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Turbulence Intensity Scaling: A Fugue

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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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fluids
Article
Turbulence Intensity Scaling: A Fugue
Nils T. Basse
Elsas väg 23, 423 38 Torslanda, Sweden; nils.basse@npb.dk
Received: 05 July 2019; Accepted: 30 September 2019; Published: 9 October 2019
Abstract:
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.
Keywords:
streamwise turbulence intensity definitions; Princeton Superpipe measurements; smooth-
and rough-wall pipe flow; friction factor; computational fluid dynamics boundary conditions
1. Introduction
The turbulence intensity (TI) is of great importance in, e.g., industrial fluid mechanics, where it
can be used for computational fluid dynamics (CFD) simulations as a boundary condition [
1
]. The TI
is at the center of the fruitful junction between fundamental and industrial fluid mechanics.
This paper contains an extension of the TI scaling research in [
2
] (smooth-wall pipe flow) and [
3
]
(smooth- and rough-wall pipe flow). As in those papers, we treat streamwise velocity measurements
from the Princeton Superpipe [
4
,
5
]. The measurements were done at low speed in compressed air
with a pipe radius of about 65 mm. Details on, e.g., the bulk Reynolds number range and uncertainty
estimates can be found in [
4
]. Other published measurements including additional velocity components
can be found in [6] (smooth-wall pipe flow) and [7,8] (smooth- and rough-wall pipe flow).
Our approach to streamwise TI scaling is global averaging; physical mechanisms include separate
inner- and outer-region phenomena and interactions between those [
9
]. Here, the inner (outer) region
is close to the pipe wall (axis), respectively.
The local TI definition (see, e.g., Figure 9 in [10]) is:
I(r) = vRMS(r)
v(r), (1)
where
r
is the radius (
r=
0 is the pipe axis and
r=R
is the pipe wall),
v(r)
is the local mean streamwise
flow velocity, and
vRMS(r)
is the local root-mean-square (RMS) of the turbulent streamwise velocity
fluctuations. Using the radial coordinate
r
means that outer scaling is employed for the position.
As was done in [
2
,
3
], we use
v
as the streamwise velocity (in much of the literature,
u
is used for the
streamwise velocity).
The measurements in [
10
] were on turbulent flow in a two-dimensional channel and similar work
for pipe flow was published in [11].
In this paper, we study TI defined using a global (radial) averaging of the streamwise velocity
fluctuations. The mean flow is either included in the global averaging or as a reference velocity.
This covers the majority of the standard TI definitions.
There is a plethora of TI definitions, which is why we use the term fugue in the title. This is
inspired by [12], where Frank Herbert’s Dune novels [13] are interpreted as “an ecological fugue”.
Fluids 2019,11, 5842; doi:10.3390/fluids11205842 www.mdpi.com/journal/fluids
Fluids 2019,11, 5842 2 of 13
The ultimate purpose of our work is to be able to present a robust and well-researched formulation
of the TI; an equivalent TI in the presence of shear flow. Our work is not adding significant knowledge
of the fundamental processes [
14
16
], but we need to understand them in order to use them as a
foundation for the scaling expressions.
The main contributions of this paper compared to [3] are:
The introduction of additional definitions of the TI
Log-law fits in addition to power-law fits
New findings on the rough pipe friction factor behaviour of the Princeton Superpipe
measurements.
Furthermore, we include a discussion on the link between the TI and the friction factor [
3
,
17
] in
the light of the Fukagata–Iwamoto–Kasagi (FIK) identity [18].
Our paper is organised as follows: In Section 2, we introduce the velocity definitions. These are
used in Section 3to define various TI expressions. In Section 4we present scaling laws using the
presented definitions. We discuss our findings in Section 5and conclude in Section 6.
2. Velocity Definitions
The friction velocity is:
vτ=pτw/ρ, (2)
where τwis the wall shear stress and ρis the fluid density.
The area-averaged (AA) velocity of the turbulent fluctuations is:
hvRMSiAA =2
R2×ZR
0vRMS(r)rdr(3)
The fit between vτand hvRMSiAA is shown in Figure 1:
hvRMSiAA =1.7277 ×vτ(4)
0 0.2 0.4 0.6 0.8
v [m/s]
0
0.2
0.4
0.6
0.8
1
1.2
vRMS AA [m/s]
Smooth pipe
Rough pipe
Fit
Figure 1. Relationship between vτand hvRMS iAA.
The velocity on the pipe axis is the centerline (CL) velocity:
vCL =v(r=0)(5)
Fluids 2019,11, 5842 3 of 13
The (area-averaged) mean velocity is given by:
vm=2
R2×ZR
0v(r)rdr(6)
The difference between the centerline and the mean velocity scales with the friction velocity.
The corresponding fit is shown in Figure 2:
vCL vm=4.4441 ×vτ, (7)
where the fit constant is close to the value of 4.28 [
19
] found using earlier Princeton Superpipe
measurements [20].
0 0.2 0.4 0.6 0.8
v [m/s]
0
1
2
3
4
vCL-vm [m/s]
Smooth pipe
Rough pipe
Fit
Figure 2. Relationship between vτand vCL vm.
