3. The LaRTE model: roughness extension

Following the derivation of the LaRTE model for smooth walls (Fowler et al. Reference Fowler, Zaki and Meneveau2022), we assume that, in the flow region between the rough wall and the wall-model interface ( $0 \lt y \lt \varDelta$ ), a horizontal qEQ mean velocity profile, $\overline {u}_{\textit{si}}$ , exists, and can be written as $\overline {\boldsymbol{u}}=\overline {u}_{s1}\boldsymbol{ i}+\overline {v}{\boldsymbol{j}}+\overline {u}_{s3}\boldsymbol{k}$ , where $\overline {u}_{\textit{si}}$ ( $i=1,3$ ) is the two-dimensional velocity in the wall-parallel plane while $\overline {v}$ is the wall normal velocity component, and $\boldsymbol{i}$ , $\boldsymbol{j}$ and $\boldsymbol{ k}$ are the usual Cartesian unit vectors. The former is expressed in terms of inner scaling according to

(3.1) \begin{equation} \overline {u}_{\textit{si}} (x,y,z,k_s^+,t) = u_{\tau i}(x,z,k_s^+,t) F(y^+,k_s^+), \end{equation}

where $u_{\tau i}(x, z, k_s^+, t) = u_{\tau } s_i$ , with $u_{\tau } = \| u_{\tau i} \|$ . The friction velocity vector $u_{\tau } s_i$ is a function of spatial coordinates along the wall, $x$ , $z$ and time $t$ , as well as surface roughness, which is characterized by the equivalent sand-grain roughness height $k_s^+ = k_s u_{\tau }/\nu$ . The inner-similarity function, $F(y^+,k_s^+)$ , depends on $y^+ = y u_{\tau }/\nu$ and contains a log-law region that merges with a linear region in the viscous sublayer. Additionally, for roughness, a dependency on $k_s^+$ must be added to the original formulation, so as to yield

(3.2) \begin{equation} F(y^+, k_s^+) = f(y^+) - f_{\textit{NW}}(y^+) \Delta U^+(k_s^+), \end{equation}

where $f(y^+)$ is the smooth inner-similarity function as invoked in Fowler et al. (Reference Fowler, Zaki and Meneveau2022), while $f_{\textit{NW}}(y^+) \Delta U^+(k_s^+)$ accounts for roughness effects through the roughness function $\Delta U^+$ . The function $f_{\textit{NW}}(y^+)$ is active in the near-wall region to drive the roughness contribution to zero in this region. When $k_s^+ = 0$ , $F(y^+, k_s^+) = f(y^+)$ and the smooth inner-similarity function is recovered. On the contrary, when $k_s^+ \gt 0$ , the velocity profile shifts downward as an effect of the increasing drag. In this work, the amount by which the profile is shifted is obtained from the empirical Colebrook formula (Colebrook Reference Colebrook1939),

(3.3) \begin{equation} \Delta U^+ = \kappa ^{-1} \log (1+0.26 \, k_s^+). \end{equation}

Equation (3.3) was preferred to other possible relations because it allows analytical evaluation of the derivatives. The dynamical evolution equation for $u_{\tau i}(x,z,k_s^+,t)$ is derived from the integration of the wall-parallel RANS momentum equation for the velocity $\overline {u}_{\textit{si}}$ , which reads

(3.4) \begin{align} \frac {\partial \overline {u}_{\textit{si}}}{\partial t} + \partial _{\!j} (\overline {u}_{\textit{si}} \overline {u}_{\textit{sj}}) + \partial _y (\overline {v} \, \overline {u}_{\textit{si}}) &= - \frac {1}{\rho } \partial _i \overline {p} + \frac {\partial }{\partial y} \left [ (\nu + \nu _T) \frac {\partial \overline {u}_{\textit{si}}}{\partial y} \right ] \nonumber \\ &\quad + \partial _{\!j} [(\nu + \nu _T) (\partial _i\overline {u}_{\textit{sj}} + \partial _{\!j} \overline {u}_{\textit{si}})] ,\end{align}

