4.1. Correlation between the mass flux and drag reduction

The influences on the skin friction can be traced to the variations of the mean velocity and Reynolds shear stress. Figure 4(a, c, e) reports the mean velocity, Reynolds shear and streamwise normal stresses at $x=45 \delta _0$ for these cases with $A=0.005$ . Under outer scales, the near-wall mean velocity is lower in cases with micro-blowing control, resulting in the decreased mean velocity gradient at the wall and, hence, the lower skin friction coefficient. In addition to cases U1 and SX, the mean velocities are abated across the boundary layer, showing no tendency to recover to the classical mean velocity defect in the outer region, in particular for case U3, probably due to the low Reynolds number considered herein. The Reynolds streamwise normal and shear stresses, however, are significantly increased, which can probably be attributed to the strong shear layer corresponding to the higher mean velocity defect in the outer region that is responsible for the more intense turbulent kinetic energy production (Sumitani & Kasagi Reference Sumitani and Kasagi1995).

Figure 4.

Wall-normal distribution of the (a, b) mean velocity, (c, d) Reynolds shear stress and (e, f) streamwise Reynolds normal stress at (a, c, e) $x=45\delta _0$ under outer scales and (b, d, f) $x=100\delta _0$ under viscous scales, with $A=0.005$ . The grey line in panel (b) is the classical logarithmic law $2.44 \log (y^+) + 5.2$ . Black symbols in panels (b, d, f) are the reference data of Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023) at ${\textit{Re}}_\tau \approx 450$ , $M_0=2$ and $T_w = 1.36 T_0$ .

Figure 4. Long description

Panel A: A line graph shows the wall-normal distribution of mean velocity under outer scales. The x-axis is labeled y/delta0 and the y-axis is labeled u/U0. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. Panel B: A line graph shows the wall-normal distribution of mean velocity under viscous scales. The x-axis is labeled y+ and the y-axis is labeled u+/u*D. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. A grey line represents the classical logarithmic law. Panel C: A line graph shows the wall-normal distribution of Reynolds shear stress under outer scales. The x-axis is labeled y/delta0 and the y-axis is labeled -rho*u’*u’/rho0U0^2. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. Panel D: A line graph shows the wall-normal distribution of Reynolds shear stress under viscous scales. The x-axis is labeled y+ and the y-axis is labeled -rho*u’*u’/rho0+. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. Black symbols represent reference data. Panel E: A line graph shows the wall-normal distribution of streamwise Reynolds normal stress under outer scales. The x-axis is labeled y/delta0 and the y-axis is labeled -rho*u’*u’/rho0U0^2. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. Panel F: A line graph shows the wall-normal distribution of streamwise Reynolds normal stress under viscous scales. The x-axis is labeled y+ and the y-axis is labeled -rho*u’*u’/rho0+. Multiple lines represent different conditions: NC, U1, U2, U3, SX, SZ, and ST. Black symbols represent reference data.

Figure 4(b, d, f) displays the van Driest transformed mean velocity

(4.2) \begin{equation} u^+_{\textit{VD}} = \frac {1}{u_\tau } \int ^{u}_0 \sqrt {\frac {\bar \rho }{\bar \rho _w}}\, \textrm {d} u, \end{equation}

and the Reynolds shear and streamwise normal stresses downstream of the control region at $x=100 \delta _0$ under viscous scales, along with the results reported by Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023) at a similar parameter ${\textit{Re}}_\tau \approx 450$ , $M_0=2$ and $T_w = 1.36 T_0$ . At this location, the linear law and the logarithmic increment in canonical wall-bounded turbulence are recovered in cases imposed with micro-blowing control, but the intercepts of the logarithmic law are slightly lower, leaving the escalation of the mean velocity in the wake region to be more evident. The Reynolds normal and shear stresses also almost recover in the near-wall region, as indicated by their nearly collapsed profiles with that in case NC. Above $y^+=30$ , both the higher Reynolds normal and shear stresses remain, and gradually decrease as it goes downstream (not shown here for brevity), indicating the long ‘history effects’ by the perturbations of the micro-blowing at the wall, in spite of the low amplitude of $A$ compared with the free stream velocity (Park & Choi Reference Park and Choi1999; Zeng et al. Reference Zeng, Qiu, Jiang, Tong, Hu and Chen2024).

