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

DNS STUDY OF TURBULENT FLOW IN A CIRCULAR PIPE SUBJECTEDTO AXIAL SYSTEM ROTATION
Zhao-Ping Zhang
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
umzha258@myumanitoba.ca
Bing-Chen Wang
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
BingChen.Wang@umanitoba.ca

ABSTRACT

circular pipe flow, and confirmed these experimental researchfindings of Reich & Beer (1989). They also observed that the
turbulence statistics were sensitive to the tilting of the nearwall axial vortical structures in the rotating direction. Orlandi& Ebstein (2000) further studied the impact of axial system
rotation on the budget balances of TKE, Reynolds stresses andenstrophy of the turbulent pipe flow using DNS. They reported
that for the Reynolds stresses, the pressure-strain and convection terms were highly dependent on the rotation number.
Based on a thorough literature review, it is noticed that detailed DNS studies of the axial-rotating circular pipe flow arestill rather limited, and in-depth understanding of the Coriolis
force effects on the flow physics and coherent structures needs
to be developed. In view of this, we aim to conduct a systematic DNS study of turbulent pipe flows subjected to axial
system rotation for a wide range of rotation numbers.

Turbulent circular pipe flows under axial system rotation
are investigated using direct numerical simulation (DNS) for awide range of rotation numbers of Rob = 0−20. The characteristics of the flow field are investigated in both physical andspectral spaces. It is observed that large-scale secondary flows
are induced by the axial rotation, which enhance the magnitudes of the radial and azimuthal Reynolds normal stresses. In
response to the axial system rotation imposed, the pressurestrain and Coriolis production terms become the leading terms
in the budget balance of these two Reynolds normal stresses.
It is also interesting to observe that the axial rotation makes
the near-wall streaky structures tilted and elongated, and suppresses the sweep events systematically which further impede
the formation of hairpin structures.

INTRODUCTION
Turbulent flow within a circular pipe subjected to system
rotation represents an interesting topic with important applications in areas such as internal blade cooling of gas turbines,
and rotary machines. The system rotation of the circular pipe
may occur either radially about a diameter of the pipe, oraxially about the centre line of the pipe. In response to either radial or axial system rotation, Coriolis force is induced
which acts on the fluid flows to dramatically alter the turbulence statistics and coherent structures, even resulting in local
laminarization of the flow. As is well known, turbulent flow
through a stationary (non-rotating) circular pipe represents aclassical research subject, which has been extensively studied using DNS by Eggels et al. (1994); Wu & Moin (2008);
Chin et al. (2010). By contrast, the number of DNS studies
on either radially and axially rotating circular pipe flows arestill very limited in the literature. Recently, Zhang & Wang(2019) carried out a DNS study of turbulent flow in a circular pipe subjected to radial system rotation. They observed
streamwise-elongated large-scale counter-rotating vortices anda Taylor-Proudman region at low rotation numbers. Following
our previous DNS study of radially-rotating pipe flow, here we
extend the research to DNS of axially-rotating pipe flows.
In their series of rotating pipe experiments, Reich & Beer
(1989) observed that both skin friction coefficient and turbulence level decreased considerably as the rotation number increased. Meanwhile, according to the trend of the mean axial velocity profile, the flow became increasingly laminarizedwith an increasing speed of axial system rotation. Orlandi &Fatica (1997) carried out a DNS study of an axially-rotating

TEST CASE AND NUMERICAL ALGORITHM
Figure 1 schematically illustrates a circular pipe under axial system rotation. Three directions of the cylindrical coordinates (i.e. radial, azimuthal and axial directions) are denoted
as r, β and z, and the corresponding velocity components are
represented as ur , uβ and uz , respectively. The pipe length isLz = 30πR, where R is the pipe radius. The flow field in thedomain is fully developed with a periodic boundary condition
applied to the axial direction and no-slip condition imposedon the solid surface. In order to study the axially-rotating effect, a constant counter-clockwise angular speed (Ωz ) is imposed in the streamwise (z) direction, and the rotation number (Rob = 2Ωz R/Ub ) ranges drastically from 0 (non-rotatingcase) to 20, at a fixed Reynolds number of Reτ = uτ R/ν = 180.
Here, Ub is the bulk mean velocity, uτ is the wall friction velocity, and ν is the kinematic viscosity of the fluid. The governing equations for an incompressible flow with respect to anaxially-rotating reference frame are∇ ·⃗u = 0,1∂⃗u+⃗u · ∇⃗u = −Πêz − ∇p + ν∇2⃗u + ⃗F
∂t
ρ

