Flow Turbulence Combust (2014) 92:731–765 DOI 10.1007/s10494-013-9512-6

Lagrangian Intermittent Modelling of a Turbulent Lifted Methane-Air Jet Flame Stabilized in a Vitiated Air Coflow Ruben Mouangue · Marcel Obounou · Laurent Gomet · Arnaud Mura

Received: 11 June 2012 / Accepted: 26 August 2013 / Published online: 25 September 2013 © Springer Science+Business Media Dordrecht 2013

Abstract The present manuscript reports results of numerical simulations of turbulent lifted jet flames of methane with special emphasis placed on the autoignition effects. The impact of dilution with burned gases on the flame stabilization is analyzed under the conditions of a laboratory jet flame surrounded by a vitiated air coflow. In this geometry, mass flow rates, temperature levels and exact chemical composition of hot products mixed with air are fully determined. The effects of both finite rate chemistry and partially premixed combustion are taken into account within a Lagrangian intermittent framework. Detailed chemistry effects are incorporated through the use of chemical time scales, which are tabulated as functions of the mixture fraction. The concept of residence time (or particle age) and the transport equation for the mean scalar dissipation rate of the mixture fraction fluctuations, i.e.,  ε Z = ρ D (∂ Z  /∂xi )(∂ Z  /∂xi )/ρ, are also considered. This allows to improve the description of turbulent mixing including both the large scales engulfment processes through the consideration of the residence time scale, as well as the small scales molecular mixing processes the intensity of which is set by the integral scalar (mixing)

R. Mouangue (B) Department of Energetic Engineering, UIT, University of Ngaoundere, Ngaoundere, Cameroon e-mail: [email protected], [email protected] M. Obounou Department of Physics, University of Yaounde 1, Faculty of Sciences, Yaounde, Cameroon e-mail: [email protected] L. Gomet · A. Mura Institut Pprime, UPR 3346 CNRS, ISAE - ENSMA, Université de Poitiers, Poitiers, France L. Gomet e-mail: [email protected] A. Mura e-mail: [email protected]

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 2 / time scale τ Z = Z ε Z . Numerical simulations of the turbulent diluted jet flame of methane studied by Cabra and his co-workers are performed. The flame liftoff height is reasonably well-captured and the computational results display a fairly satisfactory level of agreement with experimental data. They also confirm that the consideration of an additional transport equation for the mean scalar dissipation rate may significantly improve the computational results. The investigation is supplemented by a sensitivity analysis of the scalar to turbulence time scale ratio Cmix = τ Z /τt often referred to as the mixing constant. Finally, the manuscript ends with the application of the proposed closure to experimental conditions which are characterized by variations of (i) the jet exit velocity, (ii) coflow velocity and (ii) coflow temperature. Keywords Turbulent combustion modelling · Non premixed combustion · Autoignition phenomena · Lifted flames · Partially premixed conditions

Nomenclature Latin letters aT C D1 , C D2 Cmix Cμ Cp C P1 , C P2 Dα , D d D  k L eα  Z) P(ψ  Y , ψZ ) P(ψ Pτ (ψτ ) Re Sct ui xi Y Z

thermal diffusivity (m2 s−1 ) modelling constants, production terms (SDR closure) scalar mixing constant Cmix = τ Z /τt modelling parameter ( k − ε closure) heat capacity (J kg−1 K −1 ) modelling constants, stretching / dissipation terms (SDR closure) molecular diffusion coefficients (m2 s−1 ) jet exit diameter (d = 0.00457 m) coflow diameter (D = 0.210 m)   turbulent kinetic energy (m2 s−2 ),  k = ρui ui /(2 ρ) Lewis number of chemical species α (L eα = aT /Dα ) mixture fraction probability density function joint scalar probability density function mixing time scale probability density function (s−1 ) Reynolds number turbulent Schmidt number (Sct = 0.7) velocity component in direction i (m s−1 ) cartesian coordinate (m) oxygen mass fraction mixture fraction

Subscripts and superscripts BUR IEM MIL MIX

related to the equilibrium lines related to IEM mixing lines [1] related to the MIL trajectory [2] related to the pure mixing line

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st J−, J+

related to stoichiometric conditions related to jump positions

Greek letters scalar dissipation rate or SDR (s−1 ), χ = D (∂ Z /∂xi )(∂ Z /∂xi ) turbulent dissipation rate (m2 s−3 ) turbulent SDR (s−1 ),  ε Z = ρ D (∂ Z  /∂xi )(∂ Z  /∂xi )/ρ turbulent viscosity (m2 s−1 ) density (kg m−3 ) sample space variable (ψ Z , ψY , and ψτ associated with Z , Y and τ ) Dirac delta function thermal conductivity (W m−1 K −1 ) chemical production / destruction rate of Y (s−1 ) exchange time scale involved in the IEM closure (s) age or internal age (s), see reference [4] chemical time scale (s), see reference [5] self-ignition time scale (s) residence time (s) [6] turbulent integral time scale (s), τt =  k/ ε  2 / scalar mixing time scale (s), τ Z = Z εZ

χ  ε  εZ νt ρ ψ δ λ ωY τ τage τchem τign τres τt τZ

Acronyms ADF CMC IEM ISAT LIF LES LMSE MIL PCM PDF RANS SDR SLFA TCI

Approximated Diffusion Flame Conditional Moment Closure Interaction by Exchange with the Mean value [1] In Situ Adaptive Tabulation Laser Induced Fluorescence Large Eddy Simulation Linear Mean Square Estimation [3] Modèle Intermittent Lagrangien [2] Presumed Conditional Moment Probability Density Function Reynolds Averaged Navier-Stokes Scalar Dissipation Rate Steady Laminar Flamelet Assumption Turbulence Chemistry Interaction

Operators φ  φ  φ ξk  φ

Reynolds Average Favre Average,  φ = ρφ/ρ  average weighted by the mass fraction of the tracer ξk ,  φ ξk = ξ k φ/ξk   Favre fluctuation, φ = φ − φ

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1 Introduction In many practical combustion systems fuel jets merge into a hot environment of oxidizer and combustion products. The internal energy of these hot products may help to stabilize the flame; they are also useful to dilute the reaction zones, which contributes to homogenize the mixture, thus avoiding high temperature levels that are known to be responsible for the largest part of nitrogen oxides emissions. Predictions of turbulent combustion in complex recirculating flows, which are subject to autoignition phenomena, still pose great challenges to available computational combustion tools. Therefore vitiated coflow jet flames provide a simple case of turbulent reacting flows within a hot environment characterized by lower levels of oxygen concentration. From the experimental point of view, the work performed by Cabra et al. [7, 8], with either a mixture of H2 /N2 or CH4 /air, is considered as a reference test case in the field. The experimental databases gathered by Gordon et al. [9] include temperature measurements, major species—with both fuel and oxidizer concentrations—which enable the mixture fraction level to be evaluated, and also intermediate species concentrations such as OH and CH2 O. This explains why it has been retained as an experimental benchmark in several previous studies. For instance, Masri et al. [10] computed the case of a H2 /N2 mixture, with detailed chemistry taken into account by the use of the ISAT method (In Situ Adaptive Tabulation) [11]. In this respect, the transported Probability Density Function (PDF) approach provides one among the reference methods to account for Turbulence Chemistry Interaction (TCI) and finite rate chemistry effects in general. It has been retained by Cao et al. [12] and Gordon et al. [13, 14] to simulate the jet flames resulting from the combustion of the two aforementioned mixtures. The corresponding studies also gathered useful information concerning the transport contributions (convection vs. diffusion). Domingo et al. [15] carried out a Large Eddy Simulation (LES) of the same test case and introduced an interesting closure for the subgrid scale scalar dissipation rate of the reactive species. The Conditional Moment Closure (CMC) framework was retained by Patwardhan et al. [16] who analysed the sensitivity of the pre-flame zone. Michel et al. [17] resorted to the flamelet model ADF-PCM with special emphasis placed on the ignition / extinction through the consideration of the PDF of the scalar dissipation rate (SDR) at the stoichiometry P(χst ) presumed from a log-normal distribution function. In this approach, the transient flame states (ignition/extinction) are obtained from Perfectly Stirred Reactor (PSR) calculations, and molecular diffusion effects are considered through the solution of a steady laminar flamelet assumption (SLFA) where the strain rate is the only controlling parameter of the one-dimensional diffusion flame structure. Finally, Ihme and See [18] performed the numerical simulation of the Cabra flame by using an unsteady flamelet / progress variable model in which all thermochemical quantities are parameterized by the mixture fraction, a (wellchosen) reaction progress parameter, and the SDR at stoichiometry χst . Lagrangian intermittent closures, i.e., Modèles Intermittents Lagrangiens (MIL), were introduced by Borghi and his coworkers [2, 19] to address TCI effects in non-premixed flames. The original closure was then extended to the consideration of multi-step chemistry by Obounou et al. [20] and found able to describe partial extinction and re-ignition effects. The closure was subsequently applied to bluffbody stabilized flames by Fallot et al. [21] and, more recently, to lifted turbulent

