## 1. Introduction

In many technical and environmental flows, shear and buoyancy effects occur simultaneously and both may significantly contribute to the heat and mass transfer of the flow. This transport process, also known as mixed convection, represents the combination of the two extreme cases of forced and natural convection, where the former is driven purely by a pressure gradient and the latter by vertical temperature differences. While in forced convection turbulence is produced by shear and in natural convection by buoyancy, both effects contribute to turbulence production in mixed convection flows. The aforementioned flows often occur over rough surfaces, which can also strongly influence the flow properties. In the forced convection community, heterogeneous rough surfaces have received special attention in recent years, since in turbulent wall-bounded flows they can induce secondary flows of Prandtl's second kind that can significantly change the transport properties of the flow. It is therefore the objective of the present study to investigate the effects of heterogeneous surfaces on the interplay of forced and natural convection in turbulent flows.

In mixed convection flows with unstable thermal stratification, where denser fluid is bedded over less dense fluid, different turbulent large-scale flow structures have been identified and the organisation depends on the relative strength between shear and buoyancy effects, such as in turbulent channel flows (Pirozzoli *et al.* Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), turbulent Couette flows (Blass *et al.* Reference Blass, Zhu, Verzicco, Lohse and Stevens2020) and in the atmospheric boundary layer (ABL) (Deardorff Reference Deardorff1972; LeMone Reference LeMone1973; Moeng & Sullivan Reference Moeng and Sullivan1994; Khanna & Brasseur Reference Khanna and Brasseur1998). The transition between different flow organisations is accompanied by an alteration of the effective heat and momentum transfer in the flow. In the case of strong buoyancy effects and weak shear, open cells form in the flow, which resemble Rayleigh–Bénard convection, while in the case of weak to moderate buoyancy effects and strong shear, the flow organises into horizontal rolls aligned with the main flow direction (Khanna & Brasseur Reference Khanna and Brasseur1998; Pirozzoli *et al.* Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017; Salesky, Chamecki & Bou-Zeid Reference Salesky, Chamecki and Bou-Zeid2017). These streamwise rolls are illustrated in the cross-sectional plane of the channel in figure 1(*a*), with the up- and downdraft region of the roll occurring where localised buoyancy forces accumulate. At very strong shear, with negligible buoyancy effects, the flow organisation in turbulent channels resembles that of classical Poiseuille flows (Pirozzoli *et al.* Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) and in the case of neutral ABL that of flat-plate boundary layers (Khanna & Brasseur Reference Khanna and Brasseur1998).

The transition between the different flow topologies of mixed convection can be characterised by various stability parameters. In the atmospheric science community, the stability parameter $-z_i/L$ is used to classify the transitions between rolls and cellular structures, which expresses the ratio of the boundary layer depth $z_i$ and the Obukhov length $L$. The Monin–Obukhov similarity theory introduces the Obukhov length $L$ as a length scale that compares the effects of friction and buoyancy on the flow; $L$ is used for non-dimensionalisation to generalise Prandtl's mixing length theory for buoyancy effects of non-neutral stratifications (Obukhov Reference Obukhov1946; Monin & Obukhov Reference Monin and Obukhov1954; Wyngaard Reference Wyngaard2010). In this context, $|L|$ can be physically interpreted as the height at which the buoyant production of turbulent kinetic energy equals that due to shear production, such that below $|L|$ mechanical production predominates, while above $|L|$ buoyant production is dominant. In the case of convective boundary layers, large ratios of $-z_i/L$ are indicative of the formation of convective cells, while small values are typical for roll formation (Khanna & Brasseur Reference Khanna and Brasseur1998; Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017). An alternative stability parameter to characterise the relative importance of buoyancy effects and shear is the Richardson number $Ri$. For mixed convection in turbulent channel flows, low $Ri$ corresponds to pure forced convection, intermediate values of $Ri$ to roll formation and large $Ri$ values to natural convection with cell-like structures (Pirozzoli *et al.* Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). Independent of the chosen stability parameter the exact range at which the transition between the different flow regimes occurs is still under debate with recent studies focusing on the transition between rolls and cells (Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017) and the transition between neutral to moderately convective conditions (Jayaraman & Brasseur Reference Jayaraman and Brasseur2021).

Effects of surface heterogeneity, occurring for instance between urban–rural areas, on the transition between the different flow organisations in mixed convection has gained little attention, even though their impact on the quality of weather and climate prediction is important (Bou-Zeid *et al.* Reference Bou-Zeid, Anderson, Katul and Mahrt2020). Likewise, there is the chance that the flow topology may have an influence on the deformation of the surface topography, which in turn affects the flow and possibly results in a positive feedback. For example, from satellite observations of deserts in the 1960s, it was proposed that quasi-streamwise rolls in the ABL were responsible for the formation of sand dunes aligned with the mean flow direction (Hanna Reference Hanna1969; Shao Reference Shao2008). This is similar to observations from aqueous open-channel flows, where sand dunes can also form due to a positive feedback with large-scale flow structures. The formation of these large-scale flow structures is associated with irregularities in the surface topography, which leads to the formation of so-called secondary motions (Colombini Reference Colombini1993; Scherer *et al.* Reference Scherer, Uhlmann, Kidanemariam and Krayer2022), which, unlike the streamwise rolls, can occur without buoyancy effects.

These secondary motions, also named secondary motions of Prandtl's second kind (Hinze Reference Hinze1973; Anderson *et al.* Reference Anderson, Barros, Christensen and Awasthi2015), can occur in turbulent flows over surfaces with spanwise variations in the wall properties, which can significantly alter the momentum and heat transport of turbulent flows (Stroh *et al.* Reference Stroh, Schäfer, Forooghi and Frohnapfel2020*a*). Unlike streamwise rolls, secondary motions cannot be observed in instantaneous velocity fields, instead they occur only in time-averaged velocity fields, where they appear as counter-rotating vortices aligned with the main flow direction. The secondary motions introduce a spanwise variation in the mean velocity field, which, for instance in a turbulent boundary layer, leads to spanwise modulation of the boundary layer thickness (Willingham *et al.* Reference Willingham, Anderson, Christensen and Barros2014). These differences in the mean profile are associated with the formation of low- and high-momentum pathways in the mean velocity field, usually occurring at the updraft and downdraft regions of the secondary motions, respectively (Barros & Christensen Reference Barros and Christensen2014; Willingham *et al.* Reference Willingham, Anderson, Christensen and Barros2014).

The spanwise heterogeneous surfaces, triggering secondary motions, are roughly distinguished in two main surface types, namely in ridge-type and strip-type roughnesses (Wang & Cheng Reference Wang and Cheng2006). Strip-type roughness is characterised by significant spanwise differences in drag, for example alternating streamwise strips with different wall-shear stress conditions (Willingham *et al.* Reference Willingham, Anderson, Christensen and Barros2014; Anderson *et al.* Reference Anderson, Barros, Christensen and Awasthi2015; Chung, Monty & Hutchins Reference Chung, Monty and Hutchins2018) or alternating strips of smooth and rough walls, which do not feature large differences in the wall elevation (Hinze Reference Hinze1973; Stroh *et al.* Reference Stroh, Schäfer, Frohnapfel and Forooghi2020*b*; Wangsawijaya *et al.* Reference Wangsawijaya, Baidya, Chung, Marusic and Hutchins2020; Scháfer *et al.* Reference Schäfer, Stroh, Forooghi and Frohnapfel2022). Ridge-type roughness comprises notable spanwise wall elevation differences, such as streamwise-aligned ridges, which are studied numerically and experimentally for a wide range of different ridge shapes in turbulent boundary layers (Vanderwel & Ganapathisubramani Reference Vanderwel and Ganapathisubramani2015; Hwang & Lee Reference Hwang and Lee2018; Medjnoun, Vanderwel & Ganapathisubramani Reference Medjnoun, Vanderwel and Ganapathisubramani2018; Vanderwel *et al.* Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019; Medjnoun, Vanderwel & Ganapathisubramani Reference Medjnoun, Vanderwel and Ganapathisubramani2020), turbulent channel flows (Vanderwel *et al.* Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019; Stroh *et al.* Reference Stroh, Schäfer, Forooghi and Frohnapfel2020*a*) and turbulent open-channel flows (Awasthi & Anderson Reference Awasthi and Anderson2018; Zampiron, Cameron & Nikora Reference Zampiron, Cameron and Nikora2020).

The ridge-type behaviour of secondary motions is characterised by updraft regions occurring above the centre ridge position, while downdrafts descend in the valley region between adjacent ridges, as illustrated in figure 1(*b*). The spanwise distance between the ridges influences the spatial extent of the secondary motion (Vanderwel & Ganapathisubramani Reference Vanderwel and Ganapathisubramani2015; Vanderwel *et al.* Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019). For large ridge spacings, which are of the order of several boundary layer thicknesses, the flow is divided into regions disturbed by the secondary motion in the vicinity of the ridge and a homogeneous and unaffected region far from the ridges. In the case of ridge spacings of the order of the boundary layer thickness, the secondary motion affects the entire flow region and introduces a spanwise heterogeneity into the mean flow (Hwang & Lee Reference Hwang and Lee2018; Medjnoun *et al.* Reference Medjnoun, Vanderwel and Ganapathisubramani2020). Further reduction of the ridge spacing leads to a decreasing spatial extent of the secondary motions, which scales with the ridge spacing.

The objective of the present study is to investigate the influence of spanwise heterogeneous surfaces on turbulent mixed convection flows as well as their influence on the transition between the different large-scale flow organisations. Secondary motions feature, in a mean sense, some similarities to the roll motion of mixed convection, e.g. large-scale counter-rotating vortices, as depicted in figure 1. Since secondary motions can be induced by streamwise-aligned ridges, the study investigates to which extent such surface structures can influence the formation and dynamics of streamwise rolls. These questions are investigated using a simplified setting of a turbulent channel flow, similar to that of Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), augmented with streamwise-aligned ridges on the walls. The different flow regimes of mixed convection are covered by systematically varying the relative strengths of shear and buoyancy effects.

