Thickened Flame

In practice, the computational mesh is too coarse to resolve the premixed laminar flame fronts, which are usually around 0.1 to 1.0 mm thick. The flame propagation speed is determined by both diffusion of heat and radicals from the burnt downstream zone, as well as reaction within the flame. Therefore, to capture the correct flame speed, about ten cells are required within the flame.

To avoid excessive mesh refinement within the flame, Simcenter STAR-CCM+ provides the Thickened Flame model. The thickness of the computed flame front structure is artificially increased so that it can be resolved on the computational mesh, while kinetic constants are adjusted to keep the laminar flame speed unchanged.

The artificial thickness is achieved by multiplying species and heat diffusion coefficients by a factor $F$ , and decreasing the reaction rate by the same factor $F$ . This approach is based on the well-established theories of laminar premixed flame that laminar flame speed $s_L^0$ and the laminar flame thickness ${\delta }_L^0$ scale as $s_L^0\alpha \sqrt{D\dot{\omega }}$ and ${\delta }_L^0\alpha (D/s_L^0)$ , where $D$ is the diffusion coefficient and $\dot{\omega }$ the reaction rate. For example, to resolve the flame for gas turbine simulations using LES on a fine mesh requires $F$ to be approximately 10 to 100.

The thickening procedure reduces the Damkohler number $D_{\alpha }$ to $D_{\alpha }/F$ , which makes the flame less sensitive to small turbulent motions, that is, eddies that are smaller than $F\times \mathrm{\Delta }$ . This subgrid scale effect is incorporated in the modeling by using an efficiency factor $E$ . The underestimation of the flame front wrinkling by the thickened flame approach can then be corrected by increasing both diffusivities and reaction rates by the factor $E$ . Simcenter STAR-CCM+ has three models for the efficiency factor, namely Power Law, Turbulent Flame Speed, and Wrinkling Factor Ratio.

The thickening factor $F$ is applied in the flame zone only so that mixing away from the flame is unaffected. A sensor, denoted $\Omega$ is defined which is unity in the flame and zero outside.

The Thickened Flame model modifies the transport equations for the mean species and energy by increasing the diffusivity and decreasing the reaction rate. After flame thickening, the resulting transport equations are:

where $\phi$ represents enthalpy, as well as species for species transport models, and un-normalized reaction progress variable and mixture fraction for the FGM model. $E$ , $F$ , and $\mathrm{\Omega }$ are efficiency factor, flame thickening factor, and reaction zone sensor, respectively.

The thickening factor $F$ [812] takes this form:

where $F_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{l}\mathrm{o}\mathrm{c}}$ is the maximum flame thickening factor at each local cell. This value is calculated as:

where $F_{\max }$ is the global maximum flame thickening factor, $N$ is the number of cells in the flame, $\Delta$ is the cell size, and $\delta$ is the laminar flame thickness.

Reaction Zone Sensor

The following methods are available to calculate the flame sensor, $\Omega$ in Eqn. (3463):

Progress Variable

This method is available for the FGM model. It is computed based on the progress variable $c$ .

$$\Omega =\tanh \left(16.0\beta {\left[c(1-c)\right]}^2\right)$$

(3466)

Here, $\beta$ is a model input and $c$ is the normalized reaction progress.

Progress Variable Reaction Rate

This method is available for the FGM model. It is computed based on the normalized reaction progress rate $\dot{c}$ :

$$\mathrm{\Omega }=\tanh (\beta \frac{\dot{c}}{{\dot{c}}_{\max }})$$

(3467)

Here, $\beta$ is a model input, $\dot{c}$ is the normalized reaction progress rate for each cell, and ${\dot{c}}_{\max }$ is the maximum normalized reaction progress rate over all of the cells.

Arrhenius

This method is available only for EBU and has the input parameter $\Gamma$ .

For a simple one–step chemistry scheme:

$${\gamma }_{F}F+{\gamma }_{O}O={\gamma }_{P}P$$

(3468)

The reaction rate $\omega$ is formulated as:

$$\omega =Y_F^{\gamma F}{Y}_O^{\gamma O}\text{exp}\left(-\Gamma \frac{T_a}{T}\right)$$

(3469)

$\Gamma$ is a modeling parameter $(\Gamma <1)$ .

The reaction zone sensor is computed as:

$$\mathrm{\Omega }=\tanh \left(\beta \frac{\omega }{{\omega }_{\max }}\right)$$

(3470)

where ${\omega }_{\max }$ is the maximum reaction rate in the domain.

Heat Release Rate

This method is available for both EBU and the Complex Chemistry model. It computes the reaction zone sensor based on the heat release rate in each cell $\dot{h}$ and the maximum heat release rate in the domain ${\dot{h}}_{\max }$ .

$$\mathrm{\Omega }=\tanh \left(\beta \frac{\dot{h}}{{\dot{h}}_{\max }}\right)$$

(3471)

Reaction Rate

This method is available for both EBU and the Complex Chemistry model. It computes the reaction zone sensor based on the reaction rate in each cell $\omega$ and the maximum reaction rate in the domain ${\omega }_{\max }$ .

$$\mathrm{\Omega }=\tanh \left(\beta \frac{\omega }{{\omega }_{\max }}\right)$$

(3472)

Efficiency Function

There are three methods of computing the efficiency function, $E$ in Eqn. (3463), for the Thickened Flame model:

Power-Law Model

Charlette and others ([809]) proposed a formulation for estimating the efficiency function which relates the flame surface area to a cutoff length and limits the wrinkling at the smallest length-scales of the flame. Based on the asymptotic analysis, the efficiency function is evaluated using a power-law expression:

$$E={(1+\text{min}[\frac{{\Delta }_e}{{\delta }_L^0}-1,\Gamma \frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0}])}^{\alpha }$$

