Paths of energy in turbulent channel flows

Abstract

The paper describes the energy fluxes simultaneously occurring in the space of scales and in the physical space of wall-turbulent flows. The unexpected behaviour of the energy fluxes consists of spiral-like paths in the combined physical/scale space where the controversial reverse energy cascade plays a central role. Two dynamical processes are identified as driving mechanisms for the fluxes, one in the near-wall region and a second one further away from the wall. The former, stronger, one is related to the dynamics involved in the near-wall turbulence regeneration cycle. The second suggests an outer self-sustaining mechanism which is asymptotically expected to take place in the eventual log layer and could explain the debated mixed inner/outer scaling of the near-wall statistics. The observed behaviour may have strong repercussions on both theoretical and modelling approaches to wall turbulence, as anticipated by a simple equation which is shown able to capture most of the rich dynamics of the shear-dominated region of the flow.

Cascades and wall-normal fluxes in turbulent channel flows

“…Two main features are apparent in the figure, namely a blob at detached scales in the overlap layer, 100 < Y + c < 0.2Re τ , and a thin layer over the r + y = 2Y + c − 30 plane of attached scales. The intersection of the detached scale-energy source with the r + y = 0 plane, see figure 3(b), was already described in Cimarelli et al (2013) at a lower Reynolds number. Actually, the maximum of scale-energy source, which cannot be seen in the global view of figure 3(a), is highlighted in the inset of figure 3(a).…”

“…The equation has already been studied in Cimarelli et al (2013) and Cimarelli et al (2015) in the reduced space of spanwise and streamwise scales and wall-normal distances. In addition complementing the available information, the purpose here was mainly to distinguish the transport processes in wall-normal scales from those taking 434 A. Cimarelli, E. De Angelis, J. Jiménez and C. M. Casciola place in the physical wall-normal direction, a distinction that could not be addressed in the previous conditions, given the constraint r y = 0 which here we relaxed.…”

“…To provide a more complete view, the four-fifths law in the form of a balance equation for second-order structure function, originally proposed by Hill (2002), was used by Marati, Casciola & Piva (2004) to address the energy transfer in both spatial and scale spaces for a turbulent channel flow. The multidimensional and directional description provided by this equation was exploited by Cimarelli, De Angelis & Casciola (2013) to understand the formation and sustainment of long and wide turbulent fluctuations. In that paper, a model for the energy cascade was also developed to account for the dual nature of the energy transfer consisting of forward and reverse cascades ascending from the wall.…”

“…In addition to the flux in the space of scales, the generalized Kolmogorov equation features the spatial flux Φ c = v * δu 2 + 2 δpδv /ρ − (ν/2)∂ δu 2 /∂Y c . Equation (2.2) has been analysed in the hyper-plane r y = 0 in Cimarelli et al (2013). This approach allowed us to identify the reverse energy transfer as a crucial mechanism characterizing wall turbulence, responsible for the formation of the commonly observed very long and wide velocity fluctuations in the outer region of the flow.…”

A stochastic extension of the explicit algebraic subgrid-scale models

“…The amount of backscatter in the Wang and Bergstrom model in the case of turbulent channel flow can be up to 20% of the forward scatter. There have been a number of recent studies on the physical and scale space distribution of energy using the generalized balance equation of the second-order structure function for turbulent channel flow 28,29 showing that the backscatter of energy is related to the cyclic self-sustaining mechanisms of wall-turbulence. In this view, backscatter of energy is considered as a crucial element and should be accounted for by the SGS models.…”