Design of Wake Equalizing Ducts using RANSE-based SBDO

Applied Ocean Research 97 (2020) 102087 Contents lists available at ScienceDirect Applied Ocean Research journal homepage: www.elsevier.com/locate/a...

Download PDF

5MB Sizes 0 Downloads 46 Views

Report

Full Text

Applied Ocean Research 97 (2020) 102087

Contents lists available at ScienceDirect

Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Design of Wake Equalizing Ducts using RANSE-based SBDO Francesco Furcas, Giuliano Vernengo, Diego Villa, Stefano Gaggero

⁎

T

University of Genoa, Department of Electric, Electronic, Telecommunication Engineering and Naval Architecture Via Montallegro, 1, Genoa 16145, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: Energy saving devices (ESD) Wake equalizing duct (WED) Hydrodynamic shape optimization Reynolds averaged navier stokes (RANS) Simulation based design optimization (SBDO) Japab bulk carrier (JBC)

We propose a Simulation-Based Design Optimization (SBDO) approach for the design of an Energy Saving Device (ESD) based on the Wake Equalizing Duct (WED) concept. Pre-Ducts, like Wake Equalizing Ducts, reduce the wake losses, improve the propeller-hull interaction and generate an additional thrust. An integrated design approach, relying on a parametric description of the duct geometry of the WEDs and on a RANSE method, managed by a global convergence optimization algorithm, is developed to maximize the delivered thrust. The Japan Bulk Carrier (JBC) test case, for which model scale experimental data on the eﬀectiveness of the WED are available in the literature, is considered as a baseline. Results obtained by using the adaptive, fully automated proposed design framework highlight signiﬁcant improvements of the overall propulsive eﬃciency when the Pre-Duct design is tailored to the actual hull wake shape. In addition, oﬀ-design conditions are considered to verify the robustness of the proposed designs with respect to variations of the working point and discuss the opportunity of a robust optimization process.

1. Introduction Meeting the increasing demand of energy saving due to the rise of fuel oil cost and the enforcement of stricter environmental regulations has become a major concern for ship owners. Reducing operative costs and fulﬁlling the regulations imposed by the IMO such as the Energy Eﬃciency Design Index (EEDI) or the Energy Eﬃciency Operative Index (EEOI) for a ship already in use are mandatory issues from the economic and the regulatory point of view. As a consequence, the interest on Energy saving Devices (ESD) increased signiﬁcantly, since they represent a straightforward and relatively economic solution of improving the overall propulsive eﬃciency of the ship. This is particularly true in case of reﬁtting, when the margins of taking actions on the hull (local) shape, on the installed engines or on the propeller, are very limited. Several ESD solutions are available, each based on diﬀerent working principles or on their fruitful combination based on the energy losses they try to minimize. There are many sources of energy losses (due to the propeller, the hull, or to a combination of the two), which can be categorized based on their physics: axial, rotational and frictional losses. The use of ESD devices in proximity of the propeller is primarily aimed at reducing axial and rotational phenomena, which represent the largest contribution to energy losses. The propeller design, indeed, has been already optimized at the design stage by reducing the wetted surface considering the maximum eﬃciency based on reasonable

⁎

cavitation performance. Resulting full-scale frictional losses acting on the propeller surface represent no more than the 5% of the total energy losses, and hardly could be recovered by a diﬀerent propeller design [1,2]. Rotational losses (related to an outﬂow of transverse kinetic energy in the wake due to transverse velocities) are roughly of the same order while axial (or wake) losses, related to the retarded ﬂow by the hull resistance and the accelerated ﬂow by the propulsor, range between 15% to 25% of the total losses for moderately loaded propellers. Nozzles, stern ducts and Pre-Ducts [3], commercially available as Wake Equalizing Ducts (WED) [4,5] aim at reducing the separation in the aft body of the hull by accelerating the ﬂow. This would result in a reduction of the pressure resistance of the ship. By increasing the mass ﬂux through the propeller, they increase the ideal eﬃciency of the propeller (i.e. lower axial kinetic energy losses). Moreover, they positively contribute to the overall propulsive eﬃciency by increasing of the redirection of the viscous wake through the propeller disk and, then, by increasing the propeller-hull interaction contribution to the eﬃciency [2,6]. Another beneﬁcial eﬀect is the increase of uniformity of the wake, which is eﬀective for the reduction of axial losses, through a change in the radial thrust distribution of the blade towards the hub [2] and a more uniform inﬂow to the propeller (thanks again to their redirection eﬀect of the ﬂow in axial direction) which turns into a more uniform thrust distribution. A more evenly distributed blade load means, indeed, a more uniform slipstream and then less kinetic losses in the far wake. The consequent dampening of the suction peaks of the

Corresponding author. E-mail address: [email protected] (S. Gaggero).

https://doi.org/10.1016/j.apor.2020.102087 Received 9 August 2019; Received in revised form 24 December 2019; Accepted 5 February 2020 Available online 26 February 2020 0141-1187/ © 2020 Elsevier Ltd. All rights reserved.

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

easily overcome achieving better solutions. Based on this paradigm, a variety of SBDO approaches have been developed by the authors in the context of naval architecture related design problems, such as propeller design [12,18–22], supercavitating hydrofoil in 2D and 3D [23–25], diﬀerent types of Energy Saving Devices [10], hull form optimization [26] or preliminary assessment of ﬂeet design [27]. The large availability in literature of such tools both on in the case of propellers [28–31] and hulls [32,33] also using probabilistic tools [34] is an evidence of the level of maturity reached by this kind of design approaches. In light of this, we developed a speciﬁc SBDO for the design of a stern duct of WED type, using viscous RANSE analyses based on StarCCM+ [35], linked to a parametric model for the duct geometry and driven to convergence by an optimization algorithm based on a genetic algorithm of the modeFrontier optimization environment [36]. The work is organized as follow. In Section 2 the test case selected for this application, namely the Japan Bulk Carrier, is introduced. Section 3 describes the design approach, the objectives and the constraints which have been considered and how CFD calculations (Section 4) and the parametric description of the stern duct geometry (Section 5) have been used to comply with the design purposes. Results from RANSE-based SBDO with diﬀerent (and increasingly complex) parametrizations of the geometry are shown and discussed in Section 6. In addition, oﬀ-design functioning are considered in the ﬁnal veriﬁcation of the optimal WED shapes devised by the optimization processes to assess the opportunity of a multi-functioning points/robust optimization designs to better address the performance loss which may occur in oﬀ-design functioning. The entire process has been applied in model scale even it is obvious that the Reynolds eﬀect for WEDs and, more in general, for any Energy Saving Devices could play a signiﬁcant role on their performance. Since RANSE analyses allows for reliable full-scale predictions, a concrete design should be carried out in full-scale to customize the shape of the device based on the current full-scale ﬂow features of the hull wake, which can exhibit less separated ﬂow and then lead the design to a completely diﬀerent (and probably less performing) optimal conﬁguration. In spite of this, model scale has been preferred for validation purposes of the many simpliﬁcations adopted to arrange an aﬀordable design process and for the comparisons of the outcomes of the optimization process with the available measurements in order to prove the robustness, the feasibility and the ﬂexibility of such a design procedure.

