Numerical simulations of a cavitating propeller in uniform and oblique flow

Abstract

The numerical predictions of a cavitating model scale propeller working in uniform and oblique flow conditions are presented. The cavitating phenomena are numerically reproduced using a homogeneous (mixture) model where three previously calibrated mass transfer models are alternatively used to model the mass transfer rate. The turbulence effect is modelled using the Reynolds Averaged Navier Stokes (RANS) approach. The simulations are performed using an open source solver. The numerical results are compared with available experimental data. For a quantitative comparison the propeller thrust is considered, while for a qualitative comparison, snapshots of cavitation patterns are shown. The thrust values obtained with the three different mass transfer models are very close to each other, however differences in the predicted cavitation patterns are observed. Moreover, some discrepancies between the numerical results and experimental data are reported.

Keywords

Propeller cavitation uniform inflow oblique flow RANS

1. Introduction

Over the last decades due to the steady improvement of Computational Fluid Dynamics (CFD) technologies, as well as computer performances, numerical simulations have become a valuable and reliable tool for design purposes, allowing, in general, the more expensive and time consuming experimental tests to be performed only at the final stages of the project.

In the specific case of marine propellers, CFD analysis can be effectively used to predict overall machine performance as well as to investigate the effects of particular flow phenomena such as cavitation.

However, differences in the predicted results can still be observed depending on the adopted physical model and meshing strategy for instance.

Thus, benchmarks which offer the possibility to compare different CFD approaches/strategies are very valuable for both industry and academia.

In this respect, among others, we cite the E779A and PPTC model scale propellers for which, besides the availability of extended experimental data-set, a lot of results obtained using different approaches are also available in the literature ([4,19,22,24] among others).

For the sake of completeness we point out that the E779A propeller was used as a reference test case in the context of the EU-funded VIRTUE project and later in the Cooperative Research Ships SHARCS project. In both projects, open water and behind flow conditions were simulated. Comparison and analysis of the numerical results obtained by the project partners are available in [19,22].

The PPTC propeller was used as blind test case at the Workshop on Cavitation and Propeller Performance in 2011 and 2015, see [1]. In 2011 the propeller with zero shaft inclination (in the following referred to as uniform flow condition) was simulated while in 2015 the numerical investigations were carried out for the propeller inclined by 12° (namely, oblique flow condition). The numerical results obtained by the various participants are available in [17,18].

In this study, we selected the PPTC propeller to test the developed numerical strategy and in particular to evaluate the effect of the cavitation model (mass transfer model) on the accuracy of the numerical predictions. The evaluation was carried out in both uniform and oblique flow conditions.

The numerical simulations were carried out using the interPhaseChangeDyMFoam solver available in OpenFOAM-4.1, an open-source CFD toolbox [2].

The homogeneous (mixture) model was employed in combination with three different mass transfer models. More precisely, the models originally proposed by Zwart et al. [25], Singhal et al. (known as Full Cavitation Model, FCM for brevity) [20] and Kunz et al. [10] were employed, tuned according to [13,16]. The turbulence effect was modelled using the standard RANS approach in combination with the Shear Stress Transport (SST) turbulence model [12].

Regarding the results, some discrepancies between the predicted and experimental thrust values were observed.

However, the thrust values predicted by the three different mass transfer models were very close to each other. Some differences in the predicted shapes of the cavitation patterns were observed between the simulations performed using the FCM model and the simulations performed with the other two mass transfer models.

In the following, the test case is described and the mathematical model is outlined. Then the numerical strategy used to perform the simulations is presented. The results obtained for the uniform and oblique flow conditions are discussed. Finally, concluding remarks are given.

2. Test case

The PPTC model propeller, employed as the reference test case, is a five bladed, controllable pitch propeller with a diameter of $D_{P} = 0.250$ m.

A significant amount of experimental data, useful for validating CFD predictions/strategies, is currently available in [1]. The performances and local flow details were experimentally investigated for the propeller at zero shaft inclination (uniform flow condition) and at an inclination of 12° (oblique flow condition).

As far as the cavitation measurements are concerned, it is important to point out that for the uniform flow condition the experimental tests were conducted in the cavitation tunnel K15A (Kempf & Remmers) of the SVA Potsdam using a test cross section of 600 × 600 mm [8]. The propeller axis was positioned in the centre of the test section, with clearance from the side walls of $0.7 D_{P}$ .

For the oblique flow condition, a test cross section of 850 × 850 mm [3] was used. In this case the propeller was placed at mid width of the test section, while, due to the shaft inclination, the distance of the blade tips from the upper and lower walls were approximately $0.6 D_{P}$ and $1.8 D_{P}$ , respectively.

In Table 1 the main propeller characteristics are reported. $P_{0.7 R}$ is the pitch at $r / R = 0.7$ , with R being the propeller radius. $C_{0.7 R}$ is the chord length of the propeller section at radius $r / R = 0.7$ . $A_{E} / A_{0}$ is the blade area ratio, $θ_{EXT}$ the skew. $d_{H}$ is the hub diameter.

Table 1
Main geometrical propeller characteristics

$D_{P}$ $(m)$ 0.250

$P_{0.7 R} / D_{P}$ $(-)$ 1.635

$A_{E} / A_{0}$ $(-)$ 0.779

$θ_{EXT}$ (°) 18.837

$c_{0.7 R}$ $(m)$ 0.104

$d_{H} / D_{P}$ $(-)$ 0.300

In this study, for the uniform flow condition, the cavitation phenomenon was predicted at two different advance coefficients J, where $J = V / n D_{P}$ , V being the inlet free-stream velocity and n the propeller rotational velocity. The considered cavitating conditions are summarized in Table 2 where the operating regimes (cases) were labeled as in [17,18]. ρ is water density, ν water kinematic viscosity, ${Re}_{0.7 R} = (C_{0.7 R} / ν) \sqrt{V^{2} + {(0.7 D_{P} π n)}^{2}}$ the Reynolds number at $r / R = 0.7. σ_{N} = (P - P_{V}) / (0.5 ρ {(n D_{P})}^{2})$ is the cavitation number with respect to n where $P_{V}$ is the vapour pressure and P the reference pressure

Table 2

Set-ups for the considered cavitating operating conditions

		Uniform flow		Oblique flow

		Case 2.3.1	Case 2.3.3	Case 2
J	$(-)$	1.019	1.408	1.019
$σ_{N}$	$(-)$	2.024	2.000	2.024
n	(rps)	24.987	25.014	20.000
ρ	(kg/m³)	997.44	997.37	997.78
ν	(m²/s)	$9.337 \times 10^{- 7}$	$9.272 \times 10^{- 7}$	$9.567 \times 10^{- 7}$
$P_{V}$	(Pa)	2818	2869	2904
${Re}_{0.7 R}$	$(-)$	$1.6 \times 10^{6}$	$1.8 \times 10^{6}$	$1.3 \times 10^{6}$

The advance coefficients under consideration were selected before and past the maximum of the efficiency curve (see Fig. 3 in Section 5.1).

For the oblique flow condition the cavitation phenomenon was predicted only at $J = 1.019$ .

3. Mathematical model

In the last decades several CFD approaches have been developed to numerically investigate cavitating flow phenomena. A valuable review of different approaches is for instance provided by [9] and [21] and references therein.

Among all the approaches, the most widely applied today is probably the so-called homogeneous transport-equation based model. This approach was employed in the present study.

In this approach the multiphase flow is treated as a homogeneous mixture of liquid and vapour, with variable density, and the relative motion between phases is neglected. The evaluation of the variable density field is based on an equation for void ratio with the source terms modelling the mass transfer rate due to cavitation, generally known as mass transfer model.

