One-equation Model Research Articles

Improving predictions of transitional and separated flows using RANS modeling

In this paper, new concepts are presented from the aspects of viscous modeling and numerical scheme in an attempt to improve the prediction of rotor hover performance and associated viscous flow physics under a RANS statistic modeling framework. An improved viscous modeling is proposed to address the overprediction of flow separations faced by RANS modeling, which modifies the linear eddy viscosity models based on a new separation correction method. This concept is seamlessly implemented into two popular eddy viscosity models: the Spalart-Allmaras one-equation model and Menter's shear stress transport two-equation model. A new high order unstructured grid WENO scheme is adopted to improve the prediction of mean flow resolutions as well as viscous flow structures around the hovering rotor. Numerical results show significant improvements achieved in the prediction of rotor hover performance as well as viscous flow phenomena including boundary layer transitions and flow separations.

Large-Eddy Simulation of ECN Spray A: Sensitivity Study on Modeling Assumptions

In this study, various mixing and evaporation modeling assumptions typically considered for large-eddy simulation (LES) of the well-established Engine Combustion Network (ECN) Spray A are explored. A coupling between LES and Lagrangian particle tracking (LPT) is employed to simulate liquid n-dodecane spray injection into hot inert gaseous environment, wherein Lagrangian droplets are introduced from a small cylindrical injection volume while larger length scales within the nozzle diameter are resolved. This LES/LPT approach involves various modeling assumptions concerning the unresolved near-nozzle region, droplet breakup, and LES subgrid scales (SGS) in which their impact on common spray metrics is usually left unexplored despite frequent utilization. Here, multi-parametric analysis is performed on the effects of (i) cylindrical injection volume dimensions, (ii) secondary breakup model, particularly Kelvin–Helmholtz Rayleigh–Taylor (KHRT) against a no-breakup model approach, and (iii) LES SGS models, particularly Smagorinsky and one-equation models against implicit LES. The analysis indicates the following findings: (i) global spray characteristics are sensitive to radial dimension of the cylindrical injection volume, (ii) the no-breakup model approach performs equally well, in terms of spray penetration and mixture formation, compared with KHRT, and (iii) the no-breakup model is generally insensitive to the chosen SGS model for the utilized grid resolution.

Elliptic Blending with One-Equation Model

A wall-distance-free modification to Wray-Agarwal (WA) one-equation turbulence model is convoked using an elliptic relaxation approach to accurately accounting for non-local characteristics of near-wall turbulence. Model coefficients/functions are parameterized with the elliptic relaxation function to preserve the combined effects of near-wall turbulence and nonequilibrium. The characteristic length scale associated with the elliptic relaxation equation is formulated in terms of viscous and turbulent length scales in conjunction with the invariant of strain-rate tensor. Consequently, non-local effects are explicitly influenced by the mean flow and turbulent variables. A near-wall damping function is introduced to relax the viscous length-scale coefficient adhering to the elliptic relaxation model. Comparisons indicate that the new model improves the accuracy of flow simulations compared to the widely used Spalart-Allmar as model and remains competitive with the SST model.A good correlation is obtained between the current model and DNS/experimental data.

Assessment of subgrid-scale models for large-eddy simulation of a planar turbulent wall-jet with heat transfer

In the present work, five different subgrid-scale (SGS) models and implicit large-eddy simulation (LES) are evaluated and compared against the DNS for the simulation of the planar turbulent wall-bounded jet with heat transfer. The SGS models tested are the classical constant coefficient Smagorinsky model and its dynamic version, the wall-adaptive local eddy-viscosity (WALE) model, the turbulent kinetic energy one-equation model, and its dynamic version. The effects of using variable turbulent Prandtl number and the near-wall damping function are also studied in these models. The mean, second-order flow and heat-transfer statistics with the evolution of Nusselt number along the jet downstream are used to assess the different SGS models. The quality of resolution of the present LES are evaluated using the activity parameter and the index of resolution quality. Among the models tested, the constant coefficient Smagorinsky model together with Van-Driest damping predicts the solution accurately in the near-wall region as well as in estimating the thermal parameters. However, the dynamic models performed better in evaluating the Reynolds stress profiles away from the wall in the outer region. Capabilities of the models to predict the turbulent kinetic energy budgets, pressure-velocity gradient correlations and triple velocity correlations are also studied. The implemented variable Prandtl number algorithm is noted to have minimal influence on the evolution of the solution.

