# Lognormality in Turbulence Energy Spectra

## Abstract

**:**

## 1. Introduction

^{−5/3}scaling will track the power spectra. We can see in Figure 1 that due to broadening of the energy spectrum with increasing total energy, or Reynolds number, there will be an increase in the wavenumber range where k

^{−5/3}scaling will be nearly tangent to the spectrum. Additionally, the ascending part of the spectra at low wavenumbers (largest eddies) is tracked by k

^{4}-scaling [6], which is tangent to lognormal function at this range. We can see that k

^{n}-type of scaling will be tangent to lognormal at least at some point and range of the wavenumber space. However, the range of its applicability tends to be limited as tangent lines do not fully express the curved energy spectrum.

^{−5/3}scaling [13,14], which is hard to achieve in typical maximum-entropy probability distribution functions (inverse exponential). If one looks at the data from a quiescent perspective, k

^{−n}type of scaling is only partially applicable over a small subset of the spectrum, and the overall spectra tend to be parabolic in the logarithmic scale.

## 2. The Maximum Entropy Principle

_{e}), is known from η = l

_{e}Re

_{λ}

^{−3/2}[6], where Re

_{λ}is the Reynolds number based on the Taylor microscale (λ). Another important constraint is the total turbulent kinetic energy, fixed by the initial or external conditions.

_{1}, C

_{2}are constants, to be determined from the constraints, and k the wavenumber. u′(k) is empirically input as (m-log(k)). The derivation is shown in the Appendix. This incidentally resembles the lognormal function, and has asymmetrical descent to zero energy at the boundary points. An alternate deductive method for arriving at the lognormal form is to test a sequence of standard distribution function (e.g., Gaussian, exponential, lognormal, etc.) with the maximum Shannon’s entropy that still obeys the physical constraints [2]. Then, the first simplest distribution that satisfy the constraints is the most likely one [2]. In this process, uniform, Gaussian and exponential distributions do not satisfy the asymmetric boundary conditions for turbulence energy spectra, thus pointing to the lognormal as the most likely distribution. Thus, it can be deduced that the distribution function that satisfies the boundary conditions is the lognormal function due to its asymmetric decay to zero at the boundary points. Energy spectra are asymmetrical because the descent toward zero energy occurs due to physical limit of the flow scale at the low wavenumber extreme, while viscous dissipation causes the approach toward zero at the high wavenumbers. For a similar reason, the drop size distributions in spray flows take on lognormal shape [15]. As noted above the width of the distribution can then be deduced from η = l

_{e}Re

_{λ}

^{−3/2}, while the height of the distribution is set by the total integrated turbulence kinetic energy, which is proportional to the mean velocity squared and the length scales of the flow. For example, atmospheric turbulence will have a very large total integrated energy and also the ratio of largest to the smallest (Kolmogorov) scales will be very large, both of which depend on the Reynolds number. Lognormal distribution has convenient parameterization aspects for these length scale effects, as shown later.

## 3. Results and Discussion

^{−5/3}scaling is also plotted (dashed line) for comparison, and we can see that for large Reynolds numbers this scaling is tangent to the lognormal distribution in the so-called inertial subrange. This is the region that contains a large portion of the total energy, and thus Kolmogorov scaling has been useful in prescribing the power spectra [16]. Figure 1 and Figure 2 show that quantitatively and qualitatively there is a close agreement between the lognormal distribution and observed turbulent energy spectra over almost the entire length scale range. The observed energy spectra end abruptly at the low wavenumber limit, corresponding to the length scale of the turbulence-generating object or process. Since the maximum entropy distribution has no knowledge of this process, it continues its downward path toward zero. To input the information concerning the turbulence generation, a truncated lognormal function can be used at the low wavelength limit.

^{−5/3}type of scaling is a reasonable approximation for power spectra in the energy-containing inertial range.

_{λ}, while σ increases. Thus, we find a least-square fit to the following functions for these parameters using the data in Table 2.

