3.1. Set-up of virtual-origin simulations

Here, we outline the numerical method and summarise the series of simulations with Robin boundary conditions that we carry out. We conduct DNSs of turbulent channel flows in a domain doubly periodic in the wall-parallel directions, using a code adapted from García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011b) and Fairhall & García-Mayoral (Reference Fairhall and García-Mayoral2018). We solve the non-dimensional, unsteady, incompressible Navier–Stokes equations

(3.1)\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}\boldsymbol{u} ={-} \boldsymbol{\nabla} p + \frac{1}{Re}\nabla^2 \boldsymbol{u}, \end{gather}

(3.2)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}

where $\boldsymbol {u}=(u,v,w)$ is the velocity vector with components in the streamwise, wall-normal and spanwise directions, $x$, $y$ and $z$, respectively, $p$ is the kinematic pressure and ${Re}$ is the channel bulk Reynolds number. In the streamwise and spanwise directions, the primitive variables are solved in Fourier space, applying the $2/3$ dealiasing rule when computing the nonlinear advective terms. The wall-normal direction is discretised using a second-order centred finite difference scheme on a staggered grid. Time integration is carried out using a fractional step method (Kim & Moin Reference Kim and Moin1985), along with a three-step Runge–Kutta scheme. The same coefficients as Le & Moin (Reference Le and Moin1991) are used, for which semi-implicit and explicit schemes are used to approximate the viscous and advective terms, respectively.

Simulations are primarily conducted at friction Reynolds number ${Re}_\tau =\delta u_\tau / \nu \approx 180$. Even though this is a comparatively low Reynolds number, previous studies have shown that it is sufficient to capture the key physics in flows where the effect of surface manipulations is confined to the near-wall region (Martell, Perot & Rothstein Reference Martell, Perot and Rothstein2009; Busse & Sandham Reference Busse and Sandham2012; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2012). Two simulations are also conducted at ${Re}_\tau \simeq 550$ for comparison, and we will show in § 4.4 how our results are scalable to higher Reynolds numbers. In all cases, the channel half-height is $\delta =1$. For most simulations, the wall-parallel domain size is $L_x=2{\rm \pi} \delta$ and $L_z={\rm \pi} \delta$. This has been shown by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) to be sufficiently large to capture the key turbulence processes and length scales of the near-wall and log-law regions of the flow, and reproduce well the one-point statistics of domains of larger size. We show that this is the case also here by running one of the simulations at ${Re}_\tau \simeq 550$ with domain size $L_x=8{\rm \pi} \delta$ and $L_z=3{\rm \pi} \delta$. In the wall-parallel directions, the resolution in collocation points is ${\rm \Delta} x^+ \approx 6$ and ${\rm \Delta} z^+ \approx 3$ for simulations at ${Re}_\tau \simeq 180$, and ${\rm \Delta} x^+ \approx 9$ and ${\rm \Delta} z^+ \approx 4$ for the simulation at ${Re}_\tau \simeq 550$. In the wall-normal direction, the grid is stretched such that ${\rm \Delta} y^+_{min} \approx 0.3$ at the wall and ${\rm \Delta} y^+_{max} \approx 3$ at the channel centre. The flow is driven by a constant streamwise pressure gradient, in order to keep ${Re}_\tau$ fixed. The variable time step is controlled by

(3.3)\begin{equation} {\rm \Delta} t =min\left\lbrace 0.7\left[\frac{{\rm \Delta} x}{{\rm \pi}|u|},\frac{{\rm \Delta} z}{{\rm \pi}|w|},\frac{{\rm \Delta} y}{{\rm \pi}|v|}\right],\ 2.5\left[\frac{{\rm \Delta} x^2}{{\rm \pi}^2}\nu,\frac{{\rm \Delta} z^2}{{\rm \pi}^2}\nu,\frac{{\rm \Delta} y^2}{4}\nu\right] \right\rbrace, \end{equation}

which corresponds to maintaining a convective Courant–Friedrichs–Lewy (CFL) number of 0.7 and a viscous one of 2.5. In all cases, the flow was allowed to evolve until any initial transients had decayed, and then statistics were collected over a window of at least 20 largest-eddy turnover times, $\delta /u_\tau$.

Virtual origins for the three velocity components are introduced by imposing Robin, slip-length boundary conditions at the channel walls, following Gómez-de-Segura & García-Mayoral (Reference Gómez-de-Segura and García-Mayoral2020). At the bottom wall of the channel, these take the form

(3.4a–c)\begin{equation} u\rvert_{y=0} = \left.\ell_x \frac{\partial u}{\partial y}\right\rvert_{y=0},\quad v\rvert_{y=0}= \left. \ell_y \frac{\partial v}{\partial y}\right\rvert_{y=0} \quad \text{and} \quad w\rvert_{y=0}= \left.\ell_z \frac{\partial w}{\partial y}\right\rvert_{y=0},\end{equation}

so that $u$, $v$ and $w$ at the domain boundary are related to their respective wall-normal gradients by the three slip lengths, $\ell _x$, $\ell _y$ and $\ell _z$. Equivalent, symmetric boundary conditions are also applied to the top wall of the channel. The coupling between velocity components, their wall-normal gradients and the pressure is fully implicit and embedded in the lower-upper (LU) factorisation intrinsic in the fractional-step method (Perot Reference Perot1993). A detailed description of the implementation of this type of boundary conditions can be found in Gómez-de-Segura & García-Mayoral (Reference Gómez-de-Segura and García-Mayoral2019). Note that, for $v$, as mentioned in § 1, $\ell _y$ does not convey a slip effect, but, by extension, we will also refer to $\ell _y$ as the ‘slip length’ in the wall-normal direction. To prevent any net surface mass flux, the slip length for the $xz$-averaged wall-normal velocity is set to zero, and hence $\ell _y$ is only applied to its fluctuating component. Note also that a free-slip condition, e.g. $\partial u/\partial y = 0$, is equivalent, in principle, to imposing an infinitely large slip length, $\ell _x=\infty$.3

While the concepts of slip lengths and virtual origins have been used interchangeably in the literature, following Gómez-de-Segura & García-Mayoral (Reference Gómez-de-Segura and García-Mayoral2020) we make a subtle but important difference. Let us denote by $\ell _x^+$, $\ell _y^+$ and $\ell _z^+$ the slip lengths in the streamwise, wall-normal and spanwise directions, respectively, which are defined exclusively as the Robin coefficients for the simulation boundary conditions (3.4a–c). Physically, they simply correspond to the wall-normal locations where the velocity components become zero when linearly extrapolated from the reference plane, $y=0$. In order to associate the imposed slip lengths with smooth-wall data a priori, we define the virtual origins of $u$, $v$ and $w$ as the notional distance below the reference plane where each velocity component would perceive a virtual, smooth wall. To do this, we assume the shape of each r.m.s.-fluctuation profile would remain the same as over a smooth wall, independently of the others. The virtual origins would then be located at $y^+=-\ell _u^+$, $y^+=-\ell _v^+$ and $y^+=-\ell _w^+$, respectively. We note that this is not physics-based, but it simply allows us to establish an a priori correspondence between the offset in each velocity component and the slip length for that velocity while accounting for the nonlinear behaviour of the fluctuating velocities, especially for $v$, near the wall. The definition of these virtual origins is illustrated in figure 5(a). The slip lengths for the Robin boundary conditions (3.4a–c) are therefore set with the objective of yielding a prescribed set of virtual origins $\ell _u$, $\ell _v$ and $\ell _w$. Table 1 summarises the parameters of the simulations that we conduct in this study. For each case, the slip lengths $\ell _x^+$, $\ell _z^+$ and $\ell _y^+$ are given, along with the corresponding virtual origins $\ell _u^+$, $\ell _w^+$ and $\ell _v^+$. Since the virtual origins are computed from the slip lengths a priori, assuming the shape of smooth-wall velocity profiles remain unchanged, there is a one-to-one a priori relationship between slip lengths and virtual origins. The simulations are split into various ‘families’, each designed to systematically test a particular aspect of this virtual-origin framework. For example, some cases impose a virtual origin on $v$ alone (denoted by ‘$V$’), while other cases impose a virtual origin on both $u$ and $v$ (denoted by ‘$UV$’), and so on. The exact purpose of each family is explained in § 4. Note that for some of the simulations, the slip length applied to the mean flow, $\ell _{x,m}$, is different to the slip length applied to the streamwise fluctuations, $\ell _{x}$. Since we solve the flow in Fourier space in the wall-parallel directions, this can be implemented easily by imposing different slip-length boundary conditions on the different modes $\hat {u}(k_x,k_z,y)$ as required, where $k_x$ and $k_z$ are the streamwise and spanwise wavenumbers, respectively.

