# A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for Pr = 0.01–0.025

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Turbulence Model k-$\mathsf{\Omega}$-${k}_{\theta}$-${\mathsf{\Omega}}_{\theta}$

#### 2.2. Eddy Viscosity and Heat Diffusivity Model

#### 2.3. Boundary Conditions for Turbulence Models

## 3. Numerical Results

#### 3.1. Two-Dimensional Channel Tests

#### 3.2. Cylindrical Pipe Tests

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Two-dimensional channel tests ($Pr=0.01$). Non-dimensional mean axial velocity ${v}^{+}$ profiles (

**a**) and non-dimensional Reynolds stresses (

**b**) as a function of the non-dimensional wall distance ${y}^{+}$. Results obtained with KLW model are compared with DNS data [10]. The quantitative analysis (plots on bottom) show the percentage difference computed between the KLW and DNS results for ${v}^{+}$ and ${\tau}_{eff}^{+}$.

**Figure 2.**Two-dimensional channel tests ($Pr=0.01$). Non-dimensional mean temperature ${\tilde{T}}^{+}$ profiles (

**a**) and non-dimensional turbulent heat flux ${q}_{R}^{+}$ and total heat flux ${q}_{eff}^{+}$ (

**b**) as a function of the non-dimensional distance from the wall ${y}^{+}$. The results obtained with KLW model are compared with DNS data [10]. On the bottom, quantitative comparisons with DNS data for ${\tilde{T}}^{+}$ and ${q}_{eff}^{+}$.

**Figure 3.**Cylindrical pipe tests (KLW model). Non-dimensional profiles of the mean temperature ${\tilde{T}}^{+}$ (

**a**) and the root-mean-squared temperature fluctuations ${T}_{rms}^{\prime +}$ (

**b**) as a function of the non-dimensional distance from the wall ${y}^{+}$ with $Pr=0.025$ and $0.01$. The cases A, B, C, D, E, F and G are defined in Table 6.

**Figure 4.**Cylindrical pipe tests (constant heat flux). Nusselt numbers from different experimental data in Equations (40)–(45) as a function of the Peclet number.

${\mathit{C}}_{1\mathit{\epsilon}}$ | ${\mathit{C}}_{2\mathit{\epsilon}}$ | ${\mathit{C}}_{\mathit{\mu}}$ | ${\mathit{\sigma}}_{\mathit{k}}$ | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | ${\mathit{c}}_{\mathit{p}1}$ | ${\mathit{c}}_{\mathit{p}2}$ | ${\mathit{c}}_{\mathit{d}1}$ | ${\mathit{\sigma}}_{\mathit{\theta}}$ | ${\mathit{\sigma}}_{\mathit{\epsilon}\mathit{\theta}}$ |
---|---|---|---|---|---|---|---|---|---|

1.5 | 1.9 | 0.09 | 1.4 | 1.4 | 1.025 | 1.9 | 1.1 | 1.4 | 1.4 |

**Table 2.**Expansion for the components of the velocity and temperature for plane coordinates x-z around wall points and y distance from the wall.

Mean Components | Fluctuating Components |
---|---|

$u={A}_{1}y+{A}_{2}{y}^{2}+{A}_{3}{y}^{3}$ | ${u}^{\prime}={a}_{1}y+{a}_{2}{y}^{2}+{a}_{3}{y}^{3}$ |

$v=\phantom{\rule{2.em}{0ex}}+{B}_{2}{y}^{2}+{B}_{3}{y}^{3}$ | ${v}^{\prime}=\phantom{\rule{2.em}{0ex}}+{b}_{2}{y}^{2}+{b}_{3}{y}^{3}$ |

$w={C}_{1}y+{C}_{2}{y}^{2}+{C}_{3}{y}^{3}$ | ${w}^{\prime}={c}_{1}y+{c}_{2}{y}^{2}+{c}_{3}{y}^{3}$ |

$T={D}_{0}+{D}_{1}y+{D}_{2}{y}^{2}$ | ${T}^{\prime}={d}_{0}+{d}_{1}y+{d}_{2}{y}^{2}$ |

**Table 3.**Dynamical (on the left) and mixed (MX) thermal boundary conditions for the KLW model [22].

Variable | Boundary Condition | Variable | Boundary Condition | ||
---|---|---|---|---|---|

Dirichlet | Neumann | Dirichlet | Neumann | ||

${k}_{w}$ | - | $\frac{\partial {k}_{w}}{\partial y}=\frac{2{k}_{w}}{y}$ | ${k}_{\theta w}$ | - | $\frac{\partial {k}_{\theta w}}{\partial y}=\frac{2{k}_{\theta w}}{y}$ |

