Direct numerical simulation of equilibrium spatially localized structures in pipe flow

The possibility of intermittent turbulent shear flows’ description by means of nonstationary long-wave three-dimensional Navier–Stokes solutions is shown in principle. The classical problem of viscous incompressible fluid flows in a circular pipe at transitional Reynolds numbers 1800⩽Re⩽4000 is taken as a model one. By means of direct numerical simulation statistically stationary Navier–Stokes solutions which describe turbulent (at 2500⩽Re⩽4000), intermittent (at Re=2200,2350), and laminar (Re⩽2000) flow regimes are obtained. Numerical solutions at Re=2200,2350 describe equilibrium self-sustained flow regimes in which turbulent structures surrounded by almost laminar flow propagate downstream while preserving their length. Thus, theoretical confirmation of the existence of particular transitional flow regimes—equilibrium puffs—is achieved. The space-time structure of equilibrium puffs is examined with particular emphasis on flow visualization and calculation of propagation velocities. Basic turbulence statistics inside and outside the puff are computed and compared with the existing experimental data.