2.5.1. Velocity measurements

The main bias errors of the velocity measurements derived from the velocity bias error and the calibration factor ${\it\delta}_{f}={\it\lambda}/(2\sin {\it\beta})$ , where ${\it\lambda}$ is the laser wavelength and ${\it\beta}$ the beam half-angle. The laser was aimed at a distant wall and the spacing between the beams was measured to determine the half-angle. From a Monte Carlo simulation, the uncertainty of the calibration factor was determined to $\pm$ 1 %. The uncertainty from the velocity bias error was estimated to $\pm$ 0.7 % by calculating the difference between the corrected and uncorrected values $b_{vb}^{2}=(\overline{U}-(1/N)\sum _{i}U_{i})^{2}$ (see Mulu Reference Mulu2012). The random error was assessed to $\pm$ 0.65 % by calculating the standard deviation of the mean values from repeated runs of the steady-state measurements. The overall uncertainty of the velocity measurements at the 95 % confidence level was therefore $\pm 2\sqrt{1^{2}+0.65^{2}+0.7^{2}}=\pm 2.5\,\%$ .

2.5.2. Wall shear stress measurements

The uncertainty of the calculated wall shear stress was assessed by expanding a 95 % confidence interval around the calibration curve defined by (2.1). The errors involved in determining the calibration curve with corresponding uncertainty intervals are composed of two parts: (i) the estimation of ${\it\tau}$ and (ii) the uncertainties in the hot-film voltage. The main sources of uncertainties when estimating the wall shear stress are related to the inaccuracies of the friction factor, the flow rate and the pipe diameter. The random error of the estimated wall shear stress resulted from the inaccuracies of the flow meter and the inability to reach repeatable conditions. The random error varied between 0.3 % and 0.8 %, with larger uncertainties occurring at lower Reynolds numbers because of the lower accuracy of the flow meter for small flow velocities. The corresponding systematic error was estimated to be 3.2 %. The main sources of errors in the measured hot-film voltage were related to the mounting of the sensor and drifting of the calibration curve due to contamination of the sensor and changes in the water temperature. Sensitivity analyses of the hot-film voltage dependence of the water temperature and frequent updating of the calibration curve led to an estimate of the random error of the hot-film voltage within $\pm$ 0.6 %. The systematic error was estimated to 1.8 %. From the uncertainties presented above and the method of propagating uncertainties to calibration curves derived by Coleman & Steele (Reference Coleman and Steele2009), the final uncertainty at the 95 % confidence level of the measured wall shear stress used from (2.1) was $\pm$ 13 %.

3.1.1. Mean velocity profiles

Figure 4(a,b) shows the time-averaged axial velocity in cases I and II, respectively. The data are presented in wall units, i.e. $\overline{U}^{+}$ versus $y^{+}$ . No measurements were performed in the region in which $\overline{U}^{+}=y^{+}$ , which introduced ambiguity to determine the exact position of the velocity profile along the abscissa. To overcome this issue, the residuals of the steady-state velocity measurements up to $y^{+}=14$ were minimized with respect to the curve $\overline{U}^{+}=13.3\,\text{tanh}(y^{+}/13.3)$ . This curve was obtained from a least-squares fit of the pipe flow LDV measurements ( $Re=7442$ ) in the region $1<y^{+}<14$ presented in Durst, Jovanovic & Sender (Reference Durst, Jovanovic and Sender1995). The data of Durst et al. (Reference Durst, Jovanovic and Sender1995) are in close agreement with the pipe flow DNS ( $Re=11\,700$ ) of El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), which justifies the use of these data as a reference for locating the position of the measurement volume in the present measurements. The uncertainty of the position of the measured profile along the abscissa is $\pm {\it\delta}_{{\it\nu}}/2$ , considering the accuracy of the LDV measurements and the precision of the traverse. In addition to the time-averaged profiles from the unsteady measurements, the steady-state measurements, the curve $\overline{U}^{+}=13.3\,\text{tanh}(y^{+}/13.3)$ and the log law with ${\it\kappa}=0.41$ and $B=5.5$ are plotted as reference data. Despite some scatter, all points collapse onto the same curve, showing that the mean velocity is insensitive to the imposed pulsations; their profiles are similar to the reference data.

Figure 4.

Time-averaged axial velocity profiles: (a) case I, (b) case II. The dotted line is covered by the experimental data for $y^{+}>6$ .

