Efficient boundary condition-enforced immersed boundary method for incompressible flows with moving boundaries. (English) Zbl 07513815
Summary: In this work, the original boundary condition-enforced immersed boundary method (IBM) [J. Wu and C. Shu, J. Comput. Phys. 228, No. 6, 1963–1979 (2009; Zbl 1243.76081); J. Comput. Phys. 229, No. 13, 5022–5042 (2010; Zbl 1346.76164)] 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 [K. Luo et al., “Full-scale solutions to particle-laden flows: multidirect forcing and immersed boundary method”, Phys. Rev. E (3) 76, No. 6, Article ID 066709, 9.p (2007, doi:10.1103/PhysRevE.76.066709); Z. Wang, J. Fan and K. Luo, “Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles”, Int. J. Multiphase Flow 34, No. 3, 283–302 (2008; doi:10.1016/j.ijmultiphaseflow.2007.10.004)] 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 \(2^{\mathrm{nd}}\) order accuracy in space.
MSC:
| 76Mxx | Basic methods in fluid mechanics |
| 76Dxx | Incompressible viscous fluids |
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
Software:
ProteusReferences:
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