreement with relevant standards. The coatings have different
structures, giving rise to varying surface finish and thus
varying surface roughness when applied to the plates. The
following pristine test surfaces were used:
Reference - non-coated aluminium surface serving as
control surface.
SPC - plate coated with a traditional self polishing
copolymer coating.
Epoxy - plate coated with an amine-adduct cured
epoxy coating reinforced with glassflakes.
FR - plate coated with a silicone foul release coating
containing no biocides.
The epoxy coating is not a traditional antifouling, but instead applied on ship decks to create a skidproof surface. It
is included in the study to ensure distinctive difference inroughness of the surfaces. The test surfaces were numerically assessed prior to experiments using a stylus instrument with a 0.2 µm stylus tip. A 12.5 mm evaluation length
was used with a short-wavelength 0.8 mm filter with a cutoff length of 2.5 mm, allowing for determination of roughness heights, with wavelengths above 0.8 mm being disregarded. Five sample profiles at different positions on theplate surfaces were taken to ensure proper results. The obtained parameters are listed in Table 1 and contains roughness parameters Ra , Rz , and Rz,max along with the spacing
parameter RSm . Ra is an arithmetic average taken over the
15 consecutive sampling lengths. Rz is an average of thepeak-to-valley height from the 15 adjoining sample lengths,
while the Rz,max parameter determines the highest Rz related to the 15 sample lengths. Thorough definitions of theused parameters including relevant standards can be foundin Whitehouse (1994) or Thomas (2002).
The robust measuring techniques yield ambiguous results for the communal roughness parameters, since, fromvisual inspection, the non-coated surface is smoother than

EXPERIMENTAL FACILITIES AND METHODS
To achieve the objectives of the study, the 300 mm ×
300 mm × 1800 mm test section of a closed-loop wind tunnel at the Technical University of Denmark was refitted by
Ergin et al. (2016) in order to perform automated boundary
layer measurements. This allows for detailed insights in theeffects of surface roughness on near wall flow behaviour.
The design features include a sliding test section front window, computer-controlled traversing in three dimensions,angle of attack adjustment and zero-pressure-gradient alignment. The wind tunnel runs at a maximum speed of 40 m/s
with a turbulence intensity of 0.15% − 0.2%, decreasing to
0.10% − 0.15% at an operating speed of 30 m/s and below.
Boundary layer velocity measurements to evaluate different
antifouling coatings for commercial use were performed onflat plates installed in streamwise direction in the test section of the wind tunnel.
The geometry of the plates are set forth by the dimensions of the test section allowing for the length of the plates
to be 750 mm, with a height and thickness of 297 mm and
10 mm, respectively, thus preventing blockage effects. A
rack for vertically mounting the plates was installed in thewind tunnel test section, placing the plates with a distanceof 300 mm from the inlet to the leading edge. The inability of the scale-model testing to match the Reynolds of a

Table 1: Roughness amplitude spacing parameters

for all test surfaces obtained using stylus instrument.
Standard deviations based on five samples for eachsurface are up to approximately 6%.
Param. [µm]Ra
Rz
Rz,maxRSm
2

Reference
0.193
1.54
3.59
30.3

FR
0.140
0.935
2.37
66.5

SPC
1.16
6.47
8.83
112

Epoxy
2.05
9.21
14.9
388

the FR coated one. The values provide only an idea about
the size of the roughness elements for each coating.

Experimental procedureState of the art electronics, computer-controlled hot-wirecalibrator and a LabVIEW software program allowed forstraightforward data acquisition during CTA measurementson the four plate surfaces. Two 5 µm Dantec Dynamics
hot-wires were used during the experiments: one straightgeneral purpose type (55P11) to measure the instantaneous
freestream velocity, and one boundary layer type (55P15)
allowing for streamwise velocity measurements within theboundary layer1 . The use of two hot-wire probes allowed
for real-time normalization of the boundary layer velocity,
resulting in less post-processing as well as correcting measurements for minor fluctuations within the uniform flow
field. Due to the nature of the study, measurements wereconducted for turbulent boundary layers only, even though
the experimental setup can acquire both laminar and turbulent boundary layers (Ergin et al., 2016). Calibrations of
the hot-wires were performed at a constant temperature, enabling conversion from voltages to velocity readings. By
the used of an automatic air calibrator uncertainties withrespect to velocity, direction and temperature were reduced.
The relationship between the voltage and the velocity can beapproximated by a fifth order polynomial, allowing for linearisation and accurate determination of the velocity whenvoltage is recorded.
Further, temperature corrections were performed for the calibration to be valid. This was done according to Ergin et al.
(2016) prior to the linearisation. The temperature correction
ensures valid outputs across a larger range of temperatures,
given that an increase in temperature of 1 K induces a 2%
divergence in velocity.
Velocity profile measurements were conducted for fivemeasurement planes at distances x = 100 mm, 200 mm,
300 mm, 400 mm and 500 mm from the leading edge foreach of the four plate surfaces, corresponding to Rex ranging from 1.91 × 105 to 9.54 × 105 when conducted at U0 =30 m/s. Profiles are obtained for 41 points along the zaxis (cross-stream) in the interval from −50 mm to 50 mm
at each measurement plane. Measurement planes and corresponding coordinate system are illustrated in Figure 1.
A linear extrapolation of the last measurements to zero isconducted, thus locating the wall at the given point. The
amount of steps in the z-direction is required to allow for
precise estimates of the wall location using a quadratic fit to
all data points. During post-processing the fitted wall locations are subtracted from the wall distances of the measured
velocity profiles. The output is given as normalised velocity
profiles with y = 0 at the wall for each of the 41 measurement points in the cross-stream direction.

(a)

(b)

Figure 1: Illustration of two out of five measurement

planes (a) with coordinates for experimental procedures, and experimental setup (b) showing wind tunnel wall, support arm, and traversing system.
To determine skin friction drag coefficients the friction
velocity, uτ , must be estimated for the given data. Finding uτ in boundary layers is always a difficult task with noeasy way of determining it from direct measurements. A
series of indirect methods exist for this purpose, often by
assuming some sort of known dependency within the logarithmic layer. This makes it difficult to make any conclusive
statements about the logarithmic region for boundary layersthat do not follow these tendencies. In the current study,
the wall shear stress is determined by fitting the van Driest
velocity profile (2) to the obtained boundary layer velocity profiles as suggested by Sumer (2013). The van Driest
profile merges the linear profile within the viscous sublayer
with the logarithmic profile in the logarithmic region, thus abetter estimate of the friction velocity is obtained compared
to only considering the logarithmic layer.
The coordinate displacement ∆y, approximated by (1)according to Cebeci & Chang (1978), is needed when solving for uτ and ks+ .
∆y+ = 0.9

 + 
qkks+ − ks+ exp − s
.
6

(1)

(1) is valid for 5 < ks+ < 2000, thus applicable to transi-

RESULTS AND DISCUSSION
The results are presented as time-averaged steady flow
fields for each of the coated plates. One boundary layer
profile for each Reynolds number is obtained through ensemble averaging of velocity profiles in cross-stream direction on the plates yielding less data to be processed and amore general result. The obtained profiles for the raw datais plotted in Figure 2.

Figure 2: Comparison of ensemble averaged boundary

layer velocity profiles at five different measurement
planes for all plate surfaces. Obtained from experiments conducted at 30 m/s.

1 Confer Dantec Dynamics (2014) for information on the 55P11

and 55P15 hot-wire probes.

3y+ = yuτν (Schlichting, 1979), using the wall parameters uτ
and ν.

tional and rough wall flows. Using a numerical nonlinear
least-squares method (2) is fitted to each of the profiles in
Figure 2 in a non-dimensionalised domain, u+ = uuτ and+Z yu = 2uτ1 +2 )1/2
2
y + ∆y+2+
2
1 + 1 + 4κ y + ∆y
1 − exp −Ad(

0Table 2: Friction velocities, uτ [m/s], and roughness

1.91
3.82
5.72
7.63
9.54

Reference
12.3
uτ
1.62
ks+ 11.8
uτ
1.56
+
ks
11.4
uτ
1.50
ks+ 11.1
uτ
1.47
ks+ 10.9
uτ
1.44
ks+

FR
13.2
1.68
12.3
1.57
11.8
1.50
11.4
1.46
11.2
1.42

SPC
15.7
1.74
14.4
1.60
13.7
1.53
13.3
1.48
13.3
1.45

(2)

∆u+ is obtained for the epoxy coating. A (u+ , y+ )-plot for
Rex = 9.54 × 105 is given in Figure 6. No significant difference in profiles for the coated plates is observed, thusroughness effects appear less significant at higher Reynolds
numbers further downstream of the leading edge.

Reynolds numbers, ks+ [−], obtained from van Driest
fitting to each of the velocity profiles. Reynolds numbers are in the order of 105 .
Rex

dy+

Epoxy
21.4
1.90
18.9
1.68
17.7
1.58
17.1
1.52
16.7
1.48

From uτ the skin friction coefficients are readily calculated for each of the coatings across the range of Reynolds
numbers. These are given in Figure 7 as a function of the
Reynolds number Rex . Skin friction coefficients range between C f = 8.03 × 10−3 and C f = 4.45 × 10−3 , dependingon type of coating and distance from measuring point toleading edge. It is interesting to notice that even though
the roughness Reynolds numbers are estimated to be higher
for the FR coating than the non-coated plate, little difference can be observed between their respective C f curves.
On the other hand, the hot-wire anemometry method combined with the van Driest fitting allows for a clear differentiation between surface where the difference in roughness isslightly higher, e.g. epoxy vs. reference. For comparisonand benchmarking of the results the Prandtl-Schlichting interpolation formula for a flat, hydraulically smooth plate atzero angle of incidence as described by Schlichting (1979)
is applied. The relation is often used in unison with the
ITTC Performance Prediction Method, thus employed here
(Chambers et al., 1978). Prandtl-Schlichting’s relationagrees quite well with the data, except for Rex ≤ 3.82×105 ,probably because the log law applies only over a very lim-

