Convection term

This term corresponds to

(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:

(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:

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

(600)

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

(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:

\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 () a deferred correction approach is taken based on an upwind scheme, i.e.

\overrightarrow{(v_i)}_f^{(m)} \approx \overrightarrow{(v_i)}_f^{UD(m)} +\left[ \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

(A) \approx \sum_{f}^{} \dot{m}_f^{(m-1)} \left[\overrightarrow{(v_i)}_f^{UD(m)} + \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:

Inlet: (\overrightarrow{v_i})_f^{(m)} is given; contributes to the right hand side.
Wall and Sliding conditions: \dot{m}_f^{(m-1)}=0: no contribution
Outlet: (\overrightarrow{v_i})_f^{(m)}=(\overrightarrow{v_i})_f^{UD(m)}=(v_i)_P^{(m)} \; \forall m : constant velocity. Contributes to the left hand side.

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.