2 Numerical methods and simulation parameters

The numerical simulations reported in this study were all performed using standard D3Q19 (3-Dimensional, 19 discrete velocity), single relaxation time lattice Boltzmann methods (Succi Reference Succi2001) with grid embedding (Lagrava et al. Reference Lagrava, Malaspinas, Latt and Chopard2012), of grid ratio 4:1, employed in the near-wall region, between the domain boundaries and the buffer layer ( $z^{+0}\approx 30$ ), to improve the accuracy of the computations near the micro-textured walls. The resulting grid spacings were $\unicode[STIX]{x1D6E5}_{f}^{+0}\approx 0.5$ for $z^{+0}\lesssim 30$ , and $\unicode[STIX]{x1D6E5}_{c}^{+0}\approx 2$ for $z^{+0}\gtrsim 30$ , in all three directions, as shown in figure 1. In turbulent channel flows with smooth no-slip walls, the lattice Boltzmann method can fairly accurately predict the turbulent flow features with uniform grid spacings of $\unicode[STIX]{x1D6E5}^{+0}\approx 2$ throughout the channel, as shown by Lammers et al. (Reference Lammers, Beronov, Volkert, Brenner and Durst2006) and discussed in appendix A, § A.2. With micro-textured surfaces, however, the presence of a slip velocity over the surface micro-indentations leads to steep wall-normal gradients of the streamwise velocity on the no-slip tips of the surface micro-texture, as well as strong shear layers between the slip and no-slip regions at the tip of the surface micro-texture. These shear layers persist up to distances of the order of $z\sim g$ , but they become weaker with increasing distance from the wall, such that by $z^{+0}\gtrsim 30$ they can be properly resolved by the regular lattice Boltzmann grid of size $\unicode[STIX]{x1D6E5}^{+0}\approx 2$ , as shown in appendix A, § A.2. The grid-embedding scheme in the present study was employed to resolve these shear layers in the region $z^{+0}\lesssim 30$ , as well as the steep wall-normal gradients of the velocity near the tips of the wall micro-texture.

The liquid/gas interfaces in the SH longitudinal microgrooves were modelled as stationary, curved, shear-free boundaries, with the shape of the meniscus determined from an analytical solution (de Gennes, Brochard-Wyart & Quéré Reference de Gennes, Brochard-Wyart and Quéré2002) of the Young–Laplace equation

(2.1) $$\begin{eqnarray}\unicode[STIX]{x0394}\tilde{P}=-\frac{1}{We_{\unicode[STIX]{x1D70F}_{0}}}\tilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\boldsymbol{n},\end{eqnarray}$$

where $\unicode[STIX]{x0394}\tilde{P}\equiv \unicode[STIX]{x0394}P/\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}_{0}}^{2}$ is the non-dimensional Laplace pressure across the interface, $We_{\unicode[STIX]{x1D70F}_{0}}\equiv \unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}_{0}}^{2}g/\unicode[STIX]{x1D70E}$ is the Weber number, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D70E}$ is the surface tension, $\boldsymbol{n}$ is the unit normal to the interface and $\tilde{\unicode[STIX]{x1D735}}$ denotes the divergence operator in a non-dimensional local coordinate system, $(\tilde{\unicode[STIX]{x1D709}},\tilde{\unicode[STIX]{x1D702}},\tilde{\unicode[STIX]{x1D701}})=(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702},\unicode[STIX]{x1D701})/g$ , centred on the longitudinal microgroove, as shown in figure 1. Integrating (2.1), subject to the boundary conditions $\tilde{F}|_{(\tilde{\unicode[STIX]{x1D702}}=\pm 1/2)}=0$ and $(\text{d}\tilde{F}/\text{d}\tilde{\unicode[STIX]{x1D702}})|_{(\tilde{\unicode[STIX]{x1D702}}=0)}=0$ , gives the shape of the interface as

(2.2) $$\begin{eqnarray}\tilde{F}(\tilde{\unicode[STIX]{x1D702}},\tilde{\unicode[STIX]{x1D705}})=-\frac{1}{2\tilde{\unicode[STIX]{x1D705}}}\left(2\sqrt{1-\tilde{\unicode[STIX]{x1D705}}^{2}\tilde{\unicode[STIX]{x1D702}}^{2}}-\sqrt{4-\tilde{\unicode[STIX]{x1D705}}^{2}}\right),\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D705}}=(\unicode[STIX]{x0394}\tilde{P}\cdot We_{\unicode[STIX]{x1D70F}_{0}})$ is the non-dimensional curvature of the interface, which is related to the interface protrusion angle through

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D703}=\tan ^{-1}\left(\frac{\tilde{\unicode[STIX]{x1D705}}}{2\sqrt{1-(\tilde{\unicode[STIX]{x1D705}}/2)^{2}}}\right).\end{eqnarray}$$

Equations (2.2) and (2.3) were used to determine the shape of the interface in the SH longitudinal microgrooves in the present study. Static protrusion angles of $\unicode[STIX]{x1D703}=0^{\circ }$ , $-30^{\circ }$ , $-60^{\circ }$ and $-90^{\circ }$ were investigated, covering the full range of protrusion angles from flat interfaces to maximum negative protrusion angle. For comparison, the same geometries as those formed by the menisci of the SH longitudinal microgrooves were also simulated as riblets, with the free-slip boundary condition on the curved interfaces replaced with the no-slip boundary condition.

