Wake Instabilities of Tip-Loaded Propellers: Comparison between CLT and “New Generation” CLT Configurations

Abstract

Tip loading is a common strategy to increase the propulsive efficiency of propellers. Solutions such as contracted and tip-loaded (CLT) and “New generation” CLT propellers exploit the presence of an endplate (“true” or as the result of a dedicated modification of the rake distribution) to sustain the increased load at the tip of the blade, at the cost of more complex vortical structures. Their evolution, and the mutual interaction of secondary vortices originated by the endplate itself, however, has not been completely and deeply investigated. The current paper addresses this topic by improved delayed detached eddy simulations (IDDES) of the flow field around two propellers of this type at different loading conditions. The presence of secondary vortices from the endplate root (or from the bended blade at tip), partially observed in recent experiments, is evidenced by high-fidelity CFD calculations. The interaction mechanism with the primary vortices (those from the endplate tip), and the resulting strengthening of the vortical structures, also through the interaction with the blade trailing vortical wake that promote the leapfrogging phenomenon, is discussed as well, comparing the phenomena in the case of two optimally designed geometries (a CLT and a New Generation CLT propeller) exploiting the same pressure side tip-loading concept in a slightly different way. Results show a rather different instability mechanism depending on the endplate configuration and open the discussion on the effectiveness of splitting a single tip vortex into pairs of vortical structures that may induce similar (or even worse) side effects in terms of pressure minima in the wake and earlier wake destabilization.

1. Introduction

Tip loading is a common practice to increase the propulsive efficiency of marine propeller. Over the years, solutions such as the “Kappel” [1,2] or the “Tip Vortex Free” propellers [3], progressively evolved into contracted and tip-loaded propellers (CLT), represent the successful tentative to develop unconventional propulsive configurations exploiting this concept. In both cases tip loading is achieved by a modification of the outer portion of the blade, which is realized by a dedicated fin (the endplate of CLT’s) at the finite chord tip of the blade or by an exacerbated bending of the blade itself (Kappel and “New Generation” CLT, [4,5]). For Kappel geometries the bending is towards the suction side of the blade while in the case of CLT-like propellers the endplate is towards the pressure side. In any case the aim is to reduce the cross flow at the tip, which is one of the primary sources of loading loss (and consequently of efficiency), trying, in principle, not to excessively worsen the cavitating characteristics and the cavitation inception speed of the propulsor.

Over the years, most of the effort in the growth of these solutions has been dedicated to the design phase, developing appropriate strategies from very basic concepts (the “New Momentum Theory” of Gomez et al. [3]) and dedicated lifting line and lifting surfaces approaches [2] and to optimization-based methodologies [4,6,7], which have shown interesting potentialities for the design of propellers capable of increasing the propulsive efficiency while reducing the occurrence of cavitation. A certain effort was spent also on the characterization of the full-scale performances of these propellers, especially of those belonging to the CLT concept, since several studies confirmed the effectiveness of this type of tip loading in realizing a better interaction with the boundary layer in full scale, thus improving the full-scale efficiency beyond what was usually expected for conventional propulsors. Among the others, Sanchez-Caja et al. [8,9,10], or Haimov et al. [11] proposed mostly Reynolds averaged Navier–Stokes equations (RANSE) based calculations of tip-loaded propeller performances exploring the full-scale efficiency. The International Towing Tank Conference (ITTC) Propulsion Committee itself [12], recognizing the need for updated scaling rules for unconventional propellers, recently proposed a dedicated benchmark including a tip-loaded propeller (the P1727 benchmark test case) very similar to the geometries exploited by the “New Generation CLT” to study model-to-full-scale extrapolation and, indirectly, the influence of laminar-to-turbulent boundary layer transition in both experiments and numerical calculations.

But tip loading through the modification of the blade bending, and not by simply increasing the local blade pitch, has some potentially positive consequences worthy of investigation, especially in the context of radiated noise and pressure pulses, which moreover may unveil the hydrodynamic peculiarities and the reasons for the increased efficiency of this type of propeller. The presence of an endplate, or the arched tip of the blade, causes the development of multiple tip vortices which interactions could be significantly more complex than in the case of the single blade tip vortex of conventional (tip-loaded) propellers, but the occurrence of which could explain the postponed cavitation inception speed claimed for these propulsors [6,13]. This vortical system consists, mainly, in the vortex from the endplate tip that is the natural tip vortex of the propeller blade generated by the crossflow occurring between the pressure and the suction side, plus a “secondary” tip vortex, the nature of which is different depending on the type of CLT propeller under investigation. In the case of a conventional CLT propeller, it is generated by the discontinuity represented by the sharp junction between the endplate and the blade, and it is supported by the adverse pressure gradient along the leading edge that, in the end, prevents its merging with the natural tip vortex [14]. In the case of “New Generation CLT”, the secondary vortex originates instead from the severe bending of the blade. The pre-swirl induced by this blade shape, which is helpful in avoiding possible separation of the flow (i.e., efficiency losses) on the outer endplate surface when functioning conditions differ from the design, mitigates the development of the secondary tip vortex [13]. Consequently, it may appear, as seen in [4], as a locally stronger trailing wake rather than a coherent and concentrated structure.

Some of the first studies to provide evidence of the peculiar behavior of these multiple vortices system were by Bertetta et al. [15] and Gaggero et al. [4,7], where cavitation tunnel tests highlighted, through cavitation itself and dedicated laser Doppler velocimetry (LDV) measurements of the flowfield downstream the propeller, the occurrence of the secondary cavitating vortex from the endplate root in addition to the usual cavitating vortex from the (endplate) tip. Similar observations were also made by Brown et al. [6,13] using cavitation as a natural marker of the occurrence of tip phenomena. In both cases (conventional CLT with sharp junction, new generation CLT with bended blade at tip), these propellers seemed able to postpone cavitation inception. The underlying principle of this improvement seems to have been based on the avoidance of the very low pressure in the core of a single, but stronger, vortex through its splitting into these two separated vortical structures, resulting in a combination of the positive effect on efficiency of tip loading with a potentially less severe risk of cavitation. The experimental work of Amini et al. [16] supports this statement since a systematic analysis of an elliptic wing equipped with a number of winglets geometries has shown the effectiveness of these configurations in delaying the inception of cavitation but without any positive effect on the lift-to-drag ratio (i.e., propeller efficiency).

In any case, the number of works dealing with experiments on tip-loaded propellers is relatively scarce in the literature, and from the numerical point of view, analyses are not abundant. Most are based on RANS equations and on relatively coarse grids not suitable for detailed investigations on the dynamics of interacting pairs of tip vortices from which useful insights into these side effects can be obtained. Sánchez-Caja et al. [8,9,10], Shin and Andersen [17], Gao et al. [18] or Gonzalez-Adalid et al. [19], for instance, proposed calculations with the only aim of providing global performances in terms of thrust or cavitating phenomena in proximity of the blades.

Large eddy simulations (LES), or at least approaches less dissipative than RANSE, such as detached eddy simulations (DES), appear to be a fundamental method to numerically study the interactions of the tip vortices of CLT, as already successfully done in many analyses of conventional propellers. Mahesh and Kumar [20], Jang and Mahesh [21], Kumar and Mahesh [22], Asnaghi et al. [23], for example, were among the firsts to apply LES analyses to propellers in order to gain an insight into very complex phenomena, such as the wake instabilities and the propeller dynamics in very off-design functioning conditions, such as reverse-rotating propellers and crash-stop maneuvers. Similar calculations [24,25,26,27,28,29,30,31,32] were proposed in the case of marine propellers (as well as for wind turbines), testing the validity of several subgrid scale modeling in the case of rotating machinery scenarios and the possibilities to predict complex propeller/rudder/hydrofoil interactions relevant for radiated noise predictions. Also, less demanding Detached Eddy Simulations (DES) were used to address these issues, such as the propeller wake evolution and instability. Gong et al. [33] studied the wake of ducted and non-ducted propellers by using DES analyses, which were also applied to the characterization of the performance of non-conventional propellers such as pumpjets [34,35,36] or to the prediction of the radiated noise of propulsors in off-design conditions [37]. Conventional propellers and energy saving devices were investigated as well as [20,38,39,40]. Ahmed at al. [41], for instance, systematically studied the occurrence of wake instabilities in the case of the well-known E779A test case, demonstrating the reliability of such approaches that consequently may be complimentary used to enrich the knowledge of such complex phenomena.

Recently, a couple of papers following this strand of research investigated the wake of CLT propellers [29,30,42,43,44] through high-fidelity numerical analyses, extending the initial investigations and discussions on the peculiarities of the vortical wake of tip-loaded propellers [6,13]. The outcomes of these recent study were not significantly different: the increase of computational resources, mostly, makes possible to enlarge the fine mesh region, including in the numerical investigation the dynamics of the vortical structures also relatively far from the propeller. Calculations, based on LES, remarked the evolution of the secondary tip vortex, and its interaction with that from the endplate. Compared to the RANSE analyses of [15], which evidenced only a change in the pitch of the helical trajectories of the tip vortices, or of the preliminary ones by [6,13] which were even more dissipative to predict the lowest pressure in correspondence of the vortices merging point [45], with the use of large eddy simulations multiple interactions, also with the blade trailing vortical wake, were observed. This led to a possible explanation of the merging phenomenon occurring between them, through a mechanism (a modified version of that originally proposed for conventional geometries [46]) obeying a merging-bending-leapfrogging process not recognized in the case of conventional propellers. Moreover, also the intensity of the pair of trailing vortices from the endplate, after their merging process, was discussed [29] and very accurate calculations showed the occurrence of an unexpected strengthening of the resulting tip vortex (far more intense than that of an equivalent conventional propeller) which may nullify the advantages in terms of delayed cavitation discussed by Brown et al. [6,13].

In the light of these recent developments, and considering the differences in the origin of the secondary tip vortices discussed in the literature [14], the aim of the current analyses is to investigate the vortical structure and the relative destabilization process in the case of two very similar tip-loaded propellers, then enlarging the studies proposed in the literature [29,42,43] that considered only geometries that resembled, in the tip blade shape, the new generation CLT tip-loading concept. Specifically, a conventional CLT propeller [7] and a new generation CLT [4], designed for the same functioning condition, are considered and compared in order to discuss the development of the endplate tip vortices, their interaction with the trailing vortical wake of the blade, and the mechanism leading to wake destabilization. To this aim, eddy-resolving analyses, based on the improved delayed detached eddy (IDDES) formulation of the Spalart–Allmaras turbulence model are proposed on relatively fine grids suitable for capturing these phenomena also in the far wake. Results, computed using the OpenFOAM library [47], follow most of the discussion available in the literature but provide evidence of some substantial differences, ascribable to the different endplate configuration, which are worthy of interest for a deeper understanding of the dynamics of tip vortices.

2. Test Case

The geometries selected for the investigations are those presented in Figure 1. Both were obtained through a design by optimization aimed at improving the propulsive efficiency and reducing the risk of cavitation and induced pressure pulses of a reference CLT propeller initially developed by SISTEMAR S.A. for a twin-screw cruise ship. The reference geometry was designed to provide a design thrust coefficient of 0.218 at a cavitation index (revolution-based), at shaft, of about 3.5. The optimized conventional CLT (Figure 1a), for which the design and detailed characterization is given in [7], ensured an equivalent open water increase of efficiency (at almost constant delivered thrust) slightly higher than 1% with a consistent reduction of the sheet cavity extension at the leading edge (more than 60% in the worst blade angular position, based on unsteady BEM calculations). The new generation CLT (Figure 1b), of which details are given in [4], provided even higher improvements, realizing an efficiency increase (equivalent open-water analyses) higher than 2.5% and a further reduction of the cavitation extension that almost nullified the occurrence of sheet cavitation on most of the functioning conditions under investigation. Moreover, for the specific aim of the current analyses, the realization of the tip loading by the bended tip rather than by the fixed endplate contributed to a substantial change in the vortical structures of the secondary tip vortex (at least based on the preliminary RANSE analyses proposed in the study), which resulted in less strength, compared to that of the reference as well as to that of the optimized conventional CLT, but without any dangerous consequences (i.e., strengthening) on the primary tip vortex from the blade tip.

