RANS Simulation of Turbulent Diffusive Combustion Using Open Foam L

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Journal of Applied Fluid Mechanics , Vol. 9, No. 2, pp. 669-682, 2016. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.68.225.24104

RANS Simulation of Turbulent Diffusive Combustion using Open Foam L. F. Gutierrez´ 1,2†, Jose´ P. Tamagno1 and Sergio A. Elaskar1,2

1 Departamento de Aeronautica,´ Universidad Nacional de Cordoba,´ Crdoba, 5000, Argentina 2 Consejo Nacional de Investigaciones Cient´ıﬁcas y Tecnicas´ -CONICET, Argentina

† Corresponding Author Email: [email protected]

(Received Octobr 16, 2014; accepted February 24, 2015)

ABSTRACT Schemes to write the flow equations in discreet form, solution solvers, pre and post data processing utilities provided by OpenFoam libraries, are used to build a finite volume executable for simulating a low speed, turbulent and rate controlled diffusive CH4-Air combustion. Unsteady Favre’s averaged turbulent conservation equations (total mass, momentum, energy and species mass fractions), are used to describe the combustion gas dynamics, and to handle turbulence a modified k-ε model is applied. Several global kinetic mechanisms, one step, two and four steps have been considered to describe the oxidation process of CH4 in a free jet type flame. The interaction between chemistry and turbulence, is modeled according to the partially stirred reactor (PaSR) concept. To improve convergence and accuracy in solving low speed fluid dynamic equations, a pressure implicit with splitting of operators (PISO) technique extended to cover high temperature flows, is utilized. The exponential dependence of the chemical kinetics from temperature, makes stiffs the ODE’s needed to determine source average values with which the species conservation equations are solved. To deal with the stiffness issue, OpenFoam provides numerical schemes that guaranties the stability of the computation. Comparisons between results of numerical simulations and experimental data obtained with the benchmark known as flame “D”, are presented.

Keywords: Numerical simulation; Turbulent diffusive combustion; Global reaction; Flame D.

NOMENCLATURE

A0 pre-exponential factor ∆t time step D¯ k mean species molecular diffusion coefficient ε turbulent kinetic energy dissipation hs sensible enthalpy κ volume reactive fraction k turbulent kinetic energy λ thermal conductivity coefficient p pressure µ molecular viscosity coefficient Prl Prandtl number µt turbulent viscosity coefficient RR reaction rate µe f f efective viscosity coefficient T temperature ρ density Ta activation temperature τi j viscous stress tensor r pilot inlet radius τch chemical time u velocity τmix mixing time Vk diffusion velocity of specie k ω˙ k production/consumption rate of specie k Yk mass fraction of specie k ω˙ T heat released by combustion Wk molecular weight of specie k e Favre averaged quantity Vp control volume ¯ Reynolds averaged quantity

1. INTRODUCTION big toolbox that provides libraries and applications to meet nearly all tasks involved on main openFoam (Open Field Operation and Manipu- steps of finite volume CFD simulations. Is- lation Weller et al. (1998)), can be seen like a lam et al. (2009), Singh et al. (2011), Filho L. F. Gutierrez´ et al. / JAFM, Vol. 9, No. 2, pp. 669-682, 2016. et al. (2011), have show the utilization of included in the OpenFoam solvers library, has OpenFoam to treat incompressible aerodynamic been selected. problems. Turbo machinery applications have The flame D develops in a low speed envi- been reported by Beaudoin and Jasak (2008), ronment, however due to strong temperature Mangani (2008), Muntean et al. (2009) and changes associated with chemical heat release Benajes et al. (2014). Diesel engine combus- (300 ≤ T < 2500 K), compressible effects arise. tion modeling, including spray simulation are When the Mach number goes to zero, the com- presented by Nordin (2001), Karrholm¨ et al. pressible governing equations should in a con- (2008) and Novella et al. (2011). Marzouk and tinuous sense, approach their incompressible Huckaby (2010), have made studies of chemical counterpart and density tends to become inde- kinetics models related to syngas burning. pendent of pressure. Then, the continuity equa- Here are presented results for the piloted free tion is no suitable to compute the density as jet flame D, a benchmark case carefully stud- a dependent variable wherein the pressure is ied experimentally in the Combustion Research evaluated from it via an equation of state. To Facility of Sandia National Laboratories (USA). handle these issues, methods based on solving The available experimental data covers the pe- a new transport equation for pressure is for- riod 1998-2007 Barlow and Frank (1998), Bar- mulated. Here the pressure-velocity coupling low and Frank (2007). Nooren et al. (2000) solution method known as PISO (for pressure made contributions on subjects related to turbu- implicit with splitting of operators) applicable lence effects in CH4-air jet flames and on mea- to compressible flow is employed (Issa (1986), surements techniques. Temperature and species Benajes et al. (2014)). Further details about mass fractions measurements have been con- the PISO method are given in the section where ducted using Raman-Rayleigh scattering tech- fundamental aspects of numerical techniques niques. It prevents possible inconsistencies are treated. in comparing Favre averaged predictions with All openFoam codes are built for a three di- Reynolds averaged measurements. In addition, mensional (3D) space and all meshes have to the flame D has a small probability of local ex- be defined in a 3D manner. However, the tinction Barlow and Frank (2007), becoming shape of the computational domain in which suitable for comparisons with models not in- the simulations are carried out is axial symmet- cluding a flame extinction criteria. ric. In OpenFoam, an axial symmetric wedge When finite rate chemistry processes are con- shaped geometry can be generated by defin- sidered, a decision has to be made about ki- ing first a 3D mesh, and thereafter specify- netic mechanisms to be used, since it could ing front and back sides of the axial symmet- have a great impact on computing times. In ric domain as wedge patches where appropri- line with the main purpose of this work, gain- ate boundary conditions are applied. Details ing experience on the use of openFoam to build for generating axial symmetric wedge shaped executable solvers and testing its usefulness to geometries using blockMesh (utility for gen- study turbulent diffusive combustion, simplified erating simple meshes with blocks of hexahe- reaction mechanisms for the oxidation of CH4 dral cells) and employing wedge patch from fuel have been examined. The types of mech- libfiniteVolume library can be found on the anisms studied include one reaction step Bui- openFoam user guide OpenCFD (2009)1. Pham (1992), two reaction steps (Westbrook It is assumed that the Navier-Stokes equations and Dryer (1981)), with Andersen et al. (2009) are to a multi-species reacting gas also appli- rates, and the Wang et al. (2012) modifica- cable, and that conservation equations (continu- tion adding a H oxidation reaction, and four 2 ity, momentum, species and energy) in RANS reaction steps proposed by Jones and Lindst- simulations can be written in terms of mass edt (1988) where reverse parameters used in CO weighted Favre averages Favre (1969). With and H oxidation’s, are now calculated using 2 this formalism, the averaged balance equations equilibrium constants Wang et al. (2012). Two become Poinsot and Veynante (2005): and four steps mechanisms were programed for use with openFoam. In each cell of the compu- Global mass tational domain and for every flow time step, the calculation of each species source term requires ∂ρ¯ ∂ + (ρ¯ uei) = 0 (1) the integration of ordinary differential equations ∂t ∂xi (ODE’s). However, these ODE’s are stiff and 1 to guaranty stability the semi implicit numerical The wedge patches technique has been successfully applied in computing supersonic-hypersonic flows around ax- method of Bader and Deuflhard (1983) (SIBS) isymmetric blunt bodies Gutierrez´ et al. (2012)