propeller in proximity of the hull further reduces the propeller/hull interaction providing a better thrust deduction factor. Rotational losses recovery is achieved through Pre-Swirl stators, rudder ﬁns and twisted rudders (commercially available as Daewoo Pre Swirl System, SVA Fin System, Hyunday Thrust Fin System [7,8]) as well as by contra-rotating propellers, which aim is to eliminate the mean tangential ﬂow on the propeller slipstream (the so-called contrapropeller principle) by generating a pre-swirled ﬂow against the propeller rotating direction (pre-swirl stators) or by straightening the slipstream (contra-rotating propellers, rudder ﬁns). These devices decrease the rotational kinetic energy losses at the beneﬁt of increasing the axial kinetic energy and momentum ﬂux. In addition, a redistribution of the torque between stators and the propeller occurs in such a way that the propeller blades are more heavily and uniformly loaded. This results in a reduction of rotation rate of the propeller at equal thrust, and then into a reduction of viscous losses [2]. Similarly, the use of Propeller Boss Cap Fins [9,10], Kappel and CLT propellers [11,12] is aimed at reducing losses associated to hub and tip vortexes respectively. Combinations of the above mentioned ESDs led to many commercially available solutions, like the Mewis Duct [13,14], the Twisted Rudder with Costa Bulb, the Becker Twisted Fins [15] or solutions like the WAFon and the WAFon-D [16]. Among the available conﬁgurations, ESDs based on the stern duct/ pre-duct concept are rather cheap and non-invasive devices which can be easily used e.g. for reﬁtting purposes. From a computational perspective, moreover, they represent a relatively simple conﬁguration for the application of an automatic design process based on optimization (the so-called SBDO, Simulation-Based Design Optimization approach) using high-ﬁdelity RANSE-based analyses. The fundamental idea behind design approaches based on the SBDO concept, indeed, is that simulations are used to characterize the performance of a device and to iteratively modify the parameters which describe the geometric characteristics and the operating conditions of the device itself in order to improve its performance, fulﬁlling speciﬁc design constraints. The design paradigm results completely modiﬁed. Rather than developing a “direct” design code (i.e. a tool where to an input in terms of required performance corresponds an output in terms of geometry, see for instance any lifting line propeller design tool [17]) the SBDO is an “inverse” design procedure, roughly a try-and-error process, because it makes use of computational tools generally developed for analysis purposes, where to an input in terms of given geometry corresponds an output in terms of provided performance. Iteration (and, in turn, parametrization of the geometry and automation of the process) is then needed to identify those geometries which success against design objectives and constraints. Nevertheless, this “inverse” process can take advantage of the generally higher ﬁdelity (i.e. accuracy and reliability) which characterize the computational tools developed for analyses purposes. This allows to avoid as much as possible performance estimations and, then, geometrical conﬁgurations, falsiﬁed by the insuﬃcient accuracy and the inherent limitations of the low-ﬁdelity solvers usually adopted for a “direct” design, since “direct” design tools generally requires substantial simpliﬁcations to relate a performance input with a geometry output. At the same time, the use of medium- and high-ﬁdelity solvers allows to deal with high-dimensional design spaces (i.e. to increase the number of free parameters to act on to explore diﬀerent shapes and conﬁgurations) and to consider lots of decision parameters, stringent and mutually conﬂicting objectives and complex, unconventional geometries far from “usual design rules” thanks to the opportunity to have more reliable and accurate estimation of performance based on which the iterative design process is carried out. When coupled with viscous hydrodynamics RANSE calculations and robust parametric descriptions of the geometry, then, a SBDO approach can turn into a very ﬂexible design tools since many of the inherent limitations of low/medium ﬁdelity design and analysis tools (or of costconstrained experimental campaigns), which rise when non-conventional conﬁgurations are added to the investigation, can be indeed

2. Test case: The Japan bulk carrier The Japan Bulk Carrier (JBC) is the reference ship chosen for demonstrating the potentiality of the proposed design approach. This test case was speciﬁcally developed for the 7th Workshop on CFD in Ship Hyrodynamics [37,38] to verify the reliability of CFD calculations also in presence of Energy Saving Devices of Pre-Duct/Stern Duct type. For the JBC, indeed, model scale measurements of the resistance in pure towing and in self-propulsion condition, without and with a Wake Equalizing Duct, have been provided for the assessment of numerical predictions. The design speed corresponds to a Froude number of 0.142 (model scale speed of 1.179 m/s, scale factor of 40). The Japan Bulk Carrier (Fig. 1) is a Capesize bulk carrier designed by the National Maritime Research Institute (NMRI), the Yokohama University, the Shipbuilding Research Centre of Japan (SRC) and ClassNK. Its main characteristics in full-scale are summarized in Table 1. The block coeﬃcient qualiﬁes the vessel as a full-blocked ship. Measurements of the nominal wake in absence of the WED show three dimensional separation vortexes from the bilge part at the bottom of the stern. This makes the JBC an excellent test case for duct-type ESD optimization, allowing for propulsive improvements by the use of stern ducts since they are designed precisely for axial ﬂow losses reduction thanks to their ﬂow homogenization eﬀect. A detailed description of the WED equipped on the JBC is given in Fig. 2 andFig. 3. The stern duct is 2

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

3. Statement of the design problem The Wake Equalizing Duct, like any other stern ducts, is an Energy Saving Device designed mainly to recover the axial losses of the propeller. This is achieved through the positive interaction that the WED produces between the hull and the propeller. A duct having a smaller diameter than that of the propeller, positioned very close ahead of the propeller itself, provides an equalizing eﬀect on the propeller inﬂow, a redistribution of the load of the propeller (leading to a possible propeller re-design), a reduction the hub vortex losses (thanks to the higher loads at inner radii, [13,14]) and an improvement of the cavitating performance of the propulsor, with obvious advantages in terms of efﬁciency and noise. Enhancing the wake to the propeller leads to a reduction of the friction losses of large expanded area ratio conﬁgurations. The acceleration of the ﬂow provided by the duct, on the other hand, is useful to have a WED which produces thrust by itself, contributing to reduce the ship resistance and allowing for a more eﬃcient lightly loaded propeller design [2,13]. Most of the aspects contributing to the recovery eﬀect of WEDs are, then, strictly related to the propeller functioning but designing a WED by considering all of these interactions would be prohibitive. In fact, it would require the inclusion of the unsteady functioning of the actual propeller behind the hull/ESD system. This would result in extremely expensive calculations, not aﬀordable in a design-by-optimization process using RANSE analyses, which are necessary to reasonably assess the viscous interactions between the hull, the propeller and the WED at the stern of the ship. For these reasons, and to limit the computational burden for the initial design stage, the proposed design strategy considers only the generation of additional thrust by the duct, which could have a contrasting inﬂuence on the total propulsive eﬃciency of the ship, keeping unchanged the propeller geometry. Since the thrust is generated by a pressure distribution on the nozzle surface, the interaction of such a pressure ﬁeld with the stern of the hull might causes an increment of resistance on the ship aft-body. This is due to the higher values of suction generated very close to the hull by the WED, which add up to those generated by the propeller. Then, designing an optimal WED means deﬁning the appropriate geometry which could improve the wake homogeneity but in the end provide an additional thrust which is not nulliﬁed by the higher thrust deduction due to the inﬂuence of the suction of the nozzle itself on the ship stern. The inclusion of the homogenization eﬀect of the hull wake provided by the WED, indeed, could have been included as an additional design objective by a simple and inexpensive monitoring of the velocity distribution (and peaks) of the hull wake aft the WED. A meaningful quantiﬁcation of this eﬀect on the total propulsive eﬃciency of the ship, however, would have required an estimation of the response of the propeller to the change of its inﬂow, leading also in this case to unaﬀordable calculations for design purposes. To the aim of maximizing the WED net thrust, accelerating type and cambered nozzles are necessary [1], as well as using asymmetric angles of attack for the upper and the lower part of the duct could result useful to guide the ﬂow into a preferred direction and maximize the accelerating (i.e. thrust production) eﬀect of the nozzle. Similarly, non-circular nozzle shape [16] could favour the energy recovery by inﬂuencing the hull wake and by taking advantage, for the production of the thrust, of the most convenient ﬂow features of the wake itself. The quantiﬁcation of the improvements achievable by these modiﬁcation of the WED shape is, however, fair only in self-propulsion condition, when the ﬂow to the WED is accelerated by the action of the propeller (the WED itself has to be designed to maximize its functioning in the eﬀective ﬂow due to the propeller). The inclusion of the fully resolved propeller is, however, not aﬀordable and then simpliﬁed approaches are needed to make RANSE analyses usable throughout the design process. Consequently, an actuator disk model, described in Section 4, has been used to mimic the action of the propeller on the surrounding ﬂow. In addition, only double-model calculations, without

Fig. 1. The JBC reference hull with its Wake Equalizing Duct. Table 1 Main characteristics of the JBC test case. Full-scale. Hull LPP [m] BWL [m] D [m] T [m] ∇ [m3] CB

Propeller 280.0 45.0 25.0 16.5 178369.9 0.858

Diam. [m] N. of blades Boss ratio Pitch Ratio Max. Chord Ratio

8.12 5 0.18 0.75 0.2262

Fig. 2. Details of the propeller (left) and of the Wake Equalizing Duct (right) mounted on the Japan Bulk Carrier.

