On the freestream preservation of finite volume method in curvilinear coordinates

• The freestream preservation in the finite volume method in curvilinear coordinates is studied.
• The reconstruction algorithm is adopted in the Jacobian evaluation.
• The conditions of freestream preservation in the metric evaluation are proposed.

As the importance of freestream preservation is widely discussed in the finite difference (FD) method, it is found in this paper that this problem also exists in the finite volume (FV) method based on the curvilinear coordinate system. If the freestream is not well preserved, serious errors will be introduced in numerical results. In this paper, the conditions of the freestream preservation are studied in the FV method using the dimension-by-dimension reconstruction. In two space dimensions, it is demonstrated that the order of the polynomials in evaluating metrics should not be higher than the order of accuracy of the Gaussian quadrature. In addition, the Jacobian at each Gaussian point needs to be obtained by the same reconstruction algorithm. In three space dimensions, the metrics can be written in three different forms: the cross product, conservative and symmetric conservative forms. For guaranteeing the freestream preservation, it is proved that except for the conditions proposed in two space dimensions, the metrics should be evaluated by the conservative or symmetric conservative forms, but not the cross product form. Numerical tests are presented to illustrate the effect of the conditions proposed in this paper.