A New Higher-Order Super Compact Finite Difference Scheme to Study Three-Dimensional Non-Newtonian Flows

TL;DR

This work develops a higher-order super-compact (HOSC) finite difference scheme for three-dimensional non-Newtonian power-law fluids, achieving $4^{th}$-order spatial and $2^{nd}$-order temporal accuracy with a seven-point stencil at the new time level. The momentum equations are solved implicitly using a $(19,7)$ stencil, with pressure corrected by a modified artificial compressibility method and the linear systems tackled by Hybrid BiCGSTAB. Validation against established 3D benchmarks for $n \in \{0.5,1.0,1.5\}$ and $Re \in \{1,50,100,200\}$ shows excellent agreement, while results reveal how the power-law index $n$ and Reynolds number $Re$ influence vortex locations, velocity fields, viscosity distributions, and pressure contours. The approach provides a robust, efficient tool for accurate 3D non-Newtonian CFD and sets a benchmark for future studies in complex rheology-driven flows.

Abstract

This work introduces a new higher-order accurate super compact (HOSC) finite difference scheme for solving complex unsteady three-dimensional (3D) non-Newtonian fluid flow problems. As per the author's knowledge, the proposed scheme is the first ever developed finite difference scheme to solve three-dimensional non-Newtonian flow problem. Not only that, the proposed method is fourth-order accurate in space variables and second-order accurate in time. Also, the proposed scheme utilizes only seven directly adjacent grid points, at the $(n+1)^{th}$ time level, around which the finite difference discretization is made. The governing equations are solved using a time-marching methodology, and pressure is calculated using a pressure-correction strategy based on the modified artificial compressibility method. Using the power-law viscosity model, we tackle the benchmark problem of a 3D lid-driven cavity, systematically analyzing the varied rheological behavior of shear-thinning ($n=0.5$), shear-thickening $(n=1.5)$, and Newtonian $(n=1.0)$ fluids across different Reynolds numbers $(Re= 1, 50, 100, 200)$. Both Newtonian and non-Newtonian results are carefully investigated in terms of streamlines, velocity variation, pressure distributions, and viscosity contours, and the computed results are validated with the existing benchmark results. The findings demonstrate excellent agreement with the existing results. This extensive analysis, using the new HOSC scheme, not only increases our understanding of non-Newtonian fluid behavior but also provides a robust foundation for future research and practical applications.

Figures (10)

• Figure 1: (a) Illustration of the configuration in the 3D lid-driven cavity scenario and (b) View of the grids at a resolution of $81\times81\times81$.
• Figure 2: The super-compact unsteady stencil
• Figure 3: Comparison of the viscosity contours between present results and the results of Hatič et al. (2D) Hatič_2021 for shear-thinning ($n=0.5)$ and shear-thickening $(n=1.5)$ non-Newtonian fluids
• Figure 4: Comparison of the streamlines pattern between present result and benchmark result of Jin et al. (3D) Jin_2017 at $Re=100$ and $n=1.5$
• Figure 5: Comparison of $u$-velocity profiles between present results with some published results Neofytou_2005Jin_2017Thohura_2019 for shear-thinning ($n=0.5$) and Newtonian ($n=1.0$) fluids at $Re=100$