Fig. 3. Model scale arrangement of the WED, from [39].

realized with a sectional proﬁle belonging to the NACA family (namely NACA4420) with an opening/contraction angle of 20 degrees. The diameter of the duct outlet is the 55% the diameter of the propeller while the chord length of the foil is 30% of the propeller diameter. A foiled strut, placed at 12 o’clock position, is used to support the nozzle. The distance between the duct trailing edge and the propeller plane is equal to 0.25% of LPP. 3

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

• predict the model scale, double model hull resistance (Section 4.2)

the free surface, have been considered. The idea is that the wave pattern generated by the hull has only a small inﬂuence on the WED and on the propeller performance, hence negligible at least in the preliminary design phase and for the relatively low Froude number considered, equal to FR = 0.142 . This assumption allows the calculation of the total ship resistance following the simpliﬁed approach proposed in [40–42]. A “constant” wave resistance contribution, i.e. independent of the propeller working condition and of the presence of the WED, can be computed by subtracting the double model drag to the total hull resistance predicted in towing conditions with the free surface. The value obtained includes the pure wave resistance and all the double-model approximations such as the variation of the hull wetted surface. By adding it to the current drag of the double model, a reasonable estimation of the total hull drag is possible, allowing for cost-eﬀective selfpropulsion predictions:

•

With respect to the general, single phase, formulation of Eq. (2), homogeneous multi-phases ﬂows have been considered for the prediction of the wave ﬁeld and, then, the wave resistance of Eq. (1) using the Volume of Fluid approach, which requires the solution of an additional advection equation for the fraction of air (or water) of each cell. Multiple-Reference Frames, sliding meshes and additional momentum sources have been employed, respectively, for the open water propeller performance prediction, for the calculation of the self-propulsion functioning with the fully resolved propeller and for the actuator disk models used during the optimization process.

with prop . with prop . Dfree surf . = Ddouble − model + without prop . without prop . (Dfree − Ddouble − model ) surf .

(1)

A negligible stern wave due to the low Froude number, as in the present case, only slightly inﬂuences the velocity ﬁeld on the stern of the ship (see, for instance, the comparison of nominal wakes with and without free surface eﬀect in [40]). Neglecting it, then, should not signiﬁcantly condition the WED design. The proposed SBDO approach for the stern duct design is consequently formalized as the problem of minimizing the total ship-WED system resistance in self-propulsion, such that both the eﬀects of the nozzle, the production of the additional thrust and the potentially higher thrust deduction due to the local pressure ﬁeld, are considered simultaneously. Geometrical constraints on the allowed shapes and the relative positions of the nozzle with respect to the propeller, depending on the type of parametrizations used, represent the only constraints of the design approach which is free to change, in the case of the most complex parametric description available, the geometry of the nozzle in terms of its sectional and transverse shape and the local angle of attack.

4.1. Open water propeller performance Open water propeller performance have been computed only in the light of their usage for the calculation of the propulsive coeﬃcients in the case of fully resolved RANSE self-propulsion predictions. Then, no systematic analyses about the most appropriate numerical schemes or dedicated mesh sensitivity analyses have been included. Numerical schemes are those from well-established procedures [43] while the mesh, for the sake of fair analyses, is the same (local and global reﬁnements) used for self-propulsion calculations. An example is given in Fig. 4. By exploiting the periodicity of the geometry, the problem has been solved only for a blade sector, gridded with about 800k cells and modelled with appropriate periodic boundary conditions. A Moving Reference Frame was preferred to solve the equivalent steady problem. As observed in self-propulsion conditions, the agreement for the thrust is very good. Up to an advance coeﬃcient of 0.5, the discrepancy between measurements and calculations is less than 1%. At very high advance coeﬃcients, for very low values of delivered thrust, the discrepancies rise to 4%. Torque, instead, is always overestimated. Close to the design point the diﬀerence is about 4%, increased up to 8% close to the zero-thrust point. This diﬀerence with respect to measurements could justify some of the inconsistencies highlighted during self-propulsion simulations. Since, however, these computed propeller performance have been identically considered for any analyses of the results from the optimization activities, the inﬂuence of these discrepancies

4. CFD Preliminary analyses: Propeller performance, towing tank and self-propulsion of the reference JBC test case RANSE calculation are mandatory since the accurate characterization of the phenomena exploited by the WED at the ship stern, the hull wake and the hull-propeller interaction, require viscous calculations. For current analyses, we employed StarCCM+ [35], which solves continuity and momentum equations on unstructured cell-centered meshes using the Finite Volume method. For a single phase ﬂuid, the general statement of the RANSE equations is:

⎧ ∂Vi = 0 ⎪ ∂x i ∂Vj ⎞ ⎤ ∂τij ⎨ ∂p ∂ ⎡ ⎛ ∂Vi ∂Vi ⎪ ρVj ∂x = − ∂x + ∂x ⎢μ ⎜ ∂x + ∂x ⎟ ⎥ + ∂x j i j j i j ⎠⎦ ⎣ ⎝ ⎩

and the model scale ship resistance with the inclusion of the free surface (Section 4.3), which will be used to arrange, after dedicated sensitivity analyses, the most appropriate models for the self-propulsion calculations used in the optimization process and to obtain the “wave resistance” of Eq. (1); predict and validate the self-propulsion functioning with and without the WED using both the actuator disk (Section 4.4) and the fully resolved propeller (Section 4.5) respectively used throughout the optimization process and for the veriﬁcation of the optimization outcomes also in oﬀ-design conditions (Section 6);

(2)

The ﬂuid static pressure and density are, respectively, p and ρ. μ is the ﬂuid viscosity and τij are the Reynolds stresses resulting from the averaging process of the momentum equations, which are treated on the basis of the Boussinesq hypothesis, then proportional to the traceless mean strain rate tensor times the eddy viscosity μT. Current analyses were carried out using the k − ϵ two-equation turbulence model to compute the turbulent viscosity, assuming a turbulence intensity at inlet equal to 1% and a turbulent viscosity ratio μT/μ equal to 10. Throughout the work, RANSE calculations were used to:

• predict the open water propeller performance (Section 4.1), which will be used for the estimation of the self-propulsion coeﬃcients in the case of the self-propulsion calculations with the fully resolved propeller (Sections 4.4 and 6);

Fig. 4. Surface mesh for open water propeller performance prediction. Propeller seen from back. 4

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Fig. 7. Sensitivity analysis. JBC test case. Double model resistance without the reference WED.

Fig. 5. Open water propeller performance. Measurements (dots) versus RANSE calculations (solid lines).

coarser meshes were used for the sensitivity analysis by modifying the reference conﬁguration with an isotropic factor of 1.3, applied to the unreﬁned cell size (i.e. far from local surface or volume reﬁnements). The ﬁnest mesh consists of about 19.5 million cells, the coarsest of about 250 thousands elements. Results of the sensitivity analysis using the least square method proposed in [44] are summarized in Fig. 7. The iterative convergence, monitored through the L2 norm of the residuals and by the variation of the double model resistance with iterations, is shown, in the case of the reference mesh, in Figs. 8–10. An example of the inﬂuence of the mesh density on the nominal hull wake at the propeller plane is ﬁnally given in Fig. 11. The apparent order of convergence computed using the least square method is 2, which is perfectly in line with the theoretical second-order nature of the numerical schemes employed in StarCCM+. Diﬀerences between the extrapolated value and the resistance computed using the reference mesh are less than 0.1%, with an estimated uncertainty of 1.6%. Also the iterative convergence of the calculation for the reference

can be considered negligible for a mere comparative point of view. 4.2. Double model resistance prediction: Sensitivity analysis and iterative convergence Double model calculations were addressed at ﬁrst, since the SBDO makes use of this simpliﬁcation to have a computationally eﬃcient RANSE simulation. The sensitivity to grid reﬁnements and the iterative convergence of the simulations have been preliminary veriﬁed using the double model approximation. Details of the hexa-dominant reference mesh are provided in Fig. 6. The reference mesh for half the hull, without the ESD, consists of about 730k cells, arranged using local volume reﬁnements at the stern, for an average non-dimensional wall distance less than 5 obtained using 8 prism layers. Five ﬁner and two

Fig. 8. L2 norm of the residuals. Double model without the reference WED. Calculations with the reference mesh.

