Turbulence modeling

The need for a cleaner environment in urban areas and the high cost of petroleum products which are becoming scarce due to unbalanced relation between supply and demand besides air pollution of sources has led to the research for other fuels to replace fossil fuels. Ethanol from biomass waste is such an alternative to petroleum products. Most studies on optimization of process variables using Response Surface Methodology apply Central Composite Designs yet other designs exist. Optimal designs have fewer trials employed with the aim of obtaining efficient designs for fitting reduced quadratic or higher order models. Coded values of a second order optimal rotatable design in four dimensions constructed using balanced incomplete block designs (BIBD) was fit into experimental data in order to study the effects of four process variables namely; time, PH, temperature and substrate concentration on fermentation of pineapples peels using Saccharomyces cerevisiae for ethanol production. Normal probability plots and Multiple R-squared of 0.9323 and Adjusted R-squared of 0.8944 which measure model fitting reliability indicated aptness of the model. Most values of Probability F were less than 0.05, confirming that the model terms were significant and only 6.8% of the total variation could not be explained by the model ensuring good adjustment of the model to experimental data. Model adequacy was also confirmed by the good agreement between the experimental data and predicted values. The design was found reliable in modeling, and studying the effects of the four factors to the processes of fermentation of pineapples peels as substrate for ethanol production using Saccharomyces cerevisiae

Au sein d’un echangeur a air, les trompes a air permettent de creer l’ecoulement d’air froid necessairea son bon fonctionnement. Ces dispositifs, qui peuvent ^etre assimiles a des jets subsoniques confinesen conduit, peuvent contribuer au bruit rayonne par les avions lors des phases au sol. Nous proposonsdans cette these de developper un outil numerique predictif de l’acoustique rayonnee par ces dispositifsafin de pouvoir proposer des solutions de reduction de bruit appropriees. Cet outil est adapte au contexteindustriel de Liebherr Aerospace. Une methode stochastique permet, a partir d’un calcul stationnaireRANS, de generer un champ de vitesse turbulente qui autorise la formation d’un terme de forcage dansles equations d’Euler linearisees qui sont alors utilisees comme un operateur de propagation. Un nouveaumodele stochastique base sur l’hypothese de sweeping est developpe. Ce dernier permet de produiredes champs instationnaires respectant certaines proprietes aerodynamiques statis...

The flow structure of a jet at high pressure issuing normally from an orifice in a flat plate into a supersonic stream was examined using a Reynolds-averaged Navier-Stokes solver. The study confirmed the presence of secondary horseshoe vortices as well as identified new vortical structures that emanated downstream of the jet.

In order to gain insight into the one-dimensional turbulence (ODT) model of Kerstein [1] as it pertains to residual stress closure in large-eddy simulation (LES), we develop ensemble mean closure (EMC), an algebraic stress closure based on the mappings and time scale physics employed in ODT. To allow analytic determination of the stress the ODT model is simplified, conceptually, such that eddy events act upon a velocity field linearized by the local resolved scale strain. EMC can account for viscous effects, addressing the laminar flow finite eddy viscosity problem without implementation of the dynamic procedure . The algebraic form of the model lends itself to analysis and we are able to derive a theoretical value for the eddy rate constant. This value is a bound on the rate constant for full ODT subgrid closure and yields good results in LES of decaying isotropic turbulence with EMC.

This report describes an approach for extending the one-dimensional turbulence (ODT) model of Kerstein [6] to treat turbulent flow in three-dimensional (3D) domains. This model, here called ODTLES, can also be viewed as a new LES model. In ODTLES, 3D aspects of the flow are captured by embedding three, mutually orthogonal, one-dimensional ODT domain arrays within a coarser 3D mesh. The ODTLES model is obtained by developing a consistent approach for dynamically coupling the different ODT line sets to each other and to the large scale processes that are resolved on the 3D mesh. The model is implemented computationally and its performance is tested and evaluated by performing simulations of decaying isotropic turbulence, a standard turbulent flow benchmarking problem.

This report describes an approach for extending the one-dimensional turbulence (ODT) model of Kerstein [6] to treat turbulent flow in three-dimensional (3D) domains. This model, here called ODTLES, can also be viewed as a new LES model. In ODTLES, 3D aspects of the flow are captured by embedding three, mutually orthogonal, one-dimensional ODT domain arrays within a coarser 3D mesh. The ODTLES model is obtained by developing a consistent approach for dynamically coupling the different ODT line sets to each other and to the large scale processes that are resolved on the 3D mesh. The model is implemented computationally and its performance is tested and evaluated by performing simulations of decaying isotropic turbulence, a standard turbulent flow benchmarking problem.

In order to gain insight into the one-dimensional turbulence (ODT) model of Kerstein [1] as it pertains to residual stress closure in large-eddy simulation (LES), we develop ensemble mean closure (EMC), an algebraic stress closure based on the mappings and time scale physics employed in ODT. To allow analytic determination of the stress the ODT model is simplified, conceptually, such that eddy events act upon a velocity field linearized by the local resolved scale strain. EMC can account for viscous effects, addressing the laminar flow finite eddy viscosity problem without implementation of the dynamic procedure . The algebraic form of the model lends itself to analysis and we are able to derive a theoretical value for the eddy rate constant. This value is a bound on the rate constant for full ODT subgrid closure and yields good results in LES of decaying isotropic turbulence with EMC.

