*4.2. Influence of Roughness on Propeller Hydrodynamic Performance and Cavitation Extension*

Figures 5 and 6 show the change in thrust coefficient (i.e., *KT*) and torque coefficient in the presence of roughness with the different roughness length scales, as given in Table 1. Here, the smooth condition is shown with zero roughness height. With the application of roughness, the thrust coefficient decreases with an increase in roughness length scale due to the increased drag and decreased lift, as shown in Figure 5. As expected, the thrust decrease is smaller with the application of roughness on the hub compared to that of roughness application on the blades, as explored in our previous study [16]. The maximum thrust-reduction is found at around 2% with respect to the smooth condition.

**Figure 5.** Change in thrust coefficient (*KT*) with roughness.

**Figure 6.** Change in torque coefficient (10*KQ*) with roughness.

The change in torque coefficient (i.e., 10*KQ*) with roughness is also shown in Figure 6. Similar to thrust coefficient, roughness applied on the hub has a degradation effect on the torque coefficient of the propeller. The maximum reduction is found to be approximately 1.5% at the maximum roughened condition.

The decreased thrust and torque coefficients with the application of roughness lead to efficiency loss for the marine propellers, as shown in Figure 7. This is the main difference between the roughness and typical PBCF applications, as the increased thrust in PBCF enables efficiency gain due to the recovery of the energy loss. Nevertheless, the efficiency loss is not high with the application of roughness on the propeller's hub and the maximum efficiency loss is found at around 0.25%, while the cost of applying roughness and PBCF is another parameter to consider when deciding which one to implement.

**Figure 7.** Efficiency (*η*0) loss with roughness.

Figure 8 shows the comparison of wall shear stresses on the hub between smooth and rough conditions. As expected, the roughness applied to the hub increases the wall shear stresses.

**Figure 8.** Comparison of wall shear stresses between smooth and rough conditions.

The detailed flow analysis is carried out in the propeller slipstream to show the influence of roughness on the hub vortex and hub vortex cavitation. Figure 9 shows the change in turbulent kinetic energy obtained directly from the turbulence model with the application of roughness. As shown in Figure 9, the turbulent kinetic energy increases considerably with the roughness due to the transformation of the vortex's circumferential momentum into turbulent kinetic energy.

**Figure 9.** Change in turbulent kinetic energy with roughness.

The change in magnitude of the vortex structures with the roughness is shown at different sections in the propeller slipstream in Figure 10. Applying the roughness reduces the strength of the hub vortex with respect to the smooth condition. The reduced vortex strength enables the destabilisation process of the hub vortices and hence hub vortex disappears with the roughness application.

**Figure 10.** Change in the magnitude of the vortex structures in the propeller slipstream with roughness.

Figure 11 compares the distribution of the non-dimensional pressure coefficient (*Cp* = *P*/0.5*ρ*(*nD*) 2 ) between rough and smooth conditions. The roughness elements located around the hub interact with the hub vortices and change their velocity and pressure fields. The roughness decreases the velocity magnitudes, and hence, the pressure inside the vortex core and its surroundings increases significantly. With the application of roughness, the pressure inside the hub vortex increases, resulting in the reduction of hub vortex strength and hub vortex cavitation, as shown in Figure 12.

**Figure 11.** Change in non-dimensional pressure distribution with roughness.

**Figure 12.** Mitigation of hub vortex cavitation with roughness (*α<sup>v</sup>* = 0.1).

Applying roughness leads to destabilisation of the hub vortex strength, which results in the early breakdown of the hub vortex in the propeller slipstream. The reduced strength of the hub vortex due to the increased pressure inside the vortex core results in hub vortex cavitation mitigation with roughness application, as shown in Figure 12. With an increase in roughness length scale from M10 to NSM20, the hub vortex cavitation is further reduced. The maximum hub vortex cavitation volume reduction due to the roughness is computed at around 50% with respect to the smooth condition. As the roughness is solely applied to the propeller hub and boss cap, the sheet cavitation is not affected by the roughness application.

### **5. Conclusions**

This study presented an application of roughness onto the propeller hub as a novel concept to mitigate the hub vortex cavitation for marine propellers. In the numerical calculations, the RANS method—together with the k-w SST turbulence model—was utilised for the solution of cavitating flow around the benchmark model-scale propeller, INSEAN E779A, operating under uniform flow conditions. The Schnerr–Sauer cavitation model was used for modelling the sheet and hub vortex cavitation. A detailed flow-field analysis was also performed to understand the influence of roughness on the hub vortex, and hence, hub vortex cavitation. The crucial findings can be summarised as follows.


**Author Contributions:** Conceptualization; S.S., methodology; S.S.; software, S.S.; validation, S.S.; investigation, S.S., M.A.; resources; S.S., M.A.; data curation; S.S.; writing-original draft preparation; S.S.; writing-review and editing, S.S., M.A.; visualization; S.S.; supervision, M.A.; project administration, M.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The first author was sponsored by Stone Marine Propulsion Ltd. of the UK and the University of Strathclyde during his PhD study. Results were obtained using the ARCHIE-WeSt High-Performance Computer (www.archie-west.ac.uk) based at the University of Strathclyde.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

