**3.1.1 Inviscid fluxes**

354 Computational Simulations and Applications

Separate space discretization and

(Briley & McDonald, 1975) (Beam & Warming, 1976)

Multistage Runge-Kutta schme (Jameson, Schmidt & Turkel, 1981)

time integration

Implicit method

Explicit method

Flux vector splitting method (Moretti , 1979) - shock capturing (Steger & Warming, 1981)

Exact Riemann problem solution (Godunov, 1959) – First order (Van Leer, 1979) – Second order

Approximate Riemann solvers (Roe, 1981)

Explicit TVD upwind schemes (Boris & Book, 1973) (Harten, 1984) (Osher, 1984), (Oscher & Chakravarthy, 1984)

Implicit TVD upwind schemes (Yee, 1985) Central TVD scheme (Implicit or Explicit) (Davis, 1984), (Roe, 1995)

The axisymmetric time-dependent compressible Navier-Stokes equations can be written in the following conservation form. The analysis is carried out under the assumption of laminar flow. The coefficient of molecular viscosity is calculated according to Sutherland's law. The temperature is related to pressure and density by the perfect gas equation of state.

To facilitate the spatial discretization in the numerical scheme, the governing equations are be written in the integral form over a finite volume of the computational domain with the boundary domain. The contour integration around the boundary of the cell is divided in the

Combined space-time

(Lax & Friedrich, 1954) – First order (Lax & Wendroff, 1954) – second

Discretization

order

Explicit method

Two-step method

Implicit method (MacCormack, 1981)

Table 2. Numerical method for Fluid Dynamics Equations.

The ratio of the specific heats is assumed constant.

**3. Axisymmetric flow solver** 

**3.1 Finite volume method** 

(Richtmyer & Mortan, 1967) (MacCormack, 1969)

Numerical method

Spacecentered method

Upwind schemes

Highresolution (non-linear schemes)

The convective fluxes are calculated at the centre of the grid, resulting in cell-centre flux balances. The contour integration of the inviscid flux vector is approximated at the side of the computational grid. The summation is carried out over the four edges of the grid.
