Kinematic Eddy Viscosity Research Articles

CFD-PBM modelling of tailings flocculation in a lab-scale gravity thickener

Flocculation occurring in a feedwell plays a critical role in tailings slurry thickening process, which is complicated and significantly influenced by flow characteristics. This work presents a numerical approach to explore the effect of flow characteristics on flocculation performance. It combines an aggregation kernel and a breakage kernel, used to describe the polymer-bridging flocculation kinetics, with a conventional Computational Fluid Dynamics-Population Balance Model (CFD-PBM) coupling to model the complex flocculation-thickening behavior in a lab-scale gravity thickener. The solid-liquid phase interaction is described by an Euler-Euler approach with a modified Schiller-Naumann drag model. The turbulence of liquid phase is resolved by the RNG k-ε turbulence model, while the solid kinematic eddy viscosity is described by a dispersed phase zero equation model. The capability of this proposed model is validated by a good agreement between experimental and predicted results in terms of single-liquid velocity and floc size distribution. The momentum and turbulence dissipation rates are investigated in and around the feedwell over a wide range of feed velocities, showing that the momentum and turbulence dissipation rates have a positive correlation with feed velocity. The momentum and turbulence dissipation rates decrease with the increase in scaled depth in the feedwell. The formation of large vortexes in the feedwell may cause a locally low turbulence dissipation rate. A reasonable increase of feed velocity favours the flocculation, however, an excessive feed velocity can cause a decrease in mean floc size. Modelling tailings flocculation is of great significance for understanding the flocculation behavior and revealing the effect of flow characteristics on flocculation performance.

Dynamics-based classification of semienclosed basins

This study is guided by other classifications of estuaries that are based on specified physical constraints. Previous classifications have been relevant and revealing of diagnostic functioning of estuaries in terms of their stratification and their baroclinically induced residual circulation. Those classifications, however, have disregarded residual circulations driven by tides, not only in estuaries, but in semienclosed basins in general. The classification proposed here uses tidally averaged, or residual, dynamics. Residual dynamics are explored via drivers, or forcing agents, of residual flow. Drivers are tides or density gradients and are counteracted by ‘modifiers,’ i.e., Earth’s rotation or friction. The relative influence between forcing agents is characterized by a scaled non-dimensional densimetric tidal Froude number, Fr02, which hinges on tidal current amplitude, water depth and reduced gravity. The competition between modifiers is determined by a non-dimensional Ekman number, Ek, which depends on the kinematic eddy viscosity, the latitude and the depth of the basin. The classification suggested identifies all types of estuaries, whose dynamics is influenced by density gradients, but also recognizes frictionless tidal basins, tidal rivers, coastal lagoons, and basins that switch their dynamics from being baroclinically forced to being tidally forced. This classification can also be interpreted as an assessment of the strength of mixing (in Ek) versus the strength of baroclinicity (in Fr02), which determines the residual flows in semienclosed basins. Additionally, the classification incorporates basins forced by wind stress and/or baroclinicity, and basins forced by wind stress and/or tides.

Influence of turbulent Schmidt number on fugitive emissions source quantification

Abstract Finding and quantifying unknown (fugitive) releases of gases such as methane from downstream concentration data is a critical environmental problem. Many proposed solutions involve wind and gas dispersion modelling for which an assumed value of the turbulent Schmidt number (Sct), the ratio of the eddy kinematic viscosity to the turbulent diffusivity, is used to scale the estimated diffusivity. This model constant has a range of physically reasonable values. Numerical simulations were performed on multiple test cases to quantify the impact of Sct uncertainty on the ability to locate and quantify fugitive emissions sources using data from a network of gas concentration sensors. The analysis further considers ways to reduce quantification uncertainty, by either specifying detected source locations based on ancillary knowledge, or by using a two-step optimization in which the presented adjoint approach is used to locate sources and simulated annealing is subsequently used to quantify emissions for these specified locations. Multiple test cases considered both real and numerically generated controlled releases using an open-field geometry and a complex bluff-body dominated geometry based on an actual gas plant in the Alberta, Canada oil and gas sector. Results suggest that correct prediction of unknown source locations is minimally affected by Sct, but emission rate quantification can be heavily influenced. The presence of bluff-bodies was found to partially mitigate these effects, such that, if repeated analyses over a range of Sct is not tractable, open field release results (including those presented here) can be used to estimate conservative error bounds on predicted emission rates. Ultimately this paper demonstrates the under-appreciated importance of considering Sct uncertainty when seeking to quantify unknown fugitive sources from downstream concentration data.

