Wall-normal operator¶
Average¶
Average at wall-normal cell faces:
\[\begin{split}\vat{
\ave{
q
}{
\gx
}
}{
i + \frac{1}{2}
}
=
\left\{
\begin{alignedat}{2}
& \text{Negative wall:} & \vat{q}{\frac{1}{2}}, \\
& \text{Positive wall:} & \vat{q}{\nx + \frac{1}{2}}, \\
& \text{Otherwise:} & \frac{1}{2} \vat{q}{i} + \frac{1}{2} \vat{q}{i + 1}
\end{alignedat}
\right.\end{split}\]
Average at wall-normal cell centers:
\[\vat{
\ave{
q
}{
\gx
}
}{
i
}
=
\frac{1}{2} \vat{q}{i - \frac{1}{2}} + \frac{1}{2} \vat{q}{i + \frac{1}{2}}\]
Differentiation¶
Differentiation at wall-normal cell faces:
\[\begin{split}\vat{
\dif{
q
}{
\gx
}
}{
i + \frac{1}{2}
}
=
\left\{
\begin{alignedat}{2}
& \text{Negative wall:} & - \vat{q}{\frac{1}{2}} + \vat{q}{1}, \\
& \text{Positive wall:} & - \vat{q}{\nx}+ \vat{q}{\nx + \frac{1}{2}} , \\
& \text{Otherwise:} & - \vat{q}{i} + \vat{q}{i + 1}
\end{alignedat}
\right.\end{split}\]
Differentiation at wall-normal cell centers:
\[\vat{
\dif{
q
}{
\gx
}
}{
i
}
=
- \vat{q}{i - \frac{1}{2}}
+ \vat{q}{i + \frac{1}{2}}\]
Summation¶
Summation of a quantity defined at wall-normal cell faces:
\[\sumxf q
\equiv
\vat{q}{\frac{1}{2}}
+
\vat{q}{\frac{3}{2}}
+
\cdots
+
\vat{q}{\nx - \frac{1}{2}}
+
\vat{q}{\nx + \frac{1}{2}}\]
Summation of a quantity defined at wall-normal cell centers:
\[\sumxc q
\equiv
\vat{q}{1}
+
\vat{q}{2}
+
\cdots
+
\vat{q}{\nx - 1}
+
\vat{q}{\nx}\]