  This work presents a novel reduced-order model (ROM) for the incompressible
Navier-Stokes equations with time-dependent boundary conditions. This ROM is
velocity-only, i.e. the simulation of the velocity does not require the
computation of the pressure, and preserves the structure of the kinetic energy
evolution. The key ingredient of the novel ROM is a decomposition of the
velocity into a field with homogeneous boundary conditions and a lifting
function that satisfies the mass equation with the prescribed inhomogeneous
boundary conditions. This decomposition is inspired by the Helmholtz-Hodge
decomposition and exhibits orthogonality of the two components. This
orthogonality is crucial to preserve the structure of the kinetic energy
evolution. To make the evaluation of the lifting function efficient, we propose
a novel method that involves an explicit approximation of the boundary
conditions with POD modes, while preserving the orthogonality of the velocity
decomposition and thus the structure of the kinetic energy evolution. We show
that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM,
i.e., a ROM that simulates both velocity and pressure. This equivalence can be
generalized to other existing velocity-pressure ROMs and reveals valuable
insights in their behaviour. Numerical experiments on test cases with
inflow-outflow boundary conditions confirm the correctness and efficiency of
the new ROM, and the equivalence with the velocity-pressure formulation.
