Relations involving stream-wise velocity¶
Differentiations¶
\[\sumzc
\sumyf
\sumxc
\vel{2}
\dif{q}{\gcs{1}}
=
-
\sumzc
\sumyf
\left(
\vat{\left(\vel{2} q\right)}{\frac{1}{2}}
+
\sumxf
\dif{\vel{2}}{\gcs{1}}
q
-
\vat{\left(\vel{2} q\right)}{\ngp{1} + \frac{1}{2}}
\right).\]
\[\sumzc
\sumyf
\sumxc
\vel{2}
\dif{q}{\gcs{2}}
=
-
\sumzc
\sumyc
\sumxc
\dif{\vel{2}}{\gcs{2}}
q.\]
\[\sumzc
\sumyf
\sumxc
\vel{2}
\dif{q}{\gcs{3}}
=
-
\sumzf
\sumyf
\sumxc
\dif{\vel{2}}{\gcs{3}}
q.\]
Averages¶
\[\sumzc
\sumyf
\sumxc
\vel{2}
\ave{q}{\gcs{1}}
=
\sumzc
\sumyf
\left(
\vat{\vel{2}}{1}
\frac{\vat{q}{\frac{1}{2}}}{2}
+
\sum_{i = \frac{3}{2}}^{\ngp{1} - \frac{1}{2}}
\ave{\vel{2}}{\gcs{1}}
q
+
\vat{\vel{2}}{\ngp{1}}
\frac{\vat{q}{\ngp{1} + \frac{1}{2}}}{2}
\right).\]
\[\sumzc
\sumyf
\sumxc
\vel{2}
\ave{q}{\gcs{2}}
=
\sumzc
\sumyc
\sumxc
\ave{\vel{2}}{\gcs{2}}
q.\]
\[\sumzc
\sumyf
\sumxc
\vel{2}
\ave{q}{\gcs{3}}
=
\sumzf
\sumyf
\sumxc
\ave{\vel{2}}{\gcs{3}}
q.\]