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Very-large-scale fluctuations in turbulent channel flow at low Reynolds number
International Journal of Heat and Fluid Flow 62 (2016) 593–597
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Very-large-scale fluctuations in turbulent channel flow at low Reynolds number✩ Masaharu Matsubara∗, Shun Horii, Yoshiyuki Sagawa, Yuta Takahashi, Daisuke Saito Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan
a r t i c l e
i n f o
Article history: Available online 6 December 2016 Keywords: Channel flow Turbulence Structure Large-scale
a b s t r a c t It is known that there exist very large features in turbulent channel flow at high Reynolds number as well as in pipe flow and turbulent boundary layer. In addition, a low frequency peak was observed in spectra of the streamwise velocity fluctuation in transitional and low-Reynolds-number but turbulent channel flows. In this study, the large-scale fluctuation observed at the low Reynolds number has been experimentally explored with increasing Reynolds number by means of a hot wire anemometry. There are two peaks in the streamwise velocity spectra for transitional flow for 1800 ≤ Re ≤ 2600, where Re is the Reynolds number based on the bulk velocity and the channel width. The high-frequency peak corresponds to the turbulent vortices that have the same order of magnitude as the channel width, while the low-frequency peak is due to passages of the turbulent patches whose streamwise scale is greater than ten channel widths. It is surprising that the low frequency peak remains even up to Re = 30 0 0 at which the flow is fully turbulent. Furthermore, a spectral plateau around the frequency corresponding to 25 channel widths is confirmed up to Re = 40 0 0, indicating that there exist very-large-scale fluctuations in turbulent channel flow even at low Reynolds number. © 2016 Elsevier Inc. All rights reserved.
1. Introduction It has been worthy of note that there exist very large features in turbulent pipe and channel flows at high Reynolds number (Monty et al., 2007; Kim and Adrian, 1999; Guala et al., 2006), as well as in turbulent boundary layer (Hutchins and Marusic, 2007), because it may relate the turbulent properties and coherent structures near the wall. Monty et al. (2007) performed velocity measurements with hot-wire rakes, and they revealed long meandering structures that has the streamwise length up to 25 pipe radii or channel halfheights. Guala et al. (2006) also detected a very large structure in high-Reynolds-number pipe flow whose streamwise length is as long as 16 pipe radii. Linear and quasi-linear theories (Jovanovic´ and Bamieh, 2005; Cossu et al., 2009; McKeon et al., 2010; Farrell and Ioannou, 2012) predict roll-streak structures of very large streamwise length that might relate to the these very large feature in shear flows. These experimental and theoretical findings strongly suggest that very
✩ This paper is a contribution to the special issue CJWSF 2015 and CMFF’15. Guest edited by Editors-in-chief Andrew Pollard and Suad Jakirlic Guest editors (ICJWSF 2015): Laszlo Fuchs, Antonio Segalini Guest editor (CMFF’15): Janos Vad. ∗ Corresponding author. E-mail address: [email protected] (M. Matsubara).
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.11.009 0142-727X/© 2016 Elsevier Inc. All rights reserved.
large features or structures are omnipresent in turbulent wallbounded shear flows at high Reynolds number. Monty et al. (2009) made hot-wire measurements in turbulent pipe, channel and boundary layer flows. From the velocity spectra maps with different heights, it was revealed that very large scale motion (VLSM) around 0.3 δ (δ is a pipe radii or a channel half height) has the streamwise length larger than 10 δ . They strongly claim that the VLSM in internal flows should not be mixed up with the superstructures in boundary layers. Recently, Seki and Matsubara (2012) made hot-wire measurements for the transitional channel flow that was realized by decreasing the Reynolds number of turbulent channel flow in a expansion section. They found a spectral peak in the streamwise velocity fluctuation measured in transitional channel flow and observed intermittency with the turbulent patches passing through the channel, and from probability density distributions of the streamwise velocity fluctuation they asserted that the transitional range is for 1400 < Re < 2600. Reynolds number is defined as Re = Ub d/ν , where Ub is the bulk velocity, d is channel width, ν is kinematic viscosity. It is surprising that the low frequency peak is confirmed even for the fully turbulent flow at Re = 2660, suggesting existence of a large-scale flow pattern. Their investigation raises a question if the two large-scale fluctuations at low and high Reynolds numbers are the same. If they are the essentially same, it is straightforwardly inferred that the VLSM has
594
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 1. Air channel facility.
sustain and/or generation mechanisms common to the turbulent patches in traditional channel flows. As the first step to answer this question, the large-scale fluctuation has been experimentally investigated with gradually increasing Reynolds number from Re = 10 0 0, at which the flow is laminar, to Re = 40 0 0.