Figure 5. Schematics showing (a) the definition of virtual origins $\ell _u^+$, $\ell _w^+$ and $\ell _v^+$ as the shift of the root-mean-square (r.m.s.) velocity fluctuations with respect to a smooth channel; (b) the distinction between $\ell _v^+$ and $\ell _y^+$. Adapted from Gómez-de-Segura & García-Mayoral (Reference Gómez-de-Segura and García-Mayoral2020).

Table 1. Summary of simulations, including the slip lengths used for the boundary conditions, $\ell _x^+$, $\ell _z^+$ and $\ell _y^+$, and their corresponding virtual origins, $\ell _u^+$, $\ell _w^+$ and $\ell _v^+$, calculated a priori from the smooth-wall profiles. The slip length for the mean flow, $\ell _{x,m}^+$, is given only when it is different to the slip length for the streamwise velocity fluctuations. Note that, here, ${Re}_{\tau }$ is the friction Reynolds number calculated with respect to the plane $y=0$. The virtual origin for the mean flow, $\ell _U^+$, is given as the mean streamwise slip velocity, $U^+_s$, measured at $y=0$. The virtual origin for turbulence, $\ell _T^+$, is found a posteriori and compared to that predicted by (4.5), $\ell ^+_{T,pred}$. In the case names, ‘${\rm U}$’, ‘${\rm V}$’ and ‘${\rm W}$’ denote a non-zero slip-length boundary condition on $u$, $v$ and $w$, respectively, ‘${\rm M}$’ signifies that the slip applied to the streamwise velocity fluctuations is not the same as that applied to (M)ean velocity, ‘${\rm H}$’ is for the (H)igher Reynolds number cases at ${Re}_{\tau }=550$, ‘D’ is for the simulation with the larger (D)omain in the streamwise and spanwise directions and ‘${\rm L}$’ is for cases with (L)arge slip lengths. Note that the slip lengths, $\ell _x^+$, $\ell _z^+$ and $\ell _y^+$, and virtual origins, $\ell _u^+$, $\ell _w^+$ and $\ell _v^+$, are scaled with the friction velocity measured at the domain boundary, $y=0$, whereas $\ell _U^+$ and $\ell _T^+$ are scaled with the friction velocity measured at the origin for turbulence $y=-\ell _T^+$. The origin for turbulence predicted from (4.5), $\ell _{T,pred}^+$, is scaled with the friction velocity at that origin, i.e. at $y=-\ell _{T,pred}^+$.

For a virtual origin of a few wall units, we expect the slip lengths $\ell _x^+$ and $\ell _z^+$ to be approximately equal to $\ell _u^+$ and $\ell _w^+$, because the wall-parallel velocities $u^+$ and $w^+$ are essentially linear in the immediate vicinity of the wall. The case of the wall-normal velocity, however, is less straightforward. Since $v^{\prime +}$ is essentially quadratic very near the wall, the height of the virtual origin perceived by $v^+$, $y^+=-\ell _v^+$, can differ significantly from the slip length $\ell _y^+$, even for small values, as illustrated in figure 5(b). We choose $\ell _y^+$ as the ratio between $v^{\prime +}$ and $\mathrm {d} v^{\prime +} /\mathrm {d}y^+$ at a height $y^+=\ell _v^+$ above a smooth wall. From figure 5(b), $\ell _y^+$ and $\ell _v^+$ are related by $\ell _y^+=\ell _v^+-\ell _{sm}^+$, where $\ell _{sm}^+$ is obtained by linearly extrapolating the slope of the smooth-wall profile at $y^+=\ell _v^+$. Note that the value of $\ell _{sm}^+$ is a function of $\ell _v^+$, as it depends on the local slope of the profile at the height from which the extrapolation is calculated. A curvature effect can also be significant for $\ell _w^+$, since the profile of $w^{\prime +}$ becomes noticeably curved for $y^+\gtrsim 2$, but this effect is small for $u^{\prime +}$. Since the mean velocity profile is approximately linear up to $y^+ \approx 5$, the distance below the plane $y^+=0$ of the virtual origin experienced by the mean flow, $\ell _U^+$, is essentially equal to the slip velocity of the mean flow in wall units, $U_s^+$, and also to its slip length, $\ell _{x,m}^+$. It should, however, be mentioned that in general, if $\ell _x^+$ is large enough, the virtual origin perceived by the mean flow, $y^+ = -\ell _U^+$, is not necessarily coincident with the virtual origin for the streamwise fluctuations, $y^+=-\ell _u^+$, since their profiles curve differently as they approach the wall, even if $\ell _x^+ = \ell _{x,m}^+$.

3.2. Set-up of opposition-control simulations

As well as the virtual-origin simulations described above, we also investigate the effect of opposition control (Choi et al. Reference Choi, Moin and Kim1994) from the viewpoint of virtual origins. We carry out three simulations at ${Re}_\tau \approx 180$, applying opposition control to $v$ alone, $w$ alone, and both $v$ and $w$. The same DNS code as for the virtual-origin simulations is used, with the only difference being the imposed boundary conditions. The control is implemented in the code explicitly, with the measured velocity at the plane $y^+=y_d^+$ at time step $n$ is opposed at the wall at time step $n + 1$. The detection plane is set at $y_d^+ = 7.8$, with the aim of generating notional virtual origins for the controlled velocities at $y^+\approx 4$, similar to our virtual-origin simulation UWV6. A summary of the opposition-control simulations is given in table 2, including several parameters relevant to their interpretation in terms of virtual origins, which will be discussed in detail in § 5.

Table 2. Summary of opposition-control simulations. For each case, the notional virtual origins are given with respect to the reference plane $y^+=0$, assuming that the control establishes a virtual origin for the opposed velocity components at $y^+=y_d^+/2$, where $y_d^+$ is the detection plane height. The predicted virtual origin for turbulence, $\ell _{T,pred}^+$, is given, which is calculated from (4.5). The difference $\ell _U^+ - \ell _{T,pred}$ represents the predicted shift in the mean velocity profile, and ${\rm \Delta} U^+$ is the measured shift in the mean velocity profile from figure 20.

4.1. The origin for turbulence

In § 2, we introduced the idea that the quasi-streamwise vortices, and hence the turbulence, might perceive an intermediate origin between the virtual origins perceived by $v$ and $w$. We now discuss this concept in more detail. Let us postulate that the only effect of the virtual origins, particularly those perceived by $v$ and $w$, on the near-wall turbulence is to set its origin at some intermediate plane, while the flow remains otherwise the same as over a smooth wall. In this paper, we define $\ell _T^+$ as the distance between the virtual origin perceived by turbulence and the reference plane $y^+=0$. When $\ell _T^+ >0$, the virtual origin perceived by turbulence is below the reference plane, and therefore we refer to the plane $y^+ = -\ell _T^+$ as the virtual origin for turbulence. Likewise, we denote by $y^+=-\ell _U^+$ the virtual origin perceived by the mean flow. It follows from the streamwise momentum equation that the shape of the mean velocity profile in a channel is determined by the turbulence through the Reynolds stress (Pope Reference Pope2000; Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2020). If $\ell _U^+ > \ell _T^+$, the virtual origin perceived by the mean flow is deeper than that perceived by the turbulence. In this case, the mean velocity profile would be free to grow with essentially unit gradient in wall units from $y^+=-\ell _U^+$ to $y^+=-\ell _T^+$, due to the absence of Reynolds shear stress in the region $-\ell _U^+\leq y^+\leq -\ell _T^+$. Above $y^+=-\ell _T^+$, the Reynolds shear stress would be the same as over a smooth wall, and so would the shape of the mean velocity profile, but shifted by the additional velocity $U^+(y^+=-\ell _T^+) = \ell _U^+ - \ell _T^+$. Note that the above ideas apply to the virtual profile that would extend below $y^+=0$, as mentioned in § 2. The outward shift of the mean velocity profile would then necessarily be given by

(4.1)\begin{equation} {\rm \Delta} U^+{=} \ell_U^+{-} \ell_T^+, \end{equation}

which would propagate to all heights above the plane $y^+=-\ell _T^+$ (Gómez-de-Segura et al. Reference Gómez-de-Segura, Fairhall, MacDonald, Chung and Garicía-Mayoral2018a; García-Mayoral et al. Reference García-Mayoral, Gómez-de-Segura and Fairhall2019).