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subsonic jet diffusion flames by Mura and Demoulin [5]. The latest developments are concerned with turbulent reactive two-phase flows [22, 23] and combustion in high speed (supersonic) flows [24]. The main objective of the present study is to apply a recent extension of the MIL closure, where transport equations for the residence time of the fluid particles as well as the mean scalar dissipation rate of composition fluctuations are considered to improve the representation of spontaneous ignition and lift-off phenomena. The modelling developments are conducted within the Reynolds-Averaged NavierStokes (RANS) framework and the validation step relies on the Cabra et al. [8] experimental databases, which, as mentioned above, feature well-defined boundary conditions. The corresponding numerical simulations are based on two-dimensional axisymmetric computational geometries. The composition is described through the consideration of the mixture fraction variable Z , and a detailed sensitivity analysis to variations of the scalar to turbulence time scale ratio, i.e, the mixing constant Cmix = τ Z /τt , is carried out. In this respect, the mean SDR of composition fluctuations (denoted by  ε Z hereafter)—a measure of the mixing rate between fuel and oxidizer—can be evaluated either from the standard linear relaxation model, i.e.,  2 /(C  εZ = Z mix τt ), as done in previous applications of the MIL closure [2, 5, 20–25], or from the solution of a modelled transport equation for  εZ . The present manuscript is organized as follows: the experimental configuration is first presented in the next section. The main characteristics of the Lagrangian intermittent framework are subsequently recalled in the third section, where the sudden chemistry assumption, and its relevance to the present conditions, are examined in detail. This section also attempts to summarize the main extensions that have been brought to the closure since its early formulation. The computational results are gathered in the fourth section that includes several comparisons with available experimental data. Finally, a last section summarizes the main conclusions that can be drawn from the present study and also provides some prospects for future work.

2 Experimental Configuration The chemical composition of recirculating burnt gases as well as the associated mass flow rates still remain difficult to calibrate and to measure with a satisfactory level of accuracy in real combustion chambers. Therefore, to gain a better understanding of flame stabilization processes in environments where the influence of such burnt gases dominates, Cabra et al. [8] designed a laboratory vitiated burner, depicted in Fig. 1, in which the injection conditions of both fresh reactants and hot products are fully determined. It consists of a methane-air jet surrounded by a coflow of hot products that result from the combustion of a lean premixed hydrogen-air flame. The diameter of the central jet is d = 4.57 mm, while the coflow diameter is about D = 210 mm, i.e., much larger than the central jet, in such a manner that it prevents the central jet from dilution with the ambient air over a sufficiently long distance. This laboratory scale flame has been designed to obtain an experimental setup where auto-ignition effects could actively participate to the stabilization mechanism. As mentioned above, the burner was used to study two kinds of unconfined lifted flames: the main jet was either composed of hydrogen/nitrogen [7] or methane/air [8]. The fuel jets are injected in a hot coflow made up of a vitiated air mixture: the

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Fig. 1 Sketch of the experimental setup retained for validation, from Cabra et al. [26]

burner consists of a perforated plate surrounding the central injection pipe where a large number of lean premixed H2 /air flames are anchored. This produces the high temperature mixture made up of air and water vapor. For the methane/air flame, the jet is issued in this hot environment 70 mm above the perforated plate. Experimental conditions for the methane/air configuration of reference are summarized in Table 1. Experimental data gathered by Cabra et al. [8] for the methane/air case are now briefly presented. Multiscalar one-point measurements were performed using the Raman-Rayleigh scattering technique in conjunction with Laser-Induced Fluorescence (LIF). Mass fractions of CH4 , O2 , CO, CO2 , H2 , OH and temperature have been collected. These data have been made available for the researchers community and can be downloaded from a website [26]. These measurements have been published in two papers [7, 8] and in a technical report [27]. Other works have resorted to Cabra-like burners with a central jet and a vitiated coflow to reproduce combustion under burnt gases dilution. A typical example is a burner that is functionally similar to the Cabra burner but only geometric dimensions are slightly different [9, 28]. The investigations conducted on this burner gave rise to conclusions similar to those previously obtained on the Cabra burner. Finally, the recent work of Gordon et al. [9] based on two-dimensional imaging of CH2 O and OH applied to the methane/air configuration clearly assessed the presence of auto-ignition events in

Table 1 Experiments of Cabra et al. [8]: reference operating conditions (Re: Reynolds number, d and D: diameters, U: exit velocity, T: temperature, Xα : mole fraction of species α)

Fuel jet Coflow

Re

d and D (mm)

U (m/s)

T (K)

X O2

X N2

X H2 O

XC H4

28,000 23,300

4.57 210

100 5.4

320 1350

0.15 0.12

0.52 0.73

0.0029 0.15

0.33 0.0003

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this flame, which therefore appears as a well-suited test case to assess the modelling of such effects.

3 Turbulent Combustion Closure The description of non premixed turbulent reacting flows in the conditions described above obviously requires an appropriate modelling. For the purpose of the present study, the MIL framework is retained to address the possible competition that may exist between chemical reaction and scalar mixing. The corresponding closure has been previously established to be robust, easy to implement, and it features small computational costs in such a manner that it has been retained for a large variety of situations including standard non-premixed flames [20, 21], two-phase flows combustion [22, 23], and more recently combustion in non premixed high-speed flows [24, 29]. The MIL model is a Lagrangian model in the composition space which is based on the knowledge of two scalar variables, namely the mixture fraction Z and the mass fraction of a reactive species Y which follows the progress of the chemical reaction. The oxygen mass fraction has been retained as Y in the present study. The MIL model relies on the sudden chemistry assumption [20]: it is assumed that Lagrangian particles first need a finite time to ignite and, after ignition occurred, they instantaneously reach chemical equilibrium conditions. This permits a strong but clearly stated functional dependence between the two scalars to be introduced: Y =  Y , ψ Z ; x, t) Y MIL (Z ). As a result, the joint scalar probability density function1 P(ψ can be simply expressed from the knowledge of the marginal mixture fraction PDF  Y , ψ Z ; x, t) = P(ψ  Z ; x, t))P(ψY |ψ Z ; x, t) where the  Z ; x, t), since we have P(ψ P(ψ conditional PDF is obtained by considering the MIL trajectory in the composition space: P(ψY |ψ Z ; x, t) = δ(ψY − Y MIL (ψ Z )). A classical β-function is retained to  Z ; x, t). The determination of the estimate the marginal mixture fraction PDF P(ψ conditional PDF P(ψY |ψ Z ; x, t), i.e., the Lagrangian trajectory as given by δ(ψY − Y MIL (ψ Z )), is discussed below. 3.1 Lagrangian pathlines in the composition space In a two-feeding streams system, it is classical to introduce the so-called mixture fraction variable Z , which is defined as the local ratio of the mass flux originating from the fuel feeding stream to the sum of both mass fluxes (fuel and oxidizer streams). In this manner, the scalar variable Z remains bounded between zero (in the oxidizer) and unity (in the fuel). From the classical principles of mass conservation, this variable is a conserved scalar which follows an advection-diffusion equation   ∂ρ Z ∂Z ∂ρui Z ∂ + ρD (1) = ∂t ∂xi ∂xi ∂xi

quantities ψY and ψ Z denote the sample space variables associated with the oxygen mass fraction Y and mixture fraction Z , respectively. 1 The

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Equation 1 can be obtained from the transport equation for the mass fraction of any chemical species α   ∂ρYα ∂ρui Yα ∂ aT ∂Yα + = + ρ ωYα (2) ρ  

∂t

∂x ∂x L eα ∂xi     i  k 

(IV) (I)

(II)

(III)

where Fickian diffusion has been assumed and Lewis number effects not considered, i.e., L eα = 1, in such a manner that the molecular diffusivity of any species is such that Dα = D = aT , with aT = λ/(ρC p ) the thermal diffusivity defined as the ratio of the thermal conductivity λ to the product of the density ρ and heat capacity C p . The above transport equation for the species mass fraction contains classical contributions associated with temporal variation (I), convection (II), molecular diffusion (III), and a chemical production / destruction source term (IV). The sum of the two contributions that appear in the Left Hand Side (LHS) corresponds to a Lagrangian or total derivative in such a manner that, retaining a Lagrangian point of view, i.e., following the evolution of fluid particles in the physical space, the composition will evolve only through molecular diffusion effects (III) and chemical reaction (IV) which act on the Right Hand Side (RHS) of Eq. 2. In the following, the molecular diffusion term (III) that involves two-point information will be represented thanks to the Interaction by Exchange with the Mean value (IEM) or Linear Mean Square Estimation (LMSE) closure introduced in the early studies of Villermaux and Devillon [1] and Dopazo and O’Brien [3], in such a manner that the Lagrangian system associated with the set of Eqs. 1 and 2 writes −Z −Y dZ dY Z Y = = and + ωY dt τZ dt τY

(3)

where τ Z and τY denote the characteristic mixing time scales of the mixture fraction and the oxygen mass fraction, respectively. Following reference [30], the above system of Eq. 3 is strictly equivalent to the following transport equation    − ψZ  − ψY d ∂ ∂ Z Y   P(ψ Z , ψY ; t) − P(ψ Z , ψY ; t) P(ψ Z , ψY ; t) = − dt ∂ψ Z τZ ∂ψY τY −

