Resolving turbulence and drag over textured surfaces using texture-less simulations:
the case of slip/no-slip textures

This paper focuses on the effect on wall turbulence of surface texture, which alters the flow and modifies the drag compared to a smooth surface. The most common example of surface texture is roughness (Schlichting, 1937; Colebrook & White, 1937; Jiménez, 2004; Chung et al., 2021), which generally increases drag, but there are also textures such as riblets (Walsh & Lindemann, 1984), superhydrophobic surfaces (Rothstein, 2010) and anisotropic permeable substrates (Abderrahaman-Elena & García-Mayoral, 2017; Gómez-de-Segura & García-Mayoral, 2019) that can reduce drag.

According to the classical theory of wall turbulence, for small textures the only effect of the surface on the outer flow is a uniform shift in the mean velocity profile, $\Delta U^{+}$, compared to a smooth surface (Clauser, 1956; Spalart & McLean, 2011; García-Mayoral et al., 2019). The superscript ‘+’ indicates scaling in viscous units, i.e. normalisation by the the kinematic viscosity $\nu$ and friction velocity $u_{\tau}=\sqrt{\tau_{w}}$, where $\tau_{w}$ is the tangential stress at the wall. Throughout the paper, we consider upward shifts of the velocity profile, which occur for drag-reducing surfaces, positive , $\Delta U^{+}>0$, and downward shifts of the velocity profile, which occur for drag-increasing surfaces, negative, $\Delta U^{+}<0$, so our $\Delta U^{+}$ is the opposite of the usual ‘roughness function’ used for drag-increasing textures. The free-stream velocity, $U_{\delta}^{+}$, is then given by

where $c_{f}$ is the skin friction coefficient based on $U_{\delta}$, $\delta$ is the flow thickness, and the von Kármán constant, $\kappa$, and the parameter $B$, which includes the smooth-wall logarithmic intercept and the wake function, remain unchanged. When considering boundary layers, $\delta$ is the boundary-layer thickness. For channels, $\delta$ is the channel half-height and we take the centreline velocity as $U_{\delta}$, to allow for direct comparison with boundary layers (García-Mayoral & Jiménez, 2011; García-Mayoral et al., 2019). Following equation 1, the relative change in drag compared to a smooth-wall flow at the same $Re_{\tau}$ is

where the subscript ‘0’ denotes reference smooth-wall values and $\Delta c_{f}=c_{f}-c_{f0}$.

In the limit where the size of the texture elements, $L$, is vanishingly small compared to any length scales in the overlying flow, the background turbulence does not perceive the detail of each individual texture element, and only experiences the surface in an averaged sense. Such surfaces are well suited for homogenisation and can be characterised through uniform effective boundary conditions (Lācis & Bagheri, 2017; Bottaro, 2019). Modelling textured surfaces by homogeneous boundary conditions is convenient and computationally cheap, since high resolution is not required near the surface to resolve the flow around the texture elements. However, homogenisation relies on a small-parameter expansion on the ratio of $L$ to the characteristic lengthscales in the flow, so it formally ceases to apply when $L$ becomes comparable to the size of the smallest eddies in the flow, which taking for the latter the diameter of quasi-streamwise vortices would be $L^{+}\sim 10$. This limit would leave out most practical applications. Here, we aim to investigate the effect on turbulence of textures beyond this vanishingly small limit, i.e. once homogenisation breaks down. As a first step, in this paper, we focus on the particularly simple case of surface textures made of alternating slip/no-slip regions, which is a common model for superhydrophobic surfaces.

In this case, the resulting homogenised boundary conditions are $v=0$ in the wall-normal direction and $u=\ell_{x}(\partial u/\partial y)$ and $w=\ell_{z}(\partial w/\partial y)$ in the wall-parallel directions, where $\ell_{x}$ and $\ell_{z}$ are the streamwise and spanwise slip lengths (Philip, 1972). Gómez-de-Segura & García-Mayoral (2020) and Ibrahim et al. (2021) showed that the effect of these homogenised boundary conditions is a mere offset of the origins perceived by different flow components, while turbulence remains essentially the same as over smooth walls. A streamwise slip shifts the mean velocity profile by the slip velocity, reducing the drag, while a spanwise slip allows quasi-streamwise vortices, an essential part of the near-wall turbulent cycle, to move closer to the surface (Luchini, 1996; Min & Kim, 2004), increasing the drag. Fairhall & García-Mayoral (2018) showed that for surfaces modelled using slip lengths, $\Delta U^{+}$ equals the offset of the virtual origin perceived by mean flow, $\ell_{x}^{+}$, and the one perceived by turbulence, $\ell_{T}^{+}$. Gómez-de-Segura & García-Mayoral (2020) and Ibrahim et al. (2021) showed that $\ell_{T}^{+}$ is governed by the interplay between the spanwise slip, $\ell_{z}^{+}$, and the transpiration. For surfaces with no transpiration, such as the cited slip/no-slip patterns, the virtual origin of turbulence only depends on the spanwise slip length, and Fairhall & García-Mayoral (2018) proposed

