Probability Density Functions Of Curvature Research Articles

Effects of anisotropy on the geometry of tracer particle trajectories in turbulent flows

Using curvature and torsion to describe Lagrangian trajectories gives a full description of these as well as an insight into small and large time scales as temporal derivatives up to order 3 are involved. One might expect that the statistics of these observables depend on the geometry of the flow. Therefore, we calculated curvature and torsion probability density functions (PDFs) of experimental Lagrangian trajectories processed using the Shake-the-Box algorithm of turbulent von Kármán flow, Rayleigh–Bénard convection and a zero-pressure-gradient turbulent boundary layer over a flat plate. The results for the von Kármán flow compare well with experimental results for the curvature PDF and results obtained by numerical simulations of homogeneous and isotropic turbulence for the torsion PDF. Results for Rayleigh–Bénard convection agree with those measured for von Kármán flow, while results for the logarithmic layer within the boundary layer differ slightly. We provide a potential explanation for the latter. To detect and quantify the effect of anisotropy either resulting from a mean flow or large-scale coherent motions on the geometry or tracer particle trajectories, we introduce the curvature vector. We connect its statistics with those of velocity fluctuations and demonstrate that strong large-scale motion in a given spatial direction results in meandering rather than helical trajectories.

A direct numerical simulation analysis of turbulent V-shaped flames propagating into droplet-laden mixtures

Three-dimensional Direct Numerical Simulations (DNS) data has been used to analyse the gaseous phase combustion behaviour in a V-shaped flame configuration where fuel is supplied in the form of droplets such that an overall (i.e. liquid+gaseous) equivalence ratio of unity is maintained in the unburned gas. The analysis has been carried out for different initially mono-sized droplet diameters. A gaseous premixed V-shaped flame with the same flow conditions has been utilised to compare the flame propagation in V-shaped spray flames with a corresponding gaseous premixed flame case. It has been found that combustion in the gaseous phase mainly takes place under fuel-lean mode and the probability of finding fuel-lean burning increases with increasing droplet diameter. However, the mean equivalence ratio of the predominantly fuel-lean mixture increases with increasing axial distance from the flame holder. The presence of droplets has been found to give rise to dimples on the reaction progress variable isosurfaces, and the effects of droplet–induced flame wrinkling are reflected by the widening of the curvature probability density functions for large droplets. The heat release has been found to arise mainly from the premixed mode of combustion. Detailed analysis of reaction progress variable transport has been used to explain the mean variations of consumption speed and density-weighted displacement speed with the axial distance from the flame holder. The mean values of consumption speed, density-weighted displacement speed and turbulent burning velocity have been found to decrease with increasing droplet diameter. The flame speed statistics have been utilised to explain that the enhancement of the burning rate cannot be equated to the enhancement of the flame surface area in turbulent droplet-laden V-shaped flames.

Early flame propagation of hydrogen enriched methane-air mixtures at quasi laminar conditions in a rapid compression expansion machine

The temporal evolution of the kinematic properties of hydrogen enriched, lean premixed methane-air flames was studied experimentally in a rapid compression expansion machine (RCEM) during transient operation. Schlieren imaging was used to capture the flame front propagation, while the temporally evolving flow field was simultaneously acquired by particle image velocimetry (PIV). A statistical analysis based on probability density functions (PDFs) of non-dimensional formulations of the flame curvature, stretch rate and displacement speed was performed to study the effect of the onset of flame instabilities on the aerodynamic characteristics of the propagating flame kernel. It was found that initial perturbations stemming from the presence of the spark plug electrodes and from the gas expansion led to the formation of large scale cusps related to the hydrodynamic instability, which is manifested by a notable negative skewness of the curvature PDF, validating recent numerical findings. Under the influence of thermal-diffusive effects, small scale cellular structures develop along the flame front beyond a certain radius, with their appearance shifting to earlier times with increasing hydrogen content or for leaner mixtures. The increasing width of the normalized curvature and stretch rate PDFs is proposed as an indicator for the onset of this type of instability. The adopted methodology allows to track the self-acceleration of the propagating flame front effecting from hydrogen enrichment of methane. The findings of this study provide valuable physical insight and aid in the development of more accurate models, capable of capturing these complex phenomena.

