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16.2.2 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:

In FLUENT, the turbulent flame speed is computed using a model for wrinkled and thickened flame fronts [ 415]:

$\displaystyle U_t$ $\textstyle =$ $\displaystyle A (u')^{3/4} U_{l}^{1/2} \alpha^{-1/4} \ell_{t}^{1/4}$ (16.2-4)
  $\textstyle =$ $\displaystyle A u^{'} \left( \frac{\tau_t}{\tau_c} \right)^{1/4}$ (16.2-5)

  $A$ = 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

 \ell_t = C_D \frac{(u')^3}{\epsilon} (16.2-6)

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 by [ 415], 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

 {\rm Ka} = \frac{t_{l}}{t_\eta} = \frac{v^2_\eta}{U^2_{l}} (16.2-7)

  $t_{l}$ = 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  16.2-4) are replaced by their equivalent sub-grid quantities. In particular, the large eddy length scale $\ell_t$ is modeled as

 {\ell_t} = C_s \Delta (16.2-8)

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

The RMS velocity in Equation  16.2-4 is replaced by the sub-grid velocity fluctuation, calculated as

 {u'} = {\ell_t} \tau^{-1}_{sgs} (16.2-9)

where $\tau^{-1}_{sgs}$ is the sub-grid scale mixing rate (inverse of the sub-grid scale time scale), given in Equation  14.1-28.

Laminar Flame Speed

The laminar flame speed ( $U_{l}$ in Equation  16.2.1) 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 [ 241],

 U_{l} = U_{l,ref} {\left( \frac{T_{u}}{T_{u,ref}} \right)}^{\gamma} {\left( \frac{p_{u}}{p_{u,ref}} \right)}^{\beta} (16.2-10)

In Equation  16.2-10, $T_u$ 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

 U_{l,ref} = C_1 + C_2{( \phi - C_3 )}^2 (16.2-11)

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,

 \begin{array}{cc} \gamma = 2.18 - 0.8(\phi - 1) \\ \beta = -0.16 + 0.22(\phi - 1) \end{array} (16.2-12)

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  16.2.1) and unburnt thermal diffusivity ( $\alpha$ in Equation  16.2-5) are specified constants that are set in the Materials panel. 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  16.2-1) is multiplied by a stretch factor, $G$ [ 417]. 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, $\epsilon$:

 G = \frac{1}{2} {\rm erfc} \left\{ - \sqrt{\frac{1}{2\sigma}... ...\rm cr}}{\epsilon} \right) + \frac{\sigma}{2} \right] \right\} (16.2-13)

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 $\epsilon$:

 \sigma = \mu_{\rm str} \ln \left( \frac{L}{\eta} \right) (16.2-14)

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 [ 415], and is suitable for most premixed flames.

$\epsilon_{\rm cr}$ is the turbulence dissipation rate at the critical rate of strain [ 415]:

 \epsilon_{\rm cr} = 15 \nu g_{\rm cr}^2 (16.2-15)

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 [ 415], or an appropriate value can be determined from experimental data. A reasonable model for the critical rate of strain $g_{\rm cr}$ is

 g_{\rm cr} = \frac{B U_l^2}{\alpha} (16.2-16)

where $B$ is a constant (typically 0.5) and $\alpha$ is the thermal diffusivity. Equation  16.2-16 can be implemented in FLUENT using a property user-defined function. See the separate UDF Manual. for details about user-defined functions.

Preferential Diffusion

Preferential diffusion accounts for the effect of variations in fuel molecular diffusion coefficients on heat release intensity in premixed turbulent combustion. Inclusion of this effect is important for simulation of combustion with light fuels (e.g., hydrogen) or heavy fuels (e.g., evaporated oil). The model for preferential diffusion is based on the concept of leading points, formulated in [ 189]. The authors of [ 189] derived formulas for the changes in mixture composition within the combustion zone due to the difference in the molecular diffusivities of fuel, $D_{\rm fuel}$, and oxidizer, $D_{\rm ox}$. These formulas are rewritten in [ 417] as

 \lambda_{\rm lp} = \left\{ \begin{array}{l l} \frac{\lambda_... ... (1 - \lambda_0 d)} & \lambda_{\rm lp} < 1 \end{array} \right. (16.2-17)

  $C_{\rm st}$ = mass stoichiometric coefficient
  $\lambda_0$ = stoichiometric ratio of unburnt mixture composition
  $\lambda_{\rm lp}$ = stoichiometric ratio of leading-point composition


 d = \sqrt{\frac{D_{\rm ox}}{D_{\rm fuel}}} (16.2-18)

The concept of leading points is applied to the FLUENT model by using $\lambda_{\rm lp}$ instead of $\lambda_0$ for the formulation of the laminar flame speed, $U_{l}$, or the molecular heat transfer coefficient, $\alpha$. This simple approach results in reasonable agreement with the measurements of mass combustion rates in stirred bombs [ 417], without the need for additional empirical parameters.

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. [ 416].

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