Fig. 6. The reference hexa-dominant mesh for double model resistance calculations. 5

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Fig. 12. Mesh arrangement for the free surface calculations. Reference mesh arrangement. Fig. 9. Iterative convergence of the double model resistance without the WED. Calculation with the reference mesh.

the light of these results, a excessive weighting of the simulations seems not necessary and the arrangement provided by the reference mesh, which with the inclusion of the WED consists of about 1.2 million cells, is selected for the computationally reliable and eﬃcient analyses needed in the optimization process. 4.3. Model scale ship resistance prediction with the free surface Free surface analyses have been carried out to validate numerical calculations using towing tank measurements and to deﬁne the “wave resistance” of Eq. (1). The reference mesh of the double model analyses was considered as the baseline. The only modiﬁcations of the mesh concern the treatment of the free surface, across which an anisotropic reﬁnement along the vertical direction has been considered to limit as much as possible the numerical damping of the generated wave ﬁeld in proximity of the hull. For the same reason, a reﬁnement of x- and ycells dimensions has been used in a region resembling the Kelvin angle, following well established guidelines [40,45]. An example of the computational mesh (half hull) is given in Fig. 12. A sensitivity analysis (Fig. 13) has been carried out as well, using two ﬁner and two coarser meshes, ranging from 520 thousands cells (Coarsest conﬁguration) to 5 millions cells (Finest case). The reference grid, with the inclusion of the free surface treatment, consists of 1.35 millions elements. Also these calculations show a good convergence trend since the least square ﬁt exhibits an apparent order of grid convergence equal to 2 with an estimated uncertainty close to 1% in the case of the reference mesh. The unsteady nature of the free surface calculations, however, poses some additional issues. At ﬁrst, adding ﬁner conﬁgurations to the analysis is prohibitive due to the computational times associated to unsteady calculations, which were carried out with a ﬁxed time step equal to 0.001 s to ensure an acceptable Courant number on the free surface. Secondly, the non-perfectly non-reﬂective nature of the boundary conditions at inlet and outlet determines, especially at the low Froude numbers of current analyses, some issues which turn into non-periodic recorded resistance signals. Their time averaging could lead to slightly diﬀerent values of hull resistance (especially for the

Fig. 10. Iterative change of the double model resistance without the WED. Calculation with the reference mesh.

mesh is good. Residuals fall below 10−3 (and easily lower than 10−4 in case of continuity, turbulent kinetic energy and y- and z- momentum) after 1500 iterations. The iterative error, after the same number of iterations, is less than 0.15%. From a qualitative point of view, also the features of the nominal hull wake, which was considered as a monitor of the convergence, show good trends with the number of cells. The coarsest mesh considered for the sensitivity analysis already reveals the peculiarities of the ﬂow at the propeller plane, which is characterized by the presence of two bilge vortexes placed at about 90 and 270 deg. With ﬁner meshes, the numerical dampening of the vortex strength is lowered, resulting in stronger and more concentrated and decelerated axial ﬂow regions slightly moved upward. Similar analyses have been carried out with the inclusion of the Wake Equalizing Duct. The sole modiﬁcations of the mesh regard local reﬁnement across the WED position, using local control volumes, and the arrangement of appropriate surface mesh reﬁnements to account for the higher curvature at the nose and at the trailing edge of the duct. In

Fig. 11. Inﬂuence of mesh density on the nominal hull wake. Double model calculations without the WED. 6

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Fig. 14. Actuator disk radially varying axial load from BEM calculations.

modiﬁed ﬂow at the stern of the ship due to the action of the propeller. Self-propulsion conditions have been simulated, at ﬁrst, using a simple actuator disk model similar to that employed in [40] for self-propulsion predictions and in [19] for the design by optimization of the nozzles of ducted propellers. This model, indeed, does not increase the computational eﬀort of the calculations, at least in terms of cells density, and it was shown to provide reasonable induced velocity ﬁelds for the preliminary assessment of WED performance. The utilized actuator disk is, however, not the “default” model, with constant axial load distribution, available in StarCCM+. Among the developed actuator disk models [46], the current model makes use of a radially varying load distribution (Fig. 14) which, for present analyses, has been obtained from Boundary Element Method calculations of the JBC propeller geometry operating in the circumferentially averaged inﬂow represented by the nominal wake of the hull in pure towing conditions in the absence of the WED. A radially varying loading condition is a signiﬁcant step forward for an accurate modelling of the propeller action on the surrounding ﬂow ﬁeld since it is important to mimic the eﬀective action of the propeller (which has its maximum load towards the tip) with a simpliﬁed, costless, approach. The comparison of the propeller wakes in front of the propeller plane ( x / LPP = 0.02285 from the aft perpendicular) using constant or radially varying actuator disks with the fully resolved propeller using RANSE, shown in Fig. 15, supports this assertion. Moving the maximum load to the tip (the radially varying actuator disk) slightly modiﬁes the non-dimensional velocity ﬁeld in front of the propeller (i.e the ﬂow to the WED) and at least for what regards the position and the extension of the accelerated ﬂow, the distribution is a bit more similar to that from fully resolved RANSE calculations. Substantial diﬀerences can be appreciated instead when comparing the resulting slipstream [42,46]. With respect to the resolved propeller, the actuator disk , in its current formulation, cannot account for the propeller load unbalancing (at 90 deg. and 270 deg.) due to the vertical components of the hull wake, as well as for the tangential components of the induced velocities downstream due to the blades rotation. A more complex model, including also the radial distribution of the tangential load [41], could have been used, at the cost, however, of loosing the symmetry of the problem and, then, consistently weighting the computational eﬀort needed for the SBDO. The self-propulsion point is achieved using an iterative approach which continuously update the total strength of the actuator disk momentum sources based on the current estimation of the ship drag, which also in this case is the sum of the computed double model resistance, aﬀected by the presence of the actuator disk itself, plus the wave contribution. Skin friction corrections (respectively 18.2 N when the ESD is not included, 18.1 N with the ESD, as per [39]) account for model-to-

Fig. 13. Sensitivity analysis. JBC test case. Free surface calculations without the reference WED.

ﬁner mesh conﬁguration, for which a simulation time of 100 s was still not suﬃcient to have an almost periodic signal), which in turn can slightly aﬀect the outcomes of the sensitivity analysis. Since it seems inappropriate to “mix” results from the double model and from the free surface calculations using diﬀerent level of cells densities to determine the “wave resistance”, Eq. (1) was applied considering the reference mesh arrangements from both the double model and the free surface calculations since in an optimization based design it is important to rank conﬁgurations based on comparable calculations [45]. This lead to a “constant” (since it will be used regardless the WED modiﬁcations) value of wave resistance equal, in model scale at the design Froude number of 0.142, to 1.794 N (ΔCW = 0.211·10−3 ). The results of the free surface analyses and the comparison with towing tank measurements in both cases, with and without the WED, are shown in Table 2. In particular, for the case with the ESD, the total ship resistance has been computed using Eq. (1), i.e. summing to the double model resistance computed with the Wake Equalizing Duct the “wave resistance” obtained from the free surface calculation but without the ESD. Calculations are in very well agreement with measurements, even when the simpliﬁcation of Eq. (1) is used. Results are slightly underestimated but calculations foresee the eﬀect of the ESD already in towing conditions. The total resistance coeﬃcient numerically predicted is about 1% lower than measurements regardless the presence or not of the WED. Comparatively, with the inclusion of the WED, the reduction in resistance of about 0.6% seen during experiments is foreseen also by RANSE analyses which predict a reduction of the total resistance in towing conditions of about 0.5%.

4.4. Self-Propulsion prediction (actuator disk) with and without the WED Self-propulsion analyses are mandatory for the accurate characterization (and, then, for the design) of the WED since it operates in the Table 2 Comparison between the measured and the computed total resistance coeﬃcient CT · 103. Towing tank test with the reference mesh. CT · 103

Measurements

CFD

Δ%

Without WED With WED ΔWED%

4.289 4.263 -0.61%

4.239 4.217 -0.52%

-1.17% -1.09% -

7

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Fig. 15. Total velocity ﬁeld in front of the propeller/actuator disk. Self-propulsion prediction without the WED.

Eq. (2)) based on which the actuator disk produces the pressure jump associated to the propeller functioning. With the actuator disk, indeed, the pressure jump is symmetric between the pressure and the suction side. This modelling is obviously in contrast with the real functioning of a propeller, which usually delivers the thrust mainly thanks to the suction on the back side of the blades. Underestimating the role of the suction, as in the case of the symmetric pressure jump distribution of the actuator disk, is the reason of the underprediction of the ship resistance when calculated with the actuator disk models [47]. The visualization of the pressure ﬁeld close to the actuator disk and to the propeller, provided in Section 6, clariﬁes the role of the propeller suction on the propeller/hull interaction. Anyhow, calculations with the simpliﬁed actuator disk model predict reasonably well the inﬂuence (the trends, at least) of the ESD on ship performance at a negligible computational cost, which makes this simulation arrangement the preferred choice for the optimization based design process.

