Energy loss and developing length during reciprocating flow in a pipe with a free-end

Direct numerical simulations of reciprocating pipe flow in a straight pipe with a free-end are presented. The range of amplitudes and frequencies studied span the laminar regime and the beginning of transition toward a conditionally turbulent flow. Two primary results are reported: the measurement of the flow development length and the loss of energy, both due to the presence of the free-end. Two regimes of flow are identified with distinct length scales. For low frequencies, the development length scales with the pipe diameter D. However, for higher frequencies, the development length scales with the Stokes layer thickness δ=2ν/ω√. The energy loss is studied by calculating the viscous dissipation function, indicating where energy is lost, and allowing the energy lost due to the presence of the free-end to be isolated. While strong vortices are formed and convected from the exit, most of the energy they dissipate is lost within a few pipe diameters of the exit. It is shown that these trends continue even as the amplitude and frequency are increased so that the flow begins to transition from a laminar toward a turbulent state. Two modes of instability are observed in the Stokes layers near the free-end, one short wavelength mode with a wavelength set by the Stokes layer thickness and the other long wavelength mode with a wavelength set by the amplitude of the oscillatory flow. These modes are related to those observed in the fully developed oscillatory flow.