Exact solution and the multidimensional Godunov scheme for the acoustic equations

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

Numerical modeling of subsonic axisymmetric reacting gas flows

A numerical algorithm is developed and implemented for modelling axisymmetric subsonic reacting gas flows based on a previously created program for plane flows. The system of Navier-Stokes equations in the low Mach number limit is used as a mathematical model. Calculations of ethane pyrolysis for axisymmetric and plane flow of mixture at heat supply from the reactor’s walls are carried out. Through the interplay of the developed code and the code for plane flows it becomes possible to identify the geometric factor role at the presence of a large number of nonlinear physicochemical processes. We found that diffusion of synthesized molecular hydrogen mainly influences heat supply from the reactor’s walls to gas and pyrolysis products distribution along its length.

Incompressible limit of Euler equations with damping

<abstract><p>The Cauchy problem for the compressible Euler system with damping is considered in this paper. Based on previous global existence results, we further study the low Mach number limit of the system. By constructing the uniform estimates of the solutions in the well-prepared initial data case, we are able to prove the global convergence of the solutions in the framework of small solutions.</p></abstract>