3.1. Overall variations of $DR$ and $HR$

The values of HR, DR and the difference ${\textrm{HR}} - {\textrm{DR}}$ are shown in figure 5. For all cases considered, HR and DR initially increase with the frequency of forcing, but for $ \omega ^+ \gtrsim 0.04$ , they reach a maximum value before slowly decreasing at higher frequencies. The maximum DR is approximately $30\,\%$ for all Prandtl numbers, but the maximum HR increases from 30 % to approximately $40\,\%$ with increasing Prandtl number, marking a significant disparity between DR and HR. The increasing disparity towards ${\textrm{HR}} \gt {\textrm{DR}}$ occurs for $\omega ^+ \ge 0.04$ , however, at $\omega ^+ = 0.022$ , ${\textrm{HR}} \lesssim{\textrm{DR}}$ . The comparison between the reduced-domain results (filled squares) and the full-domain results (open circles) supports the reliability of the production runs. At $Pr= 7.5$ , the reduced-domain grid is more than twice as coarse as the full-domain grid, yet there is less than $2\,\%$ difference in DR and HR (Appendix A gives more details). The trends in DR seen here have been widely recorded in the previous literature, as the review by Ricco et al. (Reference Ricco, Skote and Leschziner2021) makes clear. However, to the best of the authors’ knowledge, the behaviour of HR has not been reported before. In § 3.4 we relate the observed trends between HR and DR to the attenuation in the turbulent shear stress $\overline {uv}$ and turbulent scalar flux $\overline {\theta v}$ , followed by studying their interactions with the Stokes layer (§ 3.6--3.8).

Figure 5. Values of ${\textrm{DR}}\,\%$ (blue symbols), ${\textrm{HR}}\,\%$ (red symbols) and their difference (black symbols) for the cases given in table 1. Filled squares: reduced-domain simulations; empty circles: full-domain simulations. Blue dashed line marks the maximum DR; red dashed-dotted line marks the maximum HR.

Figure 6. Plane oscillation data of ${\textrm{DR}}\,\%$ (blue symbols) and ${\textrm{HR}}\,\%$ (orange/red symbols) at $Pr = 0.7 - 1.0$ from our present study, Guérin et al. (Reference Guérin, Flageul, Cordier, Grieu and Agostini2024), and Fang et al. (Reference Fang, Lu and Shao2009).

In figure 6 we plot the plane oscillation data of DR and HR at close $Pr = 0.71 - 1.0$ from our present study (figure 5 b), Guérin et al. (Reference Guérin, Flageul, Cordier, Grieu and Agostini2024) and Fang et al. (Reference Fang, Lu and Shao2009). Unlike our data, the two cited studies obtain heat transfer increase ( ${\textrm{HR}} \lesssim 0$ ). Compared with Guérin et al. (Reference Guérin, Flageul, Cordier, Grieu and Agostini2024), our $A^+$ values differ by $2.5$ times and $\omega ^+$ values differ by $1.7$ times. Nevertheless, based on our findings from §§ 3.6to 3.8, we speculate that ${\textrm{HR}} \lt {\textrm{DR}} \lt 0$ by Reference Guérin, Flageul, Cordier, Grieu and AgostiniGuérin et al. (2024) is related to the highly protrusive Stokes layer (beyond $50$ viscous units) for $\omega ^+ \lt 0.04$ (figure 22). Compared with Fang et al. (Reference Fang, Lu and Shao2009), we have close values of $A^+$ and $\omega ^+$ . At $A^+ \simeq 12.0$ and $\omega ^+\simeq 0.06$ , the DR values differ by $9\,\%$ between Fang et al. (Reference Fang, Lu and Shao2009) ( ${\textrm{DR}} = 40\,\%$ , $Re_{\tau _0} = 180$ ) and our study ( ${\textrm{DR}} = 31\,\%$ at $Re_{\tau _0} = 590$ ); this is due to the Reynolds number difference, as discussed in the literature (Gatti & Quadrio Reference Gatti and Quadrio2016; Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Rouhi et al. 2023). However, the values of HRhave opposite signs and are different by $80\,\%$ , ${\textrm{HR}} = 30\,\%$ from our study versus ${\textrm{HR}} = -50\,\%$ from Fang et al. (Reference Fang, Lu and Shao2009). In addition to $Re_{\tau _0}$ , we speculate that such a difference is related to the different computational set-ups. We conduct DNS of incompressible turbulent channel flow with $\unicode{x1D6E5} ^+_x \times \unicode{x1D6E5} ^+_z = 8.3 \times 3.9$ , while Fang et al. (Reference Fang, Lu and Shao2009) conduct large-eddy simulation (LES) of compressible turbulent channel flow (Mach number $Ma = 0.5$ ) with $\unicode{x1D6E5} ^+_x \times \unicode{x1D6E5} ^+_z = 35 \times 12$ .

3.2. Optimal actuation frequency

Figure 5 indicates that for the considered cases, the optimal actuation frequency for DR is $\omega ^+ \approx 0.044$ for the travelling wave and $\approx 0.066$ for the plane oscillation, regardless of the Prandtl number. For HR, the optimal frequency coincides with that for DR at $Pr = 0.71$ , but it increases to $0.088$ for both types of actuation as the Prandtl number increases to 7.5.

Previous studies relate the optimal frequency for DR to the characteristic time scale of the energetic velocity scales associated with the near-wall cycle of turbulence, $\mathcal{T}^+_u$ (Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). When the actuation period $T^+_{osc} \equiv 2 \pi /\omega ^+$ matches this time scale, the wall oscillation becomes more effective in disrupting the near-wall scales, leading to the maximum DR (Ricco et al. Reference Ricco, Skote and Leschziner2021). Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) discuss this interaction using the pre-multiplied spectrum of wall shear stress $f^+ \phi ^+_{\tau \tau }$ , and the corresponding spectra for our non-actuated case are shown in figure 7(a–c) (blue lines). In agreement with Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023), the peaks in the shear stress spectra occur at $\mathcal{T}^+_u \simeq 100$ , corresponding to $\omega ^+ \simeq 0.063$ , which broadly matches the frequencies of actuation for maximum DR found here.

Figure 7. Non-actuated half-channel flow with increasing Prandtl number: (a,d,g) $Pr = 0.71$ , (b,e,h) $Pr = 4.0$ , (c,f,i) $Pr = 7.5$ . (a–c) Pre-multiplied frequency spectra of wall shear stress $f^+ \phi ^+_{\tau \tau }$ (blue line) and wall heat flux $f^+ \phi ^+_{qq}$ (red line), where $T^+ = 1/f^+$ . (d–f) Spectrograms of the streamwise velocity fluctuations (blue contour lines) and temperature fluctuations (filled contour) pre-multiplied by the spanwise wavenumber $k^+_z$ . (g–i) Same as (d–f), pre-multiplied by the streamwise wavenumber $k^+_x$ . Contour levels for $k^+_z \phi ^+_{u u}$ start from $0.2$ to $3.8$ with an increment of $0.6$ ; contour levels for $k^+_x \phi ^+_{u u}$ start from $0.2$ to $2.2$ with an increment of $0.4$ . The blue and red bullets in (d–f) locate the maximum in $k^+_z \phi ^+_{u u}$ and $k^+_z \phi ^+_{\theta \theta }$ , respectively, and in (g–i) locate the maximum in $k^+_x \phi ^+_{u u}$ and $k^+_x \phi ^+_{\theta \theta }$ , respectively.

Figure 7 also shows the corresponding spectra for the fluctuating wall heat flux, $f^+ \phi ^+_{qq}$ (red lines), as well as the pre-multiplied spectrograms of the fluctuating streamwise velocity and temperature ( $k^+_z \phi ^+_{uu}, \ k^+_x \phi ^+_{uu}, \ k^+_z \phi ^+_{\theta \theta }, \ k^+_x \phi ^+_{\theta \theta }$ ). Figure 7(a,b,c) reveals that while $f^+ \phi ^+_{qq}$ is sensitive to $Pr$ at small time scales ( $T^+ \lesssim 70$ ), the time scale of the energetic temperature scales $\mathcal{T}^+_\theta \simeq 100$ is insensitive to $Pr$ , and coincides with $\mathcal{T}^+_u$ . The streamwise and spanwise lengths of the energetic temperature scales vary somewhat with Prandtl number (see figure 7 d–i), but they also remain close to the length scales of the energetic velocity scales. However, the peak in $k^+_z \phi ^+_{\theta \theta }$ drops from $y^+ = 16$ at $Pr = 0.71$ to $y^+ = 5$ at $Pr = 7.5$ , as the conductive sublayer thins with increasing Prandtl number (Alcántara-Á vila & Hoyas Reference Alcántara-Á vila and Hoyas2021; Kader Reference Kader1981; Schwertfirm & Manhart Reference Schwertfirm and Manhart2007). The $y^+$ location of the peak in $k^+_z \phi ^+_{\theta \theta }$ scales with approximately $ Pr^{-1/2}$ , which is steeper than the scaling of approximately $ Pr^{-1/3}$ for the conductive sublayer thickness for $Pr \gtrsim 1$ (Shaw & Hanratty Reference Shaw and Hanratty1977; Kader Reference Kader1981; Schwertfirm & Manhart Reference Schwertfirm and Manhart2007; Alcántara-Á vila & Hoyas Reference Alcántara-Á vila and Hoyas2021; Pirozzoli Reference Pirozzoli2023). We also analysed the $\overline {\theta ^2}^+$ profiles of Pirozzoli (Reference Pirozzoli2023) for the DNS of the turbulent pipe flow at $Re_{\tau _0} \simeq 1100$ and $Pr = 0.5, 1, 2, 4, 16$ , and the $y^+$ locations of their inner peaks scale with approximately $ Pr^{-1/2}$ .