 ∂  Z , ψY ; t) ωY (ψY , ψ Z ) P(ψ ∂ψY

(4)

 Z , ψY ; t). Such relationships between Lagrangian written for the joint scalar PDF P(ψ equations, second-order moments as well as modelled PDF transport equations have been extensively discussed in reference [31]. Under the sudden chemistry hypothesis, the solution of Eq. 3 leads to the MIL trajectory in the composition space, i.e., it is considered that particles need a finite  , Y)  but then time to ignite during their evolution towards the mean composition ( Z instantaneously jump from the IEM mixing line to the equilibrium line, as depicted in Fig. 2. On the IEM lines, the reaction rate cancels, and the set of Eq. 3 can be written as follows:  − Y) dY (Y =  dZ (Z − Z )

(5)

Flow Turbulence Combust (2014) 92:731–765 Fig. 2 Representation of the MIL trajectory (thick lines) in the plane (Z ,Y). Note that, for the conditions of Cabra experiments, Y max = Y MIX (1)

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Pure mixing line Y MIX(Z) IEM mixing lines Y IEM(Z)

Y /Y max

Equilibrium lines Y EQU(Z) B

MIL pathline Y MIL(Z)

1

F

A D

δ YZ −

δ YZ +

E

Y

M(Z,Y )

G

C

ZJ-

Zst

Z

H

ZJ+

Z

which corresponds to the branches AM and BM in Fig. 2. It has been implied that τ Z = τY = τ but this is not restrictive since, before combustion occurs, fluctuations of concentration are related to mixture fraction fluctuations, and the differences between the corresponding mixing time scales, τ Z and τY , are therefore expected to remain negligible. The following expressions hold for the IEM trajectory  Y

IEM

(Z ) =

 /Z , Y MIX (0) − (Y MIX (0) − Y)Z

 if Z < Z

 − Z )/(1 − Z  ), otherwise Y MIX (1) − (Y MIX (1) − Y)(1

(6)

  where Y MIX (Z ) = Y MIX (0) + Z Y MIX (1) − Y MIX (0) denotes the pure mixing line AB. On the equilibrium lines, see Fig. 2, as obtained from a single step chemistry description, i.e., considering fully burned products of combustion, we simply have  Y

BUR

(Z ) =

Y MIX (0)(1 − Z /Z st ), if Z < Z st 0, otherwise

(7)

where Z st denotes the mixture fraction value at stoichiometry. This corresponds to segments AC and HC respectively. Finally, as emphasized above, within the MIL representation, it is considered that particles need a finite time before ignition occurs. They first evolve along the IEM trajectory (segments AD and BF) then, following molecular mixing, ignition occurs (points D and F), and particles reach and follow the equilibrium lines (segments DE and FG). This results in the MIL pathline depicted in Fig. 2, which is considered as

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made up of a combination of straight line segments in the composition space. The corresponding trajectory is given by ⎧ IEM (Z ), if Z < Z J− ⎨Y Y MIL (Z ) = Y BUR (Z ), if Z ∈ ]Z J− , Z J+ [ (8) ⎩ IEM (Z ), if Z > Z J+ Y The values Z J− and Z J+ represent the values of the mixture fraction Z that are associated with the ignition of particles issuing from either the oxidizer (Z = 0) or the fuel stream (Z = 1) denoted by A and B respectively in Fig. 2. These values are determined dynamically, at each instant, from a direct comparison performed between flowfield and chemical time scales, i.e., as the roots to τchem (Z ) = τ , which is performed in each cell of the computational domain. The corresponding chemical time scale values τchem (Z ) have been evaluated beforehand, for each mixture fraction value Z , by integrating the oxygen chemical consumption rate ωY from Y MIX (Z ) (fresh reactants) to Y BUR (Z ) (fully burned products), see reference [5]. These are the resulting values of 1/τchem that are tabulated as functions of the mixture fraction, and this differs from standard strategies that often tabulate the whole flamelet structure. The determination of the two values Z J− and Z J+ will be discussed in further details  Y , ψ Z ; x, t) = P(ψ  Z )δ(ψY − in Section 3.4. Once they are known, the joint PDF P(ψ MIL  Y (ψ Z )) is fully determined with P(ψ Z ) presumed to be a β-function. 3.2 Mixture fraction PDF and moments The mixture fraction PDF is presumed by using a β-function, the shape of which is fully determined provided that both the mean and variance values of the mixture fraction field are known, see Appendix. Mean and variance are calculated through the numerical solution of the following set of modelled transport equations [32]:   ∂ ν ∂ Z ∂  t −ρ ρZ + ρ ui Z =0 (9) ∂t ∂xi Sct ∂xi   2    ∂Z  ∂ ∂ Z ν νt ∂ Z ∂   t  2  2 − ρ + = 2ρ ρZ ρ ui Z − 2ρ εZ (10) ∂t ∂xi Sct ∂xi Sct ∂xi ∂xi The above equations are written in an unsteady form and contain convection, diffusion, production and dissipation contributions. The molecular diffusion terms have been neglected with respect to their turbulent counter-parts. The corresponding turbulent transport terms are closed using the turbulent eddy viscosity approximation νt = Cμ k2 / ε, with the turbulent Schmidt number set to its usual value of Sct = 0.7 [33]. The realizable  k − ε closure is retained to evaluate these contributions. The mean dissipation rate  ε Z of the scalar variance can be estimated from the standard  2 /(C linear relaxation law, i.e.,  εZ = Z mix τt ), based on the integral turbulence time scale τt , or by using a modelled transport equation for this quantity, which may appear as a more general framework. Transport equation of the mean scalar dissipation rate Among the various available forms of modelled transport equations for the mean scalar dissipation rate (SDR),

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see for instance Sanders and Gökalp [34] for a review, we retain the following one which has been introduced in the early study conducted by Mantel and Borghi [35]:  2     ρ νt ∂ εZ ρ νt ∂ Z εZ ∂ ∂ ∂ + C P1 εZ ) + ui εZ ) = (ρ (ρ  ∂t ∂xi ∂xi Sct ∂xi Sct ∂xi k       uj ∂ uj ∂ ui 2   ε Z ∂ ∂ ui + C P2 ρ νt k − νt + + δij  ∂x j ∂xi 3 ∂xi k ∂xi + C D1

ρ εZ ρ εZ − C D2 τt τZ

(11)

In the above equation, the modelling constants C P1 and C P2 have been set to unity as previously proposed by Gomet et al. [36]. The second and third terms correspond to production terms associated with mean composition and mean velocity gradients, respectively. The last two terms are turbulent stretching production and molecular dissipation terms. Computational results obtained through the solution of the above equation will be compared hereafter with those resulting from an optimal choice of the mixing constant value Cmix .  2 /τ , based Linear relaxation model A standard linear relaxation law, i.e.  εZ = Z Z on the scalar dissipation time scale τ Z can be easily recovered from a small scale equilibrium assumption between turbulent stretching production and molecular dissipation in the above Eq. 11. Hence, with the equilibrium condition, see for instance  2 /(C [37], we obtain the linear relaxation closure εZ = Z mix τt ) with Cmix = C D2 /C D1 the proportionality constant between the two characteristic integral time scales τ Z and τt . 3.3 Mean production rates  , Y)  is From the above description, and provided that the mean composition ( Z known, the instantaneous reaction rate can be expressed as a function of τ , Y and Z only, i.e., ωY = ωY (τ, Z , Y), and its expression is easily derived from Eq. 3    − Y)  − Z ) dY (Y (Z − (12) ωY (τ, Z , Y) =  − Z) τ dZ (Z where, as emphasized above, the mixing (or exchange) time scales τY and τ Z have been supposed equal and are therefore indiscrimately denoted by τ . Moreover, since the consideration of the MIL trajectory introduces a direct relationship between Y and Z , i.e., Y = Y MIL (Z ), the instantaneous chemical rate becomes a function of τ and Z only, i.e., ωY = ωY (τ, Z ), and its average value is obtained from  1 ∞  τ , ψ Z )dψτ dψ Z ωY =  ωY (ψτ , ψ Z ) P(ψ (13) 0

τmin

The lower bound for the mixing time scale integration, i.e., τmin , might be set to zero, but this would imply useless numerical integrations since the chemical source term remains zero as long as τ < min(τchem (Z )). As a consequence, a better choice consists in retaining τmin ≡ min(τchem (Z )) for the lower bound of integration, as suggested in the early studies of Borghi and Gonzalez [2] or Obounou et al. [20].

