On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow

Abstract

The asymptotic behaviour of large-scale velocity statistics in an homogeneous turbulent shear flow is investigated using direct numerical simulations (DNS) of the incompressible Navier–Stokes equations on a 5123 grid, and with viscous rapid distortion theory (RDT). We use a novel pseudo-spectral algorithm that allows us to set the initial value of the shear parameter in the range 3–30 without the shortcomings of previous numerical approaches. We find there is an explicit dependence of the early-time behaviour on the initial value of the shear parameter. Moreover, the long-time asymptotes of large-scale quantities such as the ratio of the turbulent kinetic energy production rate over dissipation rate, the Reynolds stress anisotropic tensor and the shear parameter itself depend sensitively on the initial value of the shear parameter over the range of Reynolds number we could achieve (26 ≤ Rλ ≤ 63) with the stringent resolution requirements that were satisfied. To gain further insight into the matter, we analyse the full viscous RDT. While inviscid RDT has received a great deal of attention, viscous RDT has not been fully analysed. Our motivation for considering viscous RDT is so that the energy dissipation rate enters the problem, enabling the shear parameter to be defined. We show asymptotic expansions for the short-time behaviour and numerically evaluate the integrals to determine the long-time prediction of viscous RDT. The results are in quantitative agreement with DNS for short times; however, at long times viscous RDT predicts the turbulent energy decays to zero. Through an analysis of the pressure–strain terms, we show that the nonlinear ‘slow’ terms are essential for rearranging turbulent energy from the streamwise direction to the mean shear direction, and this sustains the indefinite growth of the kinetic energy at long times. In effect, the nonlinear pressure–strain correlation maintains the three-dimensionality of the turbulence, countering the tendency of the mean shear to project the turbulence onto the two-dimensional plane of the mean-flow streamlines. We postulate that the predictions of viscous RDT at long times could be improved by introducing a model for the ‘slow’ pressure–strain term, along the lines of the Rotta model.