Diffusive terms¶
We utilise the relations listed in the prerequisite to derive the following relations.
Squared velocity¶
From the wall-normal momentum balance, we obtain
\[-
\sumzc
\sumyc
\sumxc
\dkdis{1}{1}
-
\sumzc
\sumyf
\sumxf
\dkdis{2}{1}
-
\sumzf
\sumyc
\sumxf
\dkdis{3}{1}.\]
From the stream-wise momentum balance, we obtain
\[-
\sumzc
\sumyf
\sumxf
\dkdis{1}{2}
-
\sumzc
\sumyc
\sumxc
\dkdis{2}{2}
-
\sumzf
\sumyf
\sumxc
\dkdis{3}{2}.\]
From the span-wise momentum balance, we obtain
\[-
\sumzf
\sumyc
\sumxf
\dkdis{1}{3}
-
\sumzf
\sumyf
\sumxc
\dkdis{2}{3}
-
\sumzc
\sumyc
\sumxc
\dkdis{3}{3}.\]
In total, all terms work to dissipate \(k\).
Squared temperature¶
We obtain
\[-
\sumzc
\sumyc
\sumxf
J
\frac{1}{\sqrt{Pr} \sqrt{Ra}}
\left(
\frac{1}{\sfact{1}}
\dif{T}{\gcs{1}}
\right)^2
-
\sumzc
\sumyf
\sumxc
J
\frac{1}{\sqrt{Pr} \sqrt{Ra}}
\left(
\frac{1}{\sfact{2}}
\dif{T}{\gcs{2}}
\right)^2
-
\sumzf
\sumyc
\sumxc
J
\frac{1}{\sqrt{Pr} \sqrt{Ra}}
\left(
\frac{1}{\sfact{3}}
\dif{T}{\gcs{3}}
\right)^2\]
as the dissipative terms, while
\[-
\sumzc
\sumyc
\frac{1}{\sqrt{Pr} \sqrt{Ra}}
\vat{
\left(
\frac{J}{\sfact{1}}
T
\frac{1}{\sfact{1}}
\dif{T}{\gcs{1}}
\right)
}{\frac{1}{2}}
+
\sumzc
\sumyc
\frac{1}{\sqrt{Pr} \sqrt{Ra}}
\vat{
\left(
\frac{J}{\sfact{1}}
T
\frac{1}{\sfact{1}}
\dif{T}{\gcs{1}}
\right)
}{\ngp{1} + \frac{1}{2}}\]
as the conduction on the walls.