Turbulent Kinetic Energy Balance Research Articles

Turbulence intensity in dilute two-phase flows—3 The particles-turbulence interaction in dilute two-phase flow

We propose a simplified theory for the particles-turbulence interaction in a dilute two-phase flow with particles of arbitrary sizes. The theory takes into account two sources of turbulence in particle-laden flows: (i) the carrier fluid velocity gradients; and (ii) turbulent wakes behind the coarse particles. The theoretical description is based on the modified mixing-length theory and turbulent kinetic energy balance method. The solution of the problem for particle-laden flow does not require any additional quantitative empirical data; only the standard semi-empirical parameters for pure carrier fluid are used. The dimensional analysis of the system of equations is used to reduce to the minimum the number of nondimensional parameters in turbulent particle-laden flows. In the limits of fine and coarse particles the asymptotic expression for turbulence intensity is found. It is shown that in the former case the carrier fluid fluctuations intensity is found. It is shown that in the former case the carrier fluid fluctuations depend only on the value of the total mass content of the admixture; whereas in the latter one they are determined by the total mass content of the admixture, the density ratio of the phase and the aerodynamic properties of the particles. The proposed theory is applied to predict the fluctuations intensity in various types of turbulent particle-laden flows. The results of the calculations are in fairly good agreement with the experimental data.

Near-bed turbulence and mass transfer at the sediment-water interface

A near-wall turbulence model is applied to the benthic boundary layer to analyze the diffusive transfer of dissolved substance across the sedimentwater interface. A modified turbulent kinetic energy balance is used to define eddy viscosity and eddy diffusivity in the viscous sublayer. The limiting behavior of turbulence quantities very near the bed is expressed as a power of z, the distance from the sediment-water interface. The predicted flux rate over a wide range of the Schmidt number and dissolved oxygen concentration profile near the bed are in good agreement with observations using microelectrodes.

Near‐bed turbulence and hydrodynamic control of diffusional mass transfer at the sea floor

A simple closure scheme for turbulent transport yields mean‐velocity and dissolved‐substance concentration profiles just above and solute fluxes across a sediment‐water interface underlying uniform, steady flow. In the case of near‐bed turbulence characterization, this approach retains both turbulent and viscous terms in expressions for important structural aspects of the near‐bed regions of turbulent boundary layers. Moreover, a modified turbulent kinetic energy balance is used to define eddy viscosity in the viscous‐dominated region of flow very near the bed. In the case of characterization of mass distribution and diffusional transfer rate, the approach partitions important factors that control mass transfer rate: nearbed turbulent transport, interfacial‐flux boundary condition, and reaction kinetics. Diffusive sublayers overlying smooth beds exposed to typical deep‐sea conditions are predicted to be ∼ 1 mm thick, a result in agreement with the sublayer thickness observed in situ and the inferred thickness used in many hydrodynamic models. One application of the new closure scheme indicates that typical biogenic roughness in fine‐sediment marine environments enhances solute exchange rates threefold over those expected for a smooth bed. This increase is due to enhanced turbulent transport in the vicinity of the rough bed and occurs in spite of viscous ponding of “dead water” among roughness elements.

Density Currents and Shear‐Induced Flocculation in Sedimentation Tanks

A numerical model for the computation of turbulent flows in sedimentation tanks is presented. Based on a nonorthogonal grid formulation, it can handle fairly arbitrary two-dimensional geometries. It solves the full two-dimensional elliptic governing flow equations with the \Ik\N–ϵ\N turbulence model. Effects on momentum, as well as turbulent kinetic energy balances, due to sediment-induced density currents are taken into account. Solids with a broad settling velocity distribution are treated by considering different settling velocity classes. A simple flocculation model assuming only turbulent shear-induced flocculation is formulated, and the importance of this cause of flocculation is examined. Predicted velocity and concentration profiles and removal efficiencies are compared with observations in a tertiary settling basin in Sweden. Results suggest that the flow field is significantly changed by seemingly small density differences, that shear-induced flocculation is relatively unimportant, and that the influent settling velocity distribution is critical for accurately predicting removal efficiencies.

