Reynolds Stress Gradient Research Articles

Modelling three-dimensional flow over spur-and-groove morphology

Spur-and-groove (SAG) morphology characterizes the fore reef of many coral reefs worldwide. Although the existence and geometrical properties of SAG have been well documented, an understanding of the hydrodynamics over them is limited. Here, the three-dimensional flow patterns over SAG formations, and a sensitivity of those patterns to waves, currents, and SAG geometry were characterized using the physics-based Delft3D-FLOW and SWAN models. Shore-normal shoaling waves over SAG formations were shown to drive two circulation cells: a cell on the lower fore reef with offshore flow over the spurs and onshore flow over the grooves, except near the seabed where velocities were always onshore, and a cell on the upper fore reef with offshore surface velocities and onshore bottom currents, which result in depth-averaged onshore and offshore flow over the spurs and grooves, respectively. The mechanism driving this flow results from the net of the radiation stress gradients and pressure gradient, which is balanced by the Reynolds stress gradients and bottom friction that differ over the spur and over the groove. Waves were the primary driver of variations in modelled flow over SAG, with the flow strength increasing for increasing wave heights and periods. Spur height, SAG wavelength, and the water depth at peak spur height were the dominant influences on the hydrodynamics, with spur heights directly proportional to the strength of SAG circulation cells. SAG formations with shorter SAG wavelengths only presented one circulation cell on the shallower portion of the reef, as opposed to the two circulation cells for longer SAG wavelengths. SAG formations with peak spur heights occurring in shallower water had stronger circulation than those with peak spur heights occurring in deeper water. These hydrodynamic patterns also likely affect coral and reef development through sediment and nutrient fluxes.

Wave‐Driven Mean Flow Dynamics in Submerged Canopies

AbstractThe physical roughness (canopies) formed by organisms within aquatic ecosystems (e.g., seagrass, kelp, and mangroves) modifies the local wave‐driven hydrodynamics within coastal and estuarine regions. In wave‐dominated environments, an understanding of the mean wave‐driven flows generated within and above canopies is important, as it governs material transport (e.g., of nutrients, sediment, and biota). However, until recently the effect of submerged canopies on wave‐current interactions and the resulting mean (wave‐averaged) flow dynamics has received relatively little attention. In this study, a combination of wave flume experiments and numerical modeling is used to investigate the wave‐induced mean flow profiles in the presence of a submerged canopy. The measured velocities and vegetation forces were used to derive bulk drag and inertia coefficients, and to validate a nonhydrostatic 2DV wave‐flow model. The numerical model results were used to conduct an in‐depth analysis of the mean horizontal momentum terms responsible for driving the mean (horizontal) flow within and above the submerged canopies. We show that the mean canopy hydrodynamics are driven by vertical gradients in wave and turbulent Reynolds stresses, balanced by the mean canopy drag forces. The wave Reynolds stress gradient is the dominant force driving the in‐canopy mean flow and is directly related to the vorticity that is generated when the wave orbital motions become rotational near the canopy interface. This study provides new insight in the mechanisms responsible for wave‐driven mean flows within submerged canopies and guidance for how these hydrodynamics can be predicted in coastal wave‐circulation models.

Lagrangian transport equations and an iterative solution method for turbulent jet flows

Turbulent jet flows exhibit Kelvin–Helmholtz type of instabilities at its onset. Using a Galilean transform, the Navier–Stokes equation leads to an explicit, solvable expression for the Reynolds stress gradient, which is verified through comparison with DNS (direct numerical simulation) and experimental data for canonical flows. The Reynolds stress budget shows that the momentum balance of u’2, pressure, and viscous forces forms a triad of forces to generate the Reynolds stress. For jet flows, an additional Lagrangian transport equation for u’2, also confirmed using data, constitutes a closure method, which is demonstrated and compared with experimental data and turbulence models. Similar approach is being tested in other turbulent flows.

On the interplay between turbulent forces and neoclassical particle losses in zonal flow dynamics

This study presents an investigation into the connection between radial electric fields, Reynolds stress gradients and long range correlations (LRCs), with the latter being a proxy for zonal flows (ZFs), in different plasma scenarios in the TJ-II stellarator. Monte Carlo simulations showed that radial electric fields with a similar size to those experimentally measured affect neoclassical orbit losses. Although the order of magnitude of turbulent acceleration was comparable to the neoclassical damping of perpendicular flows, we found that the dependence of this acceleration on the radial electric field did not correlate with the evolution of the LRC amplitude, indicating that turbulent acceleration alone cannot explain the dynamics of ZFs. These results are in line with the expectation that the interplay between turbulent and neoclassical mechanisms is an important ingredient of the dynamics of edge ZF.

