# Computational Efficiency Assessment of Adaptive Mesh Refinement for Turbulent Jets in Crossflow

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Measurements

#### 2.2. Numerical Setup

#### 2.3. Testing Environment and Methodology

## 3. Results and Discussion

#### 3.1. Grid Convergence

#### 3.2. Conventional Approach

#### 3.3. Adaptive Mesh Refinement

#### 3.4. Computational Efficiency and Load Balancing

#### 3.5. LES Considerations

## 4. Conclusions

- AMR is suitable for turbulent mixing problems. Accuracy is on par with the conventional approach.
- Use for complex problems with bounded regions of interest is desirable.
- AMR approach can be employed to generate problem-specific grids as opposed to generating grids using conventional methodology.
- In order to avoid abrupt changes in mesh refinement, multi-criteria approach is suggested.
- Ensuring immutability of the boundary layer is a key factor in maintaining computational efficiency. This should also ensure CFL stability.
- Refinement frequency is the principal parameter governing the computational effort in AMR. Frequent refinement steps are typically not necessary, at least for JIC problems and RANS. Accordingly, additional computational savings can be achieved.
- Load balancing is indispensable for complex problems; however, simpler problems can also benefit from it. Due to load balancing, computational times can be halved.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Results on Conventional Grids

**Figure A1.**Concentration of passive scalar c at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) for different values of $S{c}_{t}$ where $y/D=0$. Fine grid and $k-\omega $ TNT model.

**Table A1.**Performance on the fine grid for different CFL values. Assessment is made for the $k-\omega $ TNT model. Presented values are three-run averages. A total of 48 cores are used for all tests. Due to numerical instabilities, the number of outer loops had to be increased (${n}_{loops}=50$) for $C{o}_{max}>0.9$.

CFL | 0.9 | 1.5 | 3 | 5 | 10 |
---|---|---|---|---|---|

Cells, n | 3,695,236 | ||||

Total time, t [s] | 61,183 | 314,339 | 172,683 | 105,603 | 53,854 |

Avg. timestep, ${t}_{ts}$ [s] | 1.010 | 8.381 | 9.743 | 9.863 | 9.790 |

RAM, m [GB] | 9.754 | 11.190 | 10.345 | 9.937 | 9.556 |

**Table A2.**Scalability and performance on the fine grid for original setup. Assessment is made for the $k-\omega $ TNT model. Presented values are three-run averages.

Cores, c | 12 | 24 | 48 | 96 | 192 | 384 |
---|---|---|---|---|---|---|

Cells, n | 3,695,236 | |||||

Total time, t [s] | 145,500 | 131,312 | 61,183 | 27,941 | 15,671 | 10,513 |

Avg. timestep, ${t}_{ts}$ [s] | 2.410 | 2.173 | 1.010 | 0.458 | 0.238 | 0.172 |

RAM, m [GB] | 6.116 | 8.011 | 9.754 | 13.770 | 21.966 | 39.331 |

**Figure A2.**Results for different CFL values on fine grid with $k-\omega $ TNT model: (

**a**) concentration of passive scalar c at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$; (

**b**) profiles of mean velocity components for $y/D=0$ and $z/D=1$.

## Appendix B. Results on AMR Generated Grids

**Figure A3.**Jet shape for the SST model where $c=0.01$: (

**a**) results on the initial grid used for AMR after $0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ (no refinement); (

**b**) AMR results after $0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

**Figure A4.**Reynolds stress component $\overline{{u}^{\prime}{u}^{\prime}}/{U}_{cross}^{2}$ at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$. AMR approach and different turbulence models.

**Table A3.**Performance for different CFL values when using AMR approach with grid size similar to the fine grid. Testing methodology is equal to the one defined for Table A1. Refinement frequency is set to 1000 timesteps. Load balancing is performed when the imbalance reaches $10\%$.

CFL | 0.9 | 1.5 | 3 | 5 | 10 |
---|---|---|---|---|---|

Cells, n | 3,772,957 | 3,773,398 | 3,744,222 | 3,737,663 | 3,694,963 |

Total time, t [s] | 56,999 | 286,900 | 150,495 | 79,317 | 32,841 |

Avg. timestep, ${t}_{ts}$ [s] | 1.291 | 11.124 | 11.497 | 11.794 | 12.443 |

RAM, m [GB] | 13.981 | 15.358 | 14.214 | 14.154 | 13.347 |

**Table A4.**Scalability and performance for AMR approach when the grid size is similar to the fine grid. Assessment is made for the $k-\omega $ TNT model. Refinement frequency is set to 1000 timesteps. Load balancing is performed when the imbalance reaches $10\%$. Presented values are three-run averages.

