Waves and spatially localized structures in turbulent viscous fluid flows. Numerical results

Direct Navier–Stokes simulation of fully turbulent and intermittent viscous incompressible fluid flows in an infinite circular pipe is performed. Calculations were carried out at Reynolds numbers $1800\le\mathrm{Re}\le4000$, based on the mean velocity and pipe diameter $D=2R$. Numerical Navier–Stokes solutions obtained belong to the class of streamwise periodic solutions with large periods $\lambda_\mathrm{max}=16\pi R$. It is demonstrated that the most energetic Fourier components of velocity fluctuations correspond to very low nonzero longitudinal wavenumbers. The structure of turbulent and inter-mittent flows as well as associated wave-like motions are investigated. The possibility and accu-racy of the velocity field approximation by the superposition of travelling and standing waves is analysed. It is shown that the parameters of such representation (wave amplitudes, phase veloci-ties, the position of wave front) are strongly dependent on the inclusion of low longitudinal wavenumbers in the Navier–Stokes simulation. Numerical solutions at $\mathrm{Re}=2200,2350$ describe equilibrium self-sustained flow regimes in which turbulent structures (turbulent puffs) sur-rounded by almost laminar flow propagate downstream while preserving their length. Space-time structure of turbulent puffs is compared with the existing experimental data. Propagation velocity of turbulent puffs and turbulence statistics inside and outside the puff are calculated.