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1. Introduction
Turbulent skin-friction drag reduction is a subject of great interest in fluid mechanics research, due to the potential to produce significant reductions in fuel consumption and carbon emissions in many industrial scenarios. Around 
$60\,\%$ of the aerodynamic drag of a typical airliner in cruise conditions is due to frictional drag, the other components being form drag and induced drag (Leschziner, Choi & Choi Reference Leschziner, Choi and Choi2011). In the case of a typical slow merchant ship, such as a tanker or a container ship, around 
$70\,\%$ of the total hull resistance is frictional drag, the remainder being form drag and wave drag (Larsson & Raven Reference Larsson and Raven2010). For the aerospace industry, it has been estimated that a 
$1\,\%$ reduction in skin-friction drag for a long-range commercial aircraft would decrease fuel consumption by approximately 0.45 %. For a typical airline this change is equivalent to a reduction in CO
$_2$ emissions of 5.4 million equivalent tonnes per year and an annual reduction in operating costs of 0.2 % (Reneaux Reference Reneaux2004).
At the moderate Reynolds numbers considered in this study, the near-wall dynamics is responsible for the largest contribution to the skin-friction drag (Quadrio Reference Quadrio2011; de Giovanetti, Hwang & Choi Reference de Giovanetti, Hwang and Choi2016). Wall-based drag-reduction techniques therefore often aim at altering the structure of the near-wall turbulence, producing an attenuation of the ejections and sweeps that promote the transport of high-momentum fluid in the near-wall region and lead to an increased skin-friction drag.
Wall-based flow control methods for drag reduction can be broadly classified as active or passive. In active methods, the flow is often altered by wall-mounted devices that rely on an external power supply to operate. For such methods, the power required to drive the control devices must be accounted for because the net power consumption represents a critical performance metric. Passive methods instead consist of modifications of the wall surface geometry (Min & Kim Reference Min and Kim2004; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011) or compliant surfaces where some regions of the wall are allowed to move under the action of the flow (Choi et al. Reference Choi, Yang, Clayton, Glover, Atlar, Semenov and Kulik1997). Passive methods do not necessitate an external power input, although the interaction between the fluid and the complex geometry may constitute an additional source of friction. Jung, Mangiavacchi & Akhavan (Reference Jung, Mangiavacchi and Akhavan1992) first introduced in-plane wall motion as a drag-reduction technique, achieving an attenuation of the turbulent skin-friction drag by oscillating the wall in the cross-flow direction. The oscillating wall technique has proven capable of generating drag-reduction margins up to 
$40\,\%$ (Quadrio & Ricco Reference Quadrio and Ricco2004), while its generalizations in the form of steady and travelling waves of spanwise velocity at the wall have also led to drag-reduction levels as high as 
$45\,\%$ at low Reynolds numbers (Quadrio, Ricco & Viotti Reference Quadrio, Ricco and Viotti2009). The mechanism by which the spanwise wall motion suppresses the turbulent activity and leads to drag reduction has been the subject of several investigations (Choi Reference Choi2002; Quadrio Reference Quadrio2011; Skote Reference Skote2011; Ricco et al. Reference Ricco, Ottonelli, Hasegawa and Quadrio2012; Blesbois et al. Reference Blesbois, Chernyshenko, Touber and Leschziner2013). An optimal oscillation period of about 
$T^+=100$ (the superscript ‘+’ denoting scaling in viscous units) was found by Quadrio & Ricco (Reference Quadrio and Ricco2004) to produce the maximum drag reduction because it matches a characteristic temporal scale of the near-wall turbulent structures and an optimal thickness of the generated Stokes layer.
Spinning circular actuators that are flush-mounted on the wall have also been demonstrated numerically to lead to drag-reduction levels of up to 
$23\,\%$ and to net power savings of up to 
$10\,\%$ (Ricco & Hahn Reference Ricco and Hahn2013; Wise & Ricco Reference Wise and Ricco2014), where the latter is computed by subtracting the power spent to move the discs from the power saved because of drag reduction. The discs offer a few practical advantages with respect to wall oscillations and travelling waves, i.e. the local character that avoids the motion of the whole solid wall and the wider range of diameters leading to drag-reduction margins that are comparable to the maximum value (Ricco & Hahn Reference Ricco and Hahn2013; Wise, Alvarenga & Ricco Reference Wise, Alvarenga and Ricco2014). Wise et al. (Reference Wise, Alvarenga and Ricco2014) demonstrated that rotating annular actuators can deliver similar levels of drag reduction while requiring up to 
$20\,\%$ less driving power. The ring actuators can be combined efficiently with other passive and active techniques (Olivucci, Ricco & Aghdam Reference Olivucci, Ricco and Aghdam2019), attaining reductions of the skin-friction drag as large as 
$27\,\%$.
In the present work, we carry out direct numerical simulations of a turbulent channel flow over flush-mounted discs that are free to move under the action of turbulent wall-shear stress. We consider the full numerical modelling of the rotating actuators, taking into account the coupled dynamics of the turbulent flow and the discs, and the frictional losses occurring in a realistically designed disc housing.
Modelling of flow-control actuators is critical because it gives estimates of the power expenditure that are closer to those in real-world settings. This approach also allows studying the two-way coupling between the fluid mechanics and the actuators, which leads to a more accurate prediction of the flow physics and therefore of the drag-reduction performance. In the case of the discs, the two-way coupling is able to capture the detailed physics of the stress and torque fluctuations to which a disc rotating under a turbulent flow is subjected. Józsa et al. (Reference Józsa, Balaras, Kashtalyan, Borthwick and Viola2019) simulated the coupled fluid-body dynamics of passive, shear-stress-driven cylindrical actuators as a compliant-wall technique able to generate a finite in-plane wall velocity and give drag reduction. Mahfoze, Laizet & Wynn (Reference Mahfoze, Laizet and Wynn2018) simulated suction and blowing in a boundary layer and used an empirical model of the frictional losses in the electromagnetic speaker used to generate the wall-normal transpiration. To the authors’ knowledge, there have been no attempts to model the coupled dynamics between a fluid and in-plane flow-control actuators, and to estimate the mechanical losses realistically.
The dynamics of freely rotating full discs and half discs under the shearing action of wall turbulence is studied. The first case serves the purpose of analysing the unsteady response of a full disc to the wall turbulence and to verify the numerical implementation of the two-way coupled dynamics between the fluid and the disc, thereby allowing a detailed study of the fluid–structure interaction problem. The half-disc case is subsequently investigated with the primary objective of exploring the potential for turbulent skin-friction drag reduction. The half discs are realized by preventing the left spanwise halves of freely rotating discs from interacting with the flow, thereby producing a finite average shear-stress torque on the disc surfaces. The half discs thus rotate with a finite angular velocity and introduce a slip velocity at the wall, thereby affecting the wall-shear stress.
Our study is also motivated by the experimental boundary-layer study of two configurations of half discs by Koch & Kozulovic (Reference Koch and Kozulovic2013) (Reference Koch and KozulovicKK13) and Koch & Kozulovic (Reference Koch and Kozulovic2014) (Reference Koch and KozulovicKK14), shown in figure 1. The wall-shear stress was not measured on the disc surface, but it was estimated by modifying a correlation for computing the skin-friction coefficient in uncontrolled boundary layers. Koch and Kozulovic used the local streamwise slip velocity generated by the disc in the correlation, thus predicting reductions of skin-friction drag of up to 
$17\,\%$ on the disc surface. The procedure adopted by Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14 is detailed in the supplementary material S1 available at https://doi.org/10.1017/jfm.2021.533.
Experimental apparatus of Koch & Kozulovic (Reference Koch and Kozulovic2013) and Koch & Kozulovic (Reference Koch and Kozulovic2014). (a) Schematic of the disc and its supporting housing system. (b) Schematic of the wind-tunnel wall, fitted with a half disc. Taken from Koch & Kozulovic (Reference Koch and Kozulovic2013) with permission from American Society of Mechanical Engineers.