The boundary conditions on flat no-slip surfaces and flat free-slip interfaces were imposed using half-way bounce back and specular reflection methods (Succi Reference Succi2001), respectively. The boundary conditions on the curved riblet walls were imposed using the so-called ‘central linear interpolation’ (CLI) scheme (Ginzburg, Verhaeghe & d’Humières Reference Ginzburg, Verhaeghe and d’Humières2008). This is a modified version of the half-way bounce back scheme, in which interpolations between pre- and post-collision distribution functions are used to find the unknown distributions at the boundaries while accounting for the full curvature of the wall. The boundary conditions on the curved SH interfaces were imposed using local specular reflection (Ginzburg & Steiner Reference Ginzburg and Steiner2003). In this method, specular reflection is applied locally on the distribution functions in the vicinity of a stepwise realization of the curved interface, to reconstruct the unknown distributions at the boundaries. A number of verification studies were performed to ensure the accuracy of the numerical methods, the adequacy of the grid resolution, and the effect of domain size. These are described in appendix A.

All the simulations reported in the present study were performed in periodic channels of size $5h\times 2.5h\times 2h$ in the streamwise ( $x$ ), spanwise ( $y$ ), and wall-normal ( $z$ ) directions, respectively, with SH longitudinal microgrooves or riblets on both walls, as shown in figure 1. For two cases, the simulations were repeated in channels of size $20h\times 10h\times 2h$ , as discussed in appendix A, § A.3, to demonstrate the adequacy of the smaller domain size. A constant flow rate was maintained in the channels during the course of all the simulations, corresponding to bulk Reynolds numbers of $Re_{b}\equiv q/2\unicode[STIX]{x1D708}=3600$ or $Re_{b}=7860$ , where $q$ denotes the flow rate per unit spanwise width in the channel. A total of 12 simulations with SH longitudinal microgrooves and 9 simulations with riblets were performed at $Re_{b}=3600$ , corresponding to three longitudinal microgrooves of size $g^{+0}\approx 14,28,56$ and $g/w=7$ , each studied at four protrusion angles of $\unicode[STIX]{x1D703}=0^{\circ },-30^{\circ },-60^{\circ },-90^{\circ }$ as SH longitudinal microgrooves, and three protrusion angles of $\unicode[STIX]{x1D703}=-30^{\circ },-60^{\circ },-90^{\circ }$ as riblets. In addition, the simulations with SH longitudinal microgrooves of size $g^{+0}\approx 14,28$ and $g/w=7$ at interface protrusion angle of $\unicode[STIX]{x1D703}=-30^{\circ }$ were repeated at $Re_{b}=7860$ to assess the effect of the bulk Reynolds number on the drag reduction characteristics. For reference, DNS was also performed in ‘base’ turbulent channel flows with smooth, no-slip walls, at the same bulk Reynolds number as those in the channels with micro-textured walls. The cross-sectional area, $A_{0}$ , in the ‘base’ turbulent channel flow was set to the ‘nominal’ cross-sectional area of the channel with micro-textured walls, as defined by area enclosed between the tips of the micro-textures on the opposing walls of the channel. The corresponding friction Reynolds numbers in the ‘base’ turbulent channel flows were $Re_{\unicode[STIX]{x1D70F}_{0}}\equiv u_{\unicode[STIX]{x1D70F}_{0}}h/\unicode[STIX]{x1D708}\approx 222$ at $Re_{b}=3600$ , and $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ at $Re_{b}=7860$ . For comparison, all the above simulations were repeated in laminar flow at $Re_{b}=150$ . A summary of all the simulations performed in the present study is given in table 1.

Table 1. Effective wall slip velocity, $DR$ and its decomposition in laminar and turbulent channel flow with SH longitudinal microgrooves or riblets.

3 Drag reduction with micro-textured surfaces in channel flow

Drag reduction with micro-textured surfaces involves contributions from a number of different dynamical effects, some of which are drag reducing, while others are drag enhancing. To clarify the mechanism of drag reduction, these effects need to be separated, so the contribution of each element can be assessed.

For turbulent channel, pipe and plane boundary layer flows with smooth no-slip walls, Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002) derived an analytical expression for the breakdown of the skin-friction coefficient into its constitutive elements. This derivation was later generalized to more geometrically complex no-slip surfaces by Peet & Sagaut (Reference Peet and Sagaut2009). In Rastegari & Akhavan (Reference Rastegari and Akhavan2015), the formulation of Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002) was used to derive an analytical expression for the breakdown of drag reduction into its constitutive elements for laminar or turbulent channel flows with any micro-pattern of ‘flat’ SH interfaces. In this section, the formulation of Rastegari & Akhavan (Reference Rastegari and Akhavan2015) will be generalized to channel flows with any three-dimensional SH or riblet micro-texture on the walls.

For steady, fully developed, turbulent channel flow with any SH or riblet micro-texture pattern on the walls, the streamwise Reynolds-averaged momentum equation is given by

(3.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\bar{U}}{\unicode[STIX]{x2202}x}-\overline{uu}-\bar{U}\bar{U}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\bar{U}}{\unicode[STIX]{x2202}y}-\overline{uv}-\bar{U}\bar{V}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\bar{U}}{\unicode[STIX]{x2202}z}-\overline{uw}-\bar{U}\bar{W}\right)=\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x},\end{eqnarray}$$

where the overbar denotes Reynolds-averaging; $\bar{U}$ , $\bar{V}$ , $\bar{W}$ and $u$ , $v$ , $w$ are the streamwise ( $x$ ), spanwise ( $y$ ) and wall-normal ( $z$ ) components of the Reynolds-averaged mean and fluctuating velocities, respectively, and $\bar{P}$ is the Reynolds-averaged mean pressure. Equation (3.1) can also describe steady, laminar flow in a channel with any SH or riblet micro-texture pattern on the walls if $u$ , $v$ and $w$ are set equal to zero.