Evolution of Flame Curvature in Turbulent Premixed Bunsen Flames at Different Pressure Levels

The physical mechanisms underlying the curvature evolution in turbulent premixed Bunsen flames at different thermodynamic pressures are investigated using a three-dimensional Direct Numerical Simulation database. It is found that, due to the occurrence of the Darrieus-Landau instability, the high-pressure flame exhibits higher probability of developing large negative curvature values and saddle concave topologies than the low pressure cases. The terms in the curvature transport equation due to normal strain rate gradients and curl of vorticity arising from both turbulent flow and flame normal propagation play pivotal roles in the curvature evolution. The mean value of the net contribution of the flame propagation terms dominates over the net contributions arising from the background fluid motion. The net contribution of the source/sink terms tries to reduce the convexity of the flame surface in the positively curved locations. By contrast, the net contribution of the source/sink terms promotes concavity of the flame surface towards the reactants in the negatively curved regions and this effect is particularly strong for the high pressure flame, where the effects of the Darrieus-Landau instability are prominent. This also gives rise to large negative skewness of the probability density functions of curvature in the high-pressure flame with the Darrieus-Landau instability.

A direct numerical simulation analysis of spherically expanding turbulent flames in fuel droplet-mists for an overall equivalence ratio of unity

Three-dimensional direct numerical simulations with a modified single-step Arrhenius chemistry have been used to analyze spherically expanding n-heptane flames propagating into mono-sized fuel droplet mists for different droplet diameters and an overall equivalence ratio of unity. The evolutions of flame surface area and burned gas volume for both laminar and turbulent spherically expanding droplet flames have been compared to the corresponding gaseous stoichiometric premixed spherically expanding flames with the same initial burned gas radius. It has been found that the initial droplet diameter significantly affects the burned gas volume and flame area generation, which increase with decreasing droplet diameter for both laminar and turbulent cases. The droplet-flame interaction plays a key role in determining flame wrinkling under laminar conditions, which is reflected in a range of local curvatures for a given reaction progress variable isosurface, whereas each progress variable isosurface in spherically expanding laminar premixed flames exhibits a single value of curvature. The effect of droplet-induced curvature becomes less distinguishable from the flow-induced wrinkling for the turbulent cases considered here, but the reaction progress variable isosurfaces in droplet cases exhibit wider curvature probability density functions than in the corresponding turbulent premixed flame cases. It has been found that the heat release rate arises principally from premixed mode in small droplet cases, whereas the contribution of the non-premixed mode to the overall heat release rate increases with increasing droplet diameter and turbulence intensity.

Experimental study of the effect of turbulence on the structure and dynamics of a bluff-body stabilized lean premixed flame

The structure of an unconfined, lean propane–air flame stabilized on an axisymmetric bluff body is studied for different levels of turbulence intensity ranging from 4 to 30% in the approach flow. Simultaneous planar laser induced fluorescence imaging of hydroxyl radical (OH) and formaldehyde (CH2O) was performed to determine the effects of turbulence on local flame structure. In an accompanying experiment high speed particle image velocimetry at 5kHz was used to obtain the time resolved velocity field and evaluate flame surface density, brush thickness and probability density functions of curvature and strain rates conditioned on the flame front. The low turbulence intensity flame featured wrinkling and mostly symmetric flame structures with thin regions of heat release along the flame front. For moderate turbulence intensity (14%), pronounced formation of cusps and unburnt mixture fingers was observed with continuous heat release regions. However, the local flame structure was observed to be strongly altered for the 30% turbulence intensity. Localized extinction occurred along the flame surface creating isolated pockets of OH with heat release occurring along the boundary. Pockets of preheated reactants along with fresh reactants were observed inside the flame envelope. The overall flame appearance was observed to intermittently switch from varicose (symmetric) to sinuous (asymmetric) type (vortex shedding mode). The curvature and strain rates pdfs broadened with increasing levels of turbulence intensity. The flame brush thickness showed an initial increase with increasing turbulence levels but saturated beyond 24% turbulence intensity. Measurements showed reduction in flame surface density with increasing levels of turbulence intensity.