For the plane oscillation, we expect that the maximum DR and HR is achieved when the actuation period is close to $\mathcal{T}^+_u$ and $\mathcal{T}^+_\theta$ , respectively. For the travelling wave, however, the optimal actuation period cannot be directly compared with $\mathcal{T}^+_u$ or $\mathcal{T}^+_\theta$ , owing to the relative streamwise motion between the travelling wave and the advecting near-wall scales. As discussed by Quadrio et al. (Reference Quadrio, Ricco and Viotti2009), for maximum DR, $\mathcal{T}^+_u$ must be compared with a relative oscillation period $2\pi /(\kappa ^+_x \mathcal{U}^+_u + \omega ^+)$ as seen by an observer travelling with the convection speed of the near-wall energetic velocity scales $\mathcal{U}^+_u$ . Similarly, for maximum HR, $\mathcal{T}^+_\theta$ must be compared with $2\pi /(\kappa ^+_x \mathcal{U}^+_\theta + \omega ^+)$ , where $\mathcal{U}^+_\theta$ is the convection speed of the near-wall energetic temperature scales. After some recasting (using $\mathcal{T}^+_u \simeq \mathcal{T}^+_\theta \simeq 100$ , $\kappa ^+_x = 0.0014$ ), we estimate the optimum frequencies of actuation for DR and HR to be, respectively,

(3.1a) \begin{align} &\qquad\omega ^+_{\textrm{opt}, {\textrm{DR}}} = \frac {2\pi }{\mathcal{T}^+_u} - \kappa ^+_x \mathcal{U}^+_u \simeq 0.046, \end{align}

(3.1b) \begin{align} &\omega ^+_{\textrm{opt}, {\textrm{HR}}} = \frac {2\pi }{\mathcal{T}^+_\theta } - \kappa ^+_x \mathcal{U}^+_\theta \simeq 0.063 - 0.0014 \mathcal{U}^+_\theta, \end{align}

where we have assumed that $\mathcal{U}^+_u$ follows a universal curve (Kim & Hussain Reference Kim and Hussain1993; Del Á lamo & Jiménez Reference Del Á. lamo and Jiménez2009; Liu & Gayme Reference Liu and Gayme2020), where $\mathcal{U}^+_u \simeq 12$ at $y^+ \simeq 15$ (marked with blue bullets in figure 7 d–i).

The estimated optimum frequency for DR is independent of Prandtl number, at approximately $0.046$ , consistent with our DNS results for the travelling wave case (figure 5 a,c,e). However, the optimum frequency for HR depends on $Pr$ , since we expect that $\mathcal{U}^+_\theta$ decreases as the energetic temperature scales move closer to the wall with increasing Prandtl number (red bullets in figure 7 d–i). Hetsroni et al. (Reference Hetsroni, Tiselj, Bergant, Mosyak and Pogrebnyak2004) computed the profiles of $\mathcal{U}^+_\theta$ at $Pr = 1.0$ , 5.4 and 54 in a configuration similar to that used in the present study, without actuation. Using their results, we estimate that $\mathcal{U}^+_\theta$ decreases from $12$ to $6$ as $Pr$ increases from $0.71$ to $7.5$ , which gives $\omega ^+_{\textrm{opt},{\textrm{HR}}} = 0.046$ at $Pr = 0.71$ and $0.055$ at $Pr = 7.5$ . The estimated value at $Pr = 7.5$ is smaller than the value of 0.088 given by the DNS (figure 5 e), but the trends with Prandtl number are consistent. In §§ 3.6 and 3.7 we conduct a more quantitative justification for the trends in $\omega ^+_{{opt},{\textrm{HR}}}$ by considering the interaction between the Stokes layer and the near-wall thermal field.

Figure 8. Travelling wave actuation with $A^+ = 12$ , $\kappa ^+_x = 0.0014$ . Left column: ${\textrm{DR}}\,\%$ and $\overline {U}^*$ profiles, where $\unicode{x1D6E5} \overline {U}^* = \overline {U}^* - \overline {U}^*_0$ (results independent of $Pr$ ). Middle and right columns correspond to $Pr=0.71$ and $7.5$ , respectively: ${\textrm{DR}}\,\%$ (blue line), ${\textrm{HR}}\,\%$ (red line) and $\overline {\Theta }^*$ profiles, where $\unicode{x1D6E5} \overline {\Theta }^* = \Theta ^* - \Theta ^*_0$ . In the panels (d–i) the black line is the reference (non-actuated) case, the blue and red lines show the effects of increasing $\omega ^+$ on $\overline {U}^*$ and $\overline {\Theta ^*}$ , respectively, and the thick dashed line corresponds to ${\textrm{DR}}_{{max}}$ or ${\textrm{HR}}_{{max}}$ . In (d,e,f) the thin dotted lines for $y^* \gtrsim 170$ are the reconstructed profiles following § 2.3. In (j,k,l) we plot the reduced-domain profiles (red/blue) up to $y^*_{{res}} \simeq 170$ , and the thick grey profiles are from the full-domain cases ( $L_x \times L_z = 7.6 h \times \pi h$ ) from Appendix A; the dashed-dotted green profile in (j) is at matched $A^+ = 12, \kappa ^+_x = 0.0014, \omega ^+ = 0.044$ but at $Re_{\tau _0} = 950$ from Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023). The black cross symbols mark the distance $y^*$ where $\unicode{x1D6E5} \overline {U}^* = 0.9 \unicode{x1D6E5} \overline {U}^*_{170}$ ( $\unicode{x1D6E5} \overline {\Theta }^* = 0.9 \unicode{x1D6E5} \overline {\Theta }^*_{170}$ ).

Figure 9. Travelling wave actuation with $A^+ = 12$ , $\kappa ^+_x = 0.0014$ . Notation and symbols as in figure 8. Here, $\unicode{x1D6E5} \overline { u v}^* = \overline { u v }^* - \overline { u v }^*_0$ and $\unicode{x1D6E5} \overline { \theta v }^* = \overline { \theta v }^* - \overline { \theta v }^*_0$ . In (d,e,f) the numbers give the $y^*$ location of the inner peak in the $\overline { u^2 }^*$ and $\overline {\theta ^2 }^*$ profiles. In (g–l) the vertical dashed line locates the peak of non-actuated $\overline { u v }^*_0$ (g,j) and $\overline { \theta v }^*_0$ (h,i,k,l).

3.3. Mean profiles and turbulence statistics

We now consider the distributions of the mean velocity $\overline {U}^*$ ( $=\overline {U}/u_\tau$ ), the mean temperature $\overline {\Theta }^*$ ( $= \overline {\Theta }/\theta _\tau$ ) (figure 8), and the velocity and temperature statistics (figure 9) as we vary $\omega ^+$ and $Pr$ . We focus on the travelling wave case, but since the observed trends are consistent with those from the plane oscillation case (figures 28 and 29 in Appendix C), our conclusions are applicable to both types of actuation.

It is well known from the literature that DR by either plane oscillation or a travelling wave coincides with the thickening of the viscous sublayer (Choi et al. Reference Choi, DeBisschop and Clayton1998; Choi & Clayton Reference Choi and Clayton2001; Choi Reference Choi2002; Di Cicca et al. Reference Di Cicca, Iuso, Spazzini and Onorato2002; Touber & Leschziner Reference Touber and Leschziner2012; Hurst et al. Reference Hurst, Yang and Chung2014; Gatti & Quadrio Reference Gatti and Quadrio2016; Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023). This is evident also from our $\overline {U}^*/y^*$ profiles (figure 8 g), where the actuated profiles (blue profiles) depart from unity farther from the wall compared with the non-actuated profile (black profile). The conductive sublayer also thickens with wall oscillation, but it then thins substantially with increasing Prandtl number, so that at $Pr = 7.5$ the conductive sublayer is substantially thinner than the viscous sublayer (figure 8 i).