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Statistical independence is assumed between the mixing time scale τ and the  τ , ψ Z ) = Pτ (ψτ ) P(ψ  Z ), where Pτ denotes the mixing mixture fraction Z , i.e., P(ψ time scale PDF. A scale-separation argument is often invoked to justify such an assumption of statistical independence between the mixture fraction and micromixing time scale (or SDR) fluctuations [36]. This finally leads to   1  ∞  Z )dψ Z ωY =  ωY Pτ (ψτ )dψτ P(ψ (14) 0

τmin

In reference [2], the mixing time scale PDF, i.e., Pτ (ψτ ), has been assimilated to the residence time scale distribution obtained in a well-stirred reactor. In the  2 / following, it is approximated using Pτ (ψτ ) = δ(ψτ − τ ) with τ = τ Z = Z ε Z the scalar integral time scale value, i.e., we will neglect the local variations of the mixing time scale. Such a simplification should be less arbitrary than any other a priori choice of the mixing time scale PDF shape. Last but not least, this choice has been previously found sufficient to obtain a satisfactory representation of the physical phenomena, see for instance [5, 24]. On the MIL trajectory, the reaction rate appears as the sum of three contributions Z

Z

ωY (τ, Z ) = ωYIEM (τ, Z ) + ωY J− (τ, Z ) + ωY J+ (τ, Z ) + ωYBUR (τ, Z )  

 



 mixing

jumps at ZJ− and ZJ+

(15)

burned gases

As long as the mixture (Z , Y) evolves along the IEM-LMSE mixing lines AD and BF, which are characterized by Eq. 6, it is trivial to see that the use of Eq. 6 in Eq. 12 gives zero ωYIEM (τ, Z ) ≡ ωY (Y IEM (Z )) = 0. Molecular mixing takes place without any significant chemical reaction. At the jump locations DE and FG, see Fig. 2, it can be shown that the contributions to the chemical reaction rate are given by    − Y Z J− )  − Z J− ) δY Z J− (Y (Z Z J− ωY (τ, Z ) = (16) −  − Z J− ) τ δ Z J− (Z

Z ωY J+ (τ,

 − Z J+ ) (Z Z) = τ



 − Y Z J+ ) (Y δY Z J+ −  − Z J+ ) δ Z J+ (Z

 (17)

The MIL framework considers the limit δ Z J+ → δ(Z − Z J+ ) and δ Z J− → δ(Z − Z J− ) and, in the above expressions, δY Z J− and δY Z J+ denote, respectively, the differences between the IEM and the equilibrium values of the oxygen mass fraction at the jump locations Z J− and Z J+ [5]. Once weighted by the mixture fraction PDF, and using standard rules associated with the integration of distribution functions [38], Eqs. 16 and 17 give rise to Z  J− )δY Z J− ( Z  − Z J− )/τ  ωY J− = P(Z

(18)

Z  J+ )δY Z J+ ( Z  − Z J+ )/τ  ωY J+ = P(Z

(19)

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743

which expresses that a jump contribution to the mean chemical production rate is nothing but the product of local molecular flux of reactants at the jump location,  − Z J− )/τ or ( Z  − Z J+ )/τ , with the probability density function P(Z  J− ) or i.e., ( Z  J+ ), and the amplitude of the chemical conversion to be performed from the P(Z IEM state to the fully burned products state (δY Z J− or δY Z J+ ). Finally, the mean reaction rate will be obtained from the above contributions  Z J−  1 IEM   Z )dψ Z  ωY = ωY (τ, ψ Z ) P(ψ Z )dψ Z + ωYIEM (τ, ψ Z ) P(ψ 0 Z J+  

YIEM =0 ω

+

 − Z J− )  (Z  J− )δY Z J− + ( Z − Z J+ ) P(Z  J+ )δY Z J+ P(Z τ τ  

Z

 + 

Z

ωY J− +  ωY J+ Z J+ Z J−

 Z )dψ Z ωYBUR (τ, ψ Z ) P(ψ 

(20)

ωYBUR 

where the expression of ωYBUR (τ, ψ Z ) is given by Eq. 12 specialized to equilibrium; since we have Y BUR (Z ) = 0 and dY BUR (Z )/dZ = 0 on the segment GC, then  for Z ∈ [Z st ; 1], while on segment EC, since dY BUR (Z )/dZ = ωYBUR (τ, Z ) = −Y/τ   MIX  ) /τ Z st for  Z st − Y MIX (0)(Z st − Z −Y (0)/Z st , we obtain ωYBUR (τ, Z ) = − Y Z ∈ [0; Z st ]. 3.4 Discussion of the modelling proposal The above closure implies the sudden chemistry assumption and the purpose of the present section is to discuss its relevance for the present conditions. In practice, the sudden chemistry assumption retained above does not necessarily imply fast chemistry and, in this respect, it has been previously retained to describe non premixed turbulent lifted flames [5]. However, it is also clear that the above modelling picture provides only an approximate representation of TCI effects, and it would be therefore highly desirable to obtain some further assessments of its representativeness, even from a qualitative point of view or in simplified situations. In a first step of this preliminary check of the sudden chemistry assumption, selfignition calculations are conducted in the conditions of the Cabra experiments. These computations are performed with the SENKIN package [39] of the CHEMKIN2 Library [40]. To this purpose, the detailed chemical kinetics scheme GRI 3.0 is used to describe the oxidation of methane [41]. The calculations are performed as follows: the pressure is held constant and equal to p = 0.1 MPa. The initial values of the other quantities are parameterized as functions of the mixture fraction Z only, which is settled to be zero in the hot coflow and unity in the fuel injection stream: Z = (Z − Z coflow )/(Z fuel − Z coflow ). In this expression, Z is defined by Z = 2(M O2 /MCH4 )YCH4 − Y O2 . The initial composition is evaluated from this mixture fraction variable as follows Yα (Z , t = 0) = Z Yαfuel + (1 − Z ) Yαcoflow , and the enthalpy of the mixture is given by H(Z ) = Z H fuel + (1 − Z ) H coflow which allows to initialize the temperature level T0 (H, Yα ). It is worth noting that, as suggested in

744

Flow Turbulence Combust (2014) 92:731–765 500 400

1/ τ ign (s− 1 )

1/ τ ign (s− 1 )

400 300 200 100 0

300 200 100

0

0.2

0.4

0.6

0.8

1

894 700 T (K)

512

320

0

ZMR 0

0.05

0.1 Z

0.15

0.2

1350

1285

1223 T (K )

1163

1106

Z

1355

1106

Zst

Fig. 3 Left: self-ignition delay plotted as a function of the mixture fraction and its associated level of temperature. Right: detailed view in a mixture fraction domain restricted to [0:0.2]; the positions of Z MR and Z st are delinated by dotted and dashed lines respectively

the early computational studies reported in [8, 13], a small amount of OH radicals, X OH = 0.0015, has been considered to mimic the influence of the pool of radicals present in the mixture, see reference [13] for further details. Figure 3 illustrates the sensitivity of the resulting self-ignition delay to the mixture fraction Z , and initial temperature T0 . The resulting ignition time scale values are tabulated as a function of the mixture fraction only: τign (Z ). Clearly, the chemical response (chemical time scale) takes the stream temperature influence into account. Indeed, in the mixture fraction space, the minimal value of the self-ignition time scale is displaced from nearly stoichiometric conditions (Z st = 0.177) toward the preheated stream condition (Z = 0), see the right side of Fig. 3. The corresponding mixture fraction value will be denoted by Z MR in the following, since it may be associated to the most reactive mixture fraction value, see for instance [42]. On the left side of Fig. 4 is reported the temporal evolution of temperature T(t) and oxygen mass fraction Y(t) for Z = Z st . The self-ignition time τign (Z ) is compared to the characteristic time τ , required to reach the burned gases state, see Fig. 4. The latter quantity, i.e. τ , is found clearly smaller that the time requested for ignition to occur, i.e., τign . From the calculations performed for varying values of the equivalence ratio, the ratio τ/τign are obtained and plotted on the right side of Fig. 4. The obtained results show that this ratio thus remains bounded between 0 and 0.05. Whatever the value of the equivalence ratio, the time needed for the mixture to ignite remains significantly larger than the one required to reach chemical equilibrium, which provides some support to the use of the sudden chemistry assumption for the present conditions. In a second step of the analysis, we focus our attention on Partially Stirred Reactor (PaSR) conditions. It is worth noting that, in contrast with the above selfignition calculations performed in homogeneous conditions, PaSR computations may account for mixture fraction fluctuations. To this purpose, we consider the Monte Carlo simulation of the following transport equation for the composition PDF α=N α=N   ∂   in (ψ) − P(ψ) d φα − ψα    ∂  P  − ωφα (ψ) P(ψ) P(ψ) − P(ψ) = dt τres ∂ψα τφα ∂ψα α=1 α=1

Flow Turbulence Combust (2014) 92:731–765

T (K)

Δτ

τign

Y

745

Δ τ / τ ign

2500

0.25

0.06

2200

0.2

0.05

1900

0.15

1600

0.1

1300

0.05

0.04 0.03

1000

0

0.05

0.1

0.15

0.2

0.02 0.01

0 0.25

0

0

0.2

τ age (s)

(a) Temporal evolutions of temperature and oxygen mass fraction during the self-ignition of a stoichio-

0.4

0.6

0.8

1

Z

(b)

Plot of the normalized ratio Δτ / τign as a function of

the mixture fraction

metric mixture (Cabra conditions)

Fig. 4 Inspection of the sudden chemistry assumption from self-ignition calculations