In this study three previously calibrated mass transfer models were used. In particular the models originally proposed by Zwart et al. [25], Singhal et al. [20], and Kunz et al. [10] with the empirical coefficients tuned according to [13,16], were employed. At this stage, it is worth clarifying that in the case of RANS simulations additional transport equations equation have to be solved according to the selected turbulence model. In our case the transport equations for the turbulent kinetic energy, k, and turbulent frequency, ω, implemented in the Shear Stress Turbulence (SST) model were solved. For further details regarding the homogeneous model as well as its implementation in OpenFOAM we refer to [6] for convenience.

4. Numerical setup

The numerical simulations were carried out using computational domains depicted in Fig. 1. In contrast to the experimental campaign, both domains had a cylindrical cross section. The distance of the outer boundary from the propeller centre line at propeller mid plane was, in both cases, equal to $5 D_{p}$ .

Despite the differences in the domain/tunnel cross sections, the numerical simulations were performed following the experimental set-ups summarized in Table 2.

The propeller rotation was simulated using the dynamic mesh motion capabilities of the interPhaseChangeDyMFoam solver. Thus, the computational domains were subdivided into Rotating and Fixed mesh/domain regions.

For time discretization, a first order implicit time scheme was used, while for the discretization of the advective terms a second order linearUpwind scheme was employed.

Fig. 1.

Sketches of the computational domains, in longitudinal plane, used to simulate the PPTC propeller working in uniform (top), and oblique flow (bottom) conditions. Both Rotating and Fixed regions were cylinders.

In current simulations the flow was assumed to be fully turbulent and the two-equation SST turbulence model was employed in combination with the wall functions available in OpenFOAM [2]. The use of the High Reynolds/wall function approach was selected to reduce the computational effort.

For the cavitating flow regimes in question the three different calibrated mass transfer models were used.

It is worth clarifying that the different cavitating flow regimes were numerically set by varying the inflow velocity and the vapour pressure according to the given values of J and $σ_{N}$ .

As regards the computational meshes in this study, the overall computational grids had approximately 3,500,000 cells for both cases. Details of surface meshes are presented in Fig. 2. The boundary conditions, summarized in Table 3, were imposed.

Fig. 2.

Surface meshes for uniform (left) and oblique (right) flow condition.

Table 3

Boundary conditions common for both uniform and oblique flow conditions

Boundary	V (m/s)	$Pressure$ (Pa)	k (m²/s²)	ω (1/s)
Inlet	fixedValue	zeroGradient	fixedValue = 0.06	fixedValue = 2000
Outlet	zeroGradient	fixedValue = 101325	zeroGradient	zeroGradient
Outer	slip	zeroGradient	zeroGradient	zeroGradient
Propeller	movingWallVelocity	zeroGradient	kqRWallFunction	omegaWallFunction

5. Results

Below the numerical results are compared with experimental data. For the sake of clarity we remind that the thrust coefficient, $K_{T}$ , torque coefficient $K_{Q}$ and efficiency $η_{0}$ were defined as $\begin{matrix} K_{T} = \frac{T}{ρ n^{2} D_{P}^{4}}, K_{Q} = \frac{Q}{ρ n^{2} D_{P}^{5}}, η_{0} = \frac{J}{2 π} \frac{K_{T}}{K_{Q}} \end{matrix}$ where thrust, T, and torque, Q, were evaluated in the direction of the rotational axis. It is important to point out that, according to [17,18] in the case of the uniform flow condition the forces were integrated only on the blade surfaces. For the oblique flow condition the forces were integrated on blades and cap.

5.1. Uniform flow

Prior to the cavitating flow predictions, the open-water characteristics measured in the towing tank [18] were numerically predicted, in order to check briefly the capabilities of the employed numerical strategy. The same domain and computational mesh, later used for the cavitating flow predictions, were employed.

