On the accuracy of direct forcing immersed boundary methods with projection methods

Direct forcing methods are a class of methods for solving the Navier–Stokes equations on nonrectangular domains. The physical domain is embedded into a larger, rectangular domain, and the equations of motion are solved on this extended domain. The boundary conditions are enforced by applying forces near the embedded boundaries. This raises the question of how the flow outside the physical domain influences the flow inside the physical domain. This question is particularly relevant when using a projection method for incompressible flow. In this paper, analysis and computational tests are presented that explore the performance of projection methods when used with direct forcing methods. Sufficient conditions for the success of projection methods on extended domains are derived, and it is shown how forcing methods meet these conditions. Bounds on the error due to projecting on the extended domain are derived, and it is shown that direct forcing methods are, in general, first-order accurate in the max-norm. Numerical tests of the projection alone confirm the analysis and show that this error is concentrated near the embedded boundaries, leading to higher-order accuracy in integral norms. Generically, forcing methods generate a solution that is not smooth across the embedded boundaries, and it is this lack of smoothness which limits the accuracy of the methods. Additional computational tests of the Navier–Stokes equations involving a direct forcing method and a projection method are presented, and the results are compared with the predictions of the analysis. These results confirm that the lack of smoothness in the solution produces a lower-order error. The rate of convergence attained in practice depends on the type of forcing method used.