Mixing Length Approach Research Articles

Mean Velocity of Fully Developed Turbulent Pipe Flows

The mean flow profile of a turbulent flow in a smooth channel or a pipe is a classical topic in fluid mechanics [1–4]. For fully developed turbulent pipe flows, von Karman and Prandtl’s pioneering analyses indicate two dominant regions: the near-wall laminar sublayer and the turbulent region away from the wall. The two regions are connected by a small intermediate buffer zone. In both regions, u , the mean velocity scaled with the friction speed u , is a function of y , the distance from the wall scaled with the ratio of kinematic viscosity to friction speed =u . This relationship u f y is known as the classical “law-of-the-wall.” The Reynolds-averaged Navier–Stokes equation describes the mean axial speed of a fully developed turbulent flow and involves the viscous stresses and the turbulent Reynolds stresses. This equation is unclosed because there is no physical rule relating between the turbulent stresses and the mean axial speed. Yet, experiments show that the magnitudes of the stresses differ vastly for each region. Very near the wall, in the viscous sublayer where y 70 is characterized by a negligible viscous stress and a dominant Reynolds stress. Modeling the Reynolds stress in this region by the linear mixing-length approach of Prandtl [3] results in the “logarithmic velocity distribution law.” The logarithmic relationship was also derived from a dimensional analysis requiring the continuity of the velocity gradient in the buffer zone when the Reynolds number is sufficiently high and using u as the speed scale in both the near-wall and turbulent region [4]. In the buffer zone, the Reynolds and viscous stresses are of the same magnitude. Von Karman suggested that in the buffer zone u is also related to the logarithm of y . In both the buffer and turbulent zones, the logarithmic functions include two constants that may be determined experimentally. This semi-empirical approach showed a nice agreement with measurements [2]. It also resulted in “Prandtl’s universal law of friction for smooth pipes”which relates between the friction factor and the Reynolds number and agrees with data. Note that van Driest [5] proposed a damping function for the turbulent stress in the viscous sublayer and the buffer zone which resulted in a good prediction of the mean velocity in these regions. In recent years, new approaches raised controversy about the scales of both distance and speed in the outer or core region for pipe flows. The pipe radius has been proposed as an additional dominant length scale. Arguments have been made for the velocity scale to be either the maximum (centerline) speed [6] or the velocity deficit [7]. The functional form of the scaling law in the overlap region is a subject of even greater controversy. It has been suggested that a power law rather than a log law governs the relationship between velocity and distance from the wall [6–8]. Note, however, that Wosnik et al. [6], Zagarola and Smits [7], and Barenblatt [8] power laws are derived from different approaches. For a thorough review of the scales in turbulent pipe flows, see Pope [9]. Wosnik et al. [6] proposed a more detailed description of the pipe flow structure, see also Pope, [9] chapter 7. They divided the flow into the main “viscous sublayer,” “overlap,” and “outer” regions. The near-wall region, where 0 200; 000. They also found that the logarithmic law region holds for a larger range, 600< y < 0:12K. Another set of recent pipe flow experiments over a wide range of Reynolds numbers (10< Re < 1; 000; 000) was conducted by Received 10 December 2004; revision received 18 April 2006; accepted for publication 19May 2006. Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code $10.00 in correspondence with the CCC. AIAA JOURNAL Vol. 44, No. 11, November 2006

LM simulations of the Greenland boundary layer, comparison with local measurements and SNOWPACK simulations of drifting snow

