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18.2 Composition PDF Transport Theory

Turbulent combustion is governed by the reacting Navier-Stokes equations. While this equation set is accurate, its direct solution (where all turbulent scales are resolved) is far too expensive for practical turbulent flows. In Chapter  14, the species equations are Reynolds-averaged, which leads to unknown terms for the turbulent scalar flux and the mean reaction rate. The turbulent scalar flux is modeled by gradient diffusion, treating turbulent convection as enhanced diffusion. The mean reaction rate is modeled by the finite-rate, eddy-dissipation, or EDC models. Since the reaction rate is invariably highly non-linear, modeling the mean reaction rate in a turbulent flow is difficult and prone to error.

An alternative to Reynolds-averaging the species and energy equations is to derive a transport equation for their single-point, joint probability density function (PDF). This PDF, denoted by $P$, can be considered to be proportional to the fraction of the time that the fluid spends at each species and temperature state. $P$ has $N+1$ dimensions for the $N$ species and temperature spaces. From the PDF, any thermochemical moment (e.g., mean or RMS temperature, mean reaction rate) can be calculated. The composition PDF transport equation is derived from the Navier-Stokes equations as [ 289]:


 \frac{\partial}{\partial t} (\rho P) + \frac{\partial}{\part... ...{i,k}}{\partial x_i} \right\vert \psi \right\rangle} P \right] (18.2-1)

where


$P$ = Favre joint PDF of composition
$\rho$ = mean fluid density
$u_i$ = Favre mean fluid velocity vector
$S_k$ = reaction rate for species $k$
$\psi$ = composition space vector
$u^{''}_i$ = fluid velocity fluctuation vector
$J_{i,k}$ = molecular diffusion flux vector

The notation of $\langle \dots \rangle$ denotes expectations, and $\langle A\vert B \rangle$ is the conditional probability of event $A$, given the event $B$ occurs.

In Equation  18.2-1, the terms on the left-hand side are closed, while those on the right-hand side are not and require modeling. The first term on the left-hand side is the unsteady rate of change of the PDF, the second term is the change of the PDF due to convection by the mean velocity field, and the third term is the change due to chemical reactions. The principal strength of the PDF transport approach is that the highly-non-linear reaction term is completely closed and requires no modeling. The two terms on the right-hand side represent the PDF change due to scalar convection by turbulence (turbulent scalar flux), and molecular mixing/diffusion, respectively.

The turbulent scalar flux term is unclosed, and is modeled in FLUENT by the gradient-diffusion assumption


 - \frac{\partial}{\partial x_i}\left[ \rho \langle u^{''}_i ... ..._t}{ \rho {\rm Sc}_t} \frac{\partial P}{\partial x_i} \right ) (18.2-2)

where $\mu_t$ is the turbulent viscosity and Sc $_t$ is the turbulent Schmidt number. A turbulence model, as described in Chapter  12, is required for composition PDF transport simulations, and this determines $\mu_t$.

Since single-point PDFs are described, information about neighboring points is missing and all gradient terms, such as molecular mixing, are unclosed and must be modeled. The mixing model is critical because combustion occurs at the smallest molecular scales when reactants and heat diffuse together. Modeling mixing in PDF methods is not straightforward, and is the weakest link in the PDF transport approach. See Section  18.2.3 for a description of the mixing models.




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