Finite Rate Chemistry Effects in Turbulent Reacting Flows

The classical “fast chemistry” analysis by Damköhler remains a common basis for calculation methods aimed at turbulent reacting flows. Perturbation approaches can be used to introduce finite rate chemistry effects, particularly where a distinct chemical time-scale separation is present, though more comprehensive techniques, e.g. based on a transported joint probability density function (JPDF), are typically required. Potential difficulties with the JPDF technique include issues related to the intrinsic structure of turbulent flames, particularly at low Reynolds numbers, and models for molecular mixing. The ability to predict the formation of NO is particularly interesting in this context given the strong sensitivity to chemical kinetic and non-adiabatic effects. The current work initially provides an assessment of uncertainties in the formation chemistry of NO in the context of new quantitative measurements, obtained in non-premixed laminar methane/air counterflow flames using ps-LIF, and subsequently explores how these translate to turbulent flames. A consistent systematically reduced (16 independent, 4 dependent and 28 steady state scalars) reaction mechanism is applied to model the turbulent flames of Barlow and co-workers (8200 ≤Re≤ 44000). The highest Re number flame additionally permits an investigation into the ability of the transported JPDF technique to deal with emissions of nitric oxide in flames close to global extinction. The work shows that the technique has the potential to reproduce NO levels and conditional PDFs under conditions with significant local extinction/re-ignition to within the uncertainties associated with the principal elementary reaction steps.     

Statistical characteristics of the diffusion of a chemically active additive in a turbulent mixing zone

In the article a numerical solution of the connected system of the equations of turbulent transfer for the fields of the velocity and concentration of a chemically active additive is used to calculate a number of the second moments of the concentration field in a flat mixing zone. The system of transfer equations is derived from the equations for a common function of the distribution of the fields of the pulsations of the velocity and the concentration [1] and is simplified in the approximation of the boundary layer. A closed form of the transfer equations is obtained on the level of three moments, using the hypothesis of four moments [2] and its generalized form for mixed moments of the field of the velocity and the field of a passive scalar. The differential operator of the closed system of the equations of turbulent transfer for the fields of the velocity and the concentration is found by a method of closure not of the parabolic type but of a weakly hyperbolic type [3]. An implicit difference scheme proposed in [4] is used for the numerical solution. The results of the numerical solution are compared with the experimental data of [5].     

Maximum wave velocity in the moments system of a relativistic gas

We consider the system of moments associated with the relativistic Boltzmann-Chernikov equation. Using the particular symmetric form obtained by the closure procedure of Extended Thermodynamics we deduce a lower bound for the maximum velocity of wave propagation in terms of the number of moments for a non-degenerate gas. When the number of moments increases this velocity tends to the speed of light. We also give the lower bound estimate in the limit cases of ultrarelativistic fluids and in the non relativistic approximation. Received September 28, 1998     

Numerical modeling of turbulent-transfer processes in a mixing zone

The series of moments of the velocity field in a two-dimensional zone of mixing is calculated in this article by numerically solving a system of turbulent-transfer differential equations derived from an equation for a single-point distribution function of the velocity pulsation field [1] and simplified to an approximation of the boundary layer. The closed form of the transfer equation is obtained at the level of the third moments using the Millionshchikov hypothesis [2]. The differential operator of the system under this closure turns out to be weakly hyperbolic [3], and not parabolic. A difference scheme is proposed that realizes the method of matrix fitting [4]. A comparison is carried out with an experiment [5, 6].     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Validation of turbulence models in a simulated air-conditioning unit

Details are given of a study to obtain experimental data in an idealized environment for the purpose of evaluating the corresponding computational predictions and which supplement parallel measurements made in actual packaged air-conditioning units. The system consisted of a purpose-built low-speed wind tunnel with a working section constructed to reproduce particular features of the real units. In the experiment, both the mean velocity profiles and turbulence properties of the flow are obtained from triple-hot-wire anemometry measurements. A numerical model, based on finite volume methodology, was used to obtain the solution of the Reynolds-averaged Navier–Stokes equations for incompressible isothermal flow. The Reynolds stress terms in the equations are calculated using the standard k–ϵ model and second-moment closure (Reynolds stress) models. The accuracy of the two models was evaluated against the experimental measurements made 10 mm downstream of a baffle. The results show that the standard k–ϵ model gave the better agreement except in regions of strong recirculation. © 1997 John Wiley & Sons, Ltd.     