This paper discusses the modelling of natural convection in air cavities using four eddy viscosity turbulence models. Two 2D closed cavities with deferentially heated walls opposite each other and one 3D open ended tall cavity with one side is heated and adiabatic for all the others were investigated. The CFD simulation results for the two 2D cases were compared with their corresponding experimental measurements and it was found that the komega turbulence model offered the best solution among the turbulence models tested. Close agreement was also achieved between the k-omega model and the experiments in the prediction of temperature and velocity fields for the 3D case with Monte Carlo radiation model. The work has demonstrated the ability of the eddy viscosity turbulence models for modelling natural convection and radiation in different types of cavities.

This paper discusses the modelling of natural convection in air cavities using four eddy viscosity turbulence models. Two 2D closed cavities with deferentially heated walls opposite each other and one 3D open ended tall cavity with one side is heated and adiabatic for all the others were investigated. The CFD simulation results for the two 2D cases were compared with their corresponding experimental measurements and it was found that the komega turbulence model offered the best solution among the turbulence models tested. Close agreement was also achieved between the k-omega model and the experiments in the prediction of temperature and velocity fields for the 3D case with Monte Carlo radiation model. The work has demonstrated the ability of the eddy viscosity turbulence models for modelling natural convection and radiation in different types of cavities.

We investigate a hierarchy of eddy-viscosity terms in proper orthogonal decomposition (POD) Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to large eddy simulation (LES) data for a flow around a vehicle-like bluff body called an Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (J. Fluid Mech., vol. 260, 1994a, pp. 351–375;J. Fluid Mech.vol. 275, 1994b, pp. 257–283). Robustness of the model solution with respect to initial conditions, eddy-viscosity values and model order is achieved only for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528, 2011). Only the POD ...

Rectangular jet or nappe flow constitutes one of the energy dissipation methods in arch dams. The high turbulence and aeration phenomena that appear in fall jets and dissipation basins make difficult to carry out their characterization only based in classical methodologies: reduced models and empirical formulae. The Hydraulics Laboratory of the Universidad Politécnica de Cartagena (Spain) has an infrastructure designed specifically for the study of turbulent jets and energy dissipation in plunge pools. The device allows us to study air-water two-phase phenomena (aeration, spray, spread and impact). The mobile mechanism lets us to vary the discharge heights between 1.70 and 4.00 m and flows between 10 and 150 l/s. To improve the knowledge of the phenomenon of turbulent jets, we are measuring aeration rates by means of fiber optical equipment, velocities in different sections of the stilling basin with Doppler instrumentations and pressures on the bottom of the plunge pool with piezoresistive transducers. Computational Fluid Dynamics programs (CFD) are based on numerical solution of the Reynolds Averaged Navier-Stokes (RANS) equations, together with turbulence models of different degrees of complexity. The programs simulates the interaction between different fluids, such as the air-water two-phase flows, and constitute a new and powerful tool that could let contrast and complement the lab measurement. There are many studies modeling spillways with CFD methodology using different eddy viscosity turbulence models with accurate results. However, the study of overflow nappe impingement jets has not been sufficiently studied. Scale effects are important effects that need to be taking account. In this study, numerical simulations has been carry out using different scales ratio of the lab device according to Froude similarity (1:1; 1:10; 1:20 and 1:40). This paper compares the Parametric Methodology proposed by for the evaluation of hydrodynamic action in plunge pools, revised by Castillo and Carrillo (2011, 2012), with more and new laboratory measurements and the simulation results obtained with CFD program.

The Venturi tube, a widely used cavitation-generating device in the petroleum and chemical industries, is valued for its simple structure and safe stability. During cavitation-induced gas-liquid two-phase flow, collapsing bubble clusters release high temperature and pressure energy with associated effects. This study uses Fluent to investigate the Venturi tube’s internal flow field characteristics. By comparing pressure contours, velocity contours, and gas phase distributions under varying inlet-outlet pressure ratios, throat length-to-diameter ratios, and diffusion angles, it analyzes cavitation flow evolution. Results show: for a fixed-structure Venturi with constant outlet pressure, stronger cavitation effects occur with higher inlet pressure, larger throat length-to-diameter ratios, and smaller diffusion angles. These findings clarify internal flow patterns and the influence of hydraulic and structural parameters on cavitation intensity.

This study investigates the nonlinear and classical problem of Von Kármán's viscous swirling fluid flow caused by a single rotating disk. Despite over a century since this problem was first introduced, recent advancements enable more accurate calculations and practical results than previously possible. The core innovation of this paper lies in the application of the Hybrid Analytical and Numerical method (HAN method), which facilitates the derivation of a semi-analytical solution to complex nonlinear differential equations. The HAN method combines numerical and analytical approaches to solve nonlinear problems. Initially, the system of nonlinear differential equations is solved using an arbitrary numerical method. The numerical solution then aids in extracting the analytical solution, which can take forms such as polynomial solutions with constant and unknown coefficients. Since boundary conditions lack the capacity to generate a sufficient number of algebraic equations, the numerical solution provides the additional required equations. The flexibility of the HAN method stems from its ability to leverage various numerical methods, making it a robust approach for solving nonlinear differential equations. Using this methodology, the Von Kármán problem is analytically calculated with remarkable accuracy. Furthermore, this study provides highly precise calculations of several physical and practical outputs, including the thickness of the layer, the slope of flow lines at the wall in the peripheral direction, the peripheral component of wall shear stress, the moment on one side of the wetted disk, the dimensionless moment coefficient for both sides of the disk, Reynolds number as a function of the disk's finite radius, volume flux, and mechanical power. This research contributes to two main perspectives: first, the mathematical aspect, which demonstrates the ability of the HAN method to solve various nonlinear problems; second, the practical-physical perspective, showcasing the enhanced accuracy and reliability of the obtained results in analyzing fluid flow mechanics.

