# A DES-SST Based Assessment of Hydrodynamic Performances of the Wetted and Cavitating PPTC Propeller

## Abstract

**:**

## 1. Introduction

^{TM}/Marine, whose main solver is ISIS-CFD, which is a finite volume based one. Closure to turbulence is achieved through the DES model. A grid convergence test as well as verification and validation computation will be provided. Comparisons with the experimental data will also be given to size the appropriateness of the numerical approach and the robustness of the flow solver.

## 2. Geometric Setup and Computational Meshes

#### 2.1. Propeller Characteristics and Computational Conditions

#### 2.2. Computational Grids

^{TM}/Marine software suite. Since the computations refer to two distinctive regimes, the minimal grid generation requirements are substantially different. That is, because of the differences between the non-cavitating and cavitating flow set-up, in the first case only restrictions related to the propeller surface and its potentially relevant surroundings are imposed. However, for the simulation of the propeller placed inside the cavitation tunnel, the conditions change since the tunnel solid walls must be taken into consideration as well. Four different grids are generated for each computational case, so that the grid convergence test can be performed. The finest grid for the non-cavitation computations consists of 63.48 M cells, whereas the coarsest consists of 8.27 M cells. In the following, the four grids will be identified as coarse, medium, fine and very fine, respectively.

^{+}remains below 0.25 everywhere on the propeller surfaces. This condition led to a maximum cell dimension of 10

^{−6}m measured normal to the solid walls. A number of 41 layers of cells are placed inside the viscous boundary layer of the propeller in both computational cases, whereas on the solid walls of the cavitation tunnel only 22 layers are used because of the limitations imposed by the existing hardware resources, as depicted in Figure 2d. The growth ratio of the cells through the inflation layers is 1.15 for all solid surfaces of the propeller in both cases, whereas a ratio of 1.2 is imposed for the tunnel walls.

## 3. Numerical Approach

^{TM}software package, is employed in the present research. The solver is based on the finite-volume method for constructing the spatial discretization of the transport equations on mostly unstructured grids. The simulation is performed in a global approach in which the momentum and mass conservation equations are solved. Closure to the turbulence is achieved through the shear stress transport DES model, which provides the accuracy of LES for highly separated flow regions and computational efficiency of RANS in the near-wall regions. Fluxes are built using the AVLSMART bounded difference scheme, which is based on the third order QUICK scheme. The velocity field is obtained from the momentum conservation equation and the pressure field is then solved from the mass conservation constraint transformed into a pressure equation. The entire discretization is fully implicit and second order accurate both in space and time, according to which an implicit scheme is applied for the discretization in time, whereas a second order three-level time scheme is used for the time-accurate unsteady computation. Computations are performed in two successive steps. At first, the viscous flow is computed based on the $k-\omega $ SST turbulence model and it runs until the solution stabilizes within 1% limit of variation in the last 10% of the computational time. Six iterations for a time step are imposed for this first set of runs. Secondly, the computation is restarted after the turbulence model is switched to the DES for a few more seconds until the final convergence. At this stage, the imposed number of iterations per time step is augmented to 16.

## 4. Results and Discussion

#### 4.1. Wetted Propeller

#### 4.1.1. Grid Convergence Test

_{G}), the associated relative error between the K

_{T}computed on the very fine mesh M4 and the fine mesh M3, ${\epsilon}_{43}\%{K}_{T4}$, the ratio between the estimated order of convergence and the theoretical order of convergence, p

_{G}/p

_{Gth}, the grid uncertainty, U

_{G}%M

_{4}, the experimental uncertainty, U

_{D}%D, and the validation uncertainty, U

_{v}%, are tabulated in Table 5. p

_{G}

_{,th}in Table 5 is the theoretical order of accuracy, which is the order of convection scheme, whereas the validation uncertainty is ${U}_{V}=\sqrt{{\left({U}_{G}\%{M}_{4}\right)}^{2}+{\left({U}_{D}\%D\right)}^{2}}$. The relative error between the solution computed on the finest mesh M4 and the experimental data for the thrust coefficient is smaller than the Richardson-based validation numerical uncertainty, therefore the K

_{T}prediction can be considered as being validated. A similar conclusion can be withdrawn for K

_{Q}.

