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The physics of the roughness sublayer are studied by direct numerical simulations (DNS) of an open-channel flow with sandgrain roughness. A double-averaging (DA) approach is used to separate the spatial variations of the time-averaged quantities and the turbulent fluctuations. The spatial inhomogeneity of velocity and Reynolds stresses results in an additional production term for the turbulent kinetic energy (TKE) – the ‘wake production’; it is the excess wake kinetic energy (WKE), generated from the work of mean flow against the form drag, that is not directly dissipated into heat, but instead converted into turbulence. The wake production promotes wall-normal turbulent fluctuations and increases the pressure work, which ultimately leads to more homogeneous turbulence in the roughness sublayer, and to the increase of Reynolds shear stress and the drag on the rough wall. In the fully rough regime, roughness directly affects the generation of the wall-normal fluctuations, while in the transitionally rough regime, the region affected by roughness is separated from the region of intense generation of these fluctuations. The budget of the WKE and the connection between the wake and the turbulence suggest strong interactions between the roughness sublayer and the outer layer that are insensitive to the variation of the outer-layer conditions. Furthermore, the present results may have implications for the relationship between the roughness geometry and the flow dynamics in the region directly affected by roughness.

Numerical simulations provide unfettered access to details of the flow where experimental measurements are difficult to obtain. This paper summarises the progress achieved in the study of passive scalars in flows over rough surfaces thanks to recent numerical simulations. Townsend’s similarity applies to various scalar statistics, implying the differences due to roughness are limited to the roughness sublayer (RSL). The scalar field exhibits a diffusive sublayer that increasingly conforms to the roughness surface as ks+ or Pr increase. The scalar wall flux is enhanced on the windward slopes of the roughness, where the analogy between momentum and scalar holds well; the momentum and scalar fields, however, have very different behaviours downwind of the roughness elements, due to recirculation, which reduces the scalar wall flux. Roughness causes breakdown of the Reynolds analogy: any increase in St is accompanied by a larger increase in cf. A flattening trend for the scalar roughness function, ΔΘ+, is observed as ks+ increases, suggesting the possibility of a scalar fully rough regime, different from the velocity one. The form-induced (FI) production of scalar fluctuations becomes dominant inside the RSL and is significantly different from the FI production of turbulent kinetic energy, resulting in notable differences between the scalar and velocity fluctuations. Several key questions remain open, in particular regarding the existence of a fully rough scalar regime and its characteristics. With the increase in Re and Pr, various quantities such as scalar roughness function, the dispersive fluxes, FI wall flux, etc., appear to trend towards saturation. However, the limited range of Re and Pr achieved by numerical simulations only allows us to speculate regarding such asymptotic behaviour. Beyond extending the range of Re and Pr, systematic coverage of different roughness types and topologies is needed, as the scalar appears to remain sensitive to the geometrical details.

Because the cost of large-eddy simulations (LES) of wall-bounded flows that resolve all the important eddies depends strongly on the Reynolds number, methods to bypass the wall layer are required to perform high-Reynolds-number LES at a reasonable cost. In this paper the available methodologies are reviewed, and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented, mostly in the context of engineering applications.

The flow over alternating roughness strips oriented normally to the mean stream is studied using wall-modeled large-eddy simulations (WMLES) and improved delayed detached-eddy simulations (IDDES) (a hybrid method solving the Reynolds-averaged Navier–Stokes (RANS) equations near the wall and switching to large-eddy simulations (LES) in the core of the flow). The calculations are performed in an open-channel configuration. Various approaches are used to account for roughness by either modifying the wall boundary condition for WMLES or the model itself for IDDES or by adding a drag forcing term to the momentum equations. By comparing the numerical results with the experimental data, both methods with both roughness modifications are shown to reproduce the non-equilibrium effects, but noticeable differences are observed. The WMLES, although affected by the underlying equilibrium assumption, predicts the return to equilibrium of the skin friction in good agreement with the experiments. The velocity predicted by the IDDES does not have memory of the upstream conditions and recovers to the equilibrium conditions faster. Memory of the upstream conditions appears to be a critical factor for the accurate computational modeling of this flow.