Maximum Entropy Method for Solving the Turbulent Channel Flow Problem.

There are two components in this work that allow for solutions of the turbulent channel flow problem: One is the Galilean-transformed Navier-Stokes equation which gives a theoretical expression for the Reynolds stress (u′v′); and the second the maximum entropy principle which provides the spatial distribution of turbulent kinetic energy. The first concept transforms the momentum balance for a control volume moving at the local mean velocity, breaking the momentum exchange down to its basic components, u′v′, u′2, pressure and viscous forces. The Reynolds stress gradient budget confirms this alternative interpretation of the turbulence momentum balance, as validated with DNS data. The second concept of maximum entropy principle states that turbulent kinetic energy in fully-developed flows will distribute itself until the maximum entropy is attained while conforming to the physical constraints. By equating the maximum entropy state with maximum allowable (viscous) dissipation at a given Reynolds number, along with other constraints, we arrive at function forms (inner and outer) for the turbulent kinetic energy. This allows us to compute the Reynolds stress, then integrate it to obtain the velocity profiles in channel flows. The results agree well with direct numerical simulation (DNS) data at Reτ = 400 and 1000.

Flow adjustment inside homogeneous canopies after a leading edge – An analytical approach backed by LES

A two-dimensional analytical model for describing the mean flow behavior inside a vegetation canopy after a leading edge in neutral conditions was developed and tested by means of large eddy simulations (LES) employing the LES code PALM. The analytical model is developed for the region directly after the canopy edge, the adjustment region, where one-dimensional canopy models fail due to the sharp change in roughness. The derivation of this adjustment region model is based on an analytic solution of the two-dimensional Reynolds averaged Navier–Stokes equation in neutral conditions for a canopy with constant plant area density (PAD). The main assumptions for solving the governing equations are separability of the velocity components concerning the spatial variables and the neglection of the Reynolds stress gradients. These two assumptions are verified by means of LES. To determine the emerging model parameters, a simultaneous fitting scheme was applied to the velocity and pressure data of a reference LES simulation. Furthermore a sensitivity analysis of the adjustment region model, equipped with the previously calculated parameters, was performed varying the three relevant length, the canopy height (h), the canopy length and the adjustment length (Lc), in additional LES. Even if the model parameters are, in general, functions of h/Lc, it was found out that the model is capable of predicting the flow quantities in various cases, when using constant parameters. Subsequently the adjustment region model is combined with the one-dimensional model of Massman [Bound. Layer Meteorol., 83(3):407–421, 1997], which is applicable for the interior of the canopy, to attain an analytical model capable of describing the mean flow for the full canopy domain. Finally the model is tested against an analytical model based on a linearization approach.

Bifurcation and hysteresis of plasma edge transport in a flux-driven system.

Transition dynamics and mean shear flow generation in plasma interchange turbulence are explored in a flux-driven system that resembles the plasma edge region. The nonlinear evolution of the interchange mode shows two confinement regimes with different transport levels. Large amplitude oscillations in the phase space of turbulence intensity and mean flow energy are observed and investigated. Both clockwise and counterclockwise oscillations occur during the transition between the two regimes. The Reynolds stress gradients are shown to play a critical role in the generation of mean sheared flows in the edge region. Both the forward and back transitions are simulated self-consistently and a significant hysteresis is found.

Generation of a mean flow by an internal wave

We experimentally study the generation of a mean flow by a two-dimensional progressive internal gravity wave. Due to the viscous damping of the wave, a non-vanishing Reynolds stress gradient forces a mean flow. When the forcing amplitude is low, the wave amplitude is proportional to the forcing and the mean flow is quadratic in the forcing. When the forcing amplitude is large, the mean flow decreases the wave amplitude. This feedback saturates both the wave and the mean flow. The profiles of the mean flow and the wave are compared with a one-dimensional analytical model. Decreasing the forcing frequency leads to a wave and a mean flow localized on a smaller height, in agreement with the model.