This equation is equivalent to Eq. 4 but may handle a detailed representation of chemical kinetics. It describes the evolution of the joint scalar PDF in a PaSR, see for instance [43, 44]. The composition vector φ is associated with the species mass fractions Yα (α = 1, ..., N), while the vector ψ denotes the sample space associated to the random vector φ. Its dimension N depends on the detailed kinetics scheme under consideration, the GRI mechanism in the present case [41]; it involves N = 53 chemical species, and features 325 elementary reaction steps. In accordance with Eq. 3, the IEM-LMSE closure has been retained for the micromixing terms. In the above equation, τres is the mean residence time scale inside the reactor (or average particle age) and τφα is the mixing time scale associated with the scalar φα . We consider that its value is the same for all scalar species namely τφα = τ . in (ψ) denotes the joint scalar PDF at the inlet of the reactor, and if the Finally, P reactor is fed with one stream of oxidizer and one stream of fuel, it is given by in (ψ) = M δ(ψ − φ(Z = 0)) + (1 − M) δ(ψ − φ(Z = 1)), with M the inflow mass P fraction of the oxidizer stream, the quantities φ(Z = 0) and φ(Z = 1) denoting the composition in the oxidizer and fuel stream respectively. In the following, we retain the inlet conditions associated with the Cabra flame, see Table 1, and the Monte Carlo simulation is carried out with 500 stochastic particles. The results reported in Fig. 5 correspond to the steady-state solution of the above PDF transport equation obtained for three distinct sets of values of the the residence and mixing time scale. On the left, τres = 5.0 10−4 s and τ = 5.0 10−4 s have been set. This corresponds to turbulent combustion featuring a significant level of composition  ,Y)  is remote from the fluctuations, and the location of the mean composition ( Z −3 calculated Lagrangian trajectory. For τres = 5.0 10 s and τ = 5.0 10−4 s (middle of Fig. 5), the residence time value is rather large in comparison with the scalar mixing time scale, and the resulting level of residual fluctuations is quite small. As a result, the behaviour of the PaSR becomes similar to the one of a Perfectly Stirred Reactor (PSR). Combustion is mainly driven by laminar chemistry, and the notion of jump

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τres = 5.0 10− 4 s

τres = 5.0 10− 3 s

τ = 5.0 10− 4 s

Y

τ = 5.0 10− 4 s

τ = 5.0 10− 3 s

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

Z

T (K)

τres = 5.0 10− 3 s

0.6

0.8

1

0

2500

2500

2000

2000

2000

1500

1500

1500

1000

1000

1000

500

500

500

0

0

0.2

0.4

0.2

0.4

0.6

0.8

1

0

0.2

Z

0.4

0.6

0.8

1

0

0

0.2

Z

Δτ

0.6

0.8

1

0.6

0.8

1

Z

2500

0

0

Z

0.4

Z

Δτ

Δτ

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05

0.05

0

0

Y

0

0.0005

0.001

t ( s)

0.0015

0.002

0

0.0005

0.001

t ( s)

0.0015

0.002

0

0

0.005

0.01

0.015

0.02

t ( s)

Fig. 5 Lagrangian trajectory in the (Y-Z ) composition space (top), in the (T-Z ) composition space (middle) and typical temporal evolutions of the oxygen mass fraction (bottom) obtained from PaSR computations conducted for different values of the turbulent mixing time τ and residence time τres . IEM-LMSE trajectories are depicted in black dashed lines

from the (turbulent) mixing line to the equilibrium line becomes less representative. The MIL closure is not suited to such quasi-laminar situations, which do not display any significant fluctuations of composition. In constrast with the previous situations, jump locations are more easily delineated from the last set of results, which is associated to τres = 5.0 10−3 s and τ = 5.0 10−3 s (right side of Fig. 5). Here, this corresponds to turbulent combustion featuring a significant level of composition  ,Y)  is again remote from fluctuations, and the location of the mean composition ( Z the calculated Lagrangian trajectory. In comparison with the situation associated with τres = 5.0 10−4 s and τ = 5.0 10−4 s (left), it is clear that the flammable domain is wider in the mixture fraction space. This figure tends to confirm that the MIL skeleton provides a reasonable approximation of the PDF transport equation when specialized to the case of a Partially Stirred Reactor (PaSR). Finally, for each among the three conditions that were investigated, typical Lagrangian trajectories (t, Y(t)) have been reported at the bottom of Fig. 5. The reported temporal evolutions Y(t) provide some further assessement of the sudden chemistry assumption since τ remains again clearly smaller than the time required for the particle to ignite.

Flow Turbulence Combust (2014) 92:731–765

747

The above investigation thus provides some support to the MIL picture in comparison with the solution of the joint scalar PDF transport equation considered as a reference to deal with such TCI conditions. However, the above investigation also shows that, for τ = 5.0 10−4 s, variations of τres from τres = 5.0 10−3 s (middle of Fig. 5) to τres = 5.0 10−4 s (left side of Fig. 5) significantly impacts the obtained results, which emphasizes the importance of residence time representation and how it is intimately connected to the description of self-ignition phenomena. The next section is devoted to these two peculiar aspects. 3.5 Extension to the description of self-ignition The MIL closure was initially developped for steady-state turbulent non premixed flames. Chemical reaction and molecular diffusion, i.e., micromixing, as well as their possible competition—which may lead to local extinction—were therefore retained as the leading phenomena. In this manner, the model undoubtedly accounts for the competition between the SDR—through the associated characteristic time scale τ — and chemistry. In this respect, it is noteworthy that previous investigations have confirmed its ability to recover standard responses with, for intance, a S-shaped dependence of the progress variable value2 as a function of the scalar mixing frequency 1/τ [45]. However, for turbulent partially premixed jet flames featuring hot coflowing oxidizer streams with high temperature levels, this picture may be significantly altered by self-ignition processes. Indeed, in order for combustion to take place, it is no longer subject to the prerequisite presence of burned gases, as it is within elemental flamelet structures, and self-ignition description, as well as residence times representation, may therefore play a crucial role to describe such conditions. In the present work, the above features are taken into account through the identification of a self-ignition domain within the mixture fraction space. This domain is delineated by performing a direct comparison between the self-ignition delay, denoted by τign (Z ), which is tabulated as a function of the mixture fraction, and a characteristic time scale associated to the convection of fluid particles (or particle age) to be defined later on. The relevance of the latter has been already addressed for turbulent premixed flames by Bilger [46]. It seems worth noting that such a notion of residence time, or particle age, is naturally embedded within the Lagrangian Monte Carlo PDF approach. Unsteady numerical simulations may also account for these effects through suited representations, as it is in the recent LES closure proposed by Enjalbert et al. [47]. Such a convection time scale or particle age can be thought as the time needed by a Lagrangian particle issued from either Z = 0 (oxidizer stream) or Z = 1 (fuel stream) to reach a given value of the mixture fraction Z . It is evaluated from

 ξcoflow   /( Z  − Z) + τage (Z ) = τ ln Z , if Z < Z (21) τres

 ξ  − 1)/( Z  − Z) +  τage (Z ) = τ ln ( Z τresfuel , if Z > Z (22) In the above set of equations, consistently with the IEM-LMSE framework, the first term in the Right Hand Side (RHS) represents the time needed by a particle issued

2 In

a given range of mixing time scales, there is three possible values for the progress variable Y.

748

Flow Turbulence Combust (2014) 92:731–765

from Z = 0 or from Z = 1 to reach the mixture fraction value Z in the considered computational cell, while the second term in the RHS represents the average residence time needed by a Lagrangian particle issuing from either the oxidizer ξcoflow ξfuel stream,  τres , or the fuel stream,  τres , to reach the corresponding computational cell. In contrast with our previous applications of the MIL closure to combustion in high speed flows [29], this latter contribution cannot be neglected any longer in subsonic reactive flow featuring self-ignition effects. In order to evaluate this contribution, a transport equation for the quantity of residence time is considered   ∂ ∂ νt ∂ ξk ξk ξk  ξk  ξk  = ρ ρ ξk  τres + ρ ui τres −ρ τres ξk (23) ∂t ∂xi Sct ∂xi ξk = ρξk /ρξk = ρ ξ where the following notation is used:  ξk = ξ ξk . k /ρ k / The above equation relies on the work of Ghirelli and Leckner [48] but the basis for its derivation was described in the early analyses of Sandberg, see for instance [4, 49]. Once divided by ρ ξk , with  ξk corresponding to the average mass fraction of a passive tracer associated with either oxidizer or fuel injection streams, the quantity obtained from the above equation provides the values of the corresponding mean ξcoflow ξfuel residence time scale,  τres or  τres . The transport equation for the passive tracers  ξcoflow and  ξfuel is given by   ξk ∂  ∂ νt ∂ ξk − ρ =0 (24) ρ ξk + ρ ui ∂t ∂xi Sct ∂xi The comparison of the tabulated chemical induction time scale τign (Z ) with the residence time scale (or particle age τage (Z )) allows to delineate a possible selfignition domain in the mixture fraction space, as shown in Fig. 6. In this figure, various profiles of chemistry and flow time scales are illustrated. The intersections of the chemical time scales curves with those associated with flow time scales will provide the mixture fraction values associated with the jumps positions. For Fig. 6 Synoptic outline of the determination of the jumps positions from characteristic time scales direct comparisons: non premixed flame (Z J− and Z J+ ), and self-ignition (Saa− and Saa+ )