3D flame topography and curvature measurements at 5 kHz on a premixed turbulent Bunsen flame

This work reports the measurements of three-dimensional (3D) flame topography and curvature of a premixed turbulent Bunsen flame at a rate of 5 kHz, using a technique combining chemiluminescence and tomography. Line-of-sight images of chemiluminescence (termed projections) of the target flame were recorded by six cameras from different orientations simultaneously at 5 kHz. Based on these projections, a tomography algorithm reconstructed the 3D flame structure, based on which 3D curvature was then calculated. Due to the 3D nature of the data, statistics of flame properties can then be extracted both in temporal and spatial domains. Probability density function (PDF) of flame topography was extracted from a series of 3D measurements, and the PDFs of the flame at different spatial locations were examined. Furthermore, the instantaneously measured 3D flame topography also enabled the calculation of 3D flame curvature (or 2D curvature along an arbitrary orientation). The PDFs of curvature in 2D and 3D were then extracted and compared. These results provide quantification of the flame surface shape in 3D (cylindrical, elliptic, or hyperbolic), illustrating the utility of 3D diagnostics to fully resolve the dynamics of turbulent flames.

Large-eddy simulation of lean hydrogen–methane turbulent premixed flames in the methane-dominated regime

The application of large-eddy simulation (LES) to the prediction of H2-enriched lean methane–air turbulent premixed combustion is considered. A presumed conditional moment (PCM) subfilter-scale combustion model is coupled with the flame prolongation of intrinsic low-dimensional manifold (FPI) chemistry tabulation technique. The LES and PCM-FPI modelling procedures are then applied to the prediction of laboratory-scale axisymmetric Bunsen-type turbulent premixed flames. Both premixed methane–air and H2-enriched methane–air flames are considered and the predicted solutions are examined and compared to available experimental data. The enriched flame has 20% H2 in terms of mole fraction and lies in the methane-dominated regime of hydrogen–methane mixtures. The LES simulations predict similar qualitative trends to those found in the experiments for flame height and curvature. The addition of H2 decreases the flame height and broadens the curvature probability density functions, which show a Gaussian-type shape centred around zero. Moreover, the enriched flame displays a higher degree of wrinkling with sharper ridges of negative curvature and larger pockets of positive curvature. Overall, the proposed treatment for the PCM-FPI combustion model, in terms of progress variable and tabulated data, seems to perform well for the H2-enriched methane flame in the methane-dominated regime.

Turbulent premixed flame front dynamics and implications for limits of flamelet hypothesis

Turbulent premixed flames of methane–air and propane–air stabilized on a Bunsen-type burner were studied to investigate the dynamics and structure of the flame front at a wide range of turbulence intensities. The non-dimensional turbulence rms velocity, rms velocity divided by the laminar flame velocity, covered the range from about 3 to 24. The equivalence ratio was varied from 0.6 (0.7 for propane) to stoichiometric. The flame front data were obtained using planar Rayleigh imaging, and particle image velocimetry was used to measure instantaneous velocity field for the experimental conditions studied. The gradients of temperature profiles decreased noticeably with increasing non-dimensional turbulence rms velocity. Flame front curvature statistics indicated that the curvature probability density functions are highly symmetric. Frequency of crossing from negative to positive (and vice versa) curvatures did not show any clear sensitivity to non-dimensional turbulence rms velocity, but decreased by increasing fuel–air equivalence ratio. The product of curvature and diffusivity, a crucial term in the level-set equation proposed for the thin reaction zones regime, was found to be very small as compared to laminar burning velocity, but the product of rms curvature and diffusivity was higher than the laminar burning velocity. Flame surface densities integrated over the flame brush volume did not show any sensitivity to the non-dimensional turbulence rms velocity. Some of the single shot Rayleigh temperature profiles at higher turbulence intensities were radically different than those at lower intensities which are similar to laminar flame profiles. These findings question the validity of the flamelet hypothesis in the thin reaction zones regime where Karlovitz number exceeds unity.

Premixed turbulent flame front structure investigation by Rayleigh scattering in the thin reaction zone regime

Premixed turbulent flames of methane–air and propane–air stabilized on a bunsen type burner were studied using planar Rayleigh scattering and particle image velocimetry. The fuel–air equivalence ratio range was from lean 0.6 to stoichiometric for methane flames, and from 0.7 to stoichiometric for propane flames. The non-dimensional turbulence rms velocity, u′/ S L, covered a range from 3 to 24, corresponding to conditions of corrugated flamelets and thin reaction zones regimes. Flame front thickness increased slightly with increasing non-dimensional turbulence rms velocity in both methane and propane flames, although the flame thickening was more prominent in propane flames. The probability density function of curvature showed a Gaussian-like distribution at all turbulence intensities in both methane and propane flames, at all sections of the flame. The value of the term Dκ, the product of molecular diffusivity evaluated at reaction zone conditions and the flame front curvature, has been shown to be smaller than the magnitude of the laminar burning velocity. This finding questions the validity of extending the level set formulation, developed for corrugated flames region, into the thin reaction zone regime by increasing the local flame propagation by adding the term Dκ to laminar burning velocity.

