  Astrophysical turbulent flows display an intrinsically multi-scale nature,
making their numerical simulation and the subsequent analyses of simulated data
a complex problem. In particular, two fundamental steps in the study of
turbulent velocity fields are the Helmholtz-Hodge decomposition
(compressive+solenoidal; HHD) and the Reynolds decomposition (bulk+turbulent;
RD). These problems are relatively simple to perform numerically for
uniformly-sampled data, such as the one emerging from Eulerian, fix-grid
simulations; but their computation is remarkably more complex in the case of
non-uniformly sampled data, such as the one stemming from particle-based or
meshless simulations. In this paper, we describe, implement and test vortex-p,
a publicly available tool evolved from the vortex code, to perform both these
decompositions upon the velocity fields of particle-based simulations, either
from smoothed particle hydrodynamics (SPH), moving-mesh or meshless codes. The
algorithm relies on the creation of an ad-hoc adaptive mesh refinement (AMR)
set of grids, on which the input velocity field is represented. HHD is then
addressed by means of elliptic solvers, while for the RD we adapt an iterative,
multi-scale filter. We perform a series of idealised tests to assess the
accuracy, convergence and scaling of the code. Finally, we present some
applications of the code to various SPH and meshless finite-mass (MFM)
simulations of galaxy clusters performed with OpenGadget3, with different
resolutions and physics, to showcase the capabilities of the code.
