- Email: [email protected]
Contribution of large-scale motions to the Reynolds shear stress in turbulent pipe flows
International Journal of Heat and Fluid Flow 66 (2017) 209–216
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Contribution of large-scale motions to the Reynolds shear stress in turbulent pipe flows Junsun Ahn1, Jinyoung Lee1, Hyung Jin Sung∗ Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea
a r t i c l e
i n f o
Article history: Received 20 January 2017 Revised 26 May 2017 Accepted 21 June 2017 Available online 29 June 2017 Keywords: Mean velocity Reynolds shear stress Turbulent boundary layer Turbulent pipe flow
a b s t r a c t Direct numerical simulation data for turbulent pipe flows with Reτ = 544, 934, and 3008 were used to investigate the contribution of large-scale motions (LSMs) to the Reynolds shear stress. The relationship be+ tween viscous force (d2U + /dy+2 ,VF) and turbulent inertia (d−u v /dy+ ,TI) results in a transition from the inner length scale to the intermediate length scale in the meso-layer. The acceleration force of the LSMs is balanced by the deceleration force of the small-scale motions (SSMs), which makes the zero TI at the wall-normal location of the maximum Reynolds shear stress (ym + ). As the Reynolds number increases, the enhanced acceleration force of the LSMs expands the nearly zero TI region. The constantstress layer is formed in the neighborhood of the zero TI, having the relatively strong VF. For sufficiently high Reynolds number flows, the log law is established beyond the meso-layer due to the fact that VF loses its leading order. The role of the LSMs in the wall-scaling behavior of ym + is examined. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Turbulence structures and their contributions to turbulence statistics came to prominence decades ago (Adrian, 2007). The bulk motions of large-scale structures with the streamwise extent ∼O(R), where R is the pipe radius, can be divided into large- and very-large-scale motions (LSMs and VLSMs) (Adrian et al., 20 0 0; Kim and Adrian, 1999). These motions enhance momentum transfer through ejection and sweeping, which provide the dominant contributions to the Reynolds shear stress (−u v ) (Robinson, 1991). LSMs with a streamwise extent of 2−3R and VLSMs with streamwise extents up to 20R contribute 25% and 50−60% of −u v , respectively (Ganapathisubramani et al., 2003; Guala et al., 2006; Monty et al., 2007; Lee and Sung, 2011). In energy terms, the LSMs, including the VLSMs, account for a large proportion of the Reynolds shear stress, but this observation does not directly explain their statistical scaling; further insight into the link between the energetic LSMs and the Reynolds shear stress is required. In general, the mean velocity distribution of a wall-bounded turbulent flow can be divided into two layers: the inner and outer layers (Tennekes and Lumley, 1972). The viscous force dominates near the wall and vanishes far from the wall. The inner layer can be characterized in terms of the inner length scale ν /uτ , where ν is the kinematic viscosity and uτ is the friction velocity. The outer ∗
1
Corresponding author. E-mail address: [email protected] (H.J. Sung). Both authors contributed equally to this work.
http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.06.009 0142-727X/© 2017 Elsevier Inc. All rights reserved.
layer can be scaled with the shear layer thickness, i.e., the pipe radius (R). In contrast to the different length scales, the friction velocity can be used as a representative variable for both layers. The overlap layer is located where the inner and outer layers are overlapped. In the overlap layer, it has been known that the mean velocity (U) grows logarithmically with respect to the wall-normal distance in the overlap layer according to what is known as the log law: U+ = 1/κ log(y+ ) + B, where κ is the von Kármán constant and B is an additive constant. The superscript + refers to inner scaling, for example, y+ = yuτ /ν and U+ = U/uτ are the length and velocity respectively. The log law is widely applied for the description of wall-bounded turbulent flows. The log layer in the mean velocity is known as the inertial sublayer because the local viscous effect is negligible compared to the Reynolds shear stress. The log layer becomes evident for sufficiently high Reynolds number flows, such as Reτ (≡ uτ R/ν ) >50 0 0 (McKeon et al., 2004; Marusic et al., 2013). For lower Reynolds numbers (Reτ < 50 0 0), the viscous force is not negligible in the overlap layer; the power law, U+ = C(y+ )γ (where C and γ are constants independent of the Reynolds numbers), is then used instead of the log law (McKeon et al., 2004; Ahn et al., 2015). Long and Chen (1981) discussed the absence of the log law in terms of the significance of the inner and outer length scales in the overlap layer. They instead proposed the concept of the meso-layer, which is different from the classical inertial sublayer. Sreenivasan and Sahay (1997) showed that the viscous force has a persistent effect in the overlap layer; the viscous force is signifi-
210
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
cant in the neighborhood of the wall-normal location of the maximum Reynolds shear stress (ym ), and the square-root scaling ym + ∼ O(Reτ 1/2 ) is obtained empirically. This scaling can be also derived analytically from the streamwise mean momentum equation by assuming the log law at ym . Wei et al. (2005) described a four-layer structure in terms of the ratio of the gradient of the viscous stress + (d2U + /dy+2 ) to the gradient of the shear stress (d−u v /dy+ ), and showed that the neighborhood of ym moves outwards following a scaling of y+ ∼ O(Reτ 1/2 ). Chin et al. (2014b) confirmed that the onset of the log layer follows the same scaling. Afzal (1982) also analytically reasoned an intermediate length scale y+ ∼ O(Reτ 1/2 ) in the meso-layer. However, this length scale applies to a three-layer structure (consisting of inner, meso-, and outer layers) with two overlap layers and uses asymptotic expansions, in contrast to the two-layer concept. As mentioned above, the mean velocity and the Reynolds shear stress are coupled in the streamwise mean momentum equation, +
0=
1 d2U + d−u v + + , Reτ d y+ d y+ 2 PG
VF
(1)
TI
where the inverse of the friction Reynolds number, the gradient of the viscous stress, and the gradient of the Reynolds shear stress correspond to the pressure gradient (PG), the viscous force (VF), and the turbulent inertia (TI), respectively. PG is fixed by the friction Reynolds number (Reτ ), so VF is balanced by TI. By utilizing this relation, VF can be indirectly predicted by examining the spectra of TI. Guala et al. (2006) and Wu et al. (2012) showed that LSMs produce an acceleration in the mean flow up to y/R < 0.2 (which is close to the upper-bound of the overlap layer). Chin et al. (2014b) divided the TI spectrum according to the velocity–vorticity correlations. The acceleration force of the LSMs has been attributed to the correlation between the spanwise velocity and the wallnormal vorticity fluctuations (vorticity-stretching effect), which results in the change in the length scales (Tennekes and Lumley, 1972). Hwang et al. (2016) showed that the acceleration forces of LSMs in turbulent pipe flows are weaker than those in turbulent channel flows. The objective of the present study was to determine the contribution of LSMs to the Reynolds shear stress in turbulent pipe flows. Direct numerical simulation (DNS) data for turbulent pipe flows with Reτ = 544, 934, and 3008 were employed. Since VF and TI are closely coupled in the streamwise mean momentum equation, the TI can be used to monitor the behavior of the mean flow. The variations in the length scales below the overlap layer were explored by using the ratio of VF to TI. The power and log laws of the mean velocity in the overlap layer were examined by using their indicator functions. The pre-multiplied streamwise spectra of the Reynolds shear stress and the TI were obtained to determine the contribution of the LSMs to TI. The scale separations of the Reynolds shear stress and TI were used to show the development of the mean flow for higher Reynolds number flows. Finally, the growing order of the wall-normal location of the maximum Reynolds shear stress was explained in terms of the contributions of LSMs and small-scale motions (SSMs).
Crank–Nicolson and central finite-difference schemes, respectively (Park and Sung, 1995). The fully implicit fractional step method was employed to decouple the velocity and pressure (Kim et al., 2002). No-slip conditions at the wall were adopted, with periodic conditions along the axial and azimuthal directions. The axial domain length was 30 times the pipe radius (R), which was sufficiently long to describe multi-scale structures ranging from small to very large (Monty et al., 2007; Lee et al., 2015). The axial, radial, and azimuthal directions were represented by x, r, and θ , respectively, and the corresponding velocity components were represented by ux , ur , and uθ , respectively. Throughout this study, the expressions in the cylindrical coordinates have been transformed into the Cartesian coordinates to facilitate comparisons with other geometric flows: the streamwise direction is indicated by x, the wall-normal direction by y = 1 – r, the spanwise direction by z = rθ , and the corresponding velocity components are u = ux , v = −ur , and w = uθ (Wu et al., 2012). Small letters with a prime symbol are velocity fluctuations (e.g.,u ), and averaged quantities are denoted by a capital letter or with brackets (e.g., U or u u ). The details of the DNS simulations can be found in Lee and Sung (2013), Ahn et al. (2013) and Ahn et al. (2015). The numerical parameters are summarized in Table 1. Fig. 1 shows the profiles of the Reynolds stresses. All quantities increase along the wall-normal distance as the Reynolds number increases. For sufficiently high Reynolds numbers, it has been known that the Reynolds stresses of the wall-parallel fluctuations (u u + ,w w + ) behave logarithmically, and the wallnormal Reynolds stresses (v v + ) are constant in the overlap layer (Townsend, 1976). The profiles of u u + do not exhibit logarithmic behavior in the overlap layer because the Reynolds numbers are not sufficiently high. Instead, a plateau-like shoulder is evident at Reτ = 3008. For sufficiently high Reynolds numbers, a logarithmic trend is expected beyond the plateau region (Marusic et al., 2013). The profiles of w w + for Reτ = 934 and 3008 exhibit logarithmic behavior in the overlap layer, although the logarithmic constant (B3 ) could vary with the Reynolds number (Sillero et al., 2013; Chin et al., 2014a; Lee and Moser, 2015). The constant was calcu+ lated from the log law indicator function (y+ ∂ w w /∂ y+ ). The + profile of v v for Reτ = 3008 seems to be flat and to reach the value of 1.3 (Zhao and Smits, 2007). The quality of the present DNS data was validated by examining the inner peaks in the streamwise Reynolds stress, as shown in Fig. 1(d). The recent available DNS data of turbulent pipe flow (Wu and Moin, 2008; Wu et al., 2012; El Khoury et al., 2013; Chin et al., 2014a) were used to assess the present ones. Based on the present DNS data, the logarithmic growth of the inner peaks with the Reynolds number was obtained as u u + = 0.6875 log(Reτ ) + 3.2391 with a residue innerpeak of 0.9997 (Örlü and Alfredsson, 2013). This trend is in good agreement with other DNS data within a 3% tolerance limit. The slight deviations could be due to differences in the numerical conditions, such as the streamwise domain length, the grid resolution, and the numerical method (Schlatter and Örlü, 2010; Ahn et al., 2013).
3. Results and discussion 3.1. Mean velocity and Reynolds shear stress
2. Numerical data Previously obtained DNS data for turbulent pipe flows with Reτ = 544, 934, and 3008 were employed (Lee and Sung, 2013; Ahn et al., 2013; Ahn et al., 2015). In these studies, the Navier– Stokes and continuity equations were solved in the cylindrical coordinates to describe incompressible and fully developed turbulent pipe flows. The governing equations were spatially and temporally discretized over a staggered grid by using the second-order
Fig. 2(a) shows the profiles of the mean velocity (U+ ) and the Reynolds shear stress (−u v + ). The profiles of U+ are similar in the viscous sublayer and the buffer layer, where y+ is less than 30. Above the buffer layer (y+ > 30), the profile stretches upward as the Reynolds number increases. The profiles of −u v + collapse well very near the wall. As the Reynolds number increases, the maximum value in the overlap layer increases and the corresponding constant region is expanded; however, it is too short to
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
211
Table 1 Numerical parameters. ReD
Reτ
(Nx , Nr , Nθ )
x+
r+ min
r+ max
(Rθ )+
tUc /R
19,0 0 0 35,0 0 0 133,0 0 0
544 934 3008
(2049, 279, 513) (4097, 301, 1025) (12,289, 901, 3073)
8.02 6.84 7.34
0.176 0.334 0.36
4.284 9.244 9.91
6.72 5.73 6.15
0.02 0.01 0.003
Fig. 1. Profiles of the (a) streamwise, (b) wall-normal, and (c) spanwise Reynolds stresses. The log law in the spanwise Reynolds stress is evident for Reτ = 934 and 3008. (d) The inner peaks in u u + .
Fig. 2. Profiles of (a) the mean velocity and the Reynolds shear stress and (b) VF and TI.
be determined as the constant-stress layer proposed by Townsend (1976). In Section 3.2, the formation mechanism of the constantstress layer will be explored in terms of the contribution of the LSMs. Fig. 2(b) shows the profiles of VF and TI. The inset shows a magnified view of the wall-normal region (20 < y+ < 200). As the Reynolds number increases, VF is almost invariant with respect to the Reynolds number except for the small increase in its peak value for y+ less than ≈7. Above y+ = 100, the value of VF diminishes and converges to nearly zero along the wall-normal distance. As the Reynolds number increases, the value of TI decreases, and its zero-crossing point is shifted to the core region. The zerocrossing points of TI (ym + ) are ym + = 43, 52 and 81 for Reτ = 544,
934, and 3008, respectively. For Reτ = 3008, TI becomes zero and has a similar level to VF above y+ = 100. Fig. 3(a) shows the ratio of VF to TI (VF/TI). The framework of the four-layer structure is employed to interpret the present results (Wei et al., 2005). VF is relatively dominant compared to TI very near the wall (0 < y+ < 3). A counterbalance between VF and TI (VF/TI = −1) is established and confined within the buffer layer (3 ≤ y+ ≤ 30). The inset shows their results for the turbulent channel flow up to Reτ ≈ 5200 (Lee and Moser, 2015), which are similar to the present results. In the vicinity of −u v + , TI is almost peak
zero (TI ≈ 0). Although the magnitude of VF gradually decreases above y+ = 30 in Fig. 2(b), it regains its leading order in the region
212
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Fig. 3. (a) Ratio of VF to TI; (b) Wall-normal locations for VF/TI = −2, −u v + , and VF/TI = 0.5. peak
Fig. 4. Profiles of the (a) power law and (b) log law indicator functions.
y+ |VF/TI = −2 < y+ < y+ |VF/TI = 0.5 because TI is negligible near ym + . The sign change of TI at ym + leads to an infinite value of VF/TI. Above y+ |VF/TI = 0.5 , VF/TI becomes zero since VF loses its leading order and an inertial sublayer then emerges. Fig. 3(b) shows the wall-normal locations for VF/TI = −2, −u v +peak , and VF/TI = 0.5. The growth of each wall-normal location was scaled using the present DNS data of turbulent pipe flow:y+ |VF/TI = −2 = 4.71Reτ 0.318 , y+ |−u v = 4.13Reτ 0.372 , and peak
y+ |VF/TI=0.5 = 3.31Reτ 0.464 . DNS data of turbulent channel flow was included for comparison (Lee and Moser, 2015). The present scaling results show the deviations from the scaling y+ ∼ O(Reτ 1/2 ), which was obtained with the four-layer structure (Wei et al., 2005). As the wall-normal distance increases, the scaling order of the growth varies from Reτ 0.318 (≈ Reτ 1/3 ) to Reτ 0.464 (≈ Reτ 1/2 ) via Reτ 0.372 in the region y+ |VF/TI = −2 < y+ < y+ |VF/TI = 0.5 . The scaling order y+ ∼ O(Reτ 1/2 ) implies that the wall-normal location (y) can be normalized by the equal contribution of the inner and outer length scales (ν /uτ and R). For example, if the scaling order is y+ ∼ O(Reτ 1/3 ), y is normalized by (ν /uτ )2/3 R1/3 due to the strong VF. As the wall-normal distance increases beyond the buffer layer, a characteristic length scale develops from the inner length scale to the intermediate length scale due to the diminishment of VF. The region O(Reτ 0 ) < y+ < O(Reτ 1/2 ) is thus interpreted as the meso-layer (ym + ) (Sreenivasan and Sahay, 1997; Wei et al., 2005), to connect the buffer layer with the inertial sublayer. Although VF becomes weak in the meso-layer, VF is still stronger than TI. The behavior of the mean velocity in the meso-layer and the inertial sublayer was examined by the power law and log law indicator functions ( and ) shown in Fig. 4. The power and log law indicator functions are determined as = y+ /U+ ∂ U+ /∂ y+ and = y+ ∂ U+ /∂ y+ respectively. The profiles of the power law indicator function in Fig. 4(a) are similar up to y+ ≈ 80. The profiles for the lower Reynolds numbers (Reτ = 544 and 934) dis-
play gradual changes beyond the local minimum (γ ≈ 0.145) at y+ ≈ 80. The profile for Reτ = 3008 shows a plateau beyond the local minimum with γ = 0.145 in the region 90 < y+ < 150, and starts to sag down beyond y+ > 150. The meso-layer starts from y+ = 30 to y+ = 150. Since the end of the meso-layer can be scaled by y+ = 150 ≈ 3Reτ 1/2 , the transition from the inner length scale (y+ ∼ O(ν /uτ )) to the intermediate length scale (y+ ∼ O(Reτ 1/2 )) occurs, which is also observed in Fig. 3(b). The region of the mesolayer (30 < y+ < 150) is also identical to that of the critical layer (1/2 ≤ U/Uc ≤ 2/3), where Uc is the turbulent centerline velocity (Sharma and McKeon, 2013). The profiles of the log law indicator function in Fig. 4(b) are in good agreement up to y+ ≈ 40 from the wall. Beyond the local minimum at y+ ≈ 60, the profiles increase with increasing the wall-normal distance. A plateau (=1/κ ) could not be clearly identified in the overlap layer for any of the profiles. For turbulent channel flows (Reτ > 40 0 0), the profiles of the log law indicator function can be divided into two inclinations: steeply inclined or less steeply inclined, and the latter reaches a constant value of 0.38 (Lozano-Durán and Jiménez, 2014; Lee and Moser, 2015). As is the case for channel flows, the profile of Reτ = 3008 shows that the inclinations slightly change at y+ = 150, i.e., an infant log law. The log law becomes evident for sufficiently high Reynolds numbers. The corresponding value of the log law indicator function at y+ = 150 is κ = 0.392, which is slightly higher than the value for channel flows (Lee and Moser, 2015). This value is similar to the result of 0.4 ± 0.02 obtained for the Superpipe (Bailey et al., 2014), and to the value 0.39 obtained for a wall-bounded canonical turbulent flow with 2 × 104 < Reτ < 6 × 105 (Marusic et al., 2013). The present indicator functions show the successive wall-normal alignment of the power law and the infant log law along the wall-normal distance (McKeon et al., 2004). For Reτ = 3008, y+ = 150 can be expressed by not only as y+ ≈ 3Reτ 1/2 but also as y+ ≈ 10Reτ 1/3 . The
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
213
Fig. 5. Pre-multiplied streamwise spectra of (a) the Reynolds shear stress and (b) TI, with black zero-contour lines. The magenta lines in (b) are the wall-normal locations of the maximum Reynolds shear stress (ym + ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
first formulation (square-root scaling) was obtained by Marusic et al. (2013) with the independence of both the inner and the outer length scales. The second formulation (cubic-root scaling) was obtained by Tennekes and Lumley (1972), who employed a secondorder correction to the pipe flow due to the pressure gradient. 3.2. Spectra of the Reynolds shear stress and the turbulent inertia Fig. 5 shows the pre-multiplied streamwise spectra of −u v and TI. The spectrum of −u v was calculated as the Fourier transform of the two-point correlations, which are defined √ ∞ by −u v (kx , y ) ≡ 2Re{ 21π −∞ e− −1kx rx −u (x, y )v (x + rx , y )drx }, where rx is the spatial shift along the streamwise direction (Wu +
v) et al., 2012). The spectrum of TI is given by T I = ∂ (−u = ∂ y+ ∞ ∂ +−u v ∞ kx ∂ +−u v 0 ∂ y + dk x = 0 ∂ y+ d log(kx ). The streamwise wavenumber kx was pre-multiplied due to the logarithmic scaling of kx , and λx is the streamwise wavelength. The pre-multiplied streamwise spectra of the Reynolds shear stress (kx + ) are similar in the re−u v gion y+ < 30. The inner sites with the maximum energy are located at y+ = 30. The outer site for Reτ = 3008 is located at y/R ≈ 0.18 (Ahn et al., 2015). The pre-multiplied streamwise spectra of TI ( k x ∂ + /∂ y+ ) show that the bulk motion accelerates the mean −u v flow within the buffer layer (y+ < 30) (region A). For values beyond the inner site (y+ > 30), the TI spectra behave differently for different Reynolds numbers. In Fig. 5(b), the region C is bounded by λx + ≈ 30 0 0 for all three Reynolds numbers. As the Reynolds num-
ber increases (Reτ = 544, 934, 3008), the regions D and E are extended to λx + ≈ 30 0 0. A cut-off wavelength of λx + = 30 0 0 was employed to demarcate the LSMs and SSMs (Chin et al., 2014b; Ahn et al., 2015). At this wavelength, the LSMs (λx + > 30 0 0) induce the acceleration force in the overlap layer, and the sign change (region C/D) in the net force is clear (Chin et al., 2014b). The force due to the SSMs (λx + < 30 0 0) changes from acceleration to deceleration (region C), whereas the LSMs (λx + > 30 0 0) maintain their acceleration force (region D & E). The small-scale influence of λx + < 300 kept the acceleration force up to y+ = 60 (region B). In the region 30 < y+ < 60, the forces are ordered as acceleration-decelerationacceleration in the regions B→C→D. Away from the wall (y+ > 60), the acceleration force of the LSMs extends up to y/R ≈ 0.2 as the Reynolds number increases (region E) (Guala et al., 2006; Wu et al., 2012; Chin et al., 2014b; Hwang et al., 2016). As the Reynolds number increases, the energy of the LSMs becomes more intense, and they penetrate into the outer layer, although the SSMs remain unchanged. The wall-normal location of the maximum Reynolds shear stress (ym + ) increases in accordance with the increased contribution of the LSMs. The distinct border at λx + = 30 0 0 between the acceleration force of the LSMs and the deceleration force of the SSMs (region C/D) continues up to y+ ≈ 150. This location is consistent with the onset of the infant log layer for Reτ = 3008 (y+ ≈ 150 ≈ 3.9Reτ 1/2 ) shown in Fig. 3(a) (Marusic et al., 2013). Fig. 6 shows the scale separation of −u v + with a cut-off wavelength of λx + = 30 0 0 to visualize the contributions of the LSMs and SSMs. The scale separation was performed by integrat-
214
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Fig. 6. Scale separation of the Reynolds shear stress for Reτ = 544, 934, and 3008 and a cut-off wavelength of λx + = 30 0 0 in the (a) inner and (b) outer coordinates. The LSM and SSM profiles are shown as thick and thin solid lines respectively.
Fig. 7. Scale separation of TI for a cut-off wavelength of λx + = 30 0 0 and Reτ values of (a) 544, (b) 934, and (c) 3008. (d) The TI profiles for LSMs (thick) and SSMs (thin) are shown together. The vertical dashed line and the circle symbols indicate the zero-crossing points of the SSMs and LSMs respectively.
ing kx + across the entire wavelength range. The contributions −u v of the SSMs to −u v + are similar for the three Reynolds numbers in the region y+ < 200, and the profiles have maxima at yi + ≈ 40. The similarity in the profiles of the SSMs also represents the validity of the cut-off wavelength λx + = 30 0 0 (Chin et al., 2014b). The contributions of the LSMs to −u v + grow in magnitude in the region y+ > 20 as the Reynolds number increases, and the profiles have maxima at yo /R ≈ 0.18 where the outer site for Reτ = 3008 is located (Ahn et al., 2015). The constant-stress layer could be established in the region between yi + ≈ 40 and yo /R ≈ 0.18. Since the contribution of the SSMs to −u v + is independent of the Reynolds number, the presence of the constant-stress layer is affected by the contribution of the LSMs at higher Reynolds numbers (Ahn et al., 2015). The scale separation of TI is shown in Fig. 7. The profiles of the SSMs are similar for the three Reynolds numbers, and the zerocrossing point of TI for the SSMs (yi ) is located at yi + ≈ 40 in Fig. 7(d). As the Reynolds number increases, the TI of the LSMs becomes strong with the positive values. The zero-crossing point of
TI of the LSMs is located at yo /R ≈ 0.18, which is independent of the Reynolds number. Since the profiles of the SSMs are similar, the wall-normal location of the zero TI is influenced by the contribution of the LSMs. The enhanced acceleration force of the LSMs cancels out the deceleration force of the SSMs, and the counterbalanced region (TI ≈ 0) between the two opposing forces is extended. The inner and outer length-scaled structures contribute equally to the formation of the zero TI. In the neighborhood of the zero TI, the low TI means that VF is dominant. As a result, a meso-layer occurs with the power law in the mean velocity (Fig. 4a). The log law is not evident in the present results for various Reynolds numbers. According to theory, the constant-stress layer coincides with the log layer (Townsend, 1976). Since the zero TI at ym results in the dominance of VF, a meso-layer forms in the neighborhood of ym . Above the meso-layer, VF is smaller than TI, although TI is almost zero. The log layer is located in this region. This behavior is clear at higher Reynolds numbers such as Reτ > 50 0 0 (McKeon et al., 2004). In particular, the constantstress layer is the meso-layer (VF > TI ≈ 0), and the extension of the
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
215
Fig. 8. (a) The wall-normal locations of the maximum Reynolds shear stress (ym + ) with the scaling lines; (b) Variations with the Reynolds shear stress in the area fractions of the LSMs and SSMs.