Cores, c | 12 | 24 | 48 | 96 | 192 | 384 |
---|---|---|---|---|---|---|

Cells, n | 3,771,074 | 3,770,465 | 3,772,957 | 3,772,712 | 3,775,442 | 3,777,577 |

Total time, t [s] | 134,665 | 120,315 | 56,999 | 27,394 | 15,291 | 9449 |

Avg. timestep, ${t}_{ts}$ [s] | 3.035 | 2.722 | 1.291 | 0.619 | 0.333 | 0.193 |

RAM, m [GB] | 9.036 | 11.781 | 13.981 | 19.823 | 32.623 | 67.008 |

## Appendix C. Load Balancing and AMR Refinement Frequency

**Table A5.**AMR generated grids when using the $k-\u03f5$ model. Refinement frequency is set to 1000 timesteps. Load balancing is disabled. AMR specifics are defined in the manuscript. Presented values are three-run averages. A total of 48 cores are used for all tests. Due to numerical instabilities, the number of outer loops had to be increased (${n}_{loops}=20$).

Cells, n | 1,635,190 | 2,131,756 | 2,699,904 | 3,256,761 | 3,743,793 | 4,171,500 |

Total time, t [s] | 213,972 | 293,283 | 416,898 | 492,013 | 548,479 | 610,740 |

Avg. timestep, ${t}_{ts}$ [s] | 9.428 | 13.329 | 17.508 | 22.222 | 24.807 | 26.648 |

RAM, m [GB] | 7.274 | 7.723 | 8.024 | 8.744 | 9.462 | 10.271 |

**Table A6.**Results for setup identical to Table A5 with load balancing enabled and performed when the imbalance reaches $20\%$.

Cells, n | 1,674,068 | 2,160,666 | 2,682,159 | 3,256,502 | 3,746,180 | 4,182,035 |

Total time, t [s] | 121,036 | 152,235 | 191,997 | 245,430 | 282,719 | 328,035 |

Avg. timestep, ${t}_{ts}$ [s] | 5.326 | 7.003 | 8.752 | 10.991 | 12.636 | 14.474 |

RAM, m [GB] | 10.048 | 10.929 | 12.780 | 13.915 | 14.876 | 16.068 |

**Table A7.**AMR generated grids when using the $k-\omega $ TNT model. Refinement frequency is set to 1000 timesteps. Load balancing is disabled. AMR specifics are defined in the manuscript. Presented values are three-run averages. A total of 48 cores are used for all tests.

Cells, n | 1,648,562 | 2,153,262 | 2,567,886 | 3,139,800 | 3,770,395 | 4,568,486 |

Total time, t [s] | 44,703 | 56,000 | 75,137 | 93,349 | 104,631 | 122,059 |

Avg. timestep, ${t}_{ts}$ [s] | 1.014 | 1.196 | 1.696 | 2.098 | 2.364 | 2.673 |

RAM, m [GB] | 9.736 | 11.674 | 13.106 | 15.321 | 17.108 | 18.757 |

**Table A8.**Results for setup identical to Table A7 with load balancing enabled and performed when the imbalance reaches $20\%$.

Cells, n | 1,670,976 | 2,150,553 | 2,569,461 | 3,142,047 | 3,770,836 | 4,568,983 |

Total time, t [s] | 23,683 | 29,469 | 37,496 | 46,964 | 57,127 | 69,733 |

Avg. timestep, ${t}_{ts}$ [s] | 0.538 | 0.639 | 0.850 | 1.076 | 1.296 | 1.486 |

RAM, m [GB] | 9.128 | 10.184 | 11.583 | 12.558 | 13.685 | 16.707 |

**Table A9.**AMR generated grids when using the $k-\omega $ SST model. Refinement frequency is set to 1000 timesteps. Load balancing is disabled. AMR specifics are defined in the manuscript. Presented values are three-run averages. A total of 48 cores are used for all tests.

Cells, n | 1,593,696 | 1,890,216 | 2,149,496 | 2,445,512 | 2,850,847 | 4,127,374 |

Total time, t [s] | 57,857 | 72,595 | 84,600 | 98,321 | 123,465 | 161,944 |

Avg. timestep, ${t}_{ts}$ [s] | 1.180 | 1.471 | 1.728 | 2.009 | 2.463 | 3.030 |

RAM, m [GB] | 10.093 | 11.268 | 12.305 | 13.363 | 15.080 | 18.380 |

**Table A10.**Results for setup identical to Table A9 with load balancing enabled and performed when the imbalance reaches $20\%$.