Fig. 16. Actuator disk model behind the hull equipped with the WED.

full scale eﬀects. The arrangement of the actuator disk in presence of the Wake Equalizing Duct is shown in Fig. 16, where the local reﬁnements around the duct and the stern of the ship are also visible. Results of the self-propulsion measurements and of the numerical analyses with the actuator disk model are summarized in Table 3. The role of the Wake Equalizing Duct in providing an additional thrust (i.e. a reduction of the total ship resistance) is evident: with the ESD, the total ship resistance in self-propulsion is reduced of about 1% and the propeller has to provide 0.73% less thrust (22.22 N versus 22.38 N during model scale measurements). Numerical calculations resemble these trends. Even if the actuator disk is a very simple model to account for the inﬂuence of the propeller, relative diﬀerence between calculations without and with the WED are similar to those observed during measurements since the ESD allows for a 0.75% reduction of the ship resistance. Diﬀerences are higher if we look at the comparison between measurements and CFD calculations. The self-propulsion prediction with the actuator disk increases the diﬀerence with respect to experiments of about an additional 2% with respect to the 1% diﬀerence observed for pure towing calculations. This diﬀerence, as already observed in [40], could be ascribed to the mechanism (i.e the added momentum to

4.5. Self-Propulsion prediction (fully resolved propeller model) with and without the WED Self-propulsion predictions with the inclusion of the true propeller model in the RANSE simulation (Fig. 17) were considered to conﬁrm the results from the actuator disk model and the observations regarding the reasons of the discrepancies between measurements and calculations. The improvements of ESD geometries devised by the optimization, for instance, have been veriﬁed for the optimized geometries at the end of the design process using exactly this approach. Fully resolved RANSE calculations have been carried out by including the propeller in the simulations using sliding meshes. The mesh

Table 3 Comparison between the measured and the computed total resistance coeﬃcient CT · 103 and the thrust deduction factor (1 − t ) in self-propulsion condition with and without the WED. Calculations with the actuator disk model. CT · 103

Measurements

CFD

Δ%

Without WED With WED ΔWED% (1 − t ) Without WED With WED ΔWED%

4.811 4.762 -1.02% Measurements 0.812 0.822 1.2%

4.648 4.613 -0.75% CFD 0.836 0.843 0.8%

-3.39% -3.13% Δ% 3.0% 2.6% -

Fig. 17. Mesh arrangement for fully resolved propeller self-propulsion analyses. 8

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Table 4 Comparison between the measured and the computed total resistance coeﬃcient CT · 103, the propeller rate of revolution and the propulsive coeﬃcients (1 − t ), (1 − w ) and ηr in self-propulsion condition with and without the WED. Calculations with the fully resolved propeller. CT · 103

Measurements

CFD

Δ%

Without WED With WED ΔWED% Prop. rev. [rps] Without WED With WED ΔWED% (1 − t ) Without WED With WED ΔWED% (1 − w ) Without WED With WED ΔWED% ηr Without WED With WED ΔWED%

4.811 4.762 −1.02% Measurements 7.8 7.5 −3.85% Measurements 0.812 0.822 1.2% Measurements 0.550 0.478 −13.3% Measurements 1.015 1.008 −0.6%

4.752 4.706 −0.96% CFD 7.75 7.57 −2.32% CFD 0.804 0.812 1.0% CFD 0.551 0.505 −8.3% CFD 0.972 0.975 0.4%

−1.23% −1.17% Δ% −0.06% 0.93% Δ% −1.0% −1.2% Δ% −0.2% 5.6% Δ% −4.2% −3.3% -

into the inner, rotating, domain, was arranged using polyhedral cells to better account for the complex ﬂows around the propeller and it counts of about only 700k cells. The inner rotating domain, indeed, is a cylinder that encloses only the propeller, from just aft the WED trailing edge to the hub cap, inside of which all the surface and volume mesh parameters are those adopted for the open water propeller performance of Section 4.1. The outer domain (WED case) is discretized in average with 3 Million cells arranged as in the case of the reference mesh of the double model (actuator disk case) plus an additional reﬁnements in the wake of the propeller. The equivalent simulation time step corresponds to 0.5 deg. of propeller rotation. With the exception of the ﬁrst iteration, which was solved with 40 propeller revolutions to account for the initial numerical transient, each additional iterations needed to achieve the self-propulsion condition consists of 15 propeller revolutions. Performance of both the ship and the propeller were time averaged to account for the blade passing eﬀect by considering the last three propeller revolutions. Results of the self-propulsion predictions are summarized in Table 4. Compared with the same analyses carried out with the actuator disk, the improvements are evident. The predicted ship total resistance has a signiﬁcant better agreement with experiments diﬀering from measurements of less than 1.5%: modelling the true propeller functioning overcomes the limitations of the actuator disk model, allowing for a more reliable prediction of the hull/propeller interaction. The presence of the resolved propeller causes higher values of suction close to the hull which turns into a more substantial increase of resistance (Fig. 18) with respect to the actuator disk analyses. Usual propulsive coeﬃcients (thrust deduction factor (1 − t ), wake fraction (1 − w ) and relative rotative eﬃciency ηr) conﬁrm this trend. Numerical calculations predict a thrust deduction factor about 1% underestimated with respect to measurements in both conﬁgurations and, then, the trend using the WED, which lead to higher values of thrust deduction in presence of the ESD, is similarly foreseen. Wake fraction and the relative rotative eﬃciency show slightly higher discrepancies with respect to experiments but trends evidenced experimentally are conﬁrmed also by simulations. In particular, numerical calculations slightly overestimates the wake fraction (but only in presence of the WED) while the relative rotative eﬃciency is slightly underestimated, being the numerical estimation always lower than 1 while experimentally a value very near to 1 was measured. These differences, partially, may be ascribed to the diﬀerences (shown in

Fig. 18. Pressure coeﬃcient distribution around the propeller and the hull stern. Self-propulsion condition without the WED.

Section 4.1) between the numerical prediction of the propeller performance in open water conditions and the propeller performance curves available experimentally. As in the case of experimental measurements, these curves are needed by a fully resolved RANSE prediction to estimate these propulsive coeﬃcients. In addition, a non suﬃciently ﬁne grid to preserve the boundary layer wake by the nozzle far from the trailing edge, could be responsible of this slight underestimation of the wake deceleration in presence of the WED. As summarized in Table 5, indeed, in self-propulsion functioning the torque coeﬃcient is signiﬁcantly overestimated with respect, for instance, to the thrust

Table 5 Comparison between the measured and the computed propeller performance (KT and 10KQ) and the delivered power (PD) in self-propulsion condition with and without the WED. Calculations with the fully resolved propeller.

9

KT

Measurements

CFD

Δ%

Without WED With WED ΔWED% 10KQ Without WED With WED ΔWED% PD [W] Without WED With WED ΔWED%

0.217 0.233 7.4% Measurements 0.279 0.295 5.7% Measurements 28.21 26.67 −5.5%

0.217 0.227 4.5% CFD 0.301 0.309 2.6% CFD 31.24 29.71 −4.9%

−0.1% −2.7% Δ% 8.0% 4.8% Δ% 10.7% 11.3% -

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

• A generator curve, i.e. a closed line handled by a ten-points B-Spine

coeﬃcient in the same condition. Combined with the overprediction observed in open water, this discrepancy easily leads to the diﬀerences in the relative rotative eﬃciency. The role of the WED in providing an additional thrust, seen experimentally and veriﬁed with the actuator disk model, is conﬁrmed also using fully resolved RANSE calculations. The reduction of total ship resistance (-0.96%) is very close to the measured value and also the prediction of the propeller rate of rotation is very close (1%) to measurements. Both experimentally and numerically, the simultaneous reduction of the ship resistance and of the propeller rate of revolution at equilibrium achieved with the ESD lead to a substantial reduction of the delivered power: -5.5% during experiments and -4.9% from simulations. These results using the fully resolved propeller are particularly important for the ﬁnal veriﬁcation of the optimal conﬁgurations since the inﬂuence of the WED on propeller performance, which could be signiﬁcant, is neglected in the current design approach and an accurate analysis of the propeller functioning is ﬁnally needed to prove the reliability of the process. In the end, indeed, it is necessary to verify that the improvements in terms of ship resistance are not nulliﬁed by a bad functioning point of the given propeller or to assess, consequently, the need of a redesign of the propeller itself to better address the new functioning point and inﬂow. The pure towing and the self-propulsion conditions, modelled with the actuator disk or the fully resolved propeller, proposed in these sections show a very satisfactory agreement with the available measurements in relation to the inherent simpliﬁcations accepted. For the purposes of the work (i.e. a computationally eﬃcient estimation of the self-propulsion functioning during the optimization, an accurate a posteriori veriﬁcation of the results of the designs) the use of these modelling assumptions and discretization levels seem appropriate.