(1)
,

(2)

where ⃗u, ρ and p are the velocity, density and pressure of thefluid, respectively. Π represents the constant mean axial pressure gradient, and êz is the base unit vector of the z-direction,with |êz | ≡ 1. In response to the axial rotation, two componentsof the Coriolis force (⃗F) appear in the radial and azimuthal directions, i.e. Fr = 2Ωz uβ and Fβ = −2Ωz ur .

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

Figure 1. Schematic of turbulent pipe flow subjected to axial
system rotation in cylindrical coordinate system. Two Coriolis
force components are Fr and Fβ .
Table 1.
Case #
1
2
3
4
5
6
7

Flow parameters for seven test cases.
Ωz
0
1
2
3
5
7
10

Rob
0
2
4
6
10
14
20

RoAb
0
1.83
3.67
5.53
9.79
14.95
22.45

RoAτ
0
29.47
58.86
88.38
147.41
206.15295.72(a) Non-rotating pipe flow (Rob = 0)ReAτ
179.94179.84180.09179.91179.77179.96179.22The simulations were performed with a spectral-element
code “Semtex” by Blackburn & Sherwin (2004), which ishighly-accurate in algorithm suitable for conducing DNS. It
was developed using C++ programming language, and parallelized following the message passing interface (MPI) standard. For the mesh, a quadrilateral spectral-element method
was used to divide the cross-section of the pipe into 420 elements with each element discretized spatially with 8th-order
Gauss-Lobatto-Legendre Lagrange interpolants. All physicalquantities are expanded into the spectral space using Fourier
series with 1800 modes in the z-direction. The grid spacing is
uniform in the streamwise direction with ∆z+ = 9.425, andvaries in the azimuthal and radial directions with R∆β + =1.416-5.122, ∆r+ = 0.147 at the first node off the wall and∆r+ = 0.942 at the pipe centre.
Table 1 compares the nominal and actual flow parameters derived from DNS results of the different test cases.
Superscript “A” denotes the actual results. Roτ represents
rotation number defined based on the wall friction velocity
as Roτ = 2Ωz R/uτ . For each simulated case, 500 instantaneous snapshots of the flow fields over 40 large-eddy turnovertimes (LETOTs, defined as R/uτ ) were collected with approximately
p 1.1 TB data stored on a server. Here, uτ is definedas −ΠR/2ρ. All DNS calculations were conducted on the
WestGrid (Western Canada Research Grid) supercomputers.
To facilitate the analysis and discussion, an instantaneous
turbulence variable Φ is decomposed as Φ = ⟨Φ⟩ + Φ′ where⟨Φ⟩ is the temporal- and spatial-averaging component along
the homogenous (z) direction, and Φ′ represents fluctuating
component. Symbol y denotes a dimensionless coordinate
measured from the wall, i.e. y = 1 − r/R. Furthermore, inorder to conduct a near-wall analysis in analogous to a turbulent plane channel flow, a dimensionless wall coordinate is
introduced, i.e. y+ = (R − r)uτ /ν.

(b) Axially-rotating pipe flow (Rob = 20)
Figure 2.
Cross-stream distribution of instantaneous axial
velocity u+(left
half-panel) and instantaneous axial vorticityzωz+ (right half-panel) of the non-rotating (Rob = 0) and most
rapidly-rotating (Rob = 20) flows. White Arrows represent thecounter-clockwise-rotating direction. All contours are mappedat the same axial location (z/R = 15π) and time instant (20.38
LETOTs). All values are non-dimensionalized by uτ .

velocity u+
z , while the right part shows the instantaneous axial vorticity ωz+ . As is clear from Fig. 2(a), in a non-rotating
pipe flow, the flow structures show “mushroom patterns” inthe near-wall region where several couples of small counterrotating vorticities are observed. In Fig. 2(b), as the rotating
speed increases to Rob = 20, a large rotating vortical structureappears at the pipe centre, and both the “mushroom patterns”
and vorticities are stretched along the azimuthal direction.
Figure 3 compares bulk mean velocity Ub+ and volume+ of different rotation numbers Ro . In
averaged TKE kmbFig. 3(a), the Ub+ profile peaks at Rob = 2, with a magnitude
that is 10.01% higher than that of the non-rotating pipe flow.
As the rotation number increases beyond 6, the magnitude of
Ub+ decreases apparently. Finally, at Rob = 20, the Ub+ value isreduced by 10.33% in comparison with that of the non-rotating+ increases as the rotation
flow. From Fig. 3(b), The value of km
number increases, and reaches its maximum at Rob = 14. It
will be explained later that the axial rotation enhances the magnitudes of ⟨u′z u′z ⟩+ and ⟨u′β u′β ⟩+ which subsequently make a+ value.
positive contribution to the km
Figure 4 compares the profiles of the mean axial velocity ⟨uz ⟩+ , mean azimuthal velocity r⟨uβ ⟩+ /R and mean axial
vorticity ⟨ωz ⟩+ with respect to the wall-normal distance (y) of
all seven rotation numbers tested. As is evident from Fig. 4(a),the ⟨uz ⟩+ profile is axial-symmetrical for the non-rotating pipe
flow, and varies non-monotonically with an increasing Rob

RESULTS AND DISCUSSIONS
Figure 2 shows contours of the instantaneous axial veloc+
ity u+z and instantaneous axial vorticity ωz in the cross-stream
plane for the non-rotating (Rob = 0) and most rapidly-rotating(Rob = 20) flows. The left panel shows the instantaneous axial

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〈 〉

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) Bulk mean velocity Ub+〈 β〉(a) Mean axial velocity ⟨uz ⟩+(b) Mean azimuthal velocity r⟨uβ ⟩+ /R+(b) Volume-averaged TKE km

Figure 3.
Profiles of bulk mean velocity Ub+ and volume+ with respect to the rotation number Ro .
averaged TKE kmb++ have been non-dimensionalized by
The values of Ub and kmuτ and u2τ , respectively.
〈ω 〉value. At the pipe centre, the peak value of ⟨uz ⟩+ increases
when rotating speed rises from Rob = 0 to 6, and reaches its
maximum at Rob = 6 with a magnitude that is 40.05% higher
than that of the non-rotating pipe flow. As Rob increases from
6 to 20, the peak value of ⟨uz ⟩+ reduces, which however isstill 18.48% larger than that at Rob = 0. In addition, it is interesting to observe from Figs. 4(b) and (c) that both r⟨uβ ⟩+ /Rand ⟨ωz ⟩+ reach their maxima at Rob = 10, and the peak ofr⟨uβ ⟩+ /R is always located around y ≈ 0.4 where the mean
axial vorticity ⟨ωz ⟩+ is zero identically.
Figure 5 compares the profiles of Reynolds normal
stresses ⟨u′z u′z ⟩+ , ⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ at seven different rotation numbers Rob tested. As shown in Fig. 5(a), the profile of⟨u′z u′z ⟩+ peaks at y = 0.083 (or y+ ≈ 15) in the near-wall region
for a non-rotating pipe flow. As soon as the axial rotation is
imposed, the magnitude of ⟨u′z u′z ⟩+ reduces monotonically. In
response to the effect of the mean secondary flow, the peak of⟨u′z u′z ⟩+ moves towards the pipe centre as Rob increases. The
lowest and highest peak values occur at Rob = 4 and Rob = 14,
respectively, which are 15.77% and 6.87% lower than that of
the non-rotating pipe flow (at Rob = 0).
As is clear in Fig. 5(b), in contrast to the monotonic
trend of ⟨u′r u′r ⟩+ in the near-wall region of the pipe, the profile of ⟨u′r u′r ⟩+ varies non-monotonically at the pipe centre as
the rotation number increases, and reaches its maximum at
Rob = 14. Figure 5(c) shows that the profile of ⟨u′β u′β ⟩+ issimilar to that of ⟨u′r u′r ⟩+ in the azimuthal direction, with the
maximum peak also occurring at Rob = 14. In response to
the system rotation imposed, the secondary flows are induced,
which tend to suppress ⟨u′z u′z ⟩+ but enhance the magnitudesof ⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ in most pipe regions until the rotation number reaches Rob = 14. As a result, the TKE level (as(c) Mean axial vorticity ⟨ωz ⟩+Figure 4.
Profiles of mean axial velocity ⟨uz ⟩+ , mean azimuthal velocity r⟨uβ ⟩+ /R and mean axial vorticity ⟨ωz ⟩+ atseven tested rotation numbers. In panels (b) and (c), pinkvertical-dashed lines point to the location of zero ⟨ωz ⟩+ at
y = 0.4. All values have been non-dimensionalized by uτ .
shown in Fig. 3(b)) is enhanced significantly. As Rob increases
above 14, the magnitudes of ⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ decrease
with Rob , leading to a reduction in the TKE level.
Figure 6 compares the profiles of axial pre-multiplied ve+ calculated along the streamwise direclocity spectrum kz+ Eezz
tion at wall-normal position y+ ≈ 15. The 1D axial energy
spectrum of velocity fluctuations is defined as∗∗
Eei j = ub′i ub′j + ub′i ub′j,(3)

where a hat denotes Fourier transform and an overbar represents time averaging. The axial wavenumber is determined askz = nz kz0 for nz ∈ [-Nz /2, Nz /2-1] where kz0 = 2π/Lz is thesmallest positive wavenumber. The axial wavelength is defined as λz = 2π/kz , non-dimensionalized as λz+ = λz uτ /ν.
+ (correspondAs is evident from Fig. 6, the mode of kz+ Eezz
ing to the characteristic length scale of the most energetic eddies) is located in the region of λz+ ∈ [600, 2000] for a non-

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〉

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

λ(a) Axial Reynolds normal stress ⟨u′z u′z ⟩+Figure 6.
Profile of axial pre-multiplied velocity spectrum+ for 7 different rotation numbers, calculated along thekz+ Eezzstreamwise direction at wall-normal position y+ ≈ 15. Pink
arrow indicates the direction of an increasing rotation number.

〉

Π

〈

〉

〈

ε

(b) Radial Reynolds normal stress ⟨u′r u′r ⟩+〈〉〈 β β〉(a) Non-rotating pipe flow (Rob = 0)(c) Azimuthal Reynolds normal stress ⟨u′β u′β ⟩+Figure 5.
Profiles of Reynolds normal stresses ⟨u′z u′z ⟩+ ,′′+⟨ur ur ⟩ and ⟨u′β u′β ⟩+ for different rotation numbers Rob . In
panels (a) and (b), pink arrows point to the direction of an increasing rotating effect. All values are non-dimensionalizedby u2τ .

(b) Axially-rotating pipe flow (Rob = 6)
Figure 7.
Budget balance of the radial Reynolds normal stress ⟨u′r u′r ⟩+ of the non-rotating (Rob = 0) and rotating (Rob = 6) flow cases. All budget terms are nondimensionalized by u3τ /R.

rotating pipe flow. In response to the axial system rotation,
the peak value decreases dramatically as Rob increases. At thehighest rotation number Rob = 20, the magnitude is reducedby 63.02% in comparison with that at Rob = 0. Meanwhile,+ moves towards larger wavelengths as thethe mode of kz+ Eezz
rotation number increases.
To refine our investigation of the rotating impact on the
Reynolds normal stresses, the transport equation of Reynolds
stresses ⟨u′i u′j ⟩ can be further studied, which is expressed asp

Hi j − Pi j − Πi j + εi j − Dti j − Di j − Dvij −Ci j = 0 .

