Keywords: computational fluid dynamics, moving boundary, moving grid finite volume method, geometric conservation law, immersed boundary method, Cartesian cut cell method, arbitrary Lagrangian-Eulerian method, dynamic mesh method.

Today, one of the interesting problems in computational fluid dynamics is a moving boundary problem. A flow around a train moving into and passing through a tunnel and pulsating blood flow through the heart are the typical examples. Methods for solving moving boundary problems have been studied actively because of their physical as well as their engineering importance. This chapter first gives an overview of the state-of-the-art of the numerical methods for solving a moving boundary problem. For a moving boundary problem, two approaches exist from the view point of a grid. One is to use a stationary Cartesian grid where the moving boundary is represented as an immersed boundary. The other is to use a moving or deforming mesh which conforms to the moving boundary. Emphasis has been put on the moving mesh methods, such as an arbitrary Lagrangian-Eulerian method, a temporary moving transformed coordinate method, a dynamic mesh method of integral form as well as a moving grid finite-volume method. As an important law for the moving mesh method, the geometric conservation law is also discussed.

To introduce development of the moving mesh methods, the moving grid finite volume (MGFV) method is described in detail with examples in various fields of compressible and incompressible flows. Since the MGFV method is based on the governing equations written in extended divergence form in the (x,y,z,t) domain and is constructed on the discretization using an octahedral control volume in space and time unified four-dimensional domain, it inherently satisfies the geometric conservation law. Moreover, dynamic elimination and addition of the computational cell is theoretically possible. This feature has been incorporated in the moving embedded grid approach and it has been applied to a flow driven by independently moving multiple bodies. The chapter finally introduces a new moving computational domain method, which is a modification of the MGFV method and can deal with any kind of the motion of the body without limit of a region size. Application of the method to a flow around a high speed car running through a hairpin curve is demonstrated.

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