3.1.2. Mean turbulence profiles

Figure 5(a–d) presents the time-averaged r.m.s. turbulent velocity fluctuations $\overline{u^{\prime }}_{i,rms}^{+}$ , data from the steady pipe flow DNS performed by El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and the results from the pulsating channel flow LES at $l_{s}^{+}=7,14$ and 35 by Scotti & Piomelli (Reference Scotti and Piomelli2001). As observed for the mean velocity, the time-averaged turbulence profiles from the steady and unsteady cases are similar, with no sign of the imposed pulsations. For the axial component, the peak value of approximately 2.6 at $y^{+}\approx 14$ is in excellent agreement with the DNS. When $y^{+}>100$ , the measured profiles start to diverge from the DNS because of the difference in Reynolds number between the studies (see Durst et al. Reference Durst, Jovanovic and Sender1995). For the LES, the data closely overlap the present results at the highest pulsation but $\overline{u^{\prime }}_{rms}^{+}$ increases systematically with $l_{s}^{+}$ , and for $l_{s}^{+}=35$ , the peak value is approximately 15 % larger and has moved towards the wall. As discussed in § 1, the increased turbulence level is probably attributable to a larger turbulence production, stemming from the additional term $-\overline{\widetilde{u^{\prime }v^{\prime }}\partial \widetilde{U}/\partial y}$ in the transport equation for $\overline{u^{\prime 2}}$ , which becomes effective for large-amplitude, low-frequency, oscillations. Closer inspection shows that the peak value and the location of the peak are offset by roughly equal amounts. Thus, part of the difference can be attributed to scaling effects. As will be discussed in § 3.1.3, the mean wall shear stress can be expected to increase for low-frequency, large-amplitude, oscillations. The LES were however performed with a fixed mean pressure gradient, and thus at a fixed mean wall shear. The data were rescaled (not shown) with this expected increase taken into account. A closer collapse was achieved but, as observed in figure 5(a), the offset for $l_{s}^{+}=35$ is not constant and the phenomenon is thus more complicated than a matter of scaling.

Figure 5.

Time-averaged r.m.s. velocity fluctuations: (a,c) case I, (b,d) case II. Steady pipe flow DNS by El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013),  $Re=11\,700$ : solid line. Channel flow LES by Scotti & Piomelli (Reference Scotti and Piomelli2001): dotted, $l_{s}^{+}=7$ ; dashed, $l_{s}^{+}=14$ ; dashed-dotted, $l_{s}^{+}=35$ .

Close to the wall, the data of the tangential component display a larger scatter than the axial counterpart but the profiles from the unsteady cases are not appreciably different from the steady measurements. The measured peak values are slightly larger than that from the DNS, but $\overline{w^{\prime }}_{rms}^{+}$ showed a significant dependence upon the Reynolds number in the DNS (performed at $Re=5300,~11\,700,~19\,000$ and 37 700) and the experimental data fall midway between the plotted curve ( $Re=11\,700$ ) and the one at $Re=19\,000$ .

The relative turbulence intensity $\overline{u^{\prime }}_{rms}/\overline{U}$ (the steady profile of which is plotted on figure 10 e) collapses onto a single curve for all investigated cases. The intensity increases in the radial direction and the measurement closest to the wall ( $y^{+}=6$ ) attains a value of ${\sim}$ 0.35. The profiles converge towards an asymptotic constant value between 0.35 and 0.4, as expected, because both the turbulent fluctuations and the mean velocity grow linearly with $y$ close to the wall. The asymptotic turbulence intensity agrees well with the pipe flow DNS ( $Re=5300$ ) performed by Eggels et al. (Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994), and the measured r.m.s. wall shear stress fluctuations presented in figure 6(b), which are randomly scattered around 0.37.

Figure 6.

Time-averaged wall shear stress in steady and unsteady flows: (a) mean values and (b) r.m.s. turbulent fluctuations from hot-film measurements. The steady hot-film value has been used for normalization in (a). The error bars in (a) denote one standard deviation from the hot-film measurements. MH, data from Mao & Hanratty (Reference Mao and Hanratty1986); TBB, data from Tardu et al. (Reference Tardu, Binder and Blackwelder1994).

3.1.3. Mean wall shear stress and turbulent fluctuations

The steady-state, case I and case II time-averaged wall shear stress, measured with the hot-film sensor and estimated from the LDV data, is plotted in figure 6(a). The time-averaged LDV measurements in the logarithmic region (taken to be $30<y^{+}<100$ ) were used to calculate the value of $\bar{u}_{{\it\tau}}$ that yielded the optimum fit to the log law with ${\it\kappa}=0.41$ and $B=5.5$ assumed to hold. The wall shear stress was subsequently calculated from $\bar{{\it\tau}}={\it\rho}\bar{u}_{{\it\tau}}^{2}$ . The procedure of calculating the wall shear stress from the velocity was performed, not to determine ${\it\kappa}$ and $B$ , nor to investigate the existence of a universal overlap region, but merely as a means of comparison. The hot-film measurements were repeated five times at each operating point and the average of these repetitions has been used for the presented data. The error bars denote one standard deviation. To assess the repeatability of the wall shear stress measurements from the velocity data, the steady-state LDV measurements were repeated three times in the logarithmic region and displayed a similar scatter.