Averaging (3.1) over the periodic pattern of the surface micro-texture in the streamwise and spanwise directions, and integrating in the wall-normal direction from the tip of the surface micro-texture, at $z=0$ , to a height $z$ inside the channel (see figure 1), gives

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}\bar{U}}{\unicode[STIX]{x2202}z}\right\rangle -\langle \overline{uw}\rangle -\langle \bar{U}\bar{W}\rangle =-\frac{h}{\unicode[STIX]{x1D70C}}\left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle \left(1-\frac{z}{h}\right),\end{eqnarray}$$

where $\langle \,\,\rangle$ denotes averaging in the wall-parallel directions over the periodic pattern of the surface micro-texture; $2h$ is the ‘nominal’ full-height of the channel, defined as the distance between the tips of the surface micro-texture on the opposing walls of the channel; and we have used the identity $[\unicode[STIX]{x1D708}\langle \unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}z\rangle -\langle \overline{uw}\rangle -\langle \bar{U}\bar{W}\rangle ]_{z=0}=-\langle \unicode[STIX]{x2202}\overline{P}/\unicode[STIX]{x2202}x\rangle h/\unicode[STIX]{x1D70C}$ , as required by the force balance in the channel between $z=0$ and $z=2h$ .

Integration of (3.2) in the wall-normal direction, once from 0 to $z$ , and again from 0 to $h$ (Fukagata et al. Reference Fukagata, Iwamoto and Kasagi2002; Rastegari & Akhavan Reference Rastegari and Akhavan2015), gives

(3.3) $$\begin{eqnarray}-\frac{h}{\unicode[STIX]{x1D70C}U_{b_{0}}^{2}}\left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle \left(1-3\int _{0}^{1}\left(\frac{-\langle \overline{uw}\rangle -\langle \bar{U}\bar{W}\rangle }{\displaystyle -\frac{h}{\unicode[STIX]{x1D70C}}\left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle }\right)(1-\unicode[STIX]{x1D712})\,\text{d}\unicode[STIX]{x1D712}\right)=\frac{3}{Re_{b}}\left(1-\frac{Q_{g}}{Q}-\frac{U_{s}}{U_{b_{0}}}\right).\end{eqnarray}$$

Here, $U_{b_{0}}$ is the bulk velocity in a smooth, no-slip ‘base’ channel flow with the same bulk Reynolds number, $Re_{b}\equiv q/2\unicode[STIX]{x1D708}$ , hence the same flow rate per unit spanwise width, $q\equiv Q/L_{y}$ , and the same bulk flow rate, $Q$ , as the channel with micro-textured walls, and a ‘base’ channel flow cross-sectional area, $A_{0}$ , equal to the ‘nominal’ cross-sectional area of the channel with micro-textured walls, as defined by the area enclosed between the tips of the surface micro-texture on the opposing walls of the channel; $Q_{g}$ is the flow rate through the surface micro-texture indentations; $U_{s}\equiv \langle \bar{U}\rangle |_{z=0}$ is the average slip velocity at the tip of the surface micro-texture; and $\unicode[STIX]{x1D712}\equiv z/h$ is the non-dimensional wall-normal coordinate. Equation (3.3) can be rearranged as

(3.4) $$\begin{eqnarray}-\frac{h}{\unicode[STIX]{x1D70C}U_{b_{0}}^{2}}\left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle =\frac{3}{Re_{b}}\left(1-\frac{Q_{g}}{Q}-\frac{U_{s}}{U_{b_{0}}}\right)\left(\frac{1}{1-3I}\right),\end{eqnarray}$$

where $I=I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }+I_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }=[1/(-\langle \unicode[STIX]{x2202}\bar{P}/\unicode[STIX]{x2202}x\rangle h/\unicode[STIX]{x1D70C})]\int _{0}^{1}(-\langle \overline{uw}\rangle -\langle \bar{U}\bar{W}\rangle )(1-\unicode[STIX]{x1D712})\,\text{d}\unicode[STIX]{x1D712}$ denotes the contribution of the turbulent Reynolds shear stresses, $\langle \unicode[STIX]{x1D70F}_{R}\rangle \equiv -\unicode[STIX]{x1D70C}\langle \overline{uw}\rangle$ , and any mean flow shear stresses, $\langle \unicode[STIX]{x1D70F}_{c}\rangle \equiv -\unicode[STIX]{x1D70C}\langle \bar{U}\bar{W}\rangle$ , established in the channel due to the presence of the micro-texture on the walls, to the overall wall shear stress, and $I$ is bounded by $0<I<1/3$ per (3.2). For the ‘base’ channel flow with smooth no-slip walls, (3.4) reduces to

(3.5) $$\begin{eqnarray}-\frac{h}{\unicode[STIX]{x1D70C}U_{b_{0}}^{2}}\left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle _{0}=\frac{3}{Re_{b}}\left(\frac{1}{1-3I_{0}}\right),\end{eqnarray}$$

where $I_{0}=I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{0}}=[1/(-\langle \unicode[STIX]{x2202}\bar{P}/\unicode[STIX]{x2202}x\rangle _{0}h/\unicode[STIX]{x1D70C})]\int _{0}^{1}-\langle \overline{uw}\rangle _{0}(1-\unicode[STIX]{x1D712})\,\text{d}\unicode[STIX]{x1D712}$ .

