We use index notation in a Cartesian reference system. The spatially filtered Navier–Stokes equations are considered, 1∂ui∂t+uj∂ui∂xj=-1ρ∂p̃∂xi-∂τij∂xj-∂τijSGS,dev∂xj-1ρ∂P∂xi,2∂ui∂xi=0, where ui=(u,v,w) is the spatially filtered velocity field along the streamwise (x), cross-stream (y), and vertical (z) coordinate directions, respectively, t is the time, ρ is the constant fluid density (Boussinesq approximation), p̃≡p+13τkkSGS is a modified pressure term, τij is the filtered viscous stress tensor, and τijSGS,dev is the deviatoric part of the SGS stress tensor. In addition, the term -1ρ∂P∂xi is an imposed constant pressure gradient driving the flow. The spatially filtered viscous tensor is τij=-2νSij, where ν=const is the kinematic viscosity of the Newtonian fluid and Sij is the resolved (in the LES sense) rate of strain tensor. For the SGS stress tensor, the static Smagorinsky model is used, 3τijSGS,dev=-2νSGSSij=-2(CSΔ)2|S|Sij, where νSGS is the SGS eddy viscosity, CS is the Smagorinsky coefficient , Δ=(ΔxΔyΔz)1/3 is a local length scale based on the volume of the computational cell , and |S|=2SijSij quantifies the magnitude of the rate of strain. In the present work, CS=0.1, unless otherwise specified. Note that dynamic Smagorinsky models are preferred to the static one for the LES of ABL flows . Dynamic models evaluate SGS stresses via first-principles-based constraints, feature improved dissipation properties when compared to the static Smagorinsky model (especially in the vicinity of solid boundaries), and are free of explicit modeling parameters. The choice made in the present study is motivated by problematics encountered when using the available dynamic Lagrangian model in preliminary tests. However, while SGS dissipation plays a crucial role in PSFD solvers, truncation errors may overshadow SGS stress contributions in the second-order FV-based ones . The static Smagorinsky SGS model used herein might hence perform similarly to dynamic SGS models for the considered flow setup. This conjecture is supported by the results of .

The large scale separation between near-surface and outer-layer energy-containing ABL motions poses stringent resolution requirements to numerical modelers, if all the energy containing motions have to be resolved. To reduce the computational cost of such simulations, the near-surface region is typically bypassed and a phenomenological wall-layer model is leveraged instead to account for the impact of near-wall (inner-layer) dynamics on the outer-layer flow . This approach is referred to as wall-modeled large-eddy simulation (WMLES) and is used herein. An algebraic wall-layer model for surfaces in a fully rough aerodynamic regime was implemented based on the logarithmic equilibrium assumption, i.e., 4|ũ|=u*κln⁡zz0, where |ũ|≡u2+v2 is the norm of the velocity at a certain distance from the ground level, u* is the friction velocity (see Sect.  for details), κ is the von Kármán constant, z is the distance from the ground level, and z0 is the so-called aerodynamic roughness length, a length scale used to quantify the drag of the underlying surface. In this work, the values κ=0.41 and z0=0.1 m are set. The kinematic wall shear stress is assumed to be proportional to the local velocity gradient (Boussinesq hypothesis), 5τiz,w=(ν+νt)∂ui∂z|w,i=x,y, where νt is the total eddy viscosity. Employing the no-slip condition for the velocity field, the standard FV approximation of the shear stress at the wall gives 6τiz,w=(ν+νt)fui,cΔz,i=x,y, where the subscript f is used to denote the evaluation at the center of the wall face, the subscript c denotes the evaluation at the center of the wall-adjacent cell, and Δz is the distance from the wall. From the logarithmic law (Eq. ) evaluated at the first cell center, one can write u*=κ|ũ|c/ln⁡(Δzz0). Using the definition of friction velocity u*=τw2, where τw is the magnitude of the kinematic wall shear stress vector, along with Eq. (), and rearranging, the total eddy viscosity at the wall can be written as 7νt,f=κ|ũ|cln⁡Δzz02Δz|ũ|c-ν, which is the formulation implemented herein. Note that ν+νt≈νt in the boundary layer flows in the fully rough aerodynamic regime, so ν could be neglected without loss of accuracy.

