  A novel reduced order model (ROM) for incompressible flows is developed by
performing a Galerkin projection based on a fully (space and time) discrete
full order model (FOM) formulation. This 'discretize-then-project' approach
requires no pressure stabilization technique (even though the pressure term is
present in the ROM) nor a boundary control technique (to impose the boundary
conditions at the ROM level). These are two main advantages compared to
existing approaches. The fully discrete FOM is obtained by a finite volume
discretization of the incompressible Navier-Stokes equations on a collocated
grid, with a forward Euler time discretization. Two variants of the time
discretization method, the inconsistent and consistent flux method, have been
investigated. The latter leads to divergence-free velocity fields, also on the
ROM level, whereas the velocity fields are only approximately divergence-free
in the former method. For both methods, accurate results have been obtained for
test cases with different types of boundary conditions: a lid-driven cavity and
an open-cavity (with an inlet and outlet). The ROM obtained with the consistent
flux method, having divergence-free velocity fields, is slightly more accurate
but also slightly more expensive to solve compared to the inconsistent flux
method. The speedup ratio of the ROM and FOM computation times is the highest
for the open cavity test case with the inconsistent flux method.
