Immersed boundary techniques for high-order finite-difference methods

The use of immersed boundary techniques for Cartesian grid methods is becoming increasingly popular for predicting flows with complex geometries. It has been demonstrated that for cases where accuracy near immersed boundaries is not crucial, existing methods are sufficiently accurate. However, if near-wall accuracy is paramount, immersed boundary techniques for high-order methods can not be used without corrections. This assertion is corroborated by computing the flow over a backward-facing step at low Reynolds number and for Tollmien-Schlichting waves in a flat-plate boundary layer using second-order and fourth-order finite-difference methods. The immersed boundary technique is extended to the compressible Navier- Stokes equations, and both the compressible and incompressible Navier-Stokes equations are employed to evaluate immersed boundary techniques. In the compressible code, the same difference operator is applied to all derivatives computed, thus more clearly demonstrating the effects of the various corrections for the immersed boundary technique. The methods of Goldstein et al. [1] and Mohd- Yusof [2] serve as prototypical immersed boundary techniques. A correction method is suggested which eliminates the need for ad hoc adjustments and allows for the efficient use of high order finite difference methods. Its performance is compared to other recently published methods.