Scalar Time Scale Research Articles

Investigation of Modified EMST Micromixing Model Performance on Lagrangian PDF Transported in Lifted Hydrogen/Air Fames

This article presents a numerical simulation of high hydrogen/air flames using the Lagrangian transported PDF method. This method enables the calculation of fluid composition changes resulting from convection and reaction without the need for modelling, while requiring modelling for molecular mixing. Consequently, the accuracy of calculations in this L-PDF method heavily relies on an accurate representation of the mixture model term. The Euclidean model, which provides a better description of physical mixing processes, is well-suited for modelling the molecular mixing term EMST. Additionally, the accuracy of this model depends on the value of the mixing constant, representing the ratio between the mechanical time scale and the scalar time scale. Two algebraic models for the mixing constant have been implemented in the computational code, employing a well-defined function to calculate this ratio for each cell. These models contribute to memory and CPU time savings. To account for turbulence and its interaction with physical phenomena, the RSM model is employed due to its ability to identify different areas of turbulent stresses. Hence, the primary objective of this study is to evaluate the capabilities of these algebraic models in predicting scalar fields within such flames. Overall, the predictions align well with experimental data, affirming the validity of these models.

Effects of roughness on the turbulent Prandtl number, timescale ratio, and dissipation of a passive scalar

Turbulence models for scalar and momentum transport are often analogous to each other, but roughness breaks this similarity. The turbulent Prandtl number, the momentum and scalar time-scales, and the dissipation budget are among the central quantities in the modeling of scalar transport. This study examines how roughness affects them by analyzing the results of direct numerical simulations of channel flow with resolved roughness. The flow field below the crest of the elements, which is very difficult to access experimentally, provides answers to long-standing questions.

Impact of molecular mixing and scalar dissipation rate closures on turbulent bluff-body flames with increasing local extinction

Bluff-body turbulent CH4: H2 (1:1) flames at 50% (HM1), 75% (HM2) and 91% (HM3) of the blow-off velocity (235 m s−1) were studied experimentally by Masri and co-workers and found to exhibit gradually increasing periodic and shear layer instabilities. The latter are coupled with increasing levels of local extinction with subsequent re-ignition further downstream. This study provides a systematic evaluation of the sensitivity of predictions to molecular mixing and scalar dissipation rate closures. The latter include extended forms of the Euclidean Minimum Spanning Tree (EMST) and modified Curl’s (MC) models, applicable to premixed turbulent flames via a closure that accounts for local Damköhler number effects (EEMST and EMC), and a conceptually related blended scalar time-scale approach (BEMST and BMC). Computations are performed using a hybrid finite volume (FV) – transported Joint Probability Density Function (JPDF) algorithm featuring stochastic Lagrangian particles, a comprehensive 48-scalar systematically reduced C/H/N/O mechanism, and a second moment method based on the Generalised Langevin Model that provides a partial resolution of the unsteady fluid motion. The sensitivity to solution parameters affecting the temporal resolution is quantified using Fourier transforms of the time histories of velocity and scalar traces. Radial profiles, conditional means and scatter plots are compared to the experimental data along with burning indices based on the conditional mean temperature. Vortex related instabilities ∼ 1 kHz in the outer shear layer appear for all closures with EMC showing periodic local extinction and re-ignition in the neck region for HM3 and flame turbules (i.e., discrete pockets of hot gas) separating periodically at frequencies ∼ 85 Hz. Results are similar to well–resolved JPDF/LES simulations for HM1. It is shown that the EMC and (E)EMST models essentially enclose the experimental data for HM2 and HM3. For HM3, emissions of NO are controlled by local extinction events that become increasingly sensitive to the molecular mixing closure as blow-off is approached.