The variation of the drag reduction rate with the Reynolds number and mass flux at the wall can be evaluated according to law-of-the-wall of the mean velocity. If the self-similarity of the mean velocity approximately holds for cases with uniformly micro-blowing, with the van Driest transformation, the mean velocity in the logarithmic region (if the term can be applied here) obeys the following law (Jeromin Reference Jeromin1970):

(4.3) \begin{equation} u^+_{\textit{VD}} = \frac {1}{\kappa } \ln y^+ +C_1 + \frac {1}{4} \frac {\rho _w v_w}{\rho _0 u^*_\tau } \left ( \frac {1}{\kappa } \ln y^+ +C_2 \right )^2\! ,\end{equation}

as shown in figure 5(a), with $\kappa = 0.42$ , $C_1 = 5.2$ , $C_2 = 3.2$ (obtained by curve fitting) and $u^*_\tau = \sqrt {\tau _w/\bar \rho (y)}$ . Presuming this law-of-the-wall holds till the outer edge of the boundary layer, we have

(4.4) \begin{equation} \sqrt {\frac {2}{C_{\!f}}} = \frac {1}{\kappa } \ln \left ( {\textit{Re}}_\delta \sqrt {\frac {C_{\!f}}{2}}\right ) +C_1 + \frac {1}{4} \frac {v_w}{u_\tau } \left ( \frac {1}{\kappa } \ln \left ( {\textit{Re}}_\delta \sqrt {\frac {C_{\!f}}{2}}\right ) +C_2 \right )^2\!. \end{equation}

This gives the association between the micro-blowing mass flux, Reynolds number and the skin friction coefficient. Note that the van Driest transformation may not be suitable for flows at higher $M_0$ and lower wall temperature with significant wall heat transfer, so that more advanced mean velocity integral transformations are needed in such cases (Trettel & Larsson Reference Trettel and Larsson2016; Volpiani et al. Reference Volpiani, Iyer, Pirozzoli and Larsson2020; Griffin, Fu & Moin Reference Griffin, Fu and Moin2021; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023).

Figure 5.

(a) van Driest transformed mean velocity in case U1 (solid lines) and law-of-the-wall in (4.3) (dashed lines). (b) Variation of $DR$ with $F_m$ and (c) skin friction coefficient with ${\textit{Re}}_\delta$ according to (4.4). Here, $Re^*_\delta = \bar \rho _w U_0 \delta /\bar \mu _w$ .

In figure 5(b), we present the variation of the drag reduction rate $DR$ with the mass flux per unit area $F_m$ , along with our DNS results in case U1 for comparison. Equation (4.4) predicts reasonably well the behaviour of the variation of the drag reduction rate $DR$ with $F_m$ , with a slight discrepancy in magnitude, which is probably due to the low Reynolds numbers considered herein that renders the logarithmic region relatively short. The experimental measurement of Hwang (Reference Hwang2000) at the same Mach number but considerably higher Reynolds number also agrees well with the prediction given herein. The deviation at higher mass flux in such a dataset could be caused by the higher relative error due to the much smaller skin friction. The agreement between the DNS and experiments validates the capability of estimating the drag reduction rate via formula (4.4) in the regions where the micro-blowing control is imposed.

With the increasing Reynolds number, as given in figure 5(c), the drag reduction rate $DR$ further increases at a given $F_m$ . This could be attributed to the fuller mean velocity profiles and the stronger mean velocity gradients at the wall, resulting in the more efficient reduction in the skin friction. Verification is needed via DNS or experimental measurements, which is left for future work.

4.2. Differences due to micro-blowing arrangements

The previous analysis only concern the cases with uniform blowing. The differences in other types of micro-blowing arrangements will be given via an alternative approach.

Figure 6.

Sketch of the computational domain to be analysed.

