NAVIER-STOKES SOLVER AND INCORPORATION OF MOMENTUM AND ENERGY EQUATIONS

For the plasma multiphysics fluid analysis, the implementation of the conservation of momentum and energy equations for neutral gas particles dynamics, and the coupling to the mass model has been developed. A compressible Navier-Stokes solver that includes the conservation of mass, momentum and energy equations for neutral gas particles has been developed in software programming language C++ using the Finite- Element Flux-Corrected Transport method (FE-FCT). Thereafter, the Navier-Stokes solver is tested both in two- dimensional Cartesian and axisymmetric cylindrical coordinates for validation purposes. Cartesian coordinates - Shock-tube test case: The Riemann shock tube is a benchmark for the solution of the Euler equations (Navier-Stokes equations excluding viscous terms) in one dimension. Since an analytical solution is available, this test is used to exploit the response of the developed Navier-Stokes solver in shock wave propagation. Fig. 1 and 2 show the analytical and numerical solutions of neutral gas density and Mach number, respectively, for three different meshes at two different instants in time. The graphs show that the capturing of the steep gradients is improved for finer meshes and that the overshooting of the numerical results as compared to the analytical solution is also reduced, when a finer mesh is used. Cartesian coordinates - Shock wave wedge test case: The shock wave wedge test case is a benchmark for the solution of the Navier-Stokes equations in two dimensions. Fig. 3 shows the contour plot of the numerical solution of neutral gas density which is in agreement with published numerical and experimental work. Cartesian coordinates - Source term test case: To test the two-dimensional Cartesian solver for the Navier- Stokes equations in ambient air, a heating source term is introduced only for the first time step for a very short duration. Fig. 4 shows the two-dimensional plot of the velocity in the horizontal x-direction. The graph shows the formation of a series of compression and rare-fraction waves, as predicted by fluid theory. Cylindrical axisymmetric coordinates - Source term test case: The solver is tested under energy density source terms in two-dimensional cylindrical axisymmetric coordinates, emulating the momentum and Joule heating in a gas discharge situation. The results are compared to fluid dynamic theory. Fig. 5 shows the two- dimensional distribution of the radial velocity in cylindrical coordinates at time t = 9 × 10 -5 (s). The velocity changes of the air molecules involved are infinitesimally small compared to the speed of sound, suggesting that there is no shock wave, rather a sound wave propagating in space and time. This is in agreement with sound wave theory, which suggests that sound is propagated by a series of compression and rare-fraction waves. Cylindrical axisymmetric coordinates - Micro-blast wave test case: The Navier-Stokes solver response under input energy density source term and shock propagation is further validated in 2D cylindrical axisymmetric coordinates. This involves sudden releases of energy within small volumes, leading to the development of shock waves. Fig. 6 shows the one-dimensional plots of the Mach number against time along the axis of symmetry, obtained by Jiang et al. (1) and the authors (2) for different mesh sizes. The results are shown to be in good agreement with those of other authors, taking into consideration that different numerical schemes will produce different results, depending on the implementation and the approximations used.