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Momentum g” ” ” ” 2.2 Species (ui Yk ) and Enthalpy (ui hs) Tur- ∂ ∂ ∂p˜ bulent and Laminar Fluxes (ρ¯uej) + (ρ¯ueiuej)+ = (2) ∂t ∂xi ∂x j Species laminar diffusion ﬂuxes can be modeled ( ) ∂ g as τ − ρ¯ u”u” ∂x i, j i j i ∂e ρ ≈ −ρ Yk Chemical species (for k = 1,N) ¯Vk,iYk ¯D (7) ∂xi ∂ ( ) ∂ ( ) ρYe + ρ¯ ueYe = (3) on conditions that molecular diffusion follows ∂t k ∂x i k i ( ) the Fick’s law and the molecular diffusivity of ∂ g − V Y + ρ¯ u”Y ” + ω˙ species Dk are assumed equals (Dk = D). If in ∂ k,i k i k k xi addition, the transport due to molecular diffu- Energy equation in terms of the mixture sen- sion of species is assumed comparable to the sible enthalpy rate of transport due to viscous effects, then ρ ≡ ( ) ( ) ¯D µl, Chung (2006). The mixture dynamic ∂ ∂ Dp molecular viscosity is now denoted by µ . ρ¯eh + ρ¯ueeh = ω˙ + l ∂t s ∂t i s T Dt ( ) If species turbulent diffusion ﬂuxes are also ∂ ∂T λ − ρ g” ” closed using a gradient approach (Kuo (2005), + ¯ ui hs (4) ∂xi ∂xi Chung (2006), Lilleberg et al. (2013)), and ( ) turbulent diffusion transport is again assumed ∂ ∂ N τ ui − ρ comparable to the rate of transport due to tur- + i, j ∂ ∂ ∑ Vk,iYkhs,k x j xi k=1 bulent viscous effects Bird et al. (2007) then

∂Ye where ρ g” ” ≈ − k ¯ ui Yk µt (8) ∂xi ∂ ∂ Dp p¯ e p¯ = + ui (5) where µ is the turbulent viscosity. By adding Dt ∂t ∂xi t both laminar and turbulent diffusion fluxes, the Note that these equations are formally identical corresponding terms in the species conservation to the classic Reynolds averaged equations for equation (Eq. 3) can be written constant density flows. ( ) ∂ g ρ¯ ρ¯ ” ” ≈ NCLOSEDTERMSIN AVRE AV Vk,iYk + ui Yk (9) 2. U F - ∂xi ERAGED BALANCE EQUATIONS ∂e ∂e In what follows, closures for the unknown quan- Yk Yk −(µl + µt ) = −µe f f tities found in Eq. 1 to Eq. 5 are proposed. ∂xi ∂xi where µe f f is an averaged dynamic viscosity of g” ” 2.1 Reynolds Stresses (ui u j) the reaction mixture. The laminar heat diffusion expressed by Fourier Following the assumption proposed by Boussi- law can be written as nesq Wilcox (1998), modeling of turbulent Reynolds stresses require the prior assesment e ¯ e e ∂T ∂T λ ∂hs µ ∂hs of the turbulence dynamic viscosity µ . A two λ ≈ λ¯ ≈ = (10) t ∂x ∂x ∂x Pr ∂x equations turbulence model Jones and Launder i i Cp i l i (1972), Nordin (2001), Karrholm¨ (2008) that λ¯ will provide values for the flow turbulent kinetic where denotes a local mean molecular thermal = ∑N ( ) energy ek and for its dissipation eε is used to esti- diffusivity and Cp k YkCp.k T a local mean mate the turbulent viscosity as specific heat. Cp.k(T) and hs,k(T) are computed using JANAF Tables Stull and Prophet (1971). g e The unclosed term ρ¯ u”h” is by analogy with the k2 i s ρ¯ laminar case modeled as: µt = Cµ eε (6) e g µ ∂h where C (usually given the value 0.09), is one ρ¯ u”h” = − t s (11) µ i s Pr ∂x of the many cofficients needed to close the two t i equations turbulence model. These two equa- ∂ui Viscous heating, τi, j (Eq. 4), are neglected in tions are PDE’s that must be solved with the ∂x j conservation equations simultaneously. the energy equation.

