Marine propellers performance and flow-field prediction by a free-wake panel method*

GRECO Luca, MUSCARI Roberto, TESTA Claudio

CNR-INSEAN, Italian Ship Model Basin, Rome, Italy, E-mail: luca.greco@cnr.it

DI MASCIO Andrea

CNR-IAC, Istituto per le Applicazioni del Calcolo “M. Picone”, Rome, Italy

Marine propellers performance and flow-field prediction by a free-wake panel method*

GRECO Luca, MUSCARI Roberto, TESTA Claudio

CNR-INSEAN, Italian Ship Model Basin, Rome, Italy, E-mail: luca.greco@cnr.it

DI MASCIO Andrea

CNR-IAC, Istituto per le Applicazioni del Calcolo “M. Picone”, Rome, Italy

(Received August 6, 2013, Revised April 30, 2014)

A Boundary Element Method (BEM) hydrodynamics combined with a flow-alignment technique to evaluate blades shed vorticity is presented and applied to a marine propeller in open water. Potentialities and drawbacks of this approach in capturing propeller performance, slipstream velocities, blade pressure distribution and pressure disturbance in the flow-field are highlighted by comparisons with available experiments and RANSE results. In particular, correlations between the shape of the convected vortexsheet and the accuracy of BEM results are discussed throughout the paper. To this aim, the analysis of propeller thrust and torque is the starting point towards a detailed discussion on the capability of a 3-D free-wake BEM hydrodynamic approach to describe the local features of the flow-field behind the propeller disk, in view of applications to propulsive configurations where the shed wake plays a dominant role.

BEM hydrodynamics, free-wake analysis, BEM-RANSE comparison

Introduction

A hydrodynamic formulation based on a Boundary Element Method (BEM) is herein presented and applied to marine propellers in open water conditions. Aim of the work is to investigate the capabilities of a 3-D free-wake panel method in predicting the behaviour of propellers in terms of delivered thrust and torque, velocity field downstream the propeller disk, blade pressure loads and flow-field radiated noise. In the framework of potential flows, it is well known that propeller hydroloads strongly depend on the vorticity field released from the blade trailing edge (potential wake), especially for low-speed conditions, where the vortex-sheet is closer to the propeller plane. In the past there has been a growing research interest in correlating both wake pitch and tip-vortex roll up with propeller operating conditions, early relevant investigations, carried out by potential flow methods, are due to Kerwin and Lee[1], whereas semi-empirical procedures to tailor wake pitch and contraction to flow conditions and propeller loading are discussed, for instance, in Hoshino[2]. In the attempt of detecting the wake shape as solution of the potential flow hydrodynamics, Greely and Kerwin[3]and later Kinnas and Pyo[4]proposed a devoted vortex lattice method whilst Liu and Colbourne[5]suggested a BEM approach coupled with a suitable wake surface relaxation scheme. More recently BEM hydrodynamics combined with efficient wake alignment procedures have been presented to study marine propellers in open water and unsteady conditions due to manoeuvres or hull wake onset flow[6-8]. These works well highlight the capability of a free-wake BEM solver to predict propeller performance in terms of blade pressure distribution and thrust/torque. However, very few attempts of verifying flow-field velocity features, pressure signals behind the propeller disk or wake shape exist; an example is provided in Liu and Colbourne[5], where comparisons between numerical results and available experiments are shown only in terms of azimuthally-averaged velocity fields. Although satisfactory BEM propeller performance predictions may be obtained through rough vortex-sheet modelling (especially close to the design condition, where the rigid-wake approach is widely used), the study of many problems of practical interestneeds accurate wake shape predictions to yield a detailed description of the induced velocity perturbations. This is particularly true when simplified hydrodynamic propeller models, based on airfoil theories corrected with the induced-velocity field due to propeller wake, are used as fast and reliable solvers in preliminary/optimal design process[9], as well as in the hydroacoustic analysis of propellers through the Bernoulli theorem[10]or in the study of unsteady hydrodynamic loads due to body-wake interaction[11].

