Comparison Among Unstructured High Resolution Algorithms in the Solution of the Euler Equations in Two-Dimensions
Abstract
The present work compares the Yee, Warming and Harten, the Harten, the Yee and Kutler and the Hughson and Beran schemes applied to the solution of aeronautical and aerospace problems. The schemes are of TVD (“Total Variation Diminishing”) flux
difference splitting type and are second order accurate in space. The Euler equations in conservative form, employing a finite volume formulation and an unstructured spatial discretization, in two-dimensions, are solved. The time integration is performed by a Runge-Kutta method, second order accurate. The steady state physical problems of the supersonic flows along a ramp and around a blunt body configuration are studied. The results have demonstrated that the Harten scheme has presented more accurate results in the ramp problem, whereas the Hughson and Beran scheme has presented more accurate solutions in the blunt body problem. In the ramp problem, the Yee, Warming and Harten and the Yee and Kutler schemes predicted more severe pressure fields. The shock angle was best predicted by
the Harten scheme, which presented a percentage error of 4.91%. In the blunt body problem, the pressure field generated by the Hughson and Beran scheme was the most severe. The
stagnation pressure ahead of the configuration was best estimated by the Hughson and Beran scheme, which presented a percentage error of 3.7%. The best value of the lift coefficient was evaluated by the Yee, Warming and Harten and by the Yee and Kutler schemes. As can be observed, errors below 5.0% were obtained in the determination of the physical parameters of
the two problems. The Harten scheme is the cheapest one, being approximately 27.5% cheaper than the Yee, Warming and Harten scheme, the most expensive. As final conclusion, it is not possible to highlight the best scheme in terms of these two example-cases, being necessary more studies. This is the purpose of the next paper to be written by this author.
difference splitting type and are second order accurate in space. The Euler equations in conservative form, employing a finite volume formulation and an unstructured spatial discretization, in two-dimensions, are solved. The time integration is performed by a Runge-Kutta method, second order accurate. The steady state physical problems of the supersonic flows along a ramp and around a blunt body configuration are studied. The results have demonstrated that the Harten scheme has presented more accurate results in the ramp problem, whereas the Hughson and Beran scheme has presented more accurate solutions in the blunt body problem. In the ramp problem, the Yee, Warming and Harten and the Yee and Kutler schemes predicted more severe pressure fields. The shock angle was best predicted by
the Harten scheme, which presented a percentage error of 4.91%. In the blunt body problem, the pressure field generated by the Hughson and Beran scheme was the most severe. The
stagnation pressure ahead of the configuration was best estimated by the Hughson and Beran scheme, which presented a percentage error of 3.7%. The best value of the lift coefficient was evaluated by the Yee, Warming and Harten and by the Yee and Kutler schemes. As can be observed, errors below 5.0% were obtained in the determination of the physical parameters of
the two problems. The Harten scheme is the cheapest one, being approximately 27.5% cheaper than the Yee, Warming and Harten scheme, the most expensive. As final conclusion, it is not possible to highlight the best scheme in terms of these two example-cases, being necessary more studies. This is the purpose of the next paper to be written by this author.
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