Analytical and Reynolds stress transport model results for periodically oscillated turbulent flows

Abstract

Turbulent pipe flows in which the bulk velocity is oscillated harmonically about a mean level (U b = U bo [1 + γ cos ω t]) are considered from both analytical and turbulence modelling perspectives. The analytical development is concerned with the response of the phase-averaged turbulent kinetic energy (k = k0 + k 1 cos(ω t + ψ)) to imposed unsteadiness. An approximate quasi-steady analysis is used to obtain an asymptotic low frequency limit for the modulation of the turbulent kinetic energy, k 1/γ k 0 → 1.75. The analysis is extended to examine the response of ∂k/∂t, the unsteady rate-of-change term of the k-transport equation. A Reynolds stress transport model (RSTM) of turbulence is compared with data for steady and periodic pipe flows and, in most cases, satisfactory agreement is demonstrated. When the model is run for periodic flow over a wide range of frequency it is found that a “resonant” response in k 1/γ k 0 occurs at mid-frequencies; at higher frequencies the RSTM shows the turbulence to become “frozen” at a mean condition (k 1/γ k 0→0). In the final stage of the study the RSTM is used to explore departures from the quasi-steady variation of ∂k/∂t, in particular identifying the onset of frozen behaviour in the balance of the k-equation at higher frequencies.