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# Numerical Simulations

### Numerical models for combustion simulation

#### Eddy Break-up model

High Reynolds and Damkohler numbers are assumed in the model. The mean reaction rate is mainly controlled by the turbulent mixing time \tau_t defined as the ratio between turbulent kinetic energy and its dissipation rate (SpaldingDB:1971). In term of progress variable c , the expression of the mean reaction rate that is usually used is:

$\bar{\dot{\omega }}= C_{EBU} \bar{\rho }\frac{\varepsilon }{\kappa } \tilde{c}\left(1-\tilde{c}\right)$

or in term of fuel mass fraction Y_F , assuming excess of oxidizer:

$\bar{\dot{\omega }}=C_{EBU} \bar{\rho }\frac{\varepsilon }{\kappa } \frac{\tilde{Y}_{F} }{Y_{F}^{0} } \left(1-\frac{\tilde{Y}_{F} }{Y_{F}^{0} } \right)$

The eddy-break model (EBU) is available in many CFD commercial codes. The model has been extensively employed in numerical analysis because the reaction rate is simply written as a function of known quantities and therefore its use is convenient from the computational point of view. For turbulent combustion, the EBU model gives generally better results than the simple Arrhenius model. It is known that the EBU model tends to overestimate the reaction rate, especially in highly strained regions where the turbulent mixing time is small such as close to flame holders and walls (PoinsotT:2001). The accuracy of this model can be improved with the introduction in the formulation of the efficiency function of the ITNFS (Intermittent Turbulent Net Flame Stretch) model (Meneveau and Poinsot, 1991).

Other weaknesses of the original EBU model are that the reaction rate does not depend on the chemical properties of the mixture and that the turbulence is assumed to be homogeneous and isotropic. Some modifications of the constant placeCEBU have been proposed by Said and Borghi (1988) to mimic chemical features.

Spalding D.B., Stephenson P.L. and Taylor R.G. (1971) A Calculation Procedure for the Prediction of Laminar Flame Speeds. Combustion and Flame, 17:55-64.(BibTeX)
Poinsot T. and Veynante D. (2001) Theoretical and numerical combustion. Edwards, Philadelphia.(BibTeX)

Meneveau C. and Poinsot T., 1991 Stretching and quenching of flamelets in premixed turbulent combustion. Combust. Flame, 86, 311-332.

Said R. and Borghi R. (1988), A simulation with a cellular automaton for turbulent combustion modelling. 22nd Symposium (Int.) on Combustion, The Combustion Institute, placeCityPittsburgh, 569-577.

#### Flame Surface Density Models

In the flamelet assumption, the chemical reaction occurs in thin layers separating fresh gases from fully burned ones - high Damkoler number. The reaction zone may then be viewed as a collection of laminar flame elements or flamelets. Under the flamelet assumption the mean reaction rate can be described as the product of the flame surface density \Sigma (flame surface area per unit volume) and the local consumption rate per unit of flame area (Bray et al. 1989, Candel et al. 1988, Marble and Broadwell 1977, Pope 1988):

$w=\rho _{\begin{array}{l} {0} \\ {} \end{array}} u_{L} \Sigma$

where \rho_0 is the fresh gases density, u_L is the average flame consumption speed along the surface and \Sigma is the flame surface density. The average flame consumption speed uL and the unstretched laminar flame speed S_L are linked by the stretch factor I_0 [Bray 1990]:

$u_{L} =I_{0} s_{L}^{0}$

The flame surface density may be modelled using an algebraic expression or solving a balance equation.

