Convection term

This term corresponds to

$\displaystyle (A) := \int_{A}^{} \rho \boldsymbol{v} \cdot \boldsymbol{n} v_i da.$ (598)

Now, the integral is split into a sum over all element faces:

$\displaystyle (A)= \sum_{f}^{} \int_{A_f}^{} \rho \boldsymbol{v} \cdot \boldsymbol{n} v_i da,$ (599)

and the integral across a face is evaluated using the convective face value at the center of the face:

$\displaystyle (A) \approx \sum_{f}^{} \dot{m}_f \overrightarrow{(v_i)_f}.$ (600)

The flux $ \dot{m}_f$ is taken from the previous iteration:

$\displaystyle (A) \approx \sum_{f}^{} \dot{m}_f^{(m-1)} \overrightarrow{(v_i)}_f^{(m)}.$ (601)

For the first iteration ($ m=1$) $ \dot{m}_f^{(0)}$ is calculated from the initial conditions:

$\displaystyle \dot{m}_f^{(0)} = \overrightarrow{\rho }_f^{(0)} \boldsymbol{v}_f^{(0)} \cdot \boldsymbol{n}_f A_f,$ (602)

where $ \overrightarrow{\rho }_f^{(0)}$ is the convective interpolation through Upwind Difference of the density at the face centers.

For the convective face value in Equation (601) a deferred correction approach is taken based on an upwind scheme, i.e.

$\displaystyle \overrightarrow{(v_i)}_f^{(m)} \approx \overrightarrow{(v_i)}_f^{...
...t[ \overrightarrow{(v_i)}_f^{(m-1)}-\overrightarrow{(v_i)}_f^{UD(m-1)} \right].$ (603)

This approximation is exact at convergence, for which the values in iteration $ (m-1)$ and $ (m)$ coincide. By the above approximation all values at iteration $ (m)$ are element center values. Indeed, recall that $ \overrightarrow{(v_i)}_f^{UD(m)}$ is the element center value of $ v_i$, either of the element at stake (P), or its neighbor, depending on the flow direction (i.e. the sign of $ \dot{m}_f^{(m-1)}$). Consequently

$\displaystyle (A) \approx \sum_{f}^{} \dot{m}_f^{(m-1)} \left[\overrightarrow{(...
... + \overrightarrow{(v_i)}_f^{(m-1)}-\overrightarrow{(v_i)}_f^{UD(m-1)} \right].$ (604)

The terms within the square bracket with superscript $ (m-1)$ end up on the right hand side of the quation, the terms with superscript $ (m)$ contribute to the left hand side.

Since velocity boundary conditions are automatically taken into account in the calculation of $ \overrightarrow{\boldsymbol{v} }$ no special treatment is necessary. We have:

For the convective interpolation of the velocity the upwind difference scheme as well as the modified smart scheme (or other high resolution schemes) can be selected.