Rough-wall Flows Research Articles

Integral methods for friction decomposition and their extensions to rough-wall flows

A primary objective of integral methods, such as the momentum integral method, is to discern the physical processes contributing to skin friction. These methods encompass the momentum, kinetic energy and angular momentum integrals. This paper reformulates existing integrals based on the double-averaged Navier–Stokes equations, and extends their application to flows over rough walls. Our derivation yields distinct decompositions for the bottom-wall viscous friction coefficient, denoted as $C_S$ , and the roughness element drag coefficient $C_R$ . The decompositions comprise three terms: a viscous term, a turbulent term and a roughness (dispersive) term – regardless of the flow configuration, be it channel or boundary layer. Notably, when these integrals are evaluated for laminar flow scenarios, only the viscous term remains significant. In addition, we elucidate the spatial distributions of the terms within these decompositions. To demonstrate the practicality of our formulations, we apply them to analyse data from direct numerical simulations of turbulent half-channel flows. These flows feature aligned and staggered cubical roughness at various packing densities. Our analyses, based on kinetic-energy-oriented decompositions, reveal that when the surface coverage density $\lambda _p$ is small, the dominant terms within the decompositions are the viscous and turbulent terms. With increasing $\lambda _p$ , the viscous dissipation term decreases, while the turbulent production term increases and then decreases. These variations arise from a subdued near-wall cycle and the development of a shear layer at the height of the cubes.

Flow resistance over heterogeneous roughness made of spanwise-alternating sandpaper strips

The Reynolds number dependent flow resistance of heterogeneous rough surfaces is largely unknown at present. The present work provides novel reference data for spanwise-alternating sandpaper strips as one idealised case of a heterogeneous rough surface. Experimental data are presented and analysed in direct comparison with drag measurements of homogeneous sandpaper surfaces and numerical simulations. Based on the homogeneous roughness data, the related challenges and sensitivities for the evaluation of roughness functions from experiments and simulations are discussed. A hydraulic channel height is suggested as an alternative measure for the drag impact of rough surfaces in internal flows. For the investigated heterogeneous roughness, it is found that turbulent flow does not exhibit a fully rough flow behaviour, indicating that the assignment of an equivalent sand grain height as commonly applied for homogeneous roughness is not possible. A prediction of the drag behaviour of rough strips based on an average between rough and smooth drag curves appears promising, but requires further refinement to capture the impact of turbulent secondary flows and spatial transients linking smooth and rough surface parts. While turbulent secondary flow induced by the roughness strips yield significant spanwise variation of the mean velocity profile for the investigated rough strips, we show that the spanwise averaged velocity profiles collapse reasonably well with a smooth or homogeneous rough wall flow. This allows to extract a global roughness function from the spanwise averaged flow field in good agreement with the one deduced from global pressure drop measurements.

Pressure drag reduction via imposition of spanwise wall oscillations on a rough wall

The present study tests the efficacy of the well-known viscous drag reduction strategy of imposing spanwise wall oscillations to reduce pressure drag contributions in transitional and fully rough turbulent wall flow. This is achieved by conducting a series of direct numerical simulations of a turbulent flow over two-dimensional (spanwise-aligned) semi-cylindrical rods, placed periodically along the streamwise direction with varying streamwise spacing. Surface oscillations, imposed at fixed viscous-scaled actuation parameters optimum for smooth wall drag reduction, are found to yield substantial drag reduction ( $\gtrsim$ 25 %) for all the rough wall cases, maintained at matched roughness Reynolds numbers. While the total drag reduction is due to a drop in both viscous and pressure drag in the case of transitionally rough flow (i.e. with large inter-rod spacing), it is associated solely with pressure drag reduction for the fully rough cases (i.e. with small inter-rod spacing), with the latter being reported for the first time. The study finds that pressure drag reduction in all cases is caused by the attenuation of the vortex shedding activity in the roughness wake, in response to wall oscillation frequencies that are of the same order as the vortex shedding frequencies. Contrary to speculations in the literature, this study confirms that the mechanism behind pressure drag reduction, achieved via imposition of spanwise oscillations, is independent of the viscous drag reduction. This mechanism is responsible for weakening of the Reynolds stresses and increase in base pressure in the roughness wake, explaining the pressure drag reduction observed by past studies, across varying roughness heights and geometries.