where $i = 1,3$ are the wall-parallel directions, $\nu _T$ is the eddy viscosity and $\overline {p}(x,z,t)$ is the qEQ pressure that, within the boundary layer-approximation, does not depend on $y$ . As outlined in Fowler et al. (Reference Fowler, Zaki and Meneveau2022), the incorporation of the diffusion term $\partial _{\!j} [(\nu + \nu _T) (\partial _i\overline {u}_{\textit{sj}} + \partial _{\!j} \overline {u}_{\textit{si}})$ can be avoided in practical LES, as diffusion is artificially introduced by the time integration scheme. Therefore, the diffusion term is neglected in the derivation of the equation for the friction-velocity vector.

Equation (3.4), without the diffusion term, is integrated from the wall, $y=0$ , to the wall-model interface, $y=\varDelta$ . Here, only the treatment of the Eulerian time derivative term is described, whereas the advective term will be treated in Appendix A.

Following the approach used by Fowler et al. (Reference Fowler, Zaki and Meneveau2022), we begin by defining the following integrals:

(3.5) \begin{eqnarray} \delta _\varDelta ^* = \int _{0}^\varDelta \left [1-\frac {\overline {u}_{\textit{si}} (y)}{\overline {u}_{\textit{si}}(\varDelta )}\right ]\textrm{d}y \Rightarrow \int _{0}^{\varDelta } F(y^+, k_s^+)\textrm{d}y = F_\varDelta (\varDelta -\delta _\varDelta ^*), \end{eqnarray}

where $F_\varDelta = F(\varDelta ^+, k_s^+)$ , and

(3.6) \begin{align} \theta _\varDelta = \int _{0}^\varDelta \frac {\overline {u}_{\textit{si}}(y)}{\overline {u}_{\textit{si}}(\varDelta )} \left [1-\frac {\overline {u}_{\textit{si}}(y)}{\overline {u}_{\textit{si}}(\varDelta )}\right ] \textrm{d}y \Rightarrow \int _{0}^\varDelta F^2(y^+, k_s^+)\textrm{d}y = F^2_\varDelta (\varDelta -\delta _\varDelta ^*-\theta _\varDelta ). \end{align}

The above two equations determine the ‘cell displacement thickness’ $\delta _\varDelta ^*$ , and the ‘cell momentum thickness’ $\theta _\varDelta$ , respectively.

Next, consider the integration of the Eulerian time derivative,

(3.7) \begin{eqnarray} \frac {\partial }{\partial t} \int _{0}^\varDelta \overline {u}_{\textit{si}} \,\textrm{d}y &=& \frac {\partial }{\partial t} \left [ \int _{0}^\varDelta u_{\tau i} F(y^+, k_s^+) \,\textrm{d}y \right ] \nonumber \\ &=& \frac {\partial u_{\tau i}}{\partial t} \int _{0}^\varDelta F(y^+, k_s^+) \,\textrm{d}y + u_{\tau i} \int _{0}^\varDelta \frac {\partial }{\partial t} F(y^+, k_s^+) \,\textrm{d}y. \end{eqnarray}

Substituting (3.5) in the expression for the first integral, we obtain

(3.8) \begin{eqnarray} \frac {\partial }{\partial t} \int _{0}^\varDelta \overline {u}_{\textit{si}} \,\textrm{d}y &=& \frac {\partial u_{\tau i}}{\partial t} F_\varDelta (\varDelta -\delta _\varDelta ^*) + u_{\tau i} \int _{0}^\varDelta \frac {\partial }{\partial t} F(y^+, k_s^+) \,\textrm{d}y. \end{eqnarray}

The term $\int _{0}^\varDelta \partial _t F(y^+,k_s^+) \,\textrm{d}y$ can be separated into the smooth and rough contributions, as follows:

(3.9) \begin{eqnarray} \int _{0}^\varDelta \partial _t F(y^+,k_s^+) \,\textrm{d}y &=& \int _{0}^\varDelta \partial _t f(y^+) \,\textrm{d}y - \int _{0}^\varDelta \partial _t \left [f_{\textit{NW}}(y^+) \Delta U^+(k_s^+)\right ] \,\textrm{d}y. \end{eqnarray}

The first integral on the right-hand side can be handled similarly to Fowler et al. (Reference Fowler, Zaki and Meneveau2022), using the notation $f^\prime (y^+) = \textrm{d} f(y^+)/\textrm{d} y^+$ and $f_\varDelta = f(\varDelta ^+)$ ,

(3.10) \begin{eqnarray} \int _{0}^\varDelta \partial _t f(y^+) \,\textrm{d}y &=& \int _{0}^{\varDelta ^+} f^\prime (y^+) \frac {\partial y^+}{\partial t} \,\textrm{d}y^+ \frac {\nu }{u_{\tau }} \nonumber \\ &=& \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} f_\varDelta \varDelta - \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} \int _{0}^\varDelta f(y^+) \,\textrm{d}y. \end{eqnarray}

Moreover, applying the product rule to the second integral in (3.9), one obtains

(3.11) \begin{align} \int _{0}^\varDelta \partial _t \left [f_{\textit{NW}}(y^+) \Delta U^+(k_s^+)\right ] \,\textrm{d}y &= \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} f_{\textit{NW},\varDelta } \Delta U^+ \varDelta - \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} \int _{0}^\varDelta f_{\textit{NW}} \Delta U^+ \,\textrm{d}y \nonumber \\ &\quad + \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} k_s^+ \left (\Delta U^+\right )^\prime \int _{0}^\varDelta f_{\textit{NW}} \,\textrm{d}y, \end{align}

Table 1. Definitions of the momentum and displacement thicknesses required by the new LaRTE-RW model.

where $ (\Delta U^+ )^\prime = d \Delta U^+(k_s^+)/\textrm{d} k_s^+$ represents the sensitivity of the roughness function $\Delta U^+$ to the value of $k_s^+$ . Expression (3.9) then becomes

(3.12) \begin{align} \int _{0}^\varDelta \partial _t F(y^+,k_s^+) \,\textrm{d}y = \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} F_{\varDelta } \delta ^* - \frac {1}{u_{\tau }} \frac {\partial u_{\tau }}{\partial t} k_s^+ \left (\Delta U^+\right )^\prime f_{\textit{NW},\varDelta } (\varDelta - \delta ^*_{\varDelta , \textit{NW}}), \end{align}

where $\delta ^*_{\varDelta , \textit{NW}}$ is the cell displacement thickness defined using the function $f_{\textit{NW}}(y^+)$ (see table 1). Consequently, (3.7) leads to

(3.13) \begin{align} \frac {\partial }{\partial t} \int _{0}^\varDelta \overline {u}_{\textit{si}} \,\textrm{d}y& = \frac {\partial u_{\tau i}}{\partial t} F_\varDelta \left (\varDelta -\delta _\varDelta ^* \right ) \nonumber \\ &\quad + s_i \frac {\partial u_\tau }{\partial t} \left [F_\varDelta \delta _\varDelta ^* - k_s^+ (\Delta U^+)^\prime f_{\textit{NW},\varDelta } \left (\varDelta -\delta ^*_{\varDelta , \textit{NW}}\right )\right ]\!, \end{align}

that, after some manipulation, can be expressed as

(3.14) \begin{align} \frac {\partial }{\partial t} \int _{0}^\varDelta \overline {u}_{\textit{si}} \,\textrm{d}y& = \Delta F_{\varDelta , \textit{HR}} \frac {\partial u_{\tau i}}{\partial t} \nonumber \\ &\quad - u_{\tau } \left [ F_\varDelta \delta _\varDelta ^* - k_s^+ (\Delta U^+)^\prime f_{\textit{NW},\varDelta } \left (\varDelta -\delta ^*_{\varDelta , \textit{NW}}\right ) \right ] \frac {\partial s_i}{\partial t}, \end{align}