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3. TURBULENCE-CHEMISTRYINTER- 4. REACTIONMECHANISMSFOR ACTIONMODEL THEOXIDATIONOFMETHANEIN FLAMES The turbulence-chemistry interaction is modeled based on the partially stirred reactor Four simpliﬁed reaction mechanisms for the ox- (PaSR) approach where each computational cell idation of CH4 to be evaluated for a well docu- is divided into two zones: a reacting zone, mod- mented piloted ﬂame “D”, are proposed. eled as a perfectly stirred reactor (PSR) and a non reacting zone. A volume reactive fraction κ The Bui Pham one step mechanism (BP) function of the computational time step ∆t and A single forward global reaction applicable to time scales related with chemical (τch) and mix- the methane oxidation process is considered ing (τmix) processes, has been proposed Mar- Bui-Pham (1992): zouk and Huckaby (2010):

{R1} CH4 + 2O2 ⇒ CO2 + 2H2O (17) △t + τ κ = ch (12) △t + τ + τ ch mix The one step reaction is often a convenient way of approximating the effects of the many reac- τ → ∞ κ → τ → ∞ Note that if ch ; 1, and if mix ; tions which actually occur. The rate expression κ → 0. of the single reaction is expressed in terms of

The τmix time is computed according to the Arrhenius law and therefore written as ( ) √ β Ea a b µ RR1 = AT exp − [Fuel] [Oxidizer] τ e f f R T mix = Cmix ρeε (13) u ¯ (18) ( ) β Ea a b The value given to constant C is taken as 0.3 = AT exp − [CH4] [O2] mix R T Nordin (2001). u Conversion rates of fuel, or oxygen are used A is the pre-exponential factor, β the tempera- to deﬁne the characteristic chemical time τch ture exponent, Ea the activation energy, Ru the through the expression universal gas constant. The ratio Ea/Ru is called the activation temperature. Observe that the ex- ( ) ponent a and b may not be stoichiometric val- 1 −ωe˙ −ωe˙ = max Fuel , O2 (14) ues. τch ρ¯ ρ¯ The Westbrook and Dryer two steps mechanism (WD) The rate of change of the mean species mass The reaction mechanism for Westbrook and concentrations ω˙ can then, be calculated by k Dryer (1981) model are: ( ) M dCk { } ⇒ ω˙ k = κWk ∑ (15) R2 CH4 + 1.5O2 CO + 2H2O (19) i=1 dt i

{R3} CO + 0.5O2 ⇔ CO2 (20) being M the number of reactions,( W)k the molec- dCk ular weight of species k and dt the rate of i To account at least in part for the effects of information of species C from reaction i. Eq. 14 k complete conversion to CO and H O, and to computes de average source term for the species 2 2 include qualitatively the sequential nature of the k conservation equation. process of hydrocarbon oxidation, Westbrook The total heat ω˙ T released by the combustion is and Dryer (1981) develop a two steps reaction computed according to mechanism, being the last step a reversible oxidation of CO to CO2. The rate constants for N {R2} and {R3} originated from their studies for ω ω △ 0 ˙ T = ∑ ˙ k h f ,k (16) CH4 and CO oxidation reactions in a turbulent k=1 reactor. However, the original rate coefﬁcients for the {R3} reaction, were modiﬁed to secure ∆ 0 being N the number of species, h f ,k the forma- an approach to equilibrium values for CO and tion enthalpy of species k. CO2 Andersen et al. (2009).

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The modiﬁed mechanism of Westbrook and Dryer (WDM)

The initial volume fraction of CO2 inﬂuence the P process of CO oxidation, but the presence of d S H O acting as a sort of third body can also affect f 2 dfN the CO oxidation. Wang et al. (2012), modiﬁed N the Westbrook and Dryer two steps mechanism by adding a H2 oxidation rate expressed by the additional reaction:

{R4} H + 0.5O ⇔ H O (21) 2 2 2 Fig. 1. Finite volume discretization.

