Relations involving span-wise velocity¶
Differentiations¶
\[\sumzf
\sumyc
\sumxc
\vel{3}
\dif{q}{\gcs{1}}
=
-
\sumzf
\sumyc
\sumxf
\dif{\vel{3}}{\gcs{1}}
q,\]
where \(\vat{\vel{3}}{\frac{1}{2},\ccindex{j},\cpindex{k}} = \vat{\vel{3}}{\ngp{1} + \frac{1}{2},\ccindex{j},\cpindex{k}} = 0\) is assumed (i.e., the walls do not move in the \(z\) direction).
\[\sumzf
\sumyc
\sumxc
\vel{3}
\dif{q}{\gcs{2}}
=
-
\sumzf
\sumyf
\sumxc
\dif{\vel{3}}{\gcs{2}}
q.\]
\[\sumzf
\sumyc
\sumxc
\vel{3}
\dif{q}{\gcs{3}}
=
-
\sumzc
\sumyc
\sumxc
\dif{\vel{3}}{\gcs{3}}
q.\]
Averages¶
\[\sumzf
\sumyc
\sumxc
\vel{3}
\ave{q}{\gcs{1}}
=
\sumzf
\sumyc
\left(
\vat{\vel{3}}{1}
\frac{\vat{q}{\frac{1}{2}}}{2}
+
\sum_{i = \frac{3}{2}}^{\ngp{1} - \frac{1}{2}}
\ave{\vel{3}}{\gcs{1}}
q
+
\vat{\vel{3}}{\ngp{1}}
\frac{\vat{q}{\ngp{1} + \frac{1}{2}}}{2}
\right).\]
\[\sumzf
\sumyc
\sumxc
\vel{3}
\ave{q}{\gcs{2}}
=
\sumzf
\sumyf
\sumxc
\ave{\vel{3}}{\gcs{2}}
q.\]
\[\sumzf
\sumyc
\sumxc
\vel{3}
\ave{q}{\gcs{3}}
=
\sumzc
\sumyc
\sumxc
\ave{\vel{3}}{\gcs{3}}
q.\]