2 Dataset

In this paper, the streamwise–wall-normal planes extracted from direct numerical simulation (DNS) of a TBL (Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013) are examined. The friction Reynolds number $Re_{{\it\tau}}={\it\delta}U_{{\it\tau}}/{\it\nu}$ at the middle of the streamwise domain of extracted fields is 1780, where ${\it\delta}$ , $U_{{\it\tau}}$ and ${\it\nu}$ denote the boundary layer thickness, friction velocity and kinematic viscosity, respectively. The boundary layer thickness is determined by fitting the composite velocity profile of Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009) at the middle of the streamwise domain of extracted fields and it is approximately $25\,\%$ greater than the $99\,\%$ boundary layer thickness. The streamwise domain of extracted fields is $9{\it\delta}$ and it is chosen to be sufficiently long to observe the large-scale features in the flow while minimising the Reynolds-number variation within the streamwise domain. Across this streamwise domain, $Re_{{\it\tau}}$ based on the local boundary layer thickness varies from 1659 to 1888, corresponding to $Re_{{\it\theta}}={\it\theta}U_{\infty }/{\it\nu}=4000$ to 4630, where ${\it\theta}$ is the momentum thickness and $U_{\infty }$ is the free-stream velocity. The flow field is considered to have recovered from the inflow conditions at those Reynolds numbers (Sillero et al. Reference Sillero, Jiménez and Moser2013).

Figure 1. (a) Instantaneous streamwise velocity field; (b) fluctuating velocity field using the traditional Reynolds decomposition; and (c) fluctuating velocity field determined using the new decomposition method. Black lines show the turbulent/non-turbulent interface. (d) The streamwise velocity fluctuations as a function of the streamwise position at $z=0.6{\it\delta}$ . Blue and red lines represent the fluctuations under the Reynolds decomposition, $u$ , and the new decomposition, $u^{\prime }$ , respectively. Horizontal dotted line indicates the zero level, and the grey shadings are in the free stream.

3 A new decomposition

Figure 1 shows an instantaneous streamwise velocity field of a TBL. Throughout this paper, $x$ and $z$ denote the streamwise and wall-normal coordinates, respectively, and $U$ and $W$ denote the total instantaneous velocity components in the corresponding directions. Velocity quantities with an overline indicate the temporal mean, and the fluctuating velocity components about the temporal mean are represented by lower-case variables (e.g. $U=\overline{U}+u$ ; Reynolds decomposition). The superscript ‘ $+$ ’ represents normalisation by inner scales ( ${\it\nu}/U_{{\it\tau}}$ for length and $U_{{\it\tau}}$ for velocity). The heavy black lines in figure 1(a–c) represent the turbulent/non-turbulent interface (TNTI), which is determined using the kinetic energy criteria of Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014).

In the fluctuating streamwise velocity field based on the Reynolds decomposition (figure 1 b), one can observe that the region of strongly positive Reynolds fluctuation actually extends across the TNTI into the free stream when the TNTI height is low (at $x/{\it\delta}\simeq 2{-}3$ in figure 1 b). This is mathematically correct because the free stream is a region with positive velocity fluctuation relative to the temporal mean streamwise velocity at this wall-normal location. However, one should not consider it as a region of ‘turbulent’ fluctuation since it exists in the non-turbulent free stream as a consequence of the oscillation of the TNTI and does not really signify the presence of an organised turbulent motion at this wall-normal location. It happens because the intermittent nature of the TBL is ignored in the Reynolds decomposition (i.e. the mean is calculated over regions of two different states having different velocities).

Figure 2. (a) Grey lines are randomly selected instantaneous total velocity profiles at $x$ locations where the TNTI is at 68 % of the boundary layer thickness. Solid line is the ensemble average of all such instantaneous velocity profiles. (b) Conditional mean velocity profiles (blue solid lines) for a range of interface heights. Red dashed line indicates the conventional temporal mean velocity profile, $\overline{U}$ .