${\mathsf{\Omega}}_{w}$ | ${\mathsf{\Omega}}_{w}=ln\left(\right)open="("\; close=")">\frac{2\nu}{{C}_{\mu}{y}^{2}}$ | $\frac{\partial {\mathsf{\Omega}}_{w}}{\partial y}=-\frac{2}{y}$ | ${\mathsf{\Omega}}_{\theta w}$ | ${\mathsf{\Omega}}_{\theta w}=ln\left(\right)open="("\; close=")">\frac{2\alpha}{{C}_{\mu}{y}^{2}}$ | $\frac{\partial {\mathsf{\Omega}}_{\theta w}}{\partial y}=-\frac{2}{y}$ |

Properties | Value | |||
---|---|---|---|---|

Viscosity | $\mu $ | $0.00184$ | Pa s | |

Density | $\rho $ | 10340 | kg/m${}^{3}$ | |

Thermal conductivity | $\lambda $ | $Pr=0.01$ | $26.818$ | W/(m K) |

$Pr=0.025$ | $10.76896$ | |||

Heat specific capacity | ${C}_{p}$ | 146 | J/(kg K) |

Var. | ${\mathsf{\Gamma}}_{\mathit{sym}}$ | ${\mathsf{\Gamma}}_{\mathit{in}}\cup {\mathsf{\Gamma}}_{\mathit{out}}$ | ${\mathsf{\Gamma}}_{\mathit{w}}$ |
---|---|---|---|

u | $u=0$ | $\frac{\partial u}{\partial y}=0$ | $u=0$ |

v | $\frac{\partial v}{\partial x}=0$ | $\frac{\partial v}{\partial y}=0$ | $\frac{\partial v}{\partial x}=\frac{\mu}{\delta}$ |

k | $\frac{\partial k}{\partial x}=0$ | $\frac{\partial k}{\partial y}=0$ | $\frac{\partial k}{\partial x}=\frac{2k}{\delta}$ |

$\mathsf{\Omega}$ | $\frac{\partial \mathsf{\Omega}}{\partial x}=0$ | $\frac{\partial \mathsf{\Omega}}{\partial y}=0$ | $\mathsf{\Omega}=ln\left(\right)open="("\; close=")">\frac{2\nu}{{C}_{\mu}{\delta}^{2}}$ |

$\tilde{T}$ | $\frac{\partial \tilde{T}}{\partial x}=0$ | $\frac{\partial \tilde{T}}{\partial y}=0$ | $\tilde{T}=0$ |

${k}_{\theta}$ | $\frac{\partial {k}_{\theta}}{\partial x}=0$ | $\frac{\partial {k}_{\theta}}{\partial y}=0$ | $\frac{\partial {k}_{\theta}}{\partial x}=\frac{2{k}_{\theta}}{\delta}$ |

${\mathsf{\Omega}}_{\theta}$ | $\frac{\partial {\mathsf{\Omega}}_{\theta}}{\partial x}=0$ | $\frac{\partial {\mathsf{\Omega}}_{\theta}}{\partial y}=0$ | ${\mathsf{\Omega}}_{\theta}=ln\left(\right)open="("\; close=")">\frac{2\alpha}{{C}_{\mu}{\delta}^{2}}$ |

Case | A | B | C | D | E | F | G |
---|---|---|---|---|---|---|---|

$R{e}_{\tau}$ | 180 | 360 | 550 | 1000 | 3580 | 5840 | 6860 |

$Re$ | 5760 | 12,770 | 20,700 | 41,000 | 165,000 | 286,000 | 341,000 |

Var. | ${\mathsf{\Gamma}}_{\mathit{sym}}$ | ${\mathsf{\Gamma}}_{\mathit{in}}\cup {\mathsf{\Gamma}}_{\mathit{out}}$ | ${\mathsf{\Gamma}}_{\mathit{w}}$ |
---|---|---|---|

u | $u=0$ | $\frac{\partial u}{\partial y}=0$ | $u=0$ |

v | $\frac{\partial v}{\partial r}=0$ | $\frac{\partial v}{\partial y}=0$ | $\frac{\partial v}{\partial r}=\frac{\mu}{\delta}$ |

k | $\frac{\partial k}{\partial r}=0$ | $\frac{\partial k}{\partial y}=0$ | $\frac{\partial k}{\partial r}=\frac{2k}{\delta}$ |

$\mathsf{\Omega}$ | $\frac{\partial \mathsf{\Omega}}{\partial r}=0$ | $\frac{\partial \mathsf{\Omega}}{\partial y}=0$ | $\mathsf{\Omega}=ln\left(\right)open="("\; close=")">\frac{2\nu}{{C}_{\mu}{\delta}^{2}}$ |

$\tilde{T}$ | $\frac{\partial \tilde{T}}{\partial r}=0$ | $\frac{\partial \tilde{T}}{\partial y}=0$ | $\tilde{T}=0$ |