Using the definition $DR\equiv (1-C_{f}/C_{f_{0}})$ , the magnitude of drag reduction can be expressed as

(3.6) $$\begin{eqnarray}DR\equiv \left(1-\frac{C_{f}}{C_{f_{0}}}\right)=\left(1-\frac{\displaystyle \left(\frac{\unicode[STIX]{x1D70F}_{w}}{\unicode[STIX]{x1D70C}(Q/A)^{2}}\right)}{\displaystyle \left(\frac{\unicode[STIX]{x1D70F}_{w_{0}}}{\unicode[STIX]{x1D70C}(Q/A_{0})^{2}}\right)}\right)=\left(1-\frac{\displaystyle \left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle }{\displaystyle \left\langle \frac{\unicode[STIX]{x2202}\bar{P}}{\unicode[STIX]{x2202}x}\right\rangle _{0}}\left(\frac{A}{A_{0}}\right)^{3}\right),\end{eqnarray}$$

where $C_{f}\equiv \unicode[STIX]{x1D70F}_{w}\,/[\unicode[STIX]{x1D70C}(Q/A)^{2}]$ and $C_{f_{0}}\equiv \unicode[STIX]{x1D70F}_{w_{0}}/[\unicode[STIX]{x1D70C}(Q/A_{0})^{2}]$ are the skin-friction coefficients, $\unicode[STIX]{x1D70F}_{w}$ and $\unicode[STIX]{x1D70F}_{w_{0}}$ are the wall shear stresses, $A$ and $A_{0}$ are the cross-sectional areas, and $(Q/A)$ and $(Q/A_{0})$ are the bulk velocities, in the channel with micro-textured walls and the ‘base’ channel flow, respectively.

Substitution of (3.4) and (3.5) into (3.6) gives

(3.7) $$\begin{eqnarray}\displaystyle DR & = & \displaystyle \left\{\left(\frac{A}{A_{0}}\right)^{3}\left(\frac{U_{s}}{U_{b_{0}}}\right)\right\}+\left\{\left(\frac{A}{A_{0}}\right)^{3}\left(1-\frac{U_{s}}{U_{b_{0}}}-\frac{Q_{g}}{Q}\right)\left(\frac{3\unicode[STIX]{x1D700}}{1-3I}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{1-\left(\frac{A}{A_{0}}\right)^{3}\left(1-\frac{Q_{g}}{Q}\right)\right\}\nonumber\\ \displaystyle & = & \displaystyle \{DR_{slip}\}+\{DR_{\unicode[STIX]{x1D700}}\}+\{DR_{Q_{g}}\},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}=(I_{0}-I)$ is a measure of the changes in the normalized structure of turbulent Reynolds shear stresses and any mean flow shear stresses developed in the channel due to the presence of the micro-texture on the walls, compared to a ‘base’ channel flow with smooth no-slip walls.

Equation (3.7) provides an exact analytical expression for the magnitude of drag reduction in laminar or turbulent channel flow with any SH or no-slip micro-texture on the walls, while providing a decomposition of drag reduction into: (i) the contributions arising from the effective slip velocity at the wall, represented by $\{DR_{slip}\}$ ; (ii) the contributions arising from modifications to the structure of turbulent Reynolds shear stresses and any mean flow shear stresses developed in the channel due to the presence of the wall micro-texture, represented by $\{DR_{\unicode[STIX]{x1D700}}\}$ ; and (iii) the contributions arising from the fraction of the flow rate through the wall micro-texture, represented by $\{DR_{Q_{g}}\}$ .

The term $\{DR_{\unicode[STIX]{x1D700}}\}$ in (3.7) involves contributions from several different dynamical effects, and can be further decomposed. To begin with, one can decompose $\unicode[STIX]{x1D700}=(I_{0}-I)=(I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{0}}-I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }-I_{\langle \unicode[STIX]{x1D70F}_{c}\rangle })$ into

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }+\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{c}\rangle },\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }=(I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{0}}-I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle })$ represents the contributions arising from changes in the normalized structure of turbulent Reynolds shear stresses, and $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }=(-I_{\langle \unicode[STIX]{x1D70F}_{c}\rangle })$ represents the contributions arising from changes in the structure of the mean flow shear stresses, due to the presence of the wall micro-texture. Furthermore, the contributions from $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }=(I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{0}}-I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle })$ can themselves be decomposed into

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }=\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}+\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}=(I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{0}}-I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}})$ and $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}=(I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}-I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle })$ , and $I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}$ is the value of the $I$ integral in a channel flow with smooth no-slip walls at bulk Reynolds number equal to $Re_{b_{e}}=(1-U_{s}/U_{b})Re_{b}$ . The decomposition (3.9) is made in consideration of the fact that the presence of an effective slip velocity, $U_{s}$ , implies that a fraction, $U_{s}/U_{b}$ , of the flow rate can flow inviscidly (Min & Kim Reference Min and Kim2004; Fukagata et al. Reference Fukagata, Kasagi and Koumoutsakos2006). This reduces the effective bulk Reynolds number of the viscous part of the flow from $Re_{b}$ to $Re_{b_{e}}=(1-U_{s}/U_{b})Re_{b}$ . As such, $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}$ represents the changes to the normalized structure of turbulent Reynolds shear stresses in a channel flow with smooth no-slip walls when the bulk velocity, $U_{b}$ , is augmented by the presence of an effective wall slip velocity, $U_{s}$ , in effect reducing the bulk Reynolds number of the viscous part of the flow from $Re_{b}$ to $Re_{b_{e}}$ , while $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}$ represents the contributions from any other modifications to the normalized structure of the turbulent Reynolds shear stresses due to the presence of the micro-texture on the walls.

