(untitled) (Advances in Applied Mathematics and Mechanics, 2020)
In this paper, a variable correction-based immersed boundary method (IBM) for simulating viscous compressible flows over stationary and moving bodies is presented. It is the extension of the implicit velocity correction-based IBM proposed by Wu and Shu (J. Comput. Phys., 228 (2009), pp.00a0 1963-1979). Since the incompressible flow was studied in their work, only the fluid velocity field was needed to be corrected by enforcing the no-slip boundary condition. But for compressible flow problems, other flow variables around the boundary also should be corrected or updated. To simulate the flow field without the effect of immersed boundary firstly, the open source code OpenFOAM is utilized in this work. After performing the velocity correction then, the density is corrected by resolving the continuity equation with the corrected fluid velocity. After that, the temperature surrounding the boundary is corrected from the given temperature condition, which is similar to the velocity correction. Finally, the pressure is updated by using the corrected density and temperature through the equation of state. In such way, all the flow variables have been corrected and the given physical boundary conditions can be accurately implemented. To validate the proposed method, the flows over a stationary circular cylinder, a stationary airfoil and a transversely oscillating circular cylinder at different Mach and Reynolds numbers are simulated. Compared to the results in the literature, good agreement can be achieved. To further illustrate the potential of current method for dealing with complex problems, the flow over a rotating cylinder in front of a stationary cylinder is also simulated.