Mean Scalar Dissipation Rate Research Articles

Scaling CO2–brine mixing in permeable media via analogue models

Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$ , obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$ , with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.

Similarity for dissipation-scaled wall turbulence

In this paper, we put forward a hypothesis for turbulent kinetic energy, Reynolds stresses and scalar variance in wall-bounded turbulent flows, whereby these quantities, when normalized with the kinematic viscosity, mean turbulent energy dissipation rate and scalar dissipation rate, are independent of the Reynolds and Péclet numbers when they are sufficiently large. In particular, there exist two scaling ranges: (i) an inertial-convective range at sufficiently large distance from the wall over which a $2/3$ power-law scaling emerges for all quantities mentioned above; (ii) a viscous-convective range between the viscous-diffusive and inertial-convective ranges at large Prandtl number over which the normalized scalar variance is constant. The relatively large amount of available wall turbulence data either provides reasonably good support for this hypothesis or at least exhibits a trend that is consistent with the predictions of this hypothesis. The relationship between the proposed scaling and the traditional wall scaling is discussed. Possible ultimate statistical states of wall turbulence are also proposed.

Flame structure and broadening in turbulent premixed jet flames

High-resolution and high signal-to-noise (SNR) planar Rayleigh scattering thermometry measurements are performed to investigate preheat and reaction zone broadening, turbulent transport of high-temperature gases, and the internal flame structure in a series of highly turbulent, lean (ϕ = 0.75), premixed methane-air turbulent jet flames. These flames are characterized by turbulence intensities ranging from u′/SL of 35 to 150 and turbulent Reynolds numbers approaching 104, as determined from separate particle image velocimetry (PIV) measurements. Average flame profiles obtained using a gradient-based flame reconstruction method and probability density functions (PDFs) of the preheat, reaction, and thermal layer thicknesses show substantial broadening of the preheat and thermal layers. Instantaneous preheat zones are broadened by as much as a factor of 30 compared to laminar flames, with an average broadening approaching a factor of 14 for the most turbulent case. Moderate broadening of the reaction layer is observed with up to a factor of eight times that of a laminar premixed flame for instantaneous realizations and an average broadening of four times that of a laminar flame for the most turbulent case. Conditional statistics of temperature obtained at isodistance contours within and upstream of the preheat zone show large shifts towards significantly higher temperatures compared to laminar flame values. In addition, broad temperature distributions and reduced thermal gradients at each position provide strong evidence of persistent turbulent transport of high-temperature gases ahead of the preheat zone. Finally, a detailed analysis of scalar dissipation rate statistics and dissipation layer topology was conducted. The conditional mean scalar dissipation rates for the turbulent flames show significant reductions compared to calculated laminar flame values and a shift towards higher values of the reaction progress variable (CT) for increasing turbulence levels. PDFs show a transition from near-Gaussian to highly negatively skewed with increasing turbulence levels, where the most probable values of the scalar dissipation rate shift towards higher values of CT, indicating that the primary reaction zone and flame front are subjected to increased strain. In terms of structure, no significant broadening of the small-scale dissipation layers was observed with increasing turbulence intensity. However, as turbulence intensity increases, there is a transition from a sparse number of continuous layers to a large number of shorter, ”broken”, nearly straight segments. The reduction of individual layer lengths (i.e., ”breaking”) is favored over increases in dissipation layer wrinkling, where the sharp decreases in the local dissipation rate values are highly correlated with regions of high curvature. A final notable outcome of this work is that there were negligible differences in all statistical and topological results when moving between the two highest turbulent cases, even though the turbulence intensity, turbulent Reynolds number, and the turbulence Karlovitz number increased substantially (70 - 120%). This implies that there may be an asymptotic limit in which turbulence affects flame behavior for the current configuration.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

DNS analysis of turbulent vaporizing two-phase flows. Part II: Topology of the scalar field

This work is devoted to turbulence in two-phase flows with mass transfer, a fundamental research topic relevant to many applications including those related to two-phase flow combustion. The objective is to provide further insights into the influence of liquid/gas interfaces on some statistical characteristics of the fluctuating velocity and scalar fields collected in the gas phase. The second part of this study is focused on the analysis of the scalar field. The starting point is the transport equation for the mixture fraction. In most descriptions of two-phase flow combustion, the mixture fraction is indeed used to characterize the multicomponent mixture. However, considering the non-linear dependence of chemical reaction rates to composition, the single knowledge of its mean (or filtered) state is not sufficient to describe turbulent conditions. Its variance and mean (or filtered) scalar dissipation rate (SDR) must be evaluated, with the latter quantity (i.e., the SDR) driven, at leading order, by the velocity gradients relevant to straining effects and by the curvature of the mixture fraction field. In a first step of the present study, a detailed analysis of the (i) scalar field, (ii) its gradient, and (iii) the resulting dissipation rate is reported. It is subsequently completed by an exhaustive inspection of geometrical properties and alignment statistics. The corresponding statistical analysis makes use of conditional sampling based on a level-set which signs the distance to the interface, thus allowing to discriminate its influence on scalar and velocity gradients, as well as the resulting turbulence topology.

DNS analysis of turbulent vaporizing two-phase flows, Part I: Topology of the velocity field

This work is devoted to the characterization of turbulence in two-phase flows with mass transfer. The objective is to provide further insights into the influence of liquid/gas interfaces on some statistical characteristics of the fluctuating velocity and scalar fields collected in the gas phase. The first part of this study is primarily focused on the velocity field. Thus, turbulence topology is educed from the consideration of the invariants of the Reynolds stresses, velocity gradients, strain-rate, and rotation tensors. This analysis is completed by some geometrical and alignment statistics. The corresponding quantities are studied using a set of direct numerical simulation (DNS) databases featuring two distinct values of the liquid volume fraction ϕℓ. The starting point is the transport equation for the mixture fraction. In most descriptions of two-phase flow combustion, it is indeed used to characterize the multi-component mixture. However, considering the non-linear dependence of chemical reaction rates to composition, the single knowledge of its mean (or filtered) state is insufficient to describe turbulent conditions. Its variance and associated mean (or filtered) scalar dissipation rate (SDR) must be evaluated, with the latter quantity (i.e., the SDR) driven, at leading order, by the velocity gradients relevant to straining effects and by the curvature of the mixture fraction field. The present statistical analysis makes use of conditional sampling based on a level-set value which signs the distance to the interface, thus allowing to discriminate its influence on the velocity gradients and associated turbulence topology.

Spatial characteristics and modelling of mixture fraction variance and scalar dissipation rate in steady turbulent round jets

The spatial characteristics of scalar dissipation rate (SDR) are investigated in a steady-state turbulent round jet using Direct Numerical Simulation (DNS). Radial distribution of mixture fraction is found to agree well with the experimental data. Radial distributions of the turbulent diffusivities and the turbulent Schmidt numbers show different behaviours, based on how these quantities are defined. Point-wise statistics confirm that the log-normal distribution fits the probability distribution functions (PDFs) of the SDR well, except for a small positive skewness attributed to flow anisotropy. Joint PDFs of mixture fraction and SDR are symmetric with respect to mixture fraction. Self-similarity of the mean SDR, as well as its axial, radial and azimuthal components appears after a certain downstream distance. In the self-similar region, centreline SDR scales with downstream distance, x−4. Normalised SDR radial profiles show an increase near the centreline, followed by a steep decrease, after an off-centreline peak. Magnitudes of the axial, radial and azimuthal components of the mean SDR are approximately the same in the self-similar region. The budget analysis of the mean SDR transport equation reveals that the production term by scalar field stretch and the destruction term due to local curvature effects are the two dominant terms. A modification of an existing model for the self-similar mean SDR radial profiles is proposed. This is found to improve the prediction of mean SDR distributions.

Local Equilibrium and Discretization Effects on the Prediction of Scalar Dissipation Rates in Large Eddy Simulation of Turbulent Non-Premixed Combustion

ABSTRACT The effects of discretization and local equilibrium on the prediction of scalar dissipation rates are investigated by performing large-eddy simulation (LES) of a turbulent-piloted jet diffusion flame, Sandia Flame D. As a way to reduce the discretization errors on the prediction of scalar dissipation, an approach to use a higher-order finite difference scheme only for the evaluation of scalar gradients in a scalar dissipation model, while keeping the overall accuracy of the numerical method at second order, is investigated. Results indicate that the fourth-order scheme improves the accuracy with negligible addition to computational cost. With the improved evaluation of scalar gradients, the local equilibrium assumption, which is used in the algebraic model, is investigated by solving the subgrid-scale variance transport equation, where a model coefficient in a scaling-based relaxation model for subgrid-scale scalar dissipation is varied. It is found that the conditional mean scalar dissipation rates are insensitive to the model coefficient. This insensitivity is attributed to the prevalence of local equilibrium. A rapid decay of imbalance between production and dissipation of subgrid-scale scalar fluctuations is seen in both the predicted scalar dissipation and the transport budget in the variance equation. It is argued that since, at high Reynolds numbers, the transport of the scalar fluctuations occurs over a timescale larger than the subgrid-scale mixing timescale, its contribution to the evolution of subgrid-scale variance is statistically insignificant except near the nozzle exit where different streams begin to mix and scalar layers evolve rapidly. Consequently, the conditional (and unconditional) mean scalar dissipation rates in LES tend to depend solely on the production of subgrid-scale scalar fluctuations in most regions of turbulent jet flames, when the modeling is based on the Kolmogorov scaling.

