Direct numerical simulation of spatially localized structures and wave motions in turbulent shear flows. Numerical requirements

A new technique for integrating unsteady incompressible 3D Navier–Stokes equations is presented. Algorithm developed seems to be a fast, accurate, and robust tool for direct numerical simulation of both intermittent and fully developed turbulent flow regimes. The classical problem of viscous incompressible fluid flows in a circular pipe at transitional Reynolds numbers $1800\le\mathrm{Re}\le4000$ is taken as a model one. By means of direct numerical simulation statistically stationary Navier–Stokes solutions which describe turbulent (at $2500\le\mathrm{Re}\le4000$), intermittent (at $\mathrm{Re}=2200,2350$) and laminar ($\mathrm{Re}\le2000$) flow regimes are obtained. Turbulent flow regimes are characterized by complicated wave-like structure. To detect and analyze this structure mathematical model and algorithm must be specially tuned. Numerical solutions at $\mathrm{Re}=2200,2350$ 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. Accuracy and stability of the method are thoroughly tested.