Heat transfer of transitional regime with helical turbulence in annular flow

International Journal of Heat and Fluid Flow 82 (2020) 108555

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Heat transfer of transitional regime with helical turbulence in annular flow Takehiro Fukuda, Takahiro Tsukahara

⁎

T

Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan

ARTICLE INFO

ABSTRACT

Keywords: Annular Poiseuille flow DNS Subcritical transition Turbulent heat transfer Wall-bounded shear flow

Direct numerical simulations of the passive heat transfer in pressure-driven flows through concentric annuli were performed in the subcritical Reynolds-number regime. In this regime, the intermittent-turbulent velocity field would exhibit large-scale patterns of the laminar-turbulent coexistence. The main emphasis was placed on the influence of transition structures on the heat transfer at two radius ratios, rin / rout = 0.5 and 0.8, at which point a helically-shaped, turbulent transition structure was present. An incompressible Newtonian fluid was considered with fixed fluid properties and a Prandtl number of 0.71. We investigated the dependence of this heat-transfer on the Reynolds number, radius ratio, and thermal bounding condition. The state of helical turbulence was found to provide high heat-transfer rates close to those estimated by the turbulence empirical function even in the transitional regime. The present results showed that the wall-normal turbulent heat flux occurred in both longitudinal-vortex clusters around the turbulent bands as well as inside of the localized turbulent region. The non-turbulent vortex cluster provided a promotion of heat transfer that occurs in intermittent turbulent states.

1. Introduction In general, laminar flow exhibits a small frictional drag and a low heat-transfer coefficient, while turbulent flow enhances both heattransfer and mixing. Although the laminar-turbulent transitional regime of interest in this study (specifically the subcritical transition regime) exhibits properties in between those of each state, its flow characteristics significantly depend on the large-scale intermittent turbulent structures, and this has not yet been fully understood. Here, the subcritical transition regime indicates a Reynolds-number range in which localized turbulence, triggered by finite disturbances, exists stable, even below the critical value for linear-stability (Manneville, 2016; Tuckerman et al., 2020). To improve our understanding of thermofluids, the heat transfer characteristics in this regime should be examined until the Reynolds number reaches its minimum critical value (also known as the global minimum, Reg). Below this value, these turbulent regions cannot be maintained. Annular Poiseuille flow (aPf), the subject of this study, is a common, wall-bounded shear flow that is widely applied in heat exchangers, such as in the flow of coolant around a fuel rod in nuclear power plants. Because of the high demand for aPf in engineering applications, the friction and heat-transfer coefficients of aPf have been studied both numerically and experimentally using a wide range of parameters, including varied Reynolds numbers, Prandtl numbers, and radius ratios (Chung et al., 2002; Chung and Sung, 2003; Heikal et al., 1977; Kaneda

⁎

et al., 2003; Kawamura et al., 1994; Ould-Rouiss et al., 2009; Rehme, 1974; Satake and Kawamura, 1995; van Zyl et al., 2013; Yu et al., 2005a, 2005b, 2005c). Most prior studies of aPf have concerned the non-coincidence of the radial positions between the zero shear stress and maximum velocity, as was recently reviewed by RodriguezCorredor et al. (2014). As for the transition structures of aPf, the sheardriven instability that results in the formation of Tollmien–Schlichting waves has been documented (Mott and Joseph, 1968; Garg, 1980; Heaton, 2008). Recent investigations on the flow instability in grooved or eccentric annuli have also explored the possibility of drag reduction and the enhancement of mixing (Moradi and Tavoularis, 2019; Moradi and Floryan, 2019). However, the detailed heat transfer characteristics of the transition structures, as well as those of the coexisting laminarturbulent flow that occurs during the subcritical transition to turbulence, have not yet been studied. The subcritical transition to turbulence has been studied extensively in both pipe flow (cPf: circular Poiseuille flow) and channel flow (pPf: plane Poiseuille flow). Both cPf and pPf are related to the limiting aPf radius ratio:

=

rin rout

(1)

where rin and rout denote the inner and outer cylindrical radii, respectively. At η ≈ 1, aPf asymptotically approaches pPf. As η approaches 0, aPf approaches cPf, but cannot equal cPf because of the infinitesimally-

Corresponding author. E-mail addresses: [email protected] (T. Fukuda), [email protected] (T. Tsukahara).