Many parameters can be influenced by speed flow, like form resistance and speed flow. Form obstacles influence structure turbulent flowing water, so that can raise potency change speed and pattern flow around the structure building. This study considers the tilted corner prisoner structure triangle and incoming water flow to influence the pattern and speed flow that occur after the genre through the obstacle triangle using IRic simulation. IRIC is a simulation platform that supports numeric and various breaker computing problems in water science and engineering. The variations of the corner prisoner plate triangle used in this study are 30°, 45°, and 60°. The transverse channel is blocked with a ratio of β = 1/10, and upstream-downstream boundary conditions are open. Results show what will happen in the genre critical moment pass structure. Because the change is significant, Froude's number. Observations that took place showed that speed flow on each slope was stable before the pass structure and was not stable after the pass structure. Whirlpools occur on a 45° slope, so the speed flow that occurs will increase significantly. On an incline of 60°, it will have the largest Froude number, which is caused by the depth (D) created by the 60° angle being higher in comparison with another slope. For an incline of 30°, show speed, more flow, and a constant flow with a bigger Froude number.

Se presenta en este trabajo un analisis numerico por medio de volumenes finitos en casos de aplicacion aeroespacial. El estudio se realiza mediante el uso de un codigo desarrollado en la UNC con el auspicio de la UNC y de CONICET. El mencionado codigo permite simular flujos supersonicos inestacionarios y estacionarios utilizando la tecnica de volumenes finitos. Usa mallas no estructuradas en geometrias tridimensionales. En el codigo se ha implementado una nueva tecnica numerica que permite reducir la viscosidad artificial sin que el esquema “Total Variation Diminishing - TVD” pierda robustez. Para verificar el desempe˜no del esquema numerico se modelan cuatro casos; un tubo de choque, un flujo supersonico que llega a una cuna, el flujo en el interior de un difusor y el flujo alrededor del Ala ONERA M6 transonica con la finalidad de verificar la capacidad del metodo para capturar discontinuidades, asi como las condiciones del flujo consiguiendose resultados satisfactorios y concordan...

ABSTRACTPrediction of aeroengine exhaust plume near-field development requires knowledge of velocity and turbulence distributions at nozzle exit. The high Reynolds number nozzle inlet boundary layers of engineering practice are fully turbulent, but acceleration can induce re-laminarisation. Thus, to reproduce nozzle exit conditions accurately, large eddy simulation (LES) for plume prediction must be capable of capturing re-laminarisation and any subsequent boundary layer recovery. Validation is essential to establish a credible LES methodology, but previous studies have suffered from lack of nozzle inlet/exit measurements in the test cases selected. Validation data were here taken from an experiment on a convergent round nozzle with a parallel exit extension to allow boundary layer recovery. LES inlet condition generation applied a rescaling/recycling method (R2M), whose performance was validated against measurements of first and second moment statistics as well as the turbulence in...

This article analyzes (one-dimensional) nonlinear wave propagation over a disordered fluid body having a small viscosity. The lower boundary is disordered and modeled by a random process. As a pulse shaped nonlinear wave propagates over this turbulent boundary, the velocity and wave elevation are viewed as random fields. Starting from first principles the eddy viscosity is characterized and shown to depend on different scales. This is captured as the leading order pseudodifferential operator resulting from the asymptotic analysis of stochastic differential equations. A discussion is provided showing that mean-field theory would have not captured the correct attenuation rate for the large scale object. Numerical results are provided illustrating the accuracy of the eddy viscosity expression.

This investigation explores heat transfer enhancement in shell and tube heat exchangers using a tri-hybrid nanofluid suspension consisting of copper (Cu), iron oxide (Fe 3 O 4), and multiwall carbon nanotubes (MWCNT) dispersed in water as the base fluid. The numerical simulation employs a novel Galerkin finite element method (GFEM) to study turbulent convection influenced by Reynolds number, tube radius, and nanoparticle volume fraction. The complex interactions among these physical parameters are analyzed in terms of average Nusselt number, entropy generation, and flow behavior. Results demonstrate that increasing nanoparticle concentration and fluid velocity significantly improve heat exchanger efficiency. Visualization tools such as streamlines, entropy contours, and isotherms provide comprehensive insight into optimizing thermal management systems.

