Mixing Length Distributions Research Articles

Turbulence and Diffusion on Weakly Stable and Stable Nights near a 300 m Tower in a Complex Landscape

Abstract Turbulence and winds below 328 m were measured on 5 successive nights in a program to study tracer transport in the nocturnal boundary layer at a site with moderately complex terrain and mixed land use. The instruments included sonic anemometers and CO2/H2O analyzers at four levels on a 328 m tall tower, a minisodar/RASS system, a midrange sodar, a ceilometer, and an array of 61 m towers. Preliminary simulations indicated satisfactory perfluorocarbon mixing to 68 m but insufficient transport to the 328 m level on both weakly stable and stable nights, possibly due to insufficient turbulence kinetic energy and/or small vertical mixing lengths, or the presence of meso-β fronts, e.g., sea-breeze fronts, that could transport trace chemicals efficiently to 328 m. To examine the problem further, time–height distributions of turbulence kinetic energy (TKE), mixing length, Richardson number, potential temperature, and winds were derived from the observations of mean winds and temperature and the TKE budget equation, interpolated to fit the observations, under the flux/gradient and z-less scaling assumptions, and displayed with aerosol profiles. The results indicated higher and more variable levels of TKE and mixing lengths above a typical turbulence maximum at 30–50 m. Oscillations with periods of ∼2 h were common and occasional meso-β fronts and shear zones between 75 and 150 m were seen, which increased TKE aloft and in some cases led to a poorly defined boundary layer top. Significance Statement The atmosphere’s boundary layer is the interface between the free atmosphere and natural and human activity near Earth’s surface. The daytime boundary layer has been studied extensively and, because of vigorous sun-driven mixing, is well understood and readily parameterized in forecast and global climate models. In contrast, the nocturnal boundary layer is less well understood or predictable because turbulence is weak and tends to decouple it from the surface and the free atmosphere above. This paper focuses on the least-studied upper part of the nocturnal boundary layer over the southeastern United States where topography and land–sea contrast affect winds, turbulence, and chemical transport.

Mixing Behavior and Pressure Drop Analysis of Micromixer with Different Geometric Conditions

A 3-D design of and analysis of fluid flow in the micromixer with different configurations is carried out in this dissertation. The main purpose of this research is to obtain minimum mixing length as rapid mixing is essential in many of the micro-fluidic systems used in biochemistry analysis, drug delivery, sequencing, or synthesis of nucleic acids. Also effect on various parameters such as mixing behavior, volume arrow, mixing length, maximum velocity, maximum pressure, pressure drop, and velocity distribution were analyzed by changing the mixing angle between inlets. Micromixers with square cross-section rectangular mixing chamber with various types of obstacle place in fluid flow paths such as rectangular obstacles, elliptical obstacle, and circular obstacle in split and recombination manner were designed for the analysis. The micromixer has 3 inlets and 1 outlet. Water and ethanol were used as working fluids. For computational fluid dynamics analysis, COMSOL Multiphysics 5.0 is used. From various results, we have found that size, the geometry of mixing chambers and obstacles, and mixing angle effect mixing length, pressure, and velocity. With a decrease in mixing angle mixing length, pressure drop, and maximum velocity decrease i.e it gives better mixing performance. Also with an increase in the number of obstacles mixing length and maximum velocity decreases and pressure drop increases. Micromixer with mixing angle 60 degree and circular obstacles gives minimum mixing length than any other models consisting rectangular or elliptical obstacle and mixing angle greater than 60 degrees.