The main geometrical characteristics of the two propellers are given in

Table 1. Main geometrical characteristics of the test case propellers.

	Conventional CLT	New Generation CLT
Number of blades	6	6
Diameter (model scale)	0.3 m	0.3 m
rhub/R	0.2	0.2
Skew (at tip)	15°	15°
AE/AO	0.824	0.810
c/D 0.7R	0.2864	0.2862
P/D 0.7R	1.198	1.182

. Since they share the same functioning conditions and come from the same reference geometry, they are extremely similar. Both are six-bladed, right-handed propellers, with (the same) moderate skew at the tip and almost identical values of expanded area ratio, chord and pitch (reference at 0.7R), which are very similar when also looking at their radial distributions. The most relevant difference, as discussed, is in the rake used in the new generation CLT concept to realize the tip bending (towards the pressure side) convenient for the tip loading.

3. Computational Details and Grid

For the characterization of the vortical structures shed by the tip-loaded propellers under investigation, calculations were carried out using an improved delayed detached eddy simulation (IDDES) based turbulence model. Since the focus is on the development of vortical structures at the tip of the blades, their interactions and their mutual destabilization process, a model capable of accurately accounting for these unsteady phenomena must be used. Large eddy simulations would represent the obvious choice, but their computational cost suggests the use of hybrid models such as those represented by detached eddy simulation approaches. In general, the aim of DES models, indeed, is to combine the most favorable aspects of RANSE (grid requirements for boundary layer resolution) with those of LES (resolution of time-dependent, three-dimensional large eddy only far from boundaries, then avoiding the smaller structures characterizing the boundary layer itself), avoiding, or at least mitigating, the need of isotropic cells in the boundary layer. This combination is zone-based, since in DES analyses there are two regions in the computational domain that are treated in a RANSE-like and in a LES-like manner, and the switching between the two models is realized by a wall distance-based approach.

In the current analyses, among the available alternatives of detached eddy models (pure detached eddy, DES, delayed detached eddy, DDES, and improved delayed detached eddy, IDDES) simulations, the one selected is the IDDES, inspired by the original RANSE Spalart–Allmaras one equation turbulence model [48]. The Spalart-Allmaras model deals with the turbulent stresses by solving the convective equation for the modified turbulence viscosity $\widehat{\upsilon }$:

$$\frac{\partial \widehat{\upsilon }}{\partial t}+u_j\frac{\partial \widehat{\upsilon }}{\partial x_j}=C_{b1}\left(1-f_{t2}\right)\widehat{S}\widehat{\upsilon }-\left[c_{w1}{f}_w-\frac{c_{b1}}{{\kappa }^2}{f}_{t2}\right]{\left(\frac{\widehat{\upsilon }}{d}\right)}^2+\frac{1}{\sigma }\left[\frac{\partial }{\partial x_j}\left(\left(\upsilon +\widehat{\upsilon }\right)\frac{\partial \widehat{\upsilon }}{\partial x_j}\right)+c_{b2}\frac{\partial \widehat{\upsilon }}{\partial x_i}\frac{\partial \widehat{\upsilon }}{\partial x_i}\right]$$

(1)

That, in turn, is related to the turbulent eddy viscosity ${\mu }_t$, needed to compute the turbulent stresses through the Boussinesq linear assumption, by:

$${\mu }_t=\rho \widehat{\upsilon }{f}_{v1}$$

(2)

where $\rho$ is the fluid density, $\upsilon$ the molecular kinematic viscosity, $f_{v1}={\chi }^3/\left({\chi }^3+c_{v1}^3\right)$, being $c_{v1}$ a constant equal to 7.1, and $\chi =\widehat{\upsilon }/\upsilon$. The diffusion term (the last one on the right end side of Equation (1)) has the classical formulation of diffusion used in turbulence models and depends on the turbulent Prandtl number $\sigma$; the production term is assumed to rise with the increase of the magnitude of the mean vorticity included in the term $\widehat{S}$, while the destruction term depends on the proximity to the walls through the distance $d$ from the field point to the nearest wall. To mitigate the tendency to underestimation of the skin friction observed for this turbulence model, the destruction term is tuned by the use of the $f_w$ function that control its rate of decay.

A detailed description of the model and of all its constants is provided in [48] while the details of its numerical implementation in OpenFOAM are described in the library documentation [47].

In this model, initially developed for RANSE solutions, the Detaches Eddy capabilities, i.e., the zonal RANSE/LES approach, are easily included through a modification of the distance $d$ from the wall that modulates the destruction term of the turbulence viscosity. The wall distance, used in the original RANSE formulation of turbulence model (Equation (1)), is substituted by a new length scale $\tilde{d}$ [49] computed as in Equation (3):

$$\tilde{d}=min\left(\mathsf{\Psi }{C}_{DES}\mathsf{\Delta },d\right)$$

(3)

where $\mathsf{\Psi }$ is the low-Reynolds number correction function, $\mathsf{\Delta }$ a measure of the grid spacing (the largest distance between the cell center under consideration and the cell centers of the neighboring cells) and $C_{DES}$ a calibration constant. During years, this original formulation underwent several updates which were developed to mitigate some of the issues related to the purely grid-based filtering. In particular, the grid-induced separation and the log-layer mismatch were addressed, respectively, by the development of the delayed detached eddy simulation (DDES) and the improved delayed detached eddy simulation (IDDES) models.

