Equations

The incompressible Navier-Stokes equations in three-dimensional cylindrical coordinates are derived here.

Due to the azimuthal homogeneity:

\[ \begin{align}\begin{aligned}& \vel{2} \equiv 0,\\& \pder{q}{\gcs{2}} \equiv 0,\end{aligned}\end{align} \]

the equations can be further simplified, which are given below.

Incompressibility constraint

\[\frac{1}{J} \pder{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \vel{1} \right) + \frac{1}{J} \pder{}{\gcs{3}} \left( \frac{J}{\sfact{3}} \vel{3} \right) = 0.\]

This is enforced by the SMAC method.

Mass balance

\[\pder{\density}{t} + \frac{1}{J} \pder{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \density \vel{1} \right) + \frac{1}{J} \pder{}{\gcs{3}} \left( \frac{J}{\sfact{3}} \density \vel{3} \right) = 0.\]

This is solved by means of the volume-of-fluid method.

Momentum balance

\[ \begin{align}\begin{aligned}\momtemp{1} = & \momadv{1}{1} \momadv{3}{1}\\& \mompre{1}\\& \momdif{1}{1} \momdif{3}{1}\\& \momdifx,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\momtemp{3} = & \momadv{1}{3} \momadv{3}{3}\\& \mompre{3}\\& \momdif{1}{3} \momdif{3}{3}.\end{aligned}\end{align} \]

This is solved by the energy-conserving scheme.

Shear-stress tensor

The shear-stress tensor for Newtonian liquids is defined as

\[\sst{i}{j} \equiv \viscosity \vgt{i}{j} + \viscosity \vgt{j}{i},\]

where \(\vgt{i}{j}\) is a second-order tensor representing the gradient of velocity vector:

\[\sum_i \sum_j \vec{e}_i \otimes \vec{e}_j \vgt{i}{j}.\]

The components are

\[\begin{split}\begin{pmatrix} \vgt{1}{1} & \vgt{2}{1} & \vgt{3}{1} \\ \vgt{1}{2} & \vgt{2}{2} & \vgt{3}{2} \\ \vgt{1}{3} & \vgt{2}{3} & \vgt{3}{3} \\ \end{pmatrix} = \begin{pmatrix} \frac{1}{\sfact{1}} \pder{\vel{1}}{\gcs{1}} & 0 & \frac{1}{\sfact{3}} \pder{\vel{1}}{\gcs{3}} \\ 0 & \frac{1}{J} \pder{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \right) \vel{1} & 0 \\ \frac{1}{\sfact{1}} \pder{\vel{3}}{\gcs{1}} & 0 & \frac{1}{\sfact{3}} \pder{\vel{3}}{\gcs{3}} \end{pmatrix}.\end{split}\]

The dynamic viscosity is computed using the cell-faced volume-of-fluid value.