Variables

Main variables of the solver:

Geometry:

  • im (integer): number of cells in i-direction.

  • jm (integer): number of cells in j-direction.

  • gh (integer): number of ghost cells for the boundary conditions. gh depends on the order of the numerical scheme: gh = 2 (3th order), gh = 3 (5th order), gh = 4 (7th order) or gh = 5 (9th order).

  • xo (numpy array of size (im+1+2gh, jm+1+2gh)): x coordinates of the cell nodes.

  • yo (numpy array of size (im+1+2gh, jm+1+2gh)): y coordinates of the cell nodes.

  • xc (numpy array of size (im+2gh, jm+2gh)): x coordinates of the cell centers.

  • yc (numpy array of size (im+2gh, jm+2gh)): y coordinates of the cell centers.

  • vol (numpy array of size (im+2gh, jm+2gh)): volumes of the cells.

  • volf (numpy array of size (im+2gh, jm+2gh, 2)): inverse of the volume computed at the cell faces.

  • nx (numpy array of size (im+1+2gh, jm+1+2gh, 2)): normal in x-direction computed at the cell faces.

  • ny (numpy array of size (im+1+2gh, jm+1+2gh, 2)): normal in y-direction computed at the cell faces.

Physics:

  • gam (real): \(= \gamma\) constant.

  • cp (real): specific heat at constant pressure.

  • cv (real): specific heat at constant volume.

  • rgaz (real): specific gas constant.

  • muref (real): Sutherland reference viscosity.

  • tref (real): Sutherland reference tempearture.

  • cs (real): Sutherland tempearture.

  • prandtl (real): Prandtl number.

  • mach (real): Mach number.

Solution:

  • w (numpy array of size (im+2gh, jm+2gh, 5)): state. w[:,:,0] \(= \rho\) is the density, w[:,:,1] \(= \rho u\) is the momentum in x-direction, w[:,:,2] \(= \rho v\) is the momentum in y-direction, w[:,:,3] \(= \rho w\) is the momentum in z-direction and w[:,:,4] \(= \rho E\) is the total energy.

  • res (numpy array of size (im+2gh, jm+2gh, 5)): residual of Navier-Stokes equations.

Numerics:

  • k2 (real): low-order dissipation coefficient for shock sensor in the FE-MUSCL numerical scheme. Default value to 1.01.

  • k4 (real): high-order dissipation coefficient in the FE-MUSCL numerical scheme. Default value to 1.

  • sch (string): numerical scheme to set. Only one implemented. sch = “dnc”.

  • order (integer): numerical scheme order. Availables options are 3, 5, 7 or 9.

Boundary conditions:

  • interf (numpy array of size (2, 2)): range of the coordinates for one BC location. interf[0,0] = imin, interf[0,1] = jmin, interf[1,0] = imax and interf[1,1] = jmax.

  • loc (string): location of the BC. Available values are “Ilo” (i=imin), “Ihi” (i=imax), “Jlo” (j=jmin) and “Jhi” (j=jmax).

  • lf (list of strings): list of routines names for BC and numerical scheme.

Linearised operators - Jacobian:

  • IA (numpy array of size (number of non-zeros entries of the Jacobian)): lists of row indices of the non-zeros entries of the Jacobian.

  • JA (numpy array of size (number of non-zeros entries of the Jacobian)): lists of column indices of the non-zeros entries of the Jacobian.

  • Aij (numpy array of size (number of non-zeros entries of the Jacobian)): lists of values of the non-zeros entries of the Jacobian.