Viscous sublayer and conductive sublayer thickening due to the wall oscillation shifts the $\overline {U}^*$ and $\overline {\Theta }^*$ profiles. This is shown in figure 8(j,k,l) where we plot the differences $\unicode{x1D6E5} \overline {U}^* = \overline {U}^* - \overline {U}^*_0$ and $\unicode{x1D6E5} \overline {\Theta }^* = \overline {\Theta }^* - \overline {\Theta }^*_0$ between the actuated and non-actuated cases. The cases with the maximum DR (or HR) have the highest $\unicode{x1D6E5} \overline {U}^*$ (or $\unicode{x1D6E5} \overline {\Theta }^*$ ), as highlighted by the thick dashed line in the plots. The profiles in colour are from the production runs with the reduced domain (table 1), and are plotted up to $y^*_{{res}} \simeq 170$ . As a reference, the profiles in grey are from the full-domain cases from Appendix A. There is a good agreement between the reduced-domain and full-domain profiles up to $y^*_{{res}}$ . Furthermore, the variations of $\unicode{x1D6E5} \overline {U}^*$ and $\unicode{x1D6E5} \overline {\Theta }^*$ beyond $y^*_{{res}}$ are within $3\,\%$ . Therefore, it is reasonable to consider $\unicode{x1D6E5} \overline {U}^*$ and $\unicode{x1D6E5} \overline {\Theta }^*$ at $y^*_{{res}} = 170$ ( $\unicode{x1D6E5} \overline {U}^*_{170}$ and $\unicode{x1D6E5} \overline {\Theta }^*_{170}$ ) as their asymptotic values. According to Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023), the distance $y^*$ where $\unicode{x1D6E5} \overline {U}^*$ reaches a plateau indicates the extent to which the Stokes layer disturbs the $\overline {U}^*$ profile. In figure 8(j,k,l), therefore, we mark each profile at the $y^*$ location where $\unicode{x1D6E5} \overline {U}^* = 0.9 \unicode{x1D6E5} \overline {U}^*_{170}$ ( $\unicode{x1D6E5} \overline {\Theta }^* = 0.9 \unicode{x1D6E5} \overline {\Theta }^*_{170}$ ). For the present profiles with fixed $Re_{\tau _0}, A^+$ and $\kappa ^+_x$ , the distance to reach the plateau in $\unicode{x1D6E5} \overline {\Theta }^*$ depends on $\omega ^+$ and $Pr$ (compare figures 8 kand 8 l), where the plateau is reached at a lower $y^*$ with increasing Prandtl number. We expect that the $\unicode{x1D6E5} \overline {\Theta }^*$ profiles and the distances to their plateaus would also depend on $A^+$ and $\kappa ^+_x$ , but not on $Re_{\tau _0}$ . Our conjecture is based on the behaviour of its analogue $\unicode{x1D6E5} \overline {U}^*$ (figure 8 j) that, for a fixed set of actuation parameters, is almost identical between $Re_{\tau _0} = 590$ (thick blue dashed line) and $Re_{\tau _0} = 950$ (green dashed-dotted line). The Reynolds number independence of the plateau in $\unicode{x1D6E5} \overline {U}^*$ for the travelling wave is further confirmed by Gatti & Quadrio (Reference Gatti and Quadrio2016) and Gatti et al. (Reference Gatti, Quadrio, Chiarini, Gattere and Pirozzoli2024).

In figure 9 we plot the turbulence statistics for the same cases shown in figure 8. Drag reduction and viscous sublayer thickening (figure 8 a,d,g) coincide with the near-wall attenuation of $\overline { u^2}^*$ and the shift in its inner peak (figure 9 d), as known from previous work (Quadrio & Sibilla Reference Quadrio and Sibilla2000; Ricco & Wu Reference Ricco and Wu2004; Quadrio & Ricco Reference Quadrio and Ricco2011; Touber & Leschziner Reference Touber and Leschziner2012; Ricco et al. Reference Ricco, Skote and Leschziner2021; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023). Similarly, we find that at each $Pr$ , HR and conductive sublayer thickening (figure 8 b,c,e,f,h,i) coincide with the near-wall attenuation of $\overline { \theta ^2 }^*$ and the shift in its inner peak (figure 9 e,f). Interestingly, ${\textrm{DR}}_{{max}}$ occurs when the inner peak in $\overline { u^2}^*$ is located farthest from the wall, and a similar connection exists between ${\textrm{HR}}_{{max}}$ and the location of the inner peak in $\overline { \theta ^2}^*$ (compare the listed numbers in figure 9 d,e,f).

Figure 10. Travelling wave actuation with $A^+ = 12$ , $\kappa ^+_x = 0.0014$ , $\omega ^+ = 0.088$ at $Pr = 7.5$ . (a) Terms in the averaged streamwise momentum equation (3.3a ). (b) Terms in the averaged temperature equation (3.3b ).

In addition, the attenuation of the turbulent shear stress $\overline { u v }^*$ (figure 9 g) and the wall-normal turbulent temperature flux $\overline { \theta v }^*$ (figure 9 h,i) are directly linked to DR and HR, respectively. In figure 9(j,k,l) we plot the differences between the actuated and non-actuated cases $\unicode{x1D6E5} \overline { u v }^* = \overline { u v }^* - \overline { u v }^*_0$ and $\unicode{x1D6E5} \overline { \theta v }^* = \overline { \theta v }^* - \overline { \theta v }^*_0$ . Most of the attenuation occurs near the wall, with the maxima occurring near $y^* \simeq 10$ . For $y^* \gt 10$ , the attenuation declines, and $\unicode{x1D6E5} \overline { u v }^*$ and $\unicode{x1D6E5} \overline { \theta v }^*$ reach local minima at points that coincide approximately with the locations of the peaks in $\overline { u v }^*_0$ and $\overline { \theta v }^*_0$ .

By integrating the plane- and time-averaged streamwise momentum (2.2) and temperature (2.3) equations from zero to $y^*$ , we obtain

(3.2a,b) \begin{align} \frac {{\mathrm d} \overline {U}^*}{{\mathrm d}y^*} = \overline { u v }^* + \left ( 1 - y^* Re_\tau ^{-1} \right ), \qquad \frac {d \overline {\Theta }^*}{{\mathrm d}y^*} = Pr \, \left [ \overline { \theta v }^* + \left ( 1 - y^* Re_\tau ^{-1} \right ) + \mbox{Res}_{\Theta } \right ], \end{align}

where $\mbox{Res}_\Theta \equiv Re^{-1}_\tau \int _0^{y^*} (1 - \overline {U}^*/U^*_b){\mathrm d}y'$ . Hence,

(3.3a) \begin{align} &\qquad\qquad\frac {d (\unicode{x1D6E5} \overline {U}^*)}{{\mathrm d}y^*} = \unicode{x1D6E5} \overline { u v }^* - y^*\left ( {Re_\tau ^{-1}} - {Re_{\tau _0}^{-1}} \right ), \end{align}

(3.3b) \begin{align} &\frac {d(\unicode{x1D6E5} \overline {\Theta }^*)}{{\mathrm d}y^*} = Pr \, \left [ \unicode{x1D6E5} \overline { \theta v }^* - y^*\left ( {Re_\tau ^{-1}} - {Re_{\tau _0}^{-1}} \right ) + \underbrace {\unicode{x1D6E5} \mbox{Res}_{\Theta }}_{\simeq 0} \right ]. \end{align}

The difference $\unicode{x1D6E5} \mbox{Res}_{\Theta }$ between the actuated and non-actuated cases is small and can be neglected. The remaining terms in (3.3a ) and (3.3b ) are plotted in figure 10 for the travelling wave case at $Pr = 7.5$ and $\omega ^+=0.088$ . We see that the right-hand-side of (3.3a ) and (3.3b ) are dominated by $\unicode{x1D6E5} \overline {uv}^*$ and $\unicode{x1D6E5} \overline {\theta v}^*$ , up to their respective minima, but farther from the wall these terms are cancelled by the term containing the difference in Reynolds numbers $ ( {Re_\tau ^{-1}} - {Re_{\tau _0}^{-1}} )$ . That is, the net contributions to $\unicode{x1D6E5} \overline {U}^*$ and $\unicode{x1D6E5} \overline {\Theta }^*$ , hence, DR and HR, come from $\unicode{x1D6E5} \overline {uv}^*$ and $\unicode{x1D6E5} \overline {\theta v}^*$ , and then only up to the points where $\unicode{x1D6E5} \overline {uv}^*$ and $\unicode{x1D6E5} \overline {\theta v}^*$ reach their minimum values.

3.4. Source of inequality between $HR$ and $DR$

Integrating (3.3a ) and (3.3b ) once more with respect to $y^*$ gives

(3.4a,b) \begin{align} \unicode{x1D6E5} \overline {U}^*_{y^*} =\int _0^{y^*} \unicode{x1D6E5} ^*_{uv} {\mathrm d}y', \qquad \unicode{x1D6E5} \overline {\Theta }^*_{y^*} = Pr \int _0^{y^*} \unicode{x1D6E5} ^*_{\theta v} {\mathrm d}y', \end{align}

where

(3.4c) \begin{align} \unicode{x1D6E5} ^*_{uv} \equiv \unicode{x1D6E5} \overline { u v }^* - y^*(Re_\tau ^{-1} - Re_{\tau _0}^{-1}), \end{align}

(3.4d) \begin{align} \unicode{x1D6E5} ^*_{\theta v} \equiv \unicode{x1D6E5} \overline { \theta v }^* - y^*(Re_\tau ^{-1} - Re_{\tau _0}^{-1}). \end{align}

By using (3.4a ) we can relate $\unicode{x1D6E5} ^*_{uv}$ and $\unicode{x1D6E5} ^*_{\theta v}$ to $\unicode{x1D6E5} \overline {U}^*$ and $\unicode{x1D6E5} \overline {\Theta }^*$ , hence, to DR and HR, and so establish the connection between the DR and HR and the turbulence attenuation. For the DR, Gatti & Quadrio (Reference Gatti and Quadrio2016) derived the relation between DR ( $=1-R_f$ ) and the asymptotic value of $\unicode{x1D6E5} \overline {U}^*$ in the log region (3.5a ). In Appendix B we derive a similar relation between HR ( $=1-R_h$ ) and the asymptotic value of $\unicode{x1D6E5} \overline {\Theta }^*$ in the log region (3.5b ). That is, we have

(3.5a) \begin{align} &\quad\unicode{x1D6E5} \overline {U}^*_{170} = \sqrt {\frac {2}{C_{{f_0}}}}\left [ \frac {1}{\sqrt {R_f}} - 1 \right ] - \frac {1}{2\kappa _u}\ln {R_f} = \underbrace {\int ^{170}_0 \unicode{x1D6E5} ^*_{uv} {\mathrm d}y'}_{I_{uv}}, \end{align}

