Propeller Tip Vortex Mitigation By Roughness Application

In this study, the application of surface roughness on model and full scale marine propellers in order to mitigate tip vortex cavitation is evaluated. To model the turbulence, SST kOmegamodel along with a curvature correction is employed to simulate the flow on an appropriate grid resolution for tip vortex propagation, at least 32 cells per vortex diameter according to our previous guidelines. The effect of roughness is modeled by modified wall functions. The analysis focuses on two types of vortices appearing on marine propellers: tip vortices developing in lower advance ratio numbers and leading-edge tip vortices developing in higher advance ratio numbers. It is shown that as the origin and formation of these two types of vortices differ, different roughness patterns are needed to mitigate them with respect to performance degradation of propeller performance. Our findings clarify that the combination of having roughness on the blade tip and a limited area on the leading edge is the optimum roughness pattern where a reasonable balance between tip vortex cavitation mitigation and performance degradation can be achieved. This pattern in model scale condition leads to an average TVC mitigation of 37% with an average performance degradation of 1.8% while in full scale condition an average TVC mitigation of 22% and performance degradation of 1.4% are obtained.


Introduction
A hydrodynamically optimum propeller design usually does not have an optimum hydroacoustic performance as their design restrictions are contradictory [1]. This has even further importance for low-noise propellers as their operating profile requires very low radiated noise emissions mostly generated by cavitation [2]. Tip vortex cavitation (TVC) is usually the first type of cavitation that appears on a propeller, and consequently plays a key role in initiating an overall increasing sound pressure level, and determining the underwater radiated noise [3,4]. The radiated noise is of big importance because it can disturb marine wildlife and reduces comfort of people on board ship. Therefore, TVC is considered as the main cavitation characteristics to control in the in the design procedure.
The blade load distribution is a decisive parameter in the tip vortex cavitation formation [5]. With a highly loaded tip, vorticity is generated at the trailing edge of a propeller blade, resulting in a stable trailing vortex formation. With reduced tip loading, separation still occurs close to the tip, while the trailing vortex is much weaker, leading to a local tip vortex formation typical for a standard ship propeller. An unloaded tip design forces the loading towards inner radii and at these inner radii, leading edge separation, and therefore a leading edge vortex, may be formed and the trailing vortex system becomes more distributed [6]. In this condition, effects of non-uniform flow field [7,8], and blade surface roughness [9,10] should also be considered.
The start of cavitation in the tip vortex, TVC inception (TVCI), determines a break point where nuisance suddenly increase. Cavitation will occur in the core of a tip vortex only if a nucleus has enough time to reach the core and then trigger cavitation [11]. It is well reported that cavitating behaviour of a tip vortex depends on the nuclei radius, its initial location, vortex circulation, and vortex velocity [12][13][14][15]. Depending on the water quality, TVCI can be either a sudden appearance of a continuous cavity, or an intermittent appearance of an elongated bubble extending axially over a relatively small portion of the tip vortex [16,17]. At the intermittent step, formation and collapse of elongated bubbles increases noise intensity significantly compared with the fully developed TVC [18]. Looking at vortex singing, the tones are very different under different flow conditions and water quality where the vortex singing can only exist in the transferring process between strong tip vortex cavitation and weak tip vortex cavitation [19,20].
Traditional potential flow propeller design tools, in connection with designer experience, are able to provide optimal geometries in terms of efficiency, and to some extent capture the effects of on-blade cavitation, but they are not suited for the assessment of negative aspects of cavitating tip vortices. Apart from redesigning a propeller to redistribute load and consequently changing TVC properties [21], a few approaches has been proposed to suppress TVC, classified as active control and passive control methods. In active control approaches, the tip vortex flow is altered by injection of a solution, e.g. air [22], polymer [23,24], or water [25,26], into the tip vortex region. In passive control methods, the boundary layer and momentum distribution around the blade tip are altered aiming to weaken the tip vortex and its nuclei capture capability. Inclusion of an extra geometry on the blade tip [27][28][29], drilling holes on the blade [30], and roughening the blade surface [31][32][33][34] are some examples of passive TVC mitigation, of which the latter is the concern of this study.
Surface roughness affects the tip vortex roll-up as roughness elements promote transition to turbulence and growth of laminar boundary layers and thereby alter the near-wall flow structures. The vortical structures generated by the roughness elements interact with the main tip vortex and destabilize it. If size, pattern, and location of roughness elements are selected appropriately, the destabilization process leads to tip vortex breakdown, and consequently to TVC mitigation. This, however, increases the losses and leads to performance degradation [31,32,34]. To minimize this degradation, one has to optimize the roughened area which itself demands a detailed knowledge on where and how the tip vortex is formed. For propellers in behind conditions, this is even more complex as the tip vortex location and origin vary.
In the current study, different flow properties are analyzed to define effective areas in tip vortex formation and its roll-up on a propeller selected from a research series of highly skewed propellers having a low effective tip load which is typical for yachts and cruise ships. In our previous studies, numerical simulations of tip vortex flows around this propeller having smooth blades have been carried out and successfully compared with experimental measurements [35]. The aim of the present study is to provide further knowledge about the effects of the surface roughness on the TVC and the possibility of using roughness to delay the cavitation inception.
The turbulent flow field around the propellers is modelled by using the two-equation SST k − ω model of OpenFOAM on appropriate grid resolutions for tip vortex propagation, at least 32 cells per vortex diameter according to previous studies guidelines [36,37]. To prevent overprediction of turbulent viscosity in highly swirling tip regions, η 3 curvature correction method is employed [38,39]. The roughness is included in the simulations by employing two different approaches. In the first approach, rough wall functions are used to mimic the effects of roughness by modifying the turbulent properties in roughed areas [40]. The second approach modifies the mesh topology by removing cells in roughed areas to create random roughness elements. See further comment below on the resolved geometry.
To identify the areas where roughness has to be applied, three criteria based on the flow properties of the smooth propeller condition are employed. The Q-criterion is selected to identify the vortical structures and their interactions. The second criterion is the flow streamlines close to the blade surface helping to highlight from which areas on the blade the vortex momentum is provided. The third criterion is the pressure coefficient used to distinguish the areas where the cavitation incepts inside the tip vortex. These criteria are used to find a good roughness pattern that can lead to a proper tip or leading edge vortices mitigation with reasonable performance degradation.
The strategy used to find the suitable roughness pattern in model scale condition is extended to find the roughness pattern in full scale conditions. This has been done by considering the flow properties of the smooth propeller in the full scale condition as well. Then, the best roughness pattern application is evaluated at three operating conditions, i.e. J=0.82, 0.93 and 1.26. Contradictory to the model scale analysis where a rough wall function is employed to incorporate the effects of roughness, in the full scale analysis the roughness elements are included as a part of computational domain. This gives the possibility of resolving the flow around the roughness elements, and also allows to use smaller y + for the blade surfaces.
The results contain the performance and cavitation inception charts of the propeller in model and full scale conditions. Roughness application on different blade areas are examined at different operating conditions, and their impact on TVC mitigation and performance degaradtion is reported. It is investigated how having different roughness patterns alter vortical structures on the blade and in the tip vortex region. The roughness area is optimized by simultaneous consideration of the tip vortex mitigation, performance degradation and their compromise. The performance and TVC mitigation of the propeller having optimum roughness pattern are discussed.

