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]:

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

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

$$\widetilde{\phi_P} \le 0: \hspace{1 cm} \gamma=0 \;$$ (578)

$$0 < \widetilde{\phi_P} < \beta_m: \hspace{1 cm} \gamma=\widetilde{\phi_P}/\beta_m$$ (579)

$$\beta_m \le \widetilde{\phi_P} < 1: \hspace{1 cm} \gamma=1 \;$$ (580)

$$1 \le \widetilde{\phi_P}: \hspace{1 cm} \gamma=0 \;$$ (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

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

one obtains

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