Log Law Research Articles

Effect of Reynolds number, near-wall perturbation and turbulence on smooth open-channel flows

Three series of experiments were conducted to study the effect of Reynolds number, near-wall perturbation and turbulence on the velocity characteristics of smooth open-channel flows. Measurements were carried out using a laser Doppler anemometer. The variables of interest include mean velocity, turbulence, velocity probability density functions and Gram-Charlier series coefficients. For the range of depth Reynolds number studied (23×103 < Re h <72×103), the turbulence intensity and Reynolds shear stress profiles show that the effect of Reynolds number can be significant in open-channel flows. However, at identical distances from the bed, the velocity probability density functions are relatively insensitive to Reynolds number effects. The coefficients of the Gram-Charlier series expansion are also independent of Reynolds number for wall normal distances y+ <300. In the case of the flow with the near-wall perturbation, the mean velocity profile at the farthest downstream station more-or-less recovers to the undisturbed state, whereas the turbulence intensity profiles do not completely recover. From the various profiles obtained downstream of the perturbation, the wall-normal distance corresponding to 0.01<y/h<0.1 requires a greater longitudinal distance to recover to the undisturbed state. As noticed from the velocity defect profiles, the effects of near-wall disturbance begin to penetrate into the flow with increasing distance from the perturbation. The presence of higher levels of turbulence influences the skin friction coefficient, the extent of collapse with the log–law and the wake parameter.

Velocity and Surface Shear Stress Distributions Behind a Rough-to-Smooth Surface Transition: A Simple New Model

A simple new model is proposed to predict the distribution of wind velocity and surface shear stress downwind of a rough-to-smooth surface transition. The wind velocity is estimated as a weighted average between two limiting logarithmic profiles: the first log law, which is recovered above the internal boundary-layer height, corresponds to the upwind velocity profile; the second log law is adjusted to the downwind aerodynamic roughness and local surface shear stress, and it is recovered near the surface, in the equilibrium sublayer. The proposed non-linear form of the weighting factor is equal to ln(z/z 01)/ln(δ i /z 01), where z, δ i and z 01 are the elevation of the prediction location, the internal boundary-layer height at that downwind distance, and the upwind surface roughness, respectively. Unlike other simple analytical models, the new model does not rely on the assumption of a constant or linear distribution for the turbulent shear stress within the internal boundary layer. The performance of the new model is tested with wind-tunnel measurements and also with the field data of Bradley. Compared with other existing analytical models, the proposed model shows improved predictions of both surface shear stress and velocity distributions at different positions downwind of the transition.

Stratification Effects in a Bottom Ekman Layer

Abstract A stratified bottom Ekman layer over a nonsloping, rough surface is studied using a three-dimensional unsteady large eddy simulation to examine the effects of an outer layer stratification on the boundary layer structure. When the flow field is initialized with a linear temperature profile, a three-layer structure develops with a mixed layer near the wall separated from a uniformly stratified outer layer by a pycnocline. With the free-stream velocity fixed, the wall stress increases slightly with the imposed stratification, but the primary role of stratification is to limit the boundary layer height. Ekman transport is generally confined to the mixed layer, which leads to larger cross-stream velocities and a larger surface veering angle when the flow is stratified. The rate of turning in the mixed layer is nearly independent of stratification, so that when stratification is large and the boundary layer thickness is reduced, the rate of veering in the pycnocline becomes very large. In the pycnocline, the mean shear is larger than observed in an unstratified boundary layer, which is explained using a buoyancy length scale, u*/N(z). This length scale leads to an explicit buoyancy-related modification to the log law for the mean velocity profile. A new method for deducing the wall stress based on observed mean velocity and density profiles is proposed and shows significant improvement compared to the standard profile method. A streamwise jet is observed near the center of the pycnocline, and the shear at the top of the jet leads to local shear instabilities and enhanced mixing in that region, despite the fact that the Richardson number formed using the mean density and shear profiles is larger than unity.

Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers

The paper presents an in depth assessment of different similarity laws for the mean velocity profile in zero pressure gradient (ZPG) turbulent boundary layers (TBL's) in comparison with mostly experimental and few computational data. The emphasis is on the descriptions which are complete in the sense that a full representation of the mean velocity profile, its streamwise evolution and all integral parameters, including the friction factor and the shape factor, are provided as a function of Reynolds number. The first such complete description is the classical two-layer theory with its characteristic logarithmic mean velocity profile in the region where the two layers overlap, henceforth referred to as the “log law.” The main alternative scalings which have been proposed over the last decade have led to power law descriptions of the turbulent mean velocity profile. Since the different descriptions were calibrated with different data sets, the controversy over the relative merits of the different approaches has lingered on. The purpose of the present paper is to measure the principal competing theories against the same vast data set of more than 300 mean velocity profiles from more than twenty different sources. The results confirm the conclusions of numerous authors that the log law provides a fully self-consistent and accurate description of all the mean quantities and demonstrates conclusively that the same cannot be achieved by the competing power law theories. Along the way, it is also argued that the traditional description of the outer velocity profile in terms of a wall-normal coordinate normalized to unity at a hypothetical boundary layer “edge” δ and a “wake parameter” Π is not robust with respect to the fit of the outer velicity profile and should therefore not be used in theoretical arguments.

Direct numerical simulation of the Ekman layer: A step in Reynolds number, and cautious support for a log law with a shifted origin

Results at Ekman Reynolds numbers Re ranging from 1000 to 2828 expand the direct numerical simulation (DNS) contribution to the theory of wall-bounded turbulence. An established spectral method is used, with rules for domain size and grid resolution at each Reynolds number derived from the theory. The Re increase is made possible by better computers and by optimizing the grid in relation to the wall shear-stress direction. The boundary-layer thickness in wall units δ+ varies here by a factor of about 5.3, and reaches values near 5000, or 22 times the minimum at which turbulence has been sustained. An equivalent channel Reynolds number, based on the pressure gradient in wall units, would reach about Reτ=1250. The principal goal of the analysis, the impartial identification of a log law, is summarized in the local “Karman measure” d(ln z+)/dU+. The outcome differs from that for Hoyas and Jiménez [Phys. Fluids18, 011702 (2006)] and for Hu [AIAA J.44, 1541 (2006)] in channel-flow DNS at similar Reynolds numbers, for reasons unknown: Here, the law of the wall is gradually established up to a z+ around 400, with little statistical scatter. To leading order, it is consistent with the experiments of Österlund [Phys. Fluids12, 1 (2000)] in boundary layers. With the traditional expression, a logarithmic law is not present, in that the Karman measure drifts from about 0.41 at z+≈70 to the 0.37–0.38 range for z+≈500, with Re=2828. However, if a virtual origin is introduced with a shift of a+=7.5 wall units, the data support a long logarithmic layer with κ=0.38 a good fit to d(ln[z++a+])/dU+. A determination of the Karman constant from the variation of the skin-friction coefficients with Reynolds numbers also yields values near 0.38. The uncertainty is about ±0.01. These values are close to the boundary-layer experiments, but well below the accepted range of [0.40,0.41] and the experimental pipe-flow results near 0.42. The virtual-origin concept is also controversial, although nonessential at transportation or atmospheric Reynolds numbers. Yet, this series may reflect some success in verifying the law of the wall and investigating the logarithmic law by DNS, redundantly and with tools more impartial than the visual fit of a straight line to a velocity profile. At the request of the authors and editor, this article is being retracted effective 28 October 2009.

Statistical Uncertainty and the Estimation of Log Law Parameters

It is experimentally demonstrated that fitting the log law to the mean velocity profile in turbulent open-channel flow to determine the friction velocity, roughness length, and zero-plane displacement, can lead to relatively high standard errors in these parameters. It is also demonstrated that this approach, where the log law alone is used to estimate these parameters, may yield values of these parameters that are significantly different from those obtained when the friction velocity is determined independently of the log law. Although the statistical estimation of the three log law parameters from mean velocity data alone is frequently necessary, this procedure can lead to inaccurate and imprecise estimates of these quantities.

Exact rates in log law for positively associated random variables

Let { X n ; n ⩾ 1 } be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set S n = ∑ k = 1 n X k , M n = max k ⩽ n | S k | , n ⩾ 1 . Suppose σ 2 = E X 1 2 + 2 ∑ k = 2 ∞ E X 1 X k . In this paper, we study the exact convergence rates of a kind of weighted infinite series of P { M n ⩾ ε σ n log n } , P { | S n | ⩾ ε σ n log n } and I { | S n | ⩾ ε σ n log n } as ε ↘ 0 , respectively.

