Simulation of incompressible flows across moving bodies using meshless finite differencing
Authors: C. S. Chew, K. S. Yeo, C. Shu
Venue: Proceedings of the international conference on Computational methods in sciences and engineering (2003)

Computational fluid dynamics (CFD) is increasingly becoming the tool of choice for researchers and engineers facing complex fluid flow problems. One area of CFD that has attracted significant attention in recent years is the development of meshless or mesh-free methods[1]. These methods have so far been used in many applications like fracture mechanics, astrophysics and fluid flow. The key advantage of meshless methods over mesh-based methods like finite volume (FV) and finite element (FE) methods is its non-requirement of pre-specified connectivity between nodes in order to derive approximation and interpolation. This leads to the ability of finite difference (FD) methods to treat irregular domains and boundaries. A direct result of this is another advantage of mesh-free methods having the adaptability for problems involving large movement or wholesale motion of boundaries or embedded bodies. In this area, it has the edge over composite or overset grid methods in being able to use only one coordinate frame.A convecting meshless scheme for boundary driven fluid flow is described. This scheme can be applied to self-propulsion problems, or, in general, flows involving motion of rigid or deformable bodies embedded within the flow.The generalized finite difference method[2] (GFD) was chosen this scheme. A meshless cloud of nodes is embedded around the body surface along with the structured Cartesian background nodes for the entire flow field. By incorporating a subdomain for each node and its neighbours, the use of Taylor series expansion enables the derivatives to be found. Two forms of incompressible momentum equations are used: the standard Eulerian form in primitive variables for the structured nodes and the Arbitrary Lagrangian-Eulerian (ALE) form for the meshless cloud of nodes. GFD with moving least squares approximation is used for spatial discretization of the convecting meshless nodes. The projection method with Crank-Nicolson discretization is used to achieve second order accurate time discretization. Following prescribed body motion or fluid structure interaction, the meshless cloud acquire new locations in the next time step and the momentum equations in ALE form can then be solved.Test cases were performed to verify code usability and method accuracy and convergence. The simulation of the decaying vortex has been a popular benchmark for the gauging of accuracy. Basically the exact solutions of the time-dependent function values are known and both residues and absolute errors were calculated. Solving using different grid sizes proved the present method to be of second order spatial accuracy. The meshless method was then tried on the driven cavity flow at Reynolds number 1000. The results were well-compared with popular benchmark results[4]. In considering the effects of implementing the convecting meshless nodes across a normal flow field, the decaying vortex was again simulated, with a moving patch of nodes within the Cartesian field. The residues were well controlled and the errors were not significantly increased. Finally, a couple of external flows cases were simulated, basically on a flow past a stationary bluff body and a flapping ellipse inside an enclosed body of fluid.Future works in the pipeline include writing a parallel code for larger scale simulation and simulating oscillating cylinders and flapping of elliptic aerofoil(s).