Research Of Finite Volume Method On Unstructured Grid For Simulating Viscous-elastic Two-phase Flows

The viscous-viscoelastic two-phase flow exists extensively in nature and our daily life.The numerical simulation of the viscous-viscoelastic two-phase flow is of great importance in the development of many subject areas,such as fluid dynamics,aerospace,bioengineering,and material processing.This job generally contatins two main aspects,i.e.,acquiring the flow-field information and capturing the moving interface.Among the existing numerical methods,finite volume method(FVM)is one of the most effective methods for simulating viscous-viscoelastic two-phase flows due to its easy implementation,less memory space and specific physical interpretation for each term of the integral equations.However,when the traditional FVM is applied to simulate viscous-viscoelastic two-phase flows,there are some difficulties,such as irregular computational domain,inaccurate interface between two phases,non-conservation of mass,and high Weissenberg(Wi)number problem and so on.Therefore,we develop a novel finite volume algorithm on unstructured triangular grid,and simulate some typical viscous-viscoelastic two-phase flow problems by adopting the novel algorithm.The main work of this dissertation can be concluded as follows.(1)When the traditional level set solvers are applied to deal with the passive transport problems,it is generally difficult to preserve the mass conservation because of excessive numerical dissipation.Thus,a low-dissipation FVM(LDFVM)on unstructured triangular grid is proposed to solve the level set equation.This scheme borrows the idea of the spectral volume(SV)method,but in order to decrease the freedom,it utilizes the nodal values instead of the control volume(CV)-averaged values used in the SV method to perform the discretization.In this scheme,the level set function on the boundary of CV is evaluated using a linear combination of a high-order Lagrangian interpolation and a second-order upwind interpolation,where the combining weight is not constant but adaptively determined.The Zalesak's problem,single vortex flow and anisotropic motion in the normal direction are tested,and the numerical results show that LDFVM can not only capture complex interface evolution shapes accurately,but also preserve the mass conservation quite well without any mass correction technique.(2)In order to enhance the numerical instability at high Wi numbers in the simulation of viscoelastic fluid flows,a robust FVM framework on unstructured triangular grid is presented.The procedure of the new FVM framework can be briefly summarized as follows.First,it discretizes the conservation equations of mass and momentum by the cell centered FVM.For the purpose of accelerating the convergence rate of the flow field,the IDEAL(Inner Doubly iterative Efficient Algorithm for Linked equations)algorithm is used to solve the discrete equations.Second,the FVM framework adopts the high-resolution CUBISTA(Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection)scheme to evaluate the convection term of the constitutive equations.Meanwhile,in order to restrain the exponential growth of the stress tensor,the flux-based finite volume scheme is employed to deal with the deformation term.By applying this scheme,the deformation is balanced with the convection,and thus a good numerical stability is obtained in calculation.The presented FVM framework is applied to simulate the 2D lid-driven square cavity flow and abrupt 4:1 contraction flow of Oldroyd-B fluid.The numerical results demonstrate that our approach expands the range of computable Wi number compared to the conventional finite volume methods,and can effectively capture the elastic instability characteristics at high Wi numbers.(3)Based on the Navier-Stokes governing equations and the level set interface capturing technique,we simulate three classical two-phase fluid flow problems,i.e.,bubble rising,dam breaking,and Rayleigh-Taylor instability problems.Some complex interface evolution shapes,such as bubble breakup,large free surface vortices,splashing of the water surge front,and interfacial instability are observed.The numeircal results are in good agreement with those reported in literature and experimental data,which shows that the presented finite volume schemes are stable and reliable,and have a good applicability for those viscous-viscous twophase flow problems involving complicated interfacial topology evolution.(4)Based on the Navier-Stokes equations integrated with XPP constitutive model and the level set interface capturing technique,the direct dynamic numerical simulation is realized for mold filling of a floss pick.The interface evolution is obtained as well as the distribution of the physical quantities in flow field.The influence of melt elasticity on the distribution of physical quantities is analyzed in details.The best design plan of the injection gate location is discussed from the following two aspects,i.e.,mold filling time and backbone tube stretch at the end of the filling process.The numerical results of mold filling of a floss pick indicate that the mathematical model and the numerical algorithm presented in this dissertation can effectively simulate the viscous-viscoelastic two-phase flow problems with irregular computational domain,and have certain value of engineering application.