Function Of Surface Density Research Articles

Evolution of Surface Density Function in an Open Turbulent Jet Spray Flame

A three-dimensional Direct Numerical Simulation of an open turbulent jet spray flame representing a laboratory-scale burner configuration has been used to analyse the statistical behaviours of the magnitude of reaction progress variable gradient left| {nabla c} right| [alternatively known as the Surface Density Function (SDF)] and the strain rates, which affect its evolution. The flame has been found to exhibit fuel-lean combustion close to the jet exit, but fuel-rich conditions have been obtained further downstream due to the evaporation of fuel droplets, which leads to the reduction in the mean value of the SDF in the downstream direction. This change in mixture composition in the axial direction has implications on the statistical behaviours of the SDF and the strain rates affecting its evolution. The mean value of dilatation rate remains positive, whereas the mean normal strain rate assumes positive values where the effects of heat release are strong but becomes negative towards both unburned and burned gas sides. The mean values of dilatation rate, normal strain rate and tangential strain rate decrease downstream of the jet exit. However, the mean behaviours of displacement speed and its components do not change significantly away from the jet exit. The mean values of normal strain rate arising from flame propagation remain positive and thus act to thicken the flame. The mean tangential strain rate due to flame propagation (alternatively the curvature stretch rate) remains negative throughout the flame at all axial locations investigated. The mean effective normal strain rate assumes positive values throughout the flame and it increases in the downstream direction for the present case, which is consistent with the reduction in the peak mean value of the SDF in the axial direction. The mean effective tangential strain rate (alternatively stretch rate) assumes negative values throughout the flame at all axial locations.

Scalar Gradient and Strain Rate Statistics in Oblique Premixed Flame\u2013Wall Interaction Within Turbulent Channel Flows

Three-dimensional direct numerical simulations of V-flames interacting with chemically inert walls in a fully developed turbulent channel flow have been performed under adiabatic and isothermal wall boundary conditions using single-step chemistry. These simulations are representative of stoichiometric methane-air mixture at unity Lewis number under atmospheric conditions. The turbulence in the non-reacting channel is representative of the friction velocity based Reynolds number Re_{tau }=110. Differences in the statistical behaviours of the mean values of progress variable, temperature, and density have been reported for different wall boundary conditions. It is found that the mean location of the oblique flame interacting with the wall is affected by the choice of the wall boundary condition used. The influence of these differences on the flame dynamics is investigated by analysing the statistical behaviours of the surface density function (SDF) and the strain rates, which govern the evolution of the SDF. The mean variation of the SDF and the flame displacement speed are strongly affected by the wall boundary condition within the viscous sub-layer region of the boundary layer. The behaviours of the normal and tangential strain rates are found to be influenced by not only the differences in the wall boundary conditions, but also by the distance from the wall. The differences in the displacement speed statistics for different wall boundary conditions and wall distance affect the behaviours of the normal strain rate arising due to flame propagation and curvature stretch. The changes in the SDF behaviour in the near wall region have been explained in terms of the statistics of effective normal strain rate experienced by the progress variable iso-surfaces under different wall boundary conditions and wall normal distances.

Surface density function evolution and the influence of strain rates during turbulent boundary layer flashback of hydrogen-rich premixed combustion

The statistical behavior of the magnitude of the reaction progress variable gradient [alternatively known as the surface density function (SDF)] and the strain rates, which govern the evolution of the SDF, have been analyzed for boundary layer flashback of a premixed hydrogen-air flame with an equivalence ratio of 1.5 in a fully developed turbulent channel flow. The non-reacting part of the channel flow is representative of the friction velocity based Reynolds number Reτ = 120. A skeletal chemical mechanism with nine chemical species and twenty reactions is employed to represent hydrogen-air combustion. Three definitions of the reaction progress variable (RPV) based on the mass fractions of H2, O2, and H2O have been considered to analyze the SDF statistics. It is found that the mean variations of the SDF and the displacement speed Sd depend on the choice of the RPV and the distance away from the wall. The preferential alignment of the RPV gradient with the most extensive principal strain rate strengthens with an increase in distance from the cold wall, which leads to changes in the behaviors of normal and tangential strain rates from the vicinity of the wall toward the middle of the channel. The differences in displacement speed statistics for different choices of the RPV and the wall distance affect the behaviors of the normal strain rate due to flame propagation and curvature stretch. The relative thickening/thinning of the reaction layers of the major species has been explained in terms of the statistics of the effective normal strain rate experienced by the progress variable isosurfaces for different wall distances and choices of RPVs.

