Turbulent energy scale budget equations in a fully developed channel flow

Kolmogorov's equation, which relates second- and third-order moments of the velocity increment, provides a simple method for estimating the mean energy dissipation rate 〈ε〉 for homogeneous and isotropic turbulence. However, this equation is usually not verified in small to moderate Reynolds number flows. This is due partly to the lack of isotropy in either sheared or non-sheared flows, and, more importantly, to the influence, which is flow specific, of the inhomogeneous and anisotropic large scales. These shortcomings are examined in the context of the central region of a turbulent channel flow. In this case, we propose a generalized form of Kolmogorov's equation, which includes some additional terms reflecting the large-scale turbulent diffusion acting from the walls through to the centreline of the channel. For moderate Reynolds numbers, the mean turbulent energy transferred at a scale r also contains a large-scale contribution, reflecting the non-homogeneity of these scales. There is reasonable agreement between the new equation and hot-wire measurements in the central region of a fully developed channel flow.