ANSYS FLUENT 12.0 User's Guide

9.1.2 UDS Theory

Single Phase Flow

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

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

(9.1-1)

where $\Gamma_{k}$ and $S_{\phi_k}$ are the diffusion coefficient and source term supplied by you for each of the 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, ANSYS FLUENT will solve one of the three following equations, depending on the method used to compute the convective flux :

If convective flux is not to be computed, ANSYS FLUENT will solve the equation

$-\frac{\partial} {\partial x_i}(\Gamma_k \frac{\partial \phi_k}{\partial x_i}) = S_{\phi_k} \; \; \; k = 1,...,N$ (9.1-2)

where ${\Gamma}_k$ and $S_{\phi_k}$ are the diffusion coefficient and source term supplied by you for each of the scalar equations.
If convective flux is to be computed with mass flow rate , ANSYS FLUENT will solve the equation

$\frac{\partial}{\partial x_i} (\rho u_i \phi_k - \Gamma_k \f... ...tial \phi_k}{\partial x_i} ) = S_{\phi_k} \; \; \; k = 1,...,N$ (9.1-3)
It is also possible to specify a user-defined function to be used in the computation of convective flux. In this case, the user-defined mass flux is assumed to be of the form

$F = \int_{S} \rho \vec {u} \cdot d \vec{S}$ (9.1-4)

where $d \vec{S}$ is the face vector area.

User-defined scalars in solid zones do not take into account the convective term with moving reference frames.

Multiphase Flow

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

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

(9.1-5)

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 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

$F_l = \int_{S} \alpha_l \rho_l \vec {u}_l \cdot d \vec{S}$

(9.1-6)

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

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

(9.1-7)

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

$\rho_{m} = \sum_{l} \alpha_{l} \rho_{l}$

(9.1-8)

$\rho_{m} \vec {u}_m = \sum_{l} \alpha_{l} \rho_{l} \vec {u}_l$

(9.1-9)

$F_m = \int_{S} rho_m \vec {u}_m \cdot d \vec{S}$

(9.1-10)

$\Gamma^k_m = \sum_{l} \alpha_{l} \Gamma^k_l$

(9.1-11)

$S^k_m = \sum_{l} S^k_l$

(9.1-12)

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 9.1-6 and Equation 9.1-10 will need to be replaced in the corresponding scalar transport equations. For more information about the theoretical background of user-defined scalar transport equations, see this section in the separate Theory Guide.

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