Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

Abstract

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.