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Friction-based scaling of streamwise turbulence intensity in zero-pressure-gradient and pipe flows

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We explore the analogy between asymptotic scaling of two canonical wall-bounded turbulent flows, i.e. zero-pressure-gradient and pipe flows; we find that these flows can be characterised using similar scaling laws which relate streamwise turbulence intensity and friction.
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Friction-based scaling of streamwise turbulence intensity
in zero-pressure-gradient and pipe flows
Nils T. Bassea
aTrubadurens v¨ag 8, 423 41 Torslanda, Sweden
April 6, 2021
We explore the analogy between asymptotic scaling of two canonical wall-
bounded turbulent flows, i.e. zero-pressure-gradient and pipe flows; we find
that these flows can be characterised using similar scaling laws which relate
streamwise turbulence intensity and friction.
Streamwise turbulence intensity, friction-based scaling, zero-pressure
gradient and pipe flows
A recent paper [1] on zero-pressure-gradient (ZPG) flow has introduced
an asymptotic (high Reynolds number) scaling law:
is named the ”dimensionless drag” and
Email address: (Nils T. Basse)
scales as the friction Reynolds number (Reτ) squared. Note that we use
to mean ”scales as”. Here, Uτis the friction velocity, M=Rδ
0U(z)2dzis the
kinematic momentum rate through the boundary layer, νis the kinematic
viscosity, δis the boundary layer thickness, zis the distance from the wall
and Uis the mean velocity in the streamwise direction. Note the asymptotic
scaling MU2
τδhas been proposed in [1] and applied in Equations (2) and
In this paper we will show that the product ˜
δscales as the global,
i.e. radially averaged, turbulence intensity (TI) I=pu2/U , where uis the
streamwise velocity fluctuation and overbar denotes time averaging [2, 3, 4].
As a consequence, the squared product ( ˜
δ) scales as the friction factor λ.
We note that drag was addressed in [1]; the TI was not discussed.
Our paper represents an expansion of the validity of TI scaling with fric-
tion factor, since we have focused exclusively on pipe flow in previous pub-
lished work. Having a well-defined TI for ZPG flow is important for e.g.
computational fluid dynamics (CFD) simulations [5] and this is the main
motivation for this work. Another aim is to search for shared features of
canonical wall-bounded flows [6] which may lead to improved common mod-
The paper is organized as follows: In Section 1, we briefly review results
from asymptotic scaling of TI in pipe flows; these findings are related to ZPG
flows in Section 2 and we conclude in Section 3.
1. Asymptotic pipe flow scaling of the streamwise turbulence in-
The material in this section is a summary of research on pipe flow con-
tained in [2, 3, 4]. The local (streamwise) TI is defined as:
Ilocal(r) = qu2(r)
where ris the pipe radius (r= 0 is the pipe axis and r=Ris the pipe wall),
i.e. z=δr=Rr, which can then be used to define a global TI:
Iglobal =hIlocal(r)i,(5)
where h·i indicates radial averaging; see [4], where several definitions of radial
averaging have been documented. In the remainder of this paper, we treat
the global TI; for simplicity of notation, we drop the subscript ”global” and
refer to Iinstead of Iglobal.
For pipe flow, the streamwise turbulence intensity Ipipe scales roughly
with the ratio of the friction and mean velocities [3, 4]:
Ipipe Uτ
= 2 ×Reτ
where Umis the mean velocity and ReD=DUmis the bulk Reynolds
number based on the pipe diameter D. For pipe flow, Reτ=RUτ, where
Ris the pipe radius. The friction factor λscales with the square of this ratio:
λ= 8 ×U2
= 32 ×Re2
As a consequence, the streamwise turbulence intensity scales with the
square root of the friction factor:
Ipipe λ(8)
An example of the scaling using Princeton Superpipe measurements [7, 8]
is Equation (23) in [4]:
Ipipe area,AM =1
U(r)dr= 0.66 ×λ0.55,(9)
where AM is an abbreviation for the ”arithmetic mean” radial averaging.
2. Equivalence between zero-pressure-gradient and pipe flows
In [9], we have used the ”log law” for the streamwise mean velocity [10]
to derive a correction term pfZPG(Reτ) for the asymptotic scaling of drag
presented in Equation (1):
Uτ×pfZPG(Reτ) = 1.23 ט
δ0.51 1
fZPG(Reτ) = 2
ZPG (11)
+ log(Reτ)2AZPG
κZPG 2
ZPG + log(Reτ)22
Here, κZPG = 0.39 (von K´arm´an constant) and AZPG = 5.7 are constants
derived in [9] for a fit to ZPG measurements, see Figure 1.
1081010 1012
10-3 ZPG flow
Figure 1: Correction term multiplied by dimensionless drag for ZPG flow as a function of
δ. Measurements from [1].
