Stream-wise direction

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

Advection

\[\dmomadv{\vy}{\vx}\]

src/fluid/predict/uy.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-1)
                      + 0.5 * jd_xm / hx_xm * FLUXX(i  , j  );
const double mux_xp = + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j-1)
                      + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  );
const double uy_xm = + 0.5 * UY(i-1, j  )
                     + 0.5 * UY(i  , j  );
const double uy_xp = + 0.5 * UY(i  , j  )
                     + 0.5 * UY(i+1, j  );
src[cnt] -= 1. / jd_x0 * (
    - mux_xm * uy_xm
    + mux_xp * uy_xp
);

src/fluid/predict/uy.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-1, k  )
                      + 0.5 * jd_xm / hx_xm * FLUXX(i  , j  , k  );
const double mux_xp = + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j-1, k  )
                      + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  , k  );
const double uy_xm = + 0.5 * UY(i-1, j  , k  )
                     + 0.5 * UY(i  , j  , k  );
const double uy_xp = + 0.5 * UY(i  , j  , k  )
                     + 0.5 * UY(i+1, j  , k  );
src[cnt] -= 1. / jd_x0 * (
    - mux_xm * uy_xm
    + mux_xp * uy_xp
);

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

src/fluid/predict/uy.c

const double jd = JDXC(i  );
const double muy_ym = + 0.5 * jd / hy * FLUXY(i  , j-1)
                      + 0.5 * jd / hy * FLUXY(i  , j  );
const double muy_yp = + 0.5 * jd / hy * FLUXY(i  , j  )
                      + 0.5 * jd / hy * FLUXY(i  , j+1);
const double uy_ym = + 0.5 * UY(i  , j-1)
                     + 0.5 * UY(i  , j  );
const double uy_yp = + 0.5 * UY(i  , j  )
                     + 0.5 * UY(i  , j+1);
src[cnt] -= 1. / jd * (
    - muy_ym * uy_ym
    + muy_yp * uy_yp
);

src/fluid/predict/uy.c

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

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

src/fluid/predict/uy.c

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

Pressure-gradient

\[\dmompre{\vy}\]

src/fluid/predict/uy.c

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

src/fluid/predict/uy.c

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

Diffusion

\[\dmomdif{\vy}{\vx}\]

src/fluid/predict/uy.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 txy_xm = TXY(i  , j  );
const double txy_xp = TXY(i+1, j  );
src[cnt] += diffusivity / jd_x0 * (
    - jd_xm / hx_xm * txy_xm
    + jd_xp / hx_xp * txy_xp
);

src/fluid/predict/uy.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 txy_xm = TXY(i  , j  , k  );
const double txy_xp = TXY(i+1, j  , k  );
src[cnt] += diffusivity / jd_x0 * (
    - jd_xm / hx_xm * txy_xm
    + jd_xp / hx_xp * txy_xp
);

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

src/fluid/predict/uy.c

const double jd = JDXC(i  );
const double tyy_ym = TYY(i  , j-1);
const double tyy_yp = TYY(i  , j  );
src[cnt] += diffusivity / jd * (
    - jd / hy * tyy_ym
    + jd / hy * tyy_yp
);

src/fluid/predict/uy.c

const double jd = JDXC(i  );
const double tyy_ym = TYY(i  , j-1, k  );
const double tyy_yp = TYY(i  , j  , k  );
src[cnt] += diffusivity / jd * (
    - jd / hy * tyy_ym
    + jd / hy * tyy_yp
);

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

src/fluid/predict/uy.c

const double jd = JDXC(i  );
const double tyz_zm = TYZ(i  , j  , k  );
const double tyz_zp = TYZ(i  , j  , k+1);
src[cnt] += diffusivity / jd * (
    - jd / hz * tyz_zm
    + jd / hz * tyz_zp
);