(3.5b) \begin{align} &\unicode{x1D6E5} \overline {\Theta }^*_{170} = \frac {\sqrt {C_{{f_0}}/2}}{C_{{h_0}}}\left [ \frac {\sqrt {R_f}}{R_h} - 1 \right ] - \frac {1}{2\kappa _\theta }\ln {R_f} = Pr \underbrace {\int ^{170}_0 \unicode{x1D6E5} ^*_{\theta v} {\mathrm d}y'}_{I_{\theta v}}. \end{align}

These derivations assume that the profiles of $\overline {U}^*$ and $\overline {\Theta }^*$ have well-defined log regions with slopes that are not affected by the wall oscillation, which is the case for our results (figure 8). The integration was performed up to $y^*_{{res}} = 170$ , where the profiles of $\unicode{x1D6E5} \overline {U}^*$ and $\unicode{x1D6E5} \overline {\Theta }^*$ reach their asymptotic levels (figure 8 j,k,l) and their derivatives are zero (figure 10). On the right-hand sides of (3.5a ) and (3.5b ) we have the integrals of attenuation in the turbulent shear stress ( $I_{uv}$ ) and the turbulent temperature flux ( $I_{\theta v}$ ). For a fixed set of viscous-scaled actuation parameters, $I_{uv} = \unicode{x1D6E5} \overline {U}^*_{170}$ is constant (Gatti & Quadrio Reference Gatti and Quadrio2016; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) (also shown in figure 8 j), but $I_{\theta v} = \unicode{x1D6E5} \overline {\Theta }^*_{170}/Pr$ depends on $Pr$ (figure 9 k,l).

To predict DR and HR from (3.5a ) and (3.5b ), we only need $\unicode{x1D6E5} \overline {U}^*_{170}$ and $\unicode{x1D6E5} \overline {\Theta }^*_{170}$ (that is, $I_{uv}$ and $I_{\theta v}$ ) as the inputs; the other parameters are associated with the non-actuated channel flow for which semi-empirical relations exist in the literature. We choose $\kappa _u = 0.4$ and $\kappa _\theta = 0.46$ (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016); the suitability of these choices is confirmed in § 2.3 and Appendix A. Dean (Reference Dean1978)’s correlation is used for the non-actuated skin-friction coefficient $C_{f_0} = 0.073(2Re_b)^{-1/4}$ , which could be re-expressed as $C_{{f_0}} = 0.037Re^{-2/7}_{\tau _0}$ ; this correlation agrees well with the DNS data (MacDonald et al. Reference MacDonald, Hutchins and Chung2019), and yields less than $2\,\%$ difference with our DNS result $C_{f_0} = 0.0057$ at $Re_{\tau _0} = 590$ . For the non-actuated $C_{h_0}$ , in figure 11 we compare our DNS data at $Re_{\tau _0} = 590$ and different $Pr$ (cases on the left side of table 1) with several empirical relations. These relations define $C_{h_0}$ either based on the bulk temperature $\Theta _b$ or the mixed-mean temperature $\Theta _m$ . For our DNS data, there is a negligible difference between the two definitions (empty versus the filled circles). Among the relations, the latest correlation by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2022) agrees well with our DNS data (solid black line). This relation is an improvement to Kader & Yaglom (Reference Kader and Yaglom1972)’s formula, and its accuracy is supported by the recent DNS of Pirozzoli (Reference Pirozzoli2023) for $\mathcal{O}(10^{-2}) \lesssim Pr \lesssim \mathcal{O}(10^{1})$ . It is formulated as

(3.6) \begin{align} \frac {1}{C_{{h_0}}} = \frac {\kappa _u}{\kappa _\theta }\frac {2}{C_{{f_0}}} + \left ( \beta _{CL} - \beta _2 - \frac {\kappa _u}{\kappa _\theta }B \right )\sqrt {\frac {2}{C_{{f_0}}}} + \beta _3. \end{align}

Here $\beta _{CL}(Pr) = B_\theta (Pr) + 3.504 - 1.5/\kappa _\theta$ , $\beta _2 = 4.92$ , $\beta _3 = 39.6$ , $B = 1.23$ and $B_\theta (Pr)$ is the log-law additive constant for $\overline {\Theta }^*$ (Kader & Yaglom Reference Kader and Yaglom1972), as introduced in Appendix B.

Figure 11. Comparison of the non-actuated $C_{h_0}$ between our DNS cases (left side of table 1) and several empirical relations. For the DNS data, $C_{h_0}$ is calculated either based on the bulk temperature $\Theta _b$ (empty symbols) or the mixed-mean temperature $\Theta _m$ (filled symbols). The empirical relations are by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2022) (3.6), Kays et al. (Reference Kays, Crawford and Weigand1993), Kader & Yaglom (Reference Kader and Yaglom1972), Kakaç et al. (Reference Kakaç, Shah and Aung1987), Chilton--Colburn analogy (Chilton & Colburn Reference Chilton and Colburn1934; Colburn Reference Colburn1964), Prandtl analogy (see equation 11 in Schlünder Reference Schlünder1998) and the von Kármán analogy (see equation 4 in Li & Li Reference Li and Li2010).

Figure 12. Travelling wave actuation with $A^+ = 12$ , $\kappa ^+_x = 0.0014$ . Comparison between the direct HR from DNS (filled squares) and the predicted HR (empty triangles) by solving (3.5a ) and (3.5b ). In (a), for predicting HR, we use $I_{uv}$ and $I_{\theta v}$ directly from DNS. In (b), for predicting HR, we use a power-law estimate for $I_{\theta v} = I_{uv}/Pr^\gamma$ , with the values of $\gamma$ reported in figure 13(b).

Figure 13. Travelling wave at $Pr = 0.71, 4.0$ and $7.5$ (same data as in figure 5 a,c,e). The outline colour of each data point indicates its $\omega ^+$ ; $\omega ^+ = 0.022$ (black), $0.044$ (blue), $0.066$ (red), $0.088$ (green) and $0.110$ (grey). (a) Plot of $I_{\theta v}$ versus $I_{uv}$ ; the dotted line is for $I_{\theta v}=I_{uv}$ . (b) Plot of $I_{\theta v}$ versus $Pr$ ; the solid curves are the power-law fit $I_{\theta v} = I_{u v}/Pr^{\gamma }$ , with $\gamma$ as shown. (c,d) Plots of HR and ${\textrm{HR}}-{\textrm{DR}}$ versus $Pr$ ; the solid curves are the prediction of (3.5b ) using $I_{\theta v} = I_{u v}/Pr^{\gamma }$ . In (d), the blue zone marks $\gamma \gt 0.5$ and the grey zone marks $\gamma \lt 0.5$ .

In figure 12(a) we compare the values of HR from DNS with those from (3.5a ) and (3.5b ) using $I_{uv}$ and $I_{\theta v}$ as obtained from DNS. The results support the accuracy of the model for relating HR to the attenuation of the turbulent flux. In figure 12(b) we also show the agreement between the model and the DNS when we use a power-law estimate for $I_{\theta v}$ ( $=I_{uv}/Pr^{\gamma }$ ) where only $I_{uv}$ (which is independent of $Pr$ ) is obtained from DNS. We justify the power law as follows.

From (3.5b ), we note that the Prandtl number dependence of HR (hence ${\textrm{HR}}- {\textrm{DR}}$ ) is due to both $C_{{h_0}}$ and $ I_{\theta v}$ . According to the Reynolds analogy, we would expect $I_{\theta v} = I_{u v}$ when $Pr = 1.0$ . Figure 13(a) indicates that, regardless of the actuation frequency $\omega ^+$ , $I_{\theta v} \gt I_{u v}$ for $Pr = 0.71$ and $I_{\theta v} \lt I_{u v}$ for $Pr \gt 1$ , with the difference increasing with $Pr$ . In other words, increasing $Pr$ beyond unity leads to a lower attenuation of $\overline {\theta v}$ compared with $\overline {u v}$ , even though in this regime ${\textrm{HR}} \gt {\textrm{DR}}$ (figure 5 c–f). Figure 13(b) shows how $I_{\theta v}$ depends on $Pr$ , and that $I_{\theta v} = I_{uv}/Pr^\gamma$ is a good approximation to the data. For the present data with a fixed $A^+$ and $\kappa ^+_x$ , $\gamma$ depends on $\omega ^+$ but it is independent of $Pr$ . When we use this power law in (3.5b) and solve for HR by using $I_{uv}$ from DNS, figure 13(c,d) shows that the model closely matches the data. Furthermore, for $\gamma \lt 0.5$ , we can achieve ${\textrm{HR}} \gt {\textrm{DR}}$ for $Pr \gt 1$ , and the smaller $\gamma$ is the higher HR compared with DR. From figure 13(b,d), as $\omega ^+$ increases from $0.022$ (black curve) to $0.110$ (grey curve), $\gamma$ decreases from approximately $0.5$ to $0.1$ , and ${\textrm{HR}}-{\textrm{DR}}$ increases from almost zero to $10\,\%$ . Therefore, we expect ${\textrm{HR}}\lt {\textrm{DR}}$ for $Pr \gt 1.0$ if $\gamma \gt 0.5$ . This occurs for the plane oscillation at $\omega ^+ = 0.022$ , where $\gamma = 2.43$ and ${\textrm{HR}}- {\textrm{DR}} \simeq -5\,\%$ (figure 21).

