4.3.1. The $u$ spectral maps: HW data

We first analyse the streamwise velocity fluctuation time series $u(t)$ from HW measurements. We show the spectral analysis results for location S17 ($x/b=92$) and compare them across all three Reynolds numbers (flows WJ1-8, WJ2-8 and WJ3-8 in table 3).

Figure 5 shows contours of the jet-scaled, pre-multiplied power spectral density (PSD) of $u$, denoted by ${\varPhi _{uu}^{m}}$, plotted in the ${\lambda _{x}}$–$z$ plane, computed at the S17 location, for three flows WJ1, WJ2 and WJ3. Details of the computation of ${\varPhi _{uu}^{m}}$ may be found in Appendix C.1. Left and right vertical axes, respectively, plot wavelength in jet (${{\lambda _{x}}/{z_{T}}}$) and wall (${\lambda _{x}^+}={{\lambda _{x}} {U_{\tau }} /\nu }$) scalings whereas, the bottom and top horizontal axes, respectively, plot distance from the wall in jet ($\eta =z/{z_{T}}$) and wall (${z_+}=z{U_{\tau }} / \nu$) scalings. Note that the horizontal extent of each plot is fixed in $\eta$ coordinates to enable comparison of the jet-mode features. It is clear that all flows have a strong outer energy site, marked by a cross $\times $ in figures 5(a)–5(c). This site is anchored at $\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$ irrespective of Reynolds number, and thus corresponds to the jet mode of the structure of wall-jet flows. Note that significantly energetic wavelengths in this jet mode (at $\eta \approx 0.7$) range from $0.7{z_{T}}$ to $20{z_{T}}$ with the most energetic wavelength being $5{z_{T}}$. An inner energy site of wall-scaled motions, anchored at ${z_+}\approx 15$ and ${\lambda _{x}^+}\approx 1000$, is expected to be a universal feature of all wall-bounded flows, including TBLs (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009), channels and pipes (Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009) and wall jets (Gnanamanickam et al. Reference Gnanamanickam, Bhatt, Artham and Zhang2019). This inner site is marked by a plus sign $+$ in figures 5(a)–5(c) and we shall associate it with the wall mode of the structure of wall jets. Figure 5 shows that the presence of the inner site is not very clear due to overwhelmingly strong outer site, but may still be discerned from the deflection of contours in the near-wall region (see also figure 11 of Gnanamanickam et al. Reference Gnanamanickam, Bhatt, Artham and Zhang2019).

Figure 5. Contours of jet-scaled pre-multiplied PSD $\varPhi ^{m}_{uu}$ of the $u$ signal from HW data measured at the S17 location ($x/b=92$) for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and $\lambda _x/z_T$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines. Vertical dashed line shows location of velocity maximum. Wall or inner (${z_+}\approx 15$ and ${\lambda _{x}^+}\approx 1000$) and jet or outer ($\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$) energetic sites are marked, respectively, by black ‘$+$’ and ‘$\times$’ symbols. Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions.

The shape of the spectral map shows no dependence on Reynolds number. At low Reynolds number, the inner and outer sites appear spatially isolated, with the location of ${U_{max}}$ (dashed line in figure 5a) acting as the demarcation between the two. Despite this, there is non-negligible energy present at ${z_+}\approx 15$ at long wavelengths $1\lessapprox {{\lambda _{x}}/{z_{T}}} \lessapprox 5$. All Reynolds numbers show significant influence of this long-wavelength jet mode in the regions of ${U_{max}}$, the mesolayer (region between vertical dotted lines) and all the way down up to the inner site location of ${z_+}\approx 15$ (figure 5b,c). Earlier studies in the literature (Launder & Rodi Reference Launder and Rodi1979, Reference Launder and Rodi1983; George et al. Reference George, Abrahamsson, Eriksson, Karlsson, Löfdahl and Wosnik2000; Barenblatt, Chorin & Prostokishin Reference Barenblatt, Chorin and Prostokishin2005; Gnanamanickam et al. Reference Gnanamanickam, Bhatt, Artham and Zhang2019) consider the part of the wall jet below ${U_{max}}$ to be akin to a regular TBL. If this were true, then the thickness of this TBL would be ${z_{max}}$ and the longest wavelengths associated with the superstructures (${\lambda _{x}^{ss}}$) in this TBL would be ${\lambda _{x}^{ss}}\approx 6{z_{max}}$ (Hutchins & Marusic Reference Hutchins and Marusic2007a; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). Since, ${z_{T}}/{z_{max}}\approx 6$ in our wall-jet experiments (see table 3), this longest wavelength turns out to be ${\lambda _{x}^{ss}}\approx {z_{T}}$. However, figure 5 provides evidence that significant energy, even at ${z_+} \approx 15$, resides in wavelengths much longer than ${z_{T}}$. Clearly, these long wavelengths may not be simply attributed to superstructures of the (supposed) TBL that extends from the wall up to ${z_{max}}$. Therefore, these long wavelengths ought to be associated with the presence of jet-mode motions even below ${z_{max}}$.

4.3.2. The $u$ line spectra: HW data

Spectral maps of figure 5 from § 4.3.1 suggest at least three salient wall-normal locations where line spectra may be plotted to get a quantitative estimate of the spectral energy content. These locations are the inner energy site (${z_+}\approx 15$), mesolayer (${z_+}\approx 1.8\sqrt {{Re_{\tau }}}$ approximate midpoint of the mesolayer) and outer energy site ($\eta \approx 0.7$ or ${z_+}\approx 0.7{Re_{\tau }}$). Figure 6 shows line spectra of ${\varPhi _{uu}^{m}}$, taken out from the spectral maps of figure 5 and plotted at these wall-normal locations for all three Reynolds numbers. At the inner site (figure 6a), spectra show some Reynolds-number dependence, as expected, due to possible influence of viscous effects. For the remaining two locations (figure 6b,c), spectra collapse, indicating Reynolds-number similarity. At the inner site, the dominant wavelength seems to be ranging from ${\lambda _{x}}\approx {z_{T}}$ to $2.5{z_{T}}$, with lesser energy density at ${\lambda _{x}}\approx 5{z_{T}}$. As one moves out from the wall, a clear shift of the spectral peak to longer wavelengths is evident. In the mesolayer itself, the dominant wavelength ranges from $2.5{z_{T}}$ to $5{z_{T}}$, indicating jet-mode activity below ${z_{max}}$. In the outer layer, ${\lambda _{x}}\approx 5{z_{T}}$ becomes the most energetic wavelength at the outer energy site.

Figure 6. The HW line spectra ${\varPhi _{uu}^{m}}$ plotted against the jet-scaled streamwise wavelength $\lambda /z_{T}$ at three different wall-normal locations; (a) inner site $z_+ \approx 15$, (b) midpoint of the mesolayer $z_+ \approx 1.8 \sqrt {{Re_{\tau }}}$ and (c) outer site $z_+\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.3.3. Profiles of total and scale-decomposed ${\overline {u^{2}}}$: HW data

Figure 5 also suggests an empirical demarcation between long- and short-wavelength motions to ascertain the relative contributions of these spectral ranges to the total variance ${\overline {u^{2}}}$; such scale decompositions are common in TBL studies (Hutchins & Marusic Reference Hutchins and Marusic2007b; Mathis et al. Reference Mathis, Hutchins and Marusic2009; Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011; Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012). For wall jets, Gnanamanickam et al. (Reference Gnanamanickam, Bhatt, Artham and Zhang2019) have used ${\lambda _{x}}=2{z_{T}}$ as the cutoff wavelength for their HW experimental data at a single Reynolds number. Their choice is based on reasonable (yet empirical) separation between outer and inner sites with a cutoff wavelength sufficiently away from the inner site wavelength of ${\lambda _{x}^+}\approx 1000$. Our arguments in the preceding paragraph provide a heuristic basis for considering ${\lambda _{x}}={z_{T}}$ as an appropriate cutoff for scale decomposition in wall jets. Indeed, figure 5 shows that the ${\lambda _{x}}={z_{T}}$ cutoff (horizontal dotted line) separates the inner and outer sites reasonably well at all Reynolds numbers in our experiments. Therefore, we shall use ${\lambda _{x}}={z_{T}}$ as the cutoff for spectral decomposition of all statistics. It may be noted that this cutoff decomposes a statistic into contributions from long jet-mode wavelengths (${\lambda _{x}}\geq {z_{T}}$) and contributions from shorter wavelengths (${\lambda _{x}}<{z_{T}}$) comprising the wall mode and as well as the residual shorter wavelengths from the jet mode. The procedure adopted for spectral decomposition of HW data is outlined in Appendix C.1.

