^{1}and J. C. Vassilicos

^{2}

### Abstract

The dimensionless dissipation rate constant of homogeneous isotropic turbulence is such that where is a dimensionless function of which tends to 0.26 (by extrapolation) in the limit where (as opposed to just ) if the assumption is made that a finite such limit exists. The dimensionless number reflects the number of large-scale eddies and is therefore nonuniversal. The nonuniversal asymptotic values of stem, therefore, from its universal dependence on . The Reynolds number dependence of at values of close to and not much larger than 1 is primarily governed by the slow growth (with Reynolds number) of the range of viscous scales of the turbulence. An eventual Reynolds number independence of can be achieved, in principle, by an eventual balance between this slow growth and the increasing non-Gaussianity of the small scales. The turbulence is characterized by five length-scales in the following order of increasing magnitude: the Kolmogorov microscale , the inner cutoff scale , the Taylor microscale , the voids length scale , and the integral length scale .

We are grateful to Christophe Baudet, Yves Malecot, and Yves Gagne for providing us with their data.

I. INTRODUCTION

II. THE DISSIPATION CONSTANT IS PROPORTIONAL TO THE THIRD POWER OF THE NUMBER OF LARGE-SCALE EDDIES:

III. MAIN CONCLUSIONS

### Key Topics

- Turbulent flows
- 47.0
- Reynolds stress modeling
- 40.0
- Data sets
- 8.0
- Eddies
- 7.0
- Fractals
- 7.0

## Figures

Plots of (a) vs (to the left) and of (b) vs (to the right) for longitudinal velocity fluctuations obtained on the centerline of four different turbulent flows: a round air jet with nozzle diameter (streamwise distance from the nozzle , but we have confirmed the power law with data from , , , , , and ); turbulence generated by a classical grid in a square wind tunnel (mean inlet velocity and streamwise distance from grid , but we have confirmed the power law with data from and and mean inlet velocities 9 and ); turbulence generated by a fractal I grid (streamwise distance from grid , but we have confirmed the power law with data from and ); and “chunk” turbulence in the S1 wind tunnel of Modane (, but we have confirmed the power law with the other two Modane data sets). Summary descriptions of these data can be found in Tables I–III.

Plots of (a) vs (to the left) and of (b) vs (to the right) for longitudinal velocity fluctuations obtained on the centerline of four different turbulent flows: a round air jet with nozzle diameter (streamwise distance from the nozzle , but we have confirmed the power law with data from , , , , , and ); turbulence generated by a classical grid in a square wind tunnel (mean inlet velocity and streamwise distance from grid , but we have confirmed the power law with data from and and mean inlet velocities 9 and ); turbulence generated by a fractal I grid (streamwise distance from grid , but we have confirmed the power law with data from and ); and “chunk” turbulence in the S1 wind tunnel of Modane (, but we have confirmed the power law with the other two Modane data sets). Summary descriptions of these data can be found in Tables I–III.

Lin-log plots of , , , , and as functions of . To calculate we use . To calculate we use with where is estimated as described in the main text and is calculated from with . The constant .

Lin-log plots of , , , , and as functions of . To calculate we use . To calculate we use with where is estimated as described in the main text and is calculated from with . The constant .

Lin-log plots of and of as functions of . The dotted line in the plot of vs is . Explanations for how and are calculated are given in the main text and the caption of Fig. 2.

Lin-log plots of and of as functions of . The dotted line in the plot of vs is . Explanations for how and are calculated are given in the main text and the caption of Fig. 2.

Plots of vs and of as a function of . Explanations for how and are calculated are given in the main text and the caption of Fig. 2. The dotted line in the top plot is and .

Plots of vs and of as a function of . Explanations for how and are calculated are given in the main text and the caption of Fig. 2. The dotted line in the top plot is and .

Plots of vs , vs , vs and vs . Explanations for how , and are calculated are given in the main text and the caption of Fig. 2. The dotted line in the second plot is and . The dotted curve in the fourth plot is .

Plots of vs , vs , vs and vs . Explanations for how , and are calculated are given in the main text and the caption of Fig. 2. The dotted line in the second plot is and . The dotted curve in the fourth plot is .

Log-log plots of and vs and lin-log plot of vs .

Log-log plots of and vs and lin-log plot of vs .

## Tables

Round air jet data used here. They were obtained by hot-wire anemometry from the jet’s centerline by Mazellier (Ref. 17) (first line) and by Naert and Baudet (Ref. 18) (second line). In both cases, the hot wire diameter was of , the sensing length to diameter ratio was 120 for Mazellier and 330 for Naert and Baudet, and spatial velocity fluctuations were derived using the local Taylor hypothesis as described in Kahalerras *et al.* (Ref. 19). The spatial hot-wire resolution ranges between at and at in the case of Mazellier’s (Ref. 17) data and is 7.7 and , respectively, for the data of Naert and Baudet (Ref. 18). The frequency resolution ranges between 7.2 and 39.3 Kolmogorov frequencies from to in the case of Mazellier’s (Ref. 17) data and is 3.8 and 5.7 Kolmogorov frequencies, respectively, for the data of Naert and Baudet (Ref. 18).