In the present work, the computational grid is colocated, being the only colocated grid arrangement available within the OpenFOAM^® framework. Note that although advantageous in complex domains when compared to staggered grids , the colocated arrangement is known to cause difficulties with pressure–velocity coupling, hence requiring specific procedures to avoid oscillations in the solution. OpenFOAM^® offers the standard Rhie–Chow correction , which is known to negatively affect the energy-conservation properties of central schemes . In addition, when approximating the integrals over the surfaces bounding each control volume (as a consequence of the Gauss divergence theorem), the unknowns are evaluated at face centers and are assumed to be constant at each face, yielding an overall second-order spatial accuracy . Since the divergence form of the convective term is used in combination with a low-order scheme over a nonstaggered grid, the solution is inherently unstable . The present work makes use of the linear and QUICK interpolation schemes to evaluate the unknowns at face centers (more details are provided in Sect. ). The numerical solver is based on the PISO algorithm for the pressure–velocity calculation and on an implicit Adams–Moulton scheme for time integration . In Appendix , the performances of an alternative solver with a Runge–Kutta time-advancement scheme and a projection method for the pressure–velocity coupling are analyzed.

A series of WMLES of ABL flow (open-channel flow setup) is performed. Tests are carried out in the domain [0,Lx]×[0,Ly]×[0,Lz] with Lx=2πh, Ly=43πh, Lz=h, where h=1000m denotes the width of the open channel. Symmetry is imposed at the top of the computational domain, no-slip applies at the lower surface, and periodic boundary conditions are enforced along each side. A kinematic pressure gradient term -1ρ∂P∂x=1m/s2 drives the flow along the x coordinate direction, yielding u*=1m/s. The kinematic viscosity is set to a nominal value of 10-7m2/s, which results in an essentially inviscid flow.

The computational mesh is Cartesian, with a uniform stencil along each direction. Three simulations are run over 643, 1283, and 1603 control volumes, with the linear interpolation scheme for the evaluation of the unknowns at the face centers (simulations FV64, FV128, and FV160, respectively). Three additional simulations are run, at the same grid resolutions, with the linear scheme for the approximation of every term except for the nonlinear one, for which the QUICK scheme is used instead (simulations FV64*, FV128*, and FV160*). The cases span different grid resolutions at the same aspect ratio Δx/Δz=2π. Note that the chosen grid resolutions are in line with those typically used in studies of ABL flow with the pseudospectral approach see, e.g.,. All the calculations satisfy the Courant–Friedrichs–Lewy (CFL) condition C≲0.1, where C is the Courant number. Runs are initialized from a fully developed open-channel flow simulation in statistically steady state (dynamic equilibrium), and time integration is carried out for 100 eddy turnover times, where the eddy turnover time is defined as h/u*. Flow statistics are the result of an averaging procedure over the horizontal plane of statistical homogeneity of turbulence (xy) and in time over the last 60 eddy turnover times. The procedure yields well-converged statistics throughout the considered cases. In the following, the horizontal and temporal averaging operation is denoted by 〈⋅〉. The results from the present study are contrasted against the corresponding ones from the mixed PSFD code (simulations PSFD64, PSFD128, and PSFD160). The code is based on an explicit second-order accurate Adams–Bashforth scheme for time integration and on a fractional-step method for solving the system of equations. Simulations from the PSFD solver are carried out using a static Smagorinsky SGS model with CS=0.1, a rough wall-layer model with z0=0.1m, and C≲0.1. A summary of the runs is given in Table  along with the acronyms used in this study.

Figure  shows first- and second-order statistics for all the considered cases. The mean streamwise velocity is shown in Fig. a in a comparison with the phenomenological logarithmic-layer profile. The velocity at the first two cell centers off the wall is consistently underpredicted, whereas a positive log-layer mismatch (LLM) is observed in the bulk of the flow . The LLM is particularly pronounced for the cases using the QUICK interpolation scheme. This behavior could have been anticipated, as the wall shear stress is evaluated using the instantaneous horizontal velocity at the first cell center off the wall. A number of procedures has been proposed to alleviate the LLM, including modifying the SGS stress model in the near-wall region , shifting the matching location further away from the wall , and carrying out a local horizontal/temporal filtering operation . In preliminary runs, the approach of was implemented in an attempt to alleviate the LLM. However, no apparent improvement was observed and the solution became very sensitive to grid resolution and matching location. This finding suggests that alternative procedures might need to be devised to overcome the LLM in ABL flow simulations when using the considered class of FV solvers. Note that profiles from the PSFD solver also feature a positive LLM in spite of a spatial, low-pass filtering operation that is carried out on the horizontal velocity field before evaluating the surface shear stress .

