Improving the Stability of theMultiple-Relaxation-Time Lattice Boltzmann Method by a Viscosity Counteracting Approach

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method(LBM).The multiple-relaxation-time(MRT)model of the LBM can improve the accuracy and stability,but is still subject to numerical instability when simulating flows with large single-grid Reynolds number(Reynolds number/grid number).The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM.However,its effectiveness was only verified in the single-relaxation-time model of the LBM(SRT-LBM).This paper aims to propose the viscosity counteracting approach for the multiple-relaxationtime model(MRT-LBM)and analyze its numerical characteristics.The verification is conducted by simulating some benchmark cases:the two-dimensional(2D)lid-driven cavity flow,Poiseuille flow,Taylor-Green vortex flow and Couette flow,and threedimensional(3D)rectangular jet.Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBMhas better accuracy and stability than that for the SRT-LBM.

Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model

This paper aims to study the numerical features of a coupling scheme between the immersed boundary(IB)method and the lattice Boltzmann BGK(LBGK)model by four typical test problems:the relaxation of a circular membrane,the shearing flow induced by a moving fiber in the middle of a channel,the shearing flow near a non-slip rigid wall,and the circular Couette flow between two inversely rotating cylinders.The accuracy and robustness of the IB-LBGK coupling scheme,the performances of different discrete Dirac delta functions,the effect of iteration on the coupling scheme,the importance of the external forcing term treatment,the sensitivity of the coupling scheme to flow and boundary parameters,the velocity slip near non-slip rigid wall,and the origination of numerical instabilities are investigated in detail via the four test cases.It is found that the iteration in the coupling cycle can effectively improve stability,the introduction of a second-order forcing term in LBGK model is crucial,the discrete fiber segment length and the orientation of the fiber boundary obviously affect accuracy and stability,and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instability.These elaborate results shed light on the nature of the coupling scheme and may benefit those who wish to use or improve the method.

Immersed Boundary-Lattice Boltzmann Coupling Scheme for Fluid-Structure Interaction with Flexible Boundary

Coupling the immersed boundary(IB)method and the lattice Boltzmann(LB)method might be a promising approach to simulate fluid-structure interaction(FSI)problems with flexible structures and complex boundaries,because the former is a general simulation method for FSIs in biological systems,the latter is an efficient scheme for fluid flow simulations,and both of them work on regular Cartesian grids.In this paper an IB-LB coupling scheme is proposed and its feasibility is verified.The scheme is suitable for FSI problems concerning rapid flexible boundary motion and a large pressure gradient across the boundary.We first analyze the respective concepts,formulae and advantages of the IB and LB methods,and then explain the coupling strategy and detailed implementation procedures.To verify the effectiveness and accuracy,FSI problems arising from the relaxation of a distorted balloon immersed in a viscous fluid,an unsteady wake flow caused by an impulsively started circular cylinder at Reynolds number 9500,and an unsteady vortex shedding flow past a suddenly started rotating circular cylinder at Reynolds number 1000 are simulated.The first example is a benchmark case for flexible boundary FSI with a large pressure gradient across the boundary,the second is a fixed complex boundary problem,and the third is a typical moving boundary example.The results are in good agreement with the analytical and existing numerical data.It is shown that the proposed scheme is capable of modeling flexible boundary and complex boundary problems at a second-order spatial convergence;the volume leakage defect of the conventional IB method has been remedied by using a new method of introducing the unsteady and non-uniform external force;and the LB method makes the IB method simulation simpler and more efficient.