The two sets of measurements are in close agreement, with the largest deviation between the methods within 1.5 %. Furthermore, no trend is seen between the cases; the case I and the case II data are scattered within $\pm$ 1 % of the steady value. These results are somewhat different from those presented in Tardu & Binder (Reference Tardu and Binder1993) and Tardu et al. (Reference Tardu, Binder and Blackwelder1994), in which the mean wall shear stress showed a larger scatter. Their data fell between $+$ 14 % and $-$ 9 % around the steady value, although the subset plotted in figure 6(a) displays a smaller scatter. At large oscillation amplitudes, the deviations from the steady value were argued to originate from a nonlinear interaction between the oscillating flow and the mean flow. By assuming that the Blasius friction formula holds at each instant of the oscillation cycle, Tardu et al. (Reference Tardu, Binder and Blackwelder1994) derived an expression for the mean wall shear stress in the quasi-steady regime: $\bar{{\it\tau}}_{qs}=\bar{{\it\tau}}_{steady}(1+(21/64)a_{\widetilde{U}_{c}}^{2})$ . In their study, the maximum value of the normalized centreline velocity oscillations, $a_{\widetilde{U}_{c}}=A_{\widetilde{U}_{c}}/\overline{U}_{c}$ , was 0.7. This corresponds to a 16 % larger mean wall shear stress compared to the steady case and is in good agreement with the maximum value measured in their study. However, the scatter in their data was quite significant and showed no definite trend of increasing values of the time-averaged wall shear stress at large amplitudes. In the present study, $a_{\widetilde{U}_{c}}\approx 0.13$ , which results in a correction term ${<}$ 1 %, thus suggesting that the nonlinearity is negligible.

The normalized r.m.s. turbulent fluctuations ( ${\it\tau}^{\prime }/\bar{{\it\tau}}$ ) are plotted in figure 6(b). The data are scattered between 0.36 and 0.38, in agreement with the limiting value of the close-wall LDV measurements. Data from Mao & Hanratty (Reference Mao and Hanratty1986) and Tardu et al. (Reference Tardu, Binder and Blackwelder1994) at similar values of $l_{s}^{+}$ have been included in the figure. The multiple data points stem from different combinations of frequencies and Reynolds numbers. The scatter in their data is quite large, and the r.m.s. fluctuations are generally smaller than 0.36. However, recent hot-film measurements in transient pipe flows performed by He et al. (Reference He, Ariyaratne and Vardy2011) found a steady-state value of 0.38. Furthermore, ${\it\tau}^{\prime }/\bar{{\it\tau}}$ is a weak function of the Reynolds number (see Durst et al. Reference Durst, Jovanovic and Sender1995) and usually displays scatter among different studies due to the experimental difficulties involved when measuring this quantity. Consequently, the conclusion to be drawn from the present data is that ${\it\tau}^{\prime }/\bar{{\it\tau}}$ is unaffected by small-amplitude pulsations.

3.2.1. Amplitude and phase of the oscillating velocity

The amplitudes of the streamwise oscillating velocity in cases I and II are plotted in figure 7(a,b). The results are plotted versus the wall-normal coordinate normalized with the Stokes length $y_{s}~(=y/l_{s})$ to enable comparison with the analytical laminar solutions derived by Uchida (Reference Uchida1956):

In (3.1), $\text{J}_{0}$ is the Bessel function of order zero and the first kind; $A_{\widetilde{U}_{c_{n}}}$ , $l_{s_{n}}$ and ${\it\omega}_{n}$ are the $n$ th mode of the amplitude of the centreline velocity, the laminar Stokes length and the angular frequency, respectively; in case I, $n=1$ , and in case II, $n=\{1,2;1,5;1,10\}$ ; $\text{Re}[\cdots ]$ denotes the real part of the argument; $\text{i}=\sqrt{-1}$ is the imaginary unit; and $R$ is the pipe radius. Near the wall, (3.1) is approximately given by

where the wall-normal coordinate $y=R-r$ is introduced to match the coordinate system in figure 2.

Figure 7.

Amplitude of the oscillating velocity normalized with the centreline value for (a) case I and (b) case II. The solid, dotted and dashed lines in (b) are the laminar Stokes solutions for the combination of a fundamental mode along with its tenth, fifth and second harmonic, respectively. HJ, data from He & Jackson (Reference He and Jackson2009).

Figure 8.

Phase difference relative to the oscillating centreline velocity: (a) case I, (b) individual modes from case II. In (b), only one set of data has been included to increase the readability. HJ, data from He & Jackson (Reference He and Jackson2009).

In case I, the amplitude profile for $l_{s}^{+}=8$ plotted in figure 7(a) follows the Stokes solution fairly well, although the measured peak amplitude is smaller than the laminar value and attained at a larger $y_{s}$ . For $y_{s}>6$ , the velocity oscillates as a plug, similar to the Stokes solution. The thickened oscillating boundary layer indicates that turbulence diffuses the shear wave generated at the wall, thereby supporting the similarity parameter $l_{t}^{+}$ proposed by Scotti & Piomelli (Reference Scotti and Piomelli2001); see § 1. The thickening of the oscillating boundary layer is also displayed in the phase difference, ${\it\phi}_{\widetilde{U}}-{\it\phi}_{\widetilde{U}_{c}}$ , with the local minimum displaced by approximately two Stokes units into the core; see figure 8(a).

