friction factor of annular poiseuille flow in a transitional regime

91 1–10 Ó The Authors 2017 DOI: 10.1177/1687814016683358 journals.sagepub.com/home/ade Friction factor of annular Poiseuille flow in a transitional regime Takahiro Ishida and Takahiro Ts

Advances in Mechanical Engineering

2017, Vol 9(1) 1–10

Ó The Author(s) 2017 DOI: 10.1177/1687814016683358 journals.sagepub.com/home/ade

Friction factor of annular Poiseuille

flow in a transitional regime

Takahiro Ishida and Takahiro Tsukahara

Abstract

Annular Poiseuille flows in a transitional regime were investigated by direct numerical simulations with an emphasis on turbulent statistics including the friction factor that are affected by the presence of large-scale transitional structures Five different radius ratios in the range of 0.1–0.8 and several friction Reynolds numbers in the range of 48–150 were analyzed to consider various flow states accompanied by characteristic transitional structures Three characteristic structures, namely, turbulent–laminar coexistence referred to as ‘‘(straight) puff,’’‘‘helical puff,’’ and ‘‘helical turbulence’’ were observed The selection of the structures depends on both the radius ratio and the Reynolds number The findings indicated that despite the transitional state with a turbulent–laminar coexistence, the helical turbulence resulted in a fric-tion factor that was as high as the fully turbulent value In contrast, with respect to the occurrence of streamwise-finite transitional structures, such as straight/helical puffs, the friction factor decreased in a stepwise manner toward a laminar level The turbulent statistics revealed asymmetric distributions with respect to the wall-normal direction wherein the profiles and magnitudes were significantly influenced by the occurrence of transitional structures

Keywords

Annular Poiseuille flow, direct numerical simulation, helical turbulence, subcritical transition, turbulent statistics

Date received: 8 September 2016; accepted: 17 November 2016

Academic Editor: Bo Yu

Introduction

Turbulent transition in most wall-bounded shear flows

is characterized by subcritical scenarios.1,2There exists

a significant hysteresis between critical Reynolds

num-bers for global (Reg) and linear instabilities (Rel) For

instance, the cylindrical Poiseuille flow (cPf) and plane

Couette flow (pCf) are widely known to be linearly

sta-ble for any Reynolds number However, experimental

studies demonstrated that both cPf and pCf cannot

maintain these laminar states at high Reynolds

num-bers, no matter how ideal experiment is conducted,3

because unpreventable finite-amplitude disturbance

triggers a subcritical bypass transition for Re (\Rel)

The lower bound of this subcritical transitional regime

may be defined as Reg.1This corresponds to the lowest

Reynolds number to sustain turbulence even under a

spatiotemporal intermittent state, and this value is

important practically and scientifically In the plane

Poiseuille flow (pPf), Rel is known as 5772,4which is based on the channel half width (d) and the channel centerline velocity of laminar flow, and Reg is lower than Relin a manner similar to cPf and pCf The deter-mination of Reg value generally demands massive flow channel (in experiments) or computational resources (in simulations) because of the presence and large-scale patterning of localized turbulence

The flow transition to turbulence undergoes an intermittent turbulent regime around Regin the subcri-tical scenario As is widely known, a streamwise

Department of Mechanical Engineering, Tokyo University of Science, Noda, Japan

Corresponding author:

Takahiro Tsukahara, Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Email: tsuka@rs.tus.ac.jp

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localized turbulence that is referred to as a puff is