A Hybrid Rans-Rsm/Composition PDF-Transport Method for Simulation of Hydrgen-Air Turbulent Diffusion Flame

In this present study, the numerical simulations are performed for an axisymmetric turbulent jet diffusion hydrogen/air flame, by using a hybrid Finite-Volume/Composition PDF-Transport method. This method represents a rational approach for the study of turbulent reacting flows containing signficant turbulence– chemistry interactions. The major attraction of Composition PDF method is that the terms associated with chemical reaction appear in closed form, leaving only molecular mixing and turbulent transport terms to be modeled. The accuracy of Composition PDF model calculations depends on the accurate representation of the chemistry and on the mixing model including the value of the mixing-model constant CΦ (mixing-model constant CΦ is the mechanical to scalar time scale ratio (CΦ =τt/τΦ).There has been considerable development in the past three decades in PDF methods, and reviews can be found in Dopazo and O’Brien [1] and Subramaniam and Pope [2]. The EMST model, which better describes the physical processes of mixing [3], is used here. An algebraic expression obtained from DNS calculation [4] was implemented, in FLUENT code (ANSYS FLUENT 13.0), via a so-called User Defined Function (UDF) to evaluate the time-scale ratio CΦ in each cell during the simulation course.A comparison is performed with experimental measurements of R.S. Barlow [5].Overall, profile predictions of mixture fraction, flame temperature and major species show a good agreement with experimental data.

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ0, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non-hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.

Prediction of stably stratified homogeneous shear flows with second-order turbulence models

The present study investigated the role of pressure-correlation second-order turbulence modelling schemes on the predicted behaviour of stably stratified homogeneous vertical-sheared turbulence. The pressure-correlation terms were modelled with a nonlinear formulation (Craft 1991), which was compared with a linear pressure–strain model and the ‘isotropization of production’ model for the pressure–scalar correlation. Two additional modelling issues were investigated: the influence of the buoyancy term in the kinetic energy dissipation rate equation and the time scale in the thermal production term in the scalar variance dissipation equation. The predicted effects of increasing the Richardson number on turbulence characteristics were compared against a comprehensive set of direct numerical simulation databases. The linear models provide a broadly satisfactory description of the major effects of the Richardson number on stratified shear flow. The buoyancy term in the dissipation equation of the turbulent kinetic energy generates excessively low levels of dissipation. For moderate and large Richardson numbers, the term yields unrealistic linear oscillations in the shear and buoyancy production terms, and therefore should be dropped in this flow (or at least their coefficient cε3 should be substantially reduced from its standard value). The mechanical dissipation time scale provides marginal improvements in comparison to the scalar time scale in the production. The observed inaccuracy of the linear model in predicting the magnitude of the effects on the velocity anisotropy was demonstrated to be attributed mainly to the defective behaviour of the pressure-correlation model, especially for stronger stratification. The turbulence closure embodying a nonlinear formulation for the pressure-correlations and specific versions of the dissipation equations failed to predict the tendency of the flow to anisotropy with increasing stratification. By isolating the effects of the dissipation rate equations, it was demonstrated that nonlinear closure provides an improved account of the normal components of the anisotropy tensor over the linear model; however, the vertical velocity correlation and the shear stress still present too large a discrepancy. The observed shortcomings may be intrinsic to the current structure of the pressure-correlation terms. None of the models were able to predict the ultimate collapse of the turbulent fluxes and counter-gradient fluxes. However, this weakness may be beyond the modelling of the redistribution processes, but instead in fundamental shortcomings in the energy dissipation rate equations and the dissipation tensor to adequately represent the phenomenology of the transition from shear- to buoyancy-dominated flows.

Evaluation of scalar mixing and time scale models in PDF simulations of a turbulent premixed flame