from an empirical fit of the results of Busse & Sandham (2012).

Seo & Mani (2016) investigated the limits of slip-length models for slip/no-slip textures, and showed that for texture sizes $L^{+}\gtrsim 10$, the instantaneous correlation between velocity and shear at the surface was lost, which would appear to set the upper limit for the applicability of a slip-length model. Fairhall & García-Mayoral (2018) later investigated the correlation between surface velocity and shear using a spectral approach to discriminate between the slip lengths experienced by different lengthscales in the overlying flow. They found that even scales much larger than the texture size displayed an apparent loss of correlation. Fairhall & García-Mayoral (2018) argued that this observed loss of correlation of the slip length was due to the intensity of the texture-coherent flow, rather than the texture size, becoming significant. We note that Jelly et al. (2014) and Türk et al. (2014) reported that the texture-coherent flow becomes significant compared to the background turbulence for $L^{+}\gtrsim 100$, while the effect discussed by Fairhall & García-Mayoral (2018) sets in for $L^{+}\gtrsim 10$, and is caused by the discrete slip/no-slip pattern induced by the texture being broad-band in wavelength space, scattering the texture-coherent signal across the full range of lengthscales. In a follow-up work, Fairhall et al. (2019) used the amplitude-modulated flow decomposition proposed by Abderrahaman-Elena et al. (2019) to filter out the texture-coherent flow, and showed that the left-over background turbulence still exhibits a linear correlation between velocity and shear, so that a slip length can be meaningfully defined.

However, when replacing the texture by the homogeneous slip lengths perceived by the overlying turbulence, results differed for $L^{+}\gtrsim 25$, as the turbulent variables become no longer smooth-wall-like. Fairhall et al. (2019) suggested that the breakdown of the homogeneous, slip-length model is not caused by the breakdown of the effective boundary conditions, but by a different mechanism. They showed that the difference in $\Delta U^{+}$ between the flows over their textures and the flows with equivalent homogeneous boundary conditions arose from modifications to the Reynolds stress above rather than directly at the surface. For the simple case of slip/no-slip textures, this is particularly clear because the zero-transpiration boundary condition ($v=0$) yields zero Reynolds stress, $uv$, at the surface. For the textures of Fairhall et al. (2019), the differences in $uv$ occurred at heights $5\lesssim y^{+}\lesssim 25$ above the surface. Fairhall et al. (2019) argued that the extra Reynolds stress over textures was caused by the non-linear interactions between the texture-coherent flow and the background turbulence, which modify the dynamics of the latter and result in degraded drag. We note that although these observations were made in the context of drag-reducing, zero-transpiration textures, we have also observed a similar behaviour for drag-increasing surfaces such as roughness (Abderrahaman-Elena et al., 2019), small-size dense canopies (Sharma & García-Mayoral, 2020b) and porous substrates (Hao & García-Mayoral, 2024), which suggests that this may be a common mechanism across a wide variety of textures. Abderrahaman-Elena et al. (2019) argued that a virtual-origin framework alone could only account for roughness functions up to $-\Delta U^{+}\approx 2$, spanning the early stages of the transitionally rough regime. The same conclusion was reached in Habibi Khorasani et al. (2022) using homogenised boundary conditions.

In this paper, we aim to identify and characterise the physical mechanism that causes the flow to depart from its dynamics under equivalent boundary conditions in the presence of actual, resolved textures for $L^{+}\gtrsim 25$. In particular, we assess if the effect of the texture can be captured by replacing it by its corresponding homogeneous boundary conditions plus, critically, the advective terms that arise from the existence of a texture-coherent flow and its interaction with the background turbulence. Compared to Navier-Stokes, the momentum equations for the background turbulence include then additional, ‘forcing’ cross advective terms. We also explore preliminarily if this can be used for predictive modelling using a priori surrogates for the texture-coherent flow.