We follow the classical Karman momentum integral equation (Stroh et al. Reference Stroh, Hasegawa, Schlatter and Frohnapfel2016) to provide a physical insight into the spatial development of the boundary layer and to evaluate quantitatively how the boundary layer thickness as well as the skin friction vary under the influence of the micro-blowing at the wall. We consider the two-dimensional control volume sketched in figure 6, with the height $h$ higher than the boundary layer thickness, $l_x$ the streamwise length from $x_1$ to $x_2$ , and a generic distribution of the micro-blowing velocity $v_w(x)$ at the wall. The mass conservation requires the total flow rate through the four surfaces to be zero:

(4.5) \begin{align} \int ^h_0 \bar \rho (x_1,y) \tilde u(x_1,y)\, \textrm {d}y &- \int ^h_0 \bar \rho (x_2,y) \tilde u(x_2,y)\, \textrm {d}y + \int ^{x_2}_{x_1} \bar \rho _w(x) \tilde v_w(x) \,\textrm {d}x \notag\\&\quad - \int ^{x_2}_{x_1} \rho _0 \tilde v_0(x)\, \textrm {d}x =0 ,\end{align}

and with the definition of the displacement thickness $\delta ^*$ , it can be further cast as follows:

(4.6) \begin{equation} \delta ^*(x_2) - \delta ^*(x_1) = \int ^{x_2}_{x_1} \frac {\rho _0 \tilde v_0(x) - \bar \rho _w(x) \tilde v_w(x)}{\rho _0 U_0}\, \textrm {d}x,\quad \delta ^* = \int ^\infty _0 \left ( 1 - \frac {\bar \rho (y) \tilde u(y)}{\rho _0 U_0}\right )\, \textrm {d}y. \end{equation}

Under the thin boundary layer hypothesis, the momentum conservation gives

(4.7) \begin{align} &- \int ^h_0 \bar \rho (x_1,y) (\tilde u^2(x_1,y) + \widetilde {u''^2}(x_1,y))\, \textrm {d}y + \int ^h_0 \bar \rho (x_2,y) (\tilde u^2(x_2,y) + \widetilde {u''^2}(x_2,y))\,\textrm {d}y \notag\\&\quad + \int ^{x_2}_{x_1} \rho _0 U_0 \tilde v_0(x)\, \textrm {d}x = \int ^h_0 \bar p(x_1,y)\, \textrm {d}y - \int ^h_0 \bar p(x_2,y) \,\textrm {d}y - \int ^{x_2}_{x_1} \bar \tau _w(x)\, \textrm {d}x, \end{align}

similarly, with the momentum thickness $\theta$ ,

(4.8) \begin{equation} \theta = \int ^\infty _0 \frac {\bar \rho (y) \tilde u(y)}{\rho _0 U_0} \left ( 1 - \frac {\tilde u(y)}{U_0}\right )\, \textrm {d}y. \end{equation}

Equation (4.7) can be reorganised as

(4.9) \begin{equation} \begin{aligned} &\theta (x_2) - \theta (x_1) = \int ^{x_2}_{x_1} \frac {\bar \rho _w(x) \tilde v_w(x)}{\rho _0 U_0}\, \textrm {d}x + \int ^{x_2}_{x_1} \frac {\tau _w(x)}{\rho _0 U^2_0}\, \textrm {d}x \\ &\quad + \int ^h_0 \frac {(\bar p(x_2,y)-\bar p(x_1,y))}{\rho _0 U^2_0}\,\textrm {d}y + \int ^h_0 \frac {(\overline {\rho u''^2}(x_2,y)-\overline { \rho u''^2}(x_1,y))}{\rho _0 U^2_0}\,\textrm {d}y, \end{aligned} \end{equation}

in which (4.6) has been incorporated.