3. Turbulence Intensity Definitions
3.1. Local Velocity Definitions
The arithmetic mean (AM) definition is:
IPipe area, AM =1
RZR
0
vRMS(r)
v(r)dr(8)
The area-averaged definition is:
IPipe area, AA =2
R2ZR
0
vRMS(r)
v(r)rdr(9)
In [
3
] (Equation (9)),
hvRMSi
was defined as the product of
vm
and
IPipe area, AA
. Comparing the
resulting Equations (11) and (12) in [
3
] to the current Equation (4), we find a difference of less than 5%
(9/5 compared to 1.7277).
Finally, the volume-averaged (VA) definition (inspired by the FIK identity) is:
IPipe area, VA =3
R3ZR
0
vRMS(r)
v(r)r2dr(10)
Fluids 2019,11, 5842 4 of 13
3.2. Reference Velocity Definitions
As mentioned in the Introduction, we use outer scaling for the radial position
r
. For the TI,
we separate the treatment to inner and outer scaling below, see, e.g., [15].
3.2.1. Inner Scaling
For inner scaling, we define the TI using vτas the reference velocity:
Iτ=hvRMSiAA
vτ
=1.7277, (11)
where the final equation is found using Equation (4).
The square of the local version of this,
I2
τ(r) = vRMS (r)2
v2
τ
, is often used as the TI in the literature,
see also Section 5.1 in [
3
]. This is the normal streamwise Reynolds stress normalised by the friction
velocity squared.
Iτis shown in Figure 3; no scaling is observed with ReD, the bulk Reynolds number:
ReD=Dvm
νkin , (12)
where D=2Ris the pipe diameter and νkin is the kinematic viscosity.
104105106107108
ReD
1
1.5
2
2.5
3
I
Smooth pipe
Rough pipe
1.7277
Figure 3. Iτas a function of ReD.
3.2.2. Outer Scaling
For outer scaling, we use either vmor vCL as the reference velocity.
The TI using vmas the reference velocity is:
Im=hvRMSiAA
vm(13)
Finally, the TI using vCL as the reference velocity [15] is:
ICL =hvRMSiAA
vCL (14)
Fluids 2019,11, 5842 5 of 13
4. Turbulence Intensity Scaling Laws
The Princeton Superpipe measurements and the TI definitions in Section 3are used to create the
TI data points. Thereafter we fit the points using the power-law fit:
QPowerlaw fit(x) = a×xb, (15)
and the log-law fit:
QLoglaw fit(x) = c×ln(x) + d(16)
Here,
a
,
b
,
c
, and
d
are constants.
Q
is the quantity to fit and
x
is a corresponding variable. We first
apply the two fits using Q=Iand x=ReD.
The log-law fit is obtained by taking the (natural) logarithm of the power-law fit. The reason we
use these two fits is that they have been discussed in the literature [
21
,
22
] as likely scaling candidates.
We apply the two fits to the measurements and calculate the resulting root-mean-square deviations
(RMSD) between the fits and the measurements. A small RMSD means that the fit is closer to
the measurements.
Note that the smooth pipe
ReD
measurement range is much larger than the rough pipe
ReD
measurement range: a factor of 74 (9 points) compared to a factor of 6 (4 points). The consequence is a
major uncertainty in the rough pipe results, e.g., (i) fits and (ii) extrapolation.
It is also important to be aware that we only have two sets of measurements with the following
sand-grain roughnesses ks:
Smooth pipe: ks=0.45 µm [23]
Rough pipe: ks=8µm [24] (see the related discussion in Section 5.1)
The results are presented in Figures 4and 5and Tables 14.
We do not discuss the quality of fits to the rough pipe, since there are only 4 measurements for a
single ks. Thus, these values are provided as a reference.
For the smooth pipe, the power-law fits perform slightly better than the log-law fits, except for
the CL definition. The best fit is using the power-law fit to the AA definition of the TI:
IPipe area, AA =0.3173 ×Re0.1095
D, (17)
which is the same as Equation (5) in [3].
Table 1. Power-law fit constants, smooth pipe.
TI Definition a b RMSD
IPipe area, AM 0.2274 0.1004 4.0563 ×104
IPipe area, AA 0.3173 0.1095 3.5932 ×104
IPipe area, VA 0.3758 0.1134 3.6210 ×104
Im0.2657 0.1000 5.2031 ×104
ICL 0.1811 0.0837 6.9690 ×104
Table 2. Log-law fit constants, smooth pipe.
TI Definition c d RMSD
IPipe area, AM 0.0059 0.1391 6.7748 ×104
IPipe area, AA 0.0080 0.1808 8.8173 ×104
IPipe area, VA 0.0093 0.2074 1.1018 ×103
Im0.0069 0.1634 5.8899 ×104
ICL 0.0049 0.1257 5.4179 ×104
Fluids 2019,11, 5842 6 of 13
Table 3. Power-law fit constants, rough pipe.
TI Definition a b RMSD
IPipe area, AM 0.1172 0.0522 4.0830 ×104
IPipe area, AA 0.1702 0.0638 3.5784 ×104
IPipe area, VA 0.1989 0.0667 3.7697 ×104
Im0.1568 0.0610 3.4902 ×104
ICL 0.1177 0.0519 3.4317 ×104
Table 4. Log-law fit constants, rough pipe.
TI Definition c d RMSD
IPipe area, AM 0.0028 0.0960 4.3111 ×104
IPipe area, AA 0.0042 0.1291 4.0028 ×104
IPipe area, VA 0.0050 0.1477 4.2938 ×104
Im0.0039 0.1213 3.8575 ×104
ICL 0.0028 0.0967 3.6542 ×104
104105106107108
ReD
0.04
0.06
0.08
0.1
0.12
0.14
Turbulence intensity
Power-law fit, smooth pipe
Pipe area, VA
Pipe area, VA (fit)
Pipe area, AA
Pipe area, AA (fit)
Pipe area, AM
Pipe area, AM (fit)
Mean
Mean (fit)
CL
CL (fit)
104105106107108
ReD
0.04
0.06
0.08
0.1
0.12
0.14
Turbulence intensity
Log-law fit, smooth pipe
Pipe area, VA
Pipe area, VA (fit)
Pipe area, AA
Pipe area, AA (fit)
Pipe area, AM
Pipe area, AM (fit)
Mean
Mean (fit)
CL
CL (fit)
Figure 4.
Smooth pipe turbulence intensity as a function of
ReD
, left: Power-law fit, right: Log-law fit.
104105106107108
ReD
0.04
0.06
0.08
0.1
0.12
0.14
Turbulence intensity
Power-law fit, rough pipe
Pipe area, VA
Pipe area, VA (fit)
Pipe area, AA
Pipe area, AA (fit)
Pipe area, AM
Pipe area, AM (fit)
Mean
Mean (fit)
CL
CL (fit)
104105106107108
ReD
0.04
0.06
0.08
0.1
0.12
0.14
Turbulence intensity
Log-law fit, rough pipe
Pipe area, VA
Pipe area, VA (fit)
Pipe area, AA
Pipe area, AA (fit)
Pipe area, AM
Pipe area, AM (fit)
Mean
Mean (fit)
CL
CL (fit)
Figure 5. Rough pipe turbulence intensity as a function of ReD, left: Power-law fit, right: Log-law fit.
Instead of ReD, one can also express the TI fits using the friction Reynolds number [4]:
Reτ=Rvτ
νkin =vτ
2vm×ReD(18)
The relationship between
ReD
and
Reτ
can fitted using Equation (15) where
Q=Reτ
and
x=ReD
,
see Figure 6and Table 5. A log-law fit was also performed but resulted in a bad fit, i.e., a RMSD which
was between one and two orders of magnitude larger than for the power-law fit. As mentioned above,
Fluids 2019,11, 5842 7 of 13
the rough pipe fit is only provided as a reference. For channel flow, it has been found that
a=
0.09 and
b=0.88, see Figure 7.11 in [25] and associated text.
We choose to focus on the bulk Reynolds number since it is possible to determine for applications
where the friction velocity is unknown.
104105106107108
ReD
102
104
106
Re
Power-law fit
Smooth pipe
Smooth pipe fit
Rough pipe
Rough pipe fit
Figure 6. Relationship between ReDand Reτ.
Table 5. Bulk and friction Reynolds number fits.
Case a b
Smooth 0.0621 0.9148
Rough 0.0297 0.9675
5. Discussion
5.1. Friction Factor
In [
26
], the following expression for the smooth pipe friction factor has been derived based on
Princeton Superpipe measurements:
1
λSmooth
=1.930 log10 ReDpλSmooth0.537 (19)
A corresponding rough pipe friction factor has been proposed in [27]:
1
qλRough
=2 log10
ks
3.7D+2.51
ReDqλRough
(20)
The friction factor can also be expressed using the friction velocity and the mean velocity,
see Equation (1.1) in [26]:
λ=8×v2
τ
v2
m
(21)
or:
vτ
vm
=rλ
8(22)
Fluids 2019,11, 5842 8 of 13
The equations for the smooth- and rough-wall pipe flow friction factors are shown in Figure 7,
along with Princeton Superpipe measurements.
We have included additional smooth pipe measurements [
20
,
28
]. Both sets agree with
Equation (19).
Additional rough pipe measurements can be found in Table 2 (and Figure 3) in [
24
]. Here, it was
found that
ks=
8
µ
m. For our main data set [
4
,
5
],
ks=
8
µ
m does not match the measurements;
instead we get
ks=
3
µ
m for a fit to those measurements using Equation (20). It is the same pipe;
the reason for the discrepancy is not clear [
29
]. However, the difference is within the experimental
uncertainty of 5% stated in [24].
For the rough pipe friction factor, we use ks=3µm in the remainder of this paper.
104105106107108
Re D
0.005
0.01
0.015
0.02 Smooth pipe
Meas. (Hultmark et al. 2013)
Meas. (McKeon et al. 2004)
Eq. (McKeon et al. 2005)
104105106107108
Re D
0.005
0.01
0.015
0.02 Rough pipe
Meas. (Hultmark et al. 2013)
Meas. (Langelandsvik et al. 2008)
Eq. (Colebrook 1939, ks=8 m)
Eq. (Colebrook 1939, ks=3 m)
Figure 7. Friction factors, left: Smooth pipe, right: Rough pipe.
5.2. Turbulence Intensity Aspects
5.2.1. Importance for Flow
The TI is an important quantity for many physical phenomena [30], e.g.:
The critical Reynolds number for the drag of a sphere [31]
The laminar–turbulent transition [32]
Development of the turbulent boundary layer [33]
The position of flow separation [34]
Heat transfer [35]
Wind farms [36]
Wind tunnels [37].
We mention these examples to illustrate the importance of the TI for real world applications.
5.2.2. Scaling with the Friction Factor
The wall-normal [
17
] and streamwise (this paper and [
3
]) Reynolds stress have both been shown
to be linked to the friction factor
λ=
4
Cf
, where
Cf
is the skin friction coefficient. These observations
can be interpreted as manifestations of the FIK identity [
18
], where an equation for
Cf
is derived based
on the streamwise momentum equation:
Cf
is proportional to the integral over the Reynolds shear
stress weighted by the quadratic distance from the pipe axis.
An alternative formulation for
Cf
based on the streamwise kinetic energy is derived in [
38
]: here,
Cf
is proportional to the integral over the Reynolds shear stress multiplied by the streamwise mean
velocity gradient weighted by the distance from the pipe axis. It is concluded that the logarithmic
region dominates friction generation for high Reynolds number flow. The dominance of the logarithmic
region has been confirmed in [39].
Fluids 2019,11, 5842 9 of 13
The seeming equivalence between Reynolds stress (shear or normal) and the friction factor leads
us to propose that the TI scales with
vτ/vm
. Therefore we fit to Equations (15) and (16) using
Q=I
and x=vτ/vm, see Figure 8and Tables 6and 7.
In this case, the power-law fits are best for the local velocity definitions and the log-law fits are
best for the reference velocity definitions. Overall, the best fit is the power-law fit using the AM
definition of the TI:
IPipe area, AM =0.6577 ×λ0.5531 , (23)
which is a modification of Equation (14) in [3].
Figure 8. TI as a function of vτ/vm.
Table 6. Power-law fit constants.
TI Definition a b RMSD
IPipe area, AM 2.0776 1.1062 4.7133 ×104
IPipe area, AA 3.5702 1.2088 5.0224 ×104
IPipe area, VA 4.6211 1.2530 4.9536 ×104
Im2.4238 1.1039 6.8412 ×104
ICL 1.1586 0.9260 7.5876 ×104
Table 7. Log-law fit constants.
TI Definition c d RMSD
IPipe area, AM 0.0658 0.2715 5.5459 ×104
IPipe area, AA 0.0890 0.3602 6.6955 ×104
IPipe area, VA 0.1036 0.4162 8.0775 ×104
Im0.0775 0.3195 5.5182 ×104
ICL 0.0551 0.2367 5.8536 ×104
Combining Equation (22) with Equations (15) and (16) leads to:
IPowerlaw fit(λ) = a×λ
8b/2
(24)
ILoglaw fit(λ) = c
2×ln λ
8+d(25)
The predicted TI for the best case (power-law AM) is shown in Figure 9; one reason for the better
match to measurements compared to Figure 9 in [
3
] is that we use
ks=
3
µ
m instead of
ks=
8
µ
m
for the rough pipe. The correspondence between the TI and the friction factor means that the TI will
Fluids 2019,11, 5842 10 of 13
approach a constant value for rough-wall pipe flow at large
ReD
(fully rough regime). It also means
that a larger ksleads to a higher TI.
104105106107108
ReD
0.04
0.06
0.08
0.1
0.12
0.14
Turbulence intensity
Figure 9. AM definition of TI as a function of ReDfor smooth- and rough-wall pipe flow.
5.2.3. CFD Definition
Let us now consider a typical CFD turbulence model, the standard
kε
model [
40
]. Here,
k
is the
turbulent kinetic energy (TKE) per unit mass and εis the rate of dissipation of TKE per unit mass.
As an example of a boundary condition, the user provides the TI (
Iuser
) and the turbulent viscosity
ratio
µt/µ
, where
µt
is the dynamic turbulent viscosity and
µ
is the dynamic viscosity. For a defined
reference velocity vref,kcan then be calculated as:
k=3
2(vref Iuser)2(26)
As the next step, εis defined as:
ε=ρCµk2
(µt/µ)µ, (27)
where ρis density and Cµ=0.09.
An example of default CFD settings is Iuser =0.01 (1%) and µt/µ=10.
The output from a CFD simulation is the total TKE, not the individual components. If we assume
that the turbulence is isotropic, the streamwise TI we are treating in this paper is proportional to the
square root of the TKE:
vRMS =r2
3k(28)
5.2.4. Proposed Procedure for CFD and an Example
A standard definition of TI for CFD is to use the free-stream velocity as the reference velocity, i.e.,
ICL
for pipe flow. For
ICL
, we use the log-law version since this has the smallest RMSD, see Tables 6
and 7:
ICL =0.0276 ×ln(λ) + 0.1794 (29)
We note that this scaling is based on the Princeton Superpipe measurements; for industrial
applications, the TI may be much higher, so our scaling should be considered as a lower limit.
A procedure to calculate the TI, e.g., for use in CFD is:
Fluids 2019,11, 5842 11 of 13
1. Define ReD(and ksfor a rough pipe)
2.
Calculate the friction factor: Equation (19) for a smooth pipe and Equation (20) for a rough pipe
3. Use Equation (29) to calculate the TI.
As a concrete example, we consider incompressible (water) flow through a 130 mm diameter pipe.
CFD boundary conditions can be velocity inlet and pressure outlet. The steps are:
1. Define the mean velocity, we use 10 m/s
2. Calculate ReD= 1.3 ×106
3. Use Equation (19) to calculate λSmooth =0.0114
4. Use Equation (29) to calculate ICL =0.0561.
For this example, we conclude that the minimum TI is 5.6%. A code with this example is available
as Supplementary Materials, a link is provided after the Conclusions.
5.2.5. Open Questions
It remains an open question to what extent the quality of the fits (RMSD) impacts the outcome of
a CFD simulation. For the TI definitions used, the RMSD varies less than a factor of two for the fits
of TI as a function of
vτ/vm
, see Tables 6and 7. For CFD, a single TI definition is used, so it is not
possible to switch TI models in CFD and compare simulations to measurements.
To continue our measurement-based research, we would need measurements of all velocity
components, i.e., wall-normal and spanwise, in addition to the available streamwise measurements.
As mentioned earlier [
2
], it would also be interesting to have measurements for higher Mach
numbers, where compressibility will play a larger role.
In addition to pipe flow, other canonical flows, such as zero-pressure gradient flows [
14
], might
be suitable for analysis similar to what we have presented.
6. Conclusions
We have used Princeton Superpipe measurements of smooth- and rough-wall pipe flow [
4
,
5
] to
study the properties of various TI definitions. The scaling of TI with
ReD
is provided for the definitions.
For scaling purposes, we recommend the AA definition and a power-law fit: Equation (17). The TI
also scales with
vτ/vm
, where the best result is obtained with a power-law fit and the AM definition.
This fit implies that the turbulence level scales with the friction factor: Equation (23).
Scaling of TI with ReDand vτ/vmwas done using both power-law and log-law fits.
A proposed procedure to calculate the TI, e.g., CFD is provided and exemplified in Section 5.2.4.
Supplementary Materials:
The following is available online at https://www.researchgate.net/publication/
336374461_Python_code_to_calculate_turbulence_intensity_based_on_Reynolds_number_and_surface_
roughness, Supplementary Material: Python code to calculate turbulence intensity based on Reynolds number
and surface roughness.
Funding: This research received no external funding.
Acknowledgments: We thank Alexander J. Smits for making the Princeton Superpipe data publicly available.
Conflicts of Interest: The authors declare no conflict of interest.
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article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
... We discuss the relationship between the TI and friction factor and begin by stating the relationship between friction and bulk Reynolds number Re D [7]: ...
... It is not clear how well this captures the behavior for higher Reynolds numbers than the measured maximum. Complete and asymptotic expressions for the smooth pipe friction factor are [7]: ...
... The complete expression for the rough pipe friction factor is [7]: ...
Preprint
Full-text available
We have characterized a transition of turbulence intensity (TI) scaling for friction Reynolds numbers $Re_{\tau} \sim 10^4$ in the companion papers [Basse, N.T. Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 065127}] and [Basse, N.T. Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 125109}]. Here, we build on those results to extrapolate TI scaling for $Re_{\tau} \gg 10^5$, under the assumption that no further transitions exist. Scaling of the core, area-averaged and global peak TI demonstrates that they all scale inversely with the logarithm of $Re_{\tau}$, but with different multipliers. Finally, we confirm the prediction that the TI squared is proportional to the friction factor for $Re_{\tau} \gg 10^5$.
... We have studied the scaling of streamwise turbulence intensity (TI) with Reynolds number in [1,2,3] and continue our research in this paper. We define the square of the TI as I 2 = u 2 /U 2 , where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). ...
... where Re D = D U g AA /ν is the bulk Reynolds number based on the pipe diameter D = 2R. We use an equation derived in [3] to convert between Re τ and Re D : ...
... Global smooth pipe Global smooth pipe fit Global rough pipe Threshold If we were to consider e.g. equations using three parameters, we would need a third radial averaging equation, volume averaging (VA) [3]. ...
Preprint
Full-text available
We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I^2 = overbar(u^2)/U^2 , where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth-and rough-wall pipe flow. J. Fluid Mech. Vol. 728, 376-395 (2013)]. Scaling expressions are derived both for log-law and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Re_τ ∼ 11000.
... which can be fit to the Princeton Superpipe measurements [18]: ...
... Complete and asymptotic expressions for the smooth pipe friction factor are [18]: ...
Article
We have characterized a transition of turbulence intensity (TI) scaling for friction Reynolds numbers $Re_{\tau} \sim 10^4$ in the companion papers [Basse, N.T. Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 065127}] and [Basse, N.T. Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 125109}]. Here, we build on those results to extrapolate TI scaling for $Re_{\tau} \gg 10^5$, under the assumption that no further transitions exist. Scaling of the core, area-averaged and global peak TI demonstrates that they all scale inversely with the logarithm of $Re_{\tau}$, but with different multipliers. Finally, we confirm the prediction that the TI squared is proportional to the friction factor for $Re_{\tau} \gg 10^5$.
