3.1. Drag reduction

In this subsection, we experimentally examine the effects of varying microstructure size and driving strength on the global torque. The dimensionless wall shear stress is typically expressed using the skin friction coefficient, which is calculated from torque versus rotational speed data using the expressions

(3.1)\begin{equation} C_{f} =\frac{2\tau_{i}}{\rho V_{i}^{2} } =2\left( \frac{u_{\tau } }{V_{i}} \right)^{2}, \end{equation}

where $\tau _{i} = T/(2{\rm \pi} r_i^2 L)$ is average inner wall shear stress, and $u_{\tau } = (\tau _{i}/\rho )^{1/2}$ is the friction velocity. Furthermore, the microstructure dimensions and driving strength can be characterized by the width of grooves $\tilde {w}$ and $Re$ based on the inner cylinder, respectively. The magnitude of drag reduction is denoted as $DR$ and calculated as $DR=1-{{C}_{f, w}}/{{C}_{f, smooth}}$.

The impacts of varying $Re$ at different groove widths $\tilde {w}$ on the dimensionless wall shear stress ${C}_{f}$ are illustrated in figure 4(a), and the $DR$ in figure 4(b). Specifically, as the groove widens, the dimensionless wall shear stress ${C}_{f}$ declines, leading to a larger $DR$. Notably, a significant drag reduction has been observed in our experimental set-up, reaching 62 % at groove width $\tilde {w}=0.12$ and $Re=2.79\times {{10}^{4}}$. Moreover, figure 4(b) exhibits an uninterrupted increase of $DR$, in contrast to the existence of the decreasing trend observed in previous experimental studies (Rosenberg et al. Reference Rosenberg, Van Buren, Fu and Smits2016; Van Buren & Smits Reference Van Buren and Smits2017). The continuous growth of $DR$ indicates a more robust air layer captured in grooves over a wider range of $Re$ and $w$ in our research. Actually, the sustainable $DR$ at high $Re$, up to $Re=2.79\times {{10}^{4}}$, can be attributed to the hydrophilic treatment at the groove tips (see supplementary movie) to achieve a stable presence of the air layer.

Figure 4.

(a) Plot of ${{C}_{f}}$ as a function of $Re$, with various SHP groove widths $\tilde {w}$. (b) Drag reduction plotted against $Re$. In the plots, the predictions of the theoretical model (dashed lines) are very close to the experimental data (solid symbols), which proves that the theoretical model is highly accurate in predicting $C_f$ and $DR$ for different groove widths.

3.2. Angular momentum transport

To gain a deeper understanding of the drag reduction mechanism of air-filled grooves in TC flow, the angular velocity and flow organization are investigated with high-resolution PIV and the measurement methodology shown in figure 1(b). The angular velocity is measured at mid-height at $Re=1.12\times {{10}^{4}}$ for each value of $\tilde {w}$. In TC flows, the angular velocity $\omega$, rather than the azimuthal velocity $u_\theta$, demonstrates the intrinsic property of transportation and exhibits an appropriate log law (Grossmann, Lohse & Sun Reference Grossmann, Lohse and Sun2016), with the slope $\lambda ^{-1}$ of the shear- and curvature-affected logarithmic region, as shown in figure 5(a). Here, the superscript ‘$+$’ denotes non-dimensionalization in terms of inner wall units, i.e. $y^+=(r-r_i)/\delta _\nu$ and $\omega ^+=(\omega _i-\omega )/(u_\tau /r_i)$, with $u_\tau$ and $\delta _\nu =\nu / u_\tau$. In figure 5(b), the profiles of diagnostic function $y^+\,{\rm d}\omega ^+/{{\rm d}y}^+$ not only collapse but also plateau at $1/ \lambda \approx 1/0.64$, consistent with the results of Berghout et al. (Reference Berghout, Verzicco, Stevens, Lohse and Chung2020, Reference Berghout, Bullee, Fuchs, Scharnowski, Kähler, Chung, Lohse and Huisman2021). Here, the Obukhov length $L_c= u_\tau /\kappa (\omega _i+\Delta \omega )$ is introduced to rescale the wall-normal distance to incorporate the curvature effect. Notably, as $\tilde {w}$ increases, the $\omega ^+$ profiles shift upwards by $\Delta \omega ^+$ (see figure 5a), indicating a reduction in drag and the presence of a finite slip length at the inner boundary, where the shift $\Delta \omega ^+$ is calculated by the intercept difference between the mean angular velocity profile $\omega ^{+}(y^{+})$ in the logarithmic region of the grooved surfaces and that of the smooth wall.

