**3. Axisymmetric flow solver**

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. The ratio of the specific heats is assumed constant.

## **3.1 Finite volume method**

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 anticlockwise sense. The computational domain is divided into a finite number of nonoverlapping quadrilateral grids. The conservation variables within the computational cell are represented by their average values at the cell centre. When the integral governing equation is applied separately to each cell in the computational domain, we obtain a set of coupled ordinary differential equation.