The results from the intermediate regime ( $l_{s}^{+}=18$ ) are consistent with the findings of previous studies (see Tardu & Binder Reference Tardu and Binder1993; Tardu et al. Reference Tardu, Binder and Blackwelder1994; Scotti & Piomelli Reference Scotti and Piomelli2001). Near the wall for $y_{s}<1$ , the amplitude of the oscillating velocity follows the Stokes solution, although the flow is no longer in the quasi-laminar regime. Because the oscillating wall shear stress is given by $\widetilde{{\it\tau}}={\it\mu}\,\partial \widetilde{U}/\partial y$ , the observed behaviour indicates that the turbulence decreases the oscillating shear in this regime. This phenomenon will be discussed thoroughly in § 3.2.3, in which the measurements of the oscillating wall shear stress are presented.

A feature of the amplitude of the oscillating velocity in the intermediate and the low-frequency regimes ( $l_{s}^{+}=18$ and $l_{s}^{+}=25$ ) is that the amplitude profiles display similarities to the time-averaged velocity plotted on figure 4. For $y_{s}>2$ , i.e. when the direct effect of molecular viscosity is small, a logarithmic region exists in which the slopes of the two curves are linear. The normalized Stokes length; $l_{s}^{+}=l_{s}\bar{u}_{{\it\tau}}/{\it\nu}$ can be interpreted as a Reynolds number for the oscillating flow, and for sufficiently large $l_{s}^{+}$ (say $l_{s}^{+}>15$ ), features of the oscillating flow similar to the time-averaged flow could be expected. For $y_{s}>2$ and sufficiently far from $R_{s}~(=R/l_{s})$ , dimensional reasoning leads to $\text{d}\widetilde{U}/\text{d}y=C\widetilde{U}_{c}/y$ . Integrating this expression results in $\widetilde{U}/\widetilde{U}_{c}=C\,\text{ln}(y)+D$ , where $C$ and $D$ are constants that depend on $l_{s}^{+}$ . The onset of the logarithmic velocity distribution is accompanied by a reversal of ${\it\phi}_{\widetilde{U}}-{\it\phi}_{\widetilde{U}_{c}}$ ; see figure 8(a). For $y_{s}<2$ , the phase shift increases linearly on a semi-logarithmic scale whereas ${\it\phi}_{\widetilde{U}}-{\it\phi}_{\widetilde{U}_{c}}$ decreases at a nearly equal rate for $y_{s}>2$ . Amplitude and phase data at similar values of $l_{s}^{+}$ from the pipe flow study ( $Re=7000$ ) by He & Jackson (Reference He and Jackson2009) have been included in figures 7(a) and 8(a). The general trends are qualitatively similar to the present results but quantitatively there are disparities, especially in the phase difference. As for the phase difference, for $l_{s}^{+}\gtrsim 15$ , a local maximum of ${\it\phi}_{\widetilde{U}}-{\it\phi}_{\widetilde{U}_{c}}$ at $y_{s}\approx 2$ is also displayed in the data reported by Tardu et al. (Reference Tardu, Binder and Blackwelder1994) and Cervantes & Engstrom (Reference Cervantes and Engstrom2008) among others, but the peak value does not seem to scale in a simple way with $l_{s}^{+}$ .

The wall-normal development of the amplitude of the oscillating velocity in case II is plotted in figure 7(b) together with the corresponding Stokes solutions. The amplitude of the experimental and the analytical data in this figure is defined as $A_{\widetilde{U}}=(\max (\widetilde{U})-\min (\widetilde{U}))/2$ . The longest period of oscillation has been used to define the Stokes length because it is the shortest common period among the pulsations. The $l_{s}^{+}=25$ and 8 profile appear similar to the profile for $l_{s}^{+}=25$ from case I displaced vertically and with a smaller slope in the logarithmic region. The smaller slope can be explained by considering the normalized amplitude at this combination, approximately given by $(A_{8}+A_{25})/(A_{8_{c}}+A_{25_{c}})$ , where the numerator terms are the amplitudes at any wall-normal coordinate, $y_{s}$ , and the terms in the denominator are the centreline amplitudes. For $y^{+}>16$ , $A_{8}\approx A_{8_{c}}$ and the amplitude profile becomes $K+A_{25}/(A_{8_{c}}+A_{25_{c}})$ , where $K$ is a constant and the second term grows with $y_{s}$ , i.e. this profile resembles a displaced $l_{s}^{+}=25$ profile with a smaller slope in the logarithmic region. In fact, to a good approximation, the general shape of all case II profiles can be obtained by superposing the profiles from case I. For $y_{s}<0.6$ , the two profiles with a pulsation at $l_{s}^{+}=8$ follow the Stokes solutions fairly well, whereas the data at the combination $l_{s}^{+}=25$ and 18 diverge from the analytical solution. The same trend is displayed by the wall shear stress measurements presented in § 3.2.3. This does not, however, imply that the harmonic at $l_{s}^{+}=25$ tends towards the Stokes solution at the combination $l_{s}^{+}=25$ and 8 (see figure 9(a) and the discussion in the next paragraph). Rather, the resemblance to the analytical curve is the result of a favourable addition of amplitudes (with respect to the Stokes solution).