The numerical and statistical procedures are presented in § 2. Section 3 describes the dynamics of the actuators and the modelling of the torques acting on the discs. In § 4, we study the passive response of full freely rotating discs to the turbulent flow. In § 5, the simulations of arrays of half discs are discussed and compared with the experimental results of Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14. Section 6 presents the conclusions of the study.
2. Numerical procedures
In this section, we present the flow system and the numerical procedures. An incompressible turbulent flow between parallel flat walls fitted with spinning discs is investigated. The flat discs, flush-mounted on the walls, are free to rotate under the viscous action of the turbulent channel flow. The Cartesian coordinates describing the system are the streamwise coordinate 
$x^*$, the wall-normal coordinate 
$y^*$ and the spanwise coordinate 
$z^*$ (the 
$*$ superscript indicates a dimensional quantity). The respective velocity components in these directions are 
$u^*$, 
$v^*$ and 
$w^*$. The channel walls are separated by a distance 
$L_y^*=2h^*$ and the flow is driven by a streamwise pressure gradient. A constant flow rate per unit wall area 
$Q^*$ is imposed. The bulk velocity is 
$U^*_b=Q^*/2h^*$.
The fluid motion obeys the incompressible continuity and Navier–Stokes equations (NSE):
$$\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} =0, \end{gather}$$
$$\begin{gather}\frac{\partial \boldsymbol{u}}{\partial t}+(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u} ={-}\boldsymbol{\nabla} p+\frac{1}{Re_p}\nabla^2 \boldsymbol{u}. \end{gather}$$
Quantities are non-dimensionalized by the channel half-height 
$h^*$, the density of the fluid 
$\rho ^*$ and the centreline velocity of the laminar Poiseuille flow at the same flow rate, defined as 
$U_p^*=3Q^*/4h^*$. Quantities expressed in these outer units are written without any symbol. All flows are at a Reynolds number 
$Re_p = U_p^* h^* / \nu ^* = 4200$, where 
$\nu ^*$ is the kinematic viscosity of the fluid.
Equations (2.1) and (2.2) are subject to no-slip boundary conditions at the channel walls. The velocity is null over the stationary portions of the walls, while it coincides with the local disc velocity when the fluid passes over a disc. The boundary conditions are time-dependent because they are determined by the rigid-body dynamics of the discs, driven by the instantaneous wall-shear stresses exerted by the wall turbulence on the disc surfaces. The dynamics is ruled by a two-way coupling between the fluid and the freely rotating discs. The supplementary material S3 discusses an uncoupled model, where the motion of the disc does not feed back to the turbulence dynamics. Two configurations are considered: a full-disc layout where two coaxial discs, one on each wall, are completely exposed to the viscous action of the turbulent flow; and an arrangement of rows and columns of half discs, for which only the right half of each disc is wetted by the fluid. Figure 2(a) shows the configuration of a full disc for three diameters and figure 2(b) shows the configuration of the half discs.
Schematic of (a) a single freely rotating disc of increasing diameter and (b) the rows and columns of half discs.
The turbulent channel flow with the freely rotating discs is studied via direct numerical simulation. The open-source code Incompact3d (Laizet & Lamballais Reference Laizet and Lamballais2009; Laizet & Li Reference Laizet and Li2011) is used to simulate the flows. The code solves the non-dimensional, incompressible NSE using a Chorin–Temam projection method, with time advancement performed using a second-order Adams–Bashforth scheme. The time integration of the equation of motion of the discs is performed using the same numerical scheme of the fluid solver and the same temporal resolution. The spatial discretization employs sixth-order compact finite difference schemes. The Poisson equation for the projection-step pressure is solved in spectral space using fast Fourier transforms on a partially staggered grid. The computational grid is uniformly spaced along the streamwise and spanwise directions and is stretched along the wall-normal direction, ensuring that the near-wall resolution is sufficient to resolve the smallest flow scales accurately. Incompact3d is parallelized using the Message Passing Interface and exhibits excellent scalability, achieved through a two-dimensional domain decomposition (Laizet & Li Reference Laizet and Li2011). The code is run on the Cray XC30 Archer supercomputer of the National Supercomputing Service and the ShARC cluster at the University of Sheffield. The simulations are run using 1024 parallel computational cores, adopting a 
$32\times 32$ block decomposition.
The channel flow is assumed to be periodic along the 
$x$ and 
$z$ directions and the dimensions of the computational domain in these directions are 
$L_x$ and 
$L_z$. For the full-disc cases, the domain dimensions are 
$L_x=4.53{\rm \pi}$, 
$L_y=2$ and 
$L_z=2.26{\rm \pi}$ and the number of points defining the numerical grid are 
$256 \times 129 \times 256$ in the 
$x$, 
$y$ and 
$z$ directions, respectively.
All computations are initiated from the laminar Poiseuille channel flow between the two solid stationary walls, perturbed by random noise. The initial flow field evolves to the fully developed uncontrolled turbulent flow, identified by the main statistics displaying convergence and the total mean stress profile being linear (Orlandi Reference Orlandi2012). This fixed-wall fully developed turbulent flow is used as the initial flow field for the computations of the coupled fluid–disc system. The initial transient flow occurring between the activation of the fluid–disc boundary conditions and the beginning of the fully developed condition of the flow over freely moving discs is discarded to acquire meaningful statistics.
2.1. Averaging procedures
The time average of a flow variable 
$f(x,y,z,t)$ over a time interval 
$[0,T]$ during which 
$f$ is statistically stationary is defined as
\begin{equation} \bar{f}(x,y,z) = \frac{1}{T}\int^T_0 f(x,y,z,t) \, \mbox{d}t. \end{equation}
For the case where the walls are covered by rows and columns of half discs, the flow is statistically periodic along 
$x$ and 
$z$ with period 
$D_h$, which is the side of the square containing one half disc. A spatial ensemble-averaging operator is therefore defined as
\begin{equation} [f]_s(x_s,y,z_s,t) = \frac{1}{N_D} \sum_{n_x = 1}^{r_D} \sum_{n_z = 1}^{c_D} f(x_s + n_x D_h,y, z_s + n_z D_h,t), \end{equation}
where 
$0\leq x_s \leq D_h$ and 
$0\leq z_s \leq D_h$ are the ensemble spatial coordinates and 
$r_D$ and 
$c_D$ are the number of rows and columns of a total number of 
$N_D = r_D \times c_D$ discs in the computational domain. Spatial averaging along 
$x_s$ and 
$z_s$, i.e. averaging over a square of area 
$D_h^2$ defining the minimal flow unit containing one disc, is expressed as
\begin{equation} \langle\, f \rangle(y,t) = \frac{1}{D_h^2}\int^{D_h/2}_{{-}D_h/2}\int^{D_h/2}_{{-}D_h/2} [f]_s(x_s,y,z_s,t) \, \mbox{d}\kern0.06em x_s\,\mbox{d}z_s. \end{equation}
The statistical sample is doubled by averaging a quantity across the two channel halves. An uppercase letter indicates a global average, 
$F(y) = [f]_g= \langle\, \bar {f} \rangle$. For example, the global-averaged streamwise velocity is 
$U(y)=[u]_g$. The velocity field is decomposed as follows:
$$\begin{gather} \boldsymbol{u}(x,y,z,t) = \boldsymbol{U}(y) + \boldsymbol{u}_{\boldsymbol{d}}(x - \lfloor x/D_h \rfloor D_h, y, z - \lfloor z/D_h \rfloor D_h) + \boldsymbol{u}_{\boldsymbol{t}}(x,y,z,t), \end{gather}$$
$$\begin{gather}\boldsymbol{u}_{\boldsymbol{d}} = \{u_d, v_d, w_d\} = [\bar{\boldsymbol u}]_s - \boldsymbol{U}(y), \end{gather}$$
where 
$\boldsymbol {U}(y)=\{U(y),0,0\}$.