Vertical variations of coral reef drag forces

AbstractModeling flow in a coral reef requires a closure model that links the local drag force to the local mean velocity. However, the spatial flow variations make it difficult to predict the distribution of the local drag. Here we report on vertical profiles of measured drag and velocity in a laboratory reef that was made of 81 Pocillopora Meandrina colony skeletons, densely arranged along a tilted flume. Two corals were CT‐scanned, sliced horizontally, and printed using a 3‐D printer. Drag was measured as a function of height above the bottom by connecting the slices to drag sensors. Profiles of velocity were measured in‐between the coral branches and above the reef. Measured drag of whole colonies shows an excellent agreement with previous field and laboratory studies; however, these studies never showed how drag varies vertically. The vertical distribution of drag is reported as a function of flow rate and water level. When the water level is the same as the reef height, Reynolds stresses are negligible and the drag force per unit fluid mass is nearly constant. However, when the water depth is larger, Reynolds stress gradients become significant and drag increases with height. An excellent agreement was found between the drag calculated by a momentum budget and the measured drag of the individual printed slices. Finally, we propose a modified formulation of the drag coefficient that includes the normal dispersive stress term and results in reduced variations of the drag coefficient at the cost of introducing an additional coefficient.

Analytical and numerical study of the transverse Kelvin–Helmholtz instability in tokamak edge plasmas

Sheared flows perpendicular to the magnetic field can be driven by the Reynolds stress or ion pressure gradient effects and can potentially influence the stability and turbulent saturation level of edge plasma modes. On the other hand, such flows are subject to the transverse Kelvin–Helmholtz (KH) instability. Here, the linear theory of KH instabilities is first addressed with an analytic model in the asymptotic limit of long wavelengths compared with the flow scale length. The analytic model treats sheared $\boldsymbol{E}\times \boldsymbol{B}$ flows, ion diamagnetism (including gyro-viscous terms), density gradients and parallel currents in a slab geometry, enabling a unified summary that encompasses and extends previous results. In particular, while ion diamagnetism, density gradients and parallel currents each individually reduce KH growth rates, the combined effect of density and ion pressure gradients is more complicated and partially counteracting. Secondly, the important role of realistic toroidal geometry is explored numerically using an invariant scaling analysis together with the 2DX eigenvalue code to examine KH modes in both closed and open field line regions. For a typical spherical torus magnetic geometry, it is found that KH modes are more unstable at, and just outside of, the separatrix as a result of the distribution of magnetic shear. Finally implications for reduced edge turbulence modelling codes are discussed.

Evolutions of limit cycle oscillation in L-I-H transitions on HL-2A

Poloidal symmetric low frequency oscillations at the plasma edge were observed during the low-intermediate-high transitions on the HL-2A tokamak. The limit cycle oscillation (LCO) frequency gradually reduces from 3.0kHz to 1.5kHz approaching the H mode. Meanwhile a transition of phase shift between the turbulence and the radial electric field was identified. In the initial phase, the turbulence grows first, followed by the localized electric field. In the later phase, the electric field leads turbulence, and the magnitude of the Reynolds stress gradient is out of phase with the LCO flow. The increasing magnitudes of the magnetic fluctuation may be correlated with the I-H transition. In addition, the divertor heat flux is also significantly modulated by the LCOs.

Low–intermediate–high confinement transition in HL-2A tokamak plasmas

The dynamics of low–intermediate–high confinement transitions was studied using a four-step Langmuir probe in the HL-2A edge plasma. Two types (dubbed type-Y and type-J) of limit cycle oscillations (LCOs) with opposite temporal ordering between the radial electric field and turbulence were first observed. In type-Y, the turbulence grows first, followed by the localized electric field. In contrast, the electric field leads turbulence in type-J. In addition, the Reynolds stress gradient is found not enough to drive the LCO flow and the three-wave nonlinear coupling is weak there. The continuously increasing amplitude of magnetic fluctuations and the significant correlation between the magnetic fluctuation and the electron pressure gradient indicate an important role of diamagnetic drifts in the L–H transition. Mode numbers of magnetic fluctuations in the LCO frequency are identified to be m/n = 1/0.

