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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
Abstract
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.
Keywords:
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:
˜
Uτ1
p˜
δ
,(1)
where
˜
Uτ=Uτν
Mν
Uτδ=1
Reτ
(2)
is named the ”dimensionless drag” and
˜
δ=δM
ν2δ2U2
τ
ν2=Re2
τ(3)
Email address: nils.basse@npb.dk (Nils T. Basse)
1
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
(3).
In this paper we will show that the product ˜
Uτp˜
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 ( ˜
U2
τ˜
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-
els.
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-
tensity
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)
U(r),(4)
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
2
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
Um
= 2 ×Reτ
ReD
,(6)
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
τ
U2
m
= 32 ×Re2
τ
Re2
D
(7)
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
RZR
0
qu2(r)
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 ×˜
δ0.51 1
p˜
δ
,(10)
where
3
fZPG(Reτ) = 2
κ2
ZPG 2AZPG
κZPG
+A2
ZPG (11)
+ log(Reτ)2AZPG
κZPG 2
κ2
ZPG + log(Reτ)22
ZPG
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
10-7
10-6
10-5
10-4
10-3 ZPG flow
Measurements
Fit
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
λ1
Ipipe
,(12)
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:
4
0.008 0.01 0.012 0.014 0.016 0.018 0.02
20
25
30
35
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 ×˜
δ0.56,(14)
and combine it with Equation (10):
pfZPG(Reτ) = 1.23 ×˜
δ0.51
˜
Uτ
= 7.24 ×˜
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)
5
1050 10100 10150 10200
Re
100
101
102
103
104
ZPG flow
Pipe flow
1050 10100 10150 10200
Re
0.8
0.9
1
1.1
1.2
Analytical result
Fit
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 ×˜
U0.98
τ˜
δ0.50
Q0.98 ,(17)
which can be approximated as:
Ipipe ˜
Uτp˜
δ
Q(18)
By using Equations (3) and (14), we can express the product ˜
Uτp˜
δas a
function of Reτ:
˜
Uτp˜
δ˜
δ0.06 Re0.12
τ,(19)
which is scaling behaviour similar to what has been observed in pipe ﬂow
[2, 3, 4]. The exact ﬁt to Equation (19) yields:
˜
Uτp˜
δ= 0.10 ×Re0.11
τ,(20)
see Figure 4.
Based on the ﬁndings in this paper we summarise the following TI analo-
gies for ZPG and pipe ﬂows:
6
102103104105
Re
0.02
0.03
0.04
0.05
0.06
0.07 ZPG flow
Measurements
Fit
Threshold
Figure 4: The product ˜
Uτp˜
δas a function of Reτ. The vertical line at Reτ11000
indicates the pipe ﬂow transition found in [11]. Measurements from [1].
Ipipe 1
pfpipe(Reτ)˜
Uτp˜
δ
QIZPG
Q,(21)
where we have used:
IZPG 1
pfZPG(Reτ)˜
Uτp˜
δ(22)
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):
λ1
fpipe(Reτ)˜
U2
τ˜
δ
Q21
Q2fZPG(Reτ)(24)
7
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 ˜
Uτp˜
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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9
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