Reynolds Stress Equation Research Articles

CFD Simulation of an Industrial Dust Cyclone Separator: A Comparison with Empirical Models: The Influence of the Inlet Velocity and the Particle Size on Performance Factors in Situation of High Concentration of Particles

The present work is dedicated to the study of multiphase turbulent and three-dimensional rotational flow in dust cyclones, a contribution to air pollution control. Cyclones are widely used devices for the separation of constituents from solid-gas mixtures in industry. In order to improve the filtration efficiency of cyclones, and to reduce the pressure drop, parametric numerical simulation studies using the Fluent code have been conducted to characterise the effects of the parameters affecting the operation of these devices through their performance indicators. In this work, the effect of inlet velocity and the particle size on the turbulent flow air in the cyclone is presented. Numerical simulation of the flow by Fluent code using three numerical models: the first based on the dissipation of kinetic energy by viscosity (RNG) K-epsilon and standard K-epsilon as well as the last based on the solution of Reynolds stress equations (RSM), combined with the multiphase mixing model, gave interesting results in terms of the pressure and flow field in the separator, the variation of inlet velocity, and the variation of particle size. Validation with experimental and empirical results showed the advantage of the Reynolds stress turbulence model (RSM) over the standard K-epsilon and RNG K-epsilon. The RSM model better captures physical phenomena in an intense vortex flow in the presence of walls. But it is characterised by a very long calculation time and requires large machine resources. An alternative to this model is RNG K-epsilon model, which offers a reasonable calculation time with acceptable results (maximum deviation of 5 %) for speed values below 10 m/s. In the absence of numerical resources, certain empirical models such as those of First (for the evaluation of pressure drop) and Iozia and Leith (for evaluation of efficiency) may well be useful for the dimensioning of the cyclone.

Symmetry Analysis of Mean Velocity Distribution in Stratified Atmospheric Surface Layers

The mean velocity distributions of unstably and stably stratified atmospheric surface layers (ASLs) are investigated here using the symmetry approach. Symmetry groups for the mean momentum and the Reynolds stress equations of ASL are searched under random dilation transformations, which, with different leading order balances in different flow regions, lead to a set of specific scalings for the characteristic length ℓ13 (defined by Reynolds shear stress and mean shear). In particular, symmetry analysis shows that in the shear-dominated region, ℓ13 scales linearly with the surface height z, which corresponds to the classical log law of mean velocity. In the buoyancy-dominated region, ℓ13/L∼z/L4/3 for unstably stratified ASL and ℓ13/L∼const for stably stratified ASL, where L is the Obukhov length. The specific formula of the celebrated Monin–Obukhov similarity function is obtained, and hence an algebraic model of mean velocity profiles in ASL is derived, showing good agreement with the datum from the QingTu Lake observation array (QLOA) in China.

Numerical Investigation of Bed Shear Stress and Roughness Coefficient Distribution in a Sharp Open Channel Bend

The helical flow leads to the redistribution of the bed shear stress and roughness coefficient in open-channel bends, influencing the transport of sediment and riverbed evolution process. To explore the mechanisms underlying this redistribution, a 3D numerical simulation of a 180° sharp bend was performed by solving the Reynolds-averaged Navier–Stokes (RANS) equations using the Reynolds stress equation model (RSM) as the anisotropic turbulence closure approach. The results indicate that the high bed shear stress zone appeared in the inner bank region from the 50° to 110° sections, and the section maximum bed shear stress gradually shifted outwards, owing to the advective momentum transport by the circulation cells. The quantitative analyses of the terms in depth-averaged Navier-Stokes equations indicate that the contributions of the cross-stream circulation and cross-flow to the downstream bed shear stress were of the same order of magnitude, and the overall contribution rate of the cross-stream circulation was 20%. The contribution rates of the cross-flow, cross-stream circulation, turbulence, and pressure gradient to the transverse bed shear stress were approximately 30%, 7%, 3%, and 60%, respectively, indicating that the pressure gradient term arising from the transverse water surface slope played a dominant role. The Chezy resistance coefficient showed an overall decreasing trend along the bend. Therefore, an effective expression considering the streamwise variation along the centreline and transverse variation was successfully established to predict the uneven distribution of the Chezy resistance coefficient.