Power Law Velocity and Temperature Profiles in a Fully Developed Turbulent Channel Flow

Abstract The power law temperature distribution in a fully developed turbulent channel flow for large Peclet numbers has been proposed in the present work. The analysis of the power law velocity profile in a fully developed mean turbulent channel flow would be used for carrying out the analysis of the power law temperature profile. The Reynolds mean thermal energy equation in a fully developed mean turbulent channel flow has been analyzed. The mean turbulent thermal flow is divided in the inner and outer thermal layers that have been matched by Izakson–Millikan–Kolmogorov hypothesis to get the power law temperature profiles and the power law heat transfer law in the overlap region, in addition to traditional log laws for temperature profiles and heat transfer. It has been shown that the envelope of the heat transfer power law gives the heat transfer log law. Further, it is shown that the temperature power law index and prefactor are functions of the friction Peclet number, as well as function of an alternate variable, the nondimensional friction temperature. It is shown that for large Peclet numbers the power law temperature profile is equivalent to the log law temperature profile. The direct numerical simulation velocity profile data of fully developed turbulent flow provide good support for the power law temperature profile theory.

Power‐law velocity profile in a turbulent Ekman layer on a transitional rough surface

AbstractA two‐layer asymptotic theory of mean momentum in a turbulent Ekman layer without any closure model (such as eddy viscosity, mixing length, or k − ϵ) for large Rossby numbers is proposed. The flow in the inner wall layer and the outer wake layer are matched, using the Izakson–Millikan–Kolmogorov hypothesis; this leads to an open functional equation. Another open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation; this admits two functional solutions, with power‐law and log‐law velocity profiles. The envelope of the geostrophic‐drag power law leads to the log law, and determines the power‐law index and prefactor as a function of the surface Rossby number or the drag coefficient. The log laws and power laws for velocity and friction velocity, including the power‐law constants, are universal, and independent of the wall roughness. This universality is well supported by extensive experimental and laboratory data. In traditional smooth‐wall variables, there is no universality of scalings, and different expressions are needed for different types of roughness. Approximate solutions of the power‐law geostrophic drag and cross‐isobaric angle are also obtained. The power‐law geostrophic‐drag solution for each prescribed value of the power‐law index is valid for a limited domain of Rossby numbers. Copyright © 2008 Royal Meteorological Society

Near-bed turbulence models: Significance for diffusional mass transfer at the sediment/water interface

It is well-known that sediment/water interaction affects water quality in natural water bodies in a variety of ways. Diffusional mass transfer, across a sediment/water interface can contribute significantly to the mass balances of dissolved substances such as oxygen, phosphorus, nitrogen and sulfate, especially in lakes, reservoirs, detention basins, navigation canals and estuaries. At low flow velocities above a sediment bed, diffusional mass transfer across a sediment/water interface becomes limited by lack of turbulence in the boundary layer. In this paper, three boundary layer turbulence models are used to quantify the limiting effect of turbulence near a sediment bed on diffusional mass transfer across a sediment/water interface: One is Dade's formula, the second is a model by Myong and Kasagi (MK model), and the third is a model by Nagano and Tagawa (NT model). The latter two are low-Reynolds number k–ε turbulence models that have been successfully used for the prediction of turbulent heat transfer near a wall. Dade's formula gives larger eddy diffusivity/viscosity values near the sediment/water interface than the other two turbulence models, but the velocity profile obtained by Dade's formula is only slightly different from the universal velocity profile given by the linear law and the log–law. Simulated solute concentration profiles and vertical diffusive fluxes (Sherwood numbers Sh) at the sediment/water interface were found to be only weakly dependent on the choice of turbulence model when material sinks or sources in the sediments were not explicitly quantified. Sherwood numbers simulated by the NT model were less than 40% smaller than those obtained by Dade's formula and theMKmodel, when the Schmidt number Sc > 10, which is the case for most materials in aquatic solution. The mass transport modelwas also applied to the estimation of sedimentary oxygen demand (SOD) by including a microbial oxygen sink in the sediments. Simulated oxygen uptake rate, i.e. SOD (gm-2 d-1), was virtually the same for all three turbulence models, except at low shear velocity (U* < 0.5 cm), and at high microbial activity inside the sediment (Xmax > 100 mg/l). Even then the lowestSODvalues obtained with theNTturbulence model were within 20% of the higher values estimated with the other two models. One can conclude that any of the three models used to describe turbulence near the sediment bed, is appropriate forSODestimation, because microbial activity inside the sediment is as limiting as near-bed turbulence