• • •

curve. Symmetry is exploited, reducing the number of control points to six. Each control point is free to move (fulﬁlling constraints on the symmetry plane), changing the transversal shape (i.e. the average diameter) of the nozzle. In principle, port/starboard asymmetric shapes could result useful to exploit the diﬀerent loading conditions (and resulting induced ﬂow) of the propeller between 90 and 270 deg. positions in addition to top/bottom asymmetry. The use of a pure translational actuator disk naturally prevents this type of ﬂow and then symmetry is preferred; A sectional hydrofoil, which is revolved around the generator curve. The hydrofoil belongs to the NACA 4-digit series. Maximum camber, maximum thickness, their position along the chord and the chord itself are free to change; A continuous distribution of the angle of attack of the sectional hydrofoils along the generator curve, handled by a six-point BSpline curve along the curvilinear coordinate of the generator curve; The longitudinal position of the WED with respect to the propeller plane;

This parametrization realizes top/bottom asymmetric shapes with variable angle of incidence (Fig. 20) which resembles the concept of the “Asymmetric proﬁle”, of the “Eccentric duct” and of the “Tilted duct” introduced in [1] to adapt the nozzle shape to that of the local ﬂow, reducing the risk of ﬂow separation for sections operating far from the converging streamlines at the hull stern and by guiding the ﬂow into a preferred direction. Geometrical constraints concern the minimum and the maximum radius of the nozzle to avoid ducts smaller than the propeller hub or deeper than the keel line, the minimum thickness of the sectional hydrofoil (not lower than 15% of the chord), the combination of chord and longitudinal position to ensure a given clearance with the propeller and to avoid nozzles entirely pierced into the hull. In addition, control points of the generator line and of the angle of attack distribution are further constrained to avoid cusp like shape. Examples of violated constraints are shown in Fig. 21. Since it was not possible to translate these constraints, which rise from the combination of diﬀerent geometrical properties, directly into appropriate range of variations of the free parameters, a dedicated check was performed at the end of each generated geometry. If any of the constraints was not satisﬁed, the geometry was rejected and the corresponding case processed as a penalizing objective.

5. Parametric description of the stern duct The parametric description of the WED is one of the crucial aspects to arrange an automatic SBDO approach. The geometry, indeed, has to be handled automatically by few parameters, which variation allows to span a suﬃciently wide design space. The current description of the duct follows a kind of superposition of shape variations altogether modeling the geometry of the WED which is described, as shown in Fig. 19, by:

6. Results Combining the shape variations of the previous section, we designed three WEDs with increasing complexity (i.e. by increasing the number of free parameters) satisfying the same set of geometric constraints. A genetic optimization algorithm drives the design objective towards the minimization of the total ship drag by increasing the duct net thrust and/or by mitigating the propeller/hull interactions. The initial Design of Experiments always consists in a number of conﬁgurations equal to ten times the number of free variables, distributed on the design space using Sobol sequencing. At the end of each run, which consists in the “evolution” of the initial conﬁgurations of the Design of Experiments for a certain number of generations, optimal

Fig. 19. Parametric description of the WED: B-Spline control polygon of the generator curve.

Fig. 20. Possible shapes handled by the parametric description of the WED. 10

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Table 7 Range of variations of the free parameters. The (forward) longitudinal position is referenced to the WED trailing edge with respect to the aft perpendicular. Design run 2. Parameter WED chord Angle of attack [∘] longitudinal position WED radius maximum WED camber position of max. camber maximum WED thickness

cWED/D xWED/LPP rWED/D m p t/cWED

min.

max.

0.147 5 0.016 0.147 0 0 0.15

0.440 35 0.75 0.63 7 6 0.25

Table 8 Range of variations of the free parameters. The (forward) longitudinal position is referenced to the WED trailing edge with respect to the aft perpendicular. Design run 3. Parameter WED chord long. position max. WED camber pos. of max. camber max. WED thick. s2 s3 s4 s5 a2 − a5 [∘] py2/D py3/D py4/D py5/D pz1/D pz6/D pz3 pz4

Fig. 21. Geometrical constraints for the WED parametric description.

designs were analysed in self-propulsion condition with the fully resolved propeller model in order to conﬁrm the outcomes of the simpliﬁed design approach.

6.1. RANSE-Based SBDO at the design functioning point 6.1.1. Design run 1 The ﬁrst design run considers only the simplest nozzle shapes. Only circular generator lines are allowed, the sectional hydrofoil is unchanged with respect to original WED conﬁguration (NACA 4420) and the angle of attack is constant everywhere along the generator line. The four free variables listed in Table 6 have been considered for the parametric description of the geometries: the chord of the hydrofoil, its angle of attack, the longitudinal position with respect to the propeller plane and the diameter of the duct. Forty diﬀerent geometries ﬁll the design space and ten generations are considered for the evolution of the designs. Out of the four hundreds design candidates, 213 feasible designs, i.e. satisfying the set of geometric constraints, have been evaluated.

cWED/D xWED/LPP m p t/cWED

min.

max.

0.147 0.016 0 0 0.15 0.05 0.20 0.55 0.85 5 0.07 0.16 0.14 0.06 0.14 −0.14 20% of pz1 20% of pz6

0.440 0.75 7 6 0.25 0.15 0.45 0.80 0.95 35 0.32 0.69 0.62 0.27 0.63 -0.63 90% of pz1 90% of pz6

6.1.2. Design run 2 In addition to the free parameters considered in the design run 1, the second design run includes the variation of the sectional hydrofoil shape. The design process is allowed to change the parameters of the NACA four digits hydrofoil, i.e. the maximum camber m, its position along the chord p and the maximum thickness t as per Table 7. The total number of free variables is then 7, for an initial Design of Experiments ﬁlled with seventy conﬁgurations. The genetic evolution is allowed for twenty generations, resulting in 1400 overall simulated geometries, 652 of which fulﬁl the geometrical constraints.

Table 6 Range of variations of the free parameters. The (forward) longitudinal position is referenced to the WED trailing edge with respect to the aft perpendicular. Design run 1. Parameter

min.

max.

0.147 5 0.016 0.147

0.440 35 0.75 0.63

Fig. 22. Control polygon (min. and max.) of the generator line for design run 3. WED chord Angle of attack [∘] longitudinal position WED radius

cWED/D xWED/LPP rWED/D

11

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Fig. 24. Optimization histories (ship total resistance versus population generation) for the three design runs. WED 1, WED 2 and WED 3 are the optimal WED shapes from the three design cases. Fig. 23. Control polygon for the distribution of the angle of attack along the generator line for design run 3.

Table 9 Geometrical characteristics of the optimized WEDs.

6.1.3. Design run 3 The last design run adds to the free parameters of the second design activity additional 14 parameters, which are used to modify the shape of the generator line (Fig. 22) and of the local angle of attack (Fig. 23). Control points of the generator line, however, are not free to move arbitrary. Those on the symmetry plane (p1 and p6) are allowed to move only in the vertical direction. The vertical movement of p2 and p5, instead, is linked to that of p1 and p6 in order to avoid any cusps between port and starboard. The range of variation of points p3 and p4, ﬁnally, is proportional to that of p2 and p5. Similar constraints apply to points which control the distribution of the angle of attack along the generator line. In particular, a1 and a5 have variations respectively equal to those of a2 and a6, again to avoid cusps in the distribution of the angle of attack at the symmetry plane. This results in a 21-dimensions design space (Table 8), ﬁlled with 210 initial geometries. Out of the overall 3150 conﬁgurations obtained after 15 generations, 961 feasible design have been obtained.

NACA cWED/D rWED/D a.o.a. [∘]

6.1.4. Comparative analysis of the optimal WEDs The optimization histories of the three design runs, shown in Fig. 24, conﬁrm that the widening of the design space is the easiest way to identify improved geometries since designs from run 2 and run 3 are substantially better than those from run 1. These results are possible solely thanks to the ﬂexibility of the design method, which is able to reliably explore unconventional shapes by using high-ﬁdelity calculations which avoid as much as possible falsiﬁed minima. Table 9 summarizes the geometrical characteristics of the selected Wake Equalizing Ducts compared to those of the reference conﬁguration proposed by NMRI [39]. The optimization process leads, for the three optimal geometries, towards nozzles having longer chords, higher diameter and lower angles of attack (the average diameter in the case of design run 3). The angle of attack, in particular, is almost halved, and the nozzle diameter are increased of about 30% and 20%, respectively. The shape of the sectional hydrofoil has also been signiﬁcantly changed. With respect to the reference NACA4420, the position of the maximum curvature of the optimal shapes is moved towards the outermost admissible position (70% of the chord from the leading edge). The curvature is increased from the 4% (reference) to the 6% of the chord for WED 3. These results conﬁrm the observations reported in

Ref. WED

WED 1

WED 2

WED 3

4420 0.300 0.325 20

4420 0.325 0.320 9

5715 0.345 0.359 9

6716 0.443 0.404 10.45

Fig. 25. Student t-test for the three design runs.

