ANSYS FLUENT 12.0 Theory Guide - 11.4 The Eulerian Solution Method

The Lagrangian solution method solves the composition PDF transport equation by stochastically tracking Lagrangian particles through the domain. It is computationally expensive since a large number of particles are required to represent the PDF, and a large number of iterations are necessary to reduce statistical errors and explicitly convect the particles through the domain. The Eulerian PDF transport model overcomes these limitations by assuming a shape for the PDF, which allows Eulerian transport equations to be derived. Stochastic errors are eliminated and the transport equations are solved implicitly, which is computationally economical. The multi-dimensional PDF shape is assumed as a product of delta functions. As with the Lagrangian PDF model, the highly non-linear chemical source term is closed. However, the turbulent scalar flux and molecular mixing terms must be modeled, and are closed with the gradient diffusion and the IEM models, respectively.

The composition PDF of

dimension (

species and enthalpy) is represented as a collection of

delta functions (or modes). This presumed PDF has the following form:

$P(\psi;\vec{x},t) = \sum_{n=1}^{N_e} p_n(\vec{x},t) \prod_{k=1}^{N_s} \delta[\psi_k - <\phi_k>_n (\vec{x},t)]$

(11.4-1)

where

is the probability in each mode, $<\phi_k>_n$ is the conditional mean composition of specie

in the

th mode, $\psi_k$ is the composition space variable of specie

, and $\delta(\dots)$ is the delta function.

The Eulerian PDF transport equations are derived by substituting Equation 11.4-1 into the closed composition PDF transport equation (Equation 11.2-1 with Equations 11.2-2 and 11.3-6). The unknown terms,

and $<\phi_k>_n$ , are determined by forcing lower moments of this transported PDF to match the RANS lower moment transport equations, using the Direct Quadrature Method of Moments (DQMOM) approach [ 103, 218]. The resulting transport equations are:

where

is the probability of the

th mode, and $s_{k,n} = p_n <\phi_k>_n$ is the

th specie probability weighted conditional mean composition of the

th mode. $\Gamma = \mu_l+\mu_t/{{\rm Sc}_t}$ is the effective turbulent diffusivity. The terms $M_{k,n}$ , $S_{k,n}$ and $C_{k,n}$ represent mixing, reaction and correction terms respectively. Note that only

probability equations are solved and the

th probability is calculated as one minus the sum of the

solved probabilities.

The reaction source term $S_{k,n}$ in Equation 11.4-3 for the

th composition and the

th mode is calculated as,

$M_{k,n} = \frac{C_\phi}{\tau}(<\phi_k>_n - \psi_k)$

(11.4-5)

$\displaystyle M_{k,1}$	$\textstyle =$	$\displaystyle \frac{C_\phi}{\tau}(p_1 s_{k,2} - p_2 s_{k,1} )$
$\displaystyle M_{k,2}$	$\textstyle =$	$\displaystyle \frac{C_\phi}{\tau}(p_2 s_{k,1} - p_1 s_{k,2} )$	(11.4-6)

The default value of $C_\phi$ is 2, which is appropriate for gas-phase combustion. For reactions in liquids, where the diffusivities are much smaller than gases, the Liquid Micro-Mixing option interpolates $C_{\phi}$ from model turbulence [ 278] and scalar [ 103] spectra.

Using assumptions to ensure realizability and boundedness, the correction terms $C_{k,n}$ in Equation 11.4-3 for the

th composition are determined from the linear system,

$\sum_{n=1}^{N_e} <\phi_k>_n^{m_k - 1} C_{k,n} = \sum_{n=1}^{N_e}(m_k - 1)<\phi_k>_n^{m_k - 2} p_n c_{k,n}$

(11.4-7)

where

are the non-negative integer lower moments (1..

) for each component

. Note that the condition of the matrix decreases with increasing

, which reduces the stability of higher mode simulations.

$c_{k,n} = \Gamma \nabla(<\phi_k>_n ) \cdot \nabla(<\phi_k>_n )$

(11.4-8)

For the two-mode DQMOM-IEM model, the correction terms for the

th component are,

$\displaystyle C_{k,1}$	$\textstyle =$	$\displaystyle \frac{\Gamma}{<\phi_k>_1-<\phi_k>_2} \left(p_1 (\nabla <\phi_k>_1 )^2 + p_2 (\nabla <\phi_k>_2 )^2 \right)$
$\displaystyle C_{k,2}$	$\textstyle =$	$\displaystyle -C_{k,1}$	(11.4-9)

$\overline{\phi_k} = \sum_{n=1}^{N_e} {s_{k,n}}$

(11.4-10)

$\overline{\phi_k^{'2}} = (\sum_{n=1}^{N_e} {s_{k,n} <\phi_k>_n}) - \overline{\phi_k}^2$

(11.4-11)