Influence of thermal wall boundary condition on scalar statistics during flame-wall interaction of premixed combustion in turbulent boundary layers

The influence of thermal wall boundary conditions on the behaviour of scalar statistics during flame-wall interaction (FWI) of premixed flames within turbulent boundary layers has been investigated. Three-dimensional direct numerical simulations (DNS) of two different flow configurations for flames interacting with chemically inert isothermal as well as adiabatic walls in fully developed turbulent boundary layers have been performed. The first configuration is an oblique wall interaction (OWI) of a V-flame in a turbulent channel flow and the second configuration is representative of head-on interaction (HOI) of a planar flame in a turbulent boundary layer. The turbulence in the non-reacting conditions for these simulations is representative of a friction velocity based Reynolds number of Reτ=110 while the combustion characteristics are representative of stoichiometric methane–air mixture under atmospheric conditions. Differences in the mean behaviours of the progress variable and non-dimensional temperature have been observed for the two configurations in response to different wall boundary conditions. Mean scalar variance of the progress variable and temperature and their respective scalar dissipation rates have been investigated and it is found that the variances of the progress variable and temperature remain coupled in the case of adiabatic walls, while significant differences exist between these quantities in the vicinity of the wall under isothermal wall conditions for both flow configurations. The mean scalar dissipation rates of progress variable and temperature during FWI decrease towards the wall in both flow configurations under different thermal wall boundary conditions, but in the case of isothermal wall conditions a higher scalar dissipation rate for the non-dimensional temperature is observed when compared with the scalar dissipation rate for the progress variable. The behaviours of the mean turbulent scalar fluxes of progress variable and the non-dimensional temperature suggest that the scalar fluxes can show both gradient and countergradient type behaviour depending on the flow configuration and also on the proximity of the flame to the wall. In the case of V-flame OWI, the choice of the wall boundary condition has a significant influence on the behaviour of the scalar fluxes, whereas in the case of turbulent boundary layer HOI, no significant differences in the scalar fluxes are observed for different wall boundary conditions.

Experimental investigation of solute transport in variably saturated porous media using x-ray computed tomography

Solute transport through variably saturated porous media is ubiquitous in multiple subsurface flows, piquing the geoscience community's interest. This study adopts a novel experimental approach using microfocus x-ray computed tomography for real-time imaging of a three-dimensional NaI tracer plume in a partially saturated packing column. A stabilized two-phase flow field is achievable through continuous co-injection of two-phase fluids: NaCl solvent and pump oil. Thus, the critical role of the NaCl saturation Sw and Péclet number on dispersion can be fully studied by controlling the NaCl fractional flow rate and the total flow rate from the Buckley–Leverett theory. Furthermore, we study solute transport behavior based on statistical moments, the dispersion coefficient, the dilution index, and the mean scalar dissipation rate. Experimental results indicate that the solute transport is Fickian for high Sw ≥ 0.34. In contrast, anomalous transport behavior is found for Sw < 0.34, where the concentration distribution is initially left-tailed and leptokurtic before reaching a well-dispersed regime. The dispersion coefficient is 2–10 times larger for partially saturated cases compared with the fully saturated case and shows a non-monotonical dependency on Sw. Finally, the analysis of the dilution index indicates that the overall mixing strength increases when Sw decreases, whereas the mean scalar dissipation rate reveals that the time scaling of transverse mixing is the largest at an intermediate Sw. The results can be used to elucidate the solute transport behavior in a two-phase system.