Modeling Unsteady Surface Pressure Autospectra for Turbulent Boundary-Layer Flow over Small Dense Roughness

This work concentrates on analyzing and modeling the unsteady surface pressure autospectra beneath zero-pressure gradient, turbulent boundary-layer flowfields with variably rough flow surfaces. These surface conditions are qualitatively described as being for small, dense, rigid, spatially homogeneous roughness topologies, along with corresponding quantitative parameters. Turbulent boundary-layer velocity and unsteady surface pressure data from a new wind tunnel test with multiple surface conditions are analyzed in conjunction with the compiled results from five, distinct, additional references. These compiled results are utilized to develop and propose a new empirical model for the unsteady surface pressure autospectra for a range of variably rough surfaces. This new modeling framework is approached as an extension of the Goody model, which pertains to smooth-wall flows. The variable character of the newly proposed model is controlled by parameters based on viscous scaling of the effective roughness height and friction velocity such that the surface pressure model inputs are based on readily available mean properties of the turbulent boundary-layer flowfield. For the data utilized in this analysis the approximate values of viscous scaled effective roughness height range between approximately 100 and 2000. Finally, this newly proposed model is used to compare to the unsteady surface pressure data of two additional and separate experimental references for rough-wall flows, yielding further positive comparisons as well as isolated examples of the sensitivity of analysis and modeling to the determined friction velocity for a flow configuration.

Outer-layer similarity and energy transfer in a rough-wall turbulent channel flow

Direct numerical simulations (DNSs) are performed to investigate the roughness effects on the statistical properties and the large-scale coherent structures in the turbulent channel flow over three-dimensional sinusoidal rough walls. The outer-layer similarities of mean streamwise velocity and Reynolds stresses are examined by systematically varying the roughness Reynolds number $k^{+}$ and the ratio of the roughness height to the half-channel height $k / \delta$ . The energy transfer mechanism of turbulent motions in the presence of roughness elements with different sizes is explored through spectral analysis of the transport equation of the two-point velocity correlation and the scale-energy path display of the generalized Kolmogorov equation. The results show that, with increasing $k^+$ , the downward shift of the mean streamwise velocity profile in the logarithmic region increases and the peak intensities of turbulent Reynolds stresses decrease. At an intermediate Reynolds number ( $Re_{\tau }= 1080$ ), the length scale and intensity of the large-scale coherent structures increase for a small roughness ( $k^{+}=10$ ), which leads to failure of the outer-layer similarity in rough-wall turbulence, and decrease for a large roughness ( $k^{+}=60$ ), as compared with the smooth-wall case. The existence of the small roughness ( $k^{+}=10$ ) enhances the mechanism of inverse energy cascade from the inner-layer small-scale structures to the outer-layer large-scale structures. Correspondingly, the self-sustaining processes of the outer-layer large-scale coherent structures, including turbulent production, interscale transport, pressure transport and spatial turbulent transport, are all enhanced, whereas the large roughness ( $k^{+}=60$ ) weakens the energy transfer between the inner and outer regions.