Fig. 2. Time traces of the streamwise velocity at the channel center. (a) Re = 1500, (b) Re = 20 0 0, (c) Re = 260 0, (d) Re = 350 0. The red and black circles indicate high-frequency and low velocity periods, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2. Experiment setup.
3. Result and discussion Time traces of the streamwise velocity measured at the channel center are shown in Fig. 2. At Re = 1500, the signal is almost calm but high-frequency fluctuations intermittently appear, as marked
0.1
f∗
The hot wire measurement was performed in an air channel flow facility shown in Fig. 1. The facility is the same that Seki and Matsubara (2012) used. The air pressurized by a blower gushes from a perforated pipe installed inside a settling chamber whose cross section is 190 mm × 170 mm. A two-dimensional nozzle of a 34:1 contraction ratio is followed by entrance, expansion and test sections whose wall distance, d, is 5.1 mm. The entrance section is 10 0 0 mm long and 190 mm wide. Two tripping rods of 0.5 mm diameter are mounted 100 mm downstream from the outlet of the nozzle for triggering transition to turbulence, so that the flow becomes a fully developed turbulent flow at the end of the entrance section. The distance between the channel end walls is gradually widened from 190 mm to 260 mm in the 10 0 0 mm. As a result, Re drops to 73% of the entrance mean velocity in the expansion section. The 2800 mm long test section following the expansion section opens to the atmosphere at its downstream end. In a certain range of Re, the Re drop in the expansion section induces transition from upstream turbulent flow to transitional, or intermittent, flow in the test section. A constant-temperature hot-wire anemometer was used for the streamwise velocity measurement. The single hot-wire sensor is a platinum wire of 2.5 μm diameter and 1 mm length. Velocity calibrations for the sensor were performed with a jet potential core emanating from an axial nozzle connected to the blower for the main channel facility. The relation between the voltage and the velocity was fitted to a calibration curve deduced from King’s law modified for low velocity. An analog/digital converter acquired the voltage of the hot wire anemometer with a sampling frequency of 20 kHz and a sampling time of 300 s. A wedge mechanism, which could be inserted in the channel at an arbitrary streamwise position, was used for probe positioning in the wall-normal direction. The symmetry of the velocity profile was utilized for estimation of the channel center. The measurement was made at 300 mm upstream from the channel exit.
1
0.01
0.001
1000
1500
2000
2500
3000
3500
4000
Re Fig. 3. Contour map of streamwise velocity premultiplied spectra density at the channel center. Contours and colours express f∗ ∗ . White and blue lines indicate ridges for the high and low frequency peaks, respectively. The second peaks at each Re are marked as broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by red circles. This intermittent behavior of the flow is also investigated by Seki and Matsubara (2012), thus it is considered that these high-frequency fluctuations indicate the passages of turbulent patches. Appearance of the high-frequency fluctuations becomes more frequent at Re = 20 0 0, then finally at Re = 2600, the time trace are filled of high-frequency fluctuations, indicating that the flow is fully turbulent. Seki and Matsubara (2012) also confirmed from the probability density distributions that intermittency at Re = 260 0 is almost 10 0%. In spite of the turbulent state, the low velocity but high-frequency fluctuating periods (circled in black) still exist. Even at higher Reynolds number, Re = 3500, the low velocity periods are still observed though their appearance turns unclear with increase of Re. The rate of their appearance can be estimated in the premultiplied spectral diagram of the streamwise velocity fluctuation as shown in Fig. 3. The measurement was performed at 31 Reynolds numbers with the Re resolution of 100. The non-dimensional frequency f∗ and the power spectral density ∗ are normalized with the channel width d and the bulk (mean) velocity Ub as f ∗ = f d/Ub and ∗ = /Ub2 . As a consequence of this normalization, the inverse number of f∗ corresponds to a length scale in unit of d. While
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 4. Contour map of streamwise velocity premultiplied spectra density at y/d = 0.25. Contour spacing and colour mapping are the same as Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
total energy at Re = 10 0 0 is very low because the flow is laminar, a peak around f∗ = 0.01 appears for Re = 1400 showing the flow becomes transitional state. As the Reynolds number increases, the peak shifts to higher frequency and the peak value reaches the maximum at Re = 1500. Seki and Matsubara (2012) compared the peak frequency to the turbulent-patch appearance rate and they concluded that they are in good agreement within the middle of the transitional range. For 2300 ≤ Re ≤ 4000, the high-frequency peak around f ∗ = 0.4 appears, while the low-frequency peak declines. In flow visualization performed by Okumura and Matsubara (2006), the flow for Re ≥ 2010 is filled with vortical struc-
595
Fig. 5. Contour map of streamwise velocity premultiplied spectra density at y/d = 0.10. Colour mapping is the same and contour spacing is double to that in Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
tures that have a streamwise length scale similar to the channel width, indicating that the flow contains large amount of turbulent regions. Hence, it is inferred that the high-frequency peak is due to the turbulent vortical motions. It is surprising that the lowfrequency peak is still confirmed up to Re = 30 0 0 that is considerably higher than the upper marginal Re of 2600 ascertained by Seki and Matsubara (2012) as the lowest Re for fully turbulent flow. Furthermore, even for Re > 30 0 0 a plateau stays around f ∗ = 0.04, at least, in the present Re range. This corresponds to 25 channel widths taking the bulk velocity Ub as a convection velocity.
Fig. 6. Distributions of the premultiplied spectra f∗ ∗ in f∗ . (a) Re = 1800, (b) Re = 2600, (c) Re = 30 0 0, (d) Re = 40 0 0.
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M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 7. Distributions of the premultiplied spectra α ∗ ∗ in α ∗ . (a) Re = 1800, (b) Re = 2600, (c) Re = 30 0 0, (d) Re = 40 0 0.
In Fig. 3, the low-frequency peak at low Re continuously ranges from the highest peak at Re = 1500 that is due to the turbulent patch passing. This suggests that its structure is similar to the turbulent patch that spread in both stream and spanwise directions more than one order higher than the channel width. At the quarter wall distance, y/d = 0.25, there are two peaks at Re = 1800 as shown in Fig. 4. As mentioned above, the lowfrequency peak is due to the passage of the turbulent patches. The low frequency peak is continues up to Re = 2600, though for Re > 2700 there is only the one visible peak. It is not able to distinguish the low and high peaks near the wall as shown in Fig. 5; one peak in the spectral distribution at each Re remains within the bounds, 0.02 < f∗ < 0.07. The spectra distributions are compared at different heights from the wall, in Figs. 6 and 7. The low-frequency peaks in the spectra at Re = 1800 (Fig. 6 (a)) stay around f ∗ = 0.025, regardless of the height. This indicates that a structure distributes in a wide region in the wall-normal direction like a turbulent patch. The frequency independence from the height is confirmed for the higher Re in Fig. 6 (b), (c) and (d), though the peak frequency increases with increase of Re. Fig. 7 shows the spectral distribution in the normalized spatial wave number that was estimated from the frequency f and local time-mean velocity Um as α ∗ = f d/Um . The positions of the high-wave-number peaks in the spectral distributions at Re = 30 0 0 and 40 0 0 are quite close, though the positions of the low-wavenumber peak are not constant in this scaling. In comparison between Figs. 6 and 7, it is obvious that the phenomenon with the low-frequency fluctuation holds rate of its appearance not its length scale at different heights. This is the same as turbulent patches and puffs in transitional channel and pipe flows. In con-
trast to the low-frequency phenomenon, the vortical motions have tendency of invariable length scale with the measuring height, so that the high-frequency peak shifts to lower frequency with approaching to the wall. This seems to be the reason that the low and high peaks are indistinct at y/d = 0.1. In turn, the two peaks are detectable at the channel center because of their separation between the peak frequencies. In Fig. 6, the low-frequency peak in the spectral distribution at y/d = 0.5 weakens with increasing Re, and then at Re = 40 0 0 it immerses in the spectral distribution continuing from the highfrequency peak, with forming the plateau around f ∗ = 0.04. This reduction is distinct from the tendency at very high Re that the low-frequency peak increases in magnitude with increase in Re as observed by Kim and Adrian (1999) and Hutchins and Marusic (2007) in pipe flow and a turbulent boundary layer, respectively. Owing to lack of knowledge about the low-frequency phenomena at low Re, further investigations are needed to compare to those in high Re wall-bounded shear flows. Considering Re reduction before the test section and a very long distance between the measurement position and the tripping rods, it can be denied that the low-frequency peak is due to footprint of the transition occurring just downstream the rods. Therefore, it is supposed that the low-frequency flow motion is inherent nature of the turbulent channel flow. 4. Conclusion The spectral peak at the low frequency that corresponds to the streamwise length scale 25 times as large as the channel width was confirmed in the low Reynolds number but turbulent channel flow for Re ≤ 30 0 0, and the spectral plateau around that fre-
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
quency remains at least up to Re = 40 0 0. The further research would reveal nature of such large-scale fluctuation and its relation to the very large features at very high Reynolds number. That research would reveal the very-low-frequency phenomenon in turbulent channel flows. References Cossu, C., Pujals, G., Depardon, S., 2009. Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94. Farrell, B.F., Ioannou, P.J., 2012. Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196. 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. Hutchins, N., Marusic, I., 2007. Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.
597
´ M.R., Bamieh, B., 2005. Componentwise energy amplification in channel Jovanovic, flows. J. Fluid Mech. 534, 145–183. Kim, K.C., Adrian, R.J., 1999. Very large-scale motion in the outer layer. Phys. Fluids 11, 417–422. McKeon, B. J., Sharma, A. S., Jacobi, I., 2010. Predicting structural and statistical features of wall turbulence. ArXiV 1012-0426. Monty, J.P., Hutchins, N., Ng, H.C.H., Marusic, I., Chong, M.S., 2009. A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442. 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. Okumura, T., Matsubara, M., 2006. Experimental study on minimal Reynolds number with LDV measurement and visualization. In: Proceedings of 2006 Annual Meeting. Japan Society of Fluid Mechanics. AM06-03-010. Seki, D., Matsubara, M., 2012. Experimental investigation of relaminarizing and transitional channel flows. Phys. Fluids 24, 124102.
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Very-large-scale fluctuations in turbulent channel flow at low Reynolds number✩ Masaharu Matsubara∗, Shun Horii, Yoshiyuki Sagawa, Yuta Takahashi, Daisuke Saito Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan
a r t i c l e
i n f o
Article history: Available online 6 December 2016 Keywords: Channel flow Turbulence Structure Large-scale
a b s t r a c t It is known that there exist very large features in turbulent channel flow at high Reynolds number as well as in pipe flow and turbulent boundary layer. In addition, a low frequency peak was observed in spectra of the streamwise velocity fluctuation in transitional and low-Reynolds-number but turbulent channel flows. In this study, the large-scale fluctuation observed at the low Reynolds number has been experimentally explored with increasing Reynolds number by means of a hot wire anemometry. There are two peaks in the streamwise velocity spectra for transitional flow for 1800 ≤ Re ≤ 2600, where Re is the Reynolds number based on the bulk velocity and the channel width. The high-frequency peak corresponds to the turbulent vortices that have the same order of magnitude as the channel width, while the low-frequency peak is due to passages of the turbulent patches whose streamwise scale is greater than ten channel widths. It is surprising that the low frequency peak remains even up to Re = 30 0 0 at which the flow is fully turbulent. Furthermore, a spectral plateau around the frequency corresponding to 25 channel widths is confirmed up to Re = 40 0 0, indicating that there exist very-large-scale fluctuations in turbulent channel flow even at low Reynolds number. © 2016 Elsevier Inc. All rights reserved.