The paper is organised as follows. The decomposition of the flow field into background-turbulence and texture-coherent components, the resulting governing equations for the mean flow and background turbulence, and the cross non-linear terms that we will introduce in our texture-less model are presented in §2. The numerical methods are outlined in §3. The results from the model with forcing are presented, analysed and compared with those of texture-resolved and homogeneous-slip simulations in §4. Conclusions are summarised in §5.

To investigate the non-linear interaction between background turbulence and texture-coherent flow, direct numerical simulations (DNS) of turbulent channels were conducted. The numerical code is adapted from that of Fairhall et al. (2019), and is briefly summarised here. The three-dimensional incompressible Navier-Stokes equations are solved using a spectral discretisation in the streamwise and spanwise directions with the wall-normal direction discretised by second-order finite differences on a staggered grid. A fractional step method (Kim & Moin, 1985), combined with a three-step Runge-Kutta scheme, is used to advance in time, with a semi-implicit scheme used for the viscous terms and an explicit scheme for the advective terms (Le & Moin, 1991). The simulations were run with constant mean pressure gradients, adjusted to achieve the desired friction Reynolds numbers, Re_τ. The channel is of size $2\pi\delta\times\pi\delta\times 2\delta$ in the streamwise, spanwise and wall-normal directions respectively, where $\delta$ is the channel half-height. Once a statistically steady state was reached, statistics were obtained over 10 times the characteristic largest-eddy-turnover time, $\delta/u_{\tau}$.

Three sets of DNSs were conducted. The first set consists of texture-resolving simulations with patterns of alternating regions of slip/no-slip boundary conditions on the channel walls. The first set used the DNS code from Fairhall et al. (2019) unmodified. The surfaces are assumed rigidly flat, which leads to zero transpiration. This is a widely used idealisation for superhydrophobic surfaces, where the free-slip regions represent the gas pockets, and the no-slip regions the exposed tips of solid posts. The assumption of free slip is reasonable if the gas pockets are sufficiently deep (Schönecker et al., 2014). The assumption of flat, rigid interfaces is reasonable for $L^{+}\lesssim 30$ for typical applications (Seo & Mani, 2018). We note nevertheless that in this study we use this model to keep the surface texture as simple as possible, regardless of whether the model is a suitable representation or not of superhydrophobic textures. The latter question is out of the scope of this work.

We consider texture elements consisting of periodic arrays of square posts, both in collocated and staggered arrangements, as illustrated in figure 4, with a solid fraction, the ratio of post area to total surface area, $\phi_{s}\approx 1/9$. The texture unit is a square of side $L$, repeated periodically in the streamwise and spanwise directions. For our different simulations, the texture size ranges from $L^{+}\approx$ 18 to 94 for the collocated textures and $L^{+}\approx$ 33 to 66 for the staggered textures.

The second set of DNSs replaces the textured surfaces by their corresponding homogeneous slip lengths, obtained a posteriori from the corresponding texture-resolving simulations. This set also used the DNS code from Fairhall et al. (2019) unmodified. The last set of DNSs are homogeneous-slip simulations with added forcing terms, with the same slip boundary conditions of the second set. The governing momentum equations have the additional non-linear forcing terms introduced in section 2.2, with the texture-coherent velocities obtained from ensemble-averaging in the texture-resolving simulations.

In order to reduce the computational cost of the texture-resolving simulations, the code uses a multiblock layout, where the near-wall regions have finer grids compared to the channel centre. This is done in order to resolve fully not just the channel turbulence but also the texture-coherent flow, which typically requires higher resolution. In the blocks containing the walls, each texture element is resolved by $24\times 24$ grid points in the $x$ and $z$ directions. In the central block, the grid resolution is $\Delta x^{+}\approx$ 8.8 and $\Delta z^{+}\approx$ 4.4. The latter resolution is also used throughout in the simulations with homogeneous slip. In the wall-normal direction, the grid is stretched, with resolution $\Delta y_{min}^{+}\approx$ 0.3 at the surfaces and $\Delta y_{max}^{+}\approx$ 3 in the channel centre, and with the fine-$x$-$z$- resolution blocks extending a height $\approx L$ into the channel from the wall, which was verified a posteriori to exceed the height at which any fine, texture-induced flow became vanishingly small. Overall, this resulted for instance in roughly $10^{8}$ grid points for case TX18 or $2\times 10^{7}$ for TX47, which can be compared with $2\times 10^{6}$ for the corresponding smooth-wall and texture-less simulations, or $4\times 10^{8}$ for TX18 if uniform resolution in $x$ and $z$ had been applied throughout.