Figure 7(a) shows the viscous-scaled profiles of total ${\overline {u^{2}}}$ (solid lines) and its long-wavelength component (dashed lines) for the HW data of figure 5 at all Reynolds numbers; here, ${\overline {u_+^{2}}}={\overline {u^{2}}}/{U_{\tau }}^2$ and ${z_+}=z{U_{\tau }}/\nu$. Shaded vertical bands indicate the range of upper and lower limits of the mesolayer over the range of Reynolds numbers studied here. Profiles of total ${\overline {u_+^{2}}}$ collapse below ${z_+}=15$ but not anywhere else. Profiles of the long-wavelength contribution do not show any collapse; in fact, the profiles cross-over at ${z_+}=15$ with opposite Reynolds-number trends on its either side. The near-wall peak of total ${\overline {u_+^{2}}}$, although not clearly discernible due to the large outer peak, remains located around ${z_+}=15$ (dashed-dotted vertical line) and its magnitude appears to scale for the two larger Reynolds numbers. This is interesting because the HW spatial resolution deteriorates from ${l_+}=21$(WJ1-8) to $40$(WJ3-8), as seen in table 1, which suggests that the peak value should reduce. Therefore, there appears to be a Reynolds-number-dependent increasing contribution that counterbalances the ${l_+}$ effect. This contribution comes from the short-wavelength component (dashed-dotted lines) as seen in figure 7(b). Below ${z_+}=15$, the long- and short-wavelength contributions have opposite Reynolds-number trends so that the total ${\overline {u_+^{2}}}$ profiles show collapse in that region.

Figure 7. Scale decomposition of ${\overline {u^{2}}}$ profiles from HW data at location S17 (see table 1) for flows WJ1-8, WJ2-8 and WJ3-8. Solid lines in each plot show profiles of the total value of ${\overline {u^{2}}}$. Panels (a,b) respectively show long- and short-wavelength contributions to total ${\overline {u^{2}}}$ in viscous scaling i.e. ${\overline {u_+^{2}}}$ vs ${z_+}$. Panels (c,d) respectively show long- and short-wavelength contributions to total ${\overline {u^{2}}}$ in jet scaling i.e. ${\overline {u_{m}^2}}$ vs $\eta$. Dashed-dotted vertical line shows wall-normal locations ${z_+}=15$ in panels (a,b), and $\eta =0.7$ in panels (c,d). In each plot, the left (right) shaded vertical band indicates the range of lower (upper) limits of the mesolayer over the present range of Reynolds numbers.

Figures 7(c) and 7(d) show the same set of profiles as in figures 7(a) and 7(b), respectively, but now plotted in the outer or jet scaling; here, ${\overline {u_{m}^2}}={\overline {u^{2}}}/{U_{max}}^2$ and $\eta ={z/z_{T}}$. For the two higher Reynolds numbers, the profiles of total ${\overline {u_{m}^2}}$ and the long-wavelength contribution to it, scale very well from within the mesolayer all the way across the entire outer layer (figure 7c). The same is true for the short-wavelength contribution (figure 7d). The outer peak value and location ($\eta =0.7$) in both cases scale on the jet variables. This demonstrates Reynolds-number similarity of energetic motions (with respect to jet scaling) at these Reynolds numbers. The lowest Reynold-number data appears to display a low-Reynolds-number effect which is consistently observed in these as well as subsequent plots of PIV analysis.

4.4.1. The $u$ spectral maps: PIV data

Figure 8 plots contours of the jet-scaled, pre-multiplied PSD of $u$, denoted by ${\varPhi _{uu}^{m}}$, plotted in the ${\lambda _{x}}$–$z$ plane for all three Reynolds numbers WJ1, WJ2 and WJ3. These are maps of direct spatial spectra computed from the long-FOV PIV data without making use of Taylor's hypothesis of frozen turbulence. Details of the computation of ${\varPhi _{uu}^{m}}$ may be found in Appendix C.2. The extents of the mesolayer (vertical dotted lines) and counter-gradient diffusion region (vertical dashed lines), corresponding to the S15 location, are also marked. Contour colour scale and levels are same for all plots so that fair comparison is possible; contour levels help to elucidate Reynolds-number effects. The location of the inner energy site is marked only for reference since the present PIV measurements do not extend close enough to the wall to capture the inner site. The streamwise extent $L_x\approx 5.2{z_{T}}$ (${z_{T}}$ for S15 location) of the long FOV of PIV is sufficient to capture the outer site peak seen at ${\lambda _{x}}\approx 5{z_{T}}$ in the HW spectra (figure 5); wavelengths longer than $5.2{z_{T}}$ are, however, not captured.

Figure 8. Contours of jet-scaled pre-multiplied PSD $\varPhi ^{m}_{uu}$ of $u$ signal from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and $\lambda _x/z_T$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Similar to figure 5, wall or inner (${z_+}\approx 15$ and ${\lambda _{x}^+}\approx 1000$) and jet or outer ($\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$) energetic sites are marked, respectively, by black ‘$+$’ and ‘$\times$’ symbols.

Figure 8 shows that the shape of PIV spatial spectral maps shows no dependence on Reynolds number, which is consistent with the HW spectral maps of figure 5. The outer site located at ${\lambda _{x}}\approx 5{z_{T}}$ is well captured and is marked by a $+$ sign. Note that the spectral resolution is somewhat coarse at longer wavelengths; the first three long wavelengths are $5.2{z_{T}}$, $2.5{z_{T}}$ and $1.7{z_{T}}$. On the whole, the PIV spatial spectral maps of figure 8 are in excellent agreement with the HW spectral maps of figure 5. Specifically, we observe that the wall-ward expanse of the outer site broadens with Reynolds number and the PSD of long wavelengths progressively increases in the counter-gradient and mesolayer regions i.e. the jet mode grows stronger in the region below ${z_{max}}$. This is evident from the fact that contour levels in the inner and outer regions (corresponding to reddish–yellow colour) are well separated for flow WJ1, but expand and merge across the velocity maximum for flows WJ2 and WJ3. It is interesting to note that, although the band of long wavelengths carrying significant energy in these regions is $1\leq {{\lambda _{x}}/{z_{T}}} \leq 5$, the most energetic wavelength appears to be ${\approx }2.5{z_{T}}$. This indicates that, while the outer site is dominated by $5{z_{T}}$ motions, the influence of the jet mode in the regions below ${z_{max}}$ features dominance of $2.5{z_{T}}$ motions. Also, the scale-separation cutoff ${\lambda _{x}}={z_{T}}$, shown by a dotted horizontal line in each plot of figure 8, appears to be reasonable in separating inner and outer energy sites.

4.4.2. The $u$ line spectra: PIV data

Spectral maps of figure 8 from § 4.4.1 suggest three salient wall-normal locations for which line spectra may be plotted for a quantitative estimate of the spectral energy content. These locations are the mesolayer (${z_+}\approx 1.8\sqrt {{Re_{\tau }}}$ approximate midpoint of the mesolayer), the counter-gradient region (${z_+}\approx 0.12{Re_{\tau }}$ is within the counter-gradient region for the range of Reynolds number studied here) and the outer energy site ($\eta \approx 0.7$ or ${z_+}\approx 0.7{Re_{\tau }}$). Figure 9 shows line spectra of ${\varPhi _{uu}^{m}}$, taken from the spectral maps of figure 8 and plotted at these wall-normal locations for all three Reynolds numbers. For all locations in figure 8, spectra for the two larger Reynolds numbers collapse, indicating Reynolds-number similarity consistent with figures 5(b) and 5(c). In the mesolayer, the dominant wavelength is ${\lambda _{x}}=2.5{z_{T}}$ (figure 8a and figure 5b). In the counter-gradient region, ${\lambda _{x}}=2.5{z_{T}}$ and $5{z_{T}}$ both become equally energetic, whereas at the outer site ${\lambda _{x}}=5{z_{T}}$ dominates the spectrum. Also, note that, from the counter-gradient region to the outer site, the most dramatic increase in PSD values occurs for long-wavelength jet-mode motions ${\lambda _{x}}\geq {z_{T}}$.