#### Algebraic Expressions for the flame surface density

An algebraic expression for the flame surface density from Bray et al. (1989) based on a stochastic process analysis is:

$\Sigma =\frac{g\bar{c}\left(1-\bar{c}\right)}{\bar{\sigma }_{y} \hat{L}_{y} } =\frac{g}{\bar{\sigma }_{y} \hat{L}_{y} } \frac{1+\tau }{\left(1+\tau \tilde{c}\right)^{2} } \tilde{c}\left(1-\tilde{c}\right)$

where g and \sigma _y are constant with values of 1.5 and 0.5 respectively. The tilde refers to a Favre density weighted average value while the Reynolds average is denoted by an overbar. c is the progress variable and $\tau$ is the reaction heat release factor defined as:

$\tau =\frac{\rho _{u} }{\rho _{b} } -1$

L_y is the wrinkling length scale of the flame front, generally modelled as proportional to the integral length scale lt:

$L_{y} =C_{l} l_{t} \left(\frac{u_{L}^{0} }{u'} \right)^{n}$

where the constant C_l and n are of order unity. This expression generates a very fast burning rate along walls. In order to overcome this unphysical behaviour, an alternative formulation for the flame length scale was proposed by Watkins et al (1996) and applied in spark-ignition engines by Abu-Orf and Cant (2000):

$\begin{array}{l} {L_{y} =C_{l} L_{t} f\left(\frac{u'}{S_{L}^{0} } \right)} \\ {L_{L} =\frac{\upsilon }{u_{L} } } \end{array}$

where L_y is not a function of the integral length scale lt but of the laminar flamelet length scale L_L .

The flame surface density can also be expressed through a fractal analysis (Gouldin et al. 1989) as:

$\Sigma =\frac{1}{L_{outer} } \left(\frac{L_{outer} }{L_{inner} } \right)^{D-2}$

where L_{inner} and L_{outer} are respectively the inner and outer cut-off length scales, and D is the fractal dimension of the flame surface. The cut-off scales are usually derived from the turbulence Kolmogorov and integral length scale. The cut-off scales can also be obtained from DNS calculations.

#### Flame Surface Density Balance Equation

The balance equation for the flame surface density \Sigma can be found in various forms in the literature. In a general form, it can be written as:

$\frac{\partial \Sigma }{\partial t} +\frac{\partial \tilde{u}_{i} \Sigma }{\partial x_{i} } =\frac{\partial }{\partial x_{i} } \left(\frac{\upsilon _{t} }{\sigma _{c} } \frac{\partial \Sigma }{\partial x_{i} } \right)+\kappa _{m} \Sigma +\kappa _{t} \Sigma -D$

The equation is unclosed and requires modelling. Various closures of the equation are briefly summarized in Table 1 (Poinsot and Veynante 2001): the Cant-Pope-Bray (CPB) model (Cant et al., 1990), the Coherent Flame Model (CFM) (Duclos et al., 1993), the Mantel and Borghi (MB) model (1994), the Cheng and Diringer (CD) model (1991), the Choi and Huh (CH) model (1998). The table was extracted from the textbook by Poinsot and Veynante (2001). A similar table can be found also in Veynante and Vervisch (2002). In the latter reference, the terms are expressed as function of the progress variable c instead than of the fuel mass fraction Y . The D term is a destruction or consumption term while \Sigma \kappa _m and \Sigma \kappa _t are source terms due to strain rate induced by the mean flow field and the turbulence respectively. In some models (CD and Ch), the term \Sigma \kappa _m is neglected. The following parameters are model constants: \alpha _0 , \beta _0 , \gamma , \lambda , a , c , C , E and K . A_{ik} is an orientation factor and l_{tc} is an arbitrary length scale introduced for dimensional consistency. \Gamma _k is the efficiency factor in the ITNFS (Intermittent Turbulent Net Flame Stretch) model (Meneveau and Poinsot 1991).

Comparisons of various flame surface density models may be found in Duclos et al. (1993) and in Choi and Huh (1998). Duclos et al. found in their analysis that only the CFM-2 formulation is able to predict the so-called bending phenomenon, where the turbulent flame speed decreases when the turbulence level increases beyond a certain value. Prasad and Gore (1999) also compared the capabilities of CPB, CFM1, MB and CD models in predicting a turbulent premixed jet flames.

Table 1. Source ( \kappa \Sigma) and consumption ( D ) terms in the flame surface density balance equation. Poinsot and Veynante (2001).