Governing Equations
The OpenFOAM package, used in this study for numerical simulation, is an open source code written in C++ to model and simulate fluid dynamics and continuum mechanics [41]. The incompressible conservation of mass and momentum equations are solved using the PIMPLE algorithm, a merge of the SIMPLE and PISO algorithms.
The solver has been used and validated for tip vortex analysis in marine applications; see [36,42] for more details on the modelling and the numerical setup used in OpenFOAM.

Turbulence model
The turbulence is modelled by employing the SST k − ω model along with a curvature correction model [37].
In the selected model, the production term of the ω equation is multiplied by F rc , where α 1 = −0.2 and C r = 2.0. In this equation, η 3 is a velocity gradient invariants defined through the non-dimensional strain rate and rotational rate tensors [38], where the strain rate and rotational rate tensors are defined by, As can be seen, η 1 represents the non-dimensional strain rate magnitude, η 2 represents the non-dimensional vorticity magnitude, and η 3 is a linear combination of these two velocity-gradient invariants. The turbulent time scale used to non-dimensionalize these tensors is calculated by [38,39,43], where n = 1.625. The time scale is limited in order to have a correct near-wall asymptotic behaviour.
The modified rotational rate tensor incorporating the streamline curvature and frame rotation is, where C r is the constant of the equation and depends on the CC model [38]. This coefficient takes a value of 2 for bifurcation approaches. Here, Ω F ij represents the frame rotational tensor calculated from where Ω F k is the angular frame velocity about the x k -axis. The W A ij tensor which contains the effects of curvature corrections in the rotational rate tensor is defined by [44], 3.2. Roughness modelling 5

Roughness modelling
The flow around the roughness elements can be either resolved or modelled. To resolve the flow, roughness geometries have to be included into the computational domain which leads to having finer cell resolution around them compared to the rest of the domain, and consequently demand for higher computational resources.
Modelling roughness elements requires much lower number of computational cells but as it involves simplification of the roughness geometry, the flow physics may not be correctly modelled.
In the current study for modelling roughness elements, the wall function developed by Tapia (2009) for the inner region of the turbulent boundary layer or the log-law region (e.g. 11 ≤ y + in OpenFOAM wall functions) is used, with the von Karman constant κ = 0.41, the constant E=9.8, the dimensionless wall distance y + = u τ y/ν, and the velocity shift correction ∆B due to the roughness elements. In this model, the nondimensional roughness height is presented by K + s = u τ K s /ν where K s is the roughness height, u τ = τ w /ρ is the shear velocity, and τ w is the wall shear stress. As this model only affects the viscosity of the first cell adjacent to the wall, the height of the roughness elements should be smaller than the height of the cells wall normal distance, i.e. K + s ≤ y + w . Otherwise, the part of roughness elements that locates outside the adjacent cells will not be included in the modelling.
The flow regime over a rough surface depends on how roughness elements interact with different parts of the boundary layer. If the roughness elements are embedded in the viscous sublayer, the friction drag is not affected by the roughness and the flow regime is smooth. In a smooth regime represented by K + s ≤ 2.5, the correction ∆B is set to zero and the wall function recalls the smooth wall function.
In the case where the roughness element heights are much larger than the boundary layer thickness, the fully rough regime forms where the drag significantly increases. In such a condition, the pressure drag on the roughness elements dominates and the impact of roughness becomes independent of Reynolds number which means the viscous effect is no longer important. For a fully rough regime represented by 90 ≤ K + s , the ∆B correction is represented by, The transition regime happens where both viscous and pressure forces on the roughness elements contribute to the wall skin friction. In this condition represented by 2.5 < K + s < 90, the correction reads, In these equations, shape and form of roughness elements are incorporated into the modelling through the C s coefficient. However, there is no clear guideline to adjust this coefficient. It is suggested that it varies from 0.5 to 1 where C s =0.5 corresponds to the uniformly distributed sand grain roughness. If the roughness elements deviate from the sand grains, the constant roughness should be adjusted by comparing the results with experimental data.

Parameters defining the flow properties
The hydrodynamic performance of a propeller is defined by using the non-dimensional thrust and torque coefficients, and the advance ratio, In these equations, D is the propeller diameter, n is the rotational speed of the propeller in rev/sec, ρ is the fluid density, T is the propeller thrust force, Q is the propeller shaft torque, and V A is the mean inflow velocity towards the propeller plane.
To identify vortical structures in the flow, the Q-criterion representing the local balance between shear strain and rotational tensor magnitudes, is employed [45,46], To simplify the cavitation inception detection, the minimum pressure criterion is employed [35]. The criterion assumes that cavitation occurs as soon as the minimum pressure in the flow reaches the saturation pressure.
Therefore, the cavitation inception point, σ i , is determined from the pressure field of the wet flow as,