Background: Fresh concrete, fly ash and mining slurries are all frictional-visco-plastic fluids. Fresh concrete flow in Tremie pipes is used to control concrete flow rate and minimise bleeding and dilution when concrete is poured into deep submerged excavations for pile foundation construction. Slurries with very fine aggregates are used to backfill underground voids and mines to prevent subsidence and surface structural damage. Backfilling and injection of granular materials into mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control subsidence. As a viable environmental solution, mine waste and rejected materials from underground coal seams are used in both backfilling and injection mine operations. For example, during longwall mining the grout slurry is pumped into the separated beds of the fractured rock mass through a pipeline connected to a central vertical borehole, which is drilled deep into the inter-burden rock strata above the coal mine seam. Either as dilute slurry or thick paste or cake, the fill material normally needs to travel a significant longitudinal distance either in a channel, a tremie pipe, a long pipeline, or radially in a disk shaped crack in the rock mass. An undesirable blockage can occur in the central borehole, in the crack or in the transportation channel or pipeline system when the slurry velocity falls below a certain critical threshold velocity, indicating a material phase change from cohesive-viscous to cohesive-frictional. This chapter presents complete analytical solutions of the required pump pressure versus fluid volume rate for such multi-phase fluids, which are categorised as frictional Bingham-Herschel-Bulkley fluids. The theory derived can be applied to flow of such fluids in pipes, disks and open channels. Furthermore, general analytical solutions have been developed for such fluids in terms of velocity and pressure gradients and velocity and pressure, as a function of flow length (e.g. pipe length, disk radial distance, or channel length) from which special and familiar equations for simpler fluids are derivable by mathematical reduction of the general equations. The formulation is distinct in considering many new aspects including: variable shear parameters rather than fixed values; inclusion of total nonlinear behaviour; and, implementation of a friction function to mimic behaviour of the depositing and consolidating stiff slurry or paste, which can cause a significant pressure rise as a result of the increased shear resistance. Bingham-Herschel-Bulkley fluids: Recent laboratory and field experiments on mine-backfill fluids, slurries, cements, pastes and concretes proved their wide range of shear resistance www.intechopen.com Fluid Dynamics, Computational Modeling and Applications 152 and complex behaviour in response to shearing necessitating development of a general, nonlinear, cohesive, viscous, frictional, nonlinear, non-Newtonian model of shear stress versus shear strain rate, as an extension to the classical Bingham-Herschel-Bulkley fluid . Viscous plastic behaviour of such fluids are further simplified or idealised as a reduced or special case of the general nonlinear case. In practice, and for various engineering applications, this generic shear stress function is central in all mathematical formulations to describe fluid flow as a function of pressure gradient. Examples of such applications are flow of slurry, paste and concrete through pipes and tremie pipes and channels for fluid transportation and testing purposes, flow through disks, cracks, joints and rock fractures for injection and backfilling purposes. As a first approach, the shear stress vs. strain rate relation may be idealised by a simple linear function, the so called viscous-plastic Bingham line, which may be derived from a simple linear regression analysis of laboratory experimental data . The value of the shear strength function at zero shear strain rate, i.e. plastic yield strength (also called cohesion), and slope of the linear curve (viscosity) are two important parameters of the fluid property in the simple linear idealised case. The large cavern created by an underground mine may eventually lead to failure of the overburden rock, propagating layer by layer to the surface, resulting in substantial ground surface subsidence [10-13], as schematically shown in Figure ). As a major potential hazard, mining induced subsidence significantly affects mining costs where major surface structures and natural environment need to be protected, e.g. mining under river systems, gorges, cliffs, power lines, pipelines, communication cables, major roads and bridges, and other significant surface facilities . Remedial measures to manage damage caused by subsidence can often be very costly with potentially damaging impacts and irreversible consequences. Backfilling and injection of granular materials into the mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control mine subsidence. Grouts and slurries made of mine and power plant wastes and rejects are viable environmental backfill solutions to both ground stability and mine waste management problems . Like concrete paste, the flowing slurry can be categorised as a generally nonlinear viscous cohesive (Bingham Herschel-Bulkley) fluid . However, in mining applications, to reduce ground surface subsidence and control the propagation of the overburden movement to the surface, the solid particles in the injected slurry must deposit in the bed separation gaps of the coal seam over-burden strata, e.g. in longwall mining the grout slurry is pumped into the separated beds of the rock mass from a batching plant source through pipelines connected to a central vertical borehole, which is drilled deep into the over-burden rock above the coal seam (Figure ). Flow blockage can occur in the injection system, when the slurry velocity falls below a certain critical threshold velocity. The stiffening, consolidating non-flow slurry can generally be categorised as a frictional cohesive soil . In other words, a change of material phase from cohesive-viscous to cohesive-frictional will occur. Using a smaller scale model, this field injection practice has been simulated at the QCAT laboratory of the Commonwealth Scientific & Industrial Research Organisation (CSIRO) in Brisbane, Australia, to study the influence of various grout injection parameters by pumping slurries through various pipes of different sizes and diameters and for different applications (Figures ). As an important industrial application, grout injection into the inter-burden strata is used as a modern technology to control and reduce coal mine subsidence . Slurry mixes of coal mine and power plant waste materials, e.g. fly ash or any other coal wash rejects, are injected back into the inter-burden rock strata during longwall mining . To reduce subsidence and control inter-burden strata movement, the injected www.intechopen.com

We theoretically and numerically investigate the effect of temperature dependent density and viscosity on turbulence in channel flows. First, a mathematical framework is developed to support the validity of the semi-local scaling as proposed based on heuristic arguments by Huang, Coleman, and Bradshaw [“Compressible turbulent channel flows: DNS results and modelling,” J. Fluid Mech. 305, 185–218 (1995)]. Second, direct numerical simulations (DNS) of turbulent channel flows with different constitutive relations for density and viscosity are performed to assess and validate the semi-local scaling for turbulent statistics. The DNS database is obtained by solving the low-Mach number approximation of the Navier-Stokes equation. Finally, we quantify the modulation of turbulence due to changes in fluid properties. In the simulations, the fluid is internally heated and the temperature at both channel walls is fixed, such that the friction Reynolds number based on wall quantities is Reτ = 39...

