^{1,a)}, Elisabetta De Angelis

^{2}, Ellen K. Longmire

^{3}, Ivan Marusic

^{4}, Carlo M. Casciola

^{5}and Renzo Piva

^{5}

### Abstract

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 turbulentshear 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 *Re* _{τ} = 300–934 are compared to previously published results at *Re* _{τ} = 180 [N. Marati, C. M. Casciola, and R. 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 , 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 as a function of wall-normal location are observed, which was not captured at *Re* _{τ} = 180. The variations of 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.

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.

I. INTRODUCTION

II. METHODOLOGY

A. Description of DNS datasets

B. Two point scale energy budget

III. RESULTS AND DISCUSSION

A. Single point statistics of DNS datasets

B. Variation of scale energy with Reynolds number

C. Variation of terms of energy budget with Reynolds number

D. Dynamics of transfer terms: the inverse energy cascade

E. Cross-over scale between production and scale transfer terms

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 93.0
- Energy transfer
- 30.0
- Turbulent flows
- 18.0
- Viscosity
- 16.0
- Diffusion
- 9.0

## Figures

(a) Isolines of in the plane. (b) Isolines of in the plane. Solid isolines are from **Re590** and dashed isolines are from **Re934**. Numbers on the isolines indicate the value of that specific isoline. The heavy solid line is κ*y* ^{+}.

(a) Isolines of in the plane. (b) Isolines of in the plane. Solid isolines are from **Re590** and dashed isolines are from **Re934**. Numbers on the isolines indicate the value of that specific isoline. The heavy solid line is κ*y* ^{+}.

(a) Scale energy balance at *y* ^{+} = 10 (viscous sublayer/buffer region). (b) Individual contributions of terms from (a). −Π_{ e } is the effective production; −*T* _{ r } is the transfer in scale space; *E* _{ e } is the effective dissipation; filled symbols represent (Π_{ e } + *T* _{ r }). Dashed lines and filled circles are from **Re934**; solid lines and filled squares are from **Re590**; dashed-dotted-dotted lines and filled diamonds are from **Re300**; dashed-dotted lines and filled triangles are from **Re180**. −Π is the turbulent production; −*T* _{ c } is the transfer in physical space; *E* is the turbulent dissipation; *D* _{ r } and *D* _{ c } are the diffusion in r-space and physical space, respectively.

(a) Scale energy balance at *y* ^{+} = 10 (viscous sublayer/buffer region). (b) Individual contributions of terms from (a). −Π_{ e } is the effective production; −*T* _{ r } is the transfer in scale space; *E* _{ e } is the effective dissipation; filled symbols represent (Π_{ e } + *T* _{ r }). Dashed lines and filled circles are from **Re934**; solid lines and filled squares are from **Re590**; dashed-dotted-dotted lines and filled diamonds are from **Re300**; dashed-dotted lines and filled triangles are from **Re180**. −Π is the turbulent production; −*T* _{ c } is the transfer in physical space; *E* is the turbulent dissipation; *D* _{ r } and *D* _{ c } are the diffusion in r-space and physical space, respectively.

(a) Scale energy balance at *y* ^{+} = 100 (inner logarithmic region). (b) Individual contributions of terms from (a). Definitions of the various terms are the same as in Figure 2. The vertical dashed line in (a) represents the cross-over scale at *Re* _{τ} = 934.

(a) Scale energy balance at *y* ^{+} = 100 (inner logarithmic region). (b) Individual contributions of terms from (a). Definitions of the various terms are the same as in Figure 2. The vertical dashed line in (a) represents the cross-over scale at *Re* _{τ} = 934.

(a) Scale energy balance at *y* ^{+} = 250 (outer logarithmic region). (b) Scale energy balance at *y*/δ = 0.8 (outer region). Definitions of the various terms are the same as in Figure 2. Dashed lines and filled circles are from **Re934**; solid lines and filled squares are from **Re590**.

(a) Scale energy balance at *y* ^{+} = 250 (outer logarithmic region). (b) Scale energy balance at *y*/δ = 0.8 (outer region). Definitions of the various terms are the same as in Figure 2. Dashed lines and filled circles are from **Re934**; solid lines and filled squares are from **Re590**.

(a) Contour plot of *T* _{ r }(*r* ^{+},*y* ^{+}) (b) Contour plot of *T* _{ c }(*r* ^{+},*y* ^{+}). In both plots, the zero contour line is shown by the dashed-dotted line. Positive contours are shown with solid lines and negative contours are shown with dashed lines. The data shown in these plots are from **Re590**.

(a) Contour plot of *T* _{ r }(*r* ^{+},*y* ^{+}) (b) Contour plot of *T* _{ c }(*r* ^{+},*y* ^{+}). In both plots, the zero contour line is shown by the dashed-dotted line. Positive contours are shown with solid lines and negative contours are shown with dashed lines. The data shown in these plots are from **Re590**.

(a) Variation of cross-over scale with *y* ^{+} for all DNS datasets. (b) Variation of shear scale and with *y* ^{+} for Reynolds numbers in the range *Re* _{τ} = 300–2000. The straight line is the asymptotic prediction in the logarithmic layer κ*y* ^{+}. The symbols represent from the four Reynolds numbers rescaled by the same constant of order one chosen to match the shear scale for the largest available Reynolds number **Re934**.

(a) Variation of cross-over scale with *y* ^{+} for all DNS datasets. (b) Variation of shear scale and with *y* ^{+} for Reynolds numbers in the range *Re* _{τ} = 300–2000. The straight line is the asymptotic prediction in the logarithmic layer κ*y* ^{+}. The symbols represent from the four Reynolds numbers rescaled by the same constant of order one chosen to match the shear scale for the largest available Reynolds number **Re934**.

## Tables

Parameters of the DNS datasets. Details of the datasets are available in **Re180**;^{2} **Re300**;^{13} **Re590**;^{8} **Re934**.^{14} The wall-normal spacings Δ*y* ^{+} indicated are at the center of the channel.

Parameters of the DNS datasets. Details of the datasets are available in **Re180**;^{2} **Re300**;^{13} **Re590**;^{8} **Re934**.^{14} The wall-normal spacings Δ*y* ^{+} indicated are at the center of the channel.

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