Wall-normal direction

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

Advection

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

src/fluid/predict/ux.c

const double hx_xm = HXXF(i-1);
const double hx_x0 = HXXF(i  );
const double hx_xp = HXXF(i+1);
const double jd_xm = JDXF(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXF(i+1);
const double mux_xm = + 0.5 * jd_xm / hx_xm * FLUXX(i-1, j  )
                      + 0.5 * jd_x0 / hx_x0 * FLUXX(i  , j  );
const double mux_xp = + 0.5 * jd_x0 / hx_x0 * FLUXX(i  , j  )
                      + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  );
const double ux_xm = + 0.5 * UX(i-1, j  )
                     + 0.5 * UX(i  , j  );
const double ux_xp = + 0.5 * UX(i  , j  )
                     + 0.5 * UX(i+1, j  );
src[cnt] -= 1. / jd_x0 * (
    - mux_xm * ux_xm
    + mux_xp * ux_xp
);

src/fluid/predict/ux.c

const double hx_xm = HXXF(i-1);
const double hx_x0 = HXXF(i  );
const double hx_xp = HXXF(i+1);
const double jd_xm = JDXF(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXF(i+1);
const double mux_xm = + 0.5 * jd_xm / hx_xm * FLUXX(i-1, j  , k  )
                      + 0.5 * jd_x0 / hx_x0 * FLUXX(i  , j  , k  );
const double mux_xp = + 0.5 * jd_x0 / hx_x0 * FLUXX(i  , j  , k  )
                      + 0.5 * jd_xp / hx_xp * FLUXX(i+1, j  , k  );
const double ux_xm = + 0.5 * UX(i-1, j  , k  )
                     + 0.5 * UX(i  , j  , k  );
const double ux_xp = + 0.5 * UX(i  , j  , k  )
                     + 0.5 * UX(i+1, j  , k  );
src[cnt] -= 1. / jd_x0 * (
    - mux_xm * ux_xm
    + mux_xp * ux_xp
);

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

src/fluid/predict/ux.c

const double jd_xm = JDXC(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXC(i  );
const double muy_ym = + 0.5 * jd_xm / hy * FLUXY(i-1, j  )
                      + 0.5 * jd_xp / hy * FLUXY(i  , j  );
const double muy_yp = + 0.5 * jd_xm / hy * FLUXY(i-1, j+1)
                      + 0.5 * jd_xp / hy * FLUXY(i  , j+1);
const double ux_ym = + 0.5 * UX(i  , j-1)
                     + 0.5 * UX(i  , j  );
const double ux_yp = + 0.5 * UX(i  , j  )
                     + 0.5 * UX(i  , j+1);
src[cnt] -= 1. / jd_x0 * (
    - muy_ym * ux_ym
    + muy_yp * ux_yp
);

src/fluid/predict/ux.c

const double jd_xm = JDXC(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXC(i  );
const double muy_ym = + 0.5 * jd_xm / hy * FLUXY(i-1, j  , k  )
                      + 0.5 * jd_xp / hy * FLUXY(i  , j  , k  );
const double muy_yp = + 0.5 * jd_xm / hy * FLUXY(i-1, j+1, k  )
                      + 0.5 * jd_xp / hy * FLUXY(i  , j+1, k  );
const double ux_ym = + 0.5 * UX(i  , j-1, k  )
                     + 0.5 * UX(i  , j  , k  );
const double ux_yp = + 0.5 * UX(i  , j  , k  )
                     + 0.5 * UX(i  , j+1, k  );
src[cnt] -= 1. / jd_x0 * (
    - muy_ym * ux_ym
    + muy_yp * ux_yp
);

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

src/fluid/predict/ux.c

const double jd_xm = JDXC(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXC(i  );
const double muz_zm = + 0.5 * jd_xm / hz * FLUXZ(i-1, j  , k  )
                      + 0.5 * jd_xp / hz * FLUXZ(i  , j  , k  );
const double muz_zp = + 0.5 * jd_xm / hz * FLUXZ(i-1, j  , k+1)
                      + 0.5 * jd_xp / hz * FLUXZ(i  , j  , k+1);
const double ux_zm = + 0.5 * UX(i  , j  , k-1)
                     + 0.5 * UX(i  , j  , k  );
const double ux_zp = + 0.5 * UX(i  , j  , k  )
                     + 0.5 * UX(i  , j  , k+1);
src[cnt] -= 1. / jd_x0 * (
    - muz_zm * ux_zm
    + muz_zp * ux_zp
);

Pressure-gradient

\[\dmompre{\vx}\]

src/fluid/predict/ux.c

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

src/fluid/predict/ux.c

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

Diffusion

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

src/fluid/predict/ux.c

const double hx_xm = HXXC(i-1);
const double hx_xp = HXXC(i  );
const double jd_xm = JDXC(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXC(i  );
const double txx_xm = TXX(i-1, j  );
const double txx_xp = TXX(i  , j  );
src[cnt] += diffusivity / jd_x0 * (
    - jd_xm / hx_xm * txx_xm
    + jd_xp / hx_xp * txx_xp
);

src/fluid/predict/ux.c

const double hx_xm = HXXC(i-1);
const double hx_xp = HXXC(i  );
const double jd_xm = JDXC(i-1);
const double jd_x0 = JDXF(i  );
const double jd_xp = JDXC(i  );
const double txx_xm = TXX(i-1, j  , k  );
const double txx_xp = TXX(i  , j  , k  );
src[cnt] += diffusivity / jd_x0 * (
    - jd_xm / hx_xm * txx_xm
    + jd_xp / hx_xp * txx_xp
);

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

src/fluid/predict/ux.c

const double jd = JDXF(i  );
const double txy_ym = TXY(i  , j  );
const double txy_yp = TXY(i  , j+1);
src[cnt] += diffusivity / jd * (
    - jd / hy * txy_ym
    + jd / hy * txy_yp
);

src/fluid/predict/ux.c

const double jd = JDXF(i  );
const double txy_ym = TXY(i  , j  , k  );
const double txy_yp = TXY(i  , j+1, k  );
src[cnt] += diffusivity / jd * (
    - jd / hy * txy_ym
    + jd / hy * txy_yp
);

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

src/fluid/predict/ux.c

const double jd = JDXF(i  );
const double txz_zm = TXZ(i  , j  , k  );
const double txz_zp = TXZ(i  , j  , k+1);
src[cnt] += diffusivity / jd * (
    - jd / hz * txz_zm
    + jd / hz * txz_zp
);