Abstract The interactions between atmosphere and snowpack are important for modelling the boundary layer (BL) over snow surfaces. To study these interactions, the mesoscale local model (LM) of the German Meteorological Service is applied for simulations over Greenland at a horizontal resolution of about 14 km and a high vertical resolution of the BL. Since usage of the snow/soil model as implemented in LM leads to unrealistic representation of the snowpack, adjustments of the representation of snow properties in LM have been done for the simulation over the Greenland ice sheet. Simulations using LM are performed for a ten-day period in July 2002, during which a field experiment took place at station Summit. To validate the model LM, the model results are compared to these measurements. The comparison shows some deficies for the near-surface values of temperature as well as turbulence quantities. The turbulence parameterisation in the LM has deficies for very stable stratification. Therefore, a local mixing length approach and the parameterisation of a scalar roughness length over ice and snow are implemented in the LM and their influence on the model performance is tested. These sensitivity studies show an improvement of turbulent kinetic energy and heat flux with usage of scalar roughness length as well as local mixing length approach. For a first estimation of the influence of snow drift on the snow accumulation, the LM results are used to drive the one-dimensional snow model SNOWPACK in an off-line mode. SNOWPACK has a much better representation of snow than the LM snow/soil model and allows for simulating snow microphysical properties. While over large parts of the plateau area of the ice sheet the influence of snow drift on the snow accumulation is much smaller than the evaporation/sublimation, snow drift influence is of at least the same magnitude in regions of the ice sheet margins, associated with large divergence/convergence.

Limitations of Depth-Averaged Modeling for Shallow Wakes

Large-scale horizontal vortical structures are generic features of shallow flows which are often modeled using the two-dimensional (2D) depth-averaged equations. Such modeling is investigated for the well-defined case of shallow wakes of a conical island of small side slope for which a three-dimensional (3D) boundary-layer (3DBL) model has previously been validated through comparison with experiment. The 3DBL model used a 3D, two-mixing-length, eddy-viscosity turbulence model with a vertical mixing length of classical Prandtl form and a horizontal mixing length some multiple of this. A multiple of six gave good predictions. This mixing length approach is reduced to depth-averaged form, giving a horizontal mixing length of about half the water depth. The shallow wakes may be vortex shedding or steady/stable and are conventionally defined by a stability parameter. The critical value above which a stable wake is formed is considerably overestimated by the depth-averaged model (for a range of mixing lengths) and the length of stable wake bubble is considerably underestimated. It seems likely that this is because the amplification of friction coefficient due to horizontal strain rates is not represented. However, when vortex shedding is prominent the 2D and 3DBL wake structures are quite similar. These results show, for example, the limitations of depth-averaged models for the prediction of solute dispersion.

Modeling of flowing film concentrators: Part 1. Water split behaviour

A large number of mineral processing equipment employs the basic principles of gravity concentration in a flowing fluid of a few millimetres thick in small open channels where the particles are distributed along the flow height based on their physical properties and the fluid flow characteristics. Fluid flow behaviour and slurry transportation characteristics in open channels have been the research topic for many years in many engineering disciplines. However, the open channels used in the mineral processing industries are different in terms of the size of the channel and the flow velocity used. Understanding of water split behaviour is, therefore, essential in modeling flowing film concentrators. In this paper, an attempt has been made to model the water split behaviour in an inclined open rectangular channel, resembling the actual size and the flow velocity used by the mineral processing industries, based on the Prandtl's mixing length approach.

Mean flow and turbulence in vegetated open channel flow

Vegetation affects the mean and turbulent flow structure in surface water bodies, thus impacting the local transport processes of contaminants and sediments. The present paper explores the capability of two different mathematical models to predict fully developed one‐dimensional open channel flow in the presence of rigid, complex‐shaped vegetation with leaves, submerged or emergent. The flow is described by applying two different turbulence closure schemes, both of which are based on the Boussinesq eddy viscosity model: a suitably modified к − ε model and a two‐layer model based on the mixing length approach. To describe the turbulence structure within and above the canopy a turbulent kinetic energy budget equation was added to the two‐layer model. The results of the models were compared with experimental data where simple cylinders, plastic plant prototypes, or real plants, all arranged in a scattered pattern, were employed. Since good agreement between the results of the models and measurements was found in comparing velocity and turbulent shear stress, the models could potentially be used to assess vegetative resistance. Significant disagreement was found when comparing measured and computed eddy viscosity distributions, streamwise turbulence intensity, and most of the terms comprising the turbulent kinetic energy budget.

