Mixing Length Approach Research Articles

Hydrodynamic roughness induced by a multiscale topography

Turbulent flows above a solid surface are characterised by a hydrodynamic roughness that represents, for the far velocity field, the typical length scale at which momentum mixing occurs close to the surface. Here, we are theoretically interested in the hydrodynamic roughness induced by a two-dimensional modulated surface, the elevation profile of which is decomposed in Fourier modes. We describe the flow for a sinusoidal mode of given wavelength and amplitude with Reynolds-averaged Navier–Stokes equations closed by means of a mixing-length approach that takes into account a possible surface geometrical roughness as well as the presence of a viscous sublayer. It also incorporates spatial transient effects at the laminar–turbulent transition. Performing a weekly nonlinear expansion in the bedform aspect ratio, we predict the effective hydrodynamic roughness when the surface wavelength is varied and we show that it presents a non-monotonic behaviour at the laminar–turbulent transition when the surface is hydrodynamically smooth. Further, with a self-consistent looped calculation, we are able to recover the smooth–rough transition of a flat surface, for which the hydrodynamic roughness changes from a regime where it is dominated by the viscous length to another one where it scales with the surface corrugation. We finally apply the results to natural patterns resulting from hydrodynamic instabilities such as those associated with dissolution or sediment transport. We discuss in particular the aspect ratio selection of dissolution bedforms and roughness hierarchy in superimposed ripples and dunes.

Longitudinal Mixing in Flows with Submerged Rigid Aquatic Canopies

The presence of dense submerged vegetation alters mixing characteristics in open channel flows as they cause differential velocities inside and above canopies. The prediction models for longitudinal mixing in the presence of submerged canopies often use the drag coefficient to represent the canopy, which limits the usability of the models when the canopy properties are not fully understood. Here, attempts were made to present a methodology which can be used for deriving the coefficient of longitudinal dispersion in the presence of submerged vegetation based on velocity measurements, using a mixing length approach to model turbulence. An experimental study was conducted in a large-scale laboratory facility to investigate the longitudinal dispersion characteristics in open channel flow with submerged aquatic vegetation canopies. Detailed velocity and solute tracer measurements were undertaken for a representative range of flow velocities. The velocity measurements were used for deriving turbulent shear stress, mixing length, and diffusivity using established theoretical and empirical relationships to derive the longitudinal dispersion. The longitudinal dispersion measured in two locations in the water column for the two canopy submergences was discussed based on the amount of vertical mixing and differential advection. The canopy with a smaller stem length (i.e., higher submergence ratio) has a higher vertical diffusivity, resulting in increased vertical mixing in the water column. The canopy with the higher stem length (i.e., lower submergence ratio) consists of minimal vertical diffusivity, causing the longitudinal dispersion measured above the canopy to be significantly high, even though the longitudinal dispersion measured inside the canopy is much lower. The mathematical model which was adapted for calculating the coefficient of longitudinal dispersion and the tracer results show good agreement, indicating that the N-zone model can accurately predict the longitudinal dispersion in submerged aquatic canopies when used with the presented methodology.

Predicting the critical gradient of ITG turbulence in fusion plasmas

The quasilinear mixing-length approach to efficient prediction of transport in fusion devices is improved to account for the ‘Dimits’ upshift between linear and nonlinear critical pressure gradients in zonal-flow-saturated turbulence regimes. This modification uses the frequency mismatch between modes interacting turbulently to track changes in saturation efficiency. Near criticality, energy is transferred exclusively to stable eigenmodes, rapidly increasing the efficacy of the nonlinearity. The modified quasilinear model is able to predict below-threshold turbulent ion-temperature-gradient-driven transport accurately and also yields significantly improved predictions for trapped-electron-mode transport.