Alternate Scales for Turbulent Boundary Layer on Transitional Rough Walls: Universal Log Laws

The present work deals with four new alternate transitional surface roughness scales for description of the turbulent boundary layer. The nondimensional roughness scale ϕ is associated with the transitional roughness wall inner variable ζ=Z+∕ϕ, the roughness friction Reynolds number Rϕ=Rτ∕ϕ, and the roughness Reynolds number Reϕ=Re∕ϕ. The two layer theory for turbulent boundary layers in the variables, mentioned above, is presented by method of matched asymptotic expansions for large Reynolds numbers. The matching in the overlap region is carried out by the Izakson–Millikan–Kolmogorov hypothesis, which gives the velocity profiles and skin friction universal log laws, explicitly independent of surface roughness, having the same constants as the smooth wall case. In these alternate variables, just above the wall roughness level, the mean velocity and Reynolds stresses are universal and do not depend on surface roughness. The extensive experimental data provide very good support to our universal relations. There is no universality of scalings in traditional variables and different expressions are needed for inflectional type roughness, monotonic Colebrook–Moody roughness, k-type roughness, d-type roughness, etc. In traditional variables, the velocity profile and skin friction predictions for the inflectional roughness, k-type roughness, and d-type roughness are supported well by the extensive experimental data. The pressure gradient effect from the matching conditions in the overlap region leads to the universal composite laws, which for weaker pressure gradients yields log laws and for strong adverse pressure gradients provides the half-power laws for universal velocity profiles and in traditional variables the additive terms in the two situations depend on the wall roughness.

Intermediate Scaling of Turbulent Momentum and Heat Transfer in a Transitional Rough Channel

The properties of the mean momentum and thermal balance in fully developed turbulent channel flow on transitional rough surface have been explored by method of matched asymptotic expansions. Available high quality data support a dynamically relevant three-layer description that is a departure from two-layer traditional description of turbulent wall flows. The scaling properties of the intermediate layer are determined. The analysis shows the existence of an intermediate layer, with its own characteristic of mesolayer scaling, between the traditional inner and outer layers. Our predictions of the peak values of the Reynolds shear stress and Reynolds heat flux and their locations in the intermediate layer are well supported by the experimental and direct numerical simulation (DNS) data. The inflectional surface roughness data in a turbulent channel flow provide strong support to our proposed universal log law in the intermediate layer, that is, explicitly independent transitional surface roughness. There is no universality of scalings in traditional variables and different expressions are needed for various types of roughness, as suggested, for example, with inflectional type roughness, Colebrook–Moody monotonic roughness, etc. In traditional variables, the roughness scale for inflectional roughness is supported very well by experimental and DNS data. The higher order effects are also presented, which show the implications of the low Reynolds-number flows, where the intermediate layer provides the uniformly valid solutions in terms of generalized logarithmic laws for the velocity and the temperature distributions.

Corotating Disk Assembly with Turbulent Through Flow

Turbulent flow in a paired corotating disk assembly with through flow is studied numerically. A RANS equation with near wall turbulence model is used in conjunction with an isotropic eddy viscosity model to disclose the mean and turbulent flow properties. The velocities and the mean shear vectors change directions continuously as they move away from wall. The lag angle between them is up to 31° within the log layer and precludes the use of log laws to bridge the near wall flow. A wake less wall law represents the resultant velocity profile in terms of wall coordinates. The disks' moment coefficient is lower than the one found for free-disk flows but it follows the same tendency. The turbulent flow properties are in qualitative agreement with recent experimental measurements and DNS on disk flows.

