The self-preservation of homogeneous shear flow turbulence

An analysis of the equations governing homogeneous shear flow shows the possibility of solutions which are self-preserving at all scales of motion, and that these solutions are dependent on the initial conditions. The appropriate velocity scale is the one obtained from the turbulence kinetic energy, q²/2, while the length scale is the Taylor microscale, λ. Two cases of self-preserving flow are identified: one corresponding to constant mean shear, the other to a mean shear which is inversely proportional to time. For the first case (the only one considered in detail) the principal results of the postulated similarity are that λ is constant, while q² varies exponentially with time. The ratio of the turbulence energy production rate to its dissipation rate remains constant. It is also shown that the energy spectra scale over all wavenumbers with q² and λ, and that they have shapes determined by the initial conditions. The experimental evidence is generally consistent with the theory.