ANSYS FLUENT 12.0 Theory Guide - 20.3.1 Computing Surface Integrals

20.3.1 Computing Surface Integrals

Area

The area of a surface is computed by summing the areas of the facets that define the surface. Facets on a surface are either triangular or quadrilateral in shape.

$\int { dA } = \sum_{i=1}^{n}{ \vert A_i \vert }$

(20.3-2)

Integral

An integral on a surface is computed by summing the product of the facet area and the selected field variable facet value, such as density or pressure. For details on the computation of the facet values, see Section 20.3.

Area-Weighted Average

The area-weighted average of a quantity is computed by dividing the summation of the product of the selected field variable and facet area by the total area of the surface:

$\frac{1}{A} \int{\phi dA} = \frac{1}{A} \sum_{i=1}^{n} {\phi_i \vert A_i \vert }$

(20.3-3)

Flow Rate

The flow rate of a quantity through a surface is computed by summing the value of the selected field variable multiplied by the density and the dot product of the facet area vector and the facet velocity vector.

$\int {\phi \rho \vec{v} \cdot d\vec{A}} = \sum_{i=1}^{n} {\phi_i \rho_i \vec{v_i} \cdot \vec{A_i}}$

(20.3-4)

Mass Flow Rate

The mass flow rate through a surface is computed by summing the value of the selected field variable multiplied by the density and the dot product of the facet area vector and the facet velocity vector.

$\int {\rho \vec{v} \cdot d\vec{A}} = \sum_{i=1}^{n} {\rho_i \vec{v_i} \cdot \vec{A_i}}$

(20.3-5)

Mass-Weighted Average

The mass-weighted average of a quantity is computed by dividing the summation of the value of the selected field variable multiplied by the absolute value of the dot product of the facet area and momentum vectors by the summation of the absolute value of the dot product of the facet area and momentum vectors (surface mass flux):

$\frac {\displaystyle{\int}{\phi \rho \left\vert \vec{v} \cdo... ...{n}} {\rho_i \left\vert \vec{v_i} \cdot \vec{A_i}\right\vert}}$

(20.3-6)

Sum of Field Variable

The sum of a specified field variable on a surface is computed by summing the value of the selected variable at each facet:

$\sum_{i=1}^{n}{ \phi_i }$

(20.3-7)

Facet Average

The facet average of a specified field variable on a surface is computed by dividing the summation of the facet values of the selected variable by the total number of facets. See this section in the separate User's Guide for definitions of facet values.

$\frac{\displaystyle{\sum_{i=1}^{n}}{ \phi_i }}{n}$

(20.3-8)

Facet Minimum

The facet minimum of a specified field variable on a surface is the minimum facet value of the selected variable on the surface. See this section in the separate User's Guide for definitions of facet values.

Facet Maximum

The facet maximum of a specified field variable on a surface is the maximum facet value of the selected variable on the surface. See this section in the separate User's Guide for definitions of facet values.

Vertex Average

The vertex average of a specified field variable on a surface is computed by dividing the summation of the vertex values of the selected variable by the total number of vertices. See this section in the separate User's Guide for definitions of vertex values.

$\frac{\displaystyle{\sum_{i=1}^{n}}{ \phi_i }}{n}$

(20.3-9)

Vertex Minimum

The vertex minimum of a specified field variable on a surface is the minimum vertex value of the selected variable on the surface. See this section in the separate User's Guide for definitions of vertex values.

Vertex Maximum

The vertex maximum of a specified field variable on a surface is the maximum vertex value of the selected variable on the surface. See this section in the separate User's Guide for definitions of vertex values.

Standard-Deviation

The standard deviation of a specified field variable on a surface is computed using the mathematical expression below:

$\sqrt{\frac{\displaystyle{\sum_{i=1}^{n}}{(x - x{_0})}^2}{n}}$

(20.3-10)

where is the cell value of the selected variables at each facet, is the mean of

$x_0 = \frac{\displaystyle{\sum_{i=1}^{n}{x}}}{n}$

and is the total number of facets. See this section in the separate User's Guide for definitions of facet values.

Volume Flow Rate

The volume flow rate through a surface is computed by summing the value of the facet area vector multiplied by the facet velocity vector:

$\int { \vec{v} \cdot d\vec{A}} = \sum_{i=1}^{n} {\vec{v_i} \cdot \vec{A_i}}$

(20.3-11)

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