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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)}

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