https://doi.org/10.1016/j.ijheatfluidflow.2020.108555 Received 27 April 2019; Received in revised form 20 January 2020; Accepted 22 January 2020 0142-727X/ © 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

small inner cylinder. The aPf is a canonical form of wall-bounded shear flow that results in asymmetric velocity distributions in the gap between the cylindrical walls. In cPf and pPf, coexisting laminar-turbulent fields with coherent spatial patterning have been observed in the subcritical transition regime (Brethouwer et al., 2012; Prigent et al., 2002; Tsukahara et al., 2010; Wygnanski et al., 1975; Tsukahara et al., 2005). For example, the cPf is well-known to exhibit localized turbulence that is intermittently generated and maintained in the flow direction in a laminar background. This type of transition structure is called an equilibrium turbulent puff (or simply a ‘puff’) and maintains a size that is ten or more times greater than the pipe diameter (Moxey and Barkley, 2010; Samanta et al., 2011). A similar type of transition structure that demonstrates large-scale laminar-turbulent patterns has been observed in various channel flows (Brethouwer et al., 2012; Coles, 1965; Kunii et al., 2019; Tsukahara et al., 2010a, 2010b). Recent studies (Avila et al., 2011; Barkley et al., 2015; Chantry et al., 2017; Sano and Tamai, 2016; Shimizu and Manneville, 2019) have identified transition structures as keys to determining Reg. In particular, in flows with transition structures, known as a ‘turbulent stripes’, the Nusselt number is comparable to the estimated value assuming a fully-turbulent state (Fukudome et al., 2018; Tsukahara et al., 2006). With the aid of increased computing power, direct numerical simulations (DNSs) can be performed for a wider range of Reynolds numbers. Analysis using a large computational domain that is capable of simulating the above-mentioned transition structure is now also possible. Ishida et al. (2016, 2017) conducted a large-scale DNS of aPf and revealed that the shape of the transition structure depends on η. As described above, the characteristic transition structures (i.e. the largescale laminar-turbulent coexistence) of cPf and pPf were similarly shown in the transition process of aPf. Three types of transition structures, including intermediate structures, were identified. At high values of η, a spirally-stripe pattern, similar to that of pPf, has been observed. Fig. 1 shows a typical flow field accompanied by helical turbulence. Here, a turbulent band that is oblique to the flow direction is formed so that the localized turbulence swirls around the inner cylinder. The angle, α, of the oblique band is about 24∘ (Ishida et al., 2016), equivalent to = 21 of the turbulence stripe in pPf (Tsukahara et al., 2010). It is interesting to note that the annular channel geometry is uniform in the axial and azimuthal directions, while the helical turbulence exhibits a chirality in the band orientation. In the case presented in Fig. 1, a counterclockwise spiral is observed with respect to the mean flow direction, although it should be noted that a clockwise spiral may occur with similar probability. Ishida and Tsukahara, 2016 reported that the friction coefficient, Cf, depends on the type of transition structure and/or the radius ratio. Despite ongoing debate on the mechanisms for the ubiquitous obliqueness of turbulent bands in various wall-bounded shear flows, several points still remain to be better understood like the physical reasons for the self-organization, band width and α, and its dynamics. The most recent reviews are Manneville (2016) and Tuckerman et al. (2020). Very recently, Reetz et al. (2019) found an invariant solution that corresponds to the oblique stripe pattern of localized turbulence, as a result of tracking the well-known solution of Nagata (1990). This suggests the oblique-band

state, including the helical turbulence, as an essential feature in the subcritical transition. Previous DNS studies of turbulent heat transfer in aPf were limited at high or moderate Reynolds numbers, and they therefore lack information concerning the heat transfer in the subcritical transition regime. In particular, the influence of transition structures on the process of heat transfer in aPf has not yet been studied. The aforementioned coexistence of laminar-turbulent flow should differ from the homogeneous turbulence observed at higher Reynolds numbers in terms of the spatially-limited turbulent heat flux. Therefore, the quantitative nature of the turbulent heat transfer should depend on the transition structure. In this paper, we numerically investigated the turbulent heat transfer in the subcritical transition regime with helical turbulence, which exists as a helically-shaped pattern of localized turbulent region in aPf. We performed DNSs at η ≥ 0.5, at which helical turbulence may occur, and the Reynolds-number dependence of the heat-transfer rate was investigated. Our main focus concerned the enhancement of heat transfer due to the characteristic structure observed in the subcritical transition regime. The scalar (heat) transport was simulated using two thermal boundary conditions, and the dependence of the heat-transfer rate on the Reynolds number, radius ratio, and thermal boundary conditions was discussed. 2. Numerical procedures As depicted in Fig. 2, the pressure-driven flows in concentric annuli were analyzed. The cylindrical coordinate system (x, r, ϑ) was applied, and x, r, and ϑ correspond to the axial (streamwise), radial, and azimuthal directions, respectively. Another coordinate, y = r rin, was also defined to measure the wall-normal distance from the inner-cylindrical wall. The flow was driven in the direction of x by a constant uniform pressure gradient defined by:

dP = dx

2 d

+ 1+

out

in

,

(2)

where τout and τin are the mean wall shear stresses at the outer and inner walls, respectively, and d ( rout rin ) is the width of the gap between the annuli. The force balance in Eq. (2) holds for the statistically steady state, where all of dP/dx, τout, and τin are temporal averages. This balance does not necessarily hold for the instantaneous pressure and wall shear stress, and the flow rate may vary with time due to turbulent motions even under the fixed pressure gradient. Periodic boundary conditions were imposed in x and ϑ. The non-slip condition was applied to the walls. Hence, the averaged flow of interest should be

Fig. 1. Helical turbulence observed in aPf with = 0.8 at Re = 72 . Three-dimensional visualization via the iso-surfaces of the streamwise velocity fluctuation: red, ux + ( ux / u ) > 3; blue, ux + < 3 (uτ defined by Eq. (5)).

