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

TL;DR

This work presents a dimension‑independent refined meshless solution for non‑Newtonian natural convection using an RBF‑FD framework with polyharmonic splines augmented by monomials and pressure–velocity coupling via artificial compressibility. By employing a targeted, boundary‑layer–focused node refinement, the method achieves significant node‑level efficiency while preserving accuracy across 2D and 3D irregular domains, including porous filters. Results show good agreement with reference FVM data and empirical Nu correlations, and demonstrate the approach’s flexibility in handling complex geometries and high‑Ra, shear‑thinning flows. The study highlights the method’s potential for hp‑adaptivity and dimension‑independent code design, with practical implications for simulating complex non‑Newtonian convection problems.

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.

Figures (17)

• Figure 1: left: A schematic representation of the De Vahl Davis differentially heated cavity case with velocity and temperature boundary conditions. right: Relationship between viscosity and shear rate for a $n = 1$ Newtonian and a selection of shear-thinning $n < 1$ non-Newtonian cases.
• Figure 2: A comparison between node distributions in domains populated with a constant node density on the left, refined density in the middle and the obstructed domain with refined narrow channels on the right.
• Figure 3: Flow profiles for a selection of cases. Velocity magnitude is visualised with a heat-map while the overlaid contours display the changes in temperature. Each sub-figure has a distinct velocity range specified by the colourbar above.
• Figure 4: Cross-section of vertical velocity $v_y$ close to the right wall at $y = 0.5$ for a range of different Ra and $n$, calculated with $\mathrm{Pr}=100$ and $m=4$. The error bars show the locations and values of maximum vertical velocity in corresponding plots from a reference solution turan2011laminar.
• Figure 5: left: Time evolution of the average Nusselt number on the cold wall that is used as a scalar observable for the system's dynamics. right: Dimensionless values of the dynamic timestep throughout the simulation. Cases shown in this Figure use a constant discretisation density with internodal distance $h = 0.005$ corresponding to $N=35222$ computational nodes.