Effective equation to assess solute transport in two-lithology reservoirs

Understanding solute transport in heterogeneous porous media is a challenging problem due to the degree of complexities coming from the spatial variations of petrophysical properties. The developments of theoretical formulations are powerful tools used to obtain insights regarding the phenomenology involved in the transport process when analyzing large-scale systems. In this context and within the framework of the volume averaging method, in this work we present an effective one-equation model and its validity conditions for the solute transport in heterogeneous reservoirs characterized by two-lithology arrangements. For this purpose, the hypothesis of local mass equilibrium was assumed valid and what follows included the determination of physical constraints supporting such a hypothesis. In a previous work (Aguilar-Madera et al., 2019), the closure problems to calculate the effective coefficients of the two-equation model were presented, and with this base we present here the numerical estimation of the effective velocity vector and the mass dispersion tensor included in the one-equation model. As one application, the numerical solution of the one-equation model was carried out for different lithology arrangements and the breakthrough curves for two permeability ratio scenarios were analyzed. The results confirm that the permeability ratio dramatically impact not only the tracer dispersion but also the arrival time at observation points, which is strongly correlated to the velocity vector field and its spatial deviations. We found that the local mass equilibrium is met more easily in lenticular reservoirs provided that the boundary and initial conditions have no long-term effects. Finally, the dispersion coefficient was compared against reported reservoir field data, finding similarities in the orders of magnitude for certain geologies.

Wall heat transfer coefficient and effective radial conductivity of ceramic foam catalyst supports

In this work, different tubular solid-foam packed-beds were experimentally investigated under air flow in order to determine the effective conductivity and wall-to-bed heat transfer coefficient. Several foam types with different material, namely silicon carbide and alumina, and cell density were tested. Foam samples were experimentally studied using different air flow rates and wall temperatures ranging between 200 and 500°C. Axial and radial steady-state temperatures in the bed were experimentally measured and analysed with a two-dimensional, one-equation model. A Chilton–Colburn-type correlation based on the reactor diameter as characteristic dimension was proposed for the wall-to-bed heat transfer coefficient, which proved good to describe the behaviour of all tested foams. Effective radial thermal conductivity and pressure drop in the foams were also characterized.

Usefulness of comprehensive income reporting for dividend policy in the polish listed companies

This study examines the strength and character of relationship between dividend policy and comprehensive income in Polish manufacturing companies. We formulated two research questions:1) does comprehensive income determine the ability to pay out a dividend?2) what is the key role of comprehensive income in shaping a dividend policy?To receive the answers to these questions, we put a research hypothesis stating that in the Polish listed companies a positive relationship between comprehensive income and dividend policy can be observed.The research was conducted among the companies operating in the manufacturing sector (4xx of the Warsaw Stock Exchange (the WSE) classification). The analytical period is between 2009–2017 and reflects the current legal status of the financial reporting of public companies in Poland. Finally, our analysis comprised 83 manufacturing companies and 664 observations.The study procedure was carried out in two stages. The first stage was to show the results of empirical research conducted among the analysed companies and regarding the variability of comprehensive income in comparison to net income (consequently other comprehensive income). At the second stage, we presented the results of investigation on the strength and character of an impact of comprehensive income on dividends. For that purpose, two research methods were used. Firstly, we calculated the Pearson’s correlation coefficients between comprehensive income (and net income) and future dividend ratios, as well as the other company’s performance and tested for significance using the t-Student test. Secondly, we used regression analysis and two one-equation models built for comprehensive income and net income. Moreover, we used descriptive statistics, t- Student test, and F-test.The conducted research allowed us to draw two main conclusions: 1) there was a positive character of dependence between comprehensive income and future dividend, however, the correlation between them was very weak; 2) the impact of comprehensive income on future dividend ratio was positive, but we noted no statistical significance.