In view of the above considerations, this work presents a comprehensive assessment of a fully 3-D free-wake BEM formulation for marine propeller hydrodynamics. A zero-th order BEM is here combined with a devoted wake alignment procedure to detect the potential wake evolution, whereas the Bernoulli theorem is used for propeller hydroacoustics. The analysis of flow-field quantities like downstream vorticity (i.e., wake shape), velocity and pressure disturbances is investigated in details. The proposed BEM methodology has been developed during the past years and applied to study marine propellers cavitation[12]and hydroacoustics[13]. For a four-bladed propeller in open water, validation results are herein provided through comparisons with experimental data including propeller open water curves and wake-field flow measurements by particle image velocimetry (PIV) technique. In addition, computations provided by an extensively validated RANSE solver[14,15]are used as reference results.

The proposed formulation is not a novelty in the context of potential flow hydrodynamics. However, in authors’ opinion, such a thorough validation study, covering global and local propeller flow aspects, as well as experimental and computational reference data, may provide a useful guideline on the effectiveness and robustness of free-wake BEM-based approaches to face the analysis of marine configurations like propeller-rudder, pulling pods and contra-rotating propellers, where the propeller-induced wake plays a crucial role.

1. Theoretical models

1.13-D free-wake Boundary Element Method

Fig.1 INSEAN E779A model propeller: 3-D view and definition of fixed frame of reference (FFR) and rotating frame of reference (RFR) at time 0t＞ (right-handed screw)

After discretization of SBand SWinto surface panels and enforcement of Eq.(1) at the centroids of the body panels, the application of a zero-th order BEM yields a linear set of algebraic equations in terms of φ on the body surface. The pressure field upon the blade(s) is then computed by the Bernoulli equation that, written in the rotating frame of reference fixed to propeller blades (see Fig.1), reads

where wake grid points are moved parallel to the local velocity field during the pseudo-time step Δt, (4) BIE solution update (see Eq.(1)) and further ?φ computation on the updated SWshape. The procedure is iterated up to convergence. The velocity field on the wake is evaluated by taking the gradient of Eq.(1) enforced on wake grid points

Table 1 INSEAN E779A model propeller: geometry details

2. Numerical results

The proposed 3-D free-wake BEM approach is hereafter applied for the analysis of marine propeller performance and induced flow-field features in open water. By virtue of extensive CFD and experimental studies carried out at INSEAN in the past, the INSEAN E779A right-handed model propeller is considered[26]. A 3-D view of the propeller model is depicted in Fig.1 whereas basic geometry details are summarized in Table 1.

Fig.2 RANSE grid topology: overview (left) and near field (right)

Fig.3 Details of the volume mesh for the RANSE solver. Section yF/R=0 (left) and xF/R=0 (right)

Figure 2 shows the building blocks of the mesh: the one around each blade and the hub is built with an“O-topology” whereas toroidal blocks cover the whole background. An idea on the cell clustering is given in Fig.3, where two slices in the planes yF/R=0 and xF/R=0 are depicted. Note that a suitable fine grid is used to discretize the computational domain up to 4.4 radii downstream the propeller disk, within this region the RANSE solver is expected to properly capture the main flow features, like tip and hub vortices. Outside (see the buffer zone in Fig.3) a coarser mesh is used. The far field, where the inflow and outflow boundary conditions are enforced, is described by an even coarser mesh that extends about 23 radii upstream, 23 radii downstream and 16 radii in the radial direction. About 32 cells are put within the boundary layer thickness, the first point being located at a distance from the wall such that y+＜1 in wall units. The different blocks sum up to a total of 11M cells.

2.1Propeller performance

In this section BEM and RANSE hydrodynamic models are applied to predict propeller thrust T and torque Q. The following definitions of nondimensional force and moment coefficients are used: KT=T / ρn2D4, KQ=Q/ρn2D5, where n denotes the rotational speed. As previously stated, the wake shape plays a crucial role in the framework of potential flows. In this context, Fig.4 shows a comparison between propeller performance coming from the freewake (fw) and prescribed-wake (pw) BEM approach. Note that the prescribed wake pitch is here set using the experimental open water propeller thrust data as input of the Momentum Theory.

