Unified numerical study of turbulent rheological flows in regular polygonal ducts: secondary flow mechanisms and design correlations

The study of turbulent flow in polygonal ducts holds significant relevance across industries, drawing the attention of fluid dynamicists over many decades. Despite substantial research on square and triangular ducts, the fluid dynamic correspondence with total sides ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> ) and flow behaviour indices ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ) across the larger rheological spectrum of pseudoplastic, Newtonian, and dilatant fluids remains underexplored. To address the literature gap, we numerically model the rheological turbulent flow in straight ducts with regular polygonal cross-sections, offering unified insights and advancing understanding of geometric and rheological influences in internal flows. We analyse the nature of secondary flow originating inside the polygonal ducts, characterised by oppositely directed vorticity zones and curvature in in-plane streamlines. Results show that the secondary velocity magnitudes peak at approximately 1% of the bulk velocity near corners and decay downstream, influenced by both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . Dilatant fluids exhibit higher secondary velocities near the inlet but faster spatial decay. The near-wall strain rate distributions indicate higher strain rates in dilatant fluids and thickened viscous sublayers in pseudoplastic fluids. However, the net boundary-layer-thickness at the fully-developed state is predominantly influenced by the flow behaviour index rather than duct geometry, with shear-thickening fluids exhibiting nearly 10% thicker layers than shear-thinning fluids under identical Reynolds numbers. The three-dimensional illustrations bring out the influences of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> on quasi-streamwise vortices, with increased three-dimensionality sustained in shear-thickening flows. Notably, vortex persistence drops by nearly 88% when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> increases from 3 to 10, highlighting the reduced role of corner-induced secondary flows as the cross-section approaches circularity. We propose new correlations to predict the entry length, centreline-to-bulk velocity ratios, and friction factors of the fluid flow with reasonable accuracy (10%–16%). The findings and proposed correlations contribute to efficient duct design, paving the way for future studies in complex geometries and advanced fluid systems.