Parametric data-based turbulence modelling for vortex dominated flows

ABSTRACTComputational fluid dynamics simulations employing eddy-viscosity turbulence models remain the baseline numerical tool in the aerospace industry, mainly due to their numerical stability and computational efficiency. However, many industrially relevant cases require a level of accuracy that is not routinely achieved by global turbulence models. The simulation of leading-edge vortices shed at low aspect ratio wings is one such class of flows that remains a challenge for turbulence modelling. A local approach is proposed in which a parametrised eddy-viscosity turbulence model is calibrated using experimental results of configurations and flow conditions similar to the one being analysed. In this paper, the Spalart–Allmaras one-equation model is enhanced with additional source terms, which are exclusively active in the vortex field. An automatic optimisation procedure with experimental data as reference is then applied. The resulting optimised model improves the eddy viscosity distribution for a limited but relevant range of configurations and flow conditions.

Assessment of Turbulence Models in a Hypersonic Cold-Wall Turbulent Boundary Layer

In this study, the ability of standard one- or two-equation turbulence models to predict mean and turbulence profiles, the Reynolds stress, and the turbulent heat flux in hypersonic cold-wall boundary-layer applications is investigated. The turbulence models under investigation include the one-equation model of Spalart–Allmaras, the baseline k - ω model by Menter, as well as the shear-stress transport k - ω model by Menter. Reynolds-Averaged Navier-Stokes (RANS) simulations with the different turbulence models are conducted for a flat-plate, zero-pressure-gradient turbulent boundary layer with a nominal free-stream Mach number of 8 and wall-to-recovery temperature ratio of 0.48 , and the RANS results are compared with those of direct numerical simulations (DNS) under similar conditions. The study shows that the selected eddy-viscosity turbulence models, in combination with a constant Prandtl number model for turbulent heat flux, give good predictions of the skin friction, wall heat flux, and boundary-layer mean profiles. The Boussinesq assumption leads to essentially correct predictions of the Reynolds shear stress, but gives wrong predictions of the Reynolds normal stresses. The constant Prandtl number model gives an adequate prediction of the normal turbulent heat flux, while it fails to predict transverse turbulent heat fluxes. The discrepancy in model predictions among the three eddy-viscosity models under investigation is small.

Evaluation of the eddy viscosity turbulence models for the simulation of convection–radiation coupled heat transfer in indoor environment

This paper presents a numerical evaluation of the eddy viscosity turbulence models (the zero-equation, one-equation, k-ε group and k-ω group models) in terms of CFD modelling of convection-radiation coupled heat transfer in indoor environment. The results show that all the models except the zero-equation model have acceptable accuracies to predict the bulk air temperature. However, the zero-equation and one-equation models seriously overpredict the convective heat transfer coefficient at the heater surfaces while the k-ε group models severely underpredict it, ultimately resulting in the incorrect predictions of the thermal buoyancy flows and contaminant dispersion. On the contrary, the k-ω group models perform best to calculate the convection-radiation heat split, which is critical to the successful predictions of the near-wall flow field and distribution patterns of the contaminants. This study also demonstrates the importance of simultaneously considering the convective and radiative heat transfer mechanisms when evaluating the turbulence models for indoor environment simulations.

