## 9.3.2 UDS Theory

Single Phase Flow

For an arbitrary scalar , FLUENT solves the equation

 (9.3-1)

where and are the diffusion coefficient and source term supplied by you for each of the scalar equations. Note that is defined as a tensor in the case of anisotropic diffusivity. The diffusion term is thus

For isotropic diffusivity, could be written as 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 :

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

 (9.3-2)

where and 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 , FLUENT will solve the equation

 (9.3-3)

• 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

 (9.3-4)

where is the face vector area.

Multiphase Flow

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

 (9.3-5)

where , , and are the volume fraction, physical density, and velocity of phase-l, respectively. and are the diffusion coefficient and source term, respectively, which you will need to specify. In this case, scalar 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

 (9.3-6)

If the transport variable described by scalar 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, . In this case, the generic transport equation for the scalar is

 (9.3-7)

where mixture density , mixture velocity , and mixture diffusivity for the scalar are calculated according to

 (9.3-8)

 (9.3-9)

 (9.3-10)

 (9.3-11)

 (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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