9.2.2 Turbulent Flame Speed

The key to the premixed combustion model is the prediction of $U_t$, the turbulent flame speed normal to the mean surface of the flame. The turbulent flame speed is influenced by the following:

• laminar flame speed, which is, in turn, determined by the fuel concentration, temperature, and molecular diffusion properties, as well as the detailed chemical kinetics

• flame front wrinkling and stretching by large eddies, and flame thickening by small eddies

In ANSYS FLUENT, the Zimont turbulent flame speed closure is computed using a model for wrinkled and thickened flame fronts [ 391]:

$$U_t = A (u')^{3/4} U_{l}^{1/2} \alpha^{-1/4} \ell_{t}^{1/4}$$ (9.2-4)

$$= A u^{'} \left(\frac{\tau_t}{\tau_c} \right)^{1/4}$$ (9.2-5)

where
	$A$	=	model constant
	$u'$	=	RMS (root-mean-square) velocity (m/s)
	$U_{l}$	=	laminar flame speed (m/s)
	$\alpha = k/\rho c_p$	=	molecular heat transfer coefficient of unburnt
			mixture (thermal diffusivity) (m $^2$/s)
	$\ell_t$	=	turbulence length scale (m)
	$\tau_t = \ell_t/u'$	=	turbulence time scale (s)
	$\tau_c = \alpha/U_{l}^2$	=	chemical time scale (s)

The turbulence length scale, $\ell_t$, is computed from

where $\epsilon$ is the turbulence dissipation rate.

The model is based on the assumption of equilibrium small-scale turbulence inside the laminar flame, resulting in a turbulent flame speed expression that is purely in terms of the large-scale turbulent parameters. The default value of 0.52 for $A$ is recommended [ 391], and is suitable for most premixed flames. The default value of 0.37 for $C_D$ should also be suitable for most premixed flames.

The model is strictly applicable when the smallest turbulent eddies in the flow (the Kolmogorov scales) are smaller than the flame thickness, and penetrate into the flame zone. This is called the thin reaction zone combustion region, and can be quantified by Karlovitz numbers, Ka, greater than unity. Ka is defined as

where
	$t_{l}$	=	characteristic flame time scale
	$t_\eta$	=	smallest (Kolmogorov) turbulence time scale
	$v_\eta = (\nu \epsilon)^{1/4}$	=	Kolmogorov velocity
	$\nu$	=	kinematic viscosity

Lastly, the model is valid for premixed systems where the flame brush width increases in time, as occurs in most industrial combustors. Flames that propagate for a long period of time equilibrate to a constant flame width, which cannot be captured in this model.

Turbulent Flame Speed for LES

For simulations that use the LES turbulence model, the Reynolds-averaged quantities in the turbulent flame speed expression (Equation  9.2-4) are replaced by their equivalent subgrid quantities. In particular, the large eddy length scale $\ell_t$ is modeled as

where $C_s$ is the Smagorinsky constant and $\Delta$ is the cell characteristic length.

The RMS velocity in Equation  9.2-4 is replaced by the subgrid velocity fluctuation, calculated as

where $\tau^{-1}_{sgs}$ is the subgrid scale mixing rate (inverse of the subgrid scale time scale), given in Equation  7.1-28.

Laminar Flame Speed

The laminar flame speed ( $U_{l}$ in Equation  9.2.1) can be specified as constant, or as a user-defined function. A third option appears for non-adiabatic premixed and partially-premixed flames and is based on the correlation proposed by Meghalchi and Keck [ 227],

In Equation  9.2-10, $T_u$ and $p_u$ are the unburnt reactant temperature and pressure ahead of the flame, $T_{u,ref}=298K$ and $p_{u,ref}=1atm$.

The reference laminar flame speed, $U_{l,ref}$, is calculated from

where $\phi$ is the equivalence ratio ahead of the flame front, and $C_1$, $C_2$ and $C_3$ are fuel-specific constants. The exponents $\gamma$ and $\beta$ are calculated from,

The Meghalchi-Keck laminar flame speeds are available for fuel-air mixtures of methane, methanol, propane, iso-octane and indolene fuels.

