Abstract
A volume penalization method for imposing homogeneous Neumann boundary conditions in advection–diffusion equations is presented. Thus complex geometries which even may vary in time can be treated efficiently using discretizations on a Cartesian grid. A mathematical analysis of the method is conducted first for the one-dimensional heat equation which yields estimates of the penalization error. The results are then confirmed numerically in one and two space dimensions. Simulations of two-dimensional incompressible flows with passive scalars using a classical Fourier pseudo-spectral method validate the approach for moving obstacles. The potential of the method for real world applications is illustrated by simulating a simplified dynamical mixer where for the fluid flow and the scalar transport no-slip and no-flux boundary conditions are imposed, respectively.
Highlights
Computational fluid dynamics (CFD) in complex geometries which may vary in time, a problem typically encountered in fluid–structure interaction problems, is still a major challenge and requires advanced numerical techniques
There exists a large variety of immersed boundary methods, for example Lagrangian multipliers [5], level-set methods [6], fictitious domain approaches and surface [7] and volume penalization approaches [8]
In this part we present numerical simulations of flow with passive scalar in a simplified dynamical mixing device
Summary
Computational fluid dynamics (CFD) in complex geometries which may vary in time, a problem typically encountered in fluid–structure interaction problems, is still a major challenge and requires advanced numerical techniques. See, e.g., [1], yield a well adapted discretization for a given geometry. For time varying geometries the grid generation becomes even more complex and for instance elliptic grid generation techniques [2,3] are necessary which further increase the computational effort. During the last decades immersed boundary methods gained ground and became an attractive alternative. There exists a large variety of immersed boundary methods, for example Lagrangian multipliers [5], level-set methods [6], fictitious domain approaches and surface [7] and volume penalization approaches [8]. For reviews we refer the reader to [9,10]
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