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

UNIVERSALITY OF PROBABILITY DENSITY FUNCTION IN HIGH REYNOLDS-NUMBER
TURBULENT BOUNDARY LAYER
Yoshiyuki Tsuji
Department of Energy Engineering and Science
Nagoya University
Chikusa-ku, Nagoya, 464-8603, Japan
Emailc42406a@nucc.cc.nagoya-u.ac.jp

Atsushi IDO
Railway Technical Research Institutewind tunnel center,
2460 Umegahara Maibara-shi, Shiga 521-0013,
Japan

Michio Nishioka
Osaka Prefecture University, Gakuen-cho, Naka-ku,Sakai-shi, Osaka 599-8531, Japan

ABSTRACTThe probability density function (pdf) of a streamwise
velocity component is studied in zero-pressure gradientboundary layers. From analyzing the data up to𝑅𝜃 ≈ 80000,
it is found that pdfs have self-similar profiles in the log-law
region of mean velocity. In the outer part where the turbulent
intensity shows another log-law, the stream-wise velocity pdf
shows the self-similar property.
Probability density function profiles asymptote to theuniversal shape close to the Gaussian, but are positively
skewed at the core region, indicating smaller values in the tailparts. In the log-law region of turbulence intensity, however
the pdf is negatively skewed. These characteristics aresummarized depending on the Reynolds number.

In the outer region, there is another log-law of turbulent
intensity profile. It is predicted by the attached eddy model by
Townsend (1956). This relation is only confirmed in higher
Re number shear flows. We found that the pdf of streamwise
velocity indicates the self-similar profile in the outer logregion. Following these results, the universality of pdfs inmean velocity and turbulent intensity is discussed in relation
with log-law region.
EXPERIMENTAL CONDITION
The zero-pressure gradient turbulent boundary layers inhigher Reynolds numbers are studied through an experimentat the Railway Technical Research Institute (RTRI) low-noisewind tunnel in Maibara Japan. The wind tunnel wasconstructed to promote research and development on
aerodynamic noise and other subjects related to Shinkansen
and other high-speed railways (see Fig.1). The closed test
section, 5m width × 3m height × 20m length, is composed ofa windward wind tunnel (length 6.5m) and a leeward windtunnel (length 13.5m), both separately movable, and used totest and improve the characteristics of the aerodynamic force
and drag work on train models and study the flow around them.
The flat plate is placed inside the test section and thefluctuating stream-wise velocity component is measured by aconstant temperature anemometry with I-type proce at theposition 6.0, 14.4 and 18.7m from the leading edge (Fig.2).
The measurements are performed in high Reynolds number,
15000 ≤ 𝑅𝜃 ≤ 80000. The mean skin friction is determinedusing oil-film interferometry. As the detailed experimental
conditions are mentioned in Tsuji et al. (2021).

INTRODUCTION
It is clear from the Reynolds averaged equation that themean velocity is strongly related with turbulence intensity.
Thus, the mean velocity profile can be potentially discussed
from the view point of turbulence intensity distribution. Thisidea was originally confirmed by Tsuji et al. (1999, 2005) inthe case of low Reynolds numbers subject to the invariance ofpdfs in the overlap region. A logarithmic velocity profile wasderived from the pdf equation with adopting two empirical
relations. In this paper the idea is reexamined in the case ofhigher Reynolds numbers. It has been experimentallyobserved that the mean velocity in the lower part of the logregion tends to over chute, that is, the velocities here areslightly higher than if they would follow the log-law.

Fig.1 Plain view of wind tunnel facility at RTRI wind tunnel technical center (http://www.rtri.or.jp/rd/maibara-wt/INDEX.HTML).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Compared with the mean velocity profile, the log-law region
consists of two invariant pdf regions. But the second invariant
pdf region was not observed in low Reynolds numbers.

Table 1 Statistical properties of boundary layers at (a) 𝑥 = 6.0m, (b)⁡𝑥 = 14.4⁡ m, (c )⁡𝑥 = 18.7 m.

Table 1 Statistical properties of boundary layers shown in Fig.
9. (a) 𝑥 = 6.0 m, (b) 𝑥 = 14.4 m, (c ) 𝑥 = 18.7 m.