In the limit of $(x_2 - x_1) \rightarrow 0$ , (4.6) and (4.9) can be formulated as the following ordinary differential equations:

(4.10) \begin{align} \frac {\textrm {d}\delta ^*}{\textrm {d}x} &= \frac {\rho _0 \tilde v_0(x) - \bar \rho _w(x) \tilde v_w(x)}{\rho _0 U_0}, \end{align}

(4.11) \begin{align} \frac {\textrm {d} \theta }{\textrm {d}x} &= \frac {\bar \rho _w(x) \tilde v_w(x)}{\rho _0 U_0} + \frac {1}{\rho _0 U^2_0} \int ^{\delta (x)}_0 \frac {\textrm {d} \big(\bar p(x) + \overline {\rho u''^2}(x)\big)} {\textrm {d}x} \textrm {d}y + \frac {\tau _w(x)}{\rho _0 U^2_0}. \end{align}

It can be inferred from (4.10) that the growth rate of the displacement thickness $\delta ^*$ depends on the vertical mass flux at the wall and in the free stream. If the shape factor of the boundary layer remains approximately unaltered by the disturbance introduced at the wall (which we will show later), (4.11) can thus be used to interpret the variation of the skin friction. The skin friction coefficients on the right-hand side of (4.11), however, rely on the growth rate of the momentum thickness, vertical mass flux at the wall, and the streamwise derivative of the mean pressure and streamwise normal component of the Reynolds stress tensor. For simplicity, we refer to the terms on the left-hand side of (4.11) as $L_\theta$ , and those on the right-hand side as $L_v$ , $L_p$ and $L_\tau$ , respectively.

Figure 7.

Streamwise distribution of terms in (4.11), $A=0.005$ , (a) U1, (b) U2, (c) U3, (d) SX, (e) SZ, (f) ST. Symbols, $L_\theta$ ; solid lines, $L_\tau$ ; dashed lines, $L_v$ ; dash-dotted lines, $L_p$ .

Figure 7. Long description

Panel A: A line graph shows the streamwise distribution of term U1. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.010 to 0.015. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets. Panel B: A line graph shows the streamwise distribution of term U2. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.02 to 0.04. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets. Panel C: A line graph shows the streamwise distribution of term U3. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.06 to 0.06. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets. Panel D: A line graph shows the streamwise distribution of term SX. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.010 to 0.015. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets. Panel E: A line graph shows the streamwise distribution of term SZ. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.010 to 0.015. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets. Panel F: A line graph shows the streamwise distribution of term ST. The x-axis is labeled x/delta0 and ranges from 20 to 120. The y-axis is labeled Li and ranges from -0.010 to 0.015. The graph includes symbols, solid lines, dashed lines, and dash-dotted lines representing different data sets.

In figure 7, we present the streamwise variation of terms in (4.11) in cases with $A=0.005$ . These terms are balanced by each other, validating the momentum conservation and statistical convergence of the datasets used herein. From the perspective of the skin friction, we can infer from the distribution of these terms that the abrupt change of the momentum thickness gradient should be attributed to the streamwise pressure gradient, with the highest contribution reached at the commencement and termination of the micro-blowing at the wall. Intriguingly, the positive $L_p$ at the commencement is lower than that at the termination in magnitude, especially those with high amplitude of local blowing mass flux, suggesting that their overall contribution to the skin friction cannot be balanced. The streamwise gradient of the Reynolds normal stress is small by comparison. The $L_v$ term, however, is responsible for the oscillatory feature of the skin friction in cases SX, SZ and ST.

Figure 8.

Integration of terms in (4.11) within $x=(35 {-} 85) \delta _0$ , (a) $L_\theta$ , (b) $L_v$ , (c) $L_p$ . Symbols are the same as figure 3.

We now integrate these terms within $x=(35 {-} 85) \delta _0$ to further quantify the contribution of these terms to the variation of the skin friction (figure 8). The $L_\theta$ term grows almost linearly with $F_m$ , independent of the distribution of the micro-blowing imposed at the wall. This confirms our previous presumption that the shape factor of the boundary layer within this region can be regarded to be almost identical. The $L_v$ term grows linearly with $F_m$ , which is expected according to its expression. The augmentation of $L_p$ with $F_m$ , however, is comparatively scattered, showing significant dependence on the distribution of the micro-blowing region. If the pressure gradient term is antisymmetric at the commencement and termination of the micro-blowing region, the net contribution of this term should be zero. However, this is not the case for the flows considered herein, as we have shown in figure 7. Such asymmetric pressure gradient could be attributed to the different streamwise extent, and intensity of compressive and expansive waves induced by the same amplitude of disturbances, considering the free stream Mach number being higher than 1.0 in the presently considered flows. Therefore, for the purpose of reaching better effects of drag reduction, one should attempt to eliminate the influence of the above-mentioned asymmetric pressure gradient to the furthest extent. Verification of such a hypothesis requires datasets at different free stream Mach numbers and will be discussed in our future work.

