  We present a novel second-order semi-implicit hybrid finite volume / finite
element (FV/FE) scheme for the numerical solution of the incompressible and
weakly compressible Navier-Stokes equations on moving unstructured meshes using
an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a
suitable splitting of the governing PDE into subsystems and employs staggered
grids, where the pressure is defined on the primal simplex mesh, while the
velocity and the remaining flow quantities are defined on an edge-based
staggered dual mesh. The key idea of the scheme is to discretize the nonlinear
convective and viscous terms using an explicit FV scheme that employs the
space-time divergence form of the governing equations on moving space-time
control volumes. For the convective terms, an ALE extension of the Ducros flux
on moving meshes is introduced, which is kinetic energy preserving and stable
in the energy norm when adding suitable numerical dissipation terms. Finally,
the pressure equation of the Navier-Stokes system is solved on the new mesh
configuration using a continuous FE method, with $\mathbb{P}_1$ Lagrange
elements.
  The ALE hybrid FV/FE method is applied to several incompressible test
problems ranging from non-hydrostatic free surface flows over a rising bubble
to flows over an oscillating cylinder and an oscillating ellipse. Via the
simulation of a circular explosion problem on a moving mesh, we show that the
scheme applied to the weakly compressible Navier-Stokes equations is able to
capture weak shock waves, rarefactions and moving contact discontinuities. We
show that our method is particularly efficient for the simulation of weakly
compressible flows in the low Mach number limit, compared to a fully explicit
ALE scheme
