Span-wise direction

\[ \begin{align}\begin{aligned}\pder{\rho \uz}{t} = & \dmomadv{\vz}{\vx} \dmomadv{\vz}{\vy} \dmomadv{\vz}{\vz}\\& \dmompre{\vz}\\& \dmomdif{\vz}{\vx} \dmomdif{\vz}{\vy} \dmomdif{\vz}{\vz}\end{aligned}\end{align} \]

Advection

\[\dmomadv{\vz}{\vx}\]

src/fluid/predict/uz.c

const double hx_xm = HXXF(i  );
const double hx_xp = HXXF(i+1);
const double jd_xm = JDXF(i  );
const double jd_x0 = JDXC(i  );
const double jd_xp = JDXF(i+1);
const double mux_xm = + 0.5 * jd_xm / hx_xm * FLUXX(i  , j  , k-1)
                      + 0.5 * jd_xm / hx_xm * FLUXX(i  , j  , k  );
const double mux_xp = + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  , k-1)
                      + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  , k  );
const double uz_xm = + 0.5 * UZ(i-1, j  , k  )
                     + 0.5 * UZ(i  , j  , k  );
const double uz_xp = + 0.5 * UZ(i  , j  , k  )
                     + 0.5 * UZ(i+1, j  , k  );
src[cnt] -= 1. / jd_x0 * (
    - mux_xm * uz_xm
    + mux_xp * uz_xp
);

\[\dmomadv{\vz}{\vy}\]

src/fluid/predict/uz.c

const double jd = JDXC(i  );
const double muy_ym = + 0.5 * jd / hy * FLUXY(i  , j  , k-1)
                      + 0.5 * jd / hy * FLUXY(i  , j  , k  );
const double muy_yp = + 0.5 * jd / hy * FLUXY(i  , j+1, k-1)
                      + 0.5 * jd / hy * FLUXY(i  , j+1, k  );
const double uz_ym = + 0.5 * UZ(i  , j-1, k  )
                     + 0.5 * UZ(i  , j  , k  );
const double uz_yp = + 0.5 * UZ(i  , j  , k  )
                     + 0.5 * UZ(i  , j+1, k  );
src[cnt] -= 1. / jd * (
    - muy_ym * uz_ym
    + muy_yp * uz_yp
);

\[\dmomadv{\vz}{\vz}\]

src/fluid/predict/uz.c

const double jd = JDXC(i  );
const double muz_zm = + 0.5 * jd / hz * FLUXZ(i  , j  , k-1)
                      + 0.5 * jd / hz * FLUXZ(i  , j  , k  );
const double muz_zp = + 0.5 * jd / hz * FLUXZ(i  , j  , k  )
                      + 0.5 * jd / hz * FLUXZ(i  , j  , k+1);
const double uz_zm = + 0.5 * UZ(i  , j  , k-1)
                     + 0.5 * UZ(i  , j  , k  );
const double uz_zp = + 0.5 * UZ(i  , j  , k  )
                     + 0.5 * UZ(i  , j  , k+1);
src[cnt] -= 1. / jd * (
    - muz_zm * uz_zm
    + muz_zp * uz_zp
);

Pressure-gradient

\[\dmompre{\vz}\]

src/fluid/predict/uz.c

src[cnt] -= 1. / hz * (
    - P(i  , j  , k-1)
    + P(i  , j  , k  )
);

Diffusion

\[\dmomdif{\vz}{\vx}\]

src/fluid/predict/uz.c

const double hx_xm = HXXF(i  );
const double hx_xp = HXXF(i+1);
const double jd_xm = JDXF(i  );
const double jd_x0 = JDXC(i  );
const double jd_xp = JDXF(i+1);
const double txz_xm = TXZ(i  , j  , k  );
const double txz_xp = TXZ(i+1, j  , k  );
src[cnt] += diffusivity / jd_x0 * (
    - jd_xm / hx_xm * txz_xm
    + jd_xp / hx_xp * txz_xp
);

\[\dmomdif{\vz}{\vy}\]

src/fluid/predict/uz.c

const double jd = JDXC(i  );
const double tyz_ym = TYZ(i  , j  , k  );
const double tyz_yp = TYZ(i  , j+1, k  );
src[cnt] += diffusivity / jd * (
    - jd / hy * tyz_ym
    + jd / hy * tyz_yp
);

\[\dmomdif{\vz}{\vz}\]

src/fluid/predict/uz.c

const double jd = JDXC(i  );
const double tzz_zm = TZZ(i  , j  , k-1);
const double tzz_zp = TZZ(i  , j  , k  );
src[cnt] += diffusivity / jd * (
    - jd / hz * tzz_zm
    + jd / hz * tzz_zp
);