## 1 Introduction
To circumvent this drawback, [PERSON] et al. [6; 19] first attempted to solve the pressure Poisson equation only once per time step, but with the price of lowering the order of accuracy for the pressure. To retain the formal order of accuracy, several fast RK-based projection methods [8; 20; 21; 22] have been proposed to reconstruct the pressure values at intermediate stages through various interpolation/extrapolation techniques. Apart from reducing the number of Poisson equations to be solved within each time step, applying fast Poisson solvers is essential for developing cost-effective pressure-correction projection methods. For large-scale flow simulations with structured or unstructured body-conforming grids, the large linear system resulting from the discretisation of the pressure Poisson equation is usually solved by iterative methods accelerated by multigrid techniques. In cases where the use of regular grids is feasible, applying fast Fourier transforms (FFTs) to directly solve the Poisson equation is a popular choice. In general, direct FFT-based Poisson solvers are computationally less expensive than multigrid-based solvers (e.g., [23]). In the past, FFT-based Poisson solvers have been used widely in combination with either spectral or high-order compact schemes, especially in DNS/LES codes for academic re search. Recently, an FFT-based solution of the pressure Poisson equation was proposed to offer large gains in computational efficiency for the strictly incompressible SPH methodology [24]. For simulating flows over curved walls using structured body-conforming grids, [PERSON] and [PERSON] proposed a fast RK-based projection method together with a fast FFT-based Poisson solver and showed that the FFT-based solver is much faster than the multigrid-based solver [21]. However, due to the reliance on regular grids, FFT-based Poisson solvers do not lend themselves easily to being used for solving flow problems with obstacles. A common approach to overcome the difficulty is to use immersed boundary methods (IBMs) [25], which include solid boundaries as immersed boundaries within a regular grid that does not conform to the boundaries, and take special treatment at the boundaries to incorporate boundary conditions. Immersed boundary methods have been widely used in both the DNS and LES of flows around obstacles. For the former, the no-slip boundary condition can be well satisfied through reconstructing the velocity field around immersed boundaries by various interpolation/extrapolation methods. For the latter, how to incorporate wall models at immersed boundaries in an accurate and physically consistent way is still an open question. A common practice in LES with IBM is still to use the smearing approach [26; 27; 28; 29], which basically has zero order of accuracy in terms of interpolation and yields non-negligible errors in the velocity profile near the immersed boundary [30]. To our knowledge, FFT-based Poisson solvers have not been applied in flow solvers with body-conforming methods using unstructured grids. The present work proposes a fast hybrid pressure-correction algorithm for numerical simulation of incompressible flows around obstacles by projection methods combined with body-conforming methods. The focus here is to develop a fast numerical algorithm for the pressure-correction step rather than new time-integration or spatial discretisation schemes for projection methods. Therefore, for convenience and the purpose of demonstration, we adopted a classical second-order time-marching projection scheme together with a meshless method for spatial discretisation, which resembles the use of body-conforming methods with unstructured grids. In the first sub-step, the momentum equations are spatially discretised in a given flow domain of complex geometry by a standard numerical method (here, a meshless method) and boundary conditions are implemented directly at points distributed on the flow boundary like that in body-conforming methods.