Figure 5.

Angular velocity profiles based on the experiments with $\omega _i=40\ {\rm rad}\ {\rm s}^{-1}$. (a) Mean angular velocity $\omega ^+$ as a function of $y^+$ for varying $\tilde {w}$. (b) Diagnostic function $y^+\,{\rm d}\omega ^+/{{\rm d}y}^+$, with wall-normal distance rescaled by Obukhov length $L_c$.

The slip boundary achieved by the grooved SHP surface can considerably reduce the angular velocity gradient ${\partial }_{r}{\langle \omega \rangle }_{A,t}$ near the wall, and thus give rise to lower shear stress. The slip boundary affects the flow in the bulk region because of the conservation of the angular velocity current along the radius (Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007; Zhu et al. Reference Zhu, Ostilla-Mónico, Verzicco and Lohse2016). This effect manifests in the bulk region through changes in the characteristics of Taylor rolls, including their intensity and distribution. Taylor rolls, one of the typical secondary flows in a TC system, play a crucial role in the convective part of the angular velocity current along the radius (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013; Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016). Figure 6 shows the distribution of the axisymmetric azimuthal vorticity ${\omega }_\theta = \partial u_r/\partial z - \partial u_z/\partial r$, and velocity vectors for air-filled grooves of varying $\tilde {w}$ in TC flow at $Re = 1.12 \times 10^4$, plotted in the $r\unicode{x2013}z$ plane. The intensity of the Taylor rolls for the smooth inner rotor is stronger than that of the air-filled grooves rotor, and their axial position varies slightly with $\tilde {w}$.

Figure 6.

Radial–axial vorticity fields in the gap measured at $Re=1.12\times {{10}^{4}}$. The axial position is normalized by the annular gap width.

The strength of Taylor rolls can be represented quantitatively by the wind Reynolds number, denoted as ${Re}_{w}=\sigma (\langle {u_r}\rangle _t )\,d/\nu$, where $\sigma (\langle {u_r}\rangle _t )$ is the spatial standard deviation of the time-averaged radial velocity field $\langle u_{r}\rangle _t$ in the $r\unicode{x2013}z$ plane. Moreover, $\langle u_{r}\rangle _t$ values obtained in experiments at $Re = 1.12 \times 10^4$ are presented in Appendix C. The experimental results of ${Re}_{w}$ as a function of $Re$ at different $\tilde {w}$ are depicted in the inset of figure 7, which substantiates the theoretical proposition that ${Re}_{w} \propto Re^{6/7}$ (Grossmann & Lohse Reference Grossmann and Lohse2011; van der Veen et al. Reference van der Veen, Huisman, Merbold, Harlander, Egbers, Lohse and Sun2016). Meanwhile, figure 7 demonstrates that ${Re}_{w}$ of the air-filled grooves is lower than that of the smooth surface. Increasing the width of the grooves can further decrease ${Re}_{w}$, indicating a decrease in the intensity of the secondary flow. This reduction in secondary flow intensity consequently reduces the radial angular momentum transport, thereby increasing $DR$.

Figure 7.

Wind Reynolds number ${Re}_w$ compensated by $Re^{6/7}$ as a function of $\tilde {w}$, with the inset showing the original data of ${Re}_w$ varying with $Re$ at different $\tilde {w}$.

To sum up, the substantial and sustainable $DR$ in § 3.1 corresponds to declined angular momentum transport and modified flow organizations, and can be attributed primarily to the considerable and stable slip length generated by the air-filled V-grooved surface featuring a hydrophilic tip.

3.3. Theoretical model

It has been suggested that drag reduction with air-filled microstructure surfaces depends not only on the geometrical parameters of the microstructure but also on the driving strength of the system (see figure 4), i.e.

(3.2)\begin{equation} {DR}={DR}(w,\phi_s, Re). \end{equation}

As suggested by Park et al. (Reference Park, Sun and Kim2014), we scale the groove width with the viscous length scale of the flow such that $w^+ = w \nu ^{-1} \sqrt {\tau _i/\rho }$. In this subsection, an analytical expression of this relationship is presented based on angular momentum defect theory and magnitude estimate.