Fig. 2. Configuration of annular Poiseuille flow through a gap between concentric cylinders with thermal boundary conditions of uniform heat-flux heating (UHF) and a constant temperature difference (CTD). 2

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

a unidirectional shear flow with a velocity of u x = ux (y ) . The overbar denotes a value averaged with respect to time (t), x, and ϑ. The fundamental equations used to determine the velocity, u = (u x , ur , u ), and the pressure, p, were the continuity and Navier–Stokes equations. In this study, two heating systems were used as thermal boundary conditions: A UHF (uniform heat-flux heating) was applied to both walls and a CTD (constant temperature difference) of T ( Tout Tin ) was given between the walls. We considered fully-developed flow and thermal fields. The working fluid was an incompressible Newtonian fluid with a buoyancy that was assumed to be negligible; hence, the temperature was treated as a passive scalar in this study. The UHF condition, shown in Fig. 2(c), was derived from the uniform time-averaged heat flux over both surfaces, as determined by previous studies on pPf (Kawamura et al., 1998; Abe et al., 2004; Wei, 2019). The instantaneous wall heat flux was time-dependent and fluctuated as a result of turbulent motions. This heating condition provides a constant streamwise gradient of temperature, T, as follows:

qw , cp um

T 2 = x d

Table 1 Grid resolutions in some present DNSs, which are normalized by ν/uτ. = 0.8 Reτ

x+ + rmin

(4)

+ in and T = (1 + )

out

qw . cp u

(5)

In the following sections of this paper, an alternate temperature, θ, was used to represent the difference between the wall temperature, Tw = Tout = Tin, and T at a point of interest via the following equation: + (x ,

r, , t) =

Tw

T (x , r , , t ) . T

u

(6)

t

for CTD:

t

+

+ (u+· )

+

= Re 1Pr

* + (u+· ) * = Re 1Pr

1 2 +

1 2

+ u x /u m,

*.

(8) (9)

Note that the superscript + indicates a quantity normalized by the wall units (i.e., Tτ, uτ, and/or the kinematic viscosity ν), and * = / T . The boundary conditions in terms of θ are described as

for UHF:

+ (x ,

rout, , t ) =

+ (x ,

rin, , t ) = 0,

for CTD: * (x , rout , , t ) = 1and * (x , rin, , t ) = 0.

80

150

5.60 0.14

8.00 0.20

15.0 0.38

5.60 0.14

8.00 0.20

15.0 0.38

5.49

7.85

14.7

1.37

1.96

14.7

2.53

4.75

9.81

18.4

1.77 2.75

2.53 3.29

4.75 18.4

,in

=

in

andu

,out

=

out

.

(12)

The present series of DNSs were initiated at Re = 150 to generate a well-developed turbulent field, and the friction Reynolds number was gradually reduced. After the flow reached a statistical steady state in each case, the Reynolds number was decreased by one-step to a lower value or a temporal averaging was performed. For both of the tested radius ratios, the flow eventually decayed into a completely laminar state at Re = 50, and the limit for maintaining turbulent flow was determined to be Re = 52 under the present conditions. Because the simulation was performed using a constant pressure gradient based on the given Reτ value, the bulk Reynolds number Rem should increase significantly once the flow becomes laminar, as shown in Table 2. After laminarization, any turbulent motion would not arise spontaneously

(7)

The following equations for θ(x, r, ϑ, t) can then be arranged for each heating system:

for UHF:

56

3. Results and discussion

For the CTD condition, shown in Fig. 2(d), the instantaneous temperature, T, was divided into two terms:

T (x , r , , t ) = Tin + (x , r , , t ).

150

50–150 for each η. We chose to analyze two different radius ratios, = 0.5 and 0.8, at which values helical turbulence has been observed (Ishida et al., 2016). The molecular Prandtl number was set to Pr = 0.71, air was assumed to be the working medium, and any property variations were neglected. The axial domain length of L x = 51.2d was chosen empirically to capture the transition structure with a long streamwise correlation length, according to Ishida et al. (2017). With regard to the other directions, the simulated domain incorporated the entire cross section of Lr × L = d × 2 . A grid with dimensions of 1024 × 128 × 512 in the x, r, and ϑ directions, respectively, was used. As reported later, the present domain size was comparable to the width of helical turbulent band, and its aspect ratio was reasonable to demonstrate the obliqueness of the band in aPf. The grid resolutions in x = 0.0123 rad, while those and ϑ were respectively x = 0.05d and based on wall units are listed in Table 1, but not for all simulated cases. This resolution is similar to the DNS performed by Chung and Sung (2003), who employed a second-order central difference scheme using the same Prandtl number of the present study. Table 2 summarizes the resultant parameters obtained from the present DNSs. It should be noted that the local friction velocities differed between the inner and outer walls, and could be defined as:

where qout and qin are the heat fluxes at the outer and inner walls, respectively. At = 1, qw = qout = qin . The averaged friction velocity and temperature were respectively defined using:

u =

80

6.88

+ rout

(3)

qout + qin , 1+

56

1.77

+ rmax + rin

where < · > denotes a value averaged with respect to t and ϑ, ρ is the density, cp is the specific heat at a constant pressure, and um is the bulk mean velocity. The averaged wall heat flux, qw, was defined as:

qw =

= 0.5

(10)

Table 2 Resultant bulk parameters: Reτ,in and Reτ,out are the friction Reynolds number based on the friction velocity of the inner and outer walls, respectively; Rem = (u m d/ ) is the bulk Reynolds number.

(11)

This work is a continuation of the DNS carried out by Ishida et al. (2016) (although the current work differs by considering the heat transfer); hence, the present numerical procedure is only briefly described as follows. The finite difference method was adopted for the spatial discretization, while the pressure Poisson equation was solved in Fourier space. A fourth-order central scheme was employed in both x and ϑ with uniform grids, while a second-order scheme was used in r with non-uniform grids. Time advancement was carried out using the Crank–Nicolson method for the wall-normal viscous term, while the second-order Adams–Bashforth method was used for the other terms. We analyzed several values of Reτ ≡ uτd/2ν within a range of

= 0.8

3

= 0.5

Reτ

Reτ,in

Reτ,out

Rem

Reτ,in

Reτ,out

Rem

150 80 64 56 52 50

154.3 81.2 65.1 56.9 52.9 51.1

149.5 79.3 63.2 55.3 51.3 49.2

4693 2339 1844 1580 1452 1669

156.5 83.5 68.3 60.0 55.8 54.0

147.6 78.3 61.8 53.9 49.9 47.9

4638 2306 1786 1594 1485 1557

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

Fig. 3. Mean velocity profile normalized by the maximum velocity for each case. The symbols represent laminar solutions: ∘, = 0.8 ; × , 0.5.

Fig. 5. Mean temperature profile. Fig. 4. Mean velocity profile normalized by the wall units for each wall.

because of the subcritical transition regime. The mean velocity distributions of all simulated cases are shown in Fig. 3 and plotted according to the law of the wall in Fig. 4. On the horizontal axis of Fig. 3, y/ d = 0 and 1 correspond to the inner- and outer-cylindrical wall surfaces, respectively. In Fig. 4, dimensionless processing based on the friction velocity of each wall, given by Eq. (12), was performed. From these figures, a fully-turbulent profile follows well the wall law at Re = 150, and the laminar profile was confirmed at Re = 50 . Both radius ratios agree well with the obtained theoretical solution showing laminar flow and the simulation of the thermal field was similarly confirmed, thereby supporting the validity of the present calculations. At the other intermediate Reynolds numbers, the gradual asymptotic approximation of a laminar profile was noted without the presence of a logarithmic-law region. Figs. 5 and 6 show the mean temperature distribution under each thermal boundary condition. In Fig. 6, only the inner-cylindrical wall was plotted versus wall units as yin+ ; this was done using the inner-cylindrical friction temperature:

T ,in =

qin . cp u ,in

Fig. 6. Wall-normal distributions of the mean temperature on the inner-cylinder: UHF (left) and CTD (right). The same legend is used as in Fig. 5. The symbols in the UHF case show the existing DNS results for reference: × , pPf corresponding to ( , Re ) = (1.0, 150) by Kasagi et al. (1992); ∘, aPf of ( , Re ) = (0.5, 149) by Chung and Sung (2003). A red solid line representing the log law suggested by Chung and Sung (2003) for moderate Reynolds numbers is plotted in both panels.

(13)

due to the low Reynolds number. Similar confirmations were made for the temperature distributions on the outer cylinder (figure not shown). Fig. 7 shows the visualization results of the respective instantaneous fields at = 0.8 for five Reτ values from Re = 150 to 52. Images of the streamwise velocity fluctuations u x in the x plane at y/ d = 0.5 are shown, and the azimuthal domain length at that radial position corresponds to (rin + rout ) . At higher Reynolds numbers, the spatial scale of the dominant turbulent structures differs depending on the wall-normal

To demonstrate the validity of this DNS for turbulent heat transfer, the results of the spectral-method DNS for pPf, simulated by Kasagi et al. (1992), and those of aPf, simulated by Chung and Sung (2003), were used for comparison. The former agreed well with the present results obtained with ( , Re ) = (0.8, 150) regardless of the wall curvature, and consistency with the latter was also confirmed. For the CTD condition, the logarithmic-law region could not be detected 4