 MODEL \kappa _{m} \Sigma \kappa _{t} \Sigma D CPB Cant at al. (1990) A_{ik} \frac{\partial \tilde{u}_{k} }{\partial x_{i} } \Sigma \alpha _{0} C_{A} \left(\frac{\varepsilon }{\upsilon } \right)^{1/2} \Sigma \begin{array}{l} {\alpha _{0} S_{L} \frac{2+e^{-aR} }{3\tilde{Y}/Y_{0} } \Sigma ^{2} } \\ {R=\frac{\tilde{Y}\varepsilon }{Y_{0} \Sigma S_{L} \kappa } } \end{array} CFM1 Duclos et al. (1993) A_{ik} \frac{\partial \tilde{u}_{k} }{\partial x_{i} } \Sigma \alpha _{0} \frac{\varepsilon }{\kappa } \Sigma \beta _{0} \frac{S_{L} +C\kappa ^{1/2} }{\tilde{Y}/Y_{0} } \Sigma ^{2} CFM2a Duclos et al. (1993) A_{ik} \frac{\partial \tilde{u}_{k} }{\partial x_{i} } \Sigma \alpha _{0} \Gamma _{K} \frac{\varepsilon }{\kappa } \Sigma \beta _{0} \frac{S_{L} +C\kappa ^{1/2} }{\tilde{Y}/Y_{0} } \Sigma ^{2} CFM2-b Duclos et al. (1993) A_{ik} \frac{\partial \tilde{u}_{k} }{\partial x_{i} } \Sigma \alpha _{0} \Gamma _{K} \frac{\varepsilon }{\kappa } \Sigma \beta _{0} \frac{S_{L} +C\kappa ^{1/2} }{\frac{\tilde{Y}}{Y_{0} } \left(1-\frac{\tilde{Y}}{Y_{0} } \right)} \Sigma ^{2} MB Mantel and Borghi (1994) E\frac{\mathop{u''_{i} u''_{k} }\limits^{\sim } }{k} \frac{\partial \tilde{u}_{k} }{\partial x_{i} } \Sigma \alpha _{0} \left(Re_{t} \right)^{1/2} \frac{\varepsilon }{\kappa } \Sigma +\frac{F}{S_{L} Y_{0} } \frac{\varepsilon }{\kappa } \mathop{u''_{i} Y''}\limits^{\sim } \frac{\partial \tilde{Y}/Y_{0} }{\partial x_{i} } \frac{\beta _{0} S_{L} \left(Re_{t} \right)^{1/2} \Sigma ^{2} }{\frac{\tilde{Y}}{Y_{0} } \left(1-\frac{\tilde{Y}}{Y_{0} } \right)\left(1+c\frac{S_{L} }{\kappa ^{1/2} } \right)^{2\gamma } } CD Cheng and Diringer (1991) \begin{array}{l} {\alpha _{0} \lambda \frac{\varepsilon }{\kappa } \Sigma {\rm \; \; }} \\ {for{\rm \; \; }\kappa _{{\rm t}} \le {\rm \; }\alpha _{{\rm 0}} {\rm K\; }\frac{{\rm S}_{{\rm L}} }{\delta _{{\rm L}} } {\rm \; }} \end{array} \beta _{0} \frac{S_{L} }{\tilde{Y}/Y_{0} } \Sigma ^{2} CH1 Choi and Huh (1998) \alpha _{0} \left(\frac{\varepsilon }{15\upsilon } \right)^{1/2} \Sigma \frac{\beta _{0} S_{L} \Sigma ^{2} }{\frac{\tilde{Y}}{Y_{0} } \left(1-\frac{\tilde{Y}}{Y_{0} } \right)} CH2 Choi and Huh (1998) \alpha _{0} \frac{u'}{l_{tc} } \Sigma \frac{\beta _{0} S_{L} \Sigma ^{2} }{\frac{\tilde{Y}}{Y_{0} } \left(1-\frac{\tilde{Y}}{Y_{0} } \right)}

