High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows

In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided

Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity

© 2016 .A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach

A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests

In an effort to study the applicability of adaptive mesh refinement (AMR) techniques to atmospheric models an interpolation-based spectral element shallow water model on a cubed-sphere grid is compared to a block-structured finite volume method in latitude-longitude geometry. Both models utilize a non-conforming adaptation approach which doubles the resolution at fine-coarse mesh interfaces. The underlying AMR libraries are quad-tree based and ensure that neighboring regions can only differ by one refinement level. The models are compared via selected test cases from a standard test suite for the shallow water equations. They include the advection of a cosine bell, a steady-state geostrophic flow, a flow over an idealized mountain and a Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR simulations show that both models successfully place static and dynamic adaptations in local regions without requiring a fine grid in the global domain. The adaptive grids reliably track features of interests without visible distortions or noise at mesh interfaces. Simple threshold adaptation criteria for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin

A momentum-conserving, consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a consistent mass and momentum-conserving discretization of the Navier-Stokes equation. Incompressible flow with capillary forces is modelled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d'Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressiblity is satisfied which leads to a method that is both momentum and mass-conserving. To improve the stability of these methods momentum fluxes are advected in a manner "consistent" with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a falling raindrop and for an atomizing flow in air-water conditions

An adaptive octree finite element method for PDEs posed on surfaces

The paper develops a finite element method for partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in $H^1$ and $L^2$ surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection-diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces