# Grid Type and Turbulence Model Influence on Propeller Characteristics Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Geometry and Grid Generation

#### 2.2. Numerical Model

#### 2.3. Numerical Setup

#### 2.4. Hydrodynamic Theory of a Propeller

#### 2.5. Grid Sensitivity Study

## 3. Results and Discussion

#### 3.1. Open Water Performance

#### 3.2. Cavitation Test

## 4. Conclusions

- Different RANS solvers behave similarly with marginal deviations.
- Structured and unstructured grids can achieve comparable levels of accuracy. Therefore, depending on the requirements, unstructured tetrahedral grids can be an acceptable choice considering they are much simpler to generate. When employing tetrahedral grids, it is important to determine whether the results are a consequence of cumulative numerical errors and incomplete geometric representation. Imposed limitations on grid size can lead to inadequacies, which might be annulled by numerical errors. Deficiencies of tetrahedral grids could be mitigated by employing hexa-dominant or polyhedral unstructured grids. If accuracy of the results is in primary focus, structured grids should be preferred.
- Grid sensitivity study revealed steady convergence in prediction accuracy with the increase in grid size. The results obtained with coarse grids emphasize the importance of grid density and refinements. Therefore, to achieve relatively accurate results while maintaining representability, a minimum of $1.5\xb7{10}^{6}$ cells per blade should be used to properly depict a propeller.
- SST k-$\omega $ and Realizable k-$\u03f5$ models provide similar results; however, the Realizable model seems to provide both consistent and more accurate results, especially at higher advance ratios. SST k-$\omega $ could be beneficial for lower advance ratios and wall-bounded flows.
- Variances in performance predictions can be partially attributed to unresolved transition from laminar flow regime to turbulent flow along the propeller walls. Increase in Reynolds number usually leads to discernible variations of hydrodynamic characteristics at higher advance ratios, which was clearly demonstrated. Influence of Reynolds number is not incorporated in current results and cannot be properly resolved using common two-equation RANS models.
- Propeller performance predictions in cavitating conditions are also satisfactory, with errors below $2\%$ for analyzed case when using both Schnerr and Sauer and Zwart–Gerber–Belamri models. Cavitation models showed no significant differences in prediction accuracy. Tip vortex cavitation shape and extent predictions can be improved with grid refinement near the tip and in regions behind the blade, where cavitation rope is expected. Additionally, full model simulations could provide better results. Sheet cavitation needs to be further investigated, especially near the leading edge where cavitation pattern deviates the most. Employed coarse grids undoubtedly have an impact on cavity extents and thus grids should be refined appropriately in further studies.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

PPTC | Postdam Propeller Test Case |

BET | Blade Element Theory |

BEMT | Blade Element Momentum Theory |

IBLM | Integral Boundary Layer Method |

BEM | Boundary Element Method |

DNS | Direct Numerical Simulations |

LES | Large Eddy Simulations |

RANS | Reynolds-averaged Navier–Stokes |

SST | Shear Stress Transport |

RSM | Reynolds Stress Model |

TSM | Transition Sensitive Turbulence Model |

DDES | Delayed Detached Eddy Simulation |

SIMPLE | Semi-implicit Method For Pressure Linked Equations |

HRIC | High Resolution Interface Capturing |

SRF | Single Moving Reference Frame |

ZGB | Zwart–Gerber–Belamri |

## Appendix A

Unstructured | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Realizable $\mathit{k}$-$\mathit{\u03f5}$ | SST $\mathit{k}$-$\mathit{\omega}$ | Realizable $\mathit{k}$-$\mathit{\u03f5}$, Fine | SST $\mathit{k}$-$\mathit{\omega}$, Fine | |||||||||

$\mathit{J}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ |

0.000 | 0.468 | 1.069 | 0.000 | 0.587 | 1.252 | 0.000 | 0.954 | 2.063 | 0.000 | 0.937 | 2.046 | 0.000 |