Reconsideration of the overlap region in smooth shallow open channel flows

In this paper, data sets for mean velocity distributions in smooth shallow open channel flow are reconsidered to evaluate the characteristics of the overlap region and estimate the friction velocity (u∗). Both new and existing velocity measurements are used in the analysis. The velocity profiles are obtained using laser Doppler velocimetry and particle image velocimetry at typical Reynolds numbers (20 000–60 000) achieved in laboratory flumes. Validation of the estimated u∗ values using different forms of power law is established by comparing these values with the ones available in literature. Also, the Reynolds shear stress distribution based on two-dimensional measurements validate the estimated u∗. The availability of new data sets allows one to verify the usefulness of the log-law and evaluate the log-law constants. Different fitting methods; least squares, derivative, and scattered methods are used to evaluate the value of von Kármán coefficient. It is found that the value of κ obtained from the least squares method varies between 0.35 and 0.51 and depends on the Reynolds number. This refutes the conventional constant value assumption for the von Kármán coefficient (κ = 0.41). By considering the Prandtl’s mixing-length theory, the present values of the von Kármán coefficient are used to evaluate the mixing length distributions. The mixing length distributions in smooth open channel flow are found to depend on Reθ.

PIV Study of Shallow Open Channel Flow Over d- and k-Type Transverse Ribs

A particle image velocimetry was used to study shallow open channel turbulent flow over d-type and k-type transverse ribs of square, circular, and semi-circular cross sections. The ratio of boundary layer thickness to depth of flow varied from 50% to 90%. The mean velocities and turbulent quantities were evaluated at the top plane of the ribs to characterize interaction between the cavities and overlying boundary layer. It was found that the overlying boundary layer interacts more strongly with k-type cavities than observed for d-type cavities. The profiles of the mean velocities and turbulent statistics were then spatially averaged over a pitch, and these profiles were used to study the effects of rib type and cross section on the flow field. The mean velocity gradients were found to be non-negligible across the boundary layer, and the implications of this observation for momentum transport, eddy viscosity, and mixing length distributions are discussed. The results show that the skin friction coefficient, Reynolds stresses and mixing length distributions are independent of rib cross section for d-type. For the k-type ribs, significant variations in skin friction coefficient values, mean flow, and turbulence fields are observed between square ribs and circular/semi-circular ribs.

Influence of short roughness strip on the turbulent boundary layer structure

Hot-wire measurements have been undertaken in a turbulent boundary layer which is subjected to an impulse in form of a short roughness strip with the aim of examining its influence on the structure of the turbulent boundary layer. The results indicate that, while the energy containing motion is shifted from low wave number to high wave number near the wall due to the interfering of the roughness strip with the near-wall structure, the reverse is the case in the outer region. While the anisotropy at small scale changes appreciably, there is no discernable change at the large scale when distance from the wall is increased as reflected in the collapses of spectra shear correlation coefficient at the low wave number. It further shows that the roughness strip alters the flow dynamics of the boundary layer as shown in the changes in the mixing length distribution.

A mixing-length formulation for the turbulent Prandtl number in wall-bounded flows with bed roughness and elevated scalar sources

Turbulent Prandtl number distributions are measured in a laboratory boundary layer flow with bed roughness, active blowing and sucking, and scalar injection near the bed. The distributions are significantly larger than unity, even at large distances from the wall, in apparent conflict with the Reynolds analogy. An analytical model is developed for the turbulent Prandtl number, formulated as the ratio of momentum and scalar mixing length distributions. The model is successful at predicting the measured turbulent Prandtl number behavior. Large deviations from unity are shown in this case to be consistent with measurable differences in the origins of the momentum and scalar mixing length distributions. Furthermore, these deviations are shown to be consistent with the Reynolds analogy when the definition of the turbulent Prandtl number is modified to include the effect of separate mixing length origin locations. The results indicate that the turbulent Prandtl number for flows over complex boundaries can be modeled based on simple knowledge of the geometric and kinematic nature of the momentum and scalar boundary conditions.

