Equivalent Sand Grain Roughness Research Articles

Rough surfaces in underexplored surface morphology space and their implications on roughness modelling

We report direct numerical simulations results of the rough-wall channel, focusing on roughness with high $k_{rms}/k_a$ statistics but small to negative $Sk$ statistics, and we study the implications of this new dataset on rough-wall modelling. Here, $k_{rms}$ is the root mean square, $k_a$ is the first-order moment of roughness height, and $Sk$ is the skewness. The effects of packing density, skewness and arrangement of roughness elements on mean streamwise velocity, equivalent roughness height ( $z_0$ ) and Reynolds and dispersive stresses have been studied. We demonstrate that two-point correlation lengths of roughness height statistics play an important role in characterizing rough surfaces with identical moments of roughness height but different arrangements of roughness elements. Analysis of the present as well as historical data suggests that the task of rough-wall modelling is to identify geometric parameters that distinguish the rough surfaces within the calibration dataset. We demonstrate a novel feature selection procedure to determine these parameters. Further, since there is no finite set of roughness statistics that distinguish between all rough surfaces, we argue that obtaining a universal rough-wall model for making equivalent sand-grain roughness ( $k_s$ ) predictions would be challenging, and that each rough-wall model would have its applicable range. This motivates the development of group-based rough-wall models. The applicability of multi-variate polynomial regression and feedforward neural networks for building such group-based rough-wall models using the selected features has been shown.

Surface roughness effects on transonic aircraft performance: Experimental/numerical comparisons

This work highlights the necessity of taking into account surface roughness when conducting experimental tests, and when using numerical simulations to precisely calculate the turbulent lift and drag of wind-tunnel models or real aircraft in transonic conditions. The present article is a continuation of “Turbulent drag induced by low surface roughness at transonic speeds: Experimental/numerical comparisons,” Physics of Fluids, Vol. 32, 045108 (2020) by Hue and Molton. The outcomes of this former study, which was focused on flat plate samples, are here applied to a three-dimensional aircraft configuration: the Common Research Model used as a reference in the recent international Drag Prediction Workshops. Experimental campaigns have been performed in the largest ONERA wind tunnels S1MA and S2MA involving models with average surface roughness heights Ra close to 0.5 micrometers, wingspans up to 3.5 meters, Mach and Reynolds numbers up to 0.95 and 5 million respectively. Reynolds-averaged Navier–Stokes computations based on the wind-tunnel tests have then been carried out, using the equivalent sand-grain roughness height approach as well as a Musker-type correlation to determine relevant ks values. The results of both the experimental and numerical campaigns have demonstrated that the aerodynamic coefficients of the aircraft can be significantly affected by the surface roughness, even with roughness Reynolds numbers ks+ potentially below the usual threshold values sometimes considered in engineering applications (i.e. in the order of 3.5 to 5). In particular, the surface roughness effects on lift and drag have been studied using far-field analyses to evaluate the responses of friction, viscous pressure, wave and lift-induced drag components. Finally, the numerical studies have been extended to the full-scale geometry in flight conditions in order to assess the roughness effects and potential gains in realistic aircraft operating conditions.

Flow Through a Passage With Scaled Additive Manufacturing Roughness Representing Different Printing Orientations

Abstract Components with internal passages created using some laser-sintering based, additive manufacturing (AM) systems can exhibit anisotropic surface features with an appearance of three-dimensional roughness superimposed on two-dimensional, rib-like features. This paper presents an investigation of flow over roughness representing internal cooling passages printed at different angles to the AM printing plane. A roughness geometry was acquired using an X-ray tomography scan of a direct-metal-laser-sintering (DMLS) created coupon with internal cooling passages. The base surface scan was then used to create four surfaces with notional rib-like features positioned at different angles relative to the spanwise flow direction. The flow resistance of each surface was measured using the roughness internal flow tunnel. The mean flow velocity profiles for the cases with ReDh ≤ 30,000 were characterized using a four-camera, tomographic, and particle tracking system. The results demonstrate roughness orientation effects include (1) reduced bulk flow resistance as the alignment angle from the spanwise direction increases, (2) generated flow in the spanwise direction and increased tunnel flow swirl as the alignment angle increases, and (3) velocity profile changes as the flow migrates away from the rough side of the tunnel to the opposing smooth wall. The particle tracking system also demonstrates that the mean streamwise flow profiles change significantly between the 30 deg and 45 deg roughness orientations. Finally, the equivalent sandgrain roughness measurements for the four surfaces were found to follow the trends predicted using the correlations of Bons (2002, “St and cf Augmentation for Real Turbine Roughness With Elevated Freestream Turbulence,” ASME J. Turbomach., 124(4), pp. 632–644.) and Sigal and Danberg (1990, “New Correlation of Roughness Density Effect on the Turbulent Boundary Layer,” AIAA J., 28(3), pp. 554–556.).