## 2. Methodology

### 2.1. Flow configuration and numerical procedure

The balance equations for mass, momentum and energy are considered in the Boussinesq approximation

where $u_i$ and $x_i$ are the components of the velocity and position vector, with indices $1,2,3$ corresponding to the streamwise, wall-normal and spanwise directions. As an index-free notation, the velocity components are also written as $(u,v,w) = (u_1,u_2,u_3)$. The reference density is given by $\rho _0$, $p$ is the pressure, $\nu$ the kinematic viscosity, $\beta$ is the thermal expansion coefficient, $g$ the gravitational acceleration, $T$ the temperature and $\alpha$ the temperature diffusivity. A constant volume flow rate is maintained by the forcing term $\varPi = u^{2}_\tau /\delta$, where $u_\tau$ is the friction velocity and $\delta$ the half-channel height. In order to represent structured surfaces an external volume forcing term $F_{IBM,i}$ is introduced by an immersed boundary method (IBM). In the temperature equation the source term $Q_{IBM}$ represents the forcing term to maintain a constant temperature at the surface. The structured surface of the present flow configuration consists of streamwise-aligned ridges on both walls, as shown in figure 2. Due to the presence of these ridges, the half-channel height $\delta$ reduces to the effective half-channel height $\delta _{eff} = \delta - \langle \overline {h}_{s} \rangle$, where $\langle \overline {h}_{s} \rangle$ is the horizontally averaged mean height of the surface elevation of one wall.

In the case of spanwise heterogeneous surfaces, where the spanwise direction is no longer statistically homogeneous, it is a common choice to decompose the velocity field into its temporal and streamwise mean $\overline {u}_i$ and related random fluctuation $u_i''$ given by

The mean velocity can be further decomposed into its global mean $\langle \overline {u}_i\rangle$, obtained through additional spanwise averaging indicated by angular brackets $\langle \,{\cdot }\,\rangle$, and its coherent or dispersive component $\tilde {u}_i$, which is the spatial fluctuation about $\langle \overline {u}_i\rangle$, and is defined as

The sum of coherent and random fluctuations is defined as fluctuation $u_i'$ such that

In this study, the spatial averages are based on intrinsic averaging. This procedure excludes the values at the grid points inside the immersed (solid) body, while the values on the surface are included in the integration. In consequence, the average is computed through normalisation with the fluid area only. This affects the evaluation of global quantities, which are integrated in time and all three spatial directions and are defined as

where $\phi$ represents an arbitrary quantity and $\varPhi$ is its volume and time average, while $y_b(z)$ and $y_t(z)$ are the wall-normal surface elevation at the bottom and top walls, respectively. Note, that this definition applies intrinsic averaging through its spanwise dependent wall-normal integration borders.

The considered flow is characterised by three non-dimensional numbers, namely the Prandtl number $Pr$, which is the ratio of momentum and thermal diffusivity, the bulk Reynolds number $Re_b$, describing the ratio of inertial and viscous effects, and the Rayleigh number $Ra$, characterising the ratio of buoyant and viscous effects, and their definitions are given by

*a*–

*c*)\begin{equation} Pr = \frac{\nu}{\alpha},\quad Re_b = \frac{u_{b} \delta_{eff}}{\nu},\quad Ra = \frac{(2\delta_{eff})^{3} \beta g {\rm \Delta} T}{\alpha \nu}. \end{equation}

The bulk velocity is defined as $u_b= 1/(2\delta _{eff} L_z) \int _0^{L_z} \int _{y_b(z)}^{y_t(z)} \bar {u} \,\mathrm {d}y\,\mathrm {d}z$. Here, ${\rm \Delta} T$ is the imposed and constant temperature difference between the bottom and top wall surfaces ${\rm \Delta} T = T_b - T_t$. The Prandtl number is set to $Pr = 1$ for all considered cases. Following the work by Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), the bulk Richardson number is defined as

to characterise the relative importance of buoyancy and inertial effects. Please note that positive values of $Ri_b$ indicate convectively unstable conditions, while in atmospheric flows positive values commonly indicate convectively stable conditions (Wyngaard Reference Wyngaard2010). Another quantity widely used in the ABL community to categorise the flow is the ratio of the boundary layer depth $z_i$ and the Obukhov length $L$, which is known as the stability parameter $-z_i/L$ (Wyngaard Reference Wyngaard2010). For the present channel flow configuration with ridges, this translates into the ratio of $\delta _{eff}$ and $L$, with $L = - u_{\tau }^{3} / (\kappa Q \beta g )$, where $\kappa$ is the von Kármán constant with $\kappa =0.4$ (Wyngaard Reference Wyngaard2010), while $u_\tau$ and $Q$ are the friction velocity and the vertical heat flux, respectively (both defined in the next paragraph).

The drag exerted on the flow is quantified by the skin friction coefficient $C_f$ and friction Reynolds number $Re_\tau$, which are defined as $C_f =2 u_{\tau }^{2}/u_{b}^{2}$ and $Re_\tau = u_{\tau } \delta _{eff} / \nu$, with the friction velocity $u_{\tau } = (\tau _{w}/\rho )^{0.5}$ and the wall-shear stress $\tau _{w}$. The wall-shear stress is determined by extrapolating the total shear stress from the bulk region ($0.5 \le y/\delta \le 1.5$) to the virtual wall location $y_0 = \delta - \delta _{eff}$ (Chan-Braun, García-Villalba & Uhlmann Reference Chan-Braun, García-Villalba and Uhlmann2011). The heat transfer of the flow is characterised by the Nusselt number $Nu = 2\delta _{eff}Q / (\alpha {\rm \Delta} T)$, with the vertical heat flux $Q$ determined by evaluating the time and horizontally averaged temperature transport equation at the half-channel height $Q = \langle \overline {v'T'}\rangle |_\delta - \alpha \partial {\langle \bar {T}\rangle }/\partial {y}|_\delta$ (Stroh *et al.* Reference Stroh, Schäfer, Forooghi and Frohnapfel2020*a*).

The turbulence level is quantified in this study by the Reynolds number $Re_k = \sqrt {K}\delta _{eff}/\nu$, where $K$ corresponds to the time- and volume-averaged turbulent kinetic energy. This quantity is computed by applying the averaging procedure given in (2.8) $k = 0.5 \,{\cdot }\, \overline {u_i' u_i'}$. A characteristic velocity scale for natural convection is the free-fall velocity ${v_f} = (2 \delta _{eff} \beta g {\rm \Delta} T)^{1/2}$ and together with the effective channel height the free-fall time $t_f$ can be defined $t_f = 2\delta _{eff} / {v_f} = (2\delta _{eff} / (\beta g {\rm \Delta} T))^{1/2}$. The time scale characterising the forced convection processes is the bulk time unit $t_b = \delta _{eff} / u_b$. From the given definitions, the ratio of $t_b$ and $t_f$ results in the following relationship: $t_b/t_f = \sqrt {Ri_b}/2$.

The spanwise height distribution of the streamwise-aligned ridges follows a Gaussian distribution for each individual ridge, which is defined by

where $n_g$ is the total number of Gaussian ridges at one wall, $h_g$ is the maximum height of a single Gaussian ridge and $z_{c,i}$ is the spanwise centre position of each individual ridge, given by $z_{c,i} = S\ (i + 0.5)$ with $S$ as the spanwise spacing between two Gaussian ridges. The parameter $\sigma$ represents the spanwise extent of an individual Gaussian ridge. In this study, the parameters of the Gaussian ridges are set to $h_g = 0.1\delta$ and $\sigma =0.05\delta$. The cross-sectional area occupied by a single ridge is given by $A_{Gauss} = \sqrt {2{\rm \pi} } h_{g} \sigma$. Thus, the effective half-channel height is given by $\delta _{eff} = \delta - \delta _{melt}$ with the melt-down height $\delta _{melt} = n_g A_{Gauss} / L_z$.

The governing equations are numerically solved using direct numerical simulations by means of the open-source code Xcompact3d (Laizet & Lamballais Reference Laizet and Lamballais2009; Bartholomew *et al.* Reference Bartholomew, Deskos, Frantz, Schuch, Lamballais and Laizet2020) based on compact finite differences of sixth order and a third-order Runge–Kutta time integration scheme. The representation of the structured surfaces is achieved using an IBM based on polynomial reconstruction of the velocity and temperature fields inside the solid region of the ridges (Gautier, Laizet & Lamballais Reference Gautier, Laizet and Lamballais2014). The existing code was extended for the simulation of buoyancy effects and the code was validated with the data base for mixed convection and Rayleigh–Bénard flows of Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), as documented in Appendix A. The simulations were performed on a domain size $L_x \times L_y \times L_z = 16\delta \times 2\delta \times 8\delta$, which is in agreement with Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), who reported for this domain size insensitivity of the mean velocity and temperature profiles.

### 2.2. Cases

The transition between forced convection structures and streamwise rolls as well as the transition between streamwise rolls and convective cells in mixed convection flows is controlled by the mean shear and buoyancy forcing, which are determined by the imposition of the Reynolds $Re_b$ and Rayleigh number $Ra$. There are several possibilities which may be used to vary these two dimensionless numbers to achieve the same Richardson number $Ri_b$, as depicted by the black solid and dashed lines in figure 3(*a*). The simplest approach is to fix one of the dimensionless numbers, while varying the other one and *vice versa*. In the study of Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), a smooth-wall channel flow is explored for a large parameter space of $Re_b$ and $Ra$, covering all flow regimes, which is shown in terms of the resulting $Nu$ in figure 3(*b*). For fixed $Ra$, the initial reduction of $Nu$ with increasing $Re_b$ is associated with the emergence of streamwise rolls, which reduce the effective heat exchange of the convective plumes from the natural convection case. For larger values of $Re_b$ the flow transitions to the forced convection regime where $Nu$ increases with increasing $Re_b$.