For pipe flow, we can define an equivalent correction term pfpipe (Reτ);
above Reτ11000 we find κpipe = 0.34 and Apipe = 1.0 [11]. To relate this
to the friction factor we perform a fit:
qfpipe(Reτ) = 2.24 ×λ0.56 1
see Figure 2. Note that the correction term is different for smooth- and
rough-wall flow since Adepends on wall roughness [10].
The link between the ZPG and pipe flows is their correction terms, see
Figure 3. The correction terms increase monotonically with Reynolds num-
ber. To relate the two correction terms, we define their ratio Qand fit this
to a logarithmic function:
0.008 0.01 0.012 0.014 0.016 0.018 0.02
40 Pipe flow
Smooth pipe measurements
Smooth pipe fit
Rough pipe measurements
Figure 2: Correction term for pipe flow as a function of λ. Rough pipe measurements are
shown for reference. Measurements from [7, 8].
Q(Reτ) = pfpipe(Reτ)
pfZPG(Reτ)= 1.15 1.46 ×log(Reτ)0.89,(13)
where we note that the constant 1.15 = 0.39/0.34 = κZPG pipe. The ratio
approaches an asymptotic value, but the increase towards this value is a
rather slow function of Reynolds number.
For ZPG flow, we introduce Equation (10) from [9]:
Uτ= 0.17 ט
and combine it with Equation (10):
pfZPG(Reτ) = 1.23 ט
= 7.24 ט
δ0.05 (15)
For pipe flow, we combine Equations (9) and (12):
qfpipe(Reτ) = 2.24 ×λ0.56 = 1.47 ×I1.02
pipe area,AM (16)
1050 10100 10150 10200
ZPG flow
Pipe flow
1050 10100 10150 10200
Analytical result
Asymptotic value
Figure 3: Note the extreme Reynolds number range, from 102to 10200 . Left-hand plot:
Correction terms for ZPG and pipe flows as a function of Reτ, right-hand plot: Ratio of
correction terms as a function of Reτ; the horizontal black line indicates the asymptotic
value of 1.15.
Since the correction terms in Equations (15) and (16) are related by
Equation (13), we arrive at:
Ipipe area,AM = 1.19 ט
Q0.98 ,(17)
which can be approximated as:
Ipipe ˜
By using Equations (3) and (14), we can express the product ˜
δas a
function of Reτ:
δ0.06 Re0.12
which is scaling behaviour similar to what has been observed in pipe flow
[2, 3, 4]. The exact fit to Equation (19) yields:
δ= 0.10 ×Re0.11
see Figure 4.
Based on the findings in this paper we summarise the following TI analo-
gies for ZPG and pipe flows:
0.07 ZPG flow
Figure 4: The product ˜
δas a function of Reτ. The vertical line at Reτ11000
indicates the pipe flow transition found in [11]. Measurements from [1].
Ipipe 1
where we have used:
Note that it is only possible to present scaling properties of IZPG and
not an explicit equation, since velocity fluctuation measurements are not
available in [1]. We can reformulate Equation (21) to:
Ipipepfpipe IZPGpfZPG (23)
Corresponding friction factor analogies can be found by taking the square
of Equation (21):
3. Conclusions
We have explored the correspondence between zero-pressure-gradient (ZPG)
and pipe flows for asymptotic scaling of streamwise turbulence intensity with
friction. It is demonstrated that similar scalings are valid for both types of
flows; the product ˜
δfor ZPG flow is equivalent to the streamwise tur-
bulence intensity for pipe flow Ipipe. The scaling of turbulence intensity with
Reynolds number in ZPG flow closely matches the corresponding pipe flow
scaling. In addition, we have shown that the turbulence intensity is inversely
proportional to the correction term pf(Reτ) and that κand Afor the cor-
rection term are different for ZPG and pipe flows.
A source of inaccuracy of our results is that we have used measurements
carried out at all Reynolds numbers. However, in [11] we have shown that a
transition exists at Reτ11000 for pipe flow; scaling is somewhat different
below and above this threshold. Future research includes studies for higher
Reynolds numbers to characterise scaling below and above the transition.
We also plan studies of other canonical flows, e.g. channel flow [6] and plane
Couette and Poiseuille flows [12].
Acknowledgements. We are grateful to Google Scholar Alerts for making us
aware of [1] in a ’Recommended articles’ e-mail dated 14th of May 2020.
Data availability statement. Data sharing is not applicable to this article as
no new data were created or analyzed in this study.
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ResearchGate has not been able to resolve any citations for this publication.
Full-text available
We study streamwise turbulence intensity definitions using smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with the bulk (and friction) Reynolds number is provided for the definitions. The turbulence intensity scales with the friction factor for both smooth- and rough-wall pipe flow. Turbulence intensity definitions providing the best description of the measurements are identified. A procedure to calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness for rough-wall pipe flow) is outlined.
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Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We quantify the correspondence between turbulence intensity and the friction factor.
