Three-dimensional Direct Numerical Simulations Research Articles

Influence of Microscale Turbulent Droplet Clustering on Radar Cloud Observations

Abstract This study investigates the influence of microscale turbulent clustering of cloud droplets on the radar reflectivity factor and proposes a new parameterization to account for it. A three-dimensional direct numerical simulation of particle-laden isotropic turbulence is performed to obtain turbulent clustering data. The clustering data are then used to calculate the power spectra of droplet number density fluctuations, which show a dependence on the Taylor microscale-based Reynolds number (Reλ) and the Stokes number (St). First, the Reynolds number dependency of the turbulent clustering influence is investigated for 127 < Reλ < 531. The spectra for this wide range of Reλ values reveal that Reλ = 204 is sufficiently large to be representative of the whole wavenumber range relevant for radar observations of atmospheric clouds. The authors then investigate the Stokes number dependency for Reλ = 204 and propose an empirical model for the turbulent clustering influence assuming power laws for the number density spectrum. For Stokes numbers less than 2, the proposed model can estimate the influence of turbulence on the spectrum with an RMS error less than 1 dB when calculated over the wavenumber range relevant for radar observations. For larger Stokes number droplets, the model estimate has larger errors, but the influence of turbulence is likely negligible in typical clouds. Applications of the proposed model to two idealized cloud observing scenarios reveal that microscale turbulent clustering can cause a significant error in estimating cloud droplet amounts from radar observations with microwave frequencies less than 13.8 GHz.

Mach number effects on the global mixing modes induced by ramp injectors in supersonic flows

AbstractModern injectors for supersonic combustors (hypermixers) augment the fuel–air mixing rate by energizing the perturbation in the mixing layer. From an instability point of view, the increased perturbation growth is linked to the increased complexity of the equilibrium base flow when compared to the axisymmetric mixing layer. Common added features are streamwise vortex streaks, oblique recompression shocks and Prandtl–Meyer expansions. One of the main effects of such distortions of the mean flow is to transform the instability responsible for the creation of fine scales from a local amplified mode to a global self-sustained fluctuation. The focus of the present research is on the flow distortion induced by flushed ramps for free-stream Mach numbers in the range 2.5–3.5. The principal mean flow features are the recirculation region due to the recompression of the flow after the ramp, the shear layer over the recirculation region and the vortex streaks propagating from the ramp corners. A global three-dimensional stability analysis and three-dimensional direct numerical simulations of small perturbations of the mean flow are performed. The growth and energy distribution of the dominant and subdominant fluctuations supported by the three-dimensional steady laminar base flow are computed. The main results are the growth rates of the self-sustained varicose and sinuous modes and their correlation to the variation in the free-stream Mach number. The complex three-dimensional wavemaker is investigated by evaluating the three-dimensional eigenfunctions of the direct and adjoint modes, and the effects of the axial vorticity generated by the ramp corners are discussed.

Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part I: Experimental and numerical investigation

Three-dimensional Direct Numerical Simulations combined with Particle Image Velocimetry experiments have been performed on a hemisphere-cylinder at Reynolds number 1000 and angle of attack 20°. At these flow conditions, a pair of vortices, so-called “horn” vortices, are found to be associated with flow separation. In order to understand the highly complex phenomena associated with this fully three-dimensional massively separated flow, different structural analysis techniques have been employed: Proper Orthogonal and Dynamic Mode Decompositions, POD and DMD, respectively, as well as critical-point theory. A single dominant frequency associated with the von Karman vortex shedding has been identified in both the experimental and the numerical results. POD and DMD modes associated with this frequency were recovered in the analysis. Flow separation was also found to be intrinsically linked to the observed modes. On the other hand, critical-point theory has been applied in order to highlight possible links of the topology patterns over the surface of the body with the computed modes. Critical points and separation lines on the body surface show in detail the presence of different flow patterns in the base flow: a three-dimensional separation bubble and two pairs of unsteady vortices systems, the horn vortices, mentioned before, and the so-called “leeward” vortices. The horn vortices emerge perpendicularly from the body surface at the separation region. On the other hand, the leeward vortices are originated downstream of the separation bubble, as a result of the boundary layer separation. The frequencies associated with these vortical structures have been quantified.

