Relations involving wall-normal velocity¶
Differentiations¶
\[\sumzc
\sumyc
\sumxf
\vel{1}
\dif{q}{\gcs{1}}
=
-
\sumzc
\sumyc
\sumxc
\dif{\vel{1}}{\gcs{1}}
q,\]
where \(\vat{\vel{1}}{\frac{1}{2}} = \vat{\vel{1}}{\ngp{1} + \frac{1}{2}} = 0\) is used.
\[\sumzc
\sumyc
\sumxf
\vel{1}
\dif{q}{\gcs{2}}
=
-
\sumzc
\sumyf
\sumxf
\dif{\vel{1}}{\gcs{2}}
q.\]
\[\sumzc
\sumyc
\sumxf
\vel{1}
\dif{q}{\gcs{3}}
=
-
\sumzf
\sumyc
\sumxf
\dif{\vel{1}}{\gcs{3}}
q.\]
Averages¶
\[\sumzc
\sumyc
\sumxf
\vel{1}
\ave{q}{\gcs{1}}
=
\sumzc
\sumyc
\sumxc
\ave{\vel{1}}{\gcs{1}}
q.\]
\[\sumzc
\sumyc
\sumxf
\vel{1}
\ave{q}{\gcs{2}}
=
\sumzc
\sumyf
\sumxf
\ave{\vel{1}}{\gcs{2}}
q.\]
\[\sumzc
\sumyc
\sumxf
\vel{1}
\ave{q}{\gcs{3}}
=
\sumzf
\sumyc
\sumxf
\ave{\vel{1}}{\gcs{3}}
q.\]