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Multi-dimentional upwind schemes for the Euler Equations on unstructured grids

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Abstract

In the last few years, upwind methods have become very popular in the modelingof advection dominated flows and in particular those which contain strongdiscontinuities. For more than a decade, these methods have been usedsuccessfully to solve numerically the one-dimensional Euler equations.Fluctuation distribution has been recently introduced as an alternative toconventional upwinding. In contrast to standard upwinding the fluctuationdistribution approach extends naturally to multidimensional flow without requiringany splitting along coordinate directions. The technique uses a narrow-stencil,local, piecewise linear reconstruction of the flow field solution. The flow field isupdated in time by propagating a subset of eigenmodes of the convectiveoperator. Different choices of the eigenmode subset lead to different fluctuationdistribution schemes.In this paper, schemes for approximating steady solution to the two dimensionalof the inviscid fluid equations on unstructured triangular grids are presented, alsoan analysis of fluctuation splitting schemes applied to scalar advection equationshas been performed. Wave models based on Roe’s simple wave decompositionhave been further developed and tested, providing an exact solution to thelinearized equations, and decomposes the flux difference at the interface into aset of simple waves, all aligned with the grid face.In this work, the presented model of fluctuation splitting N combined with Roewave models implemented in our own Code written in C++ reached the stagewhere they can be used reliably to achieve maximal computational efficiency topractical steady state problems in aerodynamics (Supersonic oblique shockreflection, Flow in a channel with a Bump, Symmetric Constricted channel flows,flow around NACA 0012 aerofoil, flows in a turbine-blade cascade VKI LS-59 ).

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