  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 lid driven 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.