The grid-induced separation is a consequence of the “Modeled Stress Depletion”: resolved stresses are too weak as a consequence of a premature switching of the model to LES due to an incorrect (excessively large) stream-wise spacing of the grid. A mitigation of this problem is proposed by the DDES models. To prevent a premature (grid-induced) switching from RANSE to LES already in the boundary layer (where RANSE with their wall-modeling capabilities must be used), a new length-scale distance (Equation (4)) is defined thanks to a “delay function” $f_d$ that has the aim of detecting the boundary layer and maintaining, there, the RANSE nature of the model even if the grid spacing is smaller than the boundary layer thickness:

$$\tilde{d}=d-f_{d}max\left(0,d-\mathsf{\Psi }{C}_{DES}\mathsf{\Delta }\right)$$

(4)

In this formulation, when the function $f_d$ is 0, the length scale dictates RANSE mode to operate, and when the function is 1, instead, LES is applied. The most relevant consequence is that, contrary to the original DES formulation where the dissipation length scale depends solely on the grid, in the DDES it depends also on the eddy–viscosity through the $f_d$ function.

The second issue, i.e., the log-layer mismatch, is addressed in the IDDES model [50], which is the model used in current calculations. In this model an updated formulation of the length scale, making use of the so-called “elevating” function, together with new empirical formulations for the constants of the model, is proposed to include wall-modeled LES capability in the turbulence model and to realize a more robust bridge between wall-resolved and wall-modeled approaches when the grids have moderate values of spacing in the wall units.

Using this “improved and delayed” formulation of the original detached eddy simulation approach proposed by Spalart and Allmaras and based on their initial RANSE turbulence model, calculations on the two tip-loaded propellers have been carried out on a cylindrical domain which dimensions are shown in Figure 2.

The cylindrical domain extends 2D upstream the propeller plane and 7D downstream, having a radial extent of 5D that ensures a blockage well below the prescribed limits. Unform axial velocity is enforced at the inlet and assumed as fully developed at outlet and on the lateral cylindrical boundary while pressure is fixed on these boundaries. No-slip boundary conditions are enforced on propeller blades, hub and shaft, while rotation is achieved through sliding interfaces that match the inner, rotating, portion of the domain with the external, fixed, region. Boundary conditions are listed in

Table 2. Boundary conditions for OpenFOAM calculations.

Patch	Velocity	Pressure	$\widehat{\mathit{\upsilon }}$	${\mathit{\upsilon }}_{\mathit{t}}$$\mathbf{or} \left({\mathit{\mu }}_{\mathit{t}}/\mathit{\rho }\right)$
Inlet	Fixed (depending on J)	$\frac{\partial p}{\partial n}=0$	$\mathrm{Fixed} \left(4·\upsilon \right)$	$\mathrm{Fixed} \left(f_{v1}·\widehat{\upsilon }=6.07·{10}^{-7}\right)$
Outlet	$\frac{\partial U}{\partial n}=0$	$\mathrm{Fixed} (p=0)$	$\frac{\partial \widehat{\upsilon }}{\partial n}=0$	$\mathrm{Fixed} \left(f_{v1}·\widehat{\upsilon }=6.07·{10}^{-7}\right)$
Prop. Blads, hub, shaft	No-slip	$\frac{\partial p}{\partial n}=0$	0 with wallfunction (kqRWallFunction)	0 with wallfunction (nutUspaldingWallFunction)
Lateral cylindrical surface	No-slip	$\frac{\partial p}{\partial n}=0$	$\frac{\partial \widehat{\upsilon }}{\partial n}=0$	$\mathrm{Fixed} \left(f_{v1}·\widehat{\upsilon }=6.07·{10}^{-7}\right)$
Inner rotating/outer fixed region interface	Arbitrary Mesh Interface (AMI)	Arbitrary Mesh Interface (AMI)	Arbitrary Mesh Interface (AMI)	Arbitrary Mesh Interface (AMI)

. The numerical schemes adopted for the simulations, together with their OpenFOAM naming, are described in

Table 3. Adopted discretization schemes.

	Scheme
Time	$\mathrm{Crank}-\mathrm{Nicolson} \left(0.9\right), \mathsf{\Delta }{t}_{equivalent}=0.5 deg.$
Divergence	2nd order for momentum (linearUpwind)2nd order for turbulence (limitedLinear with Sweby limiter)
Gradients	Gauss with multi-dimensional limiter (cellMDLimited)
Laplacian	Gauss with linear interpolation and a limiter

.

This computational domain is discretized using a hexa-dominant grid, some details of which are given in Figure 3. The average grid size on blade surfaces is approximatively equal to 0.4% the propeller diameter while on highly curved surfaces (blades leading edge, but also trailing edge) this dimension is ten time smaller. The average cell size on the wake of the propeller is 0.8% D up to a distance 6.0D aft the propeller plane while local refinements at tip, included to better capture the tip vortex dynamics, maintain a cell size comparable to that on blade surfaces (0.4% of D). This latter refinement across the tip vortex, extends up to 3.5D downstream. Twenty prism layers are used on wall boundaries (blades and hub) in order to ensure, at the design advance coefficient, a near-wall resolution of about 4 wall-units whit a moderate stretching factor of 1.2. This results in a computational mesh of about 48 million cells, solved with a time step equivalent to 0.5 deg. of propeller rotation per step. A total of 22 propeller revolutions were solved, which corresponds to at least 1.5 flow-through times (at the lower advance coefficient of 0.6 considered in the current analyses).

4. Results and Discussion

4.1. Propeller Performance and Cavitation Risk

Prior to analyzing the characteristics and the dynamics of the vortical wake of the tip-loaded propellers under investigation, attention was focused on the propeller performances and on the predicted risk of cavitation. Both the propellers, indeed, were developed through a simulation-based design optimization method using BEM calculations with the aim of postponing as much as possible the risk of cavitation (monitored through the extension of the sheet cavity at the leading edge of the blades) at the maximum possible propulsive efficiency. In this respect, current IDDES analyses can provide a further proof, using high-fidelity calculations, of the successful outcomes of the design process. Moreover, thanks to the less dissipative nature of the proposed calculations, they provide a comparison of the risk only (since a cavitation model is not included) of tip vortex cavitation.

