Efficient boundary condition-enforced immersed boundary method for incompressible flows with moving boundaries

Abstract In this work, the original boundary condition-enforced immersed boundary method (IBM) [Wu and Shu (2009) [1] , (2010) [2] ] is improved to efficiently simulate incompressible flows with moving boundaries. The original boundary condition-enforced IBM can accurately interpret the no-slip boundary condition but becomes computationally tedious in simulating moving boundary problems due to the assembly of a large matrix at every time step and the implicit resolving process. The computational complexity of O ( N a ) grows significantly with the number of Lagrangian points N distributed on the immersed boundary. To alleviate these limitations, the conjugate gradient technique and the explicit technique are proposed to improve the efficiency of the boundary condition-enforced IBM. The IBM with the conjugate gradient technique fulfills the boundary condition in an iterative way with computational complexity of O ( N c ) , while the IBM with the explicit technique is a non-iterative approach based on error analysis with computational complexity of O ( N d ) . We also prove that the multi-direct forcing IBM [Luo et al. (2007) [7] ; Wang et al. (2008) [8] ] which is another popular IBM, is essentially a gradient descent approach to implement the boundary condition-enforced IBM with computational complexity of O ( N b ) . Detailed analyses reveal 2 = a > b > c > d = 1 , which implies the high efficiency of the improved versions of IBM, especially the explicit technique-based IBM with a linear computational complexity. For validation, the IBMs are coupled with D1Q4 lattice Boltzmann flux solver (LBFS) to simulate two-dimensional and three-dimensional flows with moving boundaries. The results show that the conjugate gradient technique-based IBM and the explicit technique-based IBM have computational complexities of O ( N 1.4 ) and O ( N ) , respectively. Both of them have 2nd order accuracy in space.