The vertical structure of turbulence intensities is also shown in Fig. , where (⋅)RMS′ denotes the root mean square (RMS) of the fluctuations. Profiles from the FV-based solver start off relatively slow at the wall when compared to those from the PSFD-based solver and to the reference profile from . This behavior is due to a combination of SGS and discretization errors, which damp the energy of high-wavenumber modes and whose accurate quantification remains an open challenge in LES see, e.g.,. Further aloft, uRMS′ (wRMS′) features relatively stronger (weaker) peak values when compared to the corresponding PSFD profile, the overprediction (underprediction) being more apparent in the simulations with the QUICK scheme. The overshoot in the peak of uRMS′ is a well-known problem of FV-based WMLES . Lack of energy redistribution via pressure fluctuation from shear generated uRMS′ to vRMS′ and wRMS′ is the root cause of said behavior, and possible mitigation strategies include allowing for wall transpiration . Grid refinement shifts the velocity RMS peaks closer to the surface and increases the magnitude of the velocity RMS therein, but leads to no improvement in the max⁡(uRMS′) and only marginally improves the estimation of the max⁡(wRMS′). A quantitative measure of the relative error on uRMS′ with respect to the reference profile uRMS,ref′ from in the z0/h≤z/h≤0.4 interval is shown in Table . The FV-based solver performs worse than the PSFD-based one and the convergence is not monotonic. Note that nonmonotonic convergence is relatively common in LES at relatively coarse resolutions and is due to the interaction between discretization and modeling errors, whose impact on the solution cannot be a priori quantified .

Skewness and kurtosis of the streamwise velocity (Suu and Kuu, respectively) are shown in Fig. . The profiles of Suu obtained with the FV-based solver and the QUICK scheme as well as those obtained with the PSFD-based solver are in good agreement with experimental results from , here taken as a reference. On the contrary, the FV-based solver overpredicts Suu when the linear interpolation scheme is used, with the skewness remaining positive throughout the whole extent of the surface layer. Note that a positive skewness of streamwise velocity represents a flow field where negative fluctuations are more likely to happen than the corresponding positive ones. The kurtosis obtained with the FV-based solver is consistently overpredicted, representing a flow field populated by a greater number of extreme events. Again, profiles from all cases feature a nonmonotonic convergence to the reference ones, as shown in Table , where the relative error on skewness and kurtosis with respect to the measurements from is reported in the interval z0/h≤z/h≤0.4.

One-dimensional spectra of streamwise velocity fluctuations (Euu) are shown in Fig. a. The profiles are contrasted against the phenomenological production range and inertial subrange power-law profiles (k-1 and k-5/3, respectively). Predictions from the PSFD-based solver feature a relatively good agreement with the phenomenological power-law profile, especially at high grid resolution. For example, the cases PSFD128 and PSFD160 exhibit a slope of -1.2 in the production range (here defined as kxz<1). Profiles from the FV-based solver, on the contrary, exhibit strong sensitivity to grid resolution and are unable to capture the expected power-law behavior. In the production range, velocity spectra from the FV solver start off relatively shallow at small wavenumber, especially when using the linear scheme. A narrow band can be identified where Euu∼(kxz)-1, followed by a rapid decay in energy density – the decay being particularly pronounced when using the QUICK interpolation scheme because of the associated numerical dissipation. Overall, the energy density in the production range and in the inertial subrange is not well captured by the FV-based solver and grid refinement does not help circumvent this limitation, at least at the considered resolutions. The authors note that this fact might limit the use of dynamic procedures based on the identity. A further characterization of the energy distribution in the wavenumber space is given in Fig. b, where premultiplied velocity spectra kxEuuu*-2 are shown. The usual reason for considering these quantities is to create a plot in semi-log scale where equal areas under the profiles correspond to equal energy. In addition, premultiplied spectra provide information on the coherence of the flow, in particular on the so-called large and very large scale motions (LSMs and VLSMs, respectively). These structures are responsible for carrying more than half of the kinetic energy and Reynolds shear stress and are a persistent feature of the surface and outer layers of both aerodynamically smooth and rough walls . The current domain is of modest dimensions and is able to accommodate only LSMs , which are identified in premultiplied spectra by a local maximum at the streamwise wavenumber kx/h≈1. The location of the peaks from the FV-based solver with linear interpolation scheme shifts toward higher wavenumber with grid refinement, with a maximum at kx/h≈4 for the FV160 case. This fact signals a flow field where the streamwise extent of energetic modes (a.k.a., coherent structures) reduces as the grid is refined. On the contrary, the FV-based solver in combination with the QUICK scheme predicts the peak in premultiplied energy density at the expected wavenumber, hence suggesting that this approach is able to capture LSMs. The PSFD-based solver features a peak at the expected wavenumber (kx=1) only at the lowest resolution (PSFD64). Profiles from the higher-resolution cases feature high energy densities at the lowest wavenumber, highlighting an artificial “periodization” of energy-containing structures in the streamwise direction. This behavior is linked to the limited horizontal extent of the computational domain. The authors have indeed verified that a larger domain (twice as large along each horizontal direction) enables one to capture LSMs with the PSFD solver at resolutions matching the one of the PSFD128 case (not shown). A corresponding single run was carried out with the FV solver over the said larger domain and premultiplied spectra were found to be in good agreement with those presented herein, supporting the conjecture that the proposed domain size suffices to capture the range of variability of FV solvers for the problem under consideration.