A numerical investigation to study the behavior of the turbulent non-newtonian flow through a channel from two expansions with constant heat flux at the lower wall is calculated. This behavior predicted for Reynolds number from 16000 to 100000 and for the range of power law index from 0.4 to 1.6. The distance ratio between the two expansions is varied from 1 to 6 with expansion ratio from 1.5 to 3 and aspect ratio from 1 to 4. These results are obtained by using ANSYS FLUENT 15.0. The results illustrate the effect of the first expansion on the second expansion and the variation of the reattachment length and Nusselt number. These results gave an idea about the ability of the step distance to control the whole flow structure.

A numerical investigation to study the behavior of the turbulent non-newtonian flow through a channel with two expansions in two directions under constant heat flux at the lower wall is presented. This study covered Reynolds numbers range of 4000 to 100000 and a power law index from 0.4 to 1.6. The expansion ratio is varied from 1.5 to 3 and aspect ratio from 1 to 4. The full Navier Stocks equations and the k-ε turbulence model besides to the energy equation are used to model the problem. These results are obtained by using ANSYS FLUENT 15.0. The results illustrate that the effect of the first expansion on the second expansion via representation the variation of the reattachment length and Nusselt number. These results gave an idea about the effect of the corner in the two-direction expansion and the whole flow structure.

While important for delivering a range of important socioeconomic services, road crossings and culverts are known to block the upstream fish passage, particularly for small-body-mass fish species. Using a combination of physical and numerical CFD modelling, design guidelines were developed for smooth box culverts without appurtenance, with a novel approach based upon the basic concepts: (a) the culvert design is optimised for fish passage for small to medium water discharges, and for flood capacity for larger discharges, and (b) lowvelocity zones are provided along the wetted perimeter in the culvert barrel, and quantified in terms of a fraction of the wetted flow area where the local longitudinal velocity is less than a characteristic fish speed linked to swimming performances of targeted fish species. The approach relies upon an accurate physicallybased knowledge of the entire velocity field in the culvert barrel, specifically the longitudinal velocity map, given that behavioural observations confirmed that fish prefer to swim upstream in low-velocity zones (LVZs) next to the walls and bottom corners. While the focus of the study is on the upstream passage of small-bodymass fish, typical of Australian native fish species, the approach and methodology are relevant to most box culvert structures.

A new dataset of uniform and steady sheet-flow experiments is presented in this paper. An acoustic concentration and velocity profiler (ACVP) is used to measure time-resolved profiles of collocated 2C velocity ($u,w$) and sediment concentration and to measure the time evolution of the bed interface position. Ensemble averaging over 11 similar experiment realisations is done to evaluate the mean profiles of streamwise velocity, concentration, sediment flux and Reynolds shear stress. The repeatability, stationarity and uniformity of the flow are carefully checked for a Shields number${\it\theta}\approx 0.5$and a suspension number of$S=1.1$. The mean profile analysis allows to separate the flow into two distinct layers: a suspension layer dominated by turbulence and a bed layer dominated by granular interactions. The bed layer can be further subdivided into a frictional layer capped by a collisional layer. In the suspension layer, the mixing length profile is linear with a strongly red...

De nombreuses applications industrielles mettent en jeu des écoulements gaz-particules. On peut citer, entre autres, les turbines aéronautiques et les réacteurs à lit fluidisé de l’industrie chimique. Dès lors, l’amélioration de ces dispositifs, imposée par les nouvelles normes européennes sur les émissions polluantes, nécessite une connaissance prédictive de la dynamique de ce type d’écoulements ainsi que l’évaluation de ses grandeurs caractéristiques telles que la ségrégation spatiale des particules. La simulation numérique est aujourd’hui largement utilisée à cet effet. Les équations de la phase gazeuse sont résolues par Simulation Numérique Directe (SND) ou par Simulation des Grandes Echelles (SGE). Le couplage avec la phase dispersée peut être envisagé de deux manières. Une première approche, dite lagrangienne, consiste à calculer les trajectoires des particules. Communément utilisée et précise, son coût numérique ne permet cependant pas d’envisager son application à des géomét...

We investigate a hierarchy of eddy-viscosity terms in POD Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to Large Eddy Simulation data for a flow around the vehicle-like bluff body called Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by . Robustness of the model solution with respect to initial conditions, eddy viscosity values and model order is only achieved for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier-Stokes based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

Classical Mechanics by Herbert Goldstein is a foundational graduate-level textbook that provides a rigorous and comprehensive treatment of classical mechanics, the study of the motion of physical systems under the influence of forces. The book covers a wide range of topics including Newtonian mechanics, Lagrangian and Hamiltonian formulations, central force motion, rigid body dynamics, small oscillations, canonical transformations, Hamilton–Jacobi theory, and an introduction to relativistic mechanics and nonlinear dynamics.

Goldstein emphasizes formal mathematical methods and principles of theoretical physics, making the text suitable for students seeking a deeper understanding of the analytical framework underlying classical physics. The book is known for its clear exposition, challenging problem sets, and its role as a bridge to more advanced topics in modern physics and quantum mechanics.

Reynolds-averaged Navier-Stokes simulations have been performed of the turbulent flow (Re = 48000) through a tight lattice (P/D = 1.194) subchannel typical of those found in nuclear thermal-hydraulic applications. A wide range of turbulence models (linear eddy-viscosity, nonlinear eddy-viscosity and stress-transport) and the use of different near-wall treatments (low-Re, standard and analytical wall-functions) have been considered. The results demonstrate both the failure of the standard wall function approach in reproducing flows with significant near-wall convection and pressure-gradient effects, and the limitations of using linear eddy-viscosity approaches in flows with anisotropic effects. Although models utilizing low-Re near-wall approaches demonstrated superior overall performance, the analytical wall-function tested offers improvements over the standard 'log-law' based formulation.