To further discern whether the pulsations interact when combined or whether the Fourier harmonics in case II are a superposition of the responses from case I, the fundamental mode in addition to either the $n=2$ , 5 or 10 harmonic were extracted from the respective Fourier series defined in (2.6). The amplitude profiles of the individual harmonics from case II, $l_{s_{IND}}^{+}$ , are plotted on figure 9(a–c) along with the case I profiles. Despite some scatter, the harmonics from case II for $l_{s_{IND}}^{+}=25$ and $l_{s_{IND}}^{+}=18$ are similar to the case I profiles, regardless of the combination of pulsations. The response of the oscillating velocity for $l_{s_{IND}}^{+}=8$ on the other hand is slightly dependent on which pulsations are combined. For $l_{s}^{+}=25$ and 8, the $l_{s_{IND}}^{+}=8$ profile is similar to a displaced Stokes solution, whereas it deviates slightly from the laminar profile when $l_{s}^{+}=18$ and 8 are imposed simultaneously. The phase difference of the individual harmonics from case II, $({\it\phi}_{\widetilde{U}}-{\it\phi}_{\widetilde{U}_{c}})_{IND}$ , is plotted on figure 8(b). As for the oscillating velocity, two sets of data from case II are available for each value of $l_{s_{IND}}^{+}$ but only one set is displayed in the figure because of the similarity between the results. Only small deviations from case I are seen for $l_{s_{IND}}^{+}=18$ and $l_{s_{IND}}^{+}=25$ , whereas the data for $l_{s_{IND}}^{+}=8$ are more scattered around the case I values. However, the profile for $l_{s_{IND}}^{+}=8$ is the fifth harmonic, and any errors in locating the phase of this pulsation imply a five-fold amplification of the error in the calculated phase difference.

Figure 9.

Oscillating velocity amplitude of individual modes from case II: (a)  $l_{s_{IND}}^{+}=25$ , (b)  $l_{s_{IND}}^{+}=18$ and (c) $l_{s_{IND}}^{+}=8$ . The data from case I have been included as reference along with the Stokes solution.

3.2.2. Amplitude and phase of the oscillating turbulent fluctuations

Figure 10 shows the development of the amplitude of the turbulent velocity fluctuations along the wall-normal coordinate from the two sets of unsteady measurements. The data have been scaled in two ways: (i) with the mean friction velocity and (ii) with the local oscillating velocity.

Figure 10.

Amplitudes of the modulation of the axial and tangential turbulent velocity fluctuations: (a,c,e) case I, (b,d,f) case II. (a–d) Scaled with the mean friction velocity. (e–f) Turbulence intensity with respect to local velocity amplitude. SP, data from Scotti & Piomelli (Reference Scotti and Piomelli2001).

The modulation amplitudes scaled with the friction velocity $(A_{\widetilde{u_{i}^{\prime }u_{i}^{\prime }}}/\bar{u}_{{\it\tau}}^{2})$ from case I are plotted on figure 10(a,c). The profiles of the axial component are similar to the time-averaged streamwise velocity fluctuations, with peak amplitudes near $y^{+}=14$ , although, in spite of the scatter in the data, it appears that the peak value moves towards the wall with increasing $l_{s}^{+}$ . The peak amplitude is smaller than the time-averaged value because of the small amplitude of the imposed oscillations. The profiles do not exhibit a monotonic trend function of the forcing frequency; the largest peak value is attained for $l_{s}^{+}=18$ , and the smallest for $l_{s}^{+}=8$ . The difference between the peak values for $l_{s}^{+}=18$ and $l_{s}^{+}=25$ is small, however. The turbulence level for $l_{s}^{+}=8$ is significantly lower than at the other two frequencies because of the small Stokes length at this frequency. As a result, most of the shear generated at the wall is attenuated in the viscous sublayer, which inhibits interactions with the steady turbulence field. For $y^{+}>100$ , the amplitude is almost zero, i.e. the modulation of turbulence is frozen. The low turbulence level at $l_{s}^{+}=8$ is displayed even more clearly in the profile of the tangential component. Here, the amplitude is small along the entire radius, and, as for the axial component, the modulation is zero for $y^{+}>100$ . At the two lower pulsation frequencies, the profiles are similar to their steady counterparts, but again of smaller magnitude. The amplitudes of the oscillating turbulent fluctuations in case II, $A_{\widetilde{u_{i}^{\prime }u_{i}^{\prime }}}=(\max (\widetilde{u_{i}^{\prime }u_{i}^{\prime }})-\min (\widetilde{u_{i}^{\prime }u_{i}^{\prime }}))/2$ , are plotted on figure 10(b,d). For both the axial and the tangential components, the amplitudes are nearly the addition of the amplitudes from case I when one pulsation is at $l_{s}^{+}=8$ ; the same is not true at the combination $l_{s}^{+}=25$ and 18. This follows from two arguments. (i) As for the oscillating velocity plotted on figure 9(a–c), the amplitudes of the turbulent fluctuations of the individual harmonics in case II are negligibly different from the case I amplitudes. (ii) A sinusoid and its fifth harmonic as well as a sinusoid and its tenth harmonic are close to peak at a common phase; the same is not true for a sinusoid and its second harmonic. These phase relationships are closely retained for the turbulent fluctuations all the way from the near-wall region to the centreline. Consequently, the profiles plotted on figure 10(b,d) are observed.