The Jones and Lindstedt four steps mechanism (JL) The discretization of spatial convection term, diffusion term and source terms, can be split { } ⇒ R5 CH4 + 0.5O2 CO + 2H2 (22) into two parts: the transformation of surface on volume integrals into discrete sums, and expres- {R6} CH4 + H2O ⇒ CO + 3H2 (23) sion that give the face values of the variables as a functions of cells values. If also it is assumed {R7} CO + H2O ⇔ CO2 + H2 (24) that the control volumes do not change in time, E. 26. can be written Jasak (1996)

{R4} H2 + 0.5O2 ⇔ H2O (25) [ ] ∫ t+∆t d ( ) Jones and Lindstedt (1988), developed this four ρϕ ϕ − ρΓϕ · ∇ϕ ( )P + ∑F f ∑ f S ( ) f dt steps mechanism applicable to non premixed t dt f f flames of hydrocarbons fuels. The rate {R5} is dominant in fuel lean mixtures and the rate ∫ ∆ {R6} is in fuel reach mixtures. The forward rate t+ t = (SuVP +SpVPϕP)dt(27) parameters of CO and H2 oxidation’s used in re- t actions {R7}and {R4}, have been proposed by Jones and Lindstedt (1988) and Marinov et al. Note that the source term has been linearized (1996), respectively. The corresponding reverse and that the volume integral is calculated as: parameters used in both oxidation reactions, are ∫ ∫ calculated by Wang et al. (2012), using tabu- Sϕ(ϕ)dV = (Su + SPϕ)dV = lated equilibrium constants fitted by polynomi- VP VP als Kuo (2005). Chemical kinetic data (Arrhenius parameters ϕ and reaction rate form) for each model used here SuVP + SPVP P (28) are presented in Tab. 1. Eq. 27 is usually call the semi-discretized form 5. FUNDAMENTAL ASPECTSOFTHE of the transport equation. Using the Euler im- NUMERICAL TECHNIQUES plicit method for time discretization the final Any unsteady transport equation for a scalar discrete form can be written: property ϕ is solved applying the Finite Volume (FV) method. The FV requires that any trans- 1 ( ) port equation be satisfied over the control vol- ρ¯ nϕn − ρ¯ n−1ϕn−1 ϕn ∆ VP + ∑F f (29) ume (Fig. 1) surrounding the point P in the t f VP ( ) n n integral form − ρΓϕ · ∇ϕ − − ϕ ∑ f S ( ) f SuVP SpVP P = 0 f ∫ [ ∫ ∫ t+∆t d − ρϕdV + ∇ · (ϕρU)dV − where n 1 and n denotes successive time lev- t dt V V els. Discrete equations for momentum and en- ∫ P ] ∫P [∫ ] t+∆t ergy can be derived from Eq. 29 by replacing ϕ ∇ · (Γϕ∇ϕ)d dt = Sϕ(ϕ)d dt V V with ue and he respectively 2. VP t VP s 2Note that difussive terms need some special considera- (26) tions

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Table 1 Chemical Kinetics Information Reaction A β Ta Rate ref. 11 R1 5 × 10 0 14950 [CH4][O2] Bui-Pham (1992) 11 0.7 0.8 R2 5.03 × 10 0 24056 [CH4] [O2] Westbrook and Dryer (1981) 6 0.25 0.5 R3f 2.24 × 10 0 5032 [CO][O2] [H2O] Andersen et al. (2009) 13 −0.25 0.5 R3b 1.14 × 10 −0.97 39452 [CO2][O2] [H2O] Andersen et al. (2009) 11 0.5 R4f 5.69 × 10 0 17609 [H2][O2] Marinov et al. (1996) 14 R4b 2.51 × 10 0 47859 [H2O] Wang et al. (2012) 10 0.5 R4f-JL 7.91 × 10 0 17609 [H2][O2] Marinov et al. (1996) 13 R4b-JL 3.48 × 10 0 47907 [H2O] Wang et al. (2012) 11 0.5 1.25 R5f 4.4 × 10 0 15095.7 [CH4] [O2] Jones and Lindstedt (1988) 8 R6f 3 × 10 0 15095.7 [CH4][H2O] Jones and Lindstedt (1988) 9 R7f 2.75 × 10 0 10063.8 [CO][H2O] Jones and Lindstedt (1988) 10 R7b 6.74 × 10 0 13688 [CO2][H2] Wang et al. (2012)

( ( ) ) ( ) −1 e −1 It is said in the introduction that the pressure- ∇· ψ(aP) H U p − ∇· ρ(aP) ∇p = 0 velocity coumpling solution algorithm PISO will be employed. Here the PISO method is out- lined and its implementatio described. Issa in- troduce de novel idea of PISO methodology Issa (33) (1986), Jasak gives a apropriate description for the openFoam enviroment Jasak (2007), Jasak It should be noted that this equation has the (1996). Starting from following semi-discrete standard form of Eq. 29 (rate of change, con- form of the momentum equation: vective and difussive terms), and discretization of any standard form equation can be han- ( ) dle in a stable, accurate and bounded man- en e − ∇ aPUP = H U p¯ (30) ner Jasak (2007). Next it is showing how the PISO method is applied within the openFoam’s environment and Fig. 2 shows corresponding with ﬂowchart. ( ) e − n 1. Momentum equations are assembled and H U = RP ∑aNUN (31) solved (ﬁrst predictor step) 3

n−1 U fvVectorMatrix UEqn( and RP = Ro + ∆t fvm::ddt(rho, U)+ fvm::div(phi, U) and ap are the center coefﬁcients of the momen-( ) + turbulence->divDevRhoReff(U) tum equations. The discrete operator H Ue == rho*g ); UEqn.relax(); e n has two contributions: (∑aNUN) the contri- if (momentumPredictor) butions of all neighbors cells to cell P, RP a {solve(UEqn == -fvc::grad(p));} source contributions( that contain) the (n−1) step n−1 ∆ −1 of transient term U ( t) and any other 2. Equations for mass fractions and enthalpy source contribution Ro (i.e, body forces). are assembled an solved. Thermophysical The state equation is written ρ = pψ, whereby properties are corrected (i.e, compressibility is updated the pressure temporal derivative can be ob- ∗ ∗ tained: ψ = ψ(T )