Recent applications of Computational Fluid Dynamics (CFD) to heat and fluid flow in the electric power generation industry are presented, with various turbulence models (including heat transfer) ranging from Reynolds Averaged Navier-Stokes Equations (RANS) with Eddy Viscosity or Second Moment Closures, to Direct and Large Eddy Simulation (DNS, LES). These simulations are clearly demonstrating the renewed importance of turbulence closure models and the need to combine all those approaches depending on the applications since the finer models require a priori estimates of the turbulence characteristics, if only to define the strategy for mesh generation. In order to address full scale industrial problems, recent developments are taking every advantage of very high performance computing facilities available. The Code_Saturne software coupled with SYRTHES for conjugated heat transfer proved to be very well suited to this kind of architecture. They are available as free software under GNU...

This study re-visits the largely overlooked, but highly promising, topic of multi-scale RANS modelling for non-equilibrium turbulent flows. The paper presents the development of RANS models which, at an effective-viscosity level, involve two transport equations for the turbulent kinetic energy; one for the energy of the large scale, energy-producing, eddies and one for that of the smaller eddies, which, through continuous break-up, transfer turbulence energy to the dissipative scales. Two transport equations for the transfer-rate of the turbulent kinetic energy are also necessary, one for the rate of energy transfer from the large to the smaller scales and one for the rate of transfer from the smaller to the dissipative scales, the latter being of course the dissipation rate of turbulence. The two-time-scale model of has been the starting point of the current developments. The coefficients of the terms which appear in these models have been determined from asymptotic analyses of decaying grid turbulence, homogeneous shear flows and local-equilibrium boundary-layer flows. The models were subsequently further developed to become more responsive to strong non-equilibrium features, such as those caused by strong shear. It has been identified that in general, models which have been optimised to satisfy equilibrium flows (as is usually done) are unable to capture the strong changes the flows are subjected to due to highly non-equilibrium effects, thus needing further development. Within the two-time-scale framework, it has been found possible to overcome this problem, by exploiting the additional information available on the turbulence spectrum. In a further departure from the development of earlier effective-viscosity two-time scale models, here the eddy viscosity expression is made sensitive to the local strain rate, as was also the case in single-scale models developed by . The original contribution of this research is the development of two widely tested two-timescale linear-eddy-viscosity models, referred to here as NT1 and NT2. The difference between these two models is primarily in the eddy viscosity formulation. The two resulting models have been tested over nine test cases and their performance has been compared to those of the standard high-Reynolds number k -ε model, the SSG Reynolds stress transport model and the two-time-scale model of . The test cases comprise of homogeneous shear and normally strained flows, adverse-pressure-gradient, favourable-pressure-gradient and oscillatory boundary layer flows, fully developed oscillatory and ramp up pipe flows and steady and pulsated backward-facing step flows. The models proposed here show considerable promise. They perform well in all cases tested and provide clear improvements over a range of single-and multi-scale models tested in an earlier study , over a set of test cases which cover a wide range of physical flow phenomena. Moreover, these improvements are achieved without any significant increase in computational requirements in comparison to the computational needs of single-time-scale eddy-viscosity models. The reason for that is the assumption behind the development process of these models, which considers that the flow can be characterized by a single time or length scale. However, this assumption is, admittedly, an approximation of the real physics of the flows, since the turbulent kinetic energy is characterized by a wide range of length (or time) scales, which build the so called turbulent kinetic energy spectrum. The turbulent kinetic energy spectrum represents the vortex stretching and cascade processes typical of turbulent motions. These phenomena become therefore important enough to be taken into account, when non-equilibrium features are present in the flow. Non-equilibrium flows include flows subjected to rapid changes, such as sudden expansions and/or contractions, rapid variations in time and/or imposed pressure gradients. Such conditions are very often found in industrial flows, which in turn emphasizes the importance of developing models that can reliably predict non-equilibrium flows. As STS models assume spectral equilibrium in order to describe the flow through a single time or length scale, they are thus not expected to be capable of reproducing such non-equilibrium features, especially capturing the lag in response of the turbulence field to changes in the mean flow field. In order to take into account some features of the turbulent kinetic energy spectrum, Multiple-Time-Scale (MTS) models arose within the RANS framework. For practical purposes, and for the sake of simplicity and reduced computational effort, these models usually employ two time (or length) scales to characterize the flow and are therefore referred to as Two-Time-Scale (TTS) models. Recent studies of the authors have investigated the performance of some TTS models against commonly used and classical STS models in a wide range of 2-D nonequilibrium flows. Both eddy viscosity and Reynolds stress transport schemes were considered for both STS and TTS models. The results confirmed the expectation of possible improvements in predicting non-equilibrium flows by the TTS models, as also indicated by

This essay explores the parallels between philosophical thought, particularly that of Immanuel Kant and Martin Heidegger and the scientific pursuit of understanding turbulence. It argues that turbulence is not merely a physical phenomenon but a manifestation of deeper ontological truths that elude empirical modeling. Through this lens, we reflect on the limits of knowledge, the nature of Being, and the presence of meaning beyond scientific grasp. Extending the analogy, we consider music as another domain where infinite creativity meets structured theory, revealing insights about meaning, presence, and human experience. This is a personal reflective essay, not a formal research article. It explores connections between turbulence and philosophical ideas from Kant and Heidegger, aiming to reflect on the limits of scientific understanding. It is shared here in the hope it sparks dialogue or resonates with others interested in the deeper meaning behind complex physical phenomena.