Fig.4 Performance results by BEM: prescribed wake (pw) and free-wake (fw) predictions compared

As shown, such ad hoc prescribed wake model is fully adequate to capture the effects induced by the vortex-sheet on propeller hydrodynamic loads, for advance ratios J=v0/nD greater than 0.5. Differently, at J＜0.5, this modeling yields slightly underestimated loads, especially in terms of torque. Unless differently specified, the following BEM computations refer to the free-wake algorithm combined with the Iterative Pressure Kutta (IPK) technique to enforce Δp=0 at blade blunt trailing edge (see Appendix A). Comparisons between BEM predictions (with and without the IPK algorithm) and available experiments are depicted in Fig.5, as shown, better agreement is achieved by BEM-IPK algorithm, especially for J＜0.88 (design value). At J＞0.88, 3D-flow effects decrease and, in turn, the application of the iterative Kutta condition is not needed. As it shall be clear in the following, increasing discrepancies at J＜0.6 are mainly due to the formation of a leading edge vortex in the outer sections of the blade not modeled by the present BEM formulation, where the detachment of the trailing vortex is enforced at the blade tip.

Fig.5 Free-wake BEM results (with and w/o IPK condition) compared to experiments

Fig.6 Performance results by BEM compared to experiments and RANSE computations

Figure 6 compares BEM-IPK outcomes with those computed by the RANSE solver and experiments. As expected, RANSE solver behaves better than the BEM-IPK at J＜0.6, whilst for J＞0.6 both approaches exhibit a good agreement with experiments. For the sake of completeness, Table 2 summarizes BEM, RANSE and experimental outcomes, at three advance ratios, namely 0.3, 0.6, and 0.88.

Table 2 BEM and RANSE thrust and torque predictions compared to experimental data

2.2Propeller flow-field features

Predictions of the flow-field downstream the propeller disk are presented in this section. Nondimensional velocities respect to nD are shown. In the Fixed Frame of Reference (FFR) shown in Fig.1, axial and radial velocities, Vxand Vr, are positive along the xFaxis and outward, respectively, whereas the tangential velocity Vtis positive clockwise, as seen from downstream.

The design condition (J=0.88)is first analyzed. The capability of a rigid-wake modelling in detecting flow-field features downstream the propeller disk is discussed by comparing results from the free-wake approach and the ad hoc prescribed wake type.

Fig.7 BEM prescribed and free-wake locations on transversal planes at xF/R=0.2 (top) and xF/R=1.15 (bottom), J=0.88

Fig.8 Perturbation velocity for J=0.88 on planes at xF/ R=0.2 and xF/R=1.15, radial position r/ R=0.7. BEM prescribed (pw) and free- wake (fw) results are compared

Outcomes, depicted in Fig.7, in terms of wake locations on transversal planes at xF/R=0.2 (top) and xF/R=1.15 (bottom), highlight the inadequacy of prescribed wake modelling, because, except close to the propeller disk, wake angular positions and overall shape do not match those predicted by the freewake solution. In addition, surface roll-up near the blade tip is completely missed. These discrepancies strongly affect the prediction of the downstream velocity field, as shown in Fig.8 where axial (top) andradial (bottom) velocity components, along a circle of radius r/ R=0.7 on the above transversal planes, are plotted. Velocity distributions at xF/R=0.2 are comparable, whereas moving further downstream the two wake models yield completely different results. Recalling propeller loads predictions in Fig.4, it may be noted that KTand KQvalues predicted by the two alternative wake models show negligible differences at J=0.88. This is not surprising because blade loading is relatively low (KT=0.157), shed vortices have a limited strength and hence only the wake portion close to the propeller affects the hydrodynamic loads. These findings confirm that a rigid wake modelling (even if tailored to the particular case), suited for the analysis of propeller performance, may be not adequate to capture the vortex-sheet shape and, in turn, the downstream flow-field velocities. For a reliable prediction of these characteristics the use of a freewake model is mandatory, in particular for lower values of the advance coefficient.

