# Numerical Analysis of Turbulent Heat Transfer in the Case of Minijets Array

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{i,j}are the mean velocity components, m/s; ρ is the density, kg/m

^{3}; P is the pressure, Pa; μ is the dynamic viscosity, Pa·s; S

_{ij}is the strain rate tensor, 1/s; $\overline{{u}_{i}{u}_{j}}$ is the Reynolds stress term, m

^{2}/s

^{2}; Θ is the mean temperature, K; a is the thermal diffusivity, and m

^{2}/s and $\overline{{u}_{j}\theta}$ is the turbulent heat flux.

^{3}; U is the average orifice velocity, m/s; D is the orifice diameter, m; and μ is the fluid dynamic viscosity, Pa·s.

^{2}K; D is the orifice diameter, m; λ is the fluid thermal conductivity, W/mK; q is the wall heat flux, W/m

^{2}; T

_{w}is the local wall temperature, K; and T

_{in}is the reference inlet temperature, K.

## 3. Jets Array

_{p}= 4182 J/kgK, density ρ = 998.2 kg/m

^{3}, thermal conductivity λ = 0.6 W/mK, dynamic viscosity μ = 0.001003 Pa·s, were applied in conducted analyses.

- incompressible flow
- constant material properties
- symmetry boundary condition
- all walls adiabatic except the heated surface
- fully developed inlet velocity profile with constant temperature

#### 3.1. Mesh

#### 3.2. Numerical Procedure

^{−6}s for all cases. All residuals below 10

^{−6}and stabilization of average and maximum Nusselt number values on the heated surface were used as the convergence criterions.

## 4. Results

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Local Nusselt number distribution along the heated wall for single jet. (

**a**) Re = 23,000, H/D = 2; (

**b**) Re = 23,000, H/D = 6. H/D—is the ratio of jet inlet to the impinged surface height to its diameter.

**Figure 4.**Geometry and boundary conditions. (

**a**) Variant A flat surface; (

**b**) Variant B concave surface.

**Figure 8.**Local Nusselt number distribution for 100 L/h, Re = 4600. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 9.**Local Nusselt number distribution for 75 L/h, Re = 3430. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 10.**Local Nusselt number distribution for 50 L/h, Re = 2260. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 11.**(

**a**) Comparison of average Nusselt number values between correlations data and numerical calculations—Variant A; (

**b**) Comparison of average Nusselt number values between Variants A and B.

**Figure 12.**Local Nusselt number distribution for: (

**a**) single jet for Re = 23,000 and H/D = 2; (

**b**) jet array—Variant A for 100 L/h. x/D is a non-dimensional distance—a ratio of x coordinate to the diameter of orifice D.

**Figure 13.**Local Nusselt number distribution for: (

**a**) Variant A, 100 L/h; (

**b**) Variant B, 100 L/h; (

**c**) Variant A, 75 L/h; (

**d**) Variant B, 75 L/h; (

**e**) Variant A, 50 L/h; (

**f**) Variant B, 50 L/h.

**Figure 14.**Magnitude of temperature gradient for: (

**a**) Variant A, 100 L/h; (

**b**) Variant B, 100 L/h; (

**c**) Variant A, 75 L/h; (

**d**) Variant B, 75 L/h; (

**e**) Variant A, 50 L/h; (

**f**) Variant B, 50 L/h.

**Figure 15.**Magnitude of temperature gradient gradT [K/m], for 100 L/h. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 16.**Magnitude of temperature gradient gradT [K/m], for 75 L/h. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 17.**Magnitude of temperature gradient gradT [K/m], for 50 L/h. (

**a**) Variant A, k-ω SST Kato-Launder; (

**b**) Variant B, k-ω SST Kato-Launder; (

**c**) Variant A, k-ε RNG Kato-Launder; (

**d**) Variant B, k-ε RNG Kato-Launder; (

**e**) Variant A, Intermittency Transition; (

**f**) Variant B, Intermittency Transition.

**Figure 18.**Local Nusselt number distribution for 100 L/h, Re = 4600, k-ε RNG Kato-Launder. (

**a**) Variant A; (

**b**) Variant B.

**Figure 19.**Magnitude of temperature gradient gradT [K/m], for 100 L/h, Re = 4600, k-ε RNG Kato-Launder. (

**a**) Variant A; (

**b**) Variant B.

Inlet Volume Flow Rate, L/h | Inlet Water Temperature, K | Heat Flux at the Surface, W/m^{2} | Outlet | ||
---|---|---|---|---|---|

50 | 75 | 100 | 293 | 150,000 | Constant pressure |

Average Reynolds number in the orifice | |||||

2260 | 3430 | 4600 |

Variant A | Variant B | |
---|---|---|

Number of elements | ~4.9 M | ~4.9 M |

Average y+ on the heated surface (for 100 L/h case) | 0.34 | 0.32 |

Maximum y+ on the heated surface (for 100 L/h case) | 0.84 | 1.12 |

Mesh Version | Number of Elements | Average Nusselt Number |
---|---|---|

v1 | 2.7 M | 74.37 |

v2 | 4.5 M | 75.35 |

v3 | 4.9 M | 75.63 |

v4 | 5.4 M | 76.11 |

v5 | 6.4 M | 75.97 |

Reference | Correlation | Limitations |
---|---|---|

Martin [22] | $\mathrm{Nu}={\mathrm{Pr}}^{0.42}KGF$ $K={({(1+\frac{H/D}{0.6/\sqrt{f}})}^{6})}^{0.05}$ $G=\frac{\left(2\sqrt{f}\right)\left(1-2.2\sqrt{f}\right)}{1+0.2\left[\left(H/D\right)-6\right]\sqrt{f}}$ $F=0.5{\mathrm{Re}}^{2/3}$ $f=\frac{\pi}{2\sqrt{3}}{(\frac{D}{S})}^{2}$ | 2000 ≤ Re ≤ 100,000 2 ≤ H/D ≤ 12 |

Robinson [23] | $\mathrm{Nu}=1.4853{\mathrm{Re}}^{0.46}{\mathrm{Pr}}^{0.4}{(\frac{S}{D})}^{0.0442}{(\frac{H}{D})}^{0.00716}$ | 650 ≤ Re ≤ 6500 3 ≤ S/D ≤ 7 |

Fabbri [24] | $\mathrm{Nu}=0.043{\mathrm{Re}}^{0.78}{\mathrm{Pr}}^{0.48}\mathrm{exp}(0.069\frac{S}{D})$ | 43 ≤ Re ≤ 3813 4 ≤ S/D ≤ 26.2 |

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**MDPI and ACS Style**

Gurgul, S.; Kura, T.; Fornalik-Wajs, E.
Numerical Analysis of Turbulent Heat Transfer in the Case of Minijets Array. *Symmetry* **2020**, *12*, 1785.
https://doi.org/10.3390/sym12111785

**AMA Style**

Gurgul S, Kura T, Fornalik-Wajs E.
Numerical Analysis of Turbulent Heat Transfer in the Case of Minijets Array. *Symmetry*. 2020; 12(11):1785.
https://doi.org/10.3390/sym12111785

**Chicago/Turabian Style**

Gurgul, Sebastian, Tomasz Kura, and Elzbieta Fornalik-Wajs.
2020. "Numerical Analysis of Turbulent Heat Transfer in the Case of Minijets Array" *Symmetry* 12, no. 11: 1785.
https://doi.org/10.3390/sym12111785