On the structure of turbulent channel flow (original) (raw)

Abstract

Hot-film measurements of the streamwise velocity component were carried out in a fully developed turbulent water-channel flow for three different Reynolds numbers (13 800, 34 600 and 48 900). The results for the first four statistical moments complement and extend the results from previous studies of turbulent channel flow. The VITA variance technique waa employed to detect deterministic events in the streamwise velocity. It waa demonstrated that the VITA technique has a band-pass-filter character. The number of events detected was found to decrerrae exponentially with the threshold level and the events occupy a wide range of timescales. This makes it impossible to define one unique frequency of occurrence or one unique duration of the events. However, by using this technique information was obtained on the amplitude and timescale distributions of the events. The chmacteristic features of the conditional iverages were found to be related to the skewness and flatness factors.

Figures (18)

Ficure 1. The water tunnel: (2) pump, (0) test section, (c) water reservoir. variations by the method used by Fahlgren et al. (1981). However, this compensation was always less than 0-4 % of the velocity. For each measurement point 230000 samples of the anemometer output signal were collected (after subtracting a d.c. voltage using the DISA D265 signal conditioner) through the 12-bit A/D converter of a DEC MINC system (PDP11/23), and stored on floppy disk. The time between consecutive samples was chosen as about half the viscous timescale. For the lowest velocity this corresponds to a sampling time of about 20 min at each measuring point. The probability density distributions of the anemometer signal were computed from the time series and transformed to the probability density distributions of the streamwise velocity by using the calibration curve. The different moments, i.e. mean velocity, r.m.s. velocity, skewness and flatness factors, were then calculated from these distributions. Assembler programs were used for most of the numerical work.

Ficure 1. The water tunnel: (2) pump, (0) test section, (c) water reservoir. variations by the method used by Fahlgren et al. (1981). However, this compensation was always less than 0-4 % of the velocity. For each measurement point 230000 samples of the anemometer output signal were collected (after subtracting a d.c. voltage using the DISA D265 signal conditioner) through the 12-bit A/D converter of a DEC MINC system (PDP11/23), and stored on floppy disk. The time between consecutive samples was chosen as about half the viscous timescale. For the lowest velocity this corresponds to a sampling time of about 20 min at each measuring point. The probability density distributions of the anemometer signal were computed from the time series and transformed to the probability density distributions of the streamwise velocity by using the calibration curve. The different moments, i.e. mean velocity, r.m.s. velocity, skewness and flatness factors, were then calculated from these distributions. Assembler programs were used for most of the numerical work.

Figure 2. The short-time variance (at t = 0) of u(t) = sin wt (—7/w < t < 7/w), 0 (otherwige).

Figure 2. The short-time variance (at t = 0) of u(t) = sin wt (—7/w < t < 7/w), 0 (otherwige).

TaBLeE 1. Characteristics of the channel flows Figure 3 shows a portion of the u-signal and the corresponding variance functions calculated for three different integration times (corresponding to 5, 10 and 20 viscous time units). The variance functions have a rather intermittent appearance, especially for short integration times. The band-pass-filter character of the VITA technique is well illustrated in this figure. For instance at A the variance function has a peak, which increases with increasing integration time. It corresponds to a rather slow decrease of the u-velocity. However, this deceleration is followed by a rapid acceleration at B, which gives the variance function its highest value for the shortest integration time. At C the acceleration of the flow has a timescale somewhere in between the other two, and the variance function has its highest value for an intermediate integration time.

TaBLeE 1. Characteristics of the channel flows Figure 3 shows a portion of the u-signal and the corresponding variance functions calculated for three different integration times (corresponding to 5, 10 and 20 viscous time units). The variance functions have a rather intermittent appearance, especially for short integration times. The band-pass-filter character of the VITA technique is well illustrated in this figure. For instance at A the variance function has a peak, which increases with increasing integration time. It corresponds to a rather slow decrease of the u-velocity. However, this deceleration is followed by a rapid acceleration at B, which gives the variance function its highest value for the shortest integration time. At C the acceleration of the flow has a timescale somewhere in between the other two, and the variance function has its highest value for an intermediate integration time.

development of turbulent channel flow. Her experiments showed that a good charac- terization of the stage of development could be made, based on the values of the skewness and flatness factors on the channel centreline. A channel flow that is not fully developed will show a large negative value of the skewness and a large positive value of the flatness factor. Comte-Bellot found that the fully developed regime, where all statistical moments are independent of the downstream distance, was

development of turbulent channel flow. Her experiments showed that a good charac- terization of the stage of development could be made, based on the values of the skewness and flatness factors on the channel centreline. A channel flow that is not fully developed will show a large negative value of the skewness and a large positive value of the flatness factor. Comte-Bellot found that the fully developed regime, where all statistical moments are independent of the downstream distance, was

is presented in the standard form of u+ vs. y+ (ut = U/u,, yt = y/l,). For this Rey- nolds number the friction velocity could be determined from the slope of the mean- velocity profile in the viscous sublayer. For the higher Reynolds numbers the friction velocities were determined from the condition of best fit to the logarithmic velocity law with von Kérmén’s constant equal to 0:41. The results show a slight reduction, with increasing Reynolds number, of the additive constant in the logarithmic velocity law (table 1). Also included in figure 4 are the maximum and minimum velocities measured at each point, showing the large span of velocities present. Ca) ee ee ee, , , : es a: is ce i or . ¢ ) nan