where $F_{\varDelta , \textit{HR}} = F_\varDelta - k_s^+ (\Delta U^+)^\prime f_{\textit{NW},\varDelta } (1 - \delta ^*_{\varDelta , \textit{NW}}/\varDelta )$ . Notice that when the wall is smooth, i.e. $k_s^+ = 0$ and $F(y^+, k_s^+) = f(y^+)$ , the previous equation reduces to

(3.15) \begin{eqnarray} \frac {\partial }{\partial t} \int _{0}^\varDelta \overline {u}_{\textit{si}} \,\textrm{d}y &=& \varDelta f_\varDelta \frac {\partial u_{\tau i}}{\partial t} - u_{\tau } f_\varDelta \delta ^*_{\varDelta S} \frac {\partial s_i}{\partial t}. \end{eqnarray}

In fact, (3.15) corresponds to the Eulerian time-derivative originally derived for smooth walls in the LaRTE model (Fowler et al. Reference Fowler, Zaki and Meneveau2022).

Using the same procedure for the advection term in (3.4), by defining the advection velocity of the friction-velocity vector $V_{\tau j}$ , and the relaxation time scale $T_s$ as

(3.16) \begin{align} V_{\tau j} &= \left ( 1- \frac {\delta _\varDelta ^*}{\varDelta }-\frac {\theta _\varDelta }{\varDelta } \right ) \frac {F_\varDelta ^2}{F_{\varDelta , \textit{HR}}} u_{\tau j} , \end{align}

(3.17) \begin{align} T_s &= F_{\varDelta , \textit{HR}} \frac {\varDelta }{u_\tau }, \end{align}

the new LaRTE-RW equation, which accounts for roughness, is given by

(3.18) \begin{eqnarray} \frac {\partial u_{\tau i}}{\partial t} &+& V_{\tau j} \partial _{\!j} ( u_{\tau i}) = \frac {1}{T_s} \!\left [ \frac {1}{u_\tau } \left (\! -\frac {\varDelta }{\rho } \partial _i \overline {p} + \overline {\tau }_{\Delta i} \!\right ) -u_{\tau i}\!\right ] + u_{\tau } \frac {F_\varDelta }{F_{\varDelta , \textit{HR}}} \frac {\delta _\varDelta ^* }{\varDelta } \frac {\partial s_i}{\partial t} + \mathcal{F}_{1 i} \frac {F_\varDelta }{F_{\varDelta , \textit{HR}}} \frac {\theta _\varDelta }{\varDelta } \nonumber \\ &+& 2 \frac {f_\varDelta }{F_{\varDelta , \textit{HR}}} \mathcal{F}_{2 i} \left ( 1 - \frac {\delta ^*_{\Delta S}}{\varDelta } - \frac {\theta _{\Delta f \, f_{\textit{NW}}}}{\varDelta }\right ) -2 \frac {f_{\textit{NW},\varDelta } \Delta U^+}{F_{\varDelta , \textit{HR}}} \mathcal{F}_{2 i} \left ( 1 - \frac {\delta ^*_{\varDelta , \textit{NW}}}{\varDelta } - \frac {\theta _{\varDelta , \textit{NW}}}{\varDelta }\right ) \nonumber \\ &-& \frac {F_\varDelta }{F_{\varDelta , \textit{HR}}} \mathcal{F}_{2 i} \left (1 - \frac {\delta ^*_{\varDelta , \textit{NW}}}{\varDelta }\right ) - u_{\tau } k_s^+ \frac { d (\Delta U^+)}{\textrm{d} k_s^+} \frac {f_{\textit{NW},\varDelta }}{F_{\varDelta , \textit{HR}}} \left (1-\frac {\delta ^*_{\varDelta , \textit{NW}}}{\varDelta } \right ) \frac {\partial s_i}{\partial t} . \end{eqnarray}

