**Figure 1:** Structured multiblock grid for Potsdam propeller blade.

*1700 words / 8 minutes read*

**Introduction**

In the previous article on open water propeller blade, we covered aspects of the domain, zones, and interfaces, boundaries conditions, grid refinement, in particular the wake, blade tip refinement to capture the cavitation vortices. In this final article in the series, we try to address aspects of grid influences on the solution flow field, including, grid uncertainty analysis for wetted vs cavitating flows conditions, mesh refinement studies using both sequential grids and adaptively refined grids and lastly model vs full-scale propeller blade performance.

**Video 2:** Cavitation CFD simulation using MARIN ReFRESCO solver.

**Uncertainty Analysis of Wetted vs Cavitation Flows **

Overall, the grid uncertainty levels of both wetted and cavitating results are low and is less than 1 percent. But the main difference between them is in the order of convergence of performance coefficients, which is low for the cavitating case than wetted flow. In wetted flow conditions, since grid sensitivities mainly occur around the blade tip, the thrust converges more slowly than torque. * Figure 2* shows the radial distribution of thrust coefficient for the two cases.

* Figures 2a and 2b* show the comparison for the two cases. The reduction in tip loading for the cavitating case is quite evident. This is mainly because of the presence of sheet cavitation which lowers the suction peak strength and leading-edge vortex. Interestingly, even though the suction peak close to the tip is absent compared to the wetted flow condition, a larger lower pressure area exists due to the cavity.

**Grid Refinement Study ( GCS and AGR)**

Grid sensitivity studies have shown a steady convergence in prediction accuracy with grid refinement. The results predicted by coarse grids emphasize the importance of grid density and refinement. Refinement can be done globally by generating a sequence of grids, while it can be done locally by applying adaptive grid refinement.
One region which shows remarkable improvement with grid refinement is the tip vortex region, where the cavity extensions are visibly significant with successive grids. One noteworthy aspect is that, even though the cavity extent is limited by the increasing eddy viscosity when using RANS, adaptive grid refinement aids in better capturing of tip vortex cavitation with lesser total cell count.

In fact, one researcher concludes in a paper that, the grid resolution is more important than cavitation modeling parameters when predicting the inception of tip vortex cavitation.

**Vapour Volume Uncertainty Analysis**

This analysis helps to examine the effect of grid refinement on cavitation flow prediction. * Figure 4* shows the cavity extents for 3 different grids refined at a rate of 1.5, visualized using iso-surfaces of vapour volume fraction.

From * Figure 4*, it can be observed that the form of the sheet cavity is less sensitive to grid refinement. This is somewhat expected, as given the low grid uncertainty of the thrust, which is directly related to the pressure distribution on the blade.

However, the tip vortex cavity shows a large grid dependency. For the first grid, there is no noticeable tip vortex cavitation. For the second refined grid, the tip vortex begins to cavitate. For the third refined grid, the cavity extension increases. On the finest grid, one can observe that the vortex rolls-up and form the characteristic ‘necked’ appearance.

**Analysis of the Cavitating Flow Field**

Even though the predicted sheet cavity shape did not appear to vary significantly with grid refinement, the variation on a 2D slice shows more subtle differences as shown in * Figure 5*. Here, the liquid-vapour interface and re-entrant jet resolution become sharper with refinement.

*, where slices of the vapour volume fraction, pressure coefficient, and normalized eddy viscosity are shown at an axial station in the propeller blade wake, 0.05D downstream of the propeller plane. As anticipated, grid refinement brings remarkable improvement in predicting tip vortex cavitation. As seen in*

**Figure 6-8***, the minimum pressure inside the vortex gets reduced significantly with refinement. Just a grid refinement ratio of 2.4 was good enough to develop a much-extended tip vortex.*

**Figure 7****Adaptive Grid Refinement**

Improvement in predicting tip vortex cavitation can be done using adaptive grid refinement with fewer grid cells.

