Abstract
In this paper, we take a numerical simulation of a complex moving rigid body under the impingement of a shock wave in three-dimensional space. Both compressible inviscid fluid and viscous fluid are considered with suitable boundary conditions. We develop a high order numerical boundary treatment for the complex moving geometries based on finite difference methods on fixed Cartesian meshes. The method is an extension of the inverse Lax-Wendroff (ILW) procedure in our works (Cheng et al., Appl Math Mech (Engl Ed) 42: 841–854, 2021; Liu et al.) for 2D problems. Different from the 2D case, the local coordinate rotation in 3D required in the ILW procedure is not unique. We give a theoretical analysis to show that the boundary treatment is independent of the choice of the rotation, ensuring the method is feasible and valid. Both translation and rotation of the body are taken into account in this paper. In particular, we reformulate the material derivative for inviscid fluid on the moving boundary with no-penetration condition, which plays a key role in the proposed algorithm. Numerical simulations on the cylinder and sphere are given, demonstrating the good performance of our numerical boundary treatments.
Highlights
In this paper, we design a high order boundary treatment combining high order finite difference scheme for both inviscid and viscous fluids in the 3D time-varying domain
We focus on the inverse Lax-Wendroff (ILW) boundary treatment for solving the moving boundary problems in 3D
Numerical results demonstrate that the ILW method can be applied to construct ghost point values when solving equations with high order finite difference methods on a Cartesian mesh in a computational domain with complex geometries
Summary
We design a high order boundary treatment combining high order finite difference scheme for both inviscid and viscous fluids in the 3D time-varying domain. The idea of ILW procedure comes from [21, 22], in which the authors simulated the traffic and pedestrian flow by solving an Eikonal equation at each time step They used the PDE repeatedly to convert the normal derivative into the tangent derivative at the boundary, and the ghost point values are defined by a Taylor expansion along the normal direction. Numerical results demonstrate that the ILW method can be applied to construct ghost point values when solving equations with high order finite difference methods on a Cartesian mesh in a computational domain with complex geometries. 3.1.1 Reformulation of the material derivative in 3D In the ILW procedure, we convert normal spatial derivatives to tangential and time derivatives of the given boundary condition using the PDE.
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