1. Introduction It has been worthy of note that there exist very large features in turbulent pipe and channel flows at high Reynolds number (Monty et al., 2007; Kim and Adrian, 1999; Guala et al., 2006), as well as in turbulent boundary layer (Hutchins and Marusic, 2007), because it may relate the turbulent properties and coherent structures near the wall. Monty et al. (2007) performed velocity measurements with hot-wire rakes, and they revealed long meandering structures that has the streamwise length up to 25 pipe radii or channel halfheights. Guala et al. (2006) also detected a very large structure in high-Reynolds-number pipe flow whose streamwise length is as long as 16 pipe radii. Linear and quasi-linear theories (Jovanovic´ and Bamieh, 2005; Cossu et al., 2009; McKeon et al., 2010; Farrell and Ioannou, 2012) predict roll-streak structures of very large streamwise length that might relate to the these very large feature in shear flows. These experimental and theoretical findings strongly suggest that very
✩ This paper is a contribution to the special issue CJWSF 2015 and CMFF’15. Guest edited by Editors-in-chief Andrew Pollard and Suad Jakirlic Guest editors (ICJWSF 2015): Laszlo Fuchs, Antonio Segalini Guest editor (CMFF’15): Janos Vad. ∗ Corresponding author. E-mail address: [email protected] (M. Matsubara).
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.11.009 0142-727X/© 2016 Elsevier Inc. All rights reserved.
large features or structures are omnipresent in turbulent wallbounded shear flows at high Reynolds number. Monty et al. (2009) made hot-wire measurements in turbulent pipe, channel and boundary layer flows. From the velocity spectra maps with different heights, it was revealed that very large scale motion (VLSM) around 0.3 δ (δ is a pipe radii or a channel half height) has the streamwise length larger than 10 δ . They strongly claim that the VLSM in internal flows should not be mixed up with the superstructures in boundary layers. Recently, Seki and Matsubara (2012) made hot-wire measurements for the transitional channel flow that was realized by decreasing the Reynolds number of turbulent channel flow in a expansion section. They found a spectral peak in the streamwise velocity fluctuation measured in transitional channel flow and observed intermittency with the turbulent patches passing through the channel, and from probability density distributions of the streamwise velocity fluctuation they asserted that the transitional range is for 1400 < Re < 2600. Reynolds number is defined as Re = Ub d/ν , where Ub is the bulk velocity, d is channel width, ν is kinematic viscosity. It is surprising that the low frequency peak is confirmed even for the fully turbulent flow at Re = 2660, suggesting existence of a large-scale flow pattern. Their investigation raises a question if the two large-scale fluctuations at low and high Reynolds numbers are the same. If they are the essentially same, it is straightforwardly inferred that the VLSM has
594
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 1. Air channel facility.
sustain and/or generation mechanisms common to the turbulent patches in traditional channel flows. As the first step to answer this question, the large-scale fluctuation has been experimentally investigated with gradually increasing Reynolds number from Re = 10 0 0, at which the flow is laminar, to Re = 40 0 0.
Fig. 2. Time traces of the streamwise velocity at the channel center. (a) Re = 1500, (b) Re = 20 0 0, (c) Re = 260 0, (d) Re = 350 0. The red and black circles indicate high-frequency and low velocity periods, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2. Experiment setup.
3. Result and discussion Time traces of the streamwise velocity measured at the channel center are shown in Fig. 2. At Re = 1500, the signal is almost calm but high-frequency fluctuations intermittently appear, as marked
0.1
f∗
The hot wire measurement was performed in an air channel flow facility shown in Fig. 1. The facility is the same that Seki and Matsubara (2012) used. The air pressurized by a blower gushes from a perforated pipe installed inside a settling chamber whose cross section is 190 mm × 170 mm. A two-dimensional nozzle of a 34:1 contraction ratio is followed by entrance, expansion and test sections whose wall distance, d, is 5.1 mm. The entrance section is 10 0 0 mm long and 190 mm wide. Two tripping rods of 0.5 mm diameter are mounted 100 mm downstream from the outlet of the nozzle for triggering transition to turbulence, so that the flow becomes a fully developed turbulent flow at the end of the entrance section. The distance between the channel end walls is gradually widened from 190 mm to 260 mm in the 10 0 0 mm. As a result, Re drops to 73% of the entrance mean velocity in the expansion section. The 2800 mm long test section following the expansion section opens to the atmosphere at its downstream end. In a certain range of Re, the Re drop in the expansion section induces transition from upstream turbulent flow to transitional, or intermittent, flow in the test section. A constant-temperature hot-wire anemometer was used for the streamwise velocity measurement. The single hot-wire sensor is a platinum wire of 2.5 μm diameter and 1 mm length. Velocity calibrations for the sensor were performed with a jet potential core emanating from an axial nozzle connected to the blower for the main channel facility. The relation between the voltage and the velocity was fitted to a calibration curve deduced from King’s law modified for low velocity. An analog/digital converter acquired the voltage of the hot wire anemometer with a sampling frequency of 20 kHz and a sampling time of 300 s. A wedge mechanism, which could be inserted in the channel at an arbitrary streamwise position, was used for probe positioning in the wall-normal direction. The symmetry of the velocity profile was utilized for estimation of the channel center. The measurement was made at 300 mm upstream from the channel exit.