Large-scale features in turbulent pipe and channel flows

In recent years there has been significant progress made towards understanding the large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers, leaving scope for further work in pipes and channels. In this article the structure of fully developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. Results reveal long meandering structures of length up to 25 pipe radii or channel half-heights. These appear to be qualitatively similar to those reported in the log region of a turbulent boundary layer. However, for the channel case, large-scale coherence persists further from the wall than in boundary layers. This is expected since these large-scale features are a property of the logarithmic region of the mean velocity profile in boundary layers and it is well-known that the mean velocity in a channel remains very close to the log law much further from the wall. Further comparison of the three turbulent flows shows that the characteristic structure width in the logarithmic region of a boundary layer is at least 1.6 times smaller than that in a pipe or channel.

Influence of PIV interrogation area on turbulent statistics up to 4th order moments in smooth and rough wall turbulent flows

The paper presents measurements performed over a smooth surface, a transitionally rough surface generated by wire mesh and a fully rough surface that comprised transverse square ribs in a channel. The flow was seeded using fluorescent particles and the high quality vectors obtained using particle image velocimetry (PIV) technique were used to compute the mean velocity, turbulent statistics up to the fourth order moments as well as the production and dissipation terms in the transport equation for the turbulent kinetic energy. By using different interrogation areas and overlaps to process the PIV images, the interrogation area sizes in wall units varied from 2.9 to 52.4. In the first part of the paper, these interrogation areas were used to study the spatial resolution effects on the mean and higher order moments over the smooth and rough surfaces. For the second part of the paper, classical scaling and the scaling proposed by George and Castillo [W.K. George, L. Castillo, Zero-pressure-gradient turbulent boundary layer, Appl. Mech. Rev. 50 (12) (1997) 689–729] and Zagarola and Smits [M.V. Zagarola, A.J. Smits, Mean-flow scaling of turbulent pipe flow, J. Fluid Mech. 373 (1998) 33–79] were used to document the salient effects of surface roughness on the flow characteristics. The results show spatial resolution effects on some of the turbulent statistics, and it is evident that these effects are more pronounced over the rough surfaces than on the smooth surface. The smaller interrogation area sizes used in this study are adequate to reasonably evaluate all the statistics and production term but not the dissipation rate. It was observed that the power law proposed by George and Castillo [W.K. George, L. Castillo, Zero-pressure-gradient turbulent boundary layer, Appl. Mech. Rev. 50 (12) (1997) 689–729] is better than the log law for modeling low Reynolds number turbulent flows over rough surfaces. The results also support the premise that the mechanisms for turbulence production and transport close to the ribs are markedly different from those for a smooth surface.

Power Law Velocity Profile in Fully Developed Turbulent Pipe and Channel Flows

The power law velocity profile has been analyzed in terms of the envelope of the friction factor which gives the friction factor log law. The power law index α and prefactor C are shown as the function of the friction Reynolds number, as well as the function of the alternate variable the nondimensional friction velocity. The fully developed turbulent superpipe flow data of McKeon et al. and fully developed channel flow data of Zanoun et al. have been analyzed and the power law index α and prefactor C data have been estimated, first as a function of the friction Reynolds number and second as function of the nondimensional friction velocity. Based on analysis, several correlations have been proposed that have been supported by the data.

Double-Averaged Open-Channel Flows with Small Relative Submergence

We investigate the turbulent structure of shallow open channel flows where the flow depth is too small (compared with the roughness height) to form a logarithmic layer but large enough to develop an outer layer where the flow is not directly influenced by the roughness elements. Since the log layer is not present, the displacement height d, which defines the position of the zero plane, and the shear velocity u* cannot be found by fitting the velocity data to the log law. However, these parameters are still very important because they are used for scaling flow statistics for the outer and roughness layers. In this paper we propose an alternative procedure for evaluating d in laboratory conditions, where d is found from additional experiments with the fully developed log layer. We also point out the appropriate procedure for evaluating the shear velocity u* for flows with low submergence. These procedures are applied to our own laboratory flume experiments with uniform sphere roughness, where velocities were measured using Particle Image Velocimetry. Results were interpreted within the framework of the double-averaged Navier–Stokes equations and include mean velocities, turbulence intensities, Reynolds stresses, and form-induced normal and shear stresses. The data collapse well and show that in flows without a developed log layer the structure of turbulence in the outer layer remains similar to that of flows with a log layer. This means that even though the roughness layer in the experiments reported herein was sufficiently high to prevent the development of the log layer, influence of the bed roughness did not spread further up into the outer layer. Furthermore, the results show that flow statistics do not depend on relative submergence except for the form-induced stresses which increase when relative submergence decreases.