Figure 14. Predicted maps of (a,c) HR and (b,d) ${\textrm{HR}}-{\textrm{DR}}$ for the travelling wave with $A^+ = 12, \kappa ^+_x = 0.0014$ and $\omega ^+ = 0.088$ . Plots (a,b) are obtained from our proposed model $I_{\theta v} = I_{uv}/Pr^{\gamma }$ with $I_{uv} = 4.28, \gamma = 0.16$ obtained from DNS, and by solving (3.5 a , 3.5b ).Plots (c,d) are obtained from the simplified model (3.7a , 3.7b ) and by solving (3.5a , 3.5b ). The white bullets represent DNS results at $Re_{\tau _0} = 590$ for $Pr = 0.71, 4.0, 7.5$ and $20$ .

3.5. Heat-transfer reduction at higher Prandtl number and Reynolds number

Gatti & Quadrio (Reference Gatti and Quadrio2016) used (3.5a ) to extrapolate their low-Reynolds-number DR data to higher Reynolds numbers, and the accuracy of this extrapolation was corroborated by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) up to $Re_{\tau _0} = 4000$ using LES and by Gatti et al. (Reference Gatti, Quadrio, Chiarini, Gattere and Pirozzoli2024) up to $Re_{\tau _0} = 6000$ using DNS. For a fixed set of actuation parameters $(A^+,\kappa ^+_x,\omega ^+)$ , $I_{uv}$ is Reynolds and Prandtl number independent, and so the only Reynolds number dependency of DR is through $C_{f_0}$ . Hence, Gatti & Quadrio (Reference Gatti and Quadrio2016) could solve (3.5a ) and predict DR at any Reynolds number. We can now do the same for HR by using the power-law relationship $I_{\theta v} = I_{uv}/Pr^{\gamma }$ . By knowing $I_{uv}$ and $\gamma$ for a fixed set of actuation parameters, we can then solve (3.5a ) and (3.5b ) to map HR and ${\textrm{HR}}- {\textrm{DR}}$ as functions of $Pr$ and $Re_{\tau _0}$ .

Figure 15. Characteristics of the Stokes layer for the travelling wave with $A^+ = 12, \kappa ^+_x = 0.0014$ and $0.022 \le \omega ^+ \le 0.110$ (same cases as in figure 5 a,c,e). Results are independent of Prandtl number. The colour of the profiles and data points change from black at $\omega ^+ = 0.022$ to light grey at $\omega ^+ = 0.110$ . (a) Profiles of $\overline { \tilde {w}^2 }^*$ . Stokes layer protrusion height $\ell ^*_{0.01}$ (Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) marked at $\overline { \tilde {w}^2}^* = 0.01$ (filled bullets); laminar Stokes layer thickness $\delta ^*_S$ marked at $\overline { \tilde {w}^2}^* = ({1}/{2}){A^*}^2\textrm{e}^{-2}$ (cross symbols). (b) Production due to the Stokes layer $P^*_{33}$ . (c)Plot of $\ell ^*_{0.01}$ (filled bullets) and $\delta ^*_S$ (cross symbols) versus the maximum value of $\tilde {P}^*_{33}$ .

In figure 14(a,b) we show these predictions for the travelling wave case with $\omega ^+ = 0.088$ , where the DNS gives $I_{uv} = 4.28$ and $\gamma = 0.16$ (data points with green outline in figure 13). The predictions agree well with the DNS data points, including the point at $Pr = 20$ ( ${\textrm{HR}} = 43\,\%, {\textrm{HR}} - {\textrm{DR}} = 13\,\%$ ) (see table 1). At a given Reynolds number, HR increases almost logarithmically with Prandtl number. For instance, at $Re_{\tau _0} = 590$ HR increases from $30\,\%$ to $40\,\%$ as the Prandtl number increases from 0.71 to 7.5, and ${\textrm{HR}} = 50\,\%$ would be obtained at a Prandtl number of approximately 75. In addition, HR decreases very slowly with increasing $Re_{\tau _0}$ . For instance, at $Pr = 7.5$ HR decreases from $40\,\%$ to approximately $36\,\%$ as $Re_{\tau _0}$ increases from $590$ to $5900$ . Such a slow decrease with $Re_{\tau _0}$ is similarly observed in the predictive model for DR (3.5a ) (Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023; Gatti et al. Reference Gatti, Quadrio, Chiarini, Gattere and Pirozzoli2024).

In figure 14(c,d) we predict HR and ${\textrm{HR}}-{\textrm{DR}}$ for the same actuated case as in figure 14(a,b), but with a simplified analytical approach, similar to that proposed by Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005). The approach assumes that turbulence is completely damped in the near-wall region, which here we consider that to be up to the Stokes layer protrusion height $\ell ^*_{0.01}$ (figure 15 a). Therefore, for $0 \le y^* \le \ell ^*_{0.01}$ , the profiles are assumed to be laminar $\overline {U}^* = y^*[1-y^*/(2Re_\tau )], \overline {\Theta }^* = Pr y^*[1-y^*/(2Re_\tau )]$ , and for $y^* \gt \ell ^*_{0.01}$ , the profiles follow the turbulent formulations (C1a ,) (C1b ). By matching the laminar and turbulent profiles at $y^* = \ell ^*_{0.01}$ , we obtain

(3.7a) \begin{align} &\,\,\,\,\unicode{x1D6E5} \overline {U}^*_{170} = \ell ^*_{0.01} \left ( 1 - \frac {\ell ^*_{0.01}}{ 2Re_\tau } \right ) - \left ( \frac {1}{\kappa _u}\ln \ell ^*_{0.01} + B_u \right ), \end{align}

(3.7b) \begin{align} &\unicode{x1D6E5} \overline {\Theta }^*_{170} = Pr \ell ^*_{0.01} \left ( 1 - \frac {\ell ^*_{0.01}}{ 2Re_\tau } \right ) - \left ( \frac {1}{\kappa _\theta }\ln \ell ^*_{0.01} + B_\theta \right ). \end{align}

We substitute for $\unicode{x1D6E5} \overline {U}^*_{170}, \unicode{x1D6E5} \overline {\Theta }^*_{170}$ in (3.5a ), (3.5b ) to predict HR and ${\textrm{HR}}-{\textrm{DR}}$ . With this approach, the predicted HR and ${\textrm{HR}}-{\textrm{DR}}$ increase with $Pr$ (figure 14 c,d), consistent with our proposed model and DNS (figure 14 a,b). However, quantitatively, this approach overpredicts HR and ${\textrm{HR}}-{\textrm{DR}}$ by two times compared with the DNS data and our proposed model, which is of no surprise. The Stokes layer does not completely attenuate the near-wall turbulence, even down to $y^* \sim \mathcal{O}(1)$ (figure 9). The level of attenuation results from the complex interaction between the Stokes layer and the near-wall velocity and thermal fields, as we discuss next.

3.6. Stokes layer interaction with the velocity and thermal fields: integral parameters

For $Re_{\tau _0} \lesssim \mathcal{O}(10^3)$ , the primary source of DR is the protrusion of the Stokes layer due to the wall oscillation that modifies the near-wall velocity field (Choi et al. Reference Choi, DeBisschop and Clayton1998; Choi & Clayton Reference Choi and Clayton2001; Choi Reference Choi2002; Choi et al. Reference Choi, Xu and Sung2002; Quadrio & Sibilla Reference Quadrio and Sibilla2000; Ricco Reference Ricco2004; Quadrio & Ricco Reference Quadrio and Ricco2011; Ricco et al. Reference Ricco, Skote and Leschziner2021), and we would expect that a similar interaction between the Stokes layer and the near-wall temperature field leads to HR. As a result, the near-wall $\overline {u^2}^*$ profiles are modified (figure 9 d), consistent with the literature (Jung et al. Reference Jung, Mangiavacchi and Akhavan1992; Baron & Quadrio Reference Baron and Quadrio1995; Choi et al. Reference Choi, DeBisschop and Clayton1998; Quadrio & Sibilla Reference Quadrio and Sibilla2000; Choi & Clayton Reference Choi and Clayton2001; Ricco & Wu Reference Ricco and Wu2004; Quadrio & Ricco Reference Quadrio and Ricco2011; Touber & Leschziner Reference Touber and Leschziner2012; Ricco et al. Reference Ricco, Ottonelli, Hasegawa and Quadrio2012; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023), and similar trends are seen in the $\overline {\theta ^2}^*$ profiles (figure 9 e,f). To determine how the Stokes layer affects $I_{uv}$ and $I_{\theta v}$ as the Prandtl number changes, we first identify the Stokes layer characteristics through the harmonic component of the spanwise velocity $\overline {\tilde {w}^2}$ , where $\tilde {w}$ is obtained by applying triple decomposition:

(3.8a) \begin{align} &W(x,y,z,t) = \overline {W}(y) + \tilde {w}(x,y,t) + w(x,y,z,t), \end{align}

(3.8b) \begin{align} &\,\,\,\tilde {w}(x,y,t) = \underbrace {\frac {1}{N}\sum _{n=0}^{N-1} W(x,y,t+nT_{osc})}_{\left \langle W \right \rangle (x,y,t)} - \overline {W}(y). \end{align}

Here, $\overline {W}$ is the plane- and time-averaged component of the spanwise velocity, $\tilde {w}$ is the harmonic component, $w$ is the turbulent component and $\left \langle W \right \rangle$ is the spanwise- and phase-averaged value of $W$ . For the streamwise and wall-normal velocities and the temperature field, the harmonic components are negligible compared with the turbulent components. Following Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023), we quantify the Stokes layer protrusion height $\ell ^*_{0.01}$ as the wall distance where $\overline {\tilde {w}^2}^* = 0.01$ , and we locate the Stokes layer thickness $\delta ^*_S$ where $\overline { \tilde {w}^2}^* = ({1}/{2}){A^*}^2\textrm{e}^{-2}$ . We also calculate the production due to the Stokes layer according to $P^*_{33} \equiv -2 \overline {\left \langle w v \right \rangle \partial \tilde {w}/\partial y }^*-2 \overline {\left \langle w u \right \rangle \partial \tilde {w}/\partial x}^*$ , where $P^*_{33}$ is the only external source term due to the Stokes layer that injects energy into the turbulent stress budgets (Touber & Leschziner Reference Touber and Leschziner2012; Umair et al. Reference Umair, Tardu and Doche2022).

