Direct Numerical Simulation Of Channel Flow Research Articles

The Basic k-ϵ Model and a New Model Based on General Statistical Descriptions of Anisotropic Inhomogeneous Turbulence Compared with DNS of Channel Flow at High Reynolds Number

Predictions are presented of mean values of statistical variables of large-scale turbulent flow of the widely used basic k-ϵ model, and of a new model, which is based on general statistical descriptions of turbulence. The predictions are verified against published results of direct numerical simulations (DNSs) of Navier–Stokes equations. The verification concerns turbulent channel flow at shear Reynolds numbers of 950, 2000, and 104. The basic k-ϵ model is largely based on empirical formulations accompanied by calibration constants. This contrasts with the new model, where descriptions of leading statistical quantities are based on the general principles of statistical turbulence at a large Reynolds number and stochastic theory. Predicted values of major output variables such as turbulent viscosity, diffusivity of passive admixture, temperature, and fluid velocities compare well with DNS for the new model. Significant differences are seen for the basic k-ϵ model.

Calibration of a Near-Wall Differential Reynolds Stress Model Using the Updated Direct Numerical Simulation Data and Its Assessment

In the article, a differential Reynolds stress model is recalibrated using turbulent channel flow direct numerical simulation data in the range of friction Reynolds numbers 550–5200. The calibration aims to produce a RANS sublayer model for use within the hybrid RANS/LES framework. The model is designed to capture the average field of a thin near-wall part of a boundary layer as accurately as possible. An a posteriori procedure is employed in which one-dimensional channel flow calculations are performed for all variations of the model coefficients at each stage of the optimization procedure. The coefficients are initialized with their original values and then optimized by minimizing the appropriately chosen norm. An improved representation of the mean velocity profile and peak Reynolds stress values is demonstrated. Both models—baseline and recalibrated—are implemented in an in-house CFD code, and several simulations, including a channel flow, a flat plate boundary layer and a boundary layer separation from a rounded step, are performed. The latter benchmark flow is also simulated in hybrid RANS/LES mode. The updated model is compared to the original one, demonstrating improvements over the baseline model in the cases it was designed for.

A Lagrangian relaxation towards equilibrium wall model for large eddy simulation

A large eddy simulation wall model is developed based on a formal interpretation of quasi-equilibrium that governs the momentum balance integrated in the wall-normal direction. The model substitutes the law-of-the-wall velocity profile for a smooth surface into the wall-normal integrated momentum balance, leading to a Lagrangian relaxation towards equilibrium (LaRTE) transport equation for the friction–velocity vector ${\boldsymbol u}_\tau (x,z,t)$ . This partial differential equation includes a relaxation time scale governing the rate at which the wall stress can respond to imposed fluctuations due to the inertia of the fluid layer from the wall to the wall-model height. A priori tests based on channel flow direct numerical simulation (DNS) data show that the identified relaxation time scale ensures self-consistency with assumed quasi-equilibrium conditions. The new approach enables us to formally distinguish quasi-equilibrium from additional, non-equilibrium contributions to the wall stress. A particular model for non-equilibrium contributions is derived, motivated by laminar Stokes layer dynamics in the viscous sublayer when applying fast-varying pressure gradients. The new wall modelling approach is first tested in standard equilibrium channel flow in order to document various properties of the approach. The model is then applied in large eddy simulation of channel flow with a suddenly applied spanwise pressure gradient (SSPG). The resulting mean wall-stress evolution is compared with DNS with good agreement. At the onset of the SSPG, the laminar Stokes layer develops rapidly while the LaRTE portion of the stress has a delayed response due to its inherent relaxation dynamics. Results also highlight open challenges such as modelling the response of near-wall turbulence occurring above the viscous sublayer and at time scales faster than quasi-equilibrium conditions.