(a)

(b)

(c)

δ*[mm]

θ[mm]Rθprocedure

1.61E+01

1.22E+01

6.85E-01

1.99E+01

1.11E+02

4.68E+02

1.578E+04⑯

1.54E+01

1.19E+01

9.81E-01

2.93E+01

1.09E+02

4.61E+02

2.258E+04⑯

1.44E+01

1.12E+01

1.29E+00

3.87E+01

1.09E+02

4.34E+02

2.787E+04⑯

1.39E+01

1.09E+01

1.56E+00

4.79E+01

1.04E+02

4.27E+02

3.346E+04⑯

δ*[mm]

θ[mm]uτ[m/s]U0[m/s]Rθprocedure

2.87E+01

2.22E+01

6.60E-01

2.01E+01

2.06E+02

8.72E+02

2.850E+04⑭2.76E+01

2.16E+01

9.63E-01

3.02E+01

2.07E+028.66E+02

4.192E+04⑭

2.66E+01

2.09E+01

1.25E+00

3.98E+01

1.98E+02

8.50E+02

5.253E+04⑭2.64E+01

2.08E+01

1.51E+00

4.91E+01

1.98E+02

8.61E+02

6.372E+04⑭

δ_*[mm]

θ[mm]uτ[m/s]uτ[m/s]U0[m/s]U0[m/s]δ99[mm]δ99[mm]δ99[mm]

Δ[mm]

Δ[mm]

Rθ

procedure

3.06E+01

2.39E+01

6.61E-01

2.02E+01

2.29E+02

9.36E+02

3.192E+04③

3.05E+01

2.40E+01

9.56E-01

3.02E+01

2.31E+02Δ[mm]9.65E+02

4.697E+04⑭2.94E+01

2.32E+011.23E+00

4.00E+01

2.24E+02

9.58E+02

6.127E+04③

2.89E+01

2.29E+01

1.52E+00

4.99E+01

2.27E+02

9.50E+02

7.297E+04③Fig.4 Mean velocity and turbulent intensity distribution(upper) and the skewness distribution at 𝑅𝜃 = 814200 .
Invariant pdf regions are indicated by (1) and (2) in lowergraph.

Fig.2 Test section for probe setting at calibration. ②Pitot
tube, ③ hot-wire, ④ ⑤ thermistor, ⑥ thermocouple, ⑦thermometer.

Turbulent intensity profiles are plotted in Fig.4. Townsend’s
original idea (Townsend, 1956 ) was extended by Perry et al.
(1986) , who reported that the distribution of eddies with apopulation density was inversely proportional to the distancefrom the wall, and consequently the intensity of the stream+ )2 is in proportion to thewise velocity component (𝑢𝑟𝑚𝑠logarithmic relation of the distance from the wall. This relation
is expressed as

RESULTS AND DISCUSSIONS
The mean veclcity profiles are plotted in Fig.3. We canobserve the clear log-law profile1

𝑈 + = ln(𝑦 + ) + 𝐵(1)

𝜅raiging from 𝑦 + ≅ 180 to 0.15𝛿 + . Upper subscript + denotes
the normalization by inner variables. K𝑎́ rm𝑎́ n constant 𝜅 andadditive constant 𝐵 are evaluated as 𝜅 = 0.385 ± 0.003, 𝐵 =4.26 ± 0.15 , respectively. The skewness (third ordermoments) of stream-wise velocity fluctuation is plotted inFig.4. The skewness decreases toward the outer edge ofboundary layer. But it shows the constant value as marked (1)and (2) in Fig.4. Therefore the pdf becomes invariant in theseregions. They are called inner and outer pdf invariant regions,for convenience. Probability distributions are plotted in Fig.5.
They are positively skewed in (1) but negatively skewed in (2).
It is confirmed that pdf profiles are self-similar in each region.