In this equation, the functions $\mathcal{F}_{1 i}$ and $\mathcal{F}_{2 i}$ are defined as

(3.19) \begin{align} \mathcal{F}_{1 i} &= u_\tau u_{\tau i} \, (\partial _{\!j} s_{\!j}) \, F_\varDelta , \end{align}

(3.20) \begin{align} \mathcal{F}_{2 i} &= u_{\tau i} u_{\tau j} \, (\partial _{\!j} \Delta U^+) \, f_{\textit{NW},\varDelta }. \end{align}

The first three terms on the right-hand side of (3.18) were already present in the previous smooth formulation (Fowler et al. Reference Fowler, Zaki and Meneveau2022), where the term $\mathcal{F}_{1 i} \theta _\varDelta / \varDelta$ was actually added by Fowler et al. (Reference Fowler, Zaki and Meneveau2023), whereas the additional four terms appear from the current roughness extension. Here, the displacement and momentum thicknesses, $\delta ^*_{\varDelta }$ and $\theta _\varDelta$ , are evaluated using the new similarity function $F(y^+, k_s^+)$ . The terms $\delta ^*_{\Delta S}$ and $\delta ^*_{\varDelta , \textit{NW}}$ are determined using the smooth function $f(y^+)$ and the near-wall function $f_{\textit{NW}}(y^+)$ , respectively. The momentum thickness $\theta _{\Delta ff_{\textit{NW}}}$ involves the product $f(y^+)f_{\textit{NW}}(y^+)$ , while $\theta _{\varDelta , \textit{NW}}$ utilizes $f_{\textit{NW}}(y^+)$ . A summary of the displacement and momentum thicknesses appearing in (3.18) is shown in table 1.

Equation (3.18) describes the qEQ development of the wall stress, $\overline {\tau }_{w i} = u_{\tau } u_{\tau i}$ , over the time scale $T_s$ , under the influence of $\overline {\tau }_{\varDelta i}$ . This equation incorporates roughness effects through a modification of $V_{\tau j}$ and $T_s$ (compared with the previous smooth formulation), and through the functions $\mathcal{F}_{2 i}$ and $k_s^+ d (\Delta U^+)/\textrm{d} k_s^+$ in the last term (accounting for roughness heterogeneity). In fact, when roughness heterogeneity is present, the gradient $\partial \Delta U^{+} / \partial x_{\!j}$ appearing in the function $\mathcal{F}_{2 i}$ (3.20) activates the fourth, fifth and sixth terms on the right-hand side of (3.18). The final (seventh) term involves the quantity $k_s^+ \, d (\Delta U^+)/\textrm{d} k_s^+$ , which represents the sensitivity of the roughness function $\Delta U^+$ to variations in $k_s^+$ in rough-wall regions (i.e. where $k_s^+ \gt 0$ ). Under the Colebrook formulation, this term is proportional to $1/k_s^+$ in the fully rough regime.

For homogeneous smooth walls, where the terms $k_s^+ \, d (\Delta U^+)/\textrm{d} k_s^+$ and $\partial _{\!j} \Delta U^+$ vanish, (3.18) simplifies to

(3.21) \begin{equation} \frac {\partial u_{\tau i}}{\partial t} + V_{\tau j} \partial _{\!j} ( u_{\tau i}) = \frac {1}{T_s} \left [ \frac {1}{u_\tau } \left ( -\frac {\varDelta }{\rho } \partial _i \overline {p} + \overline {\tau }_{\Delta i} \right ) -u_{\tau i}\right ] + u_{\tau } \frac {\partial s_i}{\partial t} \frac {\delta _\varDelta ^* }{\varDelta } + \mathcal{F}_{1 i} \frac {\theta _\varDelta }{\varDelta }. \end{equation}

This equation, which is analogous to the corresponding equation in Fowler et al. (Reference Fowler, Zaki and Meneveau2022), excluding the diffusion term and including the $\mathcal{F}_{1 i} \theta _\varDelta /\varDelta$ term, differs primarily in that $V_{\tau j}$ and $T_s$ are defined here by using the rough-wall inner-similarity function $F(\varDelta ^+,k_s^+)$ .