An Accurate Model for Prediction of Turbulent Frictional Pressure Loss of Power-Law Fluids in Eccentric Geometries

SummaryThe effect of axial flow of power-law drilling fluids on frictional pressure loss under turbulent conditions in eccentric annuli is investigated. A numerical model is developed to simulate the flow of Newtonian and power-law fluids for eccentric annular geometries. A turbulent eddy-viscosity model based on the mixing-length approach is proposed, where a damping constant as a function of flow parameters is presented to account for the near-wall effects. Numerical results including the velocity profile, eddy viscosity, and friction factors are compared with various sets of experimental data for Newtonian and power-law fluids in concentric and eccentric annular configurations with diameter ratios of 0.2 to 0.8. The simulation results are also compared with a numerical study and two approximate models in the literature. The results of extensive simulation scenarios are used to obtain a novel correlation for estimation of the frictional pressure loss in eccentric annuli under turbulent conditions. Two new correlations are also presented to estimate the maximum axial velocity in the wide and narrow sections of eccentric geometries.

A depth‐averaged two‐phase model for debris flows over erodible beds

AbstractMass exchange between debris flow and the bed plays a vital role in debris flow dynamics. Here a depth‐averaged two‐phase model is proposed for debris flows over erodible beds. Compared to previous depth‐averaged two‐phase models, the present model features a physical step forward by explicitly incorporating the mass exchange between the flow and the bed. A widely used closure model in fluvial hydraulics is employed to estimate the mass exchange between the debris flow and the bed, and an existing relationship for bed entrainment rate is introduced for comparison. Also, two distinct closure models for the bed shear stresses are evaluated. One uses the Coulomb friction law and Manning's equation to determine the solid and fluid resistances respectively, while the other employs an analytically derived formula for the solid phase and the mixing length approach for the fluid phase. A well‐balanced numerical algorithm is applied to solve the governing equations of the model. The present model is first shown to reproduce average sediment concentrations in steady and uniform debris flows over saturated bed as compared to an existing formula underpinned by experimental datasets. Then, it is demonstrated to perform rather well as compared to the full set of USGS large‐scale experimental debris flows over erodible beds, in producing debris flow depth, front location and bed deformation. The effects of initial conditions on debris flow mass and momentum gain are resolved by the present model, which explicitly demonstrates the roles of the wetness, porosity and volume of bed sediments in affecting the flow. By virtue of extended modeling cases, the present model produces debris flow efficiency that, as revealed by existing observations and empirical relations, increases with initial volume, which is enhanced by mass gain from the bed. Copyright © 2017 John Wiley & Sons, Ltd.

Intense sediment transport: Collisional to turbulent suspension

A recent simple analytical approach to the problem of steady, uniform transport of sediment by a turbulent shearing fluid dominated by interparticle collisions is extended to the case in which the mean turbulent lift may partially or totally support the weight of the sediment. We treat the granular–fluid mixture as a continuum and make use of constitutive relations of kinetic theory of granular gases to model the particle phase and a simple mixing-length approach for the fluid. We focus on pressure-driven flows over horizontal, erodible beds and divide the flow itself into layers, each dominated by different physical mechanisms. This permits a crude analytical integration of the governing equations and to obtain analytical expressions for the distribution of particle concentration and velocity. The predictions of the theory are compared with existing laboratory measurements on the flow of glass spheres and sand particles in water. We also show how to build a regime map to distinguish between collisional, turbulent-collisional, and fully turbulent suspensions.

Wake to wake interaction of floating wind turbine models in free pitch motion: An eddy viscosity and mixing length approach

Experiments were performed using two model wind turbines operated in tandem with a bottom-fixed configuration and a floating configuration with both turbines allowed to freely oscillate in the streamwise direction. Wakes of both turbines were measured using stereoscopic Particle Image Velocimetry. Turbulent characteristics of the far wake of the first turbine acting as the inflow for the downwind turbine were characterized calculating the eddy viscosity and mixing length profiles from the obtained data. The influence of the far wake on the statistical properties of the near wake of the second turbine are compared between the fixed and oscillating configurations. The incoming mixing length clearly influences the Reynolds stresses and turbulence production of the near wake in the shear layer. Below, the connection between incoming mixing length and the near wake is less evident, due to the impact of the nacelle and rotation of the rotor. For the oscillating turbine, the Reynolds stresses and turbulence production in the near wake of the downwind turbine are damped. Vertical fluctuations were found to decrease though an increase in the mean vertical component. New challenges arise in the design of a floating offshore wind farms, in terms of farm layout and load estimations.