Turbulence Measurements in a Heated, Concave Boundary Layer Under High-Free-Stream Turbulence Conditions

Turbulence measurements for both momentum and heat transfer are taken in a lowvelocity, turbulent boundary layer growing naturally over a concave wall. The experiments are conducted with negligible streamwise acceleration and a nominal freestream turbulence intensity of ∼8 percent. Comparisons are made with data taken in an earlier study in the same test facility but with a 0.6 percent free-stream turbulence intensity. Results show that elevated free-stream turbulence intensity enhances turbulence transport quantities like uv and vt in most of the boundary layer. In contrast to the low-turbulence cases, high levels of transport of momentum are measured outside the boundary layer. Stable, Go¨rtlerlike vortices, present in the flow under low-turbulence conditions, do not form when the free-stream turbulence intensity is elevated. Turbulent Prandtl numbers, Prt, within the log region of the boundary layer over the concave wall increase with streamwise distance to values as high as 1.2. Profiles of Prt suggest that the increase in momentum transport with increased free-stream turbulence intensity precedes the increase in heat transport. Distributions of near-wall mixing length for momentum remain unchanged on the concave wall when free-stream turbulence intensity is elevated. Both for this level of free-stream turbulence and for the lower level, mixing length distributions increase linearly with distance from the wall, following the standard slope. However, when free-stream turbulence intensity is elevated, this linear region extends farther into the boundary layer, indicating the emerging importance of larger eddies in the wake of the boundary layer with the high-turbulence free stream. Because these eddies are damped by the wall, the influence of the wall grows with eddy size.

Effect of scale on liquid recirculation in bubble columns

Successful scale-up of liquid hydrodynamics in bubble columns using a one dimensional model requires an adequate closure scheme for the Reynolds shear stress. The existing correlations for the prescription of the eddy viscosity or the mixing length scale are demonstrated to be applicable only for a limited range of conditions and consequently cannot be used for scale-up predictions. A method for estimating the mixing length scale has been explored and an attempt at unifying a wide range of data available in the literature within the purview of the method has been made. The futility of such an attempt is attributed to the nonreproducibility of the flow in different laboratories and the consequent lack of data obtained under identical conditions. It is demonstrated, however, that scale-up based on the mixing length distribution is possible when it is obtained from a consistent set of data for liquid velocity and gas void fraction profiles. Such a data set was obtained in the author's laboratory using CARPT (Computer Automated Radioactive Particle Tracking) and CT (Computed Tomography). Using the present method for prescribing the mixing length scale, model predictions for scale-up compare satisfactorily for the data that were obtained in the author's laboratory.

On the Turbulent Mixing Length in the Oceanic Boundary Layer

Abstract Turbulence measurements from under drifting pack ice illustrate the distribution of turbulent mixing length in the well-mixed layer of the upper ocean. Mixing length (λ ≡ K/u*, where K and u* are the local eddy viscosity and square root of Reynolds stress, respectively) is found to vary inversely with wavenumber at the peak in the weighted vertical velocity variance spectrum: λ = cλ/kmax. This relation provides an empirical tool for making local estimates of eddy viscosity if the Reynolds stress is known, or alternatively, deriving fluxes via the inertial-dissipation method in the outer part of the rotational boundary layer. The vertical structure of λ is described for conditions of (i) neutral stratification (negligible surface buoyancy flux) under thick ice. (ii) stable stratification under rapidly melting ice, and (iii) statically unstable conditions from intense freezing. The last example, obtained during the 1992 Lead Experiment, included dime measurements of turbulent salinity flux, 〈w′S′〉....

Characteristics of Velocity Profiles in Steep Open-Channel flows

Velocity measurements, using fiber-optic laser-doppler anemometer, were conducted in steep open-channel flows on smooth and incomplete rough beds. Bed slope was changed from 1/500 to 1/25, which covers Froude number from 0.5 to 3. In this condition, the law of the wall and turbulence intensity distribution were examined. The von Karman constant Kis probed to universal, but the integral constant A decreases as the bed shear stress becomes larger. This decrease of A can be expressed by the decrease of the damping factor B in the van Driest's mixing length distribution in the near-wall region. Turbulence intensity u' normalized by friction velocity U* decreases from the semiempirical universal formula with the increase of the bed shear stress, in the same manner as the velocity distribution.

