Diffusion

Wall-normal derivative

\[\dmomdif{1}{3} = \mu \frac{1}{J} \dif{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \vgt{1}{3} \right) + \mu \frac{1}{J} \dif{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \vgt{3}{1} \right)\]

Diagonal part

\[\mu \frac{1}{J} \dif{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \vgt{1}{3} \right) = \mu \frac{1}{J} \dif{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \dlxz \right)\]

src/fluid/predict/uz.c

for(size_t i = 1; i <= isize; i++){
  const double l = 1. / JDXC(i  ) * JDXF(i  ) / HXXF(i  ) / HXXF(i  );
  const double u = 1. / JDXC(i  ) * JDXF(i+1) / HXXF(i+1) / HXXF(i+1);
  const double c = - l - u;
  laplacians.LAPX(i).l = l;
  laplacians.LAPX(i).c = c;
  laplacians.LAPX(i).u = u;
}

src/fluid/predict/uz.c

BEGIN
  src[cnt] += diffusivity * (
      + LAPX(i).l * UZ(i-1, j  , k  )
      + LAPX(i).c * UZ(i  , j  , k  )
      + LAPX(i).u * UZ(i+1, j  , k  )
  );
END

Non-diagonal part

\[\mu \frac{1}{J} \dif{}{\gcs{1}} \left( \frac{J}{\sfact{1}} \vgt{3}{1} \right)\]

src/fluid/predict/uz.c

BEGIN
  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 lzx_xm = LZX(i  , j  , k  );
  const double lzx_xp = LZX(i+1, j  , k  );
  src[cnt] += diffusivity / jd_x0 * (
      - jd_xm / hx_xm * lzx_xm
      + jd_xp / hx_xp * lzx_xp
  );
END

Stream-wise derivative

\[\dmomdif{2}{3} = \mu \frac{1}{J} \dif{}{\gcs{2}} \left( \frac{J}{\sfact{2}} \vgt{2}{3} \right) + \mu \frac{1}{J} \dif{}{\gcs{2}} \left( \frac{J}{\sfact{2}} \vgt{3}{2} \right)\]

Diagonal part

\[\mu \frac{1}{J} \dif{}{\gcs{2}} \left( \frac{J}{\sfact{2}} \vgt{2}{3} \right) = \mu \frac{1}{J} \dif{}{\gcs{2}} \left( \frac{J}{\sfact{2}} \dlyz \right)\]

src/fluid/predict/uz.c

for(size_t i = 1; i <= isize; i++){
  const double l = 1. / HYXC(i  ) / HYXC(i  );
  const double u = 1. / HYXC(i  ) / HYXC(i  );
  const double c = - l - u;
  laplacians.LAPY(i).l = l;
  laplacians.LAPY(i).c = c;
  laplacians.LAPY(i).u = u;
}

src/fluid/predict/uz.c

BEGIN
  src[cnt] += diffusivity * (
      + LAPY(i).l * UZ(i  , j-1, k  )
      + LAPY(i).c * UZ(i  , j  , k  )
      + LAPY(i).u * UZ(i  , j+1, k  )
  );
END

Non-diagonal part

\[\mu \frac{1}{J} \dif{}{\gcs{2}} \left( \frac{J}{\sfact{2}} \vgt{3}{2} \right)\]

src/fluid/predict/uz.c

BEGIN
  const double hy = HYXC(i  );
  const double jd = JDXC(i  );
  const double lzy_ym = LZY(i  , j  , k  );
  const double lzy_yp = LZY(i  , j+1, k  );
  src[cnt] += diffusivity / jd * (
      - jd / hy * lzy_ym
      + jd / hy * lzy_yp
  );
END

Span-wise derivative

\[\dmomdif{3}{3} = 2 \mu \frac{1}{J} \dif{}{\gcs{3}} \left( \frac{J}{\sfact{3}} \vgt{3}{3} \right) = 2 \mu \frac{1}{J} \dif{}{\gcs{3}} \left( \frac{J}{\sfact{3}} \dlzz \right)\]

src/fluid/predict/uz.c

const double l = 1. / hz / hz;
const double u = 1. / hz / hz;
const double c = - l - u;
laplacians.lapz.l = l;
laplacians.lapz.c = c;
laplacians.lapz.u = u;

src/fluid/predict/uz.c

BEGIN
  // NOTE: tzz = "2" lzz
  src[cnt] += 2. * diffusivity * (
      + (*lapz).l * UZ(i  , j  , k-1)
      + (*lapz).c * UZ(i  , j  , k  )
      + (*lapz).u * UZ(i  , j  , k+1)
  );
END