In order to overcome this problem, a new way of decomposing the total velocity is proposed, where the TBL and non-turbulent free stream are treated separately:

(3.1) $$\begin{eqnarray}U=\widetilde{U}(z,{\it\delta}_{i})+u^{\prime }.\end{eqnarray}$$

Here $\widetilde{U}(z,{\it\delta}_{i})$ denotes the conditional mean velocity, which is a function of the wall-normal coordinate and the instantaneous TNTI height, ${\it\delta}_{i}$ ; and $u^{\prime }$ is the fluctuation about $\widetilde{U}$ . This decomposition is comparable to the triple decomposition introduced by Hussain & Reynolds (Reference Hussain and Reynolds1970), although their triple decomposition isolates the periodic variation in velocity while our decomposition removes the influence of the TNTI oscillation, which is not periodic. Strictly speaking, $\tilde{U}$ is also a function of $Re_{{\it\tau}}$ (just as $\overline{U}$ is a function of $Re_{{\it\tau}}$ ). To fully isolate the effect of the $Re_{{\it\tau}}$ variation, $\tilde{U}$ needs to be computed at every streamwise grid point available (at each streamwise grid point, $Re_{{\it\tau}}$ is different due to the spatial growth of boundary layer). However, this requires a vast number of independent velocity vector fields to achieve a reasonable degree of convergence in $\tilde{U}$ (having many $\tilde{U}$ profiles at each and every streamwise grid point). Within the streamwise domain of extracted velocity fields, the variation of $Re_{{\it\tau}}$ and ${\it\delta}$ is not excessive (deviation of $\pm 7\,\%$ and $\pm 8\,\%$ about their nominal values for $Re_{{\it\tau}}$ and ${\it\delta}$ , respectively). Therefore, as a compromise to achieve a reasonable degree of convergence, the Reynolds-number variation over the $9{\it\delta}$ domain is not accounted for in computing $\tilde{U}$ and $u^{\prime }$ for the present study. This introduces a bias to $u^{\prime }$ such that it is shifted slightly towards the positive side around the beginning of the streamwise domain (upstream end) and towards the negative side around the end of the streamwise domain (downstream end). However, the effect of this bias on the statistics presented is marginal and not significant enough to affect the conclusions to be drawn in this paper.

Figure 2 explains how $\widetilde{U}$ is calculated. In figure 2(a), some randomly selected instantaneous total streamwise velocity profiles at streamwise locations where ${\it\delta}_{i}=0.68{\it\delta}$ are shown in grey lines. The ensemble average of all such instantaneous velocity profiles gives the conditional mean profile, $\widetilde{U}$ , for this particular ${\it\delta}_{i}$ (shown as the black line in figure 2 a). In the wall-normal profile of $\widetilde{U}$ , the free stream and TBL are treated separately about ${\it\delta}_{i}$ because each $\widetilde{U}$ is computed from the instantaneous $U$ profiles with the same TNTI location. Then, a series of $\widetilde{U}$ are computed for a range of different ${\it\delta}_{i}$ . Figure 2(b) shows examples of $\widetilde{U}$ at various ${\it\delta}_{i}$ . In cases of large TNTI oscillation, the profiles of $\widetilde{U}$ can deviate substantially from the temporal mean velocity profile (red dashed line) in the intermittent region of TBL because $\overline{U}$ is averaged over two different states (TBL and free stream). However, the profiles of $\widetilde{U}$ and $\overline{U}$ almost fall on top of each other in the region close to the wall. Hence, the region close to the wall will be insensitive to the decomposition method.

The fluctuating velocity component under this new decomposition can be obtained by subtracting the profiles of $\widetilde{U}(z,{\it\delta}_{i}(x))$ from $U$ at every streamwise location. Note that $\widetilde{U}$ varies with the streamwise position since ${\it\delta}_{i}$ is a function of $x$ (and $\widetilde{U}$ is a function of ${\it\delta}_{i}$ and $z$ ). When the TNTI folds back onto itself (multiple TNTI locations at a given streamwise position), it is not appropriate to subtract a single continuous profile of $\widetilde{U}$ . In such cases, the region below the lowermost TNTI location and the region above the uppermost TNTI location are considered separately by subtracting the corresponding $\widetilde{U}$ for each respective region (below the lowermost TNTI location, the profile of $\widetilde{U}$ based on the lowermost TNTI location is subtracted, and above the uppermost TNTI location, the profile of $\widetilde{U}$ based on the uppermost TNTI location is subtracted). In the region between the upper- and lowermost TNTI locations (the actual region affected by the TNTI folding back), fluctuating velocity is computed about the zonal mean velocities of each respective region (Kovasznay et al. Reference Kovasznay, Kibens and Blackwelder1970) rather than $\widetilde{U}$ . For example, if the point of interest (residing within the TNTI folding back region) lies within the free stream, the zonal mean velocity of the free stream (the average velocity of all points that are in the free stream at a given wall-normal location) is subtracted to compute $u^{\prime }$ . Note that the actual area affected by the TNTI folding back is only approximately $1\,\%$ of the total velocity fields having the streamwise and wall-normal domain of $9{\it\delta}\times 1.4{\it\delta}$ .