(3473)

where $\Gamma$ is defined as:

$$\Gamma (\frac{{\Delta }_e}{{\delta }_L^0},\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0},{Re}_{\Delta })={[{\left({(f_u^{-a}+f_{\Delta }^{-a})}^{-1/a}\right)}^{-b}+f_{\mathrm{Re}}^{-b}]}^{-1/b}$$

(3474)

$$f_u=4{\left(\frac{27C_k}{110}\right)}^{\frac{1}{2}}\left(\frac{18C_k}{55}\right){\left(\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0}\right)}^2$$

(3475)

$$f_{\Delta }={[\frac{27C_k{\pi }^{4/3}}{110}\times \left({\left(\frac{{\Delta }_e}{{\delta }_L^0}\right)}^{4/3}-1\right)]}^{1/2}$$

(3476)

$$f_{\mathrm{Re}}={\left[\frac{9}{55}\text{exp}\left(-\frac{3}{2}{C}_k{\pi }^{4/3}{Re}_{\Delta }^{-1}\right)\right]}^{1/2}\times {Re}_{\Delta }^{1/2}$$

(3477)

The constants $a$ , $b$ , and $C_k$ control the sharpness of the transitions between the asymptotic behaviors. The suggested values are $b=1.4$ , $C_k=1.5$ :

$$a=0.6+0.2\text{exp}\left[-0.1({\mathrm{u\prime }}_{{\Delta }_e}/s_L^0)\right]-0.20\text{exp}\left[-0.01({\Delta }_e/{\delta }_L^0)\right],{Re}_{\Delta }=4\frac{{\Delta }_e}{{\delta }_L^0}\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0}$$

(3478)

Currently, $\alpha =0.5$ is used, resulting in the non-dynamic formulation.

Wrinkling Factor Ratio Model

A dimensionless wrinkling factor $\Xi$ is defined as the ratio of the flame surface to its projection in the direction of propagation. Based on DNS data and spectral analysis, Colin et al. [810] proposed the following expression for modeling the wrinkling factor:

$$\Xi =1+\beta \frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0}\Gamma (\frac{{\Delta }_e}{{\delta }_L^0},\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0})$$

(3479)

Here, ${\mathrm{u\prime }}_{{\Delta }_e}$ is the local subgrid scale turbulent velocity, $s_L^0$ is the laminar flame speed, ${\Delta }_e$ is the local filter size, and ${\delta }_L^0$ is the laminar flame thickness.

$\beta$ is computed as:

$$\beta =\frac{2\text{ln}2}{3c_{ms}\left({Re}_t^{1/2}-1\right)},{Re}_t=\frac{\mathrm{u\prime }{l}_t}{\mu }$$

(3480)

where $c_{ms}$ is a model parameter with a default value of 0.28, and ${Re}_t$ is the turbulent Reynolds number.

The function $\Gamma$ represents the integration of the effective strain rate that is induced by all scales affected due to artificial thickening, and is estimated as:

$$\Gamma (\frac{{\Delta }_e}{{\delta }_L^0},\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0})=0.75\text{exp}[-1.2{\left(\frac{{\mathrm{u\prime }}_{{\Delta }_e}}{s_L^0}\right)}^{-0.3}]{\left(\frac{{\Delta }_e}{{\delta }_L^0}\right)}^{2/3}$$

(3481)

Finally, the efficiency function takes the following form as defined by the ratio between the wrinkling factor $\Xi$ , of laminar flame $({\delta }_L={\delta }_L^0)$ to thickened flame $({\delta }_L={\delta }_L^l)$ :

$$E=\frac{\Xi {|}_{{\delta }_L={\delta }_L^0}}{\Xi {|}_{{\delta }_L={\delta }_L^l}}\ge 1$$

(3482)

Turbulent Flame Speed

The efficiency function is calculated as

$$E=\alpha \frac{S_t^{\mathrm{\Delta }}}{s_L^0}$$

(3483)

where $\alpha$ is a user constant (default of 1) and $S_t^{\mathrm{\Delta }}$ is a turbulent flame speed model at the turbulent length scale of the grid size $\mathrm{\Delta }$ .

For more information, see Flame Propagation.

Laminar Flame Thickness

There are three ways to calculate the laminar flame thickness, ${\delta }_L^0$ .

Sutherland Law Thermal Diffusivity

The laminar flame thickness is evaluated as [808]:

$${\delta }_L^0=2\frac{{\mu }_b}{\mathrm{Pr}{\rho }_u{s}_L^0}$$

(3484)

where $\mathrm{Pr}$ is the laminar Prandtl number of the burnt gas , ${\rho }_u$ is the unburnt density, and $s_L^0$ is the unstrained laminar flame speed.

${\mu }_b$ is the molecular viscosity of the burnt gas calculated from Sutherland’s law:

$${\mu }_b=\frac{{\mu }_1{T}_b^{1.5}}{{\mu }_2+T_b}$$

(3485)

where ${\mu }_1$ = 1.457e-6 ${\text{kgm}}^{-2}{K}^{-1.5}$ and ${\mu }_2$ = 110K. $T_b$ is the burnt gas temperature.

Power Law Thermal Diffusivity

The laminar flame thickness is evaluated as:

$${\delta }_L^o=2\frac{D_u}{s_L^o}{\left(\frac{T_b}{T_u}\right)}^{0.7}$$

(3486)

where $D_u$ is the unburnt thermal diffusivity, $T_b$ and $T_u$ are burnt and unburnt temperatures.