The modulation of the turbulent velocity fluctuations scaled with the local oscillating velocity $(A_{\widetilde{u^{\prime }}_{rms}}/A_{\widetilde{U}})$ from case I is plotted in figure 10(e) along with the steady-state profile ( $\overline{u^{\prime }}_{rms}/\overline{U}$ ) and the LES data from Scotti & Piomelli (Reference Scotti and Piomelli2001). For $l_{s}^{+}=18$ and $l_{s}^{+}=25$ , the data in the near-wall region collapse onto a single curve, one that is similar to the steady profile and to that for $l_{s}^{+}=35$ from the LES. The resemblance between the current measurements and the LES is noteworthy considering that the imposed amplitudes in the experiments are approximately 10 % whereas the corresponding number in the simulation is 77 %, and the difference in geometry (pipe versus channel). The modulation for $l_{s}^{+}=8$ is significantly smaller than at the lower frequencies but, again, the collapse with the LES for $l_{s}^{+}=7$ is quite good. From the current data and the data presented in Tardu et al. (Reference Tardu, Binder and Blackwelder1994), Scotti & Piomelli (Reference Scotti and Piomelli2001) and He & Jackson (Reference He and Jackson2009), the following are noted: (i) for $l_{s}^{+}\gtrsim 15$ , the oscillating velocity exhibits a logarithmic distribution and the turbulence intensities collapse onto the same curve; (ii) for $l_{s}^{+}$ of $O(10{-}15)$ , a logarithmic velocity distribution does not exist nor does the oscillating velocity follow the Stokes solution; and (iii) for $5<l_{s}^{+}<10$ , the oscillating velocity follows the Stokes solution and the turbulence level is low, which supports labelling this range of $l_{s}^{+}$ values ‘quasi-laminar’.

Figure 11.

Time lag in wall units between the modulation of the turbulent fluctuations and the modulation of the centreline velocity: (a) case I and (b) case II for $l_{s_{IND}}^{+}=25$ and $l_{s_{IND}}^{+}=18$ from the combination $l_{s}^{+}=25$ and $18$ . Open symbols, axial component; closed symbols, tangential component. HJ, data from He & Jackson (Reference He and Jackson2009).

The case II turbulence intensities are plotted in figure 10(f). At the combination of the two lower frequencies, the modulation of the turbulence intensity collapses onto the same curve as when the pulsations are imposed alone (figure 10 e). When one pulsation is at $l_{s}^{+}=8$ , the data collapse onto a single curve, which converges towards a value that agrees well with the amplitude of the turbulent oscillating wall shear stress plotted in figure 13(a).

The time lag of the modulation of the axial and tangential turbulent fluctuations relative to the modulation of the centreline velocity in case I is plotted in figure 11(a). For $l_{s}^{+}=18$ and $l_{s}^{+}=25$ , there is a region $50<y^{+}<200$ where the modulations of both components of the turbulent fluctuations diffuse at similar rates, $|\text{d}y^{+}/\text{d}{\rm\Delta}t^{+}|\approx 0.4$ , in agreement with the data of Tardu et al. (Reference Tardu, Binder and Blackwelder1994) and Scotti & Piomelli (Reference Scotti and Piomelli2001) for large $l_{s}^{+}$ . Consequently, for pulsations of sufficiently low frequency ( $l_{s}^{+}>15$ , say), there is a region in which the diffusivity of the modulation of the turbulent kinetic energy, ${\it\nu}_{\tilde{k}}^{+}$ , is of order $0.4y^{+}$ . This value is similar to the diffusivity of the turbulent kinetic energy in the overlap region of high-Reynolds-number steady pipe and channel flows, which scales with ${\it\kappa}y^{+}$ (see Pope Reference Pope2000).

In the near-wall region for $y^{+}\lesssim 50$ , the time lag of the modulation of the tangential velocity fluctuations relative to the modulation of the axial centreline velocity remain nearly constant regardless of the forcing frequency (even though the constant value differs between the pulsations). A plausible explanation for the constancy of the phase of $\widetilde{w^{\prime }w^{\prime }}$ in this region lies in the mechanism by which energy is supplied to this component. For $y^{+}<50$ , the main source term for $\widetilde{w^{\prime }w^{\prime }}$ is through redistribution by pressure strain from the modulation of the axial turbulent fluctuations. The finite time scale involved in this process causes the constant time lag of $\widetilde{w^{\prime }w^{\prime }}$ . Further out in the flow ( $y^{+}>60$ ), where turbulence diffusion becomes important in the energy budget, the modulation of the velocity fluctuations of both the tangential and axial components diffuse into the core at similar rates. Thus, as the core is approached, the time scale of the redistribution by pressure strain becomes negligible compared to that of the turbulence diffusion. The responses of $\widetilde{u^{\prime }u^{\prime }}$ and $\widetilde{w^{\prime }w^{\prime }}$ resemble those found for $\overline{u^{\prime 2}}$ and $\overline{w^{\prime 2}}$ in the study of transient accelerating pipe flow by He & Jackson (Reference He and Jackson2000). They showed that turbulence initially responds to the imposed acceleration by an increase of $\overline{u^{\prime 2}}$ in the buffer layer, whereas the time delay of $\overline{w^{\prime 2}}$ relative to the imposed acceleration is independent of position up to $y^{+}=50$ . Beyond this point, both components were shown to diffuse into the core at similar speeds.