Unburnt Density and Thermal Diffusivity

The unburnt density ( $\rho_u$ in Equation  9.2.1) and unburnt thermal diffusivity ( $\alpha$ in Equation  9.2-5) are specified constants that are set in the Materials dialog box. However, for compressible cases, such as in-cylinder combustion, these can change significantly in time and/or space. When the ideal gas model is selected for density, the unburnt density and thermal diffusivity are calculated as volume averages ahead of the flame front.

Flame Stretch Effect

Since industrial low-emission combustors often operate near lean blow-off, flame stretching will have a significant effect on the mean turbulent heat release intensity. To take this flame stretching into account, the source term for the progress variable ( $\rho S_c$ in Equation  9.2-1) is multiplied by a stretch factor, $G$ [ 393]. This stretch factor represents the probability that the stretching will not quench the flame; if there is no stretching ( $G=1$), the probability that the flame will be unquenched is 100%.

The stretch factor, $G$, is obtained by integrating the log-normal distribution of the turbulence dissipation rate, $\epsilon$:

where erfc is the complementary error function, and $\sigma$ and $\epsilon_{\rm cr}$ are defined below.

$\sigma$ is the standard deviation of the distribution of $\epsilon$:

where $\mu_{\rm str}$ is the stretch factor coefficient for dissipation pulsation, $L$ is the turbulent integral length scale, and $\eta$ is the Kolmogorov micro-scale. The default value of 0.26 for $\mu_{\rm str}$ (measured in turbulent non-reacting flows) is recommended by [ 391], and is suitable for most premixed flames.

$\epsilon_{\rm cr}$ is the turbulence dissipation rate at the critical rate of strain [ 391]:

By default, $g_{\rm cr}$ is set to a very high value ( $1 \times 10^8$) so no flame stretching occurs. To include flame stretching effects, the critical rate of strain $g_{\rm cr}$ should be adjusted based on experimental data for the burner. Numerical models can suggest a range of physically plausible values [ 391], or an appropriate value can be determined from experimental data. A reasonable model for the critical rate of strain $g_{\rm cr}$ is

where $B$ is a constant (typically 0.5) and $\alpha$ is the unburnt thermal diffusivity. Equation  9.2-16 can be implemented in ANSYS FLUENT using a property user-defined function. More information about user-defined functions can be found in the separate UDF Manual.

Gradient Diffusion

Volume expansion at the flame front can cause counter-gradient diffusion. This effect becomes more pronounced when the ratio of the reactant density to the product density is large, and the turbulence intensity is small. It can be quantified by the ratio $(\rho_u/\rho_b) (U_{l}/I)$, where $\rho_u$, $\rho_b$, $U_{l}$, and $I$ are the unburnt and burnt densities, laminar flame speed, and turbulence intensity, respectively. Values of this ratio greater than one indicate a tendency for counter-gradient diffusion, and the premixed combustion model may be inappropriate. Recent arguments for the validity of the turbulent-flame-speed model in such regimes can be found in Zimont et al. [ 392].

Wall Damping

High turbulent kinetic energy levels at the walls in some problems can cause an unphysical acceleration of the flame along the wall. In reality, radical quenching close to walls decreases reaction rates and thus the flame speed, but is not included in the model. To approximate this effect, ANSYS FLUENT includes a constant multiplier for the turbulent flame speed, $\alpha_w$, which modifies the flame speed in the vicinity of wall boundaries:

The default for this constant is 1 which does not change the flame speed. Values of $\alpha_w$ larger than 1 increase the flame speed, while values less than 1 decrease the flame speed in the cells next to the wall boundary.

ANSYS FLUENT will solve the transport equation for the reaction progress variable $c$ (Equation  9.2-1), computing the source term, $\rho S_c$, based on the theory outlined above:

$$\rho S_c = A G \rho_u I^{3/4} [U_{l}(\lambda_{\rm lp})]^{1/2} [\alpha(\lambda_{\rm lp})]^{-1/4} \ell_t^{1/4} \vert\nabla c\vert$$ (9.2-18)

$$= A G \rho_u I \left[ \frac{\tau_t}{\tau_c(\lambda_{\rm lp})} \right]^{1/4} \vert\nabla c\vert$$ (9.2-19)