The behaviour of these Stokes layer parameters is given in figure 15 for the same travelling wave cases shown in figure 5(a,c,e). The profiles do not depend on $Pr$ , that is, at a fixed $\omega ^+$ the Stokes layer structure remains unchanged as the Prandtl number changes. When $\omega ^+$ decreases, $\ell ^*_{0.01}$ and the maximum production increase proportionately (figure 15 c). In other words, the Stokes layer becomes more protrusive while injecting more energy into the turbulent field. We note that $\ell ^*_{0.01}$ is almost linearly proportional to $P^*_{{33}_{{max}}}$ , but $\delta ^*_S$ is less responsive to the rise in the production. Therefore, $\ell ^*_{0.01}$ better quantifies the Stokes layer protrusion and strength, as sensed by the turbulent field, in agreement with the observations by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023).

Figure 16. Variation of the integral parameters with the Stokes layer characteristics $(\ell ^*_{0.01}, P^*_{{33}_{{max}}})$ for the same travelling wave cases as in figure 15. Plot (a) shows DR (blue squares), HR (light orange to brick red squares) versus $\omega ^+$ . (b) Same data as in (a) versus $\ell ^*_{0.01}$ (bottom axis) and $P^*_{{33}_{{max}}}$ (top axis). Plots (c,d) correspond to (a,b) for $I_{uv}$ (blue squares) and $I_{\theta v}$ (light orange to brick red squares); in (d) we overlay $\gamma$ (grey to black triangles). In all plots, the colour of HR and $I_{\theta v}$ changes from light orange at $Pr=0.71$ to brick red at $Pr = 7.5$ . The points associated with ${\textrm{DR}}_{{max}}$ and ${\textrm{HR}}_{{max}}$ are highlighted with a larger symbol size and black outline. In (b,d) the grey region shades the range of ${\textrm{DR}}_{{max}}$ and ${\textrm{HR}}_{{max}}$ . In all plots we highlight two cases that we further analyse in figures 17 and 18: case A at $\omega ^+ = 0.088$ with $\gamma = 0.16$ and case B at $\omega ^+ = 0.022$ with $\gamma = 0.53$ .

In figure 16 we assess the relation between the integral quantities DR, HR, $I_{\theta v}$ , $I_{uv}$ and the Stokes layer characteristics. Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) showed that DR increases with $\ell ^*_{0.01}$ up to an optimal value for maximum DR, corresponding to an optimal level of $P^*_{{33}_{{max}}}$ . For our present cases, this optimal point is $\ell ^*_{0.01} = 34$ (figure 16 b), where protrusion beyond this point causes DR to decrease. We observe a similar trend in HR (figure 16 b), but the corresponding optimal value of $\ell ^*_{0.01}$ for maximum HR decreases from 34to 21 as the Prandtl number increases from 0.71to 7.5. Figure 16(c,d) shows that the variations of $I_{uv}$ and $I_{\theta v}$ with $\omega ^+$ and $\ell ^*_{0.01}$ are consistent with the trends in DR and HR. In terms of variations with $Pr$ , however, HR increases even as $I_{\theta v}$ decreases. From $Pr=0.71$ to $7.5$ , the maximum value of $I_{\theta v}$ shifts to smaller values of $\ell ^*_{0.01}$ , corresponding to a shift in $\omega ^+$ from 0.044 ( $\ell ^*_{0.01} = 34$ ) to 0.088 ( $\ell ^*_{0.01} = 21$ ). Finally, it appears that the exponent $\gamma$ in the power law increases almost linearly with $\ell ^*_{0.01}$ and $P^*_{33_{{max}}}$ (figure 16 d).

To summarise, we find that for $Pr \gt 1$ , achieving maximum HR requires a less protrusive Stokes layer than achieving maximum DR. As the Stokes layer becomes more protrusive (energetic), it loses its efficacy in attenuating $\overline {\theta v}^*$ relative to $\overline {u v}^*$ , that is, $\gamma$ increases as $\ell ^*_{0.01}$ increases.

3.7. Stokes layer interaction with the velocity and thermal fields: scale-wise analysis

To better understand our observations on the integral parameters shown in figure 16, we now examine the spectrograms of $\overline {u v}^*$ and $\overline {\theta v}^*$ and visualise the distributions of the instantaneous $u v$ and $\theta v$ near the wall. We focus on two cases at $Pr = 7.5$ , case A at $\omega ^+ = 0.088$ , where HR is maximum and DR is near maximum ( $\gamma = 0.16$ ), and case B at $\omega ^+ = 0.022$ , where HR and DR drop below their maximum values, owing to a highly protrusive Stokes layer ( $\gamma = 0.53$ ).

Figure 17 displays the various pre-multiplied spectrograms for these two cases, where $\phi$ is the spectral density, $k_x$ is the streamwise wavenumber, $k_z$ is the spanwise wavenumber and $f$ is the frequency. The differences between the actuated and non-actuated cases are denoted by, for example, $\unicode{x1D6E5} k^*_x \phi ^*_{uv} = k^*_x \phi ^*_{uv} - k^*_x \phi ^*_{{uv}_0}$ . The extent up to which $\unicode{x1D6E5} k^*_x \phi ^*_{uv}$ and $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ are non-zero (red and blue fields in figure 17 e,f,q,r) coincides with the height of the Stokes layer protrusion $\ell ^*_{0.01}$ (horizontal dashed line); this supports the robustness of $\ell ^*_{0.01}$ to measure the extent up to which the Stokes layer modifies the turbulent field. Overall, for both cases A and B, and $k^*_x \phi ^*_{uv}$ and $ k^*_x \phi ^*_{\theta v}$ , the Stokes layer protrusion (production) leads to two outcomes: (i) large-scale energy attenuation as a favourable outcome (positive $\unicode{x1D6E5} k^*_x \phi ^*_{uv}, \unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ for $\lambda ^*_x \gtrsim 350$ , shown as the red fields in figure 17 e,f,q,r), and (ii) small-scale energy amplification as an unfavourable outcome (negative $\unicode{x1D6E5} k^*_x \phi ^*_{uv}, \unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ for $\lambda ^*_x \lesssim 350$ , shown as the blue fields). The large-scale energy attenuation corresponds to the attenuation of the near-wall streaks with $\lambda ^*_x \simeq 10^3$ (see the energetic peaks in the non-actuated spectrograms). The small-scale energy amplification corresponds to the emergence of smaller scales with $\lambda ^*_x \simeq 10^2$ (figure 18 j,k,l,m,u,v,w,x). The drastic changes in the streamwise spectrograms (figure 17 e,f,q,r) are echoed in the response of the frequency spectrograms (figure 17 g,h,s,t). Changing the actuation frequency $\omega ^+$ , hence changing the Stokes layer time scale, modifies the near-wall turbulence time scale $T^*$ , which in turn modifies the streamwise length scale $\lambda ^*_x$ (of course, $T^*$ is related to $\lambda ^*_x$ through a convection speed of approximately $ \mathcal{O}(10)$ according to Taylor’s hypothesis).

Further insight can be gained by comparing the near-wall instantaneous fields of $u^* v^*$ and $\theta ^* v^*$ , as shown in figure 18. The response of $u^* v^*$ to the Stokes layer (blue intensity fields) largely follows the trends seen in $\unicode{x1D6E5} k^*_x \phi ^*_{uv}$ and $ \unicode{x1D6E5} f^* \phi ^*_{uv}$ . For case A, we see that the large scales associated with the near-wall streaks are attenuated. At the same time, sparse patches of smaller scales with $\lambda ^*_x \simeq 10^2$ ( $T^* \simeq 10$ ) emerge. For case B, the emerging smaller scales possess a similar structure and size to those in case A, but with a noticeably larger population; they appear as large and closely spaced patches that cover a significant area of the near-wall region. Consistently, negative regions of $\unicode{x1D6E5} k^*_x \phi ^*_{uv}$ and $ \unicode{x1D6E5} f^* \phi ^*_{uv}$ are small for case A (figure 17 e,g), and large for case B (figure 17 q,s). Thus, $I_{uv}$ and DR are greater for case A than for case B (figure 16). Attenuation of the large scales associated with the near-wall streaks is considered to be a primary source of DR, as extensively reported in the literature (Baron & Quadrio Reference Baron and Quadrio1995; Di Cicca et al. Reference Di Cicca, Iuso, Spazzini and Onorato2002; Karniadakis & Choi Reference Karniadakis and Choi2003; Ricco Reference Ricco2004; Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Touber & Leschziner Reference Touber and Leschziner2012; Ricco et al. Reference Ricco, Skote and Leschziner2021; Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023). However, amplification of the smaller scales as a source of drag increase, is reported to a lesser extent. Touber & Leschziner (Reference Touber and Leschziner2012) observed such amplification in the frequency spectrograms $f^*\phi ^*_{uu}$ , $f^*\phi ^*_{vv}$ and $f^*\phi ^*_{ww}$ . Consistent with our figure 17, they noted that as $\omega ^+$ changes from 0.06to 0.03, and the Stokes layer production (protrusion) increases, energy accumulates in the scales with $T^* \simeq 10$ and DR decreases. Similarly, energy amplification at $T^* \simeq 10$ is observed in $f^*\phi ^*_{\tau \tau }$ by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023), and in the near-wall $f^* \phi ^*_{uu}$ by Deshpande et al. (Reference Deshpande, Chandran, Smits and Marusic2023).

