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 𝒑 = 𝒂𝝆 and a conservative incident force 𝒌𝐨𝐛𝐬𝐫/∥ 𝐫 ∥. 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 −𝒂𝛁𝐥𝐧𝝆. Probe histories display a negative (|𝒗|, 𝝆) phase slope during the transient, and a quasi-1D azimuthal reduction explains the observed structures and yields a falsifiable ridge-thickness law 𝜹 ∼ 𝑫/𝒖. 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 𝒂𝐥𝐧𝝆+ 𝚽 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 𝒂𝐥𝐧𝝆 and add diffusion pathways. Future work should calibrate (𝒂, 𝑫, 𝝂, 𝑺(𝜽), 𝒌𝐨𝐛𝐬) 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.