The development of secondary instabilities in a boundary layer over a backward-facing step is investigated numerically. Two step heights are considered, h/δo*=0.5 and 1.0 (where δo* is the displacement thickness at the step location), in addition to a reference flat-plate case. A case with a realistic freestream-velocity distribution is also examined. A controlled K-type transition is initiated using a narrow ribbon upstream of the step, which generates small and monochromatic perturbations by periodic blowing and suction. A well-resolved direct numerical simulation is performed. The step height and the imposed freestream-velocity distribution exert a significant influence on the transition process. The results for the h/δo*=1.0 case exhibit a rapid transition primarily due to the Kelvin–Helmholtz instability downstream of step; non-linear interactions already occur within the recirculation region, and the initial symmetry and periodicity of the flow are lost by the middle stage of transition. In contrast, case h/δo*=0.5 presents a transition road map in which transition occurs far downstream of the step, and the flow remains spatially symmetric and temporally periodic until the late stage of transition. A realistic freestream-velocity distribution (which induces an adverse pressure gradient) advances the onset of transition to turbulence, but does not fundamentally modify the flow features observed in the zero-pressure gradient case. Considering the budgets of the perturbation kinetic energy, both the step and the induced pressure-gradient increase, rather than modify, the energy transfer.

Unsteady separation is a phenomenon that occurs in many flows and results in increased drag, decreased lift, noise emission, and loss of efficiency or failure in flow devices. Turbulence models for the steady or unsteady Reynolds-averaged Navier–Stokes equations (RANS and URANS, respectively) are commonly used in industry; however, their performance is often unsatisfactory. The comparison of RANS results with experimental data does not clearly isolate the modeling errors, since differences with the data may be due to a combination of modeling and numerical errors, and also to possible differences in the boundary conditions. In the present study, we use high-fidelity large-eddy simulation (LES) results to carry out a consistent evaluation of the turbulence models. By using the same numerical scheme and boundary conditions as the LES, and a grid on which grid convergence was achieved, we can isolate modeling errors. The calculations (both LES and RANS) are carried out using a well-validated, second-order-accurate code. Separation is generated by imposing a freestream velocity distribution, that is modulated in time. We examined three frequencies (a rapid, flutter-like oscillation, an intermediate one in which the forcing and the flow have the same timescales, and a quasi-steady one). We also considered three different pressure distributions, one with alternating favorable and adverse pressure gradients (FPGs and APGs, respectively), one oscillating between an APG and a zero-pressure gradient (ZPG), and one with an oscillating APG. All turbulence models capture the general features of this complex unsteady flow as well or better than in similar steady cases. The presence, during the cycle, of times in which the freestream pressure-gradient is close to zero affects significantly the model performance. Comparing our results with those in the literature indicates that numerical errors due to the type of discretization and the grid resolution are as significant as those due to the turbulence model.

Separating turbulent boundary layers over smooth and rough flat plates are studied by large-eddy simulations. A suction–blowing velocity distribution imposed at the top boundary of the computation domain produces an adverse-to-favourable pressure gradient and creates a closed separation bubble. The Reynolds number based on the momentum thickness and the free-stream velocity before the pressure gradient begins is 2500. Virtual sand grain roughness in the fully rough regime is modelled by an immersed boundary method. Compared with a smooth-wall case, streamline detachment occurs earlier and the separation region is substantially larger for the rough-wall case, due to the momentum deficit caused by the roughness. The adverse pressure gradient decreases the form drag, so that the point where the wall stress vanishes does not coincide with the detachment of the flow from the surface. A thin reversed-flow region is formed below the roughness crest; the presence of recirculation regions behind each roughness element also affects the intermittency of the near-wall flow, so that upstream of the detachment point the flow can be reversed half of the time, but its average velocity can still be positive. The separated shear layer exhibits higher turbulent kinetic energy (TKE) in the rough-wall case, the growth of the TKE there begins earlier relative to the separation point, and the peak TKE occurs close to the separation point. The momentum deficit caused by the roughness, again, plays a critical role in these changes.

Direct numerical simulations (DNS) of bypass transition due to high-amplitude free-stream turbulence (FST) are carried out for a flat-plate boundary layer. The computational domain begins upstream of the plate leading edge and extends into the fully turbulent region. Thus, there is no ad hoc treatment to account for the initial ingestion of FST into the laminar boundary layer. We study the effects of both the FST length scale and the disturbance behaviour near the plate leading edge on the details of bypass transition farther downstream. In one set of simulations, the FST parameters are chosen to match the ERCOFTAC benchmark case T3B. The inferred FST integral length scale L11 is significantly larger (RL = UL11/ν = 6580) than that employed in previous simulations of bypass transition (RL ≃ 1000). An additional set of simulations was performed at RL = 1081 to compare the transition behaviour in the T3B case with that of a smaller value of FST length scale. The FST length scale is found to have a profound impact on the mechanism of transition. While streamwise streaks (Klebanoff modes) are observed at both values of the FST length scale, they appear to have clear dynamical significance only at the smaller value of RL, where transition is concomitant with streak breakdown. For the T3B case, turbulent spots form upstream of the region where streaks could be detected. Spot precursors are traced to quasi-periodic spanwise structures, first observed as short wavepackets in the wall-normal velocity component inside the boundary layer. These structures are reoriented to become horseshoe vortices, which break down into young turbulent spots. Two of the four spots examined for this case had a downstream-pointing shape, similar to those found in experimental studies of transitional boundary layers. Additionally, our simulations indicate the importance of leading-edge receptivity for the onset of transition. Specifically, higher fluctuations of the vertical velocity at the leading edge of the plate result in higher levels of streamwise Reynolds stress inside the developing boundary layer, facilitating breakdown to turbulence.