An XFEM-based hydro-mechanical model to consider the non-Darcy flow in complex fracture networks

Geotechnical instability and failure are closely related to the hydro-mechanical coupled behavior of fractured rocks, and the existing studies often regard the fracture seepage as linear Darcy flow. Due to the neglect of non-Darcy effect caused by rough rock wall and high flow velocity, the estimated rock deformation and seepage flow can lead to incorrect assessment of geotechnical risks. Hence, this study proposed an efficient and robust hydro-mechanical (HM) coupled model to consider non-Darcy flow in complex fracture networks. The fractured rock mass's mechanical response, including the fracture surface's frictional contact, was calculated using the XFEM solver, and the widely used Forchheimer's law was adopted to describe the non-Darcy flow along the fractures. The fracture segments cut by mechanical element edges are considered one-dimensional flow elements, and the Newton-Raphson iteration method is adopted to solve the Forchheimer equation directly. To consider the influence of fracture deformation on the conductivity of fractures, the Barton-Bandis model was adopted to determine the hydraulic aperture of the fractures when the fracture was closed. Subsequently, a staggering Newton-Raphson method can be developed to decouple the fracture deformation and fluid flow process. Experimental results were adopted to validate the accuracy of the numerical model, and several numerical cases were used to illustrate the efficiency and robustness of the proposed model. The effects of inlet pressure, confining stress, and non-Darcy effect on the HM-coupled behavior of fractured rock were investigated. Numerical results suggested that the distribution of fluid pressure and failure process of rock mass based on Darcy assumption could be dramatically changed after considering the nonlinear seepage behavior.

Towards decoupling the effects of permeability and roughness on turbulent boundary layers

Boundary-layer flow over a realistic porous wall might contain both the effects of wall-permeability and wall-roughness. These two effects are typically examined in the context of a rough-wall flow, i.e. by defining a ‘roughness’ length or equivalent to capture the effect of the surface on momentum deficit/drag. In this work, we examine the hypothesis of Esteban et al. (Phys. Rev. Fluids, vol. 7, no. 9, 2022, 094603), that a turbulent boundary layer over a porous wall could be modelled as a superposition of the roughness effects on the permeability effects by using independently obtained information on permeability and roughness. We carry out wind tunnel experiments at high Reynolds number ( $14\ 400 \leq Re_{\tau } \leq 33\ 100$ ) on various combinations of porous walls where different roughnesses are overlaid over a given permeable wall. Measurements are also conducted on the permeable wall as well as the rough walls independently to obtain the corresponding length scales. Analysis of mean flow data across all these measurements suggests that an empirical formulation can be obtained where the momentum deficit ( $\Delta U^+$ ) is modelled as a combination of independently obtained roughness and permeability length scales. This formulation assumes the presence of outer-layer similarity across these different surfaces, which is shown to be valid at high Reynolds numbers. Finally, this decoupling approach is equivalent to the area-weighted power-mean of the respective permeability and roughness length scales, consistent with the approach recently suggested by Hutchins et al. (Ocean Engng, vol. 271, 2023, 113454) to capture the effects of heterogeneous rough surfaces.

History effects and wall-similarity of non-equilibrium turbulent boundary layers in varying pressure gradient over rough and smooth surfaces

Experimental investigations were conducted on non-equilibrium, turbulent boundary layers over rough and smooth surfaces. A NACA 0012 airfoil was systematically rotated to generate a family of bi-directional pressure fields on the research surface. The rough surface consisted of 2mm tall, staggered, circular cylindrical elements producing flow conditions that were in the fully rough regime. The velocity fields were measured using a custom Pitot-probe boundary layer rake and time-resolved particle image velocimetry (TR-PIV). The primary effect of the roughness was to increase the magnitude of the outer layer and skin friction parameters. Flows over both wall conditions exhibited “history effects” in the integrated boundary layer parameters, which indicates characteristic dependencies on the streamwise flow evolution. Indirect skin friction methods for non-equilibrium rough wall flows were sensitive to the regression fits of the empirical data, which were used in the absence of direct measurements. The effective sandgrain roughness parameter, ks, remained independent of pressure gradient flow history, within the uncertainty of empirical methods used for skin friction determination. Varying levels of wall-similarity were observed in the smooth and rough wall Reynolds stress profiles. Although past evaluations of wall-similarity have suggested that a sufficiently large boundary layer thickness to ks is sufficient for wall-similarity, the present results indicate a need for additional multivariable considerations including matching flow histories and considering local Reynolds numbers measurement resolution.