The texture-resolving and homogeneous-slip simulations with collocated elements with sizes $L^{+}\approx 18$ to 47 are from Fairhall et al. (2019). Additional simulations with $L^{+}\approx 71$ and 94, as well as the simulations with staggered elements, have been conducted to complete and expand the database. Simulations with homogeneous slip and added forcing have been conducted matching all the above cases to investigate the effect of the non-linear interaction. The parameters of all three sets of simulations are listed in table 1.

In this section, we present and discuss the results of the DNSs summarised in table 1. We first compare the drag predictions obtained from simulations with the texture resolved and those with its effect modelled. Figure 6 portrays those results in terms of the velocity increment $\Delta U^{+}$. For each simulation setup, the figure shows the usual increase of $\Delta U^{+}$ with texture size $L^{+}$. The results with resolved textures and with the corresponding slip boundary conditions agree well only up to texture sizes $L^{+}\approx 20$ (for smaller textures see results in Fairhall et al., 2019) for collocated layouts and $L^{+}\approx 35$ for staggered ones, although we note that the streamwise spacing between successive rows of posts is then $L_{x}^{+}\approx 25$, close to the collocated value. For larger spacings, using the equivalent homogeneous slip increasingly overpredicts $\Delta U^{+}$, by $\sim 35$% for $L^{+}\gtrsim 35$ in the case of collocated posts, and by a similar proportion for staggered ones. This difference was already reported by Fairhall et al. (2019). They argued that, given that the effective boundary conditions perceived by the overlying turbulence were the same for texture-resolved and slip simulations, the difference in drag had to arise from differences in the overlying flow. Introducing the forcing terms without amplitude modulation from §2.2.1 improves the prediction of $\Delta U^{+}$ for slip-only simulations partially, as shown in Figure 6. The error remains however large, and is only reduced by 30–50%. In turn, introducing the forcing terms with amplitude modulation from §2.2.2 yields values of $\Delta U^{+}$ in good agreement with those of texture-resolved simulations at least up to texture sizes $L^{+}\approx 70$. As the texture size increases further to $L^{+}\approx 100$, the results begin to depart. Although the deviation remains below 10%, we already expected the forcing model to break down in this range of $L^{+}$, as the assumption gradually ceases to hold that there is sufficient separation of scales between the overlying shear and the texture-coherent flow it induces, rendering the flow decomposition used in the forcing model invalid. This is further discussed in §4.1.

For the staggered-posts layouts, the values of $\Delta U^{+}$ are lower than for collocated layouts with the same $L^{+}$. This is in agreement with the Stokes-flow predictions for small textures of Sbragaglia & Prosperetti (2007). They argued that collocated textures channel the flow through streamwise-aligned channels between posts, while staggered arrangements obstruct this channelling effect, reducing the mean slip velocity. This obstruction was also important in the simulations of Seo & Mani (2018), who observed that the slip lengths measured from DNSs of surfaces with randomly distributed texture elements was reduced by approximately 30% compared to collocated elements. In any event, the results for texture-resolved, slip-only and slip-plus-forcing simulations for staggered posts follow generally the same trends observed for collocated posts, albeit with lower values of $\Delta U^{+}$ for the same $L^{+}$, and are therefore not presented in the remainder of this section. They are nevertheless included for completeness in appendix B.

