## Verification of a Cartesian grid method for compressible flow over moving structures

##### Abstract

A simplified, low order finite volume Cartesian grid method for inviscid compressible flow over rigid, moving structures is developed and tested in two spatial dimensions to assess the treatment of the moving boundary and the potential of the Cartesian grid method for solving problems with complex boundaries. The method is second order accurate when the local Lax-Friecrichs method with MUSCL and minmod-limiter is used, and first order accurate without MUSCL. The boundary conditions are imposed via ghost points located inside the structure. The values of the ghost points G1 are set based on the corresponding fluid points F1, the fluid points closest to the ghost points, shifted either in the x- or y-direction or diagonally from the ghost point. Symmetry-like boundary conditions are imposed.The density and pressure of the ghost point G1 are set equal to the density and pressure of the corresponding fluid point F1. The velocity of the ghost point is set such that the normal velocity component at the boundary, when determined by linear interpolation using the ghost point and a mirror point M1, is equal to the normal velocity component of the boundary. The mirror point M1 is located in the fluid domain, the same distance from the boundary as the ghost point, along a line passing through the ghost point G1 and the fluid point F1. The velocity at the fluid point M1 is determined by linear interpolation or extrapolation, depending on the position relative to the fluid point F1, using the fluid point F1 and the fluid point one step further into the fluid domain, F2.Emerging fluid points are points that, due to the moving boundary, were ghost points in the solid domain at the previous time level, and are fluid points at the current time level. The density and pressure of such points are kept as they were at the previous time level. The normal velocity component of the emerging fluid point is set equal to the normal velocity component of the boundary, and the tangential velocity component is set equal to the tangential velocity component that the point itself had at the previous time level. Other methods for ghost and emerging fluid point treatment have been implemented, tested and found less accurate.The method is implemented and tested for two subsonic examples in two dimensions, a moving piston and a moving cylinder. The resulting rates of convergence are as expected, two for the method with MUSCL and one for the method without MUSCL. However, as the moving boundary treatment is shown to be a limitation of the method, the accuracy of the method is expected to increase if a more sophisticated ghost point treatment is implemented. The fact that the computational effort required by boundary treatment is only a small part of the total computational effort further implies that implementing a more sophisticated method might be beneficial, yielding higher accuracy per computa- tional effort. The computational effort required for the internal solver with MUSCL is substantially larger than without MUSCL, but as the increased accuracy is even more substantial, it is beneficial to use MUSCL.