Energy balance
To see the energy balance globally and discretely, the momentum balance for each direction is integrated in the whole volume after multiplied by the corresponding local velocity.
Temporal evolution and advection
I consider the effects of
\[\pder{\rho u_i}{t}
+
\frac{1}{J}
\dif{
\left(
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\ave{
u_i
}{\gcs^j}
\right)
}{\gcs^j}\]
on the global kinetic energy balance; namely:
\[\sum_{i\,\text{face}}
J
u_i
\left\{
\pder{\rho u_i}{t}
+
\frac{1}{J}
\dif{
\left(
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\ave{
u_i
}{\gcs^j}
\right)
}{\gcs^j}
\right\}.\]
Contribution of the first term leads to
\[\sum_{i\,\text{face}}
J
u_i
\pder{\rho u_i}{t}
=
\sum_{i\,\text{face}}
J
\pder{\rho u_i u_i}{t}
-
\sum_{i\,\text{face}}
J
\rho u_i
\pder{u_i}{t},\]
while the second term gives
\[\begin{split}\sum_{i\,\text{face}}
J
u_i
\frac{1}{J}
\dif{
\left(
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\ave{
u_i
}{\gcs^j}
\right)
}{\gcs^j}
&
=
-
\sum_{i\,\text{center}}
\ave{
u_i
}{\gcs^j}
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\dif{
u_i
}{\gcs^j} \\
&
=
-
\sum_{i\,\text{face}}
J
u_i
\frac{1}{J}
\ave{
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\dif{
u_i
}{\gcs^j}
}{\gcs^j}.\end{split}\]
As a result, I obtain
\[\sum_{i\,\text{face}}
J
\pder{\rho u_i u_i}{t}
-
\sum_{i\,\text{face}}
J
u_i
\left(
\rho
\pder{u_i}{t}
+
\frac{1}{J}
\ave{
\ave{
\frac{J}{h_{\gcs^j}}
\rho u_j
}{\gcs^i}
\dif{
u_i
}{\gcs^j}
}{\gcs^j}
\right).\]
What I have inside the parentheses of the second term is the so-called gradient form of the advective terms.
Pressure gradient
The pressure-gradient terms
\[-
\frac{1}{h_{\gcs^i}}
\dif{p}{\gcs^i},\]
which contributes to the energy balance as follows:
\[-
\sumzc
\sumyc
\sumxf
J
\ux
\frac{1}{\hx}
\dif{p}{\gx}
=
\sumzc
\sumyc
\sumxc
J
p
\frac{1}{J}
\dif{
\left(
\jhx
\ux
\right)
}{\gx}\]
\[-
\sumzc
\sumyf
\sumxc
J
\uy
\frac{1}{\hy}
\dif{p}{\gy}
=
\sumzc
\sumyc
\sumxc
J
p
\frac{1}{J}
\dif{
\left(
\jhy
\uy
\right)
}{\gy}\]
\[-
\sumzf
\sumyc
\sumxc
J
\uz
\frac{1}{\hz}
\dif{p}{\gz}
=
\sumzc
\sumyc
\sumxc
J
p
\frac{1}{J}
\dif{
\left(
\jhz
\uz
\right)
}{\gz}\]
The sum is
\[\sumzc
\sumyc
\sumxc
J
p
\left\{
\frac{1}{J}
\dif{
\left(
\jhx
\ux
\right)
}{\gx}
+
\frac{1}{J}
\dif{
\left(
\jhy
\uy
\right)
}{\gy}
+
\frac{1}{J}
\dif{
\left(
\jhz
\uz
\right)
}{\gz}
\right\},\]
which is zero because the component inside the wavy parentheses is the incompressibility constraint.
Diffusion
In general, the diffusive terms are written as
\[\frac{1}{J}
\dif{}{\gcs^j}
\left(
\frac{J}{h_{\gcs^j}}
\tau_{i j}
\right),\]
which is the diffusion of the \(i\)-th momentum in the \(j\)-th direction.
The global contribution
\[\sum_{i\,\text{face}}
J
u_i
\frac{1}{J}
\dif{}{\gcs^j}
\left(
\frac{J}{h_{\gcs^j}}
\tau_{i j}
\right)
=
\sum_{i\,\text{face}}
u_i
\dif{}{\gcs^j}
\left(
\frac{J}{h_{\gcs^j}}
\tau_{i j}
\right)\]
is as follows.
\(x\) momentum contribution:
\[ \begin{align}\begin{aligned}&
\sumzc
\sumyc
\sumxf
\ux
\dif{}{\gx}
\left(
\jhx
\tau_{\vx \vx}
\right)
-
=
\sumzc
\sumyc
\sumxc
J
l_{\vx \vx}
\tau_{\vx \vx}\\&
\sumzc
\sumyc
\sumxf
\ux
\dif{}{\gy}
\left(
\jhy
\tau_{\vx \vy}
\right)
=
-
\sumzc
\sumyf
\sumxf
J
l_{\vx \vy}
\tau_{\vx \vy}\\&
\sumzc
\sumyc
\sumxf
\ux
\dif{}{\gz}
\left(
\jhz
\tau_{\vx \vz}
\right)
=
-
\sumzf
\sumyc
\sumxf
J
l_{\vx \vz}
\tau_{\vx \vz}\end{aligned}\end{align} \]
\(y\) momentum contribution:
\[ \begin{align}\begin{aligned}&
\sumzc
\sumyf
\sumxc
\uy
\dif{}{\gx}
\left(
\jhx
\tau_{\vy \vx}
\right)
=
-
\sumzc
\sumyf
\vat{
\left(
\uy
\jhx
\tau_{\vy \vx}
\right)
}{\frac{1}{2}}
+
\sumzc
\sumyf
\vat{
\left(
\uy
\jhx
\tau_{\vy \vx}
\right)
}{\nx + \frac{1}{2}}
-
\sumzc
\sumyf
\sumxf
J
l_{\vy \vx}
\tau_{\vy \vx}\\&
\sumzc
\sumyf
\sumxc
\uy
\dif{}{\gy}
\left(
\jhy
\tau_{\vy \vy}
\right)
=
-
\sumzc
\sumyc
\sumxc
J
l_{\vy \vy}
\tau_{\vy \vy}\\&
\sumzc
\sumyf
\sumxc
\uy
\dif{}{\gz}
\left(
\jhz
\tau_{\vy \vz}
\right)
=
-
\sumzf
\sumyf
\sumxc
J
l_{\vy \vz}
\tau_{\vy \vz}\end{aligned}\end{align} \]
\(z\) momentum contribution:
\[ \begin{align}\begin{aligned}&
\sumzf
\sumyc
\sumxc
\uz
\dif{}{\gx}
\left(
\jhx
\tau_{\vz \vx}
\right)
=
-
\sumzf
\sumyc
\sumxf
J
l_{\vz \vx}
\tau_{\vz \vx}\\&
\sumzf
\sumyc
\sumxc
\uz
\dif{}{\gy}
\left(
\jhy
\tau_{\vz \vy}
\right)
=
-
\sumzf
\sumyf
\sumxc
J
l_{\vz \vy}
\tau_{\vz \vy}\\&
\sumzf
\sumyc
\sumxc
\uz
\dif{}{\gz}
\left(
\jhz
\tau_{\vz \vz}
\right)
=
-
\sumzc
\sumyc
\sumxc
J
l_{\vz \vz}
\tau_{\vz \vz}\end{aligned}\end{align} \]
All terms are dissipative, except the two terms in the \(y\) contribution which are the energy throughput on the walls.