To gain better insight on the spatial coherence of the flow field, the contour lines of the two-dimensional autocorrelation of the streamwise velocity Ruu2D in the xy plane are shown in Fig. . The Ruu2D=0.1 contour is often used to identify the boundaries of coherent structures populating the flow field. The contours from the FV-based solver with linear scheme (Figs. a, d) are representative of a poorly correlated flow field with a streamwise extent of the Ruu2D=0.1 contour of 0.5h and 0.1h along the streamwise and cross-stream directions, respectively. On the contrary, the contours from the FV-based solver with the QUICK scheme (Fig. b, e) depict a flow field characterized by larger spatial autocorrelation, in line with results from the PSFD-based solver. Note that the flow statistics presented above should not be impacted by the fact that the current domain size prevents some of the contour lines (simulations FV64*, FV160*, PSFD64, and PSFD160) from closing, as discussed in .

The one-dimensional spatial autocorrelation (Ruu), shown in Fig.  along the streamwise and cross-stream directions, further corroborates the above findings. From Fig. a it is apparent that the extension of the selected domain does not enable the flow to become completely uncorrelated in the streamwise direction for the PSFD solver and for the FV solver using QUICK; Ruu remains finite in the available rx/h range across resolutions. On the other hand, profiles from the FV-based solver using the linear interpolation rapidly decay towards zero. Along the cross-stream direction (Fig. b), profiles from the PSFD-based solver feature the expected negative lobes, highlighting the presence of high- and low-momentum streamwise-elongated streaks flanking each other in the said direction. This behavior is in line with findings from previous studies on the coherence of wall-bounded turbulence and with standard turbulence theory. Profiles from the FV-based solver exhibit a similar profile, albeit featuring a more rapid decay and less prominent negative lobes, especially for the high-resolution cases using the linear interpolation scheme. A quantitative measure of the coherence of the flow field is provided in Table , where the integral lengths Λrx,u and Λry,u are reported for all the considered cases and compared against direct numerical simulations of a channel flow at Reτ=2000 from . The integral lengths in Table  are evaluated at z/h≈0.15 since the data from are available at this height. Although Λrx,u might not be meaningful across the considered cases, owing to the lack of a zero crossing of the autocorrelation function, it is apparent that the FV-based solver underestimates the integral lengths when compared to the PSFD cases and the reference DNS values, especially when the linear interpolation scheme is used.

Instantaneous snapshots of streamwise velocity fluctuations over a horizontal plane support the above findings (see Fig. ). Artificially periodized, streamwise-elongated bulges of uniform high and low momentum are indeed apparent in the snapshots from the PSFD-based solver (Fig. c, f). On the contrary, the instantaneous streamwise velocity field from the FV solver is populated by smaller regions of uniform momentum, especially when using the linear scheme, and the size of energetic structures diminishes with increasing grid resolution (see, e.g., Fig. a, d).

This section is devoted to the analysis of momentum transfer mechanisms in the ABL with a focus on quadrant analysis and on statistics of conditionally averaged flow fields.