Dependence of the aerodynamic forces on the Reynolds number with the three-dimensional flow of a vortical stream around bodies

The article discusses the dependence of the viscous stresses on the Reynolds number Re in three-dimensional flows around bodies of arbitrary form. It is shown that, with an infinite growth of a vortex with the approach to a body, singular terms appear in an asymptotic expansion in terms of ε=Re^?1/2. The infinite values of the derivatives of the velocity in flows of an incompressible liquid are due to the initial vorticity; in a supersonic flow, they can be connected with the absence of a maximum of the entropy at the critical flow line behind a leading shock wave. The singularity in the tangential stresses brings about the appearance of additional terms in the total aerodynamic forces and moments acting on the body.     

Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons.In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDEs results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik.This study has two main objectives:1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDEs. However, the intent of the paper is by no means a general justification of the maximum entropy principle.2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.Received: 15 August 2003, Accepted: 8 November 2003, Published online: 11 February 2004PACS: 02.30.Jr, 02.60.Cb, 05.30.Jp, 44.10. + i, 63.20.-e, 66.70. + f, 65.40.Gr Correspondence to: M. Herrmann     

Transported PDF Modelling of a High Velocity Bluff-Body Stabilised Flame (HM2) Using Detailed Chemistry

A transported probability density function (PDF) approach closed at the joint scalar level was used to model a bluff body stabilised turbulent diffusion flame (HM2) investigated experimentally by Masri and co-workers. The current effort extends a previous study of HM1 (Re?=?15,800) to a flame with a higher degree of local extinction (Re?=?23,900). The impact of an algebraic model that accounts for local Damköhler number effects on the time-scale ratio of scalar to mechanical turbulence is also evaluated along with the impact of improved thermochemistry. The computations have been performed using a hybrid Monte Carlo/finite volume algorithm and a systematically reduced H/C/N/O mechanism featuring 300 reactions, 20 solved and 28 steady-state species. The joint scalar PDF equations were solved using moving particles in a Lagrangian framework and the velocity field was closed at the second moment level. The redistribution terms were modelled using the Generalized Langevin model of Haworth and Pope. Results show that scalar fields are reproduced with encouraging accuracy and that the revised time scale model improves agreement with experimental data. A high sensitivity to the NO chemistry was observed and encouraging agreement was obtained for the first two moments following adoption of updated reaction rates proposed in an earlier study.     

Grid‐independent depth‐averaged simulations with a collocated unstructured finite volume scheme

In this article, the depth‐averaged transport equations are written in a new way so that it is possible to solve the transport equations for very small water depths. Variables are interpolated into the cell face with two different schemes and, the schemes are compared in terms of computational cost and accuracy. The bed source terms are computed using two different assumptions. The effect of these assumptions on numerical simulations is then investigated. Solutions of transport equations on different types of unstructured triangular grids are compared and, an appropriate choice of grid is suggested. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient approaches of determining the motion of a spherical particle in a swirling fluid flow using weighted residual methods

The motion of a spherical particle released in a swirling fluid flow is studied employing the least-squares method and method of moments. The governing equations are obtained and solved employing the two methods. The accuracy of the results is examined against the results of a fourth-order Runge–Kutta numerical method. The effects of various parameters, namely the initial radius, initial radial velocity, initial angular velocity, and drag-to-inertia ratio, on the non-dimensional velocity profiles and particle position distribution are considered. The results show that the radial velocity increases over time while the angular velocity decreases, and that an increase in the initial radial velocity increases the particle radial distance and angular velocity but decreases the radial velocity profile.     

Direct numerical simulation of turbulent heat transfer in a square duct with transverse ribs mounted on one wall

Turbulent heat transfer in a ribbed square duct of three different blockage ratios are investigated using direct numerical simulation (DNS). The results of ribbed duct cases are compared with those of a heated smooth duct flow. It is observed that owing to the existence of the ribs and confinement of the duct, organized secondary flows appear as large streamwise-elongated vortices, which intensely interact with the rib elements and four sidewalls and have profound influences on the transport of momentum and thermal energy. This study also shows that the drag and heat transfer coefficients are highly sensitive to the rib height. It is observed that as the rib height increases, the impinging effect of the flow on the windward face of the rib strengthens, leading to enhanced rates of turbulent mixing and heat transfer. The influence of sidewalls and rib height on the turbulence structures associated with temperature fluctuations are analyzed based on multiple tools such as vortex swirling strengths, temporal auto-correlations, spatial two-point cross-correlations, joint probability density functions (JPDF) between the temperature and velocity fluctuations, statistical moments of different orders, and temperature spectra.     