The separation of scales typical of wall-bounded turbulence is measured by the friction Reynolds number 
$Re_\tau = h^* / \delta ^*_\nu =180$, where 
$\delta ^*_\nu = \nu ^*/u^*_\tau$ is the near-wall viscous length scale, 
$u^*_\tau = \sqrt {[\tau _{w,x}^*]_g / \rho ^*}$ is the wall-friction velocity and 
$[\tau _{w,x}^*]_g=\nu ^*\rho ^*\,\text {d}U^*/\text {d}y^*|_{y=0}$ is the global-averaged wall-shear stress in the fixed-wall configuration. Quantities scaled in viscous units, that is, using the fixed-wall 
$u_\tau ^*$ and 
$h^*$, are denoted by a 
$+$ superscript. The skin-friction coefficient is 
$C_f=2[\tau _{w,x}]_g/U_b^2$, where 
$U_b =2/3$. The drag reduction 
$\mathcal {R}$ is defined as the percentage decrease of the skin-friction coefficient:
\begin{equation} \mathcal{R}(\%) = 100(\%) \frac{C_{f,un} - C_f}{C_{f,un}},\end{equation}
where 
$C_{f,un}$ is the skin-friction coefficient of the uncontrolled flow. We also define a spatially dependent turbulent drag reduction as follows:
\begin{equation} \mathcal{R}_{xz}(x_s,z_s)(\%) =100(\%) \frac{C_{f,un} - c_f(x_s,z_s)}{C_{f,un}},\end{equation}
where 
$c_f(x_s,z_s) = 2 Re_p^{-1} U_b^{-2}\partial (U + u_d)/\partial y|_{y=0}$. The disc drag-reduction margin 
$\mathcal {R}_d$ is defined as the reduction of wall friction computed by only taking into account the disc surface.
3. Modelling of the disc dynamics
This section presents the modelling of the dynamics of the freely rotating discs. Unless stated otherwise, this model is valid for both configurations of discs, shown in figure 2. In this section, the modelling of the two-way coupled dynamics is presented, while the modelling of the uncoupled dynamics is discussed in the supplementary material S3.1. The discs are flush-mounted on the two surfaces of the channel, have diameter 
$D^*$, and are rigid, homogeneous and free to rotate about their central axis. A schematic of the disc housing is shown in figure 3. The discs are characterized by their thickness 
$b^*$ and material density 
$\rho _d^*$. The moment of inertia of a disc about its central axis is 
$I^*= {\rm \pi}b^* D^{*4} \rho _d^* / 32$. A thin gap of width 
$0.05D^*$ is modelled around the edge of each disc to simulate the clearance region between the disc and the stationary channel walls. Within the annular gap the velocity is modelled to decay linearly from its maximum value on the disc edge to zero on the fixed portion of the wall. This treatment of the gap closely represents a real experimental set-up where such gaps would always be present, and it markedly reduces the numerical oscillations that would arise from a discontinuity in the velocity field (Ricco & Hahn Reference Ricco and Hahn2013).
Schematic of the disc and its housing. The axis of rotation of the disc is indicated by the dash-dotted line and the dark grey squares represent the ball bearing.
Each disc is fitted with a thin shaft and housed in a cavity in the wall that is deep enough for the upper surface of the disc to be flush with the channel wall. The cavity has thickness 
$d^*_h$ and is filled with the same fluid as the turbulent channel flow. The shaft is supported by a bearing mounted into a purpose-built hole at the bottom of the disc housing. The disc housing has a similar design to those realized in the experimental study of Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14, and in a preliminary experimental apparatus built at the Deutsches Zentrum für Luft- und Raumfahrt (DLR) in Göttingen to reproduce the spinning-disc system studied numerically by Ricco & Hahn (Reference Ricco and Hahn2013) (M. Rutte and U.G. Becker, personal communication).
It is important to convert the scaled quantities characterizing the turbulent flow and the disc system to physical quantities because one must simulate a flow system that can be realized in a laboratory. One risk when working solely with scaled variables is, for example, to compute a turbulent flow over unrealistically light discs. The very low inertia would lead to exceedingly large angular velocities and thus drag-reduction margins that could not be measured experimentally. Table 1 presents the characteristics of a turbulent channel flow with discs, where the working fluid is water at 20 
$^\circ$C. Water is preferred to air because the larger dynamic viscosity of water generates a larger frictional torque on the disc surface. The discs must be designed as thin and light as possible in order to minimize their inertia and thus maximize their angular acceleration for a given fluid torque. The disc thickness is chosen to be 0.5 mm and we assume the material of the discs to be aluminium. The density of aluminium 
$\rho _d^*$ is almost three times the density of water, i.e. 
$\rho _d=2.7$. If air were chosen, the scaled density of the disc would be 
$\rho _d=2200$.
Physical quantities of the turbulent water channel flow and the discs.
Three cavity depths are tested: 
$d_h=0.1$, 
$1$ and 
$10$. The smallest value is chosen to simulate more intense frictional effects caused by the flow in the cavity, while the largest value is selected to simulate a housing torque that is not influenced by the confinement of the cavity.
3.1. Dynamics of the rotating discs
The disc motion is described by its angular velocity 
$\varOmega (t)$ or by its disc-tip velocity 
$W(t) = R \varOmega (t)$, where 
$R$ is the radius of the disc. The disc-tip velocity evolves according to the following dynamical equation:
\begin{equation} \frac{2I}{D} \frac{\mathrm{d}W}{\mathrm{d}t} = T_f - T_h - T_b,\end{equation}
where 
$W$ is positive anticlockwise, 
$T_f$ is the torque exerted by the wall-shear stresses of the channel flow on the flow-facing surface of the disc, 
$T_h$ is the torque arising from the fluid contained in the housing beneath the disc and 
$T_b$ is the frictional torque given by the ball bearing. Equation (3.1) is nonlinear because the torques 
$T_f$ and 
$T_h$ depend on 
$W$. The equation of the disc dynamics (3.1) is solved together with the continuity equation (2.1) and the NSE (2.2). The equations are mathematically coupled for two reasons. First, the torque 
$T_f$, exerted by the wall-shear stresses of the wall turbulence, drives the motion of the disc and is therefore an input into the disc dynamics. Second, the disc motion fixes the boundary conditions of the fluid velocity on the disc surface. On a disc centred at 
$(x_c,z_c)$ and rotating with a disc-tip velocity 
$W$, the velocity boundary conditions for the streamwise and the spanwise velocity components read
\begin{align} u(x,y=0,2,z,t) = W(t) (z-z_c) / R, \quad w(x,y=0,2,z,t) ={-}W(t) (x-x_c) / R, \end{align}
for 
$x,y$ belonging to the surface of the disc, i.e. 
$(x-x_c)^2+(z-z_c)^2 \leq R^2$. The coupled system is visualized in figure 4 using a block diagram (Åström & Murray Reference Åström and Murray2008). The diagram features the disc-tip velocity as the main output and displays a closed-loop structure due to the two coupling elements, i.e. the fluid torque and the fluid velocity boundary conditions.
Block diagram of the two-way coupled disc–fluid system as a feedback loop. In this diagram and in figures S1 and S3, the dot indicates the derivative with respect of time and BC stands for boundary conditions.
For the full-disc cases, only the disc diameter is varied, while the disc thickness, material density and cavity depth are kept constant. For the case with 
$D=3.38$, the flow over the isolated full discs, shown in figure 2, was compared with the flow over rows and columns of discs with the same diameter and no differences in the statistics were found.
