The dynamics of an organ pipe's mouth region has been studied by numerical simulations. The investigations presented here were carried out by solving the compressible Navier-Stokes equations under suitable initial and boundary conditions using parts of the open source C++ toolbox OpenFOAM. The focus of the study is on the examination of the velocity field close to the jet. The components of the velocity field sampled at a cross-section spanning the mouth were analyzed by using methods of coarse-graining. It is shown that the dynamics of the sampled velocity field can be separated into two fractions, a velocity component mainly comprising the jet's flow velocity and a component that essentially carries particle velocity caused by back-propagating sound waves inside the resonator. The SPL-spectra calculated of the data sets of the numerical simulations are compared with the measurements on real organ pipes. The phase-space representations of sound pressure and particle velocity indicate the presence of nonlinearities in the organ pipes's mouth region. The numerical results are consistent with measurements on real organ pipes.