1
0.01
0.001
1000
1500
2000
2500
3000
3500
4000
Re Fig. 3. Contour map of streamwise velocity premultiplied spectra density at the channel center. Contours and colours express f∗ ∗ . White and blue lines indicate ridges for the high and low frequency peaks, respectively. The second peaks at each Re are marked as broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by red circles. This intermittent behavior of the flow is also investigated by Seki and Matsubara (2012), thus it is considered that these high-frequency fluctuations indicate the passages of turbulent patches. Appearance of the high-frequency fluctuations becomes more frequent at Re = 20 0 0, then finally at Re = 2600, the time trace are filled of high-frequency fluctuations, indicating that the flow is fully turbulent. Seki and Matsubara (2012) also confirmed from the probability density distributions that intermittency at Re = 260 0 is almost 10 0%. In spite of the turbulent state, the low velocity but high-frequency fluctuating periods (circled in black) still exist. Even at higher Reynolds number, Re = 3500, the low velocity periods are still observed though their appearance turns unclear with increase of Re. The rate of their appearance can be estimated in the premultiplied spectral diagram of the streamwise velocity fluctuation as shown in Fig. 3. The measurement was performed at 31 Reynolds numbers with the Re resolution of 100. The non-dimensional frequency f∗ and the power spectral density ∗ are normalized with the channel width d and the bulk (mean) velocity Ub as f ∗ = f d/Ub and ∗ = /Ub2 . As a consequence of this normalization, the inverse number of f∗ corresponds to a length scale in unit of d. While
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 4. Contour map of streamwise velocity premultiplied spectra density at y/d = 0.25. Contour spacing and colour mapping are the same as Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
total energy at Re = 10 0 0 is very low because the flow is laminar, a peak around f∗ = 0.01 appears for Re = 1400 showing the flow becomes transitional state. As the Reynolds number increases, the peak shifts to higher frequency and the peak value reaches the maximum at Re = 1500. Seki and Matsubara (2012) compared the peak frequency to the turbulent-patch appearance rate and they concluded that they are in good agreement within the middle of the transitional range. For 2300 ≤ Re ≤ 4000, the high-frequency peak around f ∗ = 0.4 appears, while the low-frequency peak declines. In flow visualization performed by Okumura and Matsubara (2006), the flow for Re ≥ 2010 is filled with vortical struc-
595
Fig. 5. Contour map of streamwise velocity premultiplied spectra density at y/d = 0.10. Colour mapping is the same and contour spacing is double to that in Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
tures that have a streamwise length scale similar to the channel width, indicating that the flow contains large amount of turbulent regions. Hence, it is inferred that the high-frequency peak is due to the turbulent vortical motions. It is surprising that the lowfrequency peak is still confirmed up to Re = 30 0 0 that is considerably higher than the upper marginal Re of 2600 ascertained by Seki and Matsubara (2012) as the lowest Re for fully turbulent flow. Furthermore, even for Re > 30 0 0 a plateau stays around f ∗ = 0.04, at least, in the present Re range. This corresponds to 25 channel widths taking the bulk velocity Ub as a convection velocity.
Fig. 6. Distributions of the premultiplied spectra f∗ ∗ in f∗ . (a) Re = 1800, (b) Re = 2600, (c) Re = 30 0 0, (d) Re = 40 0 0.
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M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
Fig. 7. Distributions of the premultiplied spectra α ∗ ∗ in α ∗ . (a) Re = 1800, (b) Re = 2600, (c) Re = 30 0 0, (d) Re = 40 0 0.
