# Non-Equilibrium Turbulent Transport in Convective Plumes Obtained from Closure Theory

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Equations of Compressible Hydrodynamic Turbulence

#### 2.1. Mean-Field Equations and Turbulent Correlations

#### 2.2. Simplest Modeling Turbulent Fluxes

## 3. Non-Equilibrium Effect

#### Non-Equilibrium Effect on Eddy Viscosity

## 4. Non-Equilibrium Effects in Jet and Plume Experiments

#### 4.1. Round Jets

#### 4.2. Buoyant Bubble Plumes

#### 4.3. Variable Density Jets

## 5. Plumes in Stellar Convection Zone

#### 5.1. Non-Equilibrium Effect in the Stellar Convection

#### 5.2. Interaction between Coherent and Incoherent Fluctuations

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Non-Equilibrium Turbulent Energy from the Two-Scale Direct-Interaction Approximation Analysis

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**Figure 1.**Schematically depicted evolution of the turbulent energy in the simulations of homogeneous-shear flow. •: DNS, $-\phantom{\rule{4pt}{0ex}}-$: standard K–$\epsilon $ model simulation; and ―: non-equilibrium model simulation. Onset of the model simulations is set at a scaled time $S{t}_{0}$ using the DNS data (S: shear rate).

**Figure 2.**The non-equilibrium effect associated with a plume motion. A plume is thermobuoyantly formed above a heat source. The non-equilibrium effect is taken into account through the Lagrangian or advective derivatives of the turbulent energy K and its dissipation rate $\epsilon $, $DK/Dt=(\partial /\partial t+\mathbf{U}\xb7\mathsf{\nabla})K$ and $D\epsilon /Dt=(\partial /\partial t+\mathbf{U}\xb7\mathsf{\nabla})\epsilon $, where $\mathbf{U}$ is the plume velocity.

**Figure 3.**Entropy distributions in the direct numerical simulations (DNSs) for the locally driven case (

**a**) and the non-locally driven case (

**b**). The horizontal cross-sections of the entropy fluctuation ${s}^{\prime}(=s-\langle s\rangle )$ at the top surface (

**Top**). In the non-locally or cooling-driven case, the horizontal extension of the cell structures is much more limited than the counterpart in the locally driven case. The vertical cross-sections of the entropy fluctuation ${s}^{\prime}$ from the horizontal mean (

**bottom**). In the non-locally driven case, the low-entropy down flow or plume structures produced at the surface are prominent in the upper region.

**Figure 4.**Spatial distributions of the non-equilibrium effect factor ${\tilde{\mathsf{\Lambda}}}_{\mathrm{D}}=\langle (\tilde{\mathbf{u}}\xb7\mathsf{\nabla})\overline{{\mathbf{u}}^{\prime}{}^{2}}\rangle $. The left axis $1-(z/d)$ is the depth from the surface ($z=d$). The magnitude of the non-equilibrium effect factor $|{\tilde{\mathsf{\Lambda}}}_{\mathrm{D}}|$ depends on the time averaging window T.

**Figure 5.**Spatial distributions of the turbulent internal-energy flux $\langle {e}^{\prime}{u}^{\prime}{}^{z}\rangle $ obtained by the direct numerical simulations (DNSs) and the model with the non-equilibrium effect. The left axis $1-(z/d)$ is the depth from the surface ($z=d$). The DNS results for the locally driven convection case (⋯), for the non-locally driven convection case ($--$) and the result from the model with the non-equilibrium effect for the non-locally driven convection case (―).

**Figure 6.**Schematic picture of the interaction between the coherent and incoherent fluctuations. Plumes (coherent fluctuations, depicted by thick black curved lines) are driven by the surface cooling. The energy of the plume motions are transferred to the energy of the random noise (incoherent fluctuation, depicted by gray circle eddies) by the interaction ${P}_{{K}^{\u2033}}$ if ${P}_{{K}^{\u2033}}>0$.

**Table 1.**Streamwise variation of normalized turbulent intensities, turbulent energy and turbulent axial velocity fluctuation normalized by the square root of the dissipation rate at the jet centerline. $x/D$: jet streamwise length, D: jet nozzle exit diameter ($D=11\phantom{\rule{4pt}{0ex}}\mathrm{mm}$), ${U}_{\mathrm{c}}$: centerline jet velocity (${U}_{\mathrm{c}}=50\phantom{\rule{4pt}{0ex}}\mathrm{mm}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$). Data are taken from Figures 7 and 8 of Lai and Socolofsky [27].