A CFD Analysis of Oscillating Airfoil with Leading Edge Roughness: Comparing Correlations for Equivalent Sand Grain Roughness

Numerical simulations express surface roughness as equivalent sand grain roughness values, so the correlation is required to convert realistic roughness into equivalent sand grain roughness values. The correlations were designed to match the wall velocity distributions for static cases with roughness, and they accurately predicted roughness effects when applied to URANS simulations. Several studies have examined the roughness effect on dynamic pitching airfoils. Still, they have only considered the presence of roughness, neglecting the process of converting the actual roughness to equivalent sand-grain roughness. Thus, in the present study, we apply a static case-based equivalent sand-grain roughness correlation to dynamic pitching analysis to validate the suitability of each correlation and verify the significance of correlation parameters. It was observed from a comparison study between numerical aerodynamic coefficients and experimental data from Ohio State University that physical parameters such as skewness and roughness height were important parameters to predict the dynamic stall behaviour more accurately. It was also observed that dynamic stall prediction accuracy is affected by the frequency and type of airfoil, and through this, it was confirmed that there are additional factors to consider when using an equivalent sand grain roughness correlation on pitching airfoils.

Slip flow and thermal characteristics in gas thrust bearings with rough surfaces

PurposeThis study aims to investigate the rarefaction effects on flow and thermal performances of an equivalent sand-grain roughness model for aerodynamic thrust bearing.Design/methodology/approachIn this study, a model of gas lubrication thrust bearing was established by modifying the wall roughness and considering rarefaction effect. The flow and lubrication characteristics of gas film were discussed based on the equivalent sand roughness model and rarefaction effect.FindingsThe boundary slip and the surface roughness effect lead to a decrease in gas film pressure and temperature, with a maximum decrease of 39.2% and 8.4%, respectively. The vortex effect present in the gas film is closely linked to the gas film’s pressure. Slip flow decreases the vortex effect, and an increase in roughness results in the development of slip flow. The increase of roughness leads to a decrease for the static and thermal characteristics.Originality/valueThis work uses the rarefaction effect and the equivalent sand roughness model to investigate the lubrication characteristics of gas thrust bearing. The results help to guide the selection of the surface roughness of rotor and bearing, so as to fully control the rarefaction effect and make use of it.

Delayed Detached-Eddy Simulations of Rough-Wall Turbulent Reactive Flows in a Supersonic Combustor

Reactive delayed detached-eddy simulations (DDESs) of hydrogen injection into a confined transverse supersonic flow of vitiated air are conducted. The corresponding conditions were studied in the LAPCAT-II combustor, which—due to thermal coating—exhibits non-negligible wall roughness. Its effects are taken into account with the equivalent sand-grain roughness model, and, to the best of the authors’ knowledge, it is the first time that this modeling framework is considered within the DDES framework to simulate turbulent reactive flows. Two operating conditions are considered, which differ in the value of the momentum ratio between the hydrogen and vitiated air inlet streams, thus leading to two distinct values of the operating equivalence ratio (ER). For its smallest value, smooth combustion is subject to a preliminary thermal runaway period, while for its largest value, combustion is more strongly intertwined with shock wave dynamics and boundary-layer separation. Since it features large subsonic regions, which are characteristic of situations close to thermal choking, the latter is referred to as the sudden or partially choked combustion mode. Computational results are assessed through comparisons with available experimental data for both operating conditions. The main mechanism through which wall roughness alters the combustion development lies in the reduction of the effective cross-sectional area that is induced by boundary-layer thickening. For the largest ER value, DDES conducted with the sand-grain roughness model does recover the partially choked combustion mode, while smooth wall (i.e., standard) DDES does not.

