Cartesian Grid Method for Moderate-Reynolds-Number Flows Around Complex Moving Objects

Abstract

A method is presented to simulate the fluid dynamics of low- to moderate-Reynolds-number flows around solid objects with complex shapes on a fixed Cartesian grid. On grid points occupied by a solid object, the method forces the fluid motion to be equal to the motion of the solid object, whereas the boundary conditions on the solid‐fluid interface are enforced through a specific treatment of grid points close to the interface. The method is fairly easily implemented in both two and three dimensions, and a grid-refinement study shows that the method is globally close to second-order accurate. Steady and unsteady flow results over a cylinder and a sphere fixed in the grid show very good agreement with previous experimental and computational investigations. Computations of a cylinder moving in the grid with a surrounding fluid initially at rest produce results that compare very well with the fixed-cylinder results, thus demonstrating the validity of the method for moving geometry simulations. The method’s applicability to investigations of the unsteady aerodynamics of microscale flapping flight is demonstrated through simulations of the fluid dynamics of a flapping cantilever beam and a free-falling two-dimensional “leaf.” I. Introduction T HE motivation for the work presented in this paper is that recent advances in the research and development of microelectromechanical system technology have encouraged research into the potential construction of micro‐air vehicles (MAVs) at millimeter to centimeter scales. One of the many challenges associated with the design of MAVs is to develop a better understanding of aerodynamics at these small scales. MAVs operate at much smaller Reynolds numbers than traditional flying machines, which means that viscous effects are more dominant, forcing the development of new flight mechanisms for the efficient generation of lift and thrust. Nature has already mastered flight at these scales and gives a good indication of the challenges that lie ahead. Insects and birds achieve flight through a wide range of flapping wing motions that vary with animal size and specialization. This type of flight is characterized by a high degree of flow unsteadiness that is exploited for aerodynamic efficiency, thus making it very hard to analyze and mimic for the purpose of designing robotic MAVs. It is clear that computational fluid dynamics can be a very powerful tool in the effort to analyze the flow physics of flapping flight, but the very nature of the unsteady aerodynamics introduces complex moving boundaries. Usually, body-fitted deforming grids are used to simulate flows involving moving boundaries because this allows for a simple and accurate enforcement of the boundary conditions. Here one has the choice of using structured or unstructured grids (or a combination of both), but these approaches have some significant disadvantages for our application. With structured grids, usually only the simplest of geometries can be represented without running into grid quality issues unless a multiblock approach is used.