constant-stress layer (TI ≈ 0 > VF) above the meso-layer becomes the log layer. As mentioned above, for three Reynolds numbers we investigated, the maximum contributions of the SSMs and LSMs to −u v + occur at yi + ≈ 40 and yo /R ≈ 0.18 respectively. The wall-normal location of the maximum Reynolds shear stress (ym ) can be expressed in terms of a combination of yi and yo , ym + ∼ (yi + )a (yo /R)b . Fig. 8(a) shows the scaling we observed, ym + = 4.13Reτ 0.372 , and the classical scaling ym + = 1.6∼2.0Reτ 1/2 as well as the numerical and experimental data (McKeon et al., 2004; Buschmann et al., 2009; Hultmark et al., 2013; Chin et al., 2014b). The trend ym + ∼ O(Reτ 0.372 ) is suitable for lower Reynolds numbers, i.e., Reτ < 50 0 0, however, the data for higher Reynolds numbers, Reτ > 50 0 0, follow the scaling trend ym + ∼ O(Reτ 1/2 ). This weak power behavior arises because the LSMs are less active for Reτ < 50 0 0. The scaling ym + ∼ O(Reτ 1/2 ) arises at higher Reynolds numbers (Reτ > 50 0 0). To show the contribution of the LSMs to the scaling ym + ∼ O(Reτ 1/2 ), the present scaling order ym + ∼ O(Reτ 0.372 ) was amended by using the variations in the contributions of the LSMs and SSMs. Fig. 8(b) shows the variations in the area fractions (AFs) of the LSMs and SSMs (AFLSM and AFSSM ) for −u v + , which were used as weighting factors in the present scaling order. As the Reynolds number increases, AFLSM increases and AFSSM decreases, i.e., the contributions of the LSMs and SSMs increase and decrease, respectively. The growth orders of AFLSM and AFSSM are AFLSM and AFSSM ∼ O(Reτ 0.590 ) and O(Reτ −0.451 ), respectively. A simple multiplication of the area fraction was performed to obtain asymptotic behavior: ym + ∼ O(Reτ 0.372 ) × AFLSM × AFSSM ∼ O(Reτ 0.372 × Reτ 0.590 × Reτ −0.451 ) ∼ O(Reτ 0.511 ). This behavior is similar to that of ym + ∼ O(Reτ 1/2 ). 4. Conclusions DNS data for turbulent pipe flows with Reτ = 544, 934, and 3008 were used to examine the contribution of LSMs to the Reynolds shear stress. The relationship between viscous force (VF) and turbulent inertia (TI) resulted in a transition from the inner length scale (y+ ∼ O(ν /uτ )) to the intermediate length scale (y+ ∼ O(Reτ 1/2 )) in the meso-layer. In the pre-multiplied streamwise spectra of TI, as the Reynolds number increases the LSMs (λx + > 30 0 0) become more intense so they penetrate into the outer layer, although the SSMs (λx + < 30 0 0) are unchanged. Since the contribution of the SSMs is similar for the three Reynolds numbers, the presence of a constant-stress layer is affected by the active contribution of the LSMs. The acceleration force of the LSMs cancels out the deceleration force of the SSMs, implying that the inner and outer length-scaled structures contribute equally to form the zero TI at ym + . As the Reynolds number increases, ym + is shifted to the core region, and the enhanced acceleration force of the LSMs
expands the nearly zero TI region. In the neighborhood of the zero TI, the low level of TI results in the dominance of VF. As a result, a meso-layer occurs in the nearly zero TI region. Above the meso-layer, VF loses its leading order and the log layer is established. This behavior becomes evident at higher Reynolds numbers. Therefore, the constant-stress layer is the meso-layer (VF > TI ≈ 0), and the extension of the constant-stress layer (TI ≈ 0 > VF) above the meso-layer is the log layer. The scaling ym + ∼ O(Reτ 1/2 ) was examined by considering the area fractions of LSMs and SSMs for −u v + . Acknowledgments This work was supported by the Creative Research Initiatives (No. 2017-013369) program of the National Research Foundation of Korea (MSIP) and the Supercomputing Center (KISTI). References Adrian, R.J., Meinhart, C.D., Tomkins, C.D., 20 0 0. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54. Adrian, R.J., 2007. Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301. Afzal, N., 1982. Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355–377. Ahn, J., Lee, J.H., Jang, S.J., Sung, H.J., 2013. Direct numerical simulations of fully developed turbulent pipe flows for Reτ = 180, 544, and 934. Int. J. Heat Fluid Flow 44, 222–228. Ahn, J., Lee, J.H., Lee, J., Kang, J.-H., Sung, H.J., 2015. Direct numerical simulation of a 30R long turbulent pipe flow at Reτ =3008. Phys. Fluids 27 (6), 065110. Bailey, S.C.C., Vallikivi, M., Hultmark, M., Smits, A.J., 2014. Estimating the value of von Kármán’s constant in turbulent pipe flow. J. Fluid Mech. 749, 79–98. Buschmann, M.H., Indinger, T., Gad-el-Hak, M., 2009. Near-wall behavior of turbulent wall-bounded flows. Int. J. Heat Fluid Flow 30, 993–1006. Chin, C., Monty, J.P., Ooi, A., 2014a. Reynolds number effects in DNS of pipe flow and comparison with channels and boundary layers. Int J. Heat Fluid Flow 45, 33–40. Chin, C., Philip, J., Klewicki, J., Ooi, A., Marusic, I., 2014b. Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows. J. Fluid Mech. 757, 747–769. El Khoury, G.K., Schlatter, P., Noorani, A., Fischer, P.F., Brethouwer, G., Johansson, A.V., 2013. Direct numerical simulation of turbulent pipe flow at moderately high Reynolds numbers. Flow Turbul. Combust. 91, 475–495. Ganapathisubramani, B., Longmire, E.K., Marusic, I., 2003. Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46. Guala, M., Hommema, S.E., Adrian, R.J., 2006. Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–542. Hultmark, M., Vallikivi, M., Bailey, S.C.C., Smits, A.J., 2013. Logarithmic scaling of turbulence in smooth and rough-wall pipe flow. J. Fluid Mech. 728, 376–395. Hwang, J., Lee, J., Sung, H.J., 2016. Influence of large-scale accelerating motions on turbulent pipe and channel flows. J. Fluid Mech. 804, 420–441. Kim, K.C., Adrian, R.J., 1999. Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422. Kim, K., Baek, S.-J., Sung, H.J., 2002. An implicit velocity decoupling procedure for the incompressible Navier–Stokes equations. Int. J. Numer. Method Fluids 38, 125–138. Lee, J.H., Sung, H.J., 2011. Very-large-scale motions in a turbulent boundary layer. J. Fluid Mech. 673, 80–120.
216
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Lee, J.H., Sung, H.J., 2013. Comparison of very-large-scale motions of turbulent pipe and boundary layer simulations. Phys. Fluids 25 (4), 045103. Lee, J., Ahn, J., Sung, H.J., 2015. Comparison of large- and very-large-scale motions in turbulent pipe and channel flows. Phys. Fluids 27 (2), 025101. Lee, M., Moser, R.D., 2015. Direct numerical simulation of turbulent channel flow up to Reτ ≈5200. J. Fluid Mech. 774, 395–415. Long, R.R., Chen, T.C., 1981. Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59. Lozano-Durán, A., Jiménez, J., 2014. Effect of the computational domain on direct simulations of turbulent channels up to Reτ =4200. Phys. Fluids 26 (1), 011702. Marusic, I., Monty, J.P., Hultmark, M., Smits, A.J., 2013. On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3. McKeon, B.J., Li, J., Jiang, W., Morrison, J.F., Smits, A.J., 2004. Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147. Monty, J.P., Stewart, J.A., Williams, R.C., Chong, M.S., 2007. Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156. Örlü, R., Alfredsson, H., 2013. Comment on the scaling of the near-wall streamwise variance peak in turbulent pipe flows. Exp. Fluids 54, 1431–1435. Park, T.S., Sung, H.J., 1995. A nonlinear low-Reynolds-number k-ε model for turbulent separated and reattaching flows - (I) Flow field computations. Int. J. Heat Mass Transfer 38 (14), 2657–2666. Robinson, S.K., 1991. Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.
Schlatter, P., Örlü, R., 2010. Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126. Sharma, A.S., McKeon, B.J., 2013. On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238. Sillero, J.A., Jiménez, J., Moser, R.D., 2013. One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to δ + ≈20 0 0. Phys. Fluids 25 (10), 105102. Sreenivasan, K.R., Sahay, A., 1997. The persistence of viscous effects in the overlap region and the mean velocity in turbulent pipe and channel flows. In: Panton, R. (Ed.), Self-Sustaining Mechanisms of Wall Turbulence. Comp. Mech. Publ, pp. 253–272. Tennekes, H., Lumley, J.L., 1972. A First Course in Turbulence. MIT Press. Townsend, A.A., 1976. The Structure of Turbulent Shear Flow, 2. Cambridge University Press. Wei, T., Fife, P., Klewicki, J., McMurtry, P., 2005. Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327. Wu, X., Moin, P., 2008. A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112. Wu, X., Baltzer, J.R., Adrian, R.J., 2012. Direct numerical simulation of a 30R long turbulent pipe flow at R+ =685: large- and very large-scale motions. J. Fluid Mech. 698, 235–281. Zhao, R., Smits, A.J., 2007. Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow. J. Fluid Mech. 576, 457–473.
