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

W. Xie 1  C. T. Fairhall 1  R. García-Mayoral 1<EMAIL_ADDRESS>1Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK

###### Abstract

We study the effect of surface texture on an overlying turbulent flow for the
case of textures made of an alternating slip/no-slip pattern, a common model
for superhydrophobic surfaces but also a particularly simple form of texture.
For texture sizes $L^{+}\lesssim 20$, we have previously reported that
turbulence remained smooth-wall-like, other than experiencing an apparent
origin offset for different flow components. For slip/no-slip textures, this
effect reduced to the flow experiencing slip conditions in the streamwise and
spanwise directions and zero transpiration at the surface. The overlying
turbulence effectively perceived such boundary conditions at least up to
texture sizes $L^{+}\approx 50$. However, beyond $L^{+}\approx 20$ the texture
interacted with the overlying turbulence in a non-homogeneous fashion,
additional Reynolds stresses arose and turbulence was no longer smooth-wall-
like. This is the typical effect of surface texture observed for rough
surfaces, and results in an increase in drag relative to smooth-wall flows. In
this paper, we argue that this occurs because the texture modifies the
overlying turbulence through non-linear, cross-advective terms between the
background turbulence and the texture-coherent flow directly induced by the
surface topology. To verify this, we conduct homogeneous-slip-length
simulations where we introduce additional, forcing terms in the Navier-Stokes
equations to capture the effect of this non-linear interaction on the
background turbulence. The interaction can then be accounted for without the
need to resolve the surface texture. We show that the additional terms
quantitatively capture the changes in the flow up to texture sizes
$L^{+}\approx 70$–$100$, including not just the roughness function but also
the flow statistics and structure.

###### keywords:

Near-wall turbulence, surface texture

##  1 Introduction

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

$U_{\delta}^{+}=\left(\frac{2}{c_{f}}\right)^{1/2}=\frac{1}{\kappa}\log\delta^{+}+B+\Delta
U^{+},$ (1)

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

$\frac{\Delta c_{f}}{c_{f0}}=\frac{1}{(1+\Delta U^{+}/U_{\delta
0}^{+})^{2}}-1,$ (2)

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

$\ell_{T}^{+}\approx\frac{\ell_{z}^{+}}{1+\ell_{z}^{+}/4}.$ (3)

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 $uv$ occured 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 further investigate the above non-linear interaction by
replacing textures by their corresponding homogeneous boundary conditions plus
the advective terms that arise from the presence of a texture-coherent flow.
Compared to Navier-Stokes, these take the form of additional forcing terms in
the governing equations for the background turbulence. 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.

##  2 Flow decomposition and governing equations

###  2.1 Amplitude-modulated triple decomposition

Conventional triple decomposition is often used to obtain a texture-coherent
and a texture-incoherent flow component. It decomposes the flow into a space-
time-averaged mean flow, a time-averaged component which is phase-locked to
the texture, and the remaining time-space fluctuations,

$\begin{split}u(x,y,z,t)&=U(y)+u^{\prime}(x,y,z,t)\\\
&=U(y)+\widetilde{u}_{u}(\widetilde{x},y,\widetilde{z})+u_{T}(x,y,z,t),\end{split}$
(4)

where $x$, $y$ and $z$ denote the streamwise, wall-normal and spanwise
directions respectively, $U(y)$ is the mean velocity profile, and
$u^{\prime}(x,y,z,t)$ is the full fluctuating field. The latter is further
decomposed into a texture-coherent component, or dispersive flow,
$\widetilde{u}_{u}(\widetilde{x},y,\widetilde{z})$, where $\widetilde{x}$ and
$\widetilde{z}$ refer to the local coordinates within the texture period, and
the remaining incoherent, background turbulent fluctuations $u_{T}(x,y,z,t)$.
The texture-coherent component can be viewed as driven by the existence of a
large-scale (mean) streamwise flow, which we denote by the subscript in
$\widetilde{u}_{u}$. The decomposition of equation 4 has been commonly used to
separate a texture-coherent contribution from the texture-incoherent,
background turbulence (Cheng & Castro, 2002; Nikora et al., 2007; Coceal et
al., 2007).

Abderrahaman-Elena et al. (2019), however, showed that this conventional
triple decomposition does not produce a $u_{T}$ free of coherence with the
surface texture. They argued that the texture-coherent flow occurs in response
to the background flow, and for any given texture element it would then be
driven by the full local signal of the overlying flow, which might be more or
less intense than the mean $U(y)$. The texture-coherent flow would thus be
modulated in intensity by the local background turbulence. Taking as an
example $\widetilde{u}_{u}$, the texture-coherent streamwise velocity driven
by an overlying streamwise velocity, it would be modulated by the streamwise
background turbulence, giving

$u(x,y,z,t)\approx
U(y)+u_{T}(x,y,z,t)+\frac{U(y)+u_{T}(x,y,z,t)}{U(y)}\widetilde{u}_{u}(\widetilde{x},y,\widetilde{z}).$
(5)

In general, the overlying flow induces a velocity field around texture
elements in all three directions, while itself has also components in all
three directions. The spanwise background flow would for instance induce a
texture-coherent, spanwise $\widetilde{w}_{w}$. Therefore, all three texture-
coherent velocity components would have contributions induced by all three
components of the overlying flow. To illustrate this idea, figure 1 shows a
sketch of the streamwise and spanwise overlying velocities inducing coherent
flow. Applied to all velocity components, the amplitude-modulated triple
decomposition of Abderrahaman-Elena et al. (2019) can be written as the
product of a matrix and a vector,

Figure 1: Sketch of the texture-coherent flow induced by an overlying velocity
in the streamwise or spanwise direction. The blue lines represent streamlines
of the induced flow.

$\boldsymbol{u}=\begin{bmatrix}\widetilde{u}_{u}/U+1&\widetilde{u}_{v}/v^{*}&\widetilde{u}_{w}/w^{*}\\\
\widetilde{v}_{u}/U&\widetilde{v}_{v}/v^{*}+1&\widetilde{v}_{w}/w^{*}\\\
\widetilde{w}_{u}/U&\widetilde{w}_{v}/v^{*}&\widetilde{w}_{w}/w^{*}+1\end{bmatrix}\begin{pmatrix}U+u_{T}\\\
v_{T}\\\ w_{T}\\\ \end{pmatrix},$ (6)

where $w^{*}$ and $v^{*}$ denote the diretion-conditional averages of $w$ and
$v$ over individual roughness elements, thus giving a norm for the intensity
of the respective texture-coherent velocities. Abderrahaman-Elena et al.
(2019) showed that this modified form of the triple decomposition was more
effective at removing the footprint of the texture from $\boldsymbol{u_{T}}$.