Thermal Radiation Effect on the MHD Turbulent Compressible Boundary Layer Flow with Adverse Pressure Gradient, Heat Transfer and Local Suction

The combined effect of magnetic field, thermal radiation and local suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient is numerically studied. The magnetic field is constant and applied transversely to the direction of the flow. The fluid is subjected to a localized suction and is considered as a radiative optically thin gray fluid. The Reynolds Averaged Boundary Layer (RABL) equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The nonlinear and coupled system of partial differential equations (PDEs) is solved using the Keller box method. For the eddy-kinematic viscosity the Baldwin Lomax turbulent model and for the turbulent Prandtl number the extended Kays Crawford model are used. The numerical results show that the flow field can be controlled by the combined effect of the applied magnetic field, thermal radiation, and localized suction, moving the separation point, xs , downstream towards the plate’s end, and increasing total drag, D . The combined effect of thermal radiation and magnetic field has a cooling effect on the fluid at the wall vicinity. The combined effect has a greater influence in the case of high free-stream temperature.

Radiation effect on the turbulent compressible boundary layer flow with adverse pressure gradient

The effects of thermal radiation and localized suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient are numerically studied. The compressible flow is subjected to a constant localized suction velocity and the fluid is considered as a radiative optically thin gray fluid. The plate is adiabatic and the flow is subjected to adverse pressure gradient.The Reynolds Averaged Boundary Layer (RABL) equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The resulting nonlinear, coupled system of partial differential equations (PDEs) is solved using the Keller box method. For the eddy kinematic viscosity, the turbulent models of Cebeci Smith and Baldwin Lomax are employed. For the turbulent Prandtl number, the extended Kays Crawford model is used. The obtained results are validated with previously published computational results and with experimentally based correlations, showing a good agreement.The results show that the flow is influenced by the combined effect of radiation and localized suction. The coupled effect moves the separation point downstream, towards the plate’s end, and increases total drag. Radiation alters the thermal boundary layer above the plate by cooling the fluid at plate’s vicinity and by slightly increasing the boundary layer maximum temperature.

Influence of fish swimming on the flow pattern of circular tanks

The effect of swimming fish on the average velocity and velocity profile of a circular tank was studied. Working with different inlet diameters and flow rates, nine different impulse forces (configurations) were evaluated. Each configuration was tested with and without fish, and the effects of two different fish sizes were compared.The velocity profiles in experiments with fish presented a considerable reduction in velocity in the centre of the tank near the outlet, which was a consequence of the increase in the kinematic eddy viscosity due to the turbulence introduced by fish swimming. A flattening of the angular velocity profile was observed in the central area of the tank, which had a radius of about 0.3m (18% of the total volume of the tank).A previous model proposed by Oca and Masaló (2013) was modified in order to better describe the distribution of velocities in the central volume of a tank with swimming fish. The proposed modification was based on Burgers⿿ proposal for a bathtub vortex, which implies the determination of the parameter (1⿿e⿿αr2), where r is the radius and the α values were experimentally obtained for each tank configuration, in which they increased with the impulse force.The average velocities in the tank were proportional to the square root of the impulse force in experiments with and without fish. Experiments with fish presented lower average velocities, which imply higher tank resistance coefficients. At similar stocking densities (14.6kg/m3), the increase in the tank resistance coefficients obtained with small fish sizes (154g) were slightly higher than those obtained with bigger fish sizes (330g).

FULLY DEVELOPED STAGE OF PERTURBATIONS IN PARALLEL TRANSVERSE SHEAR FLOWS

A predictive theory to study the nonlinear development of the perturbations in a channel with a lateral gradient in the streamwise velocity due to the presence of bank vegetation is presented. The ratio between the undisturbed parallel velocities in the vegetated and non-vegetated zones is taken as the expanded parameter at the weakly nonlinear stability analysis, along with its corresponding wavenumber at a critical condition where the perturbations nor grow nor decay. In most cases evaluated herein, we have supercritical bifurcation, where the amplitude of the perturbations is found to reach an upper bound as time tends to infinity. We find reasonable agreement between predicted and experimental results for the time-averaged velocity profile, kinematic eddy viscosity, shear layer width and maximum friction velocity.