Figure 9.

Decomposition of the skin friction coefficient $C_{\!f}$ in terms of (4.12): (a) $C_{\!fV}$ ; (b) $C_{\!fT}$ ; (c) $C_{\!fG}$ ; (d) $C_{\!fT}+C_{\!fG}$ .

4.3. Analysis via FIK identity

The contribution of the mean shear rate and Reynolds shear stresses can be quantitatively evaluated by the following generalised FIK identity, obtained by integrating the streamwise mean momentum equation (Wenzel, Gibis & Kloker Reference Wenzel, Gibis and Kloker2022),

(4.12) \begin{equation} C_{\!f} = \frac {2}{\rho _0 U^2_0 \delta ^2} \left ( \int ^\delta _0 (\delta - y) \tau _{12}\, \textrm {d}y - \int ^\delta _0 (\delta - y) \overline {\rho u'' v''}\, \textrm {d} y - \int ^\delta _0 (y-\delta )^2 I_{x} \,\textrm {d} y \right ). \end{equation}

The right-hand-side terms represent the contributions of the viscous shear stress $\tau _{12} = \bar \mu (\partial \bar u/\partial y + \partial \bar v/\partial x)$ (hereinafter denoted by $C_{\!fV}$ ), Reynolds shear stress ( $C_{\!fT}$ ) and mean convection (spatial development) of the boundary layer ( $C_{\!fG}$ ), with $I_x$ expressed as

(4.13) \begin{equation} I_x= - \frac {\partial \bar \rho \tilde u \tilde u}{\partial x} - \frac {\partial \overline {\rho u'' u''}}{\partial x} - \frac {\partial \bar \rho \tilde u \tilde v}{\partial y} - \frac {\partial \bar p}{\partial x} + \frac {\partial \bar \tau _{11}}{\partial x}, \end{equation}

and $\tau _{11}$ the viscous normal stress. The results are shown in figure 9 and the relative contributions of these terms to the skin friction are given in Appendix A. The $C_{\!fV}$ terms in cases applied with micro-blowing control deviate from those in case NC at $x=40 \delta _0$ . Only in cases U2 and U3 are the $C_{\!fV}$ terms significantly lower due to the higher blowing intensity. The slight differences of the mean velocities in cases U1, SX, SZ and ST, as observed in figure 4, can be trivial in the integral sense, except for the small oscillations in cases SX and ST. The $C_{\!fT}$ terms given in figure 9(b), which integrate the weighted Reynolds shear stress, increase significantly, but are overly compensated by the abatement of $C_{\!fG}$ displayed in figure 9(c). As a result, their summations, as given in figure 9(d), are reduced compared with that in case NC. Unlike $C_{\!fV}$ , this term is dependent on the specific distribution of the blowing region, for the turbulent motions inducing the Reynolds shear stress can be sensitive to local perturbations. Downstream of $x=80\delta _0$ , the profiles of $C_{\!fT}+C_{\!fG}$ also collapse for all cases, similar to $C_{\!fV}$ . Based on the mentioned observations, we can conclude that the decrement of the skin friction coefficient $C_{\!f}$ should be attributed to all the components in the FIK identity, instead of the mean convection term alone, as elucidated by previous studies (Kametani & Fukagata Reference Kametani and Fukagata2011; Chen et al. Reference Chen, Yu, Li and Li2016). Notably, the $C_{\!fT}+C_{\!fG}$ is the lowest for case ST downstream of the blowing region, which is the reason for the higher drag reduction rate shown in figure 3(b).

