A unified discontinuous Galerkin finite element framework on unstructured grids is developed to simulate gas-liquid twophase flows.In the framework,interior penalty discontinuous Galerkin (IPDG) method is employed to discretize imcompressible Navier-Stokes equations,while Runge-Kutta discontinuous Galerkin (RKDG) method is used to solve Level Set equation.Lid-driven cavity flow is simulated to validate IPDG method.Numerical results of bubble rising indicate the approach can be used to complex twophase flows with low computational efforts and simple implement.Moreover,RKDG method can effectively track deformations of interface without reinitialization.

To make balance between numerical accuracy and computational efficiency,a deferred correction method coupled with high order scheme is proposed for computation of convection flux.Classical benchmark problems,cavity driven flow with high Reynolds numbers,are solved numerically to verify precision and stability of the method.Finally,the method is used to solve Boussinesq fluid of natural convection heat transfer.It shows that the method overcomes numerical divergence effectively in high Rayleigh number problem.It captures accurately isothermal lines and stream lines at different eccentricities in natural convection heat transfer.

A stable and efficient element-free Galerkin method is proposed for steady convection-diffusion problems.In the method integrations are computed with a local Taylor expansion nodal integral technique.According to convection-dominated degree,integration nodes are adaptively shifted opposite to the streamline direction.Compared with conventional element-free Galerkin method with stabilization,the method exhibits better stability and higher efficiency in solving convection-dominated convection-diffusion problems.It is a pure meshfree method,which is independent of background integral.Moreover,the method is easy to be implemented.

In a FENE bead-spring chain model,a parallel algorithm for Brownian configuration fields with finite volume method is used to simulate pressure driven pipe nows, velocity-pressure driven pipe nows with same and opposite directions. Validity of the algorithm is verified with numerical examples. Velocity,stress and stretch of molecular chain of pipe nows are obtained. It shows that the proposed algorithm overcomes shortcomings of huge computation and time-consuming for Brownian configuration fields. And it has excellent expansibility. Parallel efficiency is over 85% as number of beads is greater than two. It also shows that the time of now to reach steady state is shorter with increasing number of chain segments.

A new approach。a strong form of discontinuous Galerkin method(DGM),was developed to solve level set equation on unstructured drids.A weak form is only suitable for incompressible flow,while the strong form can used to solve level set equation in any case,including incompressible and compressible flows.Thc approach allows arbitrarib higlI order accuracy through Legendre-Gauss.Lobatto nodal distribution.Several numerical tests on one-,two-and three-dimensional unstructured grids demonstrate versatility and validity of the method.Besides,implementation ofthe strong form ofDGM brings benefits,such as high order,mars conservation, dimension independence,resolving interface location at the sub-cell level,handling complex domains,avoiding reinitialization and so on.

A corrected smoothed particle hydrodynamics(SPH) method is proposed,in which standard SPH method is corrected with a density re-initialization method.A treatment for solid wall boundaries is presented to improve numerical accuracy.Drop stretching and dam-breaking are simulated numerically to show validity and reliability of the method.Filling process in a channel is investigated,and effect of Reynolds number on flow field and vortex are analyzed.It shows that the corrected SPH method can simulate filling process in a channel precisely and the flow is affected significantly by Reynolds number.

Velocity filed is decomposed into "coarse" and "fine" scales with a variational mulitiscale method.The "fine" scale is modeled by bubble functions,and solved with Petrov-Galerkin method.A stabilized term and stabilization parameter are introduced by coupling the "fine" and "coarse" scales.A variational multiscale equation which preserves properties of both "fine" and "coarse" scales is solved with a finite element method.It shows that the method is stable and accurate.It eliminates spurious oscillations caused by dominated advection term and uncoupling between velocity and pressure in numerical simulation of incompressible flows.The stabilization parameter can be applied to structure and unstructure meshes as well.

Smoothed particle hydrodynamics(SPH) method is applied to simulate transient viscoelastic flows.The method for viscoelastic flows is verified by comparing numerical solution with analytical solution of an Oldroyd-B fluid in a start-up Couette flow.An Oldroyd-B fluid in a lid-driven cavity is simulated by the method.In addition,a new treatment of solid wall boundaries is presented to prevent particles penetrating solid walls and improve numerical accuracy.It shows that SPH method is valid and stable in simulation of viscoelastic flows.