Invariant solution for an axisymmetric turbulent free jet using a conserved vector

An axisymmetric turbulent free jet described by an effective viscosity, which is the sum of the kinematic viscosity and the kinematic eddy viscosity, is investigated. The conservation laws of the jet are derived using the multiplier method. A second conserved vector, in addition to the elementary conserved vector, exists provided the effective viscosity has a special form. The Lie point symmetry associated with the elementary conserved vector is obtained and used to generate the invariant solution. The analytical solution is derived when the effective viscosity depends only on the distance along the jet. The numerical solution is obtained when the effective viscosity depends also on the distance across the jet. The eddy viscosity causes an apparent increase in the viscosity of the mean flow which produces an increase in the width of the jet due to an increase in diffusion and also a decrease in the maximum mean velocity along the axis of the jet.

Numerical investigation of the hump characteristic of a pump–turbine based on an improved cavitation model

For the safe and stable operation of a pump–turbine at pump mode, the hump characteristics must be studied and the hump region should be avoided. 3-D (three dimensional), compressible, cavitating flows in a pump–turbine at pump mode were numerically studied using SST k–ω turbulence model and mixture model. The decrease of the kinematic eddy viscosity in the region of high volume fraction of water vapor was considered in the calculation. The flow and external characteristic in the hump region were analyzed. Results show that the hump characteristic of a pump–turbine might be related to the cavity flow in the pump–turbine. It is the appearance of the cavitation that reduces the head of the pump–turbine. The cavitation incipience is thought to occur when the pump–turbine runs at the peak head and the cavitation is worse at 80% discharge of the pump–turbine. The cavitation regions locate at the inlet of the suction side. Calculation results are in good agreement with experimental data. The pressure fluctuation at the wave trough of hump characteristic is determined by the rotational speed. Numerical study of hump characteristics can provide a basic understanding for the improvement of stable operation of a pump–turbine.

Kinematic Eddy Viscosity Coefficients in Residual Current Equations with Tidal Stress

Kinematic Eddy Viscosity Coefficients in Residual Current Equations with Tidal Stress

An Alternative Scheme to Calculate the Strain Rate Tensor for the LES Applications in the LBM

Large eddy simulations (LES) based on the Smagorinsky model can be conveniently used in the lattice Boltzmann method (LBM) because the strain rate tensor, Sij, used to determine the eddy kinematic viscosity can be calculated from the second‐order moment of the nonequilibrium distribution function, and the current total nondimensional relaxation time can be determined explicitly. A new method is developed where the distribution function after the relaxation subroutine differs from that after the motion subroutine leading to a similar method to determine Sij, but its application is inconvenient due to the implicit feature. However, the derivation also leads to an alternative explicit scheme for calculating Sij based on physical analysis of the momentum transport process, where the stress tensor, Tij, is calculated first, and then Sij is determined from Tij using the constitutive relationship for Newtonian fluid. The current total nondimensional relaxation time is also given explicitly so that this LES model can be easily used in the LBM.

Combustion Analysis Using Roe's Scheme and the Spalart-Allmaras Model

of turbulent reactive flows at all speeds. The choice of the Spalart–Allmaras model, as opposed to typical twoequation models requires particular attention on the derivation of the micromixing time. Such a derivation is of fundamental importance, for the micromixing time represents the base upon which chemical source terms are built for many combustion models. Deriving a correct model for the micromixing time from the only resolved turbulent quantity kinematic eddy viscosity is challenging. To test the method, two low-Mach-number (0.1–0.2) turbulent reactive flows are simulated using the eddy dissipation model, and the results are compared with both experiments and previous numerical studies. It is established that the proposed method leads to satisfactory results.

