Cook 1997: A laminar flamelet approach to subgrid-scale chemistry in turbulent flows

## Reaction model

$$\frac{\chi}{2}\frac{\partial^2y_i}{\partial\xi^2} = -\frac{\dot{m}’’’}{\rho}$$

$$\chi = \chi_0F(\xi)$$

$$F(\xi) = \exp(-2[\text{erf}^{-1}(2\xi-1)]^2)$$

We solve the flamelet equations for $y_i$. Enthalpy is given by simple mixing:

$$h = h_{\xi=0}(1-\xi)+h_{\xi=1}(\xi).$$

All other thermochemical quantites can then be computed from $y_i$, $h$, and $P$. Denote these quantities, and $y_i$, $h$ generically as $\phi$. We then have

$$\phi = \phi(\xi,\chi_0).$$

This can be considered the flamelet reaction model.

## Mixing model

Average quantities are computed by convolving over the joint pdf:

$$\bar{\phi} = \iint\phi(\xi,\chi_0)P(\xi,\chi_0)d\chi_0d\xi.$$

The joint pdf is modeled assuming independence between $\xi$ and $\chi_0$:

$$P(\xi,\chi_0) \approx P(\xi)P(\chi_0).$$

$P(\xi)$ is modeled as a $\beta$-PDF: $P(\xi) = P_\beta(\xi;\bar{\xi}, \overline{\xi^{′′2}})$, which is parameterized by the mean and variance of $\xi$.

$P(\chi_0)$ is modeled as a delta function: $P(\chi_0) = \delta(\chi_0-\overline{\chi_0})$.

$\overline{\chi_0}$ is computed by averaging $\chi=\chi_0F(\xi)$ above:

$$\chi = \chi_0F(\xi)$$

$$\bar{\chi} = \iint\chi_0F(\xi)P(\xi,\chi_0)d\chi_0d\xi,$$

$$\phantom{X}= \iint\chi_0F(\xi)P_\beta(\xi)\delta(\chi_0-\overline{\chi_0})d\chi_0d\xi,$$

$$\phantom{X}= \overline{\chi_0}\int F(\xi)P_\beta(\xi)d\xi.$$

This gives

$$\overline{\chi_0} = \frac{\bar{\chi}}{\int P_\beta(\xi)d\xi}.$$

### RANS

$\bar{\chi}$ is commonly modeled as

$$\bar{\chi} = c_{\chi}\frac{\bar{\epsilon}}{\bar{k}}\overline{\xi’’^2},$$

where $c_{\chi}=2$. See Pitsch 1998

### LES

$\bar{\chi}$ may be modeled as

$$\bar{\chi} = \bar{\rho}(D + D_t)|\nabla\bar\xi|^2$$

The turbulent diffusivity $D_t$ may be computed using the Smagorinsky model.

- See Pierce 1998
- See also Knudsen 2012

### Dissipation pdf

- The scalar dissipation rate pdf is sometimes modeled as a lognormal distribution:
- In this case, the distribution depends on the mean and variance.
- See Heyl and Bockhorn 2001
- They used a constant variance.

# Flamelet Progress Variable (FPV) models

- Instead of parameterizing the flamelets in terms of $\xi$ and $\chi_0$, the FPV model parameterize in terms of $\xi$ and a progress variable.
- Selected references: