3.1. Drag reduction and power savings

The efficacy of the wall control, expressed by (2.2), is assessed by examining the axial friction coefficients at the inner and outer walls along with the amount of power spent to drive the flow and to enforce the inner wall control.

The axial friction coefficient at the inner and outer walls is defined, respectively, as

(3.1) \begin{align} C_{\kern-1pt f,i}=2 \;\overline {\tau }_{rz,i}=\frac {2}{Re_b}\left . \frac {{\rm d} \overline {u}}{\text{d}r} \right |_{r=r_i}\;\;\;\;\;\mbox{and} \;\;\;\;\;C_{f,o}=-2\;\overline {\tau }_{rz,o}=-\frac {2}{Re_b}\left . \frac {{\rm d} \overline {u}}{\text{d}r} \right |_{r=r_o}, \end{align}

with $\overline {\tau }_{rz,i}$ (respectively, $\overline {\tau }_{rz,o}$ ) the space- and time-averaged axial component of the wall shear stress at the inner (respectively, outer) cylinder. The uncontrolled simulation is characterised by the following axial friction coefficient values:

(3.2) \begin{align} C_{\kern-1pt f,i}^{\textit{unc}}=8.06 \times 10^{-3} \;\;\;\;\;\mbox{and} \;\;\;\;\;C_{f,o}^{\textit{unc}}=8.04\times 10^{-3}, \end{align}

which, because of the narrow-gap geometry ( $\eta = 0.98$ ), nearly coincide. Moreover, present friction coefficients lead to a resistance coefficient value:

(3.3) \begin{align} f=\frac {4}{1+\eta } \big(\eta \; C_{\kern-1pt f,i}^{\textit{unc}}+C_{f,o}^{\textit{unc}} \big) = 3.22\times 10^{-2} ,\end{align}

which agrees perfectly with the correlation reported in Yamada (Reference Yamada1962): $f_B = 0.26 \times Re_b^{0.24}=3.22\times 10^{-2}$ , $f_B$ being the value based on the Blasius boundary layer. It is worth mentioning that, in the same geometry ( $\eta =0.98$ ) and at a comparable Reynolds number ( $ \textit{Re}_b=5765$ ), the constant rotation of the inner cylinder (turbulent spiral Poiseuille flow) can lead to an increase of the resistance coefficient. Equaiton (3.3) follows directly from the definition of the resistance coefficient ((2.4) of Manna et al. (Reference Manna, Vacca and Verzicco2022)) accounting for the reference length and velocity scales.

Tables 2 and 3 report $C_{\kern-1pt f,i}$ and $C_{f,o}$ for all controlled cases. Table 2 indicates that the control only slightly affects the friction coefficient of the outer wall, because of the small $\delta _z$ values. Conversely, table 3 indicates that the wall control reduces the friction coefficient of the inner wall.

Table 2.

Axial friction coefficient at the outer wall $C_{f,o}\times 10^3$ .

Table 2. Long description

A table comparing axial friction coefficients at the outer wall for different controlled cases. The table has 6 rows and 5 columns. The column headers are lambda sub z plus, a plus equals 4, a plus equals 8, a plus equals 12, and a plus equals 16. The row labels are 399, 1203, 1801, 2399, and 3614. The values in the table are as follows: Row 1: 399, 7.89, 7.80, 7.79, 7.76. Row 2: 1203, 7.88, 7.81, 7.81, 7.83. Row 3: 1801, 7.92, 7.84, 7.84, 7.85. Row 4: 2399, 7.94, 7.87, 7.86, 7.87. Row 5: 3614, 7.99, 7.94, 7.91, 7.89.

Table 3.

Axial friction coefficient at the inner wall $C_{\kern-1pt f,i}\times 10^3$ .

Table 3. Long description

The table presents data on axial friction coefficients at the inner wall for various cases and parameters. It has 5 rows and 5 columns. The columns are labeled with lambda subscript z positive, a positive equals 4, a positive equals 8, a positive equals 12, and a positive equals 16. The rows are labeled with values 399, 1203, 1801, 2399, and 3614. Each cell contains a numerical value representing the friction coefficient. Notable trends include varying friction coefficients across different cases and parameters, with specific values provided for each combination.

Amplitude and wavelength of the control have different effects on the resistance. Indeed, keeping $\lambda _z$ constant, $C_{\kern-1pt f,i}$ progressively decreases for increasing $a$ while, for constant $a$ , $C_{\kern-1pt f,i}$ appears to have a minimum varying $\lambda _z$ .

The achieved drag reduction ( $ \textit{DR}$ ) is computed as follows:

(3.4) \begin{align} \textit{DR} = 100 \times \frac {C_{\kern-1pt f,i}^{\textit{unc}}-C_{\kern-1pt f,i}}{ C_{\kern-1pt f,i}^{\textit{unc}}}. \end{align}

Figure 5(a) presents $ \textit{DR}$ as a function of $\lambda _z^+$ , for the considered values of the control amplitude $a^+$ . For the sake of comparison, the plane (Viotti et al. Reference Viotti, Quadrio and Luchini2009) and the spatially controlled wide-gap (Zhao et al. Reference Zhao, Huang and Xu2019) data are also included.

Similarly to the plane channel case, the steady wall control is effective in producing drag reduction for all investigated $\lambda _z^+$ values with comparable magnitudes. Compared with the wide-gap configuration, the narrow-gap geometry exhibits a profitable behaviour in terms of drag reduction in the whole investigated parameter space. Conversely, the wide-gap data of Zhao et al. (Reference Zhao, Huang and Xu2019) show that an increase of resistance occurs for $\lambda ^+ \gtrsim 1000$ ( $\eta = 0.25$ ) and $\lambda ^+ \gtrsim 2500$ ( $\eta = 0.50$ ).

Figure 5(a) indicates that the maximum drag reduction ( $ \textit{DR}_{\textit{max}}$ ) takes place at $\lambda _z^+=1203$ for $a^+\leqslant 12$ and shifts to $\lambda _z^+=1801$ for $a^+=16$ . Furthermore, the $ \textit{DR}_{\textit{max}}$ value increases with $a^+$ , with the highest value approximately equal to 47 % . It is worth observing that the increase in the optimal wavelength $\lambda _z^+$ should be related to the larger amplitude of the control in agreement with the findings of Quadrio & Ricco (Reference Quadrio and Ricco2004).