The physical idea described by (4.1) was, in fact, essentially proposed by Luchini (Reference Luchini1996), who postulated that the log law would be modified by the presence of texture only through a shift ${\rm \Delta} U^+$ ‘if the structure of the turbulent eddies were unaltered in the reference frame that has the transverse equivalent wall as origin, whereas the mean flow profile obviously starts at the longitudinal equivalent wall’. In other words, ${\rm \Delta} U^+$ should be the height difference between the origin for the mean flow, at $y^+=-\ell _U^+$, and the origin for turbulence, at $y^+=-\ell _T^+$. In this framework, from the point of view of turbulence the ‘wall’ is located at $y^+=-\ell _T^+$, which, therefore, should also be the height of reference when comparing with smooth-wall data. Note that (4.1) is based on the assumption that the effect of the texture on the mean flow and the turbulence is only to change the virtual origins that they perceive, and that the dynamics of the near-wall cycle is unaffected. This requires that the flow perceives the surface in a homogenised fashion, and the direct, granular effect of the texture is negligible (García-Mayoral et al. Reference García-Mayoral, Gómez-de-Segura and Fairhall2019). In the context of superhydrophobic surfaces, for instance, Fairhall, Abderrahaman-Elena & García-Mayoral (Reference Fairhall, Abderrahaman-Elena and García-Mayoral2019) show that this is the case so long as the characteristic length scale of the texture in wall units satisfies $L^+ \lesssim 25$. Using the results from our DNSs, we will now examine the validity of (4.1), starting first with the dependence of $\ell _T^+$ on the virtual origins imposed on the three velocity components, $\ell _u^+$, $\ell _v^+$ and $\ell _w^+$.

In § 2 we have discussed the idea that the quasi-streamwise vortices of the near-wall cycle induce, as a first-order effect, a spanwise flow very near the wall and, as a second-order effect, a wall-normal one. This would explain the saturation in the effect of the spanwise slip length, $\ell _z^+$, in the absence of permeability, i.e. when $\ell _y^+=0$. Furthermore, this is consistent with the idea that when the virtual origin perceived by the wall-normal velocity is roughly at the same depth as that perceived by the spanwise velocity, $\ell _v^+ \approx \ell _w^+$, no saturation is observed (Gómez-de-Segura et al. Reference Gómez-de-Segura, Fairhall, MacDonald, Chung and Garicía-Mayoral2018a). This implies that when imposing a virtual origin on the wall-normal velocity alone, without any spanwise slip, the virtual origin perceived by the vortices should remain at the domain boundary, $y=0$, regardless of how large $\ell _v^+$ was. Since, in this case, $w=0$ at the reference plane, and the vortices induce predominantly a spanwise flow in the vicinity of the wall, transpiration alone would not allow the vortices to move any closer to the reference plane. This is the contrasting, but complementary concept to the saturation in the effect of $\ell _z^+$ in the absence of transpiration depicted in figure 3.

We assess the idea presented in the previous paragraph in simulations V1, V2, UV1 and UV2, all of which have $\ell _w^+=0$. The mean velocity profiles, r.m.s. velocity fluctuations and Reynolds stress profiles for these simulations are shown in figure 6. The figure supports the idea that the virtual origin experienced by the spanwise flow is, indeed, the most limiting in terms of setting the virtual origin for turbulence, and $\ell _T^+=0$ for all cases. For the two cases with a non-zero virtual origin for $v$ only, cases V1 and V2, there is no change in the statistics whatsoever with respect to the smooth-wall data, even for virtual origins as large as $\ell _v^+\approx 4$. When a non-zero virtual origin is also applied to the streamwise flow, such that $\ell _u^+,\ell _v^+ > 0$ but $\ell _w^+=0$, there is still no change in the wall-normal and spanwise r.m.s. velocity fluctuations, $v^{\prime +}$ and $w^{\prime +}$, or the Reynolds stress profile. We also observe that the mean velocity profile is essentially identical to the smooth-wall case, save for the shift $\ell _U^+ = U^+_s$, as shown in figure 6(a). However, the peak value of the streamwise r.m.s. velocity fluctuations, $u^{\prime +}$, increases as $\ell _u^+$ is increased, and the $u^{\prime +}$ curve does not fit well the smooth-wall data near the wall for case UV2, when $\ell _u^+ = 4$. This appears to occur independently of the mean flow and other statistics, and this will be investigated further in § 4.2.

Figure 6. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for slip-length simulations with no spanwise slip. Black lines, smooth-wall reference data; blue to red lines, cases V1, V2, UV1 and UV2. Note that, in (a), the mean streamwise slip length, $\ell _U^+$, where appropriate, has been subtracted from the mean velocity profile.

The results presented so far suggest that the quasi-streamwise vortices cannot perceive an origin deeper than the origin perceived by the spanwise velocity. We now investigate the effect on the flow when both $\ell _w^+$ and $\ell _v^+$ are non-zero, using simulations UWV1–UWV6. The mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for these simulations are included in figure 7. These simulations have non-zero slip-length coefficients for all three velocity components, with $\ell _x^+\lesssim 4$, $\ell _y^+\lesssim 2$ and $\ell _z^+\lesssim 6$. In figure 7(a), after subtracting $\ell _U^+$ from the mean flow in each case, we see that there is still a noticeable difference between the mean velocity profile of the smooth-wall and the slip-length simulations. This difference is consistent with the origin for turbulence lying below the reference plane, $y^+=0$, which acts to increase the drag. We also observe in figure 7(b,c) that the velocity fluctuations and Reynolds stress profile are shifted towards $y^+=0$ and that their qualitative shape appears to have changed.

Figure 7. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for simulations with non-zero slip-length boundary conditions applied to all three velocity components. (a–c) Scaled with the friction velocity at the reference plane, $y^+=0$; (d–f) shifted in $y^+$ by $\ell _T^+$ and scaled with the friction velocity at the origin for turbulence, $y^+=-\ell _T^+$. Black lines, smooth-wall reference data; blue to red lines, cases UWV1–UWV6.

The above is the conventional way of representing turbulence statistics in the flow-control literature. For example, the observed reduction in velocity and vorticity fluctuations above riblets has been interpreted by some authors as the quasi-streamwise vortices being modified or damped, as well as the spanwise motion of the near-wall streaks being inhibited (see e.g. Choi, Moin & Kim Reference Choi, Moin and Kim1993; Chu & Karniadakis Reference Chu and Karniadakis1993; El-Samni, Chun & Yoon Reference El-Samni, Chun and Yoon2007). Similarly, in studies on the effects of superhydrophobic surfaces, authors have reported that turbulent structures are weakened, modified or disrupted by the presence of the surface (see e.g. Min & Kim Reference Min and Kim2004; Busse & Sandham Reference Busse and Sandham2012; Park, Park & Kim Reference Park, Park and Kim2013; Jelly, Jung & Zaki Reference Jelly, Jung and Zaki2014). These interpretations would suggest that the turbulence is no longer as it would be over a smooth wall. However, following the physical arguments leading to (4.1), if the turbulence remains otherwise as it would over a smooth surface, it should be possible to account for the difference with smooth-wall data by a mere origin offset.

