Ignition simulation with an Eulerian Monte Carlo

Ignition simulation with an Eulerian Monte Carlo method in conservative formulation for solving joint scalar PDF transport equation Mathieu Ourliac 1 , V. A. Sabel’nikov 1 2

1

, G. Ordonneau

1

and O. Soulard

2

ONERA, Centre de Palaiseau, Chemin de la huni`ere, 91761 Palaiseau, France. CEA, France CEA/DAM, Ile-de-France, BP12 91680 Bruy`eres-le-Chˆatel, France.

Abstract: This study is to explore the numerical features of an Eulerian Monte Carlo (EMC) method developed for solving one point joint probability density function of turbulent reactive scalars. Conservative formulation of EMC derived earlier by Sabel’nikov and Soulard is coupled with a finite volume method and implemented in CEDRE code from ONERA. The hybrid EMC/FV solver is then applied to the computation of a turbulent premixed methane flame ignition over a backward facing step. Corrections algorithm, time averaging procedure and coupling strategy efficiency are evaluated and results are compard with experimental datas. Results for the computation of a microchamber ignition are finally presented.

1

Introduction

In turbulent reactive flows, pollutant production, soot formation and local extinctions/ignitions mainly arise from a conjunction of rare physical events (peak temperature, weak or strong mixture, . . . ) and finite rate chemistry effects. Predicting those phenomena thus requires a precise knowledge of the statistics of the species concentrations and temperature, as well as an accurate description of chemical reactions. Regarding both aspects, the one-point joint composition probability density function (PDF) appears as a promising tool: it contains the detailed one-point statistical information of the turbulent scalars and allows the chemical source terms to be treated exactly. These advantages are nonetheless counterbalanced by a severe numerical constraint. Owing to the presence of many species in practical applications, the composition PDF possesses a potentially high number of dimensions. This in turn induces heavy computational costs. Finite volume method cannot be used as its computational cost increases exponentially with dimensionality. Monte Carlo methods, on the other hand, yield linearly growing effort and are more suited to solve PDF equations. So far, in the field of turbulent combustion, Monte Carlo methods have mostly been considered under their Lagrangian form, following the seminal work of Pope [1]. Numerous publications document the convergence and accuracy of Lagrangian Monte Carlo (LMC) methods [8, 10]. They have been coupled with finite volume methods (including LES [11]), used in many configuration and they have been implemented in commercial CFD codes. However, some difficulties may be experienced when using LMC methods. In particular, the sampling error is not controlled with precision, as it depends on the non-homogeneous distribution of stochastic particles inside the physical domain. Furthermore, to limit the sampling error, it is necessary to couple the LMC method to an Eulerian (RANS or LES) solver for mean quantities. Due to the different nature of the methods involved (Lagrangian/Eulerian), it may result in a heavy tool to manipulate. The development and evaluation of a new Eulerian Monte Carlo (EMC) methods may put both approaches forward as LMC method discrepancies appears in EMC in a different way. EMC Corresponding author. E-mail: [email protected]

methods have been initiated by Valino [3] and generalized (in particular to discontinuous fields) by Sabel’nikov and Soulard [4]. This method is based on stochastic eulerian fields, which evolve from prescribed stochastic partial differential equations (SPDEs) statistically equivalent to the one point joint composition PDF equation. EMC methods have been used in non-conservative form in some works [6], [5] and [12] but at the best authors knowledge, never in conservative form. With one point joint composition PDF methods, several mean fields are required to close the PDF model equations. In the present work, the mean density, turbulent scalars and density fields are supplied by a RANS solver. This kind of hybrid method has been shown, in the frame of LMC approaches, to overcome strong statistical fluctuations arising in stand-alone PDF codes [7]. The feedback coupling is performed through an equation for mean pressure with source terms supplied by stochastic fields properties. This method, as was proved, limit stochastic oscillations [8]. However two difficulties arise: • Statistical errors due to the stochastic nature of EMC may decrease dramatically convergence properties. To this hand, a time averaging procedure is developed in this work. • In hybrid methods, there may exist duplicated fields. Here the only duplicated field is the density. Corrections algorithms are built to make density from EMC relax to density from RANS and then ensure consistency between the two apprroaches. The remaining of the paper is organized as follows. First EMC and RANS mathematical formulation are briefly reviewed and the feedback loop is described. In section 3, numerical scheme is proposed in order to solve SPDE’s. Wall boundary conditions for stochastic fields are discussed then and a numerical strategy is proposed to ensure mass conservation. Corrections algorithm and time averaging procedure developed are then presented. In the fourth section, methane premixed flame ignition calculation are shown and a detailed analysis of statistical and spatial convergence is performed. The results are compared with experimental data from ONERA [15]. Finally the first results of DLR microcombustor [2] ignition with EMC/RANS solver are detailed.

2

Mathematical model and governing equations

The hybrid scheme consists of two separate solvers - the mean flow solver (RANS) for turbulence and mean velocity fields and the EMC solver for species mass fraction and enthalpy.

