A proper orthogonal decomposition (POD)-based nonlinear estimator for fluid flow velocity fields is developed, which is capable of achieving finite-time convergence of the Galerkin coefficient estimates. Using Galerkin projection and POD-based model reduction, the incompressible Navier-Stokes equations are recast as a set of nonlinear ordinary differential equations in the Galerkin coefficients. A sliding-mode-based observer is designed to estimate the unknown time-varying Galerkin coefficients. Convergence of the estimates is proven to be achieved in finite time, and high-fidelity numerical simulation results are provided to complement the theoretical development.