Surface Density Function Evolution in Spherically Expanding Flames in Globally Stoichiometric Droplet-laden Mixtures

ABSTRACT The statistical behaviors of the magnitude of the reaction progress variable gradient (alternatively known as the Surface Density Function (SDF)) and the strain rates, which affect the SDF evolution, have been analyzed using three-dimensional Direct Numerical Simulations (DNS) of spherically expanding flames in globally stoichiometric initially mono-sized droplet-laden mixtures for different initial turbulence intensities and droplet diameters. It has been found that gaseous phase combustion predominantly takes place under fuel–lean mode and this tendency strengthens for large droplets and high turbulence intensities. The mean values of flame displacement speed, dilatation rate and normal strain rate decrease with increasing turbulence intensity and droplet diameter. By contrast, the mean tangential strain rate increases with increasing turbulence intensity for all droplet diameters. The mean normal strain rate induced by flame propagation remains negative but its magnitude decreases with increasing droplet size and turbulence intensity. The mean tangential strain rate induced by flame propagation (alternatively curvature stretch rate) assumes negative values except for the laminar flame with small droplets. The mean effective normal strain rate has been found to assume predominantly positive values and increases with increasing turbulence intensity for the droplet cases considered here. The mean effective tangential strain rate (alternatively stretch rate) is found to assume mostly negative values except for the laminar case with small droplets. The mean effective tangential strain rate decreases with increasing droplet size, which leads to a smaller extent of flame surface area generation for flames with larger droplets.

Statistical modeling of the gas–liquid interface using geometrical variables: Toward a unified description of the disperse and separated phase flows

In this work, we investigate an original strategy in order to derive a statistical modeling of the interface in gas–liquid two-phase flows through geometrical variables. The contribution is two-fold. First it participates in the theoretical design of a unified reduced-order model for the description of two regimes: a disperse phase in a carrier fluid and two separated phases. The first idea is to propose a statistical description of the interface relying on geometrical properties, such as the mean and Gauss curvatures, and to define an associated Surface Density Function (SDF). The second main idea consists in using such a formalism in the disperse case, where a clear link is proposed between local statistics of the interface and the statistics of countable objects, such as a number density function. To this end we make essential the use of topological invariants in geometry through the Gauss-Bonnet formula. This strategy strictly includes the works conducted on sprays of spherical droplets, but it also yields a statistical treatment of populations of non-spherical objects, such as ligaments, as long as they are homeomorphic to a sphere. Second, we propose an original statistical post-processing of DNS data of interfacial flows. Starting from the proposed theoretical approach, we identify a kernel for the spatial averaging of geometrical quantities which preserves the topological invariants. Coupled to a new algorithm for the evaluation of the surface and its curvatures, that also preserves these invariants, we analyze two sets of DNS results obtained with the ARCHER code from CORIA, with and without topological changes, and assess the approach. Indeed, this procedure allows us to transform the interfacial information provided by a Level-Set function into a number distribution of a collection of objects in the proper geometrical phase space.

Topology of pocket formation in turbulent premixed flames

In turbulent premixed flame propagation, the formation of isolated pockets of reactants or products is associated with flame pinch–off events which cause rapid changes in the flame surface area. Previous topological analysis of these phenomena has been carried out based on Morse theory and Direct Numerical Simulation (DNS) in two spatial dimensions. The present work extends the topological analysis to three dimensions with emphasis on the formation and subsequent burnout of reactant pockets. Singular behaviour observed previously for terms of the Surface Density Function (SDF) transport equation in the two–dimensional case is shown to occur also in three dimensions. Further singular behaviour is observed in the displacement speed close to reactant pocket burnout. The theory is compared against DNS data from hydrogen–air flames and good agreement is obtained.