Fig.9 Vorticity distribution downstream the propeller on yF/R=0 plane for J=0.88. PIV results (bottom) compared to RANSE computations (top) and trailing wake surface position predicted by BEM (black lines)

Fig.10 Tip-vortex location by BEM (left), tip-vortices location predictions by BEM (black lines) and RANSE for =J 0.88 (right)

Fig.11 Prediction of tip-vortices location on plane yF/R=0 for J =0.88

Fig.12 Vorticity distribution downstream the propeller at yF/ R=0 plane for J=0.6 (top) and J=0.3 (bottom). RANSE results are compared to trailing wake surface position predicted by BEM (black lines)

Fig.13 Total velocity distribution downstream the propeller, on yF/R=0 plane (J=0.88). BEM, RANSE and PIV data are compared. The rectangular box superimposed to RANSE contour plots identifies BEM and experimental measurement windows. From top to bottom: axial and radial velocity

Flow-field features at different operating conditions, namely =J0.6 and =J0.3, are shown in Fig.12 where RANSE and BEM results are compared, as expected, the higher blade loading induces a smaller wake pitch and a greater strength of tip-vortices. Time-resolved visualizations of the propeller wake[29]highlight that, in these working conditions (especially at =J0.3), the wake structure is dominated by spatial and temporal instabilities. In these conditions, it is well known that RANSE modelling is inadequate to capture wake evolution and stability, thus approaches like large eddy simulations (LES) or detached eddy simulations (DES) become mandatory[15]. In fact, RANSE simulations capture tip-vortex structures close to the propeller plane and identify a region of diffused vorticity downstream, axially longer for lower values of J. Despite this complex wake structure, BEM hydrodynamics is yet able to track tipvortices and predict wake contraction, as long as the wake does not coalesce into diffused vorticity.

Next, an investigation in terms of total velocity field downstream the propeller disk is performed. Figure 13 depicts a numerical/experimental comparison in terms of axial and radial velocity components for the design condition. As previously discussed for the vorticity field, BEM yields a good agreement with RANSE and PIV results. In detail, BEM capability in capturing velocity peaks near tip-vortices and describing flow regions not dominated by viscosity driven phenomena (which are smeared by RANSE computations) is highlighted. Moreover, the flow analysis on transversal planes normal to the propeller axis is shown by Fig.14where axial, radial and tangential perturbation velocities at xF/R=0.2 are depicted. Here, RANSE computations (left) and BEM predictions (right) are compared. In both figures, the position of the potential trailing wake, shown by black lines, almost perfectly matches the trace of the viscous wake. As a general statement, BEM outcomes are in good agreement with RANSE results, even though velocity peaks are a bit higher. The analysis of the radial velocity field (see Fig.14, centre) highlights the capability of the BEM-based approach to correctly describe strong flow acceleration or deceleration near blade tipvortices and slipstream contraction. Similarly, the analysis of the tangential velocity (Fig.14, bottom) shows that slipstream rotation induced by the propeller is well captured by BEM, outside the viscous wake regions. Moving further downstream, at xF/R=1.15, Fig.15 shows the same type of comparison. The good quality of BEM results is confirmed.

Fig.14 BEM and RANSE perturbation velocity distribution downstream the propeller on xF/R=0.2 plane (J= 0.88). From top to bottom: axial, radial and tangential velocity

Fig.15 BEM and RANSE perturbation velocity distribution downstream the propeller on xF/R=1.15 plane (J= 0.88). From top to bottom: axial, radial and tangential velocity

Fig.16 Perturbation velocity for J=0.88 on a transversal plane at xF/R=0.2 and radial positions (from top to bottom) r/ R=0.3, 0.5, 0.7, 0.9. BEM and RANSE results are compared to PIV data. Axial velocity (left) and radial velocity (right)

A local comparison in terms of velocity fields is provided in Fig.16, showing axial (left) and radial (right) velocity components at circles identified by r/ R=0.3, 0.5, 0.7 and 0.9 (from top to bottom) on the plane at xF/R=0.2. For each radial section, a quarter of revolution is depicted. From a general standpoint the velocity components by BEM are in satisfactory agreement with both experiments and RANSE simulations throughout the considered domain, except for the azimuthal region dominated by the viscous wake (approximately located at θ?140o). Recalling that the azimuthal location of the potential wake well matches the viscous wake (see Figs.14 and 15), differences between BEM and reference data in this narrow region can be motivated by the finite vortex core model introduced to describe the induced flow-field. As expected, RANSE computations yield a fair agreement with PIV measurements including the viscous wake region. Moving downstream, Fig.17 shows the same analysis on the plane at xF/R=1.15. Here, the quality of BEM outcomes is still satisfactory, whereas a general worsening of the agreement between RANSE computations and experiments arises because of incorrect eddy viscosity values. Heavier loading conditions at J= 0.6 are depicted in Fig.18 that refers to velocity fields on orthogonal planes placed at xF/R=0.2 and 1.15. For this condition PIV experiments are not available, thus only BEM and RANSE predictions at r/ R=0.5 and 0.9 are compared. In all cases, the azimuthal position of the wake is well captured by the BEM app-roach, as already observed in the analysis of the vorticity field; the agreement in terms of magnitude is generally reasonable, although discrepancies increase on the farthest plane.

Fig.17 Perturbation velocity for J=0.88 on a transversal plane at xF/R=1.15 and radial positions (from top to bottom) r/ R=0.3, 0.5, 0.7, 0.9. BEM and RANSE results are compared to PIV data. Axial velocity (left) and radial velocity (right)

In conclusion, the above analysis demonstrates the high quality of free-wake BEM predictions in terms of wake pitch and tip roll-up at the design conditions, yielding, in turn, reliable velocity field distributions downstream. The level of accuracy decreases beneath the design condition although it remains satisfactory closer to the propeller disk.

Fig.18 Perturbation velocity for J=0.6: BEM and RANSE results compared. Axial velocity (left) and radial velocity (right). From top to bottom: [xF/R=0.2,r/ R=0.5], [xF/R=0.2,r/ R=0.9], [xF/R=1.15, r/ R=0.5] and [xF/R=1.15,r/ R =0.9]

2.3Blade pressure distribution

Blade pressure distribution obtained by BEM is here compared to that predicted by RANSE. No experimental data are available. Chordwise amplitude of the nondimensional pressure coefficient cp=2(p-p0)/ρn2D2is depicted on blade sections located at r/ R=0.33, 0.5, 0.725, 0.83, 0.91, 0.94 at different working points, namely J=0.3, 0.6 and 0.88. BEM results are obtained by using both the free-wake algorithm (fw) and the ad-hoc prescribed wake modelling (pw). Figure 19 refers to J=0.88. As shown, at the design condition the BEM/RANSE agreement is excellent for both wake models, worse agreement is observed for the inner section loads which, however, are significantly lower in terms of peak-to-peak variations. Hence, similarly to the analysis of prope-ller thrust/torque, the ad-hoc prescribed wakemodelling confirms to be fully adequate to capture the pressure distribution upon blades. At J=0.6 (Fig.20) the BEM/RANSE comparison is still very good at the inner and mid sections. However, respect to the design condition, a first forking between fwand pwresults appears near the root of the blade, where the fwexhibits a slightly better agreement with RANSE data. On the contrary, at the outer sections (r/ R=0.91 and r/ R=0.94), the pressure distribution near the leading edge on the suction side of the blade diverges from the trend predicted by RANSE. This is caused by the formation of a leading-edge vortex, clearly shown in Fig.21 through the visualization of volume streamlines computed by RANSE, which is not modelled by the BEM approach. At higher loading conditions (=J0.3, Fig.22) both magnitude and size of the leading-edge vortex increase, (see Fig.23), thus inducing significant lower pressure peaks on the suction side of the outer blade sections not captured by the BEM solver. Note that the lower advance coefficient determines a wake-sheet closer to the propeller disk, therefore, differently from the previous cases, the use of the free-wake modelling is needed to enhance BEM results.