2.1

RANS solver and Thermochemistry

The RANS technique for simulating turbulent flows is very well known. All the scales of turbulent motion are modeled. In this work, a k-² model is employed for turbulence effects. In a Reynolds averaged framework, Navier-Stokes equations are, with ¯. denoting for Reynolds averaging and e. for Favre averaging: ej ∂ ρ¯ ∂ ρ¯u + =0 (1) ∂t ∂xj 00 u00 ∂ ρ¯ug ei u ej ∂ τ¯ij ∂ ρ¯u ei ∂ ρ¯u ∂p i j + =− + − ∂t ∂xj ∂xi ∂xj ∂xj µ ¶ ej k ∂ µt ∂k ui ∂ ρ¯k ∂ ρ¯u 00 u00 ∂e + = − ρug − ρε i j ∂t ∂xj ∂xj σk ∂xj ∂xj µ ¶ ej ε ∂ ρ¯ε ∂ ρ¯u ∂ µt ∂ε + = ∂t ∂xj ∂xj σε ∂xj

(2) (3) (4)

Reynolds stresses are computed with a Boussinesq hypothesis: µ ¶ ¶ µ ∂e uj ∂e ui 2 ∂e ul 00 u00 = −µ ρug ρk + + δ µ + t ij t i j ∂xj ∂xi 3 ∂xl where the turbulent viscosity coefficient µt is given by : µt = ρCµ k 2 /ε, with Cµ = 0.09; σk = σε = 1.

2.2

Eulerian Monte Carlo solver

In the present work, our attention is limited to the one-point Favre composition PDF fφ (ψ; x, t) of a turbulent scalar field φ. ψ = (ψ1 , ψ2 , ..., ψN ) is the ”phase space” for the scalar variable φ = (φ1 , φ2 , ..., φN ). From the Navier Stokes equation and by standard techniques [1], it is possible to derive an exact transport equation for the density weighted PDF or Favre PDF fφ . Advection and chemical source terms are treated exactly whereas the effects of molecular mixing and turbulent advection require modeling. Turbulent advection is modeled with an isotropic gradient diffusion assumption. Micromixing term represents the effect of mixing at the smallest turbulent scales. An accurate modeling of this term is a still open task. In practical applications described afterwards, the interaction by exchange with mean (IEM) model [9] will be employed. As a result, in this work, the following modeled transport equation is obtained for fφ : µ ¶ ´ ∂fφ ∂ ∂ ∂ ∂ ³ ∂ ρ¯fφ + ρ¯u ej fφ = ρ¯Γt − ρ¯ω ¯ (ψi − ψei ) − (¯ ρS(ψi )fφ ) (5) ∂t ∂xj ∂xj ∂xj ∂ψi ∂ψi In this work, the thermochemical variables involved in the PDF are the species mass fraction and the total enthapy, one have then: φ = (Y1 , ..., YNs , ht ). The PDF transport equation is a high dimensional equation. For a system with Ns species evolving in three space dimensions with temporal variations, this equation spans Ns + 5 dimensions. Conventional finite-volume based discretization schemes are not tractable for Ns À 1. To solve the PDF, in [4] a new Eulerian Monte Carlo method was formulated. In this approach, a set of stochastic fields, whose statistics reproduce the PDF fφ , are used. It is an alternative to the Lagrangian Monte Carlo method. In the EMC method, stochastic fields evolves through prescribed stochastic partial differential equation (SPDE’s). This methodology was coupled with RANS (and LES) solvers (for the turbulence) and was proved in few publication [12] to be able to simulate turbulent reacting flows. A conservative formalism for SPDEs was derived by Sabel’nikov and Soulard [5]. For an easier numerical implementation in the CFD code from ONERA, this conservative formalism is employed here. In the following, each ”realization” (subscripted ”irea = 1, Nr ”) has a set of intrinsic properties, namely the stochastic density rirea , the stochastic velocity u∗j,irea , the species mass fraction Yk,irea and the total enthalpy ht,irea . Henceforth, when no confusion is possible, the subscript irea is omitted. The Nr realizations properties evolve by : M ass f raction (k = 1, Ns ) :

¢ ∂ ¡ ∂ rYk + rYk ◦ (e uj + u∗j ) = −rω(Yk − Yek ) + rSk (Y , T ) (6) ∂t ∂xj

¢ ∂rht ∂ ¡ ∂p + rht ◦ (e uj + u∗j ) = −rω(ht − e ht ) + ∂t ∂xj ∂t 1 ∂Γt p 1 ∂ ρ¯ ˙ j, + Stochastic velocity : u∗j = Γt + 2Γt W ρ¯ ∂xj 2 ∂xj | {z } | {z } usj

T otal enthalpy :