... This Special Issue is a collection of top-quality papers from some of the Editorial Board Members of Fluids, Guest Editors, and leading researchers discussing new knowledge or new cutting-edge developments on all aspects of fluid mechanics. Research in Turbulence continues to be one of the active areas [1][2][3][4][5][6][7][8][9][10]; other papers focus on mixing [11], multiphase flows and porous media [12][13][14][15][16][17], slow (creeping) flows [18], potential flows [19], non-Newtonian fluids [20], fluid-structure interaction [21], and numerical methods [22][23][24]. ...
... Basse [9] considers the streamwise turbulence intensity definitions using smoothand rough-wall pipe flow measurements; he also presents a procedure to calculate the turbulence intensity based on the bulk Reynolds number. ...
Article
Full-text available
This Special Issue is a collection of top-quality papers from some of the Editorial Board Members of Fluids, Guest Editors, and leading researchers discussing new knowledge or new cutting-edge developments on all aspects of fluid mechanics [...]
... The baseline model is equipped with the one-equation turbulence model originally conceived by Spalart-Allmaras for aerospace applications (Spalart and Allmaras, 1992). Turbulence intensity of the incoming flow is set to 5 %, with a characteristic length scale equal to 3.8 % of the channel width (Basse, 2019). ...
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The energy harvesting performance of a hydrokinetic turbine in a confined channel has been investigated through numerical simulations. It consists of a two-bladed vertical-axis turbine with a NACA0015 profile. Fluid dynamics are reproduced by a 2D numerical model solving the unsteady RANS equations through a finite volume approach in OpenFOAM-v6. Following a numerical sensitivity analysis, the baseline model has been validated with respect to reference data for an unconfined turbine with similar design. The model has then been employed for assessing the influence of the main design parameters – namely blade-channel gap ε0, solidity σ and tip-speed ratio λ – on the performance of the turbine. First, a parametric study on the plane λ×σ revealed that best performance is obtained at high solidity and low tip-speed ratios. In a range of solidities from 0.06 up to 0.48, the optimal value of tip-speed ratio – which is considerably higher when compared with unconfined turbines – is precisely identified. Finally, the effects of the blade-channel gap ε0 on the performance of the turbine has been investigated as well. It has been found that ε0 should be set as small as possible in order to maximise energy harvesting. Moreover, numerical results allowed for estimating a limit value of ε0>9 over which the performance of a turbine operating in a confined channel should match that of an unconfined turbine.
... Basse studied TI with the friction factor from pipe flow measurements made in the Princeton Superpipe [12,13]. Wind tunnels are experimental devices to simulate certain flow conditions and to study the flow over objects of interest. ...
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Full-text available
The turbulence intensity (TI) is defined as the ratio of fluctuation from the standard deviation of wind velocity to the mean value. Many studies have been performedon TI for flow dynamics and adapted various field such as aerodynamics, jets, wind turbines, wind tunnel apparatuses, heat transfer, safety estimation of construction, etc.The TI represents an important parameter for determining the intensity of velocity variation and flow quality in industrial fluid mechanics. In this paper, computational fluid dynamic (CFD) simulation of TI alteration with increasing temperature has been performed using the finite volume method. A high-temperature—maximum 300 degrees Celsius (°C)—wind tunnel test rig has been used as theapparatus, and velocity was measured by an I-type hot-wire anemometer. The velocity and TI of the core test section were operated at several degrees of inlet temperatures at anair velocity of 20 m/s. The magnitude of TI has a relationship with boundary layer development. The TI increased as temperature increased due to turbulence created by the non-uniformities.
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During the past two decades, passive rotors have been proposed and introduced to be used in a number of different water sector applications. One of these applications is the use of a passive rotor at the outlets of pipe outfalls to enhance mixing. The main objective of this study is to develop a CFD computational workflow to numerically examine the feasibility of using a passive rotor downstream of the outlet of pipe outfalls to improve the mixing properties of the near flow field. The numerical simulation for a pipe outlet with a passive rotor is a numerical challenge because of the nonlinear water-structure interactions between the water flow and the rotor. This study utilizes a computational workflow based on the ANSYS FLUENT to simulate that water-structure interaction to estimate the variation in time of the angular speed (ω) of a passive rotor initially at rest and then subjected to time-varying water velocity (υ). Two computational techniques were investigated: the six-degrees-of-freedom (6DOF) and the sliding mesh (SM). The 6DOF method was applied first to obtain a mathematical relation of ω as a function of the water velocity (υ). The SM technique was used next (based on the deduced ω-υ relation by the 6DOF) to minimize the calculation time considerably. The study has shown that the 6DOF technique accurately determines both maximum and temporal angular speeds, with discrepancies within 3% of the measured values. A number of numerical runs were conducted to investigate the effect of the gap distance between the passive rotor and the pipe outlet and to examine the effect of using the passive rotor on the near flow field downstream of the rotor. The model results showed that as the gap distance of the pipe outlet to the passive rotor increases, the rotor’s maximum angular speed decreases following a decline power-law trend. The numerical model results also revealed that the passive rotor creates a spiral motion that extends downstream to about 15 times the pipe outlet diameter. The passive rotor significantly increases the turbulence intensity by more than 500% in the near field zone of the pipe outlet; however, this effect rapidly vanishes after four times the pipe diameter
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The value of multiple transient tests under the same experimental conditions for the purpose of leakage detection is assessed. The gain in signal-to-noise ratio and leakage detection accuracy due to the multiple tests are derived theoretically and evaluated experimentally. The role of the multiple measurements in suppressing interferences and thus minimizing false detection is proven using experimental data. A theoretical extension of existing leakage detection method that was derived for single measurements is tested. From a practical perspective, this research quantitatively shows that the multitest strategy allows detecting smaller leaks in noisier environments, and/or using smaller-amplitude waves.
