A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high- order accurate.