1. Introduction
Rough surfaces are encountered in a wide range of engineering and environmental applications. The flow in heat exchangers, the atmospheric boundary layer over urban areas, complex topography or vegetation and the leading-edge erosion of turbine blades are just a few examples of the wide range of problems where roughness plays a key role. Roughness in general leads to a drop in system performance and a huge boost in the management costs. Hence, predicting the effect of rough walls on turbulence has become an important design prerequisite for practical applications. Starting from the seminal work of Nikuradse (Reference Nikuradse1933), in the last decades, several studies have been carried out to understand the flows physics over corrugated walls. Despite extensive efforts being made by the scientific community, our knowledge cannot be considered sufficiently robust and universal. One of the first attempts to predict the main roughness effect was given by Hama (Reference Hama1954), introducing the correlation between a geometrical parameter, known as equivalent sand grain roughness 
$k_s$ (Schlichting Reference Schlichting1937), and the energy loss induced by the roughness. Hama (Reference Hama1954) observed that the main effect of the roughness is the downward shift of the mean velocity profile (scaled in inner units) in the log region, known as roughness function 
$\Delta U^+$. Hereafter, the superscript 
$^+$ denotes variables made non-dimensional with inner variables 
$u_ \tau = ( \tau _s / \rho )^{(0.5)}$ and 
$\nu /u_\tau$, where 
$u_\tau$ is the friction velocity, 
$\rho$ is the fluid density, 
$\nu$ is the kinematic viscosity and 
$\tau _s$ is the wall shear, equal to the sum of the viscous (or skin frictional) stress 
$\overline {C_f}=({1}/{L_x})\int _0^{L_x}(\mu \partial \langle U^* \rangle /\partial y^*)(1/\rho U^2_c)\,{\rm d}s$ and the form drag 
$\overline {P_d} = ({1}/{L_x}) \int _0^{L_x} \langle P \rangle \boldsymbol {n} \boldsymbol {\cdot} \boldsymbol {x}\,{\rm d}s$, (
$\boldsymbol {n}$ is the normal to the surface, 
$\boldsymbol {x}$ is the unit vector in the streamwise direction and 
$s$ is a coordinate along the surface). The symbol 
$\langle {\cdot } \rangle$ indicates quantities averaged in the spanwise direction and time, * indicates dimensional units and 
$L_x$ represents the streamwise length. The original formulation to calculate the roughness function was given by Hama (Reference Hama1954), who introduced the following correlation:
\begin{equation} \Delta U^+= \frac{1}{\kappa} \ln(k_s^+) + B, \end{equation}
where 
$\kappa$ is the von Kármán constant, 
$k_s^+=k_s{\cdot } u_\tau /\nu$ and B is a constant. Unfortunately, for a general rough wall, 
$k_s$ cannot be calculated a priori; it can be, in fact, determined once the mean velocity profile is known. In fact, as pointed out by several authors (see among others Flack, Schultz & Volino (Reference Flack, Schultz and Volino2020)), 
$k_s$ is not a physical measure of the corrugation. The prediction of the drag, as well as 
$\Delta U^+$, based on geometrical features of the wall, has received extensive attention in the past and a variety of roughness correlations have been developed in the literature (see among others Sigal & Danberg (Reference Sigal and Danberg1990), Waigh & Kind (Reference Waigh and Kind1998), Van Rij, Belnap & Ligrani (Reference Van Rij, Belnap and Ligrani2002), Bons (Reference Bons2005), Flack & Schultz (Reference Flack and Schultz2010), Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), Busse, Thakkar & Sandham (Reference Busse, Thakkar and Sandham2017), Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017), Thakkar, Busse & Sandham (Reference Thakkar, Busse and Sandham2017), Piomelli (Reference Piomelli2019), De Marchis et al. (Reference De Marchis, Saccone, Milici and Napoli2020) and Chung et al. (Reference Chung, Hutchins, Schultz and Flack2021)). Several parameters were analysed in the past, for instance the mean roughness height 
$k^+$, the peak-to-valley distance 
$k_{pv}^+$, the root mean square 
$k_{rms}^+$, the roughness solidity 
$\lambda$, the skewness 
$S_k$, the kurtosis 
$K_s$ and the effective slope (
$ES$), using both experiments or numerical simulations over two-dimensional (2-D) or three-dimensional (3-D) roughness. Schlichting (Reference Schlichting1937) introduced the term roughness solidity (
$\lambda$) to quantifies the roughness density and is defined as the total projected frontal roughness area per unit wall-parallel projected area. It has been observed that the roughness function 
$\Delta U^+$ increases with density up to 
$\lambda =0.15$, where it is maximum, and then decreases for larger 
$\lambda$ (Jiménez Reference Jiménez2004; Flack & Schultz Reference Flack and Schultz2014). This indicates qualitatively that increasing the roughness density while in the sparse regime (
$\lambda <0.15$) increases the drag due to the increased frontal area of the roughness. In the dense regime (
$\lambda >0.15$), mutual sheltering of roughness elements leads to a decrease in drag as the density is increased (Macdonald, Griffiths & Hall Reference Macdonald, Griffiths and Hall1998; Oke Reference Oke1988; Jiménez Reference Jiménez2004). Despite the utilization of solidity for distinguishing between different roughness types, it cannot solely fully characterize a rough surface. For example, Jiménez (Reference Jiménez2004) showed a dependency on 
$\lambda ^{-2}$, while 
$\lambda ^{-5}$ has been proposed by Dvorak (Reference Dvorak1969). Other geometrical parameters are required to describe the mutual sheltering of the roughness elements. The effective slope, 
$ES$, is connected to the solidity parameter, 
$\lambda$, through the relation 
$ES = 2\lambda$ (Napoli, Armenio & De Marchis Reference Napoli, Armenio and De Marchis2008; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016; Thakkar et al. Reference Thakkar, Busse and Sandham2017). MacDonald et al. (Reference MacDonald, Chan, Chung, Hutchins and Ooi2016) showed that the value of 
$ES\approx 0.35$, defined as a demarcation point between waviness regime (
$ES < 0.35$) and roughness regime (
$ES > 0.35$), is associated with a solidity value of 
$\lambda =0.175$. Moreover, Mejia-Alvarez & Christensen (Reference Mejia-Alvarez and Christensen2013) and De Marchis (Reference De Marchis2016) have reported that 
$ES=0.35$ is a limit between slope dependent and height dependent flows. Furthermore, some studies focused the attention on regular elements arranged over a flat plate (see among others Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003), Volino, Schultz & Flack (Reference Volino, Schultz and Flack2011), De Marchis (Reference De Marchis2016), Gatti et al. (Reference Gatti, von Deyn, Forooghi and Frohnapfel2020), Modesti et al. (Reference Modesti, Endrikat, Hutchins and Chung2021) and Busse & Zhdanov (Reference Busse and Zhdanov2022) for 2-D elements, and Orlandi & Leonardi (Reference Orlandi and Leonardi2008), Boppana, Xie & Castro (Reference Boppana, Xie and Castro2010), Hong, Katz & Schultz (Reference Hong, Katz and Schultz2011) and Busse & Jelly (Reference Busse and Jelly2020) for 3-D elements). Recently, Millward-Hopkins et al. (Reference Millward-Hopkins, Tomlin, Ma, Ingham and Pourkashanian2011) and Yang et al. (Reference Yang, Sadique, Mittal and Meneveau2016) have formulated drag-prediction models that use a sheltering argument to account for the interaction between roughness elements. Geometrical statistics, developed so far, correlate well with the drag of some particular type of roughness but lack universality with other generic irregular walls. This calls for an effort to develop a universal correlation to predict roughness effects. In this study, direct numerical simulations have been performed to reveal the flow features around rough elements. A new parameter, called effective distribution (
$ED$), has been introduced. The 
$ED$ is based on a modified version of the 
$ES$ (Napoli et al. Reference Napoli, Armenio and De Marchis2008) and the proposed results show a good correlation with different roughness shape. The paper is organized as follows: § 2 describes the numerical procedure adopted for direct numerical simulations, § 3 highlights flow configurations, results are presented in § 4 and conclusions are drawn in § 5.