Grain‐size distribution in open channel flow by mixing length approach

A mathematical model is developed by theoretical analysis for predicting the grain‐size distribution in suspension over sand and sand‐gravel beds. By including the viscous shear stress of sediment–fluid mixture and the Reynolds shear stress in terms of mixing length of sediment‐mixed fluid, a system of integrodifferential equations of velocity and suspension concentration is developed. The effect of hindered settling is considered in the settling velocity of sediment particle and the effect of particle–particle interaction is taken into account in the reference level. Apart from this, incipient motion probability, pick‐up probability and nonceasing probability of a sediment particle are included in the reference concentration. The proposed system of integrodifferential equations is converted into a system of first‐order differential equations, and the fourth‐order Runge–Kutta method has been adopted for solving the system of equations. The numerical solution of the proposed system reflects the behavior of fluid velocity and grain‐size distribution with changing vertical height over a sediment bed comprising nonuniform grain sizes. The computed values of velocity and grain‐size distribution are compared with a limited data set due to the unavailability of sufficient data in literature, and good agreement has been observed between them. The proposed model provides least errors in comparison to the other existing models in literature. Copyright © 2014 John Wiley & Sons, Ltd.

A signature for turbulence driven magnetic islands

We investigate the properties of magnetic islands arising from tearing instabilities that are driven by an interchange turbulence. We find that such islands possess a specific signature that permits an identification of their origin. We demonstrate that the persistence of a small scale turbulence maintains a mean pressure profile, whose characteristics makes it possible to discriminate between turbulence driven islands from those arising due to an unfavourable plasma current density gradient. We also find that the island poloidal turnover time, in the steady state, is independent of the levels of the interchange and tearing energy sources. Finally, we show that a mixing length approach is adequate to make theoretical predictions concerning island flattening in the island rotation frame.

Spectroscopic analysis of DA white dwarfs with 3D model atmospheres

We present the first grid of mean three-dimensional (3D) spectra for pure-hydrogen (DA) white dwarfs based on 3D model atmospheres. We use CO5BOLD radiation-hydrodynamics 3D simulations instead of the mixing-length theory for the treatment of convection. The simulations cover the effective temperature range of 6000 < Teff (K) < 15,000 and the surface gravity range of 7 < log g < 9 where the large majority of DAs with a convective atmosphere are located. We rely on horizontally averaged 3D structures (over constant Rosseland optical depth) to compute <3D> spectra. It is demonstrated that our <3D> spectra can be smoothly connected to their 1D counterparts at higher and lower Teff where the 3D effects are small. Analytical functions are provided in order to convert spectroscopically determined 1D effective temperatures and surface gravities to 3D atmospheric parameters. We apply our improved models to well studied spectroscopic data sets from the Sloan Digital Sky Survey and the White Dwarf Catalog. We confirm that the so-called high-log g problem is not present when employing <3D> spectra and that the issue was caused by inaccuracies in the 1D mixing-length approach. The white dwarfs with a radiative and a convective atmosphere have derived mean masses that are the same within ~0.01 Msun, in much better agreement with our understanding of stellar evolution. Furthermore, the 3D atmospheric parameters are in better agreement with independent Teff and log g values from photometric and parallax measurements.

