A freestream preserving algorithm for the flux reconstruction method implemented on tetrahedral element

In computational fluid dynamics, curvilinear elements are frequently used to fit boundaries. Metric terms are involved by the map from these curvilinear elements to the standard reference element. If these metric terms are calculated inappropriately, the solutions may be incorrect or even divergent. It is well known that the freestream solution could be used to examine whether the metrics are discretized correctly or not. Namely, the metric terms must be computed in such a way that the freestream could be preserved, which is known as the freestream preservation problem. Although this problem has been studied extensively in the finite difference approach, there are still unanswered questions in the high-order schemes for the unstructured mesh, such as the discontinuous Galerkin (DG) method and the flux reconstruction (FR) algorithm. The existing research about the freestream preservation property for these high-order schemes mainly focused on the unstructured hexahedral element. Although some algorithms had been extended to the unstructured tetrahedral element (Chana and Wilcox, 2019), it remains unclear whether the conditions to preserve freestream on the tetrahedral element agree with those of the hexahedral element. In this paper, we extend the work of Abe and his colleagues (Abe et al., 2015) on the freestream preservation problem in the FR method from the unstructured hexahedral element to the tetrahedral cell. The algorithms of both the non-conservative and symmetrical-conservative metrics occurred in their research have been extended to the tetrahedral element in our research. We also analyzed the conditions under which these algorithms could achieve freestream preservation property and found that they are not equal to those of the unstructured hexahedral cell. If $P$ and $Q$ are used to respectively represent the order of solution and grid, the algorithm constructed in the non-conservative form could preserve freestream when $P\ge 2Q$ for the hexahedral element, as depicted by Abe et al. (2015). However, this condition becomes $P\ge 2(Q-1)$ on the tetrahedral element according to our research. In the symmetrical-conservative metrics, this condition changes from $P\ge Q$ on the hexahedral element (Abe et al., 2015) to $P\ge Q-1$ on the tetrahedral cell. Namely, compared to the hexahedral element, these conditions are relaxed for the tetrahedral element. To verify our findings, two three-dimensional cases are performed. The results of these numerical simulations have confirmed that the algorithms constructed in our research are successful and the corresponding conditions are correct.