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8.6.2 Anisotropic Diffusion

You can specify anisotropic diffusion coefficients in both fluid and solid zones by defining the tensor diffusion coefficient matrix $\Gamma$ (Equation  8.6-1) on a per-scalar basis. You can use anisotropic diffusivity for UDS scalar transport equations to model species transport equations in porous media and in solids where species diffusion shows anisotropic behavior.

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Note that the anisotropic diffusion options discussed in the following sections are available with the pressure-based solver and the density-based solvers.

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UDS diffusion coefficients can be postprocessed only in those cells which have isotropic diffusivity. In all other cells, the diffusion coefficient will be zero.

In all cases, you enable anisotropic diffusion by selecting defined-per-uds under UDS Diffusivity in the Materials panel (Figure  8.6.2). This will open the UDS Diffusion Coefficients panel (Figure  8.6.3).

Define $\rightarrow$ Materials...

In the UDS Diffusion Coefficient panel, select a scalar equation (e.g., uds-0) and then choose one of the following methods under Coefficients to specify the anisotropic diffusion coefficient. These methods are described in detail below.



Anisotropic Diffusivity


For anisotropic diffusivity, you can specify $\Gamma$ in Equation  8.6-1 in the form $K\Gamma$ where $K$ is a constant $3x3$ matrix in 3D and $\Gamma$ is a scalar multiplier.

The diffusion coefficient matrix is specified as


 k_{ij} = k \hat{\bf e}_{ij} (8.6-3)

where $k$ is the diffusivity and $\hat{\bf e}_{ij}$ is a matrix (2 $\times$ 2 for two dimensions and 3 $\times$ 3 for three-dimensional problems). Note that $\hat{\bf e}_{ij}$ can be a non-symmetric matrix.

To specify anisotropic diffusion coefficients, first select a scalar equation (e.g., uds-0) from the User-Defined Scalar Diffusion list in the UDS Diffusion Coefficients panel (Figure  8.6.3). Then choose anisotropic in the drop-down list under Coefficient. This will open the Anisotropic UDS Diffusivity panel (Figure  8.6.4).

Figure 8.6.4: The Anisotropic UDS Diffusivity Panel
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In the Anisotropic UDS Diffusivity panel, enter the Matrix Components and then select the Diffusivity to be a constant, polynomial function of temperature ( polynomial, piecewise-linear, piecewise-polynomial), or user-defined. See Sections  8.2.1, 8.2.2, and 8.2.3 for details on polynomial temperature functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_DIFFUSIVITY UDF only if you have previously loaded a compiled UDF library or interpreted a UDF. Otherwise, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.



Orthotropic Diffusivity


For orthotropic diffusivity, you can specify $\Gamma$ in Equation  8.6-1 through 'principal' direction vectors and diffusion coefficients along these directions. FLUENT, in turn, computes $\Gamma$ from parameters that you supply. The principal directions are the same everywhere, but each of he directional diffusion coefficients can be specified as a constant, polynomial function of temperature, or through user-defined functions.

When orthotropic diffusivity is used, the diffusion coefficients $(k_\xi,k_\eta,k_\zeta)$ in the principal directions $(\hat{\bf e}_\xi,\hat{\bf e}_\eta,\hat{\bf e}_\zeta)$ are specified. The diffusivity matrix is then computed as


 k_{ij} = k_{\xi} e_{\xi i} e_{\xi j} + k_{\eta} e_{\eta i} e_{\eta j}+ k_{\zeta} e_{\zeta i} e_{\zeta j} (8.6-4)

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For two-dimensional problems, only the functions $(k_\xi,k_\eta)$ and the unit vector $(\hat{\bf e}_\xi)$ need to be specified.

To specify orthotropic diffusion coefficients, first select a scalar equation (e.g., uds-0) from the User-Defined Scalar Diffusion list in the UDS Diffusion Coefficients panel (Figure  8.6.3). Then choose orthotropic in the drop-down list under Coefficient. This will open the Orthotropic UDS Diffusivity panel (Figure  8.6.5).

Figure 8.6.5: The Orthotropic UDS Diffusivity Panel
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Since the directions $(\hat{\bf e}_\xi,\hat{\bf e}_\eta,\hat{\bf e}_\zeta)$ are mutually orthogonal, only the first two need to be specified for three-dimensional problems. $\hat{\bf e}_\xi$ is defined using X,Y,Z under Direction 0 Components, and $\hat{\bf e}_\eta$ is defined using X,Y,Z under Direction 1 Components. You can define Diffusivity 0 $(k_\xi)$, Diffusivity 1 $(k_\eta)$, and Diffusivity 2 $(k_\zeta)$ as constant, polynomial, piecewise-linear, piecewise-polynomial functions of temperature, or user-defined. See Sections  8.2.1, 8.2.2, and 8.2.3 for details on polynomial temperature functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_DIFFUSIVITY UDF only if you have previously loaded a compiled UDF library or interpreted a UDF. If no functions have been loaded, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.



Cylindrical Orthotropic Diffusivity


Orthotropic UDS diffusivity can also be specified on a per-scalar basis in cylindrical coordinates. This method is similar to orthotropic UDS diffusivity, except that the principal directions are specified as radial, tangential, and axial.

To specify cylindrical orthotropic diffusion coefficients, first select a scalar equation (e.g., uds-0) from the User-Defined Scalar Diffusion list in the UDS Diffusion Coefficients panel (Figure  8.6.3). Then choose cyl-orthotropic in the drop-down list under Coefficient. This will open the Cylindrical Orthotropic UDS Diffusivity panel (Figure  8.6.6).

Figure 8.6.6: The Cylindrical Orthotropic UDS Diffusivity Panel
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In three-dimensional cases, the origin and the direction of the cylindrical coordinate system must be specified along with the radial, tangential, and axial direction conductivities. In two-dimensional cases, the origin of the cylindrical coordinate system must be specified along with the radial and tangential direction conductivities. Note that in two-dimensional cases, the direction is always along the + z axis. FLUENT will automatically compute the anisotropic diffusivity matrix at each cell from this input. The calculation is based on the location of the cell in the cylindrical coordinate system specified.

You can define the Radial Diffusivity, Tangential Diffusivity, and Axial Diffusivity as constant, polynomial, piecewise-linear, piecewise-polynomial, or as user-defined functions of temperature, using the drop-down list below each of the diffusivities. See Sections  8.2.1, 8.2.2, and 8.2.3 for details on polynomial temperature functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_DIFFUSIVITY UDF only if you have previously loaded a compiled UDF library or interpreted a UDF. If no functions have been loaded, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.


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