Characteristics of the Instability Developing in the Turbulent Flow in a Plane Channel

Abstract

Disturbance evolution in developed turbulent flows in a plane channel is numerically investigated at the Reynolds numbers up to Reτ = 586. Steady turbulent flows calculated on the basis of the solution of the incompressible Navier—Stokes equations are then used as the baseline flows in studying the disturbance development process. The values of the highest Lyapunov exponent (HLE) λ1 are found and the instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are determined. Under arbitrary initial conditions the regime of the exponential disturbance growth ~ exp(λ1t) is reached for a time Δt+ < 50. It is found that the HLE value increases with the Reynolds number from λ1+ ≈ 0.021 at Reτ = 180 to λ1+ ≈ 0.026 at Reτ = 586. The LLV exhibits itself in the form of time- and space-localized spots of highly intense pulsations, concentrated in the buffer layer region. The distributions of the r.m.s. intensities of the velocity and vorticity pulsations in the LLV are qualitatively similar with the corresponding distributions in the main flow with near-wall streaks artificially extracted from it. The difference is a large disturbance concentration in the vicinity of the buffer layer, y+ = 10–30, and a relatively high (about 80% higher) vorticity pulsations amplitude. Basing upon the energy spectra of the velocity and vorticity pulsations we determined the integral spatial scales of the structures in the LLV field. It is found that the LLV structures are on average half as wide and long as the corresponding structures in the main flow. The contributions of all the terms included in the expression for the production of the kinetic energy of disturbances are determined. It is shown that the process of disturbance development is essentially controlled by the main flow inhomogeneity and the occurrence of transverse motion in it. Neglecting these factors leads to a considerable underestimation of the disturbance growth rate. Contrariwise, the presence of near-wall streaks in the main flow does not play a considerable role in the LLV disturbance development. The artificial extraction of the streaks from the main flow field does not change the nature of disturbance growth.