Consistently formulated eddy-viscosity coefficient for k-equation model

ABSTRACTAn approach to devising a consistency formulation for (production-to-dissipation ratio) is proposed to obtain a non-singular (coefficient of eddy-viscosity) embedded in the one-equation model based on the turbulent kinetic energy k. The dissipation rate ε is evaluated with an algebraically prescribed length scale having only one adjustable coefficient, accompanied by an anisotropic function enhancing the dissipation in non-equilibrium flow regions. The model accounts for the distinct effects of low Reynolds number (LRN) and wall proximity. The stress-intensity ratio is formulated as a function of local variables without resorting to a constant . The parameters and entering the turbulence production prevents presumably the overestimation of in flow regions where non-equilibrium effects could result in a misalignment between turbulent stress and mean strain rate with a linear eddy-viscosity model. A comparative assessment of the present model with the Spalart–Allmaras (SA) one-equation model and the shear stress transport (SST) k–ω model is provided for well-documented simple and non-equilibrium turbulent flows. Finally, the current model provides a proposal to compute free shear flows.

Turbulent Aeroheating Measurements on a 7-Deg Cone in High-Enthalpy CO2 Flow

A database of heating and pressure measurements on a 7 deg half-angle cone in a high-enthalpy expansion tunnel in has been generated to support development and validation of computational models to be employed in the design of future Mars missions. Laminar, transitional, and turbulent simulations were performed at the test conditions for comparisons with the data. Close agreement was obtained for both fully laminar and fully turbulent conditions. For the remaining transitional/turbulent conditions, agreement to within, or slightly more than, the estimated experimental uncertainty was demonstrated. The influence of transition intermittency and transition length models on predicted heating levels was demonstrated, as were differences in turbulent heating predictions generated using various algebraic, one-equation, and two-equation turbulence models. These comparisons provide some measure of confidence in turbulent simulation capabilities; however, because the data were not obtained on a relevant entry vehicle geometry, it is not possible to fully quantify computational uncertainties for the definition of Mars mission aerothermodynamic environments at this time.

Evaluation of Wray-Agarwal turbulence model for simulation of neutral and non-neutral atmospheric boundary layers

Wray-Agarwal (WA) turbulence model is a one-equation eddy-viscosity model derived from a blended k−ω/k−ε formulation. The model performance is better than the commonly used Spalart-Allmaras one-equation model and comparable to shear stress transport (SST) model, for a wide range of canonical flows in aerodynamics. This study proposes a Monin-Obukhov similarity theory (MOST) consistent modification of the buoyancy production/destruction term in this model to simulate the atmospheric boundary layer under different stratifications. The rough wall function is also modified to be consistent with MOST. A benchmark case for testing the streamwise homogeneity evaluated the model performance. In the tests, the wind and turbulence profiles did not decay along the stretch due to the model consistency with MOST. Adoption of the model for urban flow and wind energy simulations requires further testing.

Comparison of the one-equation LWR models for density and for speed

The macroscopic models describe road traffic flow without consideration of individual vehicles, but through aggregated traffic parameters, i.e. the flow rate, density and speed. The one-equation models are various LWR models that the most important component is the continuity equation. This equation must be supplemented by an additional equation that defines a specific relationship between two variables (traffic parameters), usually between speed and density. However, it is possible to eliminate one of those variables from the system, as a result of which the problem is limited only to one equation and one variable. In this article, two formulations of the one-equation LWR model will be considered, i.e. the density-dependent equation, and the speed-dependent equation. The question therefore arises, which of two formulations is more advantageous for a road traffic solution? Comparison of those two formulations of the LWR model is carried out only on a continuous level. Due to similar, quasi-linear form of both equations, the comparison has been carried out using their eigenvalues (characteristic speeds) for various static relationships between speed and density. Final evaluation of both one-equation formulations used spectral radiuses expressed as functions of the variables used in particular models.