Mathematical modeling of turbulent non-premixed piloted-jet flames with local extinctions

Flamelet modeling of highly stretched jet flames is combined with the use of conditional moment closuresecond-order closure procedures to evaluate a pdf, p ˜ ( θ | η ) , of the reaction progress variable, θ, that is conditioned upon a value of the mixture fraction η. Through the use of a probability of burning function. P h , the model expresses the effects of strain rate and localized flame quenching. The product of this function and another term embodying the relevant laminar flamelet source term from a flamelet library, together with p ˜ ( θ | η ) , yields the required mean turbulent source term. Both heat release rate and the formation rate of species are dealt with in this way. It was found appropriate and convenient to use the k-ω model, with a kinematic eddy viscosity, for the flow turbulence. The overall model is applied to those experimental piloted-jet methane/air flames of Barlow and Frank, in which there are pronounced extinctions and reignitions. Velocity vectors and spatial contours of mean mixture fraction, mean volumetric heat release rate, and mean strain rate clearly show the essential structures of the flames. As the flow rate increases, so does the extent of the penetration of the region of high strain rates into that of flammable mixtures. This creates localized regions of reduced mean heat release rate in which localized extinctions might be anticipated, due to the strain rate being high and the mixture being less reactive. These locations are confirmed by experiments, as is the predicted extent of the flames. There is generally good agreement between predicted and measured mean temperatures, mean mixture fractions, and mass fractions of CH 4 , O 2 , and H 2 O. Agreement is less satisfactory for transient species. Possible limitations of the model are discussed.

Numerical investigation of two-dimensional turbulent boundary-layer compressible flow with adverse pressure gradient and heat and mass transfer

The effect of heat and mass transfer on the steady turbulent compressible boundary-layer flow with adverse pressure gradient are numerically studied. The Reynolds-averaged boundary-layer equations and their boundary conditions are transformed, in a suitable form for numerical solution, by using the compressible version of the Falkner-Skan transformation and the resulting coupled and nonlinear system of partial differential equations is solved using the Keller's-box method and a modified version of it. For the eddy kinematic viscosity the model developed by Cebeci and Smith is employed whereas for the turbulent Prandlt number model a modification of the extended Kays and Crawford's model is used. Numerical calculations are carried out for the case of air, at about free stream temperature of 300°K, and for a linearly retarded flow, known as Howarth's flow when the porous limiting surface is adiabatic, heated or cooled. The porous surface is subjected to a continuous or localized suction/injection velocity and the influence of this velocity as well as of the free-stream Mach number and of the heat-transfer parameter on the turbulent boundary-layer and the separation point is examined. It is hoped that in the absence of detailed investigations into this problem, the obtained results, presented in the figures, are very interesting and give a clearer insight into the mechanism of controlling a turbulent boundary-layer compressible flow.

Open‐channel flow through simulated vegetation: Suspended sediment transport modeling

A two‐equation turbulence model, based on the k‐ϵ closure scheme, was used to determine the mean flow and turbulence structure of open channels through simulated vegetation, thus providing the necessary information to estimate suspended sediment transport processes. Dimensional analysis allowed identification of the dimensionless parameters that govern suspended sediment transport in the presence of vegetation and thus helped in the design of numerical experiments to investigate the role of different flow properties, sediment characteristics, and vegetation parameters upon the transport capacity. A reduction of the averaged streamwise momentum transfer toward the bed (i.e., shear stress) induced by the vegetation was identified as the main reason for lower suspended sediment transport capacities in vegetated waterways compared to those observed in nonvegetated channels under similar flow conditions. Computed values of kinematic eddy viscosity were used to solve the sediment diffusion equation, yielding distributions of relative sediment concentration slightly in excess of the ones predicted by the Rousean formula. A power law was found to provide a very good collapse of all the numerically generated data for suspended sediment transport rates in vegetated channels.

Improved turbulence models for estimation of wind loading

The objective of this work is to develop a cost efficient and accurate turbulence model for the estimation of wind loading on buildings. This paper presents two types of k−ε−φ turbulence model, which are stable, simple, cost efficient and high-accuracy turbulence models, for estimation of wind loading on a bluff body. The variable φ, an anisotropy parameter of the Reynolds stresses, decreases due to the deceleration effect through an adverse pressure gradient, and increases due to the acceleration effect through a favorable pressure gradient. The coefficient of the kinematic eddy viscosity, Cμ, is defined by a function of φ and the helicity H, and the coefficient of production term of turbulence energy dissipation rate in the conventional standard k−ε turbulence model, Cε1, is defined by a function of φ. The favorable pressure gradient flow and the flows around a square prism are used for adjusting each coefficient, and the flows around a low-rise building in a 14 shear flow are used for adjusting the coefficient of the helicity effect and for the verification of these models. The distributions of the mean pressure coefficient using these two types of k−ε−φ turbulence model are improved dramatically compared with those using the standard k−ε turbulence model.