Magnetohydrodynamic flow excited by rotating permanent magnets in an orthogonal container

Liquid metal magnetohydrodynamic flow driven by a system of rotating permanent magnets in a container of orthogonal cross-section has been studied. The main objective of the work is to research the impact of magnetic forcing parameters (magnetic field value, magnets arrangement, and angular velocity of their rotation) on the generated hydrodynamic structures and flow modes. On this basis, we contemplate realizing required flow features by setting certain parameters of the driving magnetic system. A numerical study of the problem in the induction-free approximation without taking into account the effect of the variable component of electromagnetic force is presented. The parameters of spin-up modes and steady-state flow regimes have been calculated by three-dimensional direct numerical simulation based on COMSOL Multiphysics 4.3a software and experimentally verified on a specially designed setup using noninvasive Doppler ultrasound technique.

Wake transition in the flow around a circular cylinder with a splitter plate

AbstractA simple way to decrease the drag and oscillating lift forces in the flow around a circular cylinder consists of positioning a splitter plate in the wake, parallel to the flow. In this paper, the effect of the splitter plate on the wake dynamics, more specifically on the wake transition, is described in detail. First, two-dimensional and three-dimensional direct numerical simulations (DNS) using the spectral element method were used to observe the behaviour of the wake in the presence of the splitter plate. Then, a linear stability analysis based on the Floquet theory was performed in order to obtain information on how the splitter plate changes the instabilities that lead to wake transition. Simulations were carried out for several gaps between the splitter plate and the cylinder, with the Reynolds number varying in the range between 100 and 350, which corresponds to the wake transition in the flow around a circular cylinder. The results of the simulations showed a discontinuity in the Strouhal number curve that is consistent with the results available in the literature. The stability analysis showed how the splitter plate modifies the transition of the flow to a three-dimensional configuration. The splitter plate has a stabilizing effect on the flow for small gaps, delaying the appearance of three-dimensional structures to higher Reynolds numbers. Mode A and a quasi-periodic (QP) mode are observed for such small gaps. As the gap is increased the discontinuity in the Strouhal number curve also caused a clear change in the characteristics of the neutral stability curve, and the existence of an unstable period-doubling mode was observed. The onset characteristics of the unstable modes are analysed and discussed in depth.

Effects of Equivalence Ratio and Turbulent Velocity Fluctuation on Early Stages of Pulverized Coal Combustion Following Localized Ignition: A Direct Numerical Simulation Analysis

This study utilized three-dimensional direct numerical simulations (DNS) in order to analyze the effects of turbulent velocity fluctuation, equivalence ratio based on volatile primary fuel in the p...

On velocity and reactive scalar spectra in turbulent premixed flames

AbstractKinetic energy and reactive scalar spectra in turbulent premixed flames are studied from compressible three-dimensional direct numerical simulations (DNS) of a temporally evolving rectangular slot-jet premixed flame, a statistically one-dimensional configuration. The flames correspond to a lean premixed hydrogen–air mixture at an equivalence ratio of 0.7, preheated to 700 K and at 1 atm, and three DNS are considered with a fixed jet Reynolds number of 10 000 and a jet Damköhler number varying between 0.13 and 0.54. For the study of spectra, motivated by the need to account for density change, which can be locally strong in premixed flames, a new density-weighted definition for two-point velocity/scalar correlations is proposed. The density-weighted two-point correlation tensor retains the essential properties of its constant-density (incompressible) counterpart and recovers the density-weighted Reynolds stress tensor in the limit of zero separation. The density weighting also allows the derivation of balance equations for velocity and scalar spectrum functions in the wavenumber space that illuminate physics unique to combusting flows. Pressure–dilatation correlation is a source of kinetic energy at high wavenumbers and, analogously, reaction rate–scalar fluctuation correlation is a high-wavenumber source of scalar energy. These results are verified by the spectra constructed from the DNS data. The kinetic energy spectra show a distinct inertial range with a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}-5/3$ scaling followed by a ‘diffusive–reactive’ range at higher wavenumbers. The exponential drop-off in this range shows a distinct inflection in the vicinity of the wavenumber corresponding to a laminar flame thickness, $\delta _L$, and this is attributed to the contribution from the pressure–dilatation term in the energy balance in wavenumber space. Likewise, a clear spike in spectra of major reactant species (hydrogen) arising from the reaction-rate term is observed at wavenumbers close to $\delta _L$. It appears that in the inertial range classical scaling laws for the spectra involving the Kolmogorov scale are applicable, but in the high-wavenumber range where chemical reactions have a strong signature the laminar flame thickness produces a better collapse. It is suggested that a full scaling should perhaps involve the Kolmogorov scale, laminar flame thickness, Damköhler number and Karlovitz number.