A two‐phase model for sheet flow regime based on dense granular flow rheology

A two‐phase model having a μ(I) rheology for the intergranular stresses and a mixing length approach for the turbulent stresses is proposed to describe the sheet flow regime of sediment transport. In the model, two layers are considered: a dilute suspension layer and a dense sediment bed layer. The concentration profile is obtained from the dilatancy law ϕ(I) in the sediment bed layer and from a Rouse profile in the suspension layer. The comparison of velocity profile, concentration profile, and macroscopic parameters (sediment transport rate, thickness, and roughness) with experimental data shows a good agreement. These comparisons demonstrate that the dense granular rheology is relevant to describe intense bed‐load transport in turbulent regime (sheet flow). The transition from the dense static bed to the dilute suspension is well described by the present model. Also, the different regimes of the dense granular rheology seems to be able to capture the transition between collision‐dominant and turbulent‐fluctuations‐dominant sheet flows, depending on the particle's characteristics.

Rocky super-Earth interiors

Aims. We present interior structure models of the recently discovered exoplanets CoRoT-7b and Kepler-10b addressing their bulk compositions, present thermal states, and internal dynamics. We investigate how mantle convection patterns are influenced by the depth-dependence of thermodynamic parameters (e.g., thermal expansivity and conductivity) caused by the extended pressure and temperature ranges within rocky super-Earths. Methods. To model the interior of rocky exoplanets, we construct a four-layer structural model solving the mass and energy balance equations in conjunction with a generalized Rydberg equation of state providing the radial density distribution within each layer. The present thermal state is calculated according to a modified mixing-length approach for highly viscous fluids. Furthermore, the obtained internal structure is used to carry out two-dimensional convection simulations to visualize the mantle convection pattern within massive exoplanets such as CoRoT-7b and Kepler-10b. Results. Both CoRoT-7b and Kepler-10b most likely have large iron cores and a bulk composition similar to that of Mercury. For a planetary radius of Rp = (1.58 ± 0.10) R⊕, a revised total mass of Mp = (7.42 ± 1.21) M⊕, and the existence of a third planet in the CoRoT-7 planetary system, calculations suggest that an iron core of 64 wt-% and a silicate mantle of 36 wt-% is produced owing to the relatively high average compressed density of ρavg = (10.4±1.8) g cm −3 . Kepler-10b’s planetary radius and total mass yield an iron core of 59.5 wt-%, which complements the silicate mantle of 40.5 wt-%. An enhanced radiogenic heating rate owing to CoRoT-7b’s young age (1.2−2.3 Gyr) raises the radial distribution of temperature by only a few hundred Kelvin, but reduces the viscosity by an order of magnitude. The planform of mantle convection is found to be strongly modified for depth-dependent material properties, with hot plumes rising across the whole mantle and cold slabs, which stagnate in the mid-mantle because of the loss of buoyancy. Conclusions. We use a new model approach to determine the detailed interior structures and present thermal states of CoRoT-7b and Kepler-10b. Both planets are found to be enriched in iron. The results imply that modest radiogenic heating does not play a significant role in determining the internal structure of rocky exoplanets. The depth-dependence of thermodynamic properties, however, strongly influences the mantle convection patterns within exoplanets such as CoRoT-7b and Kepler-10b. This may have a significant effect on the thermal evolution and magnetic field generation of close-in super-Earths.

Modeling of particle fluid interactions in a flowing film concentrator

In an effort to test the extent to which fundamental fluid mechanics can provide a prediction of particle separation in a flowing film concentrator, a study on a simple parallel-sided channel has been carried out using glass beads at various particle concentrations and flow rates. The actual particle separation taking place was compared to predictions made by first describing the slurry flow based on the Prandtl's mixing length approach. Particle concentration distributions were described using Hunt's sediment transport model. Settling velocities of particles in water were calculated using either the Richardson and Zaki or Brauer and Thiele models. The latter takes into account particle size and density distributions and solids volumetric concentration. The resulting predictions of particle separation are remarkably good.