As in the wind-tunnel study of Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14, we consider freely rotating discs that only have the right spanwise half exposed to the wall turbulence, while the other half is covered by a solid surface and is not exposed to the flow. The half discs rotate with a finite mean velocity, caused by the averaged fluid torque induced by the mean wall-shear stress on the exposed portion of disc surface. Since the steadily moving discs produce a streamwise slip velocity at the wall, a local flattening of the velocity profile and a reduction of the mean velocity gradient at the wall are expected. The half-disc simulations, summarized in table 2, differ in three ways from the experiments of Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14. In the experiments, an open free-stream boundary layer is studied, the Reynolds number is larger than in our channel-flow simulations and the disc diameters are larger in wall units. The half discs are modelled by applying the fixed-wall condition to the left-hand half of the disc. The disc model is modified by only calculating 
$T_f$ on the existing disc half and by multiplying 
$T_h$ by a factor of 
$1.5$ to account for the friction under the hypothetical plate that would cover the sheltered half of the disc. The latter change implies that the flow under the plate covering the disc half is assumed to be the same as that found in the housing cavity.
Numerical parameters of the half-disc flow simulations. The numbers of the grid size are the number of grid points along the 
$x$, 
$y$ and 
$z$ directions. The models for 
$T_h$ are presented in the supplementary material S2.
3.2. Modelling of the torques
The fluid torque 
$T_f$ results from the wall-shear stresses exerted by the turbulent flow over the surface 
$\mathcal {S}$ of a disc and is expressed by the integral
\begin{equation} T_f(t) = \int_\mathcal{S} \boldsymbol{\tau_w} \times (\boldsymbol{x}-\boldsymbol{x}_c) \, \mbox{d}\mathcal{S} = \int^{R}_{{-}R}\int^{\sqrt{R^{2}-(x-x_c)^{2}}}_{-\sqrt{R^{2}-(x-x_c)^{2}}} [ (z-z_c)\tau_{w,x} - (x-x_c)\tau_{w,z} ] \, \mbox{d}z\, \mbox{d}\kern0.7pt x, \end{equation}
where the streamwise and spanwise components of the wall-shear stress 
$\boldsymbol {\tau _w}$ are
\begin{equation} \tau_{w,x} ={-} \frac{1}{Re_p} \left.\frac{\partial u}{\partial y} \right|_{y=0}, \quad \tau_{w,z} = \frac{1}{Re_p} \left.\frac{\partial w}{\partial y}\right|_{y=0}.\end{equation}
The integral in (3.3) is evaluated at each time step and the boundary conditions on the disc are updated accordingly, realizing the fully coupled disc–fluid dynamics. The ball-bearing torque is modelled as 
$T_b = \vert T_b \vert {\mathrm {sgn}}(W)$, where 
$\vert T_b \vert$ is obtained from the specifications of an industrial bearing. The torque exerted by the cavity flow is modelled as 
$T_h=T_h(Re_p,d_h,W,D)$, using known solutions of flows over spinning discs. Details of the modelling of the torques acting below the disc surface are presented in the supplementary material S2.
4. Results for the full discs
This section presents the results of turbulent flows over the isolated full freely rotating discs, depicted in figure 2(a). The uncoupled-dynamics results are found in the supplementary material S3.2 and S3.3. For the full-disc simulations, the linear housing torque formula (S2.1) is used. The equation of motion for the disc simplifies to
\begin{equation} \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{16}{{\rm \pi} b D^3 \rho_d} [T_f(t) - T_b ] - \frac{W(t)}{Re_p d_h b \rho_d}.\end{equation} The wall turbulence exerts a finite instantaneous torque on the full disc, which results in an unsteady motion of the disc characterized by its disc-tip velocity 
$W$. The root-mean-square of the disc-tip velocities (
$W_{{rms}}$) and the chi-squared confidence intervals are presented in table 3 for the eleven disc diameters studied, ranging between 
$D^+$=50 and 
$D^+$=1200. (The subscript ‘rms’ henceforth indicates the root-mean-square of a quantity.) The time average 
$\bar {W}$ is null in all the cases because the wall-shear stress has no preferential direction over the disc surface. Small values of the disc-tip velocities are found, 
$W_{{rms}}$ never exceeding the wall-friction velocity.
Values of 
$W_{{rms}}$ for the full-disc coupled cases. The confidence intervals are at the 
$95\,\%$ level.
Time histories of 
$W$ are shown in figure 5(a) for three diameters. The discs oscillate randomly around 
$W=0$ and the variance of 
$W$ is larger at small diameters, the disc oscillations becoming less intense as the diameter increases. The time histories of the torque in the same time interval, shown in figure 5(b), behave in the opposite way, the variance of 
$T_f$ growing rapidly with the diameter.
Time evolutions of (a) the disc-tip velocity 
$W^+$ and (b) the fluid torque 
$T^+_f$ for discs with three diameters.
Because of the small angular displacements, no drag reduction occurs. Ricco & Quadrio (Reference Ricco and Quadrio2008) for spanwise-wall oscillations, Quadrio & Ricco (Reference Quadrio and Ricco2011) for wall-travelling waves and Ricco & Hahn (Reference Ricco and Hahn2013) for rotating discs demonstrated that a minimal wall velocity, about 
$W^+=1$, is required for the wall turbulence to be affected by the wall motion and to experience drag reduction. If the wall displacement is too small, the viscous effects are confined in a very thin layer because the wall-normal momentum diffusion from the wall is limited. In the freely rotating disc case, 
$W_{{rms}}$ is always smaller than the minimal wall velocity and it is thus expected that the skin-friction drag is unaffected.
The profiles of the root-mean-squares of the streamwise and spanwise velocity fluctuations are reported in figure 6. The wall-normal extension of the oscillation-induced flow is confined to the viscous sublayer and cannot modify the structure of near-wall turbulence for 
$y^+\ge 5$. The mean velocity, turbulent Reynolds stresses and root-mean-square of the wall-normal velocity fluctuations (not depicted in figure 6) are indistinguishable from those of the reference channel flow.
Wall-normal profiles of the root-mean-square of the streamwise (a–d) and spanwise (e–h) velocity fluctuations at different positions on the freely oscillating disc for the case with 
$D=0.84$ (solid red line) and the reference channel-flow case (
$\bullet$). The locations are shown on the top left of each panel. The variables are scaled with the wall units of the reference channel flow.
Figure 7 shows the values of 
$W_{{rms}}$ and 
$T_{f,{rms}}$. The blue symbols denote the coupled-dynamics results, summarized in table 3, and the red symbols denote the uncoupled-dynamics results, discussed in the supplementary material S3. For 
$D^+ < 100$, 
$W_{{rms}}$ grows with the disc diameter, while 
$W_{{rms}}$ decreases as 
$D^{-0.55}$ for 
$100 < D^+ < 900$. The maximum 
$W_{{rms}}$ at 
$D^+=100$ occurs because, for very small 
$D$, the torque is too small to exert the required power to move the discs, while for large discs, the inertia of the disc thwarts the action of the torque. The optimum diameter is comparable with the characteristic spanwise spacing of the low-speed streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). The torque 
$T_{f,{rms}}$ is instead a monotonically increasing function of the diameter. For 
$0 < D^+ < 100$, 
$T_{f,{rms}}$ grows more rapidly with 
$D$ than for larger diameters and is proportional to 
$D^{2.2}$ for 
$100 < D^+ < 900$. The uncoupled-dynamics data reveal that the two-way fluid–disc coupling has an attenuating effect on the kinetic energy of the disc.
Root-mean-square values of (a) the disc-tip velocities 
$W^+$ in the coupled case and 
$W^+_0$ in the uncoupled case and (b) the fluid torque 
$T^+_f$ as functions of the disc diameter. The time-averaging interval is 
$2500h^*/U^*_p$ or 
$19\ 300 \nu ^*/u_\tau ^{*2}$. The blue filled circles denote the coupled-dynamics data and the red open circles denote the uncoupled-dynamics data, discussed in the supplementary material S3.