New Mixing-Length Approach for the Mean Velocity Profile of Turbulent Boundary Layers

New Mixing-Length Approach for the Mean Velocity Profile of Turbulent Boundary Layers

Overshoot at the Base of the Solar Convection Zone: A Semianalytical Approach

Despite the importance of overshoot at the base of the solar convection zone for the storage of strong toroidal magnetic field produced there by the solar dynamo, uncertainties concerning the depth and mean subadiabatic stratification remain large. Overshoot models based on the nonlocal mixing-length theory generally produce a shallow, weakly subadiabatic region with a sharp transition to the radiative interior, whereas several numerical simulations lead to significantly subadiabatic overshoot with penetration depth of more than a pressure scale height. We present a semianalytical convection zone/overshoot region model based on the assumption that the convective energy flux is governed by coherent downflow structures starting at the top of the domain and continuing all the way down into the overshoot region, which allows for modeling both the parameter regime addressed by nonlocal mixing-length approach and the regime addressed by numerical simulations. It turns out that the main differences between the nonlocal mixing-length approach and numerical simulations (nearly adiabatic vs. strongly subadiabatic overshoot) are caused by the much larger energy flux used in numerical simulations as a consequence of larger thermal diffusivities required by numerical constraints. The depth of the overshoot region is determined predominantly by the mixing between downflows and upflows in the convection zone. Furthermore, our model shows that the sharp transition between the nearly adiabatic overshoot and radiative interior, a typical result of the nonlocal mixing-length approach, can be avoided by assuming an ensemble of downflows with different strength.

Flow Velocity Measurements in Vegetated Channels

Understanding of the hydraulics of flow over vegetation is very important to support the management of fluvial processes. In this paper the flow over flexible bottom vegetation is experimentally studied. A two-dimensional-acoustic doppler velocimeter is used to measure the local flow velocities for different vegetation concentrations, discharges, and flume slopes. The influence of vegetation concentration and the depth/vegetation height ratio on the measured velocity profiles is analyzed. All measured velocity distributions are S-shaped and exhibit a three-zone profile. The relationship between the velocity distribution and the turbulence intensity distribution is also analyzed. The characteristics ~inflection point, maximum value, asymptotes ! of the measured velocity distributions and the results of the previous investigators are used to select an analytical expression for the shape of the velocity profile. A theoretical velocity profile is deduced using the classical Prandtl's mixing length approach with a new expression for the mixing length. The deduced four-coefficient profile allows description of the flow both inside and above the vegetation. The physical and geometrical meaning of the four coefficients are also shown. DOI: 10.1061/~ASCE!0733-9429~2002!128:7~664! CE Database keywords: Velocity profile; Vegetation; Channel flow.

High resolution flight observations and numerical simulations: horizontal variability in the wintertime boreal boundary layer

High resolution aircraft observations made along flight tracks over inhomogeneous surface in the late wintertime boreal zone are described and compared to 2D mesoscale model simulations with surface properties defined at 2 km resolution from maps. All observations displayed the expected small-scale turbulence. On top of that, the near-surface wind speeds (but not directions) showed mesoscale variations related to local topography and roughness. Upward (but not downward) SW and LW radiative fluxes and ground temperature also displayed mesoscale variability; in SW radiation this was clearly due to local albedo changes. In the sensible heat flux there was strong horizontal variation near the surface in correlation with surface types. The above observed mesoscale along-track variations were reasonably well represented by the mesoscale model simulation. The track-averaged observed sensible and latent heat flux profiles were in rough agreement with a mixing length approach, which used the track-averaged wind, temperature and moisture profiles as input (mimicking a first-order turbulence closure scheme of a GCM).

Sensitivity of Arctic climate simulations to different boundary-layer parameterizations in a regional climate model