In the current study the spanwise spacing of the Gaussian ridges $S$ is varied in the range of $S/\delta = 0.5,1,2,4,\infty$, where $S=\infty$ corresponds to the smooth-wall case. This translates into an effective half-channel height of $\delta_{\textit{eff}}/\delta= 0.975, 0.987, 0.994, 0.997$ for the four ridge cases, respectively. Furthermore, the bulk Richardson number $Ri_b$ is varied in such a way as to cover the different flow regimes of mixed convection and their transition ranges. The variation of $Ri_b$ is achieved by two parameter sweeps, one at constant $Ra=10^{7}$ and varying $Re_b$ and the other sweep for changing $Ra$ at constant $Re_b = 2800$. These two parameter sweeps for the simulations of the present study are represented in figure 3(*a*) with green crosses, which results in a total number of 65 direct numerical simulations. These two parameter sweeps intersect in the vicinity of the minimum of $Nu$ for $Ra=10^{7}$ shown in figure 3(*b*), which allows us to study the parameter sensitivity of the transition processes from two sides. The two black dashed lines in figure 3(*a*) indicate isolines of constant $Ri_b$ at which the two transition ranges of mixed convection occur. As will be shown later, the lower isoline $Ri_b = 0.025$ lies within the transition range of forced convection structures to streamwise rolls, while the upper isoline $Ri_b = 3.19$ is in the transition range between streamwise rolls to convective cells. For the present simulations two parameter points with similar $Ri_b$ exist within the two transition ranges, allowing us to study the effect of $Re$ on the flow organisation.

From forced convection flows it is known that the strongest secondary motion occurs for $S/\delta \approx O(1)$ with a spanwise extent of $\approx \delta$ (Vanderwel *et al.* Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019). Since secondary motions of Prandtl's second kind occur only in turbulent flows, the parameter sweep with fixed $Re_b = 2800$ will accommodate these secondary motions for all $Ra$, which will allow us to study the effect of buoyancy on the secondary motions as well. In the case of the $Ra$-sweep, this will partly be the case. The spanwise extent of the convection cells and streamwise rolls found in Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) are roughly $4\delta$, such that the chosen values of $S$ cover the width of the different aforementioned flow structures.

The grid resolutions for the simulations are chosen according to those used by Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) for a second-order finite difference code. The grid requirements for mixed convection simulations in conjunction with Gaussian ridges represented by an IBM were investigated in a resolution study presented in Appendix B. It is found that the present grid resolution for smooth-wall mixed convection cases at $Ra=10^{7}$ is sufficient for the representation of streamwise Gaussian ridges and a further increase of the resolution results in no significant differences of the mean quantities and profiles. Only for lower $Ra$ does the spanwise grid resolution need to be slightly increased to achieve grid-independent statistical results for the streamwise-aligned ridge cases. The statistical time integration is carried out over at least $1500 t_b$ for cases $Ri_b\le 0.32$, except for the high $Re_b$ cases with $Ri_b = 0.025$ with time integration of at least $300t_b$. For cases with $Ri_b \ge 1.0$ the time integration comprises at least $400t_f$, while the high $Ra$ cases with $Ri_b = 3.2$ and $Ra = 10^{8}$ were averaged over at least $120t_f$.

## 3. Results

### 3.1. Global flow properties

The results of the global flow properties for the different simulations are presented in table 1, where the configurations are arranged according to the parameter triad ($Ra$, $Re_b$, $S/\delta$). The smooth-wall configurations are indicated by $S=\infty$ and the configurations with streamwise-aligned ridges are listed with decreasing $S$. Since the forced and mixed convection cases are run at constant flow rate, the presence of the Gaussian ridges will increase the drag, which translates into an increase of $C_f$ and $Re_\tau$ compared with the respective case with smooth-wall conditions. As can be seen for all considered cases, the steady decrease of $S$ leads to a monotonic increase of $C_f$ and $Re_\tau$ compared with the smooth-wall case. For forced convection the increase of $C_f$ is up to $26\,\%$, while the largest increase is found for $Ri_b = 10$ with $45\,\%$. Due to the changing friction drag, the ridge height in wall units $h_g^{+}$ changes for all cases as well, and is in the range $7 \le h_g^{+} \le 60$ for the current configurations. In the case of pure forced convection, turbulent secondary motions induced by streamwise-aligned ridges are known to increase the global friction as well as the heat transfer of the flow compared with smooth-wall conditions (Stroh *et al.* Reference Stroh, Schäfer, Forooghi and Frohnapfel2020*a*). This behaviour is also observed for the present forced convection case with streamwise-aligned Gaussian ridges, where for the densest ridge spacing $S=0.5\delta$ the heat transfer increases $16\,\%$ compared with the smooth-wall case. In a similar range $17\,\%$ is the increase for the natural convection case, and the largest increase is found for $Ri_b=3.2$ with $32\,\%$.

While the skin friction drag increases with decreasing $S$ and increasing wetted surface area, this behaviour is not found for the heat transfer for all present cases. This is illustrated for the two parameter sweeps at constant $Re_b = 2800$ in figure 4(*a*) and at constant $Ra=10^{7}$ in figure 4(*b*). The forced convection case $Ra=0$ and weak convective case $Ra=10^{5}$ in figure 4(*a*) show the successive increase of $Nu$ with decreasing ridge spacing $S$. An increase of $Ra$ or $Ri_b$ introduces an additional buoyant contribution to the vertical mixing, resulting in larger heat transfer and for large Rayleigh numbers ($Ra>10^{6}$), which represent configurations where buoyancy is comparable to shear or even stronger, the monotonic increase of $Nu$ with decreasing $S$ is also found. However, for the particular cases $Ra=7.5\times 10^{5}$ ($Ri_b=0.024$) and $Ra=10^{6}$ ($Ri_b=0.032$) heat transfer does not monotonically increase with decreasing $S$, which is visible in the inset of figure 4(*a*).

In figure 4(*b*) the natural convection case is given by $Re_b = 0$ and the influence of buoyancy is successively reduced by increasing $Re_b$. The minimum found for $Nu$ is associated with the break up of the thermal plumes of the Rayleigh–Bénard case, when shear is added, and has been reported for unstable thermal stratification in Poiseuille and Couette flows (Scagliarini, Gylfason & Toschi Reference Scagliarini, Gylfason and Toschi2014; Blass *et al.* Reference Blass, Zhu, Verzicco, Lohse and Stevens2020). The non-monotonic behaviour of $Nu$ with decreasing $S$ seen in figure 4(*a*) also occurs for the largest $Re_b = 10\,000$ case ($Ri_b = 0.025$), which is in a similar bulk Richardson number range to the former cases. Thus, all cases which depict a non-monotonic behaviour of $Nu$ with respect to $S$ fall within a range of bulk Richardson values $Ri_b = 0.016\unicode{x2013}0.032$, where shear effects are strong and buoyancy effects are weak. It will be shown that this range of Richardson numbers marks the transition from forced convection structures to streamwise rolls and the ridge spacing $S$ affects and alters this transition.

In the case of mixed convection, turbulence is driven by two generation mechanisms, the production by shear and by buoyancy, and the exact turbulence level cannot be inferred *a priori* by $Ra$ and $Re_b$. The same holds for the bulk Richardson number $Ri_b$. To rule out effects caused by the different turbulence levels or Reynolds number effects, which will be discussed in § 3.6, we separate those cases with significantly higher turbulence levels from the cases with comparable values. For the subsequent discussion and sections only cases which fall in a range of Reynolds number values $Re_k = 200\unicode{x2013}355$ are considered. This selection includes only those cases of the two parameter sweeps for which $Ra\le 10^{7}$ and $Re_b \le 2800$. Instead of using the pairs of $Ra$ and $Re_b$, the bulk Richardson number $Ri_b$ is used in the following to characterise the relative importance of buoyancy and shear effects. In analogy to $Ri=Ra/(4 Re_b^{2} Pr)$, $Nu$ is replaced by the Stanton number $St=Nu/(Re_b Pr)$. The corresponding results are presented in figure 5(*a*) and reveal an increase of $St$ with $Ri_b$.

The relative increase between heat and momentum transfer is characterised by the ratio $St/C_f$, which is shown in figure 5(*b*). For each $S$ it can be seen that larger values of $Ri_b$ induce larger $St/C_f$ values. This implies that buoyancy effects lead to a larger increase of heat transfer than of momentum transfer. In addition, a consistent influence of the ridges can be observed in this representation. A decrease of $S$ results in lower $St/C_f$ for all $Ri_b$. This is also reflected in the stability parameter $-\delta _{eff}/L$ in table 1, which is another quantity to compare the relative heat and momentum transfer. The most distinct property of figure 5(*b*) is the strong increase of $St/C_f$ in the range $Ri_b = 0.016\unicode{x2013}0.032$. This increase is delayed to larger $Ri_b$ with decreasing $S$ and can be linked to a reorganisation of the turbulent flow structures, as discussed in the following section.

