基于简化Baer-Nunziato模型的任意形状运动固体可压缩绕流的简单扩散界面方法
A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer-Nunziato model
摘要：
本文提出了一种新的扩散界面模型，用于任意形状的固定和运动固体的无粘可压缩绕流的数值模拟。固体被假定为运动的刚体，没有任何弹性。该模型是可压缩多相流的七方程Baer-Nunziato模型的简化情况，并导致具有非保守乘积的非线性双曲方程组。固体的几何形状简单地通过标量场指定，该标量场表示存在于每个控制体积中的流体的体积分数。这允许在简单的均匀或自适应笛卡尔网格上离散任意复杂的几何形状。在固体内部，流体体积分数为零，而在流体相中，流体体积分数是单位的。我们证明，在材料界面，即体积分数从1跳到零，流体速度的法向分量假设的固体速度的法向分量的值。这一结果可以直接从控制方程中导出，或者通过Riemann不变量，或者根据Dal Maso，Le Floch和Murat的理论从广义Rankine Hugoniot条件导出，这证明了使用路径保守方法来处理非保守产物的合理性。我们的新模型的控制方程求解均匀笛卡尔网格通过高阶路径守恒ADER间断伽辽金（DG）方法与后验子细胞有限体积（FV）限制器。由于数值方法是激波捕获型的，因此流固边界从来没有被数值方法明确地跟踪，既没有通过界面重构，也没有通过网格运动。一组数值试验问题，包括一维黎曼问题，以及超声速绕流固定和移动刚体的方法的有效性进行了测试。
英文摘要：
In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows, and results in a nonlinear hyperbolic system with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary in the fluid phase. We prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat, which justifies the use of a path-conservative approach for treating the nonconservative products. The governing equations of our new model are solved on uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion. The effectiveness of the proposed approach is tested on a set of numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.
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