By comparing **Figure 10** and* Figure 4* with

*, one can appreciate the differences between computations with and without adaptive grid refinement. The tip vortex cavity extension in the L1 adaptive grid is of similar length to that obtained on the finest grid in the sequential family (*

**Figure 3****). With further increase in local refinement, not only does the tip vortex cavitation becomes longer, but the roll-up of the tip vortex also gets better resolved (**

*Figure 10a**).*

**Figure 10b,10c****Model vs Full-Scale Performance**

Contrary to intuition, solutions obtained on a model scale cannot be extrapolated to the real-life full-scale propeller model. This is rightly so because with increase in geometry dimensions the Reynolds number at which computations are done increases and as a consequence, the flow patterns also change significantly. The numerical predictions performed at the model scale are more challenging than those at full scale due to the simultaneous cohabitation of laminar and turbulent flow regimes and the subsequent difficulty in the accurate prediction of the transition from laminar to turbulence in the computations.

Also, the propeller blade performance characteristics vary between the model and a full-scale geometry. Analysis of the results shows that the full-scale simulations yield a bit higher thrust coefficient, a slightly lower torque coefficient and consequently a higher open-water efficiency. It is interesting to note that, just like the model-scale, full-scale propeller efficiency does not show a great dependency on the grid density and the performance characteristics converge monotonically towards the experimental values.

Just like the solutions from the model scale cannot be extrapolated to the full-scale models, the model scale grid also cannot be scaled to fit full-scale geometry. Though the surface mesh and outer volume mesh can be scaled, the boundary layer needs to be regenerated as the Reynolds number perceived by the full model is much higher. At higher Reynolds number, the total thickness of the boundary layer reduces and so also the first spacing.* Figures 11, 12* show the near-blade and overall cavitation predicted on different grid densities for both model and full-scaled propeller blade. As can be observed from

*, near-blade cavitation is quite similar to each other on various grids. On all grids, sheet cavitation develops at the leading edge which then transforms to tip vortex cavitation. The root cavitation also grows in extent with increase in grid resolution but only marginally. Looking at*

**Figure 11***, it will be safe to say that the near-blade cavitation is mostly independent of the utilized grid densities.*

**Figure 11***. On a coarse grid, very limited vortex cavitation is seen, with the tip vortex cavitation seizing just after the blade trailing edge. In the medium grid, the tip vortex cavitation extends more than in the coarse grid but is less than one diameter in the axial direction and the modal shapes are barely distinguishable. Fine grids predict a well-established vortex cavity extension with proper modal shapes.*

**Figure 12**However, we do not see such a clear grid density influence in hub vortex cavitation. The hub vortex predicted by the medium grid matches very well with fine grid results and experiments.

*shows the pressure coefficient (Cp) comparison on various grids in the wake at the track of the tip vortex. The pressure minima is greater for the coarse and medium grids as the axial distance from the propeller increases. The Cp in the first vortex core for the coarse grid is significantly less than the vapor pressure and therefore the tip vortex cavitation terminates just after the blades. For the medium grid, the first tip vortex core on the curve maintains a low Cp level and a cavitation vortex as seen in*

**Figure 13***develops. Beyond this point, the Cp decreases and the tip vortex cavitation disappears.*

**Figure 12b****Parting Thoughts**

As these results emphasize, local refined grids bring in a significant improvement in flow feature prediction and which in turn makes the flow parameter prediction more accurate and compares well with the experimental results. Whether the refinement is through sequential grids or by adaptive grid refinement, the benefits of such an exercise are quite evident as underscored in this article.

**Further Reading:**

- Multiblock meshing of Ship Propeller
- Meshing Aspects for Open Water Marine Propeller CFD
- Cavitation in Turbines.
- CFD Modelling of Submarines

**References**

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2. “Grid Type and Turbulence Model Influence on Propeller Characteristics Prediction”, Ante Sikirica et al, Journal of Marine Science and Engineering, J. Mar. Sci. Eng. 2019, 7, 374.

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9. “Scale Effects on a Tip Rake Propeller Working in Open Water”, Adrian Lungu et al, Journal of Marine Science and Engineering, 2019, 7, 404.

10. “Cavitation on Model- and Full-Scale Marine Propellers: Steady and Transient Viscous Flow Simulations at Different Reynolds Numbers”, Ville Viitanen et al, Journal of Marine Science and Engineering, 2020, 8, 141.

11. “Energy balance analysis of a propeller in open water“, Jennie Andersson et al, Ocean Engineering, 158 (2018) 162–170.