The derivation of (3.2) involves two steps due to the inherent multi-dimensionality of this problem. First, Srinivasan et al. (Reference Srinivasan, Kleingartner, Gilbert, Cohen, Milne and McKinley2015) proposed a modified skin friction law in turbulent TC flows ($Re>Re_c \approx 14\,000$) which accounts for wall slip:

(3.3)\begin{equation} \sqrt{\frac{2}{C_f}}=M \ln \left(Re\,\sqrt{\frac{C_f}{2}}\right)+N+b^{+}, \end{equation}

where $b^+=b{/\delta }_\nu =[b/(r_o-r_i)]\,Re\sqrt {C_f/2}$ represents the dimensionless effective slip length, the geometry-dependent constants $M=5.43$ and $N=-9.59$ are determined based on the no-slip case in our experimental set-up, and $C_{f}$ is the skin friction coefficient.

Equation (3.3) illustrates the correlation between macroscopic variables, namely $DR=DR(b^+,{Re})$ or $C_f =C_f(b^+,{Re})$. Next, efforts have been made to bridge the gap between small-scale surface features and large-scale flow response. The latter is characterized primarily by the dimensionless slip length $b^+=b^+(w^+,\phi _s)$, where $\phi _s=l_{tip}/w$ denotes the solid fraction of the surface.

Drawing inspiration from the work of Rastegari & Akhavan (Reference Rastegari and Akhavan2019) on the $DR$ scaling in the grooved SHP surfaces in turbulent channel flows, $b^+(w^+,\phi _s)$ is derived from magnitude estimates on the dynamics of turbulence kinetic energy (TKE) within the layer of thickness ${\sim }w$. Instead of the common primary term $-\langle {u_\theta u_r}\rangle \,\partial \langle {U}\rangle /\partial r$ in wall-bounded flows, it is the axial shear term $-\langle {u_\theta u_z}\rangle \,\partial \langle {U}\rangle /\partial z$ that dominates the overall production of TKE in the surface layer, which arises from significant axial gradient of azimuthal velocity due to the slip/no-slip patterns of the grooved surface. Rastegari & Akhavan (Reference Rastegari and Akhavan2019) demonstrate that the additional production of TKE through these axial shear layers is dissipated by the turbulent eddies above the no-slip regions, such that

(3.4)\begin{equation} \left \{ {-\langle {u_\theta u_{z}}\rangle\,\frac{\partial \langle {U}\rangle }{\partial z} } \right \} _{shear\ layer}\sim {\left \{ \upsilon \langle {s_{ij}s_{ij}}\rangle \right \}}_{no\ slip}. \end{equation}

On the left-hand side, utilizing the mixing length theory, the Reynolds stress $-\langle {u_\theta u_z}\rangle$ is estimated as $-\langle {u_\theta u_z}\rangle \sim u^{\prime } l\,\partial \langle {U}\rangle / \partial z$, where the characteristic velocity of the largest eddies $u^{\prime }$ and their size $l$ are given by the geometric average of the corresponding scale at $r=0$ and $r \sim w$, specifically $u^{\prime } \sim \sqrt {u_{\tau n} u_\tau }=\sqrt {u_\tau / \sqrt {\phi _s}\,u_\tau }$ and $l \sim \sqrt {\delta _v \, w}$, while $\partial \langle {U}\rangle /\partial z$ is estimated as $U_{s s} /w$, where $U_{s s}=U_s /(1-\phi _s)$ is the average slip velocity of the microstructure walls. On the right-hand side, the fluctuating strain rate above the no-slip regions can be given by characteristic velocity dividing its associated inner length scale, i.e. $s_{i j} \sim u^{\prime } / \delta ^{\prime }$, where $\delta ^{\prime }=v / u^{\prime }$. Using the estimates above, the scaling is given as

(3.5)\begin{equation} U_s^{+}=b^+= K \left\{\left(1-\phi_s\right) \phi_s^{{-}3 / 8}\right\}\left\{w^{+}\right\}^{3/4}, \end{equation}

where the multiplicative coefficient $K$ is determined to be 0.34 through linear regression. Here, the experimental results of $b^+$, or equivalently $U_s^+$ under the Navier slip hypothesis, is given by (3.3) based on torque measurement.

As indicated in (3.5), contributions of geometrical features to wall slip consist of two parts. On the one hand, the decrease in solid fraction $\phi _s$ will give rise to reduced solid friction and thus enhanced slip. On the other hand, as the dimensionless groove width $\tilde {w}$ increases, the axial gradients of mean velocity and the TKE production in the shear layer decline, which also represents a larger wall slip.