Roughness effects on compressible turbulent boundary layers under different Mach numbers and wall temperature conditions

Compressible turbulent boundary layers over a zero-pressure gradient flat plate with three-dimensional sinusoidal roughness are simulated by direct numerical simulation. The roughness effects on surface drag, velocity transformation, and turbulence fluctuation characteristics are analyzed in a wide range of Mach numbers (Ma∞ = 2.25–7.25) and different ratios of wall-to-recovery temperature (Tw/Taw = 0.43 and 0.84) conditions. It is found that the roughness significantly amplifies the surface drag coefficient due to the extra pressure drag induced by roughness, and the relative increase in surface drag induced by the roughness rises by 31.1% when Ma∞ changes from 2.25 to 7.25. Current compressible velocity transformations cannot make the logarithmic region of velocity profiles independent of Tw/Taw conditions for rough cases due to the strong wall heat transfer effect below roughness peak. Therefore, a new velocity transformation (Uρt+) is proposed to make the logarithmic region of Uρt+ profiles and roughness induced a downward shift of Uρt+ profiles (ΔUρt+) in a logarithmic region independent of Ma∞ and Tw/Taw conditions. Further numerical experiments validate that, in hypersonic boundary layers, the relation between ΔUρt+ and equivalent sand-grain roughness height Reynolds number still satisfies the roughness function proposed earlier for incompressible flows. Moreover, roughness significantly changes the distribution of mean turbulent kinetic energy (TKE) in compressible turbulent boundary layers: TKE is suppressed at the bottom of roughness, while reaching its maximum at the roughness peak, which is 50%–60% larger than that in smooth case. Finally, the expansion/compression wave patterns induced by roughness alter the turbulence fluctuations in outer layer.

On predictive models for the equivalent sand grain roughness for wall-bounded turbulent flows

One of the long-standing goals of rough wall fluid dynamics research is to determine the drag penalty of surfaces based solely on their topographical parameters. The most important length scale or roughness parameter that best describes a surface in relation to friction drag has not been agreed upon yet, despite the many studies that, over the years, have attempted to identify the most appropriate surface parameter. The concept of an equivalent sand-grain roughness (ks) was introduced to standardize different types of roughness in wall-bounded turbulence, serving as an input parameter for predicting the roughness function ΔU+. To anticipate ΔU+ generated by a rough surface under turbulent flow conditions, experts use roughness correlations that establish a correspondence between the topographical characteristics of the surface and ks. Therefore, a chronological compilation of roughness correlations is presented, detailing the parameter ranges and types of roughness used in their development. This study evaluates the effectiveness of predictive correlation functions and aims to formulate a universal function by exploring a comprehensive assortment of three-dimensional (3D) surface textures available in the literature. The results suggest that a correlation based on surface height skewness (ksk) and streamwise effective slope (ESx) can predict the ratio (ks/kq), where kq is the root mean square roughness height for 3D roughness in the fully rough regime. Despite the fact that the correlation is restricted to 3D surface roughness, which is a more realistic representation, the model demonstrated a high level of accuracy in predicting ks for over 120 distinct rough surfaces.

Prediction of equivalent sand-grain size and identification of drag-relevant scales of roughness – a data-driven approach

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$ . The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$ . High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

Direct numerical simulation of turbulent flow in pipes with realistic large roughness at the wall

We carry out direct numerical simulation (DNS) of turbulent flow in rough pipes. Two types of irregular roughness are investigated, namely a grit-blasted and a graphite surface. A wide range of Reynolds numbers is tested, from the laminar up to the fully rough regime, attempting to replicate Nikuradse's pioneering study. Despite the large relative roughness, outer-layer similarity is achieved at high Reynolds number as hypothesised by Townsend, with deviations from the smooth wall case of 4 % for the grit-blasted surface and 13 % for the graphite surface. This makes it possible to define a roughness function and the equivalent sand-grain roughness. The results are compared with those obtained in plane channels, with small differences pointing to the residual influence of the duct cross-sectional shape in the presence of relatively large roughness. The computed friction factors behave similar to those Nikuradse's chart, with differences in terms of the friction factor in the laminar region and of the critical Reynolds number, which are partly absorbed by using the hydraulic radius as reference length scale. The distributions of the velocity fluctuations intensities point to a isotropisation of turbulence in the near-wall region resulting from the roughness, with influence of the roughness geometry. Comparison of the computed equivalent sand-grain roughness height suggest that existing correlations suffer from poor predictive power, at least for surfaces with large relative roughness.

