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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:

˜

Uτ∼1

p˜

δ

,(1)

where

˜

Uτ=Uτν

M∼ν

Uτδ=1

Reτ

(2)

is named the ”dimensionless drag” and

˜

δ=δM

ν2∼δ2U2

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 M∼U2

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=δ−r=R−r, 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=DUm/ν is the bulk Reynolds

number based on the pipe diameter D. For pipe ﬂow, Reτ=RUτ/ν, 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 ×˜

δ−0.51 ∼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)2/κ2

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 Reτ∼11000 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 ×λ−0.56 ∼1

√λ∼1

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 /κpipe. 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 ×˜

δ−0.56,(14)

and combine it with Equation (10):

pfZPG(Reτ) = 1.23 ×˜

δ−0.51

˜

Uτ

= 7.24 ×˜

δ0.05 (15)

For pipe ﬂow, we combine Equations (9) and (12):

qfpipe(Reτ) = 2.24 ×λ−0.56 = 1.47 ×I−1.02

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∼˜

δ−0.06 ∼Re−0.12

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 ×Re−0.11

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 Reτ∼11000

indicates the pipe ﬂow transition found in [11]. Measurements from [1].

Ipipe ∼1

pfpipe(Reτ)∼˜

Uτp˜

δ

Q∼IZPG

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):

λ∼1

fpipe(Reτ)∼˜

U2

τ˜

δ

Q2∼1

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 Reτ∼11000 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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