Figure 17. Comparisons between case A (left two columns) and case B (right two columns. (a,m) Profiles of $\unicode{x1D6E5} ^*_{uv}$ (blue) and $\unicode{x1D6E5} ^*_{\theta v}$ (red). (b,n) Profiles of $\overline { \tilde {w}^2}^*$ (dashed-dotted line) and $P^*_{33}$ (solid line). Streamwise spectrograms: (c,o) $-k^*_x\phi ^*_{uv}$ ; (d,p) $-k^*_x\phi ^*_{\theta v}$ (contour lines and contour fields represent non-actuated and actuated cases, respectively). Difference between the actuated and non-actuated cases for the wavenumber spectra (e,f,q,r) and frequency spectra (g,h,s,t). Spanwise spectrograms: (i,u) $-k^*_z\phi ^*_{uv}$ ; (j,v) $-k^*_z\phi ^*_{\theta v}$ . Difference between the actuated and non-actuated cases (k,l,w,x). In all the spectrgrams, the horizontal dashed line marks $\ell ^*_{0.01}$ .

Figure 18. Visualisations of the instantaneous fields of $u^* v^*$ (blue intensity fields) and $\theta ^* v^*$ (brick intensity fields) at $y^+ = 14$ for the same cases as in figure 17. Next to each field, we magnify the small square outlined in pink. Left: non-actuated case; middle: actuated case A; right: actuated case B. The top two rows visualise $u^* v^*$ and $\theta ^* v^*$ at one time over the $x^*$ - $z^*$ plane. (i,t) Plot of the corresponding spanwise wall velocity $W^*_s$ over $x^*$ at the same time. The bottom two rows visualise $u^* v^*$ and $\theta ^* v^*$ at one $x^*$ location over $z^*$ and time $t^*$ . (o,y) Plot of $W^*_s$ over $t^*$ at the same $x^*$ location.

As to the $\theta ^* v^*$ field (brick intensity fields in figure 18), we see that with $Pr \gt 1.0$ the Stokes layer modifies the $\theta ^* v^*$ field more than the $u^* v^*$ field. In case A, for example, the scales with $\lambda ^*_x \simeq 10^2$ and $T^* \simeq 10$ are more frequent and more densely populated in the $\theta ^* v^*$ fields (figure 18 l,m,r,s) than in the $u^* v^*$ fields (figure 18 j,k,p,q). Consistently, $\unicode{x1D6E5} k^*_x\phi ^*_{\theta v}$ (figure 17 f) is more positive than $\unicode{x1D6E5} k^*_x\phi ^*_{u v}$ (figure 17 e) for $\lambda ^*_x \gtrsim 350$ , and more negative for $\lambda ^*_x \lesssim 350$ . We can make the same observations with respect to case B (figure 17 q,r). As a result, at each $\omega ^+$ , $I_{\theta v}$ falls below $I_{uv}$ when $Pr \gt 1$ (figure 16 c,d).

At a fixed $\omega ^+$ , $u^* v^*$ and $\theta ^* v^*$ are exposed to an identical Stokes layer. However, the larger change in $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ compared with $\unicode{x1D6E5} k^*_x \phi ^*_{u v}$ for $Pr \gt 1$ implies that the energetic near-wall scales of $\theta ^* v^*$ are locally exposed to a stronger Stokes layer production. We establish this connection by evaluating the local Stokes layer production $P^*_{33p}$ at the negative peaks of $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ and $\unicode{x1D6E5} k^*_x \phi ^*_{u v}$ ; these peaks are associated with the energetic small scales with $\lambda ^*_x \simeq 10^2$ that contribute to the drop in $I_{uv}$ and $I_{\theta v}$ . Figure 19(b) demonstrates this process for case B (figure 17 r), where the negative peak of $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ is intersected with the $P^*_{33}$ profile. With increasing $Pr$ , the conductive sublayer thins and the negative peak in $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ falls closer to the wall compared with its counterpart from $\unicode{x1D6E5} k^*_x \phi ^*_{u v}$ . This means that at each $\omega ^+$ , the energetic small scales of $\theta ^* v^*$ are exposed to a larger Stokes layer production, and hence, have a higher energy. This leads to the drop in $I_{\theta v}$ compared with $I_{uv}$ , as demonstrated in figure 19(a). At $Pr = 0.71$ , the local production $P^*_{33p}$ is close between $u^* v^*$ and $\theta ^* v^*$ , and $I_{uv}$ and $I_{\theta v}$ are close to each other. However, at $Pr = 4.0$ and $7.5$ , $P^*_{33p}$ at each $\omega ^+$ is several times higher for $\theta ^* v^*$ than $u^* v^*$ , and $I_{\theta v}$ drops below $I_{u v}$ . In addition, at $\omega ^+ = 0.022$ , $P^*_{33p}$ increases by almost 3 times from $Pr = 0.71$ to $7.5$ , whereas at $\omega ^+ = 0.110$ , $P^*_{33p}$ increases by $1.5$ times. As a result, $I_{\theta v}$ has a milder decay rate at $\omega ^+ = 0.110$ ( $\gamma = 0.11$ ) than at $\omega ^+ = 0.022$ ( $\gamma = 0.53$ ).

Figure 19. Assessment of the relation between $I_{uv}$ and $I_{\theta v}$ , and the local Stokes layer production $P^*_{33p}$ at the negative peaks of $\unicode{x1D6E5} k^*_x \phi ^*_{uv}$ and $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ . (a) Same data of $I_{uv}$ and $I_{\theta v}$ as in figure 16(c), but versus $P^*_{33p}$ ; to ease the inspection, the inset shows figure 16(c). The data points at $\omega ^+ = 0.022$ and $0.110$ have black and grey outlines, respectively. Panel (b) illustrates obtaining $P^*_{33p}$ for case B (figure 17 r), by intersecting the negative peak of $\unicode{x1D6E5} k^*_x \phi ^*_{\theta v}$ with the $P^*_{33}$ profile (on the left axis).

We have seen how the behaviour of the Stokes layer leads to $I_{\theta v}\lt I_{u v}$ by examining the streamwise and frequency spectrograms. This connection is more difficult to find in the spanwise spectrograms $k^*_z \phi ^*_{u v}$ and $ k^*_z \phi ^*_{\theta v}$ (figure 17 i–l, u–x). The Stokes layer predominantly modifies the time scale $T^*$ , hence, the streamwise length scale $\lambda ^*_x$ , of the near-wall turbulence. However, $k^*_z \phi ^*_{u v}$ and $ k^*_z \phi ^*_{\theta v}$ are integrated over $T^*$ and $\lambda ^*_x$ , and so the small-scale energy amplification is cancelled by the large-scale energy attenuation. Thus, even though the Stokes layer protrusion (production) increases from case A to case B, the distributions of $k^*_z \phi ^*_{u v}$ and $ k^*_z \phi ^*_{\theta v}$ suggest that the near-wall turbulence is distorted to a lesser extent.

Figure 20. Large- and small-scale contributions to $\unicode{x1D6E5} ^*_{uv}, \unicode{x1D6E5} ^*_{\theta v}$ and their integrals $I_{uv}, I_{\theta v}$ for the travelling wave case. Left column: $\omega ^+ = 0.088$ ; middlecolumn: $\omega ^+ = 0.044$ ; rightcolumn: $\omega ^+ = 0.022$ . Blue $uv$ profiles; for $\theta v$ profiles, $Pr = 0.71$ is shown in light orange, $Pr = 4.0$ in orange and $Pr = 7.5$ in brick red. (a,b,c) Plot of $I_{uv}, I_{\theta v}$ versus $\omega ^+$ . (d,e,f) Plot of $\unicode{x1D6E5} ^*_{uv}, \unicode{x1D6E5} ^*_{\theta v}$ versus $y^*$ ; the insets plot $\overline {\tilde {w}^2}^*$ representing the Stokes layer where the number reports $\ell ^*_{0.01}$ . (d,e,f) Total $\unicode{x1D6E5} ^*_{uv} \unicode{x1D6E5} ^*_{\theta v}$ ; (g,h,i) decomposed $\unicode{x1D6E5} ^*_{uv\pm }, \unicode{x1D6E5} ^*_{\theta v\pm }$ . In (d–i) we shade under the profiles of $\unicode{x1D6E5} ^*_{uv}$ and $\unicode{x1D6E5} ^*_{\theta v}$ at $Pr = 7.5$ , as well as their decomposition.