U+30+ )2 = 𝐴 ln⁡(𝑦⁄𝛿 ) + 𝐵(𝑢𝑟𝑚𝑠𝑢99𝑢(2)Where 𝛿99 is the boundary layer thickness where the local
mean velocity is equal to 99% of 𝑈0 . The coefficients 𝐴𝑢 and𝐵𝑢 are evaluated as 𝐴𝑢 = −1.38 ± 0.05 , 𝐵𝑢 = 1.8 ± 0.1 ,respectively. The second invariant pdf region is closely relatedto the log-law region of Eq.(2). The pdf profiles are also seenin Fig.5, and they are negatively skewed around core region
and in tail parts. The skewness is positive near the wall but
decreases toward the outer region. There is a zero-crossing at
𝑦𝑧𝑒𝑟𝑜 or the skewness is zero at this position. We carefully
study the pdf shape around 𝑦𝑧𝑒𝑟𝑜 . Here, one of the measurescharacterizing pdf profiles is introduced. It is the KullbackLeibler divergence (KLD or KL-divergence) or relativeentropy. This measure, which was introduced by Kullback(1959) and adopted as a basic quantity in the information
theory, is defined as follows:(a)2010𝐷(𝑃||𝑄) ≡ ∑ 𝑃(𝑠𝑖 )ln{𝑃(𝑠𝑖 )⁄𝑄(𝑠𝑖 )}{𝒔}0 010101

102103104105where 𝑃(𝒔) and 𝑄(𝒔) , {𝒔} = {𝑠1 , 𝑠2 , 𝑠3 ⋯ } are discrete
probability distributions. KLD has a non-negative value for
any 𝑃(𝒔) and 𝑄(𝒔), and it is zero only when 𝑃(𝒔) is exactly
equal to 𝑄(𝒔). The more the 𝑃(𝒔) and 𝑄(𝒔) come to resemble+yFig.3 Mean velocity profiles measured at x=14.4m with 𝑈0 =20.1, 30.2, 39.8, 49.1 m/sec.

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

each other, the smaller is the KLD. Thus KLD indicatesquantitatively the resemblance between 𝑃(𝒔) and 𝑄(𝒔) . A
simple numerical example was presented earlier in Tsuji andNakamura (1999). In this analysis, 𝑄(𝒔) is taken to be theprobability at zero-crossing and 𝑄(𝒔) is set to the pdf that ismeasured at distance y from the wall.

Kullback, S., Information Theory and Statistics, Wiley,London, (1959).
Perry, A.E., Henbest, S., Chong, M.S., A theoretical andexperimental study of wall turbulence. J. Fluid Mech.
165(1986), 163–199.
Tsuji, Y., Nakamura, I., Probability density function in thelog-law region of low-Reynolds number turbulent boundarylayer. Phys. Fluids 11(1999), 647–658.
Tsuji, Y., Lindgren, B., and. Johansson, A. V., Self-similar
profile of probability density functions in zero-pressure
gradient turbulent boundary layers, Fluid Dynamics Research
37 (2005) 293–316.
Tsuji, Y., IDO, A., and Nishioka, M., Universality of mean
velocity profile in high Re-number turbulent boundary layer,
Transactions of the JSME, vol.87, p. 21-00280 (in Japanese),(2021).

Fig.5 Probability density function in two invariant pdf regions.
① Inner invariant region, ② Outer invariant region.
Figure 6 shows the KL-divergence distribution around zerocrossing point at 𝑦𝑧𝑒𝑟𝑜 for different Reynolds numbers. The
KL-divergence profiles are overlap sufficiently, therefore thepdf is universal independent of Reynolds numbers. In low
Reynolds numbers, there is no zero-crossing. Skewness has
positive values in the overlap region.

Townsend, A. A., The Structure of Turbulent Shear Flow.
Cambridge University Press, (1956).

Fig.5 Probability density function in two invariant pdf regions.
Different colours indicate different Reynolds numbers.
In the present high Re number data, the probe resolution
effect on the PDF can not be avoided in the inner invariant
PDF region. However the invariance of PDF in this region wasdiscussed in Tsuji et al.(2005) keeping the sufficient
resolution of hot wire.
SUMMARY
High Reynolds number data measured at RIRI wind tunnel
are analyzed. The probability density functions of stream-wisevelocity fluctuation become invariant in two different parts
across the turbulent boundary layers. Mean velocity log-law
as described by Eq. (1) consists of these two invariant pdf
regions. The turbulent intensity log-law following Eq.(2) is
consistent with the outer invariant pdf region. This region isobserved only in very high Reynolds numbers. At the zerocrossing point 𝑦𝑧𝑒𝑟𝑜 , where the skewness is zero, the pdfs areuniversal independent of Reynolds number. Zero-crossing
point locates around the center of mean velocity log-law
region.

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