Surface Density Function statistics in hydrogen-air flames for different turbulent premixed combustion regimes

ABSTRACTThe choice of reaction progress variable (RPV) on the statistical behavior of the surface density function (SDF) and the strain rates, which govern the evolution of SDF, have been analyzed using a detailed chemistry Direct Numerical Simulation (DNS) database of freely propagating statistically planar -air flames with an equivalence ratio of 0.7. The DNS database consists of three cases spanning the corrugated flamelets (CF), thin reaction zones (TRZ) and broken reaction zones (BRZ) regimes of premixed turbulent combustion. For this analysis, the RPV is defined based on the mass fractions of H2, O2 and H2O. The mean variations of the SDF and the flame displacement speed, , have also been found to be dependent on the choice of the RPV. The progressive weakening (strengthening) of the preferential alignment of the RPV gradient with the most extensive (compressive) principal strain rate with increasing Karlovitz number leads to changes in the behaviors of normal and tangential strain rates from one combustion regime to another. The differences in displacement speed statistics for different choices of RPV affect the behaviors of the normal strain rate due to flame propagation and curvature stretch. The relative thickening/thinning of the reaction layers of the major species has been explained in terms of the statistics of effective normal strain rate experienced by the -isosurfaces for different choices of RPVs.

A Direct Numerical Simulation analysis of pressure variation in turbulent premixed Bunsen burner flames-part 2: Surface Density Function transport statistics

The effects of pressure variation on the transport statistics of the magnitude of the reaction progress gradient (i.e. Surface Density Function (SDF)) have been investigated based on three-dimensional simple chemistry Direct Numerical Simulations (DNS) of Bunsen burner flames representing the flamelet regime of combustion. The large length scale separation between the nozzle diameter and flame thickness for high pressure flames makes the Darrieus–Landau (DL) instability highly likely, which in turn affects the curvature stretch. It has been found that the effective normal strain rate remains insensitive to the pressure variation for the parameter range considered here, which makes the flamelet thickness in turbulent flames comparable to the laminar flame thickness. The influences of the DL instability on the positive mean tangential strain rate counter the effects of instability on the negative mean curvature stretch and thus the effective tangential strain rate (or net flame stretch rate) remains mostly unaffected by the pressure variation within the strict flamelet regime (i.e. wrinkled flamelets and corrugated flamelets regimes) of combustion. The similarities in the SDF and the effective strain rate statistics for different values of pressure suggest that the models for the Flame Surface Density and Scalar Dissipation Rate, which were originally proposed and validated for atmospheric combustion, might remain valid also for elevated pressures.

Conditional analysis of turbulent premixed and stratified flames on local equivalence ratio and progress of reaction

Previous studies on the Cambridge/Sandia stratified burner have produced a comprehensive database of line Rayleigh/Raman/CO LIF measurements of scalars, as well as LDA and PIV measurements of velocity, for flames under non-uniform mixture fraction, under moderate turbulent conditions where the ratio of the turbulent velocity fluctuations to the laminar flame speed is of order 10. In prior work, we applied multiple conditioning methods to demonstrate how local stratification increases the levels of CO and H2, relative to the corresponding turbulent premixed flame, and enhances surface density function (SDF) and scalar dissipation rate of progress of reaction (SDR), based on extent of temperature rise, at a particular location in the flame where the mixing layer and flame brush cross. In the present study, we examine the global features of selected flames at all locations, by obtaining probability density functions (PDFs) for species concentrations, SDRs, and SDFs, conditioned on local equivalence ratio and location in the flame brush throughout the domain.We find that for most cases, species profiles as a function of temperature are well represented by laminar flame relationships at the local equivalence ratio, with some deviations attributable to either differential diffusion near the flame base and local stratification effects further downstream where the flame brush crosses the mixing layer. In particular, CO2 is significantly affected by differential diffusion, and CO and H2 by stratification. However, the stratification effects on the species are relatively minor when conditioned on local equivalence ratio, a simplifying result in the context of modeling.Measurements of the gradient of progress of reaction and scalar dissipation rates, conditioned on local equivalence ratio, show that the thermal zone of the flame is thickened by turbulence: the mean SDF and SDR values are in general lower than those of unstrained laminar flames. The effect is greater under rich conditions, with conditional mean SDR decreasing to less than half of the corresponding laminar value. The extent of flame thickening is the same in the premixed as the stratified case, once the stratified measurements are conditioned on the same equivalence ratio.

