Construction And Application Of Explicit Schemes For Immersed Boundary-Lattice Boltzmann Method

Fluid-structure interaction(FSI)is multidisciplinary and involves the interaction between fluids and solids.The numerical computational method to solve the FSI problem is an important research topic of basic disciplines.Weak decoupling methods based on non-body-fitted grids are computationally efficient,the immersed boundary method(IBM)and the lattice Boltzmann method(LBM)are typical representatives.When the conventional boundary schemes of LBM are applied to FSI problems with complex moving boundaries,it is limited by the problems of grid technology and interface computation.Therefore,IBM was introduced into LBM to form the immersed boundary-lattice Boltzmann method(IB-LBM),whose interpolation grid makes it easy to implement in complex moving boundary problems.However,these immersed boundary schemes in LBM are inferior to the traditional LBM boundary schemes regarding computational accuracy,mass conservation,and construction principle.Therefore,it is of great research significance to construct a new IB-LBM scheme.Based on the Euler-Lagrangian interpolation grid of IBM,this paper constructs two explicit and conservative schemes of IB-LBM based on the kinetic theory of LBM,which are explicit-correction-force scheme and non-equilibrium approximate force scheme,then applies them to three types of typical moving boundary problems.First,we use the multivariate Taylor expansion and the Chapman-Enskog multiscale expansion to deduce the truncation error of the IB-LBM discrete coupling equations and the continuous incompressible Navier-Stokes(N-S)equations with external force source terms in detail.And the numerical properties of the IB-LBM discrete coupled equations are determined by numerical experiments.Then,we constructed two IBLBM explicit and conservative schemes and passed multiple numerical verifications of error,precision,stability,mass conservation,unsteady condition and moving boundary.Finally,based on the proposed two IB-LBM schemes,we simulated three types of moving boundary problems with disciplinary and engineering application backgrounds by establishing fluid-structure coupling equations: free oscillation airfoils,deformed cylinder groups and flexible filaments.For the explicit-correction-force scheme of IB-LBM,this paper first uses the particle distribution function as the interpolation physical quantity from the fluid grid to the immersed boundary,and directly applies the LBM force model on the immersed boundary to obtain an explicit force with simple form.Then this paper uses the error correction matrix to correct the obtained interface force,which ensures the mass conservation of the local area of the interface.So far,the explicit-correction-force scheme of IB-LBM is constructed.The order of accuracy of this scheme is theoretically analyzed: it is shown that the order of accuracy is consistent with the bounce-back scheme when Guo’s force model is used.The results of numerical experiments verify the accuracy and stability of the scheme,and the streamline penetration phenomenon is effectively suppressed.In the test of the rotating cylinder,the change curve of the lift and drag coefficient fits the experimental curve,and the simulation results conform to the experimental phenomenon,which proves the accuracy of the proposed scheme in the moving boundary problem.For the non-equilibrium approximation force scheme of IB-LBM,this paper first reconstructs the discrete velocity distribution function from the Euler grids to the Lagrangian grids on the immersed boundary,and uses the known velocity on the interface to construct the equilibrium distribution function on the Lagrangian grids,and then construct the non-equilibrium distribution function on the Lagrangian point.Then,based on the obtained non-equilibrium distribution function and the LBM force model,a simple explicit direct force expression is derived.At the same time,an approximation force in the form of an infinite series is deduced,which ensures the mass conservation of the scheme on the immersed boundary.So far,the total non-equilibrium approximating force scheme is constructed.It is theoretically proved that the spread process using the non-equilibrium distribution function has local second-order accuracy.Numerical results verify that the streamline penetration phenomenon disappears,that is,the no-slip condition is strictly satisfied.In the moving boundary computational example of particle collision and sedimentation,the trajectory curve is basically consistent with the experiment.Using the two computational schemes proposed in this paper,the following problems are simulated: 1)Free oscillation airfoil: Based on the explicit-correctionforce scheme and the finite difference method,all coupled difference equations are given.The results of the pressure coefficient and streamline diagram are compared with other literature results,the vibration curve is obtained,and the simulation results of the unstable flow structure with Re=1000 or more are obtained.2)Deformed cylinders:The case set the constitutive relation of the cylinder as Hooke’s law,establishes an internal force model satisfying Newton’s second law,and obtains the deformation process of the cylinders combined with the explicit-correction-force scheme.3)Flexible filament: Based on the non-equilibrium state-approximation force scheme,a coupled equation system is established.The tangential tension of the filament is directly obtained by the inextensibility condition,which simplifies the work of implicitly solving the tension by adding the Poisson equation.The motion trajectory curve of the filament tail is obtained and which is similar to the simulation results of other papers.In summary,we studied the numerical properties of IB-LBM,constructed two accurate and effective explicit schemes of IB-LBM,numerically simulate three types of FSI problems with complex moving boundaries,and provides reliable numerical methods for numerical simulation of fluid problems in science and engineering.