Simulation of turbulent flows using an algebraic Reynolds stress model with universal wall functions

The paper explores the application of an Explicit Algebraic Reynolds Stress Model (EARSM) to turbulent flow simulation using universal wall functions. With universal wall functions, friction coefficients and velocity derivatives on a solid wall can be predicted with good accuracy, but the correct application of these functions entails changes in the formulation of turbulence models. Despite the fact that the use of this approach has been well investigated for RANS turbulence models, the question of how to introduce these functions in the EARSM has not been adequately addressed. A detailed consideration shows that in the case of EARSM, application of a coarser near-wall resolution leads to an unsatisfactory accuracy of the Reynolds stress gradient in the areas of rapid change in the velocity field and, as a consequence, to the oscillations of physical quantities in the boundary layer . The present paper proposes a method for eliminating these oscillations modified by calculating the velocity gradients on the inner faces of computational grid cells. A numerical implementation algorithm for the EARSM model which provides stable computation in the interface and acceptable simulation results on arbitrary unstructured grids with various level of grid-spacing near the rigid surface is presented. The effectiveness of the proposed algorithm is illustrated by solving the problem of turbulent flow over a flat plate. The results show that the algorithm is able to eliminate the spatial oscillations in velocity inside the boundary layer at any type of a grid. The analysis of two problems considering the asymmetrical flow indicates that the application of the developed EARSM algorithm gives a noticeable improvement of simulation results compared with the basic RANS model even in the case of the unstructured grid with arbitrary grid-spacing near the rigid surface.

Mixing layer in open-channel junction flows

When two open-channel flows merge in a three-branch subcritical junction, a mixing layer appears at the interface between the two inflows. If the width of the downstream channel is equal to the width of each inlet channel, this mixing layer is accelerated and is curved due to the junction geometry. The present work is dedicated to simplified geometries, considering a flat bed and a \(90^{\circ }\) angle where two configurations with different momentum ratios are tested. Due to the complex flow pattern in the junction, the so-called Serret–Frenet frame-axis based on the local direction of the velocity must be employed to characterize the flow pattern and the mixing layer as Cartesian and cylindrical frame-axes are not adapted. The analysis reveals that the centerline of the mixing layer, defined as the location of maximum Reynolds stress and velocity gradient, fairly fits the streamline separating at the upstream corner, even though a slight shift of the mixing layer towards the center of curvature is observed. The shape of the mixing layer appears to be strongly affected by the streamwise acceleration and the complex lateral confinement due to the side walls and the corners of the junction, leading to a streamwise increase of the mean velocity along the centerline and a decrease of the velocity difference. This results in a specific streamwise evolution of the mixing layer width, which reaches a plateau in the downstream region of the junction. Finally, the evaluation of the terms in the Reynolds-Averaged-Navier–Stokes equations reveals that the streamwise and normal acceleration and the pressure gradient remain dominant, which is typical of accelerated and rotational flows.

A new nonlinear analytical model for canopy flow over a forested hill

A new nonlinear analytical model for canopy flow over gentle hills is presented. This model is established based on the assumption that three major forces (pressure gradient, Reynolds stress gradient, and nonlinear canopy drag) within canopy are in balance for gentle hills under neutral conditions. The momentum governing equation is closed by the velocity-squared law. This new model has many advantages over the model developed by Finnigan and Belcher (Quart J Roy Meteorol Soc 130: 1–29 2004, hereafter referred to as FB04) in predicting canopy wind velocity profiles in forested hills in that: (1) predictions from the new model are more realistic because surface drag effects can be taken into account by boundary conditions, while surface drag effects cannot be accounted for in the algebraic equation used in the lower canopy layer in the FB04 model; (2) the mixing length theory is not necessarily used because it leads to a theoretical inconsistency that a constant mixing length assumption leads to a nonconstant mixing length prediction as in the FB04 model; and (3) the effects of height-dependent leaf area density (a(z)) and drag coefficient (Cd) on wind velocity can be predicted, while both a(z) and Cd must be treated as constants in FB04 model. The nonlinear algebraic equation for momentum transfer in the lower part of canopy used in FB04 model is height independent, actually serving as a bottom boundary condition for the linear differential momentum equation in the upper canopy layer. The predicting ability of the FB04 model is largely restricted by using the height-independent algebraic equation in the bottom canopy layer. This study has demonstrated the success of using the velocity-squared law as a closure scheme for momentum transfer in forested hills in comparison with the mixing length theory used in FB04 model thus enhancing the predicting ability of canopy flows, keeping the theory consistent and simple, and shining a new light into land-surface parameterization schemes in numerical weather and climate models.