Arctic climate simulations with the high resolution regional climate model HIRHAM showsome deviations from station data in the planetary boundary layer (PBL) during winter, whichindicates the necessity of improvements in the atmospheric PBL parameterization for a betterdescription of the vertical stratification and atmosphere–surface energy exchange. A1-dimensional single column model scheme has been used to investigate the influence of twodifferent PBL parameterizations in monthly integrations for January 1991 and July 1990. Thefirst scheme uses the boundary layer parameterization of the atmospheric circulation modelECHAM3, including the Monin–Obukhov similarity theory in the surface layer and a mixinglength approach above. The second scheme applies the Rossby-number similarity theory forthe whole PBL, connecting external parameters with turbulent fluxes and with universal functionsdetermined on the basis of Arctic data. For both schemes the heat and humidity advectionhas been determined as residual term of the PBL balance equations. Diabatic sources havebeen computed from the current model solution and local temperature and humidity changesare estimated from radiosonde data. The simulated vertical structure and the atmosphere–surface energy exchange during January strongly depends on the used PBL parameterizationscheme. These different PBL parameterization schemes were then applied for simulations of theArctic climate in the 3-dimensional regional atmospheric climate model HIRHAM, usingECHAM3 with Monin–Obukhov similarity theory, ECHAM3 with Rossby-number similaritytheory and ECHAM4 parameterizations with a turbulent kinetic energy closure. The nearsurface temperature, the large-scale fields of geopotential and horizontal wind are simulatedsatisfactorily by all three schemes, but strong regional differences occur. The results show asensitivity to the type of turbulence exchange scheme used. The comparison with ECMWFanalyses and with radiosonde data reveals that during January ECHAM3 with Rossby numbersimilarity theory more succesfully simulates the cold and stable PBL over land surfaces, whereasover the open ocean ECHAM3 with Monin Obukhov similarity works better. ECHAM3 withRossby-number similarity theory delivers a better adapted vertical heat exchange under stableArctic conditions and reduces the cold bias at the surface. The monthly mean surface turbulentheat flux distribution strongly depends on the use of different PBL parameterizations and leadsto different Arctic climate structures throughout the atmosphere with the strongest changes atthe ice edge for January.

Spatiotemporal structure of resistive pressure-gradient-driven turbulence

High-resolution turbulence calculations are needed to characterize the resistive pressuregradient-driven turbulence in toroidal geometry because the wave number spectrum is very broad with a low decay index. Here, two main issues are addressed: the spatiotemporal structure of the turbulence and the validity of the mixing-length approach for toroidal geometry. Well above the instability threshold, steady-state turbulence is established. In this case, the relevant spatial scales are the width of the individual poloidal components of the ballooning mode, instead of the toroidal scale lengths. The situation is reversed near the instability threshold, where the stationary state can be dominated by one toroidal mode.

Numerical Simulation of Drag Reducing Turbulent Flow in Annular Conduits

Inadequate transport of rock cuttings during drilling of oil and gas wells can cause major problems such as excessive torque, difficulty to maintain the desired orientation of the drill string, and stuck or broken pipe. The problem of cuttings transport is aggravated in highly inclined wellbores due to the eccentricity of the annulus which results in nonuniformity of the flowfield within the annulus. While optimum cleaning of the borehole can be achieved when the flow is turbulent, the added cost due to the increased frictional losses in the flow passages may be prohibitive. A way around this problem is to add drag-reducing agents to the drilling fluid. In this way, frictional losses can be reduced to an acceptable level. Unfortunately, no model is available which can be used to predict the flow dynamics of drag-reducing fluids in annular passages. In this paper, a numerical model is presented which can be used to predict the details of the flowfield for turbulent annular flow of Newtonian and non-Newtonian, drag-reducing fluids. A one-layer turbulent eddy-viscosity model is proposed for annular flow. The model is based on the mixing-length approach wherein a damping function is used to account for near wall effects. Drag reduction effects are simulated with a variable damping parameter in the eddy-viscosity expression. A procedure for determining the value of this parameter from pipe flow data is discussed. Numerical results including velocity profiles, turbulent stresses, and friction factors are compared to experimental data for several cases of concentric and eccentric annuli.

Turbulent particle deposition on a horizontal circular plate

Deposition of micron and submicron particles onto a circular plate was studied numerically under special consideration of turbulence at low Reynolds number flows. Therefore a standard k- ϵ model was modified with a mixing length approach to describe the flow and particle concentration field in the vicinity of the surface. The investigated conditions, such as low flow velocities from the top to a horizontal surface, low turbulence levels and almost homogeneously distributed particles, are typical for a clean room environment. The calculated data were compared with the numerical data for the laminar case. An unexpected result was found, namely that turbulence causes only a negligible transport effect under these conditions. Only at the edge of the circular plate turbulence produced by the wall lead to an increasing deposition velocity. A second series of numerical analysis changing turbulence intensities of the incoming flow was performed by modelling a grid with known boundary conditions above the plate. In the particle size range below 0.1 μm the increasing turbulence intensity caused an increase of the particle deposition rate. The desirable experimental verification of these numerical results does not seem appropriate because of the expected minimal deviation of the experimental data.