First we discuss the choice of the friction velocity $u_\tau$. As mentioned above, if as proposed by Luchini (Reference Luchini1996) turbulence perceives a virtual smooth wall at $y = -\ell _T$, it follows that the friction velocity $u_\tau$ that scales the flow would be provided by the shear stress at that height. Since the total stress in a channel is linear with $y$, the friction velocity at $y=-\ell _T$ can be found by simply extrapolating the total stress curve from the domain boundary, $y=0$. This would be given by

(4.2)\begin{equation} u_\tau(y={-}\ell_T) = u_{\tau,0}\sqrt{\frac{\delta + \ell_T}{\delta}},\end{equation}

where $u_{\tau ,0}$ is the friction velocity measured at $y=0$. Note that the friction velocity measured from the surface drag is not necessarily the same as the friction velocity that sets the scaling for the turbulence. Nevertheless, from (4.2), the ratio $u_\tau /u_{\tau ,0}$ is close to unity as long as $\ell _T/\delta \ll 1$, which will be the case at the typical Reynolds numbers of experiments and engineering applications. As we will see below, even in the cases presented in this study, which are conducted at ${Re}_\tau = 180$, $u_\tau$ measured at $y=-\ell _T$ is never more than approximately 2 % larger than $u_{\tau ,0}$.

Using the friction velocity of (4.2), we can recalculate the viscous length scale and renormalise the measured velocities and Reynolds stress. These profiles can then be shifted in $y^+$ by $\ell _T^+$, where the ‘+’ superscript now indicates scaling in wall units based on the $u_\tau$ computed from (4.2). If the turbulence dynamics is indeed unmodified compared to the flow over a smooth wall, except for this shift in origin, which affects both the wall-normal coordinate and the scaling of the flow, then the r.m.s. velocity fluctuations and Reynolds stress profile should essentially collapse to the smooth-wall data. Since ${\rm \Delta} U^+ = \ell _U^+ - \ell _T^+$, the only difference between the curve $U^+-\ell _U^+$ and the smooth-wall mean velocity profile should be $\ell _T^+$ at all heights.

We measure the virtual origin for turbulence a posteriori in cases UWV1–UWV6 by finding the shift that best fits the Reynolds stress curve to smooth-wall data in the near-wall region, $5 \lesssim y^+ + \ell _T^+ \lesssim 20$, and compute the friction velocity at this origin from (4.2). The measured value of $\ell _T^+$ is included in table 1 for each case, along with the value for all the other cases considered in this study. The figure shows that when the wall-normal coordinate is measured from the virtual origin for turbulence, $y^+=-\ell _T^+$, the wall-normal and spanwise r.m.s. fluctuations and Reynolds stress curves essentially collapse to the smooth-wall data, as shown in figure 7(e,f). This suggests that, in these cases, the turbulence remains essentially unchanged compared to the flow over a smooth wall. We will refer to this as the turbulence being essentially ‘smooth-wall-like’. Further, this implies that any apparent modifications to turbulence that might be concluded from figure 7(b,c) are actually an apparent effect caused by the way the data are portrayed. Let us note that the resulting $v^{\prime +}$ and $w^{\prime +}$ profiles appear to perceive an origin at $y^+ = - \ell _T^+$, and not the ones prescribed a priori, $y^+ = - \ell _v^+$ and $y^+ = - \ell _w^+$. This is the expected result if $v^{\prime +}$ and $w^{\prime +}$ arise from smooth-like near-wall dynamics and are thus intrinsically coupled. The offsets $\ell _v^+$ and $\ell _w^+$ are merely prescribed, a priori values to quantify the offset in $v$ and $w$ caused by the surface, but turbulence would react to their combined effect, perceiving a single origin if it is to remain smooth-wall-like. There are some small deviations from the smooth-wall data for $u^{\prime +}$, which will be discussed in § 4.2. Significantly, figure 7(d) demonstrates that the mean velocity profile is also smooth-wall-like, when plotted against $y^++\ell _T^+$, save for the difference ${\rm \Delta} U^+ = \ell _U^+-\ell _T^+$. This strongly supports the validity of (4.1).

For comparison, two other possible ways of portraying the mean velocity profiles for cases UWV1–UWV6 are included in figure 8. Once the friction velocity is computed at the origin for turbulence, $y^+=-\ell _T^+$, and the wall-normal coordinate is also measured from that height, the mean velocity profiles from the slip-length simulations are essentially parallel to the smooth-wall one for all $y^+$, as shown in figure 8(a). The only difference between the curves of $U^+ - \ell _U^+$ plotted against $y^++\ell _T^+$ from the slip-length simulations and the smooth-wall mean velocity profile is the origin for turbulence, $\ell _T^+$, as mentioned above. Alternatively, again computing the friction velocity at $y^+=-\ell _T^+$, but now leaving $y^+=0$ as the datum for the wall-normal coordinate, the profiles of $U^+ - {\rm \Delta} U^+$ collapse to the smooth-wall profile only for $y^+\gg 1$, as portrayed in figure 8(b). In other words, the profiles collapse to the smooth-wall data only above the near-wall region of the flow (Clauser Reference Clauser1956). The choice of axes in figure 8(b) would indicate that we have measured the correct ${\rm \Delta} U^+$, but would not suggest that the profiles are smooth-wall-like across the whole $y^+$ range. The only way that they will collapse immediately from $y^+ = 0$ is to measure the wall-normal coordinate from the plane $y^+=-\ell _T^+$, as already shown in figure 7(d). This also emphasises the idea that (4.1) will only hold if the origin for turbulence, i.e. the plane $y^+=-\ell _T^+$, is used as reference for the turbulence dynamics, setting their scaling for velocity and length, as well as their height origin.

Figure 8. Mean velocity profiles for cases UWV1–UWV6, scaled with the friction velocity at the origin for turbulence, $y^+ = -\ell _T^+$: (a) $U^+ - \ell _U^+$ with the wall-normal coordinate measured from the origin for turbulence, $y^+ = -\ell _T^+$; (b) $U^+ - (\ell _U^+-\ell _T^+)$ with the wall-normal coordinate measured from the boundary, $y^+=0$. Black lines, smooth-wall reference data; blue to red lines, cases UWV1–UWV6.

The collapse of the mean velocity, r.m.s. fluctuations and Reynolds stress profile to the smooth-wall data for cases UVW1–UWV6, shown in figure 7(d–f), indicates that the near-wall turbulence dynamics remain smooth-wall-like, and that $\ell _T^+$ fully describes the effect of the virtual origins on the turbulence (García-Mayoral et al. Reference García-Mayoral, Gómez-de-Segura and Fairhall2019). It could, however, be argued that energy might be organised differently yet provide the same r.m.s. values. Figure 9 portrays the premultiplied energy spectra at $y^++\ell _T^+ = 15$ for several cases along with that of a smooth-wall flow at $y^+\approx 15$. For cases UWV1–UWV6, shown in figure 9(e–h), the distribution of energy among different length scales is the same as in flows over a smooth wall, which supports the idea that the near-wall turbulence dynamics remain essentially smooth-wall-like. The same is true for cases V1, V2, UV1 and UV2, figure 9(a–d), which were discussed earlier in § 4.1 and have $\ell _T^+ = 0$. Additionally, figure 10 compares snapshots of $u^{\prime +}$ and $v^{\prime +}$ for the flow over a smooth wall at $y^+ = 5$ with those for case UWV6 at two wall-parallel planes, $y^+ = 5$ and $y^+ +\ell _T^+ = 5$. The fluctuations at $y^+ = 5$ are portrayed in figure 10(c,d), and are scaled with $u_\tau$ measured at $y^+ = 0$. On the other hand, the fluctuations at $y^+ +\ell _T^+ = 5$, shown in figure 10(e,f), are scaled with $u_\tau$ measured at $y^+ = -\ell _T^+$. The figure demonstrates that there is no qualitative visual change in the flow when the snapshots from the smooth-wall flow are compared to those from the slip-length simulation at the equivalent height, i.e. comparing the smooth-wall flow at $y^+ = 5$ with case UWV6 at $y^+ + \ell _T^+ =5$, measuring $u_\tau$ accordingly. However, if the snapshots from the slip-length simulation are compared to the smooth-wall case at the same height above the reference plane $y^+=0$, using $u_\tau$ measured at $y^+=0$ in both cases as is often done in the literature, an apparent intensification of the fluctuations relative to the smooth-wall case can be observed, particularly for $v^{\prime +}$, as shown in figure 10(c,d). This further supports the idea that the near-wall turbulence dynamics remain essentially smooth-wall-like, except for the shift of origin $\ell _T^+$. Case UWV6 is used an example, because it has the deepest virtual origin for turbulence, $y^+\approx -4$, and the effect is more pronounced, but the same can also be observed for cases UWV1–UWV5.