The torque can be written as 
$T_f = T^x_f + T^z_f$, where 
$T_f^x$ and 
$T_f^z$ are given by the streamwise and spanwise wall-shear stresses, 
$\tau _{w,x}$ and 
$\tau _{w,z}$, respectively. Figure 8 visualizes instantaneous snapshots of the torque contributions. The component 
$T_f^x$ is much larger than 
$T_f^z$ and most intense at the disc sides. Figure 9(a) (top) shows that 
$T^x_f$ and 
$T^z_f$ have opposite sign for most of their time histories. The torque variance is expressed as
\begin{equation} Var(T_f) = \overline{{T_f^x}^2} + \overline{{T_f^z}^2} + 2 \overline{T_f^x T_f^z}, \end{equation}where the last term is the covariance of the streamwise and spanwise terms. The dependence of the right-hand side terms of (4.2) on the disc diameter is shown in figure 9(b) by the open blue symbols. All three components share a very similar growth with the disc diameter. The streamwise components dominate over the other two components, contributing to more than 90 % of the total variance. The covariance is negative and larger in absolute value than the spanwise component.
Instantaneous snapshots of (a) the torque contribution given by the streamwise wall-shear stress and (b) the torque contribution given by the spanwise wall-shear stress, on a disc with 
$D=6$.
Spanwise and streamwise components of the fluid torque. (a) Time series of 
$T^{x+}_f(t)$ (black) and 
$T^{z+}_f(t)$ (colour) for the uncoupled case (top) and the coupled case (bottom). (b) Mean-square values and covariance as a function of the disc diameter. The blue series indicate the data from the coupled simulations and the red series indicate the data from the uncoupled simulations, discussed in the supplementary material S3.
The standardized histogram plots for 
$W$ and 
$T_f$ are shown in figure 10 and compared with the standard normal distribution. The disc-tip velocity 
$W$ fits the normal distribution well for all the diameters. The 
$T_f$ distribution is instead wider than the normal distribution at the tails for the small diameters. The graph of 
$T_f$ for 
$D^+=50$ has heavy tails, with an estimated value of the standardized fourth moment larger than four. This disc size is smaller than the spanwise integral length scale of the wall-shear stress, about 
$100 \delta _\nu ^*$, and thus a possible explanation for the tail behaviour is that the torque fluctuations depend on localized instantaneous extreme events, impacting on the positive tails of the wall shear-stress distribution (Hu, Morfey & Sandham Reference Hu, Morfey and Sandham2006). For larger discs, the extreme shear-stress events are likely to play a more marginal role because they are averaged over a larger wall region. Although the wall turbulence is non-Gaussian (Davidson Reference Davidson2004), the very good agreement between the 
$W$ distribution and the normal distribution is likely to be due to the filtering of the turbulence signal given by the disc dynamics equation (3.1). The extreme rare bursting and sweeping events that characterize the wall-shear stress and cause the heavy tails of the torque distribution are too short-lived to accelerate the disc so as to generate corresponding extreme values of disc-tip velocity 
$W$. It would be interesting to verify whether the 
$W$ distribution deviates from the normal one if turbulent flows at larger Reynolds numbers or lighter discs are considered.
Standardized histogram plots of (a) 
$W$ and (b) 
$T_f$ for coupled-dynamics simulations. The solid line denotes the standard normal distribution.
The power spectral densities for the disc-tip velocity and the fluid torque, shown in figure 11, reveal which temporal scales of the wall turbulence are selected by the disc dynamics. Normalizing the power spectral density graphs with the total power does not result in a collapse of the curves for all diameters. An excellent overlap for 
$W$ and 
$T_f$ is observed for the cases with the four largest diameters, for 
$D^+=800 - 1200$, at frequencies that are larger than that corresponding to the maximum energy. For 
$D^+=50 - 400$, most of the energy is contained in the higher frequencies. As the diameter increases, the spectrum shifts to lower frequencies. For these diameters, no distinct peak is found for 
$W$ in the range 
$D^+=50 - 400$ as the energy spreads over a relatively broad high-frequency range. The same discussion holds for the 
$T_f$ power spectral densities in the range 
$f^+=0.01 - 0.03$, but for 
$D^+=200$ and 
$400$ two distinct maxima are present. For moderate diameters, in the range 
$D^+=600 - 800$, a narrow peak occurs for 
$W$ and 
$T_f$ at around 
$f^+=0.005$, which corresponds to a period of about 
$200 \nu ^*/u_\tau ^{*2}$. As the diameter increases, the spectral 
$W$ peak disappears and its energy is distributed more evenly at frequencies 
$f^+$ ¡ 0.004. The torque 
$T_f$ instead maintains the peak at about 
$f^+=0.005$.
Power spectral densities, denoted by 
$S$ with subscripts indicating the quantity and shown in premultiplied form. (a) Disc-tip velocity 
$W$ and (b) fluid torque 
$T_f$. The plots are normalized by the total power.
5. Results for the half discs
The main difference between the full discs and the half discs is the finite average velocity in the half-disc case due to the driving action of the mean turbulent flow. Once the new fully developed regime is established, the disc inertia can be neglected because the half-disc 
$\bar {W}$ is determined by the steady-state balance between the fluid torque and the frictional housing torques 
$\bar {T}_f(\bar {W}) = \bar {T}_h(\bar {W}) + T_b$. In analogy with studies on drag reduction by hydrophobic walls where a local slip velocity is used (Min & Kim Reference Min and Kim2004; Busse & Sandham Reference Busse and Sandham2012), we define the average streamwise slip velocity as 
$U_{s,d}=\langle \bar {u} \rangle _d$, where the subscript 
$d$ denotes the spatial average over the moving disc surface only. The slip velocity is related to 
$\bar {W}$, i.e. 
$U_{s,d} = 4 \bar {W} / 3 {\rm \pi}$. No averaged spanwise slip velocity occurs because the discs are rigid, that is, the left-oriented spanwise velocity distribution downstream of the disc centre is antisymmetric to the right-oriented spanwise velocity distribution upstream of the disc centre. The magnitude of the average disc velocity of each half is equal to half of 
$U_{s,d}$.
5.1. Turbulent drag reduction
5.1.1. Performance and spatial distribution
As shown in table 4, cases HD1 and HD2 return small drag-reduction margins of around 
$\mathcal {R}=2\,\%$, which become almost 
$\mathcal {R}_d=5\,\%$ when the half-disc surface is considered. The disc-tip velocities and the drag-reduction margins are the same for these cases, although the disc diameters are different, suggesting a direct link between the slip velocity and the drag reduction, as for flows over hydrophobic surfaces (Min & Kim Reference Min and Kim2004). The best performing cases are HD4 and HD5, with 
$\mathcal {R}_d=20\,\%$ and 
$\mathcal {R}_d=14\,\%$, respectively.
Results from the half-disc simulations.
Figure 12 shows the drag-reduction margin as a function of the slip velocity, including the experimental data of Reference Koch and KozulovicKK13. Reference Koch and KozulovicKK13 estimated the drag-reduction level by using an empirical formula based on a correlation of the skin-friction coefficient in the fixed-wall case, as discussed in the supplementary material S1. We use this empirical formula to estimate 
$\mathcal {R}_d$ as a function of 
$U_s^+$ by numerically integrating the relation (S1.3) for our Reynolds number 
$Re_\tau =180$. As Reference Koch and KozulovicKK13 did not measure the wall-shear stress on the disc or in its proximity, no experimental data exist for a precise quantitative comparison. Nevertheless, it is useful to compare our numerical results with the predictions given by the formula of Reference Koch and KozulovicKK13. In both the experiment and the simulations, 
$\mathcal {R}_d$ is positively correlated to 
$U^+_{s,d}$, although the estimated values are lower than those obtained by direct numerical simulation. These differences may be due to different housing and bearing frictions, Reynolds numbers and geometry, i.e. a boundary-layer flow in the experiments and a channel flow in our case. Another cause of discrepancy can be the attenuation of the near-wall turbulence intensity caused by the streamwise slip, which is not accounted for by the model of Reference Koch and KozulovicKK13.