Influence of Injection Conditions on Entrainment Processes of Unsteady Gas Jets.

In unsteady gas jets, the influence of gas injection conditions on ambient gas entrainment processes was evaluated by a LIFA (laser-induced fluorescence of ambient gas technique. The influence of injection nozzle diameter, D, the ratio of nozzle length to diameter, L/D, discharge velocity, um, kinematic viscosity of the gas jet, ν, specific gravity of the gas jet, ρj and of ambient gas, ρa was investigated. Experimental results show that the influence of injection conditions on the ambient gas entrainment processes correlates with the influence of both specific gravity ratio, ρj/ρa, and eddy kinematic viscosity, νt, whhch is calculated from the discharge turbulence intensity of jets. The discharge turbulence intensity increases with increase in um and with decrease in ρj, L/D, and D. The mean jet concentration, Cjm, which was calculated by averaging Cj over the whole jet, can be predicted approximately with only one parameter, (ρj/ρa)D2/[(ν+νt)Δt].

The Effect of Free-Stream Turbulence on Spanwise Eddy Diffusivity of Heat in a Flat-Plate Turbulent Boundary Layer

Spanwise eddy diffusivity of heat was measured in a flat-plate turbulent boundary layer with a constant spanwise temperature gradient and different intensities of free-stream turbulence. Comparison was made with the usual kinematic eddy viscosity measured in parallel. According to the experimental results, the ratio of these eddy diffusivities is strongly influenced by the free-stream turbulence. However, in the vicinity of the wall, this ratio approaches a certain universal distribution which is independent of the intensity of free-stream turbulence.

An experimental study on the spanwise eddy diffusivity of heat in a flat-plate turbulent boundary layer

A flat-plate turbulent boundary layer with a constant temperature gradient in the spanwise direction is investigated experimentally to examine distribution characteristics of the spanwise eddy diffusivity of heat. The working fluid is air with a Prandtl number of 0.71. According to the experimental results, the ratio of this eddy diffusivity of heat to the usual kinematic eddy viscosity presents distinctive properties in the immediate vicinity of the wall, the remaining part of the wall region, and the outer region.

Gravity wave‐driven fluctuations in OH nightglow from an extended, dissipative emission region

The theory of gravity wave‐driven fluctuations in the OH nightglow from an extended source region is generalized to account for effects of eddy kinematic viscosity ν and eddy thermal diffusivity κ. In the nondiffusive case, the amplitudes and phases of vertically integrated normalized intensity 〈δI〉/〈Ī〉 and temperature perturbations and vertically integrated Krassovsky's ratio 〈η〉 as functions of period are influenced by the upper limit of vertical integration of the extended source, especially at long periods when vertical wavelengths λυ are small. The effects, which include oscillations in 〈δT〉/〈Ī〉, , and 〈η〉, particularly at long periods, are due to constructive and destructive interference of nightglow signals from vertically separated levels of the OH emitting region that occur when λυ is comparable to or smaller than the thickness of the main emission region. The sensitivity of these ratios to the upper limit of vertical integration occurs because of the relatively small rate of decay of the intensity of OH emission with height above the peak emission level and the exponential growth with altitude of nondissipative gravity waves. Because eddy diffusion increases λυ, especially at long periods, and reduces wave growth with height compared with the case ν = κ = 0, inclusion of eddy diffusion removes the sensitivity of 〈η〉 and the other ratios to the maximum height of vertical integration. It is essential to account for both eddy diffusion and emission from the entire vertically extended emission region to correctly predict 〈η〉, 〈δI〉/〈Ī〉, and at long gravity wave periods. Observations of Krassovsky's ratio are consistent with theoretical predictions of 〈η〉 for horizontal wavelengths larger than about 500 km, provided diffusion and emission from an extended region are both taken into account.