Steady Laminar Flamelet

In combustion theory, a turbulent flame can be considered as a composition of many thin laminar flamelet structures that are locally one-dimensional. In the tabulated approach, Simcenter STAR-CCM+ solves for reaction products using the one-dimensional model and stores these in the lookup table.

This model assumes that the thickness of the reaction zone is smaller than the smallest turbulence length scale, the Kolmogorov length scale. Therefore, the local instantaneous reaction zone structure is considered to be the same as in a quasi-steady one-dimensional laminar flamelet.

The main effect of the turbulent velocity field on the structure of the laminar flamelet is through the stretch of the reaction zone—described by the scalar dissipation rate (or strain rate). At low values of strain rate, laminar flamelet structures approach the equilibrium state; at high strain rates, flame extinction can occur. Through solving for the quasi-steady, one-dimensional laminar flames under different strain rates and with detailed reaction mechanisms, non- equilibrium and finite-rate chemistry effects can be introduced into the turbulent combustion modeling.

Simcenter STAR-CCM+ solves for an idealized axisymmetric counterflow diffusion flame, as depicted in the following diagram:

Scalar Dissipation Rate

In the context of the Steady Laminar Flamelet model, the scalar dissipation rate, $\chi$, is the rate at which turbulence-generated fluctuations in mixture fraction are dissipated. In Simcenter STAR-CCM+, the scalar dissipation rate is modeled as:

$$\chi =\frac{C_{\varphi }{Z}_{\mathrm{var}}}{{\tau }_{turb}}$$

(3527)

where ${\tau }_{turb}$ is the turbulent timescale and $C_{\varphi }$ is a constant with the default value of two.

Flame temperatures depend strongly on the scalar dissipation rate—the maximum flame temperature decreases with an increasing scalar dissipation rate. Therefore, high values of scalar dissipation rate can cause extinction.

At zero scalar dissipation, the flamelet is at chemical equilibrium. To determine the extinction scalar dissipation rate, Simcenter STAR-CCM+ starts solving with the initial scalar dissipation rate that is specified and increases this value by multiplying it with the Scalar Dissipation Multiplier until the flamelet extinguishes, or until the maximum scalar dissipation rate is reached.

When simulating a turbulent non-premixed flame, the scalar dissipation rate is highest near the inlet nozzle of a combustor (where the velocity is high). Extinction of the flame near the nozzle can cause lift-off of turbulent flames.

Tabulation for Steady Laminar Flamelet

The independent variables are:

• Mixture Fraction
• Mixture Fraction Variance
• Enthalpy

The mixture fraction can replace the space coordinate in the one-dimensional system that is shown above, to yield the following equations [767]:

$$\frac{\partial Y_i}{\partial t}=R_i+\frac{1}{2}\chi \frac{{\partial }^2{Y}_i}{\partial Z^2}$$

(3528)

$$\frac{\partial T}{\partial t}=-{\displaystyle \sum _{i=1}^N}\frac{h_i{R}_i}{C_p}+\frac{1}{2}\chi \left(\frac{{\partial }^{2}T}{\partial Z^2}\right)+\frac{1}{2}\frac{\chi }{C_p}(\frac{\partial C_p}{\partial Z}+{\displaystyle \sum _{i=1}^N}\left({C_p}_i\frac{\partial Y_i}{\partial Z}\right))\frac{\partial T}{\partial Z}$$

(3529)

where $Y_i$ is the mass fraction, $T$ is the temperature, $C_p$ the specific heat, $h_i$ is the enthalpy of species $i$ , $R_i$ is the chemical source term, and $\chi$ is the scalar dissipation rate that is specified.

The steady laminar flamelet model solves for the balance between diffusion and reaction along the one-dimensional flamelet as a function of the scalar dissipation rate, mixture fraction, and enthalpy. Simcenter STAR-CCM+ generates the flamelet table by solving Eqn. (3528) and Eqn. (3529) for different values of scalar dissipation $\chi$ and enthalpy $h$ . Results are tabulated as functions of mixture fraction, scalar dissipation, and heat loss ratio (HLR) which corresponds to the enthalpy. Therefore, any variable of interest, $\varphi$ , can be expressed as:

$$\varphi =\varphi (Z,\chi ,h)$$

(3530)

Integration

The integration of the Steady Laminar Flamelet model is similar to that of the Chemical Equilibrium model. The difference being that, given Eqn. (3530) for any state-space variable, a joint Probability Density Function of $Z$ , $\chi$ , and $h$ are required. That is, we integrate over the $Z$ , $\chi$ , and $h$ spaces:

$$\tilde{\varphi }={\varphi }_{mean}=\iint \varphi (Z,\chi ,h)P(Z,\chi ,h)dZd\chi dh$$

(3531)

where $P(Z,\chi ,h)$ is modeled as $P\left(Z\right)\times P\left(\chi \right)\times P\left(h\right)$ with a beta function for $Z$ (see Beta Probability Density Function (PDF)), and a delta function for $\chi$ and $h$ .