∂ρ ∂ = (pψ) (32) 3. Pressure equation is assembled and solved ∂t ∂t (transonic ﬂag is omited) many times as PISO corrector steps (pisocorr) are performed : By using Eq. 30, continuity (Eq. 1) and state equations the pressure equation (Eq. 33) is ob- 1 rho = thermo.rho(); tained: 3 Note that turbulence is a pointer to the member

( ) function divDevRhoReff of the RANS turbulent( models) ( ) e ∇ · ∇e − 1 − − class. This member function return G(U) = µeff U nψn − n 1ψn 1 [ ⟨( ) {( ) }⟩] p p VP+ T 2 T ∇· µ ∇Ue − I Tr ∇Ue ∆t eff 3 674 L. F. Gutierrez´ et al. / JAFM, Vol. 9, No. 2, pp. 669-682, 2016.

→ n −1 − ∇ 2 volScalarField rUA = 1.0/UEqn.A(); Up = (ap) (H(U) p) 3 U = rUA*UEqn.H(); Boundary conditions for momentum equa- 4 { tions are corrected (line 19). Finally the 5 phi = fvc::interpolate(rho)* volScalarField associated with unsteady ((fvc::interpolate(U) & mesh.Sf()) pressure term on the sensible enthalpy equation 6 + fvc::ddtPhiCorr(rUA, rho, U, phi)); is actualized (line 20). The formal order of con- 7 for (int nonOrth=0; vergence of PISO technique depends on the cor- nonOrth<=nNonOrthCorr; nonOrth++){ rector steps utilized, thus to obtain second or- 8 fvScalarMatrix pEqn( der accuracy (in discretization errors) should at 9 fvm::ddt(psi, p)+fvc::div(phi) least two steps be performed Issa (1986). - fvm::laplacian(rho*rUA, p) Appropriate solution methods need to be se- 10 ); lected to solve algebraic systems that come 11 pEqn.solve(); from discretization process. The solution al- 12 if (nonOrth == nNonOrthCorr){ gorithms can be selected as function of the 13 phi += pEqn.flux(); } system symmetry and convergence properties. 14 } For symmetric systems the pre-conjugate gra- 15 } dient method (PCG) with diagonal incomplete 16 #include "rhoEqn.H" Cholesky preconditioner (DIC) is utilized be- 17 #include "compressibleContinuityErrs cause its rapid convergence properties Con- .H" cus et al. (1985). In the other hand, con- 18 U -= rUA*fvc::grad(p); vection/diffusion equation produces asymmet- 19 U.correctBoundaryConditions(); ric matrices, for which bi-conjugate precondi- 20 DpDt = fvc::DDt(surfaceScalarField tioned method (BiPCG) with diagonal incom- ("phiU", phi/fvc::interpolate(rho)), p); plete LU pre conditioner (DILU) has proved to be efficient Venkatakrishnan and Mavriplis At line 1 density is corrected (ρn = p(n−1)ψn), (1993). For all solution algorithms a tolerance of 10−6 is fixed. ap coefficients for momentum equations (line 2) and velocity (line 3) are updated employing the All equations time evolution is done by the Eu- density obtained from first predictor step. Pres- ler implicit scheme in conjunction with the sta- sure flux is calculated (lines 5 and 6). Pressure bilized local time stepping technique (SLTS) equation is assembled from line 8 to 10: which allows solution advance in each cell with ( ) n n n−1 n−1 the maximum admissible time step, therefore fvm::ddt(psi, p) → p ψ − p ψ VP ( ( ) ) convergence to steady state is accelerated Co- −1 fvc::div(phi) → ∇· ψ ap H(U)p quel et al. (2008), Blazek (2005). Each face field (ϕ ) is evaluated using the linear fvm::laplacian(rho*rUA,( ) p) f ( )− upwind interpolation scheme (LUS) Barth and → ∇· ρ a 1 ∇p p Jespersen (1989): Here is important to note that the time advance is performed implicitly, the convective ϕ f = ϕP + (∇ϕ) · dfN (34) term is evaluated explicitly and the diffusive P term implicitly. The namespaces fvc and fvm calculate explicit and implicit terms re- Pressure and velocity gradient are evaluated us- spectively (i.e, fvc::grad(p) evaluates ∇p ing the linear scheme Blazek (2005). Dif- by using data from last time step, and fusive terms also are evaluated by the linear fvm::laplacian(rho*rUA, p) returns ma- scheme without perform orthogonality correc- tions Blazek (2005), Jasak (1996). trix( coefficients( ) ) of the discrete representation of ∇· ρ −1 ∇ ap p ). In consequence the pressure The chemistry data are supplied using the na- equation is solved implicitly at line 11 with the tive openFoam reader format OpenCFD (2009). solution method selected at run time. If the or- To obtain the source terms into energy and thogonal corrector steps imposed at run time are species equations, reaction rates are computed, fulfilled, the flux are actualized (line 13). Then and species concentrations are updated by solv- the continuity equation is solved and continu- ing a stiff system of ordinary differential equa- ity errors are computed (lines 16-17). Velocity tion (stiff ODEs) whose dimensions are propor- corrector step (momentum corrector) are per- tional to the species and reactions of the chem- formed (line 18): ical kinetic model. To solve this stiff ODEs U -= rUA*fvc::grad(p); the SIBS method is selected, this method is

