Stream-wise operator¶
Average¶
Average at stream-wise cell faces:
\[\vat{
\ave{
q
}{
\gy
}
}{
j + \frac{1}{2}
}
=
\frac{1}{2} \vat{q}{j} + \frac{1}{2} \vat{q}{j + 1}\]
Average at stream-wise cell centers:
\[\vat{
\ave{
q
}{
\gy
}
}{
j
}
=
\frac{1}{2} \vat{q}{j - \frac{1}{2}} + \frac{1}{2} \vat{q}{j + \frac{1}{2}}\]
Differentiation¶
Differentiation at stream-wise cell faces:
\[\vat{
\dif{
q
}{
\gy
}
}{
j + \frac{1}{2}
}
=
-
\vat{q}{j}
+
\vat{q}{j + 1}\]
Differentiation at stream-wise cell centers:
\[\vat{
\dif{
q
}{
\gy
}
}{
j
}
=
-
\vat{q}{j - \frac{1}{2}}
+
\vat{q}{j + \frac{1}{2}}\]
Summation¶
Summation of a quantity defined at stream-wise cell faces:
\[\sumyf q
\equiv
\vat{q}{\frac{1}{2}}
+
\vat{q}{\frac{3}{2}}
+
\cdots
+
\vat{q}{\ny - \frac{3}{2}}
+
\vat{q}{\ny - \frac{1}{2}}\]
Summation of a quantity defined at stream-wise cell centers:
\[\sumyc q
\equiv
\vat{q}{1}
+
\vat{q}{2}
+
\cdots
+
\vat{q}{\ny - 1}
+
\vat{q}{\ny}\]