A high-order immersed-boundary method for simulation of incompressible flows

Abstract

The use of immersed-boundary methods to solve complex/moving-boundary flow problems, especially those in biofluid dynamics, has been popular in recent years. Such methods typically employ a stationary structured grid, e.g., a Cartesian grid, for spatial discretization, so that grid generation is relatively simple and computation on the grid is efficient. However, existing immersed-boundary methods are usually up to second- order accurate. In this work, we have developed a high-order program to solve viscous, incompressible flows involving arbitrary boundaries by combining a fourth-order com- pact finite-difference scheme and a high-order immersed-boundary treatment based on least squares fitting. Moreover, a high-order compact scheme is developed to solve the pressure Poisson equation. In its discrete form, the new pressure solver can take ad- vantage of the same tri-diagonal structure as in the conventional second-order central finite-difference scheme, without introducing significant computational overhead. Both model equations and Navier-Stokes equations have been tested using this method. First, we use one-dimensional numerical experiments to evaluate the feasibility of the method. Then, the complete two-dimensional program is used to solve both Kovasznay flow and flow past a circular cylinder to evaluate the performance of the program. These tests show that the high-order treatment of the immersed-boundary is compatible with the compact scheme. Furthermore, a third-order accuracy is achieved for the overall program as expected. In the end, application of the method for flapping wings is demonstrated.