Quadratic QuantitiesΒΆ

Here we aim at confirming if the theoretical relations of the quadratic quantities are satisfied even after discretized.

Specifically, as discrete counterparts, we consider the discrete momentum balance multiplied by the Jacobian determinant (incorporating the volume integrals) and the velocity:

\[ \begin{align}\begin{aligned}\sumzc \sumyc \sumxf J \vel{1} \pder{\vel{1}}{t} = & - \sumzc \sumyc \sumxf J \vel{1} \dmomadv{1}{1} - \sumzc \sumyc \sumxf J \vel{1} \dmomadv{2}{1} - \sumzc \sumyc \sumxf J \vel{1} \dmomadv{3}{1}\\& - \sumzc \sumyc \sumxf J \vel{1} \dmompre{1}\\& + \sumzc \sumyc \sumxf J \vel{1} \dmomdif{1}{1} + \sumzc \sumyc \sumxf J \vel{1} \dmomdif{2}{1} + \sumzc \sumyc \sumxf J \vel{1} \dmomdif{3}{1}\\& + \sumzc \sumyc \sumxf J \vel{1} \ave{T}{\gcs{1}},\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\sumzc \sumyf \sumxc J \vel{2} \pder{\vel{2}}{t} = & - \sumzc \sumyf \sumxc J \vel{2} \dmomadv{1}{2} - \sumzc \sumyf \sumxc J \vel{2} \dmomadv{2}{2} - \sumzc \sumyf \sumxc J \vel{2} \dmomadv{3}{2}\\& - \sumzc \sumyf \sumxc J \vel{2} \dmompre{2}\\& + \sumzc \sumyf \sumxc J \vel{2} \dmomdif{1}{2} + \sumzc \sumyf \sumxc J \vel{2} \dmomdif{2}{2} + \sumzc \sumyf \sumxc J \vel{2} \dmomdif{3}{2},\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\sumzf \sumyc \sumxc J \vel{3} \pder{\vel{3}}{t} = & - \sumzf \sumyc \sumxc J \vel{3} \dmomadv{1}{3} - \sumzf \sumyc \sumxc J \vel{3} \dmomadv{2}{3} - \sumzf \sumyc \sumxc J \vel{3} \dmomadv{3}{3}\\& - \sumzf \sumyc \sumxc J \vel{3} \dmompre{3}\\& + \sumzf \sumyc \sumxc J \vel{3} \dmomdif{1}{3} + \sumzf \sumyc \sumxc J \vel{3} \dmomdif{2}{3} + \sumzf \sumyc \sumxc J \vel{3} \dmomdif{3}{3}.\end{aligned}\end{align} \]

Similarly, multiplying the discrete internal energy balance by temperature and volume-integrating it yield

\[ \begin{align}\begin{aligned}\sumzc \sumyc \sumxc J T \pder{T}{t} = & - \sumzc \sumyc \sumxc J T \dtempadv{1} - \sumzc \sumyc \sumxc J T \dtempadv{2} - \sumzc \sumyc \sumxc J T \dtempadv{3}\\& + \sumzc \sumyc \sumxc J T \dtempdif{1} + \sumzc \sumyc \sumxc J T \dtempdif{2} + \sumzc \sumyc \sumxc J T \dtempdif{3}.\end{aligned}\end{align} \]

It should be noted that, since the velocities are defined at different positions due to the staggered grid arrangement, quadratic quantities are also evaluated at different locations, which is the reason why we consider three components separately.

With some algebra, we obtain

\[ \begin{align}\begin{aligned}& \sumzc \sumyc \sumxf J \pder{k_1}{t} + \sumzc \sumyf \sumxc J \pder{k_2}{t} + \sumzf \sumyc \sumxc J \pder{k_3}{t}\\= & - \sumzc \sumyc \sumxc \dkdis{1}{1} - \sumzc \sumyf \sumxf \dkdis{2}{1} - \sumzf \sumyc \sumxf \dkdis{3}{1}\\& - \sumzc \sumyf \sumxf \dkdis{1}{2} - \sumzc \sumyc \sumxc \dkdis{2}{2} - \sumzf \sumyf \sumxc \dkdis{3}{2}\\& - \sumzf \sumyc \sumxf \dkdis{1}{3} - \sumzf \sumyf \sumxc \dkdis{2}{3} - \sumzc \sumyc \sumxc \dkdis{3}{3}\\& + \sumzc \sumyc \sumxf J \vel{1} \ave{T}{\gcs{1}},\end{aligned}\end{align} \]

and

\[ \begin{align}\begin{aligned}& \sumzc \sumyc \sumxc J \pder{h}{t}\\= & - \sumzc \sumyc \sumxf \dhdis{1} - \sumzc \sumyf \sumxc \dhdis{2} - \sumzf \sumyc \sumxc \dhdis{3}\\& - \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \left( \frac{J}{\sfact{1}} T \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right) }{\frac{1}{2}}\\& + \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \vat{ \left( \frac{J}{\sfact{1}} T \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right) }{\ngp{1} + \frac{1}{2}},\end{aligned}\end{align} \]

with respect to the squared velocity and the squared temperature, respectively.

To derive them, the individual components are focused below.

The derived two relations have source terms and sink terms, which are elaborated below as well as their implementations.