Figure 12.

Amplitude of the oscillating wall shear stress normalized by the Stokes amplitude. (a) Case I and individual harmonics from case II. (b) Case II non-decomposed. SP, data from Scotti & Piomelli (Reference Scotti and Piomelli2001); TBB, data from Tardu et al. (Reference Tardu, Binder and Blackwelder1994).

The time lag of the individual modes in case II was calculated from the respective Fourier series. The profiles for $l_{s_{IND}}^{+}=25$ and $l_{s_{IND}}^{+}=18$ , taken from the combination $l_{s}^{+}=25$ and 18, are plotted on figure 11(b). The difference from case I is small, and the same is true for the individual modes at the other combinations of pulsations as well. These data have, however, been left out from the figure.

3.2.3. Amplitude of the oscillating wall shear stress

Considering case I plotted on figure 12(a), the amplitude of the oscillating wall shear stress normalized with the laminar Stokes value $A_{\widetilde{{\it\tau}}_{s}}=\sqrt{2}A_{\widetilde{U}_{c}}{\it\mu}/l_{s}$ displays features that are similar to those of the amplitude of the oscillating velocity plotted in figure 7(a). At the highest investigated frequency, $l_{s}^{+}=8$ , the normalized amplitude is approximately 95 % of the laminar value, which indicates that the effect of the shear wave generated at the wall is mainly confined within the viscous sublayer in which the turbulence is negligible. As a result, the oscillating velocity gradient is not appreciably different from that in a laminar flow, which results in an amplitude of the wall shear stress that is similar to that in the Stokes solution.

For $l_{s}^{+}=18$ , the amplitude of the oscillating wall shear is close to that of the laminar value, even though the velocity measurements at this frequency have displayed characteristics of a turbulent flow. Several previous studies (Ronneberger & Ahrens Reference Ronneberger and Ahrens1977; Mao & Hanratty Reference Mao and Hanratty1986; Tardu et al. Reference Tardu, Binder and Blackwelder1994; Scotti & Piomelli Reference Scotti and Piomelli2001) have identified a minimum of $A_{\widetilde{{\it\tau}}}$ in the neighbourhood of $l_{s}^{+}=14$ . This effect was first noted by Ronneberger & Ahrens (Reference Ronneberger and Ahrens1977), who observed a minimum of $A_{\widetilde{{\it\tau}}}$ for $l_{s}^{+}\approx 11$ . They predicted this minimum by modelling turbulence as a rigid wall located outside the viscous sublayer at $y^{+}=15$ . For $l_{s}^{+}$ of $O(10{-}15)$ , the wavelength of the shear wave generated at the wall matches the distance to the ‘turbulent wall’. Destructive interference between the wave generated at the wall and the reflected wave was used to explain the minimum. The model showed qualitative agreement with their experimental data. However, because the LDV measurements presented in figure 7(a) follow the Stokes solution as long as $y_{s}<1$ , another potential explanation for the low value of the oscillating wall shear stress is due to an effective viscosity, ${\it\nu}_{e}$ . The similarity between the measured profile and the Stokes solution suggests that, close to the wall, a modified version of the laminar equation holds, i.e.

where the effects of the turbulence have been captured in the effective viscosity, ${\it\nu}_{e}$ , being the sum of the molecular and eddy viscosities. Equation (3.3) has solutions similar to (3.2), and thus the amplitude of the oscillating wall shear stress is approximately given by $A_{\widetilde{{\it\tau}}}\approx \sqrt{2}A_{\widetilde{U}_{c}}{\it\mu}/l_{s_{e}}$ , where $l_{s_{e}}=\sqrt{2{\it\nu}_{e}/{\it\omega}}$ is the effective Stokes length. Because $l_{s_{e}}$ grows with the square root of ${\it\nu}_{e}$ , the amplitude of the oscillating wall shear stress for $l_{s}^{+}=18$ is smaller than the purely viscous value $A_{\widetilde{{\it\tau}}_{s}}=\sqrt{2}A_{\widetilde{U}_{c}}{\it\mu}/l_{s}$ .

In the low-frequency regime, the flow tends towards a quasi-equilibrium in which the oscillating components of the centreline velocity and the wall shear stress are nearly in phase. From a phase-averaged version of the Blasius friction relation, Tardu et al. (Reference Tardu, Binder and Blackwelder1994) derived an estimate for the relative amplitude of the oscillating wall shear stress $A_{\widetilde{{\it\tau}}_{qs}}/A_{\widetilde{{\it\tau}}_{s}}=0.058l_{s}^{+}$ . For $l_{s}^{+}=25$ , the normalized amplitude becomes 1.45, thus being in good agreement with the measured wall shear stress, which overshoots the viscous Stokes value by 40 %. One reason for the small discrepancy might be that an $l_{s}^{+}$ value of 25 is too low to be considered quasi-steady, the exact limit being subtle to determine precisely. The data from Tardu et al. (Reference Tardu, Binder and Blackwelder1994) and Scotti & Piomelli (Reference Scotti and Piomelli2001) have been included in figure 12(a) for comparison – the results are in fairly good agreement.