1/ τage 1/ τign

1/ τ 1/ τchem

Saa-

ZJ-

ZJ+ Saa+

Flow Turbulence Combust (2014) 92:731–765

749

instance, the intersections between the dashed U-curve, that corresponds to the inverse of the particle age values τage , as obtained from Eqs. 21 and 22, and the solid-curve associated with the self-ignition delay τign , will determine the autoignition jumps positions Saa− and Saa+ , which are simply obtained as the roots to τage (Z ) = τign (Z ). The depth of the resulting self-ignition domain delineated by the difference between Saa+ and Saa− will increase with residence times and resulting particle ages. Jumps associated with the standard MIL closure are determined by the intersection of the solid straight line (1/τ ), with the dotted curve, associated with the chemical time scales τchem . Clearly, in the present modeling proposal, the tabulated chemical scale takes the stream temperature influence into account. Indeed, as emphasized in the previous section, in the mixture fraction space, the minimal value of the chemical time scale may be displaced from nearly stoichiometric conditions toward the composition of a preheated stream. Finally, the MIL trajectory Y MIL (Z )  ) and Z J+ = max(Z J+ , Saa+ , Z  ), is delineated by retaining Z J− = min(Z J− , Saa− , Z which are subsequently used to evaluate the mean chemical rate  ωY from Eq. 20.

4 Results and Discussion 4.1 Description of the computational details The entire model has been implemented into the commercial CFD code Fluent [50] through User-Defined Functions (UDF) [51], see Fig. 7. The chemical time scales They are stored in τchem (Z ) and τign (Z ) are read as databases  at the first iteration.    the form of two two-dimensional tables (Z )i Z , (1/τchem )i Z and (Z )i Z , (1/τign )i Z , with (i Z = 1, N Z = 100), which have been discretized using a non uniform mesh refined around stoichiometry. Seven User-Defined Scalars (UDS) have been defined ξ  2 (Eq. 10),   (Eq. 9), Z to handle the transport equations of Z ε Z (Eq. 11),  τrescoflow and ξfuel  τres (Eq. 23), and associated tracers  ξcoflow and  ξfuel (Eq. 24). The value of  ξcoflow is set to unity at the coflow inlet, and zero at the jet inlet, while a zero-flux condition is set on the other boundaries of the computational domain. Similar conditions have been retained for  ξfuel except that it is defined to be unity at the jet inlet, and zero at the coflow inlet.3 As displayed in Fig. 7, the CFD solver feeds the MIL model with (i) the  , Y),  (ii) the characteristic mixing time scale τ as obtained from mean composition ( Z ξ ξ a realizable  k − ε closure, (iii) the residence time scales  τrescoflow ,  τresfuel and associated  2 . The first  and variance Z tracers  ξcoflow ,  ξfuel , (iv) the mixture fraction mean Z two moments of the mixture fraction statistics are used to perform the numerical integration of the chemical source term as given by Eq. 20, see Appendix for further details. The resulting quantity  ωY is provided by the MIL model to the CFD solver as an input for the transport equation of the average oxygen mass fraction. The two-dimensional axisymmetric computational domain is reported in Fig. 8. The corresponding structured mesh is divided into two sub-domains so as to refine locations of interest such as the one associated with the sheared and mixing layer development. Three distinct grids have been tested to determine an optimal mesh solution for which the numerical results become independent of the details of the

3 Note

that the tracers have been introduced just for the sake of generality. For the present  and Z . application, their values could be obtained directly from 1 − Z

750

Flow Turbulence Combust (2014) 92:731–765

CFD Solver (Fluent)

MIL closure (UDF)

τ

1 1/ τchem

τ age

2

1/ τage ZJ+ = max(ZJ+ , Saa+ , Z)

1/ τ

ZJ- = min(ZJ- , Saa- , Z)

1/ τign 0

ZJ+ ZJ+

1

3

Y

Y

4

Y ZJ+

Y

Z

ZJ+

Y 0

Zst Z

1

P( ψ Z )

0

Zst Z

1

5 6

ωY

Z

2

0

Z

1

Fig. 7 Implementation of the MIL closure and couplings with the CFD solver

computational grid. The characteristics of the different meshes are reported in Table 2. Mesh 1 is the lightest one, it is used to gauge the numerical simulation. Meshes 2 and 3 give rise to similar results and may be retained to produce a gridindependent solution. However, considering the small computational costs associated with the present simulations, the finest grid (mesh 3) is selected, and it is retained in all subsequent calculations. 4.2 Results of numerical simulations ξ

ξ

We first consider the contours of the residence times,  τrescoflow and  τresfuel , which are reported in Fig. 9a (left). From this figure, we can see that, in comparison with those issuing from the hot coflowing stream, fluid particles issuing from the central jet spend less time to reach the exit of the computational domain, which appears as a consistent behaviour since the exit velocities of the two streams are 100 m/s and 5.4 m/s for the fuel and the coflow respectively. The residence time associated

Flow Turbulence Combust (2014) 92:731–765 X

Fig. 8 Description of the (two-dimensional) axisymmetric computational domain

751

Centre of jet exit plane

r

(0,0)

389.9 mm

91.4 mm

2.285 mm

228.5 mm

Coflow

with Lagrangian particles issuing from the fuel jet are indeed shorter than the one associated with particles issuing from the hot coflowing stream of burned gases. In Fig. 9b are reported the self-ignition flammability domain, i.e. (Saa+ -Saa− ), together Saa+  P(ψ Z )dψ Z . The with the probability to lie within the self-ignition domain, i.e., Saa− corresponding quantity globally increases downstream of the jet and coflow exits. On the symmetry axis, i.e., right downstream of the fuel jet exit, one can notice the presence of a maximum for this ignition probability. Since the average composition

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Flow Turbulence Combust (2014) 92:731–765

Table 2 Details of the computational domain and meshes X

From (mm) To (mm) From (mm) To (mm) X Y X Y X Y

r Mesh 1 Mesh 2 Mesh 3

Fuel jet

Coflow

−228.5 319.9 0 2.285 110 26 165 39 330 78

−70 319.9 2.285 91.4 90 40 135 60 270 120

Total number of cells

6,460 14,535 58,140

downstream of the central jet lies outside the self-ignition domain, this quantity indeed decreases after the composition fluctuations decay. The inspection of Fig. 10 may shed some additional light on the definition of the flammability domain, which is undoutedly the most important feature associated with the present modelling framework. It seems worth recalling that Z J+ and Z J− are

ξ

ξ

τresco f low, τresf uel (s)

Saa+ Saa- P( ψZ )d ψZ

(Saa+ ) − (Saa- )

(a)

(b) ξ

ξ

coflow fuel Fig. 9 Left: contour of the mean residence time fields  τres (left), and  τres (right). Right: field of the probability of auto-ignition (left) and field of the difference between the jumps superimposed  = Z st and Z  = Z MR (right) with the isolines Z

Flow Turbulence Combust (2014) 92:731–765

(S aa+ ) − (S aa- )

( Z J+ ) − (Z J- )

753

(S aa- )

(S aa+ )

 = Z MR (left) and Fig. 10 Left: fields of the difference (Saa+ − Saa− ) with superimposed isoline Z  = Z st (right). Right: fields of Saa− (left) and difference (Z J+ − Z J− ) with superimposed isoline Z  = Z MR and Z  = Z st Saa+ (right) superimposed with the isolines Z

obtained as the roots to τchem (Z ) = τ , while Saa+ and Saa− as those to τign (Z ) = τage (Z ). On the left side of this figure, the differences (Saa+ -Saa− ) and (Z J+ -Z J− ) are reported. The former is plotted together with the mean mixture fraction isoline  = Z MR , as obtained from associated with the most reactive mixture fraction, i.e., Z the self-ignition calculations reported above, while the latter is plotted together  = Z st . On the right side of Fig. 10, the values of with the stoichiometric isoline Z Saa− (left side field) and Saa+ (right side field) are also provided. In comparison with (Z J+ -Z J− ), the flammability domain that may be deduced from the difference between Saa+ and Saa− is the largest in the mixture fraction space and may reach approximately 0.8. It is indeed experienced that, for the present set of computations, the boundaries of the flammability domain are settled by self-ignition processes, i.e., by Saa+ and Saa− . This is even more striking on the additional plots of Fig. 11. The left side of this figure displays the evolution of the average residence times, ξcoflow ξfuel  τres and  τres , associated with Lagrangian particles issuing from either the oxidizer stream or the fuel stream. These quantities are plotted versus x/d along the isoline  = Z MR . The two quantities associated with the most reactive mixture fraction Z are compared to the corresponding self-ignition delay value, i.e., the smallest one, which is associated to Z MR . The figure shows that the characteristic residence time ξcoflow associated with the hot coflowing oxidizer stream  τres reaches the same order of magnitude as the self-ignition time τign (Z MR ) for a value of x/d that ranges between 25 and 30, which is in satisfactory agreement with the lift-off height H/d reported

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Flow Turbulence Combust (2014) 92:731–765

τ res (s)