Constructing wind profiles in forests from limited measurements of wind and vegetation structure

The aim of the advection initiative ADVEX within CarboEurope-IP (CEIP) was to evaluate the importance of advection on three coniferous sites. A prerequisite and crucial point for calculating the advective terms of CO 2 balance are the assumptions for the construction of wind fields. Wind measurements must be interpolated over complex control volumes. Here the plant area density is identified as a key variable. During ADVEX campaigns, the rectangular control volumes were marked by four towers at the corners, each carrying four sonic anemometers over the volume height (up to 30 m). An additional experiment was designed at the CEIP forest site Tharandter Wald to investigate the error made by interpolation between only four measurement levels. Thirteen sonic anemometers were installed on one tower up to a height of 42 m (canopy height is 30 m). In addition, plant area density profiles were measured to derive plant-specific parameterisation of the wind profiles. Profiles of velocity moments and nondimensional wind speed gradients are discussed as basic characteristics of the wind field. A new parameterisation scheme based on the mixing length approach was developed. The model was validated with the high-resolution data and applied to calculate horizontal advection for the ADVEX sites. In addition to improving the interpolation routine, this model provides the opportunity to reduce the number of large towers needed to calculate in-canopy wind fields.

Mesoscale eddy statistics and implications for parameterization refinements from a diagnosis of a high resolution model of the North Pacific

Abstract Mesoscale eddy statistics in mid-latitude wind-driven gyre interior are obtained using results of an eddy-active z-coordinate model of the North Pacific, whose horizontal resolution is about 6 km, to draw implications for possible refinements to the present eddy parameterization methods. Mainly two materials are presented. First, eddy fluxes of Ertel potential vorticity (PV), salinity, and thickness are evaluated in time-average after projection onto isopycnal coordinates. A largely rotational part, which is constrained to be small-scale (less than 3 ° × 3 ° ) and is nearly divergenceless when compared to the full flux, is removed from these fluxes. The remaining fluxes are regressed onto the along-gradient direction of the mean fields (diagonal part of a mixing tensor or Fickian diffusion) and the direction perpendicular to them (off-diagonal part or skew streamfunction). While the PV and salinity fluxes show agreement in Fickian diffusion coefficients and skew streamfunction magnitude as well as largely down-gradient features, the thickness fluxes are less well-behaved. Instead, thickness fluxes are correlated well with PV fluxes, indicating that they could be parameterized using a diffusive parameterization for PV. Overall, the present diagnosis supports the use of spatially-varying Fickian diffusion and skew streamfunction. Second, a mixing length approach using the eddy size as lengthscale is tested for explaining spatial distributions of Fickian diffusion coefficients for conservative scalars. The Fickian diffusivities are shown to be scaled with a factor of one-fifth to one-third by the ones based on the present mixing length approach. Mixing lengths are connected to instability lengthscales and Eady instability intensity. Overall, the present mixing length approach turns out to be a plausible, but incomplete, predictor of the eddy flux magnitudes.

TURBULENT COUNTERFLOW INDUCED BY SWIRL DECAY

vive wild turbulent mixing in these devices. To this end, an analytical solution to the Reynoldsaveraged Navier–Stokes equations is obtained that describes the turbulent flow in a cylindrical container. The Reynolds stresses are modeled using the Prandtl mixing length approach modified here for swirling flows. A fluid enters the container through a tangential inlet and leaves through a central exhaust, both located at the same end wall. Despite the inlet and exhaust being close, there is no shortcut flow. The fluid goes from the inlet near the sidewall to the dead end, turns around, and goes back near the axis to the exhaust. This global counterflow occurs due to swirl decay caused by friction at the sidewall. The combined effects of the end wall, swirl and friction causes pressure drops from the inlet to the dead end near the sidewall and from the dead end to the exhaust near the axis. Such a pressure distribution drives the counterflow and provides its survival against turbulent mixing. A simple experiment is performed confirming the counterflow geometry.