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Contribution of large-scale motions to the Reynolds shear stress in turbulent pipe flows Junsun Ahn1, Jinyoung Lee1, Hyung Jin Sung∗ Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea
a r t i c l e
i n f o
Article history: Received 20 January 2017 Revised 26 May 2017 Accepted 21 June 2017 Available online 29 June 2017 Keywords: Mean velocity Reynolds shear stress Turbulent boundary layer Turbulent pipe flow
a b s t r a c t Direct numerical simulation data for turbulent pipe flows with Reτ = 544, 934, and 3008 were used to investigate the contribution of large-scale motions (LSMs) to the Reynolds shear stress. The relationship be+ tween viscous force (d2U + /dy+2 ,VF) and turbulent inertia (d−u v /dy+ ,TI) results in a transition from the inner length scale to the intermediate length scale in the meso-layer. The acceleration force of the LSMs is balanced by the deceleration force of the small-scale motions (SSMs), which makes the zero TI at the wall-normal location of the maximum Reynolds shear stress (ym + ). As the Reynolds number increases, the enhanced acceleration force of the LSMs expands the nearly zero TI region. The constantstress layer is formed in the neighborhood of the zero TI, having the relatively strong VF. For sufficiently high Reynolds number flows, the log law is established beyond the meso-layer due to the fact that VF loses its leading order. The role of the LSMs in the wall-scaling behavior of ym + is examined. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Turbulence structures and their contributions to turbulence statistics came to prominence decades ago (Adrian, 2007). The bulk motions of large-scale structures with the streamwise extent ∼O(R), where R is the pipe radius, can be divided into large- and very-large-scale motions (LSMs and VLSMs) (Adrian et al., 20 0 0; Kim and Adrian, 1999). These motions enhance momentum transfer through ejection and sweeping, which provide the dominant contributions to the Reynolds shear stress (−u v ) (Robinson, 1991). LSMs with a streamwise extent of 2−3R and VLSMs with streamwise extents up to 20R contribute 25% and 50−60% of −u v , respectively (Ganapathisubramani et al., 2003; Guala et al., 2006; Monty et al., 2007; Lee and Sung, 2011). In energy terms, the LSMs, including the VLSMs, account for a large proportion of the Reynolds shear stress, but this observation does not directly explain their statistical scaling; further insight into the link between the energetic LSMs and the Reynolds shear stress is required. In general, the mean velocity distribution of a wall-bounded turbulent flow can be divided into two layers: the inner and outer layers (Tennekes and Lumley, 1972). The viscous force dominates near the wall and vanishes far from the wall. The inner layer can be characterized in terms of the inner length scale ν /uτ , where ν is the kinematic viscosity and uτ is the friction velocity. The outer ∗
1
Corresponding author. E-mail address: [email protected] (H.J. Sung). Both authors contributed equally to this work.
http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.06.009 0142-727X/© 2017 Elsevier Inc. All rights reserved.
layer can be scaled with the shear layer thickness, i.e., the pipe radius (R). In contrast to the different length scales, the friction velocity can be used as a representative variable for both layers. The overlap layer is located where the inner and outer layers are overlapped. In the overlap layer, it has been known that the mean velocity (U) grows logarithmically with respect to the wall-normal distance in the overlap layer according to what is known as the log law: U+ = 1/κ log(y+ ) + B, where κ is the von Kármán constant and B is an additive constant. The superscript + refers to inner scaling, for example, y+ = yuτ /ν and U+ = U/uτ are the length and velocity respectively. The log law is widely applied for the description of wall-bounded turbulent flows. The log layer in the mean velocity is known as the inertial sublayer because the local viscous effect is negligible compared to the Reynolds shear stress. The log layer becomes evident for sufficiently high Reynolds number flows, such as Reτ (≡ uτ R/ν ) >50 0 0 (McKeon et al., 2004; Marusic et al., 2013). For lower Reynolds numbers (Reτ < 50 0 0), the viscous force is not negligible in the overlap layer; the power law, U+ = C(y+ )γ (where C and γ are constants independent of the Reynolds numbers), is then used instead of the log law (McKeon et al., 2004; Ahn et al., 2015). Long and Chen (1981) discussed the absence of the log law in terms of the significance of the inner and outer length scales in the overlap layer. They instead proposed the concept of the meso-layer, which is different from the classical inertial sublayer. Sreenivasan and Sahay (1997) showed that the viscous force has a persistent effect in the overlap layer; the viscous force is signifi-
210
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
cant in the neighborhood of the wall-normal location of the maximum Reynolds shear stress (ym ), and the square-root scaling ym + ∼ O(Reτ 1/2 ) is obtained empirically. This scaling can be also derived analytically from the streamwise mean momentum equation by assuming the log law at ym . Wei et al. (2005) described a four-layer structure in terms of the ratio of the gradient of the viscous stress + (d2U + /dy+2 ) to the gradient of the shear stress (d−u v /dy+ ), and showed that the neighborhood of ym moves outwards following a scaling of y+ ∼ O(Reτ 1/2 ). Chin et al. (2014b) confirmed that the onset of the log layer follows the same scaling. Afzal (1982) also analytically reasoned an intermediate length scale y+ ∼ O(Reτ 1/2 ) in the meso-layer. However, this length scale applies to a three-layer structure (consisting of inner, meso-, and outer layers) with two overlap layers and uses asymptotic expansions, in contrast to the two-layer concept. As mentioned above, the mean velocity and the Reynolds shear stress are coupled in the streamwise mean momentum equation, +
0=
1 d2U + d−u v + + , Reτ d y+ d y+ 2 PG
VF
(1)
TI
where the inverse of the friction Reynolds number, the gradient of the viscous stress, and the gradient of the Reynolds shear stress correspond to the pressure gradient (PG), the viscous force (VF), and the turbulent inertia (TI), respectively. PG is fixed by the friction Reynolds number (Reτ ), so VF is balanced by TI. By utilizing this relation, VF can be indirectly predicted by examining the spectra of TI. Guala et al. (2006) and Wu et al. (2012) showed that LSMs produce an acceleration in the mean flow up to y/R < 0.2 (which is close to the upper-bound of the overlap layer). Chin et al. (2014b) divided the TI spectrum according to the velocity–vorticity correlations. The acceleration force of the LSMs has been attributed to the correlation between the spanwise velocity and the wallnormal vorticity fluctuations (vorticity-stretching effect), which results in the change in the length scales (Tennekes and Lumley, 1972). Hwang et al. (2016) showed that the acceleration forces of LSMs in turbulent pipe flows are weaker than those in turbulent channel flows. The objective of the present study was to determine the contribution of LSMs to the Reynolds shear stress in turbulent pipe flows. Direct numerical simulation (DNS) data for turbulent pipe flows with Reτ = 544, 934, and 3008 were employed. Since VF and TI are closely coupled in the streamwise mean momentum equation, the TI can be used to monitor the behavior of the mean flow. The variations in the length scales below the overlap layer were explored by using the ratio of VF to TI. The power and log laws of the mean velocity in the overlap layer were examined by using their indicator functions. The pre-multiplied streamwise spectra of the Reynolds shear stress and the TI were obtained to determine the contribution of the LSMs to TI. The scale separations of the Reynolds shear stress and TI were used to show the development of the mean flow for higher Reynolds number flows. Finally, the growing order of the wall-normal location of the maximum Reynolds shear stress was explained in terms of the contributions of LSMs and small-scale motions (SSMs).