Bray K.N.C., Champion M., Libby P.A., (1989) The interaction between turbulence and chemistry in premixed turbulent flames. In: Borghi R., Murphy SN editors. Turbulent Reactive Flows, Lecture notes in engineering, Springler Verlag, 541-563.
Candel S.M., Maistret E., Darabiha N., Poinsot T., Veynante D., Lacas F. (1988) Experimental and numerical studies of ducted flames. Marble Symposium, 209-236
Marble F.E. and Broadwell, J.E. (1977), The coherent flame model for turbulent chemical reactions. Tech. Rep. TRW-9-PU Project Squid, Perdue University West Lafayette.
Pope S.B. (1988), The evolution of surfaces in turbulence. Int. J. Engng. Science, 26, 445-469.
Watkins A.P., Li S.P. and Cant R.S., (1996) Premixed combustion modelling for spark- ignition engine applications. Sae paper No. 961190.
Abu-Orf G.M. and Cant R.S., (2000), A turbulent reaction rate model for premixed turbulent combustion in spark-ignition engines. Combust. Flame, 122, 233-252.
Goulding F.C., Bray K.N.C., Chen J.Y., (1989) Chemical closure model for fractal flamelets. Combust. Flame, 77, 241-259.
Poinsot T. and Veynante D. (2001), Theoretical and Numerical Combustion, Edwards Inc.
Cant R.S., Pope S.B. and Bray K.N.C. (1990), Modelling of flamelet surface to volume ratio in turbulent premixed combustion. 23rd Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 791-815.
Duclos J.M.,Veynante D. and Poinsot T., (1993), A comparison of flamelets models for premixed turbulent combustion. Combust. Flame, 95, 101-117.
Mantel T. and Borghi R, (1994) A new model of premixed wrinkled flame propagation based on a scalar dissipation equation. Combust. Flame, 96, 443-457.
Cheng W.K. and Diringer J.A. (1991) Numerical modelling of SI engine combustion with a flame sheet model. Int. Congress and Exposition, SAE Paper 910268.
Choi C.R. and Huh, K.Y. (1998) Development of a coherent flamelet model for a spark ignited turbulent premixed flame in a close vessel. Combust. Flame, 114, 336-348.
Veynante D. and Vervisch L. (2002), Turbulent combustion modelling, Progress in Energy and Combustion Science, 28, 193-266.
Meneveau C. and Poinsot T., 1991 Stretching and quenching of flamelets in premixed turbulent combustion. Combust. Flame, 86, 311-332.
Prasad R. and Gore J.P. (1999), An evaluation of flame surface density models for turbulent premixed jet flames. Combust. Flame, 116, 1-14.

#### The G-equation model

The flame front propagation can be modelled in term of a G-equation (Kerstein et al. 1988, Karpov et al. 1996):

$\frac{\partial G}{\partial t} +\tilde{u}_{i} \frac{\partial G}{\partial x_{i} } =S_{T} \left|\nabla G\right|,$

where the turbulent flame brush is identified by a given level G^* of the G field. One of the advantages of this approach is that the internal flame structure does not need to be resolved on the computational mesh. Only the G-field, generally much thicker, needs to be resolved. On the other hand, the above G-equation leads to both numerical difficulties and modelling problems such as the not obvious coupling between the G-equation and the mass fraction or energy balance equation and such as the quantification of the turbulent flame speed. This formulation seems to be more suitable for the LES than for RANS (Poinsot and Veynante 2001) .

A more refined formalism based on G-field has been developed by Peters (1999, 2000).

Kerstein A.R., Ashurst W., Williams F., (1988) Field equation for interface propagation in an unsteady homogeneous flow field. Phys. Rev. A., 37, 2728-2731.

Karpov V., Lipatnilov A., Zimont, V. (1996), A test of an engineering model of premixed turbulent combustion. 26th Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 249-257.

Poinsot T. and Veynante D. (2001), Theoretical and Numerical Combustion, Edwards Inc.

Peters N, (1999) The turbulent burning velocity for large-scale and small scale turbulence. J. Fluid Mech. 384, 107-132.

Peters N, (2000) Turbulent combustion, Cambridge University Press.

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