Numerical study on wind pressure characteristics of Chinese yurt building under downburst wind

SummaryInner Mongolia is a high‐frequency thunderstorm region in China, and the downburst caused by the thunderstorm weather is a severe threat to buildings. In order to study the influence of downburst on the wind pressure characteristics of the yurt building, the wind field model of the yurt building under downburst is established based on the computational fluid dynamics method, and the effect of the wall treatment method and turbulence model on the numerical simulation of wind pressure of the yurt building under downburst is analyzed. The results demonstrate that the maximum positive pressure at the windward side of the yurt building occurs at 3/4 of the yurt building height under downburst, and the maximum negative pressure at the roof of the yurt building appears at the center of the roof. Compared with the experimental results, the Shear Stress Transport (SST) k‐ω model is suitable for simulating both sides of the yurt building, while the Reynolds Stress equation Model (RSM) is suitable for simulating the windward side, roof, and leeward of the yurt building. The enhanced wall treatment is appropriate for simulating the remaining sides of the yurt building while the standard wall function is appropriate for simulating both sides of the building.

Improved modelling of interfacial terms in the second-moment closure for particle-laden flows based on interface-resolved simulation data

Correlations for the interfacial terms in the fluid dissipation rate equation and Reynolds stress equations are established for particle-laden flows, based on data from the interfaced-resolved direct numerical simulations of particle sedimentation in a periodic domain at a density ratio ranging from 0.01 to 1000, a particle concentration ranging from 2.3 % to 30.2 % and a particle Reynolds number below 250. The correlations for the mean drag and the pseudo-turbulent kinetic energy are also reported, which are used for the modelling of the interfacial term in the fluid dissipation rate equation. The interfacial term correlations obtained are then incorporated in the Reynolds stress model (RSM) (i.e. second-moment closure) for the simulation of vertical turbulent channel flows laden with the finite-size particles at relatively low particle volume fractions. The results show that the RSM with new interfacial term correlations can quantitatively predict particle-induced turbulence enhancement or suppression in vertical channel flows.

A novel particle subgrid scale modeling of large eddy simulation for swirling particle-laden turbulent flow

How to describe the effects of subgrid scale (SGS) gas flow on particle dynamics in large eddy simulation (LES) is a challenge for modeling and simulation of particle-laden swirling turbulent flow. SGS four-way coupling interaction mechanisms of gas-to-particle, particle-to-gas, and particle-particle collision have not been considered fully. A novel particle SGS kinetic turbulent energy-granular temperature (SGS-kp-θp) model is proposed firstly to model this effect on particle motions and dispersions. Second-order moment algorithm to establish the anisotropic two-phase Reynolds stress equations and their closure correlations is utilized. Validations by experiment results and Moissette (MOB) particle dispersion model built-in OpenFoam software are performed with better agreements for mean and fluctuation velocities. Anisotropic behaviors of Reynolds stresses and particle collisions are distinctively, smaller-scale particle fluctuations originating from particle collisions lead to the redistribution and dissipation of Reynolds stresses. Power spectrum density is utilized to analyze statistically the particle number density fluctuations and Reynolds stresses, dominant frequencies are roughly 2.5 Hz and 1.0 Hz marking the internal and the corner recirculation regions. Compared to those of inlet flow regions, normal and tangential Reynolds stresses at development flow regions approximately increased by twice and decreased by 3.0 times. Modeling strategy for multiphase interactions are of great importance for capturing particle dynamics in accurate.