The amplitude of the oscillating wall shear stress in case II, $A_{\widetilde{{\it\tau}}}=(\max (\widetilde{{\it\tau}})-\min (\widetilde{{\it\tau}}))/2$ , is plotted on figure 12(b). The data have been normalized with the viscous Stokes amplitude, which is defined as $A_{\widetilde{{\it\tau}}_{s}}=(\max (\widetilde{{\it\tau}}_{s})-\min (\widetilde{{\it\tau}}_{s}))/2$ , where $\widetilde{{\it\tau}}_{s}$ is approximately given by $\widetilde{{\it\tau}}_{s}\approx \sum \sqrt{2}A_{\widetilde{U}_{c_{n}}}{\it\mu}\cos ({\it\omega}_{n}t-{\rm\pi}/4)/l_{s_{n}}$ . When one pulsation is at $l_{s}^{+}=8$ , the deviations from the Stokes solutions are approximately $\pm$ 5 %, in line with the LDV measurements, which follow the Stokes solutions quite closely for $y_{s}<0.6$ . For the combination $l_{s}^{+}=25$ and 18, the oscillating wall shear stress amplitude overshoots the laminar value by 20 %, which also conforms with the LDV measurements, which are larger than the Stokes value for $y_{s}<0.4$ . Because of the relatively long period time of both oscillations for this combination of frequencies, the amplitude of the oscillatory wall shear stress can be estimated from the Blasius relation by assuming quasi-steady flow. Then, at any instant of the oscillating cycle, the phase-averaged wall shear stress is given by

where $K$ is a constant and $\langle U_{c}\rangle =\overline{U}_{c}+\widetilde{U}_{c}$ . Expanding (3.4) around $\overline{U}_{c}$ and keeping terms up to linear order yields

for the periodic component. Taking $\widetilde{U}_{c}=A_{\widetilde{U}_{c_{1}}}\sin ({\it\omega}t)+A_{\widetilde{U}_{c_{2}}}\sin (2{\it\omega}t)$ as the oscillatory part of the velocity and simplifying the analysis by assuming that $A_{\widetilde{U}_{c_{1}}}=A_{\widetilde{U}_{c_{2}}}$ , being true to within 5 % in the current measurements, it follows that

where $A[\cdots ]$ is the amplitude of the argument and the factor 0.058 results from $\bar{u}_{{\it\tau}}/\overline{U}_{c}\approx 0.048$ . The measured value corresponds well with the theoretical prediction, although only one pulsation is of low enough frequency to approach the limit of quasi-steadiness. A similar analysis was applied to the combination $l_{s}^{+}=25$ and 8 but without successful correlation because of the high frequency of the $l_{s}^{+}=8$ pulsation.

Figure 13.

Amplitude of the oscillating wall shear stress turbulent fluctuations normalized by the oscillating wall shear stress. (a) Case I and individual harmonics from case II. (b) Case II non-decomposed.

The amplitudes of the individual modes of the phase-averaged wall shear stress in case II, normalized with the corresponding Stokes values, are also plotted on figure 12(a). The results from case I serve as reference data, and, as can be seen from the figure, the amplitude of the individual modes are all within $\pm$ 5 % of the reference.

3.2.4. Amplitude of the oscillating turbulent wall shear stress

As shown in §§ 3.2.1 and 3.2.2, the oscillating velocity in case I undergoes a transition from a turbulent regime to a quasi-laminar regime in the range of $10<l_{s}^{+}<15$ . Over this range of $l_{s}^{+}$ , the turbulence intensity decreases significantly and the oscillating velocity no longer admits a logarithmic distribution. The turbulent to laminar transition is clearly manifested in the oscillating component of the r.m.s. turbulent fluctuating wall shear stress plotted on figure 13(a) as well. The amplitude is eight times smaller for $l_{s}^{+}=8$ than at the other two frequencies and in close agreement with the LDV measurements. The amplitudes at $l_{s}^{+}=18$ and $l_{s}^{+}=25$ are similar, which conforms with the collapse of the turbulence intensity profiles observed in the LDV measurements plotted in figure 10(e). The individual harmonics from case II have been included with the case I data in figure 13(a). The harmonics from case II are within $\pm$ 5 % of the values from case I, thereby conforming with the previous data showing negligible interactions between the pulsations. The full response of the case II measurement data plotted in figure 13(b) is also in line with the velocity measurements. When one pulsation is at $l_{s}^{+}=8$ , the turbulence intensities collapse to the same value, and, at the combination $l_{s}^{+}=25$ and 18, the turbulence intensity takes the same value as when these pulsations are imposed alone.