Data-driven prediction of the equivalent sand-grain roughness

Surface roughness affects the near-wall fluid velocity profile and surface drag, and is commonly quantified by the equivalent sand-grain roughness {k}_{s}. It is essential to estimate {k}_{s} for accurate fluid dynamic problem modeling. While numerous roughness correlation formulas have been proposed to predict {k}_{s} in the fully rough regime, most of them are restricted to certain roughness types, with various geometric parameters considered in each case, leading to ongoing disagreements regarding its parameterization and lack of universality. In this study, a Particle Swarm Optimized Backpropagation (PSO-BP) method is proposed to predict {k}_{s} based on the selected surface parameters from previous DNS, LES, and experimental results for flow behavior over various surface roughness. The PSO-BP model’s ability to predict {k}_{s} in the fully rough region is evaluated and compared with both the existing roughness correction formulas as well as the traditional BP model. An optimized polynomial function is also proposed to serve as a ‘white box’ for predicting {k}_{s}. It turns out that the PSO-BP method has better performance in the evaluation metrics compared to other methods, yielding a Mean Absolute Error (MAE) of 0.0390, a Mean Squared Error (MSE) of 0.0026 and a Mean Absolute Percentage Error (MAPE) of 28.12%. This novel approach for estimating {k}_{s} has practical applicability and holds promise for improving the precision and efficiency of calculations related to equivalent sand-grain roughness, and thus provides more accurate and effective solutions for CFD and other engineering applications.

Surface Roughness in RANS Applied to Aircraft Ice Accretion Simulation: A Review

Experimental and numerical fluid dynamics studies highlight a change of flow structure in the presence of surface roughness. The changes involve both wall heat transfer and skin friction, and are mainly restricted to the inner region of the boundary layer. Aircraft in-flight icing is a typical application where rough surfaces play an important role in the airflow structure and the subsequent ice growth. The objective of this work is to investigate how surface roughness is tackled in RANS with wall resolved boundary layers for aeronautics applications, with a focus on ice-induced roughness. The literature review shows that semi-empirical correlations were calibrated on experimental data to model flow changes in the presence of roughness. The correlations for RANS do not explicitly resolve the individual roughness. They principally involve turbulence model modifications to account for changes in the velocity and temperature profiles in the near-wall region. The equivalent sand grain roughness (ESGR) approach emerges as a popular metric to characterize roughness and is employed as a length scale for the RANS model. For in-flight icing, correlations were developed, accounting for both surface geometry and atmospheric conditions. Despite these research efforts, uncertainties are present in some specific conditions, where space and time roughness variations make the simulations difficult to calibrate. Research that addresses this gap could help improve ice accretion predictions.

Assessment of aerodynamic roughness parameters of turbulent boundary layers over barnacle-covered surfaces

Full-scale drag penalty predictions of flows over rough walls require surface roughness characterisation from laboratory experiments or numerical simulations. In either approach, it is necessary to determine the so-called equivalent sand-grain roughness height (k_s). There are several steps involved in determining this aerodynamic roughness lengthscale, but its procedure typically includes a combination of measurement of wall-shear stress (tau _w) using direct or indirect methods as well as analysis of velocity profiles. Indirect methods usually rely on assumptions made about flow and its scaling including the validity of universal outer-layer similarity. However, the implications of the underlying assumptions involved in full-scale drag prediction are unclear. In this work, we carry out wind tunnel measurements over a realistic rough surface (from a fouled ship-hull) to evaluate the impact of different methods with an emphasis on using the outer-layer similarity hypothesis for full-scale drag predictions. Wall-shear stress is measured using an in-house floating-element drag balance (DB), and velocity profiles are obtained using particle image velocimetry (PIV), allowing the evaluation of k_s, and the associated wake parameters through several methods. The aerodynamic roughness parameters hence obtained are used for full-scale drag penalty calculations. It is observed that the predicted drag penalty can vary by over 15% among the different methods highlighting the care that should be taken when employing such methods.