The integral $I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}=[1/(-\langle \unicode[STIX]{x2202}\bar{P}/\unicode[STIX]{x2202}x\rangle _{Re_{b_{e}}}h/\unicode[STIX]{x1D70C})]\int _{0}^{1}-\langle \overline{uw}\rangle _{Re_{b_{e}}}(1-\unicode[STIX]{x1D712})\,\text{d}\unicode[STIX]{x1D712}$ , required for the evaluation of $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}$ and $\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}$ , can be computed directly from its definition, by performing additional DNS in smooth no-slip channel flows at bulk Reynolds number equal to $Re_{b_{e}}$ . Alternatively, $I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}$ can be computed by using the relation $I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}=(1/3)[1-(3/Re_{b_{e}})(U_{b_{e}}/u_{\unicode[STIX]{x1D70F}_{e}})^{2}]$ , which can be obtained from (3.5) for any smooth-wall no-slip turbulent channel flow, where $U_{b_{e}}$ and $u_{\unicode[STIX]{x1D70F}_{e}}$ represent the bulk velocity and the wall friction velocity, respectively, in a turbulent channel flow at a bulk Reynolds number of $Re_{b_{e}}$ . The ratio $(U_{b_{e}}/u_{\unicode[STIX]{x1D70F}_{e}})$ can be evaluated from DNS, or from experimental correlations such as Dean’s correlation (Dean Reference Dean1978). In the present study, the integrals $I_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}$ were evaluated by performing additional DNS at bulk Reynolds number equal to $Re_{b_{e}}$ .

With these additional decompositions of $\{DR_{\unicode[STIX]{x1D700}}\}$ , (3.7) can be written as

(3.10) $$\begin{eqnarray}\displaystyle DR & = & \displaystyle \left\{\left(\frac{A}{A_{0}}\right)^{3}\left(\frac{U_{s}}{U_{b_{0}}}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\left(\frac{A}{A_{0}}\right)^{3}\left(1-\frac{U_{s}}{U_{b_{0}}}-\frac{Q_{g}}{Q}\right)\left(\frac{3\{\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}+\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}+\unicode[STIX]{x1D700}_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }\}}{1-3I}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{1-\left(\frac{A}{A_{0}}\right)^{3}\left(1-\frac{Q_{g}}{Q}\right)\right\}\nonumber\\ \displaystyle & = & \displaystyle \{DR_{slip}\}+\{DR_{\langle \unicode[STIX]{x1D70F}_{R}\rangle }+DR_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }\}+\{DR_{Q_{g}}\}\nonumber\\ \displaystyle & = & \displaystyle \{DR_{slip}\}+\{DR_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}+DR_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}+DR_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }\}+\{DR_{Q_{g}}\}.\end{eqnarray}$$

Equation (3.10) shows that the net drag reduction in laminar or turbulent channel flow with any SH or riblet micro-texture pattern on the walls can be decomposed into (i) contributions arising from the effective slip velocity at the wall, represented by $\{DR_{slip}\}$ ; (ii) contributions arising from modifications to the normalized structure of turbulent Reynolds shear stresses due to augmentation of the bulk velocity by the effective slip velocity at the wall, represented by $\{DR_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{Re_{b_{e}}}}\}$ ; (iii) contributions arising from other modifications to the normalized structure of turbulent Reynolds shear stresses due to the presence of the micro-texture on the walls, represented by $\{DR_{\langle \unicode[STIX]{x1D70F}_{R}\rangle _{mod}}\}$ ; (iv) contributions arising from any mean flow shear stresses developed due to presence of the micro-texture on the walls, represented by $\{DR_{\langle \unicode[STIX]{x1D70F}_{c}\rangle }\}$ ; and (v) contributions arising from the fraction of the flow rate through the wall micro-texture, represented by $\{DR_{Q_{g}}\}$ .

For SH channel flows with ‘idealized’ flat liquid/gas interfaces, $(A/A_{0})=1$ and $Q_{g}=0$ , and (3.7) or (3.10) reduce to the expression (2.5) given in Rastegari & Akhavan (Reference Rastegari and Akhavan2015). Each of the terms in (3.7) or (3.10) can be evaluated using results from DNS to clarify the mechanism of drag reduction with micro-textured surfaces.

5 Extrapolation of DNS results to high Reynolds number flows of practical interest

To explore the effect of bulk Reynolds number of the flow on the dynamics reported in § 4, two of the SH simulations, corresponding to longitudinal microgrooves of size $g^{+0}\approx 14$ and $g^{+0}\approx 28$ with $\unicode[STIX]{x1D703}=-30^{\circ }$ , were repeated at $Re_{b}=7860$ . These simulations correspond to a doubling of the friction Reynolds number of the base flow from $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ at $Re_{b}=3600$ , to $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ at $Re_{b}=7860$ . The results of these higher Reynolds number simulations are also shown in table 1 and figures 2–7. The differences between the results at $Re_{b}=3600$ and $Re_{b}=7860$ are slight, confirming that the lower Reynolds number simulations properly capture the essential features of drag reduction with micro-textured surfaces in turbulent flows at higher Reynolds numbers.

By increasing the Reynolds number of the flow from $Re_{b}=3600$ to $Re_{b}=7860$ , the magnitudes of drag reduction with SH longitudinal microgrooves of size $g^{+0}\approx 14$ and $g^{+0}\approx 28$ with $\unicode[STIX]{x1D703}=-30^{\circ }$ dropped to ${\approx}98\,\%$ and ${\approx}96\,\%$ of their respective values at $Re_{b}=3600$ , as shown in table 1. In general, as originally demonstrated by Bechert et al. (Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997) and Spalart & McLean (Reference Spalart and Mclean2011), and discussed in more detail in this section, the drag reduction performance of micro-textured surfaces, with a given geometry and size of the wall micro-texture in wall units, degrades with increasing bulk Reynolds number of the flow, due to the drop in the friction coefficient, $C_{f_{0}}$ , of the base flow with increasing bulk Reynolds number. The main source of this degradation in drag reduction performance is a drop in the magnitude of $\{DR_{slip}\}$ with increasing bulk Reynolds number. The reason for this drop can be understood by noting that the slip velocity normalized in wall units, $U_{s}^{+}$ , remains invariant with the Reynolds number for a given characteristic size of the microgrooves, $l_{g}^{+}$ , as shown in table 1. Consequently, quantities which are proportional to $(U_{s}/U_{b})$ , such as $\{DR_{slip}\}$ , will experience a degradation with increasing Reynolds number, approximately proportional to the ratios of $\sqrt{C_{f_{0}}}$ in their respective base flows. In the present study, this effect results in $\{DR_{slip}\}$ values at $Re_{b}=7860$ which are 91–92 % of the values as $Re_{b}=3600$ , which is approximately equal to the ratio of $\sqrt{C_{f_{0}}}$ in the ‘base’ flows at $Re_{b}=7860$ and $Re_{b}=3600$ . Part of this drop in drag reduction due to $\{DR_{slip}\}$ is compensated by $\{DR_{Q_{g}}\}$ , which becomes less drag enhancing with increasing bulk Reynolds number of the base flow. In general, $\{DR_{Q_{g}}\}$ represents a low Reynolds number effect and its magnitude should become negligible at high Reynolds numbers. The percentage of drag reduction arising from $\{DR_{\unicode[STIX]{x1D700}}\}$ remained approximately unchanged for the two Reynolds numbers investigated in the present study.

Aside from these adjustments to the magnitude of $DR$ , the other features of drag-reduced flows at $Re_{b}=7860$ , including the characteristics of mean velocity profiles, turbulence intensities, Reynolds shear stresses, mean flow shear stresses, streamwise vorticity fluctuations, and two-dimensional co-spectra remained similar to those described for $Re_{b}=3600$ , as shown in figures 4–7. These findings are consistent with observations by a number of other investigators (Garcia-Mayoral & Jimenez Reference Garcia-Mayoral and Jimenez2012; Park et al. Reference Park, Park and Kim2013; Seo et al. Reference Seo, Garcia-Mayoral and Mani2015), who have also found little effect of Reynolds number on the physics of drag reduction with micro-textured surfaces, and have concluded that the physics deduced from DNS studies at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 200$ contain the essential features of drag reduction found in higher Reynolds number turbulent flows.

Figure 8. Streamwise mean velocity profiles in turbulent channel flow with micro-textured walls compared to turbulent channel flow with smooth, no-slip walls: (a) general characteristics of the mean velocity profiles in the presence of micro-textured walls; (b–d) mean velocity profiles in turbulent channel flow with SH longitudinal microgrooves or riblets. Line types in (b–d) as in figure 4.

The drag reduction results obtained in DNS can be extended to higher Reynolds number flows of practical interest, using the parameterization of $DR$ in terms of the shift in the intercept of a logarithmic-law representation of the mean velocity profiles, as suggested by Bechert et al. (Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997) and Spalart & McLean (Reference Spalart and Mclean2011). In this parameterization, the mean velocity profiles in the channel flows with micro-textured walls and in the base channel flow with smooth, no-slip walls are each approximated by logarithmic velocity profiles throughout the cross-section of the channel, as shown in figure 8. These logarithmic profiles are then integrated, and set equal to the bulk velocity, to get

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{U_{b}}{u_{\unicode[STIX]{x1D70F}}}=\frac{1}{h}\int _{0}^{h}\left(\frac{1}{\unicode[STIX]{x1D705}}\ln z^{+}+B\right)\,\text{d}z=\frac{1}{\unicode[STIX]{x1D705}}\ln \left(Re_{b}\sqrt{\frac{C_{f}}{2}}\right)+\left(B-\frac{1}{\unicode[STIX]{x1D705}}\right), & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{U_{b_{0}}}{u_{\unicode[STIX]{x1D70F}_{0}}}=\frac{1}{h}\int _{0}^{h}\left(\frac{1}{\unicode[STIX]{x1D705}}\ln z^{+}+B_{0}\right)\,\text{d}z=\frac{1}{\unicode[STIX]{x1D705}}\ln \left(Re_{b}\sqrt{\frac{C_{f_{0}}}{2}}\right)+\left(B_{0}-\frac{1}{\unicode[STIX]{x1D705}}\right), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the von Kármán constant, and $B$ and $B_{0}$ are the intercepts of logarithmic-law representations of the mean velocity profiles in the channel flow with micro-textured walls and the channel flow with smooth, no-slip walls, respectively, as shown in figure 8(a).

Using the definitions $U_{b}/u_{\unicode[STIX]{x1D70F}}=\sqrt{2/C_{f}}$ and $U_{b_{0}}/u_{\unicode[STIX]{x1D70F}_{0}}=\sqrt{2/C_{f_{0}}}$ , (5.1) and (5.2) can be combined to give an expression for $DR=(1-C_{f}/C_{f_{0}})$ as

(5.3) $$\begin{eqnarray}(1-DR)=\left\{1+\left[\frac{1}{2\unicode[STIX]{x1D705}}\ln (1-DR)+(B-B_{0})\right]\sqrt{\frac{C_{f_{0}}}{2}}\right\}^{-2}.\end{eqnarray}$$

Equation (5.3) provides a parameterization of $DR$ in terms of the shift, $(B-B_{0})$ , of a logarithmic-law representation of the mean velocity profile in the flow with micro-textured walls compared to the base flow. Approximate forms of (5.3), for the limit of low drag reduction, have been earlier derived for riblets (Bechert et al. Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997; Spalart & McLean Reference Spalart and Mclean2011). In this limit, (5.3) can be approximated by

