Stencil Penalty approach based constraint immersed boundary method

Abstract

The constraint-based immersed boundary (cIB) method has been shown to be accurate between low and moderate Reynolds number (Re) flows when the immersed body constraint is imposed as a volumetric constraint force. When the IB is modelled as a zero-thickness interface, where it is no longer possible to model a volumetric constraint force, we found that cIB is not able to produce accurate results. The main source of inaccuracies in the cIB method is the distribution of the pressure field around the IB surface. An IB surface results in a jump in the pressure field across the IB. Evaluation of the discrete gradient of pressure close to the IB leads to a pressure gradient that does not satisfy the Neumann boundary condition for pressure at the IB. Furthermore, a non-zero discrete pressure gradient on the IB results in spurious flow at grid points close to the IB. We present a novel numerical formulation which adapts the cIB formulation for ‘zero-thickness’ immersed bodies. To impose the Neumann boundary condition on pressure on the IB more accurately, we introduce an additional body force to the momentum equation. A WENO based stencil penalization technique is used to define the new force term. Due to the more accurate imposition on the Neumann pressure boundary condition on the IB, spurious flow is reduced and the accuracy of no penetration velocity boundary condition on the IB is improved.