The scale energy budget utilizes a modified version of the classical Kolmogorov equation of wall turbulence to develop an evolution equation for the second order structure function [R. J. Hill, "Exact second-order structure-function relationships," J. Fluid Mech. 468, 317 (2002)]. This methodology allows for the simultaneous characterization of the energy cascade and spatial fluxes in turbulent shear flows across the entire physical domain as well as the range of scales. The present study utilizes this methodology to characterize the effects of Reynolds number on the balance of energy fluxes in turbulent channel flows. Direct numerical simulation data in the range Reτ =300-934 are compared to previously published results at Reτ=180 [N. Marati, C. M. CasciolaR. Piva, "Energy cascade and spatial fluxes in wall turbulence," J. Fluid Mech. 521, 191 (2004)]. The present results show no Reynolds number effects in the terms of the scale energy budget in either the viscous sublayer or buffer regions of the channel. In the logarithmic layer, the transfer of energy across scales clearly varies with Reynolds number, while the production of turbulent kinetic energy is not dependent on Reynolds number. An envelope of inverse energy cascade is quantified in the buffer region within which energy is transferred from small to larger scales. This envelope is observed in the range 6<y+<37, where all scales except the smallest scales display characteristics of an inverse energy cascade. The cross-over scale l+c , which indicates the transition between production dominated and scale transfer dominated regimes, increases with Reynolds number, implying a larger range of transfer dominated scales, before the dominant mechanism switches to production. At higher Reynolds numbers, two distinct regimes of l+c as a function of wall-normal location are observed, which was not captured at Reτ=180. The variations of l+c match the trends of the shear scale, which is a representation of the mean shear in the flow. Thus, this study demonstrates the utility and importance of the use of higher Reynolds number data in order to accurately characterize and understand the energy dynamics of various scales across the entire boundary layer.
Bibliographical noteFunding Information:
The authors wish to acknowledge Professor Robert Moser for providing the two higher Reynolds number DNS datasets and Dr. Nicoletta Marati for the initial development of the scale energy analysis. I.M. wishes to acknowledge the support of the Australian Research Council. This work was supported by the National Science Foundation through Grant No. CTS-0324898.