This paper presents a high-order generalized differential quadrature method with lattice Boltzmann flux solver (LBFS-GDQ) for simulating incompressible isothermal flows. In this method, high-order polynomials are adopted to approximate both the solution and fluxes globally across the computational domain. Solution derivatives and flux divergence are conveniently computed by the GDQ method. At the interior solution points, the viscous and inviscid fluxes are evaluated simultaneously via LBFS. Treatments to prevent the global accuracy from being contaminated by the streaming error of LBFS are proposed and studied, including the choice for the local streaming spacing and interpolation methods for the local reconstruction. The present method inherits the advantages of both GDQ and LBFS, i.e., global spectral accuracy, direct evolution of macroscopic variables, and convenient implementation of boundary conditions. Numerical experiments with a wide selection of incompressible flow problems confirm the excellent accuracy, efficiency, and flexibility of the proposed method.
In this paper, we present an immersed boundary-lattice Boltzmann flux solver (IB-LBFS) to simulate the interactions of viscous flow with deformable elastic structures, namely, two-dimensional (2D) and three-dimensional (3D) capsules formed by elastic membranes. The IB-LBFS is based on a finite-volume formulation and makes use of hydrodynamic conservation equations with fluxes computed by a kinetic approach, thus it is more flexible and efficient than the standard immersed boundary-lattice Boltzmann methods. The membrane of the 2D capsule is represented by a set of discrete Lagrangian points, with in-plane and bending forces acting on the membrane obtained by a finite difference method. In contrast, the membrane of a 3D capsule is discretized into flat triangular elements with membrane forces calculated by an energy-based finite-element method. The IB-LBFS is first validated by studying the deformation of a circular capsule in a linear Newtonian and a power-law shear flow. Next, the deformation dynamics of a spherical, an oblate spheroidal, and a biconcave capsule in a simple shear flow are simulated. For an initially spherical capsule, the tank-treading motion of its membrane is reproduced at the steady state; while for oblate spheroidal and biconcave capsules, the swinging and tumbling motions are observed. Furthermore, under certain parameter settings, the transient mode from tumbling to swinging motions is also found, showing a rich and complex dynamic behavior of non-spherical capsules. These results indicate that the IB-LBFS can be employed in future studies concerning the dynamics of a capsule suspension in more realistic flows.
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