The agreement exhibited by the roughness function between texture-resolved simulations and texture-less simulations with amplitude-modulated forcing extends to other flow properties. This is the case for instance of one-point turbulent statistics. Figures 7 and 8 portray the r.m.s. velocity fluctuations and the shear Reynolds stress and mean velocity profile for texture-resolved, slip-only and slip-plus-forcing simulations, both using amplitude-modulated and conventional triple decomposition, across the range $L^{+}\approx 20$–$100$. For $L^{+}\approx 20$, even slip-only simulations show good agreement with fully resolved ones. This was reported by Fairhall et al. (2019) as the limit size for which slip-lengths alone could capture the effect of the texture, as the texture-coherent flow is small in amplitude and confined to the immediate vicinity of the surface, and does therefore not alter the background turbulence significantly. The latter remains then smooth-wall like, other than by a shift in apparent origins (Ibrahim et al., 2021). In agreement with this, adding forcing to model the effect of the texture-coherent fluctuations in this $L^{+}$ range has little effect and essentially does not alter the flow. As the texture size increases from $L^{+}\approx 20$, though, the flow begins to depart from the smooth-wall-like behaviour that slip boundary conditions yield, as shown in figures 7 and 8, with a decrease of the streamwise fluctuation intensity above $y^{+}\approx 10$, and an increase throughout of the spanwise and wall-normal intensities and of the shear Reynolds stress. We note that the latter is in all cases zero at the surface, a unique feature of slip-no slip textures caused by the zero transpiration at $y=0$. In general, the above modifications, which tend to decay sufficiently away from the wall, roughly at $y^{+}\approx$ 50, are observed over slip/no-slip textures (Seo et al., 2015; Fairhall et al., 2019) but also over rough surfaces (Orlandi & Leonardi, 2006; Abderrahaman-Elena et al., 2019).

The addition of forcing using conventional triple decomposition can reproduce some of the departures from smooth-wall-like flow mentioned above for textured simulations, but forcing with amplitude modulation shows much better agreement with the resolved-texture cases. The agreement is excellent up to $L^{+}\approx 50$ and first begins to break down for the spanwise velocity. This was to be expected given the simplifications made in equation 15, and has little effect on the Reynolds stress and, therefore, on the mean velocity profile and the drag. The agreement breaks down further for $L^{+}\approx 70$, for which it begins to propagate into $\Delta U^{+}$, although it is still reasonable. For $L^{+}\approx 100$ the departures become significant. We therefore identify this as the limit beyond which the model proposed here fails.

As mentioned in section 1, the differences in the Reynolds stress profile shown in figure 8 are caused by the changes in the background turbulence. The latter are what the forcing models ultimately aim to capture. The increased Reynolds stress causes a downward shift of the mean velocity profile away from the wall, and a corresponding reduction in $\Delta U^{+}$. This relationship can be quantified by integrating the streamwise momentum equation. Here we follow Gómez-de-Segura & García-Mayoral (2019). A first integral gives

where $\tau_{uv}$ is the shear Reynolds stress, including any dispersive stress, and $\delta^{\prime}=\delta+\ell_{T}$ is the effective half-height of the channel, which accounts for the background turbulence perceiving a virtual origin at $y=-\ell_{T}$. Integrating equation 26 once more in the wall-normal direction, from the surface to a height $H$ sufficiently far above for all surface effects to have vanished, gives

where $U_{\mathrm{slip}}=U(y^{+}=0)$ is the slip velocity, and $f$ is a simple function of $H^{+}$, $\delta^{\prime+}$ and $\ell_{T}^{+}$. The same integral can be repeated for a reference smooth-wall flow at the same friction Reynolds number $\delta^{\prime+}$ between the corresponding heights, $y=\ell_{T}$ and $y=H+\ell_{T}$, yielding

where the subscript S denotes smooth-wall flow. The roughness function $\Delta U^{+}$ can then be obtained by subtracting equations 27 and 28,

For slip-only simulations, turbulence is smooth-wall like and the integral in equation 29 is essentially zero (Fairhall et al., 2019; Ibrahim et al., 2021). Near the wall, $\mathrm{d}U^{+}/\mathrm{d}y^{+}\approx 1$, so for $\ell_{x}^{+}\lesssim 10$ we have $U^{+}_{\mathrm{slip}}\approx\ell_{x}^{+}$. From equation 3, we have $\ell_{T}^{+}\lesssim 4$, so also $U_{\mathrm{S}}^{+}\left(\ell_{T}^{+}\right)\approx\ell_{T}^{+}$. The roughness function reduces then to the offset between the apparent origins for the mean flow and for turbulence, $\Delta U^{+}\approx\ell_{x}^{+}-\ell_{T}^{+}$ (Luchini, 1996). For texture-resolved simulations, however, in addition to this origin offset, the increase in Reynolds stress causes further modifications to the mean velocity profile and the roughness function. Fairhall et al. (2019) observed that for the present slip/no-slip textures, with zero transpiration and dispersive stress at $y=0$, the contribution from the latter to $\tau_{uv}$ is negligible up to at least texture sizes $L^{+}\approx 50$, as shown in figure 2. Those modifications would then be essentially caused by changes in the background, texture-incoherent turbulence alone. Figure 8 shows that the forcing model with amplitude modulation is able to capture the effect of these changes on the Reynolds stress, and thus on $\Delta U^{+}$, while the one with conventional triple decomposition captures those changes only partially.

