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18.2.3 Particle Mixing

Molecular mixing of species and heat must be modeled and is usually the source of the largest modeling error in the PDF transport approach. FLUENT provides three models for molecular diffusion: the Modified Curl model [ 162, 264], the IEM model (which is sometimes called the LSME model) [ 85] and the EMST model [ 359].



The Modified Curl Model


For the Modified Curl model, a few particle pairs are selected at random from all the particles in a cell, and their individual compositions are moved toward their mean composition. For the special case of equal particle mass, the number of particle pairs selected is calculated as


 N_{\rm pair} = \frac{ 1.5 C_{\phi} N {\Delta t} }{\tau_t} (18.2-6)

where


$N$ = total number of particles in the cell
$C_{\phi}$ = mixing constant (default = 2)
$\tau_t$ = turbulent time scale (for the $k$- $\epsilon$ model this is $k / \epsilon$)

The algorithm in [ 264] is used for the general case of variable particle mass.

For each particle pair, a uniform random number $\xi$ is selected and each particle's composition $\phi$ is moved toward the pair's mean composition by a factor proportional to  $\xi$:


$\displaystyle \phi_i^1 = (1-\xi)\phi_i^0 + \xi \frac{(\phi_i^0 m_i + \phi_j^0 m_j)}{(m_i + m_j)}$      
$\displaystyle \phi_j^1 = (1-\xi)\phi_j^0 + \xi \frac{(\phi_i^0 m_i + \phi_j^0 m_j)}{(m_i + m_j)}$     (18.2-7)

where $\phi_i$ and $\phi_j$ are the composition vectors of particles $i$ and $j$, and $m_i$ and $m_j$ are the masses of particles $i$ and $j$.



The IEM Model


For the Interaction by Exchange with the Mean (IEM) model, the composition of all particles in a cell are moved a small distance toward the mean composition:


 \phi^1 = \phi^0 - \left ( 1 - e^{-0.5 C_{\phi} / \tau_t} \right ) \left ( \phi^0 - \tilde{\phi} \right ) (18.2-8)

where $\phi^0$ is the composition before mixing, $\phi^1$ is the composition after mixing and $\tilde{\phi}$ is the Favre mean-composition vector at the particle's location.



The EMST Model


Physically, mixing occurs between fluid particles that are adjacent to each other. The Modified Curl and IEM mixing models take no account of this localness, which can be a source of error. The Euclidean Minimum Spanning Tree (EMST) model mixes particle pairs that are close to each other in composition space. Since scalar fields are locally smooth, particles that are close in composition space are likely to be close in physical space. The particle pairing is determined by a Euclidean Minimum Spanning Tree, which is the minimum length of the set of edges connecting one particle to at least one other particle. The EMST mixing model is more accurate than the Modified Curl and IEM mixing models, but incurs a slightly greater computationally expensive. Details on the EMST model can be found in [ 359].



Liquid Reactions


Reactions in liquids often occur at low turbulence levels (small Re), among reactants with low diffusivities (large Sc). For such flows, the mixing constant default of $C_{\phi}=2$ overestimates the mixing rate. The Liquid Micro-Mixing option interpolates $C_{\phi}$ from model turbulence [ 292] and scalar [ 113] spectra.


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