DB white dwarf evolution in the frame of the full spectrum turbulence theory

We present an analysis of the evolution of carbon-oxygen DB white dwarfs (helium­ rich envelope) for a wide range of effective temperatures and luminosities. To this end, we employ a full stellar evolution code, in which we include a new equation of state for helium plasmas recently developed by Saumon, Chabrier & Van Hom and new OPAL radiative opacities. The most important feature of our models is that the transport of energy by convection is described by the full spectrum turbulence theory. In particular, we have adopted two versions of this theory for stellar convection: the Canuto & Mazzitelli theory and the more recent, self-consistent theory developed by Canuto, Goldman & Mazzitelli. Both theories, which have no free parameters and account for the whole spectrum of turbulent eddies, represent a great improvement compared to the mixing-length theory approach used thus far in almost all white dwarf studies. Neutrino energy losses as well as crystallization were taken into account. In order to explore the sensitivity of our results to various input model parameters, we vary the model mass from 0.5 to 1.0 Mo in intervals of 0.1 Mo, and the helium layer mass in the interval of 10-6 ~ MHe/M* ~ 10- 2 • The emphasis is put mainly on the behaviour of the evolving outer convection zone. In particular, we analyse the dependence of the location of the theoretical blue edge of the instability strip on the various input parameters. We find that the new ingredients we have incorporated in this study - mostly the new formulations for stellar convection -lead to theoretical blue edges in agreement with observations of pulsating DB white dwarfs. In this context, the Canuto, Goldman & Mazzitelli self­ consistent theory yields theoretical blue edges somewhat hotter than those given by the Canuto & Mazzitelli theory, which is more consistent with a recent determination of the effective temperature of the hot DBV GD 358. Contrary to previous results, we find that, according to the new theories for convection, non­ variable DB white dwarfs falling within the instability strip cannot be low-mass configurations. In order to compare with previous computations, we include in our calculations the most common parametrizations of the mixing-length theory usually employed in almost all previous white dwarf studies. In this context, we find that the ML2 parametrization provides a reasonable agreement with the observed blue edge for the DB instability strip. However, the profile of the outer convective zone given by the mixing-length theory is markedly different from that given by both of the new convective formulations.

Mixing Length Approach to Dispersion Through Saturated Porous Media and its Relation to Grain Sizes.

In this study, dispersion phenomenon inside the saturated porous medium have been discussed applying mixing length theory concept, which is usually used for the turbulent mixing of the particles in surface water regimes. According to this theory, dispersion coefficient can be expressed as a product of the mixing length and the standard deviation of the pore water velocities. It is shown that the dispersion mechanism is different in saturated and unsaturated flow because the change in saturated flow velocity is attributed to the change in hydraulic gradient while the unsaturated flow velocity changes due to the different degrees of saturation of the porous medium. To check the model for dispersion coefficient and see the characteristics of mixing lengths with respect to grain sizes of the medium, miscible displacement experiments under saturated conditions were carried out using three different sizes uniform glass beads as porous media. Coefficient of variation at saturation, a parameter in saturated dispersion coefficient model, was determined from soil characteristics curves, i.e., suction-moisture content and hydraulic conductivity-moisture content relationships. Mixing lengths for a particular medium were found almost constant which assumed a value of 1.8 times the grain diameter of the medium. The applicability of the model for the dispersion coefficient, for saturated condition, based on mixing length theory concept was checked by experimental data.