_{1}× Re

_{λ}

^{2}+ a

_{2}

_{1}× Re

_{λ}

^{−3/2}+ b

_{2}

_{1}× Re + c

_{2}

^{2}, dissipation, kinematic viscosity, and/or other length scales may fine-tune the above functions. However, here we only demonstrate that turbulence energy spectra are recoverable through the lognormal distribution function by using simple functions for A, μ and σ, with the Reynolds number as the sole parameter.

^{5})

^{1/4}.

^{−5/3}scaling in the inertial range extended by a triangular peak. There have also been function fits to estimate the power spectral form [23]. The current approach is based on the fundamental Second Law in the form of the maximum entropy principle, and avoids much of the mathematical complexities, to arrive at quite good agreements with data, as shown above. Figure 7 exhibits the comparison between the current lognormal form of the energy spectra with more recent data by Kang et al. [24]. We can see that the current lognormal distribution agrees quite well across a very large set of experimental data, and appears to have some universal applicability. As noted in the introduction, the lognormal energy spectra are arrived at either through a deductive process [2] or with a mathematical derivation (see Appendix A). In hindsight, the lognormal form could have been deduced since the energy spectra in turbulence is neither random or Gaussian, but asymmetric due to different physics involved at the low (energy generation) and high (viscous dissipation) wavenumbers and sectionally monotonic (it is highly improbable to have multiple peaks). Regardless of the route, lognormal distribution appears to be the natural selection for the energy distribution in turbulence due to asymmetric boundary conditions at extreme ends of the spectrum. In addition, the spectral parameters have intuitive dependence on the turbulence Reynolds number and length scales. It may be a good utilitarian example of the entropy science, toward a high-impact subject, turbulence, as the energy spectra have important implications in engineering and atmospheric science.

## 4. Conclusions

^{−5/3}in the inertial range and k

^{4}dependence in the large-eddy length scales. This approach makes it possible to reconstruct the turbulence energy spectra, using primarily the Reynolds number that determines the width and height of the lognormal distribution. There may be secondary parameters such as dissipation, kinematic viscosity, and lengths scales that can fine-tune the energy distribution, but the fundamental turbulence energy spectra exhibit lognormal behavior that can be prescribed by the Reynolds number, as stipulated by the known properties of the energy and length scales of turbulence.

## Funding

## Conflicts of Interest

## Nomenclature

A | multiplicative constant (Equation (1)) |

a_{1}, a_{2} | constants for A (Equation (1)) |

b_{1}, b_{2} | constants for μ (Equation (2)) |

c_{1}, c_{2} | constants in σ (Equation (3)) |

C_{1}, C_{2}, C_{3} | constants in the energy spectra equation (Equation (A4)) |

dV | spatial volume |

e_{o} | total energy at wavenumber k (Equation (A1)) |

E(k) | energy density at wavenumber k |

E_{11}, E_{22} | energy density for the longitudinal (11) and lateral (22) components (Figure 7). |

E′(k) | non-normalized energy density at wavenumber k |

E^{+}_{uu} | energy density normalized by the friction velocity |

F | objective function |

FWHM | full width at half maximum |

k | wavenumber |

k_{x} | wavenumber for the longitudinal spectra |

l_{e} | integral length scale |

M | length scale |

Re | Reynolds number |

Re_{λ} | Reynolds number based on the Taylor microscale |

S | Shannon’s entropy |

t | time |

t* | tU/M normalized time |

u′ | turbulence fluctuating velocity |

U | mean velocity |

t | time interval |

δt | time interval |

ε | dissipation |

η | Kolmogorov scale |

μ | logarithmic mean in the lognormal distribution |

ν | kinematic viscosity |

## Appendix A

^{2}, is dissipated by the viscous term involving kinematic viscosity, ν [6], and thus Equation (A1) represents the conservation of energy in the k-space. The above constraint can be transposed into the energy distribution using the Lagrange multiplier method [2]. The first step is to write the objective function F so that

_{1}, C

_{2}and C

_{3}(= C

_{2}ν) are so-called Lagrange multipliers, to be determined from other constraints. For example, C

_{1}is determined by integrating E(k) to equal the total energy content in the distribution. Converting dV = d(k

^{−3}) to dk basis, we obtain the following energy distribution.