For a definite assessment of the control as a drag reduction technique, a comparison between the energy needed by the control and the saving produced by the drag reduction is necessary. For statistically steady flows such a comparison can be equivalently carried out in terms of powers.

The power spent to drive the flow through the annulus is given by

(3.5) \begin{align} \mathcal{P}_z\:=\;l_z l_\theta \frac {C_{\kern-1pt f,i}\eta +C_{f,o}}{1+\eta }, \end{align}

with $l_z=l_z^+/Re_\tau ^{\textit{unc}}$ . The power required to enforce the rotation of the inner cylinder, disregarding any other loss associated with the wall movement, is

(3.6) \begin{align} \mathcal{P}_\theta = 2 l_z l_\theta \frac {\eta }{1+\eta } \left [\frac {1}{l_z}\int _0^{l_z} \widehat {\widehat \tau }_{r\theta ,i}(z) w_i (z)\, \text{d}z \right ] \!, \end{align}

where $w_i$ is the azimuthal velocity of the inner cylinder (2.2) and $\widehat {\widehat \tau }_{r\theta ,i}$ represents the average, in time and the $\theta$ direction, of the dimensionless azimuthal component of the shear stress at the inner wall:

(3.7) \begin{align} \widehat {\widehat \tau }_{r\theta ,i}=-\frac {1}{Re_b} \left [\frac {\text{d}\widehat {\widehat w}}{\text{d}r}-\frac {\widehat {\widehat w}}{r}\right ]_{r=r_i} \!. \end{align}

Expressions (3.5) and (3.6) are obtained directly from their dimensional definitions

(3.8) \begin{align} P_z\:=\;2 L_z \; L_\theta \; U_b \frac {R_i \overline {T}_{rz,i} -R_o\overline {T}_{rz,o}}{R_i+R_o}, \\[-28pt] \nonumber \end{align}

(3.9) \begin{align} P_\theta \:=\;2 \; L_\theta \; \frac {R_i}{R_i+R_o} \int _0^{L_z}{\widehat {\widehat {T}}_{r\theta ,i}\left (Z\right ) W_i\left (Z\right ) \;\text{d}Z} \\[0pt] \nonumber \end{align}

which represent the input power necessary to ensure the prescribed flow rate and to keep the inner cylinder rotating, respectively, and follow from the equilibrium conditions. In (3.8) and (3.9) $L_\theta$ is the dimensional annulus length (referred to as the mean radius) in azimuthal direction, $T_{rz,i}$ (respectively, $T_{rz,o}$ ) is the dimensional axial component of the wall shear stress at the inner (respectively, outer) cylinder and $T_{r\theta ,i}$ (respectively, $W_i$ ) is the dimensional azimuthal component of the shear stress (respectively, velocity) at the inner wall. Indeed, expressions (3.5) and (3.6) follow from (3.8) and (3.9) taking into account the length and velocity scales along with (3.2).

Tables 4 and 5 report the powers per unit surface ${\mathcal{P}}_{zS}=\mathcal{P}_z/ (l_z l_\theta )$ and ${\mathcal{P}}_{\theta S}=\mathcal{P}_\theta / (l_z l_\theta )$ . The power saving ( $ \textit{PS}$ ) is therefore given by

Table 4.

Power (per unit surface) spent to drive the flow ${\mathcal{P}}_{zS}\times 10^3$ .

Table 4. Long description

A table with five rows and five columns. The columns are labeled with lambda subscript z positive, a positive equals 4, a positive equals 8, a positive equals 12, and a positive equals 16. The rows are labeled with values 399, 1203, 1801, 2399, and 3614. The table presents numerical values corresponding to each label intersection. Row 1: lambda subscript z positive, 399; a positive equals 4, 7.23; a positive equals 8, 6.65; a positive equals 12, 6.44; a positive equals 16, 6.31. Row 2: lambda subscript z positive, 1203; a positive equals 4, 7.05; a positive equals 8, 6.44; a positive equals 12, 6.20; a positive equals 16, 6.07. Row 3: lambda subscript z positive, 1801; a positive equals 4, 7.20; a positive equals 8, 6.53; a positive equals 12, 6.23; a positive equals 16, 6.05. Row 4: lambda subscript z positive, 2399; a positive equals 4, 7.36; a positive equals 8, 6.71; a positive equals 12, 6.28; a positive equals 16, 6.13. Row 5: lambda subscript z positive, 3614; a positive equals 4, 7.65; a positive equals 8, 7.21; a positive equals 12, 6.95; a positive equals 16, 6.92.

Table 5.

Power (per unit surface) required to enforce the rotation of the inner cylinder ${\mathcal{P}}_{\theta S}\times 10^4$ .

Table 5. Long description

The table presents power values for different parameters. It has five columns and five rows. The first column is labeled with lambda_z values: 399, 1203, 1801, 2399, and 3614. The subsequent columns are labeled with a+ values: 4, 8, 12, and 16. The table lists the corresponding power values for each combination of lambda_z and a+. Row 1: lambda_z 399, a+ 4 2.87, a+ 8 10.9, a+ 12 23.9, a+ 16 41.9. Row 2: lambda_z 1203, a+ 4 1.89, a+ 8 7.20, a+ 12 15.8, a+ 16 27.6. Row 3: lambda_z 1801, a+ 4 1.65, a+ 8 6.30, a+ 12 13.7, a+ 16 23.9. Row 4: lambda_z 2399, a+ 4 1.53, a+ 8 5.83, a+ 12 12.5, a+ 16 21.8. Row 5: lambda_z 3614, a+ 4 1.43, a+ 8 5.51, a+ 12 12.2, a+ 16 21.8.

(3.10) \begin{align} \textit{PS} = 100 \times \frac {{\mathcal{P}}_{zS}^{\textit{unc}}-\left ({\mathcal{P}}_{zS} +{\mathcal{P}}_{\theta S}\right )}{ {\mathcal{P}}_{zS}^{\textit{unc}}}, \end{align}

where ${\mathcal{P}}_{zS}^{\textit{unc}}$ is the axial power of the uncontrolled flow ( ${\mathcal{P}}_{zS}^{\textit{unc}}= 8.05 \times 10^{-3}$ ).

