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

Abstract

Traffic congestion is a persistent challenge in urban transport systems, particularly around roundabouts where nonlinear merging and circulating flows create instability and delays. Traditional microscopic, mesoscopic, and macroscopic models often neglect explicit treatment of priority rules, capacity drops, and queue spillback, limiting their ability to predict roundabout-induced congestion. To address this gap, this study develops a coupled Navier–Stokes–advection–diffusion framework for simulating traffic dynamics at roundabouts. The governing equations are discretized using the finite volume method with a QUICK scheme for spatial accuracy and advanced in time with a Crank–Nicolson integration for stability. Non-dimensionalization ensures general applicability across different traffic environments. Simulation results reproduce fundamental traffic relations, identify optimal densities for maximum throughput, and reveal oscillatory stop-and-go waves consistent with empirical observations. Scenario analysis shows that increased diffusion enhances stability but reduces flow capacity, capturing real-world trade-offs between efficiency and control. The study concludes that embedding yield laws and circulating feedback into continuum equations provides a rigorous foundation for analyzing roundabout performance. Policy recommendations include adopting geometry- and flow-calibrated entry controls, while future research should incorporate stochastic demand and adaptive control strategies to refine predictive accuracy. This contribution provides both theoretical innovation and practical insights for sustainable traffic management.