Figure 9. The PIV line spectra ${\varPhi _{uu}^{m}}$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) midpoint of mesolayer $z_+ \approx 1.8 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region $z_+ \approx 0.12 {Re_{\tau }}$ and (c) outer site $z_+\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.4.3. Profiles of total and scale-decomposed ${\overline {u^{2}}}$: PIV data

Figure 10 shows the profiles of ${\overline {u_{m}^2}}={\overline {u^{2}}}/{U_{max}}^2$ and their scale decomposition, plotted against $\eta ={z/z_{T}}$. Details of the scale decomposition calculations are given in Appendix C.2. Since profiles from all Reynolds numbers are shown, the extents of the mesolayer and counter-gradient diffusion region (measured at the S15 location) for each flow are different. As such, in this as well as all subsequent plots of scale-decomposed profiles, shaded areas are used to denote the range of wall-normal locations that covers the beginning and end of these regions over the present range of Reynolds numbers. The plot shows that all three groups of curves, namely total, ${\lambda _{x}}\geq {z_{T}}$ and ${\lambda _{x}}<{z_{T}}$, scale well (except very near the wall) for the two higher-$Re$ flows WJ2 and WJ3 displaying Reynolds-number similarity. The low-$Re$ case WJ1 shows a lack of scaling, implying a low-$Re$ effect present in the data. Moreover, this low-$Re$ effect is severe for ${\lambda _{x}}\geq {z_{T}}$ motions, with ${\lambda _{x}}<{z_{T}}$ motions virtually unaffected. That is, the low-$Re$ effect in the total curves is almost entirely contributed by ${\lambda _{x}}\geq {z_{T}}$ motions. Within each group, the curves fan out very close to the wall, indicating the increasing importance of viscous scaling in the mesolayer. For ${\overline {u_{m}^2}}$ in WJ2 and WJ3 (figure 10a), ${\lambda _{x}}\geq {z_{T}}$ motions contribute $79\,\%$ and $64\,\%$ of the total variance, respectively, at the outer peak and in the mesolayer. The corresponding values for ${\lambda _{x}}<{z_{T}}$ motions are $21\,\%$ and $36\,\%$, respectively. Note that the ${\lambda _{x}}\geq {z_{T}}$ contributions are always larger than the ${\lambda _{x}}<{z_{T}}$ contributions, with the latter showing practically no variation in the wall-normal direction.

Figure 10. Total and scale-decomposed profiles of ${\overline {u_{m}^2}}$ from PIV data plotted against the jet-scaled wall-normal distance $\eta$. Plot shows comparison of flows WJ1, WJ2 and WJ3. Cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and the counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.5.1. The $w$ spectral maps: PIV data

Figure 11 plots contours of the jet-scaled, pre-multiplied PSD of $w$, denoted by ${\varPhi _{ww}^{m}}$, plotted in a fashion similar to the contours of ${\varPhi _{uu}^{m}}$ of figure 8. Details of the computation of ${\varPhi _{ww}^{m}}$ may be found in Appendix C.2. Spectra of figure 11 show interesting features. First, as in the case of $u$ spectra, the shape of the $w$ spectra also shows no dependence on Reynolds number. Secondly, the shape of the $w$ spectra is clearly different from that of $u$ spectra; the inner site is completely absent in ${\varPhi _{ww}^{m}}$. This is due to impermeability constraint or blocking effect of the wall which strongly influences $w$ fluctuations. Third, a clear outer site (marked by a circle) for energy of $w$ fluctuations is located at $\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$. Note that, the wall-normal locations of the outer sites of $u$ and $w$ are identical ($\eta \approx 0.7$) but their wavelengths are clearly different, ${\lambda _{x}}\approx 5{z_{T}}$ for the $u$ spectra and ${\lambda _{x}}\approx 2.5{z_{T}}$ for the $w$ spectra. Thus, one may conclude that the (structural) jet mode comprises at least two dominant, distinct, energetic submodes viz. submode 1 (${\lambda _{x}}\approx 5{z_{T}}$) and submode 2 (${\lambda _{x}}\approx 2.5{z_{T}}$). The scale-separation cutoff ${\lambda _{x}}={z_{T}}$ does not seem to have a major role here since the inner site for $w$ is non-existent.

Figure 11. Contours of jet-scaled pre-multiplied PSD $\varPhi ^{m}_{ww}$ of $w$ signal from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. The jet or outer energetic site for $w$ fluctuations is located at $\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$, and is marked by a black circle; there is no wall or inner site for $w$ fluctuations. Energy sites for $u$ fluctuation ($+$ and $\times$) are also shown reference.

4.5.2. The $w$ line spectra: PIV data

Since the spectral contours of figure 11 from § 4.5.1 do not suggest any specific locations other than the outer energy site, line spectra of $w$ fluctuation are plotted in figure 12 at the same three wall-normal locations that are used in figure 9 for the line spectra of $u$ fluctuation. Figure 12 shows that the $w$ spectra show Reynolds-number similarity for the two higher Reynolds numbers only at the outer site (panel c) but not in the mesolayer and counter-gradient regions (panels a,b, respectively). In fact, except for a small portion at shorter wavelengths, the spectral density is seen to decrease with Reynolds number in the mesolayer and increase in the counter-gradient region. This is to be contrasted with the Reynolds-number-similar behaviour of the $u$ spectra at all heights (figure 9). Further, while long wavelengths are almost absent in the mesolayer (panel a), a clear and progressive shift to longer wavelengths (panels b,c) is evident and a spectral peak of submode 2 (${\lambda _{x}}=2.5{z_{T}}$) emerges at the outer site (panel c).

Figure 12. The PIV line spectra ${\varPhi _{ww}^{m}}$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) midpoint of mesolayer ${z_+} \approx 1.8 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) outer site ${z_+}\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.5.3. Profiles of total and scale-decomposed ${\overline {w^{2}}}$: PIV data

Figure 13 plots the profiles of total and scale-decomposed ${\overline {w_{m}^2}}={\overline {w^{2}}}/{U_{max}}^2$ against $\eta ={z/z_{T}}$. Details of the scale decomposition calculations may be found in Appendix C.2. Similar to figure 10, shaded areas are used to denote the range of wall-normal locations that covers the beginning and end of the mesolayer and counter-gradient regions over the present range of Reynolds numbers. It may be seen that all three groups of curves, namely total, ${\lambda _{x}}\geq {z_{T}}$ and ${\lambda _{x}}<{z_{T}}$, scale quite well throughout (even in the mesolayer, implying lack of a viscous influence on $w$ fluctuation) for the two higher-$Re$ flows WJ2 and WJ3 displaying Reynolds-number similarity; WJ1 suffers from low-$Re$ effects. This low-$Re$ effect in ${\overline {w_{m}^2}}$ is observed to be equally severe for ${\lambda _{x}}\geq {z_{T}}$ motions and ${\lambda _{x}}<{z_{T}}$ motions; for ${\overline {u_{m}^2}}$ it is severe only for ${\lambda _{x}}\geq {z_{T}}$ motions (figure 10). Thus, for ${\overline {w_{m}^2}}$, the low-$Re$ effect in the total curves is contributed by both types of scale-decomposed motions (figure 13). As to the relative contributions to the total ${\overline {w_{m}^2}}$, figure 13 shows that, at the outer peak, $58\,\%$ of the total variance is contributed by ${\lambda _{x}}\geq {z_{T}}$ and $42\,\%$ by ${\lambda _{x}}<{z_{T}}$ motions i.e. both spectral ranges contribute significantly there. Interestingly, as one moves closer to the wall from the outer region, scale-decomposed curves cross each other around $\eta \approx 0.4$ with ${\lambda _{x}}\geq {z_{T}}$ contributions rapidly going to smaller values than the ${\lambda _{x}}<{z_{T}}$ contributions below this location (figure 13). At the wall-ward end of the mesolayer, virtually all ${\overline {w_{m}^2}}$ is contributed by ${\lambda _{x}}<{z_{T}}$ motions. This is consistent with the blocking effect or the impermeability boundary condition imposed by the solid wall, which does not allow large-scale motions to produce wall-normal velocities of the same order as the wall-parallel velocities. These observations also highlight the highly unisotropic character of near-wall turbulence in wall jets.