First-order kinetic approximation for a reactive gas mixture

A multicomponent reacting gas with an arbitrary number of chemical species and one reversible reaction is studied at a kinetic level in the frame of discrete velocity models of the Boltzmann equation, with the main objective of deriving the reactive Navier Stokes equations of the model, and characterizing the dissipative terms related to shear viscosity, thermal conductivity and thermal diffusion. The closure of the system formed by conservation and chemical rate equations is based on a first-order Chapman-Enskog method, to be applied in the strong reaction regime, and on a convenient representation of the density vector space in terms of the macroscopic variables. A mathematical procedure is proposed which leads to identification of the transport coefficients, and may be applied to a quite large variety of reactive gas flows. Moreover, it allows characterization of the functional form of the transport coefficients in dependence on the local gas concentrations, once the model is specified.Received: 4 October 2004, Accepted: 3 December 2004, Published online: 18 March 2005PACS: 51.10. + y, 51.20. + d, 47.70.Fw Correspondence to: A.J. Soares     

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.     

Compressible closure models for turbulent multifluid mixing

This paper studies governing equations describing the turbulent fluid mixing behavior effectively. The goal is to propose a closure for compressible multiphase flow models with transport and surface tension, which satisfy the boundary conditions at the mixing zone edges, the conservation requirements, and an entropy inequality constraint. Implicitness of positivity for the entropy of averaging requires entropy inequality as opposed to conservation of entropy for microphysically adiabatic processes.     

联合极限状态下基于最大熵可靠度的易损性分析

考虑工程需求参数(EDP)的前四阶矩,提出基于最大熵可靠度理论的地震易损性分析方法.基于SAP2000建立钢筋混凝土框剪模型,选择最大层间位移角和最大层加速度衡量结构的联合性能极限状态,建立极限状态方程.不对EDP的分布进行人为假定,在不同峰值加速度(PGA)下计算两种EDP的前四阶矩,并作为约束条件,建立极限状态方程...     

Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations

The statistical moments of a non-linear system responding to random excitations are governed by an infinite hierarchy of equations; therefore, suitable closure schemes are needed to compute the more important lower order moments approximately. One easily implemented and versatile scheme is to set the cumulants of response variables higher than a given order to zero. This is applied to three non-linear oscillators with very different dynamic properties, and with Gaussian white noises acting as external and/or parametric excitations. It is found that the accuracy of computed second moments can be improved greatly by extending from the second order closure (Gaussian closure) to the fourth order closure, and that further refinement is unnecessary for practical purposes. Treatment of nonstationary transient response is also illustrated.     

Hydrodynamical Modeling of Charge Carrier Transport in Semiconductors

Enhanced functional integration in modern electron devices requires an accurate modeling of energy transport in semiconductors in order to describe high-field phenomena such as hot electron propagation, impact ionization and heat generation in the bulk material. The standard drift-diffusion models cannot cope with high-field phenomena because they do not comprise energy as a dynamical variable. Furthermore for many applications in optoelectronics one needs to describe the transient interaction of electromagnetic radiation with carriers in complex semiconductor materials and since the characteristic times are of order of the electron momentum or energy flux relaxation times, some higher moments of the distribution function must be necessarily involved. Therefore these phenomena cannot be described within the framework of the drift-diffusion equations (which are valid only in the quasi-stationary limit). Therefore generalizations of the drift-diffusion equations have been sought which would incorporate energy as a dynamical variable and also would not be restricted to quasi-stationary situations. These models are loosely speaking called hydrodynamical models. One of the earliest hydrodynamical models currently used in applications was originally put forward by Blotekjaer [1] and subsequently investigated by Baccarani and Wordeman [2] and by other authors [3]. Eventually other models have also been investigated, some including also non-parabolic effects [4–6, 8–20]. Most of the implemented hydrodynamical models suffer from serious theoretical drawbacks due to the ad hoc treatment of the closure problem (lacking a physically convincing motivation) and the modeling of the production terms (usually assumed to be of the relaxation type and this, as we shall see, leads to serious inconsistencies with the Onsager reciprocity relations). In these lectures we present a general overview of the theory underlying hydrodynamical models. In particular we investigate in depth both the closure problem and the modeling of the production terms and present a recently introduced approach based on the maximum entropy principle (physically set in the framework of extended thermodynamics [21, 22]). The considerations and the results reported in the paper are exclusively concerned with silicon.