Symbol

Averages, differentiations, and summations are denotes as follows.

• Wall-normal operator
• Stream-wise operator
• Span-wise operator

Discrete scale factor and Jacobian determinant

In Cartesian coordinate systems, scale factors are simply equal to the grid sizes:

\[ \begin{align}\begin{aligned}\hx & \equiv \Delta x,\\\hy & \equiv \Delta y,\\\hz & \equiv \Delta z.\end{aligned}\end{align} \]

src/domain.c

static double * allocate_and_init_hxxf(
    const int isize,
    const double * xc
){
  double * hxxf = memory_calloc(isize + 1, sizeof(double));
  for(int i = 1; i <= isize + 1; i++){
    HXXF(i  ) = XC(i  ) - XC(i-1);
  }
  return hxxf;
}

src/domain.c

static double * allocate_and_init_hxxc(
    const int isize,
    const double * xf
){
  double * hxxc = memory_calloc(isize, sizeof(double));
  for(int i = 1; i <= isize; i++){
    HXXC(i  ) = XF(i+1) - XF(i  );
  }
  return hxxc;
}

Note that the wall-normal grid sizes are halved on the walls:

\[\vat{\Delta x}{\frac{1}{2}} = \vat{\Delta x}{\nx + \frac{1}{2}} = \frac{1}{2} \Delta x_{\text{rest}}.\]

The Jacobian determinants, which are also defined at cell centers and faces, are simply the product of the local scale factors:

\[J \equiv \hx \hy \hz.\]

src/domain.c

static double * allocate_and_init_jdxf(
    const int isize,
    const double * hxxf,
    const double hy
){
  double * jdxf = memory_calloc(isize + 1, sizeof(double));
  for(int i = 1; i <= isize + 1; i++){
    JDXF(i  ) = HXXF(i  ) * hy;
  }
  return jdxf;
}

src/domain.c

static double * allocate_and_init_jdxf(
    const int isize,
    const double * hxxf,
    const double hy,
    const double hz
){
  double * jdxf = memory_calloc(isize + 1, sizeof(double));
  for(int i = 1; i <= isize + 1; i++){
    JDXF(i  ) = HXXF(i  ) * hy * hz;
  }
  return jdxf;
}

src/domain.c

static double * allocate_and_init_jdxc(
    const int isize,
    const double * hxxc,
    const double hy
){
  double * jdxc = memory_calloc(isize, sizeof(double));
  for(int i = 1; i <= isize; i++){
    JDXC(i  ) = HXXC(i  ) * hy;
  }
  return jdxc;
}

src/domain.c

static double * allocate_and_init_jdxc(
    const int isize,
    const double * hxxc,
    const double hy,
    const double hz
){
  double * jdxc = memory_calloc(isize, sizeof(double));
  for(int i = 1; i <= isize; i++){
    JDXC(i  ) = HXXC(i  ) * hy * hz;
  }
  return jdxc;
}