Figure 13. Total and scale-decomposed profiles of ${\overline {w_{m}^2}}$ plotted against the jet-scaled wall-normal distance $\eta$. Plot shows comparison of flows WJ1, WJ2 and WJ3. Cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.6.1. The $-u$ and $w$ co-spectral maps: PIV data

The use of co-spectra is quite common in the study of atmospheric flows (Stull Reference Stull1988) but not so common in laboratory flows until recently. In pipe flows, Guala, Hommema & Adrian (Reference Guala, Hommema and Adrian2006) have found that the co-spectra and force spectra of $-u$ and $w$ in the near-wall region receive significant contributions from the large- and very-large-scale motions; their study has used HW anemometry for turbulence measurements. Balakumar & Adrian (Reference Balakumar and Adrian2007) have extended this work to channel flows and zero-pressure-gradient TBLs. Recently, Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014) have investigated force spectra in pipe-flow DNS data in terms of velocity–vorticity correlations. The present work stands out compared with these earlier laboratory studies in two respects. First, we report co-spectra and force spectra of $-u$ and $w$ from PIV measurements; we are not aware of any prior work that reports these spectra using PIV measurements. Secondly, the behaviour of the co-spectra and force spectra of $-u$ and $w$ in wall jets has never been reported so far in the literature. Our study therefore makes these two important contributions.

Figure 14 plots the contours of jet-scaled, premultiplied, power co-spectral density (PCSD) ${\varPhi _{(-uw)}^{m}}$ of the $-u$ and $w$ fluctuations constructed from PIV data for flows WJ1, WJ2 and WJ3. Details of the procedure for computing ${\varPhi _{(-uw)}^{m}}$ from spatial PIV data are given in Appendix C.3. Several new interesting features may be observed. First, we consider the outer part of the flow beyond the velocity maximum i.e. $\eta >0.15$. Here, the PCSD values are negative so that ${(-\overline {uw})}<0$, which is consistent with diffusion of mean momentum down the mean-velocity gradient in this region of ${\partial U /\partial z}<0$. A strong outer site for PCSD is observed at $\eta \approx 0.7$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$ (marked by a $\times$). This site matches well with the corresponding outer site for the PSD of $u$ fluctuation (figure 8). At any height $\eta >0.15$, the PCSD is unimodal, with wavelengths carrying significant Reynolds shear stress in the range $0.1\leq {{\lambda _{x}}/{z_{T}}}\leq 5$ irrespective of Reynolds number (upper limit is fixed due the FOV constraint). This indicates that the contribution to ${(-\overline {uw})}$ in this region of the flow comes purely from the jet-mode wavelengths that scale on ${z_{T}}$. Next, we consider the mesolayer marked as the near-wall region between vertical dotted lines. In the mesolayer, PCSD values are positive so that ${(-\overline {uw})}>0$, which is consistent with diffusion of mean momentum down the mean-velocity gradient in this region of ${\partial U /\partial z}>0$. Towards the wall end of the mesolayer, the PCSD curves appear unimodal with a broad plateau-like region around ${\lambda _{x}}\approx 0.7{z_{T}}$. Also, the wavelengths carrying significant Reynolds shear stress here range from ${\lambda _{x}^+}\geq 100$ to ${{\lambda _{x}}/{z_{T}}}\leq 5$ at all Reynolds numbers. This indicates that both wall and jet modes produce significant Reynolds shear stress in this region. It is known that the wall mode is located at ${z_+}\approx 30$ and ${\lambda _{x}^+}\approx 700$ in channel flows (see figure 8 from Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). Assuming that the wall mode is universal, its location is marked in figure 14 with a five-pointed star. As one moves away from the wall, this bimodal character of the PCSD becomes readily evident at the outer edge of the mesolayer. At this height, a clear separation between wall and jet modes may be seen with ${{\lambda _{x}}/{z_{T}}}\approx 1$ as a reasonable demarcation between the two. The wall mode is seen to peak around ${\lambda _{x}^+}\approx 500$ while the jet mode peaks at ${{\lambda _{x}}/{z_{T}}}\approx 2.5$ (submode 2). Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) have envisioned the mean-velocity scaling in wall jets to consist of wall and jet scaling regions that overlap in the mesolayer, which retains a Reynolds-number dependence. Results of figure 14 emphatically confirm that the dynamical aspects also follow this structure, with the wall and jet structural modes overlapping in the mesolayer.

Figure 14. Contours of jet-scaled pre-multiplied PCSD $\varPhi ^{m}_{(-uw)}$ of $u$ and $w$ signals from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Horizontal dashed dotted line marks submode 2 of the jet mode. The wall or inner site for the co-spectra of $u$ and $w$ is shown by a five-pointed star at ${z_+}\approx 30$ and ${\lambda _{x}^+}\approx 700$. The jet or outer energy site for $u$ fluctuation is also shown for reference, which coincides well with the location of the outer co-spectral peak.

Results presented until now indicate that the jet-mode activity apparently resides on both sides of the velocity maximum. Therefore, it might be apt to classify it as the inner and outer parts of the jet mode (inner jet mode and outer jet mode for brevity). As one moves out further from the mesolayer into the counter-gradient diffusion region (region between vertical dashed lines), it is seen that a large negative contribution is received from the outer jet mode (${{\lambda _{x}}/{z_{T}}}\geq 0.3$) whereas the positive contribution from the wall mode remains rather limited. The spectral character changes from bimodal in the mesolayer to weakly trimodal in the counter-gradient region. For all Reynolds numbers, contours of negative PCSD appear to cross the velocity maximum ($\eta \approx 0.15$) line at ${{\lambda _{x}}/{z_{T}}}\approx 0.3$ (scaling on ${z_{T}}$) demonstrating intrusion of the outer jet mode into the counter-gradient diffusion region, consistent with the suggestions in the earlier wall-jet studies (Gnanamanickam et al. Reference Gnanamanickam, Bhatt, Artham and Zhang2019; Bhatt & Gnanamanickam Reference Bhatt and Gnanamanickam2020). Moreover, within this region, the dominant wavelengths of the outer jet mode are ${{\lambda _{x}}/{z_{T}}}\approx 1$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$ (submode 1). In fact, the bimodality of the mesolayer PCSD (${\lambda _{x}^+}\approx 500$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$ or submode 2 of the inner jet mode) seems to split the outer jet mode in the counter-gradient region into two dominant wavelengths ${{\lambda _{x}}/{z_{T}}}\approx 1$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$ (submode 1). These observations, for the first time, lend strong quantitative support to the influence of ‘long arm of the jet’ in the counter-gradient region of wall jets anticipated by Narasimha (Reference Narasimha1990).

Finally, if one focuses on jet-mode wavelengths ${{\lambda _{x}}/{z_{T}}}\geq 1$ in the contour map, then the structure of the jet mode is a dipole extending across the height of the flow. This dipole has positive PCSD values in the near-wall (inner jet mode) region up to the beginning of the counter-gradient region and negative values from the counter-gradient region all the way into the outer region (outer jet mode). This dipole structure is typical of free jets (Deo, Mi & Nathan Reference Deo, Mi and Nathan2008) but in the case of wall jets, the pole in the near-wall region appears to be somewhat crammed and sandwiched between the wall and the beginning of counter-gradient region. With increasing Reynolds number, the outer jet mode appears to encroach deeper down below the velocity maximum, consistent with what has been suggested in the literature Gnanamanickam et al. (Reference Gnanamanickam, Bhatt, Artham and Zhang2019) and Bhatt & Gnanamanickam (Reference Bhatt and Gnanamanickam2020). This suggests that the asymptotic state of a wall jet in the limit of infinite Reynolds number may well be a half-free jet, as proposed by Gersten (Reference Gersten2015). However, more data at higher Reynolds numbers are needed to confirm this trend conclusively.

