We introduce a widely applicable tensor network-based framework for developing reduced order models describing wall-bounded fluid flows. As a paradigmatic example, we consider the incompressible Navier-Stokes equations and the lid-driven cavity in two spatial dimensions. We benchmark our solution against published reference data for low Reynolds numbers and find excellent agreement. In addition, we investigate the short-time dynamics of the flow at high Reynolds numbers for the liddriven and doubly-driven cavities. We represent the velocity components by matrix product states and find that the bond dimension grows logarithmically with simulation time. The tensor network algorithm requires at most a few percent of the number of variables parameterizing the solution obtained by direct numerical simulation, and approximately improves the runtime by an order of magnitude compared to direct numerical simulation on similar hardware. Our approach is readily transferable to other flows, and paves the way towards quantum computational fluid dynamics in complex geometries