(7) (8)

udj

˙ j are independent standard Brownian processes, j = 1 − 3, Γt = µt /(ρSct ) is a turbulent where W diffusion coefficient and Sct = Let = 1 in this work. The consistency condition : r = ρ is added

to the set of equations 6, 7 and 8. In the total enthalpy equation Eq. 7, viscous dissipation was neglected. The stochastic fields are not physical fields but have their Favre PDF following the same PDF transport equation as the physical turbulent reacting scalars PDF. With the same initial and boundary conditions for the fY ,ht and fφ , one have then : fY ,ht = fφ . In [4], stochastic scalar evolving through SPDEs had their Reynolds PDF equivalent to the Favre PDF of turbulent scalars. In the present work the equivalency is directly made upon Favre PDF. Mean velocity u ej , turbulent diffusion coefficient Γt and turbulent mixing frequency ω = Cφ ε/k stem from RANS solver. The Favre valuesPof thePspecies mass P fractionPand total P averaged P enthalpy are computed by: Yek = rYk / r, e ht = rht / r and Te = rT / r, where the sum are taken over the Nr realizations. The source term Sk in the species mass fraction SPDE’s stands for combustion process. Here is the key aspect of PDF approaches. Transported PDF methods allows this term to be treated exactly with an Arrhenius form for S. All the one-point moments arising from the SPDEs and from the solutions of Eq. 5 will be identical. As a consequence, each stochastic mass fraction field will satisfy mass conservation and bound properties.

2.3

Feedback loop

As pointed in [8, 11], due to the noisy nature of the stochastic fields, the mean pressure calg, where R = R0 P Yk /Wk , R0 being the culated directly from the equation of state : p = ρRT absolute gaz constant, is strongly infected by statistical fluctuations. Even for simple flows, this statistical noise will render the simulation completely inaccurate and even numerically unstable. To overcome this difficulty, an approximate equation is used for the mean pressure calculation. One defines the scalar field Φγ by : Φγ =

p u e2 + (γ − 1) ρ 2

(9)

Knowing the transport equation for mean pressure and mean kinetic energy, one can derive a transport equation for Φγ : ∂ρΦγ ∂ ∂ + (ρΦγ u ej ) = ∂t ∂xj ∂xj

µ ¶ X Ns ∂Φγ ρΓt − (γ − 1)hk Sk , ∂xj |k=1 {z }

(10)

S Φγ

where hk is the specific enthalpy of species k and γ is the specific heat ratio. The source terms S Φγ in Eq. 10 are obtained from stochastic fields properties. This way of coupling permits to improve stability and statistical convergence taking into account pressure/composition correlations in the mean pressure field.

Figure 1: Feedback loop

In summary, the equations Eq. 1, 2, 3, 4, 6 and 7 for the hybrid approach have been provided. The RANS flow solver supplies mean Favre velocity and turbulence fields that are used by the PDF solver to advance the stochastic fields. The stochastic fields properties are used to evaluate the source term in the equation for Φγ . Knowing Φγ the mean pressure is calculated at the next time step with the modified equation of state 9.

3

Numerical solution procedure

3.1

RANS solver

The momentum, turbulent kinetic equation and dissipation equations corresponding to the RANS equations are solved with the eulerian gas phase solver named CHARME from ONERA’s CFD code CEDRE. The CHARME solver is a cell centered multi-domains solver written for nonstructured grid in conservative formalism. A preconditioning technique [13] is employed to overcome the well known numerical fluctuations arising in low Mach number flows simulation. Numerical fluxes are decentered with Roe technique and temporal integration is performed with third order Runge Kutta integration scheme.

3.2

SPDEs scheme

Due to its Eulerian nature, SPDE’s solver is easier to implement to RANS code than a particle PDF solver [10]. There is no need here for projection of stochastic particles properties on the mean solver mesh. Only one mesh is used in this work, the same for the numerical resolution of SPDE’s and RANS equations. Moreover the conservative formalism for SPDE’s allows us to use directly a large part of the RANS code routines. As pointed in [5] Eq. 6 and 7 are hyberbolic stochastic partial differential equations in which the stochastic velocity field is white in time and is interpreted in the Stratonovitch sense (see [4] and references therein). In order to be consistent with Stratonovitch interpretation, we must respect a mid point rule in temporal integration . Therefore, an explicit first order scheme is chosen, with a predictor-corrector procedure generalizing the Heun scheme. This procedure is easy to implement (there is no need for additional terms compared to implicit schemes) and was proved to give satisfactory results [14]. For spatial discretization, scalar numerical fluxes are flux difference splitting type, and an ordinary differential flux procedure is used for decentering. A CFL criterion is built on the stochastic velocity. As shown in [22], this guarantee the linear stability of the overall scheme. The stochastic velocity appears in the CFL criteria as a convective/diffusive CFL criterion. It may seem astonishing in regards to Eq. 6 but it is not. The stochastic advection term is statistically equivalent to the diffusion term in the PDF equation. It is therefore natural to have such a advection/diffusion CFL criterion: ∆t = CF L

∆x2 |e u|∆x + 2Γt

(11)

A ”strong” coupling strategy is employed in this work : one iteration for each solver is performed at each global solver iteration. Therefore we will only take care of CFL criterion from Eq. 11. Fully explicit schemes would be consistent with an Ito interpretation of the stochastic product [4].