Cells, n | 1,607,775 | 1,891,996 | 2,150,862 | 2,445,944 | 2,851,503 | 4,128,115 |

Total time, t [s] | 31,972 | 39,160 | 42,714 | 50,833 | 66,171 | 89,405 |

Avg. timestep, ${t}_{ts}$ [s] | 0.663 | 0.807 | 0.872 | 1.110 | 1.333 | 1.689 |

RAM, m [GB] | 9.036 | 9.834 | 10.559 | 11.013 | 12.314 | 14.863 |

**Table A11.**Influence of the refinement frequency on the computational time. Setup is identical to Table A7 with load balancing enabled and performed when the imbalance reaches $5\%$.

Frequency | 10 | 50 | 100 | 500 | 1000 |
---|---|---|---|---|---|

Cells, n | 3,793,285 | 3,788,252 | 3,775,575 | 3,768,449 | 3,773,790 |

Total time, t [s] | 147,421 | 91,853 | 65,316 | 58,514 | 57,493 |

Avg. timestep, ${t}_{ts}$ [s] | 2.595 | 1.562 | 1.438 | 1.315 | 1.306 |

RAM, m [GB] | 16.164 | 15.697 | 14.702 | 15.162 | 13.985 |

**Table A12.**Influence of the refinement frequency on the computational time. Setup is identical to Table A7 with load balancing enabled and performed when the imbalance reaches $20\%$.

Frequency | 10 | 50 | 100 | 500 | 1000 |
---|---|---|---|---|---|

Cells, n | 3,789,764 | 3,785,494 | 3,774,413 | 3,767,392 | 3,770,836 |

Total time, t [s] | 149,921 | 105,332 | 68,027 | 58,432 | 57,127 |

Avg. timestep, ${t}_{ts}$ [s] | 2.679 | 1.909 | 1.520 | 1.311 | 1.296 |

RAM, m [GB] | 16.430 | 15.755 | 15.106 | 14.989 | 13.685 |

**Table A13.**Influence of the load balancing (imbalance adjustment) on the computational time. Setup is identical to Table A7 with refinement frequency set to 10 timesteps.

Imbalance | $2.5\%$ | $5\%$ | $10\%$ | $20\%$ | $40\%$ |
---|---|---|---|---|---|

Cells, n | 3,794,307 | 3,793,285 | 3,791,325 | 3,789,764 | 3,788,651 |

Total time, t [s] | 128,477 | 147,421 | 142,727 | 149,921 | 146,456 |

Avg. timestep, ${t}_{ts}$ [s] | 1.225 | 2.595 | 2.483 | 2.679 | 2.605 |

RAM, m [GB] | 15.263 | 16.164 | 16.355 | 16.430 | 16.493 |

**Table A14.**Influence of the load balancing (imbalance adjustment) on the computational time. Setup is identical to Table A7 with refinement frequency set to 1000 timesteps.

Imbalance | $2.5\%$ | $5\%$ | $10\%$ | $20\%$ | $40\%$ |
---|---|---|---|---|---|

Cells, n | 3,777,773 | 3,773,790 | 3,772,957 | 3,770,836 | 3,770,892 |

Total time, t [s] | 57,932 | 57,493 | 56,999 | 57,127 | 57,755 |

Avg. timestep, ${t}_{ts}$ [s] | 1.333 | 1.306 | 1.291 | 1.296 | 1.311 |

RAM, m [GB] | 14.234 | 13.985 | 13.981 | 13.685 | 13.577 |

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**Figure 2.**Grids used in grid convergence study. Images are fragments of the computational domain near the jet centerline: (

**a**) coarse grid; (

**b**) medium grid; (

**c**) fine grid.

**Figure 3.**Comparison between the results presented in [15] and results obtained on the fine grid: (

**a**) scalar concentration at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$; (

**b**) profiles of mean velocity components for $y/D=0$ and $z/D=1$.

**Figure 4.**Velocity component $U/{U}_{cross}$ at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) for $y/D=0$.

**Figure 5.**Two–dimensional plots of the scalar c at $y/D=0$ (top) and $z/D=1.5$ (bottom). (

**a**,

**d**) steady simulation using the $k-\omega $ TNT model; (

**b**,

**e**) transient simulation using the $k-\omega $ TNT model; (

**c**,

**f**) transient simulation using the $k-\omega $ SST model.

**Figure 6.**Grids generated using the AMR approach. Variances in mesh refinement are a direct consequence of innate differences between turbulence models. Images are fragments of the computational domain: (

**a**) $k-\u03f5$ model; (

**b**) $k-\omega $ TNT model; (

**c**) $k-\omega $ SST model.

**Figure 7.**Results obtained for the AMR approach for cases where mesh size is similar to the fine grid: (

**a**) scalar concentration at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$; (

**b**) profiles of mean velocity components for $y/D=0$ and $z/D=1$.

**Figure 8.**Velocity component $U/{U}_{cross}$ when using AMR at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) for $y/D=0$.