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

z (or x Fig. 7. Two-dimensional contours of the instantaneous velocity and thermal fields in the gap-central x ) plane of aPf with = 0.8 . From top to bottom, each row has respective Reτ values of 150, 80, 64, 56, and 52. (Left column) ux / u , (center) θ′/Tτ for UHF, and (right) θ′/Tτ,in for CTD. The three fields in each row were obtained at the same instant in the simulation. The main flow direction is from left to right. The entire domain of L x × (rin + rout ) = 51.2d × 28.3d is visualized.

height (Abe et al., 2004). However, at the present low Reynolds numbers, the dependency of the visualization position (in r) can be negligible in this respect, because our main target is the laminar-turbulent coexistence extending into the x space. In fact, the localized turbulent regions cover the entire gap in the r direction, as will be shown by the visualization of the flow cross-section later in this paper. Fig. 7 also presents the distributions of the temperature fluctuation, which are visualized at the same time and position in accordance with each velocity field. The results obtained using the UHF condition are arranged in the center column, and those with the CTD condition are in the right-hand column. Each field will be discussed in descending order

in terms of the Reynolds number, starting at the top of the figure. At Re = 150 in Fig. 7(a)–(c), the homogeneous turbulent field exists in the x directions, and no distinctive large-scale patterns are observed. The same results were observed for the temperature fields; particularly, the fluctuation in the CTD condition was shown to be strongly generated throughout the entire domain. Note that the color bars are unified at all contours from 2 to + 2 for the frictional velocity and frictional temperatures (inner-cylindrical values for the CTD condition). Since the CTD condition has a mean temperature gradient, / r , even in the gapcenter, the temperature variation in this condition is much larger than that of the UHF condition (see Fig. 8). 5

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

direction is probabilistic and may change over a long period of time. Because the time required to change its orientation is sufficiently long compared to the time scale of the intrinsic turbulent vortex, the oblique band pattern can be regarded as robust and steady. The localized turbulence appears to form a well-defined diagonal band and its thickness gradually decreases when the laminarization is close to Re = 52–56 (Fig. 7(j) and (m)). The localized turbulence region refers to a region in which positive and negative fluctuations are mixed and are spatially fine (on the scale of about d). In Fig. 7(j)–(o), an oblique-band-like localized-turbulent region is observed to exist diagonally in each visualized domain; however, the banded pattern propagates downstream at approximately the bulk velocity. External to the localized turbulence, a spatially-large positive variation occurs, indicating the presence of a laminar region. Note that at the visualized gap center, the laminar velocity profile and the UHF-temperature profile exhibit higher values than those of the turbulent flow. In the CTD images, fine-scale temperature fluctuations are observed in the areas corresponding to the localized turbulence. Slightly thicker, streaky structures that extend in the x-direction can be observed both the upstream and downstream to sandwich the localized turbulence (see Fig. 9 for a guide to these structures). These streaky structures in the CTD condition are presumed to be a result of a cluster of non-turbulent longitudinal vortices generated at the interface between the localized turbulence and the laminar flow region. This laminar-turbulent boundary is of interest because a wallnormal overhang profile of the local ux occurs, as observed in the turbulent bands by Duguet and Schlatter, 2013, as well as a spanwise (azimuthal in aPf) secondary flow, as observed by Couliou and Monchaux (2015). The longitudinal vortices appearing at this laminarturbulent boundary have a diameter of approximately the gap-width, d, and induce flow across the gap-center. At Re = 64 –80, such longitudinal vortex-induced temperature variations are observed in the CTD condition, instead of calm laminar regions (Fig. 7(f) and (i)). A clear laminar region (without temperature fluctuations in the CTD condition) appears at the much lower Reτ values of 52–56. The transition process and structures until laminarization can be summarized as follows. From Re = 80, intermittency occurs in the turbulent field as non-turbulent longitudinal-vortex clusters are first formed around the localized turbulent regions. The longitudinal-vortex clusters induce flow across the gap-center, but the effect this has on the velocity and UHF-temperature fields is small, resulting in an apparent laminar flow region. As the Reynolds number further decreases, the laminar regions expand by an amount corresponding to the narrowing of the turbulent band. At this time, the longitudinal-vortex clusters are maintained around the turbulent bands. From the above results, it is conjectured that even in the subcritical transition regime, the longitudinal-vortex cluster provides a large heat transfer coefficient comparable to that of turbulent flow. Similar observations were made at = 0.5, and the above trends were similarly confirmed in Fig. 10. The turbulent band at = 0.5 was almost on a diagonal of the computational domain, and the oblique angle was therefore small: = 30 –40∘ for = 0.8, whereas = 10 –20∘ for = 0.5. The flow geometry of aPf has infinite degrees of freedom in x, but is limited in ϑ depending on η. It should be noted that the exact spacing of the turbulent band cannot be determined by the current DNS alone because of the finite periodic computational domain in x. To determine a more natural pattern interval free from the periodic boundary, simulations of a much larger domain would be required in future studies. Figs. 11 and 12 show the Reynolds-number dependence of the Nusselt number, Nu, and the Stanton number, St, as defined by the following equations:

Fig. 8. Temperature variance.