#### 4.1.2. Propeller Performances and Flow Analysis

^{−5}s, so that the Courant numbers could be kept below unity regardless of the value of the oncoming flow velocity. Needless to say, the small time-step value led to high CPU time costs. However, they were finally justified by the good agreement with the available experimental data provided in [36].

^{−1}. Both the axial velocity and vorticity manifest periodic pulses that correspond to the cores of the vortices. Their intensity decreases in the wake because of the viscous dissipation. Strong tip-released structures are shed in the wake, correlating with local maxima of turbulent kinetic energy. The good agreement between the velocity and vorticity is explainable since they derive from the other. Because the hybrid LES model DES-SST only works with no-slip conditions for velocity on the solid boundaries, it proves to be more reliable since it does not employ any wall functions.

#### 4.2. Cavitating Propeller

^{−5}s so that the Courant number should be less than 0.3. Eight advance coefficients ranging from 0.7676 to 1.5708 are considered for the grid convergence test, as tabulated in Table 7 and Table 8. Table 7 contains the validations for the thrust coefficients, while Table 8 tabulates the corresponding data for the torque coefficients. All the computations are performed for a rotation speed of 1500 rpm, therefore the advance coefficients result from the variable advance velocity which is imposed for each computation. The experimental data used for comparisons are provided in [46].

#### 4.2.1. Propeller Cavitation Simulation for ${\sigma}_{n}=2.024$ (test case 2.3.1 in [46])

#### 4.2.2. Propeller Cavitation Simulation for ${\sigma}_{n}=1.424$ (test case 2.3.2 in [46])

#### 4.2.3. Propeller Cavitation Simulation for ${\sigma}_{n}=2.000$ (test case 2.3.3 in [46])

## 5. Concluding Remarks

^{TM}/Marine package, uses the finite volume of fluid approach. All the parallel computations were performed on 120 processers. Non-cavitating and cavitating flow regimes were computed for a significant range of advance coefficients, and the comparisons with the available experimental data validated the numerical solutions. Grid convergence tests were performed, and the computed solutions proved not only a satisfactory accuracy of the numerical approach, but also the solver robustness. Based on the discussions provided in the preceding sections, the following conclusions may be put forward:

- −
- extensive comparisons with the available experimental data have proven the accuracy of the numerical treatment;
- −
- the automatic grid refinement based on a regularized version of the Hessian of the pressure allowed convergence achievement at a reasonable CPU cost;
- −
- flow analysis of the non-cavitating propeller working regime revealed that the hybrid DES-SST turbulence model allows very good progression of the vortical structures in the wake;
- −
- comparisons of the numerical solution with the LDV measurements revealed that the numerical solution slightly under-predicted the intensity of the radial and tangential wake components and, on the other hand, it insignificantly over-predicted its axial component;
- −
- the Kunz cavitation model was able to provide a satisfactory estimation of the cavitation sheet extension but failed in capturing the bubble cavitation.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Computational domain and boundary conditions formulation. (

**a**): non-cavitation computations; (

**b**): side view of the computational domain for the cavitation computation; (

**c**): computational domain for the cavitation computation—view from outlet; (

**d**): same as previous—view from inlet; (

**e**): side view of the cavitation tunnel grid.

**Figure 2.**Computational mesh on the PPTC propeller. (

**a**): side view detail; (

**b**): mesh in the longitudinal plane of symmetry; (

**c**): transversal mesh in the propeller plane seen from the suction side; (

**d**): viscous boundary layer on the cavitation tunnel walls; (

**e**,

**f**): details of the sliding grid used for the cavitation computation.

**Figure 3.**Computational error versus the cubic root of the cell numbers (N

_{C}). (

**a**): K

_{T}; (

**b**): 10K

_{Q}.

**Figure 4.**Propeller performance diagrams. Comparison between the numerical solutions and experimental data provided in [36].