Velocity Distributions in Spatially Varied Flow with Increasing Discharge

The experimental study presented in this paper explores the distribution of the mean velocities for a channel receiving spatially varied (SV) inflow from directly above the centerline of the channel via sets of nozzles. The velocity components u , v , and w have been measured using a laser Doppler velocimeter. Initially the validity of the law of the wall in the channel receiving SV inflow was investigated. It was found that the measured velocity profile within the inertial sublayer region near the channel bed falls below the line representing the log law. Further, if the SV inflow enters at the center of the channel, the degree that the data depart from the log law increases from the sidewall toward the center of the channel. The structures of the velocity distributions are also presented in detail. The results include both near field, that is the area close to where the flow enters the channel, and far field measurements. It was found that the influence of the SV inflow on velocity profiles is largely c...

Neural network modelling for mean velocity and turbulence intensities of steep channel flows

AbstractThe main purpose of this study is to evaluate the potential of simulating the profiles of the mean velocity and turbulence intensities for the steep open channel flows over a smooth boundary using artificial neural networks. In a laboratory flume, turbulent flow conditions were measured using a fibre‐optic laser doppler velocimeter (FLDV). One thousand and sixty‐four data sets were collected for different slopes and aspect ratios at different locations. These data sets were randomly split into two subsets, i.e. training and validation sets. The multi‐layer functional link network (MFLN) was used to construct the simulation model based on the training data. The constructed MFLN models can almost perfectly simulate the velocity profile and turbulence intensity. The values of correlation coefficient (γ) are close to one and the values of root mean square error (RMSE) are close to zero in all conditions. The results demonstrate that the MFLN can precisely simulate the velocity profiles, while the log law and Reynolds stress model (RSM) are less effective when used to simulate the velocity profiles close to the side wall. The simulated longitudinal turbulence intensities yielded by the MFLN were also fairly consistent with the measured data, while the simulated vertical turbulence intensities by the RSM were not consistent with the measured data. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Wall skin friction and mean velocity profiles of fully developed turbulent pipe flows

The friction factor λ( Re) and the mean velocity U + = f( y +) measurements of Nikuradse [J. Nikuradse, Gesetzmässigkeite der turbulenten Strömung in glatten Rohren, Forschg. Arb. Ing.-Wes. No. 356 (1932); J. Nikuradse, Strömungsgesetze in rauhen Rohren, Forschg. Arb. Ing.-Wes. No. 361 (1933)] of fully developed turbulent flows in smooth and rough pipes are of vital consideration since they provided the data by which established theories have been developed in the last few decades. The pressure gradient and the resultant friction factor of Nikuradse’ smooth pipe agree well with the authors’ own results. On the other hand, the Nikuradse’s corresponding mean velocity profile measurements show differences from measurements presented in this. The differences might be attributed to the state of Nikuradse’s flow at the location of velocity profile measurements in addition to differences in the applied measuring techniques. It is concluded that pitot tubes do usually not have the needed spatial resolution in the near-wall region and produces therefore velocity overshoots under y + = 300 when used in turbulent pipe shear flows. Hence, when the lower limit for the log-range starts at y + ⩽ 50, which was common for almost all previous work up to the late 1990s, a higher value for the so-called von Kármán constant ( κ) of the logarithmic velocity profile resulted. In addition to pitot tube velocity measurements, hot-wire measurements are provided, showing that the slope (i.e., 1/ κ) of the logarithmic velocity profile is inconsistent with value deduced from the λ( Re) measurements performed by the authors. Nikuradse’s pitot tube velocity data also yield log law constants that are not reflected by their corresponding pipe friction measurements. However, the authors observed that the hot wire and pitot tube results are about the same if the inner limit of the log range of the logarithmic velocity profile is y + ⩾ 300 and the effect of the mean shear gradient is minimal under the same condition.