4.6.2. The $-u$ and $w$ line co-spectra: PIV data

Co-spectral contours of figure 14 from § 4.6.1 suggest specific locations where line co-spectra may be plotted to gain quantitative information about the spectral density of wall and jet-mode motions that contribute to the wall-normal turbulent flux of streamwise mean momentum. Figure 15 shows these line spectra. It may be observed that, at the outer end of the mesolayer (panel a), line spectra show clear bimodal character with equally strong, positive wall mode (${\lambda _{x}^+}\approx 500$) and inner jet mode (submode 2, ${\lambda _{x}}=2.5{z_{T}}$) spectral peaks. Both modes transport momentum down the mean velocity gradient. As one moves out into the counter-gradient region (panel b), the character changes to weakly trimodal; the positive wall-mode is vanishing to very low values (gradient transport) and negative outer jet mode seems to be splitting into ${\lambda _{x}}={z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$ (counter-gradient transport). This splitting is due to the positive inner jet mode (gradient transport) located at ${\lambda _{x}}=2.5{z_{T}}$ as seen in contour plots of figure 14. Finally, at the outer site (panel c), only negative outer jet mode (gradient transport) persists.

Figure 15. The PIV line co-spectra ${\varPhi _{(-uw)}^{m}}$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) outer end of the mesolayer ${z_+} \approx 3 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) outer site ${z_+}\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.6.3. Profiles of total and scale-decomposed ${(-\overline {uw})}$: PIV data

Figure 16 shows the total and scale-decomposed profiles of ${(-\overline {uw})_m}={(-\overline {uw})}/{U_{max}}^2$ plotted against $\eta$ (see Appendix C.3 for scale decomposition procedure). The low-$Re$ effect in flow WJ1 and Reynolds-number similarity of flows WJ2 and WJ3 discussed before are evident in this plot too. For ${(-\overline {uw})_m}$ in WJ2 and WJ3 (figure 16), ${\lambda _{x}}\geq {z_{T}}$ contributions amount to ${\approx }84\,\%$ and ${\approx }33\,\%$ of the total momentum flux, respectively, at the outer peak and in the mesolayer (see zoomed inset in the plot). The corresponding contributions from ${\lambda _{x}}<{z_{T}}$ motions are ${\approx }16\,\%$ and ${\approx }66\,\%$, respectively. Thus, the ${\lambda _{x}}\geq {z_{T}}$ contribution to the mesolayer momentum flux is not negligible. This may seem counter-intuitive in view of the results of figure 13, where the ${\lambda _{x}}\geq {z_{T}}$ contributions to ${\overline {w_{m}^2}}$ go to zero rapidly in the mesolayer as the wall is approached. However, the large-scale ${\lambda _{x}}\geq {z_{T}}$ motions could amplitude modulate the near-wall $w$ signals, generating a significant fraction of the large-scale momentum flux which would otherwise be absent in the case of a linear superposition of scales without a nonlinear interaction. Although a detailed account of amplitude modulation in our wall-jet experiments is out of the scope of this work, this could potentially explain the generation of $33\,\%$ of the total momentum flux in the mesolayer by the large-scale ${\lambda _{x}}\geq {z_{T}}$ motions. It may further be noted that both the scale-decomposed contributions (${\lambda _{x}}\geq {z_{T}}$ and ${\lambda _{x}}<{z_{T}}$) are positive in the mesolayer and negative in the outer layer, with the changeover taking place somewhere close to the wall-ward end of the counter-gradient region. It is interesting that most of the counter-gradient momentum flux is contributed by ${\lambda _{x}}\geq {z_{T}}$ motions with very little contribution coming from ${\lambda _{x}}<{z_{T}}$ motions. This clearly supports the present finding that the outer jet mode plays a crucial role in reversing the gradient transport of momentum in the region below the velocity maximum.

Figure 16. Total and scale-decomposed profiles of ${(-\overline {uw})_m}$ plotted against the jet-scaled wall-normal distance $\eta$. Each plot shows comparison of flows WJ1, WJ2 and WJ3. The cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Inset in plot shows zoomed view of the mesolayer region. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.7.1. Force spectral maps: PIV data

Wall-normal gradient of the PCSD of $-u$ and $w$ highlights spectral contributions to ${\partial {(-\overline {uw})}/\partial z}$ at various heights from the wall. Thus, ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ represents the so-called force spectrum (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014) or turbulent shear force PSD, and is plotted in figure 17 in the form of a contour map; note that the wall-normal gradient is with respect to the jet or outer coordinate $\eta$. Further, it may be noted that ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ must satisfy ${\partial {(-\overline {uw})_m} /\partial \eta }=\int _{0}^{\infty }[{\partial \varPhi _{(-uw)}^{m}/\partial \eta }]\,\textrm {d}(\ln {k_{x}})$. Similar to Chin et al., contours of ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ have been thresholded in the range $\pm 0.008$ to bring out salient features of various accelerating and decelerating regions. Blue regions correspond to decelerations (momentum sink) while red regions correspond to accelerations (momentum source).

Figure 17. Contours of jet-scaled, turbulent shear force PSD ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Horizontal dashed dotted line marks submode 2 of the jet mode. The wall or inner site for the co-spectra of $u$ and $w$ is shown by a five-pointed star at ${z_+}\approx 30$ and ${\lambda _{x}^+}\approx 700$. The jet or outer energy site for $u$ fluctuation is also shown for reference.

In each plot of figure 17 various important regions in the force spectral map of a wall jet are marked by letters B through G. Out of these, regions B through F have been identified earlier by Chin et al. in the force spectra of pipe flows (DNS data). Although wall jets and pipe flows are quite different types of flows, it is useful to compare these regions of force spectra to understand their similarities and differences. There is also a near-wall deceleration region A identified by Chin et al. at ${z_+}\approx 10$. Since our FOV does not extend down to ${z_+}<10$, this region A in wall jets is not captured in our experiments. Further, it may be noted that regions B through E are essentially near-wall regions seen to occur below ${z_+}\approx 200$ in pipe flows (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). For wall jets also, we note that these regions occur below ${z_+}\approx 200$, as seen in figure 17. Regions F and G are in the outer region, and region G is specific to wall jets with no counterpart for it in pipe flows. It may be noted that Chin et al. further investigate these regions in pipe flows using velocity–vorticity correlations. In our planar PIV measurements, the wall-normal vorticity component $\omega _z$ is not accessible. Also, as mentioned before, the limited spatial resolution of our measurements does not allow accurate estimation of the spanwise fluctuating vorticity component $\omega _y$. Therefore, velocity–vorticity correlations cannot be used to further study these regions in our measurements. Nonetheless, we discuss below the similarities and differences of these accelerating and decelerating regions in wall jets with respect to those observed in pipes.

Region B in wall jets (figure 17) is quite similar to that in pipe flows (figure 8 of Chin et al.) and occurs at ${z_+}\approx 40$ and ${\lambda _{x}^+}\approx 200$. This region corresponds to acceleration i.e. momentum source. In pipe flows, regions C and D appear detached from each other, and respectively appear on the right side and above region B. In wall jets, however, regions C and D cannot be clearly distinguished but in fact belong to a larger region of deceleration. The deepest wall-ward end of this region seems to be anchored around ${z_+}\approx 40$ in wall jets; the corresponding location for region D in pipe flows is ${z_+}\approx 30$. Similarly, the smallest wavelength in this large decelerating region in wall jets is ${\lambda _{x}^+}\approx 100$, which is identical to the lower-bound wavelength for region C in pipes. This confirms that regions C and D, which are distinct in pipes, coalesce in wall jets to form a single region of deceleration; in fact, Chin et al. have suggested that deceleration regions A, C and D could actually be a single deceleration region with an island of acceleration region B. Since the extents of regions B, C and D scale on the wall variables, these regions may be considered to be dominated by the wall mode. Region E in wall jets represents near-wall acceleration caused by long-wavelength motions (submodes 1 and 2) with a clear submode 2 peak of the inner jet mode. This region extends from the wall up to $\eta \approx 0.025$ independent of Reynolds number, indicating that these motions belong to the inner jet mode. In pipe flows, region E is very widespread and does not show any distinct peak. The location of co-spectral inner site (marked by a pentacle taken from figure 14) appears to separate regions B and E. Region F represents broadband deceleration in the outer layer that extends across the velocity maximum and counter-gradient region to amalgamate with regions C and D. At velocity maximum, the lower bound of this broadband decelerating region appears to be fixed at ${\lambda _{x}}\approx 0.1{z_{T}}$ irrespective of Reynolds number i.e. scaling on the jet variables. Region G is new to wall jets and is not seen in pipes. This region represents outer jet-mode accelerations in the region $\eta \approx 1$ and beyond. This possibly relates to the entrainment and acceleration of ambient fluid by the outer jet-mode eddies. Finally, we note that the contour colours fade with increasing Reynolds number, implying weakening of acceleration and deceleration spectral densities in the outer scaling; the same is true for wall scaling (not shown).