A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

Approach to the 4/3 law for turbulent pipe and channel flows examined through a reformulated scale-by-scale energy budget

In this study, we propose a scale-by-scale (SBS) energy budget equation for flows with homogeneity in at least one direction. This SBS budget represents a modified form of the equation first proposed by Danaila et al. (J. Fluid Mech., vol. 430, 2001, pp. 87–109) for the channel centreline – the primary difference is that, here, we consider the role of pressure along with the errors associated with the isotropic approximations of the interscale divergence and Laplacian of the squared velocity increment. The term encompassing the effects of mean shear is also characterised such that the present analysis can be extended straightforwardly to locations away from the centreline. We show, based on a detailed analysis of previously published channel flow direct numerical simulations and pipe flow experiments near the centreline, how several terms in the present SBS budget equation (including the third-order velocity structure function) behave with increasing Reynolds number. The behaviour of these terms is shown to imply a rate of emergence and subsequent growth of the 4/3 law scale subrange at the channel centreline and pipe axis. The analysis also suggests that the peak magnitude of the third-order velocity structure function occurs at a scale that is fixed in proportion to the Taylor microscale at sufficiently high Reynolds number.

Effect of limited near-wall inlet data on the direct numerical simulation of turbulent channel flow

Direct numerical simulation (DNS) of turbulent flows require a large computational domain and a long simulation time to capture and evolve the large-scale structures and attain a statistically stationary state. In contrast, experimental measurements can relatively easily capture the large-scale structures, but struggle to resolve the dissipative flow scales. This study investigates the spatial extent required for the DNS of a turbulent channel flow to recover turbulent fluctuations and energy when using experimental inlet data which is typically unable to capture fluctuations down to the viscous sub-layer. Synthetic experimental fields from streamwise periodic channel flow DNS at Reτ = 180 are used as an inlet for a channel flow DNS with inlet-outlet boundary conditions. The effect of limited near-wall data at the inlet is examined by removing the near-wall energy and fluctuations in all but the zeroth Fourier mode. The influence of limited near-wall data on the convergence of mean and streamwise fluctuating velocity profiles is less significant when the fluctuations are removed at the inlet up to y+ = 5. However, the spanwise fluctuations are slightly weakened. The spanwise energy spectra suggest that at 1/16 of the domain length (x/h ≍ π/4) the flow scales are recovered. When the fluctuations are removed up to y+ = 17 or greater, recovery of full range of flow scales requires a domain larger than x/h = 4π.

On the accuracy of measurement of turbulent velocity gradient statistics with hot-wire probes

A very high resolution minimal channel flow direct numerical simulation was used to examine, virtually, the ability of various multi-sensor hot-wire probe configurations to measure the statistics of velocity gradient components. Various array and sensor configurations and the spatial resolution of probes with these configurations were studied, building on designs and investigations of various authors. In contrast to our previous studies, which focused on turbulent vorticity, vorticity-velocity correlations, dissipation and production rate, here the measurement accuracy of each component of the velocity vector gradient tensor is analyzed separately. The results of the study show that the virtual experiments compare well with a physical experiment, and that such virtual experiments are a powerful tool to examine the accuracy of velocity gradient measurements. The cross-stream gradients needed to determine the vorticity components can be measured with sufficient accuracy with most of the array and sensor configurations of vorticity probes used so far. A systematic error of some of the gradient measurements can appear due to the array or sensor configurations. None of the examined probe designs can measure, with sufficient accuracy, the streamwise velocity gradients, directly or indirectly, using the continuity equation.

Tracking of large-scale structures in turbulent channel with direct numerical simulation of low Prandtl number passive scalar