Fig. 9. Nomenclature for the laminar-turbulent coexistence, sampled from the CTD-temperature fields presented in Fig. 7(i,o): top, ( , Re ) = (0.8, 64) ; bottom, ( , Re ) = (0.8, 52) . The distinct localized turbulent region and surrounding longitudinal-vortex cluster are highlighted.

Large-scale patterns with sizes comparable to the computational domain emerge at Reτ values less than or equal to 80, and the localization of laminar and turbulent regions is clearly shown at Re = 64 (Fig. 7(g)–(i)). The velocity field exhibits helical turbulence, such as that shown in Fig. 1. The temperature and velocity fields of the UHF are very similar because they have similar boundary conditions. In particular, it can be seen that a band-shaped, localized turbulence that is oblique to the streamwise direction also appears in the thermal field. The direction of inclination differs between (g) and (j,m) because this

Nu

6

qw m/d

=

2Re Pr + m

,

(14)

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

Fig. 10. Same conditions as Fig. 7, but for

= 0.8 . From top to bottom, the Reτ values for each row are 80 (a–c), 56 (d–f), and 52 (g–i).

Fig. 11. Nusselt number as a function of the bulk Reynolds number. The dotted line indicates an empirical correlation function of the channel flow (Kays, 1980).

Fig. 13. Wall-normal distributions of the wall-normal turbulent heat flux, normalized by uτTτ: UHF (top) and CTD (bottom). Fig. 12. Stanton number as a function of the bulk Reynolds number, not including the results of fully-laminar flow. The friction coefficient is also shown.

St

2Nu . Re m Pr

the transitional regime (1400 < Rem < 2400). The visualizations in Fig. 7 shows that the area of the localized turbulence is roughly half that of the entire flow path, while the mean Nusselt-number index is comparable to the estimated value assuming that the entire flow path is turbulent. The high Nu at Re m = 1700 –2400 (equivalent to approximately Re = 64 –80) under the CTD condition is also noteworthy. This supports the fact that the influence of the longitudinal-vortex cluster, rather than laminar flow patches, is dominant. In Fig. 12, comparisons of varied Stanton numbers and friction coefficients, Cf, are shown. The reduction of St at the transitional regime (1400 < Rem < 2400) is more pronounced than that of Cf. This tendency must also depend on the Prandtl number, as the current condition is Pr ≠ 1, and a discussion of the thermal and flow dissimilarities is excluded from this study. Next,

(15)

Here, λ is the thermal conductivity, and m = ux r dr / ux r dr re0.5 presents the bulk temperature. In these plots, a line of Nu Re 0.8 is m Pr drawn according to the empirical function proposed by Kays (1980). Here, the coefficient of the empirical function for each condition was determined using the Nu value for ( , Re ) = (0.8, 150), which was fully turbulent. The parenthesized plot points indicate instances of laminar flow. From Fig. 11, it can be seen that Nu values close to those of the turbulence correlation function were obtained for both radius ratios in 7

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

Fig. 14. Cross-sectional views of the velocity and temperature fluctuations for the two thermal boundary conditions, and the azimuthal instantaneous distribution of plane at an arbitrary x value, visualized the wall-normal integrated turbulent heat flux at ( , Re ) = (0.8, 56) . All contours show the cross-sectional views of the r at the same position and same instant. Left: color contour visualization of ux and the in-plane vectors of (ur , u ) ; right-top: color contour visualization of θ′ and the vectors of (ur , u ) ; right-bottom:

r in + d /2 r in

ur dr . The magnitudes are normalized using the friction temperature and/or the friction velocity. Two thermal fields are

obtained from the same velocity field.

Fig. 15. Same conditions as Fig. 14, but with

the wall-normal flow structures and turbulent heat fluxes that produce a high heat-transfer effect in the transitional regime will be discussed. The heat-transfer characteristics of the helical turbulence and the heattransfer enhancement effect will also be examined. Referring to the Fukagata–Iwamoto–Kasagi (FIK) identity for scalar heat transfer (Fukagata et al., 2002; 2005), the turbulent-contribution term to the mean Nusselt number is known to be the weighted wallnormal turbulent heat flux. Fig. 13 shows the mean profiles of the wall-normal turbulent heat flux, ur . The distributions differ greatly depending on the heating condition, while the η-dependency is not as remarkable. However, because the temperature in this plot is dimensionless based on the frictional temperature of the inner-cylinder, the dependency on the radius

= 0.5 .