Modelling of Conditional Scalar Dissipation Rate in Turbulent Premixed Combustion

In turbulent premixed flames, for the mixing at a molecular level of reactants and products on the flame surface, it is crucial to sustain the combustion. This mixing phenomenon is featured by the scalar dissipation rate, which may be broadly defined as the rate of micro-mixing at small scales. This term, which appears in many turbulent combustion methods, includes the Conditional Moment Closure (CMC) and the Probability Density Function (PDF), requires an accurate model. In this study, a mathematical closure for the conditional mean scalar dissipation rate, <Nc|ζ>, in Reynolds, Averaged Navier–Stokes (RANS) context is proposed and tested against two different Direct Numerical Simulation (DNS) databases having different thermochemical and turbulence conditions. These databases consist of lean turbulent premixed V-flames of the CH4-air mixture and stoichiometric turbulent premixed flames of H2-air. The mathematical model has successfully predicted the peak and the typical profile of <Nc|ζ> with the sample space ζ and its prediction was consistent with an earlier study.

Turbulence is an Ineffective Mixer when Schmidt Numbers Are Large.

We solve the advection-diffusion equation for a stochastically stationary passive scalar θ, in conjunction with forced 3D Navier-Stokes equations, using direct numerical simulations in periodic domains of various sizes, the largest being 8192^{3}. The Taylor-scale Reynolds number varies in the range 140-650 and the Schmidt number Sc≡ν/D in the range 1-512, where ν is the kinematic viscosity of the fluid and D is the molecular diffusivity of θ. Our results show that turbulence becomes an ineffective mixer when Sc is large. First, the mean scalar dissipation rate ⟨χ⟩=2D⟨|∇θ|^{2}⟩, when suitably nondimensionalized, decreases as 1/logSc. Second, 1D cuts through the scalar field indicate increasing density of sharp fronts on larger scales, oscillating with large excursions leading to reduced mixing, and additionally suggesting weakening of scalar variance flux across the scales. The scaling exponents of the scalar structure functions in the inertial-convective range appear to saturate with respect to the moment order and the saturation exponent approaches unity as Sc increases, qualitatively consistent with 1D cuts of the scalar.

Multi-particle models of molecular diffusion for Lagrangian simulation coupled with LES for passive scalar mixing in compressible turbulence

Lagrangian simulation coupled with large eddy simulation (LES) is studied for turbulent scalar mixing in compressible flows, where Lagrangian simulation solves advection-diffusion equations with computational particles. In Lagrangian simulation, a molecular diffusion term needs to be modeled with a so-called mixing model, which also requires modeling the dissipation rate of scalar fluctuations. The present study extends a particle-based subgrid-scale model for the scalar dissipation rate to compressible flows, and its validity in compressible turbulence is examined with direct numerical simulation databases of a temporally evolving turbulent planar jet with a jet Mach number of 0.6 or 1.6. A priori test confirms that the model can well predict the mean scalar dissipation rate in the subsonic and supersonic turbulent jets. The model hardly depends on the spatial distribution of the particles when the number of particles used in the model is about 12. However, the scalar dissipation rate tends to be overestimated by the model in regions with large dilatation fluctuations although such regions with strong compressibility effects occupy only a small part of the flow. Lagrangian simulation coupled with LES of the turbulent jet is also performed with the scalar dissipation model combined with the mixing volume model. Lagrangian simulation with these models effectively predicts passive scalar statistics, such as averages, rms fluctuations, and turbulent fluxes, in the turbulent jet with both subsonic and supersonic jet velocities. The scalar dissipation model well predicts the coarse-grained scalar dissipation rate in Lagrangian simulation. The present results confirm that the mixing volume model combined with the particle-based model of the scalar dissipation rate is useful in Lagrangian simulation coupled with LES, which is a promising approach for simulating high-speed turbulent reacting flows as reaction terms appear in Lagrangian simulation in a closed form.

Scalar fluctuation and its dissipation in turbulent reacting flows

The dissipation rate of a scalar variance is related to the mean heat release rate in turbulent combustion. Mixture fraction is the scalar of interest for non-premixed combustion, and a reaction progress variable is relevant for premixed combustion. A great deal of work is conducted in past studies to understand the spectra of passive scalar transport in turbulent flows. A very brief summary of these studies to bring out the salient characteristics of the passive scalar spectrum is given first. Then, the classical analysis of the reactive scalar spectrum is revisited in the lights of recent understanding gained through analyzing the scalar spectrum deduced from direct numerical simulation data of both non-premixed and premixed combustion. The analysis shows that the reactive scalar spectral density in premixed combustion has a dependence on Karlovitz and Damköhler numbers, which comes through the mean scalar dissipation rate appearing in the spectral expression. In premixed combustion, the relevant scale for the scalar dissipation rate is shown to be of the order of the chemical length scale, and the dissipation rate is not influenced by the scales in the inertial-convective range unlike for the passive scalar dissipation rate. The scalar fluctuations produced near the chemical scales cascade exponentially to larger scales. These observations imply that the passive scalar models cannot be extended to premixed combustion.