[1]. Optimized geometries show the lowest possible thickness corresponding to 15% of the chord considering the admissible range of variation between 15% and 25%. In fact, the avoidance of ﬂow separation and the consequent increase in resistance favour thinner hydrofoils. The results of Fig. 25 from the Student t-test performed on the sets of feasible conﬁgurations identiﬁed during the three design runs conﬁrm 12

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

the trends observed for the selected geometries. Since the samples are quite limited for reliable statistical analyses, these results can not be considered conclusive (compare, for instance, the larger diameter of the optimal WED 3, Table 9, with the trend for the same parameter of Fig. 25). However, they highlight the trends the optimization process followed to improve the propulsive performance and, then, they provide useful design guidelines since the Student t-test essentially provides a measure of the strength of the relationship between the output and the input variables, and of the kind of relationship (direct or inverse) among inputs and outputs. Positive values on the bar diagram of Fig. 25 stand for a direct relationship, while the importance of the relationship is measured by the bar size. The chord shows a opposite inﬂuence between design 1 and designs 2 and 3, while the average radius of design 3 seems not to signiﬁcantly aﬀect the performance of the WED. Despite those two parameters, all the other parameters, regardless the complexity of the parametric description from design 1 to design 3, show similar inﬂuence. The total ship resistance is reduced by lowering the angle of attack, which in turn seems the most relevant parameter affecting this design objective. Also the WED radius and thickness have an inverse eﬀect on the ship resistance (lower sectional thickness, for instance, reduce the drag) while it is important that higher curvatures (the m parameter) are shifted towards the trailing edge (the p parameter) to improve the WED performance. For a better understanding of the reasons of the improvements observed for the optimized geometries, the three optimal Wake Equalizing Ducts have been further analysed including the actual inﬂuence of the propeller by fully resolved self-propulsion RANSE analyses. This ﬁnal veriﬁcation is identical to those used for the validation of the reference test case and the investigation on the opportunity of self-propulsion analyses based only on the actuator disk of Section 4.5. The results with the fully resolved propeller model have been compared with the same calculations already available for the ship with and without the reference ESD in terms of forces, pressure ﬁelds on the nozzles and on the hull and self-propulsion coeﬃcients. Fully resolved propeller calculations in self-propulsion conﬁrm the improvements already observed with the simpler actuator disk analyses proposed in the optimization processes. Results of Fig. 26 compare the resistance reductions with respect to the bare hull case. As expected, calculations with the fully resolved propeller foresee higher values of ship resistance, which are closer to measurements in the case of the reference conﬁguration. This is due to a better predicted hull/propeller

interaction with respect to the simpliﬁcation of the actuator disk model. For instance, in the case of the optimal design of run 1, i.e. WED 1, the total predicted resistance using the fully resolved propeller is 0.8 N higher than the resistance computed with the actuator disk, i.e. 39.39 N vs. 38.55 N. In the case of WED 3, the diﬀerence is about 0.4 N (38.09 N with the actuator disk, 38.51 N with the fully resolved propeller). The slightly more extended suction on the stern of the ship in the case of the fully resolved propeller, appreciable in Fig. 28, is the reason of these diﬀerences. Despite this, the relative reductions of resistance provided by the WEDs from design run 1 to 3 are similarly predicted, as well as the reduction trend associated with larger and more complex design spaces. These results are of particular importance, since they provide an additional a posteriori validation of the design process based on simpliﬁed, computationally eﬃcient, calculations. Compared to the reference nozzle, WED 3 provides a resistance reduction more than three times higher. The ﬁrst design too, WED 1, accounting only standard variation of the geometry, doubles the resistance reduction by changing only the angle of attack and its diameter. Considering the focus on the total hull resistance reduction, disregarding any consideration about possible improvements of the propeller performance that cannot be modelled by the actuator disk approach, these improvements can be only explained by the improved hull/propeller interaction and the higher net thrust provided by the nozzle. By looking at the pressure distributions over the optimized WED shapes (Figs. 27 and 28), the “accelerating duct behaviour” of these new geometries is more evident. The optimized ducts have substantially higher values of suction at the leading edge on the inner surface of the nozzle thanks to lower angles of attack and higher cambers, which move the stagnation point on the external nozzle surface. Combined to the local shape of the nozzles, such a suction positively contributes to generate additional thrust. The increase in net thrust provided by the WEDs is highlighted in Table 10. With respect to the reference conﬁguration, the optimized nozzles produce more than twice the thrust provided by the original ESD and their net contribution to the propulsive action in the end is positive. In fact, higher values of suction at the leading edge of the ducts, and then close to the hull, could also have a detrimental eﬀect on the ship performance. This inﬂuence can be observed in Fig. 28. In particular the suction ﬁeld at the stern of the ship is more aﬀected as the WED is closer to the hull (see e.g. WED 2 and 3, respectively). This ampliﬁes the negative interaction between the

Fig. 26. Total ship resistance reduction. Self-propulsion condition with the actuator disk and the fully resolved propeller with respect to the bare hull case.

Fig. 27. Pressure contours on the optimized WED shapes. Self-propulsion calculations with the fully resolved propeller. 13

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

Table 10 Variation of the double-model resistance ΔRDM% (with respect to the bare hull case) and WED delivered thrust TWED% (percentage of the bare hull double model resistance). Calculations with the actuator disk and with the fully resolved propeller. ΔRDM%

Ref. WED WED 1 WED 2 WED 3

TWED%

act. disk

prop.

act. disk

prop.

0.76% 0.08% 1.12% 1.92%

1.19% −0.47% 0.49% 1.45%

1.14% 2.36% 4.38% 5.40%

1.78% 2.70% 5.06% 6.06%

Table 11 Self-propulsion functioning at the design speed. Comparison of the performance of the selected WEDs computed with the fully resolved RANSE propeller.

Bare hull Reference WED WED 1 WED 2 WED 3

(1 − w )

(1 − t )

n

KT

10KQ

ηo

ηr

ΔPD[%]

0.5509 0.5052

0.804 0.812

7.75 7.57

0.2169 0.2266

0.3014 0.3091

0.486 0.463

0.972 0.976

− −4.9

0.5368 0.5621 0.5554

0.838 0.861 0.862

7.59 7.62 7.59

0.2177 0.2106 0.2121

0.3019 0.2967 0.2975

0.484 0.499 0.497

0.973 0.969 0.971

−5.9 −6.4 −7.2

of the double model with the WEDs is compared to the double model resistance of the bare hull. The presence of the WED essentially worsen the interaction with the hull. Indeed, the reference duct itself, as well as most of the optimized ones, increases the sole resistance of the double model hull. Among the devised geometries, only WED 1 has a positive inﬂuence on hull performance: if analysed with the actuator disk, it is almost neutral while it positively contributes to the reduction of the hull resistance when instead the fully resolved propeller calculations are considered. The self-propulsion coeﬃcients listed in Table 11 prove the discussion regarding the eﬀectiveness of the ESD, shedding a light into its inﬂuence on the propeller functioning. Adding any WED signiﬁcantly changes the self-propulsion condition. With the reference WED, as already discussed during the comparison with measurements, the propeller rate of rotation is signiﬁcantly reduced as well as the effective wake fraction. In this new operating condition, the propeller eﬃciency (calculated using the open water propeller curves of Fig. 5) decreases of about 2%. The positive combination of a less severe propeller/hull interaction, keeping higher thrust reductions, and the lower propeller rotation rate at the equilibrium leads to a 4.9% saving on the delivered power. With the optimized WEDs, instead, the functioning points of the propeller are found for slightly higher propeller rate of revolution. The eﬀective wake fraction is less severe thanks to the more intense accelerating action of the optimized nozzles, i.e. 0.5621 for WED 2 versus 0.5052 for the reference WED. The higher rotation rate is compensated by overall unloaded propeller functioning conditions due to the additional thrust provided by the ESD. This leads to a more efﬁcient propeller working point, up to 3% more eﬃcient than with the reference WED geometry in the case of WED 3. The overall saving, in terms of delivered power, is higher than 7%, which is about 2.5% higher (and solely thanks to a redesign of the WED) than the saving achieved by the reference conﬁguration.