Crank–Nicolson and central finite-difference schemes, respectively (Park and Sung, 1995). The fully implicit fractional step method was employed to decouple the velocity and pressure (Kim et al., 2002). No-slip conditions at the wall were adopted, with periodic conditions along the axial and azimuthal directions. The axial domain length was 30 times the pipe radius (R), which was sufficiently long to describe multi-scale structures ranging from small to very large (Monty et al., 2007; Lee et al., 2015). The axial, radial, and azimuthal directions were represented by x, r, and θ , respectively, and the corresponding velocity components were represented by ux , ur , and uθ , respectively. Throughout this study, the expressions in the cylindrical coordinates have been transformed into the Cartesian coordinates to facilitate comparisons with other geometric flows: the streamwise direction is indicated by x, the wall-normal direction by y = 1 – r, the spanwise direction by z = rθ , and the corresponding velocity components are u = ux , v = −ur , and w = uθ (Wu et al., 2012). Small letters with a prime symbol are velocity fluctuations (e.g.,u ), and averaged quantities are denoted by a capital letter or with brackets (e.g., U or u u ). The details of the DNS simulations can be found in Lee and Sung (2013), Ahn et al. (2013) and Ahn et al. (2015). The numerical parameters are summarized in Table 1. Fig. 1 shows the profiles of the Reynolds stresses. All quantities increase along the wall-normal distance as the Reynolds number increases. For sufficiently high Reynolds numbers, it has been known that the Reynolds stresses of the wall-parallel fluctuations (u u + ,w w + ) behave logarithmically, and the wallnormal Reynolds stresses (v v + ) are constant in the overlap layer (Townsend, 1976). The profiles of u u + do not exhibit logarithmic behavior in the overlap layer because the Reynolds numbers are not sufficiently high. Instead, a plateau-like shoulder is evident at Reτ = 3008. For sufficiently high Reynolds numbers, a logarithmic trend is expected beyond the plateau region (Marusic et al., 2013). The profiles of w w + for Reτ = 934 and 3008 exhibit logarithmic behavior in the overlap layer, although the logarithmic constant (B3 ) could vary with the Reynolds number (Sillero et al., 2013; Chin et al., 2014a; Lee and Moser, 2015). The constant was calcu+ lated from the log law indicator function (y+ ∂ w w /∂ y+ ). The + profile of v v for Reτ = 3008 seems to be flat and to reach the value of 1.3 (Zhao and Smits, 2007). The quality of the present DNS data was validated by examining the inner peaks in the streamwise Reynolds stress, as shown in Fig. 1(d). The recent available DNS data of turbulent pipe flow (Wu and Moin, 2008; Wu et al., 2012; El Khoury et al., 2013; Chin et al., 2014a) were used to assess the present ones. Based on the present DNS data, the logarithmic growth of the inner peaks with the Reynolds number was obtained as u u + = 0.6875 log(Reτ ) + 3.2391 with a residue innerpeak of 0.9997 (Örlü and Alfredsson, 2013). This trend is in good agreement with other DNS data within a 3% tolerance limit. The slight deviations could be due to differences in the numerical conditions, such as the streamwise domain length, the grid resolution, and the numerical method (Schlatter and Örlü, 2010; Ahn et al., 2013).
3. Results and discussion 3.1. Mean velocity and Reynolds shear stress
2. Numerical data Previously obtained DNS data for turbulent pipe flows with Reτ = 544, 934, and 3008 were employed (Lee and Sung, 2013; Ahn et al., 2013; Ahn et al., 2015). In these studies, the Navier– Stokes and continuity equations were solved in the cylindrical coordinates to describe incompressible and fully developed turbulent pipe flows. The governing equations were spatially and temporally discretized over a staggered grid by using the second-order
Fig. 2(a) shows the profiles of the mean velocity (U+ ) and the Reynolds shear stress (−u v + ). The profiles of U+ are similar in the viscous sublayer and the buffer layer, where y+ is less than 30. Above the buffer layer (y+ > 30), the profile stretches upward as the Reynolds number increases. The profiles of −u v + collapse well very near the wall. As the Reynolds number increases, the maximum value in the overlap layer increases and the corresponding constant region is expanded; however, it is too short to
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
211
Table 1 Numerical parameters. ReD
Reτ
(Nx , Nr , Nθ )
x+
r+ min
r+ max
(Rθ )+
tUc /R
19,0 0 0 35,0 0 0 133,0 0 0
544 934 3008
(2049, 279, 513) (4097, 301, 1025) (12,289, 901, 3073)
8.02 6.84 7.34
0.176 0.334 0.36
4.284 9.244 9.91
6.72 5.73 6.15
0.02 0.01 0.003
Fig. 1. Profiles of the (a) streamwise, (b) wall-normal, and (c) spanwise Reynolds stresses. The log law in the spanwise Reynolds stress is evident for Reτ = 934 and 3008. (d) The inner peaks in u u + .
Fig. 2. Profiles of (a) the mean velocity and the Reynolds shear stress and (b) VF and TI.
be determined as the constant-stress layer proposed by Townsend (1976). In Section 3.2, the formation mechanism of the constantstress layer will be explored in terms of the contribution of the LSMs. Fig. 2(b) shows the profiles of VF and TI. The inset shows a magnified view of the wall-normal region (20 < y+ < 200). As the Reynolds number increases, VF is almost invariant with respect to the Reynolds number except for the small increase in its peak value for y+ less than ≈7. Above y+ = 100, the value of VF diminishes and converges to nearly zero along the wall-normal distance. As the Reynolds number increases, the value of TI decreases, and its zero-crossing point is shifted to the core region. The zerocrossing points of TI (ym + ) are ym + = 43, 52 and 81 for Reτ = 544,
934, and 3008, respectively. For Reτ = 3008, TI becomes zero and has a similar level to VF above y+ = 100. Fig. 3(a) shows the ratio of VF to TI (VF/TI). The framework of the four-layer structure is employed to interpret the present results (Wei et al., 2005). VF is relatively dominant compared to TI very near the wall (0 < y+ < 3). A counterbalance between VF and TI (VF/TI = −1) is established and confined within the buffer layer (3 ≤ y+ ≤ 30). The inset shows their results for the turbulent channel flow up to Reτ ≈ 5200 (Lee and Moser, 2015), which are similar to the present results. In the vicinity of −u v + , TI is almost peak
zero (TI ≈ 0). Although the magnitude of VF gradually decreases above y+ = 30 in Fig. 2(b), it regains its leading order in the region
212
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Fig. 3. (a) Ratio of VF to TI; (b) Wall-normal locations for VF/TI = −2, −u v + , and VF/TI = 0.5. peak
Fig. 4. Profiles of the (a) power law and (b) log law indicator functions.
y+ |VF/TI = −2 < y+ < y+ |VF/TI = 0.5 because TI is negligible near ym + . The sign change of TI at ym + leads to an infinite value of VF/TI. Above y+ |VF/TI = 0.5 , VF/TI becomes zero since VF loses its leading order and an inertial sublayer then emerges. Fig. 3(b) shows the wall-normal locations for VF/TI = −2, −u v +peak , and VF/TI = 0.5. The growth of each wall-normal location was scaled using the present DNS data of turbulent pipe flow:y+ |VF/TI = −2 = 4.71Reτ 0.318 , y+ |−u v = 4.13Reτ 0.372 , and peak
y+ |VF/TI=0.5 = 3.31Reτ 0.464 . DNS data of turbulent channel flow was included for comparison (Lee and Moser, 2015). The present scaling results show the deviations from the scaling y+ ∼ O(Reτ 1/2 ), which was obtained with the four-layer structure (Wei et al., 2005). As the wall-normal distance increases, the scaling order of the growth varies from Reτ 0.318 (≈ Reτ 1/3 ) to Reτ 0.464 (≈ Reτ 1/2 ) via Reτ 0.372 in the region y+ |VF/TI = −2 < y+ < y+ |VF/TI = 0.5 . The scaling order y+ ∼ O(Reτ 1/2 ) implies that the wall-normal location (y) can be normalized by the equal contribution of the inner and outer length scales (ν /uτ and R). For example, if the scaling order is y+ ∼ O(Reτ 1/3 ), y is normalized by (ν /uτ )2/3 R1/3 due to the strong VF. As the wall-normal distance increases beyond the buffer layer, a characteristic length scale develops from the inner length scale to the intermediate length scale due to the diminishment of VF. The region O(Reτ 0 ) < y+ < O(Reτ 1/2 ) is thus interpreted as the meso-layer (ym + ) (Sreenivasan and Sahay, 1997; Wei et al., 2005), to connect the buffer layer with the inertial sublayer. Although VF becomes weak in the meso-layer, VF is still stronger than TI. The behavior of the mean velocity in the meso-layer and the inertial sublayer was examined by the power law and log law indicator functions ( and ) shown in Fig. 4. The power and log law indicator functions are determined as = y+ /U+ ∂ U+ /∂ y+ and = y+ ∂ U+ /∂ y+ respectively. The profiles of the power law indicator function in Fig. 4(a) are similar up to y+ ≈ 80. The profiles for the lower Reynolds numbers (Reτ = 544 and 934) dis-