${k}_{\theta}$ | $\frac{\partial {k}_{\theta}}{\partial r}=0$ | $\frac{\partial {k}_{\theta}}{\partial y}=0$ | $\frac{\partial {k}_{\theta}}{\partial r}=\frac{2{k}_{\theta}}{\delta}$ |

${\mathsf{\Omega}}_{\theta}$ | $\frac{\partial {\mathsf{\Omega}}_{\theta}}{\partial r}=0$ | $\frac{\partial {\mathsf{\Omega}}_{\theta}}{\partial y}=0$ | ${\mathsf{\Omega}}_{\theta}=ln\left(\right)open="("\; close=")">\frac{2\alpha}{{C}_{\mu}{\delta}^{2}}$ |

**Table 8.**Cylindrical pipe tests. Mean values across the transverse section of the pipe: RMS temperature fluctuations ${T}_{rms,m}^{\prime +}$, mean temperature ${\tilde{T}}_{m}^{+}$ and the ratio ${T}_{rms,m}^{\prime +}/{\tilde{T}}_{m}^{+}$.

Case | ${\mathit{Re}}_{\mathit{\tau}}$ | $\mathit{Pr}=0.01$ | $\mathit{Pr}=0.025$ | ||||
---|---|---|---|---|---|---|---|

${\tilde{\mathit{T}}}_{\mathit{m}}^{+}$ | ${\mathit{T}}_{\mathit{rms},\mathit{m}}^{\prime +}$ | ${\mathit{T}}_{\mathit{rms},\mathit{m}}^{\prime +}$/${\tilde{\mathit{T}}}_{\mathit{m}}^{+}$ | ${\tilde{\mathit{T}}}_{\mathit{m}}^{+}$ | ${\mathit{T}}_{\mathit{rms},\mathit{m}}^{\prime +}$ | ${\mathit{T}}_{\mathit{rms},\mathit{m}}^{\prime +}$/${\tilde{\mathit{T}}}_{\mathit{m}}^{+}$ | ||

A | 180 | 0.62 | 0.025 | 0.041 | 1.46 | 0.112 | 0.076 |

B | 360 | 1.17 | 0.075 | 0.064 | 2.53 | 0.275 | 0.108 |

C | 550 | 1.71 | 0.134 | 0.078 | 3.44 | 0.402 | 0.117 |

D | 1000 | 2.82 | 0.268 | 0.095 | 5.16 | 0.609 | 0.117 |

E | 3578 | 6.35 | 0.711 | 0.111 | 9.30 | 0.895 | 0.096 |

F | 5843 | 7.85 | 0.814 | 0.103 | 10.82 | 0.896 | 0.082 |

G | 6852 | 8.34 | 0.836 | 0.100 | 11.32 | 0.918 | 0.081 |

Pr | Source | Reynolds Number | ||||||
---|---|---|---|---|---|---|---|---|

341,360 | 285,800 | 165,400 | 41,000 | 20,680 | 12,760 | 5760 | ||

0.025 | KLW | 29.4092 | 26.1685 | 18.5608 | 9.2123 | 7.5382 | 6.6523 | 5.6329 |

Kays | 42.9137 | 38.1888 | 27.0105 | 12.4689 | 9.3894 | 7.9416 | 6.3323 | |

Kirillov | 29.6295 | 26.3008 | 18.5779 | 9.1108 | 7.1690 | 6.3135 | 5.4593 | |

Cheng | 28.7295 | 25.4008 | 17.6779 | 9.0885 | 7.1690 | 6.3135 | 5.4593 | |

0.01 | KLW | 15.9311 | 14.4070 | 10.8496 | 6.7213 | 6.0596 | 5.7411 | 5.3430 |

Kays | 22.9363 | 20.7126 | 15.4458 | 8.6137 | 7.1788 | 6.5008 | 5.7193 | |

Kirillov | 16.6110 | 15.0068 | 11.2848 | 6.7221 | 5.7863 | 5.3740 | 4.9623 | |

Cheng | 15.7110 | 14.1068 | 10.6900 | 6.7221 | 5.7863 | 5.3740 | 4.9623 |

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**MDPI and ACS Style**

Da Vià, R.; Giovacchini, V.; Manservisi, S.
A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for *Pr* = 0.01–0.025. *Appl. Sci.* **2020**, *10*, 4337.
https://doi.org/10.3390/app10124337

**AMA Style**

Da Vià R, Giovacchini V, Manservisi S.
A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for *Pr* = 0.01–0.025. *Applied Sciences*. 2020; 10(12):4337.
https://doi.org/10.3390/app10124337

**Chicago/Turabian Style**

Da Vià, Roberto, Valentina Giovacchini, and Sandro Manservisi.
2020. "A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for *Pr* = 0.01–0.025" *Applied Sciences* 10, no. 12: 4337.
https://doi.org/10.3390/app10124337