Karman Constant Research Articles

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.

Local Imbalance of Turbulent Kinetic Energy in the Surface Layer

We utilize experimental data collected in 2002 over an open field in Hanford, Washington, USA, to investigate the turbulent kinetic energy (TKE) budget in the atmospheric surface layer. The von Karman constant was determined from the near-neutral wind profiles to be 0.36 ± 0.02 rather than the classical value of 0.4. The TKE budget was normalized and all terms were parameterized as functions of a stability parameter z/L, where z is the distance from the ground and L is the Obukhov length. The shear production followed the Businger–Dyer relation for −2 < z/L < 1. Contrary to the traditional Monin–Obukhov similarity theory (MOST), the shear, buoyancy and dissipation terms were found to be imbalanced due to a non-zero vertical transport over all stabilities. Motivated by this local imbalance, modified parameterizations of the dissipation and the turbulent transport were attempted and generated good agreement with the experimental data. Assuming stationarity and horizontal homogeneity, the pressure transport was estimated from the residual of the TKE budget.

Local Similarity in the Stable Boundary Layer and Mixing-Length Approaches: Consistency of Concepts

In stably stratified flows vertical movement of eddies is limited by the fact that kinetic energy is converted into potential energy, leading to a buoyancy displacement scale z B . Our new mixing-length concept for turbulent transport in the stable boundary layer follows a rigid-wall analogy, in the sense that we assume that the buoyancy length scale is similar to neutral length scaling. This implies that the buoyancy length scale is: ? B = ? B z B , with ? B ? ?, the von Karman constant. With this concept it is shown that the physical relevance of the local scaling parameter z/? naturally appears, and that the ? coefficient of the log-linear similarity functions is equal to c/? 2, where c is a constant close to unity. The predicted value ? ? 1/? 2 = 6.25 lies within the range found in observational studies. Finally, it is shown that the traditionally used inverse linear interpolation between the mixing length in the neutral and buoyancy limits is inconsistent with the classical log-linear stability functions. As an alternative, a log-linear consistent interpolation method is proposed.

Parameterization of the logarithmic layer of double-averaged streamwise velocity profiles in gravel-bed river flows

The logarithmic layer of double-averaged (in time and space) streamwise velocity profiles obtained from field measurements made in the Swiss rivers, Venoge and Chamberonne is parameterized and discussed. Velocity measurements were made using a 3D Acoustic Doppler Velocity Profiler. Both riverbeds are hydraulically rough, composed of coarse gravel, with relative submergences ( h/ D 50) of 5.25 and 5.96, respectively. From the observations, the flow may be divided into three different layers: a roughness layer near the bed, an equivalent logarithmic layer and a surface or outer layer. It was found that a logarithmic law can describe the double-averaged profiles in the layer 0.30 < z/ h < 0.75. The parameterization of the logarithmic law is discussed. Special emphasis is given to the geometric parameters roughness and zero-displacement heights and to the equivalent von Karman constant.

Modeling the influence of wave‐enhanced turbulence in a shallow tide‐ and wind‐driven water column

The ability of one‐dimensional hydrodynamic models to reproduce dissipation of turbulent kinetic energy and velocity profiles for conditions of whitecapping waves in a shallow water, tide‐ and wind‐forced environment was assessed. The models were forced with the conditions experienced during a month‐long field experiment in a shallow estuarine embayment, and the results were compared with the observed dissipation and mean velocity profiles. Three turbulence models were assessed: the k‐ω model and two k‐l models, with different prescribed bilinear relationships for the turbulent length scale, l. The k‐ω turbulence model was found to best replicate the measured decay of dissipation with depth with a surface roughness length, z0s = 1.3Hs, and wave energy parameter, α = 60. The k‐l model achieved equally as good reproduction of the observations as the k‐ω model when the proportionality constant in the prescribed linear length scale relationship for the upper half of the water column was modified from the traditionally employed von Karman's constant, κ = 0.4, to 0.25. The model results show that the whitecapping waves often supplied the dominant source of turbulent kinetic energy over the majority of the water column in the shallow, tide‐ and wind‐forced system.