The friction velocity relaxes to $ -\varDelta /\rho \, \partial _i \overline {p} + \overline {\tau }_{\varDelta i}$ on the right-hand side of (3.18), which requires a closure that accounts for surface roughness. We follow the approach proposed by Fowler et al. (Reference Fowler, Zaki and Meneveau2023), where the stress at the interface is computed using (2.9), and the pressure is taken directly from the LES field, i.e. $ \overline {p} = \hat {p}_{\textit{LES}}$ . In contrast to Fowler et al. (Reference Fowler, Zaki and Meneveau2023), where the contribution of the fast laminar Stokes layer was subtracted from the LES pressure input, we retain the full LES pressure. This choice is justified by the rough-wall configuration considered here, in which a laminar Stokes layer is not expected to exist.

Moreover, while Fowler et al. (Reference Fowler, Zaki and Meneveau2022) use a filtered pressure gradient – obtained by applying a temporal filter over scales of order $ 3T_s$ – as input to the wall stress model, in order to simplify implementation, we use the instantaneous LES pressure gradient. We have verified that the simulation results are not sensitive to this modelling choice. Consequently, using the mild pressure gradients fit (Meneveau Reference Meneveau2020), the closure for the stress at the interface reads

(3.22) \begin{equation} -\frac {\varDelta }{\rho } \, \partial _i \overline {p} + \overline {\tau }_{\varDelta i} = \frac {1}{2} \, c_{\!f}^{w\textit{m}}({\textit{Re}}_\varDelta , \chi , k_s/\varDelta ) \, U_{\textit{LES}}^2 \, s_i, \end{equation}

where $ c_{\!f}^{w\textit{m}}$ is the modelled skin-friction coefficient evaluated with the mild-pressure gradients fit utilizing the non-dimensional pressure gradient parameter $\chi$ that is defined in Appendix C, where the algorithm used to compute $ c_{\!f}^{w\textit{m}}$ is detailed.

It is worth noting that the equilibrium model $ U^+ = \kappa ^{-1} \log (\varDelta /k_s) + 8.5$ assumes fully rough conditions. While this model is well-suited for fully rough regimes, the GMD model offers greater flexibility as it can be applied to transitionally rough cases. The GMD model includes a transition between smooth wall and fully rough behaviour, with the associated ‘Moody diagram’ displaying $ C_{\!f} \sim {\textit{Re}}^{-1}$ behaviour at low Reynolds numbers and becoming horizontal in the fully rough regime (see figure 10 of Meneveau Reference Meneveau2020). The LaRTE-RW model uses the GMD formulation to compute stress at the interface, making it applicable to transitionally rough conditions as well. However, it is important to note that the relationship between $ \Delta U^+$ and $ k_s^+$ is based on the Colebrook formula, which is universal in the fully rough regime, but it is not in transitionally rough cases.

The LaRTE-RW evolution (3.18) is discretized in time using a forward Euler method, allowing for the explicit update of $u_{\tau i}$ . The inherent numerical diffusion from this discretization scheme led to the omission of the wall-parallel diffusion term. Spatial gradients resulting from the advection term in the Lagrangian time derivative are computed using a semi-Lagrangian approach, where the right-hand side of the equation is evaluated at the upstream position with bilinear spatial interpolation. Some more details on the numerical solution of (3.18) can be found in the discussion provided in Fowler et al. (Reference Fowler, Zaki and Meneveau2022) for the corresponding evolution equation for smooth walls.