In this study a comparison of three turbulence closure models (two isotropic and one anisotropic) with experimental data is performed. The interaction between the main channel (MC) flow and the floodplain (FP) generates a complex flow structure. A shallow mixing layer develops between the MC flow and the slower FP flow generating a high horizontal shear layer, streamwise and vertical vortices, momentum transfer and other phenomena, related to velocity retardation and acceleration. This phenomenon dissipates part of the kinetic energy and contributes to the reduction of the velocity differences between the MC and the FP. The large scale vortices that are generated in the shear layer are anisotropic, provoking the formation of secondary flow cells that influence the primary velocity distribution. These three-dimensional turbulent structures can be reasonable well reproduced by a simple anisotropic model (Algebraic Stress Model). The isotropic models are capable of simulating the bound...

In recent years, CFD has become an increasingly used tool in the design of blood-based devices. However, the estimation of red blood cells damage (hemolysis) remains a very important challenge due to the complex rheology of blood and the turbulence present in most pumping devices. The objective of this study was to identify an appropriate turbulence model suitable for predicting hemolysis in Hemodialysis cannula. Several modern turbulence models were evaluated in comparison to Direct Numerical Simulation (DNS), which was used as the gold standard. The fluid dynamics in the cannula was modeled as a coaxial jet in which Reynolds’ number approached 2800. Based on comparison of velocity and stress time-averaged profiles, the Shear Stress Transport (SST) model with Gamma-Theta transition was identified as an optimal compromise between accuracy and computational cost.

In this paper, verification and validation of a turbulence closure model is performed for an experimental compound channel flow, where the velocity and turbulent fields were measured by a Laser Doppler Velocimeter (LDV). Detailed Explicit Algebraic Reynolds Stress Model (EARSM) simulations are reported. There are numerous methods and techniques available to evaluate the numerical uncertainty associated with grid resolution. The authors have adopted the Grid Convergence Index (GCI) approach. The velocity components, the turbulence kinetic energy (TKE), the dissipation rate and the Reynolds stresses were used as variables of interest. The GCI results present low values for the u velocity component, but higher values in what concerns the v velocity component and w velocity component (representing secondary flows) and for Reynolds stresses RSxy and RSyz. This indicates that the mean flow has converged but the turbulent field and secondary flows still depend on grid resolution. Based on ...

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He's polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations.

Modern economic theory, both at the micro and macro levels, and ever-increasing complexity of economic processes have led to the need to create and improve the use of mathematical modeling and quantitative analysis. On the basis of the latter, one of the areas of economic research emerged and was formed-econometrics. Econometric models and methods are now not only powerful tools for obtaining new knowledge in economics, but also a widely used apparatus for making practical decisions in forecasting, banking, and business. The development of information technologies and special application programs, the improvement of analysis methods have made econ-ometrics a powerful tool for economic research. The paper presents the problem of numerical calculation of turbulent flow in a channel with a minor obstacle based on the COMSOL MULTIPHYSICS®software package. The SA and SST models were used for the numerical calculation of turbulent flow in the channel. The results obtained using the COMSOL MULTIPHYSICS®package were compared with experimental data and also analyzed in term of computational load. The results are obtained for Reynolds number Re = 20,000. CCS CONCEPTS • Reynolds-averaged Navier-Stokes equations; • SA and SST models; KEYWORDS econometrics •mathematical modeling •turbulence models •Reynolds-averaged Navier-Stokes equations •SA and SST models

The mechanism of drag reduction due to spanwise wall oscillation in a turbulent boundary layer is considered. Published measurements and simulation data are analyzed in light of Stokes’ second problem. A kinematic vorticity reorientation hypothesis of drag reduction is first developed. It is shown that spanwise oscillation seeds the near-wall region with oblique and skewed Stokes vorticity waves. They are attached to the wall and gradually align to the freestream direction away from it. The resulting Stokes layer has an attenuated nature compared to its laminar counterpart. The attenuation factor increases in the buffer and viscous sublayer as the wall is approached. The mean velocity profile at the condition of maximum drag reduction is similar to that due to polymer. The final mean state of maximum drag reduction due to turbulence suppression appears to be universal in nature. Finally, it is shown that the proposed kinematic drag reduction hypothesis describes the measurements sig...

Most existing multivariate models in finance are based on diffusion models. These models typically lead to the need of solving systems of Riccati differential equations. In this paper, we introduce an efficient method for solving systems of stiff Riccati differential equations. In this technique, a combination of Laplace transform and homotopy perturbation methods is considered as an algorithm to the exact solution of the nonlinear Riccati equations. The resulting technique is applied to solving stiff diffusion model problems that include interest rates models as well as two and three-factor stochastic volatility models. We show that the present approach is relatively easy, efficient and highly accurate.

Most existing multivariate models in finance are based on diffusion models. These models typically lead to the need of solving systems of Riccati differential equations. In this paper, we introduce an efficient method for solving systems of stiff Riccati differential equations. In this technique, a combination of Laplace transform and homotopy perturbation methods is considered as an algorithm to the exact solution of the nonlinear Riccati equations. The resulting technique is applied to solving stiff diffusion model problems that include interest rates models as well as two and three-factor stochastic volatility models. We show that the present approach is relatively easy, efficient and highly accurate.

