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

机译：高阶非结构拉格朗日一步WENO有限体积格式   非保守双曲线系统的应用：可压缩的应用   多相流

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In this article we present the first better than second order accurateunstructured Lagrangian-type one-step WENO finite volume scheme for thesolution of hyperbolic partial differential equations with non-conservativeproducts. The method achieves high order of accuracy in space together withessentially non-oscillatory behavior using a nonlinear WENO reconstructionoperator on unstructured triangular meshes. High order accuracy in time isobtained via a local Lagrangian space-time Galerkin predictor method thatevolves the spatial reconstruction polynomials in time within each element. Thefinal one-step finite volume scheme is derived by integration over a movingspace-time control volume, where the non-conservative products are treated by apath-conservative approach that defines the jump terms on the elementboundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian(ALE) method, where the mesh velocity can be chosen independently of the fluidvelocity. The new scheme is applied to the full seven-equation Baer-Nunziato model ofcompressible multi-phase flows in two space dimensions. The use of a Lagrangianapproach allows an excellent resolution of the solid contact and the resolutionof jumps in the volume fraction. The high order of accuracy of the scheme inspace and time is confirmed via a numerical convergence study. Finally, theproposed method is also applied to a reduced version of the compressibleBaer-Nunziato model for the simulation of free surface water waves in movingdomains. In particular, the phenomenon of sloshing is studied in a moving watertank and comparisons with experimental data are provided.
机译：在本文中，我们提出了一种优于二阶精度的非结构化Lagrangian型单步WENO有限体积方案，用于求解具有非保守积的双曲型偏微分方程。该方法使用非结构化三角形网格上的非线性WENO重建算子，在空间上具有很高的精度，同时具有基本的非振荡性。通过局部拉格朗日时空Galerkin预测器方法可以获得高时间精度，该方法在每个元素中随时间演化空间重构多项式。最终单步有限体积方案是通过在运动时空控制体积上进行积分而得出的，其中非保守乘积通过定义元素边界上跳跃项的路径保守方法进行处理。整个方法被公式化为任意拉格朗日欧拉（ALE）方法，其中网格速度可以独立于流体速度进行选择。该新方案应用于二维空间中可压缩多相流的完整七方程Baer-Nunziato模型。拉格朗日方法的使用可实现出色的固体接触分辨率和体积分数跳跃的分辨率。通过数值收敛研究证实了该方案在空间和时间上的高准确性。最后，该方法还应用于可压缩的Baer-Nunziato模型的简化版本，用于模拟运动域中的自由表面水波。特别是在移动的水箱中研究了晃荡现象，并与实验数据进行了比较。

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