2. Numerical procedure
Direct numerical simulations have been performed for a fully developed turbulent channel flow with roughness on the bottom wall. The non-dimensional Navier–Stokes and continuity equations for incompressible, neutrally stable flows can be expressed as
$$\begin{gather} \frac{\partial U_i}{\partial t} + \frac{\partial U_i U_j}{\partial x_j} ={-} \frac{\partial P}{\partial x_i} + \frac{1}{Re} \frac{\partial^{2} U_i}{\partial x^{2}_j} + \varPi \delta_{i1} , \end{gather}$$
$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{U} = 0, \end{gather}$$
where 
$Re$ is the Reynolds number based on the bulk velocity (
$U_b = 1/h \int _0^{h} U\, {{\rm d}y}$), which is held constant in time, 
$h$ is the channel half-height, 
$\delta _{ij}$ is the Kronecker delta, 
$U_i$ is the 
$i$th component of the velocity vector, 
$x_i$ is the 
$i$th coordinate direction and P is the pressure. The quantity 
$\varPi$ is the pressure gradient which varies with time in order to keep the flow rate constant. The Navier–Stokes equations were discretized in an orthogonal coordinate system using the staggered central second-order finite difference approximation. The surface roughness was treated using the immersed boundary technique, which allows solution over complex geometries without the need for intensive body-fitted grids. It consists of imposing 
$U_i=0$ on the body surface, which does not necessary coincide with the grid. Full details about the immersed boundary method and the numerical schemes can be found in Orlandi & Leonardi (Reference Orlandi and Leonardi2006).
3. Flow configuration
Periodic boundary conditions have been applied in the streamwise (
$x$ or 
$x_1$) and spanwise (
$z$ or 
$x_3$) directions while a no-slip condition has been imposed in the wall-normal direction (
$y$ or 
$x_2$). The computational box in the 
$x$, 
$y$, 
$z$ direction is 
$6.4 h \times 2.2 h \times {\rm \pi}h$, respectively; a sketch of two of the roughness cases considered here is shown in figure 1. The computational domain has been discretized using 
$512 \times 256 \times 256$ grid points. The mesh is uniform in the streamwise and spanwise directions, with 
$\Delta x / h = 0.0125$ and 
$\Delta z / h = 0.009$. On the other hand, a non-uniform mesh has been used in the 
$y$ direction. Specifically, in the wall-normal direction the points are clustered near the wall within the cavity 
$\Delta y_ {min} / h = 0.002$. The mesh increases towards the channel centreline, with 
$\Delta y_{max} / h= 0.026$. The Reynolds number is 
$Re=4300$ and corresponds to the friction Reynolds number 
$Re_\tau =240$ when both walls are smooth. Details of the computational box and resolution are summarized in table 1.
The 3-D computational domain for the two set of rough surfaces considered: (a) set 1 having 16 triangles and 
$k/h\approx 0.1$; (b) set 2 having eight triangles and 
$k/h\approx 0.2$.
Legend, sketch of the geometrical shape, computational box and grid resolution of the different walls studied here.
For a fixed pitch to height ratio 
$w/k=4$, which is below the value for which transverse bars can be considered virtually isolated (
$w/k=8$, Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003)), two sets of simulations have been analysed varying the roughness height. The first set is made of 16 triangular transverse bars equally spaced in the streamwise direction 
$w/h=0.4$ (figure 1a). The baseline case (
case 
$A1_1$), has a constant roughness height 
$k/h=0.1$. In the second set of simulations (figure 1b), we halved the number of triangular bars in streamwise direction but doubled the roughness height to 
$k/h=0.2$ (
case 
$A1_2$). The subscript indicates the roughness height. Other cases are considered as a modification of the baseline to highlight specific geometrical features, such as a protuberance above the roughness layer and the wake of larger elements affecting the downstream roughness. The height is slightly adjusted in each case to keep constant either the value of 
$ES$, kurtosis and skewness. In cases 
$B1_1$ (
) and 
$B1_2$ (
) we doubled the vertical size of one element; in cases 
$B2_1$ (
), 
$B2_2$ (
) and 
$B2_{2b}$ (
) we removed the element immediately downstream of the tallest one. The set of simulations labelled with 
$C$ present two taller triangles, with streamwise distances gradually increasing from 
$C1_1$ to 
$C4_1$ (
, 
, 
, 
) and from 
$C1_2$ to 
$C4_2$ (
, 
, 
, 
). The geometrical and flow properties are summarized in table 2. According to the values of the friction Reynolds number and 
$k/h$ here considered, a fully rough regime is ensured (see among others Bandyopadhyay (Reference Bandyopadhyay1987) and Leonardi, Orlandi & Antonia (Reference Leonardi, Orlandi and Antonia2007)).
Geometrical and flow properties: 
$k/h$, roughness height; 
$k_{max}/h$, big element roughness height (equal to 2*
$k/h$); 
$w$, cavity width; 
$\lambda$, distance between big elements; 
$ES$, effective slope; 
$ED$, effective distribution; 
$K_u$, kurtosis; 
$S_k$, skewness; 
$D/\rho U_b^2$, total drag; 
$\Delta U^+$, roughness function.
4. Results and discussion
The effect of the roughness is to shift downward the mean velocity 
$U^+$ profile, with respect to that on a smooth wall, by an increment 
$\Delta U^{+}$, i.e.
\begin{equation} U^{+} = \kappa^{{-}1} \ln y^{+} + C - \Delta U^{+}. \end{equation} The roughness function in the present paper has been computed as the distance of the log region with respect to the ideal smooth wall with 
$C=5.6$. The friction velocity on the rough wall is computed as the sum of the form drag and frictional drag so there are no uncertainties due to the asymmetry of the channel. The velocity profiles on the upper smooth wall (not shown here because beyond the scope of the paper), when scaled with the proper friction velocity (from the shear on that wall and not from the pressure drop) agree well with the law of the wall although the extent of the log region is shorter due to the smaller local turbulent Reynolds number. The virtual origin in 
$y$ is chosen to have a slope of 
$\kappa =0.41$. Other choices could have been made for the virtual origin, but since the goal here is to calculate the roughness function for a large number of different cases, fixing the slope of the log region to 
$\kappa ^{-1}$ allowed a consistent calculation of the roughness function. In figure 2 the total drag and the roughness function are plotted as function of the 
$ES$, skewness and kurtosis. For the same 
$ES$, or 
$K_u$ or 
$S_k$, the drag varies significantly, up to 
$300\,\%$ and the roughness function up to 
$40\,\%$. The value of the drag has been included in the analysis because it does not present the uncertainties that the roughness function has in its definition (virtual origin, the slope of the log region and the value of the constant 
$C$). Even grouping data relative to the same mean roughness height or root mean square does not eliminate the scatter in the data.
Drag and roughness function dependence on the geometrical features of the rough wall (
$ES$, skewness and kurtosis). Symbols as in table 1.