Fig.19 Blade pressure by BEM and RANSE, J=0.88

Fig.20 Blade pressure by BEM and RANSE, J=0.6

Fig.21 Leading edge vortex at the blade suction side by streamlines visualization. RANSE computations for J=0.6

Fig.22 Blade pressure by BEM and RANSE, =0.3J

Fig.23 Leading edge vortex at the blade suction side by streamlines visualization. RANSE computations for =0.3J

2.4Propeller noise signature

Fig.24 Time history of pressure disturbance in the flow-field for J=0.88. BEM and RANSE predictions compared at xF/R=0.2(left) and xF/R=1.15 (right) planes. From top to bottom: hydrophones at zF/R=1.0, 1.1 and 1.3

3. Conclusions

In this paper, drawbacks and capabilities of a Boundary Element Method for the analysis of marine propeller hydrodynamics in uniform onset flow are investigated through comparison with experimental data and RANSE simulations. The proposed panel method is combined both with a non-linear alignment technique to describe the free-wake shape, and an Iterative Pressure Kutta (IPK) condition to assure no finite pressure jump at blades trailing edges. A comprehensive investigation on propeller loads, slipstream velocities, wake shape, blade pressure distribution and pressure disturbance in the flow-field is presented. The validation study is performed by considering a four-bladed model propeller. Numerical results confirm the capability of free-wake/BEM-IPK hydrodynamics in capturing propeller thrust and torque over a wide range of operating conditions.

Correlation of open water loads predictions with analysis of induced velocity and vorticity fields, demonstrates how a simplified prescribed wake model is generally sufficient to describe thrust, torque, blade pressure distributions and pseudo-noise signals when propeller is moderately loaded or unloaded, whereas a free-wake modeling is necessary to well capture the wake pitch that strongly affects propeller performance at high loading conditions. Note that other effects related to roll-up and wake contraction are of minor relevance. Differently, a correct evaluation of slipstream velocity and vorticity distribution requires the wake alignment at any loading conditions. As a matter of fact, the trailing wake surface determined by BEM is accurate in terms of pitch distribution, slipstream contraction and tip-vortices location; this is especially true close to the design advance ratio, where the agreement with RANSE computations and experimental data is very good, as long as diffusive or dissipative driven effects make the comparison meaningful. Beneath the design working point, an overall worsening of BEM results is observed; this fact is not surprising since the more complex shape of the rolledup wake makes the wake alignment procedure prone to numerical instabilities during the convergence solution seeking. Furthermore, the inception of a leadingedge tip vortex (not modeled by the present BEM formulation) and the excessive eddy viscosity associated to RANSE computations contribute to the lower level of agreement with both RANSE and experiments (if available). In conclusion, the quality of the numerical results make the proposed 3-D free-wake BEM hydrodynamic approach suited for the analysis of marine propellers; in particular the capability to capture a reasonable wake structure makes the algorithm appealing for the study of configurations with significant wakebody interactions.

Aknowledgement

The authors wish to thank Dr. Francesco Salvatore for his valuable contribution in the development of the BEM hydrodynamic solver.

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Appendix A

Let us consider a Z-bladed propeller working in open water conditions. Two frames of reference, depicted in Fig.1, are introduced: the rotating frame of reference (RFR) (O, x, y, z) fixed with the reference blade and the fixed frame of reference (FFR) (O, xF,yF,zF) rigidly connected to the propeller shaft housing. By enforcing the integral solution of the Laplace equation on the propeller surface, the application of a zero-th order BEM yieldswhere Smnis the body or wake panel surface. To solve Eq.(A1) the evolution equation for Δφ is combined with the Kutta-Morino hypothesis[17], yielding

with τmndenoting the time-delay required for a material wake grid point to be convected from the mn trailing-edge panel to xW. Equations (A1) and (A3) provide the velocity potential field upon SB; then, the application of the Bernoulli equation (see Eq.(2)) yields the pressure distribution upon the blades. However, for blade shapes with blunt trailing-edges, the Kutta-Morino condition fails and, in turn, Eq.(A3) is no longer valid. In these cases, the assumption of no vortex filament at the blade trailing-edge is assured by imposing directly a zero pressure jump

Fig.1 A Blade (left) and propeller (right) computational grids for BEM calculations: definition of indices convention

10.1016/S1001-6058(14)60087-1

* Biography: GRECO Luca (1976-), Male, Ph. D., Researcher