Numerical simulation results obtained with a transported scalar probability density function (PDF) method are presented for a piloted turbulent premixed flame. The accuracy of the PDF method depends on the scalar mixing model and the scalar time scale model. Three widely used scalar mixing models are evaluated: the interaction by exchange with the mean (IEM) model, the modified Curl’s coalescence/dispersion (CD) model and the Euclidean minimum spanning tree (EMST) model. The three scalar mixing models are combined with a simple model for the scalar time scale which assumes a constant C ϕ = 12 value. A comparison of the simulation results with available measurements shows that only the EMST model calculates accurately the mean and variance of the reaction progress variable. An evaluation of the structure of the PDF’s of the reaction progress variable predicted by the three scalar mixing models confirms this conclusion: the IEM and CD models predict an unrealistic shape of the PDF. Simulations using various C ϕ values ranging from 2 to 50 combined with the three scalar mixing models have been performed. The observed deficiencies of the IEM and CD models persisted for all C ϕ values considered. The value C ϕ = 12 combined with the EMST model was found to be an optimal choice. To avoid the ad hoc choice for C ϕ , more sophisticated models for the scalar time scale have been used in simulations using the EMST model. A new model for the scalar time scale which is based on a linear blending between a model for flamelet combustion and a model for distributed combustion is developed. The new model has proven to be very promising as a scalar time scale model which can be applied from flamelet to distributed combustion.

On the turbulent Prandtl number in homogeneous stably stratified turbulence

In this paper, we derive a general relationship for the turbulent Prandtl number Prt for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Prt, is developed in terms of a mixing length scale LM and an overturning length scale LE, the ratio of the mechanical (turbulent kinetic energy) decay time scale TL to scalar decay time scale Tρ and the gradient Richardson number Ri. We show that our formulation for Prt is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio LM/LE ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, TL/Tρ. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1–20; J. Fluid Mech., vol. 525, 2005, pp. 193–214) to propose a new parameterization for Prt in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Prt that will be useful for turbulence closure schemes in numerical models.

Spatial and Temporal Characteristics of OH in Turbulent Opposed-Jet Double Flames

Simultaneous high repetition-rate, two-point hydroxyl (OH) time-series measurements with associated PLIF/PIV measurements are employed to investigate spatio–temporal scales and flame-velocity interactions in turbulent opposed jets sustaining methane-air double flames. For a fuel-side equivalence ratio, ϕB = 1.2, a rich premixed flame exists on the fuel side while a diffusion flame exists on the air side of the stagnation plane. The bulk Reynolds number (Re) and strain rate (SR) can be adjusted to generate flames at ϕB = 1.2 with both well separated and completely merged flame fronts. Simultaneous PLIF/PIV measurements highlight distinct spatial OH structures of the premixed and diffusive fronts corresponding to variations in the flow field. The self-propagating tendency of the rich premixed front causes large-scale wrinkling, thereby enhancing the OH contour length by 15% as compared to the diffusive front. Two-point OH time-series measurements are implemented to quantify both spatial and temporal fluctuations via study of radial length and time scales. In general, these integral length and time scales follow similar trends and reach a minimum at the axial location of peak [OH]. In comparison to merged double flames having higher Re and SR, greater OH fluctuations are observed in the rich-premixed front as compared to the diffusive front for a well separated double flame. Because of the developing turbulence, the OH length scales exhibit reduced axial gradients across the reaction zone for higher Re in comparison to lower Re. A stochastic time-series simulation, using a state relationship based on a joint mixture fraction and progress variable, is utilized to extract estimated scalar time scales from those of measured OH. The simulations indicate that the hydroxyl fluctuations in double flames are only twice those of the underlying conserved scalar.

Thermal/Turbulence Time Scale Ratios in Low Exothermic Reacting and Non-reacting Flows

The aim of this work is to contribute to the study of chemistry effects on the turbulent scalar field that is, here, the temperature, through a better knowledge of the ratio between the scalar time scale and the mechanical time scale (Rt) in the distributed reaction regime characterized by a Damköhler number lower than unity. The numerical studies are performed by calculating the order of magnitude of Rt in both low exothermic reacting and non-reacting case. Propane has been chosen as a typical hydrocarbon for this study. The molar fractions are computed by the IEM (Interaction Exchange with the Mean) model in which the probability density function is calculated from its transport equation. The models and the numerical simulations are used to describe a jet stirred reactor with subsonic jet injection. The predictions are validated by comparison with experimental data for temperature and concentrations fields. It is pointed that in the non-reacting case, Rt parameter is around unity; the temporal scales dynamics and scalar are coupled. In this case, the transport of the passive scalar is ensured by the dynamic field. In the reactive case, however, this coupling is not assured any more. Indeed, the chemical reaction, characterized by a non-linear source term, affects the scalar field by decreasing its characteristic scales, which results in Rt lower than unity. The CFD-PDF predictions were within engineering accuracy of experimental data. It can be summarized that the results of exercise are satisfactory, and the CPU-time and RAM memory savings encouraging.

Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal CO/H 2 kinetics

Direct numerical simulations of three-dimensional turbulent temporally evolving plane CO/H 2 jet flames are performed with detailed chemistry at Reynolds numbers of up to 9000 and with up to 500 million grid points. The effect of Reynolds number on turbulent mixing properties and flame structure is quantified for low Damköhler number flames. These flames exhibit strong flame–turbulence interactions resulting in local extinction followed by re-ignition. The probability density of the stoichiometric scalar dissipation rate is found to be nearly log-normal with some negative skewness. Conditional statistics of the hydroxyl radical reveal increasing levels of extinction and longer re-ignition times with increasing Reynolds number. The mechanical-to-scalar mixing time scale ratio, a key quantity in transported probability density function (pdf) modeling, is investigated for both conserved and reacting scalars. The conserved scalar timescale ratio is found to be consistent with prior studies. For reacting scalars, the effects of molecular diffusivity and chemical reaction on the timescale ratio are quantified.

The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. Linear theory and direct numerical simulation

The effect of rotation on a homogeneous turbulent shear flow has been studied by means of a series of direct numerical simulations with different rotation numbers. The evolution of passive scalar fields with mean gradients in each of the three orthogonal directions in the flow was investigated in order to elucidate the effect of rotation on turbulent scalar transport. Conditions of the near-wall region of a boundary layer were approached by using a rapid shear and therefore, comparisons could be made with rapid distortion theory based on the linearized equations of the flow and scalar transport. Reynolds stresses, pressure–strain correlations and two-point velocity correlations were computed and turbulent structures were visualized. It is shown that rotation has a strong influence on the time development of the turbulent kinetic energy, the anisotropy of the flow and on the turbulent structures. Furthermore, rotation significantly affects turbulent scalar transport. The transport rate of the scalar and the direction of the scalar flux vector show large variations with different rotation numbers, and a strong alignment was observed between the scalar flux and the principal axes of the Reynolds stress tensor. The ratio of the turbulent and scalar time scales is influenced by rotation as well. The predictions of the linear theory of the turbulent one-point statistics and the scalar flux agreed fairly well with direct numerical simulation (DNS) results based on the full nonlinear governing equations. Nonetheless, some clear and strong nonlinear effects are observed in a couple of cases which significantly influence the development of the turbulence and scalar transport.

Tensorial time scale in turbulent gradient transport of Reynolds stresses

On the notion that the Reynolds stresses are transported with different time scales depending on the transport direction, the third-order velocity correlations are represented by a new turbulent gradient transport model with a tensorial Lagrangian time scale. In order to verify the proposed model, DNS data are first obtained in a turbulent channel flow at Reτ=180 and 300, and tensorial Lagrangian time scales are computed. The present model predictions are compared with direct numerical simulation data and those predicted by the third-order turbulent transport model of Hanjalic and Launder that uses a scalar time scale. The result demonstrates that the Reynolds stresses are indeed transported with different time scales depending on the transport direction.

레이놀즈 응력의 난류구배수송을 위한 텐서시간척도

On the notion that the Reynolds stresses are transported with different tune scale depending on the transport direction, the third order velocity correlations are represented by a new turbulent gradient transport model with tensorial Lagrangian time scale. In order to verify the proposed model, DNS data are first obtained in a turbulent channel flow at Re = 180 and tensorial Lagrangian time scales are computed. The present model predictions are compared with DNS data and those predicted by the third-order turbulent transport model of Hanjalic and Launder that uses a scalar time scale. The result demonstrates that the Reynolds stresses are indeed transported with different time scale depending on the transport direction.