_{1}, C

_{2}, and C

_{3}are determined from the constraints of the turbulence energy content, limiting length scales, and viscosity, respectively. C

_{1}is the multiplier to the energy distribution, so that C

_{1}increases for large turbulence energy content. For example, for large-scale high-speed flows, such as atmospheric flows, C

_{1}will be a very large number (see Figure 2). C

_{2}determines the width of the energy spectrum. The limiting length scales are the Kolmogorov dissipation length scale and the maximum length scale that exists in the flow, so that if either of these length scales are extended, then the energy spectrum will broaden. C

_{2}is the parameter that reflects this width or the length scale effect. C

_{3}is the viscous dissipation parameter, and depends strictly on the kinematic viscosity.

^{−1/3}is obtained in the inertial subrange. However, in the current maximum entropy formalism this is an unknown element or a lack of a piece of information. The maximum entropy principle gives the most probable energy distribution under the given physical constraints, but it does not produce unknown information. Thus, the missing pieces of information need to be supplied from observational data, and Equation (A4) provides a framework for testing various kinematic scaling for u′(k). Comparison with observational data can then be used to deduce the empirical form for u′(k) ~ (μ-log(k)). When this scaling is input in Equation (A4), we obtain a lognormal form in k.

## References

- Planck, M. Distribution of energy in the spectrum. Ann. Phys.
**1901**, 4, 553–560. [Google Scholar] [CrossRef] - Cover, T.; Thomas, J. Elements of Information Theory; John Wiley and Sons, Inc.: Hoboken, NJ, USA, 1991. [Google Scholar]
- Li, X.; Tankin, R.S. Derivation of droplet size distribution in sprays by using information theory. Combust. Sci. Technol.
**1988**, 60, 345–357. [Google Scholar] - Kolmogorov, N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech.
**1962**, 13, 82–85. [Google Scholar] [CrossRef] [Green Version] - Kraichnan, R.H. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech.
**1959**, 5, 497–543. [Google Scholar] [CrossRef] [Green Version] - Hinze, J.O. Turbulence, McGraw-Hill Series in Mechanical Engineering; McGraw-Hill: New York, NY, USA, 1975. [Google Scholar]
- Comte-Bellot, G.; Corrsin, S. Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated isotropic turbulence. J. Fluid Mech.
**1971**, 48, 273–337. [Google Scholar] [CrossRef] - Champagne, F.H.; Friehe, C.A.; La Rue, J.C.; Wyngaard, J.C. Flux measurements and fine-scale turbulent measurement in the surface layer over land. J. Atm. Sci.
**1977**, 34, 515–530. [Google Scholar] [CrossRef] [Green Version] - Pearson, B.; Fox-Kemper, B. Log-normal turbulence dissipation in global ocean models. Phys. Rev. Lett.
**2018**, 120, 094501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saddoughi, S.G.; Veeravalli, S.V. Local isotropy in turbulent boundary layers at high Reynolds numbers. J. Fluid Mech.
**1994**, 268, 333–372. [Google Scholar] [CrossRef] - Frisch, U. Turbulence; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Mouri, H.; Hori, A.; Takaoka, M. Large-scale lognormal fluctuations in turbulence velocity fields. Phys. Fluids
**2009**, 21, 065107. [Google Scholar] [CrossRef] [Green Version] - Brown, T.M. Information theory and the spectrum of isotropic turbulence. J. Phys. A.
**1982**, 15, 2285. [Google Scholar] [CrossRef] - Verkley, W.T.M.; Lynch, P. Energy and enstropy spectra of geostrophic turbulent flows derived from a maximum entropy principle. J. Atmos. Sci.
**2009**, 66, 2216. [Google Scholar] [CrossRef] - Bevensee, R.M. Maximum Entropy Solutions to Scientific Problems; Prentice Hall: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Lee, T.W. Quadratic formula for determining the drop size in pressure atomized sprays with and without swirl. Phys. Fluids
**2016**, 28, 063302. [Google Scholar] [CrossRef] - Tennekes, H.; Lumley, J.L. First Course in Turbulence; MIT Press: Cambridge, MA, USA, 1976. [Google Scholar]
- Uberoi, M.S.; Freymuth, P. Turbulence energy balance and spectra of the axisymmetric wake. Phys. Fluids
**1970**, 13, 2205. [Google Scholar] [CrossRef] - Sanborn, V.A.; Marshall, R.D. Local Isotropy in Wind Tunnel Turbulence; Rep. CER 65 UAS-RDM71; Colorado State University: Fort Collins, CO, USA, 1965. [Google Scholar]
- Tieleman, H.W. Viscous Region of Turbulent Boundary Layer; Rep. CER 67-68 HWT21; Colorado State University: Fort Collins, CO, USA, 1967. [Google Scholar]
- Coantic, M.; Favre, A. Activities in, and preliminary results of, air-sea interactions research at I.M.S.T. Adv. Geophys.
**1974**, 18, 391–405. [Google Scholar] - Moser, R.D.; Kim, J.; Mansour, N.N. Direct numerical simulation of turbulent channel flow up to Re τ = 590. Phys. Fluids
**1999**, 11, 943–945. [Google Scholar] [CrossRef] - Orszag, S. Analytical theories of turbulence. J. Fluid Mech.
**1970**, 41, 363. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Kang, H.; Stuart, C.; Meneveau, C. Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation. J. Fluid Mech.
**2003**, 480, 129–160. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Lognormal distribution (normalized by the peak value) plotted in a semi-logarithmic scale. The data are from the same references as in Figure 2.