Figure 9. Premultiplied two-dimensional spectral densities of $u^2$, $v^2$, $w^2$ and $uv$ at $y^+ + \ell _T^+ = 15$, normalised by $u_\tau$ at the origin for turbulence, $y^+ = -\ell _T^+$, for various slip-length simulations (line contours), compared to smooth-wall data (filled contours) at $y^+ = 15$. The shift $\ell_T^+$ is given in table 1 for each case. (a–d) Cases V1, V2, UV1 and UV2, with line colours as in figure 6. (e–h) Cases UWV1–UWV6, with line colours as in figure 7. (i–l) Cases UM1–UM6, with line colours as in figure 11. (a,e,i), $k_x k_z E_{uu}^+$; (b,f,j), $k_x k_z E_{vv}^+$; (c,g,k), $k_x k_z E_{ww}^+$; (d,h,l), $-k_x k_z E_{uv}^+$. The contour increments for each column are 0.3224, 0.0084, 0.0385 and 0.0241, respectively.

Figure 10. Streamwise (a,c,e) and wall-normal (b,d,f) instantaneous velocity fluctuation flow fields. (a,b) Smooth-wall reference case at $y^+= 5$, scaled with $u_\tau$ at $y^+ = 0$; (c,d) slip-length simulation UWV6 at $y^+ = 5$, scaled with $u_\tau$ at $y^+ = 0$; (e,f) the same snapshot as (c,d), but now for the wall-parallel plane $y^+ + \ell _T^+= 5$, scaled with $u_\tau$ at the origin for turbulence, $y^+ = - \ell _T^+$.

4.2. Separating the effect of the virtual origin experienced by the mean flow from that experienced by streamwise velocity fluctuations

In § 4.1, we observed for cases V1 and V2 that increasing $\ell _u^+$ resulted in an increase in the maximum value of $u^{\prime +}$ and a deviation of its profile away from the smooth-wall data. This leads to the question of why this is the case, to what extent the peak will continue to increase on increasing $\ell _u^+$, and what effect this has on the flow beyond simply modifying $u^{\prime +}$. More importantly, since this behaviour appears to occur independently of the mean flow, we also wish to address the question of whether or not the virtual origin perceived by the streamwise velocity fluctuations, and their apparent intensification, has any significant effect on ${\rm \Delta} U^+$. To answer these questions, we carry out a series of simulations with no slip on the mean flow, i.e. $\ell _U^+=0$, and gradually increase the slip length applied to the streamwise fluctuations from $\ell _x^+=5$ to $\ell _x^+=\infty$, the latter being equivalent to a free-slip condition. These are simulations UM1–UM6. They would highlight any effects caused by deepening the virtual origin experienced by the streamwise fluctuations. Results for these simulations are shown in figure 11. Note that the virtual origin experienced by the streamwise velocity fluctuations has no significant effect on the mean velocity profile, $v^{\prime +}$, $w^{\prime +}$ or the Reynolds stress profile, even for an infinite slip length. This demonstrates that the streamwise fluctuations play a negligible role in setting the origin for turbulence, i.e. $\ell _T^+ = 0$, and ${\rm \Delta} U^+ = 0$. The peak value of $u^{\prime +}$ increases with the streamwise slip, but, even when $\ell _x^+=\infty$, is not much larger than the smooth-wall value. The $y$-location of this peak also does not change significantly. The gradual increase in the peak value is likely due to the greater $y$-range over which the streamwise fluctuations near the wall are brought to zero by viscosity. As $\ell _x^+$ is increased, the streamwise fluctuations experience a deeper virtual origin and have more room to decay to zero more slowly. This changes the slope of $u^{\prime +}$ near the wall and results in the gradual increase observed in the peak value, but has no other effect on the turbulence. This can, again, be confirmed from the premultiplied energy spectra for these cases, given figure 9(i–l). Except for case UM6, the simulation with infinite slip, the spectra for all cases match very well to the smooth-wall reference data. There are some deviations in the contours for case UM6, but the peak location and overall distribution of energy among length scales still remains essentially smooth-wall-like. It is perhaps surprising that the Reynolds stress, $-\overline {u'v'}^+$, exhibits no change at all, given that $u^{\prime +}$ changes quite noticeably near the wall. However, since $v^{\prime +}$ is so small in the immediate vicinity of the wall, and the shape of its profile remains unmodified, the change in the Reynolds stress is negligible.

Figure 11. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for simulations with slip on the streamwise fluctuations but not on the mean flow, i.e. $\ell ^+_{x,m} = 0$. Black, smooth-wall reference data; blue to red, cases UM1–UM6 with increasing slip on the streamwise fluctuations.

The above behaviour can be discussed in terms of the near-wall-cycle structures (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997). The quasi-streamwise vortices and streaks interact in a quasi-cyclic process in which the vortices act to sustain the streaks through sweeps and ejections of high- and low-speed fluid, respectively. In terms of the r.m.s. velocity fluctuations, the streaks are related to $u^{\prime +}$, while the quasi-streamwise vortices generate mainly $v^{\prime +}$ and $w^{\prime +}$. Since applying a slip length in the streamwise direction has no effect on the spanwise or wall-normal velocities, this means that the $y$-location of the quasi-streamwise vortices cannot change with respect to the domain boundary. If the vortices are unaffected, and the streaks are sustained by the vortices, there cannot, therefore, be a substantial change in the location or magnitude of the peak value of $u^{\prime +}$. This would explain why the origin for turbulence seems to be independent of the origin for the streamwise velocity fluctuations, at least in the regime where $\ell _u^+ \geq \ell _T^+$.

So far, we have shown that applying a slip length to the streamwise flow appears to have an effect that is independent of the spanwise and wall-normal velocities. Further, applying a slip length to the streamwise fluctuations alone has no effect on the mean flow or the turbulent fluctuations other than the effect on $u^{\prime +}$ that has no further consequence discussed above. This suggests that the virtual origin experienced by the mean flow is independent of the virtual origin experienced by the streamwise fluctuations. The only streamwise origin relevant to ${\rm \Delta} U^+$ would therefore be that experienced by the mean flow, and not the origin experienced by the near-wall streaks, and the streaks appear not to play a significant role in determining the drag. We shall refer to this as the streaks being ‘inactive’ with respect to the change in drag. This is consistent with the idea proposed by Luchini (Reference Luchini1996) that turbulence as a whole has one origin, and the other important origin is the one for the mean flow, as we discussed in § 4.1. We check this using simulations UWV6 and UWV6M, with the same set of slip-length coefficients for the fluctuating velocity components, but a different slip on the mean velocity. As shown in table 1, for the velocity fluctuations $(\ell _x^+,\ell _z^+,\ell _y^+)=(4.0,6.0,2.0)$ in both cases, but for the mean flow is $\ell _{x,m}^+=4.0$ and 10.0, respectively. The statistics for these simulations are portrayed in figure 12. The figure shows that once $\ell _U^+$ is subtracted from the mean velocity, the mean velocity profile and the other statistics portrayed are identical for both simulations. This demonstrates that if the mean flow experiences a virtual origin different from the one of the streamwise fluctuations, this causes no change to the turbulence itself. This is because the additional mean velocity in case UWV6M corresponds simply to a Galilean shift of the flow compared to case UWV6. In combination with the fact that changing the virtual origin perceived by the streaks has no effect on the drag, this confirms that the important parameter in determining ${\rm \Delta} U^+$ is $\ell _U^+$, and not $\ell _u^+$. Note that actual textures may not impose different virtual origins on the mean flow and the streamwise fluctuations, as is done in case UWV6M, but its comparison with case UWV6 demonstrates which streamwise origin is physically relevant to ${\rm \Delta} U^+$, and hence the drag.

Figure 12. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for simulations UWV6 (blue lines) and UWV6M (red dashed lines), which have the same slip lengths applied to the velocity fluctuations but different slip lengths applied to the mean flow.