Case description
The basic design of the propeller is from a research series of five-bladed highly skewed propellers having low effective tip load for vessels where it is very important to suppress and limit propeller-induced vibration and noise. The main, or first occuring source of noise, for this type of propellers, is cavitation in the tip region. In the previous studies conducted by the authors [35,36,42], the computational guideline to successfully model the tip vortex flow around this propeller having smooth surface within OpenFOAM was investigated. In this guideline, the turbulence modelling impact, minimum required spatial mesh resolution for modelling the tip vortex in the near field region and the numerical set up that can provide low numerical dissipation and high stability are discussed. Here, the same guideline for the computational domain and mesh specifications is employed.
The computational meshes were generated by StarCCM+ and then converted into the OpenFOAM format.
As the open water conditions was of interest, simulation of the flow around one blade and then using cyclic boundaries on the sides were possible. However, when the numerical analysis of the model scale propeller was conducted, there were some stability issues with the cyclic boundaries making it very cumbersome to converge.
Therefore, the whole propeller is modelled and in order to keep the number of cells low enough, the tip vortex refinement was applied on one blade only. The utility used to convert StarCCM+ mesh to the OpenFOAM format has a limitation on the number of cells or faces which forced using the Trimmer mesher to create the model scale propeller mesh. For the full scale propeller, using the Trimmer mesher led to some bad quality cells, e.g. wrongly oriented faces or face non-orthogonality more than 85 degrees. This problem is believed to be related to the low numerical precision of reading or writing the points storing the locations of cells and faces.
However, during the study it was not possible to find out where this low reading or writing precision occurs. At the same time as we were able to improve the stability of employing cyclic boundaries, the polyhedral mesher is used to create the mesh around the full scale blade.
The computational domain used for the model scale propeller is presented in Figure 1a. to specify the desired resolution at this region.
In Table 1, the operating conditions and normalised resolution details of the blade and tip vortex refinements are presented. As the inlet velocity is kept constant, the rotational rate is adjusted according to the propeller advance ratio. Similar to the propeller surface resolution, the tip vortex refinement resolutions are presented by the non-dimensionalized terms, i.e. H + 1 = u τ ∆H 1 /ν, H + 2 = u τ ∆H 2 /ν and H + 1 = u τ ∆H 3 /ν. In these equations, ∆H 1 , ∆H 2 and ∆H 3 are the specified cell resolutions in the helical tip refinement regions of H1, H2 and H3.
Based on our previous studies for mitigation of back side tip vortices [34,47], the tip region of the refined blade,     The study consist of the roughness modelling on the blade tip sides, i.e. the back side and the Front side.
For one case where the roughness is only applied on the back side tip region, BS Tip, the mesh topology is modified by removing cells to include the roughness elements into the simulations, Figure 3c. This will provide

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the opportunity to resolve the flow around these roughness elements.  At higher J values, e.g. J=1.26, the main vortex appears as a leading edge vortex formed on the front side of the blade, and it is thus expected the area covered with roughness is different. Therefore, the roughness pattern optimization for this vortex consists of investigation of radial areas defined in Figure 4a. In order to find which part of radial areas will have more impact on TV mitigation, also smaller areas along the leading edge are considered, e.g. as in Figure 4b. The summary of evaluated roughness patterns and arrangements are presented in Table 2. Following our previous studies [34,47], all of the analysis is performed by considering a fixed value for the roughness height, K s =250 µm in model scale condition; this corresponds to K + s = 35. The roughness height is extended into the full scale condition by considering the geometrical scale ratio. This gives K s =3.75 mm in full scale conditions corresponding to K + s ≈ 925.

Results
Open water performance of the propeller in the model scale and full scale conditions are presented in Figure 5.

Mitigation of back side TVC
Performance of the propeller for different surface roughness conditions are presented in Table 3 where the roughness is modelled via the rough wall function. Thrust and torque coefficients as well as the efficiency are presented relative to the smooth propeller condition at J=0.82 where the tip vortex is formed on the back side. In all of the tested roughness arrangements, the results indicate an increase in the torque coefficient when roughness is included. The thrust coefficient, however, is more dependent on the roughness pattern. For the FR blade, the maximum thrust decrease, -13.4 %, and efficiency drop, -16.6 %, are observed. Having roughness on the FS tip leads to higher K t but it also requires a higher K q . This eventually results in a lower propeller efficiency, around -2.5 %. When roughness is only applied on the BS tip, the variation of the thrust and torque is smallest. A true quantitative justification of these results demands uncertainty analysis, but it is anticipated that the trends are correctly capture at this grid refinement level. The results, however, clearly confirm that in order to minimise the negative effects of roughness on the propeller performance, the roughness area should be optimised.  In Figure 14, the predicted cavitation inception based on the minimum pressure criterion is presented for different roughness patterns defined for the back side TVC at J=0.82. As the propeller was not tested at this operating condition, the experimental data is extrapolated to this condition. As expected, the FR condition has the lowest cavitation inception compared to other roughness patterns. The predicted cavitation inception in the BS tip and BS+FS tip patterns is close to each other, and the difference between them is believed to lie in the uncertainty of the numerical results in the current simulations.
Even though the TVC mitigation of BS+FS tip pattern is higher, the BS tip pattern is selected to be the outcome of the roughness area optimisation for the back side TVC as it has a much lower performance degradation . where by resolving the flow field around them more flow physics can be captured. These, however, demands for a finer computational resolution around the roughness elements and also an accurate tomography of the roughened area. As the main objective here is to discuss how results of the roughness wall function approach related to the resolving flow field approach, an arbitary roughness tomography is employed that satisfies the same averaged roughness elements height used in the wall function approach, i.e. K s =250 µm.

Smooth
In Figure 12  For the resolved flow around the roughness elements, the cavitation inception is found to be around 3.28 while with wall-modelling approach the predicted inception point is 4.53. Lower propeller performance is noted for  the resolved flow as well, Table 4. When conducting a comparative analysis, e.g. comparing different patterns, the large difference between the cavitation inception predictions has less importance. But when it comes to find the balance between the cavitation tip vortex and the blade cavitation, the accurate prediction of flow around roughness elements is inevitable.  Very little improvement in TVC mitigation is observed when the roughness is applied on BS of the blade compared to the smooth condition, while the results of FS and FR conditions are found to be similar. This clearly indicates that in order to mitigate the front side TVC, roughness should be applied on the front side.
This agrees with our findings from the back side TVC mitigation where it is noted roughness should be applied on the same side of the blade where TVC forms.
In order to find on which radial distance roughness should be applied to effectively mitigate TVC, different radial patterns are considered, e.g. R7080, R8090, R90100 and R70100. These patterns are illustrated in Figure   4 and described in Table 2. The TVC inception of R7080 pattern is found to be close to the smooth condition Among the tested criterion, the pressure coefficient is found to be the most effective one. Based on the pressure coefficient distribution close to the blade, different roughness patterns are created, Figure 4b. The performance of these patterns relative to the smooth condition is presented in Table 5. The results show in all of the patterns both K T and K Q increase compared to the smooth condition. This has led to higher efficiency in these patterns as well. Even though the variation of efficiency is relatively small and can be assumed to lie in the uncertainty of the numerical results, it indicates that none of the patterns would have a negative impact on the propeller performance. As a result, the pattern that has the highest TVC mitigation is selected as the optimum roughness pattern for the front side TVC mitigation, i.e. RE80100.

Optimised roughness pattern
The optimized roughness pattern is achieved by combining BS Tip pattern obtained from back side TVC mitigation study and RE80100 pattern obtained from front side TVC mitigation study. In Figure 15, the open water performance of the model scale propeller in smooth and optimized roughness pattern (ORP) is presented.
In the presented results, the roughness is modelled via the rough wall function. For J<1.125, similar torque coefficients are predicted in smooth and ORP while the thrust coefficient is lower in the ORP. For larger values of J, the produced thrust in smooth and ORP conditions are similar while more torque is needed in the ORP condition. This leads to having a lower efficiency in ORP condition across all of the operating conditions.
Interestingly, the efficiency curves are found to be similar in smooth and ORP conditions with a small shift downward in ORP.
In Figure 16, the cavitation inception diagram of the model scale propeller in smooth and ORP conditions are presented. The general impression is that application of roughness leads to a wider cavitation free bucket on the side, and the impact on the centre area, e.g. J=1.05, is small. This is expected as in 1.0<J<1.1 operating conditions, the dominant TVC switches from one side of the blade to the other one. This corresponds to have a weak TVC and therefore small impact of roughness on its strength.
More detailed comparison of TVC mitigation and performance degradation is presented in Figure 17 for the back side TVC and Figure 18 for the front side TVC. It can be noted that the average performance degradation for ORP in back side TVC mitigation is around 1.4% and the average TVC mitigation is 14%. The lowest impact of roughness on TVC mitigation is found to be around J=0.93.
Compared to ORP results of the back side TVC, the impact of roughness on mitigation and performance degradation is found to be larger in the front side TVC where the average performance degradation is around 1.8% and the average TVC mitigation is 37%.

Optimized roughness pattern in full scale condition
In Figure 19, TVC mitigation and performance degradation of the full scale propeller at three different operating conditions are provided. These simulations are performed on the propeller with y + = 50 where the roughness elements having K + s = 925 are incorporated into the computational domain, Table 1. Similar to the model scale results, the lowest TVC mitigation and highest performance degradation are found to be at the design point, on the same side of the vortex roll-up is found to be effective. The optimized pattern that can be used across different operating conditions are obtained by simultaneous application of roughness on these two areas.
It is noted that application of roughness leads to a wider cavitation free bucket where its impact in the operating conditions close to the design point is lower compared to the conditions with higher loads on the blade. The

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We showed if the roughness pattern is optimized with respect to the tip vortex flow properties, a reasonable balance between the TVC mitigation and performance degradation can be achieved.

Acknowledgements
Financial support for this work has been provided by VINNOVA through