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Fig. 2. PISO algorithm flowchart. based on Richardson extrapolation of the ap- • Main jet diameter: d = 0.0072m proximated solution. The solver needs to solve • ODEs in every time step to determine the chem- Pilot annulus outer diameter: 0.0182m ical species concentrations at the end of the • Wind tunnel exit section: 0.300m × timestep. The Richardson extrapolation of the 0.300m function ([C]) assumes that as the interval (in our case the computational time step ∆t ) is split Since the walls of the burner are very thin, no up in to increasing number of sub-steps, the so- thickness was allocated to walls. The main jet, lution will converge to some value ([C(t+∆t)]). pilot and air compositions expressed in mass However, the solver will never apply enough fractions, are given in Tab. 2. Because of the sub-steps to find it. Instead, it will approximate, wind tunnel exit dimensions, it is assumed that depending on the solution using large sub-steps. the simulated flame D develops inside a co- The analytical function used to approximate flowing air free jet. [C(t + ∆t)] is a polynom, and the error function of the method contains only even terms of the step size Bader and Deuflhard (1983) (for more Table 2 Incoming Flow Mass Fractions details on solving stiff systems see Hairer and Air Jet pilot-bp pilot-WD pilot-WDM-JL Wanner (2005)). N2 77 64.73 74.2 73.79 73.72 O2 23 19.66 5.4 5.4 5.4 6. DESCRIPTIONOFTESTCASE CH4 0 15.61 0 0 0 The benchmark case selected to test the exe- H2O 0 0 9.42 9.42 9.42 cutable code is the Sandia flame D (Sandia Na- CO2 0 0 10.98 10.98 10.98 tional Laboratories, Ca, USA). This flame D is CO 0 0 0 0.407 0.407 a piloted non premixed methane and air flame H2 0 0 0 0 0.0129 with a main fuel jet Reynolds number (Re j) at the burner exit of 22400. The pilot is a lean 6.1 Mesh Generation mixture (ϕ = 0.77) of methane and air and in the calculations it is assumed to behave like an The computation domain, discretized using the injection of hot burnt gases. The burner exit is openFoam utility blockMesh has a length of positioned approximately 0.15m above the end 0.6m and a radius of 0.15m. The mesh is con- section of a vertical wind tunnel which provides structed assuming symmetry about the center the air flow. line of the main jet, and as Fig. 3 shows it is divided into three blocks. The burner dimensions are Barlow and Frank The first block is defined to include the main (2007): jet, the second the pilot region, and the third block covers the air co-flow. An axial stretching

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outerWall (3/2) −1 ε = 0.1643k (Lt ) (36)

Air block3 outlet Properly determined initial values are important to ensure starting stability and adequate accu-

Pilot block2 racy in solving the conservation equations, and Fueljet block 1 SymAxis also the effectiveness of the pilot. Inlet con- Fig. 3. Computational domain. ditions for the species equations are set in ac- cordance with flow compositions described in Table. 2. However in the numerical simulation factor of 10 is applied, so that cells have mini- a time dependent technique is used, and initial mum length of 0.383mm and maximum length properties of the flow in the whole computa- of 3.83mm. The shorter cells are packed close tional domain must be defined. They are listed to inlet sections. In the air co-flow the cells below Barlow and Frank (2007): are stretched in radial direction with a minimum size of 1.2mm for cells adjacent to block 2. The  simulation was run with a mesh arrangement of u(0) = 0; T(0) =300; K p(0) = 94.536 kPa  40000 cells.  Y (0) = 0.77; Y (0) = 0.23; Y (0) = 0 6.2 Boundary and Initial Conditions N2 O2 k In Tab. 3 boundary conditions applied to flow variables are listed. The following abbrevia- (37) tions are used: fV for fixed value and zG for zero gradient. To apply boundary conditions in a non-limited by solid surfaces domain, the zero Note that only for the species O2 and N2 mass gradient approach is imposed. It has been nu- fractions initial values have been imposed, but merically verified that when the flow is reaching not for any other species. This implies that only steady state conditions, the assigned outer limit air is initially present in the computational do- becomes a streamline parallel to the axis of the main. It should be noted that when the air mo- flow and its velocity approaches the fixed value tion in the computational domain starts, the tur- (0.9 m/s) of the air co-flow. This can be inter- bulence properties for the air co-flow are set in preted as proof that the free jet approach applied terms of the intensity and characteristic length to the domain where the flame D develops, is listed in Table 3. valid. 7. RESULTS AND DISCUSSION The results of openFoam simulations are com- 4 Table 3 Boundary conditions pared with experimental data of temperature ε Ux p T Y k and species. It should be noted that all simpli- MJ 49.6 zG 291 fV Eq. 35 Eq. 36 fied reaction mechanisms tested are capable of Air 0.9 zG 294 fV Eq. 35 Eq. 36 sustaining the diffusive flame started by the pi- Pilot 11.4 zG 1880 fv Eq. 35 Eq. 36 lot. Also note that all results, numerical and ex- OL zG zG zG zG zG zG perimental, have been plotted in term of the ra- Outlet zG 100615 zG zG zG zG tio x/r, being x the axial distance and r = 7.2 × 10−3m, the pilot radius (accordingly with ex- It Lt perimental data presented by Barlow and Frank Main jet 4.569 × 10−2 5.04 × 10−4 (2007), Barlow (2003)). In Fig. 4, a compari- Air 7.492 × 10−2 2.10 × 10−2 son between experimental and predicted center Outer Limit 6.030 × 10−2 7.77 × 10−4 line temperatures is shown. It can be observed that the one step mechanism has produced ac- Starting values for k and ε of Main Jet, Air and ceptable comparisons only in regions where the Pilot are expressed as functions of turbulent in- chemical activity is not the strongest. By as- tensity (It) and characteristic lengths (Lt) re- suming that the reaction products are CO2 and spectively. Values for these parameters are com- H2O the total heat of reaction is over predicted. puted by using available correlations listed be- At adiabatic flame temperatures typical of hy- low Wilcox (1998): drocarbon fuel (∼ 2000K), substantial amounts of CO and H2 exist in equilibrium with CO2 and 2 k = 1.5(uIt ) (35) H2O. The same is true to a lesser extent with other species such as H, O and OH. This equi- 4MJ: Main Jet; OL: Outer Limit librium lowers the total heat of reaction and the