We performed large-eddy simulations of flow over a series of three-dimensional dunes at laboratory scale (Reynolds number based on the average channel depth and streamwise velocity was 18 900) using the Lagrangian dynamic eddy-viscosity subgrid-scale model. The bedform three-dimensionality was imposed by shifting a standard two-dimensional dune shape in the streamwise direction according to a sine wave. The statistics of the flow are discussed in 10 cases with in-phase and staggered crestlines, different deformation amplitudes and wavelengths. The results are validated qualitatively against experiments. The three-dimensional separation of flow at the crestline alters the distribution of wall pressure, which in turn may cause secondary flow across the stream, which directs low-momentum fluid, near the bed, toward the lobe (the most downstream point on the crestline) and high-momentum fluid, near the top surface, toward the saddle (the most upstream point on the crestline). The mean flow is characterized by a pair of counter-rotating streamwise vortices, with core radius of the order of the flow depth. However, for wavelengths smaller than the flow depth, the secondary flow exists only near the bed and the mean flow away from the bed resembles the two-dimensional case. Staggering the crestlines alters the secondary motion; the fastest flow occurs between the lobe and the saddle planes, and two pairs of streamwise vortices appear (a strong one, centred about the lobe, and a weaker one, coming from the previous dune, centred around the saddle). The distribution of the wall stress and the focal points of separation and attachment on the bed are discussed. The sensitivity of the average reattachment length, depends on the induced secondary flow, the streamwise and spanwise components of the channel resistance (the skin friction and the form drag), and the contribution of the form drag to the total resistance are also studied. Three-dimensionality of the bed increases the drag in the channel; the form drag contributes more than in the two-dimensional case to the resistance, except for the staggered-crest case. Turbulent-kinetic energy is increased in the separated shear layer by the introduction of three-dimensionality, but its value normalized by the plane-averaged wall stress is lower than in the corresponding two-dimensional dunes. The upward flow on the stoss side and higher deceleration of flow on the lee side over the lobe plane lift and broaden the separated shear layer, respectively, affecting the turbulent kinetic energy.

We performed large-eddy simulations of the flow over a series of three-dimensional (3D) dunes at laboratory scale. The bedform three-dimensionality was imposed by shifting a standard two-dimensional (2D) dune shape in the streamwise direction according to a sine wave. The turbulence statistics were discussed in Part 1 of this article (Omidyeganeh & Piomelli, J. Fluid Mech., vol. 721, 2013, pp. 454–483). Coherent flow structures and their statistics are discussed concentrating on two cases with the same crestline amplitudes and wavelengths but different crestline alignments: in-phase and staggered. The present paper shows that the induced large-scale mean streamwise vortices are the primary factor that alters the features of the instantaneous flow structures. Wall turbulence is insensitive to the crestline alignment; alternating high- and low-speed streaks appear in the internal boundary layer developing on the stoss side, whereas over the node plane (the plane normal to the spanwise direction at the node of the crestline), they are inclined towards the lobe plane (the plane normal to the spanwise direction at the most downstream point of the crestline) due to the mean spanwise pressure gradient. Spanwise vortices (rollers) generated by Kelvin–Helmholtz instability in the separated shear layer appear regularly over the lobe with much larger length scale than those over the saddle (the plane normal to the spanwise direction at the most upstream point of the crestline). Rollers over the lobe may extend to the saddle plane and affect the reattachment features; their shedding is more frequent than in 2D geometries. Vortices shed from the separated shear layer in the lobe plane undergo a three-dimensional instability while being advected downstream, and rise toward the free surface. They develop into a horseshoe shape (similar to the 2D case) and affect the whole channel depth, whereas those generated near the saddle are advected downstream and toward the bed. When the tip of such a horseshoe reaches the free surface, the ejection of flow at the surface causes ‘boils’ (upwelling events on the surface). Strong boil events are observed on the surface of the lobe planes of 3D dunes more frequently than in the saddle planes. They also appear more frequently than in the corresponding 2D geometry. The crestline alignment of the dune alters the dynamics of the flow structures, in that they appear in the lobe plane and are advected towards the saddle plane of the next dune, where they are dissipated. Boil events occur at a higher frequency in the staggered alignment, but with less intensity than in the in-phase alignment.