###  2.2 Governing equations for the background turbulence in flows over
textures

Let us now derive the governing equations for the background-turbulence
component, and in particular the terms that account for the non-linear
interaction with the texture-coherent flow. In a texture-resolving DNS, the
incompressible flow within the channel is governed by continuity,
$\nabla\cdot\boldsymbol{u}=0$, and the Navier-Stokes equations,

$\partial_{t}\boldsymbol{u}+\nabla\cdot\boldsymbol{u}\boldsymbol{u}=-\nabla
p+\nu\nabla^{2}\boldsymbol{u},$ (7)

where $\boldsymbol{u}$ is the velocity vector with components $u$, $v$, $w$.
Using the velocity decompositions introduced in section 2.1, the governing
momentum equations for the background turbulence can be derived by subtracting
the momentum equation for texture-coherent flow from the triple-decomposed
Navier-Stokes equations. We apply this first for the case of conventional
triple decomposition, as a simpler example that serves us to illustrate the
concept, and then apply it for the case of amplitude-modulated decomposition.

####  2.2.1 Texture-coherent flow obtained from conventional triple
decomposition

Using conventional triple decomposition, the velocity vector can be written as

$\boldsymbol{u}=\begin{pmatrix}U+\widetilde{u}_{u}+u_{T}\\\
\widetilde{v}_{u}+v_{T}\\\ \widetilde{w}_{u}+w_{T}\\\ \end{pmatrix},$ (8)

and the pressure as $p=P+\tilde{p}+p_{T}$, where $P$ is the contribution that
provides the mean pressure gradient driving the flow. Substituting the triple-
decomposed velocities and pressure into the Navier-Stokes equations and taking
the temporal and spatial average, we obtain the momentum equation for the mean
flow $\boldsymbol{U}=(U,0,0)$,

$\nabla\cdot\left<\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\right>+\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>=-\nabla
P+\nu\nabla^{2}\boldsymbol{U},$ (9)

where $\left<\left(\cdot\right)\right>$ denotes spatial averaging in the wall-
parallel directions and $\overline{\left(\cdot\right)}$ denotes temporal
averaging. This has the usual form of the mean-flow equation for rough-wall
flows, with the Reynolds and the dispersive stress on left hand side. While
the dispersive stress can be significant for other textures, for the present
slip/no-slip textures it is negligible compared to the Reynolds stress at
least for $L^{+}\lesssim 50$, only becoming relevant for texture sizes
$L^{+}\gtrsim 70$, as shown in figure 2.

Figure 2: Reynolds and dispersive stress for slip/no-slip collocated square
posts at $Re_{\tau}\approx 180$. (a) to (f) $L^{+}\approx 18$, 24, 35, 47, 71
and 94.  , Reynolds stress;  , dispersive stress.

The momentum equation for the texture-coherent fluctuation,
$\boldsymbol{\widetilde{u}_{u}}$, can be obtained by taking the temporal
average of the triple-decomposed Navier-Stokes equations and subtracting the
momentum equation for $\boldsymbol{U}$, equation 9, which gives

$\boldsymbol{N_{c}}=-\nabla\widetilde{p}+\nu\nabla^{2}\boldsymbol{\widetilde{u}_{u}},$
(10)

where

$\boldsymbol{N_{c}}=\nabla\cdot(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})-\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>+\nabla\cdot\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}-\nabla\cdot\left<\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\right>.$
(11)

Given that the background turbulence is homogeneous in the wall-parallel
directions and statistically steady in time, we have
$\nabla\cdot\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}-\nabla\cdot\left<\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\right>\approx
0$ in the non-linear term $\boldsymbol{N_{c}}$, and the resulting momentum
equation for the texture-coherent fluctuation,
$\boldsymbol{\widetilde{u}_{u}}$, simplifies then to

$\nabla\cdot(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})-\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>=-\nabla\widetilde{p}+\nu\nabla^{2}\boldsymbol{\widetilde{u}_{u}}.$
(12)

Finally, the momentum equation for the background turbulence can be obtained
by subtracting the above momentum equation for
$\boldsymbol{\widetilde{u}_{u}}$ from the full triple-decomposed Navier-stokes
equation, yielding

$\partial_{t}\boldsymbol{u_{b}}+\nabla\cdot\boldsymbol{u_{b}}\boldsymbol{u_{b}}+\boldsymbol{N_{b}}=-\nabla
p_{b}+\nu\nabla^{2}\boldsymbol{u_{b}},$ (13)

where $\boldsymbol{u_{b}}=\boldsymbol{U}+\boldsymbol{u_{T}}$ includes the mean
$\boldsymbol{U}$ profile and the background turbulent fluctuations, and is
thus the flow we are interested in solving for in our texture-less DNSs, and
${p_{b}}$ is the corresponding background pressure, ${p_{b}}=P+p_{T}$. The
additional non-linear terms, $\boldsymbol{N_{b}}$, are

$\begin{split}\boldsymbol{N_{b}}=\nabla\cdot\boldsymbol{\widetilde{u}_{u}}\boldsymbol{u_{T}}+\nabla\cdot\boldsymbol{u_{T}}\boldsymbol{\widetilde{u}_{u}}+\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>,\end{split}$
(14)

where
$\nabla\cdot\boldsymbol{\widetilde{u}_{u}}\boldsymbol{u_{T}}+\nabla\cdot\boldsymbol{u_{T}}\boldsymbol{\widetilde{u}_{u}}$
are the non-linear interactions of the background turbulence and the texture-
coherent flow and
$\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>$
is the dispersive stress, which acts only on the mean flow $\boldsymbol{U}$.
The presence of surface texture thus modifies the background turbulence
through these non-linear terms.