Drag-reduction margins 
$\mathcal {R}_d$ and 
$\mathcal {R}$ as functions of the slip velocity 
$U_{s,d}^+$. The simulations refer to cases HD1–HD5 in table 4 and the boundary-layer experimental data are from Reference Koch and KozulovicKK13.
As 
$U^+_{s,d}$ increases, 
$\mathcal {R}$ first increases linearly and then levels off at higher disc velocities. On the stationary wall region (amounting to 
$63\,\%$ of the total wall area) neither drag reduction nor drag increase is measured for 
$U_{s,d}^+\approx 1$ (case HD5), while a 
$3\,\%$ drag increase is observed for 
$U_{s,d}^+\approx 2$ (case HD4). Although 
$\mathcal {R}_d$ on the rotating half disc increases by 
$6\,\%$ from HD5 to HD4, the non-negligible drag increase on the rest of the wall surface causes 
$\mathcal {R}$ to only grow by about 
$0.6\,\%$.
Figure 13(a) reports 
$\mathcal {R}_{xz}$, the spatial distribution of the drag-reduction margin, for the best-performing experiment of Reference Koch and KozulovicKK13, obtained by using the empirical correlation (S1.1). The estimated wall-shear-stress reduction is streamwise homogeneous and proportional to the local streamwise velocity, increasing linearly with 
$z$. The prediction of Reference Koch and KozulovicKK13 thus fails to capture the streamwise dependence because the assumption is that the drag reduction is solely caused by the streamwise wall slip induced by the spinning disc.
Spatial distribution of the drag-reduction margin 
$\mathcal {R}_{xz}$ on a single, ensemble-averaged flow unit. (a) Estimation from experimental data of Reference Koch and KozulovicKK13 (fastest rotating case, the region where no data are available is in white), (b) case HD5 and (c) case HD4.
The maps of 
$\mathcal {R}_{xz}$ for the numerical cases HD4 and HD5, shown in figure 13(b,c), reveal a non-homogeneous spatial distribution along both 
$x$ and 
$z$. The drag-reduction margin increases moving away from the centreline as predicted by the model of Reference Koch and KozulovicKK13 because the local streamwise slip velocity grows as the right-hand side of the disc is approached. The maximum wall-shear-stress reduction occurs over the lower part of the rotating disc, while the maximum drag increase occurs in a region that is narrow in the spanwise direction, near the centreline, and upstream of the disc centre. This increase of wall friction might be an effect of the modelled discontinuity at the centreline. As the slip velocity doubles from 
$U_{s,d}^+=1$ for HD5 to 
$U_{s,d}^+=2$ for HD4, the drag-increase region on the disc surface and near the centreline becomes wider and more intense, while the drag-reduction margin also grows, especially near the disc tip. The improvement of drag reduction as the disc spins faster is more significant than the increase of drag. These two contrasting effects are the reason why the drag-reduction margin over the disc surface does not double as does the disc-tip velocity, as shown in figure 12. In the HD5 case with low 
$U_{s,d}^+$, the wall-shear stress is almost unchanged over the fixed portion of the surface. In the faster HD4 case, a narrow region of drag reduction is instead observed close to the centreline, while a thin area of increased drag is observed around the circumference and immediately downstream of the disc, where the coupled-dynamics boundary conditions change to fixed-wall no-slip. This increased wall-shear stress is the reason for the enhanced drag over the stationary portion of the wall, which balances the drag reduction over the disc. No drag reduction is found downstream of the disc, as claimed by Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14.
5.1.2. Analogies with other drag-reduction techniques
An interesting case for comparison is the oil-channel experimental campaign of Bechert, Hage & Brusek (Reference Bechert, Hage and Brusek1996). In these experiments, a section of the channel wall was replaced with a tensioned steel belt, free to move in the streamwise direction under the action of the wall-shear stress. By recording the shear force on the passive wall section and the speed of the belt, Bechert et al. (Reference Bechert, Hage and Brusek1996) related the wall slip velocity to the drag reduction. Slip velocities in the range 
$U_s \approx 0.05 - 0.1$ resulted in drag reductions of between 5 % and 10 %. Our simulations achieve higher 
$\mathcal {R}$ for a similar slip velocity (i.e. for case HD5 with 
$U_{s,d}=0.05$, 
$\mathcal {R}_d=14\,\%$, while for case HD4 with 
$U_{s,d}=0.08$, 
$\mathcal {R}_d=20\,\%$). The differences are likely due to the higher Reynolds numbers in the experiments and to the experimental challenges associated with ensuring that the moving belt remained taut and horizontal.
It is also useful to compare the disc results with those produced by hydrophobic surfaces. A relationship exists between the hydrophobic-wall 
$U_s^+$ and the drag-reduction level (Rastegari & Akhavan Reference Rastegari and Akhavan2018). By using the data of Min & Kim (Reference Min and Kim2004), 
$U_s^+=0.6$ leads to 
$\mathcal {R}=7.8\,\%$ and 
$U_s^+=1$ gives 
$\mathcal {R}=12.4\,\%$. These hydrophobic-wall values pertain to the whole wall, but they must be compared with our 
$\mathcal {R}_d$ values because in our case the slip is confined to the disc surface. The hydrophobic-wall 
$\mathcal {R}$ values are in good agreement with our 
$\mathcal {R}_d$ values in table 4. Rastegari & Akhavan (Reference Rastegari and Akhavan2018) also used a formula, their (5.3), that relates the drag-reduction level to the shift of the quantity 
$B$, the intercept of logarithmic-law formula for the mean velocity profile when scaled using the drag-reducing friction velocity, due to the change of wall-shear stress. By using the shifts 
$\varDelta B$ for our cases HD4 and HD5 in their formula, we obtain 
$\mathcal {R}=5.6\,\%$ for the HD4 case and 
$\mathcal {R}=5.1\,\%$ for the HD5 case, in perfect agreement with our numerical values in table 4.
The spanwise disc velocity varies linearly from its negative minimum at the upstream disc edge to its positive maximum at the downstream disc edge. A similar spanwise-wall velocity distribution was used by Viotti, Quadrio & Luchini (Reference Viotti, Quadrio and Luchini2009), where the wall forcing was spanwise only, sinusoidal and distributed over the whole wall. Large drag-reduction margins exceeding 40 % were obtained. It is therefore reasonable to compare the passive disc technique with Viotti et al.'s (Reference Viotti, Quadrio and Luchini2009) case and ask whether the spanwise disc velocity affects the drag-reduction performance and whether it is the reason behind the drag reduction computed via direct numerical simulation being larger than that obtained by the streamwise-only Reference Koch and KozulovicKK13 model, as shown in figure 12. To address this question, two further numerical cases are considered by modifying case HD4. In one case, the disc streamwise velocity is assigned the value corresponding to the velocity arising passively in case HD4, while the spanwise velocity is artificially set to zero. In the other case, the spanwise velocity is instead prescribed and the streamwise velocity is null. The results in table 5 show that, in terms of the overall drag-reducing performance, the streamwise-only case leads to larger drag reduction than the HD4 case, both over the disc surface and by averaging over the entire wall. The purely spanwise forcing instead results in very small drag reduction on the disc surface and drag increase over the entire wall. One reason for the small drag reduction in the spanwise-only case is that only part of the wall is forced along the spanwise direction, whereas, in Viotti et al.'s (Reference Viotti, Quadrio and Luchini2009) case, the wall velocity is spanwise homogeneous. Another reason is the low value of spanwise wall velocity, which is not large enough to generate a sufficiently thick spanwise viscous layer, required to alter the wall turbulence effectively and to lead to drag reduction (Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Quadrio & Ricco Reference Quadrio and Ricco2011). The analysis of the flow statistics of these two cases, found in § 5.3, further elucidates the dynamics of the spanwise-only case. We conclude that the two forcing effects are not additive, the streamwise disc slip is the main responsible cause for drag reduction and the spanwise wall velocity only plays a marginal role in the overall drag-reduction dynamics. We note that the role of the spanwise wall velocity in the half-disc technique is thus different from that in hydrophobic surfaces, where a spanwise slip is always detrimental to the overall drag-reduction effect.