The determined values for the friction velocities and theroughness Reynolds numbers are given in Table 2. Fromthe values it is apparent that neither of the surfaces giverise to a fully rough flow. Concurrently the reference plateappears not to result in hydraulically smooth flow with ks+
ranging between 12.3 and 10.9. Nonetheless, a communal
comparison of ks+ yields proper results compared to visualinspection and Table 1, thus supporting the method.
The friction velocities are used to non-dimensionalisethe velocity profiles. Figures 3 through 6 depict the effectsof Reynolds number as well as roughness on the boundary
layer velocities. From Figures 3 and 4 it is clear that there
is a tendency for the velocity profiles to collapse onto thesame logarithmic layer profile for the same surface roughness, as one would expect (Schultz, 2004). This is the casefor Reynolds number above Rex > 1.91 × 105 , thus further
downstream of the leading edge. At the measurement pointclosest to the leading edge a deviation from this tendency
is observed. The method of fitting requires the presence ofa significant logarithmic layer to yield proper results. For
the case of lower Reynolds numbers the logarithmic layer
is fairly thin, inducing uncertainties to the method of skinfriction estimation. Figure 3 shows the fit of the van Driest
profile to constant stress layer of the measured velocity data,thus y < (0.2 − 0.3)δ (Sumer, 2013), with u (δ ) = 0.99U0 .
When comparing the velocity profiles for each of the coatings at the same Reynolds number in a (u+ ,y+ )-domain(confer Figures 5 and 6) a downward shift of the size ofthe roughness function ∆u+ for increased ks+ is observedin accordance with the work of Coles (1956). The largest

Figure 3: Effect of Reynolds number on mean bound-ary layer velocity profiles u+ with respect to y+ for
non-coated plate. Plotted with a semilogarithmic xaxis. The van Driest profile is plotted for the case of
Rex = 1.91 × 105 from the obtained values in Table 2.
4

Figure 6: Comparison of mean boundary layer veloc-

Figure 4: Effect of Reynolds number on mean bound-

ity profiles u+ with respect to y+ for Rex = 9.54 × 105
in a semilogarithmic plot. Data for each coating isshown..

ary layer velocity profiles u+ with respect to y+ for
FR coated plate. Plotted with a semilogarithmic xaxis.

Figure 7: Skin friction coefficients as a function of

Reynolds number for all plates. Obtained by fittingof the van Driest velocity profile. Errorbars estimating uncertainty to 2.5%, and 4.0% for lower Rex , havebeen added for the reference plate.

Figure 5: Comparison of mean boundary layer veloc-

ity profiles u+ with respect to y+ for Rex = 1.91 × 105
in a semilogarithmic plot. Data for each coating isshown.

Conclusions
Constant temperature anemometry hot-wire measurements were conducted in a wind-tunnel at the Technical
University of Denmark to obtain better estimates for skinfriction drag coefficients produced by distributed surface
roughness of pristine marine antifouling coatings. Boundary layer velocity profiles were obtained for three types ofmarine coatings (foul release (FR), self-polishing copolymer (SPC), and epoxy coating) applied to aluminium plateswith a rounded leading edge. One plate was left uncoatedfor reference. Measurements spanned a Reynolds number
of Rex = 1.91 × 105 to Rex = 9.54 × 105 allowing for estimation of roughness Reynolds numbers, ks+ , and skin friction drag coefficients, C f , for each of the coatings as a function of Rex by fitting with the van Driest velocity profile.
The obtained skin friction drag coefficients were found toagree well with the Prandtl-Schlichting formula for the uncoated reference plate as well as the slightly rougher FR
coated plate. The highest skin friction coefficients were obtained for the epoxy coated plate. Uncertainties with respect to C f were estimated to approximately 2.5%, yielding

ited range of wall distances at lower Reynolds numbers.
The fact that the empirical relation is consistent with thedata provides an interesting result in relation to the experimental setup, since the reference plate is not hydraulicallysmooth. Later numerical investigations proved that the geometry of the leading edge of the plates caused boundary
layer separation in the vicinity of the leading edge, slightly
contaminating the results for the ’zero-incidence’ skin friction coefficients (Zafiryadis, 2016). Additional sources oferror arising from calibrations, temperature variations, accuracy of probe positioning, variations in freestream velocity in wind tunnel, and minor coatings flaws contribute tothe overall uncertainty of approximately 2.5% for each velocity profile. Errorbars are plotted for each of the measurement points for the reference plate in Figure 7 for representation. Due to the mentioned drawback of the method
for estimating C f , the uncertainty estimate is increased for
Rex = 1.91 × 105 .

