Multizone shell model for turbulent wall bounded flows.

We suggested a multizone shell (MZS) model for wall-bounded flows accounting for the space inhomogeneity in a piecewise approximation, in which the cross-sectional area of the flow, S, is subdivided into j zones. The area of the first zone, responsible for the core of the flow, S1 approximately S/2, and the areas of the next j zones, S(j), decrease toward the wall like S(j) proportional, variant 2(-j). In each j zone the statistics of turbulence is assumed to be space homogeneous and is described by the set of shell velocities u(nj)(t) for turbulent fluctuations of the scale proportional to 2(-n). The MZS model includes a set of complex variables V(j)(t), j=1,2, em leader, infinity, describing the amplitudes of the near-wall coherent structures of the scale s(j) approximately 2(-j) and responsible for the mean velocity profile. The suggested MZS equations of motion for u(nj)(t) and V(j)(t) preserve the actual conservation laws (energy, mechanical, and angular momenta), respect the existing symmetries (including Galilean and scale invariance), and account for the type of nonlinearity in the Navier-Stokes equation, dimensional reasoning, etc. The MZS model qualitatively describes important characteristics of the wall-bounded turbulence, e.g., evolution of the mean velocity profile with increasing Reynolds number Re from the laminar profile toward the universal logarithmic profile near the flat-plane boundary layer as Re--> infinity.