Prediction of interfacial shear stress and pressure drop in vertical two-phase annular flow

The interfacial shear stress and pressure drop of an upward vertical annular flow of nitrogen–water, HFC134a–water, and nitrogen–95 % ethanol solution were comprehensively investigated considering the effect of the liquid–gas density ratio and surface tension. A direct link between the disturbance wave height and equivalent sand-grain roughness was noted through the analogy with the famous Moody chart for single-phase turbulent flows. A predictive model of the interfacial friction factor was developed based on this finding. To predict the pressure drop of the annular flow, a new model with good predictive performance for annular flows of various working fluids including steam–water under boiling water reactor operating condition (286 °C and 7 MPa) was proposed.

Modeling the Surface Pressure Spectrum on Rough Walls in Pressure Gradients

Abstract Models for surface pressure spectra beneath rough wall boundary layers are assessed, with particular emphasis on prediction from steady, Reynolds-averaged Navier–Stokes (RANS) data. RANS roughness boundary conditions are shown to have qualitatively good trends between roughness function and roughness Reynolds number, but model-to-model discrepancies remain and the universality of an equivalent sandgrain roughness height for turbulence models is doubtful. Existing empirical models for the surface pressure spectrum show good agreement in some spectral regions and a newly proposed model shows good matching across the spectrum in a variety of pressure gradient conditions. Adjustments are made to existing TNO analytical models to incorporate rough wall effects, including changes to the velocity spectrum model and the inclusion of a wall-shift, shown to be independent of local Reynolds number, pressure gradient, or turbulence model. The mathematical character of the rough wall spectrum has been revealed, but challenges remain to implement both flow and spectral modeling without a priori knowledge of the flow.

Scaling of the roughness effects in turbulent flows over systematically-varied irregular rough surfaces

To clarify the effects of surface topology on the scaling of roughness effects, we conduct experiments on turbulent rough-walled channel flows from transitionally to fully rough regimes. We considered three-dimensional irregular rough surfaces with different effective slope ES, skewness factor Sk, and fixed roughness height scales. The ES values are varied from 0.09 to 0.72 for positively and negatively skewed surfaces with Sk=±0.4. It is revealed that the transitional behavior to the fully rough regime is not influenced by Sk, but rather by ES. The transition to the fully rough regime for surfaces with ES≥0.18 is insensitive to the ES values, whereas wavy surfaces with ES=0.09 lead to a moderate transition. The equivalent sand-grain roughness ks steeply increases with increasing ES values from 0.09 to 0.36, whereas a further increase in the ES value does not significantly influence ks. It is also found that positively skewed surfaces with Sk=+0.4 yield larger ks values compared to surfaces with Sk=−0.4; this Sk effect is more pronounced for wavy surfaces with small ES values.

Experimental Study of Impact of In-Service Deterioration on Aerodynamic Performance of High-Pressure Nozzle Guide Vanes

AbstractIn this paper we experimentally evaluate the impact of in-service deterioration on the aerodynamic performance of heavily film-cooled high-pressure nozzle guide vanes from large civil jet engines. We study 15 mid-life to end-of-life parts removed from operational engines, and compare their performance to those of new parts. Deterioration features included: increased surface roughness; thermal barrier coating spallation; damaged film cooling holes; and trailing edge burn-back. We characterize and present statistics for the surface roughness. Aerodynamic measurements were performed in the high technology readiness level Engine Component AeroThermal (ECAT) facility at the University of Oxford, at engine-representative conditions of exit Mach number, exit Reynolds number, coolant-to-mainstream pressure ratio, and turbulence intensity. We present detailed experimental measurements of the coolant capacity characteristics, downstream loss, and downstream flow structures. The results show that service time has the following effects on high-pressure nozzle guide vanes: increased equivalent sandgrain roughness of (up by 1056% change); reduced coolant flow capacity (maximum change of −6.27% for film cooling holes and −24.7% for the trailing edge slot); increased overall mixed-out kinetic energy loss coefficient by (up to 33% change); leads to greater downstream flow angle variation (change of −6 deg). This is one of the first significant studies of its type in the open literature, and is an important step towards whole-life engine performance assessment.