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.

Supersonic turbulent flows over sinusoidal rough walls

Direct numerical simulations were performed to characterize fully developed supersonic turbulent channel flows over isothermal rough walls. The effect of roughness was incorporated using a level-set/volume-of-fluid immersed boundary method. Turbulence statistics of five channel flows are compared, including one reference case with both walls smooth and four cases with smooth top walls and bottom walls with two-dimensional (2-D) and three-dimensional (3-D) sinusoidal roughnesses. Results reveal a strong dependence of the turbulence on the roughness topography and the associated shock patterns. Specifically, the 2-D geometries generate strong oblique shock waves that propagate across the channel and are reflected back to the rough-wall side. These strong shocks are absent in the smooth-wall channel and are significantly weaker in cases with 3-D roughness geometries, replaced by weak shocklets. At the impingement locations of the shocks on the top wall in the 2-D roughness cases, localized augmentations of turbulence shear production are observed. Such regions of augmented production also exist for the 3-D cases, at a much weaker level. The oblique shock waves are thought to be responsible for a more significant entropy generation for cases with 2-D surfaces than those with 3-D ones, leading to a higher irreversible heat generation and consequently higher temperature values in 2-D roughness cases. In the present supersonic channels, the effects of roughness extend beyond the near-wall layer due to the shocks. This suggests that outer layer similarity may not fully apply to a rough-wall supersonic turbulent flow.

Impact of spanwise effective slope upon rough-wall turbulent channel flow

Whereas streamwise effective slope ( $ES_{x}$ ) is accepted as a key topographical parameter in the context of rough-wall turbulent flows, the significance of its spanwise counterpart ( $ES_{y}$ ) remains largely unexplored. Here, the response of turbulent channel flow over irregular, three-dimensional rough walls with systematically varied values of $ES_{y}$ is studied using direct numerical simulation. All simulations were performed at a fixed friction Reynolds number 395, corresponding to a viscous-scaled roughness height $k^{+}\approx 65.8$ (where $k$ is the mean peak-to-valley height). A surface generation algorithm is used to synthesise a set of ten irregular surfaces with specified $ES_{y}$ for three different values of $ES_{x}$ . All surfaces share a common mean peak-to-valley height and are near-Gaussian, which allows this study to focus on the impact of varying $ES_{y}$ , since roughness amplitude, skewness and $ES_{x}$ can be eliminated simultaneously as parameters. Based on an analysis of first- and second-order velocity statistics, as well as turbulence co-spectra and the fractional contribution of pressure and viscous drag, the study shows that $ES_{y}$ can strongly affect the roughness drag penalty – particularly for low- $ES_{x}$ surfaces. A secondary observation is that particular low- $ES_{y}$ surfaces in this study can lead to diminished levels of outer-layer similarity in both mean flow and turbulence statistics, which is attributed to insufficient scale separation between the outer length scale and the in-plane spanwise roughness wavelength.

Effects of form-induced velocity in rough-wall turbulent channel flows

Wall roughness induces form-induced (or dispersive) velocity and pressure perturbations inside the roughness sublayer of a wall-bounded turbulent flow. This work discusses the role played by the form-induced velocity in influencing turbulence statistics and structure, using existing direct numerical simulation data of transient half channels in response to an impulse acceleration (Mangavelli et al. Effects of surface roughness topography in transient channel flows. J Turbul 2021;22:434–460). Focuses are given to (i) reshaping of turbulent coherent motions by the rate-of-strain of the mean velocity, and (ii) contributions of different velocity sources to turbulent pressure fluctuations. Half-channel flows in both fully-developed and non-equilibrium, transient states are discussed. Results show that form-induced velocity gradients not only form an important source of turbulent pressure in an equilibrium flow, but also lead to turbulence production and potentially direct structural change of turbulent eddies in a non-equilibrium flow under acceleration.