The propeller characteristics curves were reproduced within the range of $J = 0.6 - 1.5$ ( ${Re}_{07} = 7.9 - 9.0 \times 10^{5}$ ).

The different operating conditions were set varying the inlet velocity while keeping the propeller rotational speed equal to $n = 15$ rps. Figure 3 shows the predicted performances.

Fig. 3.

Open water characteristics for uniform flow condition.

The numerical results compared well with the experimental data for $J = 0.6$ to 1.0. For the higher values of the advance coefficient the relative errors of the computed thrust and torque values were higher. The errors on the predicted thrust and torque values were very similar to each other, therefore the differences in open water efficiency are small, even up to a J-value of about 1.4.

As shown in Table 2, for the cavitating flow predictions, we considered a loaded propeller condition corresponding to $J = 1.019$ and an unloaded propeller condition corresponding to $J = 1.408$ .

In this case, the cavitation phenomenon was evaluated under the thrust identity assumption at non-cavitating conditions. Since with $J = 1.019$ and $J = 1.408$ the predicted thrust values were underestimated, the inlet velocity was progressively reduced in order to obtain the required thrust. For this reason the advance coefficients, effectively used in CFD simulations for cases 2.3.1 and 2.3.3, were 1.000 and 1.405, respectively.

Observing Table 4 we can see that for case 2.3.1 the thrust coefficients predicted at fully wetted flow and cavitating flow regimes were very similar to each other. Thus, the current numerical simulations were unable to properly predict the loss of thrust due to cavitation for this operating condition.

Table 4

Thrust values for the uniform inflow condition

Case	$K_{T}$ (Exp.)	$K_{T}$ (Zwart)	$K_{T}$ (FCM)	$K_{T}$ (Kunz)
2.3.1 (wetted flow)	0.387	0.386	0.386	0.386
2.3.1 (cavitating flow)	0.372	0.388	0.390	0.388
2.3.3 (wetted flow)	0.167	0.166	0.166	0.166
2.3.3 (cavitating flow)	0.136	0.145	0.148	0.145

It is worth pointing out that a similar trend was also reported by [6,11] and seems to be related to the solver-grid combination. In this respect, recently, [6] using different solvers but the same computational mesh, reported quite scattered results. In particular with StarCCM+ the delivered thrust in cavitating conditions was slightly over-predicted, conversely, with OpenFOAM the reduction of thrust due to cavitation was over-predicted for case 2.3.1.

For case 2.3.3 the numerical simulations predicted in a proper manner the negative influence of the cavitation on the delivered thrust. However, from the quantitative point of view, the predicted thrust reduction was of the order of 11–12% while experimentally a 22% reduction was measured.

Considering the cavitation patterns depicted in Fig. 4, it is possible to note that the current numerical predictions did not properly reproduce the cavitating tip vortex flow observed experimentally for case 2.3.1.

Fig. 4.

Comparison of the cavitation patterns for uniform inflow. The numerical cavitation patterns, obtained using three different mass transfer models, are depicted as isosurfaces of vapour volume fraction $α = 0.2$ . Experimental sketches adapted from [18].

The premature suppression of the tip vortex flow can be associated with the over-estimation of the turbulent viscosity [14].

At this stage it is important to clarify that the detailed resolution of the cavitating tip vortex flow was beyond of the scope of this work. However, we imagine that within the proposed numerical strategy the resolution of the tip vortex flow could be improved using a much finer computational grid as highlighted in [5] for instance. We expect that the results could also be further improved by employing the less diffusive hexa-structured grids [15].

Comparing the results obtained with three different mass transfer models, it is possible to note that the cavitation patterns predicted using the FCM model differed slightly from those obtained with the Zwart and Kunz models. The differences seem to be related to the fact that in the Zwart and Kunz models the cavitation inception criterion does not take into account the effect of turbulence, while in the FCM model the turbulence kinetic energy, a parameter representative of the turbulence level, is present [10,20,23,25].