### 3.2. Turbulent flow structures

The different flow organisations of mixed convection flows observed over smooth-wall conditions can be visualised by instantaneous velocity or temperature fluctuations in horizontal planes (Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017). The influence of heterogeneous surfaces on this flow organisation is shown for various pairs of $Ri_b$ and $S$ for the instantaneous temperature fluctuations in wall-parallel planes located at the half-channel height ($y=\delta$) in figure 6 and slightly above the top of the ridges ($y=0.15\delta$) in figure 7. Please note that the discussion for the near-wall region refers to the bottom wall, unless stated otherwise. Both horizontal planes display the same instantaneous realisation of the flow field and comprise cases that fall into a similar range of turbulent Reynolds number $Re_k$. The $Ri_b$ increases from top to bottom, starting with the forced convection case $Ri_b = 0$ ($Re_b=2800$, $Ra=0$) in the top panel and the natural convection case $Ri_b = \infty$ ($Re_b=0$, $Ra=10^{7}$) in the bottom panel. The ridge spacing $S/\delta$ decreases from left to right, with the smooth-wall case displayed in the first column of figures 6 and 7.

Considering the smooth-wall cases $S=\infty$ first, the flow topology of the forced convection case has a spotty organisation which is also the case for the mild convective case $Ri_b = 0.003$ ($Re_b=2800$, $Ra=10^{5}$) in figure 6. The transition to streamwise rolls takes place for slightly larger buoyant forcing for case $Ri_b = 0.016$ ($Re_b=2800$, $Ra=5\times 10^{5}$), but the rolls still display some patchiness. This transition also results in a change of the near-wall structures as can be seen in figure 7, where strong elongated temperature fluctuations are preferentially located in regions of streamwise roll updrafts, and are less pronounced in the downdraft region. These elongated temperature fluctuations coincide with the near-wall low-speed streaks (not shown here), since for neutral and moderately convective cases the temperature behaves like a passive scalar with strong correlation with the streamwise velocity (Khanna & Brasseur Reference Khanna and Brasseur1998). The formation mechanism of streamwise rolls is strongly linked to localised buoyancy forces, which concentrate in low-speed streaks and thereby create linear updrafts (Khanna & Brasseur Reference Khanna and Brasseur1998). Multiple updrafts can merge to a strong buoyancy-enhanced streak, forming the updraft region of the streamwise roll in figure 6. This updraft reaches the opposing wall and reduces or destroys the coherence of the low-speed streaks there. At the same time, between these impingement regions of the updrafts at the opposite wall, buoyancy-enhanced low-speed streaks can form, which in turn generate a strong localised downdraft and in combination with the updrafts result in a large-scale streamwise roll motion in the cross-section.

In contrast to case $Ri_b = 0.016$, the streamwise roll of case $Ri_b = 0.024$ in figure 6 is more articulated in its structure, which is associated with a sudden increase of $St/C_f$ in figure 5(*b*). The streamwise rolls persist up to $Ri_b = 1$ in figure 6 with a spanwise wavelength of approximately $8\delta$, such that the chosen domain size is able to accommodate a single pair of counter-rotating rolls as reported by Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). For the cases $Ri_b=0.32$ and $Ri_b=1.0$ in figure 6 the rolls show a strong waviness of the thermal up- and downdrafts, which is also seen in the near-wall region where the updraft region encompasses spanwise inclined near-wall streaks in figure 7. When buoyancy forces become more important, the streamwise roll is more disrupted and reduce its streamwise coherence, since thermal plumes become dynamically more important (see case $Ri_b = 10$ in figure 6) (Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017). The increased influence of buoyancy also modifies the near-wall structures, where cell-like structures appear in the updraft region, which still depict some streamwise coherence, as seen in figure 7. For the Rayleigh–Bénard case $Ri_b=\infty$ ($Re_b=0$, $Ra=10^{7}$) in figure 6, the streamwise roll is not present any more and the flow organises into convective cells. These structures have a preferential roll orientation in the $x$- and $z$-directions, also seen in the near-wall region in figure 7, which is due to the rectangular domain size (Pirozzoli *et al.* Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017)

The introduction of the ridges displays no significant differences of the flow structures in the channel centre for the forced convection case $Ri_b=0$ and weak buoyancy case $Ri_b=0.003$ compared with the smooth-wall case. However, the elongated high temperature fluctuations in the near-wall region are more coherent in the streamwise direction in the case of ridge spacings $S=\delta, 2\delta, 4\delta$, for which they preferentially occur at the spanwise ridge position. For mild buoyancy effects, these elongated temperature regions collapse with low-speed streaks (not shown here), forming low-momentum pathways directly above the ridges. As such, the preferential position of the low-speed streaks coincides with the mean upward motion of the secondary motions, which will be shown in § 3.3. For the densest ridge spacing $S=0.5\delta$ these elongated structures still occur at the ridges, however, they appear less coherent in the streamwise direction.

Significant effects of the ridges can be seen for the transition between forced convection structures and streamwise rolls in figure 6 for cases $Ri_b = 0.016\unicode{x2013}0.032$. As shown before, the transition towards streamwise rolls for smooth-wall conditions takes place at $Ri_b=0.016$ and this transition can be also observed for the coarsest ridge spacing $S=4\delta$, which, however, is more interrupted by individual turbulent spots than the smooth-wall case. This is also reflected in a change of the near-wall structures, where, for case $S=4\delta$, the elongated high temperature fluctuations still favour the updraft region, but in contrast to the smooth-wall case, also occur inside the downdraft region at the ridge position (figure 7). For denser ridge spacing $S<4\delta$ the preferred concentration of low-speed streaks is not observed anymore. Therefore, the streamwise roll is not visible in the channel core and the flow structures resemble those seen for the forced convection and weak buoyancy cases $Ri_b = 0.003$. For slightly larger bulk Richardson number $Ri_b = 0.024$ the streamwise roll is now clearly visible for the two coarsest ridge spacings $S=4\delta$ and $S=2\delta$ in figure 6, and strong enough to reorganise the near-wall structures seen in figure 7, while for the denser ridge spacings the streamwise roll is not present. Eventually, the streamwise roll is observed for all ridge spacings $S$ at $Ri_b = 0.032$, while for the denser ridge spacings still some spot-like structures overlap with the rolls. Considering figure 5(*b*), the delayed transition between forced convection structures and streamwise rolls with decreasing $S$ can be related to the increased drag introduced by the ridges. Denser ridge spacings introduce more drag and shear in the near-wall region, and in consequence larger buoyancy forces are required to form the streamwise rolls which in turn induce an increase in heat transfer.

The streamwise roll, present for the intermediate Richardson number $Ri_b=0.32$ and $Ri_b=1.0$ cases (figure 6), displays no significant influence of the ridges. This is likewise the case for the near-wall region in figure 7, where it can be seen that the formation of high temperature fluctuations occur inside the updraft region of the roll.

At higher convective conditions for $Ri_b = 10.0$ the streamwise roll is present for the smooth-wall case and the two coarsest ridge spacings $S =2\delta, 4\delta$, while it completely disappears for denser ridge spacings $S \le \delta$. For $S = \delta$ the streamwise roll is replaced by convection cells, resembling that found for the Rayleigh–Bénard case with a spacing of $S=4\delta$. For the densest ridge spacing $S=0.5\delta$, rolls with a preferential orientation in the spanwise direction occur, which have similarities to the densest ridge spacing $S=0.5\delta$ of the Rayleigh–Bénard case. This transition from roll to cell structures is also reflected in a transition of the near-wall structures in figure 7. The roll-to-cell transition is also observed for the lower bulk Richardson case $Ri_b = 3.2$ (higher $Re_k$) for the same ridge spacings $S$ (not shown). This result is remarkable, since the transition from roll to cell structures over homogeneous wall conditions in ABL is expected to begin at larger values of the stability parameter $-z_i/L \approx 26$ (Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017), while the stability parameter for the cases $Ri_b = 3.2$ and $Ri_b = 10.0$ are in the range $-\delta _{eff}/L = 3.4\unicode{x2013}9.7$. This illustrates that streamwise-aligned ridges reduce significantly the range of $Ri_b$ or $-\delta _{eff}/L$ in which streamwise rolls appear. This suggests that heterogeneous rough surfaces can trigger the roll-to-cell transition with smaller buoyancy forces.

As can be seen for the Rayleigh–Bénard case $Ri_b = \infty$ in figure 6, decreasing the ridge spacings $S$ results in an increasingly preferential orientation of the convective cell towards the spanwise direction $z$. The rolls with orientation in the $x$-direction experience increasing lateral drag as $S$ decreases, and for $S\le 2\delta$ these rolls can eventually no longer emerge and only rolls in the $z$-direction, whose circulation is along the ridge direction, occur. This observation will be discussed further in § 3.4. The increase of $Nu$ for smaller $S$ is also reflected by intensified thermal up- and downdrafts in the channel centre plane. We note that additional simulations for $S=\infty$ and $S=0.5\delta$ in a wider domain ($L_x = L_z = 16\delta$) do not indicate a domain size dependence of the obtained results.

### 3.3. Mean properties

The effects of ridge spacing $S$ and the relative strength of the shear and buoyancy effects on the time and horizontally averaged mean streamwise velocity and temperature profiles are shown in figure 8. The mean temperature is represented as the difference from the bottom wall temperature $T_b$ and scaled by the friction temperature $T_\tau = Q/\rho c_{p} u_\tau$. Starting from forced convection, the trend of the mean profiles to become flatter with increasing $Ri_b$ is consistent with the results of Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017). The logarithmic region of $\langle \bar {u} \rangle$ found for a weak convective condition starts to deviate at $Ri_b = 0.032$. The reduction of the spanwise ridge spacing $S$ leads to a decrease of the mean streamwise velocity and temperature profile, which is in agreement with increased surface drag (see table 3 and figure 5*b*). However, temperature profiles within the transition range from forced convection structures to streamwise rolls show deviations from this behaviour close to the channel core, which is highlighted by the inset of figure 8(*b*). In this figure, it can be seen for $Ri_b=0.024$ and $Ri_b = 0.032$ that the temperature takes larger values with decreasing $S$ at the channel centre and at the same time the slope of the temperature increases. This indicates that the thermal mixing of the flow is increasingly weakened by the ridges. The transition from streamwise rolls to spot-like structures for case $Ri_b = 0.024$ can be also inferred from the similar slope of $S=\delta$ and $S=0.5\delta$ to that of the forced convection cases. Interestingly, for case $Ri_b =0.032$, where all $S$ feature streamwise rolls, the slope of the temperature profile for $S=0.5\delta$ also resembles that of the forced convection cases, which indicates that the transition point is already close. As can be seen, the influence and effects of the ridge spacing $S$ are of the same order as a change of the bulk Richardson number $Ri_b$.

