# Statistical Descriptions of Inhomogeneous Anisotropic Turbulence

## Abstract

**:**

## 1. Introduction

^{2}/s] is the kinematic viscosity of the fluid and L [m] is the spatial dimension of the flow configuration, e.g., the diameter of a tube, length of an air foil, or height above the earth’s surface. Values for $\nu $ in the cases of water and air are typically 10${}^{-6}$ and 10${}^{-5}$ m

^{2}/s, with the corresponding velocities 0.1 and 1 m/s. A configuration where $L=0.1$ m results in a value of $Re$ of ${10}^{5}$. This exceeds, by far, the critical value of approximately ${10}^{3}$, where turbulence starts to occur.

## 2. Langevin Equation including Kolmogorov Similarity

**x**is used to denote a fixed position in the non-moving fixed coordinate system, while $\mathbf{y}\left(t\right)$ is the position of a moving particle. The turbulent flow field is considered to be stationary in a fixed frame of reference. Statistical averages of Eulerian flow variables can be calculated by time averaging, which is indicated by angled brackets or superscript ${}^{0}$. The white-noise amplitude can be specified by implementing the Lagrangian version of Kolmogorov’s similarity theory of 1941, also referred to as K-41 theory: [13] and [1] Section 21.3. This yields

## 3. Specification of Damping Function by ${\mathbf{C}}_{\mathbf{0}}^{-\mathbf{1}}$-Expansion

## 4. Higher-Order Formulation of the Langevin Equation

## 5. The Diffusion Limit

## 6. Statistical Descriptions of Momentum Flux

## 7. Statistical Descriptions of Scalar Flux

## 8. Decaying Grid Turbulence

## 9. Turbulent Channel Flow

#### 9.1. Exact Results

#### 9.2. Results from the ${C}_{0}^{-1}$-Expansion

#### 9.3. Comparison with the DNS Results

#### 9.3.1. Statistical Values of Fluctuations

#### 9.3.2. Statistical Values of Turbulent Fluxes

#### 9.3.3. Diffusion Coefficients

#### 9.3.4. Kolmogorov Constant

## 10. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Monin, A.S.; Yaglom, A.M. Statistical Fluid Mechanics; Dover: New York, NY, USA, 2007; Volumes 1–2. [Google Scholar]
- Bernard, P.S.; Wallace, J.K. Turbulent Flow: Analysis, Measurement and Prediction; Wiley: New Jersey, NJ, USA, 2002. [Google Scholar]
- Hanjalić, K.; Launder, B. Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar] [CrossRef]
- Stratonovich, R.L. Topics in the Theory of Random Noise; Gordon and Breach: New York, NY, USA, 1967; Volume 1. [Google Scholar]
- van Kampen, N.G. Stochastic Processes in Physics and Chemistry, 3rd ed.; Elsevier: New York, NY, USA, 2007. [Google Scholar]
- Brouwers, J.J.H. Langevin and diffusion equation of turbulent fluid flow. Phys. Fluids
**2010**, 22, 085102. [Google Scholar] [CrossRef] [Green Version] - Brouwers, J.J.H. Statistical description of turbulent dispersion. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2012**, 86, 066309. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brouwers, J.J.H. Statistical Models of Large Scale Turbulent Flow. Flow Turbul. Combust.
**2016**, 97, 369–399. [Google Scholar] [CrossRef] [Green Version] - Brouwers, J.J.H. Statistical Model of Turbulent Dispersion Recapitulated. Fluids
**2021**, 6, 190. [Google Scholar] [CrossRef] - Hoyas, S.; Jiménez, J. Scaling of the velocity fluctuations in turbulent channels up to Reτ = 2003. Phys. Fluids
**2006**, 18, 011702. [Google Scholar] [CrossRef] [Green Version] - Hoyas, S.; Oberlack, M.; Alcántara-Ávila, F.; Kraheberger, S.V.; Laux, J. Wall turbulence at high friction Reynolds numbers. Phys. Rev. Fluids
**2022**, 7, 014602. [Google Scholar] [CrossRef] - Kuerten, J.; Brouwers, J.J.H. Lagrangian statistics of turbulent channel flow at Reτ = 950 calculated with direct numerical simulation and Langevin models. Phys. Fluids
**2013**, 25, 105108. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers. Akad. Nauk. Sssr Dokl.
**1941**, 30, 301–305. [Google Scholar] - Borgas, M.S. The Multifractal Lagrangian Nature of Turbulence. Philos. Trans. Phys. Sci. Eng.
**1993**, 342, 379–411. [Google Scholar] - Thomson, D.J. Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech.
**1987**, 180, 529–556. [Google Scholar] [CrossRef] - Sawford, B.L. Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids A Fluid Dyn.
**1991**, 3, 1577–1586. [Google Scholar] [CrossRef] - Reichl, L.E. A Modern Course in Statistical Physics; Wiley-VCH: Hoboken, NJ, USA, 2004. [Google Scholar]
- Brouwers, J.J.H. On diffusion theory in turbulence. J. Eng. Math.
**2002**, 44, 277–295. [Google Scholar] [CrossRef] - George, W.K. The decay of homogeneous isotropic turbulence. Phys. Fluids A
**1992**, 4, 1492–1509. [Google Scholar] [CrossRef]

**Figure 1.**The root mean square values of the velocity fluctuations versus the dimensionless distance from the wall. The root mean square values obtained from DNS are represented by full lines. The root mean square values of the ${C}_{0}^{-1}$ expansion are represented by broken lines. They result from Equations (75) and (76) in which the right-hand sides were evaluated using the DNS values. Differences between full and broken lines can be ascribed to truncation of the ${C}_{0}^{-1}$ expansion.

**Figure 2.**The mean momentum fluxes versus the dimensionless distance from wall. The values of ${\sigma}_{11},{\sigma}_{22}$ and ${\sigma}_{12}$ are obtained using DNS and are represented by full lines. The values of ${D}_{12}(d/dx){u}_{1}^{0}$ and ${D}_{22}(d/dx){u}_{1}^{0}$ result from the ${C}_{0}^{-1}$ expansion and are represented by broken lines. They follow from Equations (69), (71) and (72) in which the right-hand sides were evaluated using the DNS values.

**Figure 3.**Coefficients of diffusion in the wall normal direction for momentum transport $<{u}_{1}^{\prime}{u}_{2}^{\prime}>$ and heat transport $<{\theta}^{\prime}{u}_{2}^{\prime}>$ versus dimensionless distance from wall. The values of $<{u}_{1}^{\prime}{u}_{2}^{\prime}>/(d/dx){u}_{1}^{0}$ and $<{\theta}^{\prime}{u}_{2}^{\prime}>/(d/dx){\theta}^{0}$ are obtained using DNS and are presented by full lines. The values of ${D}_{22}$ result from the ${C}_{0}^{-1}$ expansion. They follow from Equation (74) in which the right-hand side was evaluated using the DNS values. They are represented by a broken line.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brouwers, J.J.H.
Statistical Descriptions of Inhomogeneous Anisotropic Turbulence. *Mathematics* **2022**, *10*, 4619.
https://doi.org/10.3390/math10234619

**AMA Style**

Brouwers JJH.
Statistical Descriptions of Inhomogeneous Anisotropic Turbulence. *Mathematics*. 2022; 10(23):4619.
https://doi.org/10.3390/math10234619

**Chicago/Turabian Style**

Brouwers, J. J. H.
2022. "Statistical Descriptions of Inhomogeneous Anisotropic Turbulence" *Mathematics* 10, no. 23: 4619.
https://doi.org/10.3390/math10234619