Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

We have introduced a modified penalty approach into the flow-structure interaction solver that combines an immersed boundary method (IBM) and a multi-block lattice Boltzmann method (LBM) to model an incompressible flow and elastic boundaries with finite mass. The effect of the solid structure is handled by the IBM in which the stress exerted by the structure on the fluid is spread onto the collocated grid points near the boundary. The fluid motion is obtained by solving the discrete lattice Boltzmann equation. The inertial force of the thin solid structure is incorporated by connecting this structure through virtual springs to a ghost structure with the equivalent mass. This treatment ameliorates the numerical instability issue encountered in this type of problems. Thanks to the superior efficiency of the IBM and LBM, the overall method is extremely fast for a class of flow-structure interaction problems where details of flow patterns need to be resolved. Numerical examples, including those involving multiple solid bodies, are presented to verify the method and illustrate its efficiency. As an application of the present method, an elastic filament flapping in the Kármán gait and the entrainment regions near a cylinder is studied to model fish swimming in these regions. Significant drag reduction is found for the filament, and the result is consistent with the metabolic cost measured experimentally for the live fish.

The locomotion of a flapping flexible plate in a viscous incompressible stationary fluid is numerically studied by an immersed boundary-lattice Boltzmann method for the fluid and a finite element method for the plate. When the leading-edge of the flexible plate is forced to heave sinusoidally, the entire plate starts to move freely as a result of the fluid-structure interaction. Mechanisms underlying the dynamics of the plate are elucidated. Three distinct states of the plate motion are identified and can be described as forward, backward, and irregular. Which state to occur depends mainly on the heaving amplitude and the bending rigidity of the plate. In the forward motion regime, analysis of the dynamic behaviors of the flapping flexible plate indicates that a suitable degree of flexibility can improve the propulsive performance. Moreover, there exist two kinds of vortex streets in the downstream of the plate which are normal and deflected wake. Further the forward motion is compared with the flapping-based locomotion of swimming and flying animals. The results obtained in the present study are found to be consistent with the relevant observations and measurements and can provide some physical insights into the understanding of the propulsive mechanisms of swimming and flying animals.

Locomotion of a passively flapping flat plate has been studied numerically by means of a multiblock lattice Boltzmann method. A flexible plate is modelled by a rigid plate with a torsion spring acting about the pivot at the leading edge of the plate. A dynamic model of this kind is called a lumped-torsional-flexibility model. When the leading edge is forced to heave sinusoidally, the plate pitches passively and propels itself in the horizontal direction as a result of the fluid–plate interaction. We have investigated various aspects of the mechanics behind the behaviour of the flapping plate, including the periodic- and non-periodic-flow states, the spontaneous motion of the plate, vortical structure and how they compare to similar propulsion systems in animals. In the periodic-flow regime, two dynamical responses of the passively pitching plate (forward and backward movements) are observed. Which movement will occur depends only on the frequency ratio F of the natural frequency of the system and the heaving frequency associated with the lumped torsional flexibility. It is found that the plate will select the forward movement when F > 1 and the backward movement when F ≤ 1. In the forward-movement regime, analysis of the dynamical behaviours and propulsive properties of the passively pitching plate indicates that the torsional flexibility can remarkably improve the propulsive performance. In addition, four kinds of vortex structures in the near wake are identified, which mainly depend on the forward speed of the plate. Finally the forward movement is compared to the flapping-based locomotion of swimming and flying animals. The results obtained in this study are consistent with the observations and measurements of swimming and flying animals; thus, they may provide physical insights into understanding of the propulsive mechanisms of the flapping wings and fins of animals.

Flow over traveling wavy foils in a side-by-side arrangement has been numerically investigated using the space-time finite element method to solve the two-dimensional incompressible Navier-Stokes equations. The midline of each foil undergoes lateral motion in the form of a streamwise traveling wave, which is similar to the backbone undulation of swimming fish. Based on the phase difference between the adjacent undulating foils, two typical cases, i.e., in-phase and anti-phase traveling wavy movements, are considered in the present study. The effects of lateral interference among the foils on the forces, power consumption, propeller efficiency, and flow structures are analyzed. It is revealed that the lateral interference is of benefit to saving the swimming power in the in-phase case and enhancing the forces in the anti-phase case. Some typical vortex structures, e.g., vortex-pair row, single vortex row, and in-phase and anti-phase synchronized vortex-street, are observed in the wake of the traveling wavy foils. The results obtained in this study provide physical insight into the understanding of hydrodynamics and flow structures for flow over the traveling wavy foils and swimming mechanisms relevant to fish schooling.