**Figure 9.**Two–dimensional plots of the scalar c at $y/D=0$ (top) and $z/D=1.5$ (bottom) for cases that employed the AMR approach: (

**a**,

**d**) $k-\u03f5$ model; (

**b**,

**e**) $k-\omega $ TNT model; (

**c**,

**f**) $k-\omega $ SST model.

**Figure 10.**Results for different refinement steps when using AMR and $k-\omega $ SST model. Cases 1, 3 and 5 are considered. Case specifics are discussed in Section 2.2 and are noted in Appendix C: (

**a**) concentration of passive scalar c at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$; (

**b**) profiles of mean velocity components for $y/D=0$ and $z/D=1$.

**Figure 11.**Results obtained using the WALE model and AMR: (

**a**) concentration of passive scalar c at $z/D=1.5$, $z/D=3$, $z/D=4.5$ (top to bottom) where $y/D=0$; (

**b**) profiles of mean velocity components for $y/D=0$ and $z/D=1$.

**Figure 12.**Results obtained using the WALE model with AMR: (

**a**) isosurface of the passive scalar $c=0.01$; (

**b**) distribution of scalar c at $y/D=0$.

**Table 1.**Grid details and GCI for the conventional approach. Assessment is made at three locations downstream of the jet centerline for the passive scalar c.

Mesh | Cells | Ratio | $\mathit{x}/\mathit{D}=0.4$ | $\mathit{x}/\mathit{D}=1.2$ | $\mathit{x}/\mathit{D}=2.0$ | |||
---|---|---|---|---|---|---|---|---|

n | r | e [%] | GCI [%] | e [%] | GCI [%] | e [%] | GCI [%] | |

Coarse | $7.179\times {10}^{5}$ | - | - | - | - | - | - | - |

Medium | $1.638\times {10}^{6}$ | 1.3 | 3.419 | 0.359 | 13.900 | 9.157 | 10.762 | 0.424 |

Fine | $3.695\times {10}^{6}$ | 1.3 | 0.266 | 0.022 | 5.039 | 3.320 | 0.330 | 0.013 |

**Table 2.**Computational efficiency on fine grid and when using the AMR approach without load balancing.

Case | Fine | AMR | AMR |
---|---|---|---|

$\mathit{k}-\mathit{\omega}$ TNT | $\mathit{k}-\mathit{\omega}$ TNT | $\mathit{k}-\mathit{\omega}$ SST | |

Cells, n | 3,695,236 | 3,770,395 | 2,850,847 |

Total time, t [s] | 61,183 | 104,631 | 123,465 |

Avg. timestep, ${t}_{ts}$ [s] | 1.010 | 2.364 | 2.463 |

RAM, m [GB] | 9.754 | 17.108 | 15.080 |

Case | Balancing Disabled | Balancing Enabled | ||||
---|---|---|---|---|---|---|

$\mathit{k}-\mathit{\u03f5}$ | TNT | SST | $\mathit{k}-\mathit{\u03f5}$ | TNT | SST | |

Cells, n | 3,743,793 | 3,770,395 | 2,850,847 | 3,746,180 | 3,770,836 | 2,851,503 |

Total time, t [s] | 548,479 | 104,631 | 123,465 | 282,719 | 57,127 | 66,171 |

Avg. timestep, ${t}_{ts}$ [s] | 24.807 | 2.364 | 2.463 | 12.636 | 1.296 | 1.333 |

RAM, m [GB] | 9.462 | 17.108 | 15.080 | 14.876 | 13.685 | 12.314 |

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## Share and Cite

**MDPI and ACS Style**

Sikirica, A.; Grbčić, L.; Alvir, M.; Kranjčević, L. Computational Efficiency Assessment of Adaptive Mesh Refinement for Turbulent Jets in Crossflow. *Mathematics* **2022**, *10*, 620.
https://doi.org/10.3390/math10040620

**AMA Style**

Sikirica A, Grbčić L, Alvir M, Kranjčević L. Computational Efficiency Assessment of Adaptive Mesh Refinement for Turbulent Jets in Crossflow. *Mathematics*. 2022; 10(4):620.
https://doi.org/10.3390/math10040620

**Chicago/Turabian Style**

Sikirica, Ante, Luka Grbčić, Marta Alvir, and Lado Kranjčević. 2022. "Computational Efficiency Assessment of Adaptive Mesh Refinement for Turbulent Jets in Crossflow" *Mathematics* 10, no. 4: 620.
https://doi.org/10.3390/math10040620