The occurrence of secondary motions over streamwise-aligned ridges in forced convection flows is observed in time- and streamwise-averaged velocity fields in the channel cross-section, and is shown for case $Ri_b=0$ in figure 9. The relative strengths of the coherent motion among the different ridge spacings can be directly compared, due to the same scaling of the cross-sectional velocity components in bulk units. The smooth channel flow exhibits no coherent motion in the cross-section (not shown here), while for streamwise-aligned ridges the secondary motions appear in the mean flow field as counter-rotating vortices at each ridge in figure 9. These vortices introduce an upward motion above each ridge and a downward motion is located to the side of each ridge. As can be seen, the secondary motion induces a bulging of the mean streamwise velocity above the ridge, transporting low momentum into the bulk region, and for large spacings $S=4\delta$ and $S=2\delta$ almost reaching the half-channel height. The spacing between the ridges of case $S=4\delta$ in figure 9(*a*) is large enough, so that a homogeneous region unaffected by the secondary motion can form between the ridges. Decreasing the spanwise spacing $S$, the secondary motions fill almost the entire channel domain for case $S=2\delta$ (figure 9(*b*) and case $S=\delta$ (figure 9(*c*). However, for case $S=\delta$ the wall-normal extent of the secondary motions is slightly reduced compare with case $S=2\delta$, which indicates that the secondary motions of adjacent ridges affect each other at a spacing of $S=\delta$. Further decrease of the ridge spacing to $S=0.5\delta$ (figure 9*d*) shows a significant reduction of the spatial extent of the secondary motion in the wall-normal direction.

The investigation of the horizontal fields of the instantaneous temperature have shown that the transition of the flow topology for smooth-wall conditions is affected by the introduction of the streamwise-aligned ridges. This reorganisation is also reflected in the mean streamwise velocity and temperature fields in the cross-section, which are shown in figure 10 for the transition from forced convection structures to streamwise rolls at a ridge spacing of $S=4\delta$. The strengths of the cross-sectional velocity components are represented by the arrows in the mean streamwise velocity field, while their flow topology is visualised by streamlines in the mean temperature field. The weak buoyancy case $Ri_b = 0.003$ is shown in figure 10(*a*) and, similar to the forced convection case in figure 9, secondary motions are present in the mean velocity field which also leads to a bulging of the mean temperature at the ridges.

The transition towards streamwise rolls has been seen to occur for smooth-wall conditions at $Ri_b=0.016$, while for $S=4\delta$ a slight tendency towards rolls was present. Figure 10(*b*) shows that secondary motions can still occur at the ridges, although two diminished roll structures emerge that extend to the opposite wall. This in turn replaces the local bulging of the mean temperature at the ridge by a significant wider bulging of the mean temperature. As $Ri_b$ increases to $Ri_b = 0.024$ the secondary motions are replaced by a streamwise roll. The upward and downward motions of the convective rolls are located in the valley between two Gaussian ridges and each roll has a spanwise extent of four half-channel heights. As can be seen in the streamwise mean velocity field, the rolls induce a stronger cross-sectional velocity in the entire channel, which introduces a recirculation zone at the leeward side of the Gaussian ridges. The cross-sectional velocities of the streamwise roll further intensify as $Ri_b$ increases, as can be seen for case $Ri_b = 0.32$ in figure 10(*e*).

Figure 11 shows the effect of $S$ on the mean temperature and mean streamwise velocity for case $Ri_b =10$, which features the transition between streamwise rolls and convective cells. The bulging of $\bar {T}$ and $\bar {u}$ due to the streamwise roll is found for smooth-wall conditions and $S\ge \delta$, while this is not found for the densest ridge spacing $S=0.5\delta$. For the latter case this reflects the transition from streamwise rolls to convective cells with preferential orientation in the $z$-direction found in the instantaneous temperature fields in figures 6 and 7, which results in the disappearance of the cross-sectional motion in figure 11(*e*). The up- and downdrafts of the streamwise rolls for $S=4\delta$ and $S=2\delta$ are located in the valleys between adjacent ridges. For the former case the lateral movement of the roll encounters the ridge in the middle between up- and downdrafts, where large spanwise velocities of the roll occur. The ridges close to the up- and downdrafts for case $S=2\delta$ support the wall-normal motion of the roll by the upward deflection at the ridges, which results in stronger bulging of $\bar {u}$ at the up- and downdraft region compared with $S=4\delta$. As can be seen for case of $S=\delta$ in figure 11(*d*) the strength of the cross-sectional motion is reduced compared with the coarser $S$ cases, since the roll experiences more lateral drag on crossing the ridges due to decreasing $S$.

### 3.4. Turbulent properties

The mean velocity and temperature fields presented in the previous section have shown that secondary motions and streamwise rolls manifest as large-scale coherent motion in the cross-sectional plane. The energetics of these structures is further analysed by applying the decomposition procedures of (2.6) and (2.7) to the turbulent kinetic energy $k= 0.5 \overline {u'_i u'_i}$. This separates the turbulent kinetic energy $k$ into its coherent contribution $\tilde {k}$ and random contribution $k''$ given by $k = \tilde {k} + k''$. In order to extract the influence of the cross-sectional motion, the coherent turbulent kinetic energy $\tilde {k}$ is decomposed into its cross-sectional $\tilde {k}_c = 0.5 (\tilde {v}\tilde {v} + \tilde {w}\tilde {w})$ and streamwise $\tilde {k}_s = 0.5 \tilde {u}\tilde {u}$ parts. Since the global mean velocity components $\langle \bar {v} \rangle$ and $\langle \bar {w} \rangle$ are zero, the coherent components $\tilde {v}$ and $\tilde {w}$ represent the mean velocity motion in the cross-sectional plane (seen for instance in figure 10). The coherence of the large-scale motion is quantified by $K_c$, which is the volume average of the coherent turbulent kinetic energy of the cross-sectional components $\tilde {k}_c$ obtained by the integration formula of (2.8). While $K_c$ is a good measure of coherence for the majority of cases considered here, it will be shown later that there are two cases for which $K_c$ is not a useful measure. The first case applies when the coherent motion involves a strong temporal dynamics leading to a reduction of the coherent velocities $\tilde {v}$ and $\tilde {w}$ by long time averages. The second case concerns any coherent motion in the $x$–$y$ plane, so that this coherence is masked in the random velocity variance contribution.

The influence of the transition between forced convection structures and streamwise rolls on $K_c$ is illustrated in figure 12(*a*). As can be seen, the forced convection case $Ri_b = 0$ and weakly convective case $Ri_b = 0.003$ for smooth-wall conditions display no coherent energy in the cross-sectional components due to the missing presence of coherent motion. However, the introduction of the Gaussian ridges and the appearance of secondary motions results in a coherent kinetic energy contribution $K_c$ with the highest value for a ridge spacing of $S=\delta$, consistent with recent studies (Medjnoun *et al.* Reference Medjnoun, Vanderwel and Ganapathisubramani2020; Wangsawijaya *et al.* Reference Wangsawijaya, Baidya, Chung, Marusic and Hutchins2020). At $Ri_b = 0.016$ the streamwise rolls emerge for the smooth-wall case, which is reflected by an increase of $K_c$. This increase induced by the streamwise rolls is eventually also present for ridge spacings $S=4\delta$ and $S=2\delta$ for case $Ri_b = 0.024$. As shown in the previous sections, the two densest ridge spacing cases display secondary motions at higher $Ri_b$, which leads to an unchanged and constant value of $K_c$ up to $Ri_b = 0.024$ for $S=\delta$ and $S=0.5\delta$. This behaviour is consistent with the observation of the delayed increase in $Nu$ with increasing $Ra$ for these two cases in figure 4(*a*), illustrating the importance of the flow structures in the scaling of global quantities. Due to the presence of streamwise rolls for all cases at $Ri_b = 0.032$, $K_c$ also increases for all $S$, whose values almost double compared with $Ri_b = 0.024$.

The change of coherence due to the transition between streamwise rolls and convective cells with increasing $Ri_b$ is shown in figure 12(*b*). For this range of values $Ri_b$, $K_c$ is scaled in free-fall units, which eases the comparison with the natural convection case $Ri_b = \infty$. For the smooth-wall condition the coherence increases up to $Ri_b = 1$ and subsequently decreases to the natural convection case. This maximum of $K_c$ occurs for a value of $-\delta _{eff}/L = 1.23$, which is consistent with recent findings in ABL, for which the maximum coherence of streamwise rolls is found at $-z_i/L = 1.08$ (Jayaraman & Brasseur Reference Jayaraman and Brasseur2021). For the rough-wall cases the coherence decreases monotonically with decreasing $S$ only for $Ri_b =0.032$ and $Ri_b=\infty$, while this behaviour is not found for values in between. Comparing with the smooth-wall cases the introduction of coarsely spaced ridges $S=4\delta$ yields a large drop of $K_c$ for $Ri_b = 1$ and $Ri_b =3.2$. The reason for this reduction is the aforementioned temporal variability of the streamwise rolls, which causes the up- and downdrafts to slowly move in the spanwise direction over a long period of time instead of being fixed, thereby reducing averaged values of $\tilde {v}$ and $\tilde {w}$ and thus $K_c$. This will be discussed in more detail in § 3.5.