A positive value of $ \textit{PS}$ indicates that the active flow control procedure is advantageous even when taking into account the power needed by the control itself.

Figure 5(b) shows $ \textit{PS}$ versus $\lambda ^+$ , changing $a^+$ , along with the data of Viotti et al. (Reference Viotti, Quadrio and Luchini2009). The latter have been rescaled to allow a proper comparison between a channel with two controlled walls and an annulus with only the inner cylinder actuated. On account of the large $\lambda ^+$ values, the upper and lower spatial Stokes layers do not interfere and thus the annulus $ \textit{PS}$ is half of the channel value. Figure 5(b) shows that the inner wall control in the annular geometry allows a maximum $ \textit{PS}$ value of about 11 % ( $a^+=8$ , cases 2 and 3), which agrees well with the planar geometry value of Viotti et al. (Reference Viotti, Quadrio and Luchini2009). While at the highest $a^+$ value the wall control is always unprofitable, a small gain is obtained for $a^+=12$ at intermediate $\lambda ^+$ values. For $a^+=4$ and $a^+=8$ , $ \textit{PS}$ is always positive in the investigated wavelengths range. For the sake of conciseness, in what follows only case 2 ( $\lambda ^+=1203$ , $4\leqslant a^+ \leqslant 16$ ), for which the maximum power saving has been found, will be discussed.

Before concluding this section, it is worth pointing out that in order to assess the performance improvements of the whole annular flow system, as appropriate in engineering terms, the resistance reduction and the power saving should be compared in terms of overall data. This entails comparing the single wall controlled annulus against the channel with both walls controlled in terms of $f$ and ${\mathcal{P}}_{zS} +{\mathcal{P}}_{\theta S}$ . The end result could be roughly summarised in a drop of a factor two in the percentage reductions (results not shown herein).

Figure 5.

Percentage of drag reduction $ \textit{DR}$ (a) and power saving $ \textit{PS}$ (b) as a function of the forcing wavelength. Square, present calculations (black, $a^+ =4$ ; red, $a^+ =8$ ; green, $a^+ =12$ ; blue, $a^+ =16$ ). Circle, Viotti et al. (Reference Viotti, Quadrio and Luchini2009) (magenta, $a^+ =6$ ; green, $a^+ =12$ ). Blue triangle, Zhao et al. (Reference Zhao, Huang and Xu2019), $a^+ =16$ (left, $\eta =0.25$ ; right, $\eta =0.50$ ). The $ \textit{PS}$ channel data are rescaled.

3.2. Near-wall structures

Before analysing the turbulence statistics, it is useful to observe the modifications of near-wall structures caused by the wall control. Figure 6 shows contours of axial velocity fluctuations in the $z{-}r\theta$ plane, at $y^*=5$ . In every panel, also contours on two $y{-}r\theta$ planes, at $z^*=500$ and $z^*=2000$ , are included. A similar figure, for the reference uncontrolled case, is given in figure 7; here the presence of streaks is evident with a spacing of roughly $100$ wall-units.

Figure 6 evidences the progressive azimuthal tilting of the streaks as the amplitude of the control $a^+$ is increased. At the same time the length and width of the streaks reduce. The wall control also induces an enhanced lift-up of the streaks from the wall (see the $y{-}r\theta$ planes of figure 6).

Figure 6.

Instantaneous contour plot in actual inner coordinates of $u^\prime$ in $z-\theta$ (at $y^* = 5$ ) and $\theta -y$ (at $z^* = 500$ and $z^* = 2000$ ): (a) $a^+=4$ ; (b) $a^+=8$ ; (c) $a^+=12$ ; (d) $a^+=16$ .

Figure 6. Long description

Panel A: A 3D heat map with axes labeled rθ* (0 to 3000), y* (0 to 400), and z* (0 to 1000). The color scale ranges from blue (-3) to red (3), indicating the value of u′*. The map shows varying intensities of color, with higher values concentrated in certain regions. Panel B: A 3D heat map with the same axes and color scale as Panel A. The distribution of values is similar but with slight variations in the intensity and location of the highest and lowest values. Panel C: A 3D heat map with the same axes and color scale as Panel A. The pattern of data distribution is consistent with Panels A and B, with some differences in the specific regions of high and low values. Panel D: A 3D heat map with the same axes and color scale as Panel A. The data distribution shows a similar pattern to the previous panels, with notable regions of high and low values.

Figure 7.

Instantaneous contour plot in inner coordinates of $u^\prime$ in $z-\theta$ (at $y^+ = 5$ ) and $\theta -y$ (at $z^+ = 500$ and $z^+ = 2000$ ) planes. Uncontrolled case.

3.3. Mean velocity and Reynolds stresses profiles

Figure 8 shows the profiles of the mean axial velocity component, averaged over space ( $\theta$ and $z$ directions) and time, versus $y$ . The inner scaling is performed using the actual (figure 8 a) and nominal (figure 8 b) friction velocity. Consistent with the previous drag reduction studies in plane channel flows, e.g. Viotti et al. (Reference Viotti, Quadrio and Luchini2009) and Touber & Leschziner (Reference Touber and Leschziner2012), figure 8(a) shows that the wall control causes an extension of the viscous sublayer with a shortening and upward shift of the log-law portion. Moreover, in agreement with the $ \textit{DR}$ results reported in figure 5(a), the $a^+$ increase enhances the upward shift, in the actual inner representation. Figure 8(b) evidences that the cylinder control significantly reduces the mean axial velocity gradient in the viscous sublayer, complying with the skin friction coefficient reduction.

Figure 8.

Mean profiles of axial velocity in actual (a) and nominal (b) inner coordinates. Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line. Dashed black line, $\overline {u}=2.5 \ln(y)+5.5$ ; dot-dashed black line, $\overline {u}=y$ .

Figure 8. Long description

Panel A: A line graph shows mean profiles of axial velocity in actual inner coordinates. The x-axis is labeled y* and ranges from 10^-1 to 10^2. The y-axis is labeled u* and ranges from 0 to 30. The graph includes multiple lines: a solid black line for uncontrolled flow, a dashed red line, a dot-dashed green line, a dotted blue line, a long-dashed violet line, a dashed black line, and a dot-dashed black line. Panel B: A line graph shows mean profiles of axial velocity in nominal inner coordinates. The x-axis is labeled y+ and ranges from 10^-1 to 10^2. The y-axis is labeled u+ and ranges from 0 to 20. The graph includes the same set of lines as Panel A.

To investigate the origin of the mean velocity profile alteration, we introduce the decomposition (2.3) into the axial component of the momentum equation averaged in time and both in $\theta$ and $z$ directions:

(3.11) \begin{align} 0\;=\; - \frac {1}{r} \frac {d \left (\;\left \langle \tilde {u} \tilde {v}\right \rangle _z\; r\right ) }{\text{d}r} - \frac {1}{r} \frac {d \left (\;\overline {u^\prime v^\prime }\; r\right ) }{\text{d}r} +\frac {1}{Re} \frac {1}{r} \frac {d }{\text{d}r}\left (\frac {{\rm d} \overline {u}}{\text{d}r} \; r\right )\;=\; -2 \frac {\overline {\tau }_{rz,o} \; r_o -\overline {\tau }_{rz,i} \; r_i }{r_o^2 -r_i^2}. \end{align}

The effect of the axial variability of the inner cylinder rotation (2.2) results in the generation of an additional shear stress $-\left \langle \tilde {u} \tilde {v}\right \rangle _z$ which adds to the turbulent one $-\overline {u^\prime v^\prime }$ . However, the $-\left \langle \tilde {u} \tilde {v}\right \rangle _z$ stress has been found to be negligible compared with $-\overline {u^\prime v^\prime }$ (results not shown). Indeed, the modifications of $\overline {u}$ , and therefore of $\overline {\tau }_{w,i}$ , are only indirectly caused by the inner cylinder rotation and only through the decline of the turbulent shear stress.

Figure 9(a–c) report the radial distributions of the normal stresses $\overline {u^{\prime }_i u^{\prime }_i}$ , $i=1,2,3$ , with $(u_1,u_2,u_3)=(u,v,w)$ , respectively, and the turbulent shear stress $-\overline {u^{\prime }_1 u^{\prime }_2}$ (in figure 9 d). The general response to the wall control is a progressive reduction, especially close to the wall, of all Reynolds stresses as $a^+$ increases.

Figure 9.

Radial distribution in nominal inner coordinates of (a) $R_{uu}^+=\overline {u^\prime u^\prime }^+$ , (b) $R_{vv}^+=\overline {v^\prime v^\prime }^+$ , (c) $R_{ww}^+=\overline {w^\prime w^\prime }^+$ and (d) $R_{uv}^+=\overline {u^\prime v^\prime }^+$ . Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line.

3.4. Reynolds stress budgets

In this section the Reynolds stress budgets are presented following the methodology proposed by Moin et al. (Reference Moin, Shih, Driver and Mansour1990), Xu & Huang (Reference Xu and Huang2005) and Touber & Leschziner (Reference Touber and Leschziner2012) as applied to plane channel flows. Using tensor notation, the $\overline {u^{\prime }_i u^{\prime }_j}$ Reynolds stresses are governed by the following equations:

(3.12) \begin{align} P_{\textit{ij}}+C_{\textit{ij}}+T_{\textit{ij}}+\varPi _{\textit{ij}}+D_{\textit{ij}}-\epsilon _{\textit{ij}}=0, \end{align}

which are production, convection, turbulent transport, velocity pressure gradient, viscous diffusion and dissipation terms, respectively. The production terms are

(3.13) \begin{align} P_{zz}\;=\; \underbrace {-2 \overline { u^\prime v^\prime } \;\frac {\text{d} \overline {u}}{\text{d}r}}_{P_{zz,1}}\underbrace {-2 \left \langle \widetilde {u^\prime v^\prime } \;\frac {\partial \tilde {u}}{\partial r}\right \rangle _z}_{P_{zz,2}}\underbrace {-2 \left \langle \widetilde {u^\prime u^\prime } \;\frac {\partial \tilde {u}}{\partial z}\right \rangle _z}_{P_{zz,3}}, \\[-28pt] \nonumber \end{align}

(3.14) \begin{align} P_{\textit{rr}}\;=\; \underbrace {-2 \left \langle \widetilde {u^\prime v^\prime }\; \frac {\partial \tilde {v}}{\partial z}\right \rangle _z}_{P_{rr,1}}\underbrace {-2 \left \langle \widetilde {v^\prime v^\prime }\; \frac {\partial \tilde {v}}{\partial r}\right \rangle _z}_{P_{rr,2}}+\underbrace {4 \left \langle \widetilde {v^\prime w^\prime }\frac { \tilde {w}}{r} \right \rangle _z}_{P_{rr,3}}, \\[-28pt] \nonumber \end{align}

(3.15) \begin{align} P_{\theta \theta }\;=\; \underbrace {-2 \left \langle \widetilde {v^\prime w^\prime }\; \frac {\partial \tilde {w}}{\partial r}\right \rangle _z}_{P_{\theta \theta , 1}} \underbrace {-2 \left \langle \widetilde {u^\prime w^\prime }\; \frac {\partial \tilde {w}}{\partial z}\right \rangle _z}_{P_{\theta \theta , 2}} \underbrace {-2 \left \langle \widetilde {w^\prime w^\prime }\; \frac { \tilde {v}}{ r}\right \rangle _z}_{P_{\theta \theta , 3}} \underbrace {-2 \left \langle \widetilde {v^\prime w^\prime }\; \frac { \tilde {w}}{ r}\right \rangle _z}_{P_{\theta \theta , 4}}, \\[-28pt] \nonumber \end{align}

(3.16) \begin{align} P_{zr}\;=\; \underbrace {- \overline { v^\prime v^\prime } \frac {d \overline {u}}{\text{d}r}}_{P_{zr,1}}\; +\underbrace {2 \left \langle \widetilde {u^\prime w^\prime } \frac {\tilde {w}}{r} \right \rangle _z }_{P_{zr,2}}\underbrace {- \left \langle \widetilde {u^\prime v^\prime } \frac {\partial \tilde {u}}{\partial z}\right \rangle _z}_{P_{zr,3}}\underbrace {-\left \langle \widetilde {v^\prime v^\prime } \frac {\partial \tilde {u}}{\partial r}\right \rangle _z}_{P_{zr,4}}\underbrace {- \left \langle \widetilde {u^\prime u^\prime } \frac {\partial \tilde {v}}{\partial z}\right \rangle _z}_{P_{zr,5}}\underbrace {-\left \langle \widetilde {u^\prime v^\prime } \frac {\partial \tilde {v}}{\partial r}\right \rangle _z}_{P_{zr,6}}. \\[0pt] \nonumber \end{align}

The expressions of the remaining terms involved in (3.12) are given in the Appendix A.1.