A delayed detached eddy simulation model with low Reynolds number correction for transitional swirling flow in a multi-inlet vortex nanoprecipitation reactor

The objective of the presented work is to verify a delayed detached eddy simulation (DDES) model for simulating transitional swirling flow in a micro-scale multi-inlet vortex reactor (MIVR). The DDES model is a k-ω based turbulence model with a low Reynolds number correction applied to the standard k-ω model such that the Reynolds-averaged Navier-Stokes (RANS) component of the DDES model is able to account for low Reynolds number flow. By limiting the dissipation rate in the k-equation, the large-eddy simulation (LES) part of the DDES model behaves similarly to a one-equation sub-grid model. The turbulent Reynolds number is redefined to represent both modeled and resolved turbulence level so that underestimation of the RANS length scale in the LES range can be reduced. Applying the DDES model to simulate both laminar and transitional flow in the micro-scale MIVR produces an accurate prediction of mean velocity and turbulent intensity compared with experimental data. It is demonstrated that the proposed DDES model is capable of simulating transitional flow in the complex geometry of the micro-scale MIVR. These simulation results also help to understand the flow and mixing patterns in the micro-scale MIVR and provide guidances to optimize the reactor for the application of producing functional nanoparticles.

Differences in Men’s and Women’s Wages in the Education Sector in the Baltic Sea Region States

Abstract In the presented study, we analyse the men and women wage differences in the education sector and in the group of people employed as professional teachers in the Baltic Sea Region countries. For the analysis, we have applied one-equation econometric models and the Oaxaca-Blinder decomposition technique in the expansion proposed by Oaxaca and Ransom. The data are taken from the EU Structure of Earnings Survey of 2014. The obtained results show that the men and women wage gap for the analysed groups varies considerably among the analysed countries. We can also note that the GPG ratios in the education sector are usually lower than those at a national level. In addition, we can remark that in the states where remuneration systems are more transparent (Germany, Sweden), the level of explanation of ‘raw’ wage gaps is much higher than in other countries.

A Contribution for the Assessment of Discretization Error Estimators Based on Grid Refinement Studies

This paper presents grid refinement studies for statistically steady, two-dimensional (2D) flows of an incompressible fluid: a flat plate at Reynolds numbers equal to 107, 108, and 109 and the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg with Re = 6 × 106. Results are based on the numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations supplemented by one of three eddy-viscosity turbulence models of choice: the one-equation model of Spalart and Allmaras and the two-equation models k – ω SST and k−kL. Grid refinement studies are performed in sets of geometrically similar structured grids, permitting an unambiguous definition of the typical cell size, using double precision and an iterative convergence criterion that guarantees a numerical error dominated by the discretization error. For each case, different grid sets with the same number of cells but different near-wall spacings are used to generate a data set that allows more than one estimation of the numerical uncertainty for similar grid densities. The selected flow quantities include functional (integral), surface, and local flow quantities, namely, drag/resistance and lift coefficients; skin friction and pressure coefficients at the wall; and mean velocity components and eddy viscosity at specified locations in the boundary-layer region. An extra set of grids significantly more refined than those proposed for the estimation of the numerical uncertainty is generated for each test case. Using power law extrapolations, these extra solutions are used to obtain an approximation of the exact solution that allows the assessment of the performance of the numerical uncertainty estimations performed for the basis data set. However, it must be stated that with grids up to 2.5 (plate) and 8.46 (airfoil) million cells in two dimensions, the asymptotic range is not attained for many of the selected flow quantities. All this data is available online to the community.

Viscous flow simulations at high Reynolds numbers without wall functions: Is [formula omitted] enough for the near-wall cells?

