A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by join- ing the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an im- plicit second-order scheme is utilized to enhance the com- putational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method （FVM） and the pressure Poisson equation is solved by the Galerkin finite el- ement method （FEM）. The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both veloc- ity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature(2-D 3-T)radiation diffusion equa-tions are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons,ions and photons.In this paper,we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes.The vertex unknowns are treated as primary ones for which the finite volume equations are constructed.The edgemidpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns,which makes the final scheme a pure vertex-centered one.By comparison,most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal.Here,the conormal decomposition is not convex in general,leading to a fixed stencil of the flux approximation and avoiding a certain search algo-rithm on complex grids.Moreover,the new scheme effectively alleviates the nu-merical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations.Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids.For the problem without analytic solution,the contours of the nu-merical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

A Discontinuous Galerkin Extension of the Vertex-Centered Edge-Based Finite Volume Method

The finite volume(FV)method is the dominating discretization technique for computational fluid dynamics(CFD),particularly in the case of compressible fluids.The discontinuous Galerkin(DG)method has emerged as a promising highaccuracy alternative.The standard DG method reduces to a cell-centered FV method at lowest order.However,many of today’s CFD codes use a vertex-centered FV method in which the data structures are edge based.We develop a new DG method that reduces to the vertex-centered FV method at lowest order,and examine here the new scheme for scalar hyperbolic problems.Numerically,the method shows optimal-order accuracy for a smooth linear problem.By applying a basic hp-adaption strategy,the method successfully handles shocks.We also discuss how to extend the FV edge-based data structure to support the new scheme.In this way,it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

Comparison and Modification: TVD Schemes for Scalar Transport on An Unstructured Grid

In most TVD schemes, the r-factors were proposed according to the cell-centered（CC） finite volume method（FVM） framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered（VC） FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner（1996）, Darwish and Moukalled（2003） and Cassuli and Zanolli（2005） can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely Фu in the paper, was validated by the tests to effectively dissipate spurious oscillations.

海上溢油漂移扩散的数值模拟

该文基于三维自由水面垂向分层动网格的Euler-Lagrangian模式,采用VC方式(Vertex-Centered)的非结构化有限体积方法离散三维浅水方程,模拟渤海的潮汐水流运动过程。以此为水动力背景场,采用油、水分离的Lagrangian粒子追踪模型,对渤海海域发生的溢油漂移扩散问题进行模拟。其油品的蒸发、乳化和溶解等风化作用以引入综合衰减系数的方式考虑。模型对油膜的漂移路径、厚度和面积进行了实时模拟和预测,用多个测站的同步连续实测值对水动力模式进行了参数率定与模型验证,油膜输运扩散的计算结果也与相关报告数据进行了对比,均吻合良好,说明模型具有较高的可靠性与重现性。

Multi-Mesh-Scale Approximation of Thin Geophysical Mass Flows on Complex Topographies

This paper is devoted to a multi-mesh-scale approach for describing the dynamic behaviors of thin geophysical mass flows on complex topographies.Because the topographic surfaces are generally non-trivially curved,we introduce an appropriate local coordinate system for describing the flow behaviors in an efficient way.The complex surfaces are supposed to be composed of a finite number of triangle elements.Due to the unequal orientation of the triangular elements,the distinct flux directions add to the complexity of solving the Riemann problems at the boundaries of the triangular elements.Hence,a vertex-centered cell system is introduced for computing the evolution of the physical quantities,where the cell boundaries lie within the triangles and the conventional Riemann solvers can be applied.Consequently,there are two mesh scales:the element scale for the local topographic mapping and the vertex-centered cell scale for the evolution of the physical quantities.The final scheme is completed by employing the HLL-approach for computing the numerical flux at the interfaces.Three numerical examples and one application to a large-scale landslide are conducted to examine the performance of the proposed approach as well as to illustrate its capability in describing the shallow flows on complex topographies.