axial direction, the Coriolis production term is zero identically for ⟨u′z u′z ⟩. Therefore, we only analyze budget balanceof ⟨u′r u′r ⟩ and ⟨u′β u′β ⟩ here, and the Coriolis production terms
are Crr = 4Ωz ⟨u′r u′β ⟩ and Cβ β = −4Ωz ⟨u′r u′β ⟩, respectively.
As is shown in Fig. 7(a), the budget balance of ⟨u′r u′r ⟩+p++
is mostly dominated by −Π+
rr , εrr and −Drr in the near-wallregion of the pipe for the non-rotating pipe flow. Specifically,p+−Drr and −Π+
rr behave as source terms in the inner region
for y ≤ 0.071 and in the outer region for y ≥ 0.071, respectively. In contrast to the non-rotation pipe flow patterns shownin Fig. 7(a), it is apparent in Fig. 7(b) that the budget balancep+of ⟨u′r u′r ⟩+ is dominated by −Π+
rr , −Drr and −Crr at Rob = 6.
Owing to the system rotation imposed, the Coriolis production
term −Crr appears as a new source term which pumps a large
amount of turbulence energy into ⟨u′r u′r ⟩+ . The pressure-strain(4)pHere, Hi j , Pi j , Πi j , εi j , Dti j , Di j , Dvij and Ci j represent theconvection, production, pressure-strain, viscous dissipation,
turbulent diffusion, pressure diffusion, viscous diffusion andCoriolis production terms, respectively. From Fig. 1, it is understood that the two components of the Coriolis force arein the radial and azimuthal directions under the axial rotation. Because no Coriolis force component appears in the412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

〈 β β〉

ββ

εββ

ββ

ββ

Πββ

ββ

ββ
ββ

(a) Non-rotating pipe flow (Rob = 0)

〈 β β〉(a) Non-rotating pipe flow (Rob = 0)(b) Axially-rotating pipe flow (Rob = 6)
Figure 8.
Budget balance of the azimuthal Reynolds normal stress ⟨u′β u′β ⟩+ of the non-rotating (Rob = 0) and rotating (Rob = 6) flow cases. All budget terms are nondimensionalized by u3τ /R.

(b) Axially-rotating pipe flow (Rob = 20)
Figure 9. Contours of JPDF P(u′z , -u′r ) at wall-normal position y+ ≈ 15 of the pipe for the non-rotating (Rob = 0) androtating (Rob = 20) flows. All contours have been split into
four quadrants (Q) by black solid lines.

term functions completely as a sink term in the near-wall region up to y = 0.872. It is noticed that the peak values of −Π+rrp+
and −Drr increase almost ninefold in the inner region compared with those of the non-rotating pipe flow.
Figure 8 compares budget balances of the azimuthal
Reynolds normal stress ⟨u′β u′β ⟩+ of the non-rotating (Rob = 0)
and rotating (Rob = 6) pipe flow cases. As is clear in Fig. 8(a),for the non-rotating pipe flow, the budget balance of ⟨u′β u′β ⟩+(demonstrated based on u′z + ) in the β -z plane at the wallnormal position y+ ≈ 15 for the non-rotating (Rob = 0) androtating (Rob = 20) flows. The axial domain is arbitrarily selected at z/R ∈ [0, 10]. In Fig. 10(a), the positively (indicatedby red color) and negatively (indicated by blue color) fluctuating streaks alternate in the azimuthal direction in a nonrotating pipe flow. It is observed that the axial length scales of
the negatively fluctuating streaks are apparently longer thanthose of the positively fluctuating streaks. By contrast, theazimuthal length scales of the positively fluctuating streaks
are wider than those of the negatively fluctuating streaks. As
shown in Fig. 10(b), at Rob = 20, the negatively fluctuating
streaks are elongated along the axial direction. The period (orspacing between two adjacent negatively fluctuating streaks)
becomes greater in the azimuthal direction in response to thesystem rotation imposed. It is also interesting to observe thatthe streaky structures are tilted, a physical feature that relates
to the secondary flows shown previously in Fig. 2(b). In contrast with the negatively fluctuating streaks, a reduction on thepositively fluctuating streaks is observed at Rob = 20. It indicates that axial system rotation tends to impede the formationof the hairpin structures by suppressing the sweep events andto laminarize the flow field which is consistent with the observation in Fig. 9(b).
Figure 11 compares instantaneous axial vorticity fluctuation (ωz′ + ) patterns in the β -z plane at y+ ≈ 15 for the nonrotating (Rob = 0) and rotating (Rob = 20) flows. For the nonrotating pipe flow shown in Fig. 11(a), numerous small-scale
structures are seen in the β -z plane. However, in response to
the axial system rotation, these small-scale turbulence structures are stretched along the axial direction at Rob = 20 andtilted slightly along the azimuthal direction, a pattern that is
consistent with the observation of Fig. 10(b).