τ (s) 0.004

0.005

τ chem (Zst )

ξ

τresco f low

0.004

0.003

τ ign (ZMR )

0.003

0.002 0.002 0.001

ξ

τresf uel

0.001 0 10

20 H d 30

40

0 10

50

20

30

x d

40

50

x d ξ

ξ

coflow fuel Fig. 11 Left: profiles of the residence times  τres (solid line) and  τres (dashed line) plotted along  = Z MR and self-ignition time scale τign (Z MR ). Right: profiles of the mixing time scale the isoline Z  = Z st chemical time scale τchem (Z st ) τ plotted along the isoline Z

in the experiments. The second plot, which is reported on the right side of Fig. 11, displays the evolution of the turbulent mixing time scale τ versus x/d. This quantity  = Z st and it is compared to has been evaluated along the stoichiometric isoline Z the chemical time scale evaluated at stoichiometry τchem (Z st ). This figure confirms that, within the present framework, and for the present conditions, the width of the flammability domain—in the mixture fraction space—is determined rather by selfignition and residence time scales, than chemical and mixing time scales. However, this does not mean that the turbulent mixing time scale τ does not play a crucial role in the modelling. Indeed, it also plays a central role since it fixes the mixing rate between the fuel and the hot coflowing stream of oxidizer and therefore controls, to a large extent, the mean chemical rate level, see Eq. 20 and below. 4.2.1 Sensitivity to the mixing constant value The influence of variations of the mixing constant4 value Cmix = τ/τt is evaluated by conducting the numerical simulations with four distinct values of Cmix = 0.5, 1.0, 1.5 and 2.0. Figure 12 reports the effects of the corresponding variations on the computational profiles of the mean temperature at three different locations in the turbulent jet flame. It appears that the maximum value observed on the mean temperature profiles decreases with Cmix . This decrease is more pronounced at x/d = 50, i.e. downstream of the location at which the combustion stabilizes, in the core of the turbulent lifted flame. This may be explained by the intensification of the combustion processes that takes place as the integral mixing time scale is decreased. Indeed, the smaller is the mixing time scale, the narrower becomes the combustion zone. In addition to this, the amount of fresh reactants that are brought at the flame location through molecular mixing processes (micro-mixing) is increased since the mixing frequency is increased. As a result, this strong and narrow combustion zone leads to higher temperature levels. Finally, the results of this preliminary analysis

4 Note

 2 that the ratio Cmix differs from C D , which is standardly defined by  εZ = CD Z ε/ k.

Flow Turbulence Combust (2014) 92:731–765 2000

x/D=50 1500

T (K)

 at Fig. 12 Radial profiles of T three distinct locations downstream of the injection pipe x/d = 30, 40 and 50. Measurements (symbols), numerical simulations: Cmix = 0.5 (solid line), Cmix = 1.0 (dashed line), Cmix = 1.5 (dashdot line), Cmix = 2.0 (dotted line)

755

Cmix=0.5 Cmix=1.0 Cmix=1.5 Cmix=2.0

1000

500 0

2

4

6

8

10

r/d 2000

x/D=40

T (K)

1500

Cmix=0.5 Cmix=1.0 Cmix=1.5 Cmix=2.0

1000

500 0

2

4

6

8

10

r/d 2000

x/D=30

T (K)

1500

Cmix=0.5 Cmix=1.0 Cmix=1.5 Cmix=2.0

1000

500 0

2

4

6

8

10

r/d

confirm the relevance of the classical value of the scalar to turbulent time scales ratio, i.e., Cmix = 0.5, introduced in the early study of Béguier et al. [52]. The use of this peculiar value gives rise to a fairly satisfactory agreement between numerical simulations and available experimental data, it has been therefore retained for the rest of the present work. 4.2.2 Linear relaxation of scalar f luctuations with Cmix = 0.5 The results of the numerical simulation performed with the present closure are now compared with those resulting from the use of the standard MIL closure

756

Flow Turbulence Combust (2014) 92:731–765 0.2

3000 2500

0.15

T (K)

Y

2000 0.1

1500 1000

0.05 500 0 0

20

40

60

80

0 0

100

20

40

60

80

100

x/d

x/d

O (left), T  (right). Measurements (symbols), modified MIL closure (solid Fig. 13 Axial profiles of Y 2 line), standard MIL closure (dashed line)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

x/D=30 EXP Trans Relax

0

2

4

r/d

6

8

Z

Z

[20] which, unlike the present extended Lagrangian closure, does not incorporate the auto-ignition phenomena as well as improved closures for the residence times and SDR. Figure 13 reports the axial profiles of the average mass fraction of  and the mean temperature T.  Figure 14 displays the radial profiles of the oxygen Y average mixture fraction at two locations downstream of the injection. The results of numerical simulations display a reasonable agreement with available measurements.  computed with the modified MIL model presents two distinct The axial profiles of Y zones, which are similar to those observed on the experimental profiles. Below the abscissa x/d = 30, turbulent mixing between the fuel jet and the vitiated coflow dominates without any significant chemical reaction, which results in rather low temperature levels. After the stabilization zone, which is located approximately 30 diameters downstream of the fuel inlet, the values of the average temperature increase significantly. In Fig. 15, radial profiles of the mean mass fraction of oxygen are plotted together with the Favre-averaged values of the temperature at different locations downstream of the injection pipe. A quick overview of the results shows that the main tendency is correctly recovered by using the modified MIL model, and it also confirms that the auto-ignition phenomena is one of the leading order driving processes in the present configuration. Indeed, the location as well as the order of magnitude of both mean temperature and mean oxygen mass fraction are rather well captured with the modified MIL closure, while they are not by using the classical representation that

10

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

x/D=40

EXP Trans Relax

0

2

4

6

8

10

r/d

O at x/d = 30 (left) and x/d = 40 (right). Measurements (symbols), Fig. 14 Radial profiles of Y 2 modified MIL with linear relaxation Cmix = 0.5 (solid line), modified MIL with a modelled transport equation for the mean scalar dissipation rate (dotted line)

Flow Turbulence Combust (2014) 92:731–765 2000 x/D=50

x/D=50

T (K)

1500

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

757

500 0

2

4

r/d

6

8

10

2

4

r/d

6

8

10

2000 x/D=40

T (K)

1500

1000

500 0

2

4

r/d

6

8

10

0

2

4

r/d

6

8

10

2000 x/D=30

x/D=30

T (K)

1500

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

x/D=40

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

1000

1000

500 0

2

4

r/d

6

8

10

0

2

4

r/d

6

8

10

O (left) and T  (right) at three distinct locations downstream of the Fig. 15 Radial profiles of Y 2 injection pipe x/d = 30, 40 and 50. Measurements (symbols), modified MIL (solid line), standard MIL (dashed line)

does not incorporate auto-ignition phenomena. Finally, the order of magnitude of the flame liftoff height seems to be quite well estimated with the modified MIL model. 4.2.3 Transport equation for the mean scalar dissipation rate A particular attention is now paid to the evaluation of the mean scalar dissipation rate (SDR). Therefore, a transport equation for the mean SDR has been implemented, see Eq. 11, and the corresponding computational results have been compared to those obtained by resorting to linear relaxation closure using Cmix = 0.5. In Fig. 14, the radial profiles of the mixture fraction are displayed. As expected, the observed differences are not very important but it seems that a slight improvement of the reactive mixing layer description is observed by using a modelled transport equation for the mean SDR. Moreover the computational results differ only in the near field of the jet, and the difference between the computational results becomes negligible further downstream in the far field, where the turbulent flow is fully devel-

758

2000 x/D=50

x/D=50

T (K)

1500

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Flow Turbulence Combust (2014) 92:731–765

EXP Trans Relax

0

2

4

6

8

1000 EXP Trans Relax

500 0

10

2

4

10

x/D=40

T (K)

1500

EXP Cabra Trans Relax

1000

EXP Trans Relax

500 0

2

4

6

8

10

0

2

4

6

8

10

r/d

r/d 2000

x/D=30

x/D=30

T (K)

1500

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

8

2000 x/D=40

Y

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

6

r/d

r/d

EXP Trans Relax

1000

EXP Trans Relax

500 0

2

4

6

r/d

8

10

0

2

4

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r/d

O (left) and T  (right) at three distinct locations downstream of the Fig. 16 Radial profiles of Y 2 injection pipe x/d = 30, 40 and 50. Measurements (symbols), modified MIL with linear relaxation Cmix = 0.5 (solid line), modified MIL used in conjunction with a modelled transport equation for the mean scalar dissipation rate (dotted line)

 oped. Figure 16 displays the radial profiles of the average mass fraction of oxygen Y  and the mean temperature T. Again, the closure retained to evaluate the mean SDR plays a role in the near field but it becomes unimportant further downstream. The effects observed on the mean temperature profiles are also significant, especially at x/d = 50. Figure 17a reports the mean temperature profiles plotted against the average mixture fraction value at three distinct locations x/d = 30, x/d = 40, and x/d = 50. The figure confirms the quality of the agreement obtained between the present numerical simulations and the experimental data. A comparison performed with temperature profiles associated with equilibrium conditions considering the mixture  Z ), or neglecting it, i.e. P(ψ  Z ) ≈ δ(ψ Z − Z  ), unfraction fluctuations through P(ψ doubtedly shows that accounting for both mixture fraction fluctuations and finite rate chemistry effects is required to deal with the present conditions. This is confirmed by Fig. 17b, which displays the mixture fraction variance field together with the average