ratio for the CTD condition is slightly more apparent. It was determined that the turbulent heat flux monotonically decreases as the Reynolds number decreases at any radius ratio and heating condition. The turbulent heat flux is zero at Re = 50, at which value the flow is laminar. In the following, since the spatial features in the transitional turbulence region are not visible when the turbulence heat flux is spatially averaged over the entire computational domain, the local turbulent-heatflux distributions in the r dimension are considered. Fig. 14 shows the instantaneous temperature fields for the UHF and CTD conditions, in addition to the instantaneous velocity field. Typical aspects of the flow path cross-section are shown under the conditions where helical turbulence is formed, as shown in Fig. 7(j)–(l). For the velocity field, the r flow cross-section is plotted using a cylindrical 8

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

Fig. 16. Contours of instantaneous wall-normal turbulent heat flux in the gap central plane of a raw field, and its high- and low-pass , and ur , ranging from 1 (blue) to + ( , Re ) = (0.8, 56) . From top to bottom, each row indicates respective contour of ur + +, ur threshold is set as a streamwise mode number of mx = 10 . (Left column) UHF, and (right) CTD. Half of the azimuthal direction of L x × (rin + rout )/2 = 51.2d × 14.1d, is visualized at the same position and same instant. The bottom row shows profiles of the wall-normal r in + d /2 r in

(·)dr on the dash-dotted line marked in the contours.

coordinate system based on the actual aspect-ratio (d: rin: rout). The instantaneous fluctuation of the temperature is shown at the same position and time. This temperature field is shown using Cartesian coordinates to allow for comparison with the integrated turbulent heatflux distribution that is shown with respect to the ϑ coordinate. The rdirection integrals of the wall-normal turbulent heat flux, ur , were further calculated for these instantaneous thermal fields, and the distributions were plotted as a function of ϑ. The same in-plane velocity distribution was superimposed on each contour map of θ′. At the instant presented in Fig. 14, an circumferentially-localized turbulent region that is a section of the helical turbulence can be observed at = 7 /8–11π/8 in the figure, wherein fine and seemingly random vortices are generated. On the other hand, no fine eddies are observed around = 2 , and the integrated value of the turbulent heat flux approaches zero. A high turbulent heat-flux integral value is shown at the position where the turbulent region exists under both thermal boundary conditions. Noteworthy quasi-periodic structures occur around = /2 : See the enlarged view in Fig. 14, where the characteristic flow patterns are highlighted by the dotted-line vectors. In this outlined area, a flow in the clockwise direction is observed while it meanders in the gap between the cylinders, and the streamline of this flow in the cross section is snake-like. In addition to this meandering motion, longitudinal vortices exist between the snake-like streamline and cylinder surfaces, and this is also shown in the enlarged view in Fig. 14. Such a well-organized street of vortices should not be turbulent. This is a manifestation of the aforementioned cluster of non-turbulent longitudinal vortices. The vortex cluster is located between the turbulent and the laminar regions. In the region of the vortex cluster, positive and negative fluctuations of θ′ occur regularly in the ϑ direction, as shown in the visualization results of the respective thermal fields. The integral of the turbulent heat-flux in this region is comparable to that in the

filtered fields, at 1 (red). The filter the domain, i.e., integral values of

turbulent region, particularly with the CTD condition. Comparing the UHF and CTD conditions, significant turbulent heat flux in the CTD condition is widely generated due to the longitudinal-vortex cluster and the localized turbulent region. Fig. 15 shows similar plots as in Fig. 14 at the same Reτ, but for a smaller η of 0.5. Because of the moderate radius ratio, the azimuthal space is narrowed with respect to d. In the flow field (Re = 56) where the helical turbulence is formed, three regions of the turbulent part, the laminar part, and the longitudinal-vortex cluster coexist compactly. The longitudinal-vortex cluster is therefore difficult to distinguish in the flow cross-section, unlike in the previous case with = 0.8. Such a structural bias does not occur at a sufficiently high Reynolds number (Re = 150 ), but fine turbulent eddies in the vicinity of both the inner and outer walls are observed (figure not shown). In such a fully-turbulent flow, these fine-scale eddies are much smaller than d and the azimuthal space, and therefore it is considered that there is no dependency on the radius ratio. The distribution of the temperature fluctuation in a low-Reτ turbulent flow, shown on the right-hand side of Fig. 15, however, appears to fluctuate strongly for the CTD condition throughout the entire azimuthal direction. The heat transfer in the laminar region may therefore be promoted by the cluster of longitudinal vortices. Next, to distinguish the turbulent and laminar parts and the vortex clusters in a quantitative manner, we applied a low-pass Fourier filter in the streamwise direction with a filter cut-off at m x = 10 and assumed that the low-pass field contained the vortex clusters and the high-pass field the turbulent parts. The cut-ff threshold of the streamwise mode number, m x = 10, corresponds to the streamwise wavelength of 5d ( L x /m x = 5.12d), which should be larger than typical fine-scale eddies in the turbulent channel flow (Toh and Itano, 2005; Abe et al., 2018). Two-dimensional spectral analysis including the azimuthal component 9