4.7.2. Force line spectra: PIV data

Figure 18 plots the line spectra of ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ at locations suggested by the force spectral contours of figure 17. In the mesolayer (panel (a), ${z_+}\approx \sqrt {{Re_{\tau }}}$), wall mode (${\lambda _{x}^+}\approx 200$) and inner jet mode (submode 2, ${\lambda _{x}}=2.5{z_{T}}$) are both observed to accelerate the mean flow (see regions B and E of figure 17). Wavelengths in between these modes, however, contribute to mean flow deceleration (region D of figure 17). Except for the short-wavelength motions, line spectra show no collapse in the mesolayer. Interestingly, the zero crossings of the force line spectra in the mesolayer occur at ${\lambda _{x}}\approx 0.2{z_{T}}$ and ${\lambda _{x}}\approx {z_{T}}$ for all Reynolds numbers, implying that these wavelengths neither accelerate nor decelerate the mean flow; this observation provides additional support for the scale decomposition cutoff ${\lambda _{x}}={z_{T}}$. In the counter-gradient region (panel (b), ${z_+}\approx 0.12{Re_{\tau }}$), line spectra show Reynolds-number similarity for WJ2 and WJ3 flows. All wavelengths are seen to decelerate mean flow in this region with the outer jet submode 2 (${\lambda _{x}}=2.5{z_{T}}$) being dominant here. Beyond the location of outer co-spectral peak (panel (c), ${z_+}\approx {Re_{\tau }}$), line spectra show mean flow acceleration by outer jet-mode wavelengths (${\lambda _{x}}>{z_{T}}$). This acceleration appears to be related to the entrainment of the quiescent ambient fluid by the outer jet eddies.

Figure 18. The PIV line spectra of turbulent shear force PSD ${\partial \varPhi _{(-uw)}^{m}/\partial \eta }$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) inside the mesolayer ${z_+} \approx \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) beyond the outer site at ${z_+}\approx {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.7.3. Profiles of total and scale-decomposed ${\partial {(-\overline {uw})}/\partial z}$: PIV data

Spectral decomposition of the jet-scaled Reynolds shear force ${\partial {(-\overline {uw})_m} /\partial \eta }$ is plotted in figure 19 and shows several interesting features. Towards the wall-ward end of the mesolayer, the force ${\partial {(-\overline {uw})_m} /\partial \eta }$ switches sign from positive to negative i.e. ${(-\overline {uw})}$ reaches maximum value there. This implies that the meso-overlap layer proposed by Gupta et al., whose empirical limits are obtained from mean-velocity scaling, matches well with the momentum balance mesolayer (this is also shown by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020), using wall-jet DNS data from the literature). Most of the wall jet experiences a broadband retarding force due to turbulence i.e. ${\partial {(-\overline {uw})_m} /\partial \eta }<0$, except for a small region at the beginning of the mesolayer and a region near the outer edge of the wall-jet flow ($\eta >0.8$), where accelerations are experienced i.e. ${\partial {(-\overline {uw})_m} /\partial \eta }>0$. Within the mesolayer, the retardation force due to ${\lambda _{x}} < {z_{T}}$ motions is more than that due to ${\lambda _{x}}\geq {z_{T}}$ motions. Beyond the mesolayer, however, the opposite is true. Scale-decomposed force profiles show interesting variations in the wall-normal direction. At a given Reynolds number, the retardation force due to ${\lambda _{x}}<{z_{T}}$ motions reaches a maximum value within the mesolayer and starts reducing further outward. Curves for different Reynolds numbers (especially WJ2 and WJ3) show similarity beyond the location of maximum retardation in the mesolayer. Below this location, the curves fan out, implying a dominant influence of the viscous effects close to the wall. The retardation force at a given Reynolds number due to ${\lambda _{x}}\geq {z_{T}}$ motions, on the other hand, steadily increases until it peaks farther from the wall near the velocity maximum. Reynolds-number similarity is seen to hold to a larger degree with the curves fanning out less at the wall-ward end than for the shorter wavelengths. The outer edge of the mesolayer corresponds well to the cross-over between the two sets of curves. The counter-gradient region is marked by maximum retardation of mean flow with the ${\lambda _{x}}\geq {z_{T}}$ contribution dominating there. Beyond velocity maximum, the outer part of the flow experiences dominantly large-scale ${\lambda _{x}}\geq {z_{T}}$ retardation with the ${\lambda _{x}}<{z_{T}}$ contribution becoming negligibly small beyond $\eta \approx 0.5$. It is interesting to note that, at $\eta \approx 0.8$, there is a cross-over from negative to positive values of ${\partial {(-\overline {uw})_m} /\partial \eta }$ (figure 19), consistent with the maximum negative value of ${(-\overline {uw})_m}$ at that location (figure 16). This location agrees very well with that reported by Gnanamanickam et al. (Reference Gnanamanickam, Bhatt, Artham and Zhang2019) in figure 9(a) of their paper. As mentioned before, this large-scale acceleration may be attributed to the entrainment of quiescent ambient fluid by the outer jet eddies.

Figure 19. Total and scale-decomposed profiles of ${\partial {(-\overline {uw})_m} /\partial \eta }$ plotted against the jet-scaled wall-normal distance $\eta$. Each plot shows comparison of flows WJ1, WJ2 and WJ3. The cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.8.1. Turbulence production spectral maps: PIV data

Figure 20 shows jet-scaled, pre-multiplied production spectra ${\varPhi _{prod}^{m}}$ for all three Reynolds numbers (panels a–c). Procedure adopted to compute these spectra from PIV data is given in Appendix C.4. Two positive production sites are seen, one in the outer region and the other close to the wall. Both these sites correspond very well respectively to the inner and outer energy sites (marked by $+$ and $\times$) of $u$ fluctuation seen in figures 5 and 8. The outer production site is dominated by long wavelengths typical of the outer jet mode. Inner production site occupies the mesolayer and possesses a bimodal character with contributions ranging from shorter wavelengths typical of the wall mode to longer wavelengths typical of the inner jet mode. The short-wavelength end of the inner site ${\lambda _{x}^+}\approx 50$ scales on the wall variables whereas the long-wavelength end ${{\lambda _{x}}/{z_{T}}}\approx 5$ scales on the jet variables. This supports the present contention that the structure of the mesolayer comprises structural contributions from the wall as well as (inner) jet modes. In the counter-gradient diffusion region (between vertical dotted lines), turbulence production is negative and the patch of negative contours appears only at long wavelengths (${\lambda _{x}}$ of the order of ${z_{T}}$). An appeal to figure 14 reveals that this negative production patch is due to intrusion of the outer jet mode into the region below velocity maximum. It may be seen that the negative production patch becomes stronger with increasing Reynolds number. We shall revisit the counter-gradient region in some detail in Part 2 of this study.

Figure 20. Contours of jet-scaled turbulence production PSD ${\varPhi _{prod}^{m}}$ from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Horizontal dashed dotted line marks submode 2 of the jet mode. Wall (inner) and jet (outer) energy sites for $u$ fluctuation are also shown for reference.

4.8.2. Turbulence production line spectra: PIV data

Figure 21 plots the line spectra of ${\varPhi _{prod}^{m}}$ at locations suggested by the spectral contours of turbulence production of figure 20. In the mesolayer (panel (a), ${z_+}\approx 1.8\sqrt {{Re_{\tau }}}$), production is broadband with contributions from the wall mode as well as the inner jet mode. No scaling is observed in jet coordinates, implying the importance of wall-scaled processes in the mesolayer. In the counter-gradient region (panel (b), ${z_+}\approx 0.12{Re_{\tau }}$), overall PSD values are significantly lower. However, the outer jet mode is seen to contribute significantly to negative production, which extends over wavelengths $0.2{z_{T}}\leq {\lambda _{x}} \leq 5{z_{T}}$. Curves for WJ2 and WJ3 show Reynolds-number similarity at longer wavelengths, as expected. Finally, at the outer spectral peak (panel (c), ${z_+}\approx 0.7 {Re_{\tau }}$), the outer jet mode contributes the strongest positive PSD in the entire flow with WJ2 and WJ3 showing reasonable Reynolds-number similarity.