Figure 1(c) shows the fluctuating streamwise velocity field determined using the new decomposition method. Under this new decomposition, the large-scale positive fluctuations extending into the free stream are removed (see $x/{\it\delta}\simeq 2{-}4$ in figure 1 c). In addition, some large-scale negative fluctuations in the boundary layer (at $x/{\it\delta}\simeq 4{-}5$ ) are also removed by the new decomposition. As previously illustrated in figure 2(b), $\widetilde{U}$ for large ${\it\delta}_{i}$ is substantially smaller than $\overline{U}$ in the outer region because a large proportion of free stream, which is faster than the flow in the boundary layer, is included in calculation of $\overline{U}$ . Hence, the Reynolds decomposition (subtraction of $\overline{U}$ ) causes large-magnitude negative fluctuations in the intermittent region inside the TBL when the boundary layer is instantaneously thicker. An example illustrating the difference between the fluctuating streamwise velocity component under the two decompositions is shown in figure 1(d). It shows the streamwise velocity fluctuations under the Reynolds decomposition and our decomposition as a function of the streamwise position at $z/{\it\delta}=0.6$ . Here, the regions of the TBL that are outside of the TNTI (i.e. in the free stream) are shaded in grey and the fluctuating velocity under our decomposition (red dashed) in this region only shows the fluctuation about the free stream, so their magnitude is close to zero whereas the fluctuation is positive under the Reynolds decomposition (blue solid) in the free stream. Also, the excessive negative fluctuation in $u$ , caused by an instantaneously thicker boundary layer, is removed in $u^{\prime }$ . It should be noted that the fluctuation under our decomposition is ensured to have zero temporal mean.

Overall, under the Reynolds decomposition, the oscillation of the TNTI can contaminate the fluctuating velocity components in the wake region, and it is demonstrated that the new velocity decomposition method removes this contamination. In contrast, the fluctuating velocity fields near the wall (log region and below) are not significantly altered by applying the new decomposition, suggesting that the fluctuating velocity fields near the wall are independent of the velocity decomposition method (as expected from the profiles of $\widetilde{U}$ and $\overline{U}$ being almost the same near the wall). The other velocity components (wall-normal and spanwise) are not significantly affected by the decomposition method as compared with the streamwise velocity component, so they are not presented in this paper.

Figure 3. Intermittency profile of the TBL.

4 Effect on typical statistics

In order to quantify the degree of intermittent behaviour, the intermittency, ${\it\gamma}$ , is plotted in figure 3. It is defined as the proportion of time that the flow spends in the turbulent region (Corrsin & Kistler Reference Corrsin and Kistler1955; Kovasznay et al. Reference Kovasznay, Kibens and Blackwelder1970); the intermittency profile of the present study agrees well with Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014). Up to approximately $z/{\it\delta}=0.4$ , the flow is practically non-intermittent ( ${\it\gamma}=1$ ). With increasing wall-normal distance from the wall, the TBL starts to exhibit the intermittent behaviour at wall-normal positions above $z/{\it\delta}=0.4$ . This suggests that the decomposition method should not affect the near-wall region since the decomposition method is only important in the intermittent region.

Figure 4. Pre-multiplied power spectral densities of the streamwise velocity fluctuation determined under Reynolds decomposition (blue solid) and our new decomposition (red dashed) at the wall-normal position of (a)  $z=0.1{\it\delta}$ , (b)  $z=0.4{\it\delta}$ and (c)  $z=0.66{\it\delta}$ . (d) Pre-multiplied power spectral density of ${\it\delta}_{i}$ fluctuation.