4.1. Inconsistency in geometrical parametrization
The flow structure and the pressure around the roughness elements have been analysed to understand why the drag and roughness function vary despite having the same geometrical statistics (mean roughness height, 
$ES$, 
$S_k$ or 
$K_u$). In particular, comparing 
$A1_1$ (
uniform triangles) and 
$B1_1$ cases, (
same as 
$A1_1$ with an element 
$\Delta k$ higher), with a slight change of 
$ES$ corresponds to a major difference in drag. For uniform roughness, the cavities are filled with a recirculating flow (figure 3a). The non-dimensional form drag (normalized with 
$\rho U_b^2$) of each element is 
$P_d=0.005$. On the other hand, in 
$B1_1$, the streamlines impinge on the tallest element (the element ‘0’ in figure 3b) generating a stagnation point and high-pressure differences with respect to the leeward side of the wedge. The form drag is 
$P_d=0.04$, approximately eight times larger than that of uniform triangles. This shows how sensitive the drag is to pinnacles emerging outside the roughness layer, which, instead, is not accounted for in 
$ES$, skewness and kurtosis. The form drag of the upstream triangle (labelled ‘
$-1$’) is slightly smaller because the streamlines are tilted upward by the taller element. The large recirculation closes on the second element downstream (labelled ‘2’) at a distance of approximately 
$8k$. The pressure drag of the two roughness elements in the wake is very small and negative, meaning that the pressure on the leeward side is higher than that on the windward side. The drag of the other roughness elements is to a good approximation unaffected implying that a perturbation to the geometrical topography of the surface affects the flow slightly upstream (up to 
$4k$) and a bit more downstream (
$8k$). The surface 
$B2_1$ is obtained by removing the triangle (labelled ‘1’ in figure 3b) downstream of the highest roughness element. The mean streamlines are very similar with a main recirculation originated on the highest peak and closing approximately 
$8k$ downstream (figure 3c). The drag on each element is approximately the same. The overall drag and roughness function are to a good approximation the same as those of 
$B1_1$ despite variations of 
$ES$ and 
$K_u$. This suggests that roughness elements located in the wake region of higher elements must be weighted differently in the geometrical statistics of the surface. Adding a second taller roughness element to 
$B1_1$, immediately downstream (labelled ‘1’ in figure 3d), surface 
$C1_1$, increases the mean surface height, as well as the higher moments statistics, but reduces the drag (and roughness function) instead of increasing it (as it would have been expected by having a higher mean roughness). The wake of the first higher pinnacle shields the second, as already considered in various prior studies (see among others Raupach, Antonia & Rajagopalan (Reference Raupach, Antonia and Rajagopalan1991), Shao & Yang (Reference Shao and Yang2005), Shao & Yang (Reference Shao and Yang2008) and Yang et al. (Reference Yang, Sadique, Mittal and Meneveau2016)). These studies denoted, as volumetric sheltering, the momentum reduction in the wakes of roughness elements and its effect on the drag of neighbouring roughness elements. The recirculation shrinks compared with 
$B1_1$ and 
$B1_2$, filling the cavity formed by the two higher triangles (elements ‘0’ and ‘1’ of figure 3d). Increasing the distance between the two highest pinnacles (cases 
$C3_1$ 
and 
$C4_1$ 
, figure 4a,b), leads to an increase of drag and 
$\Delta U^+$ because the streamlines tend to reattach on the lower array of triangles with a consequent increase of pressure drag on the large element. These results suggest that the position of the roughness elements, affects the flow physics and the drag despite the geometrical feature 
$K_u$ or 
$S_k$ are the same. A single pinnacle much higher than the others has a major effect on the flow. Its contribution is not proportional to the wet area, or exposed area to the flow; it is much higher if the upstream elements are smaller. It could be interpreted mathematically into a geometry height gradient, which was partially taken into account by the 
$ES$. However, cases 
$C1_1$–
$C4_1$ highlight how it is important the presence of other tall roughness elements upstream, and their wake. The distance between two consecutive highest peaks is a key parameter to determine the influence of roughness on turbulent flow.
Streamlines superposed to colour contours of pressure: (a) 
$A1_1$ (
); (b) 
$B1_1$ (
); (c) 
$B2_1$ (
); (d) 
$C1_1$ (
). The pressure drag of each triangle is indicated below them, i.e. 
$P_d(4)=0.005$. Definition of 
$\Delta k$ is included in the figure.
Streamlines superposed to colour contours of pressure: (a) 
$C3_1$ (
); (b) 
$C4_1$ (
).
4.2. Effective distribution
The analysis of § 4.1 showed that any parametrization based on geometrical features of the walls needs to be consistent with the following points.