Channel flow DNS (Direct Numerical Simulation) at friction Reynolds number 180 and with passive scalars of Prandtl numbers 1 and 0.01 was performed in various computational domains. The “normal” size domain was ∼2300 wall units long and ∼750 wall units wide; size taken from the similar DNS of Moser et al. The “large” computational domain, which is supposed to be sufficient to describe the largest structures of the turbulent flows was 3 times longer and 3 times wider than the “normal” domain. The “very large” domain was 6 times longer and 6 times wider than the “normal” domain. All simulations were performed with the same spatial and temporal resolution. Comparison of the standard and large computational domains shows the velocity field statistics (mean velocity, root-mean-square (RMS) fluctuations, and turbulent Reynolds stresses) that are within 1%-2%. Similar agreement is observed for Pr = 1 temperature fields and can be observed also for the mean temperature profiles at Pr = 0.01. These differences can be attributed to the statistical uncertainties of the DNS. However, second-order moments, i.e., RMS temperature fluctuations of standard and large computational domains at Pr = 0.01 show significant differences of up to 20%. Stronger temperature fluctuations in the “large” and “very large” domains confirm the existence of the large-scale structures. Their influence is more or less invisible in the main velocity field statistics or in the statistics of the temperature fields at Prandtl numbers around 1. However, these structures play visible role in the temperature fluctuations at low Prandtl number, where high temperature diffusivity effectively smears the small-scale structures in the thermal field and enhances the relative contribution of large-scales. These large thermal structures represent some kind of an echo of the large scale velocity structures: the highest temperature-velocity correlations are not observed between the instantaneous temperatures and instantaneous streamwise velocities, but between the instantaneous temperatures and velocities averaged over certain time interval.

Effect of mean pressure gradient on the turbulent wall pressure–velocity correlations

Two-point correlations of the wall fluctuating pressure and turbulent pressure and velocities in the flow field are computed in three dimensions using direct numerical simulation (DNS) databases. The mean pressure gradient in the flow has strong influence on these correlations. The DNS () in a converging/diverging channel with streamwise curvature near the lower wall is used to compute the statistics. The correlations in favourable pressure gradient and adverse pressure gradient are compared with the correlation in flat channel flow DNS (). The results show that these pressure–pressure and pressure–velocity correlations are not only sensitive to the local pressure gradient and wall curvature, but also to the pressure gradient history.

Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

AbstractWe reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

Using direct numerical simulation to analyze and improve hot-wire probe sensor and array configurations for simultaneous measurement of the velocity vector and the velocity gradient tensor

Multi-sensor, hot-wire probes of various configurations have been used for 25 years to simultaneously measure the velocity vector and the velocity gradient tensor in turbulent flows. This is the same period in which direct numerical simulations (DNS) were carried out to investigate these flows. Using the first DNS of a turbulent boundary layer, Moin and Spalart [“Contributions of numerical simulation data bases to the physics, modeling and measurement of turbulence,” NASA Technical Memorandum 100022 (1987)] examined, virtually, the performance of a two-sensor X-array probe with the sensors idealized as points in the numerical grid. Subsequently, several investigators have used DNS for similar studies. In this paper we use a highly resolved minimal channel flow DNS, following Jiménez and Moin [“The minimal flow unit in near-wall turbulence,” J. Fluid Mech. 225, 213 (1991)], to study the performance of an 11-sensor probe. Our previous studies of this type have indicated that, on balance, a probe of the design described here may provide the most accurate measurements of many of the statistics formed from the velocity vector and the velocity gradient tensor (rms and skewness values of the velocity and vorticity components as well as the Reynolds shear stress and the dissipation and production rates). The results of the present study show that, indeed, the sensor and array configurations of a probe of this design are considerably better than previous designs that have been used, and they are likely to give reasonably satisfactory results for such measurements with a real probe in a real bounded flow.

Spatial averaging of velocity measurements in wall-bounded turbulence: single hot-wires

Recent advancements in velocity measurements to understand high Reynolds number (Re) wall turbulence have pushed the boundaries of sensor size required to resolve the smallest scales. We present here a framework for studying the effect of finite sensor size on velocity measurements, and scrutinize in detail the behaviour of single-wire hot-wires. Starting with a general linear filter, expressions for the filtered correlation, spectrum and the corresponding variance are derived. Considering the special case of a box-type filter and a simple model for the two-point correlation, theoretical results are developed, which are favourably compared with the numerical simulation of hot-wires based on the turbulent channel flow direct numerical simulation databases. The results clarify the reason why previous studies found the approximate shape of the spectra not resolved by hot-wires as Gaussian. The length scale based on the correlation over the sensor length is found to be the appropriate length scale for characterizing averaging due to finite sensor size. The efficacy of the linear box filter is established by comparing the numerical simulation of hot-wires with experiments conducted at matched sensor lengths and Re in a channel flow, at least for hot-wire lengths of less than 40 in viscous scaling. Finally, a model of the streamwise two-point correlation is presented, which is employed to estimate the filtering effect on the peak of the streamwise velocity variances for a range of Re, and the model results compare favourably with that obtained from measurements. Even though the theoretical results are compared here in the case of wall turbulence, they are suitable for hot-wire measurements in turbulent flows in general.