5Dantec Dynamics 2014 Probes for Hot-wire Anemometry, www.dantecdynamics.com, Publication No.:
238v11.
Ergin, F. G., White, E. B., and Meyer, K. E. 2016 Designand construction of DTU Wind Tunnel test section forautomated boundary-layer stability experiments. American Institute of Aeronautics and Astronautics, Dantec
Dynamic contribution to NATURAL EU project.
Grigson, C. 1992 Drag Losses of New Ships caused by HullFinish. Journal of Ship Research, 36, No. 2, pp. 182–192.
Monaghan, R. J. 1953 A Review and Assessment of Various Formulae for Turbulent Skin Friction in Compressible Flow. Technical Note No. Aero. 2182, Aeronautical
Research Council, p. 11.
Schlichting, Dr. H. 1979 Boundary Layer Theory. McGrawHill Book Company. 7th edition pp. 201–204, 641–642.
Schultz, M. P. 2004 Frictional Resistance of Antifouling
Coating Systems. Department of Naval Architecture andOcean Engineering, United States Naval Academy, Annapolis. Journal of Fluids Engineering, 126.
Schultz, M.P., Bendick, J.A., Holm, E.R., and Hertel, W.M.
2010 Economic impact of biofouling on a naval surface
ship. Biofouling, 27, No. 1.
Sumer, B. M. 2013 Lecture Notes on Turbulence. Technical
University of Denmark.
Thomas, T.R. 2002 Rough Surfaces. Longman Group Limited, ISBN: 0-582-46816-7. pp. 21–25, 72–90.
Vorburger T. V., Scire, F. E., and Teague, E. C. 1982 Surface
Roughness Measurements of Circular Disks and Their
Correlation with Hydrodynamic Drag. Center for Manufacturing Engineering, National Engineering Laboratory,
National Bureau of Standard, Washington.
Whitehouse, D. J. 1994 Handbook of Surface Metrology.
IOP Publishing Ltd, ISBN: 0-7503-0039-6. pp. 7–21,
102–114.
Yigit, K. D., Khorasanchi, M., Turan, O., Incecik, A., andSchultz, M. P. 2014 A CFD model for the frictional resistance prediction of antifouling coatings. Elsevier, Ocean
Engineering.
Zafiryadis, F. L. 2016 Frictional resistance prediction of antifouling coatings. B.Sc. thesis, Technical University ofDenmark.

satisfying repeatability of the data with the possibility of
distinguishing the velocity profiles and skin friction coefficients for rough surfaces. Despite attempts to reach adequately high Reynolds numbers, an incomplete similarity
was realised with ks+ outside of the fully rough regime inwhich marine coatings operate in full-scale. Thus, extrapolation was deemed idle due to unreal thickness-to-roughness
height ratios compared to a full-scale case. Nonetheless,knowledge of the impact of roughness on fluid flow is obtained without information of the characteristics of the surface roughness, a task which is unobtainable using other experimental methods as well as numerical approaches. The
technique yielded applicable and accurate C f predictions
for lower Re, thus providing a solid foundation for further
work on expanding the measurement setup to acquire higher
Reynolds numbers suitable for full-scale skin friction dragpredictions of pristine and even biofouled antifouling coatings.

REFERENCES
Benson, J. 2005 Boundary-layer response to a change insurface roughness. The University of Reading, Department of Meteorology.
Bixler, G. D. and Bhushan, B. 2013 Fluid Drag Reduction
with Shark-Skin Riblet Inspired Microstructured Surfaces. Advanced Functional Materials, 23, 2013.
Candries, M., Atlar, M., Guerrero, A., and Anderson, C.D.
2001 Lower Frictional Resistance Characteristics of Foul
Release Systems. The 8th International Symposium onpractical Design of Ships and other Floating Structures.
Cebeci. T and Chang, K.C. (1978) Calculation of incompressible rough-wall boundary-layer flows. AIAA Journal 16, p. 730.
Chambers, L. D., Stokes, K. R., Walsh, F. C., and Wood,
R.J. 1978 ITTC Performance Prediction Method. ITTC Recommended Procedures. Effective date 1999.
Claire, H. and Yebra, D. Advances in marine antifouling
coatings and technologies. Woodhead Publishing Limited, 1. edition, 2009, pp. 61–67.
Coles, D. 1956 The law of the wake in the turbulent boundary layer. Guggenheim Aeronautical Laboratory, California Institute of Technology, Pasadena.

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