Simulating Pulsatile Flows Through a Pipe Orifice by an Immersed-Boundary Method

A numerical study of the fundamental hydrodynamic effects in complex geometries is a challenging task when discretizing Navier‐Stokes equations in the vicinity of complex geometry boundaries. The use of boundary-fitted, structured or nonstructured grids can help to deal with this problem, although the numerical algorithms implementing such grids are usually inefficient in comparison to those using simple rectangular meshes. This disadvantage is particularly pronounced when simulating nonsteady incompressible flows if the Poisson equation for the pressure has to be solved at each time step. Iterative methods used for complex meshes have low convergence rates, especially for fine grids. On the other hand, very efficient and stable algorithms for solving Navier‐Stokes equations in rectangular domains have been developed. These algorithms use fast direct methods for solving the Poisson’s equation for the pressure @1#. These difficulties led to the development of various approaches that rely on formulating complex geometry flows on simple rectangular domains. One approach is based on the immersed-boundary ~IB! method as introduced by Peskin @2# during the early seventies. At present, immersed-boundary-based methods are considered to be a powerful tool for simulating complex flows. In the present study, we applied an immersed-boundary method for simulating timedependent flows through a pipe orifice. Immersed-boundary methods were originally used to reduce simulating complex geometry flows to those defined on simple ~rectangular! domains. To understand the basic method, consider a flow of an incompressible fluid around an obstacle V~ S is its boundary! placed onto a rectangular domain ~P!. The flow is governed by the Navier‐Stokes and incompressibility equations with the no-slip boundary condition on S. The fundamental idea behind IB methods is to describe a flow problem, defined in P2V ,b y solving the governing equations inside an entire rectangular P without an obstacle, which allows using simple rectangular meshes. To impose the no-slip condition on an obstacle surface S ~which becomes an internal surface for the rectangular domain where the problem is formulated!, a forcing term f ~an artificial body force! is added to the Navier‐Stokes equations as follows: ]u ]t 52~ u„!u2„ p1n„ 2 u1f. (1) The purpose of the forcing term in Eq. ~1! is to impose the no-slip boundary condition at the point xS which defines the immersed boundary S. Formally, the solution to Eq. ~1! is identical to that without the source term only if f[0 everywhere outside the obstacle. This requirement is very difficult to satisfy when implementing IB numerical schemes. Therefore, one can expect that only an approximate equivalency of the two solutions will be obtained. The peculiarity of IB methods is that the no-slip boundary condition is not imposed at the initial stage but instead is gradually attained during the time-advanced computing procedure. In other words, the obstacle’s boundary ‘‘gets built up’’ inside the surrounding fluid. For this reason, IB methods are sometimes called ‘‘virtual body’’ methods. Introducing an artificial force in ~1! is crucial for implementing immersed-boundary approaches. In addition, the boundary S does not coincide with the grid points of a rectangular mesh where the velocity values are computed. This means that in order to impose the no-slip boundary condition, numerical algorithms require that the node velocity values be interpolated onto the boundary points. Thus, the performance and effectiveness of any IB method depends on both the source force ~f! and the computation data exchange ~inter- and extrapolation! between the grid and the immersed ~virtual! boundary points.