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This Ph.D. thesis contains theoretical and experimental work on plasma turbulence measurements using collective light scattering. The motivation for measuring turbulence in hot fusion plasmas is, along with the method used and results obtained, the subject of chapter 1. The theoretical part is divided into three chapters. Chapter 2 contains a full analytical derivation of the expected dependency of the detected signal on plasma parameters. Thereafter, spatial resolution of the measurements using different methods is treated in chapter 3. Finally, the spectral analysis tools used later in the thesis are described and illustrated in chapter 4. The experimental part is divided into four chapters. In chapter 5 transport concepts relevant to the thesis are outlined. Main parameters of the Wendelstein 7-AS (W7-AS) stellarator in which measurements were made are collected in chapter 6. The setup used to study fluctuations in the electron density of W7-AS plasmas is covered in chapter 7. This localised turbulence scattering (LOTUS) diagnostic is based on a CO2 laser radiating at a wavelength of 10.59 µm. Fast, heterodyne, dual volume detection at variable wavenumbers between 14 and 62 cm−1 is performed. The central chapter of the thesis, chapter 8, contains an analysis of the measured density fluctuations before, during and after several confinement transition types. The aim was to achieve a better understanding of the connection between turbulence and the confinement quality of the plasma. Conclusions and suggestions for further work are summarised in chapter 9.
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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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Despite a growing body of recent evidence on the hierarchical organization of the self-similar energy-containing motions in the form of Townsend's attached eddies in wall-bounded turbulent flows, their role in turbulent skin-friction generation is currently known very little. In this paper, the contribution of each of these self-similar energy-containing motions to turbulent skin friction is explored up to Re τ ≃ 4000. Three different approaches are employed to quantify the skin-friction generation by the motions, the spanwise length scale of which is smaller than a given cutoff wavelength: 1) FIK identity in combination with the spanwise wavenumber spectra of the Reynolds shear stress; 2) confinement of the spanwise computational domain; 3) artificial damping of the motions to be examined. The near-wall motions are found to continuously lose their role in skin-friction generation on increasing the Reynolds number, consistent with the previous finding at low Reynolds numbers. The largest structures given in the form of very-large-scale and large-scale motions are also found to be of limited importance: due to a non-trivial scale-interaction process, their complete removal yields only 5 ∼ 8% of skin-friction reduction at all the Reynolds numbers considered, although they are found to be responsible for 20 ∼ 30% of total skin friction at Re τ ≃ 2000. Application of all the three approaches consistently reveals that the largest amount of skin friction is generated by the self-similar motions populating the logarithmic region. It is further shown that the contribution of these motions to turbulent skin friction gradually increases with the Reynolds number, and that these coherent structures are eventually responsible for most of turbulent skin-friction generation at sufficiently high Reynolds numbers.
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The intrinsically unsteady heat transfer on the surface of a cylinder in crossflow has been investigated in detail by numerical simulation as a function of the freestream turbulence intensity and the Reynolds number. After a brief startup transient, a periodic steady state is established at all circumferential locations. The resulting timewise fluctuations were seen to be of different phase depending on where on the circumference they occur. On one side of the cylinder, maxima occurred at the same moment in time as minima occurred on the other side. This finding and comparisons of the magnitudes of the local heat transfer coefficients showed that side-to-side symmetry does not prevail in the presence of the unsteadiness. The fluctuation frequencies were found to be virtually uniform over the entire circumference of the cylinder and varied only slightly with the Reynolds number and the turbulence intensity. The full slate of results included: (a) timewise and circumferential variations of the local heat transfer coefficient, (b) timewise variations of the all-angle spatial-averaged heat transfer coefficient, (c) spatial variations of the timewise-averaged heat transfer coefficient, (d) spatial- and timewise-averaged heat transfer coefficients as a function of turbulence intensity and Reynolds number, (e) timewise fluctuation frequencies, (f) comparisons with the experimental literature, and (g) effect of the selected turbulence model. As expected, the magnitude of the heat transfer coefficient increases as the turbulence intensity increases.
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The classical view of wall-bounded turbulence considers a near-wall inner region where all velocity statistics are universally dependent on distance from the wall when scaled with friction velocity and the kinematic viscosity of the fluid. This is referred to as an inner scaling and leads to Prandtl's law of the wall. Data from numerical simulations and experiments over the past decade or so, however, have provided compelling evidence that statistics of the fluctuating streamwise velocity do not follow inner scaling in this near-wall region and an interaction of outer and logarithmic regions exists, resulting in a Reynolds number dependence. In this paper we briefly review some of these studies and discuss the Reynolds number dependence of the streamwise turbulence intensity near the wall in terms of an inner-outer interaction. An established model for such an interaction between near-wall and logarithmic region turbulence is considered that comprises two mechanisms: superposition and modulation. Here outer-region motions, of which a fraction is wall-attached, are superimposed onto the near-wall dynamics, and concurrently the near-wall motions are modulated by this superimposed signature. We discuss to what extent the superposition effect can relate changes in the inner-scaled near-wall peak value of streamwise turbulence intensity to logarithmic region turbulence resembling features of attached eddies.