Three-dimensional direct numerical simulation study of conditioned moments associated with front propagation in turbulent flows

In order to gain further insight into (i) the use of conditioned quantities for characterizing turbulence within a premixed flame brush and (ii) the influence of front propagation on turbulent scalar transport, a 3D Direct Numerical Simulation (DNS) study of an infinitely thin front that self-propagates in statistically stationary, homogeneous, isotropic, forced turbulence was performed by numerically integrating Navier-Stokes and level set equations. While this study was motivated by issues relevant to premixed combustion, the density was assumed to be constant in order (i) to avoid the influence of the front on the flow and, therefore, to know the true turbulence characteristics as reference quantities for assessment of conditioned moments and (ii) to separate the influence of front propagation on turbulent transport from the influence of pressure gradient induced by heat release. Numerical simulations were performed for two turbulence Reynolds numbers (50 and 100) and four ratios (1, 2, 5, and 10) of the rms turbulent velocity to the front speed. Obtained results show that, first, the mean front thickness is decreased when a ratio of the rms turbulent velocity to the front speed is decreased. Second, although the gradient diffusion closure yields the right direction of turbulent scalar flux obtained in the DNS, the diffusion coefficient Dt determined using the DNS data depends on the mean progress variable. Moreover, Dt is decreased when the front speed is increased, thus, indicating that the front propagation affects turbulent scalar transport even in a constant-density case. Third, conditioned moments of the velocity field differ from counterpart mean moments, thus, disputing the use of conditioned velocity moments for characterizing turbulence when modeling premixed turbulent combustion. Fourth, computed conditioned enstrophies are close to the mean enstrophy in all studied cases, thus, suggesting the use of conditioned enstrophy for characterizing turbulence within a premixed flame brush.

Local volumetric dilatation rate and scalar geometries in a premixed methane–air turbulent jet flame

The local volumetric dilatation rate, namely, the rate of change of an infinitesimal fluid volume per unit volume, ∇·u, is an important variable particularly in flows with heat release. Its tangential and normal strain rate components, aT and aN, respectively, account for stretching and partially for separation of iso-scalar surfaces. A three-dimensional direct numerical simulation (DNS) of a turbulent premixed methane–air flame in a piloted Bunsen burner configuration has been performed by solving the full conservation equations for mass, momentum, energy and chemical species using tabulated chemistry. Results for the volumetric dilatation rate as a function of the iso-scalar surface geometry, characterized by the mean and Gauss curvatures, km and kg, are obtained in several zones (reactants, preheat, reacting and products) of the computational domain. Flat iso-scalar surfaces are the most likely geometries in agreement with previous DNS. The relationship between density and a reaction progress variable, under a low Mach number flamelet assumption, leads to an expression for ∇·u with contributions from progress variable source and molecular diffusion budget, with a significant contribution from the latter; this approximate expression for the volumetric dilatation rate is studied with DNS results. The joint pdf of aN and aT confirms that the line aT+aN=0 separates mostly expansive flow regions from compressive zones.

Reaction Zones and Their Structure in MILD Combustion

Three-dimensional direct numerical simulation (DNS) of turbulent combustion under moderate and intense low-oxygen dilution (MILD) conditions has been carried out inside a cuboid with inflow and outflow boundaries on the upstream and downstream faces, respectively. The initial and inflowing mixture and turbulence fields are constructed carefully to be representative of MILD conditions involving partially mixed pockets of unburned and burned gases. The combustion kinetics are modeled using a skeletal mechanism for methane-air combustion, including non-unity Lewis numbers for species and temperature-dependent transport properties. The DNS data is analyzed to study the MILD reaction zone structure and its behavior. The results show that the instantaneous reaction zones are convoluted and the degree of convolution increases with dilution and turbulence levels. Interactions of reaction zones occur frequently and are spread out in a large portion of the computational domain due to the mixture non-uniformity and high turbulence level. These interactions lead to local thickening of reaction zones yielding an appearance of distributed combustion despite the presence of local thin reaction zones. A canonical MILD flame element, called MIFE, is proposed, which represents the averaged mass fraction variation for major species reasonably well, although a fully representative canonical element needs to include the effect of reaction zone interactions and associated thickening effects on the mean reaction rate.

