Validation of immersed boundary method for the numerical simulation of flapping wing flight

摘要

The immersed boundary method (IBM) is now used extensively in many simulations involving moving bodies, such as flapping wings. The unique feature of this method is that the entire simulation is performed on a Cartesian grid and the bodies in the simulation do not conform to the grid. Hence, there is no complicated grid movements needed and grid quality issues are avoided. Despite the clear advantages of the IBM, it has its problems as well. One of the most important disadvantages with regard to flapping micro aerial vehicle (FMAV) simulations is the increase in grid size with increasing Reynolds number (Re). Unlike conforming grids which has the flexibility to adjust the cells very close to the wall, the Cartesian grid used in the IBM is more rigid as it is not aligned with the wall of the body. Hence it will require more grids to capture the boundary at the same accuracy as the conforming grid. FMAVs generally operate in the Re range of 5,000 to 10,000, hence, a large number of grid cells will be required, especially in 3D, to accurately simulate and capture the boundary layer. Moreover, there are two classifications of IBM - the continuous and discrete forcing approach, depending if the forcing is applied to the equation before or after discretization. Due to the smoothing of the forcing function inherent in the continuous approach, it has been argued that this approach is not suitable for relatively high Reynolds number flow. At that regime, the discrete forcing approach is thought to be more suitable because it is able to provide a sharp representation of the immersed boundary of the bodies. The objective of this paper is to validate the suitability of both the continuous and discrete forcing approach for flows at Re = 5,000 to 10,000, by comparison with numerical and experimental results of plunging airfoils in literature. Results obtained with conforming grid flow solvers are included for comparison. Results show that both the continuous and discrete forcing IBM solvers are able to give reasonably accurate results in terms of thrust coefficient, although the discrete forcing IBM solver is able to give smoother results at a lower grid resolution. This indicates that it is able to capture the thin boundary layer better. The vorticity contours plots of both IBM solvers are visually similar compared to the referenced results. There is disagreement in the lift coefficient, but it applies to all the numerical solvers. This could be due to the transition from laminar to turbulent flow. Lastly, the discrete forcing IBM solver is used to simulate a 3D NACA0012 wing undergoing plunging motion at a Re of 10,000. The results compare relatively well, given the lower resolution due to limited computational resources. This shows that the IBM solvers are capable of doing 3D flapping wing simulations.