The Structure of Turbulence in the Near-neutral Atmospheric Boundary Layer

Abstract In this paper, the idea of local similarity, previously used to study turbulence in the stably stratified boundary layer, is used to investigate the structure of turbulence in the neutral atmospheric boundary layer and to relate this structure to the turbulent kinetic energy balance. It is found that in the lower half of the boundary layer, the turbulence exhibits local similarity and that the turbulence kinetic energy budget is essentially a balance between shear production and dissipation. In the upper half of the boundary layer, where turbulent transport is a significant term in the turbulence kinetic energy balance, local similarity breaks down. This is consistent with the second-order moment equations, which suggest that local similarity arises in situations where the turbulent and pressure transport terms are small. Results from a large-eddy simulation of the neutral boundary layer are also investigated using local similarity. The simulated turbulence agrees well with the observations for t...

On the Spectra of Turbulent Velocity Field in a Stably Stratified Flow

Abstract This paper describes a method to solve the spectral equation for the balance of turbulent kinetic energy in a stably stratified turbulent shear flow. The cospectra of vertical momentum and heat flux arc modelled with the aid of a basic eddy-viscosity (or turbulent exchange coefficient) function. For the term representing the inertial transfer of turbulent kinetic energy, Pao's [Phys. Fluids 8 (1965)] form is assumed. Analytical expressions for the three-dimensional kinetic energy spectrum as well as the cospectra of momentum and heat flux are obtained over the range of wave numbers k≥kb, which includes the inertial subrange kb≪k≪ks and the viscous subrange k>ks (kb and ks are the buoyancy and Kolmogorov wavenumbers, respectively). The two one-dimensional spectra, e.g., the kinetic energy spectra of the horizontal and vertical components of turbulence are derived from the three-dimensional kinetic energy spectrum. These one-dimensional spectra are compared with the measured data of Gargett et al. [J. Fluid Mech. 144 (1984)] for the case I ( = ks/kb) = 630. Finally, we compute the basic eddy-viscosity function and discuss its behavio

Turbulent kinetic energy balance in a LEBU modified turbulent boundary layer

An experimental study of the turbulent kinetic energy balance is performed in a LEBU manipulated turbulent boundary layer. The estimation of almost all the terms of the k-equation is obtained by hot wire anemometry. Near the manipulating device, strong alterations are observed, when compared with the natural conditions. The wake of the manipulator imposes two distinct zones. The lower part is characterized by negligible production compensated by diffusion, while, in the outer part, a large excess of production is balanced by diffusion and dissipation. The excess of dissipation rapidly vanishes downstream. The relaxation process is slower for production and diffusion.

EXPERIMENTAL STUDY ON TURBULENCE STRUCTURE IN TURBULENT PLANE FORCED PLUME

Experimental results on the mean and turbulent properties, notably the Reynolds shear stress and turbulent heat fluxes of turbulent plane jets and plumes are presented. All the quantities have been measured using two-component hot-film anemometry and resistence thermometry in the developed flow region of hot water issuing into low speed coflowing stream from a sharp-edged rectangular slot of 0.4cm wide. The results show that buoyancy increases the turbulence levels in plumes when compared with jets. For example, the maximum Reynolds shear stress is nearly 60% higher in plumes. The constant eddy viscosity model is found to be applicable to describe the turbulence structure in both flows. Furthermore, the differences in turbulent kinetic energy balance between both flows are examined. The buoyant production of turbulent energy is about 30% of the shear production in plumes.

The structure of radiatively driven convection in stratocumulus

AbstractData from a series of five research flights in marine stratocumulus are used to investigate the structure of radiatively driven, free convective layers. Conditional sampling methods are used to determine the properties of the primary convective elements: negatively buoyant downdraughts. The turbulent velocity data are also subjected to spectral analysis and the turbulent kinetic energy (TKE) balance is evaluated for each case. Results from each case are found to be very similar if scaled using appropriate mixed layer quantities.Downdraughts are selected using a criterion based on vertical velocity. The distributions of their intersected widths at cloud top, and the velocity spectra, are consistent with downdraughts occupying narrow regions (∼0.1‐0.15 h wide, h being the mean mixed layer depth) around the periphery of larger, regular (∼0.5 h‐0.75 h diameter) updraughts as suggested by cellular patterns observed in the cloud tops. Downdraughts are found to carry over half of the total turbulent fluxes of heat and water substance within cloud, and also transport variance down the mean gradients. Their thermodynamic properties are consistent with their containing a small fraction of air from above cloud top although their negative buoyancy is almost entirely due to the incorporation of radiatively cooled air. The properties of the downdraughts near cloud top suggest they are formed primarily as a result of the local horizontal convergence of upwelling motions constrained by the inversion. While being cooled radiatively, these quasi‐horizontal motions scour the base of the inversion, incorporating some drier air before being forced downwards in relatively narrow zones. Comparisons with other convective layers over land or sea suggest that convection in stratocumulus proceeds with a greater mass flux and correspondingly reduced differences between convective elements and the environment, possibly reflecting the improved ventilation possible at an inversion interface compared with much rougher surfaces.The TKE balance divides into three main regions and is consistent with the interpretations given above. The pressure transport is implied to be the largest source term close to cloud top. Comparisons with model results reveal some important differences.

Measurement of various terms in the turbulent kinetic energy balance within a flame and comparison with theory

Direct measurements have been made of four of the six terms in the exact form of the turbulent kinetic energy (TKE) equation within a premixed turbulent flame, in order to understand how the combustion interacts with the turbulence. Results are compared to the predictions of the Bray-Moss-Libby (B-M-L) model. The mean reaction ( w ) also was estimated from the data, using a local control volume concept. The measurements were made possible by the simultaneous use of laser velocimetry and Rayleigh scattering. Increasing the heat release was found to increase the “flame generated” turbulence as well as the “apparent” turbulence and the turbulent flame speed. Part of this flame-generated turbulence is due to vorticity created by the curved flamelets, as evidenced by the unexpected existence of low speed products. The flame also is found to exert an upstream influence that causes an increase in the turbulence of the pure reactants before these reactants ever reach a flamefront. The measured diffusion term in the TKE balance indicates a countergradient diffusion of TKE toward the center of the flame where TKE is maximum; however, this is not true diffusion since the TKE balance includes apparent turbulence. The B-M-L model correctly predicts quantities which are dominated by intermittency due to flame motion such as the diffusion term in the TKE, and fluxes ϱu″c″ and ϱu″ 2c″ which change sign within the flame. This is because the model successfully accounts for intermittency by separating the physics of reactants and products. The model does not successfully predict even the trends associated with the more subtle “true turbulence” associated with products ( u′ p). It is felt that additional work is needed to remove “apparent turbulence” from the TKE balance, and eventually to replace one closure relation in the model to account for vorticity produced by curved flamelets.

The interaction of turbulence and pressure gradients in a baffle-stabilized premixed flame

Simultaneous measurements of time-resolved velocity and temperature have been obtained by laser-Doppler anemometry and numerically compensated fine-wire thermocouples in the near wake of a premixed flame stabilized on a disk baffle located on the axis, and at the exit, of a confining pipe. The diameter of the disk was 0.056 m, the diameter of the pipe was 0.080 m, the volumetric equivalence ratio with natural gas as the fuel was 0.79 and the Reynolds number, based on pipe diameter and upstream pipe bulk velocity of 9 m/s, was 46 800. The purpose of the measurements is to quantify the relative magnitudes of terms involving the mean pressure gradient and Reynolds stresses in the balance of turbulent kinetic energy and heat flux in a strongly sheared, high-Reynolds-number, reacting flow. The latter term has been associated with non-gradient diffusion in other flows. Source terms involving the mean pressure gradient are large in the conservation of turbulent heat flux but not in the conservation of Reynolds stress. The ‘thin-flame’ model of burning suggests that the sign and magnitude of the heat flux is closely related to the conditioned mean velocities. The mean axial velocity of the reactants is larger (by up to 0.27 of the reference velocity) than that of the products on the low-velocity side of the shear layer that surrounds the recirculation bubble but the reverse is true on the high-velocity side. These observations are related to the sign of the axial pressure gradient, which is associated with the streamline curvature, and the consequent preferential acceleration of the low-density products. Generally, the Reynolds stresses of the products are higher than those of the reactants and, in contrast to previously reported measurements, the contribution to the unconditioned stresses by the difference in the mean velocity between products and reactants, the so-called ‘intermittent’ contribution, is small. This is a consequence of the high Reynolds number of our flow.

