Study Of Free Element Method For Computational Fluid Dynamics

With the development of computer science and the increase of the complexity of engineering,the collocation methods become more and more popular because of the flexity and simplicity.The advantage of flexity has been attracking reseachers.Free Element method combines the collocation method and isoparametric interpolation and has been successfully applied in thermal conduction problems and analyzing stresses near cracks,which indicates the method has great potential in solving partial difference equations.The thesis developed a new Free Element scheme to solve fluid dynamics,which also was applied in simulating turbulent flows and fluid-solid conjugate heat tansfer.Free Element method is a numerical method for solving partial difference method based on elements generated locally,where the variables are approximated by shape functions of elements.The freedoms of Free Element method locate at nodes and the elements are attached to nodes.This work introduced virtual elements that consisted of midpoints to discretize spatial terms and resolved the instability caused by convection terms.Meanwhile,this thesis employed polygonal elements in Free Element method at the first time.Taking in the ideal of numerical fluxes and reconstruction procedures employed in unstructured Finite Volume method,a stable discretization scheme was designed.The proposed method was tested by the cases of shock tube et al.The method predicted right positions of shock and pressure distributions.The matrixfree LU-SGS was employed in this work to improve the efficiency of the method without additional memory demand.Compared with the least square colloction method in literature,Free Element Method was more accurate and efficient,espetially in simulating visous flow.With different types of element and node distributions,the proposed method was tested and eventually convergent to the right results,which indicated the method was robust.In incompressible Navier-Stokes equations,there is no equation for pressure and the only the gradients of pressure appear in the source terms of momentum equations,which causes that the oscillation of pressure can’t be detected.Accourding to the ideal of momentum interpolation,a free element pressure correction procedure was proposed.By the way of momentum interpolation,the difference of pressure gradients was added to the velocity of midpoints.Therefore,the gradients of pressure were couple with velocity and the oscillation of pressure could be detected.Then,an equation for correction of pressure were derived to update pressure.According the analysis in this work,the results of this work are independent on relaxation factor and time step size.The numerical results showed that the proposed method is second order accuracy and works in both two-and three-dimensional problems.Furthermore,the new version of Free Element Method was employed to solve Raynold-Averaged Navier-Stokes equations,which coupled with the two-equation turbulent models.Several classical cases were employed to verify the rightness of Free Element Method.At last,based on the former method,a procedure was introduced to solve conjugate heat transfer of solid and fluid materials.The velocity of flow was solved by the proposed pressure correction method and the equation of energy was discretized based on the elements that consisted of midpoints.In order to treat non-matched grids,the constinuous equations of temperature and heat fluxes were discretized in aid of radial basis function interpolation.The test case of heat transfer between plates showed that the results of non-matching grids would convergent to the results of matched grids with the increase of the number of interpolation points.Furthermore,the proposed method was employed to simulate manifold micro channel sinks.The solutions recovered the results of experiment,which indicated the validity of the proposed method.This work developed the Free Element method in simulating flow problem,verified the accuracy and validity of Free Element method and reduced the compelixty of the preprocess in simulating flow problems.