12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

SCALING OF TURBULENT BOUNDARY LAYER PROFILES IN AIRFLOWOVER YOUNG WIND-WAVES

Meital Geva
School of Mechanical Engineering
Tel-Aviv University
Tel-Aviv 69978, Israel
meitalgeva@mail.tau.ac.il

Lev Shemer
School of Mechanical Engineering
Tel-Aviv University
Tel-Aviv 69978, Israel
shemerl@tauex.tau.ac.il

ABSTRACT

and references therein. Moreover, z0 cannot be directly evaluated from the water surface features (Csanady, 2001), it isoften estimated using the Charnock relation, αch = z0 g/u2∗(Charnock, 1955). Here αch is presumably constant and gis the gravitational acceleration. It was found however that
αch is in fact not constant and may vary by two orders ofmagnitude (Bye et al., 2010). Although there were numerous ongoing attempts to relate z0 to wind-waves characteristic,no universal formulation was found so far. The wind-wave
field varies continuously in time and space; thus the water surface has time-dependent random three-dimensional topography (Zavadsky & Shemer, 2017). Although water waves statistics can be obtained relatively easily, measurements of the air
flow over a moving stochastic three-dimensional wavy surface
are more challenging (Zavadsky & Shemer, 2012). While thedata on air flow over wind-waves is limited, more extensive
information is available on flow over rough solid surface withdiverse topographies, see Chung et al. (2021) for a recent review. These studies indicate that when the roughness effects
are limited to the near wall region, the boundary layers infree and bounded flows experience similarity, e.g., Shockling
et al. (2006); Schultz & Flack (2007) and references therein.
Jimenez (2004) asserted that similarity may be found when theratio between the roughness height and boundary layer thickness is above 40, whereas Castro (2007) observed that wall
similarity holds up to a limit of 5. Flack & Schultz (2014)
showed that this limit depends on the shape of the roughness
elements. Albeit fundamental differences between the solidrough surface and the wavy water surface exist, we found essential common features characterizing the turbulent boundary
layers in these cases, providing an alternative insight on the
air-water coupling.

Understanding the air velocity profile over wind-waves is
crucial for evaluation of momentum and energy transfer between air and water. It is well accepted that the airflow above
water waves has logarithmic profile that is usually expressedin terms of an effective roughness parameter. This parameter however cannot be evaluated directly from the wave measurements at the water surface. Determination of accurate air
velocity profile in field conditions is nearly impossible; the existing estimates are based on rough data. We suggest an alternative approach based on the extensive experimental data
accumulated in our wind-wave facility. Combined measurements of the finely resolved mean air velocity profile above
water surface and of the characteristics of the wind-wave field
were performed at 6 wind forcing conditions and at 7 fetches.
We demonstrate that the spatially developing boundary layerover the complicated three-dimensional, random and moving
wind-wave surface is characterized by wall similarity observedin turbulent flow above rough solid surfaces. The results enable more accurate estimations of the actual drag coefficient.

INTRODUCTION
The coupling between air and water in the presence ofwind-waves determines mass and momentum transfer between
atmosphere and ocean and thus controls the weather and, toa large extent, determines the life conditions, in particular inoceans. In wind waves studies, the generally accepted air velocity profile is:U(z) =

 
z
u∗
ln
κ
z0

(1)

WIND-WAVE EXPERIMENTS
The facility

where u∗ is the friction velocity determined by the shear
stressat the water surface, τw , and the air density ρ, u∗ =pτw /ρ. The friction parameter z0 is a function of the roughness density. Both u∗ and z0 are obtained from experiments;
however, the available field and laboratory data lack sufficient
resolution, e.g., Hristov et al. (2003); Caulliez et al. (2008)

Experiments were conducted at the Tel-Aviv University
wind-wave facility described schematically in Figure 1. The
facility consists of a closed-loop wind tunnel atop of a 5 mlong rectangular test section. The test section is 0.4 m wide1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1.
ity.

Schematic illustration of the TAU wind-wave facil-and 0.5 m high; it is filled with water to depth of 0.2 m. The
airflow uniformly enters the test section after passing througha large settling chamber, honeycomb and converging nozzle
with an area reduction ratio of about 4. The instantaneous surface elevation η(x,t) is recorded at several downstream locations using capacitance-type wave gauges that are positioned
at the centreline of the test section. The mean airflow is measured using a pitot tube with an outer diameter of 1 mm. The
vertical profiles of the turbulent fluctuations are obtained using
an X-hot film. The pitot tube and the hot-film are supported
by a vertical computer-controlled stage that allow precise positioning. To keep both sensors dry, measurements above thewater surface were restricted to elevations above the maximum
possible wave crest. The accuracy of the probe positioning is
0.05 mm; measurements were performed at up to nearly 100
vertical locations z. The data acquisition lasted from few minutes for wind velocity measurements at each z, to an hour ormore for surface elevation η(x,t). The acquisition duration
was thus larger by orders of magnitude than the characteristic
wind-wave periods that range from 0.1 s to 1.5 s. The experiments were performed at 6 wind forcing conditions. For
each maximum wind velocity U0 , the data were gathered at 7
downstream distances ranging from 1 m to 3.4 m. For moredetails on the experimental facility and the inherent difficulties
in this kind of measurements see Liberzon & Shemer (2011)
and Zavadsky & Shemer (2012).