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Considerable discussion over the past few years has been devoted to the question of whether the logarithmic region in wall turbulence is indeed universal. Here, we analyse recent experimental data in the Reynolds number range of nominally 2×10 4 <Re τ <6×10 5 for boundary layers, pipe flow and the atmospheric surface layer, and show that, within experimental uncertainty, the data support the existence of a universal logarithmic region. The results support the theory of A. A. Townsend [The structure of turbulent shear flow. 2nd ed. Cambridge Monographs on Mechanics and Applied Mathematics. London etc.: Cambridge University Press. XI (1976; Zbl 0325.76063)], where, in the interior part of the inertial region, both the mean velocities and streamwise turbulence intensities follow logarithmic functions of distance from the wall.
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We review wall-bounded turbulent flows, particularly high–Reynolds number, zero–pressure gradient boundary layers, and fully developed pipe and channel flows. It is apparent that the approach to an asymptotically high–Reynolds number state is slow, but at a sufficiently high Reynolds number the log law remains a fundamental part of the mean flow description. With regard to the coherent motions, very-large-scale motions or superstructures exist at all Reynolds numbers, but they become increasingly important with Reynolds number in terms of their energy content and their interaction with the smaller scales near the wall. There is accumulating evidence that certain features are flow specific, such as the constants in the log law and the behavior of the very large scales and their interaction with the large scales (consisting of vortex packets). Moreover, the refined attached-eddy hypothesis continues to provide an important theoretical framework for the structure of wall-bounded turbulent flows.
A new asymptotic −1/2 power-law scaling is derived from the momentum integral equation for the drag in flat-plate turbulent boundary layers. In the limit of infinite Reynolds number, the appropriate velocity scale for drag is found to be M/ν, where M is the boundary layer kinematic momentum rate and ν is the fluid kinematic viscosity. Data covering a wide range of Reynolds numbers remarkably collapse to a universal drag curve in the new variables. Two models, discrete and continuous, are proposed for this universal drag curve, and a robust drag estimation method, based on these models, is also presented.
Orlandi et al. ( J. Fluid Mech. , vol. 770, 2015, pp. 424–441) present direct numerical simulations over a very wide Reynolds number range for plane Couette and Poiseuille flows. The results reveal new information on the abrupt nature of transition in these flows, and the comparisons between Couette and Poiseuille flows help to provide a clearer picture of Reynolds number trends, especially with regard to inner/outer layer interactions. The stress distributions give strong support to Townsend’s attached eddy hypothesis, particularly for the wall-parallel component where there has been little experimental data available. The results pose some intriguing questions regarding the reconciliation of the present results with data at higher Reynolds numbers in different canonical flows.
Measurements of the streamwise component of the turbulent fluctuations in fully developed smooth and rough pipe flow are presented over an unprecedented Reynolds number range. For Reynolds numbers \$R{e}_{\tau } \gt 20\hspace{0.167em} 000\$, the streamwise Reynolds stress closely follows the scaling of the mean velocity profile, independent of the roughness, and over the same spatial extent. This observation extends the findings of a logarithmic law in the turbulence fluctuations as reported by Hultmark, Vallikivi & Smits (Phys. Rev. Lett., vol. 108, 2012) to include rough flows. The onset of the logarithmic region is found at a location where the wall distance is equal to ∼100 times the Kolmogorov length scale, which then marks sufficient scale separation for inertial scaling. Furthermore, in the logarithmic region the square root of the fourth-order moment also displays logarithmic behaviour, in accordance with the observation that the underlying probability density function is close to Gaussian in this region.
The distribution of the streamwise velocity turbulence intensity has recently been discussed in several papers both from the viewpoint of new experimental results as well as attempts to model its behaviour. In the present paper numerical and experimental data from zero pressure-gradient turbulent boundary layers, channel and pipe flows over smooth walls have been analyzed by means of the so called diagnostic plot introduced by Alfredsson & Örlü [P.H. Alfredsson, R. Örlü, The diagnostic plot–a litmus test for wall bounded turbulence data, Eur. J. Mech. B Fluids 29 (2010) 403–406]. In the diagnostic plot the local turbulence intensity is plotted as function of the local mean velocity normalized with a reference velocity scale. Alfredsson et al. [P.H. Alfredsson, A. Segalini, R. Örlü, A new scaling for the streamwise turbulence ntensity in wall-bounded turbulent flows and what it tells us about the outer peak, Phys. Fluids 23 (2011) 041702] observed that in the outer region of the boundary layer a universal linear decay of the turbulence intensity independent of the Reynolds number exists. This approach has been generalized for channel and pipe flows as well, and it has been found that the deviation from the previously established linear region appears at a given wall distance in viscous units (around 120) for all three canonical flows. Based on these results, new empirical fits for the streamwise velocity turbulence intensity distribution of each canonical flow are proposed. Coupled with a mean streamwise velocity profile description the model provides a composite profile for the streamwise variance profile that agrees nicely with existing numerical and experimental data. Extrapolation of the proposed scaling to high Reynolds numbers predicts the emergence of a second peak of the streamwise variance profile that at even higher Reynolds numbers overtakes the inner one.