Multi-particle model of coarse-grained scalar dissipation rate with volumetric tensor in turbulence

A multi-particle model is proposed for a coarse-grained scalar dissipation rate, where a coarse-grained quantity is defined with an ensemble average of spatially-distributed fluid particles within a finite volume. The model computes the coarse-grained scalar dissipation rate from the coarse-grained scalar gradient with a subgrid scale model of the scalar dissipation rate, which requires length-scale estimation for particle distribution. A volumetric tensor that characterizes the particle distribution is used in the model for computing the length scale and the coarse-grained scalar and velocity gradients from the particles. The model is examined in a priori and posteriori tests. A priori test with direct numerical simulation (DNS) databases of turbulent planar jets shows that the present model works well for a wide range of the length scale of particle distribution when the number of particles NM is about 10-16. The model with NM≲10 overestimates the coarse-grained scalar dissipation rate, while NM≳16 causes stronger dependence of the model on the length scale of particle distribution. The proposed model is tested in hybrid large-eddy-simulation/Lagrangian-particle-simulation (LES/LPS) of planar jets, where the coarse-grained scalar dissipation rate appears as an unknown variable in a mixing volume model that computes a molecular diffusion term based on a multi-particle interaction. LES/LPS and DNS yield a similar profile of the root-mean-squared scalar fluctuation, which strongly depends on the scalar dissipation rate. Comparison of the mean scalar dissipation rate between the model and the DNS shows that the present model applied to the LES/LPS well predicts the coarse-grained scalar dissipation rate at various jet Reynolds numbers.

Evolution of scalar and velocity dynamics in planar shock-turbulence interaction

Due to the short residence time of air in supersonic combustors, achieving efficient mixing in compressible turbulent reactive flows is crucial for the design of supersonic ramjet (Scramjet) engines. In this respect, improving the understanding of shock-scalar mixing interactions is of fundamental importance for such supersonic combustion applications. In these compressible flows, the interaction between the turbulence and the shock wave is reciprocal, and the coupling between them is very strong. A basic understanding of the physics of such complex interactions has already been obtained through the analysis of relevant simplified flow configurations, including propagation of the shock wave in density-stratified media, shock-wave–mixing-layer interaction, and shock-wave–vortex interaction. Amplification of velocity fluctuations and substantial changes in turbulence characteristic length scales are the most well-known outcomes of shock–turbulence interaction, which may also deeply influence scalar mixing between fuel and oxidizer. The effects of the shock wave on the turbulence have been widely characterized through the use of so-called amplification factors, and similar quantities are introduced herein to characterize the influence of the shock wave on scalar mixing. One of the primary goals of the present study is indeed to extend previous analyses to the case of shock-scalar mixing interaction, which is directly relevant to supersonic combustion applications. It is expected that the shock wave will affect the scalar dissipation rate (SDR) dynamics. Special emphasis is placed on the modification of the so-called turbulence–scalar interaction as a leading-order contribution to the production of mean SDR, i.e., a quantity that defines the mixing rate and efficiency. To the best of the authors’ knowledge, this issue has never been addressed in detail in the literature, and the objective of the present study is to scrutinize this influence. The turbulent mixing of a passive (i.e., chemically inert) scalar in the presence of a shock wave is thus investigated using high-resolution numerical simulations. The starting point of the analysis relies on the transport equations of the variance of the mixture fraction, i.e., a fuel inlet tracer that quantifies the mixing between fuel and oxidizer. The influence of the shock wave is investigated for three distinct values of the shock Mach number M, and the obtained results are compared to reference solutions featuring no shock wave. The computed solutions show that the shock wave significantly modifies the scalar field topology. The larger the value of M, the stronger is the amplification of the alignment of the scalar gradient with the most compressive principal direction of the strain-rate tensor, which signifies the enhancement of scalar mixing with the shock Mach number.