For $Ri_b = 10$ the value of $K_c$ for $S=4\delta$ is below $S=2\delta$, indicating that the coherence of the streamwise roll is more affected by the coarser ridge spacing. In this case, the reduction of $K_c$ is not related to time variability of the streamwise rolls, but to the relative position of the ridges to the up- and downdrafts. As can be seen in figure 11 the up- and downdrafts for $Ri_b = 10$ occur between the ridges, which for $S=4\delta$ results in the roll encountering a ridge in the middle of its lateral motion, which causes stronger lateral drag and thereby weakens the roll motion. For $S=2\delta$, the ridges do not interfere with the roll motions at their strongest lateral velocity. Instead, the adjacent ridges at the up- and downdrafts support the upward motion of the roll by its wall-normal deflection at the ridges. Even though denser ridge spacings contribute to more drag, the support of the deflections compensates in part for the losses in $K_c$ for $S=2\delta$, while this does not occur for $S=4\delta$.

For $Ri_b = 3.2$ and $Ri_b = 10$ the value of $K_c$ vanishes for the densest ridge spacing $S=0.5\delta$, which reflects that streamwise rolls are not present for these cases, as can be seen in the instantaneous temperature fields in figure 6. Also, for the natural convection case, the value of $K_c$ approaches zero for $S\le 2\delta$, even though the instantaneous temperature fields in figure 6 suggest an increase of the coherence in the $x$-direction due to rolls aligned in the $z$-direction. This reflects the property of $K_c$ that only coherent motion in the $z$-direction can be detected, while any coherence in the $x$-direction is masked. Consequently, for the current ridge cases, a reduction of $S$ leads to a weakening of the coherence in the $z$-direction, which is equivalent to a weakening of rolls with orientation in the $x$-direction. Note that, for a further reduction of $S$ down to the limit of $S\rightarrow 0$, the surface will approach a smooth wall again with a reduced cross-sectional area. Since $Re_b$ and $Ra$ are kept constant while varying $S$ we expect the flow for $S\rightarrow 0$ to be similar to the present smooth-wall case $S=\infty$.

The strength of the different large-scale coherent motions is associated with different wall-normal regions, where coherence is dominant. This is shown for the transition between the forced convection structures and streamwise rolls for the horizontally averaged velocity stresses and temperature variance in figure 13 for three different $Ri_b$ cases. The dashed line represents the Reynolds stresses (e.g. $\langle \overline {u'u'}\rangle$), while the solid line indicates the coherent stress (e.g. $\langle \tilde {u}\tilde {u}\rangle$). The difference of both contributions results in the random stress, e.g. $\langle \overline {u'' u''} \rangle = \langle \overline {u' u'} \rangle - \langle \tilde {u}\tilde {u}\rangle$. For the forced convection case $Ri_b = 0$ and rough-wall condition the coherent streamwise stress $\langle \tilde {u}\tilde {u}\rangle$ is concentrated close to the wall, with increasing peak values with decreasing $S$ down to $S=\delta$. The densest ridge spacing $S=0.5\delta$ has a similar peak value as $S=\delta$, however, extending less into the bulk region, consistent with the reduced spatial extent of the mean secondary motions for this ridge spacing seen in figure 9. This is also reflected in the wall-normal location of the peak values of $\langle \tilde {v}\tilde {v}\rangle$ and $\langle \tilde {w}\tilde {w}\rangle$, which are located closer to the wall for $S=0.5\delta$. The coherent temperature variance displays a similar trend to $\langle \tilde {u}\tilde {u}\rangle$ with respect to $S$, since temperature is a passive scalar for this case resulting in a strong correlation between the streamwise velocity and temperature.

As discussed before, streamwise rolls are present for case $Ri_b = 0.024$ and $S=\infty, 4\delta, 2\delta$, which can be seen most clearly by increased values of $\langle \tilde {T}\tilde {T}\rangle$ and $\langle \overline {T' T'} \rangle$ within the entire bulk region in figure 13(*k*). For these cases the coherent temperature variance $\langle \tilde {T}\tilde {T}\rangle$ contributes a large fraction of the temperature variance $\langle \overline {T' T'} \rangle$, which reflects the strong bulging of the mean temperature seen in figure 10(*c*). The induced coherence by streamwise rolls is also seen in the coherent velocity stresses but is less pronounced. Among them, this is most noticeable for the spanwise coherent velocity stress $\langle \tilde {w}\tilde {w}\rangle$ in figure 13(*h*), where stronger spanwise coherent stresses are observed with respect to the forced convection cases (figure 13*g*). A slight increase of $\langle \tilde {u}\tilde {u}\rangle$ and $\langle \tilde {v}\tilde {v}\rangle$ can be also found for these cases in the bulk region, which is illustrated by the insets in figure 13(*b*,*e*). Although the coherent velocity stresses of the streamwise rolls are for this case rather weak, this motion is sufficient to cause a strong imprint in the coherent temperature variance. For the two densest ridge spacings $S=\delta$ and $S=0.5\delta$ where secondary motions occur, the velocity stresses and temperature variances remain similar to the forced convection cases at $Ri_b = 0$. This further supports the fact that the increased drag and vertical mixing due to the ridges for these two cases are strong enough to inhibit the formation of streamwise rolls.

For case $Ri_b = 0.032$, when all ridge configurations exhibit streamwise rolls, the two densest ridge spacings $S$ now also show a significant increase in the coherent temperature variance $\langle \tilde {T}\tilde {T}\rangle$, although their values are lower than for the streamwise roll cases of $Ri_b = 0.024$. Likewise, the increase of coherent velocity stresses, which was initiated at $Ri_b=0.024$, continues, which is clearly seen by $\langle \tilde {w}\tilde {w}\rangle$ in figure 13(*i*). As can be seen, the successive reduction of the ridge spacing $S$ results in a decrease of the coherent velocity stresses, indicating that the streamwise rolls are damped by the presence of the ridges. For $S=0.5\delta$ only a mild increase of $\langle \tilde {u}\tilde {u}\rangle$ and $\langle \tilde {v}\tilde {v}\rangle$ is found in the bulk region (figure 13*c*,*f*), and the near-wall peak of $\langle \tilde {v}\tilde {v}\rangle$, introduced by the secondary motions, is still visible. The persistence of stronger wall-normal coherent motions near the wall for $S=0.5\delta$, similar to the forced convection cases, is consistent with the streamwise rolls to appear more spot like, as seen in figure 6.

Figure 14 shows the velocity stresses and temperature variance for the transition from streamwise rolls to natural convection. For case $Ri_b = 1$ the streamwise roll displays the strongest coherent cross-sectional motion, which is illustrated by larger values of $\langle \tilde {w}\tilde {w}\rangle$ compared with the $\langle \tilde {u}\tilde {u}\rangle$. At the same time the wall-normal Reynolds stresses have comparable magnitude to the streamwise Reynolds stress in the channel centre region. The decrease of coherent velocity stresses with decreasing $S$ is found for the three densest ridge spacings, while case $S=4\delta$ displays significantly lower values due to the time variability of the streamwise rolls. At the same time, the coherent temperature variance in the bulk region increases with decreasing $S$ (inset figure 14*j*) and $\langle \overline {T' T'} \rangle$ exhibits larger values in the near-wall region compared with the smooth-wall case.

As discussed in relation to figure 12, the coherence of streamwise rolls is reduced when increasing $Ri_b$ beyond values of $Ri_b \approx 1$, which is reflected in a reduction of the coherent velocity stresses for $Ri_b = 10$. For $Ri_b = 10$ and dense ridge spacings $S=\delta$ and $S=0.5\delta$ the instantaneous temperature fields in figure 6 show a transition from the streamwise rolls to convection cells, which is reflected here in significant lower coherent stresses $\langle \tilde {v}\tilde {v}\rangle$ and $\langle \tilde {w}\tilde {w}\rangle$ compared with the smooth-wall case. The increased peak value of $\langle \overline {T'T'} \rangle$ with decreasing $S$ indicates that ridges are more efficient in mixing temperature close to the wall. For the densest ridge spacing $S=0.5\delta$ the convection cells are oriented along the spanwise direction, which is in agreement with the observation of zero coherent spanwise stresses $\langle \tilde {w}\tilde {w}\rangle$ in figure 14(*h*).

The observation that the coherence of convective cells oriented along the $x$-direction is significantly reduced for $S\le 2\delta$ is also reflected in the coherent velocity stresses and coherent temperature variance. The coherent contribution of $\langle \tilde {w}\tilde {w}\rangle$ for $S=4\delta$ is significantly reduced compared with smooth-wall conditions, as can be seen in figure 14(*i*). The only increase of coherence can be found for $\langle \tilde {T}\tilde {T}\rangle$ in the near-wall region with decreasing $S$ in figure 14(*l*), while it vanishes in the bulk region for dense spacings. The preferred orientation of the convective cells in the spanwise direction, as seen in figure 6, needs to result in larger streamwise motion, which is reflected in the steady increase of $\langle \overline {u'u'}\rangle$ with $S$ (figure 14*c*). This is also accompanied by a steady increase of $\langle \overline {T'T'}\rangle$ with decreasing $S$, which suggests that the aligned ridges induce stronger thermal plumes. Note that, in a square domain with smooth-wall conditions, the streamwise and spanwise stresses have the same distributions due to the directional invariance of the cells (Pandey, Scheel & Schumacher Reference Pandey, Scheel and Schumacher2018). However, the smooth-wall case already displays slightly larger values for $\langle \overline {u' u'} \rangle$ than $\langle \overline {w' w'} \rangle$ and this indicates that the convection cells are slightly more oriented in the spanwise direction before introducing the aligned ridges.