play gradual changes beyond the local minimum (γ ≈ 0.145) at y+ ≈ 80. The profile for Reτ = 3008 shows a plateau beyond the local minimum with γ = 0.145 in the region 90 < y+ < 150, and starts to sag down beyond y+ > 150. The meso-layer starts from y+ = 30 to y+ = 150. Since the end of the meso-layer can be scaled by y+ = 150 ≈ 3Reτ 1/2 , the transition from the inner length scale (y+ ∼ O(ν /uτ )) to the intermediate length scale (y+ ∼ O(Reτ 1/2 )) occurs, which is also observed in Fig. 3(b). The region of the mesolayer (30 < y+ < 150) is also identical to that of the critical layer (1/2 ≤ U/Uc ≤ 2/3), where Uc is the turbulent centerline velocity (Sharma and McKeon, 2013). The profiles of the log law indicator function in Fig. 4(b) are in good agreement up to y+ ≈ 40 from the wall. Beyond the local minimum at y+ ≈ 60, the profiles increase with increasing the wall-normal distance. A plateau (=1/κ ) could not be clearly identified in the overlap layer for any of the profiles. For turbulent channel flows (Reτ > 40 0 0), the profiles of the log law indicator function can be divided into two inclinations: steeply inclined or less steeply inclined, and the latter reaches a constant value of 0.38 (Lozano-Durán and Jiménez, 2014; Lee and Moser, 2015). As is the case for channel flows, the profile of Reτ = 3008 shows that the inclinations slightly change at y+ = 150, i.e., an infant log law. The log law becomes evident for sufficiently high Reynolds numbers. The corresponding value of the log law indicator function at y+ = 150 is κ = 0.392, which is slightly higher than the value for channel flows (Lee and Moser, 2015). This value is similar to the result of 0.4 ± 0.02 obtained for the Superpipe (Bailey et al., 2014), and to the value 0.39 obtained for a wall-bounded canonical turbulent flow with 2 × 104 < Reτ < 6 × 105 (Marusic et al., 2013). The present indicator functions show the successive wall-normal alignment of the power law and the infant log law along the wall-normal distance (McKeon et al., 2004). For Reτ = 3008, y+ = 150 can be expressed by not only as y+ ≈ 3Reτ 1/2 but also as y+ ≈ 10Reτ 1/3 . The
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
213
Fig. 5. Pre-multiplied streamwise spectra of (a) the Reynolds shear stress and (b) TI, with black zero-contour lines. The magenta lines in (b) are the wall-normal locations of the maximum Reynolds shear stress (ym + ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
first formulation (square-root scaling) was obtained by Marusic et al. (2013) with the independence of both the inner and the outer length scales. The second formulation (cubic-root scaling) was obtained by Tennekes and Lumley (1972), who employed a secondorder correction to the pipe flow due to the pressure gradient. 3.2. Spectra of the Reynolds shear stress and the turbulent inertia Fig. 5 shows the pre-multiplied streamwise spectra of −u v and TI. The spectrum of −u v was calculated as the Fourier transform of the two-point correlations, which are defined √ ∞ by −u v (kx , y ) ≡ 2Re{ 21π −∞ e− −1kx rx −u (x, y )v (x + rx , y )drx }, where rx is the spatial shift along the streamwise direction (Wu +
v) et al., 2012). The spectrum of TI is given by T I = ∂ (−u = ∂ y+ ∞ ∂ +−u v ∞ kx ∂ +−u v 0 ∂ y + dk x = 0 ∂ y+ d log(kx ). The streamwise wavenumber kx was pre-multiplied due to the logarithmic scaling of kx , and λx is the streamwise wavelength. The pre-multiplied streamwise spectra of the Reynolds shear stress (kx + ) are similar in the re−u v gion y+ < 30. The inner sites with the maximum energy are located at y+ = 30. The outer site for Reτ = 3008 is located at y/R ≈ 0.18 (Ahn et al., 2015). The pre-multiplied streamwise spectra of TI ( k x ∂ + /∂ y+ ) show that the bulk motion accelerates the mean −u v flow within the buffer layer (y+ < 30) (region A). For values beyond the inner site (y+ > 30), the TI spectra behave differently for different Reynolds numbers. In Fig. 5(b), the region C is bounded by λx + ≈ 30 0 0 for all three Reynolds numbers. As the Reynolds num-
ber increases (Reτ = 544, 934, 3008), the regions D and E are extended to λx + ≈ 30 0 0. A cut-off wavelength of λx + = 30 0 0 was employed to demarcate the LSMs and SSMs (Chin et al., 2014b; Ahn et al., 2015). At this wavelength, the LSMs (λx + > 30 0 0) induce the acceleration force in the overlap layer, and the sign change (region C/D) in the net force is clear (Chin et al., 2014b). The force due to the SSMs (λx + < 30 0 0) changes from acceleration to deceleration (region C), whereas the LSMs (λx + > 30 0 0) maintain their acceleration force (region D & E). The small-scale influence of λx + < 300 kept the acceleration force up to y+ = 60 (region B). In the region 30 < y+ < 60, the forces are ordered as acceleration-decelerationacceleration in the regions B→C→D. Away from the wall (y+ > 60), the acceleration force of the LSMs extends up to y/R ≈ 0.2 as the Reynolds number increases (region E) (Guala et al., 2006; Wu et al., 2012; Chin et al., 2014b; Hwang et al., 2016). As the Reynolds number increases, the energy of the LSMs becomes more intense, and they penetrate into the outer layer, although the SSMs remain unchanged. The wall-normal location of the maximum Reynolds shear stress (ym + ) increases in accordance with the increased contribution of the LSMs. The distinct border at λx + = 30 0 0 between the acceleration force of the LSMs and the deceleration force of the SSMs (region C/D) continues up to y+ ≈ 150. This location is consistent with the onset of the infant log layer for Reτ = 3008 (y+ ≈ 150 ≈ 3.9Reτ 1/2 ) shown in Fig. 3(a) (Marusic et al., 2013). Fig. 6 shows the scale separation of −u v + with a cut-off wavelength of λx + = 30 0 0 to visualize the contributions of the LSMs and SSMs. The scale separation was performed by integrat-
214
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Fig. 6. Scale separation of the Reynolds shear stress for Reτ = 544, 934, and 3008 and a cut-off wavelength of λx + = 30 0 0 in the (a) inner and (b) outer coordinates. The LSM and SSM profiles are shown as thick and thin solid lines respectively.
Fig. 7. Scale separation of TI for a cut-off wavelength of λx + = 30 0 0 and Reτ values of (a) 544, (b) 934, and (c) 3008. (d) The TI profiles for LSMs (thick) and SSMs (thin) are shown together. The vertical dashed line and the circle symbols indicate the zero-crossing points of the SSMs and LSMs respectively.
ing kx + across the entire wavelength range. The contributions −u v of the SSMs to −u v + are similar for the three Reynolds numbers in the region y+ < 200, and the profiles have maxima at yi + ≈ 40. The similarity in the profiles of the SSMs also represents the validity of the cut-off wavelength λx + = 30 0 0 (Chin et al., 2014b). The contributions of the LSMs to −u v + grow in magnitude in the region y+ > 20 as the Reynolds number increases, and the profiles have maxima at yo /R ≈ 0.18 where the outer site for Reτ = 3008 is located (Ahn et al., 2015). The constant-stress layer could be established in the region between yi + ≈ 40 and yo /R ≈ 0.18. Since the contribution of the SSMs to −u v + is independent of the Reynolds number, the presence of the constant-stress layer is affected by the contribution of the LSMs at higher Reynolds numbers (Ahn et al., 2015). The scale separation of TI is shown in Fig. 7. The profiles of the SSMs are similar for the three Reynolds numbers, and the zerocrossing point of TI for the SSMs (yi ) is located at yi + ≈ 40 in Fig. 7(d). As the Reynolds number increases, the TI of the LSMs becomes strong with the positive values. The zero-crossing point of
TI of the LSMs is located at yo /R ≈ 0.18, which is independent of the Reynolds number. Since the profiles of the SSMs are similar, the wall-normal location of the zero TI is influenced by the contribution of the LSMs. The enhanced acceleration force of the LSMs cancels out the deceleration force of the SSMs, and the counterbalanced region (TI ≈ 0) between the two opposing forces is extended. The inner and outer length-scaled structures contribute equally to the formation of the zero TI. In the neighborhood of the zero TI, the low TI means that VF is dominant. As a result, a meso-layer occurs with the power law in the mean velocity (Fig. 4a). The log law is not evident in the present results for various Reynolds numbers. According to theory, the constant-stress layer coincides with the log layer (Townsend, 1976). Since the zero TI at ym results in the dominance of VF, a meso-layer forms in the neighborhood of ym . Above the meso-layer, VF is smaller than TI, although TI is almost zero. The log layer is located in this region. This behavior is clear at higher Reynolds numbers such as Reτ > 50 0 0 (McKeon et al., 2004). In particular, the constantstress layer is the meso-layer (VF > TI ≈ 0), and the extension of the
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
215
Fig. 8. (a) The wall-normal locations of the maximum Reynolds shear stress (ym + ) with the scaling lines; (b) Variations with the Reynolds shear stress in the area fractions of the LSMs and SSMs.