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2400 exp ing Jones and Lindstedt four steps mechanism 2200 BP with experiments, have shown no mayor dif- WD 2000 WDM ferences when the temperature raises, a small 1800 JL improvement where the heat release should 1600 be stronger and some deterioration when the burned gases are diluted by mixing with the 1400 cold air coﬂow. Calculated center line distri- T[K] 1200 butions of CH4,O2, CO2 and H2O mass frac- 1000 tions, are plotted in Fig. 6, and are compared 800 with the Flame “D” experimental data. It is 600 found that CH4 fuel consumption (Fig. 6(a), is 400 fairly well predicted by all combustion mech- 200 anisms proposed. As shown in Fig. 6(b), the 0 10 20 30 40 50 60 70 80 90 O2 is consumed faster than experimental data x/r and after reaching almost a null value, it start Fig. 4. Temperature center line distribution. to grow because of mixing with the air coﬂow. This growing is predicted by all’s the combus-

0.01 tion mechanisms considered. Regarding CO2, 0.1 exp exp WD WDM Fig. 6(c) shows that best approximation to ex- WDM JL JL 0.008 perimental data is obtained with Westbrook and 0.08 Dryer original two steps mechanism. Results obtained with the four steps reactions of Jones 0.006 0.06 ]

2 and Lindstedt, Fig. 6(d), have shown tendency Y[H Y[CO] to follow the ascending branch of the exper- 0.004 0.04 imental curve, but a greater divergence in its descending branch. In relation to the behav- 0.02 0.002 ior of the H2O simulation, it seems that the one step mechanism provides the best results 0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 (Fig. 6(d)). The CO and H2 mass fractions cal- x/r x/r Fig. 5. H2 and CO center line distributions. culated through the Westbrook and Dryer modified two steps mechanism by adding a H2 oxidation rate, seem to be best approach to experi- adiabatic flame temperature below values pre- mental values (Fig. 5). dicted by Bui Pham one step reaction mecha- Simulation times needed to practical reach nism. steady state conditions are derived after apply- In addition to the fact that the burned gas con- ing the ||L2|| norm to selected variables residu- tains incompletely oxidized species, it is also als. Typical results obtained for flow parameters recognized that typical hydrocarbons burn in a (p,T), species O2 (reactant) and H2O (product) sequential manner, that is, the fuel is partially in one and four steps combustion mechanisms, oxidized to CO and H2. To account for this ef- are plotted inf Fig. 7(a) and Fig. 7(b). It can fect of incomplete conversion of CH4 and O2 be concluded that in each of plotted variables −6 reactants to products CO2 and H2O, Westbrook a 10 precision value is achieved with a com- and Dryer have introduced the two steps mech- putation time of not more that one second. anism adding a CO oxidation reaction. This two 8. CONCLUSIONS reactions model is also applied to the methane’s flame and temperature results are included in This work has two main proposes. First, to Fig. 4. It can be seen that the addition of the show how an executable code to numerically CO - CO2 equilibrium provides a somewhat simulate a rate controlled and turbulent diffu- better adiabatic flame temperature. Suppos- sive combustion process, can be built using the edly, further refinements in expressing dissoci- set of libraries provided by openFoam. Second, ated effects on burned gas temperature would to demonstrate what so good are simulations lead to additional improvements. In this sense, carried out using the code. It is estimated that the H2 oxidation in the Westbrook and Dryer the first purpose has been accomplished but the was added, however the flame temperature re- second has been only partially because it pro- sults did not substantially improved, if com- duced results that should be conceptualized as pared with the original two steps mechanism. not completely satisfactory. Going from one step to two steps, and even to four steps hy- Comparisons of plotted simulations results us-

678 L. F. Gutierrez´ et al. / JAFM, Vol. 9, No. 2, pp. 669-682, 2016.

exp 0.2 BP 0.2 WD WDM 0.15 0.15 ] JL ] 4 2 Y[O

Y[CH 0.1 0.1

0.05 0.05

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 x/r (a) x/r (b)

0.2 0.2

0.15 0.15 ] 2 O] 2 Y[CO 0.1 Y[H 0.1

0.05 0.05

0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 x/r (c) x/r (d) Fig. 6. CH4,O2,H2O, CO2 center line distributions.