Flame surface density distribution in turbulent flame kernels during the early stages of growth

Direct numerical simulations (DNS) of flame kernels with single-step chemistry are used to asses the influence of the initial turbulent energy spectra on the global evolution of the kernel, and the distribution of flame surface density (FSD) in the flame brush. A high mean positive curvature in relation to the flame thickness during these early stages is found to result in significant variation in the mean velocity and displacement experienced by individual isosurfaces of the reaction progress variable. A bias is shown to exist in the distribution of the surface density function (SDF) in the direction of kernel displacement, with a magnitude corresponding qualitatively to the initial turbulent Reynolds number. Characteristics of the decaying turbulent fields were evaluated from an independent ensemble of cold-flow simulations, and time dependent turbulence intensities and integral length scales are presented with appropriate confidence intervals. Alternative approaches for determining the combustion regime for flame kernels are discussed, and an analysis of the global flame development is given in terms of total flame surface area and integrated burning rate.

Effects of global flame curvature on surface density function transport in turbulent premixed flame kernels in the thin reaction zones regime

Three-dimensional simplified chemistry-based Direct Numerical Simulations (DNS) have been used to study the effects of global flame curvature (i.e. inverse of mean kernel radius) on the transport of the reaction progress variable gradient magnitude |∇c| which is alternatively known as the Surface Density Function (SDF). In order to understand the influences of global curvature on the statistical behaviour of SDF and the different terms of the SDF transport equation, turbulent premixed flame kernels with different initial radii have been simulated. In addition to this, a statistically planar flame has been simulated as a special case of the flame kernel with infinite radius. It has been found that the maximum mean value of the SDF decreases with decreasing kernel radius. The statistical behaviour of the different terms of the SDF transport equation has been studied in detail in order to understand the influence of curvature on the SDF transport. It has been observed that the tangential strain rate contribution to the SDF transport acts as a source term but its strength decreases with decreasing kernel radius. By contrast, the curvature contribution to the SDF transport acts to dissipate SDF in all the flames but the sink contribution becomes increasingly weak with decreasing kernel size. The SDF propagation term acts as a source term towards the fresh gas side and a sink term towards the burned gas side but the source contribution decreases with decreasing kernel size. The local curvature effects on the different terms of the SDF transport equation have been studied in terms of their joint pdfs with local curvature. It is shown that the local curvature dependence of displacement speed plays a key role in the statistics of the curvature and propagation terms of the SDF transport equation.

Influence of Lewis number on the surface density function transport in the thin reaction zone regime for turbulent premixed flames

The effects of tangential strain rate and curvature on the surface density function (SDF) and on different terms within the SDF transport equation in the thin reaction zone regime are studied for statistically planar turbulent premixed flames with global Lewis numbers Le=0.8, 1.0, and 1.2 by using three-dimensional direct numerical simulations with simplified chemistry. A positive correlation is observed between the SDF and tangential strain rate, and this is explained in terms of the local statistical behaviors of tangential strain rate and dilatation rate. Curvature is shown to affect the SDF through the curvature response of both tangential strain rate and dilatation rate on a given flame isosurface. The correlation between the curvature and SDF is positive in the Le=0.8 flame and negative in the Le=1.2 flame. The curvature dependence of the SDF is weak in the case of unity Lewis number. Strain rate and curvature are found to have an appreciable effect on different terms of the SDF transport equation. The SDF strain rate term arising from tangential strain rate contribution in all the flames is positively correlated with tangential strain rate as expected and is also negatively correlated with curvature. For the Le=1.0 and 1.2 flames, the SDF propagation term is found to negatively correlate with flame curvature toward the reactant side of the flame and positively toward the product side. By contrast, for the Le=0.8 flame, the SDF propagation term is negatively correlated with curvature throughout the flame brush. The variation of the SDF curvature term with local flame curvature for all the flames is found to be nonlinear due to the additional stretch induced by the tangential diffusion component of the displacement speed. Physical explanations are provided for all of these effects, and the modeling implications are discussed in detail.