Steady flamelet modelling of a turbulent non-premixed flame considering scalar dissipation rate fluctuations

A piloted turbulent methane/air jet flame (Sandia Flame D) is calculated by the combination of a multiple-time scale (MTS) k – ε turbulence model, the steady laminar flamlet model (SLFM), and the probability density function (PDF) transport equation approach. A joint PDF transport equation of mixture fraction and turbulent kinetic energy dissipation rate is computed by the node-based Monte Carlo particle method. The fluctuations of scalar dissipation rate at stoichiometric condition are deduced, in an ad hoc manner, from the distribution of turbulent kinetic energy dissipation rate based on their similar PDF distribution, viz. approximately lognormal. The SLFM is employed to decouple the interaction between turbulent fluctuations and chemistry through the instantaneous mixture fraction and the instantaneous stoichiometric scalar dissipation rate. The MTS k – ε turbulence model produces better prediction of the centerline profiles of mean velocity and mixture fraction than the standard k – ε model. Within the current modelling scope, the value of 1.5 for the ratio of mechanical to scalar timescales is better than the usual value of 2.0 since it gives better profiles of the root-means-square (rms) mixture fraction and mean CO, H 2 mass fractions. Compared to the presumed-PDF for scalar dissipation rate, the present ad hoc model improves the predictions of flame structure to some extent, especially the conditional means of CO and H 2 mass fractions in the fuel-rich region of the flame.

Comparison of OH time-series measurements and large-eddy simulations in hydrogen jet flames

Hydroxyl time-series measurements have been obtained using picosecond time-resolved laser-induced fluorescence in a turbulent H 2 /N 2 jet diffusion flame. The flame is well characterized, as it is part of the Turbulent Nonpremixed Flame Workshop. Large-eddy simulations (LES) of the same flame were used to predict OH, density, and mixture fraction temporal statistics. The two sets of results are for the first time compared to examine in detail the LES-predicted time scales for OH fluctuations. These time scales come from complete time series and cannot be recovered from more common single-time measurements or time-averaged simulations. The measured and predicted power spectral densities for OH are found to collapse to the same shape when normalized by the local integral time scale. Hence, LES predicts the correct scale distribution for scalar fluctuations. The measured time scales range from 0.4 to 1.1 ms along the jet centerline. The LES results are quantitatively low (predicted fluctuations are too fast) by a factor of two. Grid resolution, the chemical submodel, and procedures for time-series processing are found to have insignificant effects on the computed time scales. However, the turbulent fluctuations prescribed on the inflow plane strongly influence scalar time scales, indicating that separate validation of mixture fraction and velocity time scales are necessary for further model development.

On the scalar probability density function transport equation for binary mixing in isotropic turbulence at supercritical pressure

Effects of Soret cross-diffusion, whereby concentration diffusion occurs in the presence of either temperature or pressure gradients, are investigated in the context of the binary isotropic turbulent mixing problem at supercritical pressure with emphasis on results relevant to probability density function (PDF) modeling. Direct numerical simulations (DNS) are conducted for compressible isotropic turbulent binary mixing of heptane with nitrogen at mean pressure and temperature equal to 45 atm and T=700 K, respectively. The formulation is based on a cubic real gas state equation, and includes generalized forms for heat and mass diffusion derived from nonequilibrium thermodynamics and fluctuation theory. Results from two simulations at 1923 resolution are compared. One case is based on the complete diffusion formulation, whereas in the second simulation only “standard” Fickian and Fourier mass and heat flux terms are considered as a basis for comparisons. The evolutions of the two mixing processes are shown to be nearly the same at early times; however, at long times the pressure gradient based Soret diffusion acts as a forcing function resulting in a statistically stationary scalar fluctuation distribution. In this case the scalar variance achieves a stationary value, in contrast to the exponential decay associated with purely Fickian diffusion. The mechanical to scalar time scale ratio, CY=τu/τY, is increased by a factor of approximately 3 by Soret effects during this latter mixing regime. Soret diffusion is shown to substantially alter the forms of the conditional expected diffusion and the conditional expected dissipation at these long times. A commonly used model based on an assumed proportionality between the conditional diffusion and the scalar fluctuation is tested, and is found to perform poorly in the presence of cross diffusion at long mixing times. Several new conditional expectations resulting from cross diffusion are measured. Implications of the results for extension of PDF models to the high pressure regime are discussed.

