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

TL;DR

This paper introduces a novel fourth-order spatial and second-order temporal compact scheme for solving incompressible Navier-Stokes equations on deformable, time-varying domains, enabling high-accuracy simulations on complex moving geometries.

Contribution

It presents the first higher-order compact method capable of directly solving the non-conservative Navier-Stokes equations on deformable, multi-block, time-dependent regions.

Findings

Achieved high-order accuracy on evolving geometries.

Validated the scheme's ability to handle complex, deformable domains.

Ensured geometric conservation law compliance in simulations.

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 [J. Comput. Phys. 251 (2013) 251-271] 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 non-conservative form of 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.