On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries

In recent years, the immersed-boundary method that is based on the structured mesh has gained considerable popularity in computational fluid dynamics for solving complex and moving-boundary problems. Despite its wide applications, so far there is not a unified definition of the method, possibly because there are many variations in the existing implementations. Here we follow the classification approach by Mittal & Iaccarino [1] where the immersed-boundary method is in general classified into two types. One type involves a diffused boundary whose effect on the flow field is incorporated as a volumetric force spread into the bulk fluid, typically within the distance of a few grid cells from the physical boundary [2, 3]. The volumetric force may be determined from the constitutive law in case of an elastic boundary [2, 4], or by a feedback mechanism in which the force depends on the difference between the interpolated velocity at the interface and the desired boundary condition [3]. The other type of the immersed-boundary method retains the singular representation of the physical boundary and thus the nature of the surface force exerted by the boundary on the adjacent fluid. This type of “sharp-interface” methods can typically achieve higher order of accuracy than the “diffuse-interface” methods. Several distinct sharp-interface approaches have been formulated in the past to treat the boundary conditions at the fluid–solid interface. For example, in the “cut-cell” approach [5], a finite-volume scheme is designed to represent the conservation equations for the irregular cells cut through by the boundary, whereas the bulk flow is discretized using the standard finite-difference method. In the method presented by Li and coworkers [6, 7], the solution experiences discontinuities across the physical interface immersed in the domain, and the finite-difference formulae involving the nodes across the interface are corrected by taking into consideration of the discontinuities. In another type of sharp-interface methods, an unknown forcing term is introduced only at the nodal points immediately next to the fluid–solid interface, whose direction and magnitude are such that the boundary conditions at the location of the fluid–solid interface are satisfied. The forcing does not have to be explicitly calculated but can be incorporated through a local flow field reconstruction around the forcing points. To reconstruct the flow locally, an interpolation scheme is applied, and the pressure and velocity information at the fluid–solid interface are included as input data in the scheme. Therefore, the boundary conditions at the interface are enforced through the interpolation, and actual evaluation of the forcing is never needed. Since there is no feedback iteration involved, this method is also termed “direct forcing” approach. Many existing implementations fall into this category [8–15]. In the direct-forcing approach, the construction of the interpolation stencil is flexible and may take several topological forms. Figure 1 shows some of the examples of the stencil. For simplicity, we only use a non-staggered grid for illustration. The interpolation points may be located either on the fluid side of the interface (Fig. 1(a,c)), or on the solid side (Fig. 1(b,d)). In the latter case, the values of the flow variables