Turbulent flow in curved channels

Fully developed turbulent flow in channels with mild to strong longitudinal curvature is studied by direct numerical simulations. The Reynolds based on the bulk mean velocity and channel half-width $\delta$ is fixed at $20\,000$ , resulting in a friction Reynolds number of approximately 1000. Four cases are considered with curvature varying from $\gamma = 2\delta /r_c = 0.033$ to 0.333, where $r_c$ is the curvature radius at the channel centre. Substantial differences between the mean wall shear stress on the convex and concave walls are already observed for $\gamma = 0.033$ . A log-law region is absent and a region with nearly constant mean angular momentum develops in the channel centre for strong curvatures. Spanwise and wall-normal velocity fluctuations are strongly amplified by curvature in the outer region of the concave channel side. Only near the walls, where curvature effects are relatively weak, do the mean velocity and velocity fluctuation profiles approximately collapse when scaled by wall units based on the local friction velocity. Budgets of the streamwise and wall-normal Reynolds-stress equations are presented and turbulence structures are investigated through visualizations and spectra. In the case with strongest curvature, the flow relaminarizes locally near the convex wall. On the concave channel side, large elongated streamwise vortices reminiscent of Taylor–Görtler vortices develop for all curvatures considered. The maximum in the premultiplied two-dimensional wall-normal energy spectrum and co-spectrum shifts towards larger scales with increasing curvature. The large scales substantially contribute to the wall-normal velocity fluctuations and momentum transport on the concave channel side.

Perspective on the Response of Turbulent Pipe Flows to Strong Perturbations

Pipe flow responds to strong perturbations in ways that are fundamentally different from the response exhibited by boundary layers undergoing a similar perturbation, primarily because of the confinement offered by the pipe wall, and the need to satisfy continuity. We review such differences by examining previous literature, with a particular focus on the response of pipe flow to three different kinds of disturbances: the abrupt change in surface condition from rough to smooth, the obstruction due to presence of a single square bar roughness elements of different sizes, and the flow downstream of a streamlined body-of-revolution placed on the centerline of the pipe. In each case, the initial response is strongly influenced by the pipe geometry, but far downstream all three flows display a common feature, which is the very slow, second-order recovery that can be explained using a model based on the Reynolds stress equations. Some future directions for research are also given.

An Analytically Derived Shear Stress and Kinetic Energy Equation for One-Equation Modelling of Complex Turbulent Flows

The Reynolds stress equations for two-dimensional and axisymmetric turbulent shear flows are simplified by invoking local equilibrium and boundary layer approximations in the near-wall region. These equations are made determinate by appropriately modelling the pressure–velocity correlation and dissipation rate terms and solved analytically to give a relation between the turbulent shear stress τ/ρ and the kinetic energy of turbulence (k=q2/2). This is derived without external body force present. The result is identical to that proposed by Nevzgljadov in A Phenomenological Theory of Turbulence, who formulated it through phenomenological arguments based on atmospheric boundary layer measurements. The analytical approach is extended to treat turbulent flows with external body forces. A general relation τ/ρ=a11−AFRiFq2/2 is obtained for these flows, where FRiF is a function of the gradient Richardson number RiF, and a1 is found to depend on turbulence models and their assumed constants. One set of constants yields a1= 0.378, while another gives a1= 0.328. With no body force, F ≡ 1 and the relation reduces to the Nevzgljadov equation with a1 determined to be either 0.378 or 0.328, depending on model constants set assumed. The present study suggests that 0.328 is in line with Nevzgljadov’s proposal. Thus, the present approach provides a theoretical base to evaluate the turbulent shear stress for flows with external body forces. The result is used to reduce the k–ε model to a one-equation model that solves the k-equation, the shear stress and kinetic energy equation, and an ε evaluated by assuming isotropic eddy viscosity behavior.

