Diffusion

Momentum

By utilising the relations listed in the prerequisite, the discrete diffusive terms in the momentum balances yield the following relations.

From the wall-normal momentum balance, we obtain

\[ \begin{align}\begin{aligned}& - \sumzc \sumyc \sumxc J \vgt{1}{1} \sst{1}{1},\\& - \sumzc \sumyf \sumxf J \left( \frac{1}{\sfact{2}} \dif{\vel{1}}{\gcs{2}} \right) \sst{2}{1},\\& - \sumzf \sumyc \sumxf J \vgt{3}{1} \sst{3}{1},\\& - \sumzc \sumyc \sumxc J \left\{ \frac{1}{J} \dif{ \left( \frac{J}{\sfact{1}} \right) }{\gcs{1}} \ave{ \vel{1} }{\gcs{1}} \right\} \sst{2}{2}.\end{aligned}\end{align} \]

From the stream-wise momentum balance, we obtain

\[ \begin{align}\begin{aligned}& - \sumzc \sumyf \vat{ \left( \frac{J}{\sfact{1}} \vel{2} \sst{1}{2} \right) }{i = \frac{1}{2}} - \sumzc \sumyf \sumxf J \left( \frac{1}{\sfact{1}} \dif{\vel{2}}{\gcs{1}} \right) \sst{1}{2} + \sumzc \sumyf \vat{ \left( \frac{J}{\sfact{1}} \vel{2} \sst{1}{2} \right) }{i = \ngp{1} + \frac{1}{2}},\\& - \sumzc \sumyc \sumxc J \left( \frac{1}{\sfact{2}} \dif{\vel{2}}{\gcs{2}} \right) \sst{2}{2},\\& - \sumzf \sumyf \sumxc J \vgt{3}{2} \sst{3}{2},\\& \sumzc \sumyf \vat{J}{\frac{1}{2}} \left\{ \frac{1}{\vat{J}{\frac{1}{2}}} \vat{ \dif{ \left( \frac{J}{\sfact{1}} \right) }{ \gcs{1} } }{\frac{1}{2}} \frac{\vat{\vel{2}}{1}}{2} \right\} \vat{ \sst{2}{1} }{\frac{1}{2}} + \sumzc \sumyf \sum_{i = \frac{3}{2}}^{\ngp{1} - \frac{1}{2}} J \left\{ \frac{1}{J} \dif{ \left( \frac{J}{\sfact{1}} \right) }{ \gcs{1} } \ave{\vel{2}}{\gcs{1}} \right\} \sst{2}{1} + \sumzc \sumyf \vat{J}{\ngp{1} + \frac{1}{2}} \left\{ \frac{1}{\vat{J}{\ngp{1} + \frac{1}{2}}} \vat{ \dif{ \left( \frac{J}{\sfact{1}} \right) }{ \gcs{1} } }{\ngp{1} + \frac{1}{2}} \frac{\vat{\vel{2}}{\ngp{1}}}{2} \right\} \vat{ \sst{2}{1} }{\ngp{1} + \frac{1}{2}}.\end{aligned}\end{align} \]

From the span-wise momentum balance, we obtain

\[ \begin{align}\begin{aligned}& - \sumzf \sumyc \sumxf J \vgt{1}{3} \sst{1}{3},\\& - \sumzf \sumyf \sumxc J \vgt{2}{3} \sst{2}{3},\\& - \sumzc \sumyc \sumxc J \vgt{3}{3} \sst{3}{3}.\end{aligned}\end{align} \]

In total, we have

\[ \begin{align}\begin{aligned}& - \sumzc \sumyc \sumxc J \vgt{1}{1} \sst{1}{1} - \sumzc \sumyf \sumxf J \vgt{2}{1} \sst{2}{1} - \sumzf \sumyc \sumxf J \vgt{3}{1} \sst{3}{1},\\& - \sumzc \sumyf \sumxf J \vgt{1}{2} \sst{1}{2} - \sumzc \sumyc \sumxc J \vgt{2}{2} \sst{2}{2} - \sumzf \sumyf \sumxc J \vgt{3}{2} \sst{3}{2},\\& - \sumzf \sumyc \sumxf J \vgt{1}{3} \sst{1}{3} - \sumzf \sumyf \sumxc J \vgt{2}{3} \sst{2}{3} - \sumzc \sumyc \sumxc J \vgt{3}{3} \sst{3}{3}\end{aligned}\end{align} \]

as the dissipative terms, while

\[- \sumzc \sumyf \vat{ \left( \frac{J}{\sfact{1}} \vel{2} \sst{1}{2} \right) }{i = \frac{1}{2}} + \sumzc \sumyf \vat{ \left( \frac{J}{\sfact{1}} \vel{2} \sst{1}{2} \right) }{i = \ngp{1} + \frac{1}{2}}\]

as the energy transfer on the walls, respectively.

Scalar

The global diffusive effects of the scalar transport on the quadratic quantity lead to

\[\sumzc \sumyc \sumxc J T \left\{ \dscalardif{1} \dscalardif{2} \dscalardif{3} \right\},\]

giving

\[- \sumzc \sumyc \sumxf J \kappa \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} - \sumzc \sumyf \sumxc J \kappa \frac{1}{\sfact{2}} \dif{T}{\gcs{2}} \frac{1}{\sfact{2}} \dif{T}{\gcs{2}} - \sumzf \sumyc \sumxc J \kappa \frac{1}{\sfact{3}} \dif{T}{\gcs{3}} \frac{1}{\sfact{3}} \dif{T}{\gcs{3}}\]

as the dissipative terms, while

\[- \sumzc \sumyc \vat{ \left( \frac{J}{\sfact{1}} \kappa T \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right) }{\frac{1}{2}} + \sumzc \sumyc \vat{ \left( \frac{J}{\sfact{1}} \kappa T \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right) }{\ngp{1} + \frac{1}{2}}\]

as the transport on the walls.