International Journal of Heat and Fluid Flow 82 (2020) 108555

T. Fukuda and T. Tsukahara

conclusion that the non-turbulent longitudinal-vortex clusters produce wall-normal heat fluxes comparable to those of turbulent regions. 4. Conclusion DNSs of the turbulent heat transfer in annular Poiseuille flow using a computational domain with dimensions of 51.2d × d × 2π were carried out, and the heat transfer characteristics of flow exhibiting helical turbulence in the subcritical transition region were investigated. The dependency of this heat transfer on the Reynolds number, radius ratio, and thermal boundary conditions was also examined, and our findings are described below. The instantaneous distributions of temperature fluctuations revealed transition structures similar to those seen in the velocity field under both UHF and CTD conditions. Under the UHF condition, similarity with the velocity field was confirmed, while under the CTD condition longitudinal vortices appeared between the laminar and turbulent regions. Both of the examined radius ratios ( = 0.8 and 0.5) resulted in high heat transfer rates close to those estimated by the empirical turbulence function, even in the transitional regime. Both the UHF and CTD conditions demonstrated the most active heat transfer in the localized turbulence region. In addition, because there is a significant turbulent heat flux in the non-turbulent longitudinal-vortex cluster around the localized turbulence region, heat transfer was even promoted in the intermittent turbulent states. CRediT authorship contribution statement Takehiro Fukuda: Data curation, Formal analysis, Investigation, Software, Validation, Visualization. Takahiro Tsukahara: Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Writing - original draft, Writing review & editing. Fig. 17. Spatially-averaged wall-normal turbulent heat flux for a raw field, and its high- and low-pass filtered fields (with threshold of a streamwise mode number, mx = 10 ), at ( , Re ) = (0.8, 56) . A gray dashed line represents ur + + for the non-filtered field and a purple solid line does the sum of ur , and ur .

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

mϑ also confirmed that m x = 10 is adequate for separating two peaks due to fine-scale eddies and vortex clusters (figure not shown). With this as thresholds, the low-pass filtered velocity fluctuation, represented by u , would mainly contain variations due to vortex clusters, and the would exhibit only turbulent parts. In high-pass filtered field of u the middle panels of Fig. 16, we show the wall-normal turbulent heat flux distributions in each of filtered fields. Some clustered longitudinal vortices are successfully extracted in the low-pass field, while the highpass field clearly distinguishes the laminar and turbulent parts. One may find that the addition of the two fields of ur and ur should coincide with the non-filtered field shown in the top of the figure. As shown in the bottom of the figure by plotting the streamwise distribution at arbitrary ϑ (indicated by the dash-dotted line), a large value can certainly be confirmed at the location corresponding to the vortex cluster in the low-pass field. The contributions of the high-pass field show significant values at the turbulent parts, but are smaller than the contribution of the low-pass field. The trend is similar between both thermal boundary conditions. Finally, Fig. 17 shows the average value of the turbulent heat flux at each filtered field visualized in Fig. 16. The same filter as in Fig. 16 is used here. Since the spatial average covers the entire computational ur ur domain, the simple sum of and should be close to the ensemble-average values that was presented in Fig. 13. It can be seen that, in the low-pass field whose main fluctuating component is due to the vortex cluster, the values are significantly large enough to exceed that in the high-pass field. This statistical result supports our

Acknowlgedgments This work was supported by the Grant-in-Aid for JSPS (Japan Society for the Promotion of Science) Fellowship 16H06066, 16H00813, and 19H02071. Numerical simulations were performed on SX-ACE supercomputers at the Cybermedia Centre of Osaka University and the Cyberscience Centre of Tohoku University. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.ijheatfluidflow.2020.108555 . References Abe, H., Antonia, R.A., Toh, S., 2018. Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit. J. Fluid Mech. 850, 733–768. Abe, H., Kawamura, H., Matsuo, Y., 2004. Surface heat-flux fluctuations in a turbulent channel flow up to Reτ = 1020 with Pr = 0.025 and 0.71. Int. J. Heat Fluid Flow 25 (3), 404–419. Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D., Hof, B., 2011. The onset of turbulence in pipe flow. Science 333 (6039), 192–196. Barkley, D., Song, B., Mukund, V., Lemoult, G., Avila, M., Hof, B., 2015. The rise of fully turbulent flow. Nature 526 (7574), 550–553. Brethouwer, G., Duguet, Y., Schlatter, P., 2012. Turbulent-laminar coexistence in wall flows with Coriolis, buoyancy or Lorentz forces. J. Fluid Mech. 704, 137–172. Chantry, M., Tuckerman, L.S., Barkley, D., 2017. Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1. Chung, S.Y., Rhee, G.H., Sung, H.J., 2002. Direct numerical simulation of turbulent

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