Keywords: semi-Lagragian Galerkin method, conservation of mass, conservation of momentum, characteristic method, Hermitian type element, implicit scheme, intersecting flow.

The purpose of the study, presented in this paper, is to express the change of flow pattern in the intersecting adiabatic flows by the difference of the flow channel and of the degrees using the Semi-Lagragian Galerkin finite element method. An incompressible viscous fluid is the most used assumption in the field of computational fluid dynamics in recent years. However, only a few phenomena can be assumed as incompressible in natural phenomena. In fact, it is necessary to consider compressibility if much broader phenomena are to be considered. In order to consider the compressibility, it is usual to use the full compressible equations including the energy conservation equation. However, since a lot of variables should be treated and formulation is complicated, solution of the full compressible equation system is computationally heavy. In this study, the equation system, in which density is considered as a variable assuming adiabatic state, is derived. In order to obtain the pressure, the Poisson's law which is an expression of relation of pressure and density is applied for the equation of state. The conservation of mass and momentum are used as the basic equation. The governing equation is assumed to consist of both advection and non-advection terms. To calculate the governing equations, the Semi-Lagragian method is applied as an analytical technique. A characteristic of this method is based on the fact that the governing equations can be divided into two phases which are advection and non-advection at an instanteneous time of calculation. In case of the calculation equation of advection term, sometimes, instability of the comptation occurs. Thus, the characteristic method which is useful for the calculation of advection term is applied. As the temporal discretization, the advection phase is calculated by the characteristic method, and the non-advection phase is calculated by the implicit method. In the spatial discretization the Galerkin method is adopted. In this study, the unknown function at the upstream side on the characteristic curve is approximated by the Hermitian type element. This element has ten degrees of freedom, which consists of function values of nodes of the triangle and first order derivatives for each node and a function value of centre of the triangle. The present interpolation is used in a high precision fluid flow analysis of the intersecting flow. Also in the calculation of non-advection terms, the same-order interpolation for velocity and density is applied for each element. This method is called the Semi-Lagragian Galerkin method. Finally, the intersecting flow is analysed using the Semi-Lagrasian Galerkin method. The flows in different crossing angles are confirmed. In addition, the symmetric and asymmetric flow patterns are also confirmed.

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