In terms of propeller performances, calculations using the current model mainly confirm the results of previous RANSE analyses ($SST k-\omega$ model, steady simulations using a moving reference frame on a mesh of about 500k cells per blade passage) that are collected and compared in

Table 4. Propellers performances. Comparison with the preliminary RANSE calculations.

	Conventional CLT						New Generation CLT
	RANSE, from [7]			IDDES (Current Analysis)			RANSE, from [4]			IDDES (Current Analysis)
J	KT	10KQ	${\eta }_o$	KT	10KQ	${\eta }_o$	KT	10KQ	${\eta }_o$	KT	10KQ	${\eta }_o$
0.6	0.3648	0.6682	0.5212	0.3756	0.6974	0.5142	0.3612	0.6498	0.5308	0.3657	0.6638	0.5261
0.7	0.3096	0.5882	0.5864	-	-	-	0.3065	0.5758	0.5930	-	-	-
0.8	0.2514	0.5030	0.6365	-	-	-	0.2487	0.4931	0.6422	-	-	-
0.9	0.1911	0.4107	0.6664	0.1940	0.4328	0.6419	0.1892	0.4030	0.6723	0.1894	0.4154	0.6531

. The largest differences can be appreciated for the torque coefficient, which is approximatively 4% overestimated by IDDES with respect to the RANSE (conventional CLT case). Also, the thrust is slightly higher when computed with the IDDES model and this mitigates the influence of the increased torque on the propulsive efficiency, that is in average 3% lower than the RANSE values available in the literature for these two propellers. In any case, a relative improvement close to 2% is observed for the new generation CLT geometry, which is a result perfectly in accordance with the outcomes of the optimization activity and the consequent design process.

Also, from the point of view of (risk of) cavitation, calculations using IDDES qualitatively confirm the features observed in the optimization-based design process. At the design advance coefficient of Figure 4 (but the comparison is easier at the reduced advance of Figure 5, where the increased load exacerbates the suction on the back side of the blade) it is possible to observe a certain risk of cavitation (i.e., by the extension of the low pressure region, $-C_{PN}>3.5$, with $C_{PN}=2\left(p-p_{ref}\right)/\rho N^2{D}^2$) that in the case of the conventional CLT affects the leading edge and the endplate root while in the case of the new generation CLT is limited to the leading edge of the blade only. This is coherent with the load distributions, which are very similar (as it can be seen by comparing the pitch and camber distributions available in [4,7]) and globally determine a similar risk of leading-edge phenomena, and with the way the tip loading concept is realized. The presence of a “true” endplate in the case of conventional CLT determines, indeed, a substantial different flow field at the tip. This is particularly evident on the face side of the propeller, where the presence of the endplate locally increases significantly the value of pressure compared to the smooth transition observable when the new generation CLT concept is exploited, and on the suction side where the sharp geometry of the endplate supports lower values of suction. This different pressure jumps between suction and pressure sides in proximity of the endplate (or of the equivalent endplate of the new generation CLT), moreover, is a possible explanation of the different behavior of the tip vortices observed in the next.

However, the adoption of eddy resolving models, together with their less dissipative behavior, allows for the verification, as discussed in [45], of the strengthening of the suction on the core of the vortices when they merge with each other downstream in the wake of the propeller. Despite the grid resolution not yet being sufficient (and far from the billions of cells used in [29] to accurately analyze these phenomena) to capture the suction on the core of the tip vortices far downstream the propeller blades, the isosurfaces at different pressure coefficients proposed in Figure 6 and Figure 7 (and the detailed view of Figure 8 and Figure 9), respectively, for the loaded and the design functioning conditions, highlight the occurrence of this phenomenon and permit a preliminary discussion on the vortices dynamics and intensity of these two propellers.

The overall view of Figure 6 and Figure 7 denotes the improved performance granted by the new generation CLT geometry, thus confirming the results of the design activity of [4] verified, at that time, only through simpler RANSE calculations not devoted to wake dynamics. At both advance coefficients, the isosurfaces at constant suction are significantly less extended and thick compared to those developing from the conventional CLT endplate, which is a clear proof of the reduced risk of cavitation achieved by the new tip configuration. At the same time, they reveal some peculiar behavior of the tip vortices that are further investigated, in more detail, in the dedicated section.

At both loading it is possible, indeed, to appreciate the strengthening of the tip vortex in the wake of the propellers, as a result of different phenomena. For the conventional CLT, at the design advance coefficient of 0.9, tip vortices identified by the low pressure at their cores “reappear” in the wake approximatively 1D aft the propeller plane, i.e., when the roll-up process between the endplate and the secondary vortex (particularly weak and then not highlighted by the pressure isosurface) is finalized, and the two are finally merged into one, single, tip vortex (see, for instance Figure 8). That is a phenomenon which is not observable in the case of the new generation CLT configuration due to the lower intensity of the tip vortex, not even captured by the highest threshold on −C_PN.

In correspondence with the loaded condition of Figure 7 and Figure 9, the isosurfaces of pressure highlight the peculiar nature of tip-loaded propellers through (equivalent) endplates. The conventional CLT clearly shows the occurrence of both endplate and secondary vortices, while in the case of the new generation CLT only the vortex from the endplate tip is visible (even if the secondary cannot be ignored, when it is tracked using other criteria, such as the second invariant of the velocity gradient tensor). Also, the strengthening effect is clearly observable and ascribable to the leapfrogging and the resulting merging process. In the case of the conventional CLT the overall higher strength of tip vortices is evident from the persistent nature of the tip vortex (after the complete endplate/secondary vortices roll-up), which also in this case has a strength higher than that of the correspondent vortex from the tip of the new generation CLT. For this geometry, indeed, only hints of vortices are visible in the range 0.75D to 2.0D in the propeller wake, since the pressure is reduced (and then trackable by the isosurface) only in correspondence with the intersections between the endplate and the secondary vortices. The strengthening, described in [22] is definitely clear when leapfrogging takes place. For both configurations, the lower advance coefficient corresponds to helical vortical filaments with a reduced pitch and then more prone to interaction because of more close each other. After the roll-up between two (sometimes three) vortices emanated from consecutive blades, the resulting structure loses the coherence of typical of helical filaments and undergoes a reduction of the pressure in its core, measured by the thicker −C_PN isosurface originated far in the propeller wake.