####  2.2.2 Texture-coherent flow obtained from amplitude-modulated triple
decomposition

Let us now derive the momentum equations for the background turbulence using
the amplitude-modulated triple decomposition of equation 6. For the present
case of slip/no-slip textures, we make the following simplifications. At the
surface, the wall-normal velocity $v$ is zero throughout, and therefore any
flow induced by the background $v$ is small, as reported in Fairhall et al.
(2019), so we neglect $\widetilde{u}_{v}$, $\widetilde{v}_{v}$ and
$\widetilde{w}_{v}$ in equation 6. Let us however note that this $v$-induced
flow could be important for protruding textures such as pyramids, cones, and
roughness in general. In addition, we also neglect the flow induced by the
background $w$, since the latter is weak compared to the background streamiwse
flow for slip/no-slip textures, as also reported in Fairhall et al. (2019).
The dominant texture-coherent effect is then the flow induced by streamwise
velocity $U+u_{T}$, and the modulated triple decomposition simplifies then to

$\begin{split}\boldsymbol{u}&=\left(\boldsymbol{I}+\begin{bmatrix}\widetilde{u}_{u}/U&0&0\\\
\widetilde{v}_{u}/U&0&0\\\
\widetilde{w}_{u}/U&0&0\end{bmatrix}\right)\cdot\begin{pmatrix}U+u_{T}\\\
v_{T}\\\ w_{T}\\\ \end{pmatrix},\\\
&=\left(\boldsymbol{I}+\boldsymbol{C}\right)\cdot\boldsymbol{u_{b}},\end{split}$
(15)

where we assume that $\widetilde{\boldsymbol{u}}_{u}$ can still be obtained by
ensemble averaging (Abderrahaman-Elena et al., 2019; Fairhall et al., 2019)
and is therefore still governed by equation 10. Similarly, the pressure is
decomposed into

$p=P+p_{T}+\frac{U+u_{T}}{U}\widetilde{p}_{u},$ (16)

which can be further written as

$p=p_{b}+\boldsymbol{Q}\cdot\boldsymbol{u_{b}},$ (17)

where $\boldsymbol{Q}=[\widetilde{p}_{u}/U\quad 0\quad 0]$, corresponding to a
linear expansion about $U$.

The Navier-Stokes equations can then be written as

$\begin{split}\partial_{t}\left(\left(\boldsymbol{I}+\boldsymbol{C}\right)\cdot\boldsymbol{u_{b}}\right)+&\nabla\cdot\left[\left(\left(\boldsymbol{I}+\boldsymbol{C}\right)\cdot\boldsymbol{u_{b}}\right)\left(\left(\boldsymbol{I}+\boldsymbol{C}\right)\cdot\boldsymbol{u_{b}}\right)\right]\\\
&=-\nabla\left(p_{b}+\boldsymbol{Q}\cdot\boldsymbol{u_{b}}\right)+\nu\nabla^{2}\left(\left(\boldsymbol{I}+\boldsymbol{C}\right)\cdot\boldsymbol{u_{b}}\right),\end{split}$
(18)

Following the procedure in section 2.2.1, the momentum equation for the
background flow can be obtained by subtracting the governing equation for
texture-coherent flow, equation 10, from equation 18,

$\partial_{t}\boldsymbol{u_{b}}+\nabla\cdot\boldsymbol{u_{b}}\boldsymbol{u_{b}}+\boldsymbol{N_{b}^{\prime}}=-\nabla
p_{b}+\nu\nabla^{2}\boldsymbol{u_{b}}+\boldsymbol{R},$ (19)

where

$\begin{split}\boldsymbol{N_{b}^{\prime}}&=\nabla\cdot\left[\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)\boldsymbol{u_{b}}+\boldsymbol{u_{b}}\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)\right]+\nabla\cdot\left[\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)\right]\\\
&-\nabla\cdot(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})(\boldsymbol{U}+\boldsymbol{\widetilde{u}_{u}})+\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>,\end{split}$
(20)

and

$\begin{split}\boldsymbol{R}=\partial_{t}\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)+\nabla\left(\boldsymbol{Q}\cdot\boldsymbol{u_{b}}\right)-\nabla\widetilde{p}-\nu\nabla^{2}\left(\boldsymbol{C}\cdot\boldsymbol{u_{b}}\right)+\nu\nabla^{2}\boldsymbol{\widetilde{u}_{u}}.\end{split}$
(21)

Here, we have drawn a parallel with the result using conventional triple
decomposition of equation 13 by grouping the additional advective terms into
$\boldsymbol{N_{b}^{\prime}}$, and any other additional terms arising from the
amplitude modulation into a residual $\boldsymbol{R}$. In appendix A we report
on the magnitude of the latter, and show that $\boldsymbol{N_{b}^{\prime}}$ is
the dominant term. We will thus neglect the residual $\boldsymbol{R}$ in our
model.

Figure 3:  Comparison of viscous stress ( ), shear Reynolds stress ( ),
averaged forcing term ( ) and total stress ( ) for FA47, FA71 and FA94.

Finally, we can derive the momentum equation for the mean velocity
$\boldsymbol{U}$ analogously to equation 9 by taking the temporal average of
equation 19 particularised for the zero mode. This gives

$\nabla\cdot\left<\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\right>+\boldsymbol{N_{m}}=-\nabla
P+\nu\nabla^{2}\boldsymbol{U},$ (22)

where the forcing term $\boldsymbol{N_{m}}$ is

$\boldsymbol{N_{m}}=\nabla\cdot\left<\boldsymbol{\widetilde{u}_{u}}\boldsymbol{\widetilde{u}_{u}}\right>+\nabla\cdot\left<\boldsymbol{C}\cdot\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}+\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\cdot\boldsymbol{C}^{T}\right>+\nabla\cdot\left<\boldsymbol{C}\cdot\overline{\boldsymbol{u_{T}}\boldsymbol{u_{T}}}\cdot\boldsymbol{C}^{T}\right>.$
(23)

Here, $\boldsymbol{u_{T}}\left(\boldsymbol{C}\cdot\boldsymbol{u_{T}}\right)$
has been rearranged as
$\boldsymbol{u_{T}}\boldsymbol{u_{T}}\cdot\boldsymbol{C}^{T}$ before averaging
in time. Integrating the above mean momentum equation in $y$ gives the
following stress balance

$\nu\frac{dU}{dy}-\left<\overline{u_{T}v_{T}}\right>-N_{m,x}=u_{\tau}^{2}\frac{\delta^{\prime}-y}{\delta^{\prime}},$
(24)

where the role of the dispersive Reynolds stress in equation 9 is now played
by an extended term $N_{m,x}$,

$N_{m,x}=\left<\widetilde{u}_{u}\widetilde{v}_{u}+\frac{\widetilde{u}_{u}\widetilde{v}_{u}}{U^{2}}\overline{u_{T}v_{T}}+\frac{\widetilde{u}_{u}}{U}\overline{u_{T}v_{T}}+\frac{\widetilde{v}_{u}}{U}\overline{u_{T}u_{T}}\right>.$
(25)

$N_{m,x}$ includes contributions from the background turbulence, and can thus
only be calculated from the results in §4 a posteriori in order to verify that
the sum of the three stresses in equation 24 is indeed linear. Results of such
measurement are portrayed for reference in figure 3 for the collocated
textures of sizes $L^{+}\gtrsim 50$, as for smaller textures the contribution
of $N_{m,x}$ is negligible. For the sizes portrayed its intensity increases
with $L^{+}$, although it remains small compared to the viscous and background
Reynolds stress.

