A Study Of Mesh Adaptation For Simulating Unsteady Moving-Boundary Flows

Although the numerical methods and the capacity of modern computers have been developing continuously,computations of complex flows encountered in science and engineering are still extremely time-consuming.Mesh adaptation can improve the accuracy and efficiency of numerical simulation,because the adapted mesh is always dense in the regions containing characteristic physical phenomena.The technique of mesh adaptation for the simulation of steady flow is relatively mature.However,further research on this aspect for unsteady flow is still needed.As the physical phenomena may evolve arbitrarily with time in the entire computational domain,mesh adaptation becomes more important for unsteady flow problems.Without mesh adaptation,a uniform and fine grid is usually needed to ensure the accuracy of the numerical results.The use of such a grid is computationally prohibitive,especially for three-dimensional problems.Therefore,the study of the technique of dynamic mesh adaptation is necessary for the simulation of unsteady flows.In this paper,mesh adaptation for the simulation of unsteady flow is researched deeply.A dynamic mesh adaptation method for complex three-dimensional incompressible unsteady moving-boundary flows is established,and the adaptive time stepping is combined with the dynamic mesh adaptation to further improve the efficiency of computation.The main work and contributions of this paper are shown as follows:(1)A P~1-conservative interpolation reconstruction approach for numerical solution on three-dimensional tetrahedral mesh is constructed,which ensures the conservation and accuracy of interpolation between two meshes in dynamic mesh adaptation.The conservation property is achieved by a local mesh intersection technique and the mass of a tetrahedron of the current mesh is calculated by the integral on its intersection with the background mesh.A tetrahedron intersection detection method was developed to detect all the overlapped background tetrahedrons for each current tetrahedron.A mesh intersection algorithm is proposed to construct the intersection region of a current tetrahedron with all the overlapped background tetrahedrons,and quickly discretize the intersection region into a tetrahedral mesh.An efficient localization algorithm is employed to search the host units in background mesh for each vertex of the current mesh.In order to enforce the maximum principle and avoid the loss of monotonicity,correction of nodal interpolated solution on tetrahedral meshes is given,and the corresponding proof is presented.The reliability of the method for numerical solution reconstruction is verified by numerical experiments on several analytic functions and numerical solutions of real flow fields.(2)An approach of dynamic mesh adaptation for simulating complex three-dimensional incompressible moving-boundary flows by the immersed boundary methods is proposed.A predictor-corrector scheme is introduced to avoid the frequent mesh adjustment and eliminate the phase lag between adapted mesh and unsteady solution,which occurs frequently in process of the mesh adaptation for unsteady flow.A simple vorticity-based adaptive criterion is presented to capture the characteristic phenomena that evolve with time.A hierarchical refining/coarsening technique for automatically adjusting tetrahedral meshes is proposed.Regular refinement is accomplished by dividing one tetrahedron into eight sub-cells,and irregular refinement is used to eliminate the hanging points.Merging the eight sub-cells obtained by regular refinement,the mesh is coarsened.Based on the hierarchical refining/coarsening strategy,mesh adaptivity can be achieved by adjusting the mesh only one time for each adaptation period.The level difference between two neighboring cells never exceeds one,and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases.The error caused by each solution transferring from the old mesh to the new adapted one is small,because most of the nodes on the two meshes are coincident.Compared with the adaptive mesh refinement(AMR)of the Cartesian mesh,there is no need for the present method to address the issues associated with maintaining conservation across interface between meshes of different levels,because there is no hanging point.The mesh adaptation method proposed in this paper is not only applicable to the immersed boundary method,but can also be easily extended to the conventional numerical methods.A series of numerical experiments show that the proposed mesh adaptation method can reduce the number of mesh nodes greatly and the accuracy of numerical solutions can be preserved.(3)A technique of adaptive time stepping is developed and combined with dynamic mesh adaptation to simulate unsteady flows.The second order Backward Differentiation Formula(BDF2)of variable step size for time discrete is derived.The adaptation of time step is based on the estimation of temporal error.Via a PID(Proportional Integral Derivative)controller or a classical heuristic controller,the size of time step is determined adaptively by the estimate of temporal error and the specified tolerance.The efficiency and reliability of the present adaptive time and space discretization approach are validated by the numerical experiments for two-and three-dimensional flows.In the numerical experiments,the behaviors and effects of different error estimators and step-size controllers have also been compared and discussed.(4)The present mesh adaptation method for three-dimensional moving-boundary flows is applied to the preliminary numerical investigation of self-propelled swimming of eel-like multi-fish,and the availability of the method in complex situations is further verified.The flows of multi-fish swimming in different formations(eg,tandem,parallel and diamond configurations of two fish and three fish configurations)are simulated,and for various configurations,the interactions between individuals and the effects of environmental vortices were analyzed.With the application of dynamic mesh adaptation,the computational cost is greatly reduced,and the immersed boundary method can become a potential and efficient tool for the research of biological fluid mechanics.