typi-cally observed in cPf.5 Recent studies considered the

thermodynamic limits of puff decaying and splitting

time scales to determine the value of Reg for cPf.6

However, this is not true for both pPf and pCf, as they

reveal a more complex transition scenario given the

existence of an additional free dimension in the

span-wise direction This contrasts to cPf, which is closed in

the azimuthal, or spanwise, direction The transitional

structure observed in pPf and pCf around Reg often

forms a two-dimensional (2D) pattern, referred to as

‘‘turbulent stripe.’’7,8 The turbulent stripe is very large

in both the streamwise and the spanwise directions

This stripe pattern is inclined with a certain degree

against the streamwise direction However, the

inclina-tion angle of pattern has not been unresolved, because

it is influenced by both the computational/experimental

horizontal domain sizes and the Reynolds number In

addition, the decaying and splitting processes of

turbu-lent stripe that are very important phenomena for

determined Regas in cPf are still an open issue to date

This study focuses on the subcritical transition of

pressure-driven flows between two concentric cylinders

This is termed as annular Poiseuille flow (aPf) In

addi-tion to its merit of a closed system in the spanwise

(azi-muthal) direction, the aPf may be an ideal flow system

to understand canonical wall-bounded shear flows in a

comprehensive manner based on the radius ratio,

denoted by h = ri=ro(where riand roare the inner and

outer radii, respectively) The limiting conditions of

h! 0 and 1 represent the flow geometries of a cylinder

and a channel between two parallel plates, respectively,

despite the existence of a thin inner cylinder for h! 0

Actually, a linear stability analysis by Heaton9

demon-strated that aPf should connect two important

canoni-cal flows, namely, pPf as h! 1 and cPf as h ! 0

Hysteresis is also present between Rel and Reg In

terms of the transitional structure, our previous study10

revealed smooth alternations, with dependence on h, of

the localized turbulent patterns: helical turbulence,

heli-cal puff, and straight puff Robust heliheli-cal turbulence

corresponding to the turbulent stripe in pPf has been

identified for high h 0:5 With decreasing h, the pitch

angle of helical turbulence decreases and the flow

rela-minarizes partly at marginally low Reynolds number

for h = 0:3 Finally, a new form of streamwise

loca-lized helical turbulence, named helical puff, emerges

For much smaller h 0:1, the organized structure

becomes axisymmetric, and finite-length streamwise

localized structures similar to equilibrium puffs in cPf

emerges They reveal that the helical turbulence for

high h and the straight puff for low h are linked by the

helical puff for intermediate h Detailed expressions of

each structure will be shown later with a discussion of

the Reynolds number dependency

Ishida et al.10 focused only on transitional struc-tures, and statistical properties in aPf were not exam-ined Hence, this article would describe statistical results including the friction factor This work uses direct numerical simulation (DNS) of a quenching study from fully developed turbulence to a laminar regime in order to investigate how the transitional structure would affect turbulent statistics We discuss rather low-dimensional statistical properties in transi-tional and fully developed turbulent regimes

Patel and Head11measured the skin friction to scru-tinize the transition regime and how turbulence can be sustained in cPf and pPf, both of which are the limiting cases of aPf They found that Reg occurred at

RemjcPf= umD=n ’ 2000 for cPf and at

RemjpPf= 2umd=n ’ 1350 for pPf (where umdenotes the bulk mean velocity and D is the pipe diameter) Recently, Samanta et al.12 considered the dependence

of the friction factor on initial conditions in cPf focus-ing on localized structures of puff and spot Their data provided a well-defined connection between the lami-nar and turbulent laws and predicted well the upper bound of transitional regime Since the aPf is widely used in engineering applications, such as heat exchan-ger, several previous studies examined the friction fac-tor as typified by Rothfus and colleagues.13–15 Those studies suggested the critical Reynolds numbers of the onset and end of the transitional regime,14which were defined by a slight progressive departure from the theo-retical analysis and the empirical equation of cPf (applied for aPf using the hydraulic diameter), respec-tively The obtained onset Reynolds numbers of transi-tion for h = 0 and 1 almost agree with those obtained

by the recent studies for the global instability of cPf6 and pPf.11 These studies imply that both laminar and turbulent flow patterns exist in the transitional region Hanks and Bonner16employed a theoretical analysis

to suggest that the first transition occurs near the inner cylinder and the second transition occurring near the outer cylinder follows the first one A smaller radius ratio h led to larger differences in the value of the criti-cal Reynolds number between the first and second tran-sitions Moreover, due to the existence of singularity (inner cylinder), the first critical Reynolds number indi-cated that the flow was always unstable for any Reynolds number as h approached zero In contrast, there was no difference between the first and second critical Reynolds numbers at h = 1 Their result also suggested that the dual-flow regime consisting of lami-nar and turbulent flows near the outer and inner cylin-ders, respectively, separated at the radial position of maximum velocity, and this typically occurred between the first and the second critical Reynolds numbers The second critical Reynolds number (on the outer-cylinder side) obtained experimentally14 and theoretically16was