### 3.5. Variability of streamwise rolls

In the previous section it was found that the coherence of the streamwise rolls for coarse ridge spacings $S=4\delta$ drops significantly for case $Ri_b = 1$ and $Ri_b = 3.2$ compared with the denser values of $S$, indicating weaker cross-sectional motion of the roll. However, the instantaneous temperature visualisations in figures 6 and 7 for this specific case do not indicate weaker streamwise rolls, suggesting that a time-varying behaviour of the streamwise rolls might be present. For this purpose, the volume-averaged coherent turbulent kinetic energy of the cross-sectional components $K_c$, which is based on the average of the entire time series, is now averaged for shorter time windows. The short-time-averaged coherent turbulent kinetic energy $K_c^{s}$ (superscript $s$ indicates the short-time average) is computed over a time range of ${\rm \Delta} t_s \approx 3.4t_f$. Note that the value of the short-time average is the shortest available data for the present simulations. The time evolution of $K_c^{s}$ for consecutive short-time intervals is shown for case $Ri_b = 1$ in figure 15. As can be seen, all cases feature a relatively slow dynamics and for $S=\infty$ and $S\le 2\delta$ the time variations vary mildly around their full time-averaged values $K_c$ in figure 12(*b*). For $S=4\delta$ the time variation is more pronounced and the dynamics shows a clearly visible periodic reduction of $K_c^{s}$ with a period of $O(t) \approx 100t_f$, which corresponds to $O(t) \approx 200 t_b$. For this analysis an additional simulation with $S=8\delta$ is performed and it shows similar time variation compared with $S=4\delta$, however, with smaller amplitude. This observed dynamics is significantly slower than observations in ABL where the dynamics of the flow reaches a statistically quasi-steady state in roughly $6t_f$ (Moeng & Sullivan Reference Moeng and Sullivan1994). The time mean value of $K_c^{s}$ is significantly larger than the value $K_c$, which suggests that the streamwise rolls of $S=4\delta$ and $S=8\delta$ feature some time variability, which is masked by considering quantities based on the average of the entire time series. This time-varying behaviour is also found for $Ri_b =3.2$ and $S=4\delta$, which also displayed a significant reduction of $K_c$ in figure 12.

The variability of the streamwise rolls is illustrated by the time evolution of the short-time- and streamwise-averaged temperature $\overline {T}^{s}$ at the wall-normal channel centre $y=\delta$ along the spanwise direction $z$ in figure 16. The spanwise position of the thermal up- and downdrafts of the streamwise roll is represented by the higher and lower temperature values, respectively. While the spanwise location of the up- and downdrafts remain at the same position for $S=\infty$ and $S\le 2\delta$, the spanwise location of the up- and downdrafts of case $S=4\delta$ and $S=8\delta$ is strongly varying in time. The up- and downdrafts for these two cases exhibit strong lateral movement. While this movement is almost periodic for $S=8\delta$ and remains between the ridges, the up- and downdrafts of $S=4\delta$ are able to cross the ridges at some time instances, e.g. $t/t_f \approx 100$, while they are not able to cross them at other time instances, e.g. $t/t_f \approx 350$. The large values of $K_c^{s}$ of $S=4\delta$ and $S=8\delta$ in figure 15 correspond to occasions when the up- and downdrafts are located close to or directly at the ridges, e.g. $t = 700\unicode{x2013}800t_f$, while small values of $K_c^{s}$ correspond to locations of the up- and downdrafts in between the ridges. This increase of $K_c^{s}$ can be interpreted by the formation mechanism of streamwise rolls proposed by Khanna & Brasseur (Reference Khanna and Brasseur1998), which relates them to the organisation of localised buoyancy forces within near-wall streaks. When the up- and downdrafts are located at ridges, the ridges support the formation of strong localised buoyancy forces, leading to strong local up- and downdrafts. Due to the symmetric arrangement of the ridges at both walls, the up- and downdrafts impinge at another ridge on the opposite wall, which is supposed to counteract the impinging roll by the formation of localised buoyancy forces with opposing direction. Thus, for $S=4\delta$ the ridges support the formation of strong up- and downdrafts, however, they cannot remain at the spanwise location due to the counteraction of the opposing ridge, leading to the lateral evasion. For case $S=8\delta$ this enhancement at the ridges occurs only for the up- or the downdraft and not simultaneously as for $S=4\delta$, which might be an explanation for the observation that up- and downdrafts are not able to cross the ridges.

For case $S=2\delta$ a short-time interval $t \approx 120t_f$ with large values of $K_c^{s}$ is present, which corresponds to a time interval in which the up- and downdrafts are located above ridges. However, most of the time the up- and downdrafts remain in between adjacent ridges, as has been shown in figure 11, and only a slight meandering within this range is observed. The examination of time series of cases $Ri_b = 0.32$ and $Ri_b = 10$ does not reveal this strong lateral movement of the up- and downdraft locations, while for $Ri_b = 3.2$ and $S=4\delta$ a similar lateral movement is found. The results suggest that the dynamics of the streamwise rolls is very sensitive to ridge spacings of the order of the spanwise rolls’ width, as seen for $S=4\delta$ and $S=8\delta$. For denser ridge spacings $S\le 2\delta$ several adjacent ridges contribute by localised buoyancy forces to the formation of the up- and downdrafts which might be strong enough to inhibit disturbances by the opposing ridges and thereby prevent lateral movement of the streamwise rolls. Future investigations with a staggered ridge arrangement or ridges placed only at the bottom or top wall might further shed light on the influence of ridges on the roll formation. For instance, the comparison of a symmetric and staggered arrangement of streamwise-aligned ridges in forced convection flows (Stroh *et al.* Reference Stroh, Schäfer, Forooghi and Frohnapfel2020*a*) has shown that a staggered arrangement promotes the coherence of the large-scale secondary motion, which might also be valid for rolls and may lead to a fixation of the rolls for $S=4\delta$.

### 3.6. Reynolds number effects

In the previous section the flow organisation of mixed convection flows was considered in terms of varying bulk Richardson number $Ri_b$ and ridge spacing $S$, while the Reynolds number $Re_k$ was approximately constant. The effect of $Re_k$ on the transition between forced convection structures and streamwise rolls is shown in figure 17, which presents the cross-sectional mean temperature and flow topology for case $Ri_b=0.024$ with values of $Re_k = 252\unicode{x2013}263$ and case $Ri_b=0.025$ with threefold larger values of $Re_k = 826\unicode{x2013}841$. As can be seen figure 17(*a*) both cases feature a streamwise roll down to a ridge spacing of $S=2\delta$ (not shown for $S=\infty$ and $S=4\delta$). The comparison of the temperature fields between the low and high $Re_k$ cases depicts that the thermal boundary layer is reduced for higher $Re_k$ due to the more efficient mixing of the flow in the near-wall region. As shown in the previous sections, the streamwise rolls are replaced by secondary motions for the lower $Re_k$ cases with $Ri_b = 0.024$ and $S\le \delta$, while for the larger $Re_k$ cases the streamwise roll remains for these ridge spacings. However, the streamwise roll appears more distorted and affected by the ridges, as can be seen for $S=\delta$ and $S=0.5\delta$. In addition, secondary motion in the form of one pair of counter-rotating vortices emerges at one ridge at the bottom wall lying in the downdraft region of the roll. This illustrates that the ridges on the opposing wall of the up- and downdraft regions are able to form coherent structures that counteract the large-scale roll formation. For the densest ridge spacing $S=0.5\delta$ in figure 17(*c*) the streamwise roll for the large $Re_k$ case is now confined to a smaller wall-normal region in the bulk flow. This is associated with the recirculation zones at the leeward side of the ridges, which are connected between the up- and downdraft regions, thereby forming a roughness sublayer which inhibits the attachment of the lateral movement of the streamwise roll at the wall. These results suggest that the transition range between forced convection structures and streamwise rolls with heterogeneous rough surfaces is not solely determined by the pair of $Ri_b$ and $S$, but also by the value of the Reynolds number $Re_k$. Due to the increased turbulent mixing for larger values of $Re_k$ the streamwise rolls can counteract the additional shear by the ridges, such that the transition between forced convection structures and streamwise rolls is shifted towards smaller values of $Ri_b$.

The influence of $Re_k$ on the roll-to-cell transition is illustrated for two cases with $Ri_b = 3.2$ and different $Re_k$ by the temperature fluctuation in the horizontal mid-plane in figure 18. As can be seen, both cases exhibit streamwise rolls for smooth-wall conditions and $S\ge 2\delta$, while differences in the flow structures start to appear at $S=\delta$. At this ridge spacing the lower $Re_k$ case still shows streamwise rolls (figure 18*a*), while for the larger $Re_k$ case the streamwise rolls are disturbed by strong thermal plumes spanning almost the entire spanwise domain. For the lower $Re_k$ case with $S=0.5\delta$ no coherent streamwise rolls can be observed and, similar to $S=\delta$, for the larger $Re_k$ case thermal plumes emerge, which indicates the beginning of the transition to convective cells. This shows that increasing $Re_k$ has a comparable effect on the flow organisation to the reduction of the ridge spacing $S$. While smaller values of $S$ increase the friction of the flow and weaken the lateral motion of the streamwise rolls, the higher $Re_k$ increases the thermal mixing near the wall, and both effects promote the formation of thermal plumes. The loss of coherence of the streamwise rolls for case $S=0.5\delta$ in figure 18(*a*) is also supported by a vanishing value of $K_c$ in figure 12(*b*). Also the inspection of the time series of $K_c^{s}$ reveals only a very weak contribution, which is an order magnitude lower than for case $S=\delta$. The observation that only convective cells with preferential orientation in the $z$-direction occur for the natural convection cases $S\le 2\delta$ and for $Ri_b=10$ with $S=0.5\delta$ is also seen for the large $Re_k$ case with $S=0.5\delta$. The influence of $Re_k$ on the transition between streamwise rolls and convective cells is such that higher values of $Re_k$ initiate this transition to convective cells at smaller $Ri_b$, thus reducing the range of streamwise rolls.