4.3. Predicting the origin for turbulence from the virtual origins experienced by the three velocity components

In the preceding discussion, we have shown that it is possible to displace the turbulence to its virtual origin at $y^+=-\ell _T^+$ by imposing virtual origins for the three velocity components. We have demonstrated that the turbulence statistics and mean velocity profile essentially collapse to the smooth-wall data when rescaled by the friction velocity at $y^+=-\ell _T^+$ and measured from that same height. Since we have shown that $\ell _u^+$ has no effect on ${\rm \Delta} U^+$, at least in the regime where $\ell _u^+ \geq \ell _T^+$, and that $\ell _U^+$ has no effect on the origin perceived by the turbulence, it follows that $\ell _T^+$ depends only on $\ell _v^+$ and $\ell _w^+$. We have also observed that the resulting location of the origin for turbulence exhibits two distinct regimes, with respect to the virtual origins experienced by $v$ and $w$. The first is when the virtual origin for $v$ is deeper than or equal to the virtual origin for $w$, i.e. $\ell _v^+ \geq \ell _w^+$. With reference to table 1, if $\ell _v^+ \geq \ell _w^+$, then $\ell _T^+\approx \ell _w^+$, which can be observed for cases V1, V2, UV1, UV2, UWV1, UWV2 and UWV4. The second regime occurs when the origin for $w$ is deeper than the origin for $v$, i.e. $0 < \ell _v^+< \ell _w^+$, for example, in cases UWV3, UWV5, UWV6 and UWV3H. We then find that turbulence perceives an origin intermediate between $\ell _v^+$ and $\ell _w^+$, such that $\ell _v^+< \ell _T^+<\ell _w^+$. This is in agreement with the physical arguments presented in § 2. We now wish to infer an expression that can be used to predict the virtual origin for turbulence a priori from the virtual origins for $v$ and $w$.

Since we are interested in finding an expression for $\ell _T^+$ in terms of the virtual origins perceived by $v$ and $w$, and not the slip-length coefficients, it would be more appropriate to express the saturation in terms of $\ell _w^+$ rather than $\ell _z^+$. Following Fairhall & García-Mayoral (Reference Fairhall and García-Mayoral2018), but now taking into account the curvature of the $w^{\prime +}$ profile, we revisit the expression for the effective spanwise slip (2.6), and arrive at the following empirical relation

(4.3)\begin{equation} \ell_{w,eff}^+{\approx} \frac{\ell_w^+}{1+\ell_w^+{/}5},\end{equation}

which is analogous to (2.6) for $\ell _z^+$, but asymptotes to a value of 5 instead of 4. Figure 13 is an alternative portrayal of the data from Busse & Sandham (Reference Busse and Sandham2012) presented earlier in figure 2(b), but this time using $\ell _{w,eff}^+$ instead of $\ell _{z,eff}^+$ to calculate ${\rm \Delta} U^+$. The figure shows excellent agreement between ${\rm \Delta} U^+$ and the difference $\ell _x^+ - \ell _{w,eff}^+$, with the data for low ${Re}_{\tau ,0}$ and high ${\rm \Delta} U^+$ deviating again as discussed for figure 2(b). The results show that when $\ell _v^+=0$, (4.3) gives an accurate prediction for the origin for turbulence, $\ell _{T,pred}^+$.

Figure 13. Alternative portrayal of the data from Busse & Sandham (Reference Busse and Sandham2012) presented in figure 2(b), with ${\rm \Delta} U^+$ now a function of $\ell _x^+ - \ell _{w,eff}^+$. Triangles, simulations at ${Re}_{\tau ,0} = 180$; circles, simulations at ${Re}_{\tau ,0} = 360$. From blue to red, increasing $\ell _{w,eff}^+$. The dashed line represents ${\rm \Delta} U^+ = \ell _x^+ - \ell _{w,eff}^+$.

However, $\ell _{T,pred}^+$ should also depend on $\ell _v^+$, because the saturation in the effect of $\ell _w^+$ is a direct result of the plane at which $v$ perceives an impermeable wall. Therefore, if the saturation of the effect of $\ell _w^+$ is evaluated with respect to the plane at which $v$ appears to vanish, i.e. $y^+=-\ell _v^+$, rather than $y^+=0$, it should be possible to predict the virtual origin for turbulence in the regime $0<\ell _v^+<\ell _w^+$ (Gómez-de-Segura et al. Reference Gómez-de-Segura, Fairhall, MacDonald, Chung and Garicía-Mayoral2018a). This regime is sketched in figure 14(a), where the virtual origin for turbulence lies between the virtual origins for $v$ and $w$, with $\ell _v^+ < \ell _{T,pred}^+ < \ell _w^+$. Then, $\ell _{T,pred}^+$ would be given by

(4.4)\begin{equation} \ell_{T,pred}^+{\approx} \ell_v^+{+} \frac{(\ell_w^+{-} \ell_v^+)}{1+(\ell_w^+{-} \ell_v^+)/5}.\end{equation}

On the other hand, as demonstrated by Gómez-de-Segura et al. (Reference Gómez-de-Segura, Fairhall, MacDonald, Chung and Garicía-Mayoral2018a), if $\ell _v^+\approx \ell _w^+$ then no saturation in the effect of $\ell _w^+$ occurs and $\ell _{T,pred}^+\approx \ell _w^+$, as sketched in figure 14(b). If we increase $\ell _v^+$ further, such that $\ell _v^+>\ell _w^+$, we would not expect the quasi-streamwise vortices to approach the surface further, since their first-order effect is to induce a spanwise velocity at the reference plane. Even if $v$ was allowed to penetrate freely through the reference plane, the quasi-streamwise vortices would require some amount of spanwise slip in the first place to approach this plane. Therefore, when $\ell _v^+\geq \ell _w^+$, we would expect $\ell _T^+\approx \ell _w^+$. This is confirmed in our simulations UWV1, UWV2 and UWV4. This regime is sketched in figure 14(c). Combining (4.4) and the preceding argument for the regime where $\ell _v^+\geq \ell _w^+$, a general expression for approximating $\ell _T^+$ from $\ell _w^+$ and $\ell _v^+$ would be

(4.5)\begin{equation} \ell_{T,pred}^{+}{\approx} \begin{cases} \ell_{v}^{+} + \dfrac{(\ell_{w}^{+}- \ell_{v}^+)}{1+(\ell_{w}^{+}- \ell_{v}^+)/5} & {\rm if}\ \ell_{w}^{+} \gt \ell_{v}^+,\\ \ell_{w}^+ & {\rm if}\ \ell_{w}^+{\leq} \ell_{v}^+.\end{cases} \end{equation}

When $\ell _{T,pred}^+$ is predicted from the values of $\ell _v^+$ and $\ell _w^+$, which are known a priori from $\ell _y^+$ and $\ell _z^+$, it shows excellent agreement with the value of $\ell _T^+$ measured a posteriori for the cases presented thus far, as shown in table 1. In all these cases, ${\rm \Delta} U^+$ is given by the linear law (4.1), that is, the difference $\ell _U^+-\ell _T^+$.

Figure 14. Schematics of the location of the origin for turbulence, $y^+ = -\ell _T^+$, when imposing different origins for the spanwise and wall-normal velocities. The planes where $v^{\prime +}=0$ and $w^{\prime +}=0$ correspond to the imposed virtual origins, $y^+ = - \ell _v^+$ and $y^+=-\ell _w^+$, respectively. The origin for turbulence, $y^+ = -\ell _T^+$, is represented by the red line. (a) $\ell _v^+<\ell _w^+$, (b) $\ell _v^+ = \ell _w^+$, (c) $\ell _v^+>\ell _w^+$. Note that in each case, the distance between the centre of the quasi-streamwise (Q-S) vortices and the plane $y^+=-\ell _T^+$ is the same.