(i) The roughness elements in the wake of larger elements have a negligible contribution to the drag. As a consequence, the geometrical quantities used to parameterize the roughness should be filtered by the contribution of those elements in the wake length.
(ii) The contribution to the drag of each roughness element depends on its pattern and distance from previous elements (figure 3b).
(iii) The distance between two consecutive rough elements affects the velocity distribution, the momentum in the cavity and as a consequence the intensity of the stagnation point on the windward roughness element.
These features have been included in a new geometrical parametrization, 
$ED$, as a revision of the 
$ES$ introduced by Napoli et al. (Reference Napoli, Armenio and De Marchis2008), as follows:
\begin{equation} \begin{cases}
\displaystyle ED = \left(ES - \sum ^m_{j=1} \sum ^n_{i=1}
\alpha_{i,j} \,{\cdot}\, ES_{i} + \sum ^m_{j=1} \sum ^m_{i=1}
\beta_{i,j} \,{\cdot}\, ES_{\Delta k_i}\right) + \sum ^n_{i=1}
\frac {w_{i,i+1}}{L_x} \frac {k}{\delta_k}\\
\displaystyle\alpha_{i,j} = \min\left(1,\frac{\Delta
k_j}{w_{i,j}}\right) \quad \text{for } 1\leqslant
\frac{w_{i,j}}{k}<8, \quad \alpha_{i,j} = 0 \quad
\text{for} \ \frac{w_{i,j}}{k}>8 \\
\displaystyle\beta_{i,j} =
\frac{\lambda_{i,j}}{wake_j} \quad \text{for }
\frac{\lambda_{i,j}}{wake_j}<1, \quad \beta_{i,j}
= 1 \quad \text{for }
\frac{\lambda_{i,j}}{wake_j}>1 \end{cases}
\end{equation}
where 
$ES$ is the overall 
$ES$ calculated as in the original formulation of Napoli et al. (Reference Napoli, Armenio and De Marchis2008) (
$({1}/{L_{x_1}} )\int _{L_{x_1}} {| {\partial k(x_1)}/{\partial x_1} | } \,{\rm d}\kern0.7pt x_1$), 
$ES_i$ is the 
$ES$ of the 
$i$th element, weighted by 
$\alpha _{i,j}=\min (1,{\Delta k_j}/{w_{i,j}}$) with 
$w_{i,j}$ the distance between the 
$j$th higher peak, emerging 
$\Delta k_j$ over the crest plane, and the 
$i$th roughness element. The index 
$n$ in the summations indicates the number of roughness elements while 
$m$ is the number of pinnacles above the crests plane. The second term on the right-hand side of (4.2) subtracts from the 
$ES$ the contribution of the elements in the wake of the higher peaks. The coefficient 
$\alpha _{i,j}$ takes into account the effect of the wake of the 
$j{\rm th}$ pinnacle above the crests plane on the 
$i{\rm th}$ roughness element. This term is unity when the 
$i{\rm th}$ roughness element is close to the higher peak ‘j’ and in its wake, and decreases to zero as it is farther apart from it. In fact, when the cavity between two roughness elements is narrow, the flow has a d-type behaviour and the drag is almost unaffected by that roughness element so its contribution to the 
$ES$ is removed. By increasing 
$w_{i,j}$, the distance of the roughness element from the peak, its contribution to the drag gradually increases and then just a fraction of its 
$ES$ is subtracted from the overall 
$ES$. Previous papers showed that for a pitch to height ratio larger than 
$8$ the roughness elements act as isolated with a reattachment of the flow on the flat wall of the cavities. Therefore the wake length can be approximated to 
$wake_j=8 k_j$, which is this dataset could be simplified to 
$wake_j=8 k_{max}$ since all the pinnacles above the crests plane have the same height. However, to keep a general formulation of (4.2) 
$wake_j=8 k_j$ is used allowing to have a non-uniform distribution of elements above the crests plane. For 
$w_{i,j}/k_j>8$ (resulting in 
$w_{i,j}>wake_j$), indicating the roughness element outside the wake region, 
$\alpha _{i,j}=0$. A similar concept is used to account for the pattern of the higher pinnacles, the 
$m$ elements emerging above the crests plane. When the downstream pinnacle is too close to that upstream, its contribution to the drag is smaller than when it is isolated. Therefore, the 
$ES_{\Delta k_j}$ is weighted by 
$\beta _{i,j}= {\lambda _{i,j}}/wake_j$, where 
$\lambda _{i,j}$ is the distance between the pinnacles 
$i$ and 
$j$ that are higher than the crests plane. When 
$\lambda _{i,j}< wake_j$, the downstream pinnacle is in the wake region of the upstream pinnacle. This means that the flow around the downstream pinnacle (i) will be affected by the wake of the upstream pinnacle (j), which results in a reduced contribution to the overall drag. To account for this effect, a coefficient 
