# An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region

## Abstract

**:**

## 1. Introduction

**L**ength scale, turbulence

**I**ntensity, turbulent

**K**inetic energy and turbulent dissipation rate $\epsilon $ (

**E**).

## 2. The High Reynolds Number Transition Region

## 3. Model Overview

#### 3.1. Basic Model

#### 3.2. Turbulent Mixing Length Scales

#### 3.3. Turbulence Intensity

## 4. Model Results

#### 4.1. Model Output

#### 4.1.1. Quantities Depending on TKE, but Not Mixing Length

#### 4.1.2. Quantities Depending on Mixing Length, but Not TKE

## 5. Discussion

#### 5.1. Turbulence Isotropy

#### 5.2. Physical Mechanism

#### 5.3. Scaling of ${C}_{\mu}$

#### 5.4. Scaling of ${\nu}_{t}$ and I

#### 5.5. Time Scales

#### 5.5.1. $k-\epsilon $ Turbulence Model with Two Time Scales

#### 5.5.2. Eddy-Turnover Time Scales

#### 5.5.3. Time Evolution of TKE

- Decaying Turbulence

#### 5.6. A Plasma Physics Analogy

#### 5.7. Recommendations for CFD Practitioners

#### 5.7.1. Equilibrium Usage as Inlet Boundary Conditions for CFD Simulations

- L: The expressions are taken from [11]. Note that ${\ell}_{\mathrm{LIKE}}/{\ell}_{\mathrm{AA}}=1/{\kappa}_{g}\sim 3$.
- E: For the equilibrium model, we define ${C}_{\mu ,\mathrm{AA}}^{3/4}={\left(\right)}^{{B}_{g}}-3/2$, see Equation (34). There is a difference of a factor ${C}_{\mu}^{1/4}$ between the TKE dissipation rates, which (partially) compensates for the difference between the length scales. Note that ${C}_{\mu ,\mathrm{AA}}$ is a function of $R{e}_{\tau}$, whereas the LIKE algorithm uses the fixed standard value ${C}_{\mu ,\mathrm{B}}$ [11].

#### 5.7.2. Non-Equilibrium Usage as a Standalone Model

#### 5.8. Known Model Issues

## 6. Conclusions

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Left-hand plot: Functions for the square of the normalised fluctuating velocity as a function of friction Reynolds number; right-hand plot: area-averaged (AA) turbulence production-to-dissipation ratio as a function of friction Reynolds number. For both plots, the smooth and rough pipe lines are identical and cannot be distinguished.

**Figure 2.**Left-hand plot: Friction factor and friction velocity as a function of $R{e}_{\tau}$. For the friction factor, Superpipe measurements are included; right-hand plot: AA turbulence intensity as a function of $R{e}_{\tau}$.

**Figure 4.**Left-hand plot: The ratio of the absolute value of the Reynolds stress to the turbulent kinetic energy as a function of $R{e}_{\tau}$ for $\beta =1$ and $\beta =1.5$; right-hand plot: ${C}_{\mu}$ as a function of $R{e}_{\tau}$ for $\beta =1$ and $\beta =1.5$. For both plots, smooth and rough pipe lines are identical and cannot be distinguished [16].

**Figure 5.**Left-hand plot: The turbulence-to-mean shear time scale ratio as a function of $R{e}_{\tau}$ for $\beta =1$ and $\beta =1.5$. Right-hand plot: AA length scale ratio as a function of $R{e}_{\tau}$ for $\beta =1$ and $\beta =1.5$. For both plots, smooth and rough pipe lines are identical and cannot be distinguished [16].

**Figure 7.**Turbulent viscosity (left-hand plot) and turbulent viscosity ratio (right-hand plot) as a function of $R{e}_{\tau}$ for $\beta =1$ and $\beta =1.5$.

**Figure 8.**${C}_{\mu ,\mathrm{R}}$ from [25] compared to ${C}_{\mu ,\mathrm{B}}/{\langle \mathcal{P}/\epsilon \rangle}_{\mathrm{AA}}$ and our AA expressions. Left-hand plot: As a function of ${\langle \mathcal{P}/\epsilon \rangle}_{\mathrm{AA}}$. Right-hand plot: As a function of $R{e}_{\tau}$. Smooth and rough pipe lines are identical and cannot be distinguished.

**Table 1.**A comparison between the LIKE algorithm [10] and the proposed equilibrium model.

Source | LIKE Algorithm | Equilibrium Model |
---|---|---|

L | ${\ell}_{\mathrm{LIKE}}={\ell}_{m,\mathrm{N},\mathrm{CL}}=0.14\times \delta $ | ${\ell}_{\mathrm{AA}}={\langle {\ell}_{m,\mathrm{G}-\mathrm{H}}\rangle}_{\mathrm{AA}}=0.14{\kappa}_{g}\times \delta $ |

I | ${I}_{\mathrm{LIKE}}={I}_{\mathrm{mix}}=0.16\times R{e}_{D}^{-1/8}$ | ${I}_{\mathrm{AA}}=\sqrt{\left(\right)open="["\; close="]">{B}_{g}+\frac{3}{2}{A}_{g}-\frac{8{C}_{g}}{3\sqrt{R{e}_{\tau}}}\times \frac{\lambda}{8}}$ |

K | ${k}_{\mathrm{LIKE}}={k}_{\mathrm{mix}}={U}_{m}^{2}{I}_{\mathrm{mix}}^{2}$ | ${k}_{\mathrm{AA}}={U}_{m}^{2}{I}_{\mathrm{AA}}^{2}$ |

E | ${\epsilon}_{\mathrm{LIKE}}={C}_{\mu ,\mathrm{B}}\times \frac{{k}_{\mathrm{LIKE}}^{3/2}}{{\ell}_{\mathrm{LIKE}}}$ | ${\epsilon}_{\mathrm{AA}}={C}_{\mu ,\mathrm{AA}}^{3/4}\times \frac{{k}_{\mathrm{AA}}^{3/2}}{{\ell}_{\mathrm{AA}}}=\frac{{k}_{\mathrm{AA}}^{3/2}}{{L}_{\mathrm{AA}}}$ |

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Basse, N.T.
An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region. *Water* **2023**, *15*, 3234.
https://doi.org/10.3390/w15183234

**AMA Style**

Basse NT.
An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region. *Water*. 2023; 15(18):3234.
https://doi.org/10.3390/w15183234

**Chicago/Turabian Style**

Basse, Nils T.
2023. "An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region" *Water* 15, no. 18: 3234.
https://doi.org/10.3390/w15183234