**Figure 5.**Pressure distribution on the propeller blades computed at $J=0.6418$. (

**a**): pressure side; (

**b**): suction side.

**Figure 6.**${y}^{+}$distribution on the propeller blades computed at $J=0.6418$. (

**a**): pressure side; (

**b**): suction side.

**Figure 7.**Non-dimensional pressure field computed in the domain longitudinal plane of symmetry. (

**a**): $J=0.6418$; (

**b**): $J=1.4422$.

**Figure 8.**Streamwise vorticity field computed in the longitudinal plane of symmetry. (

**a**): $J=0.6418$; (

**b**): $J=1.602$.

**Figure 9.**Non-dimensional axial velocity field computed in the domain longitudinal plane of symmetry. (

**a**): $J=0.6418$; (

**b**): $J=1.602$.

**Figure 10.**Comparison of the computed vortical structures released by the VP 1304 propeller. (

**a**): $J=1.4422$, (

**b**): $J=1.6021$.

**Figure 11.**Comparisons between the experimental [44] and computed wake. (

**a**): $x/D=0.1$; (

**b**): $x/D=0.2$.

**Figure 12.**Comparisons between the wake components measured [44] and computed at x/D = 0.1. (

**a**): r/R = 0.7; (

**b**): r/R = 0.9; (

**c**): r/R = 0.97; (

**d**): r/R = 1.0.

**Figure 13.**Comparisons between the wake components measured [44] and computed at x/D = 0.2. (

**a**): r/R = 0.7; (

**b**): r/R = 0.9; (

**c**): r/R = 0.97; (

**d**): r/R = 1.0.

**Figure 14.**Comparison between the normalized axial velocities measured [44] and computed at x/D = 0.1. (

**a**): measured values; (

**b**): computed values.

**Figure 15.**Comparison between the normalized axial velocities measured [44] and computed at x/D = 0.2. (

**a**): measured values; (

**b**): computed values.

**Figure 16.**Grid convergence test for the propeller working in cavitation tunnel. Computational error versus the cubic root of the cell numbers (N

_{C}). (

**a**): K

_{T}; (

**b**): 10K

_{Q}.

**Figure 18.**Comparison between the computed cavitation extension for ${\sigma}_{n}=2.024$ and the experimental cavitation sketch [46] reproduced with the permission of L. Lubke (SVA). (

**a**): computation; (

**b**): experiment.

**Figure 19.**Comparison between the computed cavitation extension for ${\sigma}_{n}=1.424$ and the experimental cavitation sketch [46] reproduced with the permission of L. Lubke (SVA). (

**a**): computation; (

**b**): experiment.

**Figure 20.**Comparison between the computed cavitation extension for ${\sigma}_{n}=2.000$ and the experimental cavitation sketch [46] reproduced with the permission of L. Lubke (SVA). (

**a**): computation; (

**b**): experiment.

Parameter | Symbol | Units | Values |
---|---|---|---|

Diameter | D | [m] | 0.250 |

Pitch ratio | P_{0.7}/D | [–] | 1.635 |

Area ratio | A_{E}/A_{0} | [–] | 0.779 |

Chord length | c_{0.7} | [m] | 0.1042 |

Skew angle | θ | [°] | 18.837 |

Hub ratio | d_{h}/D | [–] | 0.300 |

Computation | Parameter | Units | Values |
---|---|---|---|

W/O cavitation | ${\rho}_{w}$ | kg/m^{3} | 998.2 |

${\mu}_{w}$ | kg/m·s | 1.003 × 10^{−3} | |

W cavitation | ${\rho}_{w}$ | kg/m^{3} | 997.44 |

${\rho}_{v}$ | kg/m^{3} | 0.01927 | |

${\mu}_{w}$ | kg/m·s | 1.003 × 10^{−3} | |

${\mu}_{v}$ | kg/m·s | 8.8 × 10^{−6} |

Parameter | K_{T} | ||||||||
---|---|---|---|---|---|---|---|---|---|

J | 0.6418 | 0.8021 | 0.961 | 1.1212 | 1.283 | 1.4422 | 1.6021 | 1.6563 | |

EFD | 0.6084 | 0.517 | 0.4293 | 0.3354 | 0.246 | 0.1576 | 0.0491 | 0.009 | |

CFD | Coarse | 0.5772 | 0.4906 | 0.4085 | 0.3215 | 0.2364 | 0.1514 | 0.0471 | 0.0086 |

│ε│ % | 5.12 | 5.11 | 4.84 | 4.14 | 3.89 | 3.93 | 4.02 | 4.47 | |

Medium | 0.5821 | 0.4960 | 0.4123 | 0.3235 | 0.2392 | 0.1533 | 0.0476 | 0.0087 | |

│ε│ % | 4.32 | 4.07 | 3.95 | 3.54 | 2.77 | 2.72 | 2.98 | 3.35 | |

Fine | 0.5891 | 0.5010 | 0.4165 | 0.3267 | 0.2417 | 0.1546 | 0.0480 | 0.0087 | |

│ε│ % | 3.17 | 3.09 | 2.98 | 2.58 | 1.76 | 1.91 | 2.17 | 2.91 | |

Very fine | 0.5973 | 0.5271 | 0.4198 | 0.3397 | 0.2435 | 0.1551 | 0.0483 | 0.0092 | |

│ε│ % | 1.82 | 1.95 | 2.21 | 1.28 | 1.02 | 1.59 | 1.63 | 2.22 |

Parameter | 10 K_{Q} | ||||||||
---|---|---|---|---|---|---|---|---|---|

J | 0.6418 | 0.8021 | 0.961 | 1.1212 | 1.283 | 1.4422 | 1.6021 | 1.6563 | |

EFD | 1.3669 | 1.1994 | 1.0399 | 0.8606 | 0.6832 | 0.5029 | 0.2678 | 0.1786 | |

CFD | Coarse | 1.2973 | 1.1397 | 0.9876 | 0.8235 | 0.6505 | 0.4801 | 0.2563 | 0.1693 |

│ε│ % | 5.09 | 4.98 | 5.03 | 4.31 | 4.78 | 4.54 | 4.28 | 5.23 | |

Medium | 4.8645 | 4.7624 | 4.8127 | 4.1410 | 4.5998 | 4.3788 | 4.1336 | 5.0140 | |

│ε│ % | 4.43 | 4.37 | 4.32 | 3.92 | 3.77 | 3.55 | 3.42 | 4.13 | |

Fine | 1.3221 | 1.1615 | 1.0065 | 0.8342 | 0.6620 | 0.4879 | 0.2600 | 0.1725 | |

│ε│ % | 3.28 | 3.16 | 3.21 | 3.07 | 3.11 | 2.99 | 2.92 | 3.42 | |

Very fine | 1.3291 | 1.1682 | 1.0111 | 0.8412 | 0.6657 | 0.48845 | 0.2607 | 0.1731 | |

│ε│ % | 2.77 | 2.60 | 2.77 | 2.25 | 2.56 | 2.87 | 2.65 | 3.08 |

Parameters | ${\mathit{r}}_{\mathit{G}}$ | ${\mathit{\epsilon}}_{43}\mathit{\%}{\mathit{K}}_{\mathit{T}4}$$\mathbf{or}\text{}{\mathit{\epsilon}}_{43}\mathit{\%}{\mathit{K}}_{\mathit{Q}4}$ | ${\mathit{p}}_{\mathit{G}}/{\mathit{p}}_{\mathit{G},\mathit{t}\mathit{h}}$ | ${\mathit{U}}_{\mathit{G}}\mathit{\%}{\mathit{M}}_{4}$ | ${\mathit{U}}_{\mathit{D}}\mathit{\%}\mathit{D}$ | ${\mathit{U}}_{\mathit{V}}\mathit{\%}$ |
---|---|---|---|---|---|---|

${K}_{T}$ | 2.042 | 0.740 | 1.023 | 1.583 | 1.0 | 1.872 |

$10{K}_{Q}$ | 2.042 | 0.550 | 1.023 | 1.583 | 1.0 | 1.872 |

**Table 6.**Comparison between the open water propeller performance coefficients measured and computed.

J | K_{T} | │ε│ % | 10K_{Q} | │ε│ % | η_{O} | │ε│ % | |||
---|---|---|---|---|---|---|---|---|---|

EFD | CFD | EFD | CFD | EFD | CFD | ||||

0.6418 | 0.6084 | 0.5973 | 1.8245 | 1.3669 | 1.3291 | 2.7654 | 0.4550 | 0.4593 | 0.9451 |