In the foregoing we try to relate the Reynolds stress tensor $R_{\textit{ij}}$ depression to the production $P_{\textit{ij}}$ tensor. To this aim we closely follow the analysis of Touber & Leschziner (Reference Touber and Leschziner2012) in the plane channel case with a temporally controlled wall.

Figure 10.

Radial distribution in nominal inner coordinates of (a) $P_{zz}^+$ , (b) $-P_{zr}^+$ and (c) $P_{\theta \theta }^+$ . Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line.

Figure 10. Long description

Three line graphs depict the radial distribution in nominal inner coordinates. Panel A: The line graph shows the distribution of P zz+ against y+. The solid black line represents the uncontrolled data, while the dashed red, dot-dashed green, dotted blue, and long-dashed violet lines represent different controlled conditions. Panel B: The line graph shows the distribution of negative I P zr+ against y+. The solid black line represents the uncontrolled data, while the dashed red, dot-dashed green, dotted blue, and long-dashed violet lines represent different controlled conditions. Panel C: The line graph shows the distribution of P θθ+ against y+. The solid black line represents the uncontrolled data, while the dashed red, dot-dashed green, dotted blue, and long-dashed violet lines represent different controlled conditions.

The root cause of the reduction of the $\overline {u^{\prime } u^{\prime }}$ and $-\overline {u^{\prime } v^{\prime }}$ Reynolds stresses can be related to the drop of the $P_{zz}$ and $-P_{zr}$ production terms (see figures 10 a and 10 b). In more detail, the production $P_{zz}$ is the sum of three contributions (see (3.13)) of which only the last two terms ( $P_{zz,2}$ and $P_{zz,3}$ ) are caused by the axial variability of the inner cylinder rotation. A similar conclusion applies to $P_{zr}$ (see (3.16)) consisting of six contributing terms of which the last five are directly related to the wall control. Figure 11 shows the marginal direct contribution of the wall control on the decline of $P_{zz}$ and $-P_{zr}$ .

Figure 11.

Individual contributions to the production terms ( $a^+=12)$ : (a) $P_{zz}^+$ ; (b) $-P_{zr}^+$ . Terms not reported in(a) (respectively, b) are indistinguishable from $P^+_{zz,2}$ (respectively, $-P^+_{zr,2}$ ) within plot accuracy.

The production term of $\overline {v^{\prime } v^{\prime }}$ budget (3.14) is negligible in all cases (results not shown). This is because $\tilde {v}$ is small ( $P_{rr,1}$ and $P_{rr,2}$ ) and because of the narrow-gap geometry ( $P_{rr,3}$ ). Finally, $P_{\theta \theta }$ (3.15) is not zero due to wall control (2.2) and increases with $a^+$ (see figure 10 c). Comparing figures 10(a) and 10(c), it follows that $P_{zz}\gt \gt P_{\theta \theta }$ . These results are similar to the plane channel data of Touber & Leschziner (Reference Touber and Leschziner2012).

The leading term in the production of the turbulent shear stress $\overline {u^{\prime } v^{\prime }}$ ( $P_{zr,1}$ , see (3.16)) scales with $\overline {v^{\prime } v^{\prime }}$ . This normal stress is fed by the energy coming from the other two components through the pressure strain term (Tennekes & Lumley Reference Tennekes and Lumley1972).

In order to analyse the intercomponent energy transfer, we split the velocity pressure gradient into the pressure strain ( $\varPhi _{ii}$ ) and pressure diffusion ( $\varPsi _{ii}$ ) contributions

(3.17) \begin{align} \varPi _{zz}=\varPhi _{zz}+\varPsi _{zz}=2\; \overline {p^\prime s^\prime _{zz}}+0, \\[-28pt] \nonumber \end{align}

(3.18) \begin{align} \varPi _{\textit{rr}}=\varPhi _{\textit{rr}}+\varPsi _{\textit{rr}}=2\;\overline {p^\prime s^\prime _{\textit{rr}}} -2 \frac {\partial \overline {\left (p^\prime v^\prime \right )}}{\partial r}, \\[-28pt] \nonumber \end{align}

(3.19) \begin{align} \varPi _{\theta \theta }=\varPhi _{\theta \theta }+\varPsi _{\theta \theta }=\frac {2}{r} \;\overline {p^\prime s^\prime _{\theta \theta }}-2\; \frac {\overline {p^\prime v^\prime }}{r}, \\[0pt] \nonumber \end{align}

with

(3.20) \begin{align} s^\prime _{zz}= \frac {\partial u^\prime } {\partial z}, \;\;\; \;\;\; \;\;\; s^\prime _{\textit{rr}}=\frac {\partial v^\prime } {\partial r}, \;\;\; \;\;\; \;\;\; s^\prime _{\theta \theta }=\frac {1}{r} \left ( \frac {\partial w^\prime }{\partial \theta } +v^\prime \right ), \end{align}

the fluctuating strain rates. The following analysis shares some similarities with that of Touber & Leschziner (Reference Touber and Leschziner2012) which led the authors to relate the reduction of $\overline { v^\prime v^\prime }$ to the pressure gradient component in the normal to the wall direction. However, differently from Touber & Leschziner (Reference Touber and Leschziner2012), we concentrate on the pressure strains $\varPhi _{ii}$ which represent the true energy redistributive terms (Xu & Huang Reference Xu and Huang2005).