$\mathit{x}/\mathit{D}$ | $\sqrt{\langle {\left({\mathit{u}}^{\prime}{}^{\mathit{x}}\right)}^{2}\rangle}/{\mathit{U}}_{\mathbf{c}}$ | $\sqrt{\langle {\left({\mathit{u}}^{\prime}{}^{\mathit{y}}\right)}^{2}\rangle}/{\mathit{U}}_{\mathbf{c}}$ | $\sqrt{\langle {\left({\mathit{u}}^{\prime}{}^{\mathit{z}}\right)}^{2}\rangle}/{\mathit{U}}_{\mathbf{c}}$ | $\mathit{K}/{\mathit{U}}_{\mathbf{c}}^{2}$ | $\langle {\left({\mathit{u}}^{\prime}{}^{\mathit{x}}\right)}^{2}\rangle /\sqrt{{\mathit{\epsilon}}_{\mathbf{iso}}}$ |
---|---|---|---|---|---|

31 | 0.26 | 0.20 | 0.19 | 0.1437 | $3.25\phantom{\rule{4pt}{0ex}}\times {10}^{-2}$ |

37 | 0.26 | 0.20 | 0.19 | 0.1437 | $2.65\phantom{\rule{4pt}{0ex}}\times {10}^{-2}$ |

(−20%) |

**Table 2.**Evolutions of turbulent energy and its dissipation rate along the plume flow. Case A is the asymptotic phase and Case B is the adjustment phase. $z/D$: vertical streamwise height; D: dynamic length scale ($D=6.8\phantom{\rule{4pt}{0ex}}\mathrm{cm}$ for Case A and $D=20.4\phantom{\rule{4pt}{0ex}}\mathrm{cm}$ for Case B); ${W}_{\mathrm{c}}$: plume centerline velocity; and ${b}_{g}$: Gaussian plume radius. Data are taken from Figures 20 and 25 of Lai and Socolofsky [28].

Case A (Asymptotic Phase) | |||||||
---|---|---|---|---|---|---|---|

$z/D$ | ${W}_{\mathrm{c}}$ ($\mathrm{cm}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$) | ${b}_{g}$ | $K/{W}_{\mathrm{c}}^{2}$ | $(\epsilon /{W}_{\mathrm{c}}^{3}){b}_{g}\times {10}^{3}$ | K (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$) | $\epsilon $ (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-3}$) | ${K}^{2}/\epsilon $ (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$) |

6.6 | 20.64 | 5.46 | 0.135 | 0.04 | 57.51 | 64.42 | 51.34 |

11.0 | 17.90 | 8.00 | 0.165 | 0.086 | 52.87 | 61.65 | 45.34 |

(−8.1%) | (−4.3%) | (−11.7%) | |||||

Case B (Adjustment Phase) | |||||||

$z/D$ | ${W}_{\mathrm{c}}$ ($\mathrm{cm}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$) | ${b}_{g}$ | $K/{W}_{\mathrm{c}}^{2}$ | $(\epsilon /{W}_{\mathrm{c}}^{3}){b}_{g}\times {10}^{3}$ | K (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$) | $\epsilon $ (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-3}$) | ${K}^{2}/\epsilon $ (${\mathrm{cm}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$) |

2.19 | 28.50 | 5.27 | 0.15 | 0.035 | 121.84 | 153.74 | 96.56 |

3.66 | 25.14 | 8.53 | 0.225 | 0.0896 | 142.20 | 165.78 | 121.97 |

(+16.7%) | (+7.8%) | (+26.3%) |

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**MDPI and ACS Style**

Yokoi, N.
Non-Equilibrium Turbulent Transport in Convective Plumes Obtained from Closure Theory. *Atmosphere* **2023**, *14*, 1013.
https://doi.org/10.3390/atmos14061013

**AMA Style**

Yokoi N.
Non-Equilibrium Turbulent Transport in Convective Plumes Obtained from Closure Theory. *Atmosphere*. 2023; 14(6):1013.
https://doi.org/10.3390/atmos14061013

**Chicago/Turabian Style**

Yokoi, Nobumitsu.
2023. "Non-Equilibrium Turbulent Transport in Convective Plumes Obtained from Closure Theory" *Atmosphere* 14, no. 6: 1013.
https://doi.org/10.3390/atmos14061013