# Direct Numerical Simulation of Thermal Turbulent Boundary Layer Flow over Multiple V-Shaped Ribs at Different Angles

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Geometric Model

#### 2.2. Numerical Method

#### 2.3. Double-Averaging Method

#### 2.4. Data Analysis

#### 2.5. Validation of the Numerical Method

## 3. Results

#### 3.1. Mean Statistics

#### 3.2. Secondary Motion

#### 3.3. Characteristics of Shear Stress and Heat Flux

#### 3.4. Decomposition of Velocity and Thermal Field

#### 3.5. Global and Local Reynolds Analogy

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Grid Refinement

**Figure A1.**Grid refinement study: (

**a**,

**b**) streamwise velocity $\langle \overline{u}\rangle $ and (

**c**,

**d**) dimensionless temperature $\langle \overline{\mathsf{\Theta}}\rangle $ in position A and B, respectively.

## Appendix B. Decomposition of Reynolds Stress and Heat Flux

**Figure A2.**Spatial derivatives of Reynolds stress and heat flux in case RC: (

**a**) $\partial \overline{{u}^{\prime}{u}^{\prime}}/\partial x$ and $\partial \overline{{u}^{\prime}{v}^{\prime}}/\partial y$ underneath the ribs ($y/H=0.5$); (

**b**) $\partial \overline{{u}^{\prime}{u}^{\prime}}/\partial x$ and $\partial \overline{{u}^{\prime}{v}^{\prime}}/\partial y$ above the ribs ($y/\delta =0.3$); (

**c**) $\partial \overline{{u}^{\prime}{\mathsf{\Theta}}^{\prime}}/\partial x$ and $\partial \overline{{v}^{\prime}{\mathsf{\Theta}}^{\prime}}/\partial y$ underneath the ribs ($y/H=0.5$); (

**d**) $\partial \overline{{u}^{\prime}{\mathsf{\Theta}}^{\prime}}/\partial x$ and $\partial \overline{{v}^{\prime}{\mathsf{\Theta}}^{\prime}}/\partial y$ above the ribs ($y/\delta =0.3$).

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**Figure 1.**Schematic of surface topographies discussed in the present study. Transverse bars were configured in case RA, and V-shaped cases with angles equal to ${60}^{\xb0},{45}^{\xb0},{30}^{\xb0}$ were set in RB, RC and RD, respectively.

**Figure 2.**Turbulent statistics of smooth wall case at $R{e}_{\theta}=2000$ in inner units: (

**a**) Mean streamwise velocity; (

**b**) root mean square of turbulence intensities and Reynolds shear stress.

**Figure 3.**Comparison of the vertical profiles of (

**a**) mean streamwise velocity and (

**b**) temperature against the DNS data of Nagano et al.

**Figure 4.**Profiles of (

**a**) momentum thickness ($\theta $), (

**b**) drag coefficient (${C}_{f}$) and (

**c**) Stanton number (${C}_{h}$) in the streamwise direction for all cases.

**Figure 6.**Comparison of the mean streamwise velocity profiles along the wall-normal direction with (

**a**) viscous scaling, and (

**b**) outer scaling.

**Figure 7.**Comparison of the mean temperature profiles along the wall-normal direction in (

**a**) inner coordinate, and (

**b**) outer coordinate.

**Figure 8.**Difference of (

**a**) mean streamwise velocity and (

**b**) mean temperature between V-shaped and transverse rib cases.

**Figure 9.**Contours of time-averaged normalized streamwise velocity and dimensionless temperature in diverging and converging areas in the $x-y$ coordinate.

**Figure 10.**Contours of time-averaged normalized statistics in the $y-z$ coordinate of the V-shaped cases. (

**a**,

**d**,

**g**): mean streamwise velocity, (

**b**,

**e**,

**h**): the intensity of the secondary flow motions, and (

**c**,

**f**,

**i**): mean dimensionless temperature. Lines with arrows, which represent the direction of secondary flows, are superposed to highlight the vortical structures. The first, second and third rows represent cases RB, RC and RD, respectively.

**Figure 11.**Profiles of double-averaged (

**a**) total shear stress, (

**b**) dispersive shear stress and (

**c**) turbulent shear stress along wall-normal direction.

**Figure 12.**Contours of time-averaged stress distributions in the wall-normal-spanwise planes: (

**a**–

**c**) dispersive shear stress; (

**d**–

**f**) turbulent shear stress.

**Figure 13.**Profiles of double-averaged (

**a**) total heat flux, (

**b**) dispersive heat flux, and (

**c**) turbulent heat flux along the wall-normal direction.

**Figure 14.**Contours of time-averaged heat flux in the wall-normal-spanwise planes: (

**a**–

**c**) dispersive heat flux; (

**d**–

**f**) turbulent heat flux.

**Figure 15.**Comparison of contributions of different terms towards the total shear stress and heat flux.

**Figure 16.**(

**a**) Mean Reynolds analogy s of each case, (

**b**) Mean Reynolds analogy in the ${C}_{f}/{C}_{fs}-{C}_{h}/{C}_{hs}$ coordinate.

**Figure 17.**Investigation of local ${C}_{f}/{C}_{fs}$ and ${C}_{h}/{C}_{hs}$. (

**a**,

**c**,

**e**,

**g**) are the local Reynolds analogies at the positions that crossed over the half range of the V-shaped ribs in the second-row figures. Only one point is plotted in the transverse rib case for the periodicity in the spanwise direction. (

**b**,

**d**,

**f**,

**h**) are the contours of ${C}_{f}/{C}_{fs}$ and ${C}_{h}/{C}_{hs}$ in the $x-z$ coordinate for each case.

Case | $\mathit{\gamma}{(}^{\xb0})$ | ${\mathit{Re}}_{\mathit{\theta}}$ | $\mathit{nx}\times \mathit{ny}\times \mathit{nz}$ | ${\mathit{u}}_{\mathit{\tau}}$ | ${\mathbf{\Theta}}_{\mathit{\tau}}$ | $\mathit{\delta}/{\mathit{\delta}}_{0}$ |
---|---|---|---|---|---|---|

RA | 90 | 2090 | 499 × 309 × 192 | 0.0735 | 0.0585 | 4.55 |

RB | 60 | 2124 | 499 × 309 × 192 | 0.0927 | 0.0615 | 4.62 |

RC | 45 | 2296 | 739 × 309 × 168 | 0.1025 | 0.0712 | 4.79 |

RD | 30 | 2160 | 989 × 309 × 128 | 0.0911 | 0.0790 | 4.87 |

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**MDPI and ACS Style**

Ji, F.; Ding, J.; Lu, J.; Wang, W.
Direct Numerical Simulation of Thermal Turbulent Boundary Layer Flow over Multiple V-Shaped Ribs at Different Angles. *Energies* **2023**, *16*, 3831.
https://doi.org/10.3390/en16093831

**AMA Style**

Ji F, Ding J, Lu J, Wang W.
Direct Numerical Simulation of Thermal Turbulent Boundary Layer Flow over Multiple V-Shaped Ribs at Different Angles. *Energies*. 2023; 16(9):3831.
https://doi.org/10.3390/en16093831

**Chicago/Turabian Style**

Ji, Feng, Jing Ding, Jianfeng Lu, and Weilong Wang.
2023. "Direct Numerical Simulation of Thermal Turbulent Boundary Layer Flow over Multiple V-Shaped Ribs at Different Angles" *Energies* 16, no. 9: 3831.
https://doi.org/10.3390/en16093831