Figure 2. (a) Schematic illustration of wind-waves growth
and the developing boundary-layer in air; (b) snapshots of the
wave field taken through the side window of the test section at
three locations at maximum wind velocity U0 =11.2 m/s.

served in Zavadsky & Shemer (2012) that the variation of u∗with fetch did not exceed about 7%. The non-trivial experimental observation that u∗ remains constant along the test section was also reported elsewhere (Plant & Wright, 1977; Hsu
et al., 1982; Caulliez et al., 2008). Note that the water drift
velocity is less than 0.5u∗ (Caulliez et al., 2008; Zavadsky &Shemer, 2017).

Characteristics of young wind-waves

Table 1. Characteristic mean wind velocities: U0 - the maximum wind velocity and u∗ - the friction velocity. Subscript
ZS represents the data from Zavadsky & Shemer (2012) andsubscripts MI is the momentum integral estimations.

Under steady wind forcing, the wave height, h, and thewavelength, λ , grow with the downstream distance, x, as
shown schematically in Figure 2(a). Boundary layer withthickness δ (x) is formed and develops with x. Snapshots of
the air-water interface at different downstream distances presented in Figure 2(b) demonstrate the changing topography of
the water-surface. Wind-waves are stochastic and three dimensional, losing their coherence fast in time as well as in space(Kumar et al., 2022). The characteristic wave amplitude is defined as the rms value of the local instantaneous surface elevation η(x,t). Due to wave breaking, the steepness ηrms k p at allwind forcing conditions is limited to about 0.2÷0.25, wherek p is the peak wave number (Zavadsky & Shemer, 2012; Hsu
et al., 1982; Buckley et al., 2020). The average friction velocities determined for each wind forcing by two independent
methods, from the velocity profiles and by measuring vertical
distribution of Reynolds stresses (Zavadsky & Shemer, 2012),
are presented in Table 1. Both methods to determine the friction velocity rely on a proper fitting that should be limited to
the logarithmic layer (z/δ <0.2). For the extreme wind conditions, Zavadsky & Shemer (2012) were forced to extend thefitting domain to z/δ <0.3. They acknowledge that insertionof some portion of the wake in the fit results in an unavoidable
overestimate of the friction velocity. Their estimations of thefriction velocities u∗ are listed in Table 1 and agree well with
Plant & Wright (1977) and Buckley et al. (2020). It was ob-

U0 (m/s)u∗,ZS (m/s)

u∗,MI (m/s)

5.50.29 ± 0.010

0.30 ± 0.009

6.6

0.37 ± 0.013

0.39 ± 0.011

7.7

0.45 ± 0.016

0.47 ± 0.012

8.9

0.56 ± 0.019

0.54 ± 0.013

10.05

0.68 ± 0.024

0.61 ± 0.011

11.2

0.85 ± 0.03

0.77 ± 0.014

RESULTS
Integral boundary layer parameters
The measured velocity profiles at seven downstream locations in the range 1 m ≤ x ≤ 3.4 m for the wind forcing

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3.

Mean vertical velocity profiles in air.

Figure 5. (a) Dimensionless mean velocity profiles at various
wind velocities and fetches; (b) the downshift ∆U + as a function of the characteristic surface elevation.

Boundary layer similarity
The dimensionless profiles U + = U/u∗ as a function ofz+ = zu∗ /ν are presented in Figure 5(a) together with the velocity profile over smooth water surface represented by a boldline. Although the vertical coordinate of the measurement at
the two extreme wind velocities exceeds z+ =300, it is apparent that the slopes of U + (z+ ) in the logarithmic part of the
profiles in all cases are close to that over a smooth surface. It
is also evident that the profiles over the wavy water surface
are shifted downward relative to the bold line. The shift ∆U +increases with wind velocity and downstream distance. The
increase in U0 and in x results in wave amplification that increases the surface’s roughness. In turbulent flows over roughsolid surfaces, the amount of the downshift from the smoothvelocity profiles, ∆U + , depends on the dimensionless characteristic sand grain roughness height ks+ . Hudson et al. (1996)
analyzed the boundary layer over a deterministic solid wavy
surface and used the surface amplitude as ks . For a moving stochastic air-water interface, we select the representative
wave amplitude ηrms (x) as the characteristic roughness, ks (x).
Figure 5(b) demonstrates the values of ηrms (x) vs. the valuesof ∆U + from figure 5(a); data obtained in other facilities areadded. The contribution of the surface drift velocity cancels
out since ∆U + represents the difference in velocities over thesmooth and wavy water surfaces. Nearly all markers in Figure 5(b) follow the fully rough asymptote. It is thus evident
that the shapes of the mean velocity profiles in the overlap region are unaffected by the roughness ηrms which determines
only their vertical shift. The velocity profile may be therefore
approximated by