Three-component LDA measurements in a turbulent boundary layer

An experimental study of a three-dimensional, pressuredriven, attached turbulent boundary-layer flow was made at Mach 0.4. Both the mean velocities and the full Reynolds stress tensor were measured simultaneously by a three-component LDA system. Value of the resultant shear stress to turbulent kinetic energy ratio varied between 0.1 and 0.2 and did not remain constant across the boundary-layer. Slopes of the streamwise and azimuthal mixing length distributions in the wall region were around 0.4 and 1.2, respectively. Skew angle of the turbulent shear stress was larger than skew angle of the velocity gradient.

Local phenomenological properties of turbulence in the suspension flow of solid particles-gas system

An adaptive mathematical model was developed to determine the local phenomenological properties of turbulence in two phase solid particles-gas system. Equations for predicting the dependence of the eddy viscosity and mixing length on the dimensionless radial position in the pipe were obtained. The proposed equations have been tested using own experimental measured data as well as data available in literature devoted to Newtonian turbulent system. Distributions of eddy viscosity and mixing length over the cross-section are given. In deriving the set of equations for the calculation of eddy viscosity and mixing length we assumed the non-Newtonian character of the turbulent two phase solid particles-gas flow.

Distribution of eddy viscosity and mixing length in a non-Newtonian flow of a solid particle-gas suspension

The present work concerns the study of the radial distribution of eddy viscosity and mixing length and their dependence upon the Reynolds number and the concentration of the solid phase in a non-Newtonian flow of a suspension of solid particles in a gas. The investigated systems have a pseudoplastic character and the deviation from Newtonian behaviour increases with an increase in the concentration of the dispersed phase. Relations are presented for the eddy viscosity and mixing length in the flow of pseudoplastic fluids. From the analysis of results it follows that the mixing length and eddy viscosity increases with an increase in the Reynolds number. In contrast, an increase in the concentration of the solid phase and consequently of the pseudoplasticity causes a decrease in the investigated quantities. The radial distribution of the mixing length and the eddy viscosity is characterized by a maximum, after which the investigated quantities vary only slightly. This enables the area of the core of the turbulent flow to be defined.

Open‐channel Flow Measurements with a Laser Doppler Anemometer

A powerful two‐color Laser Doppler Anemometer (LDA) system, with direct digital signal processing has been used to measure accurately the longitudinal and vertical velocity components in two‐dimensional, fully‐developed open‐channel flow over smooth beds. The law of the wall and the velocity defect law were re‐examined because the log‐law has been often applied to open channels without detailed verification. It was found that the log‐law can be applied strictly only to the near‐wall region. In this region, the von K´rm´n constant κ and the integral constant A are truly universal, having values of κ=0.412 and A=5.29 irrespective of the Reynolds and Froude number. As the Reynolds number becomes larger, the deviation from the log‐law cannot be neglected in the outer region. This deviation can be expressed well by Coles' wake function which involves a Reynolds‐number dependent parameter Π. The distributions of eddy viscosity and mixing length were evaluated and found to depend on Π. All the data including the...

Full-coverage film cooling. Part 2. Prediction of the recovery-region hydrodynamics

Hydrodynamic data are reported in the companion paper (Yavuzkurt, Moffat & Kays 1980) for a full-coverage film-cooling situation, both for the blown and the recovery regions. Values of the mean velocity, the turbulent shear stress, and the turbulence kinetic energy were measured at various locations, both within the blown region and in the recovery region. The present paper is concerned with an analysis of the recovery region only. Examination of the data suggested that the recovery-region hydrodynamics could be modelled by considering that a new boundary layer began to grow immediately after the cessation of blowing. Distributions of the Prandtl mixing length were calculated from the data using the measured values of mean velocity and turbulent shear stresses. The mixing-length distributions were consistent with the notion of a dual boundary-layer structure in the recovery region. The measured distributions of mixing length were described by using a piecewise continuous but heuristic fit, consistent with the concept of two quasi-independent layers suggested by the general appearance of the data. This distribution of mixing length, together with a set of otherwise normal constants for a two-dimensional boundary layer, successfully predicted all of the observed features of the flow. The program used in these predictions contains a one-equation model of turbulence, using turbulence kinetic energy with an algebraic mixing length. The program is a two-dimensional, finite-difference program capable of predicting the mean velocity and turbulence kinetic energy profiles based upon initial values, boundary conditions, and a closure condition.