Micro scalar timescales in premixed turbulent combustion

Time series measurements of gas density have been obtained in lean (=0.65–0.8) premixed turbulent flames stabilized on a Bunsen-type burner (U0=2.3, 3.5 m/s). Two Eulerian scalar timescales were obtained from spectral analyses of these time series. The first, T^, characterizes the flame front wrinkling process, describable by random telegraph signal statistics, and the second is a small scalar timescale, τD. This was shown to be due to the transitions of the flame fronts as they pass through the probe volume. The ratio between these two Eulerian scales τD/T^=0.4 in the present flames, is a measure of the scale range of scalar fluctuations and is potentially an interesting factor at high turbulent Reynolds numbers when broadening of the preheat zone by turbulent mixing affects this ratio directly by broadening the “dissipation range” in the spectrum. A further timescale, a direct measure of the flame front transit time, was also determined from these data. Estimates of the local mean scalar dissipation were obtained from this data using flamelet assumptions and an interior distribution of the scalar, tested by fitting to numerical simulations of laminar flames, for the local scalar gradients. A simple expression was derived which related the mean scalar disspation to the burning fraction of the scalar probability density function, γ, and a chemical time characteristic of the flame front. It was found that the dissipation rate is affected more by the local scalar gradients than by γ. A dissipation time scale was deduced from this rate and was found to be proportional to the ratio of the scalar integral timescale and the flame transit time.

A comparison between dynamic and scalar timescales in lean premixed turbulent flames

A comparison between the dynamic and scalar timescales in turbulent premixed flames is presented. Methane-air turbulent Bunsen-type flames are used in this study. They are all lean flames and basically in the flamelet regime. Laser Doppler velocimetry (LDV), Mie scattering and Rayleigh scattering techniques allowed the determination of dynamic (both conditional and nonconditional) and scalar (density) timescales. For both dynamic and scalar temporal scales, we used the autocorrelation function of the LDV signal and the Rayleigh scattering signal, respectively. The main results of the paper indicate that there are significant differences between the conditional (in the reactants) dynamic timescales and those evaluated from nonconditional velocity measurements. The nonconditional velocity timescales are much closer to the scalar timescales, which indicates that they are strongly influenced by the flame front dynamics. Therefore, to evaluate the turbulent premixed flame regimes, only turbulence parameters based on conditional measurement are meaningful, as the nonconditional ones incorporate part of the flame response to the turbulence structure in the reactants. Comparisons between scalar and dynamic timescales, as well as between integral and dissipation timescales, are made. They can be useful in turbulent premixed combustion models for the mean reaction term. In general, the results presented here indicate a complex effect of the u′/S L , ratio, which seems to depend on the way this parameter is varied.

Etude numérique des jets turbulents à température élevée

Numerical prediction of the structure of high temperature axisymmetric turbulent jets. Turbulent axisymmetric jets at high temperature are studied numerically by using first and second order turbulence models. Regarding the temperature fields, on which we concentrate in this work, predictions with both types of models do not show large differences. In general, predictions agree well with the measurements; the existing differences are usually favorable for the second order model. The effect of solving a transport equation for the scalar dissipation rate on the prediction of the mechanical to scalar time scale ratio and on the prediction of the scalar fluctuations is studied. The influence of varying the density ratio on parameters such as the axial decay rates of the temperature and velocity and the turbulence intensity are studied. Two definitions of the mixing efficiency are introduced. According to both definitions, the mixing efficiency decreases with increasing effects of buoyancy.