Numerical and experimental investigation of turbulent DME jet flames

Results are presented here from a three-dimensional direct numerical simulation of a temporally-evolving planar slot jet flame and experimental measurements within a spatially-evolving axisymmetric jet flame operating with DME (dimethyl ether, CH3OCH3) as the fuel. Both simulation and experiment are conducted at a Reynolds number of 13050. The Damköhler number, stoichiometric mixture fraction and fuel and oxidizer compositions also are matched between simulation and experiment. Simultaneous OH/CH2O PLIF imaging is performed experimentally to characterize the spatial structure of the turbulent DME flames. The simulation shows a fully burning flame initially, which undergoes partial extinction and subsequently, reignition. The scalar dissipation rate (χ) increases to a value much greater than that calculated from near-extinction strained laminar flames, leading to the observed local extinction. As the turbulence decays, the local values of χ decrease and the flame reignites. The reignition process appears to be strongly dependent on the local χ value, which is consistent with previous results for simpler fuels. Statistics of OH and CH2O are compared between simulation and experiment and found to agree. The applicability of OH/CH2O (formaldehyde) product imaging as a surrogate for peak heat release rate is investigated. The concentration product is found to predict peak heat release rate extremely well in the simulation data. When this product imaging is applied to the experimental data, a similar extinction/reignition pattern also is observed in the experiments as a function of axial position. A new 30-species reduced chemical mechanism for DME was also developed as part of this work.

Lagrangian Mixing Dynamics at the Cloudy–Clear Air Interface

Abstract The entrainment of clear air and its subsequent mixing with a filament of cloudy air, as occurs at the edge of a cloud, is studied in three-dimensional direct numerical simulations that combine the Eulerian description of the turbulent velocity, temperature, and vapor fields with a Lagrangian cloud droplet ensemble. Forced and decaying turbulence is considered, such as when the dynamics around the filament is driven by larger-scale eddies or during the final period of the life cycle of a cloud. The microphysical response depicted in nd − 〈r3〉 space (where nd and r are droplet number density and radius, respectively) shows characteristics of both homogeneous and inhomogeneous mixing, depending on the Damköhler number. The transition from inhomogeneous to homogeneous mixing leads to an offset of the homogeneous mixing curve to larger dilution fractions. The response of the system is governed by the smaller of the single droplet evaporation time scale and the bulk phase relaxation time scale. Variability within the nd − 〈r3〉 space increases with decreasing sample volume, especially during the mixing transients. All of these factors have implications for the interpretation of measurements in clouds. The qualitative mixing behavior changes for forced versus decaying turbulence, with the latter yielding remnant patches of unmixed cloud and stronger fluctuations. Buoyancy due to droplet evaporation is observed to play a minor role in the mixing for the present configuration. Finally, the mixing process leads to the transient formation of a pronounced nearly exponential tail of the probability density function of the Lagrangian supersaturation, and a similar tail emerges in the droplet size distribution under inhomogeneous conditions.

Harmonic mechanical excitations of steady convective instabilities: A means to get more uniform heat transfers in mixed convection flows?

In laminar mixed convection flows, steady thermoconvective patterns generate non uniform heat and/or mass transfers at walls that can be detrimental in some industrial processes. For instance the longitudinal thermoconvective patterns of Poiseuille–Rayleigh–Bénard (PRB) flows generate non uniform thin films or coatings when they are present in cold wall horizontal Chemical Vapor Deposition (CVD) reactors. The aim of this paper is to show that, when the basic steady flow is convectively unstable against an unsteady flow regime, introducing small harmonic mechanical excitations in the basic flow may enable to obtain more uniform time averaged heat transfers. More specifically, three-dimensional direct numerical simulations are used to characterize the temperature field and wall heat transfer associated with unsteady wavy convective instabilities of PRB flows that result from harmonic excitations of the longitudinal thermoconvective rolls at channel inlet. A design of experiments is used to build cubic response surfaces of the different quantities analyzed (growth length of the wavy rolls, magnitude of their spanwise oscillations, wall Nusselt number …) on a wide range of the flow parameters. Air PRB flows (Pr=0.71) in channels of aspect ratios equal to Width/Height=10 and 150⩽Length/Height⩽300, for Reynolds numbers 100⩽Re⩽300 and Rayleigh numbers 5000⩽Ra⩽16,000 are considered. Comparisons with experiments are presented and a good agreement is obtained. The optimal conditions to have uniform heat transfers on the horizontal walls of PRB flows correspond to the minimal growth length of the wavy rolls until saturation and the maximum magnitude of their spanwise oscillations. They are approximately obtained for moderate Reynolds number (Re≈150), high Rayleigh numbers (Ra≈15,000), low excitation frequency and rather high excitation magnitude. A discussion of these results for the applications to CVD in horizontal rectangular reactors at atmospheric pressure is finally proposed.