A key point epitomised by (4.5) is that the only relevant parameters are the relative positions of the virtual origins of $u$ and $w$ relative to the plane where $v$ appears to vanish. The classical understanding, as first proposed by Luchini et al. (Reference Luchini, Manzo and Pozzi1991), is that the only relevant parameter is the difference between streamwise and spanwise protrusion heights. However, the results of Busse & Sandham (Reference Busse and Sandham2012) show that this is not the case. We argue that the plane where $v$ appears to vanish (or alternatively, how much the flow can transpire through the reference plane from which the tangential virtual origins are measured) is also important. The result is an extension of Luchini's theory where, rather than on the difference between the virtual origins perceived by the tangential velocities, ${\rm \Delta} U^+$ depends on their positions relative to that perceived by the wall-normal velocity, regardless of the plane taken as reference.

4.4. Scaling with Reynolds number and domain size

As discussed in § 2, the universal parameter for quantifying the performance of drag-reducing surfaces is ${\rm \Delta} U^+$. The idea is that so long as the texture size of a given surface, $L^+$, is fixed in wall units, then so would be $\ell _u^+$, $\ell _v^+$ and $\ell _w^+$, and ${\rm \Delta} U^+$ should remain essentially independent of ${Re}_\tau$. In our simulations, we impose different virtual origins on each velocity component, $\ell _u^+$, $\ell _v^+$ and $\ell _w^+$, and so we wish to determine whether this effect scales in wall units for varying Reynolds numbers. To verify this, we conduct two simulations at different Reynolds numbers, ${Re}_\tau = 180$ and 550, but keep the virtual origins constant in wall units, see cases UWV3 and UWV3H in table 1. For the two cases considered here, the slip lengths $\ell _x^+$, $\ell _y^+$ and $\ell _z^+$ are identical for both Reynolds numbers. Note that, in general, this will not necessarily be the case, since the one-to-one a priori relationship discussed in § 3.1 between the Robin slip-length coefficients in (3.4a–c), $\ell _x^+, \ell _y^+$ and $\ell _z^+$, and the resulting virtual origins, $\ell _u^+, \ell _v^+$ and $\ell _w^+$, will slightly change with the Reynolds number. However, these differences are consistent with the change in the turbulence statistics over a smooth wall as a result of varying the Reynolds number (see e.g. Moser, Kim & Mansour Reference Moser, Kim and Mansour1999), as shown figure 15. After shifting the mean velocity profile and r.m.s. velocity fluctuations by $\ell _T^+$ and rescaling them by the friction velocity at $y^+=-\ell _T^+$, they essentially collapse to the smooth-wall data. For both Reynolds numbers, the value of ${\rm \Delta} U^+ = \ell _U^+ - \ell _T^+$ measured a posteriori is 1.3 (see table 1), indicating that ${\rm \Delta} U^+$ is indeed independent of the Reynolds number for fixed values of the virtual origins $\ell _u^+$, $\ell _v^+$ and $\ell _w^+$ in wall units. However, the measured drag reduction, ${DR}$, varies between the two cases, as expected. From (2.3), ${DR}$ is smaller at higher ${Re}_\tau$, due to the increase in $U_{\delta _0}^+$ with ${Re}_\tau$, even though we observe no change in ${\rm \Delta} U^+$.

Figure 15. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for slip-length simulations UWV3, UWV3H and UWV3HD. These simulations have the same virtual origins, in wall units, for each velocity component, $(\ell _u^+,\ell _w^+,\ell _v^+) = (3.6,2.9,1.9)$, but UWV3 is conducted at ${Re}_\tau \simeq 180$, whereas UWV3H and UWV3HD are conducted at ${Re}_\tau \simeq 550$. UWV3HD has a larger domain size in the wall-parallel directions, $8{\rm \pi} \times 3{\rm \pi}$ instead of $2{\rm \pi} \times {\rm \pi}$. (a–c) Scaled with the friction velocity at the reference plane, $y^+=0$; (d–f) shifted in $y^+$ by $\ell _T^+$ and scaled with the friction velocity at the origin for turbulence, $y^+=-\ell _T^+$. Smooth-wall reference data are portrayed at (----) ${Re}_\tau \simeq 180$ and (——) ${Re}_\tau \simeq 550$; (——, blue), case UWV3; (——, magenta), case UWV3H; (${\cdot \cdot \cdot \cdot \cdot \cdot}$, red), case UWV3HD.

Our simulation domains, $2{\rm \pi} \times {\rm \pi}$, are sufficiently large to capture the key turbulence processes and length scales of the near-wall and log-law regions of the flow (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014), but scales larger than this will be unresolved. To verify that virtual origins interact only with the smaller scales that reside near the wall, we conduct an additional simulation at ${Re}_\tau \simeq 550$, UWV3HD, with the same parameters as UWV3H but a domain size $8{\rm \pi} \times 3{\rm \pi}$. The results shown in figure 15 are indistinguishable, suggesting that the origin-offset mechanism does not interact with the larger, outer turbulence scales, other than by the shift in origin.

4.5. Departure from smooth-wall-like turbulence

The fundamental idea behind the proposed virtual-origin framework, as discussed in § 3.1, is that when we impose virtual origins on each velocity component, we assume that the shape of each r.m.s. velocity profile remains smooth-wall-like independently of the others. For this assumption to hold, the near-wall turbulence cycle should be left essentially unaltered. Otherwise, these profiles will no longer be smooth-wall-like. As a guide, we can say that the virtual origins should be smaller than the smallest eddies of near-wall turbulence. As discussed in § 1, this would be the quasi-streamwise vortices, with diameter and distance to the surface both of order 15 wall units (Robinson Reference Robinson1991; Schoppa & Hussain Reference Schoppa and Hussain2002). This would then serve as a rough limit for the applicability of this framework. The cases presented thus far have all been within this regime, and we have demonstrated that the flow remained essentially smooth-wall-like, once the virtual origin for turbulence, $\ell _T^+$, was accounted for. It was also possible to predict $\ell _T^+$ from the virtual origins a priori.

To better understand the limits of this framework, we conduct a series of simulations where the imposed virtual origins are relatively large, e.g. up to $\ell _u^+,\ell _w^+ \approx 6$ and $\ell _v^+ \approx 11$. These are cases UWVL1–UWVL4, and their mean velocity profiles, r.m.s. velocity fluctuations and Reynolds stress profiles are shown in figure 16. We see that when the imposed virtual origins become too large, the r.m.s. velocity fluctuations and Reynolds stress profile no longer remain smooth-wall-like. In these cases, as we increase the depth of the virtual origins, specifically for $v$ and $w$, the quasi-streamwise vortices approach the reference plane $y^+=0$ to such an extent that they are, in fact, ingested by the domain boundary. The whole near-wall cycle is then fundamentally disrupted, changing the nature of the flow near the wall. This is most apparent for cases UWVL3 and UWVL4. The premultiplied energy spectra for these cases, shown in figure 17(a–d), indicate that there can be a dramatic change in the distribution of energy among length scales, compared to the smooth-wall case, when the imposed virtual origins are large. For example, in case UWVL4 there is a significant redistribution of energy in the wall-normal velocity to larger spanwise and shorter streamwise wavelengths. This also occurs when the transpiration triggers the appearance of Kelvin–Helmholtz-like spanwise rollers (see e.g. García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011b; Gómez-de-Segura & García-Mayoral Reference Gómez-de-Segura and García-Mayoral2019) or in the presence of roughness large enough to disrupt the near-wall cycle (Abderrahaman-Elena, Fairhall & García-Mayoral Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019). This increased spanwise coherence of $v^{\prime +}$ can also be observed in the snapshots of case UWVL4, which are compared to those of the smooth-wall reference case in figure 18. A similar behaviour was also observed by Gómez-de-Segura et al. (Reference Gómez-de-Segura, Fairhall, MacDonald, Chung and Garicía-Mayoral2018a) when using a Stokes-flow model for the virtual layer of flow below $y^+=0$, rather than the Robin slip-length boundary conditions used here.

Figure 16. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for simulations with slip-length boundary conditions applied to all three velocity components. Here, the values of the slip-length coefficients are relatively large, e.g. up to $\ell _x^+,\ell _y^+,\ell _z^+ \approx 10$. (a–c) Scaled with the friction velocity at the reference plane, $y^+=0$; (d–f) shifted in $y^+$ by $\ell _T^+$ and scaled with the friction velocity at $y^+=-\ell _T^+$. Black lines, smooth-wall reference data; blue to red lines, cases UWVL1–UWVL4.