Direct Numerical Simulation and Large-Eddy Simulation of Supersonic Channel Flow

Compressible isothermal-wall channel flows are studied with direct numerical simulation and large-eddy simulation tools. Computations are carried out using a high-order, low-dissipation, bandwidth-optimized weighted essentially nonoscillatory numerical scheme to describe the hyperbolic terms of the Navier–Stokes equations. Periodic supersonic channel flow direct numerical simulation (, , , and ) is used to validate the procedure and the numerical scheme; a new subgrid term contribution based on pressure drop is proposed for the driving term required in momentum and energy equations for large-eddy simulation. Coherent structures of the flowfield are analyzed with scatter plots, criterion, and vorticity fields. As expected, the strong Reynolds analogy is not valid for this nonadiabatic flow. Streaks and horseshoe-like structures are highlighted and detailed. The authors propose a scenario for the formation of horseshoe-like structures. With large-eddy simulation tools, a dynamic procedure to evaluate the turbulent Prandtl number is required because results are found more accurate and computations more stable. Wall temperature and pressure impact are also emphasized on the normalized van Driest velocity profile in the logarithmic region. The classical log law is recovered: with , and a constant depending on and . An analog law is also recovered for the normalized temperature with the maximum of the Prandtl number . An a priori study on mesh requirements determination for a large range of pressure levels is realized through highly near-wall resolved large-eddy simulation.

The influence of the arrangements of multi-sensor probe arrays on the accuracy of simultaneously measured velocity and velocity gradient-based statistics in turbulent shear flows

A highly resolved turbulent channel flow direct numerical simulation (DNS) with Res = 200 has been used to investigate the influence of the arrangements of the arrays (array configurations), within the sensing area of a multi- array hot-wire probe on the measurement accuracy of velocity and velocity gradient-based statistics. To eliminate all effects related to the sensor response and array charac- teristics (such as sensor dimensions, overheat ratio, thermal cross talk, number and orientations of the sensors and uniqueness range) so that this study could be focused solely on the effects of the array configurations (positions and separations), a concept of a perfect array was introduced, that is, one that can exactly and simultaneously measure all three velocity components at its center. The velocity component values, measured by these perfect arrays, are simply the DNS values computed at these points. Using these velocity com- ponents, the velocity and velocity gradient-based statistics were calculated assuming a linear velocity variation over the probes' sensing areas. The calculated values are compared to the DNS values for various array arrangements to study the influence of these arrangements on the measurement accu- racy. Typical array configurations that previously have been used for physical probes were tested. It is demonstrated that the array arrangements strongly influence the accuracy of some of the velocity and velocity gradient-based statistics and that no single configuration exists, for a given spatial

Dissipation element analysis in experimental and numerical shear flow

Tomographic particle-image velocimetry (TPIV) measurements and direct numerical simulations (DNS) are performed to analyze the small and large scale structure of channel flow data, i.e., the fields of the fluctuations of the three velocity components and the turbulent kinetic energy, respectively, are investigated by dissipation element analysis which is a new method to gain statistical information from turbulent scalar fields based on distinct, space-filling elements.In the inertial range, the conditional mean of the experimental turbulent kinetic energy difference reveals a scaling in the range of the expected characteristic 2/3-Kolmogorov scaling. This is in excellent agreement with the numerical results from the channel flow DNS and comparative data from the DNS of a homogeneous shear flow. Furthermore, the shape of the marginal pdfs of the normalized element length and the scalar difference, i.e., the linear rise and the exponential decay, is shown to agree with the findings from the associated DNS. On average, the dissipation elements possess an orientation over a certain bandwidth in the mean flow direction.