$\beta$ is introduced, which scales the contribution of the downstream pinnacle to the overall drag. When 
$\lambda _{i,j}< wake_j$, 
$\beta _{i,j}$ is less than 1 to reflect the reduced contribution of the downstream pinnacle. As 
$\lambda _{i,j}$ increases and the downstream pinnacle (i) moves out of the wake region, 
$\beta _{i,j}$ increases as well, until it reaches a constant value of 
$1$ when 
$\lambda _{i,j}>wake_j$. This indicates that both pinnacles have the same contribution to the overall drag and can be treated as isolated peaks. In figure 5 a sketch of these distances is depicted. The fourth term in (4.2) accounts for the distance between two consecutive roughness elements which, as observed by Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003), affects the velocity profile, stagnation pressure and then the drag. By definition, the 
$ES$ does not take into account how wide are the cavities between roughness elements; it is, in fact, calculated only in the domain region characterized by roughness elements, 
$ES$ being zero everywhere else. Nevertheless, looking at the streamlines depicted in figure 4, the distance between two consecutive elements 
$w_{i,i+1}$ has considerable effect on the fluid flow, showing the importance to taking into account these features. The dependence on the scale of roughness is accounted for with 
$k/\delta _k$, where 
$\delta _k$ is the outer layer length scale (channel half-height or boundary layer thickness). In the present research two roughness heights were simulated, thus further geometrical configurations are required to confirm the role of the fourth term of (4.2). Figure 6 shows the correlation between the 
$ED$ and the total drag or roughness function. The novel parametrization proposed in this paper, 
$ED$, correlates with the drag significantly better than the 
$ES$ (shown in figure 2). A large variation in terms of drag and roughness function for the same value of 
$ES$ was observed in figure 2, suggesting that the 
$ES$ alone may not fully capture the impact of geometrical features on turbulent flows. On the other hand, the 
$ED$, taking into account the geometrical features which affect the turbulent flows discussed above, varies smoothly with the drag and roughness function. To further corroborate our previous finding and validate the 
$ED$, we applied it to a more complex irregular rough wall (figure 7). The irregular surface shape was generated through the superimposition of sinusoidal functions with random amplitudes and four different wavelengths, see De Marchis, Milici & Napoli (Reference De Marchis, Milici and Napoli2019). To use the 
$ED$, it is necessary to define a crests plane and then the pinnacles emerging above it. While this is obvious in the simplified cases discussed before, it is not straightforward for a more generic surface with peaks of variable height as that in (figure 7). The method we used consists in identifying the local peaks of the surface and then calculating the probability density function of those values using a number of bins equal to one third of the samples to have some statistical convergence. The crests plane was taken as the most probable peak height. The calculation of the distances between peaks, and the 
$ED$ is then straightforward. Since the value of 
$ED$ of the original case in De Marchis et al. (Reference De Marchis, Milici and Napoli2019) was very small, the mean roughness height was increased to have a higher value of the 
$ED$. Specifically, the new geometry (reported as ‘crest roughness type’ using the ‘
’ marker in the caption of figure 6) has a value of 
$ED=0.37$ quite similar to the range here analysed. Data from both cases have been included in figure 6(b). Despite the surface in De Marchis et al. (Reference De Marchis, Milici and Napoli2019) not being used to develop (4.2) but used as an independent validation only, results are consistent with the correlation between 
$ED$ and roughness function obtained with our database of triangular roughness elements. This result suggests that the 
$ED$ has the potential to be generalized to more complex and realistic 2-D geometries. More work is needed to extend it to 3-D roughness.