4. Validation

To provide a preliminary assessment of the LaRTE-RW model’s accuracy, WMLESs were carried out for statistically steady open-channel (or half-channel) flow, considering both smooth and homogeneous rough-wall conditions. The computational domain was $L_x \times L_y \times L_z$ = $6.4 \delta \times \delta \times 3.2 \delta$ , with a uniform grid resolution of $192 \times 60 \times 94$ points. Periodic boundary conditions were applied in the streamwise and spanwise directions, while the top boundary was treated with symmetry conditions. The bottom-wall boundary condition depended on the specific wall stress model employed. In particular, the LaRTE-RW model, which computes wall stress according to (3.18), was compared with both the GMD model and log-law EQWM.

The wall-model interface was located at $ \varDelta /\delta = 0.05$ , ensuring that three grid points were below this interface, following the recommendations of Kawai & Larsson (Reference Kawai and Larsson2012). Although the flow below the interface is unphysical, due to the limited resolution, having points in this region helps facilitate eddy generation and allows the mixing length to exceed the grid size.

In both GMD and LaRTE-RW models, the input velocity from the LES field, $ U_{\textit{LES}}$ , was filtered in time using a single-sided exponential filter of the form

(4.1) \begin{equation} U_{\textit{LES}} = \alpha \, U_{\textit{LES}}^{n} + (1 - \alpha ) \, U_{\textit{LES}}^{n-1}, \end{equation}

where $ U_{\textit{LES}}^{n}$ is the instantaneous horizontal velocity at the wall-model height at time step $ n$ , and $ U_{\textit{LES}}^{n-1}$ is the temporally filtered velocity at previous time step. The smoothing coefficient is defined as

(4.2) \begin{align} \alpha = \frac {\Delta t}{T_{\textit{avg}}}, \end{align}

where $ \Delta t$ is the LES time step, and the averaging window is $ T_{\textit{avg}} = 3 T_s$ . This exponential filter stabilizes the input to the wall model by attenuating under-resolved, high-frequency fluctuations, while still preserving physically relevant unsteadiness in the large-scale motions. In contrast, the log-law model utilized the instantaneous velocity at the wall-model interface. Simulations with a sand-grain roughness height of $k_s^+ \simeq 130$ were performed, along with smooth-wall simulations, for comparison with the results from Lee & Moser (Reference Lee and Moser2015).

Figure 1. Inner-scaled mean velocity profiles for smooth and rough homogeneous surfaces, with the inset showing the corresponding roughness function. The DNS data from Lee & Moser (Reference Lee and Moser2015) are used for the smooth wall at ${\textit{Re}}_{\tau } = 5200$ , while rough wall simulations are conducted at a roughness Reynolds number of $k_s^+ = 130$ .

The simulated inner-scaled mean-velocity profiles are shown in figure 1. Under equilibrium conditions, both the LaRTE-RW and GMD models produced similar results, with no notable deviations. All velocity profiles closely follow the smooth log-law solution, represented by the solid black line. The smooth-wall simulations show a slight overprediction of the wake region amplitude compared with reference DNS data (Lee & Moser Reference Lee and Moser2015), a discrepancy also observed by Fowler et al. (Reference Fowler, Zaki and Meneveau2022) for the smooth LaRTE model, likely due to numerical factors such as grid resolution and SFS model approximation (Bae et al. Reference Bae, Lozano-Durán, Bose and Moin2019; Wang, Hu & Zheng Reference Wang, Hu and Zheng2020), rather than to the wall models themselves.

For the homogeneous rough surface, the velocity profiles again display a log-law region. The measured roughness function ( $\Delta U^+ \approx 8.5$ ) is consistent with the Colebrook formula, as well as Nikuradse’s data, as shown in the inset of figure 1. It is noted that this agreement with known results is a verification of the model’s consistency and correct implementation.

Figure 2. Sketch of the computational domain for heterogeneous-wall simulations (not to scale).