0.001 p 0.001 p T T O2 O2 0.0001 0.0001 H2O H2O

1e-05 1e-05

||R|| 1e-06 1e-06 2 L

1e-07 1e-07

1e-08 1e-08

1e-09 1e-09 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t[s] (a) t[s] (b) Fig. 7. Residuals of O2, H2O, T and p for BP (a) and JL (b) chemical models.

679 L. F. Gutierrez´ et al. / JAFM, Vol. 9, No. 2, pp. 669-682, 2016. drocarbon combustion mechanisms, carried out Barth, T. J. and D. C. Jespersen (1989). simulations do not correctly describe the chem- The design and application of upwind ical activity and subsequent heat release, in flow schemes on unstructured meshes. areas where it is most intense. It seems too sim- Beaudoin, M. and H. Jasak (2008). Develop- plified kinetics models are not capable of de- ment of a generalized grid interface for scribing the recognized sequential manner of turbomachinery simulations with open- the process of hydrocarbons oxidation, given foam. In Open source CFD International that a detailed combustion mechanism would conference, Volume 2. need around 53 species and 400 elementary reactions. It is to note that losses by radiation Benajes, J., J. Galindo, P. Fajardo and are also cause for discrepancies and to take it R. Navarro (2014). Development of a into account a radiative flux in the energy equa- segregated compressible flow solver for tion should be incorporated, which has not been turbomachinery simulations. Journal of done. There are to many unanswered ques- Applied Fluid Mechanics 7(4). tions about the analytical formulation of radia- Bird, B. B., W. E. Stewart and E. N. Light- tive flux in diffusion flames. foot (2007). Transport phenomena. John Wiley & Sons. 8.1 Future Work Blazek, J. (2005). Computational Fluid Marzouk and Huckaby (2010), recommend to Dynamics: Principles and Applica- work with more complex reaction mechanisms tions:(Book with accompanying CD). El- for the oxidation of hydrocarbons fuels in sevier. flames, for instance the westbrook1988 model Bui-Pham, M. N. (1992). Studies in struc- uses 10 species, one global reaction and 21 ele- tures of laminar hydrocarbon flames. Ph. mentary reversible reactions. It is hope that with D. thesis, Ph.D Thesis,. UCSD. more realistic kinetics models, a significant step to reduce differences between experiments and Chung, K. L. (2006). Combustion physics. simulations will be given. Cambridge University Press. Concus, P., G. H. Golub and G. Meurant ACKNOWLEDGMENTS (1985). Block preconditioning for the To CONICET, MINCyT Crdoba and Aeronau- conjugate gradient method. SIAM Jour- tical departament of UNC for the financial and nal on Scientific and Statistical Comput- facilities support to this research. ing 6(1), 220–252. Coquel, F., Q. L. Nguyen, M. Postel and REFERENCES Q. H. Tran (2008). Local time stepping for implicit-explicit methods on time Andersen, J., C. Rasmussen, T. Giselsson varying grids. Springer. and P. Glarborg (2009). Global combustion mechanisms for use in cfd modeling Favre, A. (1969). Statistical equations of tur- under oxy-fuel conditions. Energy & Fu- bulent gases. Problems of hidrodynamics els 23(3), 1379–1389. and continuum mechanics, 231–266. Bader, G. and P. Deuflhard (1983). A semi- Filho, P. M. F., L. da Silva Vieira, , implicit mid-point rule for stiff systems W. D. P. do Nascimento, L. G. Noleto of ordinary differential equations. Num- and M. Barcelos (2011). Development of ber. Math 41, 373–398. an analysis methodology of aerodynamic profiles applied to design of unmanned Barlow, R. and J. Frank (2007). Piloted aerial vehicles. CH4/Air flames C, D, E, and F. Sandia National Laboratories. Gutierrez,´ L. F., J. P. Tamagno and S. A. Elaskar (2012). High speed flow simu- Barlow, R. A. and J. H. Frank (1998). Ef- lation using openfoam. Mecanica´ Com- fects of turbulence on species mass frac- putacional, Volume XXXI. Number 16. tions in methane/air jet flames. In Sym- Aerospatiale Technology (A). posium (International) on Combustion, Volume 27, pp. 1087–1095. Elsevier. Hairer, E. and G. Wanner (2005). Solving ordinary differential equations ii. stiff and Barlow, R. S. (2003). Sandia TUD differential-algebraic problems. Piloted CH4-Air Jet Flames. http://www.sandia.gov/TNF/ Islam, M., F. Decker, E. D. Villiers, A. Jack- DataArch/FlameD.html . son, J. Gines, T. Grahs, A. Gitt-Gehrke, I. Font and J. Comas (2009). Application 680 L. F. Gutierrez´ et al. / JAFM, Vol. 9, No. 2, pp. 669-682, 2016.

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