3.3 3.3.1

Boundary condition for stochastic fields Inlet/Outlet boundary conditions

The stochastic velocity in Eq. 6 and 7 has no physical meaning. It is not a turbulent velocity fluctuation but is statistically equivalent to the turbulent diffusion term in Eq. 5. One may remark that at time t for a realization numbered irea , the norm of u∗irea depends on the position through Γt but its direction is the same in the whole computational domain. Moreover, its norm may exceed several times the value of the mean flow velocity. Therefore at a physical inlet, one may have a stochastic flux leaving the domain. The same numerical process is involved at a physical outlet where stochastic mass may enter inside the domain. Therefore, a special care was focused on the inlet and outlet boundary conditions. An actualization procedure of the boundaries nature is performed at each iteration according to the Wiener vector noise direction. 3.3.2

Wall boundary conditions

Numerical treatment of stochastic differential equation has been the subject of numerous studies [17], [18], [19], [20]. These studies are based on a Lagrangian formalism. Mirror reflection is commonly used for simulating a zero-flux boundary condition, however the transposition to eulerian formalism is not obvious. At the wall the flux of any scalar φ is equal to zero (wall is supposed impermeable and adiabatic). One has for stochastic component due to non slip condition : p ˙ n φ − Γt ∇n φ = 0, 2Γt W

(12)

˙n =W ˙ .n, n is the normal to the wall directed outward, ∇n is the normal to the wall where W derivative. As is well known Γt is zero at the wall (Γt ∝ nm , m = 3 − 4 with n is the distance to the wall). Thus we have : lim us ∝ nm/2 → 0, (13) n→0

and Eq. 12 is satisfied. Classical condition : ∇n φ|s = 0 is used at the wall. But what happens if the mesh involved is not fine enough to resolve the boundary layer, as it is the case in the present work? The configuration presented in figure 2 is considered. To calculate the flux, the value of Γt is taken at the cell face (a) at the midpoint A. Since Γt 6= 0 an accumulation of stochastic mass will arise. To circumvent this difficulty, we propose to use stochastic flux passing through the wall.

˙ 1 > 0, W ˙ 2 < 0). Figure 2: SPDEs numerical behavior near an impermeable wall (W This procedure is presented in figure 3. The stochastic flux fl crossing the wall is decentred and calculated with first order interpolation for Γt (x), r(x), Yj (x), ht (x). This procedure was proved to be stable (with moderately stiff gradient) for 1D SPDE computation between two walls. Everything occurs as if (for a specific realization at time step n) the stochastic field is translated in block by the stochastic velocity.

Figure 3: Stochastic flux crossing the wall

3.4

Correction algorithms

As discussed above, the hybrid algorithm is consistent with RANS at the level of the governing equations. However due to the noisy nature of fields and the accumulation of numerical errors, the solutions may be not consistent at a numerical level. In the present work, only the mean density is duplicated. We will call ”physical density” denoted ρ the mean density obtain from continuity equation 1 and ”mean stochastic density” denoted r the one obtained as Reynolds P averaging of stochastic densities r = Nr r/Nr . The stochastic density is a very noisy field, its calculation is then less accurate than calculation of the physical density field. The consistency condition is then identified as [5] : r = ρ (14) At initial state, consistency condition is satisfied. As explained above, summation on Ns species and Nr stochastic realization of Equation 6 leads to continuity equation if and only if : u e∗i = 0

(15)

At a theoretical level, if consistency is satisfied at initial state, then the condition 14 is satisfied by construction of u∗ (see [21]). So as to respect Eq. 14 at the numerical level, the stochastic velocity u∗j,irea for realization irea , is corrected : u∗j,icrea = u∗j,irea − u e∗j /Nr

(16)

However as pointed out by Muradoglu et al. [10] this simple correction of the stochastic velocity is not sufficient to ensure consistency. In order to limit the amplitude of statistical error, a diffusion term is added to the mass fractions SPDEs. The corrected SPDEs for mass fractions are : ¢ ∂ ¡ ∂ ∂ ∂ rYk + rYk ◦ (e uj + u∗j c ) = −rωk (Yk − Yek )+rSk (Y , T )− ρΓt ∂t ∂xj ∂xj ∂xj

·µ

r−ρ ρ

¶

¸ Yk , (17)

The correction diffusion term is equal to zero once r = ρ. Those two corrections (Eq. 17 and 16) have been implemented into CEDRE and will be used in future calculations.

3.5

Time Averaging technique and algorithm

Time averaging is a powerful tool to reduce the statistical noise in the mean fields extracted from stochastic fields without increasing Nr . The time averaging method employed in the present work

is defined, for mean field Q at time step n: n+1 Qn+1 + αQnT A T A = (1 − α) Q

This procedure is applied to quantities transfered from EMC to RANS, that is to say, the following mean quantities: Yek , Te and SeΦγ . Time averaging is performed when the calculation is deemed to be close to a stationary state. In this work the criteria for time averaging enforcement f n+1 /ln δQ f n < 1. When time averaging is turned on, α is computed by is : ²T A = ln δQ √ the expression proposed by Pope et al. [8]: α = 1/ Nit where Nit is the number of iterations performed. The overall solution sequence can be summarized as follows. The EMC code runs for one iteration. At the end of the iteration, mean quantities and source terms for Φγ are calculated. Those quantities are transferred to the RANS code. If ²T A is decreased to the specified value ²0 , time averaging is performed. RANS code is then run for one iteration. The coupling is done here in a way to be able to simulate unstationnary flows (time averaging is then urned off) . Mean velocity, k and ε are transferred to the EMC code for the following iteration.