Modelling of the Curvature Term in the Flame Surface Density Transport Equation: A Direct Numerical Simulations Based Analysis

A simple chemistry based three-dimensional Direct Numerical Simulations (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of Karlovitz number Ka, turbulent Reynolds number Re t, heat release parameter τ and global Lewis number Le has been used for the modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Reynolds Averaged Navier Stokes (RANS) simulations. The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed (i.e. T1) and the term arising due to the tangential diffusion component of displacement speed (i.e. T2). Subsequently, the sub-terms (i.e. T1 and T2) of the curvature contribution to the FSD transport have been split into the closed (i.e. T1 r and T2 r) and unclosed (i.e. T1 ur and T2 ur) components. It has been found that T2 remains deterministically negative throughout the flame brush. However, the qualitative behaviour of T1 changes significantly depending upon the values of Ka, Re t and Le. Detailed physical explanations have been provided for the observed behaviours of the components of the curvature term. Moreover, it has been observed that the closed contributions of T1 and T2 (i.e. T1 r and T2 r) remains negligible in comparison to the unclosed contributions (i.e. T1 ur and T2 ur). Suitable model expressions have been identified for T1 ur and T2 ur in the context of RANS simulations, which are shown to perform satisfactorily in all cases considered in the current analysis, accounting for the variations in Ka, Re t, τ and Le.

Marangoni induced force on a drop in a Hele Shaw cell

We analyse the force balance on a cylindrical drop in a Hele-Shaw cell, subjected to a Marangoni flow caused by a surface tension gradient. Depth-averaged Stokes equations, called Brinkman equations, are introduced and a general closed form solution is obtained. The validity of the averaging procedure is ascertained by considering a linear surface tension gradient acting on a cylindrical flattened drop. The Marangoni-driven flow field and resulting force predicted by the Brinkman model are seen to match well a full three-dimensional direct numerical simulation. A closed form expression of the force acting on the drop is obtained, calculated from contributions due to the normal viscous stress, tangential viscous stress, and pressure fields, integrated on the drop perimeter. This expression is used to predict the force balance when a stationary droplet is submitted to both a carrier flow and a Marangoni flow. We show that previous results in the literature had underestimated by a factor two the Marangoni-induced force.

Local flow topologies and scalar structures in a turbulent premixed flame

A three-dimensional direct numerical simulation of a propagating turbulent premixed flame is performed using one-step Arrhenius model chemistry. The interaction of the flame thermochemical processes with the local geometries of the scalar field and flow topologies is studied. Four regions (“fresh reactants,” “preheating,” “burning,” and “hot products”), characterized by their reaction rate and mass fraction values, are examined. Thermochemical processes in the “preheating” and “burning” regions smooth out highly contorted iso-scalar surfaces, present in the “fresh reactants,” and annihilate large curvatures. Positive volumetric dilatation rates, −P = ∇ · u, display maxima for elliptic concave and minima for convex scalar micro-structures. Constant average tangential strain rates, aT, with large fluctuations, occur throughout the flow domain, whereas normal strain rates, aN, follow the trends of volumetric dilatation rates. Focal topologies, present in the “fresh reactants,” tend to disappear in favor of nodal structures as moving towards the “hot products.” The vorticity vector is predominantly tangential to the iso-scalar surfaces. The Unstable Node/Saddle/Saddle and Stable Focus/Stretching topologies, present in the “fresh reactants,” correlate with large values of aN and aT providing hints on the flow topologies fostering scalar mixing.