Two‐phase model for sand transport in sheet flow regime

We introduce a two‐phase model for sand transport in sheet flow regime. This model uses a collisional theory and a k – ɛ fluid turbulence closure to respectively model the sediment and fluid phase stresses. The sediment stress closure adopts a balance equation of sediment particle fluctuation energy based on kinetic theory that incorporates two‐way interactions between fluid and sediment phases. The fluid turbulence closure also considers the two‐way interaction between fluid turbulence and sand particles. Model‐data comparisons for the sheet layer for oscillatory flows in a U‐tube and for open channel flows demonstrate the model's predictive skill. For steady open channel flows the fluid phase velocity follows closely the law of wall (i.e., the log‐profile) in which the von Karman constant is reduced and the equivalent roughness is increased, compared to the clear fluid flow conditions. The model also provides information in the near‐bed region where the transition from the solid‐like to the fluid‐like behavior of sediment particles is resolved and both the bed load layer thickness and bed load transport rate can be evaluated. For unsteady flows, this model can predict time evolutions for sediment transport throughout the water column.

Reply to comment by Edgar L Andreas on “Vertical coarse aerosol fluxes in the atmospheric surface layer over the North Polar Waters of the Atlantic”

[1] We are grateful for the interest Edgar L Andreas took in our work. His comments [Andreas, 2007] on our paper [Petelski and Piskozub, 2006] are insightful and confirm the methodology we chose, the aerosol concentration gradient, as a step forward toward achieving a reliable estimation of the aerosol production function. The difference between our approach and the recalculation of our results by Andreas [2007] is the value of the von Karman constant analogue for the aerosol flux. [2] It is not obvious that the dimensionless parameter of the logarithmic profile of aerosol concentration needs to be identical to the von Karman constant. The physics of momentum flux and aerosol flux are identical because we assume Monin-Obukhov similarity, but that implies only the existence of such a constant, not its value. Andreas gives theoretical reasons for its value to be identical to the Karman constant (approximately 0.40). We implicitly assumed it to be 1.0. Hence the approximate 2.5-fold difference of calculated fluxes in our original paper and in the comment by Andreas (approximate because we presented best fit flux functions for each wind speed bin data separately, which makes each of them differ from values of the ‘‘universal’’ function, namely, equation (5) of Petelski and Piskozub [2006], derived from the complete data set). However, we believe that the only way this problem can be solved is by experimental determination of the constant value. Until better data are available (one possibility is eddy correlation measurements in the open sea), our aerosol concentration data processed using the dry deposition method [Smith et al., 1993] may be the best calibration of the coefficient. Results presented in Figures 9 and 10 of Petelski and Piskozub [2006] (the ‘‘Eq(10)’’ lines) seem to suggest that the coefficient may be higher than the von Karman constant, possibly even closer to the value we implicitly chose (1.0). Also, the only eddy correlation measurements of aerosol concentration [Nilsson and Rannik, 2001] in the open sea, measured in the same areas as our data, seem to suggest that the flux functions favored by Andreas [2002] may be underestimated, further strengthening our estimation of the coefficient value. [3] Even with the above disagreements with Andreas [2007], we believe that his analysis of the influence of the flux coefficient choice on the results moves us forward toward a consensus on the marine aerosol source function. Even a 2.5-fold difference in the function value is minor compared to the several orders of magnitude range of functions reviewed by Andreas [2002]. However, further studies using more than one method of aerosol flux estimation are needed to narrow the von Karman constant–related uncertainty.

A mathematical determination of von Karman's constant, Κ

One of the few fundamental constants in fluid mechanics is von Karman's constant governing many turbulent flows. Using an energy argument and mathematical symmetry it is argued here that von Karman's constant is κ = 0.414 (with derivable corrections for specific applications)

Effect of stratification due to suspended sand on velocity and concentration distribution in unidirectional flows

Sediment‐induced stratification effects on velocity profiles and sediment concentration distribution in a steady, uniform turbulent flow are examined in this paper. The early work concerning sediment stratification relates this to the von Karman constant's variability. Subsequent attempts to account for stratification were based on the stratified flow analogy, introducing the parameters α and β, whose values were assumed to be those obtained for thermally stratified flows. Following these investigators, we assume stratification effects to be expressed through these parameters. We solve the governing equations for velocity and sediment concentration for a parabolic neutral eddy viscosity model. Analytically closed‐form solutions are obtained. We run our model against experimental data to obtain the optimal set [α, β]. For neutral conditions, β = 0 by definition, and we obtain α = 1. For stratified conditions we obtain [α = 0.8, β = 4.0]. This is the first time both α and β have been obtained from sediment‐laden flow observations. Accounting for stratification improves the prediction of velocity and concentration. For predictive purposes, we need to know the movable bed roughness and the reference concentration. Analyses of experimental data sets provide predictive relationships for these in terms of sediment and flow parameters.