4

Ignition behind a backward facing step

In this section, the hybrid method described above is applied to ignition simulation of a premixed methane flame over a backward facing step. Statistical and spatial convergence are evaluated before comparison with experimental data [15]. The primary purpose of this study is to validate the hybrid EMC/RANS method for turbulent flows ignition simulation in terms of temporal evolution, numerical accuracy and efficiency. The second task is to evaluate the performance of the feedback coupling method, corrections algorithms and time averaging technique.

4.1

Configuration and chemical scheme

The physical domain is shown in figure 4 with inlet and outlet conditions in table 5. This configuration is well known in the turbulent combustion community. A mixture is injected and the combustion process is stabilized by the recirculation zone behind the step. It was experimentally studied at ONERA with temperature PDF measurements: this is the reason for the choice of the configuration for the hybrid EMC/RANS method validation. Furthermore, the interaction of the flame with the upper wall will permit to validate the numerical model for stochastic fluxes at the wall. At the upper and lower wall, wall functions are applied for k and ε, and walls are assumed to be adiabatic. This condition is a simplified one, it overestimate temperature at the lower wall and underestimate the temperature at the upper wall. One may keep this remark in mind when comparing computed and experimental data. Complete combustion process of methane is described with a 5 species 1 reaction scheme [16]. CH4 + 2O2 + M → CO2 + 2H2 O + M Chemical source terms follow the Arrhenius law, it expresses for the advancement variable: a S = AYCH Y b e−Ta /T 4 O2

where : a = 0.2, b = 1.3, A = 6.7 109 mole/s, and Ta = 24 300 K [16]. It is well known that such a simplified chemical scheme is inadequate to be physically accurate. However it is adequate for code validation performed here. For all calculation the same CF L criteria (equation 11) is employed : CF L = 0.1. In the following, calculations are done with mesh 3 (see table 12) and the number of realization is kept unchanged Nr = 50.

Figure 4: Backward facing step configuration Inlet

Outlet

velocity U0 temperature T0 turbulent kinetic energy k0 turbulent dissipation ε0 equivalence ration pressure

58 m/s 525 K 60 m/s2 800 m2 /s3 1 1 bar

Figure 5: Inlet and outlet conditions.

4.2

Ignition simulation

A source term in the SPDE for total enthalpy is turned on for t ∈ [0; 5 10−6 s], which correspond to 5 global iterations. The calculation is 2D and the volume of the energy source is 0.05m×0.01m. The amount of energy deposited inside the computational domain is 30 mJ. Fig. 6 shows mean temperature and root mean square temperature time evolution. The first image represent the state at when enthalpy source is tuned off. The physical time between two images is 10−4 s. Due to the energy source, large pressure oscillations occur. The flame is highly disturbed by the propagation of pressure waves, however the flame remains anchored behind the step.

4.3

Stationnary Steady state

At a physical time of t = 2L/U0 the time averaging procedure is turned on. This physical time nearly correspond to the criteria presented in section 3.5. The time averaging procedure is very efficient to dissipate the acoustic waves. Due to the stochastic noise, the residus of mean quantities cannot exactly converge to zero. There still exist oscillations around the the converged state. Therefore, our convergence criteria is based on the steadiness of the mean value on 100 iterations of the residus of the momentum equation. Before analyzing in details the results, one can see on fig. 8 what the mean temperature fields look like when convergence is obtained. With the flame angle one can calculate the flame velocity ST : ST = U0 sin α, where α is the flame angle. Here the flame velocity obtained is 6.1 m/s. The value obtained experimentally by DRASC at ONERA [15] is 5% smaller. This error may be explained by the simplicity of the chemistry employed here. Figure 9 shows a sample stochastic profiles of mean temperature. Those profiles are steep and have all the same shape. Shifting between the profile recover the diffusion for the mean profile.

0.25

0.25

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840 1140 1440 1740 2040 2340

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840 1140 1440 1740 2040 2340

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5

85 165 245 325 405 485 565 645

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5

55 105 155 205 255 305 355 405

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840 1140 1440 1740 2040 2340

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5

75 145 215 285 355 425 495 565

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45 85 125 165 205 245 285

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Figure 6: Te and Trms time evolution

2500

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T

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500 0.02

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Y

Figure 7: Stochastic temperature profiles

4.3.1

Analysis of the statistical error

In order to analyze the statistical errors, we let the number of realizations Nr vary, while the mesh is kept unchanged. In this work 5 calculations are performed on mesh 3 : Nr = 5, Nr = 10, Nr = 25, Nr = 50 and Nr = 100. We consider that convergence is attained for a computational time τc of 10 times the residence time L/U0 : τc = 0.15 s. Figure 10 shows the mean temperature and rms temperature profiles at two axial positions with different number of stochastic fields. Those figures tend to indicate the existence of a convergence of the mean and rms temperature profile when Nr increases. For the two smallest values of Nr , the error is large as much for the mean temperature as for the rms temperature. Moreover, the influence of isolated stochastic fields can be evidenced, it disrupt the profile. This effect disappears for larger values of Nr .