Eddy Diffusivities for Momentum and Heat in the Upper Troposphere and Lower Stratosphere Measured by MU Radar and RASS, and a Comparison of Turbulence Model Predictions

Abstract Recently, middle- and upper-atmosphere Doppler radar (MU radar) has enabled the measurement of middle-atmosphere turbulence from radar backscatter Doppler spectra. In this work, eddy diffusivities for momentum Km in the upper troposphere and lower stratosphere during clear-air conditions were derived from direct measurements of the Reynolds stress and vertical gradient of mean wind velocity measured by MU radar. Eddy diffusivity for heat Kh below 8 km was determined from measurements of temperature fluctuations by the Radio Acoustic Sounding System (RASS) attached to the MU radar. The eddy diffusivity for momentum was on the order of 10 m2 s−1 in the upper troposphere and decreased gradually in the stratosphere by an order of magnitude or more. The eddy diffusivity for heat was almost of the same order of magnitude as Km. Estimates of eddy diffusivity from the radar echo power spectral width give fairly good values compared with the direct measurement of Km. Applicability of three turbulence models—the spectral width method, the k–ɛ model modified for stratified flows, and the algebraic stress model—were also examined, using radar observation values of turbulent kinetic energy k and turbulent energy dissipation rate ɛ together with atmospheric stability observations from rawinsonde data. It is concluded that the algebraic stress model shows the best fit with the direct measurement of Km, even in the free atmosphere above the atmospheric boundary layer once k and ɛ values are obtained from observations or a model.

Mean dynamics of transitional boundary-layer flow

The dynamical mechanisms underlying the redistribution of mean momentum and vorticity are explored for transitional two-dimensional boundary-layer flow at nominally zero pressure gradient. The analyses primarily employ the direct numerical simulation database of Wu & Moin (J. Fluid Mech., vol. 630, 2009, p. 5), but are supplemented with verifications utilizing subsequent similar simulations. The transitional regime is taken to include both an instability stage, which effectively generates a finite Reynolds stress profile, −ρuv(y), and a nonlinear development stage, which progresses until the terms in the mean momentum equation attain the magnitude ordering of the four-layer structure revealed by Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303). Self-consistently applied criteria reveal that the third layer of this structure forms first, followed by layers IV and then II and I. For the present flows, the four-layer structure is estimated to be first realized at a momentum thickness Reynolds number Rθ = U∞ θ/ν ≃ 780. The first-principles-based theory of Fife et al. (J. Disc. Cont. Dyn. Syst. A, vol. 24, 2009, p. 781) is used to describe the mean dynamics in the laminar, transitional and four-layer regimes. As in channel flow, the transitional regime is marked by a non-negligible influence of all three terms in the mean momentum equation at essentially all positions in the boundary layer. During the transitional regime, the action of the Reynolds stress gradient rearranges the mean viscous force and mean advection profiles. This culminates with the segregation of forces characteristic of the four-layer regime. Empirical and theoretical evidence suggests that the formation of the four-layer structure also underlies the emergence of the mean dynamical properties characteristic of the high-Reynolds-number flow. These pertain to why and where the mean velocity profile increasingly exhibits logarithmic behaviour, and how and why the Reynolds stress distribution develops such that the inner normalized position of its peak value, ym+, exhibits a Reynolds number dependence according to $y_m^+ {\,\simeq\,} 1.9 \sqrt{\delta^+}$.

Mean dynamics of transitional channel flow

The redistribution of mean momentum and vorticity, along with the mechanisms underlying these redistribution processes, is explored for post-laminar flow in fully developed, pressure driven, channel flow. These flows, generically referred to as transitional, include an instability stage and a nonlinear development stage. The central focus is on the nonlinear development stage. The present analyses use existing direct numerical simulation data sets, as well as recently reported high-resolution molecular tagging velocimetry measurements. Primary considerations stem from the emergence of the effects of turbulent inertia as represented by the Reynolds stress gradient in the mean differential statement of dynamics. The results describe the flow evolution following the formation of a non-zero Reynolds stress peak that is known to first arise near the critical layer of the most unstable disturbance. The positive and negative peaks in the Reynolds stress gradient profile are observed to undergo a relative movement toward both the wall and centreline for subsequent increases in Reynolds number. The Reynolds stress profiles are shown to almost immediately exhibit the same sequence of curvatures that exists in the fully turbulent regime. In the transitional regime, the outer inflection point in this profile physically indicates a localized zone within which the mean dynamics are dominated by inertia. These observations connect to recent theoretical findings for the fully turbulent regime, e.g. as described by Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) and Klewicki, Fife & Wei (J. Fluid Mech., vol. 638, 2009, p. 73). In accord with momentum equation analyses at higher Reynolds number, the present observations provide evidence that a logarithmic mean velocity profile is most rapidly approximated on a sub-domain located between the zero in the Reynolds stress gradient (maximum in the Reynolds stress) and the outer region location of the maximal Reynolds stress gradient (inflection point in the Reynolds stress profile). Overall, the present findings provide evidence that the dynamical processes during the post-laminar regime and those operative in the high Reynolds number regime are connected and describable within a single theoretical framework.