Vorticity Confinement Applied to Accurate Prediction of Convection of Wing Tip Vortices and Induced Drag

The vorticity confinement (VC) method was applied to tip vortices shed by edges of wings in order to predict induced drag using far-field integration. The optimal VC parameter was determined by its application to tip vortices shed by 3-D stationary and rotating wings. The VC was used with a total variation diminishing (TVD) approach to reduce the needed confinement. The TVD minmod and differentiable flux limiters were tested to evaluate their effect on the VC parameter. The 3-D inviscid simulations were post-processed using the wake-integral technique to determine lift-induced drag force. Dependence of the VC parameter on the flight Mach number and the angle of attack was evaluated. Grid convergence studies were conducted for induced drag generated by 3-D wing. The VC approach was combined with the Reynolds stress equation turbulence model, and the results were compared to experimental data of tip vortex evolution.

Blending the Eddy-Viscosity and Reynolds-Stress Models Using Uniform High-Order Discretization

Accuracy, robustness, and efficiency are the three major goals of turbulence closure. Generally, eddy-viscosity models (EVMs) with the Boussinesq approximation are numerically stable and efficient. However, the accuracy of EVMs for aerodynamically complex configurations is inadequate. Reynolds-stress models (RSMs) are perceived as the most advanced turbulence models, but their robustness and efficiency could be improved. Here, a hybrid closure is proposed that combines a novel eddy-viscosity equation and six Reynolds-stress equations. The eddy-viscosity equation with natural boundary conditions on the walls is designed to implement a transition from eddy-viscosity mode (one-equation mode) to Reynolds-stress mode (seven-equation mode). The one-equation mode, where the eddy-viscosity equation is solved independently with the Boussinesq approximation and the Bradshaw relation, is used to establish the initial flowfield. After a given number of iterations, the seven-equation mode takes over and continues until convergence. The eddy-viscosity equation acts as a length-scale equation to close the RSM. The benchmark results show that the numerical stability is better than for traditional -based RSMs. Moreover, the stability allows a uniform high-order discretization of the mean-flow and turbulence equations (12 partial differential equations), which can reduce the cost needed to achieve a given level of accuracy. This improves the efficiency.

The Stability Functions and Realizability of the Turbulent Kinetic Energy\u2013Scalar Variance Closure for Moist Atmospheric Boundary-Layer Turbulence

The problem of realizability of the second-order turbulence closure models (parametrization schemes) is addressed through the consideration of the so-called “stability functions”. The emphasis is on the turbulence kinetic energy–scalar variance (TKESV) closure scheme that carries prognostic transport equations for the turbulence kinetic energy (TKE) and for the variances and covariance of two quasi-conservative scalars suitable for describing moist atmospheric boundary-layer turbulence. Stability functions appear within the framework of truncated closure schemes, where (i) the Reynolds-stress and scalar-flux equations (and, within the framework of one-equation TKE schemes, also equations for scalar variances and covariance) are reduced to the diagnostic algebraic formulations by neglecting the substantial derivatives and the third-order transport terms, and (ii) simplified linear parametrizations of the pressure-scrambling terms are used. The stability functions are ill-behaved (tend to infinity or become negative) over a certain range of governing parameters, e.g., mean velocity shear and buoyancy gradient. Using the approach of Helfand and Labraga (J Atmos Sci 45:113–132, 1988), we develop regularized stability functions for the TKESV scheme that reveal no pathological behaviour over their entire parameter space. The physical meaning of the regularization procedure and its relation to non-linear parametrizations of the pressure-scrambling terms and to weak non-equilibrium hypothesis are discussed. Finally, realizability of turbulence closures is considered within a more general framework of the moments problem of the probability theory.

