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29.5.1 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 } (29.5-2)



Integral


An integral on a surface is computed by summing the product of the facet area and the selected field variable, such as density or pressure. Each facet is associated with a cell in the domain. If the facet is the result of an isovalue cut through the cell, the field variable assigned to the facet is the associated cell value. If the facet is on a boundary surface, an interpolated face value is used for the integration instead of the cell value. This is done to improve the accuracy of the calculation, and to ensure that the result matches the boundary conditions specified on the boundary and the fluxes reported on the boundary.


 \int { \phi dA } = \sum_{i=1}^{n}{ \phi_i \vert A_i \vert } (29.5-3)



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 } (29.5-4)



Flow Rate


The flow rate of a quantity through a surface is computed by summing the product of density and the selected field variable with 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}} (29.5-5)



Mass Flow Rate


The mass flow rate through a surface is computed by summing the product of density with 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}} (29.5-6)



Mass-Weighted Average


The mass-weighted average of a quantity is computed by dividing the summation of the product of the selected field variable and 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}} (29.5-7)



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 } (29.5-8)



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 Section  30.1 for definitions of facet values.


 \frac{\displaystyle{\sum_{i=1}^{n}}{ \phi_i }}{n} (29.5-9)



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 Section  30.1 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 Section  30.1 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 Section  30.1 for definitions of vertex values.


 \frac{\displaystyle{\sum_{i=1}^{n}}{ \phi_i }}{n} (29.5-10)



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 Section  30.1 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 Section  30.1 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}} (29.5-11)

where $x$ is the cell value of the selected variables at each facet, $x_0$ is the mean of $x$


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

and $n$ is the total number of facets. See Section  30.1 for definitions of facet values.



Volume Flow Rate


The volume flow rate through a surface is computed by summing the dot product of the facet area vector and the facet velocity vector:


 \int { \vec{v} \cdot d\vec{A}} = \sum_{i=1}^{n} {\vec{v_i} \cdot \vec{A_i}} (29.5-12)


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