a similar to that for the pPf, although they did not

exactly correspond The second critical Reynolds

num-ber for aPf was indicated to connect between cPf and

pPf with the variance of h Hanks and Peterson17also

performed an experimental study to verify the

theoreti-cal analysis, measured the flow rate by the oscilloscope

traces at a low radius ratio (h = 0:0416), and observed

the first and the second transitions Despite the

occur-rence of the first transition, no oscillation was observed

in the dual-flow regime where oscillations were expected

to exist, while disturbed oscillations were detected for

the second transition

Based on these results of existing studies, this study

on aPf investigates its subcritical transition scenario

from the developed turbulent state to the laminar flow

and scrutinizes the presence of the transitional structure

in detail to illuminate the manner in which transitional

structures affect flow statistics including the friction

factor

Numerical methods

In this study, DNS of pressure-driven flows was

per-formed in two concentric cylinders A cylindrical

coor-dinate system (x, r, u) was adopted as shown in

Figure 1 Additionally, we define y = r ri, which

cor-responds to the wall-normal distance from the inner

wall, and z = 2pru, which corresponds to the azimuthal

length The aPf was driven by a constant uniform

pres-sure gradient (equation 1) in the axial direction x

dp

dx = 2

d

to+ hti

Here, to and ti correspond to the mean wall shear

stresses at the outer and inner walls, respectively, and

d = ro ri is the gap width between the inner and

outer cylinders A periodic boundary condition was

imposed in the x and u directions A non-slip condition

was applied on the walls The fully developed states of

flows were considered The working fluid is an

incom-pressible Newtonian fluid The friction velocities on the

inner and outer cylinder, respectively, are defined as

follows

ut i=

ffiffiffiffi

ti r

r and ut o=

ffiffiffiffiffi

to r

r

ð2Þ

The averaged friction velocity is defined by the fol-lowing expression

ut=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

to+ hti

r 1 + hð Þ

s

ð3Þ

The fundamental equations for the velocity

u = (ux, ur, uu) and the pressure p are given by the equa-tion of continuity (equaequa-tion (4)) and the Navier–Stokes equation (equation (5))

∂tu++ u + r

u+=  rp++ 1

Ret

r2u+ ð5Þ The superscript + indicates the quantities normalized

by the wall unit (i.e the friction velocity ut and/or the kinematic viscosity n)

The finite difference method was adopted for the spatial discretization The fourth-order central scheme was employed in both x and u with uniform grids, and the second-order central scheme was in r with non-uniform grids The non-non-uniform spacing with fine grids near the walls was given as done by Moin and Kim.18 Further information with respect to the numerical methods we employed can be found in the previous reports.10,19

Ret= utd=2n = 48 to 150 and selected five different values of h = 0.1, 0.2, 0.3, 0.5, and 0.8 Table 1 sum-marizes the simulation parameters In this study, long streamwise domains of Lx= 51.2d–166.0d were set to capture intermittent structures with long streamwise extents As the Reynolds number decreased, the com-putational domain size was elongated in x so that its length would be almost constant in terms of the wall units As for the azimuthal direction, the present simu-lation covered the complete domain of u = 2p Its nor-malized azimuthal length Lz=d = rLu=d should depend

on h and r

Results and discussions Flow regimes and transitional structures

The alternations of dominant transitional structures depending on h and Ret are discussed in this subsec-tion First, we introduce the turbulent fraction to demonstrate the laminar–turbulent coexistence Then,

we would show rather simpler transition processes for high and low h, where observed structures are reminis-cent of well-known transition scenarios in pPf and cPf Finally, we describe the transition process for inter-mediate h = 0.3, which seems to combine the features

of transitional structures for lower and higher h, as an anomalous case of aPf

Figure 1 Configuration of the annular Poiseuille flow (aPf).