0.25 0.2

T001:

0.15

600

1031.25

1462.5

1893.75

2325

0.1 0.05 0

0

0.2

0.4

Figure 8: Mean EMC/RANS solver

0.6

0.8

temperature

from

Figure 9: Mean temperature from experiment [15]

800

2400 2200 2000

Mean Temperature

Temperature RMS

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400

N=5 N = 10 N = 25 N = 50 N = 100

200

1800 1600 1400 1200

N=5 N = 10 N = 25 N = 50 N = 100

1000 800 600

0.02

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Y

Figure 10: Influence of the number of stochastic fields Nr on the rms temperature and mean temperature vertical profiles x = 250mm The solution for Nr = 100 is the reference solution subscripted ”ref ”. Then the error in mean E1 and rms temperature E2 , relative to this solution, are calculated : R |Te − Teref |dxdy E1 = D R Teref dxdy D

¯ ¯ ¯ R ¯p g q g ¯ T 002 − T 002 ¯ dxdy ref ¯ D¯ E2 = R qg 002 dxdy Tref D

where D is the whole computational domain. Figure 11 shows the evolution of E1 and E2 with the number of realizations. The convergence is observed : the rate of convergence is 0.9 for the mean and 0.7 for the rms temperature. This decay is slightly larger than the theoretical 0.5 rate expected. This is surely due to the correction algorithms and time averaging procedure. The precision recovered by using 100 realization rather than 50 is only 4% of the mean temperature and 7% of the rms temperature. By multiplying the number of realizations by 2, the CPU cost per iteration is multiplyed by 1.9. Nr = 50 is a good compromise between computational cost and precision.

Figure 11: Evolution of the error E1 and E2 against the number of realization

4.3.2

Analysis of the spatial error

Spatial discretization error is due to the finite number of mesh cells. The RANS part of the solver is second order accurate in space. Previous simulations with hybrid RANS/EMC [22] or hybrid RANS/PDF [23] on the same configuration showed that a grid resolution given by ∆x = 10−2 m and 2.4 10−3 m was sufficient to capture the flame structure. This grid was chosen to be the coarser. In order to analyze the spatial convergence of the method, we compared the results obtained with 3 finer grids. Table 12 recapitulates the resolutions of the 4 grids. The same refinement coefficient was applied to every mesh in the near step nose region. Mesh Mesh Mesh Mesh

1 2 3 4

∆xmin (m) 1.5 10−2 6 10−3 4 10−3 1.5 10−3

∆ymin (m) 5 10−3 5 10−3 3 10−3 1 10−3

Nx ∗ Ny 900 1600 4000 8000

Figure 12: Computational meshes. Figure 13 shows the spatial error obtained with 50 realizations as a function of the number of computational cells. We consider mesh 4 as the reference solution. A convergence is clearly obtained but the improvement of the solution is small compared to the increase of the computational cost.

Figure 13: Evolution of the error E1 and E2 against the number of computational cells

4.3.3

Comparison with experimental datas

Our results are compared with those of [15]. Level of temperature are closed to those of the experiment. The profiles are qualitatively similar between experiment and calculation. However, the temperature in burnt gazes is bigger in the calculation. This may be explained again by the chemistry which gives an overestimated adiabatic temperature. In addition, in the calculation, the walls are supposed to be adiabatic, which is actually not the case. The main point of this comparison is the good agreement concerning the position and width of the RMS temperature peak. At the position x = 0.25 m the large width is well predicted by EMC/RANS solver. That means that rare events like a large or small flame angle are predicted.

4.4 4.4.1

Corrections effect Coupling strategy for stability

First calculations were performed with a direct feedback coupling, meaning without a new equag. This way of coupling gave poor results in term of numerical tion for Φγ but with p = ρRT stability. An example of pressure profile after 1 000 iterations is shown fig. 19 for different ways of coupling. One can see that the direct way of coupling leads to large oscillations in the near flame region. Actually due to the noisy nature of temperature, steep gradient of density and temperatures are inconsistent leading to large pressure fluctuations and preventing convergence. One can remark that density is a duplicated field, so we may use mean density r for pressure g. With this approach, shifting between density and temperature calculation with : p = rRT profile is diminished but stochastic oscillations are increased leading to even worst results. As a conclusion, an additional mean equations is always needed for smoothing information from Monte Carlo solver. This observation, already evidenced by Pope et al. [8] for Lagrangian PDF methods is recovered here. 4.4.2

Corrections for consistency

As presented in 3.4, the first correction ensure that u e∗ = 0 and the second make density r relax to the physical density. The influence of those terms is evaluated independently. Figure 19 shows the profile of Q = (r − ρ)/ρ. Large gradients of Q appear in the flame region (y = 0.03m). This

