Improving the conditioning and accuracy of the projection-based immersed boundary methods

Projection-based immersed boundary methods are continuous forcing variants of the immersed boundary method, used to model flow over arbitrary surfaces without conforming to the underlying computational fluid grid. These approaches remove ad-hoc parameters and complex stencil modifications used in other immersed methods, and impose the no-slip condition through a first-kind integral equation for the stress distribution along the immersed surface. However, projection-based immersed approaches are generally limited to first-order spatial accuracy, and the integral equation is ill-conditioned and yields spurious stress computations. This work addresses both deficiencies through a Taylor expansion of the flow field within the support of the smooth delta functions used in the governing and constraint equations. This modification allows the unknown surface stresses to be incorporated into the constraint equation, transforming the integral equation from an ill-posed Fredholm equation of the first-kind into a well-posed second-kind equation. This approach does not require heuristic regularization parameters used in other approaches to solve the ill-conditioning problem. Moreover, it offers the potential for achieving higher-order accuracy, with only a slight increase in algorithmic cost. We validate the method on a range of incompressible flow problems.