In order to quantitatively investigate the effect of the TNTI oscillation on the Reynolds fluctuation, pre-multiplied power spectral densities of the streamwise velocity fluctuation determined using the Reynolds decomposition ( $k_{x}{\it\phi}_{uu}$ , blue solid) and our new decomposition ( $k_{x}{\it\phi}_{u^{\prime }u^{\prime }}$ , red dashed) at various wall-normal positions are plotted in figure 4 as a function of energetic length scale, ${\it\lambda}_{x}$ . An example of the streamwise velocity fluctuation signals used for computation of the spectra is shown in figure 1(d). Figure 4(a) shows the spectra at $z/{\it\delta}=0.1$ , which is within the log region. At this wall-normal position, the spectra of $u$ and $u^{\prime }$ are very close to each other. This is consistent with the observation that the instantaneous fluctuating streamwise velocity fields near the wall are not notably affected by the method of velocity decomposition. Up to approximately $z/{\it\delta}=0.4$ (figure 4 b), the spectra of $u$ and $u^{\prime }$ do not differ from each other significantly because the flow is almost non-intermittent up to this wall-normal position, as previously mentioned. However, as we move to the wake region, where the intermittent nature of the flow is significant, the spectra of $u^{\prime }$ deviate significantly from the spectra of $u$ . At $z/{\it\delta}=0.66$ (where ${\it\gamma}=0.5$ ), a significant amount of large-scale energy is removed from the streamwise velocity spectra when the influence of TNTI oscillation is removed by deploying our new decomposition (figure 4 c). Under our new decomposition, the peak wavelength of the streamwise velocity fluctuation is shifted from ${\it\lambda}_{x}=(2{-}3){\it\delta}$ (the typical wavelength of large-scale motions (LSMs)) to ${\it\lambda}_{x}\simeq 0.6{\it\delta}$ , which is of the order of the wall-normal distance from the wall. In fact, the dominant wavelength of the streamwise velocity fluctuation under the Reynolds decomposition coincides with the dominant wavelength of the TNTI oscillation ( ${\it\lambda}_{x}\simeq 3{\it\delta}$ ), as shown by the pre-multiplied power spectral density of ${\it\delta}_{i}$ in figure 4(d). In addition, the streamwise velocity fluctuation under the Reynolds decomposition is strongly anticorrelated with the fluctuation of TNTI height (the correlation coefficient between $u$ and the fluctuating component of ${\it\delta}_{i}$ is approximately $-0.65$ at $z=0.63{\it\delta}$ ), which supports the idea that in this region the streamwise velocity fluctuation under the Reynolds decomposition is strongly influenced by the TNTI oscillation. A similar remark was made by Kovasznay et al. (Reference Kovasznay, Kibens and Blackwelder1970) from the anticorrelation between $u$ and the intermittency function (a boxcar function whose value is 1 inside the turbulent region and 0 otherwise). Hence, in the intermittent region, the significant portion of what appears to be the large-scale energy in the spectra of streamwise velocity fluctuations under the traditional Reynolds decomposition could be just a mathematical artefact of the movement of the TNTI rather than an indication of the organised large-scale turbulent motions (with the proviso that the motion of the TNTI could, to some degree, be driven by turbulent motions). However, it needs to be made clear that the existence of the LSMs near the wall and the conclusions drawn from previous studies on them are not questioned by these results, since the effect of the velocity decomposition method on flow fields near the wall is almost negligible.

Figure 5. Pre-multiplied spectrogram of the streamwise velocity fluctuation as a function of energetic length scale and distance from the wall. The solid lines indicate the contour levels from 0.1 to 1.8 with an increment of 0.15. (a,b) Fluctuating velocity component computed using the Reynolds decomposition. (c) Fluctuating velocity component computed using the new decomposition.

The pre-multiplied spectrograms of the streamwise velocity shown in figure 5 permit us to look at the distribution of energy over a range of energetic length scale and distance from the wall. The spectrograms of $u$ agree well with Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009), particularly in the outer region, although the large-scale outer peak (at $z\approx 0.06{\it\delta}$ with ${\it\lambda}_{x}\approx 6{\it\delta}$ ) is absent due to the low Reynolds number of this study. The common features of the spectrograms of the Reynolds fluctuations in the outer region includes the peak in the power spectra at ${\it\lambda}_{x}=(2{-}3){\it\delta}$ (commonly referred to as LSM). The spectrograms of $u^{\prime }$ (figure 5 c) reaffirm that our new decomposition removes the significant portion of energies at those very large scales and the usual LSM peak wavelength is shifted to $0.6{\it\delta}$ throughout the intermittent region ( $z>0.4{\it\delta}$ ). The effect of this ‘added’ fluctuation due to the oscillation of the TNTI is better represented by looking at the difference between the power spectral densities of the streamwise velocity fluctuations under Reynolds decomposition and our decomposition (figure 6). It shows that the influence of the TNTI oscillation on the energy distribution of streamwise velocity is confined mostly within the intermittent region only at large-scale wavelengths (maximum around $z=0.6{\it\delta}$ and ${\it\lambda}_{x}=3{\it\delta}$ ).