An engineering model of a developed turbulent flow in a Couette device

An engineering model describing a developed turbulent flow in a Couette device with hydraulically smooth walls is developed. The empirical wall functions are employed for calculation of the velocity distributions in both the laminar and the buffer boundary layers. The core flow is modeled by the Prandtl mixing length approach. In comparison to majority of published investigations the streamline curvature is taken into account, therefore the model developed is valid for wide gap Couette devices. An analytical relation between the non-dimensional torque applied to the inner (rotating) cylinder G and the Reynolds number Re is obtained: 1.103 η Re / ( ( 1 + η 2 ) ( 1 − η ) ) / G = ln G + 2 η / ( 1 + η 2 ) − ln ( ( 1 + η ) / ( 1 − η ) ) + 0.406 , where η is the inner-to-outer radius ratio. In contrast to recently published relations the achieved solution does not contain coefficients, which are different for each radius ratio η and have to be determined experimentally. The analytical expression is in a good agreement with the experimental data of Lewis and Swinney [1] measured at the fixed radius ratio η = 0.724. The derived relation is also compared with the empirical Wendt correlation, obtained in 1933, at the different radius ratios.

The broad Hα, [O III] line wings in stellar supercluster A of NGC 2363 and the turbulent mixing layer hypothesis

Context. SuperclusterA in the extragalactic HII region NGC2363 is remarkable for the hypersonic gas seen as faint extended broad emission lines with a full width zero intensity of 7000km/s. Aims. We explore the possibility that the observed broad profiles are the result of the interaction of a high velocity cluster wind with dense photoionized clumps. Methods. The geometry considered is that of near static photoionized condensations at the surface of which turbulent mixing layers arise as a result of the interaction with the hot wind. The approximative treatment of turbulence is carried out using the mixing length approach of Canto & Raga. The code mappings Ic is used to derive the mean quantities describing the flow and to compute the line emissivities within the turbulent layers. The velocity projection in three dimensions of the line sources is carried out analytically. Results. A fast entraining wind of up to ~4300km/s appears to be required to reproduce the faint wings of the broad H-alpha and [O III] profiles. A slower wind of 3500km/s, however, can still reproduce the bulk of the broad component and does provide a better fit than an ad hoc Gaussian profile. Conclusions. Radial acceleration in 3D (away from supercluster A) of the emission gas provides a reasonable first order fit to the broad line component. No broad component is predicted for the [N II] and [S II] lines, as observed. The wind velocity required is uncomfortably high and alternative processes that would provide comparable constant acceleration of the emission gas up to 4000km/s might have to be considered.

Velocity distribution of flow with submerged flexible vegetations based on mixing-length approach

By choosing a PVC slice to simulate flexible vegetation, we carried out experiments in an open channel with submerged flexible vegetation. A 3D acoustic Doppler velocimeter (micro ADV) was used to measure local flow velocities and Reynolds stress. The results show that hydraulic characteristics in non-vegetation and vegetation layers are totally different. In a region above the vegetation, Reynolds stress distribution is linear, and the measured velocity profile is a classical logarithmic one. Based on the concept of new-riverbed, the river compression parameter representing the impact of vegetation on river is given, and a new assumption of mixing length expression is made. The formula for time-averaged velocity derived from the expression requires less parameters and simple calculation, and is useful in applications.

A Simple Model for Spatially-averaged Wind Profiles Within and Above an Urban Canopy

This paper deals with the modelling of the flow in the urban canopy layer. It critically reviews a well-known formula for the spatially-averaged wind profile, originally proposed by Cionco in 1965, and provides a new interpretation for it. This opens up a number of new applications for modelling mean wind flow over the neighbourhood scale. The model is based on a balance equation between the obstacle drag force and the local shear stress as proposed by Cionco for a vegetative canopy. The buildings within the canopy are represented as a canopy element drag formulated in terms of morphological parameters such as λ f and λ p (the ratios of plan area and frontal area of buildings to the lot area). These parameters can be obtained from the analysis of urban digital elevation models. The shear stress is parameterised using a mixing length approach. Spatially-averaged velocity profiles for different values of building packing density corresponding to different flow regimes are obtained and analysed. The computed solutions are compared with published data from wind-tunnel and water-tunnel experiments over arrays of cubes. The model is used to estimate the spatially-averaged velocity profile within and above neighbourhood areas of real cities by using vertical profiles of λ f .