Figure 12 shows, in nominal inner coordinates, the radial distributions of $\varPhi _{zz}^+$ , $\varPhi _{\textit{rr}}^+$ and $\varPhi _{\theta \theta }^+$ . A positive (negative) value of $\varPhi _{ii}$ denotes a gain (loss) of energy from the ith component towards the other two (Hinze Reference Hinze1975). The general effect of the inner cylinder control is an attenuation of all $\varPhi _{ii}$ preserving the shape of the radial profiles of the uncontrolled case. The energy transfer essentially takes place from the axial component towards the azimuthal and radial ones. The radial component of the pressure strain $\varPhi _{\textit{rr}}$ is negative close to inner cylinder wall (figure 12 b). This result is usually attributed to the splatting phenomenon, i.e. to the occurrence of the sweep events (Kim et al. Reference Kim, Moin and Moser1987).

Figure 12.

Radial distribution in nominal inner coordinates of (a) $\varPhi ^+_{zz}$ , (b) $\varPhi ^+_{\textit{rr}}$ and (c) $\varPhi ^+_{\theta \theta }$ . Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line.

3.5. Pressure and strain rates fluctuations

The mechanism determining the reduction of $\varPhi _{zz}$ and $\varPhi _{\textit{rr}}$ is analysed next through the radial distribution of the root-mean-square (r.m.s.) of the fluctuating pressure $p^\prime$ and strain rates $s^\prime _{zz}$ and $s^\prime _{\textit{rr}}$ . Indeed, the behaviour of the correlation $\overline {p^\prime s^\prime _{ii}}$ , can be explained in terms of the single r.m.s. values, on account of the Cauchy–Schwarz inequality (Pope Reference Pope2000). Figure 13, showing the radial distribution of the r.m.s. of $s^\prime _{zz}$ , $s^\prime _{\textit{rr}}$ and $p^\prime$ , indicates a monotonic reduction of the single r.m.s. values when $a^+$ is increased.

Figure 13.

Radial distribution in nominal inner coordinates of (a) $rms(s^{\prime}_{zz})^+$ , (b) $rms(s^{\prime}_{\textit{rr}})^+$ and (c) $rms(p^\prime )^+$ . Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line.

The direct role played by the wall control on the pressure fluctuations can be evidenced considering the pressure Poisson equation obtained by taking the divergence of the momentum equation:

(3.21) \begin{align} \Delta p=-\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\boldsymbol{u} \;\boldsymbol{\nabla }\boldsymbol{u} \right ). \end{align}

Substituting in (3.21) the velocity decomposition (2.3), the pressure Poisson equation can be decomposed as follows:

(3.22) \begin{align} \Delta \overline {p} = \mathcal{S}^{\overline {p}}, \\[-28pt] \nonumber \end{align}

(3.23) \begin{align} \Delta p^\prime = \mathcal{S}^{p^\prime }. \\[0pt] \nonumber \end{align}

The source term of (3.23) has been split in two parts, one involving the time-averaged velocity gradient ( $\mathcal{S}^{p^\prime }_{r}$ ) and the other without it ( $\mathcal{S}^{p^\prime }_{s}$ ). Solving (3.23) with $\mathcal{S}^{p^\prime }_{r}$ (respectively $\mathcal{S}^{p^\prime }_{s}$ ) as source term, yields the fluctuating rapid (respectively, slow) pressure field (Kim Reference Kim1989):

(3.24) \begin{align} \Delta p_r^\prime = \mathcal{S}^{p^\prime }_r, \\[-28pt] \nonumber \end{align}

(3.25) \begin{align} \Delta p_s^\prime = \mathcal{S}^{p^\prime }_s. \\[0pt] \nonumber \end{align}

The source terms $\mathcal{S}^{p^\prime }_{r}$ and $\mathcal{S}^{p^\prime }_{s}$ are detailed in Appendix A.2.

Figure 14.

Radial distributions of the mean-square of $\mathcal{S}^{p^\prime }_{r}$ (red dashed line) and $\mathcal{S}^{p^\prime }_{s}$ (black line): (a) uncontrolled case; (b) $a^+=4$ ; (c) $a^+=8$ ; (d) $a^+=12$ ; (e) $a^+=16$ .

Figure 14. Long description

Five line graphs depict the radial distributions of mean-square values for different cases. Panel A: The line graph shows the mean-square of S_p’_r and S_p’_s for the uncontrolled case. The x-axis is labeled y+ and ranges from 0 to 50. The y-axis is labeled Mean-square of S_p’_r and S_p’_s and ranges from 0 to 20. The red dashed line represents S_p’_r, and the black line represents S_p’_s. Panel B: The line graph shows the mean-square of S_p’_r and S_p’_s for a specific case. The x-axis is labeled y+ and ranges from 0 to 50. The y-axis is labeled Mean-square of S_p’_r and S_p’_s and ranges from 0 to 15. The red dashed line represents S_p’_r, and the black line represents S_p’_s. Panel C: The line graph shows the mean-square of S_p’_r and S_p’_s for another specific case. The x-axis is labeled y+ and ranges from 0 to 50. The y-axis is labeled Mean-square of S_p’_r and S_p’_s and ranges from 0 to 10. The red dashed line represents S_p’_r, and the black line represents S_p’_s. Panel D: The line graph shows the mean-square of S_p’_r and S_p’_s for yet another specific case. The x-axis is labeled y+ and ranges from 0 to 50. The y-axis is labeled Mean-square of S_p’_r and S_p’_s and ranges from 0 to 4. The red dashed line represents S_p’_r, and the black line represents S_p’_s. Panel E: The line graph shows the mean-square of S_p’_r and S_p’_s for a final specific case. The x-axis is labeled y+ and ranges from 0 to 50. The y-axis is labeled Mean-square of S_p’_r and S_p’_s and ranges from 0 to 2. The red dashed line represents S_p’_r, and the black line represents S_p’_s.

The source term of the rapid pressure fluctuations can be further decomposed into four contributions $\mathcal{S}^{p^\prime }_{r,12}$ , $\mathcal{S}^{p^\prime }_{r,31}$ , $\mathcal{S}^{p^\prime }_{r,32}$ and $\mathcal{S}^{p^\prime }_{r,23}$ , see (A21). In the uncontrolled case all but the first term $\mathcal{S}^{p^\prime }_{r,12}$ vanish. While $\mathcal{S}^{p^\prime }_{r,31}$ and $\mathcal{S}^{p^\prime }_{r,32}$ have a planar counterpart, the $\mathcal{S}^{p^\prime }_{r,23}$ term is peculiar of cylindrical geometries. The contribution of each term appearing in the $\mathcal{S}^{p^\prime }_{r}$ and $\mathcal{S}^{p^\prime }_{s}$ definition is addressed in Appendix A.2 (figure 20), with reference to the $a^+=12$ case.