Figure 6. Difference between pre-multiplied spectrograms of streamwise velocity fluctuation under the Reynolds decomposition and the new decomposition. The solid lines indicates the contour levels from 0 to 0.2 with an increment of 0.04.

Figure 7. Two-point correlation map of the streamwise velocity fluctuation under the Reynolds decomposition (a,c,e) and the new decomposition (b,d,f) at the reference wall-normal position of (a,b)  $z=0.1{\it\delta}$ , (c,d)  $z=0.4{\it\delta}$ and (e,f)  $z=0.6{\it\delta}$ . The black solid lines indicate the contour levels from 0.1 to 0.7 with an increment of 0.1, and the grey solid lines indicate the contour levels of zero and below with a decrement of 0.1.

The examination of the spectra of the streamwise velocity shows that the scale of turbulence is altered in the outer region when the influence of the free stream is removed by employing our decomposition. Therefore, it suggests that the large-scale coherence might not extend as far into the outer region as previously thought. In order to investigate this conjecture, the two-point correlation maps of the streamwise velocity under the Reynolds decomposition ( $\mathscr{R}_{uu}$ ) and the new decomposition ( $\mathscr{R}_{u^{\prime }u^{\prime }}$ ) at various reference wall-normal positions are plotted in figure 7. At the reference wall-normal position of $z=0.1{\it\delta}$ (figure 7 a,b), the two-point correlation maps under both decompositions are very similar. However, there is a hump extending into the intermittent region (the hump starts at $z\simeq 0.4{\it\delta}$ and extends up to $z\simeq 0.64{\it\delta}$ in the region of ${\rm\Delta}x\simeq (1{-}2){\it\delta}$ ) on the downstream ( ${\rm\Delta}x>0$ ) head of the contour line at $\mathscr{R}_{uu}=0.1$ and it is removed under our new decomposition, indicating that this hump could be due to the contamination of velocity fluctuation by TNTI oscillation. Nevertheless, the average spatial organisation of coherent velocity fluctuations in the region close to the wall is not affected by the oscillation of the TNTI, which is consistent with the instantaneous observation and spectral analysis. When $\mathscr{R}_{uu}$ and $\mathscr{R}_{u^{\prime }u^{\prime }}$ about the reference position of $z=0.4{\it\delta}$ are compared (figure 7 c,d), the upstream ( ${\rm\Delta}x<0$ ) tail of the correlation map remains almost unchanged. The inclined and wall-attached nature of the upstream tail indicates that it is associated with the flow structures that grow from the wall. In contrast, the downstream head of $\mathscr{R}_{uu}$ , which extends into the outer region, is removed by employing our decomposition (in $\mathscr{R}_{u^{\prime }u^{\prime }}$ ). This downstream head appearing under the Reynolds decomposition (and its disappearance under the new decomposition) provides some indication that there is a connection between the motion of the TNTI and the velocity fluctuation at the reference point of $0.4{\it\delta}$ (the motion of the TNTI appears as large-scale velocity fluctuations in the outer region). Albeit weak, this relationship even extends close to the wall ( $z=0.1{\it\delta}$ ) as observed by the existence of the ‘hump’ described earlier (figure 7 a). The difference between $\mathscr{R}_{uu}$ and $\mathscr{R}_{u^{\prime }u^{\prime }}$ becomes more dramatic when the reference wall-normal position is in the middle of the intermittent region. Figure 7(e,f) shows $\mathscr{R}_{uu}$ and $\mathscr{R}_{u^{\prime }u^{\prime }}$ at the reference wall-normal position of $z=0.6{\it\delta}$ . Here, the overall shape of $\mathscr{R}_{uu}$ and $\mathscr{R}_{u^{\prime }u^{\prime }}$ is similar but $\mathscr{R}_{u^{\prime }u^{\prime }}$ extends much less spatially. The inclined upstream tail of $\mathscr{R}_{u^{\prime }u^{\prime }}$ indicates that inclined flow structures do extend up to the intermittent region, but the average extents of these structures are exaggerated under the Reynolds decomposition. In addition, the contour line of $\mathscr{R}_{uu}=0.1$ has a flat top and this is a likely consequence of the regions of positive streamwise velocity fluctuation existing into the free stream (figure 1 b), which also have a flat top as a result of subtracting the gradually varying $\overline{U}$ profile from the constant free-stream velocity.