4.2. Wake Evolution and Instability

A detailed description of the instability phenomena affecting the two CLT propellers is given by the isosurfaces of the second invariant of the velocity gradient tensor (Q factor) of Figure 10, Figure 11, Figure 12 and Figure 13 and by the visualization of the instantaneous vorticity field (on longitudinal and transverse sections) collected in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20. Isosurfaces of Q permit a general overview of the most relevant phenomena on the wake of the propellers, with particular emphasis on the tip vortices, while the vorticity fields highlight the vortex sheets from the blades and their interaction (and influence on the stability) with the tip vortices themselves.

The analyses of Figure 10 and Figure 12, plus the detailed view at tip (Figure 11 and Figure 13), clearly show a slightly different behavior of the tip and secondary vortices between the two endplate solutions, that partially agree with the observations of Posa [29] and Wang et al. [42,43], that indeed analyzed a tip-loaded propeller more similar to the new generation CLT concept than to a conventional CLT, but with a bit more exacerbated bending of the blade tip. At the design advance coefficient, tip vortices from both the propellers maintain a stable nature also relatively far in the wake. In the case of the conventional CLT, a clear roll-up process occurring between the endplate and the secondary vortex is well visible and both vortices have their clear and distinct nature when they detach from the endplate root and tip. The merging process, that takes place about 0.4D aft the propeller plane, strengthens the resulting vortex (as seen in Figure 8 looking at the pressure isosurfaces) that consequently increases its stability [29,41] and maintains a clear and coherent helical shape up to 3.0D in the wake regardless the relatively high number of blades (6) that naturally reduces the distances between consecutive vortical filaments with obvious consequences on their (easier) interactions through mutual inductance and leapfrogging [51,52]. Vortical structures of the new generation CLT are not particularly different at this loading condition, except for the secondary tip vortex that is not distinctly evidenced by the isosurface of Q since it appears more like a thick and more intense blade trailing wake rather than a pure vortex. Moreover, it has a rather different (higher) pitch that, on one side, prevents its complete roll-up with the endplate tip vortex with the consequent formation of a more intense merged vortex (the isosurface of Q = 50,000 s⁻² of this propeller, in this regard, is about one rotation shorter). On the other hand, it supports the interaction with the trailing vortex sheet [22,29,41,43,53] and the formation of the secondary vortical structures that, by bridging tip consecutive tip vortices (Figure 14b), in the end anticipate the destabilization of the vortical filaments.

The typical occurrence of short waves, long waves and mutual inductance [51,52] as precursor to the destabilization of tip vortices can be appreciated too for both geometries. Short waves instability, also as a consequence of the not yet sufficient grid resolution, are barely visible as the small oscillations of the helical vortex occurring approximately between 1.5 and 2.0D from the propeller plane. Long waves are, instead, well distinguishable. Their occurrence, between 2 and 3 complete revolutions (about 3.0D aft the propeller), trigger a mutual inductance instability that rapidly leads to leapfrogging [46,51] and further merging of the vortices in the wake.

The loaded condition (J = 0.6) of Figure 11 and Figure 13 definitely clarify the instability process and the differences occurring between the two endplate configurations. In this case the presence of an endplate tip plus a secondary tip vortex is obvious for both geometries but their initial interaction, i.e., their mutual roll-up process observed for this type of tip-loaded geometries [29,43] is moderately different. The secondary tip vortex of the conventional CLT is the dominant vortical structure around which the vortex from the endplate tip rolls-up. This is also consistent with the pressure observations, that highlighted the presence of the most severe suction in correspondence of the endplate root, where the sudden pressure jump favors the generation of intense vortices. The merging process is similar to what observed at the design functioning and coherent with the calculations of Wang et al. [43] and Posa [29], but also with the preliminary calculations, using a much coarser grid and RANS Equations, aimed at validating the outcomes of the design activity of the propeller itself [4,7]. The secondary vortex has, moreover, a higher pitch and this determines more stretched trajectories that finally contributes to its more persistent stability (being consecutive vortices more distant each other) observed for this conventional CLT configuration.

The merging process of vortices from the new generation CLT is, instead, only partial and takes place not solely between the endplate and the secondary vortices trailed by the same blade. Differently from the conventional CLT, the smooth bending of the blade towards the pressure side is responsible of the generation of a pre-swirl flow that mitigates the strength of the secondary tip vortex [2,14]. This, incidentally, is the phenomenon exploited to improve the performance of the new generation CLT propeller [5]. Even if visible, indeed, the strength of this vortex appears lowered if compared with that (also in terms of the vorticity magnitude of Figure 15) of the conventional CLT and, as observed at the design advance coefficient, it looks more similar to an intense helical sheet trailing vortical wake rather than a single, concentrated, vortex. The vortex from the endplate tip, consequently and contrary to the conventional CLT case, is the dominant vortical structure of this blade but it has a significantly lower pitch compared to that of its secondary vortex (and that of the equivalent conventional CLT). The consequences are illustrated in Figure 13. The higher pitch of the secondary vortex determines a partial roll-up very close to the trailing edge of the blade (also as a consequence of the higher load and the significant contraction of the wake as a whole) and several interactions with the endplate vortices from (in some cases) the (two) previous blades. The new generation CLT propeller, at least with this choice of moderate bending of the blade tip and at this loading condition, acts more like a conventional propeller with a unique tip vortex interacting with the intense trailing wake of the blade, following the well-established trailing wake/previous tip vortex interaction mechanism illustrated in [46] or in [53].

The occurrence of short- and long-wave instability and their precursor effect on mutual tip vortex interactions, leapfrogging and breakup of vortical structures, is anticipated with respect to the design condition and quite different between the two propellers. Short-wave instabilities of the conventional CLT are visible approximatively 1.0D from the propeller. Long-wave instabilities follow the short ones fairly quickly, activating a leapfrogging and merging between two pairs of consecutive filaments that, by attracting the remaining non-merged vortices, realizes a further leapfrogging and the final breakup of the tip vortical structures. For this propeller, the mutual inductance of tip vortices, as described in [51], seems to play the major role, compared to the complete roll-up and interaction with the sheet trailing vortical wake [22,53], for the wake destabilization.

Short-wave instabilities in the case of the new generation CLT seem, instead, clearly induced by the interactions with the secondary vortex/trailing wake of the blade at tip. As described in [41] for the conventional E779A propeller under heavy loading condition, the short-wave instabilities of the endplate tip vortex occur after its interaction with the secondary vortex/helical trailing wake detached by the subsequent blades, which sometimes, after detaching from their parent tip vortices, seems to “skip” the interaction with the next ones. This is exactly observed for this tip-loaded propeller where the secondary tip vortices, “moving” downstream with a much higher pitch, influence the endplate tip vortices of at least two previous blades.

The reduced distance between vortices, fostered by the reduced pitch of endplate tip vortex, promotes the mutual inductance that, at least for this instantaneous snapshot of the vortical wake, results in the sudden pairing of three filaments quickly leading to vortex breakup.

The destabilization of the hub vortex, instead, seems not particularly influenced by the dynamics at the tip, but is significantly different from that observed at the design advance coefficient. In this loaded condition, a relatively thick hub vortex undergoes destabilization through pure wave oscillation, of which the occurrence is reasonably related to the commencement of the instabilities of the tip vortices. At the design advance coefficient of Figure 10, instead, spiraling is the phenomenon that characterizes the hub vortex as a consequence, probably, of the interaction with several blade root vortices not yet merged in a single, stronger (and then, stabler) hub vortex.

A final insight into the dynamics of the vortical systems (tip but also blade trailing sheet vortices) of the two propellers is given by the longitudinal and transverse distribution of vorticity that allow for the analysis not only of the dynamics of the (stronger) tip vortices but also of the features of the (less intense) helical sheets of vorticity from the blades, their evolution, and their influence on the destabilization of tip vortices.

A general overview provided in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 reveals the distinctive features of these propellers. Slipstream contraction, especially at the lower advance coefficient, is observable while the vorticity magnitude provides a further measure of the intensity of vortices shed by the different endplate configurations. The conventional CLT always shows stronger tip vortices (the endplate tip vortex at design condition of Figure 14, the secondary tip vortex at the reduced advance of Figure 15) than the new generation CLT, which are thicker and more persistent in the wake. The convergent nature of its endplate [7,9] causes the stronger wake contraction. Moreover, a progressive bending [46] of the trailing wake sheets moving from the near to the transition and finally to the far wake, is distinguishable. This bending, discussed also for conventional propellers [22,41] and observed experimentally in several measurements [53], is one of the responsible of the destabilization process of the trailing vortices, especially in the case of the new generation CLT, following a mechanism comparable to those very similarly outlined in Posa [29] or Wang et al. [43]. In the near wake, indeed, it is possible to observe the helical sheets wakes well attached to their tip vortex. Moving to the transition wake, and with the increase of the propeller loading, the trailing wakes are convected and distorted, accelerating their separation from their parent tip vortex. This process is obviously facilitated at lower advance coefficients, when the propeller-induced velocities are higher in the slipstream, but also the interactions with tip vortices promote the deformation of the trailing wake which is attracted in the outer region by the intensified merged vortices, triggering and feeding a mutual influence that finally leads to destabilization of the wake. The progressive separation of the trailing wake from its tip vortex and its bending toward the outer region of the slipstream determines the trailing wake/previous tip vortex interaction described in [53], which in turn is responsible for the secondary vortical structures bridging the tip vortices. This phenomenon is particularly evident for the new generation CLT, already at the design advance coefficient of Figure 14, thanks to the nature of the secondary tip vortex of this configuration which, as discussed above, is more similar to a locally intense trailing blade wake rather than to a concentrated tip vortex. For this propeller, the trailing wake interaction with tip vortices seems, then, the most important driver in wake instability. The separated sheet, that “moves” downstream quickly thanks to a significantly higher pitch (Figure 13), easily interacts with multiple tip vortices, inducing destabilization earlier than other cases. The trailing wake, indeed, is composed of two layers of opposing signs due to the boundary layer on blade pressure and suction side that generates separated sheet layers [41,53]. The earlier interactions, already in the near wake, take place between the tip vortex and the opposite strength layer from the pressure side, thus fostering the destabilization phenomena (Figure 16). This bending process is visible also for the conventional CLT, but the deformation of the trailing wake, despite the more intense tip vortex and approximatively the same load, is less severe. The secondary bridging vortical structures between consecutive tip vortices, even if present especially at the lower advance coefficient, are less intense and suggest, as proposed in [51], the usual tip vortices mutual interaction/leapfrogging mechanism as the driver of wake instability for the conventional geometry. A further proof of the trailing wake/previous tip vortex interaction is given in Figure 17, Figure 18, Figure 19 and Figure 20, where the transverse sections of the streamtube show the tangential/circumferential deformation of the trailing wake, that is maximum again in the case of the new generation CLT. In this respect, the new generation CLT propeller under investigation behaves similarly to the calculations of Kumar and Mahesh [22] and Wang et al. [43] which observed fully rolled up helical sheets also in transverse direction contributing to the destabilization of tip vortices. Especially compared to the analyses of the P1727 ITTC Benchmark test case geometry [43], which indeed is a tip-loaded propeller realized with the progressive bending of the blade tip, the new generation CLT is characterized by a very similar merging/stretching process, well visible in Figure 18 and Figure 20: the merged tip vortex is stretched by the outer trailing wake of the blade which progressively breaks into outer vortices promoting the tip vortex instability, and inner vortices interacting with the hub vortex.

This latter phenomenon is, for the propellers under investigation, less evident. The analyses by Ahmed et al. [41] on conventional (but heavily root-loaded) propellers, and also by Posa [29,30] (even if with a shaft downstream) on comparable tip-loaded propellers, evidenced the occurrence of some inner trailing wake interactions, that in the case of the conventional propeller [41] were identified as the earlier cause of wake instability. For the conventional and the new generation CLT propeller of this study, only a hint of these interactions between helical sheets in the inner portion of the streamtube can be evidenced, and especially in correspondence of the lightly loaded condition at the design coefficient. The instantaneous snapshots of the vortical structures, indeed, show some bending of the trailing wake close to the hub (or the persistence of some vortical structures from the blades roots, as per Figure 11), contributing to the spiraling nature of the hub vortex, that completely disappear at the reduced advance coefficient where trailing wakes are completely stretched along the “wavy” hub vortex, in a way similar to that observed in [41] at the highest loading of their conventional propeller. The fragmented nature of the trailing wake (i.e., the smaller vortical structures in place to the continuous distribution of vorticity) is absent as well. The non-occurrence of this phenomenon, discussed in [22] as a possible explanation of the instability of the wake and also evidenced for the tip-loaded propeller of [29], can be related to the reduced load at the root and to its spanwise distribution, which is more uniform for these types of propellers compared to conventional geometries. For the same reason, the coherent and long-standing mid-span vortices observed in [29,43] (or the additional vortices from the suction side of the blade, shown in [26]), consequent to the roll-up process of the inner trailing wake, which in this case is very weak, are hardly visible in the longitudinal and transverse sections of the streamtube and can be evidenced only by the very low threshold on the Q-factor.

5. Conclusions

Improved delayed detached eddy simulations were used to analyze and compare the tip vortices generated by two tip-loaded propellers realizing the tip loading by two different choices of endplate, i.e., by the use of a conventional endplate (conventional CLT configuration) or by the use of a smooth bending of the blade tip (a sort of additional/increased rake at the tip of the new generation CLT configuration). Calculations at two advance coefficients were in agreement with the experiments discussed in [6,15], which both reported, through the visualization of vortical structures by cavitation, the occurrence of a pair of tip vortices, one from the endplate tip, the other from the endplate root (i.e., the endplate/blade junction in the case of the conventional CLT, the location with the highest curvature of the blade tip for the new generation CLT propeller).

Moreover, thanks to the less dissipative nature of the adopted turbulence model, some peculiar features of these propellers, in particular related to the destabilization process of the trailing wake, were evidenced and some relevant differences between the two configurations (and those discussed in the literature) were identified. These differences concern the strength of the tip vortices and their interaction process, including also the role of the helical trailing wake. If, for both the propellers, the merging process between the endplate and the secondary tip vortices was observed to occur relatively early, already in the near wake within half a diameter even at the highest advance coefficient, the relative importance of the endplate tip vortex compared to the secondary was different, as was the strength of (merged) vortices shed by the two propellers.

The conventional CLT evidenced a stronger endplate tip vortex at the design condition that instead behaved as a non-essential vortical structure at the loaded condition, where the secondary tip vortex from the endplate root had the highest intensity and drove the destabilization process mainly by tip vortex mutual inductance and leapfrogging.

The new generation CLT propeller, despite providing the same thrust with a very similar spanwise load distribution, was characterized by a completely different secondary tip vortex, which was always less intense than the vortex from the endplate tip. Moreover, this secondary vortex was prone to multiple interactions with the tip vortex through skipping of several tip vortices from the preceding endplate blades and by bridging secondary vortical structures among these vortices. For this endplate configuration, the destabilization process seems initiated by the interactions between the trailing wake and the previous tip vortex, supported by the progressive bending of the trailing wake itself under the action of the increased merged tip vortex intensity.

As a whole, under the same loading conditions, the tip vortices of the conventional CLT propeller were more intense, resulting in a slower instability process. In this respect, calculations proved the effective pre-swirl action of the secondary vortex in the case of the new generation CLT propellers (and, more generally, of the tip-loaded propellers realized with a smooth bending of the blade tip), that was positive for the mitigation of tip vortex intensity (i.e., cavitation) at the cost of an earlier and more chaotic wake destabilization.

These outcomes were discussed also in terms of the pressure distributions in the propeller wake, which were useful in highlighting the occurrence of vortex merging (endplate and secondary vortex but also mutual inductance/leapfrogging) through the sudden reduction of the pressure in the core of the vortices. This feature is of particular importance since it opens the discussion on the effectiveness of tip loading through endplates to delay the inception of cavitation by splitting the single tip vortex into a pair of weaker structures. A detailed comparison should be carried out with an equivalent conventional propeller designed with the same inputs, but the behavior of the new generation CLT, that in its current configuration with a moderate bending of the tip is relatively close to a conventional geometry (see, for instance, the discussion on the nature of the secondary endplate tip vortex), could provide useful insights. In off-design conditions, the presence of the endplate, having a convergent shape to accommodate the streamtube contraction, could induce, through separation, much more intense tip vortices, in the end nullifying the potentialities of tip loading. The adoption of the bended tip, despite the similar issues that could arise in the case of very extreme tip blade deformations, seems more robust against variations of the functioning condition. These could be the subject of additional investigations aimed at confirming, with a wider and systematic comparison, the benefit of tip-loaded propellers or to provide the necessary design guidelines to be somehow included into simplified methods such as the optimization-based ones used to define these geometries. To this aim, and by exploiting the higher level of fidelity of such methods successfully applied for wake dynamics characterization, truly cavitating analyses of tip vortex inception, as well as the monitoring of pressure pulses and radiated noise, could shed further light on the consequences of the multiple vortical structures at blade tip and on the overall effectiveness of tip-loaded geometries for applications where propulsive efficiency is only one of the design requirements.