Figure 17. Premultiplied two-dimensional spectral densities of $u^2$, $v^2$, $w^2$ and $uv$ at $y^+ + \ell _T^+ = 15$, normalised by $u_\tau$ at $y^+ = -\ell _T^+$, for various slip-length simulations (line contours), compared to smooth-wall data (filled contours) at $y^+ = 15$. The shift $\ell _T^+$ is given in table 1 for each case. (a–d) Cases UWVL1 and UWVL4, with line colours as in figure 16; (e–h) cases WV1 and WV3, with line colours as in figure 19. (a,e), $k_x k_z E_{uu}^+$; (b,f), $k_x k_z E_{vv}^+$; (c,g), $k_x k_z E_{ww}^+$; (d,h), $-k_x k_z E_{uv}^+$. The contour increments for each column are 0.3224, 0.0084, 0.0385 and 0.0241, respectively.

Figure 18. Streamwise (a,c) and wall-normal (b,d) instantaneous velocity fluctuation flow fields. (a,b) Smooth-wall reference case at $y^+= 15$, scaled with $u_\tau$ at $y^+ = 0$; (c,d) slip-length simulation UWVL4 at $y^+ +\ell _T^+ = 15$, scaled with $u_\tau$ at $y^+ = -\ell _T^+$.

The results of cases UWVL1–UWVL4, suggest that the virtual-origin framework holds only for $\ell _T^+ \lesssim 5$. Beyond this point, the Reynolds stress profiles presented in figure 16(c,f) indicate that the near-wall turbulence is no longer smooth-wall-like, and the underlying assumptions of the framework are no longer valid. As discussed above, the origin for turbulence, $\ell _T^+$, should depend only on $\ell _v^+$ and $\ell _w^+$. Note, however, that it is more difficult to impose limits on $\ell _w^+$ and $\ell _v^+$ independently, because both spanwise slip and transpiration are required to increase $\ell _T^+$ beyond five wall units, as encapsulated by (4.5). For very large spanwise slip lengths without transpiration (e.g. Busse & Sandham Reference Busse and Sandham2012), the virtual-origin framework still holds, and a saturation in the effect of $\ell _z^+$ is observed, as discussed in § 2. In turn, as we have seen in cases V1, V2, UV1 and UV2, increasing $\ell _v^+$ beyond $\ell _w^+$ bears no consequence on $\ell _T^+$, no matter how large $\ell _v^+$.

The cases considered so far satisfy $\ell _u^+ \gtrsim \ell _T^+$, i.e. the streaks perceive a virtual origin at least as deep as that perceived by the quasi-streamwise vortices. In this regime, as discussed § 4.2, the streamwise fluctuations have a greater $y$-range in which they are brought to zero by viscosity, as shown in figure 11, but otherwise the quasi-streamwise vortices and the turbulence remain smooth-wall-like. There is sufficient room for the near-wall-cycle structures to reside, and no change in the turbulence dynamics is observed. In contrast, we now consider the opposite regime, where $\ell _u^+ < \ell _T^+$. This would arguably be the case of interest for roughness, and has been shown to be the case when roughness is sufficiently small (Abderrahaman-Elena et al. Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019). We fix the origin for the streamwise velocity at $y^+=0$, i.e. $\ell _u^+ = 0$, and progressively increase the depth of the origin for turbulence below this plane, i.e. $\ell _T^+>0$. As portrayed in figure 19, for cases WV1–WV3, we then observe a gradual departure from smooth-wall-like turbulence. Note that $\ell _U^+<\ell _T^+$ in these cases, and so ${\rm \Delta} U^+<0$, which would correspond to an increase in drag. Case WV1, with $\ell _T^+\approx 2$, appears to be the limiting case, in which turbulence still remains essentially smooth-wall-like, as can be observed in figure 19(d–f), and $\ell _{T,pred}^+$, calculated from (4.5), still provides a reasonable estimate for the origin for turbulence, as shown in table 1. However, increasing $\ell _T^+$ further results in clear differences between the r.m.s. velocity fluctuations and Reynolds stress profiles compared to the smooth-wall data. This can also be seen in the premultiplied energy spectra shown in figure 17(e–h), where the distribution of energy among length scales is no longer smooth-wall-like. For example, the spectra of case WV3 indicates that the energy in the streamwise and spanwise velocity components is now shifted, on average, to shorter streamwise wavelengths. Schematically, the quasi-streamwise vortices would approach the reference plane, but the streaks would be constrained by the condition that $u^{\prime +}=0$ at $y^+=0$. The streaks, which are sustained by the vortices, would then no longer have sufficient room to reside above $y^+=0$, compared to the flow over a smooth wall, and would become squashed in $y$ and weakened. This, in turn, would restrict the whole near-wall turbulence dynamics, and cause the flow not to remain smooth-wall-like. From our simulations, this breakdown appears to occur when the virtual origin for turbulence is more than approximately 2 wall units deeper than the origin perceived by the streaks. Therefore, an additional constraint on the present virtual-origin framework would be that the imposed virtual origins should satisfy $\ell _T^+ \lesssim \ell _u^+ + 2$. This is in agreement with the observation in Abderrahaman-Elena et al. (Reference Abderrahaman-Elena, Fairhall and García-Mayoral2019) that a virtual-origin framework alone cannot capture the effect of roughness on the flow once the roughness size is large enough that ${\rm \Delta} U^+ \simeq -2$. Further, it highlights the limitations of modelling the effects of drag-increasing surfaces, such as roughness, with virtual origins alone.

Figure 19. Mean velocity profiles, r.m.s. velocity fluctuations and Reynolds shear stress profiles for simulations with slip-length boundary conditions applied to the spanwise and wall-normal velocity components only. (a–c) Scaled with the friction velocity at the reference plane, $y^+=0$; (d–f) shifted in $y^+$ by $\ell _T^+$ and scaled with the friction velocity at $y^+=-\ell _T^+$. Black lines, smooth-wall reference data; blue to red lines, cases WV1–WV3.

The breakdown of the virtual-origin framework and the subsequent departure from smooth-wall-like turbulence require further discussion. The homogeneous slip-length boundary conditions (3.4a–c) are an approximation of the apparent boundary conditions that real textured surfaces impose on the flow. They are a reasonable model so long as the characteristic texture size is small compared to the length scales of the turbulent eddies in the flow (García-Mayoral et al. Reference García-Mayoral, Gómez-de-Segura and Fairhall2019). As the texture size, $L^+$, is increased, we expect the apparent virtual origins that a given surface imposes on the flow to become deeper. However, on increasing $L^+$ further, flows over real textured surfaces eventually exhibit additional dynamical mechanisms, typically drag-degrading, such that the effect of the texture can no longer be approximated by a simple virtual-origin model. For example, as $L^+$ increases for superhydrophobic surfaces, the flow begins to perceive the texture as discrete elements, as opposed to a homogenised effect (Seo & Mani Reference Seo and Mani2016; Fairhall et al. Reference Fairhall, Abderrahaman-Elena and García-Mayoral2019), and the entrapped gas pockets can also be lost (Seo, García-Mayoral & Mani Reference Seo, García-Mayoral and Mani2018), both of which fundamentally change the apparent boundary conditions imposed by the surface on the flow. For riblets, in turn, increasing the texture size can trigger the onset of Kelvin–Helmholtz-like rollers, which can have a strong drag-increasing effect on the flow (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011b). Importantly, the depth of the virtual origin for turbulence at which the present framework breaks down, i.e. $\ell _T^+ \approx 5$, could imply a texture size that would place a corresponding real surface in a regime beyond the onset of the failure mechanisms just mentioned. For instance, the onset of the Kelvin–Helmholtz-like instability in riblets can occur for $\ell _T^+ \gtrsim 1$ (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011b). In this case, the limits imposed by the operating window of the real surface are the most restrictive, and not the theoretical limits of the virtual-origin framework. Therefore, if the goal of a given simulation is to use virtual origins to model the effect on the flow of a real surface, it is crucial to keep in mind the texture size, and hence the magnitude of the virtual origins, at which the flow no longer perceives the surface in a homogenised fashion. This could, in many instances, be the actual limit up to which the virtual-origin framework can be feasibly applied.