T K

Flow Turbulence Combust (2014) 92:731–765 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 500 0

759

x/D=50 EXP Trans Teq Teq-MOY

0.1

0.2

0.3

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T K

Z 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 500 0

x/D=40 EXP Trans Teq Teq-MOY

Z Z 0.1

0.2

0.3

2

0.4

T K

Z 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 500 0

x/D=30 EXP Trans Teq Teq-MOY

0.1

0.2

0.3

0.4

Z  at three distinct locations downstream of the injection pipe Fig. 17 Left: radial profiles of T x/d = 30, 40 and 50. Measurements (symbols), modified MIL used in conjunction with a modelled transport equation for the mean scalar dissipation rate (dotted line), equilibrium temperature at  ), (bold solid line), and average equilibrium temperature, i.e. the average composition, i.e. Teq ( Z eq = 1 Teq (ψ Z ) P(ψ  Z )dψ Z , (solid line). Right: mean and variance of the mixture fraction field T 0

mixture fraction field. It clearly shows that the mixture fraction variance field differs from zero on a rather large section of the computational domain. Finally, combustion in a non-premixed jets geometry has been successfully computed, and the turbulent lifted flames obtained with the present approach display a satisfactory agreement with the experimental data in terms of liftoff heights, average mass fractions of chemical species, mean temperature and mixture fraction profiles. The liftoff height results from a quite complex and critical balance between different physical mechanisms and, from the numerical modelling point of view, the results will obviously depend on the corresponding modelling constant values. The sensitivity analysis which has been performed confirms that the value Cmix = 0.5 can be retained

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as a relevant one. Consideration of a modelled transport equation for the mean scalar dissipation rate (SDR) also improves the computational results. 4.2.4 Lift-of f height sensivity to velocity and temperature The level of agreement between numerical results and experimental data reported above may be altered as experimental conditions are varied. Therefore, several complementary numerical simulations are conducted to investigate the liftoff height sensitivity to various flow-field parameters. As emphasized in the early studies of Cabra et al. [7, 8], the principal parameters that control the flame liftoff height characteristics in this burner are: (i) fuel jet velocity, (ii) coflow velocity, and (iii) coflow temperature. The sensitivity to coflow temperature is first investigated, see Fig. 18, and the computational results indeed confirm that the liftoff height displays a crucial sensitivity to this parameter. Such results may be compared to those reported by Gordon et al. [13]. Based on the Monte-Carlo resolution of the PDF transport equation, the computational results of Gordon et al. [13] display significantly delayed ignition times if used in conjunction with the Smooke mechanism: the normalized liftoff height H/d lies between 25 and 50 for coflow temperature values ranging between 1575 K and 1425 K. In contrast, reduced ignition delays are obtained with the five-steps mechanism of Chen and coworkers, which emphasizes how critical is the coflow temperature and chemical kinetics description for these conditions. The level of agreement obtained from the present investigation is reported in Fig. 19, it is not perfect but allows to recover the experimental trend and confirms that the

1000

Temperature (K) 500

2000

1500

3000 2500

Tco f low = 1000K

Tco f low = 1350K

Tco f low = 1500K

Flame blowoff

Lifted flame

Liftoff height reduction

Fig. 18 Mean temperature field as obtained for three distinct values of the coflow temperature

Flow Turbulence Combust (2014) 92:731–765 80

761 80

80

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Computations Experiments

60

H d 40

40

40

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Tco f low

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0 4

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Computations Experiments

60

7

0 80

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150

185

220

U jet

Fig. 19 Comparisons between measured and calculated normalized liftoff heights H/d: sensitivity to coflow temperature Tcoflow , coflow and jet exit velocities, U coflow and U jet

flame liftoff height exhibits the greatest sensitivity to this parameter, i.e., the coflow temperature. To complete our present investigation of liftoff height sensitivity, we conducted further numerical simulations for varying values of both jet and coflow velocity. The sensitivity with respect to these parameters has been also investigated in previous computational studies devoted to the Cabra flame. For instance, Michel et al. [17] reported such sensitivity analyses. The obtained results are gathered in Fig. 19. The level of agreement with experimental data is comparable to the one obtained from previous numerical investigations, which confirms the relevance of the present modelling proposal to deal with such conditions.

5 Conclusions The extended modified Lagrangian Intermittent Model has been retained to take into account auto-ignition phenomena in non-premixed turbulent reactive flows. It has been used in conjunction with a transport equation for the mean scalar dissipation rate (SDR) which allows a better estimation of the scalar mixing time scale. This closure has been implemented in the commercial code Fluent and applied to the computation of the methane/air turbulent lifted flame studied by Cabra and coworkers. The obtained results display a satisfactory agreement with experimental data and the liftoff height is reasonably well captured. Its sensitivity with respect to both jet and coflow velocity as well as its variation with the coflow temperature have been systematically investigated and, in comparison with previous studies, the obtained results display a reasonable level of agreement with experimental data. In a future work, it would be interesting to generalize the evaluation of the mixture fraction PDF for instance through the consideration of the partial presumed PDF method introduced in [53]. This may allow to improve the description of mixture fraction statistics and eventually to describe the influence of dilution by external air which takes place in the farfield downstream of the central jet exit. Such an objective will also require some generalizations of the mixture fraction concept, which are currently investigated. Finally, further improvements of the results could be expected from the use of a LES closure able to simulate the unsteady behaviour of the large scale mixing events. Acknowledgements The financial support from the France Embassy in Cameroon through its Cooperation Service and Cultural Action (SCAC) with EGIDE is gratefully acknowledged. We

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are also deeply indebted to Ricardo Cabra and Professor Robert Dibble (Berkeley University) for providing experimental data. The reviewers are thanked for their constructive comments about the manuscript.

Appendix A Mixture Fraction PDF The β-distribution has long been used to represent two-feeding streams mixing in turbulent flows, it can be expressed with the -function (m1 + m2 ) m1 −1  2 ) =  ) = β(ψ Z ; Z ; Z P(Z ψ (1 − ψ Z )m2 −1 (m1 ) (m2 ) Z where the -function is defined as, see [54],  ∞ (m) = xm−1 exp(−x)dx

(25)

(26)

0

The expectation or mean value of the distribution will be given by  = m1 /(m1 + m2 ) Z

(27)

m1 m2 (m1 + m2 )2 (m1 + m2 + 1)

(28)

and its variance by  2 = Z

which enables to determine the two parameters m1 and m2 as follows    1 − SZ m1 = Z SZ   1 − SZ  m2 = (1 − Z ) SZ

(29)

 2 /( Z  (1 − Z )) has been introduced. It is worth where the segregation level S Z = Z noting that m1 and m2 are positive defined. It is trivial to show that the above  Z ) follows the positivity criterion P(ψ  Z ) ≥ 0, ∀ ψ Z ∈ [0 : 1], and distribution P(ψ normalization properties of a PDF, 

1

 Z )dψ Z = 1 P(ψ

(30)

0

As defined above, the PDF shape is given by the values of the two parameters m1 and m2 , and it is therefore fully determined provided that both the mean and variance values of the mixture fraction field are known.

Appendix B Numerical Integration with PDF Algorithms To reduce the computational costs associated with numerical integrations, optimized routines are exploited from the Numerical Recipes library [55], and integral

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763

quantities are evaluated directly from the continued fraction representation of the incomplete β-function Ix (m1 , m2 ) =

xm1 (1 − x)m2 Cf m1 B(m1 , m2 )

(31)

with B(m1 , m2 ) the β-function value and C f the continued fraction defined by Cf =

1 1+

d1 d 1+ 1+2 ...

(32)

where odd and even coefficients, d2n+1 and d2n , are given by d2n+1 = −

(m1 + n)(m1 + m2 + n)x n(m2 − n)x , d2n = − (m1 + 2n)(m1 + 2m + 1) (m1 + 2n − 1)(m1 + 2n)

C √f converges rapidly for x < (m1 + 1)/(m1 + m2 + 2), taking in a worst case O( max(m1 , m2 )) iterations. Since Ix (m1 , m2 ) = 1 − I1−x (m1 , m2 ) an equivalent behaviour is obtained for x > (m1 + 1)/(m1 + m2 + 2). Finally, Eq. 31 is evaluated through a modified Lentz’s method [56], and the integral of any polynomial of ! degree k (with k an integer), kj=0 c j Z j where c j denote real constant values, can be effectively evaluated as  a

b

k 

 Z )dψ Z = c j (ψ Z ) j P(ψ

j=0

k  j=0

cj

B(m1 + j, m2 ) (Ib (m1 + j, m2 ) − Ia (m1 + j, m2 )) B(m1 , m2 ) (33)

without any resort to CPU-expensive computational integrations. Since the MIL model relies on a trajectory made up of broken lines, this peculiar property of the β-distribution is fully exploited.

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