a1 Institute of Mechanics, Moscow State University, 1 Michurinsky prospect, 119899 Moscow, Russia
The four-dimensional (4D) incompressible Navier–Stokes equations are solved numerically for the plane channel geometry. The fourth spatial coordinate is introduced formally to be homogeneous and mathematically orthogonal to the others, similar to the spanwise coordinate. Exponential growth of small 4D perturbations superimposed onto 3D turbulent solutions was observed in the Reynolds number range from Re = 4000 to Re = 10000. The growth rate of small 4D perturbations expressed in wall units was found to be λ+4D = 0.016 independent of Reynolds number. Nonlinear evolution of 4D perturbations leads either to attenuation of turbulence and relaminarization or to establishment of a self-sustained 4D turbulent solution (4D turbulent flow). Both results on flow evolution were obtained at the lowest Reynolds number, depending on the grid resolution, pointing to the proximity of Re = 4000 as the critical Reynolds number for 4D turbulence. Self-sustained 4D turbulence appeared to be less intense compared with 3D turbulence in terms of mean wall friction, which is about 55% of that predicted by the empirical Dean law for turbulent channel flow at all Reynolds numbers considered. Thus, the law of resistance of 4D turbulent channel flow can be expressed as Cf = 0.04Re−0.25.
(Received February 22 2010)
(Revised February 24 2011)
(Accepted March 21 2011)
(Online publication May 04 2011)