High-order structure functions for passive scalar fed by a mean gradient

Transport equations for even-order structure functions are written for a passive scalar mixing fed by a mean scalar gradient, with a Schmidt number Sc=1. Direct numerical simulations (DNS), in a range of Reynolds numbers Rλ ∈ [88, 529] are used to assess the validity of these equations, for the particular cases of second-and fourth-order moments. The involved terms pertain to molecular diffusion, transport, production, and dissipative-fluxes. The latter term, present at all scales, is equal to: i) the mean scalar variance dissipation rate, ⟨χ⟩, for the second-order moments transport equation; ii) non-linear correlations between χ and second-order moments of the scalar increment, for the fourth-order moments transport equation.The equations are further analysed to show that the similarity scales (i.e., variables which allow for perfect collapse of the normalised terms in the equations) are, for second-order moments, fully consistent with Kolmogorov–Obukhov–Corrsin (KOC) theory. However, for higher-order moments, adequate similarity scales are built from ⟨χn⟩. The similarity is tenable for the dissipative range, and the beginning of the scaling range.

Relevance of Two Basic Turbulent Premixed Combustion Models for the Numerical Simulations of V-Shaped Flames

ABSTRACTThe present study is devoted to the analysis of basic turbulent premixed combustion closures applied to the numerical simulations of V-shaped flames. It is well known that an important parameter for the numerical simulation of such premixed turbulent flames is the description of the departure from the bimodal limit (thin flame limit), which is associated to the maximum value of the progress variable segregation rate, i.e., S = 1. The evolution of this segregation rate is often deduced from a modeled transport equation written for the progress variable variance. However, the closure of such a transport equation does involve many additional sub-models, which are related to the mean and variance progress variable fluxes and mean scalar dissipation rate of the progress variable variance. In the present work two original closures for the mean chemical rate are considered. Special emphasis is also placed on algebraic closures for S that circumvent the difficulty associated to the modeling of the second moment of the progress variable. In the first step of the analysis these closures are analyzed in the case of the propagation of a one-dimensional turbulent premixed flame brush. They are subsequently applied to the numerical simulation of premixed V-shaped flames that have been studied experimentally by Galizzi (2003) and by Degardin et al. (2006). It is found that these closures provide a satisfactory representation of the turbulent premixed flames.

Boundedness of the mixed velocity-temperature derivative skewness in homogeneous isotropic turbulence

The transport equation for the mean scalar dissipation rate ϵ¯θ is derived by applying the limit at small separations to the generalized form of Yaglom’s equation in two types of flows, those dominated mainly by a decay of energy in the streamwise direction and those which are forced, through a continuous injection of energy at large scales. In grid turbulence, the imbalance between the production of ϵ¯θ due to stretching of the temperature field and the destruction of ϵ¯θ by the thermal diffusivity is governed by the streamwise advection of ϵ¯θ by the mean velocity. This imbalance is intrinsically different from that in stationary forced periodic box turbulence (or SFPBT), which is virtually negligible. In essence, the different types of imbalance represent different constraints imposed by the large-scale motion on the relation between the so-called mixed velocity-temperature derivative skewness ST and the scalar enstrophy destruction coefficient Gθ in different flows, thus resulting in non-universal approaches of ST towards a constant value as Reλ increases. The data for ST collected in grid turbulence and in SFPBT indicate that the magnitude of ST is bounded, this limit being close to 0.5.

Transported Joint Probability Density Function Simulation of Turbulent Spray Flames Combined with a Spray Flamelet Model Using a Transported Scalar Dissipation Rate

ABSTRACTNumerical simulations of a turbulent ethanol/air spray flame are performed using a combined formulation of a joint gas mixture fraction-enthalpy probability density function (PDF) and a spray flamelet model, where the scalar dissipation rate of the mixture fraction is described with a transport equation. Spray evaporation is included in the flamelet model through the pre-calculation of counterflowing laminar spray flames with a detailed reaction mechanism of 38 species and 337 elementary reactions for ethanol/air combustion. A transport equation for the mean scalar dissipation rate of the mixture fraction is derived, taking into account the spray evaporation. The computed results are compared with the experimental data provided by A. Masri, University of Sydney, Australia. All major spray flame characteristics, including the Sauter mean diameter, the droplet velocity, and its turbulent fluctuations as well as gas temperature, generally show good agreement with experiments. Different models for the scalar dissipation rate of the mixture fraction are not significantly different for the flame under consideration except for inside the chemical reaction zone, where the new model shows some improvement compared to the experiment.