Figure 8(a) presents the experimental results of $b^+$ varying with $\tilde {w}$ and $Re$. The wall slip is enhanced with increasing $\tilde {w}$ and $Re$ within the investigated parameter space, aligning with the observed trend of $DR$ depicted in figure 4. Figure 8(b) shows $\{\phi _s^{3 / 8} /(1-\phi _s)\} b^+$ computed from these experimental databases, which can be seen to scale as $\{w^{+}\}^{3 / 4}$ in agreement with (3.5). We also explored the scaling dependence between the shift $\varDelta \omega ^+$ and $w^+$ as well as $\phi _s$, as shown in figures 9(a) and 9(b), and the multiplicative coefficient is determined to be 0.2 through linear regression.

Figure 8.

(a) Plots of $b^+$ as a function of $Re$ and $\tilde {w}$, where a solid symbol denotes experimental. (b) The logarithmic plot verifying the scaling of $b^+$, with the dashed lines showing the prediction of (3.5).

Figure 9.

(a) Plots of $\Delta \omega ^+$ as a function of $Re$ and $\tilde {w}$, where a solid symbol denotes experimental. (b) The logarithmic plot verifying the scaling of $\Delta \omega ^+$, with the dashed lines showing the prediction of the shift $\Delta \omega ^+$.

Accurate prediction of the friction coefficient $C_f$ is essential when dealing with turbulent boundary layers affected by surface roughness. The joint equations (3.3) and (3.5) are effective in predicting $C_f$, i.e.

(3.6)\begin{equation} \sqrt{\frac{2}{C_f}}=M \ln \left(Re\,\sqrt{\frac{C_f}{2}}\right)+N + K \left\{\left(1-\phi_s\right) \phi_s^{{-}3 / 8}\right\}\left\{w^{+}\right\}^{3/4}. \end{equation}

Initially, (3.5) is used to determine $b^{+}$ from the known values of $w^{+}$ and $\phi _s$. The calculated $b^{+}$ is then substituted into (3.3) to calculate $C_f$. This method requires only the parameters $w$, $\phi _s$ and $Re$ to predict the friction coefficient. As shown in figure 4, comparing the theoretical model with the experimental values shows that both the friction coefficient $C_f$ and $DR$ are predicted with high accuracy. The theoretical model can accurately predict friction and drag behaviour across surface roughness levels and flow conditions. In this way, theoretical models help engineers and researchers to understand complex multiscale interaction between the microstructural parameters ($w$ and $\phi _s$) and the flow parameter ($Re$) on the magnitude of $DR$, predicting complex flow phenomena that may be encountered in practical applications, providing a scientific basis for design optimization.

In terms of errors, the theory overpredicts $b^{+}$ at large $w$ with high $Re$, and underpredicts it at lower $Re$. In the former cases, the plastrons over grooves undergo fluctuations and become challenging to maintain, leading to partial wetting and deterioration of the SHP surfaces, primarily adjacent to the hydrophilic tip of the grooves, as shown in figure 2(b). The partial deterioration of the plastron, or in other words, the depinning of the liquid–gas interface, will increase the solid fraction $\phi _s$. According to (3.5), as $\phi _s$ increases, the dimensionless slip of the surface declines, indicating the overestimate of $b^{+}$. Nevertheless, at lower $Re$, the underprediction of $b^{+}$ can be attributed to the curvature effect of the TC flow, which is more pronounced at finite $Re$ where the boundary layer is rather thick and curved (Berghout et al. Reference Berghout, Verzicco, Stevens, Lohse and Chung2020). Specifically, the curvature-induced TKE production ($P_{{curvature }}$) in the cylindrical coordinate should be taken into account in (3.4) (Berghout et al. Reference Berghout, Verzicco, Stevens, Lohse and Chung2020), i.e.

(3.7)\begin{equation} P_{{shear\ layer }}+P_{{curvature }}=\varepsilon_{{no\text{-}slip }}, \end{equation}

where $P_{{curvature }}=(1/r)\langle u_\theta u_r\rangle \langle U\rangle =-u^{\prime } l (\langle U\rangle /r) (\partial \langle U\rangle /\partial r)>0$. This additional positive term will emerge on the left-hand side of (3.5) at finite $Re$ in TC flow, resulting in the overprediction of $b^{+}$ if the curvature effect is neglected. However, compared to the results from Van Buren & Smits (Reference Van Buren and Smits2017) where the optimum groove size before surface failure is $w^+\approx 35$, the counterpart in our experimental set-up can reach $w^+\approx 75$ or even higher due to the stabilization provided by hydrophilic tips of SHP grooves.