(5.4) $$\begin{eqnarray}DR\approx \frac{(B-B_{0})}{\displaystyle \frac{1}{2\unicode[STIX]{x1D705}}+\sqrt{\frac{1}{2C_{f_{0}}}}}\approx (B-B_{0})\sqrt{2C_{f_{0}}}.\end{eqnarray}$$

It has been suggested that, in drag reduction with micro-textured surfaces, $(B-B_{0})$ is independent of Reynolds number, and only a function of the geometrical parameters of the surface micro-texture in wall units (Clauser Reference Clauser1956; Bechert et al. Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997; Spalart & McLean Reference Spalart and Mclean2011; Garcia-Mayoral & Jimenez Reference Garcia-Mayoral and Jimenez2012). As such, (5.3) can be used to investigate the effect of the bulk Reynolds number of the flow on drag reduction with micro-textured surfaces.

Figure 9. Parameterization of $DR$ in turbulent channel flow with SH longitudinal microgrooves or riblets in terms of the shift, $(B-B_{0})$ , of a logarithmic-law representation of the mean velocity profile and the Reynolds number of the base flow: (a) $DR$ as a function of $(B-B_{0})$ and the friction Reynolds number of the base flow; (b) $(B-B_{0})$ as a function of $l_{g}^{+}$ ; (c) average streamwise slip velocity, $U_{s}^{+}$ , as a function of $l_{g}^{+}$ ; (d) $\{(B-B_{0})-U_{s}^{+}\}$ as a function of $l_{g}^{+}$ . Line types in (a): ——, $Re_{b}=3600$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ ); — —, $Re_{b}=7860$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ); – – –, $Re_{b}=2\times 10^{4}$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 10^{3}$ ); — ⋅ —, $Re_{b}=2.7\times 10^{5}$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 10^{4}$ ); — ⋅ ⋅ —, $Re_{b}=3.8\times 10^{6}$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 10^{5}$ );  $\cdots \cdots$ ,  $Re_{b}=5.3\times 10^{7}$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 10^{6}$ ). Symbols as in figure 3.

Figure 9(a) shows the magnitudes of $DR$ as a function of $(B-B_{0})$ from DNS of turbulent channel flow with SH longitudinal microgrooves or riblets in the present study, compared to the predictions of (5.3). The appearance of $\sqrt{C_{f_{0}}}$ in (5.3) leads to a degradation of $DR$ with increasing bulk Reynolds of the flow. The magnitude of this degradation, as predicted by (5.3), is consistent with the degradation of $DR$ observed in DNS when the $Re_{b}$ was increased from $3600$ to $7860$ . Figure 9(a) also shows the predictions of (5.3) for higher Reynolds number flows of practical interest, up to $Re_{b}=7.3\times 10^{8}$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 10^{6}$ ). It can be seen that, for a given value of $(B-B_{0})$ , the drag reduction performance of SH longitudinal microgrooves and riblets degrades considerably with increasing bulk Reynolds number of the flow. Nevertheless, with a proper choice of the surface micro-texture geometry, drag reductions of up to 40–50 % should still be feasible with SH longitudinal microgrooves, assuming the liquid/gas interfaces in SH microgrooves remain stable and do not collapse.

Figure 9(b) shows the variation of $(B-B_{0})$ with the characteristic size of the microgrooves, $l_{g}^{+}$ , from DNS of turbulent channel flow with SH longitudinal microgrooves or riblets in the present study. Good agreement can be seen in the $(B-B_{0})$ values at different bulk Reynolds numbers for SH longitudinal microgrooves of size $g^{+0}\approx 14$ and $28$ at $\unicode[STIX]{x1D703}=-30^{\circ }$ , confirming the Reynolds number independence of $(B-B_{0})$ . However, the $(B-B_{0})$ values for SH longitudinal microgrooves with different protrusion angles do not collapse and show different trends compared to those for riblets.

To gain a better understanding of the source of these differences, one can decompose $(B-B_{0})$ into

(5.5) $$\begin{eqnarray}(B-B_{0})=\{U_{s}^{+}\}+\{(B-B_{0})-U_{s}^{+}\},\end{eqnarray}$$

where $\{U_{s}^{+}\}$ is the average streamwise slip velocity at the tip of the surface micro-texture, as shown in figure 8, representing all the drag reducing contributions to $DR$ arising from streamwise slip, while $\{(B-B_{0})-U_{s}^{+}\}$ represents all the drag enhancing contributions to $DR$ arising from spanwise slip, mean flow shear stresses developed due to the presence of the surface micro-texture and the faction of the flow rate through the surface micro-texture. Figure 9(c,d) shows the breakdown of $(B-B_{0})$ into $\{U_{s}^{+}\}$ and $\{(B-B_{0})-U_{s}^{+}\}$ from DNS of SH longitudinal microgroove and riblets in the present study. With riblets, $\{U_{s}^{+}\}$ follows the scaling $U_{s}^{+}\approx 0.4\{l_{g}^{+}\}^{0.8}$ for all microgroove depths, while with SH longitudinal microgrooves, it follows scalings which range from $U_{s}^{+}\approx 2.0\{l_{g}^{+}\}^{0.85}$ at $\unicode[STIX]{x1D703}=-30^{\circ }$ , to $U_{s}^{+}\approx 1.1\{l_{g}^{+}\}^{0.8}$ at $\unicode[STIX]{x1D703}=-90^{\circ }$ , indicating that with increasing protrusion angle, the scaling of $\{U_{s}^{+}\}$ with SH longitudinal microgrooves begins to approach that of riblets. The drag enhancing contributions represented by $\{(B-B_{0})-U_{s}^{+}\}$ , however, show a common scaling with both SH longitudinal microgrooves and riblets, as shown in figure 9(d), once again indicating that, aside from their magnitudes of streamwise surface slip, SH longitudinal microgrooves and riblets share a common mechanism of drag reduction. The collapse of the data from SH longitudinal microgrooves and riblets in figure 9(d) is not as good as the results shown in figures 2(k) and 3(b,d), due to the approximations used in the derivation in (5.3). Nevertheless, the characterization of $DR$ in terms of $(B-B_{0})$ given by (5.3), combined with the results shown in figure 9, provides a simple method for predicting the magnitude of drag reduction with micro-textured surfaces in high Reynolds number turbulent flows of practical interest.

6 Implications for design of more stable SH surfaces

The presence of a common mechanism of drag reduction with SH longitudinal microgrooves and riblets suggests that the two approaches can be combined synergistically to reduce the pressure loads on SH interfaces, and enhance their longevity and sustainability in turbulent flow environments.

Figure 10(a–c) shows the r.m.s. pressure fluctuations in turbulent channel flows with SH longitudinal microgrooves investigated in the present study. The presence of interface curvature at the protrusion angle of $\unicode[STIX]{x1D703}=-30^{\circ }$ leads to drops of 13 %, 15 % and 28 % in the magnitudes of r.m.s. pressure fluctuations at the wall with microgrooves of size $g^{+0}\approx 14$ , $28$ and $56$ , compared to same-size microgrooves with flat SH interfaces, respectively. The drops in instantaneous pressure fluctuations are even more dramatic, as shown in figure 10(d–g) for the case $g^{+0}\approx 14$ . The instantaneous pressure fluctuations on the micro-textured wall with SH longitudinal microgrooves at $\unicode[STIX]{x1D703}=-30^{\circ }$ are nearly a factor of two smaller than the micro-textured wall with SH longitudinal microgrooves at $\unicode[STIX]{x1D703}=0^{\circ }$ .

Figure 10. Turbulent pressure fluctuations in channel flow with SH longitudinal microgrooves: (a–c) r.m.s. turbulent pressure fluctuations; (d–g) contour plots of instantaneous turbulent pressure fluctuations on the channel boundaries for $g^{+0}\approx 14$ ; $w^{+0}\approx 2$ and $\unicode[STIX]{x1D703}=0^{\circ },-30^{\circ },-60^{\circ },-90^{\circ }$ , respectively. Line types as in figure 4.

These results suggest that one can improve the stability of SH surfaces by developing hybrid designs of SH surfaces, in which the SH surface is embedded within the microgrooves of shallow ( $\unicode[STIX]{x1D703}=-30^{\circ }$ ) scalloped riblets. Indeed, this may be the design adopted by nature in shark scales, which has gone unnoticed to date.

7 Summary and conclusions

Skin-friction drag reduction with SH longitudinal microgrooves and riblets has been investigated by mathematical analysis and DNS in channel flow. An exact analytic expression is derived for the magnitude of drag reduction with micro-textured surfaces, which allows the net drag reduction to be decomposed into its constitutive elements. It is shown that in laminar or turbulent channel flow with any SH or riblet micro-texture pattern on the walls, five elements contribute to the net drag reduction: (i) the effective slip velocity at the wall, which augments the bulk velocity and allows a fraction $(U_{s}/U_{b})$ of the flow rate to flow inviscidly; (ii) modifications to the normalized structure of turbulent Reynolds shear stresses due to this augmentation of the bulk velocity by the effective slip velocity at the wall; (iii) other modifications to the normalized structure of turbulent Reynolds shear stresses in the presence of the wall micro-texture; (iv) modifications to the mean flow shear stresses due to the presence of the wall micro-texture; and (v) the flow rate through the wall micro-texture.

Comparison to DNS results in turbulent channel flow at $Re_{b}=3600$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ ) and $Re_{b}=7860$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ) shows that SH longitudinal microgrooves and riblets share a common mechanism of drag reduction, in which 100 % of the drag reduction arises from effects (i) and (ii), namely, the effective slip velocity at the wall, and modifications to the normalized structure of turbulent Reynolds shear stresses due the presence of this effective slip velocity. All other modifications to the normalized structure of turbulent Reynolds shear stresses and the structure of the mean flow shear stresses due to the presence of the wall micro-texture, as well as the effect of the flow rate through the wall micro-texture, are drag enhancing, and follow common scalings for SH longitudinal microgrooves and riblets when expressed as a function of the characteristic size of the wall micro-texture, $l_{g}^{+}=\sqrt{A_{G}^{+}}$ .

It is shown that DNS studies at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 200$ properly represent the physics of drag reduction with micro-textured surfaces at higher Reynolds numbers. Extrapolation of drag reduction data from DNS to high Reynolds number flows of practical interest is discussed. It is shown that, for a given geometry and size of the surface micro-texture in wall units, there is significant degradation in the drag reduction performance of micro-textured surfaces with increasing Reynolds number of the flow.

Curved SH interfaces with small protrusion angle ( $\unicode[STIX]{x1D703}=-30^{\circ }$ ) are shown to reduce the r.m.s. and instantaneous pressure fluctuations on the SH interface compared to flat SH surfaces. This suggests that the longevity of SH surfaces in turbulent flow environments may be improved by using hybrid designs, in which the SH surface is embedded within the microgrooves of shallow ( $\unicode[STIX]{x1D703}=-30^{\circ }$ ) scalloped riblets.