Distorted turbulence and secondary flow near right-angled plates

In many turbulent flows near obstacles or in ducts, the turbulence is inhomogeneous in two directions perpendicular to the main flow direction. In convective flows, there may initially be no mean motion. In both types of flow the gradients of Reynolds stresses drive mean motions in directions of inhomogeneity. Using the method of rapid distortion theory developed by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, p. 209), we analyse these gradients where homogeneous isotropic turbulence is impinging onto two semi-infinite flat rigid surfaces intersecting at right angles. The mean velocity is assumed to be uniform (i.e. the surfaces move at free-stream velocity, or the boundary layers are very thin). The inhomogeneous spectra, variances and Reynolds-stress gradients are evaluated. For isotropic free-stream turbulence with mean square velocity u2∞1, the mean square velocity fluctuation at a high Reynolds number in the corner is u21(X, 0, 0) = 2.121u2∞1, independent of the form of the spectrum. This is explained by estimating how the free-stream eddies are blocked by the two walls. The gradients of Reynolds stresses force a mean secondary flow to develop; its direction is into the corner, and its magnitude at time t is of order tu2∞1/L∞, where L∞ is the integral scale. These results are tested in a wind tunnel experiment. A turbulence-generating grid installed at the entrance to the test section generates nearly isotropic, grid-generated turbulence. A corner plate with faces parallel to the mean flow and sharp edges is placed downstream of the grid so that shear-free turbulence impinges onto the corner plate. The turbulent Reynolds number based on $\(L_\infty\), \(Re_L\,{=}\,\sqrt{\overline{u^2_{\infty 1}}}L_\infty/\nu\)$, is 1400 at the leading edge of the plate. A hot-wire anemometry is used to measure instantaneous velocities. The experimental results are consistent with the rapid distortion theory estimates for the variances and the secondary mean motion, which is in the same direction and has the same order of magnitude as Prandtl's analysis of shear-driven secondary flow (of the second kind). We conclude that the blocking mechanism adds to the shear effects and has a significant and sometimes dominant contribution to the crossflows wherever it acts in two non-parallel directions, such as convection in a corner. Consequently, mean transport into corners occurs for most kinds of distorted flow with weak viscous stresses, which has many engineering and environmental implications. There are also implications for the chaotic nature of many confined flows near corners.

Mean momentum balance analysis of rough-wall turbulent boundary layers

Mean momentum balances (MMB) are carried out for zero pressure gradient rough-wall turbulent boundary layer flows. The balance characterizes the mean dynamical mechanisms and reveals dominant terms in the governing equation, which constitutes a necessary step in the derivation of scaling behaviors. The Reynolds stress profiles in rough-wall boundary layers are, however, quite scattered and the uncertainty in the data poses challenges for educing the MMB. The present study employs a method that invokes theoretical constraints to more reliably reveal Reynolds stress gradient behaviors in the presence of data scatter. Properties of the rough-wall mean momentum balances are compared to those of the smooth-wall case. Important qualitative features of the layer structure that exists for the smooth-wall are shown to also exist for rough-wall boundary layers. Specifically, the present analysis reveals the existence of a stress gradient balance layer, and thus the importance of the viscous force term well above the roughness crests. The smooth-wall Reynolds stress peak position scales in proportion with the geometric mean of inner–outer characteristic lengths. Roughness, however, imposes new dynamical length scales and evidence is provided to indicate that the scale separations between the inner length, roughness length, peak Reynolds stress length and outer length are important. The failure of the rough-wall Reynolds stress profiles to merge under smooth-wall meso-scalings clearly reveals the additional richness of the problem. Although more data are required to gain a complete characterization, the present results provide evidence that the combined roughness-Reynolds number problem exhibits significantly greater complexity than captured by the prevalent scheme for characterizing and classifying roughness regimes.