Schematic representation of the parameter defined in the determination of the 
$ED$.
Drag and roughness function as function of the 
$ED$.
Contour plot of the mean streamwise velocity for the irregular rough wall of De Marchis et al. (Reference De Marchis, Milici and Napoli2019).
5. Conclusions
Direct numerical simulations have been performed to analyse a set of 2-D rough surfaces using triangle-shaped elements. One of the main challenges of the last decade was the prediction of a drag and roughness function based on surface topography, which requires a parametrization. Therefore, 17 geometries with different shapes but similar 
$ES$, skewness and kurtosis have been investigated. The study found that for most of the data, different shapes with the same geometrical quantities can result in different drag and roughness functions. To address this issue, a new geometrical parameter, called 
$ED$, was introduced. The 
$ED$ is a geometrical parameter that accounts for the physical behaviour of the fluid around roughness elements. It is calculated based on flow features typically occurring in rough flow. The first suggested that roughness in the wake of large pinnacles has a negligible contribution to the drag and should not be included in the calculation of geometrical statistics of the surface to predict the drag. In addition, 
$ED$ recognizes that roughness elements have contributions to the drag and 
$\Delta U^+$ based on their size and pattern. The 
$ED$ has been calculated as a modification to the 
$ES$, by subtracting the contribution of roughness elements located in the wake region and adding the contribution of pinnacles above the crest plane. Additionally, if the surface is characterized by more than a peak emerging above the crests plane, the distance between two subsequent pinnacles has to be considered. When the downstream pinnacle is in the wake region of the upstream pinnacle its contribution to the drag and then to the calculation of 
$ED$ is reduced. On the other hand, as the separation increases and the downstream pinnacle moves out of the wake region, it is no longer influenced by the flow dynamics of the upstream pinnacle, and can be treated as an isolated peak. The 
$ES$ lacks details regarding the separation between individual elements, particularly evident in a flat section where 
$ES$ equals zero. As pointed out by Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003), the velocity profile undergoes significant influence from the flat section located downstream of each element. To address the influence of the flat section, the final term in (4.2) was incorporated. Overall, the 
$ED$ provides a representative geometrical parameter of the entire roughness configuration, taking into account the peaks above the mean roughness, the wake region induced by the higher elements, and the distance between two consecutive elements. The 
$ED$ was shown to correlate well with drag and roughness function for the large dataset used to develop it as well as for a more realistic irregular rough wall generated with random sinusoidal functions. Overall, the 
$ED$ improves previous correlations between the drag and geometrical features of the wall. More work is still required to further generalize it to roughness surfaces irregular in the spanwise direction.
Funding
The authors greatly appreciate the financial support provided by the following projects. S.L. and F.B. were partially supported by National Science Foundation grant no. 2202710. RETURN Extended Partnership and received funding from the European Union Next-GenerationEU (National Recovery and Resilience Plan – NRRP, Mission 4, Component 2, Investment 1.3 – D.D. 1243 2/8/2022, E0000005). The Texas Advanced Computing Center, and High Performance Computing at UT Dallas are acknowledged for providing computational time. TiSento – SENSORIALIZED COMPOSITE PIPE FOR HYDRAULIC APPLICATIONS, n. 084221000550 CUP G18I18001710007. Funded under measure 1.1.5 of the PO FESR SICILY 2014-2020. This research has been partially supported by the European Union – NextGenerationEU – National Sustainable Mobility Center CN00000023, Italian Ministry of University and Research Decree n. 1033–17/06/2022, Spoke 3, CUP B73C22000760001.
Declaration of interests
The authors report no conflict of interest.
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