A dynamic approach to sediment‐laden turbulent flows

An approach that studies the effects of the dynamic coupling between turbulent flow and sediment in suspension on the mean velocity and the sediment concentration profiles of sediment‐laden open‐channel flows is systematically developed. The analysis is based on the Boussinesq approximation to the governing equations that treats the mixture of the liquid and sediment as a single fluid with changing density over the flow depth. As a result, it permits the direct systematic comparison between the turbulent flow of clear water and that with suspended sediment. By further employing the mixing length approach for the related turbulent closure problem, the coupling effects on mean velocity and concentration profiles are evaluated, and the resulting distributions are compared with available experimental data. The derived mean velocity distribution contains profiles of existing models as special cases and thus unifies the results of existing major models conceptually and quantitatively. Moreover, the analysis clearly demonstrates the importance of the coupling effect on the suspended sediment concentration profile and, consequently, a more generalized formula of concentration profile is obtained. Although the mean velocity and concentration profiles are the main concern of this paper, the approach establishes a framework for further study of the effects of the dynamic interaction on other physical quantities.

Thermal diffusivity induced by resistive pressure-gradient-driven turbulence

The calculation of the thermal diffusivity induced by resistive pressure-gradient-driven turbulence is revisited. The relevant correlation length for the magnetic fluctuations is shown to be different from estimates made with a mixing-length approach and with dimensional analysis. This leads to different scaling predictions for the electron heat diffusivity. The root of this difference is the fact that the flux and current perturbations are related by two scale lengths, the poloidal wavelength and the mode width.

Variation of magnetic fields and flows during the solar cycle

We have studied dynamo models with realistic assumptions for the induction effects and compare the resulting magnetic fields with observations. We make use of recent results of helioseismology to infer the differential rotation and adopt mixing length approach to get the form of the α-effect, which is in general anisotropic. Dynamo models are computed numerically using an eigenvalue method. Finally, we discuss the resulting Lorentz force, which is relevant for explaining observed flows such as the torsional waves, as a back reaction of the dynamo magnetic fields. Comparison is made in particular with Mt. Wilson/Kitt Peak magnetograms and synoptic charts showing the magnetic flux, torsional oscillation pattern, and the distribution of ephemeral regions, coronal activity, and polar faculae.

Determination of bottom boundary layer parameters at two shallow sea sites using the profile method

Time series of mean current profiles within 3.5 m of the bottom were obtained in two hydrographically different regions using a moored self-recording measuring device. The shear velocity, calculated by the profile method, reached maximum values of u * = 6cm s −1. Tests of the significance of the logarithmic fit yielded good results with values for the correlation coefficient r > 0.9 most of the time. Values of z 0 are very high (0(cm)) and show a systematic variability with the velocity. The profiles are slightly concave, thus z 0 and u * decrease with the profile height considered. Both results indicate differences between observations and the Prandtl theory. Factors influencing the shape of the profile, like nonstationarity, bottom forms, and stratified suspended matter are discussed. Considering the different hydrography and morphology of the two sites it is concluded that suspended matter can account for a part of the profile deformation observed. The large roughness lengths and their relation to velocity can be explained by a modification of Prandtl's mixing length approach which is valid even for stationary unstratified flow.

Dry deposition to a uniform canopy: Evaluation of a first-order-closure mathematical model

A mathematical model is developed for the transport of momentum and matter within a canopy consisting of identical elements protruding vertically from a smooth substrate. Turbulent flux is modelled using a mixing-length approach. The loss of momentum (or matter) to individual elements is related to the mean wind speed, and the element-element interaction via the turbulent wind field is represented by a sheltering factor. Careful consideration is given to the formulation of lower boundary conditions. The model assumptions are compared with those of other models. The model predictions are compared with measurements on a vertically- and horizontally-uniform artificial canopy in a wind-tunnel. The model reproduces well the observed relationship between the parameters of the logarithmic wind speed profile above the canopy and the observed deposition velocities of thorium-B (ThB) atoms and particles in the diameter range 0.08–32 μm, using a sheltering factor which is little dependent on wind speed and has the same magnitude for momentum, gas and particles. The predicted dependences of deposition velocity on friction velocity and, for particles, on diameter shed light on the performance of semi-empirical correlations proposed in the literature. For ThB atoms, the calculated deposition velocities are compared with those of other mathematical canopy models: a comparable degree of agreement is obtained here with fewer free parameters. The fraction of deposit on the substrate is underpredicted by an order of magnitude in some cases, pointing to the limitations of the modelling of conditions near the substrate in terms of quasi-shear flow.