Relationship between momentum of an impinging drop and intensities of vortex rings generated below free surface

The characteristics of vortex rings induced below the free water surface due to the impingement of a falling single drop with small drop diameters and two impact angles were investigated by means of a three-dimensional direct numerical simulation (DNS) with a Level-Set method. The results show that the present DNS well captures the structures and transport processes of a crater vortex-ring (CV) and a water-column vortex-ring (WCV) due to the vertical impingement of a drop, and that the intensity of the WCV is larger than that of the CV. In the oblique impingement of a drop, a CV is similarly generated, but a vortex tube (VT) creeping along the free surface is also generated instead of the WCV, and the intensity of the VT is much smaller than that of the CV. The relationship between the momentum of a drop and the intensity of the dominant vortex ring is determined independently of diameter and impact angle and is also consistent with the previous measurements (Takagaki and Komori, 2014).

Three-Dimensional Direct Numerical Simulation of Unsteady Transitional Round Fountains in a Homogeneous Fluid

Fountains are common both in nature and in industrial and environmental settings. These are jet flows with a negative buoyancy force acting in the direction opposite to the jet direction. The onset of asymmetry and unsteadiness, which occurs in transitional fountains with intermediate Reynolds (Re) and Froude (Fr) numbers, is the key to shed light on the turbulence generation mechanism in fountains. In this study, a series of three-dimensional direct numerical simulations are carried out for transitional round fountains with Re and Fr in the ranges of 1 ≤ Fr ≤ 8, 50 ≤ Re ≤ 500 to reveal their unsteady flow dynamics, in particular the onset of asymmetry, three-dimensionality, and unsteadiness. The numerical results show that the onset of asymmetry and unsteadiness can be detected and quantified by the tangent velocity on the interfacial surface between the fountain fluid and the ambient fluid. The results also demonstrate that a critical Re is at about 165 for Fr =2 fountains and is reduced to about 65 for Fr =3 fountains. Similarly, a critical Fr exists between 2 and 3 for the Re = 100 fountains, whereas for Re = 200 fountains it reduces to be between 1.8 and 2.

Near-field local flame extinction of oxy-syngas non-premixed jet flames: A DNS study

An investigation of the local flame extinction of H2/CO oxy-syngas and syngas-air nonpremixed jet flames was carried out using three-dimensional direct numerical simulations (DNS) with detailed chemistry by using flamelet generated manifold chemistry (FGM). The work has two main objectives: identify the influence of the Reynolds number on the oxy-syngas flame structure, and to clarify the local flame extinction of oxy-syngas and syngas-air flames at a higher Reynolds number.Two oxy-syngas flames at Reynolds numbers 3000 and 6000 and one syngas-air flame at Reynolds number 6000 were simulated. The scattered data, probability density function distributions and fully burning probability provide the local flame characteristics of oxy-syngas and syngas-air nonpremixed jet flames. It is found that the H2/CO oxy-syngas flame burns well compared to the syngas-air flame and the high Reynolds number causes more flow straining, resulting in higher scalar dissipation rates which lead to lower temperatures and eventually local flame extinction. The oxy-syngas flames burns more vigorously than the syngas-air flame with the same adiabatic flame temperature of approximately 2400K.

Modelling paradigms for MILD combustion

Three-dimensional Direct Numerical Simulation (DNS) data of methane-air MILD combustion is analysed to study the behaviour of MILD reaction zones and to identify a suitable modelling paradigm for MILD combustion. The combustion kinetics in the DNS was modelled using a skeletal mechanism including non-unity Lewis number effects. The reaction zones under MILD conditions are highly convoluted and contorted resulting in their frequent interactions. This leads to combustion occurring over a large portion of the computational volume and giving an appearance of distributed combustion. Three paradigms, standard flamelets, mild flame elements (MIFEs) and PSR, along with a presumed PDF model are explored to estimate the mean and filtered reaction rate in MILD combustion. A beta function is used to estimate the presumed PDF shape. The variations of species mass fractions and reaction rate with temperature computed using these models are compared to the DNS results. The PSR-based model is found to be appropriate, since the conditional averages obtained from the DNS agree well with those obtained using the PSR model. The flamelets model with MIFEs gives only a qualitative agreement because it does not include the effects of reaction zone interactions.