Some Budgets of Turbulent Stresses in Round Jets and Diffusion Flames

Abstract Measurements are presented of budgets of shear stress, normal stresses and turbulent kinetic energy in a round air jet and in two H2 diffusion flames, all in a co-flowing air stream. The results are based on simultaneous two-component LDA and Mie-scattering measurements, and are presented in Favre averaged form. The flames show basically similar features to the cold flow, but overall magnitudes of the normalized terms are lower, and the advection contribution to turbulent kinetic energy is much reduced. The turbulence dissipation rate e is obtained from the budgets of turbulent kinetic energy by difference, and is used later to obtain values of the velocity pressure-gradient correlations u ∂p/∂x and ν ∂p/∂r, again by difference. The results are consistent with other published estimates of these terms; whilst they seem small in the turbulent kinetic energy balance, they dominate in the budgets for the stresses, and are largely balanced by production. A comparison with a well-publicized numerical m...

Velocity, temperature, and species characteristics of the flow in a gas-turbine combustor

The isothermal and combusting flow characteristics of a model can-type gas-turbine combustor are reported. In isothermal flow a laser-Doppler velocimeter was used to measure the longitudinal and tangential velocity characteristics, and the mean concentration of a tracer of helium gas, injected through the fuelling device, was used to obtain the mixture fraction distribution. In combusting flow, with propane as fuel, the velocimeter measured density weighted velocity characteristics, thin digitally compensated bare-wire thermocouples measured unweighted temperature characteristics, and a sampling probe transported gas samples to various instruments which measured near density weighted concentrations of O 2, UHC, H 2, CO 2, CO, and NO x . The mean velocity results show that, relative to isothermal flow, combustion increases the strength and decreases the width of the primary vortex and, further downstream, attenuates the magnitude and strength of the swirl by the longitudinal acceleration of the flow. Turbulence measurements indicate that the production of turbulent kinetic energy in the dilution zone of the combustor is due to “conventional” mechanisms by the interaction of shear stress with shear strain. In the recirculation zone, however, gradient transport is negligible and the effect of the mean pressure gradient is likely to be important in the balance of turbulent kinetic energy. The scalar field is dominated by the primary jets. An increase of 34% in the air-fuel ratio causes a similar increase in the pattern factor at the exit plane and a 14% decrease in the combustion efficiency. The pattern factor is improved by 15% and optimal combustion efficiency is obtained by increasing the inlet air temperature by 432K. In the primary zone, combustion is controlled more by physical than chemical kinetic processes and the formation of pollutants is well described by a partial equilibrium model. Downstream of the primary holes the fuel breakdown and the CO oxidation are reaction-rate limited, and the scalar field may be described by a constrained equilibrium model.

Experiments on turbulent boundary layers over a concave surface. 3rd Report. Transport of turbulence kinetic energy.

Turbulent shear flows over a concave surface are subjected to streamline curvature. The objective of the present paper is to clarify experimentally the turbulence kinetic energy and shear stress balance in a boundary layer having a mild surface curvature without substantial pressure gradient. Each term in the transport equation for the turbulence kinetic energy and shear stress is deduced from the data. The extra rate of mean strain 2kU due to curvature makes a positive contribution to the production of v^-2. The effect of the concave curvature on the production of the Reynolds stress - (uv)^^^ is larger than the production of the turbulence kinetic energy. The structure of the turbulent boundary layer over a concave surface is prescribed by the energy balance.