Direct Numerical Simulation of a Turbulent Boundary Layer Encountering a Smooth-to-Rough Step Change

Using a direct numerical simulation (DNS), we investigate the onset of non-equilibrium effects and the subsequent emergence of a self-preserving state as a turbulent boundary layer (TBL) encounters a smooth-to-rough (STR) step change. The rough surface comprises over 2500 staggered cuboid-shaped elements where the first row is placed at 50 θ0 from the inflow. A Reθ=4500 value is attained along with δk≈35 as the TBL develops. While different flow parameters adjust at dissimilar rates that further depend on the vertical distance from the surface and perhaps on δSTR/k, an equilibrium for wall stress, mean velocity, and Reynolds stresses exists across the entire TBL by 35 δSTR after the step change. First-order statistics inside the inner layer adapt much earlier, i.e., at 10–15 δSTR after the step change. Like rough-to-smooth (RTS) scenarios, an equilibrium layer develops from the surface. Unlike RTS transitions, a nascent logarithmic layer is identifiable much earlier, at 4 δSTR after the step change. The notion of equivalent sandgrain roughness does not apply upstream of this fetch because non-equilibrium advection effects permeate into the inner layer. The emergent equilibrium TBL is categorized by a fully rough state (ks+≈120–130; ks/k≈2.8). Decomposition of wall stress into constituent parts reveals no streamwise dependence. Mean velocity in the outer layer is well approximated by Coles’ wake law. The wake parameter and shape factor are enhanced above their smooth-wall counterparts. Quadrant analysis shows that shear-stress-producing motions adjust promptly to the roughness, and the balance between ejections and sweeps in the outer layer remains impervious to the underlying surface.

An empirical model for velocity in rough turbulent oscillatory boundary layers

A novel empirical model for the streamwise velocity in oscillatory boundary layer flow, valid in the rough turbulent regime, is presented. The model consists of simple expressions that require only the free-stream velocity time-series and equivalent sand-grain roughness length of the bed to be known a priori. A frequency-independent attenuation and phase lead is assumed for all flow harmonics, expressions for which are extracted from data from previous experimental studies made in 3 different oscillatory flow tunnels. Only the oscillating component of the flow is considered in the model; steady streaming is neglected. Errors in kinematics predicted by the model are typically two orders of magnitude smaller than the maximum oscillatory velocity in the free-stream. Hence, it is well-suited to engineering application due to its simplicity and accuracy.

Effects of Wall Topology on Statistics of Cube-Roughened Wall Turbulence

In this work, we carried out roughness-resolved direct numerical simulations of cube-roughened turbulent channel flows to investigate the effects of aligned and staggered arrangements, element spacings, and element orientations on the statistics of rough-wall turbulence. The results show that the equivalent sandgrain roughness, Reynolds stresses and dispersive stresses are affected by element spacings and arrangements in different ways depending on the cube orientations. Placing the roughness elements in a staggered way in general increases the equivalent sandgrain roughness, unless when the element–element interaction is insignificant for the cube orientation with wakes of short length. As for the Reynolds normal stresses and the streamwise component of the dispersive stresses (both are normalized by the total friction velocity), placing the cube elements in a staggered way decreases their maximal values when compared with the aligned arrangements. As for the effects of element spacing on the equivalent sandgrain roughness and the maximums of the Reynolds normal stresses, similar trends are observed for different element orientations and arrangements when increasing the element spacing l/r (where r is the cube height) from 2.0 to 2.8. When further increasing the element spacing l/r from 2.8 to 3.5, however, different trends are observed for different element orientations. As for the dispersive stresses, greater maximal values of their streamwise components are observed for larger element spacing for all the considered cube-roughened surfaces. For the turbulence statistics in the wake of the cube element, certain similarities are observed for different element spacings.