##  3 Numerical Method

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), 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, Reτ. 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. Statistics were obtained by averaging over a period of 10 eddy-
turnovers after statistical convergence had been reached.

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.

Figure 4: Schematic of collocated- and staggered-post arrangements for
periodic 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.

Case |  $Re_{\tau}$ |  $L^{+}$ |  $N_{x,w}$ |  $N_{z,w}$ |  $\ell_{x}^{+}$ |  $\ell_{z}^{+}$ |  $\Delta U^{+}\\!\\!\\!\\!\\!$
---|---|---|---|---|---|---|---
TX18 | 179.7 | 17.7 | 1576 | 768 | 5.8 | 4.0 | 3.7
TX24 | 180.2 | 23.6 | 1152 | 576 | 6.9 | 4.3 | 4.1
TX35 | 179.9 | 35.3 | 768 | 384 | 8.5 | 5.0 | 4.8
TX35H | 407.0 | 35.3 | 1728 | 864 | 8.4 | 5.1 | 5.4
TX47 | 180.1 | 47.1 | 576 | 288 | 10.0 | 6.3 | 5.6
TX71 | 180.0 | 70.7 | 384 | 192 | 12.7 | 7.3 | 7.5
TX94 | 179.7 | 94.2 | 288 | 144 | 14.6 | 8.5 | 9.1
TX94H | 360.7 | 94.2 | 576 | 288 | 14.2 | 10.0 | 9.7
sTX35 | 181.0 | 33.3 | 576 | 288 | 6.0 | 4.4 | 3.7
sTX47 | 180.7 | 49.9 | 384 | 192 | 7.3 | 4.7 | 4.1
sTX71 | 179.8 | 66.6 | 288 | 144 | 8.5 | 3.2 | 4.7
SL18 | 179.9 | 17.7 | 128 | 128 | 5.8 | 4.0 | 4.0
SL24 | 179.9 | 23.6 | 128 | 128 | 6.9 | 4.3 | 5.0
SL35 | 180.3 | 35.3 | 128 | 128 | 8.5 | 5.0 | 6.5
SL35H | 405.4 | 35.3 | 256 | 256 | 8.4 | 5.1 | 6.5
SL47 | 179.7 | 47.1 | 128 | 128 | 10.0 | 6.3 | 7.8
SL71 | 179.9 | 70.7 | 128 | 128 | 12.7 | 7.3 | 10.2
SL94 | 179.9 | 94.2 | 128 | 128 | 14.6 | 8.5 | 12.1
SL94H | 359.7 | 94.2 | 256 | 256 | 14.2 | 10.0 | 12.1
sSL35 | 180.1 | 33.3 | 128 | 128 | 6.0 | 4.4 | 4.2
sSL47 | 180.4 | 49.9 | 128 | 128 | 7.3 | 4.7 | 5.3
sSL71 | 180.0 | 66.6 | 128 | 128 | 8.5 | 3.2 | 6.9
FA18 | 179.9 | 17.7 | 128 | 128 | 5.8 | 4.0 | 3.6
FA24R | 180.1 | 23.6 | 128 | 128 | 6.9 | 4.3 | 4.8
FA24 | 180.0 | 23.6 | 128 | 128 | 6.9 | 4.3 | 4.0
FA24S | 180.2 | 23.6 | 128 | 128 | 6.9 | 4.3 | 4.0
FA35R | 180.1 | 35.3 | 128 | 128 | 8.5 | 5.0 | 6.0
FA35 | 179.8 | 35.3 | 128 | 128 | 8.5 | 5.0 | 4.8
FA35S | 180.0 | 35.3 | 128 | 128 | 8.5 | 5.0 | 5.0
FA35H | 404.7 | 35.3 | 256 | 256 | 8.4 | 5.1 | 5.4
FA47R | 180.3 | 47.1 | 128 | 128 | 10.0 | 6.3 | 7.0
FA47 | 180.2 | 47.1 | 128 | 128 | 10.0 | 6.3 | 5.7
FA71R | 180.0 | 70.7 | 128 | 128 | 12.7 | 7.3 | 8.9
FA71 | 180.1 | 70.7 | 128 | 128 | 12.7 | 7.3 | 7.9
FA94R | 179.9 | 94.2 | 128 | 128 | 14.6 | 8.5 | 10.5
FA94 | 180.0 | 94.2 | 128 | 128 | 14.6 | 8.5 | 9.9
FA94H | 359.5 | 94.2 | 256 | 256 | 14.2 | 10.0 | 10.2
sFA35 | 179.8 | 33.3 | 128 | 128 | 6.0 | 4.4 | 3.6
sFA47 | 180.3 | 49.9 | 128 | 128 | 7.3 | 4.7 | 4.1
sFA71 | 180.1 | 66.6 | 128 | 128 | 8.5 | 3.2 | 5.4
Table 1: Simulation parameters for the texture-resolving, slip-only and slip-
plus-forcing simulations. For the case names, TX and sTX indicate resolved
collocated and staggered textures, SL and sSL slip-only simulations, and FA
and sFA simulations with forcing. The number in the case name is approximately
the texture size in wall units, $L^{+}$, listed also for texture-less
simulations as a reference to the corresponding textured ones. An appended R
indicates forcing based on conventional triple decomposition; otherwise the
forcing is based on amplitude-modulated decomposition. An appended S indicates
the use of an a priori surrogate model for the texture-coherent flow. An
appended H indicates higher friction Reynolds number, $Re_{\tau}$. $N_{x,w}$
and $N_{z,w}$ are the number of grid points in the streamwise and spanwise
directions in the refined blocks that contain the channel walls.
$\ell_{x}^{+}$ and $\ell_{z}^{+}$ are the streamwise and spanwise slip
lengths, and $\Delta U^{+}$ is the resulting velocity increment away from the
wall.

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.

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.

###  3.1 Dealiasing for the forcing terms

When calculating the product of two variables, such as in the advective term,
in discrete Fourier space, the problem of aliasing arises. The convolution of
two discrete functions, $\hat{f}_{M}$ and $\hat{f}_{N}$, containing $M$ and
$N$ discrete modes, results in a function containing $M+N$ modes. However, if
the product function allocated for the convolution, $\hat{f}_{P}$, is of size
$P<M+N$, the excess modes cannot be correctly represented. This additional,
high frequency information is then reflected into the resolved modes of the
product function.