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This paper reports on near-wall two-component–two-dimensional (2C–2D) particle image velocimetry (PIV) measurements of a turbulent pipe flow at shear Reynolds numbers up to $Re_{\unicode[STIX]{x1D70F}}=40\,000$ acquired in the CICLoPE facility of the University of Bologna. The 111.5 m long pipe of 900 mm diameter offers a well-established turbulent flow with viscous length scales ranging from $85~\unicode[STIX]{x03BC}\text{m}$ at $Re_{\unicode[STIX]{x1D70F}}=5000$ down to $11~\unicode[STIX]{x03BC}\text{m}$ at $Re_{\unicode[STIX]{x1D70F}}=40\,000$ . These length scales can be resolved with a high-speed PIV camera at image magnification near unity. Statistically converged velocity profiles were determined using multiple sequences of up to 70 000 PIV recordings acquired at sampling rates of 100 Hz up to 10 kHz. Analysis of the velocity statistics shows a well-resolved inner peak of the streamwise velocity fluctuations that grows with increasing Reynolds number and an outer peak that develops and moves away from the inner peak with increasing Reynolds number.
Article
A theoretical decomposition of mean skin friction generation into physical phenomena across the whole profile of the incompressible zero-pressure-gradient smooth-flat-plate boundary layer is derived from a mean streamwise kinetic-energy budget in an absolute reference frame (in which the undisturbed fluid is not moving). The Reynolds-number dependences in the laminar and turbulent cases are investigated from direct numerical simulation datasets and Reynolds-averaged Navier–Stokes simulations, and the asymptotic trends are consistently predicted by theory. The generation of the difference between the mean friction in the turbulent and laminar cases is identified with the total production of turbulent kinetic energy (TKE) in the boundary layer, represented by the second term of the proposed decomposition of the mean skin friction coefficient. In contrast, the analysis introduced by Fukagata et al. ( Phys. Fluids , vol. 14 (11), 2002, pp. 73–76), based on a streamwise momentum budget in the wall reference frame, relates the turbulence-induced excess friction to the Reynolds shear stress weighted by a linear function of the wall distance. The wall-normal distribution of the linearly-weighted Reynolds shear stress differs from the distribution of TKE production involved in the present discussion, which consequently draws different conclusions on the contribution of each layer to the mean skin friction coefficient. At low Reynolds numbers, the importance of the buffer-layer dynamics is confirmed. At high Reynolds numbers, the present decomposition quantitatively shows for the first time that the generation of the turbulence-induced excess friction is dominated by the logarithmic layer. This is caused by the well-known decay of the relative contributions of the buffer layer and wake region to TKE production with increasing Reynolds numbers. This result on mean skin friction, with a physical interpretation relying on an energy budget, is consistent with the well-established general importance of the logarithmic layer at high Reynolds numbers, contrary to the friction breakdown obtained from the approach of Fukagata et al. ( Phys. Fluids , vol. 14 (11), 2002, pp. 73–76), essentially based on a momentum budget. The new decomposition suggests that it may be worth investigating new drag reduction strategies focusing on TKE production and on the nature of the logarithmic layer dynamics. The decomposition is finally extended to the pressure-gradient case and to channel and pipe flows.
Article
A more poetic long title could be 'A voyage from the shifting grounds of existing data on zero-pressure-gradient (abbreviated ZPG) turbulent boundary layers (abbreviated TBLs) to infinite Reynolds number'. Aided by the requirement of consistency with the Reynolds-averaged momentum equation, the 'shifting grounds' are sufficiently consolidated to allow some firm conclusions on the asymptotic expansion of the streamwise normal stress , where the indicates normalization with the friction velocity squared. A detailed analysis of direct numerical simulation data very close to the wall reveals that its inner near-wall asymptotic expansion must be of the form , where , and , are functions fitted to data in this paper. This means, in particular, that the inner peak of does not increase indefinitely as the logarithm of the Reynolds number but reaches a finite limit. The outer expansion of , on the other hand, is constructed by fitting a large number of data from various sources. This exercise, aided by estimates of turbulence production and dissipation, reveals that the overlap region between inner and outer expansions of is its plateau or second maximum, extending to , where the outer logarithmic decrease towards the boundary layer edge starts. The common part of the two expansions of , i.e. the height of the plateau or second maximum, is of the form with and constant. As a consequence, the logarithmic slope of the outer cannot be independent of the Reynolds number as suggested by 'attached eddy' models but must slowly decrease as . A speculative explanation is proposed for the puzzling finding that the overlap region of is centred near the lower edge of the mean velocity overlap, itself centred at with the Reynolds number based on free stream velocity and displacement thickness. Finally, similarities and differences between in ZPG TBLs and in pipe flow are briefly discussed.