Abstract
Direct Navier–Stokes simulation of the fully developed turbulent viscous incompressible fluid flows in an infinite circular pipe is performed. Calculations were carried out at Reynolds number Re=Ū2R/ν=4000, based on the mean velocity and pipe diameter. Numerical Navier–Stokes solutions obtained belong to the class of streamwise periodic solutions with large periods λmax=16πR≂7460ν/uτ, where uτ is the friction velocity. It is demonstrated that the most energetic Fourier components of velocity fluctuations correspond to very low nonzero longitudinal wave number α=2π/(0.5λmax)=0.25R−1≂1.7×10−3uτ/ν. These long-wavelength motions were not resolved before in numerous channel and pipe flow simulations carried out with periodicity lengths λmax<2500ν/uτ. Space–time structure of the turbulent flow as well as associated wave-like motions are investigated. The possibility and accuracy of the velocity field approximation by the superposition of traveling and standing waves is analyzed. It is shown that the parameters of such a representation (wave amplitudes, phase velocities, the position of wave front, etc.) are strongly dependent on the inclusion of low longitudinal wave numbers in the Navier–Stokes simulation.
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