  In this work, the authors introduce a generalized weak Galerkin (gWG) finite
element method for the time-dependent Oseen equation. The generalized weak
Galerkin method is based on a new framework for approximating the gradient
operator. Both a semi-discrete and a fully-discrete numerical scheme are
developed and analyzed for their convergence, stability, and error estimates. A
generalized {\em{inf-sup}} condition is developed to assist the convergence
analysis. The backward Euler discretization is employed in the design of the
fully-discrete scheme. Error estimates of optimal order are established
mathematically, and they are validated numerically with some benchmark
examples.