The method typically used to address this aliasing is to pad $\hat{f}_{M}$ and
$\hat{f}_{N}$ with additional modes, with zero value, before the
multiplication (Canuto et al., 2012). For the non-linear term
$\left(\boldsymbol{u_{T}}\boldsymbol{u_{T}}\right)$ in equation 14, we use the
standard ‘2/3 rule’ for dealiasing. All functions have the same size, $M=N=P$,
so they need to be padded with an additional $N/2$ modes to prevent aliasing
in $\hat{f}_{P}$ in modes 0 to $N$. Aliasing still occurs from the reflection
of modes greater than $3N/2$ into modes from $N$ to $3N/2$, but this is of no
consequence as these modes are dismissed once the convolution product has been
calculated. In contrast with the standard advection, for the product
$\boldsymbol{\widetilde{u}_{u}}\boldsymbol{u_{T}}$, the convolution of the
background turbulence and the texture-coherent flow, the sizes of the two
components need not be the same. We have for instance $N=128$ for
$\boldsymbol{u_{T}}$, with $M$ different for cases with different texture
size, and typically $M\geq N$. Since we are interested in solving only the
background turbulence, that is, modes up to $N$, for dealiasing in this case
we need at least $P=N+M/2$, as illustrated in figure 5. Taking case FA35 as an
example, in the streamwise direction the texture-coherent component
$\boldsymbol{\widetilde{u}_{u}}$ has $M=768$ points in discrete Fourier space,
so $X=128+768/2=512$ is required for dealiasing.

Figure 5: Sketch of the dealiasing strategy for the forcing terms. The
spectral region shaded in yellow with modes larger than $N+M/2$ is reflected
into the shaded region under the dashed line, which includes only modes larger
than $N$, and thus avoids aliasing of modes below $N$.

##  4 Results and discussion

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.

Figure 6: Comparison of $\Delta U^{+}$ obtained from texture-resolved, slip-
only and slip-plus-forcing simulations. Full symbols are for $Re_{\tau}\approx
180$, and open symbols for $Re_{\tau}\approx 350$–$400$. Solid and dashed
lines are for simulations with collocated and staggered texture arrangements,
respectively. $\bullet$, texture-resolved simulations; $\bullet$, simulations
with homogeneous slip boundary conditions only; $\blacktriangledown$,
simulations with homogeneous slip plus forcing based on conventional triple
decomposition; $\blacktriangledown$, simulations with homogeneous slip plus
forcing based on amplitude-modulated triple decomposition.

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.

Figure 7: R.m.s velocity fluctuations for collocated textures of sizes
$L^{+}\approx 20$–$100$ at $Re_{\tau}\approx 180$.  , texture-resolved
simulations;  slip-only simulations;  , simulations with forcing based on
conventional triple decomposition;  , simulations with forcing based on
amplitude-modulated decomposition. Figure 8: Shear Reynolds stress and mean
velocity profile for collocated textures of sizes $L^{+}\approx 20$–$100$ at
$Re_{\tau}\approx 180$. Line styles are as in figure 7.

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 thorughout 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

$\frac{\mathrm{d}U^{+}}{\mathrm{d}y^{+}}+\tau_{uv}^{+}=\frac{\delta^{\prime+}-y^{+}}{\delta^{\prime+}},$
(26)

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

$U^{+}\left(H^{+}\right)-U^{+}_{\mathrm{slip}}+\int_{0}^{H^{+}}\tau_{uv}^{+}\left(y^{+}\right)\mathrm{d}y^{+}=f,$
(27)

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

$U_{\mathrm{S}}^{+}\left(H^{+}+\ell_{T}^{+}\right)-U_{\mathrm{S}}^{+}\left(\ell_{T}^{+}\right)+\int_{\ell_{T}^{+}}^{H^{+}+\ell_{T}^{+}}\tau_{uv,\mathrm{S}}^{+}\left(y_{S}^{+}\right)\mathrm{d}y_{S}^{+}=f,$
(28)

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

$\begin{split}\Delta
U^{+}&=U^{+}\left(H^{+}\right)-U_{S}^{+}\left(H^{+}+\ell_{T}^{+}\right)\\\
&=U^{+}_{\mathrm{slip}}-U_{\mathrm{S}}^{+}\left(\ell_{T}^{+}\right)+\int_{0}^{H^{+}}\\!\\!\left[\tau_{uv}^{+}\left(y^{+}\right)-\tau_{uv,\mathrm{S}}^{+}\left(y^{+}+\ell_{T}^{+}\right)\right]\mathrm{d}y^{+}.\end{split}$
(29)

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.

Figure 9: Spectral energy densities of the velocity fluctuations and the shear
Reynolds stress at $y^{+}\approx 15$ for collocated textures of sizes
$L^{+}\approx 20$–$100$ at $Re_{\tau}\approx 180$. Shaded contours, texture-
resolved simulations;  slip-only simulations;  , simulations with forcing
based on conventional triple decomposition;  , simulations with forcing based
on amplitude-modulated decomposition.

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.

Figure 10: Instantaneous realisations of the fluctuating $u$ (left), $v$
(centre) and $w$ (right) velocity components at $y^{+}\approx 15$ for
$L^{+}\approx 50$. Top row, texture-resolved simulation, TX47; middle row,
slip-only simulation, SL47; bottom row, simulation with amplitude-modulated
forcing, FA47. From blue to yellow, values are from -4.0 to 4.0 for $u$, from
-1.2 to 1.2 for $v$, and from -2.2 to 2.2 for $w$, all in viscous units.

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.

###  4.1 The limit of the forcing model

The preceding discussion has shown how the model based on the amplitude-
modulated decomposition works well up to $L^{+}\approx 70$, and how for
$L^{+}\approx 70$ and $L^{+}\approx 100$ deviations from the fully resolved
simulations are increasingly apparent. We have also mentioned that this is to
be expected, as the assumption of separation of scales between the two flow
components ceases to hold. The decomposition hinges on the idea that one
component, the overlying background turbulence, excites the other, the
texture-coherent signature, as the response flow in the immediate vicinity of
the texture, which does not necessarily cease to hold once the lengthscales of
the two become comparable. The algebraic form, however, as laid out in
equation 5 or more generally in equation 6, assumes that the exciting
overlying flow is on a much larger scale than the excited one, in the spirit
of Luchini et al. (1991). As $L^{+}$ increases, the breakdown of scale
separation occurs first for the cross flow components, as these are mainly
produced by the overlying quasi-streamwise vortices, with length and span of
order $200\times 20$ wall units, while the streamwise component is mainly
produced by the streamwise streaks, with length and span of order $1000\times
100$ wall units (Kline et al., 1967; Blackwelder & Eckelmann, 1979; Smith &
Metzler, 1983).