Aircraft observations of the ekman layer during the joint air‐sea interaction experiment

AbstractResults are presented from an observational study of the mid‐latitude atmospheric boundary layer over the sea. the data were obtained mainly by instrumented aircraft as part of the Joint Air‐Sea Interaction Experiment (JASIN).The eight occasions chosen for study are characterized by small surface buoyancy fluxes and the conditions are well described as near neutral and barotropic. In the absence of low‐level inversions, a well‐mixed Ekman layer is observed on each occasion but is limited to a depth of approximately 0.2u */f. Accurate measurements of horizontal gradients (including pressure) together with improved wind observations and turbulent flux measurements enabled each of the terms of the momentum balance to be evaluated throughout the depth of this layer. These terms were found to balance quite closely, were well described by Ekman scaling and were consistent with the requirements of the measured turbulent kinetic energy balance. the latter suggests that the main shear production terms are dissipated locally and that little is exported to upper levels in the Ekman layer to enable significant deepening by entrainment although only a slight increase in instability appears to be needed to alter this balance. Conditions in which such boundary layer structure might be observed are suggested. Values of various coefficients used in schemes for relating surface fluxes to mean quantities (CD, CE, Cg, A, B) are derived and compared with previous measurements.Spectral analysis reveals that most of the turbulent transport is confined to a distinct high wavenumber region whose characteristics vary as a function of Ekman layer depth and stability parameters. This is superimposed on larger‐scale fluctuations which do not vary appreciably within the Ekman layer and which therefore dominate the variances in the upper regions as the intensity of the smaller‐scale turbulence decreases strongly with height.Finally, further implications of this interpretation are discussed with particular reference to the heat and water vapour balance. These imply that removal of water vapour from the Ekman layer is accomplished by transfer related to cloud activity, marking a significant change in the mechanism of turbulent transport at the top of this layer. the relationship between these and similar results obtained concurrently by different methods is also discussed.

Scalar-velocity correlations in a turbulent premixed flame

Measurements are described of the correlations between fluctuations in velocity, static pressure and temperature in a premixed turbulent jet flame. Two component laser velocimetry, thermocouples and a cooled microphone Pitot probe are employed along with appropriate response compensation and spectral analysis techniques to obtain the correlations from time-resolved measurements. Axial and radial surveys of the flame are presented. Evidence is found of counter-gradient diffusion in some regions of the flame. The pressure-velocity correlation is seen to be negative, and its values are comparable to those of other quantities entering the turbulent kinetic energy balance.

The dynamics of stratocumulus: aircraft observations and comparisons with a mixed layer model

Results are presented from a detailed case study of daytime stratocumulus over the North Sea using an instrumented aircraft. The measurements include turbulence fluctuation data, radiation fluxes and droplet spectra and were made both in and out of cloud. The mean structure is discussed and a diagnostic, one-dimensional, mixed layer model is formulated to predict the variation of turbulent fluxes with height and to assess the sensitivity of these solutions to various physical processes. These predictions are compared with observations enabling the effectiveness of some widely used model assumptions to be tested. Water transport by gravitational settling is found to be important throughout the depth of the cloud. Mixing within the cloud is driven by convection generated primarily by radiative effects: strong cooling from cloud top with warming beneath although the net heating of the cloud layer is close to zero. Turbulent kinetic energy is exported downwards via negatively buoyant elements from the zone of intense cooling near cloud top into the body of the cloud. This appears to be achieved by the combined action of the turbulent transport and velocity-pressure correlation terms in the turbulent kinetic energy balance equation which are therefore important in maintaining mixing in the lower part of the cloud. The cloud and sub-cloud layers are found to be decoupled, i.e. they appear as two separated mixed layers. Reasons for this are examined and some further consequences for cloud evolution are investigated. In the particular case studied, potential instability could be generated in the lower layer leading to low-level cumulus formation. These rise into the stratocumulus layer thereby reconnecting the two previously separated regions. The implications for stratiform cloud modelling are discussed and some recommendations for future work made.

The dynamics of stratocumulus: Aircraft observations and comparisons with a mixed layer model

AbstractResults are presented from a detailed case study of daytime stratocumulus over the North Sea using an instrumented aircraft. The measurements include turbulence fluctuation data, radiation fluxes and droplet spectra and were made both in and out of cloud. The mean structure is discussed and a diagnostic, one‐dimensional, mixed layer model is formulated to predict the variation of turbulent fluxes with height and to assess the sensitivity of these solutions to various physical processes. These predictions are compared with observations enabling the effectiveness of some widely used model assumptions to be tested.Water transport by gravitational settling is found to be important throughout the depth of the cloud. Mixing within the cloud is driven by convection generated primarily by radiative effects: strong cooling from cloud top with warming beneath although the net heating of the cloud layer is close to zero. Turbulent kinetic energy is exported downwards via negatively buoyant elements from the zone of intense cooling near cloud top into the body of the cloud. This appears to be achieved by the combined action of the turbulent transport and velocity‐pressure correlation terms in the turbulent kinetic energy balance equation which are therefore important in maintaining mixing in the lower part of the cloud.The cloud and sub‐cloud layers are found to be decoupled, i.e. they appear as two separated mixed layers. Reasons for this are examined and some further consequences for cloud evolution are investigated. In the particular case studied, potential instability could be generated in the lower layer leading to low‐level cumulus formation. These rise into the stratocumulus layer thereby reconnecting the two previously separated regions. The implications for stratiform cloud modelling are discussed and some recommendations for future work made.

Averaging procedures for flow within vegetation canopies

Most one-dimensional models of flow within vegetation canopies are based on horizontally averaged flow variables. This paper formalizes the horizontal averaging operation. Two averaging schemes are considered: pure horizontal averaging at a single instant, and time averaging followed by horizontal averaging. These schemes produce different forms for the mean and turbulent kinetic energy balances, and especially for the ‘wake production’ term describing the transfer of energy from large-scale motion to wake turbulence by form drag. The differences are primarily due to the appearance, in the covariances produced by the second scheme, of dispersive components arising from the spatial correlation of time-averaged flow variables. The two schemes are shown to coincide if these dispersive fluxes vanish.

The simulation of a fair weather marine boundary layer in GATE using a three‐dimensional model

AbstractA detailed three‐dimensional model has been used to simulate a cumulus‐topped tropical oceanic boundary layer. A particularly dense set of observational data from surface‐based and airborne instrumentation collected during one day of the GARP Atlantic Tropical Experiment (GATE) has been used enabling comprehensive initial and boundary conditions to be specified. This also permits detailed comparisons to be drawn between the model and observations.The model is found to reproduce the structure of the observed undisturbed boundary layer quite realistically. The clouds developed in the model are similar to those observed in both amount and physical size and the modelled turbulent fluxes in the subcloud layer agree well with observations. Also, the evolution of the modelled boundary layer is similar to that in which the observations were made. The model confirms that the major effects of the turbulent mixing are to warm the subcloud layer but to moisten and cool the cloud layer. The limited scale of the model (2 km x 2 km x 2 km) does, however, lead to systematic differences between measured and modelled variances.The model's turbulent kinetic energy balance for the coupled cloud‐subcloud layers is presented. Although reliable measurements are only available in the subcloud layer, they confirm that the model's values are qualitatively correct except when very close to the bottom boundary.The model's temperature and humidity profiles are strongly affected by mixing occurring across cloud‐base throughout much of the subcloud layer. This is reflected in the shape of the θv profile which displays a characteristic minimum midway through the subcloud layer. When scaled using cloudbase parameters, the resulting profiles are similar to those obtained in more unstable conditions over land.