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15.5.1 The Eulerian Unsteady Laminar Flamelet Model

The Eulerian unsteady laminar flamelet model can be used to predict slow-forming intermediate and product species which are not in chemical equilibrium. Typical examples of slow-forming species are gas-phase pollutants like NOx, and product compounds in liquid reactors. By reducing the chemistry computation to one dimension, detailed kinetics with multiple species and stiff reactions can be economically simulated in complex 3D geometries.

The model, following the work of Barths et al. [ 23], postprocesses an unsteady marker probability equation on a steady-state, converged flow field. In FLUENT, the steady flow solution must be computed with the steady laminar flamelet model (see Section  15.4). Since the unsteady flamelet equations are postprocessed on a steady-state, steady flamelet solution, the effect of the unsteady flamelet species on the flow-field are neglected.

The transport equation for the unsteady flamelet marker probability, $I$, is


 \frac{\partial}{\partial t} \left(\rho I \right) + \nabla \c... ... = \nabla \cdot \left( \frac{\mu_t}{\sigma_t} \nabla I \right) (15.5-1)

Equation  15.5-1 is always solved unsteady, and is initialized as


 I = \left\{ \begin{array}{cl} 1 & {\rm for} \; \overline{f} ... ...& {\rm for} \; \overline{f} < f_{init} \\ \end{array} \right. (15.5-2)

where $\overline{f}$ is the mean mixture fraction and $f_{init}$ is a user supplied constant, which should be set greater than the stoichiometric mixture fraction. At inlet boundaries, FLUENT always sets $I$ toward zero, and hence the $I$ field decreases to zero with time as $I$ is convected and diffused out of the domain (for cases with outlet boundaries).

The unsteady flamelet species equations (Equation  15.3-6) are integrated simultaneously with the marker probability equation, $I$. For liquid-phase chemistry, the initial flamelet field is the mixed-but-unburnt flamelet, as liquid reactions are assumed to proceed immediately upon mixing. However, gas-phase chemistry invariably requires ignition, so the initial flamelet field is calculated from a steady flamelet solution. All slow-forming species, such as NOx, are zeroed in this initial flamelet profile since, at ignition, little time has elapsed for any significant formation. The slow-forming species are identified by the user before solution of the unsteady flamelet equations.

The scalar dissipation at stoichiometric mixture fraction ( $\chi_{\rm st}$) is required by the flamelet species equation. This is calculated from the steady-state FLUENT field at each time step as a probability-weighted volume integral:


 \chi_{\rm st}(t) = \frac{ \int_V I({\vec x},t) \, \rho({\vec... ...vec x}) \, {\overline{\chi_{\rm st}}}^{1/2}({\vec x}) \, dV} (15.5-3)

where $\overline{\chi_{\rm st}}$ is defined in Equation  15.5-3, and $V$ denotes the fluid volume. FLUENT provides the option of limiting $\overline{\chi_{\rm st}}$ to a user-specified maximum value, which should be approximately equal to the flamelet extinction scalar dissipation (the steady flamelet solver can be used to calculate this extinction scalar dissipation in a separate simulation).

The unsteady flamelet energy equation is not solved in order to avoid flamelet extinction for high scalar dissipation, and to account for non-adiabatic heat loss or gain. For adiabatic cases, the flamelet temperature $T^{ad}(f,t)$ is calculated at each time step from the steady flamelet library at the probability-weighted scalar dissipation $\chi_{\rm st}$ from Equation  15.5-3. For non-adiabatic cases, the flamelet temperature at time $t$ is calculated from


 T(f,t) = T^{ad}(f,t) \, \xi(f,t) (15.5-4)

where


 \xi(f,t) = \frac{ \int_V I \, \rho \, \overline{T}({\vec x \vert f}) / T_{ad}(f,t) \, dV } {\int_V I \, \rho \, dV} (15.5-5)

In Equation  15.5-5, $\overline{T}({\vec x \vert f})$ denotes the FLUENTsteady-state mean cell temperature conditioned on the local cell mixture fraction.

Unsteady flamelet mean species mass fractions in each cell are accumulated over time as


 \overline{Y_k}^{ufla} = \frac{ \int_0^t I \, \rho \, \left[ ... ...,t) \, P(f) \, df \right] \, dt } {\int_0^t I \, \rho \, dt} (15.5-6)

where $Y_k(f,t)$ is the $k$'th species unsteady flamelet mass fraction, and $P(f)$ denotes the Beta PDF.

The probability marker equation (Equation  15.5-1) and the flamelet species equation (Equation  15.3-6) are advanced together in time until the probability marker has substantially convected and diffused out of the domain. The unsteady flamelet mean species, calculated from Equation  15.5-5, reaches steady-state as the probability marker $I$ vanishes.



Liquid Reactions


Liquid reactors are typically characterized by:

The Eulerian unsteady laminar flamelet model can be used to model liquid reactions. When the Liquid Micro-Mixing model is invoked, FLUENT uses the volume-weighted-mixing-law formula to calculate the density.

The effect of high Sc is to decrease mixing at the smallest (micro) scales and increase the mixture fraction variance, which is modeled with the Turbulent Mixer Model  [ 17]. Three transport equations are solved for the inertial-convective ( $\overline{f^{'2}_{ic}}$), viscous-convective ( $\overline{f^{'2}_{vc}}$) and viscous-diffusive ( $\overline{f^{'2}_{vd}}$) subranges of the turbulent scalar spectrum,


 \frac{\partial}{\partial t} \left(\rho \overline{f^{'2}_{ic}... ...\right)^2 - C_2 \rho \frac{\epsilon}{k} \overline{f^{'2}_{ic}} (15.5-7)


 \frac{\partial}{\partial t} \left(\rho \overline{f^{'2}_{vc}... ...}{k} \overline{f^{'2}_{vc}} - C_3 \sqrt{ \frac{\epsilon}{\nu}} (15.5-8)


 \frac{\partial}{\partial t} \left(\rho \overline{f^{'2}_{vd}... ... \frac{\epsilon}{\nu}} \left( 1 - C_4 - \frac{C_5}{Sc} \right) (15.5-9)

where the constants $C_1$ through $C_5$ have values of 2, 1.86, 0.058, 0.303 and 17050 respectively. The total mixture fraction variance is the sum of $\overline{f^{'2}_{ic}}$, $\overline{f^{'2}_{vc}}$ and $\overline{f^{'2}_{vd}}$.

In Equation  15.5-9, the cell Schmidt number, $Sc$, is calculated as $Sc = \mu / \rho D$ where $\mu$ is the viscosity, $\rho$ the density and $D$ the mass diffusivity as defined for the pdf-mixture material.

Since liquid species are invariably in chemical non-equilibrium, the unsteady laminar flamelet model should be used, postprocessing on a converged, steady-state, steady flamelet FLUENTcase and data.


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