Figure 11: Correlation over multiple instantaneous realisations of the
streamwise velocity $u$ and shear $\partial u/\partial y$ for texture size
$L^{+}\approx 70$, for wavelengths $\lambda_{x}^{+}\approx 110$ to $1100$ and
$\lambda_{z}^{+}\approx 110$ to $550$ from red to blue. Left panels,
magnitudes of $u$ and $\partial u/\partial y$; centre and right panels, phases
between the two in $x$ and $z$. Top, full velocity signal; bottom, background
turbulence component. Dashed lines mark slip-lengths obtained from linear
regression of the data displayed.

Figure 12: Correlation over multiple instantaneous realisations of the
streamwise velocity $w$ and shear $\partial w/\partial y$ for texture size
$L^{+}\approx 70$, for wavelengths $\lambda_{x}^{+}\approx 110$ to $1100$ and
$\lambda_{z}^{+}\approx 110$ to $550$ from red to blue. Left panels,
magnitudes of $w$ and $\partial w/\partial y$; centre and right panels, phases
between the two in $x$ and $z$. Top, full velocity signal; bottom, background
turbulence component. Dashed lines mark slip-lengths obtained from linear
regression of the data displayed.

Figure 13: Correlation over multiple instantaneous realisations of the
streamwise velocity $u$ and shear $\partial u/\partial y$ for texture size
$L^{+}\approx 95$, for wavelengths $\lambda_{x}^{+}\approx 110$ to $1100$ and
$\lambda_{z}^{+}\approx 110$ to $550$ from red to blue. Left panels,
magnitudes of $u$ and $\partial u/\partial y$; centre and right panels, phases
between the two in $x$ and $z$. Top, full velocity signal; bottom, background
turbulence component. Dashed lines mark slip-lengths obtained from linear
regression of the data displayed.

Figure 14: Correlation over multiple instantaneous realisations of the
streamwise velocity $w$ and shear $\partial w/\partial y$ for texture size
$L^{+}\approx 95$, for wavelengths $\lambda_{x}^{+}\approx 110$ to $1100$ and
$\lambda_{z}^{+}\approx 110$ to $550$ from red to blue. Left panels,
magnitudes of $w$ and $\partial w/\partial y$; centre and right panels, phases
between the two in $x$ and $z$. Top, full velocity signal; bottom, background
turbulence component. Dashed lines mark slip-lengths obtained from linear
regression of the data displayed.

This difference can be observed in the earlier loss of correlation at the wall
between velocity and shear in the spanwise direction. Fairhall et al. (2019)
showed that the correlation held well for the background turbulence up to
$L^{+}\approx 50$ for both $u$ and $w$. Figures 12 to 14 extend their analysis
to the textures of size $L^{+}\approx 70$ and $L^{+}\approx 95$. The figures
show that the decomposition using equation 6 still results in an excellent
recovery of the correlation between $u$ and $\partial u/\partial y$ for both
texture sizes, both in terms of proportion and phase between them, which makes
the value of $\ell_{x}$ estimated from this data meaningful. The same is
however not true for the correlation between $w$ and $\partial w/\partial y$,
which exhibits a considerable scatter both in terms of phase and magnitude. As
a result, it is difficult to define a spanwise slip length $\ell_{z}$
meaningfully. We note nevertheless that this is of minor importance with
regards to setting the effective boundary condition for $w$. At these large
texture sizes, we have $\ell_{z}^{+}\gtrsim 7$, and thus the effect of the
slip length in the boundary condition for $w$ in equation 3 is significantly
saturated, with variations in $\ell_{z}$ not resulting in significant
variations in $\ell_{T}$. In any event, the loss of correlation is indicative
of the background turbulence component obtained still being contaminated by
some amount of texture-induced signal, and thus of the decomposition produced
using equation 6 breaking down.

Figure 15: Spectral energy denisties at $y=0$ for the full streamwise and
spanwise velocity components, $u$ and $w$, and for the corresponding
background turbulence components as obtained using the amplitude-modulated
decomposition of equation 6, $u_{T}$ and $w_{T}$, for collocated square posts
of sizes $L^{+}\approx 35$, 47, 71 and 94.

This breakdown is perhaps even clearer in the spectral densities of the
tangential velocity components in the immediate vicinity of the texture, both
for the background turbulence and the full texture-resolved flow, as portrayed
at $y=0$ in figure 15. Up to texture size $L^{+}\approx 50$ the main energetic
region that can be observed in the spectra of $u$ and $w$ is recovered in the
spectra of the background-turbulence components. However, for $L^{+}\gtrsim
70$, the decomposition can successfully filter out the contributions in the
vicinity of the texture harmonics, which is fully attributable to the texture-
coherent flow and its modulation in amplitude (Abderrahaman-Elena et al.,
2019; Fairhall et al., 2019), but the spectral signature of the background
turbulence is visibly altered by the decomposition. This is indicative of
equation 6 failing to produce the correct signal for the background turbulence
in the latter size range. Figure 15 shows that this happens once the
lengthscales of the harmonics induced by the texture begin to overlap
significantly with the main spectral region of the background turbulence, that
is, once scale separation ceases to hold.

The results for texture sizes $L^{+}\gtrsim 50$ portrayed in figures 6 to 9
indicate that the deviations from the texture-resolved results first occur for
$w^{\prime}$, and soon extend to the other velocity components. The
contribution of long structures of $u^{\prime}$ and $uv^{\prime}$ is
particularly overpredicted, ultimately leading to an overprediction of the
shear Reynolds stress and an underprediction of the drag. We note however
that, even at $L^{+}\approx 100$, the amplitude-modulated model performs
significantly better than the state-of-the-art slip-only models, even if they
fail to capture fully the dynamics at play in texture-resolved simulations.

The present break down of the model for $L^{+}\gtrsim 70$ does not completely
rule out its applicability for larger textures, and rather limits its validity
in the present form of equations 15 and 19. It is entirely possible that a
formulation of the flow decomposition not requiring scale separation could
extend its validity to larger sizes. The present formulation also neglects any
texture-coherent flow induced by cross-plane fluctuations. Such contributions
to the texture-coherent flow would become increasingly important for larger
texture sizes, leading to additional cross-advective forcing terms on the
background turbulence. Their absence from the present formulation likely also
contributes to its breakdown. Dropping the above simplifications would lead to
a significantly more complex formulation, however, and is left for future
work.