Local scalar flame properties of freely propagating premixed turbulent flames at various Lewis numbers

Local scalar flame properties of freely propagating turbulent premixed flames including the flame curvature h, and local displacement speed relative to the fresh gases S d u , have been measured simultaneously for methane, propane, and hydrogen/air flames at Lewis numbers varying from 0.33 to 1.4 and ratios of rms turbulent velocity to unstretched laminar burning velocity u′/ S L,0 u varying from 0 to 3.1. Three different mixtures were separately spark-ignited in a vertical wind tunnel. The expanding flame freely propagated in a grid-generated decaying turbulent flow. An advanced field-imaging technique based on high-speed laser tomography measured the temporal evolution of local flame properties. Local flame curvature and local displacement speed were calculated from flame-front contours. Curvature probability density functions (pdfs) were negatively skewed, especially for nonunity Lewis numbers, and displacement speed distributions underlined the influence of local stretch and thermodiffusive effects on flame speed variations. The temporal evolution of the mean flamelet radius of curvature converges towards half of the integral length scale measured in the cold flow. Flame response in terms of displacement speed to curvature is found to be statistically dependent, and a linear relationship is observed. For propane/air flames, the displacement speed can be assumed to be independent of local flame curvature at each stage of flame propagation, whereas a very strong increase of displacement speed with positive curvatures can be observed along the wrinkled flame contour for hydrogen/air flames.

The curvature of material lines in a three-dimensional chaotic flow

Folding of material filaments was examined computationally in the three-dimensional flow in a cylindrical duct with helical deflectors by tracking the curvature of line elements in the flow. Two geometries were analyzed: a configuration in which the flow is globally chaotic, and an alternative geometry which has a mixture of chaotic and regular motion. The behavior of the curvature field in this complex flow geometry was in agreement with that previously observed for much simpler two-dimensional model flows [Phys. Fluids 8, 75 (1996)]. Curvature profiles along individual element trajectories indicate that an inverse relationship exists between the rates of stretching and curvature. Material elements are compressed when they are folded. After an initial transient, the mean curvature oscillates within a finite range with a periodicity matching that of the flow geometry. The spatial structure of the curvature field becomes period-independent, and the probability density functions of curvature computed for different numbers of periods collapse to an invariant, self-similar distribution without the need for scaling.

Effects of stretch on the local structure of preely propagating premixed low-turbulent flames with various lewis numbers

An experimental investigation of the flame response to strain rate in the case of unsteady premixed low-turbulent flames is presented. In order to point out the fundamental aspects of the mutual interaction between combustion and turbulence, measurements of local flame properties (curvature, displacement speed) and tangential strain rate were performed under varying conditions of Lewis number and turbulence. Three different mixtures (methane/air, propane/air, and hydrogen/air) were successively spark ignited in a vertical wind tunnel. The expanding flame freely propagated in a grid-generated decaying turbulent flow. An advanced field imaging technique coupling high-speed laser tomography and cross-correlation particle image velocimetry (PIV) was used to measure the temporal evolution of local flame stretch exerted by the turbulent cold flow. Local flame curvature and local displacement speed were calculated from flame-front contours. Curvature probability density functions (PDFs) were negatively skewed, especially for nonunity Lewis numbers, and displacement speed distributions underlined the influence of local stretch and thermodiffusive effects on flame-speed variations. Tangential strain rate was determined by using the velocity field in the neighborhood of the flame front and appears to be independent of the Lewis numbers. A strong correlation between local flame curvature and tangential strain rate was demonstrated, underlining the cold flow effects on the local flame structure. The influences of turbulence and Lewis number were evaluated and compared with numerical simulations. Then, local flame stretch distributions were determined versus time, indicating that a significant proportion of the flame was under compression.