The modifications in the background turbulence can be observed in more detail in the spectral density maps of the different variables, as those portrayed in figure 9. Such maps show the contributions to the statistics shown in figures 7 and 8 at a given height $y$ from different streamwise and spanwise lengthscales. Figure 9 displays the energy densities at $y^{+}\approx 15$, a height of intense r.m.s. fluctuations of the background turbulence and also sufficiently above the surface for the texture-coherent flow to be negligible. As observed by Ibrahim et al. (2021), the signature of turbulence in slip-only simulations is essentially the same as over smooth walls. Compared to those cases, for fully resolved textures there is additional energy in shorter streamwise scales, and also in wider spanwise scales particularly for $v$. This effect is negligible for $L^{+}\lesssim 20$, but becomes increasingly marked for greater $L^{+}$. Figure 9 shows that slip-only models fail to capture this gradual change in the dynamics of the background turbulence for $L^{+}\gtrsim 25$. The forcing model based on conventional triple decomposition is able to generate some additional energy in shorter and wider scales, but not to the full extent observed in texture-resolved simulations. In contrast, the forcing model based on amplitude modulation can generate energy in all the necessary scales, and results in a good overall collapse of the spectra with that for resolved textures up to $L^{+}\approx 70$. For larger $L^{+}$, deviations appear first in the core regions of the maps for $v$ and $w$ and in streamwise long scales of $u$ and $uv$. These spectra indicate again that the additional forcing terms based on the amplitude-modulated decomposition can capture the effect of the texture on the background turbulence, while forcing based on conventional triple decomposition shows only a limited improvement compared to slip-only simulations.

To illustrate the general appearance of the flow, instantaneous realisations of the three velocity components for texture-resolved, slip-only, and amplitude-modulated-forcing simulations are shown in figure 10 at a height $y^{+}\approx 15$ for texture size $L^{+}\approx 50$. This size is large enough to exhibit significant differences across models, yet not so large that the forcing model begins to fail. While the slip-only simulation exhibits canonical, smooth-wall-like-turbulence structures, both the texture-resolving and the forcing-model simulations show a similar disruption of the latter, with a reduction in the coherence of the streamwise velocity for scales of order 1000 wall units, as indicated also by the energy spectra. For the wall-normal and spanwise velocities, the lengthscales of the turbulent eddies are also shorter compared to those over homogeneous slip. At this $L^{+}\approx 50$, the lengthscales and magnitude of the texture-coherent flow near the wall become comparable to those of the background turbulence. The spectra and instantaneous flow fields suggest a modification of the near-wall dynamics through the disruption of streaks and quasi-streamwise vortices by this texture-coherent flow. The similarity of the amplitude-modulated-forcing flow field and the texture-resolved one suggests that the alteration of the background turbulence by the presence of the texture can also be captured by the forcing model.

Finally, to verify that the expected scaling in viscous units for textures whose effect is confined to the vicinity of the wall (García-Mayoral & Jiménez, 2012) also holds for the corresponding models, in addition to the simulations at $Re_{\tau}\approx 180$ we have conducted simulations at $Re_{\tau}\approx 400$ for the collocated textures of size $L^{+}\approx 35$ and $L^{+}\approx 100$. The results support this scaling, and are presented and discussed in appendix C.

The simulation and prediction of turbulence over textured surfaces requires a spatial resolution that can be more demanding than that needed to resolve the turbulence itself, and can become prohibitive for texture sizes in the range $L^{+}\approx 5$-$100$. This is a relevant range for applications, and includes the working range for drag-reducing textures and the transitionally rough range for drag-increasing ones, beyond which the effect on drag usually asymptotes. In this paper we have investigated the effect of the texture detailed geometry on the background turbulence, with the ultimate aim to sidestep the need to resolve that geometry directly. As a first approach, we have focused on a particularly simple, idealised texture, consisting of a slip/no-slip pattern on an otherwise flat surface, a popular model for superhydrophobic surfaces. Such slip/no-slip textures have some unique properties that simplify their analysis, namely that the effective boundary conditions for the overlying turbulence are well known and characterised, reducing to slip, Robin conditions for the tangential velocities and zero transpiration; and that the wall-normal velocity and Reynolds stress are zero at the reference, interfacial plane, and remain negligible in its vicinity. This allowed Fairhall et al. (2019) to identify that drag degraded due to additional Reynolds stresses occurring above the surface, rather than arising at the surface and propagating into the flow above. They proposed that such Reynolds stresses had to originate from the non-linear interaction of the texture-coherent flow with the background turbulence. This interaction became significant and could not be neglected for $L^{+}\gtrsim 25$, and would need to be accounted for in any model in addition to the effective slip boundary conditions, which still held up in this range of sizes and at least up to $L^{+}\approx 50$,