The effects of strain rate and curvature on surface density function transport in turbulent premixed methane–air and hydrogen–air flames: A comparative study

The effects of tangential strain rate and curvature on the surface density function (SDF) and on source terms within the SDF transport equation are studied for lean methane–air and hydrogen–air flames using two-dimensional direct numerical simulations with detailed chemistry. A positive correlation is observed between the SDF and the tangential strain rate, and this is explained in terms of the interaction between the local tangential strain rate and the dilatation rate due to heat release. Curvature is also seen to affect the SDF through the curvature response of both tangential strain rate and dilatation rate on a given flame isosurface. Strain rate and curvature are found to have an appreciable effect on several terms of the SDF transport equation. The SDF straining term in both methane and hydrogen flames is correlated positively with tangential strain rate, as expected, and is also correlated negatively with curvature. For methane flames, the SDF propagation term is found to correlate negatively with flame curvature toward the reactant side of the flame and positively toward the product side. By contrast, for hydrogen flames the SDF propagation term is negatively correlated with curvature throughout the flame brush. The variation of the SDF curvature term with local flame curvature for both methane and hydrogen flames is found to be nonlinear due to the additional stretch induced by the tangential diffusion component of the displacement speed. Physical explanations are provided for all of these effects, and the modeling implications are considered in detail.

Influence of Lewis number on curvature effects in turbulent premixed flame propagation in the thin reaction zones regime

The influence of differential diffusion on the statistical behavior of the local displacement speed (Sd) in relation to flame curvature is studied based on three-dimensional compressible direct numerical simulations (DNS) of statistically planar flames with single-step Arrhenius-type chemistry. Three different Lewis number cases (Le=0.8, 1.0, and 1.2) are considered. In order to study the influence of differential diffusion on curvature effects in flame propagation, temperature statistics are presented in terms of standard probability density functions (pdfs) and also joint pdfs with curvature for the nonunity Lewis number cases. Temperature statistics are found to be consistent with previous incompressible combustion DNS studies. It is found that both dilatation and tangential strain rate are negatively correlated with curvature. The relative strength of these two correlations determines the nature of the correlation between surface density function (SDF) (∣∇c∣) and curvature. It is also found that the variations of temperature and SDF on an isosurface of reaction progress variable (c) have significant influence on local displacement speed behavior. Displacement speed statistics are presented in terms of standard pdfs as well as joint pdfs with curvature for the three Lewis number cases. The curvature response of the displacement speed and its different components is found to be nonlinear which is consistent with previous two-dimensional DNS with detailed chemistry. The observed nonlinear behavior in the present study in the absence of a detailed chemical mechanism is explained through the influence of differential diffusion.

Effects of strain rate and curvature on surface density function transport in turbulent premixed flames in the thin reaction zones regime

Strain rate and curvature effects on Surface Density Function (SDF) transport in the thin reaction zones regime are studied using a three-dimensional direct numerical simulations (DNS) with a single-step Arrhenius type chemistry. It is shown that if the tangential strain rate on a flame isosurface exceeds a critical value, then a negative normal strain rate is induced, which acts to bring the isoscalar lines closer to each other and hence leads to a higher value of SDF. This is reflected in a positive correlation between SDF and tangential strain rate. Curvature is also found to affect SDF through the correlation between tangential strain rate and curvature on a given flame isosurface. Strain rate and curvature are found to have an appreciable effect on various terms of the SDF transport equation. The SDF straining term is correlated positively with tangential strain rate as expected and is also correlated negatively with the curvature. The combined SDF curvature and propagation terms operate as a source near the fresh gas side of the flame and as a sink towards the burned gas side. The SDF propagation term is found to correlate negatively with flame curvature towards the fresh gas side and positively towards the burned gas side. The variation of the SDF curvature term with local flame curvature is found to be nonlinear due to the additional stretch induced by the tangential diffusion component of the displacement speed.

Pocket formation and the flame surface density equation

The occurrence and properties of singularities in the equation for the surface density function σ≡|∇| are analyzed analytically and numerically using data from two-dimensional direct numerical simulation (DNS) of pocket formation in a premixed methane-air flame. The various stages and the relevant time scales associated with pocket formation were determined in a previous study. It was found that isolated pockets form if and only if a nondegenerate critical point of a saddle point type appears. The appearance of a singularity in the isoline representing the flame front may have implications to modeling of the terms in the surface density function (SDF) approach during such transient events as pocket formation. The sink and source terms in SDF are evaluated in the neighborhood of a critical point using DNS data during pocket formation and an analytic representation of a scalar in the vicinity of the critical point that allows for the computation of all kinematic properties. The analytic and computational results show that the normal restoration and dissipation terms in the SDF become singular at the critical point when the pocket emerges. Furthermore, the analytic results show that the singularities exactly cancel, and therefore, the main conclusion is that it is unnecessary to model the singular behavior of these terms at critical points. However, closure of their sum is recommended.