유사입자에 의한 개수로 난류 유속 분포의 변화에 대한 재검토

Vanoni가 1946년에 유사를 포함한 흐름에서는 Karman상수가 감소한다고 제안한 이후로 개수로 흐름에서 유사 입자가 평균 유속 분포를 변화시킨다는 것은 매우 잘 알려진 사실이다. 그러나 유사 입자가 어떤 역할을 하는지, 어떤 매개 변수가 변화할 것인지에 대해서는 현재까지도 논쟁이 계속되고 있다. 한편, 어떤 연구자들은 난류중에 유사입자가 투입되더라도 Karman 수는 변하지 않는다고 주장하였다. 본 연구에서는 Karman 수의 변화에 대한 기존의 연구들을 재평가하였다. 이를 통하여 기존 연구들의 방법에서 몇 가지 오류들을 밝혀 내었으며, 왜 이 사항이 오랜 동안 논쟁이 되어왔는가 하는 이유를 찾아내었다. 또한 분석 결과 유사 입자의 투입에 의해 Karman 수는 감소되지만 그 감소량은 기존의 연구들에서 제시된 것보다 훨씬 작음을 밝혔다. 또한 본 연구는 유사의 투입에 의해 Karman 수가 감소되는 기구를 제시하였다. It is well known that sediment particles introduced in open-channel turbulent flows change mean velocity profile, since Vanoni suggested the reduction of the Karman constant in 1946. However, how the sediment particles take such a role and what parameters would be changed have been debated up to now. Some researchers, on the other hand, have insisted that the constant would not be changed regardless of introducing sediment particles. The present study is a careful re-evaluation of the previous studies on this issue. The study revealed some questionable approaches or methods in the decision of the previous researches and found the reason why this issue has been debated for a long time. The result indicated that the Karman number is reduced by adding sediment particles, but the amount of reduction is much smaller than the previous researches insisted. Finally, the present study proposes a mechanism of the Karman number reduction due to sediment particles.

Hydrogen Interaction with Dissolved Atoms and Relaxation Properties of Metal Solid Solutions

The H(D) atom’s interaction with one another, ‘heavy’ interstitial atoms (O, N, C), and substitutional atoms is analyzed on the basis of strain-induced (elastic) interaction. The interaction energies are calculated for bcc, fcc, and hcp metal solid solutions with regard to the discrete atomic structure of the host lattice. The elastic constants, Born-von Karman constants of the host lattice, and concentration expansion coefficients of the solid solution lattice due to solute atoms, are used as the parameters for numerical input. It is shown that the interaction is long-range, oscillating, and anisotropic. In all cases, the coordination shells of both types - with attraction and with repulsion - exist. The interaction energy dependence on the distance is due mainly to the crystal lattice type. The strain-induced interaction should be supplemented by repulsion in the nearest coordination shells for the case of interstitial-interstitial interaction and by chemical interaction in the case of H-substitutional interaction. Two examples are given for the use of the strain-induced interaction energies in calculations relaxation processes.

Direct Numerical Simulation of Turbulent Flow Around a Rotating Circular Cylinder

Abstract Turbulent flow around a rotating circular cylinder has numerous applications including wall shear stress and mass-transfer measurement related to the corrosion studies. It is also of interest in the context of flow over convex surfaces where standard turbulence models perform poorly. The main purpose of this paper is to elucidate the basic turbulence mechanism around a rotating cylinder at low Reynolds numbers to provide a better understanding of flow fundamentals. Direct numerical simulation (DNS) has been performed in a reference frame rotating at constant angular velocity with the cylinder. The governing equations are discretized by using a finite-volume method. As for fully developed channel, pipe, and boundary layer flows, a laminar sublayer, buffer layer, and logarithmic outer region were observed. The level of mean velocity is lower in the buffer and outer regions but the logarithmic region still has a slope equal to the inverse of the von Karman constant. Instantaneous flow visualization revealed that the turbulence length scale typically decreases as the Reynolds number increases. Wavelet analysis provided some insight into the dependence of structural characteristics on wave number. The budget of the turbulent kinetic energy was computed and found to be similar to that in plane channel flow as well as in pipe and zero pressure gradient boundary layer flows. Coriolis effects show as an equivalent production for the azimuthal and radial velocity fluctuations leading to their ratio being lowered relative to similar nonrotating boundary layer flows.

On the Kinetic Energy Budget of the Unstable Atmospheric Surface Layer

We present a new account of the kinetic energy budget within an unstable atmospheric surface layer (ASL) beneath a convective outer layer. It is based on the structural model of turbulence introduced by McNaughton (Boundary-Layer Meteorology, 112: 199–221, 2004). In this model the turbulence is described as a self-organizing system with a highly organized structure that resists change by instability. This system is driven from above, with both the mean motion and the large-scale convective motions of the outer layer creating shear across the surface layer. The outer convective motions thus modulate the turbulence processes in the surface layer, causing variable downwards fluxes of momentum and kinetic energy. The variable components of the momentum flux sum to zero, but the associated energy divergence is cumulative, increasing both the average kinetic energy of the turbulence in the surface layer and the rate at which that energy is dissipated. The tendency of buoyancy to preferentially enhance the vertical motions is opposed by pressure reaction forces, so pressure production, which is the work done against these reaction forces, exactly equals buoyant production of kinetic energy. The pressure potential energy that is produced is then redistributed throughout the layer through many conversions, back and forth, between pressure potential and kinetic energy with zero sums. These exchanges generally increase the kinetic energy of the turbulence, the rate at which turbulence transfers momentum and the rate at which it dissipates energy, but does not alter its overall structure. In this model the velocity scale for turbulent transport processes in the surface layer is (kzɛ)1/3 rather than the friction velocity, u*. Here k is the von Karman constant, z is observation height, ɛ is the dissipation rate. The model agrees very well with published experimental results, and provides the foundation for the new similarity model of the unstable ASL, replacing the older Monin–Obukhov similarity theory, whose assumptions are no longer tenable.

A new friction factor relationship for fully developed pipe flow

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Karman constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

Modified log-wake law for zero-pressure-gradient turbulent boundary layers

This paper shows that the turbulent velocity profile for zero-pressure-gradient boundary layers is affected by the wall shear stress and convective inertia. The effect of the wall shear stress is dominant in the so-called overlap region and can be described by a logarithmic law in which the von Karman constant is about 0.4 while the additive constant depends on a Reynolds number. The effect of the convective inertia can be described by the Coles wake law with a constant wake strength about 0.76. A cubic correction term is introduced to satisfy the zero velocity gradient requirement at the boundary layer edge. Combining the logarithmic law, the wake law and the cubic correction produces a modified log-wake law, which is in excellent agreement with experimental profiles. The proposed velocity profile law is independent of Reynolds number in terms of its defect form, while it is Reynolds number dependent in terms of the inner variables. The modified log-wake law can also provide an accurate equation for skin friction in terms of the momentum thickness. Finally, by replacing the logarithmic law with van Driest's mixing-length model in which the damping factor varies with Reynolds number, the modified log-wake law can be extended to the entire boundary layer flow.

Turbulence Structure of the Unstable Atmospheric Surface Layer and Transition to the Outer Layer

