Citation Context

...onal fractional step algorithm, the entire IBM itself has not been regarded as a fractional step (projection) method, as reported here. We follow the algebraic approach of Perot [30], where the fractional step method is written as a block-LU decomposition. In the next section, we review the traditional fractional step method as it is the fundamental basis for our IBM. In Section 3, we introduce the immersed boundary projection method. This formulation is compared to previous methods in Section 4; namely the original IBM [31], the direct forcing method [28], the immersed interface method (IIM) [23], and the distributed Lagrange multiplier (DLM) method [16]. In Section 5, numerical examples are considered to assess the temporal and spatial accuracy of the current method. Flows over stationary and moving cylinders are simulated and results are compared to previous experimental and numerical studies. Section 6 summarizes the current formulation. Further details on the discretization of the immersed boundary projection method are placed in the Appendix.2. Fractional step method We consider the incompressible Navier–Stokes equationsou ot þ u ru rp þ 1 Re r2u; ð1Þ r u 0; ð2Þwhere u, ...

Citation Context

...f the interface (i.e. we assume that n × τ = k ≡ constant, so that n can point either towards, or outwards from, the center of curvature). The proof for expressions (8)-(11) can be found in detail in =-=[23, 24, 25, 21]-=-. We note that from expressions (8)-(11) the values of the jumps of the first and second derivatives of velocity and pressure with respect to the (x, y) coordinates are easily obtained by a simple coo...

Citation Context

... class. This method applies boundary conditions by introducing a body force into the equations [30] and thus eectively smearing the interface. In the immersed interface method and related approaches =-=[31,32]-=-, the interface is kept sharp with special discretization stencils applied where the grid meets the boundary. Fictitious domain methods [33,34] impose boundary conditions through the use of constraint...

Citation Context

...dvanced by many other researchers as well, for hyperbolic equations (e.g., [16], [17], [19], [20]), elliptic equations (e.g., [27], [38], [39]), and incompressible Navier-Stokes equations (e.g., [2], =-=[32]-=-). In this paper, we focus our attention on simple domains in two space dimensions where the use of a mapped coordinate system is natural and often easier to work with computationally than embedded bo...

Citation Context

...heir close-range interactions. For simulation methods requiring boundary-fitted meshes, such as arbitrary Lagrangian–Eulerian methods, the mesh movement and regeneration for the complicated flow geometry is challenging and computationally expensive. Immersed boundary or immersed interface methods can avoid this by representing the changing geometry on a fixed mesh. These methods account for the cell–fluid interactions by diffusing the membrane force to the nearby fluid domain mesh points [28] or by modifying the local discrete differential operators when their stencils cross the cell membrane [21]. Both cell–cell interactions and cellular flow in small blood vessels have been simulated by the immersed boundary method in two dimensions [4,3]. The immersed finite-element method [23] and the lattice-Boltzmann method [24] similarly model the cell membranes as added body forces and have been used for studying cell aggregation and blood rheology. These fixed-mesh methods avoid re-meshing, but still require either a fine volumetric mesh or adaptive meshing to resolve close interactions between cells [39]. The mesh-based Lagrangian–Eulerian and immersed boundary or interface methods start with...

Citation Context

...conserva28tion” [25], which is not implemented here, since most of our applications do not involve enclosed volumes. For other approaches that also lead to better conservation of volume, see [4] and =-=[16]-=-. 5 Applications 5.1 Flapping Filament in a Flowing Soap Film The first application is a flapping flexible filament in a flowing soap film. This replicates a simulation done by Luoding Zhu [27,28] by ...