Round air jet data used here. They were obtained by hot-wire anemometry from the jet’s centerline by Mazellier (Ref. 17) (first line) and by Naert and Baudet (Ref. 18) (second line). In both cases, the hot wire diameter was of , the sensing length to diameter ratio was 120 for Mazellier and 330 for Naert and Baudet, and spatial velocity fluctuations were derived using the local Taylor hypothesis as described in Kahalerras *et al.* (Ref. 19). The spatial hot-wire resolution ranges between at and at in the case of Mazellier’s (Ref. 17) data and is 7.7 and , respectively, for the data of Naert and Baudet (Ref. 18). The frequency resolution ranges between 7.2 and 39.3 Kolmogorov frequencies from to in the case of Mazellier’s (Ref. 17) data and is 3.8 and 5.7 Kolmogorov frequencies, respectively, for the data of Naert and Baudet (Ref. 18).

(i) Wind tunnel grid-generated (except in Modane) turbulence data used here. They were obtained by hot wire anemometry from the tunnel’s centerline by Mazellier (Ref. 17) with a classical grid (first line), by the current authors with a classical grid (second line) and a fractal I grid (Ref. 16) corresponding to Fig. 17e in Hurst and Vassilicos (Ref. 16) (fourth line), by Hurst and Vassilicos (Ref. 16) with a fractal cross grid corresponding to Fig. 3(c) in their paper (third line) and by Malecot and Gagne (Refs. 19 and 20) in wind tunnel S1 at Modane (fifth line). The Modane data are of so-called “chunk” turbulence (Refs. 19 and 20) which is not grid-generated. The section size and length are the wind tunnel test section's. The mesh size is the mesh size of the classical grid (square bars) or the effective mesh size of the fractal grid (Ref. 16). The blockade ratio is that of the grid. We used the same hot wire as Hurst and Vassilicos (Ref. 16), i.e., wire diameter and sensing length to diameter ratio of 200. Mazellier (Ref. 17) and Malecot and Gagne (Refs. 19 and 20) used a diameter wire with a sensing length to diameter ratio of 120. The spatial hot-wire resolution ranges between 1.3 and for Mazellier’s (Ref. 17) data, between 2.1 and for the data of Hurst and Vassilicos (Ref. 16) between 1.5 and for our data and is about for the Modane data. The frequency resolution varies from 5 to 90 Kolmogorov frequencies for Mazellier’s (Ref. 17) from 3 to 5 Kolmogorov frequencies for the data of Hurst and Vassilicos (Ref. 16) and is around 2 Kolmogorov frequencies for the Modane data. For our data, it ranges between 5 and 100 Kolmogorov frequencies. We used a DISA 55M10 anemometer which has a frequency response with the wire we used, well above the highest resolvable frequencies for our wire and flow speeds. Our acquisition card is NI9215 (USB NI Compact DAQ) with resolution. Our signal to noise ratio ranges between 36 and . Our calibration procedure was made using a Pitot tube before and after each run and King’s law was used for the conversion from voltage to velocity. We also monitored the temperature during measurements to make sure that no compensation for temperature drift was needed. The local Taylor hypothesis described in Kahaleras *et al.* (Ref. 19) was used to derive spatial velocity fluctuations in all the flows of this table.

(i) Wind tunnel grid-generated (except in Modane) turbulence data used here. They were obtained by hot wire anemometry from the tunnel’s centerline by Mazellier (Ref. 17) with a classical grid (first line), by the current authors with a classical grid (second line) and a fractal I grid (Ref. 16) corresponding to Fig. 17e in Hurst and Vassilicos (Ref. 16) (fourth line), by Hurst and Vassilicos (Ref. 16) with a fractal cross grid corresponding to Fig. 3(c) in their paper (third line) and by Malecot and Gagne (Refs. 19 and 20) in wind tunnel S1 at Modane (fifth line). The Modane data are of so-called “chunk” turbulence (Refs. 19 and 20) which is not grid-generated. The section size and length are the wind tunnel test section's. The mesh size is the mesh size of the classical grid (square bars) or the effective mesh size of the fractal grid (Ref. 16). The blockade ratio is that of the grid. We used the same hot wire as Hurst and Vassilicos (Ref. 16), i.e., wire diameter and sensing length to diameter ratio of 200. Mazellier (Ref. 17) and Malecot and Gagne (Refs. 19 and 20) used a diameter wire with a sensing length to diameter ratio of 120. The spatial hot-wire resolution ranges between 1.3 and for Mazellier’s (Ref. 17) data, between 2.1 and for the data of Hurst and Vassilicos (Ref. 16) between 1.5 and for our data and is about for the Modane data. The frequency resolution varies from 5 to 90 Kolmogorov frequencies for Mazellier’s (Ref. 17) from 3 to 5 Kolmogorov frequencies for the data of Hurst and Vassilicos (Ref. 16) and is around 2 Kolmogorov frequencies for the Modane data. For our data, it ranges between 5 and 100 Kolmogorov frequencies. We used a DISA 55M10 anemometer which has a frequency response with the wire we used, well above the highest resolvable frequencies for our wire and flow speeds. Our acquisition card is NI9215 (USB NI Compact DAQ) with resolution. Our signal to noise ratio ranges between 36 and . Our calibration procedure was made using a Pitot tube before and after each run and King’s law was used for the conversion from voltage to velocity. We also monitored the temperature during measurements to make sure that no compensation for temperature drift was needed. The local Taylor hypothesis described in Kahaleras *et al.* (Ref. 19) was used to derive spatial velocity fluctuations in all the flows of this table.

Wind tunnel turbulence data used here, presented in the same order as in Table II. See caption of Table II for details.

Wind tunnel turbulence data used here, presented in the same order as in Table II. See caption of Table II for details.

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