Our observations regarding the large-scale attenuation and small-scale amplification can be extended by applying scale-wise decomposition to $\unicode{x1D6E5} ^*_{uv}$ and $ \unicode{x1D6E5} ^*_{\theta v}$ . For instance, we can write $\unicode{x1D6E5} ^*_{uv} = \unicode{x1D6E5} ^*_{uv+} + \unicode{x1D6E5} ^*_{uv-}$ , where

(3.9a) \begin{align} &\unicode{x1D6E5} ^*_{uv+} = \int _{\lambda ^*_x \gt 350} \unicode{x1D6E5} \phi ^*_{uv}(k^*_x,y^*) {\mathrm d}k^*_x - y^*\left (Re^{-1}_\tau - Re^{-1}_{\tau _0}\right ), \end{align}

(3.9b) \begin{align} &\quad\qquad\qquad \unicode{x1D6E5} ^*_{uv-} = \int _{\lambda ^*_x \le 350} \unicode{x1D6E5} \phi ^*_{uv}(k^*_x,y^*) {\mathrm d}k^*_x. \end{align}

We set the threshold of $\lambda ^*_x = 350$ for partitioning because the spectrograms for all cases show $\unicode{x1D6E5} k^*_x \phi ^*_{uv} \gt 0$ for $\lambda ^*_x \gtrsim 350$ and $\unicode{x1D6E5} k^*_x \phi ^*_{uv}\lt 0$ for $\lambda ^*_x \lesssim 350$ (see figure 17 e,q). As noted earlier (figure 10 a and (3.3a )), we need to subtract the term $y^* ( Re^{-1}_\tau - Re^{-1}_{\tau _0} )$ from $\unicode{x1D6E5} \overline {u v}^*$ to obtain its net contribution $\unicode{x1D6E5} ^*_{uv}$ . Considering figure 17(e,q), this contribution appears in the large scales of $\unicode{x1D6E5} k^*_x \phi ^*_{uv}$ ( $\lambda ^*_x \gt 350$ ), and therefore, we subtract $y^* ( Re^{-1}_\tau - Re^{-1}_{\tau _0} )$ from the large-scale integration (3.9a ). The same reasoning is applied to the partitioning of $\unicode{x1D6E5} ^*_{\theta v}$ , and we calculate $\unicode{x1D6E5} ^*_{\theta v+}$ and $\unicode{x1D6E5} ^*_{\theta v-}$ in a manner that is identical to that expressed by (3.9a ) and (3.9b ).

Figure 20 shows how changing $\omega ^+$ or $Pr$ modifies the balance between the large-scale attenuation ( $\unicode{x1D6E5} ^*_{uv+}, \unicode{x1D6E5} ^*_{\theta v+}$ ) and the small-scale amplification ( $\unicode{x1D6E5} ^*_{uv-}, \unicode{x1D6E5} ^*_{\theta v-}$ ). The net attenuations $\unicode{x1D6E5} ^*_{uv}$ and $\unicode{x1D6E5} ^*_{\theta v}$ (figure 20 d,e,f) depend on the disparity between their decomposed parts (figure 20 g,h,i). Decreasing $\omega ^+$ (increasing the Stokes layer protrusion) simultaneously increase $ \unicode{x1D6E5} ^*_{u v+}, \unicode{x1D6E5} ^*_{\theta v+}$ and $\unicode{x1D6E5} ^*_{u v-}, \unicode{x1D6E5} ^*_{\theta v-}$ . However, below the optimal $\omega ^+$ , $ \unicode{x1D6E5} ^*_{\theta v-}, \unicode{x1D6E5} ^*_{u v-}$ increase more than $ \unicode{x1D6E5} ^*_{\theta v+}, \unicode{x1D6E5} ^*_{u v+}$ , leading to the drop in $I_{uv}, I_{\theta v}$ . We demonstrate this by shading the areas under $\unicode{x1D6E5} ^*_{\theta v}$ and its decompositions at $Pr=7.5$ light orange. From the optimal $\omega ^+ = 0.088$ (figure 20 g) to $0.022$ (figure 20 i), $\unicode{x1D6E5} ^*_{\theta v-}$ increases more than $\unicode{x1D6E5} ^*_{\theta v+}$ , and $I_{\theta v}$ decreases. At each $\omega ^+$ , increasing $Pr$ increases $\unicode{x1D6E5} ^*_{\theta v-}$ more than $\unicode{x1D6E5} ^*_{\theta v+}$ , hence, the drop in $I_{\theta v}$ . This is due to the thinning of the conductive sublayer and the increase in the local Stokes layer production, as discussed with respect to figure 19.

3.8. Comparison between the travelling wave and the plane oscillation

Figure 21. Similar plots as in figure 13(b,d), but for the plane oscillation with $A^+ = 12$ (figure 3 c). The outline colour of each data point indicates its $\omega ^+$ ; $\omega ^+ = 0.022$ (black), $0.044$ (blue), $0.066$ (red), $0.088$ (green) and $0.110$ (grey).

All our findings based on the travelling wave motion (§§ 3.3--3.7) remain broadly valid for the case of plane oscillation. By comparing figures 21 and 22 with figures 13 and 16, we see that, for example, the power-law relation $I_{\theta v} = I_{uv}/Pr^{\gamma }$ fits well with the plane oscillation data (figure 21 a), and the division ${\textrm{HR}}\gt {\textrm{DR}}$ ( $\gamma \lt 0.5$ ) and ${\textrm{HR}} \lt {\textrm{DR}}$ ( $\gamma \gt 0.5$ ) is also valid (figure 21 b). The value of $\gamma$ at each $\omega ^+$ is close to its counterpart for the travelling wave case, with the exception of $\omega ^+ = 0.022$ , where $\gamma = 2.48$ for the plane oscillation (black lines in figure 21), almost five times larger than the value of 0.53 found for the travelling wave motion. Such a strong decay rate drops $I_{\theta v}$ to almost zero by $Pr = 7.5$ for the plane oscillation, leading to ${\textrm{HR}}-{\textrm{DR}} \simeq -5\,\%$ . This strong decay is associated with the appearance of a highly protrusive Stokes layer (figure 22 a,c), in agreement with the discussion given in § 3.7.

Figure 22. Same plots as in figure 16, except the data points (filled squares) are from the plane oscillation with $A^+ = 12$ . The range of variations in the travelling wave data points are shaded in green.

Figure 22 shows that, for $0.066 \le \omega ^+ \le 0.110$ , DR, HR, $I_{u v}$ , $I_{\theta v}$ and $\ell ^*_{0.01}$ for the plane oscillation are close to the values found for the travelling wave. However, as $\omega ^+$ decreases to $ 0.044$ and then to $0.022$ , significant differences appear. For the plane oscillation, the Stokes layer becomes highly protrusive, reaching up to $\ell ^*_{0.01} \simeq 120$ at $\omega ^+ = 0.022$ . As discussed in § 3.7, such a protrusive Stokes layer loses its efficacy in producing a net attenuation of the near-wall turbulence. As a result, $I_{uv}$ and $I_{\theta v}$ significantly drop and $\gamma$ rises significantly. Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) showed that when $\ell ^*_{0.01} \sim \mathcal{O}(10^2)$ , the Stokes layer departs from its laminar-like structure, the mean velocity and temperature profiles become highly disturbed (figure 28 j,k,l), and the predictive relations for DR (3.5a ) and HR (3.5b ) begin to fail. Similarly, we observe that at $\omega ^+ = 0.022$ the predictive relation for HR (3.5b ) is much less accurate compared with its performance at higher frequencies (figure 21 b).

As a final note, it appears that $\gamma$ , a $Pr$ -independent quantity, increases almost linearly with $\ell ^*_{0.01}$ and $P^*_{{33}_{{max}}}$ , and decreases inversely with increasing $\omega ^+$ following

(3.10) \begin{equation} \gamma \simeq 4.2 P^*_{{33}_{{max}}} \simeq 0.008 \ell ^*_{0.01} \simeq {1}/({80\omega ^+}). \end{equation}

This is supported by the results shown in figure 23, where we compile all our data for the travelling wave and the plane oscillation cases (with the exception of the plane oscillation at $\omega ^+ = 0.022$ where the Stokes layer is particularly protrusive). Our present data are at a fixed $A^+ = 12$ for a plane oscillation ( $\kappa ^+_x = 0$ ) and a travelling wave ( $\kappa ^+_x = 0.0014$ ). The relation for $\gamma$ (3.10) is expected to also depend on $A^+$ and $\kappa ^+_x$ . However, for the upstream travelling wave with $\omega ^+ \gtrsim 0.04$ , we anticipate $\gamma$ to weakly depend on $\kappa ^+_x$ . Our conjecture is partly based on the validity of (3.10) for $\kappa ^+_x = 0$ and $\kappa ^+_x = 0.0014$ for $\omega ^+ \gtrsim 0.04$ . Furthermore, $\gamma$ relates $\unicode{x1D6E5} \overline {\Theta }^*_{170}$ to $\unicode{x1D6E5} \overline {U}^*_{170}$ , which $\unicode{x1D6E5} \overline {U}^*_{170}$ is seen to weakly depend on $\kappa ^+_x$ for the upstream travelling wave with $\omega ^+ \gtrsim 0.04$ (see figure 13 in Gatti & Quadrio Reference Gatti and Quadrio2016), and we anticipate $\unicode{x1D6E5} \overline {\Theta }^*_{170}$ to behave similarly.

Figure 23. Assessment of the relations (3.10) between $\gamma$ , $P^*_{33_{{max}}}$ , $\ell ^*_{0.01}$ and $\omega ^+$ for the travelling wave (filled squares) and the plane oscillation (empty circles); $\omega ^+ = 0.022$ (black), $0.044$ (blue), $0.066$ (red), $0.088$ (green) and $0.110$ (grey).