In Fig. 3, the low-frequency peak at low Re continuously ranges from the highest peak at Re = 1500 that is due to the turbulent patch passing. This suggests that its structure is similar to the turbulent patch that spread in both stream and spanwise directions more than one order higher than the channel width. At the quarter wall distance, y/d = 0.25, there are two peaks at Re = 1800 as shown in Fig. 4. As mentioned above, the lowfrequency peak is due to the passage of the turbulent patches. The low frequency peak is continues up to Re = 2600, though for Re > 2700 there is only the one visible peak. It is not able to distinguish the low and high peaks near the wall as shown in Fig. 5; one peak in the spectral distribution at each Re remains within the bounds, 0.02 < f∗ < 0.07. The spectra distributions are compared at different heights from the wall, in Figs. 6 and 7. The low-frequency peaks in the spectra at Re = 1800 (Fig. 6 (a)) stay around f ∗ = 0.025, regardless of the height. This indicates that a structure distributes in a wide region in the wall-normal direction like a turbulent patch. The frequency independence from the height is confirmed for the higher Re in Fig. 6 (b), (c) and (d), though the peak frequency increases with increase of Re. Fig. 7 shows the spectral distribution in the normalized spatial wave number that was estimated from the frequency f and local time-mean velocity Um as α ∗ = f d/Um . The positions of the high-wave-number peaks in the spectral distributions at Re = 30 0 0 and 40 0 0 are quite close, though the positions of the low-wavenumber peak are not constant in this scaling. In comparison between Figs. 6 and 7, it is obvious that the phenomenon with the low-frequency fluctuation holds rate of its appearance not its length scale at different heights. This is the same as turbulent patches and puffs in transitional channel and pipe flows. In con-
trast to the low-frequency phenomenon, the vortical motions have tendency of invariable length scale with the measuring height, so that the high-frequency peak shifts to lower frequency with approaching to the wall. This seems to be the reason that the low and high peaks are indistinct at y/d = 0.1. In turn, the two peaks are detectable at the channel center because of their separation between the peak frequencies. In Fig. 6, the low-frequency peak in the spectral distribution at y/d = 0.5 weakens with increasing Re, and then at Re = 40 0 0 it immerses in the spectral distribution continuing from the highfrequency peak, with forming the plateau around f ∗ = 0.04. This reduction is distinct from the tendency at very high Re that the low-frequency peak increases in magnitude with increase in Re as observed by Kim and Adrian (1999) and Hutchins and Marusic (2007) in pipe flow and a turbulent boundary layer, respectively. Owing to lack of knowledge about the low-frequency phenomena at low Re, further investigations are needed to compare to those in high Re wall-bounded shear flows. Considering Re reduction before the test section and a very long distance between the measurement position and the tripping rods, it can be denied that the low-frequency peak is due to footprint of the transition occurring just downstream the rods. Therefore, it is supposed that the low-frequency flow motion is inherent nature of the turbulent channel flow. 4. Conclusion The spectral peak at the low frequency that corresponds to the streamwise length scale 25 times as large as the channel width was confirmed in the low Reynolds number but turbulent channel flow for Re ≤ 30 0 0, and the spectral plateau around that fre-
M. Matsubara et al. / International Journal of Heat and Fluid Flow 62 (2016) 593–597
quency remains at least up to Re = 40 0 0. The further research would reveal nature of such large-scale fluctuation and its relation to the very large features at very high Reynolds number. That research would reveal the very-low-frequency phenomenon in turbulent channel flows. References Cossu, C., Pujals, G., Depardon, S., 2009. Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94. Farrell, B.F., Ioannou, P.J., 2012. Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196. 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. Hutchins, N., Marusic, I., 2007. Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.
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´ M.R., Bamieh, B., 2005. Componentwise energy amplification in channel Jovanovic, flows. J. Fluid Mech. 534, 145–183. Kim, K.C., Adrian, R.J., 1999. Very large-scale motion in the outer layer. Phys. Fluids 11, 417–422. McKeon, B. J., Sharma, A. S., Jacobi, I., 2010. Predicting structural and statistical features of wall turbulence. ArXiV 1012-0426. Monty, J.P., Hutchins, N., Ng, H.C.H., Marusic, I., Chong, M.S., 2009. A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442. 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. Okumura, T., Matsubara, M., 2006. Experimental study on minimal Reynolds number with LDV measurement and visualization. In: Proceedings of 2006 Annual Meeting. Japan Society of Fluid Mechanics. AM06-03-010. Seki, D., Matsubara, M., 2012. Experimental investigation of relaminarizing and transitional channel flows. Phys. Fluids 24, 124102.