6. Summary and conclusions

This paper introduces a significant extension of the LaRTE model for rough walls, enabling seamless prediction of flow behaviour across smooth and fully rough regimes. The extended model’s performance was evaluated against traditional models, including the log-law EQWM and the GMD model, under various conditions, including homogeneous and heterogeneous roughness, and varying pressure gradients. The model was successfully applied to WMLES of flow over patches with smooth-to-rough and rough-to-smooth transitions, showing improved accuracy compared with the traditional log-law wall model. In particular, the case of rough to smooth transition highlighted the importance of including a turbNEQ component into the wall model to accelerate convergence towards the smooth wall asymptote.

Furthermore, the study of pressure gradient cases revealed important insights into wall-model responses to free stream forcing. While wall models accurately capture the general trends of increasing friction in FPG regions, and decreasing friction in APG regions, inaccuracies emerge downstream of pressure-gradient reversals. Notably, in FPG–APG cases, wall models tend to react too quickly to free stream disturbances, whereas in FPG–APG–FPG cases, the response is overly delayed. Transitions between FPG and APG regions further revealed limitations in the current modelling framework. In particular, when the flow transitions from FPG to APG, wall models tend to underpredict the skin-friction coefficient due to their inability to capture the re-energization of near-wall turbulence. Conversely, in APG to FPG transitions, experimental observations show a rapid recovery in wall shear stress, attributed to the realignment of turbulent structures with the wall. However, the LaRTE-RW model, like other WMLES approaches, underpredicts this response due to its dependence on the slowly evolving outer-layer signal. These findings reinforce the need for wall models that incorporate both non-equilibrium dynamics and a more faithful representation of near-wall turbulence memory and response times. Therefore, future work will focus on further refining the LaRTE-RW model with enhanced non-equilibrium components to address this particular need.

Findings from smooth-wall simulations add further support to these observations, suggesting that the present inaccuracies observed in rough-wall configurations are not caused by the incorporation of roughness effects, but are instead rooted in limitations of the underlying wall-modelling methodologies themselves. This study also highlights the impact of SFS models on $C_{\!f}$ predictions, particularly highlighting that inaccuracies in upstream boundary layer predictions can propagate downstream, underscoring the importance of accurately resolving boundary layer states at early stages.

However, it is worth noting that a key advantage of the newly proposed formulation is in the proven ability to separate the qEQ component of wall stress from its non-equilibrium counterpart. This modular structure facilitates targeted refinement and tuning. In contrast, models like the log-law EQWM and GMD integrate these dynamics inseparably, making such improvements more difficult to implement. We note that although the assumption of the mixing length for deriving the mean velocity profile from the simplified boundary layer momentum equation may not adequately apply in the vicinity of rough-to-smooth transitions or under varying pressure gradients and it represents a drawback in the log-law and GMD models, in the LARTE-RW context is logical, as it aligns with the qEQ conditions the LaRTE-RW equation is designed to represent. Departures from equilibrium effects must be considered through the non-equilibrium portions of the model.

Overall, the current results emphasize the benefits of the new LaRTE-RW model in WMLES for rough-wall flows under non-equilibrium conditions. The model demonstrates superior predictive capabilities, particularly during the initial stages of FPG development, also when paired with the LDSM. The over-predictions of wall stress observed in the GMD model are effectively mitigated by the LaRTE-RW model, leading to slightly improved performance. The LaRTE-RW introduces advantageous time delays where standard EQWMs provide excessively fast responses to imposed flow conditions. However, the time delays prove detrimental in situations requiring more swift responses, such as for boundary layers developing below accelerating–decelerating–accelerating free stream conditions. Finally, the LaRTE-RW model surpasses traditional models like the log-law EQWM in handling complex configurations such as heterogeneous roughness transitions. The present research findings demonstrate the LaRTE-RW model’s potential for advancing the predictive accuracy of WMLES in complex turbulent flow configurations, but that additional research is needed to fine-tune response times and mechanisms to further improve wall modelling in LES.