To further elucidate the mechanism underlying the enhanced drag reduction observed in case ST, we examine the conditionally averaged flow fields associated with intense sweeping and ejection events at $y^+=20$ . The events are identified based on the criteria $u''\gt u''_{rms}, v''\lt -v''_{rms}$ and $u''\lt -u''_{rms}, v''\gt v''_{rms}$ , corresponding to Q2 (ejection) and Q4 (sweeping) events in the framework of quadrant analysis. The conditional average, denoted as $\langle \boldsymbol{\cdot }\rangle _c$ , is computed over the streamwise range $x = (100 {-} 120) \delta _0$ and is expressed as a function of $\Delta x$ , $y$ and $\Delta z$ . Figure 10(a, b) presents a three-dimensional visualisation of the conditionally averaged streamwise velocity fluctuation $\langle u'' \rangle _c$ for the uncontrolled case NC. As expected, for Q2 events, the isosurface of negative $\langle u'' \rangle _c$ is centred at $y^+=20$ , extends in the streamwise direction and is flanked by positive fluctuations in the spanwise direction, reminiscent of the near-wall velocity streaks. For Q4 events, the opposite trend is observed. The positive $\langle u'' \rangle _c$ isosurface is comparatively taller and exhibits more elongated streamwise extensions. Such observations are consistent with previous findings in incompressible wall-bounded turbulence. In figure 10(c,d), we compare the results across different micro-blowing configurations in the $\Delta x=0$ plane. For Q2 events, the magnitude of negative velocity fluctuations is slightly attenuated under micro-blowing control, though the effect shows little sensitivity to the spatial arrangement of the blowing regions. For Q4 events, however, the positive velocity fluctuations associated with sweeping events decay more slowly with distance from the statistical centre in the micro-blowing cases, with the notable exception of case ST with staggered blowing slots. This trend implies that the reduced intensity of sweeping events in case ST could be a key contributor to its higher drag reduction rate downstream.

Figure 10.

Conditionally averaged streamwise velocity fluctuations $\langle u'' \rangle _c$ surrounding (a, c) Q2 and (b, d) Q4 events at $y^+=20$ . (a, b) Three-dimensional visualisation for case NC, (c, d) two-dimensional distributions in the $\Delta x^+=0$ plane for cases NC (black lines), U1 (red), SX (green), SZ (blue) and ST (purple).

Figure 10. Long description

Panel A: A three-dimensional visualization for case NC showing streamwise velocity fluctuations around Q2 events. The visualization includes color-coded regions indicating different velocity values, with blue representing -0.1 and red representing +0.05. Panel B: A three-dimensional visualization for case NC showing streamwise velocity fluctuations around Q4 events. The visualization includes color-coded regions indicating different velocity values, with blue representing -0.05 and red representing +0.1. Panel C: A two-dimensional distribution in the plane for cases NC (black lines), U1 (red), SX (green), SZ (blue), and ST (purple) showing streamwise velocity fluctuations around Q2 events. The x-axis is labeled Δz/δ and the y-axis is labeled y/δ. Panel D: A two-dimensional distribution in the plane for cases NC (black lines), U1 (red), SX (green), SZ (blue), and ST (purple) showing streamwise velocity fluctuations around Q4 events. The x-axis is labeled Δz/δ and the y-axis is labeled y/δ.

It should be noted, however, that the mechanisms for drag reduction inferred from the Karman momentum integral equation and the FIK identity are not contradictory. Rather, they represent complementary descriptions of mean momentum transport from two distinct perspectives. The Karman momentum integral equation is derived from the mean momentum balance and mass conservation of the boundary layer. It primarily captures the streamwise evolution of mass and momentum fluxes induced by factors acting both on the boundaries and within the control volume. In contrast, the FIK identity is formulated under a quasi-one-dimensional hypothesis. It focuses on the local mean momentum balance and its direct relationship to the skin friction at a given streamwise station, with the influence of spatial development accounted for by the $C_G$ term. If we assume that the mean velocity and total shear stress in the turbulent boundary layer obey self-similarity, an assumption reasonably valid upstream and downstream of the blowing region considered here, then an increase in boundary layer thickness (or a more rapid growth rate) directly implies a reduction in skin friction. Under this framework, the lower skin friction is physically equivalent to a reduced integral contribution from both the viscous shear stress term, and the combined Reynolds shear stress and streamwise inhomogeneity terms appearing in the FIK identity. Neverthless, neither of these methods have taken the historical effects from the upstream flow into consideration.