The quadrant hole analysis is a technique based on the decomposition of the velocity fluctuations into four quadrants: the first and third quadrants, outward interactions (u′>0, w′>0) and inward interactions (u′<0, w′<0), respectively, are negative contributions to the momentum flux, whereas the second and fourth quadrants, a.k.a. ejections of low-speed fluid outward from the wall (u′<0, w′>0) and sweeps of high-speed fluid toward the wall (u′>0, w′<0), represent positive contributions. A range of flow statistics can be defined based on this decomposition and used to provide insight on the mechanisms supporting momentum transfer in the ABL.

Figure  features the quadrant-hole analysis, where the notation is the same as in , with H being the hole size, Si,H the resolved Reynolds shear stress contribution to the ith quadrant at hole size H, and Si,Hf is the corresponding quadrant fraction. Stress fractions are presented for values of the hole size H ranging from 0 to 8, where larger hole sizes correspond to contributions to the resolved Reynolds shear stress from more extreme events. Clearly, the FV-based solver with the linear scheme underpredicts ejections (Fig. a), outward interactions (Fig. b), and inward interactions (Fig. c), and overpredicts sweeps at large hole size H (Fig. d). On the contrary, the FV solver with the QUICK scheme underpredicts all the profiles except for the ejections, which are captured fairly well instead (see Fig. a). Note that ejections are violent events, concentrated over a very thin region in the cross-stream direction of the ABL .

To gain insight on the vertical structure of momentum transfer mechanisms, the exuberance ratio and the ratio of sweeps to ejections are analyzed in the following. Figure a shows the exuberance ratio, defined as the ratio of negative to positive contributions to the momentum flux, (S1,0+S3,0)/(S2,0+S4,0) . The exuberance ratios from the PSFD-based solver are larger in absolute value than the correspondent ones from the FV-based solver across the whole surface layer except very close to the surface. Profiles highlight that outward and inward interactions have a significant impact on the resolved Reynolds stress in the PSFD-based solver, whereas the flow simulated with the FV-based solver is characterized by a predominance of sweeps and ejections. This behavior is consistent throughout the ABL. Figure b shows the ratio of sweeps to ejections at the lowest portion of the ABL (z/h≤0.4). Profiles obtained with the QUICK scheme are in line with predictions from the PSFD-based solver and with findings from measurements of surface-layer flow over rough surfaces, where ejections are identified as the dominant momentum transport mechanism in the ABL . On the contrary, the FV-based solver with a linear scheme tends to favor sweeps over ejections as the mechanisms for momentum transfer in the surface layer.

To conclude the analysis on the mechanisms responsible for momentum transfer, velocity statistics from a conditionally averaged flow field are discussed next. The approach of is adopted to compute the conditionally averaged flow field, where the conditional event is a positive streamwise velocity fluctuation at Δx/h=0, Δy/h=0, and z/h=0.5. Figure  features a pseudocolor and vector plot of the conditionally averaged velocity field in a cross-stream vertical plane for selected cases, whereas Fig.  displays a three-dimensional isosurface thereof. The flow structure in the equilibrium surface layer is known to be characterized by counter-rotating rolls and low- and high-momentum streamwise-elongated streaks flanking each other in the cross-stream direction. Rolls and streaks are indeed the dominant flow mechanism responsible for tangential Reynolds stress . As apparent from Fig. , the PSFD conditionally averaged velocity field exhibits counter-rotating patterns associated with positive and negative streamwise velocity fluctuations (corresponding to the aforementioned streaks). Throughout the ABL, the roll modes feature a diameter that is consistent with findings from the literature (d≈h). Moreover, positive and negative velocity fluctuations are approximately of the same magnitude (≈u*). From Fig. , it is apparent that the considered isosurfaces extend about 4h along the streamwise direction. Quite surprisingly, the FV-based solver is not able to predict the roll modes, irrespective of the interpolation scheme and grid resolution, and severely underpredicts the magnitude of the low-momentum streaks. Further, Figs.  and both depict a FV conditionally averaged flow field that is poorly correlated along the cross-stream and streamwise directions, resulting in significantly smaller momentum-carrying structures. This fact supports previous findings from the two-dimensional spatial autocorrelation (Fig. ). The lack of roll modes implies that the FV-based solvers used here are not able to capture the fundamental mechanism supporting momentum transfer in the ABL, at least at the considered grid resolutions. This limitation is likely to be the root cause of several of the observed problematics associated with the FV-solver solution, including the relatively high (low) streamwise-velocity skewness when using linear (QUICK) schemes (see Fig. ,a) and the observed imbalance between sweeps and ejections (Figs.  and ).