  This paper presents robust discontinuous Galerkin methods for the
incompressible Navier-Stokes equations on moving meshes. High-order accurate
arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework
for both monolithic as well as projection or splitting-type Navier-Stokes
solvers. The framework is flexible, allows implicit and explicit formulations
of the convective term, and adaptive time-stepping. The Navier-Stokes equations
with ALE transport term are solved on the deformed geometry storing one
instance of the mesh that is updated from one time step to the next.
Discretization in space is applied to the time discrete equations so that all
weak forms and mass matrices are evaluated at the end of the current time step.
This design ensures that the proposed formulations fulfill the geometric
conservation law automatically, as is shown theoretically and demonstrated
numerically by the example of the free-stream preservation test. We discuss the
peculiarities related to the imposition of boundary conditions in intermediate
steps of projection-type methods and the ingredients needed to preserve
high-order accuracy. We show numerically that the formulations proposed in this
work maintain the formal order of accuracy of the Navier-Stokes solvers.
Moreover, we demonstrate robustness and accuracy for under-resolved turbulent
flows.