Figure 11.

Distributions of the pre-multiplied spanwise cospectra $k_z E^+_{uv}(k_z)$ , lines, case NC at $x=95 \delta _0$ ; flooded, case U1 at (a) $x=85 \delta _0$ , (b) $x=95 \delta _0$ , (c) $x=105 \delta _0$ and (d) $x=115 \delta _0$ .

The history effects can be further elucidated by inspecting the downstream evolution of the flow structures at different scales, which can be manifested by the pre-multiplied spanwise co-spectra of $u''$ and $v''$ at different streamwise stations, denoted as $k_z E^+_{uv}(k_z)$ , as shown in figure 11. Here, the spectra are weighted by the local mean density. We only present the comparison between the results of case U1 and NC for brevity. Other cases with micro-blowing control exhibit similar trends. Compared with case NC, the spectral intensities are significantly higher in the outer region, especially at $x=85 \delta _0$ . As it approaches downstream, these outer-region peaks gradually diminish. It can be inferred that sufficiently far downstream of the control region, the spectra will recover to a distribution indistinguishable from that of case NC. In that far-field region, the FIK decomposition would again yield results similar to the uncontrolled case, with the $C_{\!fV}$ and $C_{\!fT}$ terms dominating, and the $C_{\!fG}$ term remaining positive but relatively small. However, this does not change our previous conclusion that the increase in boundary layer thickness, resulting from the injection of low-momentum fluid at the wall, remains the direct cause of the reduction in the skin-friction coefficient.

We would also like to remark on the higher reduction rate of the wall heat transfer compared with that of the skin friction downstream of the blowing control region. Following the work of Wenzel et al. (Reference Wenzel, Gibis and Kloker2022), the wall heat transfer can be decomposed as the following formula, which is obtained by the two-fold integration of the mean temperature transport equation

(4.14) \begin{equation} C_h = C_{\textit{hB}} + C_{\textit{hT}} + C_{\textit{hV}} + C_{\textit{hP}} + C_{\textit{hM}}, \end{equation}

where the terms on the right-hand-side of (4.14) are expressed as

(4.15) \begin{align} C_{\textit{hB}} &= \frac {1}{\rho _0 U_0 C_p (T_r - T_w) \delta ^2} \int ^\delta _0 (\delta - y) \overline {\kappa \frac {\partial T}{\partial y}}\, \textrm {d} y, \end{align}

(4.16) \begin{align} C_{\textit{hT}} &= - \frac {C_v}{\rho _0 U_0 C_p (T_r - T_w) \delta ^2} \int ^\delta _0 (\delta - y) \overline {\rho v'' T''}\, \textrm {d} y, \end{align}

(4.17) \begin{align} C_{\textit{hV}} &= \frac {1}{\rho _0 U_0 C_p (T_r - T_w) \delta ^2} \int ^\delta _0 (\delta - y)^2 \overline {\tau _{\textit{ij}} \frac {\partial u_i}{\partial x_{\!j}}}\, \textrm {d} y, \end{align}

(4.18) \begin{align} C_{\textit{hP}} &= - \frac {C_v}{\rho _0 U_0 C_p (T_r - T_w) \delta ^2} \int ^\delta _0 (\delta - y)^2 \overline {p \frac {\partial u_k}{\partial x_k}}\, \textrm {d} y, \end{align}

