Transport equation for the mean turbulent energy dissipation rate on the centreline of a fully developed channel flow

The transport equation for the mean turbulent energy dissipation rate ε along the centreline of a fully developed channel flow is derived by applying the limit at small separations to the two-point budget equation. Since the ratio of the isotropic energy dissipation rate to the mean turbulent energy dissipation rate εᵢₛₒ/ε is sufficiently close to 1 on the centreline, our main focus is on the isotropic form of the transport equation. It is found that the imbalance between the production of ε due to vortex stretching and the destruction of ε caused by the action of viscosity is governed by the diffusion of ε by the wall-normal velocity fluctuation. This imbalance is intrinsically different from the advection-driven imbalance in decaying-type flows, such as grid turbulence, jets and wakes. In effect, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative S1,1 and the destruction coefficient G of enstrophy in different flows, thus resulting in non-universal approaches of S1,1 towards a constant value as the Taylor microscale Reynolds number, R_λ , increases. For example, the approach is slower for the measured values of S1,1 along either the channel or pipe centreline than along the axis in the self-preserving region of a round jet. The data for S1,1 collected in different flows strongly suggest that, in each flow, the magnitude of S1,1 is bounded, the value being slightly larger than 0.5.