Keywords: finite element method, adiabatic flows, conservation of mass, conservation of momentum, semi-Lagrange Galerkin method, characteristic method, Hermite interpolation function.

The purpose of the study, described in this paper, is an analysis of adiabatic flows using the semi-Lagrange Galerkin method. Adiabatic flows are compressible flows assuming an adiabatic state. In this study, conservations of mass and momentum of the adiabatic flows are employed as the governing equations. The fluid is assumed as liquid. Therefore, the Birch-Murnaghan equation of state is applied to the governing equation. The semi-Lagrange method is used, in which the governing equations are divided into the advection and non-advection calculations. The advection calculation is approximated by the characteristic method. In both advection and non-advection calculations the Hermite interpolation function, which is a complete third order approximation, with triangular elements is used for both velocity and density. In the governing equation of the flow problems, the advection term and the diffusion term are included. For the case of either term being superior, the characteristic of flows are different. Depending on the characteristic of flows, a suitable appropriate technique is required. If the advection term is superior, the solution becomes unstable. To prevent this problem, the semi-Lagrange Galerkin method is introduced. The terms of temporal differentiation and advection are shown in the form of material differentiation and transformed using the characteristic method. In addition, in the semi-Lagrange method, the advection and non-advection calculations must be performed. After having calculated the advection term using the semi-Lagrange method, the non-advection calculation is calculated using the implicit method. This technique is called the semi-Lagrange Galerkin method. In the advection and non-advection calculations, the Hermite interpolation function is used for the velocity and density in this study. The Hermite interpolation function is composed of ten degrees of freedom. Therefore, the Hermite interpolation function provides a complete third order triangular element. As an example of the numerical analysis in this study, a cavity flow is carried out to show that the semi-Lagrange Galerkin method is effective. In addition, as an example of numerical analysis, the adiabatic flows are analysed using the semi-Lagrange Galerkin method in a circular computational area and a circular computational area which has a body at the centre.

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