**Figure 4.**Temporal decay of energy spectrum. Data are from Comte-Bellot and Corrsin [7], and t* normalized time, t* = tU/M. Local Reynolds numbers are 71.6 (

**circle**), 65.3 (

**diamond**) and 60.7 (

**square**).

**Figure 5.**Turbulence energy spectra at various distances from the wall for channel flows, for Rel = 180 (

**top left**), 395 (

**top right**) and 590 (

**bottom**). Lines are the lognormal distribution, compared with DNS data [21] with symbols (

**circle**, y + = 5;

**square**, y + = 10;

**diamond**, y + = 20;

**triangle up**, y + 30;

**triangle down**, y + = 180 or 300.

**Figure 7.**Comparison of the lognormal energy distribution with more recent data for E

_{11}(

**circle symbols**) and E

_{22}(

**diamond**), by Kang et al. [25]. Current theoretical lines are straight for E

_{11}and dashed for E

_{22}.

**Table 1.**Parameters of the energy spectra for data of Comte-Bellot and Corrsin [7].

t* | Re_{λ} | E_{max} | E_{total} | k(E_{max}) [cm^{−1}] | FWHM [cm^{−1}] |
---|---|---|---|---|---|

42 | 71.6 | 461.2 | 774.6 | 0.058 | 1.24 |

98 | 65.3 | 212.6 | 342.8 | 0.12 | 0.98 |

171 | 60.7 | 123.9 | 174.9 | 0.22 | 0.84 |

Re_{λ} | E_{max}/(εν^{5})^{1/4} | (kη)_{Emax} | Reference |
---|---|---|---|

37 | 130 | 0.0222 | Comte-Bellot and Corrsin [7] |

72 | 878 | 0.00439 | Comte-Bellot and Corrsin [7] |

308 | 53,100 | 0.0002084 | Uberoi and Freymuth [17] |

600 | 199,000 | 0.0000578 | Saddoughi and Veeravalli [9] |

850 | 312,900 | 0.000024 | Coantic and Favre [20] |

1500 | 1,446,000 | 0.0000179 | Saddoughi and Veeravalli [9] |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, T.
Lognormality in Turbulence Energy Spectra. *Entropy* **2020**, *22*, 669.
https://doi.org/10.3390/e22060669

**AMA Style**

Lee T.
Lognormality in Turbulence Energy Spectra. *Entropy*. 2020; 22(6):669.
https://doi.org/10.3390/e22060669

**Chicago/Turabian Style**

Lee, Taewoo.
2020. "Lognormality in Turbulence Energy Spectra" *Entropy* 22, no. 6: 669.
https://doi.org/10.3390/e22060669