###  4.2 Using a priori estimates for the texture-coherent flow

The amplitude-modulated forcing model proposed in this paper shows good
agreement with texture-resolved simulations at least up to texture sizes
$L^{+}\approx 70$, both in terms of drag prediction and turbulent statistics
and structure. This strongly suggests that the model has successfully
identified the key physical mechanisms at play, i.e. the non-linear
interactions between the texture-coherent flow and the background turbulence,
and is able to reproduce them without resolving the texture elements. However,
the model as is does not have a truly predictive character, as it requires
information on the texture-coherent flow obtained by ensemble averaging of the
texture-resolved simulations – the need for which it aims to prevent. In order
to be fully predictive, the model would need a priori estimates of the
texture-coherent flow.

As mentioned in §1, Abderrahaman-Elena et al. (2019) observed that the
ensemble average flow over small textures resembles the flow induced by a
steady homogeneous overlying shear at matching $L^{+}$, which can be obtained
from laminar steady simulations of a single periodic unit of texture. This is
in essence an extension of the Stokes-flow computations used by Luchini et al.
(1991), Kamrin et al. (2010) and Luchini (2013) to predict protrusion heights
or slip lengths, with the addition of inertial terms. The laminar computations
of Abderrahaman-Elena et al. (2019) were extended for the slip/no-slip
textures considered in this paper in Adams (2021). In the latter, a synthetic
eddy-viscosity obtained from smooth-wall DNSs was added to represent the
effect of the background turbulence on the shape of the mean velocity profile
under a driving pressure gradient. Compared to the texture-resolved DNSs of
Fairhall et al. (2019), the resulting flow fields for $L^{+}\approx 35$
overpredicted the coherent $\widetilde{u}_{u}$ by roughly 20%, and showed good
agreement in $\widetilde{v}_{u}$ and $\widetilde{w}_{u}$, yielding overall a
reasonable estimate for the texture-coherent flow.

Figure 16: Mean velocity profile, r.m.s velocity fluctuations and Reynolds
shear stress for simulations with $L^{+}\approx 24$ (dashed lines) and
$L^{+}\approx 35$ (solid lines) at $Re_{\tau}\approx 180$.  , texture-resolved
simulations, TX24 and TX35;  , slip-only simulations, SL24 and SL35;  ,
simulations with amplitude-modulated forcing, FA24 and FA35;  , simulations
with amplitude-modulated forcing using an a priori surrogate for the texture-
coherent flow, FA24S and FA35S. Figure 17:  Spectral energy densities of the
velocity fluctuations and the Reynolds shear stress at $y^{+}\approx 15$ for
$L^{+}\approx 35$ at $Re_{\tau}\approx 180$. Filled contours, texture-resolved
simulation, TX35;  , slip-only simulation, SL35;  , simulation with amplitude-
modulated forcing, FA35;  , simulation with amplitude-modulated forcing using
an a priori surrogate for the texture-coherent flow, FA35S.

As a first step to explore the predictive capabilities of the amplitude-
modulated forcing model, we study its performance when the texture-coherent
flow from DNSs is replaced by the a priori surrogates of Adams (2021) for
$L^{+}\approx 24$ and $L^{+}\approx 35$. Results are shown in figures 16 and
17. Figure 16 shows excellent agreement in the r.m.s. velocity fluctuations,
the Reynolds shear stress and $\Delta U^{+}$, except for an overshoot in
$u^{\prime}$ for the larger texture $L^{+}\approx 35$ in the immediate
vicinity of the wall, $y^{+}\lesssim 15$. This is likely caused by the
excessive intensity in $\widetilde{u}_{u}$ in the surrogate texture-coherent
flow, as mentioned above. The overshoot does nevertheless not propagate into
the Reynolds stress or any of the other velocity components, and is also
confined to a narrow region near the texture, so it essentially does not alter
the drag or the near-wall dynamics and dissipation, mainly governed by cross-
plane velocities induced by bursts and quasi-streamwise vortices (Ibrahim et
al., 2021; Habibi Khorasani et al., 2022; Jiménez, 2022). Similar results can
also be observed in the spectral densities, shown for the three velocity
components and the Reynolds shear stress at $y^{+}\approx 15$ in figure 17 for
$L^{+}\approx 35$. The surrogate texture-coherent flow is able to produce an
energetic signature in the same short wavelengths exhibited by texture-
resolved simulations and those with forcing based on the a posteriori texture-
coherent flow, and not present for smooth-wall or slip-only simulations.
Although there is room for improvement, especially in the prediction of the
near-wall peak in $u^{\prime}$, these results suggest that the need to conduct
simulations that resolve the texture can be fully circumvented, and that it is
possible to resolve the background turbulence, and predict the drag and other
flow properties using only a priori data on the texture detail, significantly
reducing computational costs.

##  5 Conclusions

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 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.

##  Appendix A Magnitude of the forcing term and the residual in the momentum
equations

Figure 18: Time-r.m.s intensity $\psi_{x}^{\prime}$ of the terms in the
streamwise component of the momentum equation derived for the background
turbulence assuming amplitude-modulated texture-coherent flow, equation 19,
for the case of collocated texture with $L^{+}\approx 35$.  , advective term;
, viscous term;  , pressure-gradient term;  , temporal-derivative term;  ,
non-linear forcing term $\boldsymbol{N_{b}^{\prime}}$;  , residual
$\boldsymbol{R}$. Figure 19: Time-r.m.s intensity $\psi_{y}^{\prime}$ of the
terms in the wall-normal component of the momentum equation derived for the
background turbulence assuming amplitude-modulated texture-coherent flow,
equation 19, for the case of collocated texture with $L^{+}\approx 35$.  ,
advective term;  , viscous term;  , pressure-gradient term;  , temporal-
derivative term;  , non-linear forcing term $\boldsymbol{N_{b}^{\prime}}$;  ,
residual $\boldsymbol{R}$. Figure 20: Time-r.m.s intensity $\psi_{z}^{\prime}$
of the terms in the spanmwise component of the momentum equation derived for
the background turbulence assuming amplitude-modulated texture-coherent flow,
equation 19, for the case of collocated texture with $L^{+}\approx 35$.  ,
advective term;  , viscous term;  , pressure-gradient term;  , temporal-
derivative term;  , non-linear forcing term $\boldsymbol{N_{b}^{\prime}}$;  ,
residual $\boldsymbol{R}$.

This appendix portrays evidence of the relative importance of the forcing term
$\boldsymbol{N_{b}^{\prime}}$ and the residual $\boldsymbol{R}$ in equation
19. The residual is neglected in the governing equations of the simulations
presented in §4 with forcing based on the amplitude-modulated decomposition,
and it therefore needs to be small for the proposed model to hold. Conversely,
if the magnitude of $\boldsymbol{N_{b}^{\prime}}$ was small, we would expect
that retaining this term had little effect on the dynamics of the background
turbulence.