is presented in the standard form of u+ vs. y+ (ut = U/u,, yt = y/l,). For this Rey- nolds number the friction velocity could be determined from the slope of the mean- velocity profile in the viscous sublayer. For the higher Reynolds numbers the friction velocities were determined from the condition of best fit to the logarithmic velocity law with von Kérmén’s constant equal to 0:41. The results show a slight reduction, with increasing Reynolds number, of the additive constant in the logarithmic velocity law (table 1). Also included in figure 4 are the maximum and minimum velocities measured at each point, showing the large span of velocities present. Ca) ee ee ee, , , : es a: is ce i or . ¢ ) nan

two Channels (v7/2b0 = o24 tor Kreplin and #%/206 = Ou tor the present stuay). Frequency spectra were calculated by means of standard FFT routines. The data are presented as fH, i.e. the frequency times the energy density function, in a linear scale, versus fin a logarithmic scale. The ordinate is normalized so that the total non- dimensional energy is unity. Spectra for the three Reynolds numbers at y+ ~ 50 and further out collapse when the frequency is scaled with the outer timescale. This scaling was also found by Perry & Abell (1975) to be appropriate in the outer region of turbu- lent pipe flow. However, the outer scaling works less well with decreasing distance from the wall. At y+ = 13 neither outer (figure 6) nor inner scaling works satisfac- torily. In figure 6 a clear Reynolds-number trend is seen in the high-frequency range. Not even at the edge of the viscous sublayer (y+ = 5) do the spectra for all three Rey- nolds numbers collapse with inner scaling.

two Channels (v7/2b0 = o24 tor Kreplin and #%/206 = Ou tor the present stuay). Frequency spectra were calculated by means of standard FFT routines. The data are presented as fH, i.e. the frequency times the energy density function, in a linear scale, versus fin a logarithmic scale. The ordinate is normalized so that the total non- dimensional energy is unity. Spectra for the three Reynolds numbers at y+ ~ 50 and further out collapse when the frequency is scaled with the outer timescale. This scaling was also found by Perry & Abell (1975) to be appropriate in the outer region of turbu- lent pipe flow. However, the outer scaling works less well with decreasing distance from the wall. At y+ = 13 neither outer (figure 6) nor inner scaling works satisfac- torily. In figure 6 a clear Reynolds-number trend is seen in the high-frequency range. Not even at the edge of the viscous sublayer (y+ = 5) do the spectra for all three Rey- nolds numbers collapse with inner scaling.

the reference time will also be referred to as having positive slope. The number of such events detected per unit time will be denoted by mpos. The corresponding quantity for the events with negative slope is denoted by mneg. A large portion of the results presented in the following were obtained for y+ positions of about 18, where the turbulence production and intensity are largest. sc De eM es es i OO er ee: at es cee ge ey Oi oe a A

the reference time will also be referred to as having positive slope. The number of such events detected per unit time will be denoted by mpos. The corresponding quantity for the events with negative slope is denoted by mneg. A large portion of the results presented in the following were obtained for y+ positions of about 18, where the turbulence production and intensity are largest. sc De eM es es i OO er ee: at es cee ge ey Oi oe a A

are not very distinct, which implies that the events occupy a wide range of timescales for all threshold levels. The relation between 7' and the duration of the events will become evident from the conditional averages presented later. Information is thus obtained on the timescale distribution of the events from plots as those in figure 8. For this case (yt = 13) the 7-value for which pos has its maximum corresponds to about 16 viscous time units or about 1-0 in outer units and is only about one-third of that for which mpeg has its maximum. This means that the timescale for the retarda- tions is typically several times as large as the timescale for the accelerations. This is probably related to the large positive values of the skewness factor of du/dt as mea- sured by Comte-Bellot (1965) and also to the shape of the u-signal patterns found by Wallace et al. (1977). These patterns are characterized by a relatively slow retardation followed by a rapid acceleration. A comparison between the curves in figure 8 and the corresponding power spectrum in figure 6 reveals that most of the turbulent energy is found in a frequency range that corresponds to timescales for which many events are detected.

are not very distinct, which implies that the events occupy a wide range of timescales for all threshold levels. The relation between 7' and the duration of the events will become evident from the conditional averages presented later. Information is thus obtained on the timescale distribution of the events from plots as those in figure 8. For this case (yt = 13) the 7-value for which pos has its maximum corresponds to about 16 viscous time units or about 1-0 in outer units and is only about one-third of that for which mpeg has its maximum. This means that the timescale for the retarda- tions is typically several times as large as the timescale for the accelerations. This is probably related to the large positive values of the skewness factor of du/dt as mea- sured by Comte-Bellot (1965) and also to the shape of the u-signal patterns found by Wallace et al. (1977). These patterns are characterized by a relatively slow retardation followed by a rapid acceleration. A comparison between the curves in figure 8 and the corresponding power spectrum in figure 6 reveals that most of the turbulent energy is found in a frequency range that corresponds to timescales for which many events are detected.

FicureE 12. The data of figure 11 separated into events with (a) positive slope, (b) negative slope.

FicureE 12. The data of figure 11 separated into events with (a) positive slope, (b) negative slope.

Ficure 16. Conditional averages for events having positive slope for different integration times; Re = 13800, y+ = 12-9, & = 1-0.

Ficure 16. Conditional averages for events having positive slope for different integration times; Re = 13800, y+ = 12-9, & = 1-0.

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