0.160 | 0.476 | 1.139 | 0.106 | 0.540 | 1.163 | 0.118 | 0.915 | 1.936 | 0.120 | 0.869 | 1.883 | 0.118 |

0.322 | 0.846 | 1.783 | 0.243 | 0.844 | 1.772 | 0.244 | 0.821 | 1.753 | 0.240 | 0.791 | 1.712 | 0.237 |

0.482 | 0.754 | 1.620 | 0.357 | 0.750 | 1.608 | 0.357 | 0.716 | 1.568 | 0.350 | 0.696 | 1.536 | 0.347 |

0.642 | 0.653 | 1.446 | 0.462 | 0.654 | 1.439 | 0.464 | 0.609 | 1.380 | 0.451 | 0.602 | 1.372 | 0.448 |

0.802 | 0.553 | 1.277 | 0.553 | 0.554 | 1.268 | 0.558 | 0.504 | 1.198 | 0.537 | 0.507 | 1.208 | 0.535 |

0.961 | 0.455 | 1.105 | 0.629 | 0.454 | 1.093 | 0.636 | 0.406 | 1.026 | 0.605 | 0.412 | 1.037 | 0.607 |

1.121 | 0.358 | 0.927 | 0.688 | 0.357 | 0.912 | 0.698 | 0.313 | 0.856 | 0.652 | 0.321 | 0.860 | 0.665 |

1.283 | 0.261 | 0.733 | 0.726 | 0.260 | 0.720 | 0.737 | 0.219 | 0.671 | 0.667 | 0.230 | 0.673 | 0.699 |

1.442 | 0.162 | 0.524 | 0.707 | 0.162 | 0.517 | 0.720 | 0.124 | 0.470 | 0.608 | 0.140 | 0.504 | 0.639 |

Structured, Fluent | ||||||
---|---|---|---|---|---|---|

Realizable $\mathit{k}$-$\mathit{\u03f5}$ | SST $\mathit{k}$-$\mathit{\omega}$ | |||||

$\mathit{J}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ |

0.000 | 0.960 | 2.047 | 0.000 | 0.936 | 2.080 | 0.000 |

0.160 | 0.907 | 1.947 | 0.119 | 0.915 | 2.010 | 0.116 |

0.322 | 0.815 | 1.777 | 0.235 | 0.808 | 1.788 | 0.232 |

0.482 | 0.702 | 1.576 | 0.342 | 0.701 | 1.578 | 0.341 |

0.642 | 0.598 | 1.391 | 0.439 | 0.596 | 1.383 | 0.441 |

0.802 | 0.496 | 1.212 | 0.523 | 0.494 | 1.196 | 0.527 |

0.961 | 0.400 | 1.038 | 0.589 | 0.396 | 1.015 | 0.596 |

1.121 | 0.310 | 0.866 | 0.638 | 0.303 | 0.837 | 0.646 |

1.283 | 0.222 | 0.685 | 0.662 | 0.214 | 0.652 | 0.672 |

1.442 | 0.135 | 0.490 | 0.633 | 0.128 | 0.473 | 0.621 |

Structured, STAR-CCM+ | ||||||
---|---|---|---|---|---|---|

Realizable $\mathit{k}$-$\mathit{\u03f5}$ | SST $\mathit{k}$-$\mathit{\omega}$ | |||||

$\mathit{J}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ | ${\mathit{K}}_{\mathit{t}}$ | $\mathbf{10}\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ |

0.000 | 0.974 | 2.055 | 0.000 | 0.971 | 2.045 | 0.000 |

0.160 | 0.897 | 1.907 | 0.120 | 0.931 | 2.032 | 0.117 |

0.322 | 0.827 | 1.781 | 0.238 | 0.830 | 1.794 | 0.237 |

0.482 | 0.723 | 1.582 | 0.350 | 0.722 | 1.585 | 0.349 |

0.642 | 0.615 | 1.392 | 0.452 | 0.616 | 1.389 | 0.453 |

0.802 | 0.510 | 1.205 | 0.541 | 0.509 | 1.201 | 0.541 |

0.961 | 0.407 | 1.024 | 0.609 | 0.408 | 1.019 | 0.612 |

1.121 | 0.312 | 0.849 | 0.656 | 0.311 | 0.842 | 0.660 |

1.283 | 0.218 | 0.663 | 0.672 | 0.217 | 0.656 | 0.675 |

1.442 | 0.123 | 0.463 | 0.611 | 0.122 | 0.454 | 0.615 |

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**Figure 1.**Computational grids and cell distribution at the blade center: structured grid (left); unstructured grid (right).

**Figure 2.**Grids with ≈$1.7\times {10}^{6}$ cells used for open water calculations with radial cross sectional detail at $r=0.07$ m: structured grid (

**a**); unstructured grid (

**b**).

**Figure 5.**Relative errors in performance predictions for grids used in open water grid sensitivity study, Fluent.

**Figure 6.**Relative errors in performance predictions for grids used in cavitation grid sensitivity study, STAR-CCM+.

**Figure 7.**Comparison between experimental measurements and CFD results for structured grid attained with $SST$ k-$\omega $ model in: Fluent (left); STAR-CCM+ (right).

**Figure 8.**Open water results for unstructured grids using: Realizable k-$\u03f5$ (left); SST k-$\omega $ model (right).

**Figure 9.**Pressure distribution along the blade (

**a**) and around the leading edge (

**b**) for radial section at $0.08$ m.

**Figure 10.**Refined unstructured grid (left) and open water results for refined unstructured grid using SST k-$\omega $ and Realizable k-$\u03f5$ model (right).

**Figure 11.**Calculated open water characteristics for structured grids and Realizable k-$\u03f5$ model in: Fluent (left); STAR-CCM+ (right).

**Figure 12.**Relative errors in hydrodynamic coefficient predictions for structured grids attained with SST k-$\omega $ and Realizable k-$\u03f5$ models in: Fluent (

**a**); STAR-CCM+ (

**b**).

**Figure 13.**Cavity extents for $50\%$ vapor volume fraction: STAR-CCM+ with Schnerr and Sauer model (

**a**); Fluent with Schnerr and Sauer model (

**b**); Fluent with Zwart–Gerber–Belamri model (

**c**); experimental observations (

**d**).

**Table 1.**Hydrodynamic characteristics of a cavitating propeller for $J=0.995$ calculated using STAR-CCM+ and Fluent.

Test Case | ${\mathit{K}}_{\mathit{t}}$ | ${\mathit{K}}_{\mathit{q}}$ | $\mathit{\eta}$ |
---|---|---|---|

Experimental | 0.374 | 0.970 | 0.610 |

Fluent, Sauer | 0.380 | 0.961 | 0.642 |

STAR-CCM+, Sauer | 0.377 | 0.968 | 0.632 |

Fluent, Zwart | 0.379 | 0.956 | 0.643 |

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**MDPI and ACS Style**

Sikirica, A.; Čarija, Z.; Kranjčević, L.; Lučin, I. Grid Type and Turbulence Model Influence on Propeller Characteristics Prediction. *J. Mar. Sci. Eng.* **2019**, *7*, 374.
https://doi.org/10.3390/jmse7100374

**AMA Style**

Sikirica A, Čarija Z, Kranjčević L, Lučin I. Grid Type and Turbulence Model Influence on Propeller Characteristics Prediction. *Journal of Marine Science and Engineering*. 2019; 7(10):374.
https://doi.org/10.3390/jmse7100374

**Chicago/Turabian Style**

Sikirica, Ante, Zoran Čarija, Lado Kranjčević, and Ivana Lučin. 2019. "Grid Type and Turbulence Model Influence on Propeller Characteristics Prediction" *Journal of Marine Science and Engineering* 7, no. 10: 374.
https://doi.org/10.3390/jmse7100374