is primarily dominated by εβ+β and −Dv+in the inner regionββof the pipe. However, in the outer region, −Π+and εβ+β
ββmake primary contributions to ⟨u′β u′β ⟩+ . The system rotation
changes the budget balance of ⟨u′β u′β ⟩+ considerably. As is evident from Fig. 8(b), at Rob = 6, the Coriolis production term−Cβ β becomes the dominant sink term to balance with −Π+.
ββ
By comparing Fig. 7(b) with Fig. 8(b), it is clear that the Coriolis production term is able to transfer turbulence energy from⟨u′β u′β ⟩+ to ⟨u′r u′r ⟩+ , simply because Crr ≡ −Cβ β .
Figure 9 compares contours of joint probability density
function (JPDF) P(u′z , -u′r ) at the wall-normal position y+ ≈15 of the pipe for the non-rotating (Rob = 0) and rotating(Rob = 20) flows. To compare with the familiar turbulent
plane-channel flows, the negative of the instantaneous radial
velocity fluctuation u′r is used here. In Figs. 9(a) and (b), theQ2 and Q4 quadrants relate to the ejection (featuring u′z < 0and -u′r > 0) and sweeping (featuring u′z > 0 and -u′r < 0)events, respectively. It is evident that as the rotation number
increases from Rob = 0 to 20, the ejection events tend to produce high intensities with a low probability, while the sweep
events show an opposite trend. Lower intensities of the sweep
events further result in a suppression in the formation of hairpin structures under the influence of the axial system rotation.
Figure 10 compares instantaneous streaky structures5

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

(a) Non-rotating pipe flow (Rob = 0)(a) Non-rotating pipe flow (Rob = 0)(b) Axially-rotating pipe flow (Rob = 20)
Figure 10. Contours of instantaneous axial velocity fluctuations (u′z + ) in the β -z plane at wall-normal position y+ ≈ 15
for the non-rotating (Rob = 0) and rotating (Rob = 20) flows.

(b) Axially-rotating pipe flow (Rob = 20)
Figure 11. Contours of instantaneous axial vorticity fluctuations (ωz′ + ) in the β -z plane at wall-normal position y+ ≈ 15
for the non-rotating (Rob = 0) and rotating (Rob = 20) flows.

CONCLUSIONS

smaller than that in the non-rotating case, indicating that theaxial system rotation tends to suppress the sweep events to further restrain the formation of the hairpin structures, a trend thatis further evidenced by the JPDF pattern of u′z and -u′r .

Turbulent flow confined within a circular pipe subjected
to axial rotation has been studied using DNS. To conduct athorough investigation of the Coriolis force effect on the flow
field, a wide range of rotation numbers varying from Rob = 0
to 20 have been tested. In response to the axial system rotation tested, secondary flow in form of a single counter-rotating
vortex is induced at the pipe centre. As rotation number Rob
increases, a maximum of ⟨uz ⟩+ is observed at the pipe centre at Rob = 6. As Rob further increases above 6, the system
rotation tends to weaken ⟨uz ⟩+ . It is also interesting that thepeak position of the mean azimuthal velocity r⟨uβ ⟩+ /R is independent of Rob , which occurs at y ≈ 0.4 where ⟨ωz ⟩+ is zero
identically.
As the rotation number increases, the secondary flows adversely influence the magnitude of ⟨u′z u′z ⟩+ in the near-wallregion and shift the ⟨u′z u′z ⟩+ peak towards the pipe centre. The
values of ⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ increase in most pipe regions
but decrease when Rob goes beyond 14, making the dominant
contributions to the TKE level of the flow. The magnitude of+ starts to decrease as soon as the axial rotation is imkz+ Eezz
posed. This indicates that the axial rotation tends to suppress
the strength of axial turbulent motions, and the characteristic
wavelength of the axial turbulent structures enlarges as the axial system rotation speeds up.
It is observed that the Coriolis production and pressurestrain terms are the dominant terms in the budget balances of⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ in the outer region of the pipe under anaxial system rotation. The function of the Coriolis force is
to shift TKE between ⟨u′r u′r ⟩+ and ⟨u′β u′β ⟩+ , simply because
Crr = −Cβ β . In response to the system rotation imposed, thenear-wall streaks extend in the streamwise direction and tilt in
the azimuthal direction. It is observed that the magnitude of
the positively fluctuating streaks in the rotating case is much

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