Figure 14 reports the profiles of the mean-square of $\mathcal{S}^{p^\prime }_{r}$ and $\mathcal{S}^{p^\prime }_{s}$ versus $y^+$ , which are made dimensionless using $U_b$ and $D/2$ as velocity and length scales, respectively (Kim Reference Kim1989); we wish to stress that the use of $D/2$ , rather than $D$ , as length scale allows the direct comparison of the uncontrolled case with the turbulent channel flow. Data show that the wall control causes a reduction of both terms. Unlike the uncontrolled case, the rapid term exceeds the slow one in the first 10 wall units, particularly for $a^+=12$ and $a^+=16$ . Therefore, close to the wall the pressure fluctuation is essentially formed with the rapid term $p^\prime _r$ . Because of the wall control, the leading term of the right-hand side of (3.24) is $\mathcal{S}^{p^\prime }_{r,32}$ (see figure 20 a) particularly in the first 10 wall units. Consequently, the amplitude and wavelength of the controlled wall directly affect $p^\prime _r$ and thus $p^\prime$ within its penetration length. This conclusion agrees with the conjectures of Hack & Zaki (Reference Hack and Zaki2015) and Lozano-Duran et al. (Reference Lozano-Duran, Giometto, Park and Moin2020) in the context of transitional boundary layer flows subjected to spanwise wall oscillations and channel flows subjected to sudden transverse pressure gradient, respectively. Likewise, the damping of the pressure fluctuations due to the effect of the rapid contribution has been shown to play a key role in causing flow relaminarisation in a turbulent boundary layer (Piomelli & Yuan Reference Piomelli and Yuan2013).

The direct effect of the wall control on $ s^\prime _{zz}$ and $s^\prime _{\textit{rr}}$ is investigated analysing their balance equations and, in particular, the production terms $\mathcal{P}_{s^\prime _{zz}}$ and $\mathcal{P}_{s^\prime _{\textit{rr}}}$ (Huang et al. Reference Huang, Wang and Fu2021). In cylindrical geometries, it is easy to verify that these production terms read

(3.26) \begin{align} \mathcal{P}_{s^\prime _{zz}}\;=\; \underbrace {-\frac {\text{d} \overline {u}}{\text{d} r} \frac {\partial v^\prime }{\partial z}}_{\mathcal{P}_{s^\prime _{zz,1}}} \underbrace {-\frac {1}{r}\frac {\partial \tilde {w}}{\partial z}\frac {\partial u^\prime }{\partial \theta }}_{\mathcal{P}_{s^\prime _{zz,2}}}, \\[-28pt] \nonumber \end{align}

(3.27) \begin{align} \mathcal{P}_{s^\prime _{\textit{rr}}}\;=\; \underbrace {-\frac {\text{d} \overline {u}}{\text{d} r} \frac {\partial v^\prime }{\partial z}}_{\mathcal{P}_{s^\prime _{rr,1}}} \underbrace {-\frac {1}{r}\frac {\partial \tilde {w}}{\partial r}\frac {\partial v^\prime }{\partial \theta }}_{\mathcal{P}_{s^\prime _{rr,2}}} +\underbrace { \frac {\tilde {w}}{r^2}\frac {\partial v^\prime }{\partial \theta }}_{\mathcal{P}_{s^\prime _{rr,3}}}. \\[0pt] \nonumber \end{align}

Figure 15.

Radial distributions of the mean-square of (a) $\mathcal{P}_{ s^\prime _{zz}}$ and (b) $\mathcal{P}_{ s^\prime _{\textit{rr}}}$ . Uncontrolled, solid black line; $a^+=4$ , dashed red line; $a^+=8$ , dot-dashed green line; $a^+=12$ , dotted blue line; $a^+=16$ , long-dashed violet line.

In the uncontrolled condition only the first term on the right-hand side of both (3.26) and (3.27) is different from zero, implying that $\mathcal{P}_{s^\prime _{zz}}=\mathcal{P}_{s^\prime _{\textit{rr}}}$ . The presence of the control modifies both production terms, generating $\mathcal{P}_{s^\prime _{zz,2}}$ in the $s^\prime _{zz}$ governing equation and the two additional contributions $\mathcal{P}_{s^\prime _{rr,2}}$ and $\mathcal{P}_{s^\prime _{rr,3}}$ , in the $s^\prime _{\textit{rr}}$ equation. The production term $\mathcal{P}_{s^\prime _{rr,3}}$ is peculiar of the cylindrical geometry.

Figure 15 reports the profile of the mean-square of $\mathcal{P}_{ s^\prime _{zz}}$ and $\mathcal{P}_{ s^\prime _{\textit{rr}}}$ versus $y^+$ , again made dimensionless using $D/2$ and $U_b$ as reference length and velocity scales, respectively. Data show that the control induces a considerable reduction of both production terms. With reference to the $a^+=12$ case, figure 16 reports the mean-square of each individual contribution of the right-hand side of (3.26) and (3.27) along with the total production terms. The comparison between figures 16(a) and 16(b) indicates that while the production term $\mathcal{P}_{s^\prime _{zz,2}}$ contributes only marginally to the total value (see figure 16 a), $\mathcal{P}_{s^\prime _{rr,2}}$ is comparable to $\mathcal{P}_{ s^\prime _{rr,1}}$ . Therefore, the wall cylinder rotation directly causes a substantial modification of the $s^\prime _{\textit{rr}}$ radial distribution.