## 4. Discussion and conclusion

The present study shows that heterogeneous surfaces in form of streamwise-aligned Gaussian ridges have a significant influence on the flow organisation of mixed convection flows. The appearance of streamwise rolls is considerably reduced for dense ridge spacings $S$, which is related to the increased drag introduced by the ridges. Therefore, the formation of the rolls requires larger buoyancy forces, such that the transition from forced convection structures to streamwise rolls is delayed by the ridges towards higher $Ri_b$ values than expected for smooth-wall conditions. Specifically, this transition occurs for the smooth channel at $Ri_b = 0.016$, while for large ridge spacings of $S\ge 2\delta$ this transition occurs first at $Ri_b=0.024$ and for denser ridge spacings $S\le \delta$ at $Ri_b = 0.032$.

The strongest influence of the heterogeneous surface on the flow organisation occurs between the roll-to-cell transition range, where a change of the surface properties has a comparable effect to a change of $Ri_b$ for homogeneous wall conditions. This behaviour is observed by the inspection of the instantaneous and mean cross-sectional velocity and temperature fields. In the range of $Ri_b =3.2\unicode{x2013}10$, where streamwise rolls are present for smooth-wall conditions, dense ridge spacings already trigger the transition from roll to cell structures. This is surprising, since this range of bulk Richardson number, which corresponds to a range of stability parameter $-\delta _{eff}/L = 3.4\unicode{x2013}9.7$, is below the range where commonly cell structures are observed in the ABL (Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017). The results show that the increased lateral drag introduced by the densely spaced ridges diminishes the coherence of the streamwise rolls, and eventually leads to the transition to convective cells at smaller $Ri_b$. In addition to the earlier roll-to-cell transition the ridges also affect the orientation of the convection cells for denser ridge spacings. While the convective cells have no preferential orientation for the smooth-wall natural convection case, they increasingly prefer to orient perpendicular to the ridges with decreasing $S$. This is also explained by the additional drag, which is experienced by ridge-aligned convective cells, such that the lateral near-wall motion of these cells is increasingly disturbed for smaller $S$. This will eventually lead to their breakdown and the flow prefers to stream only along the ridges, resulting in the occurrence of spanwise coherent convective cells.

For the moderate values of Reynolds numbers that can be afforded for the present simulations, we find that an increase in $Re$ favours the transition from forced convection structures to streamwise rolls at smaller $Ri_b$, which is associated with the increased thermal vertical mixing at larger $Re$. At the roll-to-cell transition range, an increase of $Re$ promotes the transition towards convective cells, such that convective cells appear for larger $S$ if $Re$ is increased.

One particular observation is that the dynamics of streamwise rolls is very sensitive to ridge spacings of the order of the rolls’ width, which is found for $Ri_b = 1$ and $Ri_b = 3.2$. For the specific ridge spacing $S=4\delta$ the up- and downdraft regions move over the entire channel slowly in time, with time periods of approximately 100 free-fall time units or 200 time bulk units, which is in contrast to denser ridge spacings and smooth-wall conditions, where the spanwise location of the rolls is fixed. Due to this variation of streamwise rolls in the former case, some statistical features of the rolls are masked by long time integration. This is seen for example for the strength of the roll's coherence, which almost vanishes for long time intervals. Inspection of consecutive short-time averages reveals that the strength of the roll's coherence depends on the spanwise location of the up- and downdraft regions. The coherence is reduced if the up- and downdraft regions occur in the valley of adjacent ridges, and is increased if they occur in the vicinity of the ridges. In the former case the rolls experience stronger lateral drag due to their horizontal movement above the ridges, while in the latter case the ridges support the formation of localised buoyancy forces at the ridges, which in turn strengthens the up- and downdraft regions. Although the ridges reinforce the rolls, they do not reside there permanently. This is likely due to the symmetric arrangement of the ridges at both walls, since the up- and downdrafts impinge on an opposing ridge, which disturbs the roll formation. While the formation mechanism of streamwise rolls is still not clear and under debate (Etling & Brown Reference Etling and Brown1993; Salesky *et al.* Reference Salesky, Chamecki and Bou-Zeid2017), the present observations indicate that the formation and the dynamics of streamwise rolls are very sensitive to heterogeneous surfaces.

## Acknowledgements

This work was performed on the computational resources of HOREKA and used the storage facility LSDF funded by the Ministry of Science, Research and the Arts Baden-Württemberg, and Deutsche Forschungsgemeinschaft (DFG) within the framework programme bwHPC.

## Funding

K.S. and B.F. acknowledge funding through DFG project number 423710075. J.P.M. acknowledges support through project PID2019-105162RB-I00 funded by MCIN/AEI/10.13039/501100011033.

## Declaration of interests

The authors report no conflict of interest.

## Data availability statement

Mean statistics of all simulations are published as open-access dataset in the KITopen repository, DOI: https://doi.org/10.5445/IR/1000149685.

## Author contributions

K.S. designed the computational framework, carried out the numerical simulations, performed the data post-processing and statistical analysis with supervision by B.F. and J.P.M. All authors contributed to the conceptualisation of the study as well as to the discussion and interpretation of the data. K.S. wrote the original draft of the paper with review and editing support from B.F. and J.P.M. Funding was acquired by B.F.

## Appendix A. Validation of code implementation

The implementation of the active scalar in Xcompact3d is validated against the Rayleigh–Bénard and mixed convection cases of Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) at $Ra = 10^{6}$ and $10^{7}$. For a direct comparison the same grid resolution is used as in Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017), which is given in table 2. The mean differences in skin friction coefficient and Nusselt number with respect to the reference data are indicated by $\varepsilon _{C_f}$ and $\varepsilon _{Nu}$. While for the skin friction coefficient the two low Reynolds number cases at $Re=10^{6}$ show deviations of up to $3.7\,\%$, this is reduced to below $1.7\,\%$ for the higher Rayleigh number cases. The Nusselt number is in very good agreement for both chosen Rayleigh numbers and stays below $0.8\,\%$ for all simulation cases. The mean velocity and mean temperature profiles, as well as the variances $\overline {u'u'}$- and $\overline {T'T'}$-profiles, are shown in figure 19 and the comparison with the reference data shows very good agreement between the considered flow cases.

## Appendix B. Grid study with Gaussian ridges

The grid resolution requirements for the simulations with Gaussian ridges are studied for different flow configurations to show that the chosen grid resolution is sufficiently fine to capture the investigated flow physics. The grid refinement study is performed for three different flow configurations, namely pure forced convection, mixed convection and pure Rayleigh–Bénard flow. The domain size for this study was reduced to $L_x \times L_y \times L_z = 8\delta \times 2\delta \times 4\delta$ to keep the grid study computationally affordable. In all cases the spanwise spacing of the Gaussian ridges is $S/\delta =1$, corresponding to four Gaussian ridges at each sidewall. The different grid resolutions of the simulation cases and the resulting global flow properties are given in table 3.

For the pure forced convection case the mean variations in skin friction coefficient and Nusselt number from the coarsest to the finest grid simulation are within a range of $0.5\,\%$ and $0.7\,\%$, respectively. The grid refinement does not reveal any significant changes in the mean velocity, temperature and covariance profiles between all considered cases (not shown here). In order to satisfy the grid requirements proposed by Pirozzoli *et al.* (Reference Pirozzoli, Bernardini, Verzicco and Orlandi2017) for pure forced convection flows and being conservative with the spanwise grid resolution for the representation of the Gaussian ridges, the grid $N_x \times N_y \times N_z = 256 \times 193 \times 192$ is chosen to be appropriate. This results for the large domain simulation ($L_x \times L_y \times L_z = 16\delta \times 2\delta \times 8\delta$) in a grid of $N_x \times N_y \times N_z = 512 \times 193 \times 384$ for pure convection flows with Gaussian ridges.

The grid refinement study for the mixed convection case is performed at $Ra=10^{7}$, which requires a finer grid compared with the grid study of the pure forced convection case at $Re_b = 2800$ according to smooth-wall cases (see table 3). Furthermore, we increase the bulk Reynolds number to $Re_b = 5000$ in order to make this grid study more demanding in terms of the requirements of the shear induced turbulence. The mean difference of all cases in $C_f$ and $Nu$ with respect to the finest grid case lies below $1.81\,\%$ for the skin friction coefficient and $0.45\,\%$ for Nusselt number. This demonstrates, that the grid resolution for the plane wall mixed convection cases is already sufficient for the additional numerical representation of Gaussian ridges by the IBM based on polynomial reconstruction. Similar results are obtained for the pure Rayleigh–Bénard case at $Ra=10^{7}$, where the mean difference in $Nu$ with respect to the finest grid case is below $0.35\,\%$ for all simulation cases. Consequently, for the investigation of mixed and natural convection at $Ra=10^{7}$ a grid of $N_x \times N_y \times N_z = 1024 \times 257 \times 512$ for the large domain cases is chosen. For lower $Ra$ cases, the chosen grid resolution of the pure forced convection study marks the lower bound to sufficiently represent the Gaussian ridges in these cases.