A hot-wire probe configuration and data reduction method to minimize velocity gradient errors for simultaneous measurement of three velocity components in turbulent flows

A highly resolved turbulent channel flow direct numerical simulation (DNS) with Reτ = 200 has been used to investigate the influence of the velocity gradients on the measurement accuracy of a hot-wire probe capable of measuring all three velocity components simultaneously. A new proposed sensor arrangement has been tested. First, the effective cooling velocity was determined for each sensor of the idealized probe, where the influence of the velocity component tangential to the sensors and flow blockage by the presence of the probe are neglected. Then, velocity component statistics were calculated, neglecting the velocity gradients over the probe sensing area, and they were compared to the DNS database values. It has been shown that the influence of the velocity gradients on the new proposed arrangement is minimized. Its accuracy was compared to existing three- and four-sensor configurations as well as to two-sensor X- and V-array probes.

On the accuracy of simultaneously measuring velocity component statistics in turbulent wall flows with arrays of three or four hot-wire sensors

A highly resolved turbulent channel flow direct numerical simulation (DNS) with Reτ = 200 has been used to investigate the ability of probes made up of arrays of three or four hot-wire sensors to simultaneously and accurately measure statistics of all three velocity components in turbulent wall flows. Various virtual sensor arrangements have been tested in order to study the effects of position, number of sensors and spatial resolution on the measurements. First, the effective cooling velocity was determined for each sensor of an idealized probe, where the influence of the velocity component tangential to the sensors and flow blockage by the presence of the probe are neglected. Then, simulating the response of the virtual probes to obtain the effective velocities cooling the sensors, velocity component statistics have been calculated neglecting the velocity gradients over the probe sensing area. A strong influence of both mean and fluctuation velocity gradients on measurement accuracy was found. A new three-sensor array configuration designed to minimize the influence of the velocity gradients is proposed, and its accuracy is compared to two-sensor X- and V-array configurations.

Polymer-laden homogeneous shear-driven turbulent flow: a model for polymer drag reduction

Drag reduction (DR) under a turbulent boundary layer implies the suppression of turbulent momentum flux to the wall, a large-eddy phenomenon. Our hypothesis is that the essential mechanisms by which dilute concentrations of long-chain polymer molecules reduce momentum flux involve only the interactions among turbulent velocity fluctuations, polymer molecules and mean shear. Experiments indicate that these interactions dominate in a polymer-active ‘elastic layer’ outside the viscous sublayer and below a Newtonian inertial layer in a polymer-laden turbulent boundary layer. We investigate our hypothesis by modelling the suppression of momentum flux with direct numerical simulation (DNS) of homogeneous turbulent shear flow (HTSF) and the finite extensible nonlinear elastic with Peterlin approximation (FENE-P) model for polymer stress. The polymer conformation tensor equation was solved using a new hyperbolic algorithm with no artificial diffusion. We report here on the equilibrium state with fixed mean shear rateS, where progressive increases in non-dimensional polymer relaxation timeWeS(shear Weissenberg number) or concentration parameter 1 − β produced progressive reductions in Reynolds shear stress, turbulence kinetic energy and turbulence dissipation rate, concurrent with increasing polymer stress and elastic potential energy. The changes in statistical variables underlying polymer DR with 1 − β,WeS, %DRand polymer-induced changes to spectra are similar to experiments in channel and pipe flows and show that the experimentally measured increase in normalized streamwise velocity variance is an indirect consequence of DR that is true only at lower DR. Comparison of polymer stretch and elastic potential energy budgets with channel flow DNS showed qualitative correspondence when distance from the wall was correlated toWeS. AsWeSincreased, the homogeneous shear flow displayed low-DR, high-DR and maximum-DR (MDR) regimes, similar to experiments, with each regime displaying distinctly different polymer–turbulence physics. The suppression of turbulent momentum flux arises from the suppression of vertical velocity fluctuations primarily by polymer-induced suppression of slow pressure–strain rate correlations. In the high-Weissenberg-number MDR-like limit, the polymer nearly completely blocks Newtonian inter-component energy transfer to vertical velocity fluctuations and turbulence is maintained by the polymer contribution to pressure–strain rate. Our analysis from HTSF with the FENE-P representation of polymer stress and its comparisons with experimental and DNS studies of wall-bounded polymer–turbulence supports our central hypothesis that the essential mechanisms underlying polymer DR lie directly in the suppression of momentum flux by polymer–turbulence interactions in the presence of mean shear and indirectly in the presence of the wall as the shear-generating mechanism.

