Traffic Flow Modelling Around Roundabouts Using Navier–Stokes and Advection–Diffusion Equations

Abstract: Urban roundabouts are safer than signalised junctions yet remain prone to congestion under unbalanced demand and operational incidents. Existing microscopic and macroscopic approaches rarely capture, within a single framework, the tightly coupled evolution of traffic velocity and density needed for robust design and control. We address this gap with a coupled Navier–Stokes and Advection–Diffusion (NS–AD) model on an annular domain, adopting a barotropic closure and a conservative incident force . Convection is discretised with a QUICK/TVD flux (MUSCL with a van–Leer limiter), while diffusion, sources and viscosity are advanced by a semi-implicit Crank–Nicolson step, enabling high-resolution fronts without spurious oscillations over long horizons. Simulations reproduce two robust signatures: a persistent annular congestion ridge in density coincident with the circulating ring, and a co-located speed amplification that exhibits an upstream/downstream braking/acceleration asymmetry governed by . Probe histories display a negative phase slope during the transient, and a quasi-1D azimuthal reduction explains the observed structures and yields a falsifiable ridge-thickness law . Difference maps between incident and baseline scenarios isolate the causal footprint of the incident as outward mass migration to the ring and momentum gain along preferred circumferential paths. We conclude that a composite potential succinctly explains outward push, pressure support and viscous/diffusive regulation, while the numerical pairing QUICK + TVD with Crank–Nicolson provides a stable, accurate basis for design studies. For policy and practice, the results support rapid incident clearance near the central island, targeted entry metering on feeder approaches, and low-cost geometric/control provisions, such as short bypasses and advisory speeds, that reduce the effective barrier and add diffusion pathways. Future work should calibrate against field data using ridge width, phase slope, relaxation time and entry/exit counts, extend to multi-lane and multi-class settings, assimilate real-time data for model-predictive control, and couple to emissions and safety surrogates to assess broader impacts. Keywords: Advection–diffusion of density; Crank–Nicolson time integration; Incident (stalled-vehicle) modelling; Navier–Stokes traffic modelling; QUICK/TVD convective scheme; Roundabout congestion. Title: Traffic Flow Modelling Around Roundabouts Using Navier–Stokes and Advection–Diffusion Equations Author: Momanyi Mogire Krifix, Shichikha Maremwa, Kandie Joseph, Lucy Jerop Ngetich International Journal of Recent Research in Mathematics Computer Science and Information Technology ISSN 2350-1022 Vol. 12, Issue 1, April 2025 - September 2025 Page No: 104-115 Paper Publications Website: www.paperpublications.org Published Date: 29-September-2025 DOI: https://doi.org/10.5281/zenodo.17225615 Paper Download Link (Source) https://www.paperpublications.org/upload/book/Traffic%20Flow%20Modelling%20Around%20Roundabouts-29092025-7.pdf