(4.19) \begin{align} C_{\textit{hM}} &= - \frac {C_v}{\rho _0 U_0 C_p (T_r - T_w) \delta ^2} \int ^\delta _0 (\delta - y)^2\notag\\&\quad\times \left ( \bar \rho \tilde u \frac {\partial \tilde T}{\partial x} + \bar \rho \tilde v \frac {\partial \tilde T}{\partial y} + \frac {\partial \overline {\rho u'' T''}}{\partial x} - \overline {\frac {\partial }{\partial x} \left ( \kappa \frac {\partial T}{\partial x} \right )} \right )\,\textrm {d} y, \end{align}

representing the contribution from the molecular heat conduction, turbulent heat flux, viscous dissipation of kinetic energy, the work of pressure on dilatation and the mean flow convection. It should be noted that the purpose of employing this decomposition is not to evaluate the wall heat transfer reduction directly, but to identify the physical origin of the streamwise variation in the Reynolds-analogy factor $s$ . Amongst these terms, the $C_{\textit{hP}}$ term remains small at the relatively low Mach number considered here. The mean flow convection $C_{\textit{hM}}$ term is expected to behave similarly to its counterpart in the skin-friction decomposition. Therefore, the following discussion focuses exclusively on the remaining three terms.

Figure 12.

Distributions of (a) mean temperature against mean velocity, (b) turbulent Prandtl number and (c) mean kinetic energy viscous dissipation normalised by viscous scales, at $x=95 \delta _0$ for cases with $A=0.005$ .

Figure 12. Long description

Panel A: A line graph shows the distribution of mean temperature against mean velocity. The x-axis is labeled with a dimensionless velocity unit, and the y-axis is labeled with a dimensionless temperature unit. Multiple lines represent different cases, including NC, UI, U2, U3, SX, SZ, and ST, each with distinct colors and markers. The lines show a general trend where the mean temperature decreases as the mean velocity increases. Panel B: A line graph illustrates the turbulent Prandtl number against the dimensionless wall-normal coordinate. The x-axis is labeled with the dimensionless wall-normal coordinate, and the y-axis is labeled with the turbulent Prandtl number. Multiple lines represent different cases, showing fluctuations around a value of 1. Panel C: A line graph depicts the mean kinetic energy viscous dissipation normalized by viscous scales against the dimensionless wall-normal coordinate. The x-axis is labeled with the dimensionless wall-normal coordinate, and the y-axis is labeled with the normalized mean kinetic energy viscous dissipation. Multiple lines represent different cases, showing a decreasing trend as the dimensionless wall-normal coordinate increases.

The integral of $C_{\textit{hB}}$ term can be reformulated as

(4.20) \begin{equation} \overline {\kappa \frac {\partial T}{\partial y}} \approx \bar \kappa \frac {\partial \bar T}{\partial y} = \left ( \bar \mu \frac {\partial \bar u}{\partial y} \right ) \left ( \frac {C_p}{\textit{Pr}} \frac {\partial \bar T}{\partial \bar u} \right )\!, \end{equation}

with the first factor being identical to the integral of the $C_{\!fV}$ term if the flow can be regarded to be one-dimensional, while the second factor is determined by the correlation between the mean temperature and mean velocity. Recalling the results in figure 9(a), as well as the distributions of the mean temperature $\bar T$ against the mean velocity $\bar u$ reported in figure 12(a), the integral of the $C_{\textit{hB}}$ term is essentially irrelevant of the arrangement of the micro-blowing control. This indicates that this term can be directly analogised to its counterpart $C_{\!fV}$ in the skin friction decomposition. Similarly, the integral of $C_{\textit{hT}}$ term can be recast as

(4.21) \begin{equation} \overline {\rho v'' T''} = \frac {1}{{\textit{Pr}}_T} \frac {\partial \bar T}{\partial \bar u} \overline {\rho u'' v''} \end{equation}

based on the definition of the turbulent Prandtl number. Both the factors $1/{\textit{Pr}}_T$ (given in figure 12 b) and $\partial \bar T/\partial \bar u$ in this formula are not subject to the influences of the spatial distribution of the micro-blowing, rendering the analogy between $C_{\textit{hT}}$ and $C_{\!fT}$ terms equally straightforward. This leaves the viscous dissipation term $C^V_h$ , which accounts for the dominant portion of heat generation within the boundary layer. As indicated in figure 12(c), the heat production associated with mean kinetic energy is reduced in the presence of micro-blowing control. Based on these observations, we conclude that the attenuation of the mean velocity gradient in the near-wall region is the crucial mechanism responsible for the decrease in wall heat transfer and, hence, for the reduction in the Reynolds-analogy factor downstream of the blowing region.