With the aim of providing a physical interpretation of the effects of the control onto the rapid pressure source $\mathcal{S}^{p^\prime }_{r}$ and strain production $\mathcal{P}_{s^\prime _{\textit{rr}}}$ terms, it is helpful to recognise that $\mathcal{P}_{s^\prime _{rr,1}}=\mathcal{S}^{p^\prime }_{r,12}/2$ and $\mathcal{P}_{s^\prime _{rr,2}}=\mathcal{S}^{p^\prime }_{r,32}/2$ may be rewritten as

(3.28) \begin{align} \mathcal{P}_{s^\prime _{rr,1}}=-\frac {\partial }{\partial z} \left (v^\prime \frac {\text{d}\overline {u}}{\text{d}r}\right ),\;\;\; \;\;\; \mathcal{P}_{s^\prime _{rr,2}}=-\frac {1}{r}\frac {\partial }{\partial \theta } \left (v^\prime \frac {\text{d}\tilde {w}}{\text{d}r}\right )\!, \end{align}

respectively. Assuming that the terms in parentheses in (3.28) are connected with the generation cycle of near-wall streaks (Chernyshenko & Baig Reference Chernyshenko and Baig2005) and to their azimuthal tilting (Touber & Leschziner Reference Touber and Leschziner2012), it may be inferred that the modulation of the above quantities in the homogeneous directions determined by the control causes a general depression of the near-wall turbulence.

Present results suggest that the spatially varying inner cylinder rotation directly causes an alteration of both $p^\prime$ and ${s^\prime }_{\textit{rr}}$ close to the wall. At the same time, figure 13 demonstrates that the r.m.s. values of the above mentioned quantities drop when the control amplitude is increased. Interestingly, the correlation between $p^\prime$ and ${s^\prime }_{\textit{rr}}$ , namely $\varPhi _{\textit{rr}}=\overline {p^\prime {s^\prime }_{\textit{rr}}}$ , also drops with $a^+$ (see figure 12). Consequently, it must be inferred a fortiori that the two results are strongly interconnected, that is, the net effect of the flow control onto the pressure strain term is a direct consequence of the instantaneous $p^\prime$ and ${s^\prime }_{\textit{rr}}$ alteration ultimately leading to an attenuation of their intensities. The reduction of the pressure strain term causes a drop of the energy transferred towards $\overline {v^\prime v^\prime }$ . As a consequence of the $\overline {v^\prime v^\prime }$ decay, the production term $-P_{zr}$ declines and so does the $-\overline {u^\prime v^\prime }$ Reynolds stress. Therefore, the effectiveness of the wall control is connected to its ability to depress the radial distribution of the radial velocity component fluctuations close to the wall.

The mechanism proposed above directly links the wall motion to the reduction of energy transfer towards the wall-normal velocity component, which is widely recognised to play a key role in drag reducing flows. Present results provide additional support to the conclusions of Xu & Huang (Reference Xu and Huang2005) and Touber & Leschziner (Reference Touber and Leschziner2012) for turbulent channel flows.

Figure 16.

Radial distributions of the mean-square of the single contributions and the total value of (a) $\mathcal{P}_{ s^\prime _{zz}}$ and (b) $\mathcal{P}_{ s^\prime _{\textit{rr}}}$ .

3.6. Artificial damping of the radial velocity fluctuations

To assess this conclusion, we carried out an additional numerical experiment where the radial velocity fluctuations $v^\prime$ have been artificially damped close to the inner cylinder. This goal has been achieved by including in the Navier–Stokes equations a body force (Jiménez & Pinelli Reference Jiménez and Pinelli1999). Such a kind of manipulation has been widely used to investigate the mechanisms of drag reduction in turbulent channel flows, see for instance Lee & Kim (Reference Lee and Kim2002) and Umair & Tardu (Reference Umair and Tardu2023). The algorithm used for artificially damping the radial component is similar to that employed by Umair & Tardu (Reference Umair and Tardu2023) and includes in the radial component of the momentum equation the following source term:

(3.29) \begin{align} \mathcal{S}_{\text{d}}\left (r,z,\theta ,t\right ) =-\frac {1-F( y^+)}{\Delta t} v^\prime (r,z,\theta ,t), \end{align}

in which $\Delta t$ is the computational time step and $F ( y^+ )$ is the filter function given by

(3.30) \begin{align} F\left ( y^+ \right ) =\frac {1}{2} \left \{ 1 + \tanh \left [\alpha \left (y^+-\delta _d^+ \right ) \right ] \right \}, \end{align}

with $\alpha$ being the filter function steepness.

Computations have been carried out with $\alpha =0.3$ and $\delta _d^+=5$ . Such a choice allows us to effectively suppress $v^\prime$ in the viscous sublayer region up to $y^+ \simeq 10$ .

The box sizes and the spatial discretisation parameters, i.e. the number of the Fourier and Chebyshev modes along with the subdomains width and number, are the same as in the uncontrolled case. Data were collected by discarding the initial transient and only once the time- and space-averaged wall shear stress and total kinetic energy reached statistically constant values.

The following values of the skin-friction coefficient at the inner and outer cylinder

(3.31) \begin{align} C_{f_i}=2.86 \times 10^{-3}\;\;\;\mbox{and}\;\;\;\;C_{f_o}=3.90 \times 10^{-3}. \end{align}

have been found, thus entailing a drag reduction at the inner cylinder of $ \textit{DR}=28.8\,\%$ .

The $v^\prime$ -damped mean velocity profiles in actual (figure 17 a) and in nominal (figure 17 b) inner coordinates are similar to those with a controlled wall (figure 8). Figures 18(a) and 18(b) indicate that the artificial damping of the radial fluctuations induces a decline of all turbulent stresses close to the wall. The magnitude of the reduction and the associated shape are very much similar to those occurring in the controlled case (see figure 9).

Figure 17.

Mean profiles of axial velocity in actual (a) and nominal (b) inner coordinates: dashed black line, $\overline {u}=2.5 \ln(y)+5.5$ ; dot-dashed black line, $\overline {u}=y$ . Uncontrolled case, black solid lines; $v^\prime$ -damped case, red dashed lines.

Figure 18.

Radial distribution in nominal inner coordinates of (a) normal Reynolds stresses $R_{uu}^+=\overline {u^\prime u^\prime }^+$ (solid line), $R_{vv}^+=\overline {v^\prime v^\prime }^+$ (dot-dashed line), $R_{ww}^+=\overline {w^\prime w^\prime }^+$ (dashed line) and (b) shear stress $R_{uv}^+=\overline {u^\prime v^\prime }^+$ . Uncontrolled case, black lines; $v^\prime$ -damped case, red thick lines.