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9.3.2 UDS Theory

Single Phase Flow

For an arbitrary scalar $\phi_k$, FLUENT solves the equation

 \frac{\partial \rho \phi_{k}}{\partial t} + \frac{\partial}{... ...rtial \phi_k}{\partial x_i}) = S_{\phi_k} \; \; \; k = 1,...,N (9.3-1)

where $\Gamma_{k}$ and $S_{\phi_k}$ are the diffusion coefficient and source term supplied by you for each of the $N$ scalar equations. Note that $\Gamma_{k}$is defined as a tensor in the case of anisotropic diffusivity. The diffusion term is thus $\nabla \cdot \left( {\bf\Gamma_k} \cdot \phi_k \right)$

For isotropic diffusivity, $\Gamma_k$ could be written as $\Gamma_k I$ where I is the identity matrix.

For the steady-state case, FLUENT will solve one of the three following equations, depending on the method used to compute the convective flux :

Multiphase Flow

For multiphase flows , FLUENT solves transport equations for two types of scalars: per phase and mixture. For an arbitrary $k$ scalar in phase-1, denoted by $\phi^k_l$, FLUENT solves the transport equation inside the volume occupied by phase-l

 \frac{\partial \alpha_{l} \rho_{l} \phi^k_l}{\partial t} + \... ...a_{l} \Gamma^k_l \nabla \phi^k_l) = S^k_l \; \; \; k = 1,...,N (9.3-5)

where $\alpha_{l}$, $\rho_{l}$, and $\vec {u}_l$ are the volume fraction, physical density, and velocity of phase-l, respectively. $\Gamma^{k}_{l}$ and $S^k_l$ are the diffusion coefficient and source term, respectively, which you will need to specify. In this case, scalar $\phi^k_l$ is associated only with one phase ( phase-l) and is considered an individual field variable of phase-l.

The mass flux for phase-l is defined as

 F_l = \int_{S} \alpha_l \rho_l \vec {u}_l \cdot d \vec{S} (9.3-6)

If the transport variable described by scalar $\phi^k_l$ represents the physical field that is shared between phases, or is considered the same for each phase, then you should consider this scalar as being associated with a mixture of phases, $\phi^k$. In this case, the generic transport equation for the scalar is

 \frac{\partial \rho_{m} \phi^k}{\partial t} + \nabla \cdot (... ...i^k - \Gamma^k_m \nabla \phi^k) = S^{k_m} \; \; \; k = 1,...,N (9.3-7)

where mixture density $\rho_{m}$, mixture velocity $\vec {u}_m$, and mixture diffusivity for the scalar $k$ $\Gamma^k_m$ are calculated according to

 \rho_{m} = \sum_{l} \alpha_{l} \rho_{l} (9.3-8)

 \rho_{m} \vec {u}_m = \sum_{l} \alpha_{l} \rho_{l} \vec {u}_l (9.3-9)

 F_m = \int_{S} rho_m \vec {u}_m \cdot d \vec{S} (9.3-10)

 \Gamma^k_m = \sum_{l} \alpha_{l} \Gamma^k_l (9.3-11)

 S^k_m = \sum_{l} S^k_l (9.3-12)

To calculate mixture diffusivity , you will need to specify individual diffusivities for each material associated with individual phases.

Note that if the user-defined mass flux option is activated, then mass fluxes shown in Equation  9.3-6 and Equation  9.3-10 will need to be replaced in the corresponding scalar transport equations.

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Up: 9.3 User-Defined Scalar (UDS)
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© Fluent Inc. 2006-09-20