Contributions to pressure drag in rough-wall turbulent flows: Insights from force partitioning

The Force Partitioning Method is employed to decompose the hydrodynamic drag in rough-wall turbulent channel flows. The contributions of vortex and strain dominated regions on the pressure drag are quantified using an auxiliary surface-dependent potential field $\ensuremath{\phi}$. Different sources of drag are identified, and their relative importance quantified. The \ensuremath{\mathit{Q}}-induced force (where \ensuremath{\mathit{Q}} is the second invariant of the velocity gradient tensor) is responsible for about 50% of the rough-wall drag, and is mainly generated by the strain-dominated (\ensuremath{\mathit{Q}} 0) regions before each roughness element. The equivalent sand-grain height ${k}_{s}$ is also characterized using $\ensuremath{\phi}\ensuremath{-}$dependent norms.

Direct numerical simulation of supersonic turbulent flows over rough surfaces

We perform direct numerical simulation of supersonic turbulent channel flow over cubical roughness elements, spanning bulk Mach numbers $M_b=0.3$ – $4$ , both in the transitional and fully rough regime. We propose a novel definition of roughness Reynolds number which is able to account for the viscosity variations at the roughness crest and should be used to compare rough-wall flows across different Mach numbers. As in the incompressible flow regime, the mean velocity profile shows a downward shift with respect to the baseline smooth wall cases, however, the magnitude of this velocity deficit is largely affected by the Mach number. Compressibility transformations are able to account for this effect, and data show a very good agreement with the incompressible fully rough asymptote, when the relevant roughness Reynolds number is used. Velocity statistics present outer layer similarity with the equivalent smooth wall cases, however, this does not hold for the thermal field, which is substantially affected by the roughness, even in the channel core. We show that this is a direct consequence of the quadratic temperature–velocity relation which is also valid for rough walls. Analysis of the heat transfer shows that the relative drag increase is always larger than the relative heat transfer enhancement, however, increasing the Mach number brings data closer to the Reynolds analogy line due to the rising relevance of the aerodynamic heating.

Outer turbulent boundary layer similarities for different 2D surface roughnesses at matched Reynolds number

The outer layer similarity in zero pressure gradient (ZPG) boundary layers developing over different geometries of 2D roughness elements is assessed, using single hotwire anemometry. Two types of 2D roughness are used: circular rods and sinewave surfaces with two different heights, k = 1.6, and 2.4 mm, and three different streamwise spacings, i.e. sx=8k,12k, and 16k. These roughnesses cover a range of ratios of the boundary layer thickness (δ) to the roughness height (k) from δ/k = 21 to 45. As expected, all roughnesses caused a downward shift on the wall-unit normalised streamwise mean velocity profiles compared with smooth wall profiles, with a maximum shift observed for rods with a spacing of sx=8k, while the minimum shift is noticed for a sinewave surface with a spacing of sx=16k. The defect velocity profiles collapse entirely for all smooth and rough wall flows when normalised by the friction velocity. It was found that the shape factor, H is a suitable scaling parameter for improving the collapse of the data when using the diagnostic plot. The inner peak of the turbulence intensity profiles for the sinewave roughness is reduced gradually with increasing Reynolds number while the turbulent boundary layer (TBL) develops from a transitionally to a fully rough regime. Meanwhile, this inner peak disappears completely for all rod roughnesses, as the TBL is always in a fully rough regime, and the profiles exhibit only an outer peak, located at a wall-normal location y/δ≈0.06. The results suggest that Townsends similarity hypothesis for 2D surface roughness is relatively well approximated in the outer region of the flow as reflected by the collapse of the distributions of velocity defect, turbulence intensity, skewness, and flatness when scaled with δ.

Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region

We study the global, i.e., radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I2=u2¯/U2, where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark et al., “Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow,” J. Fluid Mech. 728, 376–395 (2013)]. Scaling expressions are derived both for log- and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Reτ∼11 000.

A review on turbulent flow over rough surfaces: Fundamentals and theories

• Townsend hypothesis can explain large-scale motions near rough and smooth walls. • Is unable to describe all roughness regimes and does not scale d-type roughness. • Wall similarity hypothesis faces both supporting and conflicting evidence. • Roughness function can be scaled with height and effective slope of roughness. • Available models do not point to agreed conclusions or trends for all roughness types, Surface roughness can significantly influence the fluid dynamics and heat transfer in convective flows by inducing perturbations in the velocity profile which affect surface drag, turbulent mixing and heat transfer. While surface roughness can often negatively affect the performance of systems, it can also lead to performance improvements, such as in convective flows where roughness elements have been shown to enhance heat transfer. Turbulent flows over rough surfaces have been studied for about a century leading to significant developments in this field. Direct Numerical Simulation (DNS) has made significant contributions to the knowledge of turbulent flows over rough surfaces as well as evaluation of the developed theories. Moreover, the turbulent closures model has seen wide use for simulation of rough-wall turbulent flows in practical applications where DNS is hindered by its complexity and computational resources. Despite a significant number of experimental and CFD studies and the latest advances in this field, a recent review was not available. Therefore, this review surveys the past and recent experimental and numerical studies to address the fundamentals and theories related to the structure of turbulent flows over rough walls. This study chiefly investigates the structure of mean flow profile over rough surfaces, and its correlation with smooth wall mean flow profile. This review study can contribute to prospective experimental and CFD work, and for characterising rough-wall turbulent flows and heat transfer in different academic and engineering applications such as aerodynamics, hydraulics, meteorology, and manufacturing. The review concludes that despite significant progress, the structure of turbulent flow is still not fully understood. This is mainly due to a lack of systematic studies on the structure of turbulent flow and also due to the variety of roughness which influence the dynamics of the flow in the roughness sublayers. The current roughness scale (sand-grain roughness height) fails to completely characterise roughness in many cases. Therefore, there is a need for a universal roughness scale that can describe every type of roughness and be used in any rough-flow regimes, including fully rough and transitionally rough regimes.

Data-driven prediction of the equivalent sand-grain height in rough-wall turbulent flows

Abstract

Asymptotic turbulent friction in 2D rough-walled flows

The friction f is the property of wall-bounded flows that sets the pumping cost of a pipeline, the draining capacity of a river, and other variables of practical relevance. For highly turbulent rough-walled pipe flows, f depends solely on the roughness length scale r, and the f - r relation may be expressed by the Strickler empirical scaling f ∝ r 1/3 Here, we show experimentally that for soap film flows that are the two-dimensional (2D) equivalent of highly turbulent rough-walled pipe flows, f ∝ r and the f - r relation is not the same in 2D as in 3D. Our findings are beyond the purview of the standard theory of friction but consistent with a competing theory in which f is linked to the turbulent spectrum via the spectral exponent α: In 3D, α = 5/3 and the theory yields f ∝ r 1/3; in 2D, α = 3 and the theory yields f ∝ r.

Constrained large-eddy simulation of turbulent flow over inhomogeneous rough surfaces

In this work we extend the method of the constrained large-eddy simulation (CLES) to simulate the turbulent flow over inhomogeneous rough walls. In the original concept of CLES, the subgrid-scale (SGS) stress is constrained so that the mean part and the fluctuation part of the SGS stress can be modelled separately to improve the accuracy of the simulation result. Here in the simulation of the rough-wall flows, we propose to interpret the extra stress terms in the CLES formulation as the roughness-induced stress so that the roughness inhomogeneity can be incorporated by modifying the formulation of the constrained SGS stress. This is examined with the simulations of the channel flow with the spanwise alternating high/low roughness strips. Then the CLES method is employed to investigate the temporal response of the turbulence to the change of the wall condition from rough to smooth. We demonstrate that the temporal development of the internal boundary layer is just similar to that in a spatial rough-to-smooth transition process, and the spanwise roughness inhomogeneity has little impact on the transition process.