0.8021 | 0.5170 | 0.5271 | 1.9536 | 1.1994 | 1.1682 | 2.6013 | 0.5500 | 0.5763 | 4.5636 |

0.9610 | 0.4293 | 0.4198 | 2.2129 | 1.0399 | 1.0111 | 2.7695 | 0.6310 | 0.6353 | 0.6768 |

1.1212 | 0.3354 | 0.3397 | 1.2821 | 0.8606 | 0.8412 | 2.2542 | 0.6950 | 0.7210 | 3.6061 |

1.2830 | 0.2460 | 0.2435 | 1.0163 | 0.6832 | 0.6657 | 2.5615 | 0.7350 | 0.7473 | 1.6459 |

1.4422 | 0.1576 | 0.1551 | 1.5863 | 0.5029 | 0.4884 | 2.8733 | 0.7190 | 0.7292 | 1.3853 |

1.6021 | 0.0491 | 0.0483 | 1.6293 | 0.2678 | 0.2607 | 2.6512 | 0.4670 | 0.4726 | 1.1849 |

1.6563 | 0.0090 | 0.0092 | 2.2222 | 0.1786 | 0.1731 | 3.0795 | 0.1330 | 0.1402 | 5.1355 |

Parameter | K_{T} | ||||||||
---|---|---|---|---|---|---|---|---|---|

J | 0.7676 | 0.8851 | 1.0132 | 1.1432 | 1.2467 | 1.3776 | 1.4875 | 1.5708 | |

EFD | 0.493 | 0.4363 | 0.3708 | 0.3057 | 0.2492 | 0.1837 | 0.1187 | 0.0633 | |

CFD | Coarse | 0.4697 | 0.4140 | 0.3529 | 0.2930 | 0.2395 | 0.1765 | 0.1139 | 0.0611 |

│ε│ % | 4.72 | 5.11 | 4.84 | 4.14 | 3.89 | 3.93 | 4.02 | 3.47 | |

Medium | 0.4712 | 0.4188 | 0.3568 | 0.2947 | 0.2413 | 0.1783 | 0.1155 | 0.0616 | |

│ε│ % | 4.42 | 4.02 | 3.78 | 3.61 | 3.17 | 2.93 | 2.72 | 2.61 | |

Fine | 0.4744 | 0.4225 | 0.3603 | 0.2980 | 0.2433 | 0.1796 | 0.1162 | 0.0620 | |

│ε│ % | 3.77 | 3.17 | 2.84 | 2.53 | 2.38 | 2.21 | 2.14 | 2.02 | |

Very fine | 0.4791 | 0.4247 | 0.3619 | 0.2992 | 0.2447 | 0.1808 | 0.1169 | 0.0626 | |

│ε│ % | 2.82 | 2.65 | 2.41 | 2.12 | 1.82 | 1.59 | 1.53 | 1.13 |

Parameter | 10 K_{Q} | ||||||||
---|---|---|---|---|---|---|---|---|---|

J | 0.7676 | 0.8851 | 1.0132 | 1.1432 | 1.2467 | 1.3776 | 1.4875 | 1.5708 | |

EFD | 1.224 | 1.1018 | 0.9693 | 0.8361 | 0.7194 | 0.5795 | 0.4472 | 0.3326 | |

CFD | Coarse | 1.1668 | 1.0524 | 0.9273 | 0.8009 | 0.6893 | 0.5555 | 0.4297 | 0.3202 |

│ε│ % | 4.67 | 4.48 | 4.33 | 4.21 | 4.18 | 4.14 | 3.91 | 3.73 | |

Medium | 1.1698 | 1.0537 | 0.9307 | 0.8042 | 0.6930 | 0.5589 | 0.4319 | 0.3215 | |

│ε│ % | 4.43 | 4.37 | 3.98 | 3.82 | 3.67 | 3.55 | 3.42 | 3.35 | |

Fine | 1.1839 | 1.0670 | 0.9401 | 0.8113 | 0.6988 | 0.5633 | 0.4355 | 0.3245 | |

│ε│ % | 3.28 | 3.16 | 3.01 | 2.97 | 2.86 | 2.79 | 2.62 | 2.43 | |

Very fine | 1.1864 | 1.0697 | 0.9415 | 0.8131 | 0.7004 | 0.5650 | 0.4362 | 0.3253 | |

│ε│ % | 3.07 | 2.91 | 2.87 | 2.75 | 2.64 | 2.51 | 2.45 | 2.18 |

**Table 9.**Comparison between the open water performance coefficients measured and computed for the propeller working in the cavitation tunnel.