Counter-rotating dual-stage swirling combustion characteristics of hydrogen and carbon monoxide at constant fuel flow rate

In this paper, experimental and numerical methods were used to study the combustion characteristics of a counter-rotating double-stage swirling syngas combustor at constant fuel flow rate, and the effect on it of hydrogen content of syngas. In the experiment, the speed and temperature in the combustor were respectively obtained with PIV and temperature rake, while Reynolds stress equation model and the detailed chemical reaction mechanism of syngas were adopted in the numerical method. The calculation results were in good agreement with the experimental data. Research results indicated that in the working conditions of different hydrogen contents, the flow field structures in the combustor are almost the same, and the maximum temperatures at the outlet remain almost the same. However, as hydrogen content in the fuel increases, the axial velocity in the central area of flow field is increasing, and the outlet temperature distribution coefficient decreases first and then increases. In addition, it was also found in the study that the distribution structure of temperature on the central section of the combustor is almost impervious to the changes in hydrogen content, but with numerical differences, i.e. the higher hydrogen content in the fuel, the farther the stabilization position of flames in the central area is away from the head. It was also indicated in the study that the conventional combustor is no longer applicable to the combustion of syngas, especially the hydrogen-rich fuel. And the work provided the improvement scheme of hydrogen-containing fuel for gas turbine combustor.

CFD analysis of the combustion in the BERL burner fueled with a hydrogen-natural gas mixture

The regulatory restrictions, currently acting, impose a significant reduction of the Greenhouse Gas (GHG) emissions. After the coal-to-gas transition of the last decades, the fossil fuel-to-renewables switching is the current perspective. However, the variability of energy production related to Renewable Energy Sources requires the fundamental contribution of thermal power plants in order to guaranty the grid stability. Moving toward a low-carbon society, the industry is looking at a reduction of high carbon content fuels, pointing to Natural Gas (NG) and more recently to hydrogen-NG mixtures. In this scenario, a preliminary study of the BERL swirled stabilized burner is carried out in order to understand the impact of blending natural gas with hydrogen on the flame morphology and CO emissions. Preliminary 3D CFD simulations have been run with the purpose to assess the best combination of combustion model (Non Premixed and Partially Premixed Falmelets), turbulence model (Realizable k ɛ and the Reynolds Stress equation model) and chemical kinetic mechanism (GriMech3.0, GriMech 1.2 and Frassoldati). The numerical results of the BERL burner fueled with natural gas have been compared with experimental data in terms of flow patterns, radial temperature profiles, O2, CO and CO2 concentrations. Finally, a 30% hydrogen in natural gas mixture has been considered, keeping fixed the thermal power output of the burner and the global equivalence ratio.

A two-fluid model for gas-particle turbulent flows based on the probability density function approach

According to experimental observations, laden particles in turbulence may attenuate or augment the carrier phase turbulence. But until now, there are no widely recognized models for estimating the so-called turbulence modulation phenomenon. In this paper, a novel two-fluid model is proposed based on the probability density function (PDF) approach. The Reynolds stress equation of the present model contains both production and dissipation terms due to the presence of particles, the turbulence modulation phenomenon can be well explained with the new model. To further explore the two-fluid model, a comparative study on PDF and Reynolds-averaged approaches is carried on, the differences and relations between the present model and the classical two-fluid Reynolds averaged Navier–Stokes (RANS) model are analyzed in the paper. Theoretical and numerical analysis indicates that the proposed model shows particular promise for predicting particle-laden turbulent flows.

Numerical simulation of windless-air-induced added mass and damping of vibrating bridge decks

The windless-air-induced added mass/mass moment of inertia (ma/Ia) and damping (ca) effects on mechanical parameters of a vibrating bridge deck are usually ignored in wind tunnel tests. In this paper, for three typical deck sections, computational fluid dynamics simulations are carried out to study the vertical/torsional single degree-of-freedom forced vibration under windless conditions, and further to reveal the effects of ma, Ia, and ca on the modal parameters. The influences of turbulence model, computational domain size, grid resolution, and time step size are analyzed. For the addressed issue, the Reynolds stress equation model (RSM) is found to be more suitable than the shear stress transportation (SST) k-ω model. A mathematical model for aerodynamic forces at zero wind speed is presented by using the dimensionless ma, Ia, and ca, and they can be extracted by combining the motions and the numerically simulated aerodynamic forces using the least squares method. The numerically simulated results for an ideal plate under small amplitudes are very close to the theoretical values, and consequently verify their accuracy. The causes for the non-zero values of flutter derivatives H4∗ and A3∗ at zero wind speed are revealed. In the windless air, the added damping almost linearly increases with the vibration amplitude. Five existing long-span bridge deck models are taken as examples to investigate the effects of ma, Ia, and ca on structural frequencies and damping ratios.