Following up from Fairhall et al. (2019), we have analysed how the decomposition of the flow into a texture-coherent and a background-turbulence components propagates into the governing equations of the two components separately. We have first decomposed the flow using conventional triple decomposition, which produces a much simpler set of governing equations. However, as observed by Abderrahaman-Elena et al. (2019), this decomposition results in a residual texture-coherent signature in the background turbulence. To separate fully the texture-coherent signal and obtain a texture-incoherent background turbulence, Abderrahaman-Elena et al. (2019) proposed a decomposition in which the latter modulates in amplitude the former. Here we have analysed the governing equations for the background turbulence that result from this decomposition. For both the conventional and the amplitude-modulated triple decomposition, we have argued that the background turbulence is governed by Navier-Stokes equations with additional, forcing terms, arising from the cross advective terms between both flow components.

To assess the above hypothesis, we have conducted a series of simulations in which the background turbulence was fully resolved, but the texture was replaced by the corresponding effective boundary conditions plus the forcing terms in the momentum equations mentioned above. For the formulation based on conventional triple decomposition, the results are partially improved compared to using effective boundary conditions alone. For the formulation based on the amplitude-modulated decomposition, the results show excellent agreement up to texture sizes $L^{+}\approx 50$, and begin to deviate for $L^{+}\approx 70$. This is the case both for the prediction of drag and $\Delta U^{+}$, and for the statistical properties of turbulence, including r.m.s. velocity fluctuations, Reynolds stresses and mean velocity profiles. In particular, the addition of forcing terms can accurately generate fluctuations at shorter wavelengths than smooth-wall flows, which are present for geometry-resolved simulations but which the effective boundary conditions are unable to generate on their own. Once deviations begin to occur, for texture sizes $L^{+}\gtrsim 70$, they first manifest in the spanwise velocity component. This is also the component for which the flow decomposition first begins to break down, likely due to the span of background-turbulence eddies becoming comparable to the texture size earlier than their length, which disrupts the implicit assumption of separation of scales embedded in the amplitude-modulated decomposition. Concurrently, the texture-coherent motions excited not only by the overlying streamwise velocity, but by the cross velocity components, cease to be negligible, and therefore our simplifications in the forcing also cease to hold.

The above results strongly suggest that the key effects of surface texture on the overlying turbulence are imposing a set of effective boundary conditions but, critically, also altering the momentum equations through the non-linear interaction with the texture-coherent flow in the advective terms. While this is useful in terms of identifying physical mechanisms, it falls short in terms of predictive power, as quantifying that interaction requires prior knowledge of the texture-coherent flow, but the latter only becomes available from the postprocessing of fully resolved flows. We have therefore also explored the possibility of using a priori surrogates for the texture-coherent flow, based on the laminar steady flow about a single periodic unit of texture. Preliminary results show good agreement, motivating further work.

The present study has only considered a particularly simple type of surface texture. The results are encouraging, but would need to be further tested in more complex and general textures, mainly roughness. Many of the simplifications possible here, e.g. the effective boundary conditions reducing to homogeneous slip and zero transpiration, or accounting only for the texture-coherent flow induced by the overlying streamwise velocity, will likely not apply in the general case, calling for a more complete and complex framework.

[Acknowledgements] Computational resources were provided by the University of Cambridge Research Computing Service under EPSRC Tier-2 grant EP/P020259/1, and by the UK ’ARCHER2’ system under PRACE (DECI-17) project pr1u1702 and EPSRC project e776. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.

[Declaration of interests]The authors report no conflict of interest.

[Author ORCIDs]
Chris Fairhall, https://orcid.org/0000-0002-2529-3650;
R. García-Mayoral, https://orcid.org/0000-0001-5572-2607.