Pressure-gradientΒΆ
The pressure-gradient terms
\[-
\frac{1}{\sfact{i}}
\dif{p}{\gcs{i}},\]
which contribute to the energy balance as follows:
\[ \begin{align}\begin{aligned}\newcommand{\tmp}[1]{
J
\vel{#1}
\frac{1}{\sfact{#1}}
\dif{p}{\gcs{#1}}
=
\sumzc
\sumyc
\sumxc
J
p
\frac{1}{J}
\dif{
\left(
\frac{J}{\sfact{#1}}
\vel{#1}
\right)
}{\gcs{#1}}
}
-
\sumzc
\sumyc
\sumxf
\tmp{1},\\-
\sumzc
\sumyf
\sumxc
\tmp{2},\\-
\sumzf
\sumyc
\sumxc
\tmp{3}.\end{aligned}\end{align} \]
The sum of these three relations is
\[\sumzc
\sumyc
\sumxc
J
p
\left\{
\frac{1}{J}
\dif{
\left(
\frac{J}{\sfact{1}}
\vel{1}
\right)
}{\gcs{1}}
+
\frac{1}{J}
\dif{
\left(
\frac{J}{\sfact{2}}
\vel{2}
\right)
}{\gcs{2}}
+
\frac{1}{J}
\dif{
\left(
\frac{J}{\sfact{3}}
\vel{3}
\right)
}{\gcs{3}}
\right\},\]
which is zero because the component inside the wavy parentheses is the incompressibility constraint.