It is interesting to observe that, for case 2.3.1, the best reproduction of the cavitation pattern observed experimentally was obtained with the FCM. As a matter of fact, with the other two mass transfer models a layer of sheet cavitation on the blade leading edge, not observed experimentally, was obtained.

Conversely, for case 2.3.3, the cavitation patterns were closer to the experimental evidence using Zwart and Kunz model. In this case, the tendency of the calibrated FCM to suppress the leading edge cavitation, observed in this study, lead to a less accurate reproduction of the cavitation pattern.

From the current results it seems that, at least for the test case in question and within the scope of the proposed numerical strategy, the mass transfer model can affect the accuracy of the predicted shape of the cavitation phenomenon, while for the prediction of the thrust only minor negligible differences can be observed.

Finally, we must remember that the current numerical simulations were carried out assuming fully turbulent flow over the blades while in experiments a portion of laminar flow can be present, as pointed out in [22,23] for instance.

At this stage it is not easy to relate the differences in the predicted cavitation patterns to the fully turbulent flow assumption since, for the loaded condition, the cavitation pattern obtained with the FCM model was very similar to the experimental evidence.

This point requires further work. In particular, it could be interesting to quantify/rank the influence of a more advanced turbulence model, capable of taking into account laminar-to-turbulent flow transition, in respect of the influence of the mass transfer model.

5.2. Oblique flow

For the oblique flow condition, except for the propeller arrangement, the simulations were carried out similarly to the uniform flow case. It is important to point out that, also in this case, in contrast to the experimental set-up, an unbounded (very large) domain was used. This approach was assumed to be representative of the actual operating condition of the propeller despite the presence, during experiments, of the cavitation tunnel walls.

The simulations were carried out at $J = 1.019$ . In this case the thrust identity at wetted flow condition was not enforced similarly to the CFD simulations from other sources described in [17]. Table 5 shows the thrust coefficients predicted for both non-cavitating and cavitating flow conditions.

Table 5
Thrust values for oblique flow conditions

J $σ_{n}$ $K_{T}$ (Exp.) $K_{T}$ (Zwart) $K_{T}$ (FCM) $K_{T}$ (Kunz)

1.019 (non cavitating) 0.392 0.411 0.411 0.411

1.019 2.024 0.363 0.383 0.385 0.387

J	$σ_{n}$	$K_{T}$ (Exp.)	$K_{T}$ (Zwart)	$K_{T}$ (FCM)	$K_{T}$ (Kunz)
1.019	(non cavitating)	0.392	0.411	0.411	0.411
1.019	2.024	0.363	0.383	0.385	0.387

From Table 5 we observe that for both non-cavitating and cavitating flow regimes the thrust predicted by the CFD simulations was overestimated. However, it is important to observe that in this case the amount of the thrust reduction due to cavitation was properly reproduced. The CFD simulations predicted, as observed experimentally, a loss of thrust of about 7%.

Figure 5 shows that the cavitation patterns were reasonably well captured considering the adopted numerical strategy. Probably even employing an extremely fine mesh, within the current simulation framework, it would still be impossible to resolve the bubble cavitation observed experimentally. Some important mechanism for bubble break up (such as surface tension, bouncing and coalescence) are not modelled.

As a matter of fact, with the adopted approach the bubble cluster is usually reproduced as sheet cavitation [7,17].

From the current snapshots it is interesting to note that, as for the uniform flow condition, also here the predicted cavitation pattern was influenced by the mass transfer model. From Fig. 5 it is possible to note that the overall cavitation pattern obtained with the FCM differ from those obtained with the other mass transfer models. According to the uniform flow condition the tendency of the calibrated FCM to suppress the leading edge cavitation was confirmed. It is important to point out that the cavitation patterns, obtained with the Zwart and Kunz models, probably better represent the cavitation patterns potentially observed at full-scale for the oblique flow condition as well as uniform flow condition.

Fig. 5.

Comparison of the cavitation patterns at inclined flow for $J = 1.019$ , $σ_{n} = 2.024$ . The numerical cavitation patterns, obtained using the three different mass transfer models, are depicted as isosurfaces of vapour volume fraction $α = 0.2$ . Experimental data from [17].

6. Conclusions

In this study the cavitating flow around the PPTC model propeller was numerically predicted by means of CFD simulations. The study was carried out for the propeller at zero shaft inclination (uniform flow) and inclined shaft (oblique flow) conditions. Selected operating regimes from those proposed at Workshop on Cavitation and Propeller Performance in 2011 [18] and 2015 [17] were used.

The numerical simulations were carried out using the interPhaseChangeDyMFoam solver available in OpenFOAM-4.1. The propeller rotation was simulated taking advantage of the dynamic mesh motion capabilities of the solver. The homogeneous (mixture) model was employed in combination with three different mass transfer models. In particular, beyond the Kunz et al. model available in the standard OpenFOAM distribution, the models proposed by Zwart et al. and Singhal et al. (FCM) were additionally implemented and used. All models were tuned using the empirical coefficients suggested in [13]. The turbulence effect was taken into account using the workhorse SST turbulence model in combination with the available wall functions. As a matter of fact, in this study the flow was assumed to be fully turbulent and thus the boundary layer transition, potentially present in experiments carried out at model scale, was neglected.

The numerical results were compared with available experimental data. In particular, the accuracy of the predicted cavitation phenomenon was quantitatively evaluated considering the delivered thrust. Moreover, for a qualitative comparison, the available experimental sketches of the cavitation patterns were used.

As far as the thrust predictions are concerned, for the uniform flow condition we observed that the current results were, in line with those of [11] obtained using a different numerical strategy.

For the loaded condition (case 2.3.1), in contrast to the experimental campaign, the current simulations predicted a similar thrust for both non-cavitating and cavitating flow conditions. For unloaded condition (case 2.3.3) the thrust at cavitating flow regime was significantly over-predicted compared to the experimental value.

For case 2.3.1, the current result considering [6] seem to be related to the adopted solver-mesh combination, while the significant discrepancy observed for case 2.3.3 is not clear and requires further investigation.

Instead, for oblique flow condition the amount of thrust reduction due to cavitation was properly captured.

In general the differences between the thrust values obtained using alternatively the three different mass transfer models were minimal. However, the cavitation patterns obtained with the FCM differed slightly from those obtained with the other two mass transfer models. In particular, the tendency of the FCM to suppress the leading edge cavitation was observed, a feature which, in this study, revealed to be beneficial depending, naturally, on the operating regime in question.

This characteristic observed in the FCM model, seems to be related to a parameter, representative of the turbulence level, present in this model. As a matter of fact in both Zwart and Kunz models such a parameter is not present.

Thus, as a next step and within the current numerical strategy, it could be interesting to perform additional investigations varying the turbulence model in order to: i) verify the possible influence of the turbulence model on the accuracy of the numerical predictions carried out using alternatively the three different mass transfer models; ii) try to rank the influence of the turbulence modelling in respect to the influence of the cavitation modelling.

Footnotes

Acknowledgements

This work was performed in the context of the UBE 2 (Underwater Blue Efficiency 2) project supported by the Regional Program POR FESR 2014 2020, 1.3.b – Bando DGR 1489/2017, Ricerca e sviluppo – Aree di specializzazione tecnologie marittime e smart health of Regione Friuli-Venezia Giulia. The research was also funded by the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n°612279.

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$D_{P}$	$(m)$	0.250
$P_{0.7 R} / D_{P}$	$(-)$	1.635
$A_{E} / A_{0}$	$(-)$	0.779
$θ_{EXT}$	(°)	18.837
$c_{0.7 R}$	$(m)$	0.104
$d_{H} / D_{P}$	$(-)$	0.300