We present a new model of the structure of turbulence in the unstable atmospheric surface layer, and of the structural transition between this and the outer layer. The archetypal element of wall-bounded shear turbulence is the Theodorsen ejection amplifier (TEA) structure, in which an initial ejection of air from near the ground into an ideal laminar and logarithmic flow induces vortical motion about a hairpin-shaped core, which then creates a second ejection that is similar to, but larger than, the first. A series of TEA structures form a TEA cascade. In real turbulent flows TEA structures occur in distorted forms as TEA-like (TEAL) structures. Distortion terminates many TEAL cascades and only the best-formed TEAL structures initiate new cycles. In an extended log layer the resulting shear turbulence is a complex, self-organizing, dissipative system exhibiting self-similar behaviour under inner scaling. Spectral results show that this structure is insensitive to instability. This is contrary to the fundamental hypothesis of Monin--Obukhov similarity theory. All TEAL cascades terminate at the top of the surface layer where they encounter, and are severely distorted by, powerful eddies of similar size from the outer layer. These eddies are products of the breakdown of the large eddies produced by buoyancy in the outer layer. When the outer layer is much deeper than the surface layer the interacting eddies are from the inertial subrange of the outer Richardson cascade. The scale height of the surface layer, z s, is then found by matching the powers delivered to the creation of emerging TEAL structures to the power passing down the Richardson cascade in the outer layer. It is z s = u * 3 /kes, where u * is friction velocity, k is the von Karman constant and es is the rate of dissipation of turbulence kinetic energy in the outer layer immediately above the surface layer. This height is comparable to the Obukhov length in the fully convective boundary layer. Aircraft and tower observations confirm a strong qualitative change in the structure of the turbulence at about that height. The tallest eddies within the surface layer have height z s, so z s is a new basis parameter for similarity models of the surface layer.

Measurements and CFD Modeling of Drag-Reduction Effects

Summary Computational-fluid-dynamics (CFD) modeling of drag reduction (DR) by polymer additives dissolved in hydrocarbon was carried out in pipe flows and in a rotating shear viscometer. A two-layer turbulence model was applied that described turbulence damping by the molecules of drag-reducing additives (DRAs) in the near-wall regions. The von Karman constant in the near-wall region was used as the model parameter. Extensive measurements of the DR effect for a rotating viscometer were performed at different Reynolds numbers, apparent molar masses, and DRA concentrations. The model with only one fit parameter was able to reproduce the experimental results in both pipe flows and the rotating viscometer. Experimental results were used in relating the model parameter to the relevant physical properties.

Friction and mixing in the Faroe Bank Channel outflow

Three hydrographic sections made during RRS Discovery cruise 242, from 7 September to 6 October 1999, are used to study the Faroe Bank Channel overflow. Each section, made perpendicular to the plume direction, comprised 10–14 stations measuring the outflow properties and velocity. The overflow defined as water with potential temperatures below 3 °C has densities greater than 28.1 kg m –3, salinities between 34.9 and 35.1, with velocities as high as 80 cm s –1 approximately 20 km downstream from the sill. We use these measurements to describe the characteristics of the outflow and to estimate the amount of friction and mixing as the plume travels along the continental slope into the Iceland Basin. From Lowered Acoustic Doppler Current Profiler (LADCP) measurements we estimate stress in the bottom Ekman layer. Our stress results are best related to the overflow velocity with a drag coefficient of 0.5 × 10 –3 and to the height of the bottom Ekman layer with a von Karman constant of 0.75. Below a potential temperature of 8 °C, the isotherms sink with distance downstream. Calculating transport below different isotherms, we make an estimate for the amount of entrainment. In the region downstream of the sill, entrainment is initially maximum near the 0 °C isotherm just above the cold dense outflow and turbulent diffusivities reach as high as 500 cm 2 s –1. Fifty kilometre downstream of the sill the depth of maximum entrainment has moved upwards to the 2 °C isotherm and turbulent diffusivity has decreased to 50 cm 2 s –1. Froude numbers are found to exceed one in the centre of the overflow on all three sections. Calculating Richardson numbers we see evidence that mixing takes place above the fast flowing outflow core.

Resistance law for taylor-couette turbulent flow at very high reynolds numbers

New expressions for the resistance law and dimensionless moment of force are derived for a Taylor-Couette turbulent flow starting from the generalized model of local balance for the turbulent energy. In the case of extremely high Reynolds numbers, the formulas derived involve a single empirical (Karman) constant.

Effects of turbulent dispersion on the wind speed profile in the surface layer

New Reynolds mean momentum equations including turbulent viscosity and dispersion are used to analyze the vertical profile of wind speed in the surface layer. It is demonstrated that the wind profile of the surface layer including turbulent dispersion has a logarithmic modification on the classical power law. Under the condition of unstable stratification, the effect of dispersion is stronger than under stable stratification. Under neutral stratification, the power law degenerates to the logarithmic law, but the von Karman constant is replaced by k 1 = (1+ k / 4)−1 k, which can also be obtained by similarity theory.