  In this article we design a finite volume semi-implicit IMEX scheme for the
incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a
time discretization technique that separates explicit and implicit terms which
encompass both slow and fast scales. The finite volume approach for both
explicit and implicit terms allows to encode into the nonlinear flux the
velocity of displacement of the Chimera mesh via integration on moving cells.
To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta
schemes. These novel schemes are combined with a fractional-step method, thus
the governing equations are eventually solved using a projection method to
satisfy the divergence-free constraint of the velocity field. The implicit
discretization of the viscous terms allows the CFL-type stability condition for
the maximum admissible time step to be only defined by the relative fluid
velocity referred to the movement of the frame and not depending also on the
viscous eigenvalues. Communication between different grid blocks is enabled
through compact exchange of information from the fringe cells of one mesh block
to the field cells of the other block. Taking advantage of the continuity of
the solution and the definition of a minimal compact stencil, the numerical
solution of any system of differential equations is characterized by continuous
data extrapolation. Free-stream preservation property, i.e. compliance with the
Geometric Conservation Law (GCL), is respected. The accuracy and capabilities
of the new numerical schemes is proved through an extensive range of test
cases, demonstrating ability to solve relevant benchmarks in the field of
incompressible fluids.