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Figure 15: TRM S profile at x = 250mm

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Figure 18: Te profile at x = 460mm pathological behavior is due to statistical error. All the means calculated by the EMC solver contains statistical errors. The errors are confined in the flame region and are small (less the 5%). However during the calculation the inconsistency between stochastic and physical density may decrease convergence properties. In order to overcome this difficulty, the idea was to diffuse

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Figure 19: Horizontal profile of density difference with and without correction term

this error into the whole domain (see Eq. 17). The error summation in the whole domain is kept unchanged, but it is not confined in the flame region anymore, and the maximum value is decreased. The consistency is not fully satisfied when the correction term is turned off, while it is successfully enforced by the correction algorithm. The stochastic noise being binomial, with sufficiently high number of stochastic fields u e∗ ' 0 would be respected. However for reason of computational cost a moderately number of realization is required. This is the reason for the addition of the correction term in Eq. 16. In figure 20 time evolution of the ratio of longitudinal components of non corrected stochastic velocity ues x to the mean flow velocity u ex is shown for the calculation with 50 realizations. This error is therefore not negligible, the correction term is necessary to ensure consistency between the two solvers. 4.4.3

Wall boundary conditions

Without the wall correction presented in section 3.3.2 large gradients of density arise near the wall and the calculation crashes. For a specific realization, if the stochastic velocity is directed toward the wall, density is accumulated near the wall. By adding the stochastic flux, the mass accumulation is removed. This procedure of stochastic flux crossing the wall was proven, in previous work, to be able to guarantee mean stochastic mass conservation.

5 5.1

Laser ignition in a micro-combustor with EMC/RANS solver Main features

The configuration of interest is a chamber filled by a coaxial H2 /O2 injector. Physical processes involved in this micro-combustor are of various types. We have to study here the ignition of a diffusion flame which is characterized by a reaction zone between oxidizer and fuel. Moreover, the large volume of the chamber where oxygen and hydrogen are injected at high velocity enables large recirculation zones, promoting mixing of the propellants. The energy deposited into the chamber by the laser pulse produces locally a rapid pressure and temperature rise which generates a first pressure wave and modify locally the flow. Then

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the flame expands and this expansion modifies more globally the flow which interacts with the chamber dynamics. Here, we intend to evaluate the new EMC/RANS solver by comparing the experimental results in [2] with the simulation. The real phenomenon is 3D, but for reason of computational cost, we have performed a 2D calculation.

5.2

Boundary conditions and chemical scheme

Injection mass flow have been chosen according to table 21. A nozzle diameter of 4 mm has been assumed in order to avoid supersonic flow in the oxygen pipe. The actual injector and chamber geometries were used except for the chamber exit where a straight tube was replaced by a classical convergent-divergent nozzle in order to set the location of the sonic line and to avoid numerical difficulties. Species Mass flow (g/s) k Temperature

O2 1.15 2 300

H2 0.575 4 300

Figure 21: Inlet conditions For the outlet condition, a constant pressure equal to 1 bar is imposed. In y = 0, there is a symmetry condition. An adiabatic condition is prescribed at the wall. As initial conditions for the steady-state cold flow, a RANS calculation was performed. The whole computational domain is filled with hydrogen and oxygen at a pressure is of 2.5 bar. The stationary cold flow is used as initial conditions for transient computation. For hydrogen and oxygen, several kinetic schemes are available. In this work, the Jachimowski chemistry was chosen for its good predictivity of ignition delays. The mesh is fully unstructured and uses trihedral cells. It counts around 63 000 cells. Due to

the CFL condition (Eq. 11) the time step is very small. In the present work, CF D = 0.1 so the : ∆t = 5 10−10 s. Actual physical phenomena involved in the laser ignition cannot be reproduced with the current model. Ionisation in particular is not modelled. So, laser pulse is represented by a source term in the enthalpy equation for cells corresponding to the laser location x = 0.065 m, y = 0.005 m with diameter D = 0.01 m. Then source term is switched off for t = 10−7 s. The subsequent energy deposited in the flow is 30 mJ.

5.3

Results

The calculation is performed with 20 realizations.

Figure 22: Time evolution of mean temperature and temperature rms At the initial cold state, the chamber is filled with O2 /H2 mixture. The energy deposit creates an hot kernel and an acoustic wave. The reaction occur nearly instantaneously in the kernel and give rise to a second acoustic wave. One can see on figure 22 that the flame propagation inside the chamber by consuming the mixing H2 /O2 for t = 5 10−5 s. The calculation is still running.

6

Conclusions

A new conservative form was proposed for SPDEs. The EMC based based on conservative SPDEs was implemented into CEDRE CFD code and coupled with RANS solver. Corrections algorithms and boundary conditions for stochastic fields are proposed. The EMC/RANS method is then applied to the calculation of a premixed flame ignition behind backward facing step. The corrections properties allows to obtain good convergence properties and good agrement with experimental results. The simulation of the first stage of ignition in DLR microchamber is done. The calculation is still running. The hybrid method developed in the present paper may be improved in two directions. First, the PDF approach may be extended to joint velocity-composition PDF. Second, the implementation of an implicit numerical method for solving SPDEs would permit to increase the com-

putational time step (no more CFL condition to respect) and thus decrease the computational cost.