Figure 4. Boundary layer characteristics: (a) boundary layer
thickness δ , (b) displacement thickness δ1 , (c) momentum
thickness√ δ2 and (d) normalized roughness height defined asa/δ = 2ηrms /δ . Colours correspond to those in Figure 3.

specified in Table 1 are plotted in Figure 3(a). It should be
stressed that due to the movement and randomness of the water surface, there is an inherent difficulty to measure the windvelocity profiles adjacent to the water surface. This difficulty
is more severe at the two strongest wind forcings as the wave
heights increase; to keep the air velocity sensors dry, no measurements were carried out in the close vicinity of the interface
at U0 =10.05 m/s and 11.2 m/s.
The boundary layer thickness δ was defined as elevationfrom the interface where the wind velocity reached 0.99U0 .
Estimation of the integral values, the displacement thickness
δ1 and momentum thickness δ2 , require extrapolating thelower part of each velocity profile to the water surface. The fitting at elevations smaller than the available experimental data
was based on the linear-logarithmic profile suggested by Miles(1957), which consists of a linear segment in the viscous sublayer followed by a logarithmic part. The boundary layer characteristics are shown in Figure 4(a)-(c) and will be used in sequence.



z
1
+ 8.5
U (x, z) = lnκηrms (x)+3(2)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
boundary layers implies that the deficit depends only on theroughness height. We utilize the momentum integral relationin the presence of pressure gradient:
u2∗
dδ2
1 dp
=
− (2δ2 + δ1 )dx
U02
ρU02 dx(5)

that can be approximated asdδ2 (2δ2 + δ1 ) u2∗
u2∗
≈
+
dx
δ
U02
U02

For a given wind velocity U0 , substituting the values of δ ,δ1 and δ2 (Figure 4a-c) into equation (6) enables additional independent estimation of the friction velocity u∗ at any x. These
values are listed in the right column in Table 1 and are consistent with estimates by Zavadsky & Shemer (2012). Evaluationof the friction velocities via the integral relations is more robust and less sensitive to the profile details in the lower part ofthe boundary layer. The friction velocities determined by thismethod remain nearly constant along the test section.
Equation (6) can be further used to relate the skin friction
at any surface elevation. The integral values of displacement
and momentum thickness, δ1 and δ2 respectively, can be explicitly evaluated using the vertical velocity profile. We haveshown the applicability of the fully-rough velocity profile in
the overlap region; the following derivation also considers theouter region by accounting for the favorable pressure gradientFigure 6. Evidence of wall similarity in (a) the velocity defect law and (b) normalized Reynolds stresses. Colors andmarkers as in Figure 3.

that unlike equation (1) contains only directly measurable
quantities. The fact that the overlap and outer layers of theboundary layers are not affected by the surface roughness may
also be evident in the collapse of the velocity defect profiles and the similarity in the distribution of the normalized+
Reynolds shear stress −u′ w′ (Figure 6). Since the air in arectangular tank is subjected to a favorable pressure gradient,outer-layer similarity may not be always expected. The appropriate values of Clauser parameter β and of the defect shapefactor G are evaluated:

β=

(6)

U + (z) =1
z2Π 2  πz ln
+ 8.5 +sinκηrms (x)κ2δ(7)

where Π is Coles’ parameter. Equation (6) can be expressed as:∂ G dΠ

δ1 d pτ dx

F − δ ∂ Π dx
dδ
=
dx
G

(3)

(8)

Π2 −4−4Π(1+Π)−4ΠQ/π

u∗
G=
δ1U0

Z δ
0

U0 −U(z)
u∗

+ 1+Π
,
where
G =
2
κU0+
2(κU0+ )R
F = U1+ 1 − 2δ2δ+δ1
and Q = 0π sin(z)/z dz = 1.85.
0This equation
can be integrated numerically for a givenpU0+ = 2/c f and Π. The result of the equation is anexpression for the boundary layer thickness δ = Φ(x, c f , Π) .
Based on equation (7) at z = δ , we obtain a relation between
the surface elevation and the skin friction:

2
dz

(4)

The pressure drop d p/dx was measured directly in Liberzon & Shemer (2011) for the airflow conditions listed in Table
1; it agrees with the approximation based on Navier-Stokes re+∂ (u′ w′ )lation as β = − δδ1 ∂ (z/δ ) ≈ − δδ1 .The values of β and G do
not vary significantly for all operational conditions, β =-0.1÷0.17 and G=5.13±0.33. This provides further support for thewall similarity. Note that for all cases,√ the ratio between theroughness height, defined as a = 2ηrms and the boundary
layer thickness is above 5 (see figure 4d) consistent with thecritical value required for wall similarity (Castro, 2007).

ηrms =

Φ(x, c f , Π)+κ(Ue 0 −8.5−2Π(x)/κ)(9)

The reader is referred to Geva & Shemer (2022) for detailed
derivation.
For all cases listed in Table 1, equation (8) was solved
numerically by a 4th order Runge-Kutta routine with the experimentally obtained dependence Π(x) and the initial valueof δ ≈ 0.05 m at x = 1 m. Estimates of spatial evolution ofηrms for a given wind forcing were found using equation (9)Skin friction estimation and the coupling between air and young wind-waves
The vertical downshift ∆U + represents deficit due to momentum exchange between air and water. The similarity of the4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 7. The coupling between air flow and the growth of young wind waves: (a) surface elevation as a function of x for various
wind velocities and (b) Skin-friction coefficient dependence on x/ηrms .

integral equation was used to provide an independent robust
evaluation of friction velocities. These values were found toagree well with estimates in previous study that used different techniques. The friction velocities determined by the integral method remain nearly constant along the test section eventhough the boundary layer develops with x. More importantly,
we use the momentum integral equation to express the dragcoefficients that characterize the coupling between airflow andwater surface. The experiments show that for a given wind
forcing, the rms value of the surface elevation grows nearlylinearly with downstream distance. This linear growth of thecharacteristic wave amplitude is consistent with the case of linear increase in solid roughness height which results in constantskin-friction coefficients. The skin friction coefficients in thisstudy were found to agree well with the relation of skin friction dependence with x/ks over rough solid surface. This study
provides an alternative approach to study the coupling between
wind and waves and offers a new understanding of the relation
between the drag coefficients and the local surface conditions.

and plotted in figure 7(a) with broken lines. The results compare reasonably well with measurements of the surface elevation for various wind velocities (dot markers). Linear fits wereadded as well, demonstrating that at each wind velocity U0 , thecharacteristic amplitude of the surface elevation grows nearlylinearly with x, so that x/ηrms remains approximately constant.
This supports the observation of constant with x skin-friction
for a given wind velocity reported by previous studies; it issimilar to the case of the developing boundary layer over solid
surface where the effective roughness grows linearly with x(Sridhar et al., 2017). Estimates of x/ηrms based on the slopesof the linear fits (Figure 7a) for a given c f are displayed bycolored markers in Figure 7(b). This figure compares resultswith skin friction estimation based on:

c f = [2.87 + 1.58 log10 (x/ks )]−2.5(10)that was obtained over solid rough surface with constantks and without pressure gradient (Schlichting, 1979). Surprisingly, reasonable agreement was found. It should be mentioned that we do not present the skin friction values as afunction of Rex as is common for flows over rough solid surface. Instead, in wind waves studies, the nondimensional fetch,xg/u2∗ is customarily used. The dependence of the drag coefficients on the nondimensional fetch is presented in Geva &Shemer (2022) and compared with numerous results available
elsewhere. The values of the drag coefficients in our experiments agree well with those data.

Acknowledgement
This study was supported by grant # 508/19 from Israel
Science Foundation.

REFERENCES
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Castro, I. P. 2007 Rough-wall boundary layers: Mean flow universality. J. Fluid Mech. 585, 469–485.
Caulliez, G., Makin, V. & Kudryavtsev, V. 2008 Drag of thewater surface at very short fetches: Observations and modeling. J. Phys. Oceanogr. 38 (9), 2038–2055.
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CONCLUSIONS
In wind-waves studies, the coupling between air flow andthe developing wave field is crucial in estimation of mass, momentum and energy transfer between both phases. So far, themajority of wind-waves studies had focused mainly on the water parameters. This study was dedicated to the analysis of
the developing boundary layers over wind waves in relations
to the local wave conditions. We have examined 42 different
cases covering 6 wind forcings and 7 spatial locations. It wasdemonstrated that the boundary layers over wind waves maintain wall similarity; the local characteristic surface elevation
serves as the equivalent sand grain roughness. The momentum

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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