Developing theory of probability density function for stochastic modeling of turbulent gas-particle flows

Turbulent gas-particle flows are studied by a kinetic description using a probability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are directly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.

A compressibility corrections of the pressure strain linear part models

The main focus of this paper is the analysis of the compressibility effects and the validation of some recent Reynolds stress models for computing compressible turbulent flows. The pressure strain correlation is one of the several terms appearing in the Reynolds stress equation which directly reflect the compressibility effects on the turbulence. For this reason, a special attention is paid to the modeling of this term in order to account for compressibility effects at high-speed. The models developed by Speziale Sarkar and Gatski (SSG) and Fu, Launder and Tselepidakis (FLT) for the pressure strain correlation are examined to be extended to compressible turbulent flows. A compressibility corrections of these models using the turbulent Mach number are proposed. The calculations have been performed for the compressible homogeneous shear flow and the turbulent plate mixing-layers. The comparison of the proposed compressibility modifications of the SSG and FLT models with its universal version shows some important ameliorations in results for the majority characteristic parameter of the structural compressibility effects. It?s found that the predicted results from the modified SSG and FLT models are in reasonable agreement with the accepted data.

Modeling The Turbulent Scalar Fluxes In An Internal Wave Breaking Region In Stratified Flow With Obstacle

Based on averaging of the direct numerical simulation data obtained earlier, statistical moments are calculated in a turbulent patch arising after internal wave overturning in a flow with obstacle and stable stratification. Temporal evolution and spatial behavior of the turbulent-scalar-flux transport equation budget have been studied. A priori estimations of approximations for the pressure-scalar-gradient correlation and turbulent-diffusion processes in this equation have been carried out. The performed analysis is helpful to explore the turbulent patch in terms of statistical moments, and to verify closure hypotheses in turbulence models. It is shown that, in the global balance of the turbulent-scalar-flux equation, the mean-shear and buoyancy productions and the pressure-scalargradient correlation are roughly balanced, as for the shear Reynolds stress equation, and in contrast to the scalar variance and normal Reynolds stress equations. The algebraic turbulent-scalar-flux relations of the gradient diffusion type, as well as those derived in local-equilibrium and nonequilibrium assumptions are incorrect; therefore the use of the full turbulent-scalar-flux equation is justified.

Evaluation Study of Pressure-Strain Correlation Models in Compressible Flow

This paper is devoted to the second-order closure for compressible turbulent flows with special attention paid to modeling the pressure-strain correlation appearing in the Reynolds stress equation. This term appears as the main one responsible for the changes of the turbulence structures that arise from structural compressibility effects. The structure of the gradient Mach number is similar to that of turbulence, therefore this parameter may be appropriate to study the changes in turbulence structures that arise from structural compressibility effects. Thus, the incompressible model (LRR) of the pressure-strain correlation and its corrected form by using the turbulent Mach number, fail to correctly evaluate the compressibility effects at high shear flow. An extension of the widely used incompressible model (LRR) on compressible homogeneous shear flow is the major aim of the present work. From this extension the standard coefficients Ci became a function of the compressibility parameters (the turbulent Mach number and the gradient Mach number). Application of the model on compressible homogeneous shear flow by considering various initial conditions shows reasonable agreement with the DNS results of Sarkar. The ability of the models to predict the equilibrium states for the flow in cases A1 and A4 from DNS results of Sarkar is examined, the results appear to be very encouraging. Thus, both parameters Mt and Mg should be used to model significant structural compressibility effects at high-speed shear flow.