References [1] Pope S.B., (1985), PDF methods for turbulent reactive flows, Prog. Energy Combust. Sci., vol. 11, pp. 119-192. [2] Schmidt, V., Klimenko, D., Haidn, O., Oschwald, M., Nicole, A., Ordonneau, G., Habiballah, M., (26-30 october 2003), Experimental investigation and modeling of the ignition transient of a coaxial H2 /02 -injector, 5th International Symposium on Space Propulsion, Chattanooga (USA) [3] Valino L., (1998), A fiel Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent flow, Flow Turbul. Combust., vol. 60, pp. 157-172. [4] Sabelnikov V.A., Soulard O., (2006), White in time scalar advection as a tool for solvint joint composition PDF equations, Flow. Turbul. Combust. Sci., vol. 77, pp. 333-357. [5] Soulard O., Sabelnikov V.A., (2006): Eulerian Monte Carlo methods for solving joint velocityscalar PDF equations in turbulent reacting flows, European Conference on Computational Fluid Dynamics, Delft, Netherlands. [6] Mustata R., Valino L., Jimenez C., Jones W.P., Bondi S., (2006), A probability density Eulerian Monte Carlo field method for large eddy simulations: Application to a turbulent piloted methane/air flame (Sandia D), Combustion and flame, vol. 145, pp. 88-104. [7] Jenny P., Pope S.B., Muradoglu M. and Caughey D.A., (2001), Ahybrid algotrithm for the joint PDF equation of turbulent reactive flows, Journal of Computational Physics, vol. 145, pp. 88-104. [8] Muradoglu M., Lui K., Pope S.B., (2003), PDF modeling of a bluff-body stabilized turbulent flame, Combustion and flame, vol. 132, pp. 115-137. [9] Villermaux J, Devillon J.C., (1972), Repr´esentation de la redistribution des domaines de s´egr´egations dans un fluide par un mod`ele d’int´eraction ph´enom´enologique, 2nd International Symposium on Chemical Reaction Engeneering, Amsterdam, Netherlands. [10] Muradoglu M., Pope S.B., Caughey D.A. (2001), The hybrid method for the PDF equations of turbulent reactive flows: consistency conditions and correction algorithms, Journal of Computational physics, vol. 172, pp. 841-878. [11] Venkatramanan R., Pitsch H., Fox R.O., (2005), Hybrid large eddy simulation/Lagrangian filtered-density-function approach for simulating tubrulent combustion, Combustion and Flame, vol. 143, pp. 56-78. [12] Jones W.P., Navarro-Martinez S., (2007), Large eddy simulation of autoignition with a subgrid probability density function method, Combustion and Flame, vol. 150, pp. 170-193. [13] Turkel E., (1993), Review of Preconditioning Methods for Fluid Dynamics, Applied Numerical Mathematics, vol. 12, pp. 257-284. [14] Carillo O., Ibanes M., Garcia-Ojalvo J., Casademunt J. and Sancho J.M., (2003), Intrinsinc noise induced phase transition: beyond the noise interpretation., Physical review. E, statistical, nonlinear, and soft matter physics, vol. 67.

[15] Magre P. and Collin G., (1994), Application de la DRASC `a l’op´eration A3C, Technical Report ONERA [16] Westbrook C.K., Dryer F.L. (1984), Chemical kinetic modeling of hydrocarbon combustion, Prog. Energy Comb. Sci., vol. 10 [17] Szymczak P., Ladd, A.J.C. (2003) Boundary conditions for stochastic solutions of the convection-diffusion equation Physical Review, E 68 [18] Szymczak P., Ladd, A.J.C. (2004) Stochastic bondary conditions to the convection diffusion equation including chemical reactions at solid surfaces, Physical Review, E 69 [19] Dreeben T., Pope S.B. (1997) Probability density function/Monte Carlo simulation of nearwall turbulent flows, Journal of Fluid Mechanics, vol. 357 [20] Minier J.-P., Pozorski J. (1999) Wall boundary conditions in PDF methods and application to a turbulent channel flow, Physic of Fluids, vol. 11 [21] Soulard O., Sabel’nikov V.A. (2006) Eulerian Monte Carlo methods for solving joint velocity and mass fraction PDF in turbulent reactive gazs flows, Combustion, Explosion and Schock Waves, vol. 11 [22] Soulard O. (2006) Approches PDF pour la combustion turbulente: Prise en compte d’un spectre d’echelles turbulentes dans la mod´elisation du micromelange et elaboration d’une methode de Monte Carlo Eulerienne, Phd Thesis, Rouen University [23] Thauvoye C. (2005) Simulation num´erique d’´ecoulements turbulents r´eactifs par une m´ethode hybride `a fonction densit´e de probabilit´es., Phd Thesis, Poitier University