constant-stress layer (TI ≈ 0 > VF) above the meso-layer becomes the log layer. As mentioned above, for three Reynolds numbers we investigated, the maximum contributions of the SSMs and LSMs to −u v + occur at yi + ≈ 40 and yo /R ≈ 0.18 respectively. The wall-normal location of the maximum Reynolds shear stress (ym ) can be expressed in terms of a combination of yi and yo , ym + ∼ (yi + )a (yo /R)b . Fig. 8(a) shows the scaling we observed, ym + = 4.13Reτ 0.372 , and the classical scaling ym + = 1.6∼2.0Reτ 1/2 as well as the numerical and experimental data (McKeon et al., 2004; Buschmann et al., 2009; Hultmark et al., 2013; Chin et al., 2014b). The trend ym + ∼ O(Reτ 0.372 ) is suitable for lower Reynolds numbers, i.e., Reτ < 50 0 0, however, the data for higher Reynolds numbers, Reτ > 50 0 0, follow the scaling trend ym + ∼ O(Reτ 1/2 ). This weak power behavior arises because the LSMs are less active for Reτ < 50 0 0. The scaling ym + ∼ O(Reτ 1/2 ) arises at higher Reynolds numbers (Reτ > 50 0 0). To show the contribution of the LSMs to the scaling ym + ∼ O(Reτ 1/2 ), the present scaling order ym + ∼ O(Reτ 0.372 ) was amended by using the variations in the contributions of the LSMs and SSMs. Fig. 8(b) shows the variations in the area fractions (AFs) of the LSMs and SSMs (AFLSM and AFSSM ) for −u v + , which were used as weighting factors in the present scaling order. As the Reynolds number increases, AFLSM increases and AFSSM decreases, i.e., the contributions of the LSMs and SSMs increase and decrease, respectively. The growth orders of AFLSM and AFSSM are AFLSM and AFSSM ∼ O(Reτ 0.590 ) and O(Reτ −0.451 ), respectively. A simple multiplication of the area fraction was performed to obtain asymptotic behavior: ym + ∼ O(Reτ 0.372 ) × AFLSM × AFSSM ∼ O(Reτ 0.372 × Reτ 0.590 × Reτ −0.451 ) ∼ O(Reτ 0.511 ). This behavior is similar to that of ym + ∼ O(Reτ 1/2 ). 4. Conclusions DNS data for turbulent pipe flows with Reτ = 544, 934, and 3008 were used to examine the contribution of LSMs to the Reynolds shear stress. The relationship between viscous force (VF) and turbulent inertia (TI) resulted in a transition from the inner length scale (y+ ∼ O(ν /uτ )) to the intermediate length scale (y+ ∼ O(Reτ 1/2 )) in the meso-layer. In the pre-multiplied streamwise spectra of TI, as the Reynolds number increases the LSMs (λx + > 30 0 0) become more intense so they penetrate into the outer layer, although the SSMs (λx + < 30 0 0) are unchanged. Since the contribution of the SSMs is similar for the three Reynolds numbers, the presence of a constant-stress layer is affected by the active contribution of the LSMs. The acceleration force of the LSMs cancels out the deceleration force of the SSMs, implying that the inner and outer length-scaled structures contribute equally to form the zero TI at ym + . As the Reynolds number increases, ym + is shifted to the core region, and the enhanced acceleration force of the LSMs
expands the nearly zero TI region. In the neighborhood of the zero TI, the low level of TI results in the dominance of VF. As a result, a meso-layer occurs in the nearly zero TI region. Above the meso-layer, VF loses its leading order and the log layer is established. This behavior becomes evident at higher Reynolds numbers. Therefore, the constant-stress layer is the meso-layer (VF > TI ≈ 0), and the extension of the constant-stress layer (TI ≈ 0 > VF) above the meso-layer is the log layer. The scaling ym + ∼ O(Reτ 1/2 ) was examined by considering the area fractions of LSMs and SSMs for −u v + . Acknowledgments This work was supported by the Creative Research Initiatives (No. 2017-013369) program of the National Research Foundation of Korea (MSIP) and the Supercomputing Center (KISTI). References Adrian, R.J., Meinhart, C.D., Tomkins, C.D., 20 0 0. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54. Adrian, R.J., 2007. Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301. Afzal, N., 1982. Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355–377. Ahn, J., Lee, J.H., Jang, S.J., Sung, H.J., 2013. Direct numerical simulations of fully developed turbulent pipe flows for Reτ = 180, 544, and 934. Int. J. Heat Fluid Flow 44, 222–228. Ahn, J., Lee, J.H., Lee, J., Kang, J.-H., Sung, H.J., 2015. Direct numerical simulation of a 30R long turbulent pipe flow at Reτ =3008. Phys. Fluids 27 (6), 065110. Bailey, S.C.C., Vallikivi, M., Hultmark, M., Smits, A.J., 2014. Estimating the value of von Kármán’s constant in turbulent pipe flow. J. Fluid Mech. 749, 79–98. Buschmann, M.H., Indinger, T., Gad-el-Hak, M., 2009. Near-wall behavior of turbulent wall-bounded flows. Int. J. Heat Fluid Flow 30, 993–1006. Chin, C., Monty, J.P., Ooi, A., 2014a. Reynolds number effects in DNS of pipe flow and comparison with channels and boundary layers. Int J. Heat Fluid Flow 45, 33–40. Chin, C., Philip, J., Klewicki, J., Ooi, A., Marusic, I., 2014b. Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows. J. Fluid Mech. 757, 747–769. El Khoury, G.K., Schlatter, P., Noorani, A., Fischer, P.F., Brethouwer, G., Johansson, A.V., 2013. Direct numerical simulation of turbulent pipe flow at moderately high Reynolds numbers. Flow Turbul. Combust. 91, 475–495. Ganapathisubramani, B., Longmire, E.K., Marusic, I., 2003. Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46. Guala, M., Hommema, S.E., Adrian, R.J., 2006. Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–542. Hultmark, M., Vallikivi, M., Bailey, S.C.C., Smits, A.J., 2013. Logarithmic scaling of turbulence in smooth and rough-wall pipe flow. J. Fluid Mech. 728, 376–395. Hwang, J., Lee, J., Sung, H.J., 2016. Influence of large-scale accelerating motions on turbulent pipe and channel flows. J. Fluid Mech. 804, 420–441. Kim, K.C., Adrian, R.J., 1999. Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422. Kim, K., Baek, S.-J., Sung, H.J., 2002. An implicit velocity decoupling procedure for the incompressible Navier–Stokes equations. Int. J. Numer. Method Fluids 38, 125–138. Lee, J.H., Sung, H.J., 2011. Very-large-scale motions in a turbulent boundary layer. J. Fluid Mech. 673, 80–120.
216
J. Ahn et al. / International Journal of Heat and Fluid Flow 66 (2017) 209–216
Lee, J.H., Sung, H.J., 2013. Comparison of very-large-scale motions of turbulent pipe and boundary layer simulations. Phys. Fluids 25 (4), 045103. Lee, J., Ahn, J., Sung, H.J., 2015. Comparison of large- and very-large-scale motions in turbulent pipe and channel flows. Phys. Fluids 27 (2), 025101. Lee, M., Moser, R.D., 2015. Direct numerical simulation of turbulent channel flow up to Reτ ≈5200. J. Fluid Mech. 774, 395–415. Long, R.R., Chen, T.C., 1981. Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59. Lozano-Durán, A., Jiménez, J., 2014. Effect of the computational domain on direct simulations of turbulent channels up to Reτ =4200. Phys. Fluids 26 (1), 011702. Marusic, I., Monty, J.P., Hultmark, M., Smits, A.J., 2013. On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3. McKeon, B.J., Li, J., Jiang, W., Morrison, J.F., Smits, A.J., 2004. Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147. Monty, J.P., Stewart, J.A., Williams, R.C., Chong, M.S., 2007. Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156. Örlü, R., Alfredsson, H., 2013. Comment on the scaling of the near-wall streamwise variance peak in turbulent pipe flows. Exp. Fluids 54, 1431–1435. Park, T.S., Sung, H.J., 1995. A nonlinear low-Reynolds-number k-ε model for turbulent separated and reattaching flows - (I) Flow field computations. Int. J. Heat Mass Transfer 38 (14), 2657–2666. Robinson, S.K., 1991. Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.
Schlatter, P., Örlü, R., 2010. Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126. Sharma, A.S., McKeon, B.J., 2013. On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238. Sillero, J.A., Jiménez, J., Moser, R.D., 2013. One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to δ + ≈20 0 0. Phys. Fluids 25 (10), 105102. Sreenivasan, K.R., Sahay, A., 1997. The persistence of viscous effects in the overlap region and the mean velocity in turbulent pipe and channel flows. In: Panton, R. (Ed.), Self-Sustaining Mechanisms of Wall Turbulence. Comp. Mech. Publ, pp. 253–272. Tennekes, H., Lumley, J.L., 1972. A First Course in Turbulence. MIT Press. Townsend, A.A., 1976. The Structure of Turbulent Shear Flow, 2. Cambridge University Press. Wei, T., Fife, P., Klewicki, J., McMurtry, P., 2005. Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327. Wu, X., Moin, P., 2008. A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112. Wu, X., Baltzer, J.R., Adrian, R.J., 2012. Direct numerical simulation of a 30R long turbulent pipe flow at R+ =685: large- and very large-scale motions. J. Fluid Mech. 698, 235–281. Zhao, R., Smits, A.J., 2007. Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow. J. Fluid Mech. 576, 457–473.