Figure 2 shows turbulent fraction Ft, modified from

Ishida et al.,10that is employed to specify the transition

process in aPf The turbulent fraction illustrates the

content rate of turbulence in the flow field When

Ft= 1, the flow regime corresponds to featureless

tur-bulence without any laminar patch The zero Ft

corre-sponds to a laminar flow Therefore, intermediate

values of Ft indicate the occurrence of turbulent–

laminar coexistence Fully developed or featureless

tur-bulence was observed in the high Reynolds number

regime With decreasing Rem, Ft decreased gradually

for all values of h Once the flow undergoes a

transi-tional Reynolds number regime, the laminar region

intermittently occurred in the turbulent region, and Ft

decreased toward zero

As typical flow regimes describing transitional

structures, Figures 3–5 show the 2D contours of

wall-normal velocity fluctuations in the xz plane for

differ-ent values of h First, let us focus on the case of a

rather high h of 0.5 As visualized in Figure 3(a),

fea-tureless turbulence was observed at Ret 80 despite

the occasional appearance of small laminar patches

with wispy obliqueness at Ret= 80 Helical turbulence

represented by robust oblique laminar–turbulent

pat-terns emerged at Ret= 64 and 56 This helical

turbu-lence corresponded to a stripe pattern observed in the

plane channel flows (pCf and pPf) The laminar region

expanded with a decrease in Ret, and the helical

turbu-lence was marginally modulated and collapsed at

Ret= 52, as shown in Figure 3(d) Therefore, several

helical puffs (with finite streamwise lengths) and

turbulent spots were observed for a very low and nar-row Reynolds number range near Reg Whereas the previous study10 suggested that the helical puff occurred only for h = 0.3, we newly observed the heli-cal puff even for high h = 0:5 Xiong et al.20 also reported similar observations in the pPf at very low Reynolds numbers, demonstrating the occurrence of a localized oblique pattern around Reg even for pPf (h! 1) In Figure 3(e), a three-dimensional (3D) visualization of the helical turbulence in aPf was pre-sented to aid in easy understanding In contrast to the stripe pattern observed in pPf, the helical turbulence coils around the inner cylinder because the flow system

of aPf is closed in the azimuthal direction The transi-tion process for h = 0.8 was similar to that for h = 0:5

Table 1 Computational conditions for DNS: Rem= (umd=n), the bulk Reynolds number; Lxand Lzdenote the streamwise and azimuthal lengths (Lzi= 2pr i and Lzo= 2pr o ); N x , Ny, and Nzdenote the corresponding grid numbers (y = r  ri, z = ru).

Lx=d 51.2–74.0 51.2–80.0 51.2–166.0 102.4–166.0 51.2–180.0

Lzi(h)=d, Lzo(h)=d 25.1, 31.4 6.28, 12.6 2.69, 8.98 1.57, 7.85 0.70, 6.98

DNS: direct numerical simulation.

Figure 2 Turbulent fraction as a function of the bulk Reynolds

number Error bar denotes the magnitude of time variance.

Figure 3 Two-dimensional xz-plane contours of wall-normal velocity fluctuations u0rat Re t = (a) 80, (b) 64, (c) 56, and (d) 52 for h = 0:5 The mean flow direction is from left to right and (e) three-dimensional visualization of wall-normal velocity fluctuations at Ret= 56 (u 0  0:75, red; u 0 <  0:75, blue).

Therefore, clear helical turbulence can be observed for

h 0:5

A small radius ratio must be expected to reveal

tran-sitional states similar to those in cPf Figure 4 shows

the flow fields at h = 0:1 The featureless turbulence

without any laminar patch was confirmed at

Ret= 150 and 80 Even at Ret= 64, almost the whole

field is dominated by the turbulent region and

some-times small laminar patches occur, as given in Figure

4(b), although helical turbulence emerges for high h A

streamwise localized laminar–turbulent pattern

corre-sponding to the puff in cPf was found below Ret= 60

Even for small h of 0.1, oblique interfaces between

tur-bulent and laminar region were rarely observed at

slightly high values of Ret= 60 and 56 As Ret

decreased from 60 to 48, the number of puffs or

streamwise length of the puff was decreased The puff

in aPf exhibited splitting, decaying, and combining

pro-cesses, which were similar to those of the puff in

cPf.21,22 At low Ret, the combining process could not

be detected and the probability of splitting decayed

analogous to the previous study.22 Although the puffs

split, one of the separated puffs immediately decayed in

aPf at Ret= 52 and 50 At Ret= 48, any puff splitting

was not observed, the streamwise size of the puff

chan-ged slightly and increased/decreased with time The

flow relaminarized at Ret= 46, and Figure 4(h) shows

an instant field before the relaminarization completed

As well as the other cases shown above, the

inter-mediate h of 0.3 resulted in the flow state of featureless

turbulence at Ret 80 (see Figure 5(a)) At Ret= 64,

the helical turbulence similar to that observed for

h 0:5 occurred, as shown in Figure 5(b) There was a

breaking-off point of helical turbulence, and stream-wise localized helical turbulence (called helical puff) was observed at Ret= 60 Figure 5(c) shows the flow field at Ret= 56, where mixed helical and straight puffs can be observed The obliqueness of the helical puff decayed, the straight puff infrequently occurred at lower Ret, and the probability of occurrence of a straight puff was increased compared to that of the helical puff, with decreasing Ret At Ret= 50, the transitional structure immediately decays and the flow becomes laminar This complex transition process was also found for h = 0:2 The occurrence ratio of straight puff increases more compared to h = 0:3 Additionally, helical turbulence could not be recognized for h = 0:2 even at high values of Ret A future study will discuss the separation of these complex transition processes based on both h and Ret

Friction factor

The friction factor, denoted by Cf, is defined by follow-ing equation as shown in Figure 6

Cf =2tw

u2 m

ð6Þ

The Blasius empirical friction law in the turbulent regime is also shown for a comparison The laminar solution depends on h In the laminar solution of

Cf= 8Ch=Rem, the definition of Chis as follows

Ch= 1 2h + h2

1 + h2+1hlnh2 ð7Þ

Figure 4 Two-dimensional xz-plane contours of wall-normal

velocity fluctuations u0rat Re t = (a) 80, (b) 64, (c) 60 (d) 56,

(e) 54, (f) 52, (g) 50, and (h) 48 for h = 0:1 The color range is

the same as that in Figure 3.

Figure 5 Two-dimensional xz-plane contours of wall-normal velocity fluctuations u0rat Re t = (a) 80, (b) 64, (c) 56, (d) 54, and (e) 52 for h = 0:3 The color range is the same as that in Figure 3.

With respect to cPf and pPf as the limiting cases of

aPf, Ch values are 1.0 and 1.5, respectively In order to

facilitate an easy understanding of the transition

pro-cess, the product RemCf as a function of Rem is also

plotted in Figure 6(b) and (c) The friction factor at

high Reynolds numbers are in agreement with the

empirical function Furthermore, Cf maintained high

value even at lower Reynolds numbers in the

transi-tional regime, when the flow is accompanied by the

helical turbulence (for h = 0:5 and 0:8) However,

given the localization of helical turbulence in the x

direction and the emergence of a helical puff, Cf

deviated from the empirical function toward a laminar

solution, as seen at Ret= 52 (Rem= 1460) for h = 0.5

With respect to intermediate h of 0.2–0.3, Cf

main-tained a high value similar to that for high h This must

be also because of the presence of helical turbulence or

helical puff A further decrease in Retinduces a sudden

drop of Cf, retaining the same values of Rem and

gra-dually approaches a laminar solution It should be

noted again that the present flow system is driven by a

fixed-mean pressure gradient and Rem depends on the

flow state Under such a condition, once a

relaminari-zation occurs, the bulk Reynolds number Rem should

increase significantly with a fixed Ret and lowered Cf

In this sense, there exists an overlapping region around

Reg for h = 0:2 and 0:3 This must be caused by the

alternations of transitional structures from helical

tur-bulence to puffs through helical puff, due to decrease

in Re for moderate h With respect to h = 0:1, C

gradually decreased with the Reynolds number after the occurrence of the straight puff There is no overlap-ping region, unlike that observed for intermediate h (h = 0:2 and 0:3) because the transitional structure does not change from straight puff at any Reynolds number for h = 0.1