The Reynolds-Averaged Navier Stokes equations supplemented by eddy-viscosity models are still the most common mathematical model for the simulation of wall-bounded viscous flows at high Reynolds numbers. Viscous flow simulations around complex geometries are more accurate with the direct application of the no-slip condition at walls, i.e. avoiding the use of wall functions. Determination of the shear-stress at the wall from its definition requires the use of near-wall cells which are typically requested to present a non-dimensional height of y+≃1. However, the effect of this widespread rule of thumb on the numerical accuracy of the solutions has not been established and it may be dependent on the turbulence model choice.In this paper, we discuss the numerical accuracy of viscous flow simulations performed with a RANS solver supplemented by three eddy-viscosity models in the calculation of statistically steady, incompressible flows: the one-equation model of Spalart & Allmaras and the two-equation models SST k–ω and k–kL. The selected test cases are: a flat plate at Reynolds numbers Re equal to 107, 108 and 109; the NACA 0012 airfoil at angles of attack of 0, 4 and 10 degrees with Re=6×106 and the KVLCC2 tanker at model and full scale Reynolds numbers, Re=4.6×106 and Re=2.03×109. Calculations are performed in sets of geometrically similar grids with different sizes of the near-wall cells height to assess the influence of this choice on the numerical uncertainty of the solutions.The results show that the numerical accuracy of the simulations is dependent on the selected turbulence model. For y+<1, there is no significant effect of the near-wall cell height on the numerical uncertainty of the selected flow quantities for the Spalart & Allmaras and k–kL models and so y+≃1 is sufficient to obtain numerical uncertainties of friction resistance coefficients smaller than 1% . On the other hand, with the SST k–ω model, y+≃1 leads to numerical uncertainties of friction resistance coefficients larger than 5% for all the present test cases. To attain the same uncertainty of the other turbulence models with the SST k–ω, one must use y+≃0.1.

Numerical-based theoretical analysis on effects of weak fluid acceleration of free-stream due to wind-tunnel blockage on grid-generated turbulence

We present a numerical-based theoretical analysis for effects of weak fluid acceleration of free-stream due to wind-tunnel blockage on grid-generated turbulence. The increasing of displacement thickness in the developing of the boundary layers due to wind-tunnel blockage reduces the effective area of the test section of the wind tunnel. Wind tunnel blockage, which often occurs in small wind tunnels, could accelerate the freestream in the wind tunnel and thereby affect the decay of grid-generated turbulence. A previous study proposed a mathematical form that validated the blockage effects on the grid-generated turbulence by demonstrating that the fluid acceleration is too weak to affect the anisotropy of the grid-generated turbulence. The previous study used the governing equation of the two-equation model based on the k−ϵ modeling. The present study have derived a mathematical form to describe the effects of weak fluid acceleration by using the governing equation used in the previous study with simple numerical simulation. The previous mathematical form has an applicable limit of St<0.2. Here, S is the streamwise nondimensional acceleration rate and is defined as S=(dU′/dx′)/(Uo/M), where U′, x′, Uo, and M are the streamwise mean velocity, the streamwise direction, the inflow velocity, and the mesh size, respectively. Moreover, t is the convection time normalized by the characteristic time M/Uo, which may be comparable to the normalized streamwise direction x/M for small S. The applicable limit in the present mathematical form is increased thirty-fold compared with that of the previous mathematical form. The effectiveness of the present scheme is then discussed using a scheme derived from the one-equation model as well as the previous scheme.

A New One‐Equation Model of Fluid Drag for Irregularly Shaped Particles Valid Over a Wide Range of Reynolds Number

AbstractA new drag law for irregularly shaped particles is presented here. Particles are described by a shape factor that takes into account both sphericity and circularity, which can be measured via the most commonly used image particle analysis techniques. By means of the correlation of the drag coefficient versus the particle Reynolds number and the shape factor, a new drag formula, which is valid over a wide range of Reynolds number (0.03–10,000), is obtained. The new model is able to reproduce the drag coefficient of particles measured in terminal velocity experiments with a smaller scatter compared to other laws commonly used in multiphase flow engineering and volcanology. Furthermore, the new formula uses only one equation, whereas previous models, to insure validity over an ample range, made use of a step function that introduces a discontinuity at the switch of equations. Finally, this drag law works in the whole range of variation, from extremely irregular particles to perfect sphere. A code of the iterative algorithm for the drag coefficient calculation and terminal velocity is included in the supporting information both as a Fortran routine and a Matlab function.