Gamma Method

For incompressible flows a scheme was proposed for the convective facial values consisting of a linear combination of the upwind scheme and the facial values $ \overline{\phi }_f$ [36]:

$\displaystyle \overrightarrow{\phi }_f = \gamma \overline{\phi }_f + (1- \gamma) \phi_P.$ (577)

Here, $ \gamma$ is a piecewise linear function of $ \widetilde{\phi }_P$:

$\displaystyle \widetilde{\phi_P} \le 0: \hspace{1 cm}$ $\displaystyle \gamma=0 \;$   (Upwind Difference) (578)
$\displaystyle 0 < \widetilde{\phi_P} < \beta_m: \hspace{1 cm}$ $\displaystyle \gamma=\widetilde{\phi_P}/\beta_m$ (579)
$\displaystyle \beta_m \le \widetilde{\phi_P} < 1: \hspace{1 cm}$ $\displaystyle \gamma=1 \;$   (Central Difference) (580)
$\displaystyle 1 \le \widetilde{\phi_P}: \hspace{1 cm}$ $\displaystyle \gamma=0 \;$   (Upwind Difference) (581)

Due to the second term in the above equation $ \overrightarrow{\phi }_f$ is a nonlinear function of $ \phi_P$. $ \beta_m$ should be in the range $ 0.1 \le \beta_m \le
0.5$. In CalculiX $ \beta_m=0.1$.

For the velocity $ \boldsymbol{v}$ the scalar $ \vert\vert\boldsymbol{v} \vert\vert$ is used to calculate $ \gamma$. Since

$\displaystyle \vert\vert\boldsymbol{v}\vert\vert,_j = \frac{v_i}{\vert\vert\boldsymbol{v}\vert\vert } v_{i,j}$ (582)

one obtains

$\displaystyle \widetilde{\vert\vert\boldsymbol{v}\vert\vert}_P = 1 - \frac{(\ve...
...symbol{v}_P \cdot \nabla \boldsymbol{v}_P^T \cdot \boldsymbol{j}_\xi d(P,E) } .$ (583)