Figures 18 to 20 portray the time-r.m.s. magnitude of the different terms in
equation 19 from the texture-resolved simulation of collocated posts with
texture size $L^{+}\approx 35$ (case TX35). Figures 18, 19 and 20 portray
results for the streamwise, wall-normal and spanwise components of the
momentum equation respectively, for a representative set of $x$-$z$
wavenumbers and as a function of height $y$. The figures show that the
dynamics are mostly dominated by the interplay between temporal and advective
terms, but in the vicinity of the wall, for $y^{+}\lesssim 10$, the forcing
term, which is neglected in slip-only simulations, can become comparable to,
and even larger that, the latter two. This is particularly the case for the
streamwise momentum equation, as illustrated by figure 18, and across the
whole range of wavelengths. For the spanwise and wall-normal momentum
equations, the explicit contribution of $\boldsymbol{N_{b}^{\prime}}$ appears
to be less significant. In turn, the residual is also negligible in the
spanwise and wall-normal momentum equations. It is typically a fraction of
$\boldsymbol{N_{b}^{\prime}}$ of order 1/4-1/3 in the streamwise one, and thus
reasonably smaller than the leading-order terms. These results back up
neglecting $\boldsymbol{R}$ when implementing the amplitude-modulated forcing
model in §4.

##  Appendix B Simulations for staggered-posts textures

Figure 21: R.m.s. velocity fluctuations, shear Reynolds stress and mean
velocity profile for staggered-posts textures with $L^{+}\approx 35$-$70$.  ,
texture-resolved simulations;  , slip-only simulations;  , simulations with
forcing based on amplitude-modulated decomposition. Figure 22: Spectral energy
densities of the velocity fluctuations and the Reynolds shear stress at
$y^{+}\approx 15$ for staggered texture arrangements. Shaded contours,
texture-resolved simulations;  slip-only simulations;  , simulations with
forcing based on conventional triple decomposition;  , simulations with
forcing based on amplitude-modulated decomposition.

Figures 21 and 22 portray results from the simulations for staggered-posts
configurations, as sketched in figure 4. The aim of these simulations was to
verify if the forcing model using amplitude-modulated decomposition, first
tested for a collocated-posts texture, would also be applicable to other
surface arrangements. In additions to simulations fully resolving the texture
and simulations with forcing, we have also conducted simulations implementing
slip boundary conditions alone, to serve as a benchmark for comparison. These
simulations complement and expand the data set of Fairhall et al. (2019).

We have conducted simulations for texture sizes $L^{+}\approx 35$, $50$ and
$70$, aiming to span the size range for which slip-only simulations begin to
deviate from texture-resolved ones, yet the texture size is not large enough
for the decomposition and the forcing model to break down. The simulation
parameters and resulting roughness functions are listed in table 1.

The results follow the same trends of the collocated-textures cases. Figure 21
shows that the r.m.s. velocity fluctuations for texture-resolved simulations
exhibit differences with the slip-only ones, which display smooth-wall-like
turbulence (Fairhall et al., 2019; Ibrahim et al., 2021). These differences
are small for $L^{+}\approx 35$, but become increasingly pronounced for larger
texture sizes. Their most salient features are a decrease in near-wall peak
$u^{\prime}$ and an increase in $v^{\prime}$ and $w^{\prime}$. These
differences eventually result in a significant increase in the shear Reynolds
stress, and a corresponding downward shift of the mean velocity profile away
from the wall. The forcing model is able to accurately capture these
differences, and produces results very close to those of texture-resolved
simulations, with small deviations first appearing in $w^{\prime}$, as
discussed in §4 for collocated textures.

The spectral densities portrayed in figure 22 are also consistent with the
observations in §4 for collocated textures. Compared to smooth-wall
turbulence, in texure-resolving simulations have additional energy
particularly in shorter scales in the streamwise direction. Slip-only
simulations are not able to produce this modification in turbulence, and their
spectrum remains essentially smooth wall like as discussed in Fairhall et al.
(2019) and Ibrahim et al. (2021). In contrast, the forcing model is able to
produce the spectral energy densities of texture-resolved simulations with
excellent agreement.

##  Appendix C Effect of Reynolds number

Figure 23: R.m.s. velocity fluctuations, shear Reynolds stress and mean
velocity profile for simulations at $Re_{\tau}\approx 180$ (solid) and
$Re_{\tau}\approx 400$ (dashed).  , TX35;  , TX35H;  , SL35;  , SL35H;  ,
FA35;  , FA35H. Figure 24: Spectral energy densities of the velocity
fluctuations and the Reynolds shear stress at $y^{+}\approx 15$ for
$L^{+}\approx 35$ and $L^{+}\approx 100$ at different $Re_{\tau}$. Grey filled
contours, texture-resolved simulations at $Re_{\tau}\approx 180$;  , at
$Re_{\tau}\approx 400$. Blue filled contours, simulations with forcing based
on amplitude-modulated decomposition at $Re_{\tau}\approx 180$;  , at
$Re_{\tau}\approx 400$.

The main set of simulations discussed in this paper were conducted at
$Re_{\tau}\approx 180$. To verify that our observations would scale out to
other Reynolds numbers, we have conducted a reduced set of simulations at
$Re_{\tau}\approx 400$, otherwise matching in inner scaling setups from the
main set, with results shown in figures 23 and 24. We have chosen the
collocated layouts with $L^{+}\approx 35$ and $L^{+}\approx 100$, as one
texture size for which the texture-resolved and the forcing simulations show
good agreement between them, but differences with the slip-only simulation,
and one texture size for which the forcing model shows signs of breaking down.

The results are consistent with our usual observations for other surface
topologies (García-Mayoral & Jiménez, 2012; Fairhall et al., 2019; Sharma &
García-Mayoral, 2020a, b; Hao & García-Mayoral, 2024). Near the wall, the same
effects are produced in inner units for matching simulations, with results
eventually collapsing to smooth wall data at equal $Re_{\tau}$ for
$y^{+}\gtrsim 50$. Differences in this region are observable, but entirely
attributable to the variations in $Re_{\tau}$, as they are the same observed
for smooth-wall flows (Moser et al., 1999). These are evidenced for instance
in the higher peak levels of the r..m.s. velocity fluctuations and the shear
Reynolds stress near $y^{+}\gtrsim 25$, as shown in figure 23. The spectral
densities portrayed in 24 confirm these results. The spectral energy content
at each wavelength matches when scaled in inner units for corresponding
simulations, with larger-scale content only resolved for the higher
$Re_{\tau}\approx 400$, as made possible by the larger simulation domain.
Overall, the results suggest that the effect of the model, like the effect of
the texture, scales in inner units as expected, and can be extrapolated to and
used at different Reynolds numbers.

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