J | K_{T} | │ε│ % | 10K_{Q} | │ε│ % | η_{O} | │ε│ % | |||
---|---|---|---|---|---|---|---|---|---|

EFD | CFD | EFD | CFD | EFD | CFD | ||||

0.7676 | 0.4930 | 0.4791 | 2.82 | 1.224 | 1.1864 | 3.07 | 0.492 | 0.4923 | 0.06 |

0.8851 | 0.4363 | 0.4247 | 2.65 | 1.1018 | 1.0697 | 2.91 | 0.558 | 0.5581 | 0.02 |

1.0132 | 0.3708 | 0.3619 | 2.41 | 0.9693 | 0.9415 | 2.87 | 0.617 | 0.6172 | 0.03 |

1.1432 | 0.3057 | 0.2992 | 2.12 | 0.8361 | 0.8131 | 2.75 | 0.665 | 0.6656 | 0.09 |

1.2467 | 0.2492 | 0.2447 | 1.82 | 0.7194 | 0.7004 | 2.64 | 0.687 | 0.6877 | 0.10 |

1.3776 | 0.1837 | 0.1808 | 1.59 | 0.5795 | 0.5650 | 2.51 | 0.695 | 0.6954 | 0.05 |

1.4875 | 0.1187 | 0.1169 | 1.53 | 0.4472 | 0.4362 | 2.45 | 0.628 | 0.6287 | 0.11 |

1.5708 | 0.0633 | 0.0626 | 1.13 | 0.3326 | 0.3253 | 2.18 | 0.476 | 0.4760 | 0.01 |

**Table 10.**Cavitation observations. Test case 2.3.1 in [46].

Parameter | Symbol | UM | Value |
---|---|---|---|

Advanced coefficient | J | [-] | 1.019 |

Thrust coefficient (non-cavitating) | ${K}_{T}$ | [-] | 0.387 |

Cavitation number | ${\sigma}_{n}$ | [-] | 2.024 |

Number of revolutions | n | [s^{−1}] | 24.987 |

**Table 11.**Cavitation observations. Test case 2.3.2 in [46].

Parameter | Symbol | UM | Value |
---|---|---|---|

Advanced coefficient | J | [-] | 1.269 |

Thrust coefficient (non-cavitating) | ${K}_{T}$ | [-] | 0.245 |

Cavitation number | ${\sigma}_{n}$ | [-] | 1.424 |

Number of revolutions | n | [s^{−1}] | 24.986 |

**Table 12.**Cavitation observations. Test case 2.3.3 in [46].

Parameter | Symbol | UM | Value |
---|---|---|---|

Advanced coefficient | J | [-] | 1.408 |

Thrust coefficient (non-cavitating) | ${K}_{T}$ | [-] | 0.167 |

Cavitation number | ${\sigma}_{n}$ | [-] | 2.000 |

Number of revolutions | n | [s^{−1}] | 25.014 |

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## Share and Cite

**MDPI and ACS Style**

Lungu, A.
A DES-SST Based Assessment of Hydrodynamic Performances of the Wetted and Cavitating PPTC Propeller. *J. Mar. Sci. Eng.* **2020**, *8*, 297.
https://doi.org/10.3390/jmse8040297

**AMA Style**

Lungu A.
A DES-SST Based Assessment of Hydrodynamic Performances of the Wetted and Cavitating PPTC Propeller. *Journal of Marine Science and Engineering*. 2020; 8(4):297.
https://doi.org/10.3390/jmse8040297

**Chicago/Turabian Style**

Lungu, Adrian.
2020. "A DES-SST Based Assessment of Hydrodynamic Performances of the Wetted and Cavitating PPTC Propeller" *Journal of Marine Science and Engineering* 8, no. 4: 297.
https://doi.org/10.3390/jmse8040297