The organization of near-wall turbulence: a comparison between boundary layer SPIV data and channel flow DNS data

The vortical structures of near-wall turbulence at moderate Reynolds number are analyzed and compared in datasets obtained from stereoscopic particle image velocimetry (SPIV) in a turbulent boundary layer and from direct numerical simulations (DNS) in a turbulent channel flow. The SPIV data are acquired in the LTRAC water tunnel at Re=+=820 in a streamwise/wall-normal plane, and in the LML wind tunnel at Re=+=2590 in a streamwise/wall-normal plane and in a plane orthogonal to the mean flow. The DNS data is taken from DelAlamo et al. (Self-similar vortex clusters in the turbulent logarithmic region, J. Fluid Mech. 561 (2006), pp. 329-358), at Re=950. The SPIV database is validated through an analysis of its mean velocity profile and power spectra, which are compared with reference profiles. A common detection algorithm is then applied to both the SPIV and DNS datasets in order to retrieve the streamwise and spanwise vortices. The algorithm employed is based on the 2D swirling strength and on a fit of an Oseen vortex model, which allows to retrieve the vortex characteristics, including its radius, vorticity, and position of the center. At all Reynolds numbers, the near-wall region is found to be the most densely populated region, predominantly with streamwise vortices that are on average smaller and more intense than spanwise vortices. In contrast, the logarithmic region is equally constituted of streamwise and spanwise vortices having equivalent characteristics. Two different scalings were employed to analyze the vortex radius and vorticity: the wall unit scaling and the Kolmogorov scaling. In wall unit scaling, a good universality in Reynolds numbers of the vortices radius and vorticity is observed in the near-wall and logarithmic regions: the vorticity is found to be maximum at the wall, decreasing first rapidly and then increasing slowly with wall-normal distance; the radius is increasing slowly with wall-normal distance in both the regions, except for the streamwise vortices, for which a sharp increase in radius is observed in the near-wall region. However, wall units scaling is found to be deficient in the outer region, where Reynolds number effects are observed. The Kolmogorov scaling appears to be universal both in Reynolds number and wall-normal distance across the investigated three regions, with a mean radius of the order of 8 and a mean vorticity of the order of 1.5-1, but for the SPIV data only. In the DNS dataset, the radius in the Kolmogorov scaling slowly decreases with wall-normal distance. This difference between the SPIV and DNS datasets may be linked to a difference between the boundary layer flow and the channel flow, rather than to the techniques themselves. Finally, the distribution of the vorticity of the detected vortices seems to follow faithfully a log-normal distribution, in good agreement with Kolmogorov's theory (A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13 (1962), pp. 82-85).

Kolmogorov scaling of turbulent flow in the vicinity of the wall

How to scale even the simplest of turbulent flows continues to be a cause for considerable controversy. In the present research, a data base compiling results from channel flow direct numerical simulations and turbulent boundary layer experiments is employed to investigate the properties of shear and normal Reynolds stresses very close to the wall. Two types of scaling based on Kolmogorov length and velocity scales are analyzed. It is shown that it is highly likely that large length scales of the order of the channel half-width or the boundary layer thickness play an important role even in the innermost regions of wall-bounded turbulent flows, which hints at the persistence of Reynolds number effects in even high Reynolds number flows.