Figure 21. The PIV line spectra of turbulence production PSD ${\varPhi _{prod}^{m}}$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) inside the mesolayer ${z_+} \approx 1.8 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) outer site ${z_+}\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$. Note the change in ordinate scale for the plots.

4.8.3. Profiles of total and scale-decomposed ${(-\overline {uw})}{\partial U /\partial z}$: PIV data

Figure 22 plots the profiles of total and scale-decomposed jet-scaled turbulence production term ${(-\overline {uw})_m}{\partial U_m /\partial \eta }$ at all Reynolds numbers (see Appendix C.4 for scale decomposition procedure). At the beginning of the mesolayer, ${\lambda _{x}}<{z_{T}}$ motions (dashed-dotted lines) contribute approximately $60\,\%$ of the total production while ${\lambda _{x}}\geq {z_{T}}$ motions (dashed lines) contribute the remaining $40\,\%$. The production term and its decompositions rapidly drop with height from the wall and reach negative values in the counter-gradient region, as seen in the zoomed inset. Specifically, we note that the negative production is mainly contributed by ${\lambda _{x}}\geq {z_{T}}$ motions consistent with their major contribution to the counter-gradient momentum flux seen in figure 16; ${\lambda _{x}}<{z_{T}}$ motions contribute only little negative production. Beyond the velocity maximum, the production term profile displays Reynolds-number similarity and is almost entirely contributed by ${\lambda _{x}}\geq {z_{T}}$ motions of the jet mode. The maximum value of these terms occurs at $\eta \approx 0.8$, the location where ${(-\overline {uw})_m}$ (figure 16 reaches its maximum in the outer region. This implies that the inflection point in the outer-region mean velocity profile, characterized by the maximum value of ${\partial U_m /\partial \eta }$, is also located close to $\eta \approx 0.8$.

Figure 22. Total and scale-decomposed profiles of ${(-\overline {uw})_m}{\partial U_m /\partial \eta }$ plotted against the jet-scaled wall-normal distance $\eta$. Each plot shows comparison of flows WJ1, WJ2 and WJ3. The cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Inset shows zoomed view of the counter-gradient region. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.9.1. The ${(-\overline {q^{2}w})}$ spectral maps: PIV data

The present PIV measurements in the streamwise–wall-normal plane enable computation of triple products of velocity fluctuations $u$ and $w$. Of particular interest is the quantity ${(-\overline {q^{2}w})}$, where ${{\overline {q^{2}}}={\overline {u^{2}}}+{\overline {w^{2}}}}$. This is because ${\partial {(-\overline {q^{2}w})}/\partial z}$ signifies the turbulent transport term occurring on the right side of the TKE equation (Tennekes & Lumley Reference Tennekes and Lumley1972). Therefore, it is of interest to see how jet and wall modes contribute to the spectral structure of ${(-\overline {q^{2}w})}$ and ${\partial {(-\overline {q^{2}w})}/\partial z}$. We shall call these, respectively, the triple-product spectra and transport spectra. Further, we are not aware of any work in the literature that presents these spectra from DNS or experiments in any type of flow. This aspect, therefore, makes the present study unique.

Figure 23 shows the spectral contributions to ${(-\overline {q^{2}w})}$ in flows WJ1, WJ2 and WJ3. The procedure adopted to compute these spectra from PIV data is given in Appendix C.5. Inner ($+$ sign) and outer ($\times$ sign) energy sites for $u$ fluctuation are marked for reference since the triple product ${(-\overline {q^{2}w})}$ features in the TKE budget equation, as mentioned before. A clear dipole structure is evident in the outer region of the flow and appears to exhibit Reynolds-number similarity. The positive pole is marked by a six-pointed star in each plot and its location is fixed in jet scaling at $\eta \approx 0.35$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$. This indicates dominant submode 2 contributions to the triple product at and around the positive pole. The negative pole, although not captured well by the limited height of the FOV, appears to be located at $\eta \approx 1.2$ and ${{\lambda _{x}}/{z_{T}}}\approx 5$, suggesting (albeit inconclusive at the moment) dominant submode 1 contributions in the region of negative pole. The shortest significantly contributing wavelength for both poles appears to scale on ${z_{T}}$ (${\lambda _{x}} \approx 0.04{z_{T}}$). This indicates that the dipole is a characteristic feature of the outer jet mode. The positive pole region becomes unsymmetric (in logarithmic axes) near the velocity maximum and continues to occupy the extent of the counter-gradient region at all Reynolds numbers. At the wall-ward end of the counter-gradient region, the positive pole structure develops a kink at ${{\lambda _{x}}/{z_{T}}}\approx 1$, forming two lobes on either side that extend down into the mesolayer. The short-wavelength lobe is located at ${\lambda _{x}^+}\approx 300\unicode{x2013}400$ and seems to remain close to the upper edge of the mesolayer at all Reynolds numbers. The long-wavelength lobe, on the other hand, seems to encroach progressively deeper into the mesolayer with increasing Reynolds number.

Figure 23. Contours of jet-scaled triple-product PSD ${\varPhi _{(-q^{2}w)}^{m}}$ from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of the counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Horizontal dashed dotted line marks submode 2 of the jet mode. Positive pole of the jet or outer site of triple-product spectra is shown by a six-pointed star located at $\eta \approx 0.35$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$. Wall (inner) and jet (outer) energy sites for $u$ fluctuation are also shown for reference.

4.9.2. The ${(-\overline {q^{2}w})}$ line spectra: PIV data

Figure 24 shows the line spectra of ${\varPhi _{(-q^{2}w)}^{m}}$ plotted at locations suggested by the triple-product PSD contours of figure 23. In the mesolayer (panel (a), ${z_+}\approx 1.8\sqrt {{Re_{\tau }}}$), the triple product is mainly contributed by wavelengths intermediate to the wall and inner jet modes. Absence of scaling in jet coordinates underscores importance of wall-scaled processes in the mesolayer. In the counter-gradient region (panel (b), ${z_+}\approx 0.12{Re_{\tau }}$), a weakly bimodal PSD is observed with peaks corresponding to the intermediate scales and the jet-mode scales. Curves for WJ2 and WJ3 exhibit Reynolds-number similarity at longer wavelengths. At the triple-product outer spectral peak (panel (c), ${z_+}\approx 0.35 {Re_{\tau }}$), the outer jet mode contributes an overwhelming PSD with WJ2 and WJ3 showing Reynolds-number similarity.

Figure 24. The PIV line spectra of triple-product PSD ${\varPhi _{(-q^{2}w)}^{m}}$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) inside the mesolayer ${z_+} \approx 1.8 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) triple-product outer site ${z_+}\approx 0.35 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$. Note the change in ordinate scale for the plots.

4.9.3. Profiles of total and scale-decomposed ${(-\overline {q^{2}w})}$: PIV data

Figure 25 plots the profiles of jet-scaled total and scale-decomposed triple product ${(-\overline {q^{2}w})_m}$ (see Appendix C.5 for scale decomposition procedure). Reynolds-number similarity of flows WJ2 and WJ3, noted earlier, is evident in figure 25 as well. It is seen that ${\lambda _{x}}\geq {z_{T}}$ motions contribute almost $75\,\%$ of the triple product at the outer peak location $\eta \approx 0.35$; the remaining $25\,\%$ comes from ${\lambda _{x}}<{z_{T}}$ motions. All three groups of curves show Reynolds-number similarity in the outer layer for flows WJ2 and WJ3. Also, all of these groups cross the zero line at the location of the outer energy site of $u$ fluctuation. In the mesolayer, the total curves attain negative values and most contribution there comes from ${\lambda _{x}}<{z_{T}}$ motions since the ${\lambda _{x}}\geq {z_{T}}$ contribution almost goes to zero (see zoomed inset in figure 25). Some Reynolds-number effect is also observed there, which is consistent with the fact that the mesolayer dynamics is indeed Reynolds-number dependent.

Figure 25. Total and scale-decomposed profiles of ${(-\overline {q^{2}w})_m}$ plotted against the jet-scaled wall-normal distance $\eta$. Each plot shows comparison of flows WJ1, WJ2 and WJ3. The cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Inset shows zoomed view of the meso-overlap region. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.

4.10.1. The ${\partial {(-\overline {q^{2}w})}/\partial z}$ spectral maps: PIV data

Transport spectral PSDs in jet variables are defined as $\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta$ based on the wall-normal gradient of the triple-product spectra. This PSD obeys $\partial {(-\overline {q^{2}w})_m} / \partial \eta =\int _0^{\infty } [\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta ]\, \textrm {d}(\ln k_x)$. Transport spectra enable examination of spectral contributions to the transport of TKE by turbulent fluctuations. Figure 26 shows transport spectra for all three Reynolds numbers. Positive values (red colour) of $\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta$ indicate gain for TKE along a mean streamline while negative values (blue colour) indicate loss. In the outer region, transport spectra scale on jet variables, indicating strong loss of TKE at the location of the outer $u$ energy site ($\times$ sign). This transport takes place at long wavelengths and away from the location of the positive pole of ${\varPhi _{(-q^{2}w)}^{m}}$ (marked by the six-pointed star) towards the outer edge of the wall jet. Similarly, long-wavelength transport towards the wall takes place from this positive pole, leading to a strong gain of TKE in the regions of velocity maximum, counter-gradient region and even the mesolayer. The TKE gain in the mesolayer at long wavelengths (specifically submode 2 wavelength ${\lambda _{x}}\approx 2.5{z_{T}}$) steadily increases with Reynolds number. The lobe of the positive pole in figure 23 that extends to shorter wavelengths in the mesolayer drives transport towards the wall at shorter wavelengths, which is seen as a gain of TKE in figure 26. The shortest wavelength of significant gain in the mesolayer remains fixed at ${\lambda _{x}^+}\approx 300$ scaling on the wall variables. The near-wall end of the mesolayer region shows two loss regions flanking this gain region on either side. Of these, the short-wavelength loss region is very weakly captured by the present measurements due to limited access to low values of ${z_+}$. The long-wavelength (${\lambda _{x}}$ of the order of ${z_{T}}$) loss region is reasonably captured for the lowest Reynolds number but it progressively moves out of the FOV and closer to the wall in $\eta$ units with increasing Reynolds numbers.

Figure 26. Contours of jet-scaled turbulent transport PSD $\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta$ from spatial 2-D PIV data for flows (a) WJ1, (b) WJ2 and (c) WJ3. Contours are plotted on a plane with wall-normal distance on the horizontal axis and wavelength on the vertical axis. The extent of each plot is fixed in outer coordinates $\eta$ and ${{\lambda _{x}}/{z_{T}}}$. The meso-overlap layer $0.7\leq \eta \sqrt {{Re_{\tau }}}\leq 3$ proposed by Gupta et al. (Reference Gupta, Choudhary, Singh, Prabhakaran and Dixit2020) is marked by vertical dotted lines; these limits correspond to location S15 ($x/b=77$). Vertical dashed lines mark the limits of counter-gradient momentum diffusion region corresponding to location S15 ($x/b=77$). Horizontal dotted line at ${{\lambda _{x}}/{z_{T}}}=1$ marks the empirical cutoff used for scale decompositions. Horizontal dashed dotted line marks submode 2 of the jet mode. Positive pole of the jet or outer site of triple-product spectra is shown for reference by a six-pointed star located at $\eta \approx 0.35$ and ${{\lambda _{x}}/{z_{T}}}\approx 2.5$. Wall (inner) and jet (outer) energy sites for $u$ fluctuation are also shown reference.

4.10.2. The ${\partial {(-\overline {q^{2}w})}/\partial z}$ line spectra: PIV data

Figure 27 shows the transport line spectra $\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta$ plotted at locations suggested by the contours of figure 26. These spectra are somewhat noisy to enable robust conclusions. However, some broad interesting features may be noted. In the mesolayer (panel (a), ${z_+}\approx 1.8\sqrt {{Re_{\tau }}}$), alternate TKE loss and gain patches are seen. Very short wavelengths show loss of TKE whereas ${\lambda _{x}}$ of the order of $0.1{z_{T}}$ show gain. With increasing Reynolds number a loss region seems to appear around ${\lambda _{x}}={z_{T}}$ whereas gain is seen at further longer wavelengths. Absence of scaling in jet coordinates underscores the importance of wall-scaled processes in the mesolayer. In the counter-gradient region (panel (b), ${z_+}\approx 0.12{Re_{\tau }}$), gain in TKE is rather broadband, with jet-mode scales (${\lambda _{x}}\geq {z_{T}}$) contributing equal densities. The opposite is true at the outer energy site (panel (c), ${z_+}\approx 0.7 {Re_{\tau }}$) where these outer jet-mode scales contribute strong loss of TKE. Flows WJ2 and WJ3 show that Reynolds-number similarity may prevail in this region.

Figure 27. The PIV line spectra of turbulent transport $\partial {\varPhi _{(-q^{2}w)}^{m}} / \partial \eta$ plotted against the jet-scaled streamwise wavelength ${\lambda _{x}}/{z_{T}}$ at three different wall-normal locations; (a) inside the mesolayer ${z_+} \approx 1.8 \sqrt {{Re_{\tau }}}$, (b) counter-gradient region ${z_+} \approx 0.12 {Re_{\tau }}$ and (c) outer energy site ${z_+}\approx 0.7 {Re_{\tau }}$. Flows WJ1, WJ2 and WJ3 are shown. Vertical dotted, dashed-dotted and dashed lines, respectively, show streamwise wavelengths ${\lambda _{x}}={z_{T}}$, ${\lambda _{x}}=2.5{z_{T}}$ and ${\lambda _{x}}=5{z_{T}}$.

4.10.3. Profiles of total and scale-decomposed ${\partial {(-\overline {q^{2}w})}/\partial z}$: PIV data

Figure 28 plots the total and scale-decomposed profiles of the jet-scaled turbulent transport ${\partial {(-\overline {q^{2}w})_m} /\partial \eta }$. The procedure of scale decomposition is outlined in Appendix C.5. Total turbulence transport peaks in the region of counter-gradient momentum diffusion. The broadband contribution of turbulent transport in this region serves as a gain for the TKE there. This explains the non-negligible values of ${\overline {u^{2}}}$ near the velocity maximum in figures 7 and 10, even though the turbulence production is close to zero (or in fact negative in the counter-gradient region) in that region, as seen in figure 22. Interestingly, the transport by ${\lambda _{x}}<{z_{T}}$ motions peaks at the wall-ward end of the counter-gradient region or the outer end of the mesolayer. The transport by ${\lambda _{x}}\geq {z_{T}}$ motions, on the other hand, peaks at the velocity maximum. Towards the outer edge of the mesolayer, transport by ${\lambda _{x}}<{z_{T}}$ motions is $66\,\%$ while that by ${\lambda _{x}}\geq {z_{T}}$ motions is approximately $33\,\%$ of the total. In the outer layer for $\eta >0.35$, transport becomes negative, implying loss for TKE at those locations. It may be noted that, unlike force spectra (figure 19) and production spectra (figure 22), transport spectra in the outer layer contain a non-negligible contribution from ${\lambda _{x}}<{z_{T}}$ motions although ${\lambda _{x}}\geq {z_{T}}$ motions dominate there. Profiles of flows WJ2 and WJ3 show Reynolds-number similarity.

Figure 28. Total and scale-decomposed profiles of ${\partial {(-\overline {q^{2}w})_m} /\partial \eta }$ plotted against the jet-scaled wall-normal distance $\eta$. Each plot shows comparison of flows WJ1, WJ2 and WJ3. The cutoff wavelength used is ${\lambda _{x}}={z_{T}}$. Shaded vertical bands denote the range of wall-normal locations that covers the beginning and end of the mesolayer (bands between dotted lines) and counter-gradient region (bands between dashed lines) over the present range of Reynolds numbers.