On the incorporation of obstacles in a fluid flow problem using a Navier–Stokes–Brinkman penalization approach

Abstract

Simulating the interaction of fluids with immersed moving solids is playing an important role for gaining a better quantitative understanding of how fluid dynamics is altered by the presence of obstacles and, vice versa, which forces are exerted on the solids by the moving fluid. Such problems appear in various contexts, ranging from numerous technical applications such as e.g. turbines to medical problems such as the regulation of cardiovascular hemodynamics by valves. Typically, the numerical treatment of such problems is posed within a fluid structure interaction (FSI) framework. General FSI models are able to capture bidirectional interactions, but are challenging to solve and computationally expensive. Simplified methods offer a possible remedy by achieving better computational efficiency to broaden the scope to demanding application problems with focus on understanding the effect of solids on altering fluid dynamics. In this study we report on the development of a novel method for such applications. In our method rigid moving obstacles are incorporated in a fluid dynamics context using concepts from porous media theory. Based on the Navier–Stokes–Brinkman equations which augments the Navier–Stokes equation with a Darcy drag term our method represents solid obstacles as time-varying regions containing a porous medium of vanishing permeability. Numerical stabilization and turbulence modeling is dealt with by using a residual based variational multiscale (RBVMS) formulation. The additional Darcy drag term and its respective stabilization are easily accommodated in any existing finite-element based Navier–Stokes solver. The key advantages of our approach – computational efficiency and ease of implementation – are demonstrated by solving a standard benchmark problem of a rotating blood pump posed by the Food and Drug Administration Agency (FDA). Validity is demonstrated by conducting a mesh convergence study and by comparison against the extensive set of experimental data provided for this benchmark.