Statistics of the single-component simulations and of case HD4.
5.1.3. Water or air as working fluid
As discussed in § 3, it would be better to use water instead of air as working fluid because of the larger dynamic viscosity of water, which would cause larger wall-shear stresses on the disc surface, slip velocities and drag-reduction margins. Nevertheless, it is of interest to compare the behaviour of the half discs in the two fluids. To conduct this comparison, it must be decided which parameters are kept constant. We assume that the same discs are used (same diameter, thickness and material), the half-height of the channel 
$h^*$ is the same and the flows are at the same 
$Re_p$.
The ratio of the angular accelerations for the water and air flows can be estimated by using (3.1) in dimensional form, neglecting the friction losses:
\begin{equation} \frac{\mathrm{d}W^*/\mathrm{d}t^*|_{{water}}}{\mathrm{d}W^*/\mathrm{d}t^*|_{{air}}} = \frac{\nu^*|_{{water}} \ \mu^*|_{{water}}}{\nu^*|_{{air}} \ \mu^*|_{{air}}} \approx 3, \end{equation}
where 
$\mu ^*$ is the dynamic viscosity.
For the full discs, once the oscillatory regime has established, the different angular accelerations would not impact on the flow dynamics because the typical disc tip velocity would be smaller than the friction velocity in both air or water and drag reduction would not occur in either case. The different accelerations would instead impact on the transient dynamics of the half discs from the stationary condition to the drag-reduction regime. The half discs in water would accelerate at about three times the angular velocity of the half discs in air and it would thus take less time in water for the half discs to reach the fully developed conditions. Once the fully developed condition of the half discs is established, the averaged angular acceleration is zero because the fluid torque is balanced by the friction torque below the surface of the disc. By estimating the fluid torque as 
$T_f^* \sim \tau _w^* R^{*3}$, the friction torque as 
$T_h^* \sim W^* R^*$ and by extracting the wall-shear stress from the friction Reynolds number, it follows that
\begin{equation} \frac{W^*|_{{water}}}{W^*|_{{air}}} = \frac{\nu^*|_{{water}} \ \mu^*|_{{water}}}{\nu^*|_{{air}} \ \mu^*|_{{air}}} \approx 3, \end{equation}in agreement with (5.1) for the acceleration. In viscous units, the ratio becomes
\begin{equation} \frac{W^+|_{{water}}}{W^+|_{{air}}} = \frac{\mu^*|_{{water}}}{\mu^*|_{{air}}} \approx 50. \end{equation}It follows from figure 12 that, for the same conditions discussed earlier, the drag reduction in air would be almost negligible compared with the drag reduction in water.
5.2. Flow visualizations
Figure 14 shows instantaneous flow visualizations of the fine-scale vortical structures using the 
$\lambda _2$ invariant method (Jeong & Hussain Reference Jeong and Hussain1995). A persistent reduction in the size and number of the vortices occurs along the streamwise direction above the spinning discs, in contrast with the stationary-wall part that is visually indistinguishable from the reference fixed-wall flow. The effect is more pronounced for case HD4 as the slip velocity 
$U_{s,d}$ is double that of case HD5 and the drag-reduction level is larger. An analogous reduction of the number of vortices was observed by Olivucci et al. (Reference Olivucci, Ricco and Aghdam2019) for flow over actively spinning rings.
Instantaneous iso-surfaces of the invariant 
$\lambda _2^+=-0.009$ (Jeong & Hussain Reference Jeong and Hussain1995), coloured by the streamwise velocity component. (a) Case HD5 and (b) case HD4.
5.3. Flow statistics
Figure 15 depicts the flow statistics computed by averaging over the entire wall surface. The mean flow is affected in the viscous sublayer, where the gradient is reduced. The peaks of the Reynolds stresses and the root-mean-square of the streamwise velocity are less intense, while the statistics of the wall-normal and spanwise velocities are unvaried.
Wall-normal profiles of flow statistics for case HD4 (red solid line), case HD5 (green solid line) and the reference channel flow (
$\bullet$), averaged on the whole wall. The variables are scaled with the wall units of the reference channel flow.
Figure 16 shows the wall-normal profiles of time- and ensemble-averaged statistical quantities at different locations in the proximity of and on the half-disc surface. The streamwise mean velocity 
$[\bar {u}]_s^+$ is unaffected on the left fixed-wall part and immediately downstream of the disc. It is instead altered significantly up to 
$y^+=5$ in the middle of the disc surface, where the streamwise slip velocity has the direct effect of reducing the wall-shear stress. In the buffer layer and above, the flow is instead almost unaltered compared with the fixed-wall case. This result gives support to the main assumption used by Reference Koch and KozulovicKK13 and Reference Koch and KozulovicKK14 for estimating the drag reduction, discussed in the supplementary material S1, i.e. the bulk mean flow above the disc coincides with the reference case, so that the boundary-layer thickness remains constant.
Wall-normal profiles of several flow statistics at four different positions on the wall: (a–d) the period-averaged streamwise velocity 
$[\bar {u}]_s^+$, (e–h) the turbulent Reynolds stress 
$[\overline {u_t v_t}]_s^+$, (i–l) the root-mean-square of the streamwise velocity, (m–p) the root-mean-square of the wall-normal velocity and (q–t) the root-mean-square of the spanwise velocity. The positions are identified by blue dots in the sketches. The statistics of case HD4 (red solid line) are compared with the reference channel-flow statistics (
$\bullet$). The variables are scaled with the wall units of the reference channel flow.
On the left part, the statistics are largely unaffected, with the streamwise-velocity root-mean-square decreasing slightly for 
$y^+>5$ and the Reynolds stresses 
$[\overline {u_t v_t}]_s^+$ being indistinguishable from the reference case. The wall-normal profiles of all the fluctuating velocities are attenuated in the same way up to 
$y^+=80$ on and downstream of the disc surface. This behaviour chimes with the flow response to discs and rings spinning at a constant angular velocity, studied by Olivucci et al. (Reference Olivucci, Ricco and Aghdam2019). They showed that the flow statistics are largely independent of the streamwise location, i.e. the attenuation of the wall turbulence brought about by the disc rotation persists downstream even where the portion of the wall is stationary. Both 
$[\overline {u_t v_t}]_s^+$ and 
$[\overline {u_t u_t}]_s^+$ decrease significantly, shifting upward and reducing by a maximum of 
$30\,\%$.
Figure 17 reports the flow statistics of the cases with streamwise-only and spanwise-only wall forcing, discussed in § 5.1.2. The statistics are computed on the half disc along a line parallel to the streamwise axis of symmetry of the disc. The line is positioned close to the axis of symmetry because in that region the spanwise motion amplitude is relatively large. The mean velocity profiles of the streamwise-only case and case HD4 are very close, which is expected given that the mean slip velocities are similar. The local peak intensity of the Reynolds stresses is more weakened by the full case HD4 than either one of the streamwise-only or spanwise-only cases. The latter is, however, more efficient (at least locally) at pushing the peak of the stresses away from the wall and is also locally more efficient at weakening all three turbulence intensity components, as shown in figure 17(i–t). The large reductions of turbulence quantities given by the spanwise-only case along this streamwise line do not explain the small drag-reduction margin reported in table 5 when the whole disc surface is considered. We thus resort to the spanwise distributions of the local skin-friction coefficient, based on our version of the Fukagata–Iwamoto–Kasagi identity developed for spinning-disc flows (Olivucci et al. Reference Olivucci, Ricco and Aghdam2019). The budget equation is
\begin{equation} \langle c_f \rangle_x (z_s) = C^{lam}_f + \langle c^d_f \rangle_x (z_s) + \langle c^t_f \rangle_x (z_s), \end{equation}
where 
$C^{lam}_f$ is the laminar skin-friction coefficient, 
$c^d_f$ and 
$c^d_f$ are the local contributions to 
$C_f$ from 
$u_t v_t$ and 
$u_d v_d$, respectively, and the subscript 
$x$ denotes averaging along the streamwise direction. Averaging (5.4) along 
$z$ gives the budget identity for the global skin-friction coefficient 
$C_f = C^{lam}_f + C^{d}_f + C^{t}_f$. The terms of (5.4) are depicted in figure 18. In the spanwise-only case, the wall-shear stress is reduced on the disc near the centreline, but it also increases as the right disc edge is approached (blue line in figure 18c). These two effects balance each other, causing the low drag reduction. The detrimental contribution of 
$c_f^d$ is absent when the spanwise component is eliminated, as the flat red curve in figure 18(b) indicates.
Wall-normal profiles of several period-averaged wall-normal statistics at different positions on the half disc. The locations are shown in the sketches. Case HD4 statistics (red solid line) are compared to the reference channel-flow (
$\bullet$) and to the streamwise-only (blue solid line) and spanwise-only (yellow solid line) test cases. The variables are scaled with the wall units of the reference channel.
Distribution of the local skin-friction coefficients according to (5.4). (a) Case HD4, (b) streamwise only and (c) spanwise only. Budget terms 
$c_f^d$ (red solid line), 
$c_f^t$ (blue solid line), 
$c_f^{lam}$ (green solid line) and reference channel 
$c_{f,0}^t$ (dashed line).
5.4. Power consumed by the moving discs
Use of passive discs can be classified as a passive-absorbing drag-reduction technique, following the definition first proposed by Aghdam & Ricco (Reference Aghdam and Ricco2016), who considered the channel as the control volume for the energy balance. Passive-absorbing methods do not require external power to cause drag reduction, but they consume power as the wall turbulence interacts with the wall, which is fitted with half discs in our case. Another example of a passive-absorbing method is compliant surfaces that deform under the action of wall turbulence. The other group of passive techniques is called passive-neutral methods, such as riblets (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011) or dimples (van Nesselrooij et al. Reference van Nesselrooij, Veldhuis, van Oudheusden and Schrijer2016) that instead do not transfer power from the wall turbulence to the exterior of the fluid control volume because they are rigid and stationary. Passive-neutral devices nonetheless involve a power penalty within the fluid control volume with respect to the reference case, caused by a larger wetted area for riblets and dimples, or detrimental pressure drag for dimples, wavy walls (Ghebali, Chernyshenko & Leschziner Reference Ghebali, Chernyshenko and Leschziner2017) and baffles (Marensi et al. Reference Marensi, Ding, Willis and Kerswell2020).
In the half-disc case, the power extracted from the wall turbulence by the discs is consumed through friction effects below the disc surface. The power 
$\mathcal {P}_x=C_f U_b^3 D^2$ required to pump the fluid in a single-disc flow unit along the streamwise direction satisfies the balance
\begin{equation} \mathcal{P}_x = \mathcal{P}_\varepsilon + \mathcal{P}_f, \end{equation}
which follows from the integration of the mean kinetic energy equation on the flow unit volume (Hinze Reference Hinze1975). The power-budget terms are visualized in figure 19 by arrows that represent the power transfer direction. The power 
$\mathcal {P}_\varepsilon$ represents the total dissipation in the fluid, i.e. the volume integral of the pointwise viscous dissipation rate. The power 
$\mathcal {P}_f$ is the power outflow to the moving discs, calculated by integrating the pointwise power on the wall surface, which is equivalent to 
$\mathcal {P}_f = \bar {T}_f \varOmega$ in accordance with (3.3), where 
$\varOmega$ is the disc angular velocity. In the fixed-wall case, the power budget is 
$\mathcal {P}_{x,un}=\mathcal {P}_{\varepsilon ,un}$. Since the disc rotates steadily, 
$\mathcal {P}_f = \mathcal {P}_h = (\bar {T}_h +T_b) \varOmega$, i.e. the power extracted by the disc from the wall turbulence is all dissipated by the frictional losses occurring in the disc housing. In the best drag-reduction case HD4, 
$\mathcal {P}_\varepsilon$ amounts to 
$99\,\%$ of 
$\mathcal {P}_x$, so 
$\mathcal {P}_f$ accounts for the remaining 
$1\,\%$.
Schematic representation of the mean power budget for the disc–fluid system. The size of the arrows is not proportional to the power amount.
6. Conclusions
Turbulent channel flows over discs that rotate passively under the action of wall-shear stress have been studied via direct numerical simulations. The dynamics of the discs is driven by the instantaneous torque produced by the overriding wall turbulence and the NSE are solved with modified boundary conditions that synthesize the motion of the discs. They are therefore two-way-coupling simulations of the equations of motion of the discs and the fluid. The disc dynamics is driven actively by the fluid torque and thwarted by two resisting torques, i.e. the cavity torque that depends on the flow regime occurring in the cavity and the torque given by the ball bearing that supports a disc.
We have considered discs of different diameters that are either fully exposed to the wall turbulence or partially covered by the wall. In the full-disc cases, the root-mean-square of the amplitude of the disc velocity is always smaller than the wall-friction velocity. The wall displacement is so small that the minimal forcing conditions needed to alter the wall turbulence are not met and no drag reduction occurs. An optimal diameter of 
$D^+=100$, matching the characteristic spanwise spacing of the near-wall low-speed streaks, generates the largest disc displacement and velocities. A reduced-order model, discussed in the supplementary material S3, has helped clarify the spectral filtering of the wall-turbulence signal to the disc motion. In the fastest half-disc case, drag-reduction levels up to 
$20\,\%$ occur over the disc surface and up to 
$5.6\,\%$ when the entire channel walls are considered. We also computed the power extracted by the discs, which is dissipated by the friction effects below the disc surface. The only experimental results on passively rotating half discs were analysed (Koch & Kozulovic Reference Koch and Kozulovic2013, Reference Koch and Kozulovic2014). The drag-reduction margin was not measured in those experiments, but estimated to be 
$17$% through a fixed-wall skin-friction correlation. It was therefore not possible to carry out quantitative comparisons. Nevertheless, the experimental trend of drag reduction as a growing function of an equivalent slip velocity matches our numerical data qualitatively and the angular velocities are of comparable amplitude. Further work is needed to investigate the dependence of the disc-slip velocity and the drag-reduction margin on the Reynolds number.
Supplementary material
Supplementary material is available at https://doi.org/10.1017/jfm.2021.533.
Acknowledgements
The authors would like to thank the reviewers for useful comments, Dr M. Rütten at DLR in Göttingen for providing the technical specifications of his disc-fitted boundary-layer experimental set-up, Professor R. Dwyer-Joyce at the University of Sheffield for useful discussions on the bearing torque, Professor N. Qin also at the University of Sheffield for pointing out the experimental study of Koch and Kozulovic to us and Professor K. Taira at University of California, Los Angeles for interesting discussions about the disc response dynamics.
Funding
The authors acknowledge EPSRC grant no. EP/L000261/1 and the UK Turbulence Consortium for providing computer time. P.O. and D.W. thank the Department of Mechanical Engineering at the University of Sheffield for providing financial support, and P.O. thanks the Gordon Franklin Scholarship for sponsoring his PhD studies. P.R. was partially funded by the H2020 EU ‘DRAGY’ project.
Declaration of interests
The authors report no conflict of interest.
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