The curvature of material lines in chaotic cavity flows

Material line folding is studied in two-dimensional chaotic cavity flows. Line folding is measured by the local curvature k=l×l′/‖l‖3, where l(q) is an infinitesimal vector in the tangential direction of the line, q is a coordinate along the line, and l′ is the derivative of l with respect to q. It is shown both analytically and numerically that folding is always accompanied by compression. The vector l′ plays a crucial role as a driving force for the stretching and folding processes. A material line is stretched when l′ is tangential to the line and it is folded when l′ is normal to the line. The spatial structure of the curvature field is computed numerically. The short-time structure of the curvature field is similar to the structure of unstable manifolds of periodic hyperbolic points, and closely resembles patterns observed in tracer mixing experiments and in stretching field computations. The long time structure of the field asymptotically approaches an entirely different time-independent structure. Probability density functions of curvature are independent of both time and initial conditions.

Flame Curvature Statistics in Axisymmetric Turbulent Jet Flames

ABSTRACT Flame curvature in axisymmetric turbulent jet flames are obtained experimentally for u'/S L from 0.8–1.5. The measurements indicate that the flame curvature statistics exhibit a substantial change as a function of the axial location in turbulent jet flames primarily due to the evolution of flame wrinkle structures. In particular, it is found that the flame curvature tends to be very small near the burner exit, while development of the flame wrinkle structures further from the burner exit results in an increase in magnitude and asymmetry in flame curvature probability density functions (pdΓs). The mean flame curvature in the vertical plane is close to zero in spite of the asymmetry in the flame curvature pdΓs except near the flame tip where the mean flame curvature becomes negative due to the frequent occurrences of cusps. In contrast, the flame curvature in the horizontal plane are strongly affected by the burner dimension and geometry in that the magnitude of both positive and negative flame cur...

Direct numerical simulation of H2/O2/N2 flames with complex chemistry in two-dimensional turbulent flows

Premixed H2/O2/N2 flames propagating in two-dimensional turbulence have been studied using direct numerical simulations (DNS: simulations in which all fluid and thermochemical scales are fully resolved). Simulations include realistic chemical kinetics and molecular transport over a range of equivalence ratios Φ (Φ = 0.35, 0.5, 0.7, 1.0, 1.3). The validity of the flamelet assumption for premixed turbulent flames is checked by comparing DNS data and results obtained for steady strained premixed flames with the same chemistry (flamelet ‘library’). This comparison shows that flamelet libraries overestimate the influence of stretch on flame structure. Results are also compared with earlier zero-chemistry (flame sheet) and one-step chemistry simulations. Consistent with the simpler models, the turbulent flame with realistic chemistry aligns preferentially with extensive strain rates in the tangent plane and flame curvature probability density functions are close to symmetric with near-zero means. For very lean flames it is also found that the local flame structure correlates with curvature as predicted by DNS based on simple chemistry. However, for richer flames, by contrast to simple-chemistry results with non-unity Lewis numbers (ratio of thermal to species diffusivity), local flame structure does not correlate with curvature but rather with tangential strain rate. Turbulent straining results in substantial thinning of the flame relative to the steady unstrained laminar case. Heat-release and H2O2 contours remain thin and connected (‘flamelet-like’) while species including H-atom and OH are more diffuse. Peak OH concentration occurs well behind the peak heat-release zone when the flame temperature is high (of the order of 2800 K). For cooler and leaner flames (about 1600 K and for an equivalence ratio below 0.5) the OH radical is concentrated near the reaction zone and the maximum OH level provides an estimate of the local flamelet speed as assumed by Becker et al. (1990).

Surface properties of turbulent premixed propane/air flames at various Lewis numbers

Surface properties of turbulent premixed flames including the wrinkled flame perimeter, fraction of the flame pocket perimeter, flame curvature, and orientation distributions have been measured for propane-air flames at Lewis numbers ranging from 0.98 to 1.86 and u′ S L = 1.42–5.71 . The wrinkled flame perimeter is found to be greater for the thermodiffusively unstable Lewis number (Le < 1) by up to 30% in comparison to the most stable condition (Le = 1.86) tested, while the fraction of the flame pocket perimeter shows a similar tendency to be greater for Le < 1. The flame curvature probability density functions are nearly symmetric with respect to the zero mean at all Lewis numbers throughout the range of u′ S L tested, and show a much stronger dependence on the turbulence condition than on the Lewis number. Similarly, the flame orientation distributions show a trend from anisotropy toward a more uniform distribution with increasing u′ S L at a similar rate for all Lewis numbers. Thus, for turbulent premixed propane/air flames for a practical range of Lewis number from 0.98 to 1.86, the effect of Lewis number is primarily to affect the flame structures and thereby flame surface areas and flame pocket areas, while the flame curvature and orientation statistics are essentially determined by the turbulence properties.