Refined radial basis function-generated finite difference analysis of non-Newtonian natural convection

Abstract: In this paper we present a refined Radial Basis Function-generated Finite Difference (RBF-FD) solution for a non-Newtonian fluid in a closed differentially heated cavity. The non-Newtonian behaviour is modelled with the Ostwald-de Waele power law and the buoyancy with the Boussinesq approximation. The problem domain is discretised with scattered nodes without any requirement for a topological relation between them. This allows a trivial generalisation of the solution procedure to complex irregular three dimensional (3D) domains, which is also demonstrated by solving the problem in a two dimensional (2D) and 3D geometry mimicking a porous filter. The results in 2D are compared with two reference solutions that use the Finite volume method in a conjunction with two different stabilisation techniques, where we achieved good agreement with the reference data. The refinement is implemented on top of a dedicated meshless node positioning algorithm using piecewise linear node density function that ensures sufficient node density in the centre of the domain while maximising the node density in a boundary layer where the most intense dynamic is expected. The results show that with a refined approach, more than 5 times fewer nodes are required to obtain the results with the same accuracy compared to the regular discretisation. The paper also discusses the convergence with refined discretisation for different scenarios for up to $2 \cdot 10^5$ nodes, the impact of method parametres, the behaviour of the flow in the boundary layer, the behaviour of the viscosity and the geometric flexibility of the proposed solution procedure.