This study investigates how the inclusion of porous block influences the flow development in a serpentine passage, which consists of two straight square-sectioned ducts connected with a square-ended bend. Aluminium porous foam blocks have been attached to two opposite walls of the straight sections normal to the bend in a staggered manner, used as turbulence promoters. The investigation considers flows in the passage in stationary conditions, and in orthogonal rotation. Particle image velocimetry (PIV) is used to map out the flow development. The wall static pressure distribution along two sides of the passage (the leading and trailing under rotating conditions) is also provided. Tests are carried out at Reynolds numbers of 16,000, 26,000 and 36,000 and rotation numbers of 0.32 and 0.64. Preliminary tests, in a smooth (without porous blocks) passage of identical geometry, establish the reference conditions and enable the identification and quantification of the effects of the porous blocks on the flow development. PIV results show the meandering manner of the flow both upstream and downstream of the bend region, due to the presence of the porous blocks. Within the bend, in contrast to the case with a smooth upstream section, a single vortex dominates the flow. While rotation does not change the overall flow character, it does force more fluid through the blocks along the trailing (pressure) side of the duct. For both stationary and rotating conditions, the upstream section is long enough for the flow to become periodic over successive rib intervals, which produces very attractive data for CFD validation. These data include mean flow as well as second-moment profiles. The pressure coefficient is found to decrease linearly along the channel, due to the high blockage by the porous blocks. Rotation does not change the shape of the pressure coefficient profile but causes a greater reduction along the passage.

This study investigates how the inclusion of porous block influences the flow development in a serpentine passage, which consists of two straight square-sectioned ducts connected with a square-ended bend. Aluminium porous foam blocks have been attached to two opposite walls of the straight sections normal to the bend in a staggered manner, used as turbulence promoters. The investigation considers flows in the passage in stationary conditions, and in orthogonal rotation. Particle image velocimetry (PIV) is used to map out the flow development. The wall static pressure distribution along two sides of the passage (the leading and trailing under rotating conditions) is also provided. Tests are carried out at Reynolds numbers of 16,000, 26,000 and 36,000 and rotation numbers of 0.32 and 0.64. Preliminary tests, in a smooth (without porous blocks) passage of identical geometry, establish the reference conditions and enable the identification and quantification of the effects of the porous blocks on the flow development. PIV results show the meandering manner of the flow both upstream and downstream of the bend region, due to the presence of the porous blocks. Within the bend, in contrast to the case with a smooth upstream section, a single vortex dominates the flow. While rotation does not change the overall flow character, it does force more fluid through the blocks along the trailing (pressure) side of the duct. For both stationary and rotating conditions, the upstream section is long enough for the flow to become periodic over successive rib intervals, which produces very attractive data for CFD validation. These data include mean flow as well as second-moment profiles. The pressure coefficient is found to decrease linearly along the channel, due to the high blockage by the porous blocks. Rotation does not change the shape of the pressure coefficient profile but causes a greater reduction along the passage.

The laminar 2-D blended convection of the nanofluids at different volume fractions has gained interest in the last decade due to an enormous application in technology. The laminar-flow stream system can be further modified by changing the geometry of the channel, adding an external heating source, and changing the initial conditions at which the stream is being influenced. The investigation of this system includes the variation of the geometrical parameters of the channel, Reynolds number, Nusselt number, and type of the nanoparticles used in preparing the nanofluid with water as the base fluid. These parameters constitute a very successful leading to utilize the numerical solutions by using a finite volume method. Regarding heat flow, one side of the channel was supplied by the heat while the temperature of the other side was kept steadily. The upstream walls of the regressive confronting step were considered as adiabatic surfaces. The nanofluids were made by adding aluminum oxide ...

Systems Tl-Sb-S(Se) are characterized by the formation of isoformula compounds Tl 3 SbS ( Se ) 3 , TlSbS ( Se ) 2 , and TlSb 3 S ( Se ) 5 , which possess semiconductor properties . Investigation of phase equilibria of complex semiconductor systems over stable joins, which contain analogue compounds, is of interest due to the possibility of formation of broad regions of solid solutions in them and controlling their composition and properties. Taking into account these facts, we investigated phase equilibria in reciprocal system 3 Tl 2 S + Sb 2 Se 3 3 Tl 2 Se + Sb 2 S 3 (A). Physicochemical characteristics of the starting compounds of system A are presented in 11]. Compounds Tl 2 S , Tl 2 Se , Sb 2 S 3 , and Sb 2 Se 3 melt congruently at 723, 663, 819, and 863 K, respectively. The structure of Tl 2 S is rhombohedral (antitype CdI 2 ): a = 12.22 Å, c = 18.21 Å, space group R -. The compound Tl 2 Se crystallizes in the tetragonal crystal system with the unit cell parameters a = 8.52Å, c = 12.68 Å, space group P 4/ ncc . The compounds Sb 2 S 3 and Sb 2 Se 3 have orthorhombic structures (space group Pbmn -):

Numerical modelling of turbulence plays a crucial role in providing accurate microscale wind fields which are necessary to predict transport and dispersion of pollutants in the vicinity of buildings reliably. Although more elaborate turbulence models have been developed, two-equation turbulence models are widely applied in atmospheric flow and dispersion models. The large CPU time required for direct numerical simulation (DNS) and large eddy simulation (LES) still prohibits their application to microscale wind flow simulations of practical interest. In this study, the results obtained by the microscale model MIMO for the three-dimensional flow around a surface mounted cubical obstacle, using different two-equation turbulence models, are discussed. The accuracy of the investigated turbulence models is assessed by comparing the numerical results with wind tunnel measurements.