As described above, Cf in the transitional regime considerably depends on the form of transitional struc-ture Similar aspects can be confirmed from the turbu-lent fraction that is given in Figure 2 Both Cf and Ft would maintain magnitudes as high as those of feature-less turbulence, if the transitional structures forms heli-cal pattern Such an enhancement of turbulence contributions is weakened in the following order: heli-cal turbulence! helical puff ! straight puff

Figure 7 shows the local friction velocities at the inner and outer cylinders (denoted by uti and uto, respectively) and the ratio of the friction Reynolds number on the inner and outer cylinders (denoted by

Reti and Reto, respectively) The difference between uti and uto increases at low Ret and small h when com-pared to the difference at high Ret and large h Specifically, the difference enlarges noticeably when

Ret= 80! 64, which corresponds to a shift of the regime from the featureless turbulence to a transitional state of helical/straight puff On the outer-cylinder side,

uto does not depend much on both h and Ret, and it is approximate to the global friction velocity ut defined

by equation (3) In contrast, utiis significantly different from u and depends on h and Re The distribution of Figure 6 (a) Friction factor Cf, (b) product of RemCf, and (c) extended figure of RemCf around the transitional regime.

the ratios of the friction Reynolds number for all values

of h was almost a linear distribution, as shown in

Figure 7(b) This implies a less dependency of the ratio

Reti=Retoon the global Reynolds number (Ret)

Here, let us briefly describe a background

mechan-ism of the high friction velocity, that is, the large

velo-city gradient, on the inner-cylinder wall, which can be

seen in Figures 7(a) and 8 In aPf, the inner cylinder

should be associated with a large number of intensive

sweep and ejection events that can be attributed to the

transverse curvature effect.23 In addition, elongated

streaky structures near the inner wall are more active

than those along the outer wall The turbulent motions

on the outer wall are similar to those near the flat wall

of cPf and pPf These phenomena have also been

stud-ied by Satake and Kawamura.24They discovered that a

high-speed fluid impinging against and across the inner

rod would form a wake-like region behind the rod and

a large-scale wall-normal motion in the large

low-pressure region This distinctive turbulent event

increases the friction Reynolds number on the inner

cylinder, and this phenomenon leads an asymmetric

profile especially for low h The asymmetric properties

are discussed also in the following sections

Mean flow statistics

The mean velocity profiles are presented in Figure 8 In

the figure, the profiles of u are slightly tilted toward

the inner cylinder (y/d = 0) showing asymmetric distri-butions with respect to the gap center For large values

of h, the distributions are rather symmetric at any Ret and similar to the mean velocity profile in pPf The peak position shifts to the inner-cylinder side, in partic-ular, for low h As for the Reynolds number effect, the peak of uxdoes not move so much when Ret decreases from 56 to 52, but both the peak value and the bulk velocity (normalized by ut) increase noticeably This is because those flows are already in the intermittent state and a decrease in the Reynolds number would expand laminar regions In terms of transitional structures among the helical turbulence and the helical/straight puffs, any significant difference cannot be observed in the uxprofiles However, with respect to the decreasing

Ret from 80 to 56, the peak position of ux moved toward the gap center (y/d = 0.5) and the asymmetric property is suppressed particularly for low h = 0:1 This implies that the asymmetric property of the core region would be moderated in the transitional regime accompanied by large-scale intermittent structures, while featureless turbulence at a high Reynolds number reveals an asymmetric ux profile due to the aPf geome-try Because of this, the ux profile in the transitional aPf even for h = 0:1 is more dissimilar to cPf, in which

ux has a peak at y=d = 1 and decreases monotonically with y=d! 1 However, both flows exhibit straight puffs, as shown in Figure 4(f)–(h) It is interesting to note that the flow statistics including the mean velocity profile in the subcritical regime were dissimilar to cPf, although the flow structure is analogous in both flows For reference, Table 2 assembles the types of domi-nant transitional structures that are found in each case

we focus in this section As seen in the table, Ret= 80 gives rise to the featureless turbulence at any h ( 0:5) Below this Reynolds number, the transitional structure

is formed and depends on h At h = 0:1, only the straight puff can be formed, as discussed with Figure 4 However, we have observed the helical shape of struc-tures at h = 0:5 in Figure 3 For (Re, h) = (56, 0:3),

Figure 7 (a) Friction velocity on the inner and outer cylinders

(u ti and uto) as a function of Re m (b) Comparison of the friction

Reynolds number based on the inner friction velocity versus the

friction Reynolds number based on the outer friction velocity If

h = 1, it should be Re ti = Re to

Figure 8 Mean streamwise velocity profiles The line types indicate different values of h, and the colors correspond to different values of Re t The inner-cylinder wall corresponds to y=d = 0, while the outer one is at y=d = 1.

the helical puff seems unstable and might cycle through

breakup and reshaping

Figure 9 shows the root mean square value of each

fluctuating velocity component In Figure 9(a), the

streamwise turbulent intensity u0xrms shows asymmetric

distributions with two clear peaks near the inner and

outer cylinders The difference between the two peaks

increases with smaller values of h, and the peak near

the inner cylinder is larger than that near the outer

cylinder It should be noted that, if scaled with each

friction velocity (either uti or uto), the inner peak of

u0xrms=uti is much lower than the outer one of u0xrms=uto

(figure not shown here) In a manner dissimilar to the

ux, the asymmetricity in a profile is increased at low

values of Retas well as h The two near-wall peaks for

the high Ret of 80 exhibit the almost same magnitude

At this Reynolds number, the fully turbulent state

might reduce the intensity gap between both sides and,

as a result, three curves with different h are roughly

matched This aspect can be similarly seen in the other

components An interesting distribution of u0 is

observed at Ret= 52 for h = 0.1 and 0.3, and a third peak is observed around the gap center This centerline peak must be attributed to localized (straight) puffs with very large laminar regions, as shown in Figures 4(f) and 5(e)

In Figure 9(b) and (c), u0

rrms and u0

urms indicate skewed profiles with higher peaks near the outer cylin-der at a high value of Ret= 80 Given decrease in Ret, the difference between the peaks near the inner and outer cylinder of u0

rrmsdecreases and an almost plateau region is observed around the gap center With respect

to this plateau region, we found that the cases in the presence of helical turbulence always provide large

u0rrms and u0urms compared to the cases of puff For instance, u0

urmsfor h = 0:5 is 10%–20% larger than that for h 0:3, as given in Figure 9(c) These results sug-gesting an enhanced turbulent intensity in the helical turbulence are in consistency with the above-mentioned high Cf in the transitional regime A high-value distri-bution of u0

urmsnear the inner cylinder is the same trend with that of u0 for low h

Table 2 Transitional structure observed in the selected cases The value sets of the parameters (Re t , h) are corresponding to those shown in Figures 8–11.

Ret= 80 Featureless Turbulence Featureless Turbulence Featureless Turbulence

Ret= 56 Helical Turbulence (Helical) Puff Straight Puff

Figure 9 Root-mean-square values of: (a) streamwise, (b) wall-normal, and (c) azimuthal velocity fluctuations.

Figure 10 shows the Reynolds shear stress ( u0

xu0

r)

The radial position of the zero Reynolds shear stress

moves slightly toward the inner cylinder for low values

of h, while the zero value for pPf should be located at

the center of the gap The zero position shifted to the

channel center with a decrease in the value of Retfrom

80 to 56, as does the peak position of ux(Figure 8) The

zero position approaches the center of the gap even for

low values of h with decrease in Ret With respect to

the magnitude of Reynolds shear stress, we may detect

tendencies similar to those in the turbulent intensity:

the larger peak near the inner cylinder for low h, the

weakening due to decay from featureless turbulence,

and the enhancement by the presence of helical

turbulence

Figure 11 shows the turbulent energy, denoted by

k = (u0

xu0

x+ u0

ru0

r+ u0

uu0

u)=(2u2

t) As widely known, turbulent motion is vigorous near the wall With

respect to the aPf, the turbulence is more activated near

the inner cylinder than near the outer cylinder In

con-trast, the peak of k near the outer cylinder is slightly

higher only at the high value of Ret= 80 Therefore,

this enhanced turbulent fluctuation near the inner

cylinder was prominent when the transitional structure

occurred Note again that in terms of normalization by

each wall units, the magnitude of k becomes lower on

the inner-cylinder side than on the outer side It may

imply that the near-wall coherent structure common to

plane-wall turbulence may tend to be absent near the

inner cylinder, in particular, under the transitional

state Hanks and Bonner16 proposed a picture of the dual flow, that is, the first transition from the inner-cylinder side, in aPf In contrast, our DNSs have demonstrated no dual flow and shown a less active tur-bulence near the outer-cylinder wall For h = 0:1, the streamwise localized puff with a large laminar regime led to high fluctuations around the gap center, as dis-cussed in the distribution of ux

Conclusion

This study on aPfs has investigated the Reynolds num-ber (Ret) and radius ratio (h) dependencies of the sub-critical transition process, friction factor, and turbulent statistics, by means of DNSs

The results indicated that the transition process of aPf depended on both Ret and h Clear helical turbu-lence was observed for high h (h 0:5) Even for such

a high h, helical puffs emerged in a very narrow range

of Reynolds number around Reg In contrast, with respect to low h (h = 0:1), straight puffs similar to those in cPf occurred in low Ret ranges With respect

to intermediate values of h (h = 0.2 and 0.3), given decrease in Ret, the transitional structure varied and included helical turbulence, helical puff, and straight puff

The occurrence of the transition structures consider-ably influenced the friction factor Cf and turbulent fraction Ft When the flow regime corresponding to helical turbulence, the friction factor and turbulent fraction maintained a high value that was similar to that for fully turbulent regime However, the friction factor immediately decreased, once the transitional structure was localized in the streamwise direction and formed the helical/straight puff For low h 0:1, only the straight puff was organized independently of Ret, and Cf decreased toward laminar solution monotoni-cally In contrast of this monotonic trend, an overlap-ping region existed around Reg for intermediate

h= 0:2 and 0:3 This non-monotonic trends of Cf and

Ft must be caused by the alternations of transitional structures As mentioned above, the shape of transi-tional structure depended not only on h but also on

Ret For h = 0.3, helical turbulence was sustained only

in a small range of high Ret, where the intermittent regime occurred With decrease in Ret, the helical tur-bulence changed into the straight puff through the heli-cal puff Attributed to these alternations of transitional structures for a fixed intermediate h, both Cf and Ft followed two different paths of the helical turbulence and the straight puff

The turbulent statistics in aPf corresponded to an asymmetric distributions with respect to the gap center With respect to the streamwise mean velocity uxprofile, the asymmetricity was strong when Retwas high and h was small Even for low values of h (h = 0:1), the Figure 10 Reynolds shear stress.

Figure 11 Turbulent energy.

distributions became symmetric once the transition

from featureless to intermittent turbulence occurred

The transitional structure for low h was a puff (similar

to that in the cPf but different from the oblique pattern

in pPf) Hence, in the subcritical regime, it was

interest-ing to note that uxexhibited dissimilarity with the cPf,

although the flow structure was analogous for both

flows

In contrast, the distributions and magnitude of

bulent intensities exhibited complex behaviors The

tur-bulent intensities displayed strong peaks near the inner

cylinder with decrease in Ret, whereas the peak near

the outer cylinder was higher at high Ret This is due

to the difference in the local friction velocity between

the outer and inner cylinder, which becomes large

sig-nificantly after a shift from featureless turbulence to

the transitional regime With respect to the transitional

regime, the turbulent intensities were enhanced, in

par-ticular, when the helical turbulence occurred The third

peak appeared around the gap center, when there

existed the straight puffs among which large laminar

regions emerged

Acknowledgements

The present simulations were performed on NEC-SX

super-computers at the Cyberscience Centre of Tohoku University

and at the Cybermedia Centre of Osaka University.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of this

article.

Funding

The author(s) disclosed receipt of the following financial

sup-port for the research, authorship, and/or publication of this

article: T.I was supported by a Grant-in-Aid from Japan

Society for the Promotion of Science (JSPS) Fellowship

#26-7477 This work was partially supported by Grant-in-Aid for

Young Scientists (A) #16H06066 from JSPS.

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