Research On Nonuniform Cartesian Grid Based Direct-forcing Immersed Boundary Method And Multigrid Method

Stationary or moving boundary problems with complex geometries are often encountered in marine science and its applications.Traditionally,sophisticated body-fitted grids should be generated and dynamic mesh technique should be employed.However,based on fixed cartesian grid,the immersed boundary method can easily impose the no-slip boundary conditions for both moving and stationary boundaries by adding properly designed Eulerian forcing terms in the discretized momentum equotions to reconstructe the velocity field in the vicinity of the immersed boundaries,thus avoiding frequent remeshing operations and taking the advantages of high computational efficiency and better adaptivity.A Fortran numerical code based on staggered non-uniform cartesian grid is developed to solve the 2-D N-S equations of the imcommpressible fluid.By introducing non-uniform cartesian grid,the numerical code can refine the grid in the region of special interests without rapid increasing of grid numbers which is a great shortcoming of the uniform cartesian grid.In this code,second order finite difference schemes on non-uniform cartesian grid are constructed for convection terms and duffusion terms by using a taylor expansion technique;Fractional projection method is used to perform unsteady computation and the no-slip velocity boundary condition is impose by using a explicit iterative direct forcing immersed boundary method.Considering the slow convergence speed of the Possion equation,a V-cycle geometrical multigrid method code based on non-uniform cartesian grid is developed.A novel grid coarsing strategy is proposed.The new strategy is easy to implement and will not resulting in great difference of the grid size near the computational domain.A area-weighted restriction operator and a linear interpolation operator are used to transferring the data between different grid layers.The tolerance of iteration is numerically examed and the results show that the difference of iteration solution and the exact solution of discretized passion equation is negligible when the tolerance is lower than10^-5,indicating no sense to use a tolerance too low.The efficiency of multigrid solver is very sensitive to the choice of various parameters,like the smoother types,the layers of coarse grids,and the iterations in both restriction and interpolation procedures.The effect of four major parameters mentioned above on efficiency is analyzed by orthogonal experiment method and variance analysis theory.Quantitative results are obtained and a general procedure is recommended to get the parameter values with hignest computational efficiency.Various numerical simulations are conducted.Flow past a stationary circular cylinder and typical airfoil at different Reynolds number are used to validate the solver for stationary boundary problems.Flow past a transversely oscillating circular cylinder with different oscillating frequencies are used to validate the solver for moving boundary problems.Finally,the pure wake interaction between two sedimentation circular particles with different density and different initial offset positions are numerically simulated,indicating the solver have the potential to simulate two-way fluid-structure interaction?FSI?problems.