ビンガム流体の圧力損失に関する研究

In order to predict the relation between friction factor and Reynolds number forBinghain plastic fluid flow in circular smooth pipes, an improved mixing-length distribution is proposed in this paper. The distribution (i.e. Eq.(18)) is a modification of Van Driest's model for Newtonian fluid flow that is obtained through the results of the other researchers' studies about Newtonian and Bingham plastic fluid flow and the authors' mixing-length distribution for pseudo-plastic fluid flow in circular smooth pipes.The proposed mixing-length distribution makes it possible to predict the pressure loss of Bingham plastic fluid flow in circular smooth pipes since 163 out of 194 experimental data shown in Fig. 8 are within ±7% of the theoretical values. A set of theoretical design curves of friction factor versus Reynolds number is calculated and shown in Fig. 9.

Pressure Loss of Pseudo-Plastic Fluid Flow in Pipes

In order to predict both the velocity profile and the relationship between friction factor and Reynolds number for pseudo-plastic fluids flow in smooth pipes, and improved mixing-length distribution is proposed in this paper. This distribution is a modification of Van Driest's model for Newtonian fluids flow obtained by referring to the results of the other researcher's studies about Newtonian and pseudo-plastic fluids. It seems that this mixing-length distribution is suitable for estimating and calculating both the velocity profile and the pressure loss for Newtonian and pseudo-plastic fluids flow in circular smooth pipes through the analyzed results as compared with experimental results. A set of design curves of friction factor versus Reynolds number is presented.

Experimental and Analytical Study of Axial Turbulent Flows in an Interior Subchannel of a Bare Rod Bundle

Reactor fuel elements generally consist of rod bundles with the coolant flowing axially through the bundles in the space between the rods. Heat transfer calculations form an important part in the design of such elements, which can only be carried out if information of the velocity field is available. A one-equation statistical model of turbulence is applied to compute the detailed description of velocity field (axial and secondary flows) and the wall shear stress distribution of steady, fully developed turbulent flows with incompressible, temperature-independent fluid, flowing through triangular arrays of rods with different aspect ratios (P/D). Also experimental measurements of the distributions of the axial velocity, turbulence kinetic energy, and Reynolds stresses were performed using a laser Doppler anemometer (LDA), operating in a “fringe” mode with forward scattering, in a simulated interior subchannel of a triangular rod array with P/D = 1.123 and L/DH = 77. From the experimental results, a new mixing length distribution is proposed. Comparisons between the analytical results and the results of this experiment as well as other experimental data available in the literature are presented. The results are in good agreement.

Evaluation of Mach 5 Nozzle Wall Turbulent Transport Models

The turbulent boundary-layer flow developed on a two-dimensional deLaval nozzle wall was systematically evaluated in terms of its transport properties. The turbulent shear stress, eddy viscosity, mixing length, and turbulent Prandtl number distributions were computed by the time-averaged conservation equations using measured mean-flow inputs. The measurements were obtained in the NOL Boundary-Layer Channel and contained controlled variations of heat transfer, Reynolds number, and pressure gradients. Definite trends relating upstream cooling, pressure gradient, Reynolds number, and heat transfer to the shear stress, eddy viscosity, mixing length, and Prandtl number were observed. In addition, a strong influence of the upstream history effect was observed in all the turbulence terms except the Prandtl number. This effect can be related to the removal of thermal energy upstream. Comparison of the turbulent shear stress with the laser-doppler velocimeter measurements showed disagreement between the measured and computed values in the inner portion of the boundary layer.

Turbulent Couette Flow

Turbulent flow between concentric rotating cylinders is analyzed using an energy balance hypothesis proposed by Zagustin. Predictions of the mean velocity profiles are obtained without any additional assumptions regarding the distribution of mixing length. The theoretical results compare favorably with previous observations with only the inner or outer cylinder rotating, and with new data for counter-rotating cylinders. The theoretical results are also consistent with observed features of plane Couette flow.