Fig. 28. Pressure distribution on the stern of the ship. Self-propulsion calculations with the actuator disk and the fully resolved propeller.

6.2. Oﬀ-design functioning

propulsor system and the hull, since suction on the stern easily turns into additional drag. A quantiﬁcation of the interaction between the WEDs and the hull is provided in Table 10 where the resistance in self-propulsion condition

The results of the self-propulsion functioning conditions at the design speed suggest the opportunity to enrich the analysis including also oﬀ-design points, i.e. at both higher and lower speeds for all the WEDs under investigation. The improvements achieved at the design speed, indeed, where possible with a signiﬁcant modiﬁcation of the reference WED and of the relative functioning of the propeller. The issue is to 14

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

verify if the original shape, signiﬁcantly diﬀerent from the optimal ones (compare it, for instance, with WED 1 obtained only by a modiﬁcation of the global parameters) was selected as a trade-oﬀ among diﬀerent operative conditions which require diﬀerent and contrasting conﬁgurations rather than as the best geometry for the ship design speed. We want to verify if a robust optimization, i.e. by accounting perturbations in the functioning conditions, or merely an optimization at diﬀerent operating conditions is required or, alternatively, if and to what extent the improvements achieved at the design functioning allows for savings also in oﬀ-design. Regarding the propeller functioning, in addition, these additional analyses would allow to verify if possible improvements in the propeller/hull interaction provided by the WEDs are not nulliﬁed by a worsening of the propeller performance. To this aim, self-propulsion points with the fully resolved propeller have been computed, for any choice of ESD (bare hull, the reference WED and the three optimized WEDs) at two additional ship speeds, corresponding to ± 20% the design speed. The self-propulsion point at the lower speed for the bare hull required a model scale propeller rate of revolution of 5.92 rps and a delivered power of 14.9 W. At the higher speed, the equilibrium was at 9.39 rps for a model scale delivered power of 55.5 W. These are the values with respect to which percentage improvements, in the case of WEDs, are calculated in Figs. 31 and 32. Results of these self-propulsion calculations are summarized in Fig. 30 in terms of thrust deduction factor and wake fraction. Propeller performance, i.e. thrust, torque and eﬃciency, are shown in Fig. 29. Figs. 31 and 32 summarize the results for what regards propeller delivered power and rate of revolution, respectively. The inﬂuence of the diﬀerent WED geometries on propeller performance and self-propulsion coeﬃcients observed at the design ship speed is veriﬁed also in oﬀ-design conditions since the reference WED and the optimized ones are equally “ranked” at each ship speed. Considering the bare hull conﬁguration at the design speed, both the reference WED and WED 1 are responsible of higher wake fractions while with WED 2 and WED 3 the average inﬂow to the propeller is faster. This behaviour is observed also at low and high speeds, and relative diﬀerences are roughly the same. Thrust deduction factor behaves similarly. The reference WED provides only a small increase with respect to the bare hull for any ship speed while the most signiﬁcant improvements are possible with WED 1 and WED 2, being WED 3 very similar to WED 2. These trends, in turn, aﬀect the predicted propeller

Fig. 30. Self-propulsion coeﬃcients at diﬀerent ship speeds for the bare hull and for any WED choice.

Fig. 31. Variation in the delivered power at diﬀerent ship speeds for the bare hull and for any WED choice.

performance in self-propulsion functioning. The reference WED is responsible of a lower propeller eﬃciency which is conﬁrmed also at higher and lower speeds. WED 2, which allows the highest propeller eﬃciency at the design point, outdoes all the ESDs also in oﬀ-design conditions. The most signiﬁcant results are those of Fig. 31, which summarize the eﬀectiveness of the devised ESDs in terms of reduction of the delivered power and, then, in terms of fuel saving. All the conﬁgurations allow for signiﬁcant savings at any ship speed and, also in this case, the “ranking” in performance among the WED is conﬁrmed. The best conﬁguration at the design point, that of WED 3, which is able to a 7.2% reduction in delivered power, provides the highest saving of about 8% at the higher speed and an outstanding improvement of about the 9% at the lower speed. With the exception of WED 2, which in any case conﬁrms the saving achieved at the design speed, all the conﬁgurations when working in oﬀ-design conditions allow for savings higher than those achieved at the design point. In particular, all the geometries

Fig. 29. Propeller performance at diﬀerent ship speeds for the bare hull and for any WED choice. 15

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

simply acting on the angle of attack and the diameter of the WED (design run 1), while the maximum saving (about 4%) was achieved when non-symmetric conﬁgurations were considered (design run 3). These improvements turned into an additional 1% saving in delivered power with respect to the reference ESD in the case of WED 1 while WED 3 allowed for a model scale saving higher than 2%. Considering the simpliﬁcations and the assumptions of the proposed design approach, the most signiﬁcant role of these ESD is the production of an additional thrust. Its net contribution to the reduction of the ship resistance is positive since the delivered additional thrust is greater than the increase of hull drag usually associated to higher suctions on the stern of the ship. The self-propulsion analyses with the fully resolved propeller, in any case, demonstrated that the action of the WEDs modiﬁes the functioning point of the propellers. With the exception of the reference conﬁguration, this turned into a more eﬃcient operating condition, with an increase of the propeller eﬃciency of about 2%. The ﬁnal veriﬁcation of the eﬀectiveness of designed WEDs in oﬀ-design conditions conﬁrmed the reliability of the design process and the validity of the assumptions made. All the optimized WEDs, as well as the reference geometry, still provide energy savings in oﬀ-design conditions, conﬁrming the robustness of this concept of ESD with respect to quite signiﬁcant variations of the functioning condition. In the end, the proposed design approach demonstrated its ﬂexibility and reliability for early design stage applications by automatically handling thousands of geometries in a reasonable computational time, in the extent of few days. The design process, which has been carried out in model scale for the sake of validation with the few experiments available, can be easily applied to full scale analyses, provided that reliable estimation of full scale hull “functioning” including hull wake, bilge vortexes, propeller induced velocities and wake are available. This would allow to design truly custom WED geometries tailored to the ship full-scale wake and, by exploiting the potentialities of the design approach, to take advantage of reasonable energy savings also in the case of the non-favourable (i.e. fast, nonseparated, ﬂows) functioning conditions represented by the full-scale ship. Furthermore, the data provided by the high-ﬁdelity analyses of selfpropulsion functioning with the fully resolved propeller prove that a redesign of the propeller itself operating together with the custom WED could even provide additional energy recovery, encouraging a simultaneous design of the WED and of the propeller to exploit the maximum from the mutual interactions: on one side a WED shape designed for the current inﬂow, on the other an unloaded propeller blade adapted to wake from the WED.

Fig. 32. Variation in the propeller rate of revolution at self-propulsion equilibrium at diﬀerent ship speeds for the bare hull and for any WED choice.

seem to work even better at the lower speed with an average improvement with respect to the design functioning higher than 1%. A possible explanation of this behaviour relies on the nature of the ﬂow at the stern of the ship at the very low Froude (i.e. speed) numbers considered in the analysis. At such low ship speed the risk of ﬂow separation is obviously higher and the hull wake, as shown in the wake fractions of Fig. 30, results even slower. These are the functioning conditions in correspondence of which any Wake Equalizing Duct, regardless its optimized shape, gives its best since its accelerating eﬀect is suﬃcient to recover a suﬃciently large portion of the losses associated to ﬂow separation at the stern of the hull. All the results of the optimization processes currently carried in model scale, then, have to be considered in the light of these observations, since moving to full-scale makes the hull boundary layer proportionally thinner with less risk of ﬂow separation, which easily reduces the margin of improvements achievable even with a WED shape directly optimized for the full-scale hull wake.

7. Conclusions

CRediT authorship contribution statement

We developed a SBDO approach for the design of Energy Saving Devices based on the concept of the Wake Equalizing Ducts. The tool relies on a parametric description of the geometry able to handle nonsymmetric top/bottom duct shapes with a continuously varying angle of attack, a genetic algorithm and a RANSE-based ﬂow solver. Viscous calculations are mandatory for this type of problem/optimization due to dominant viscous hull/ESD/propeller interactions, which exploitation provides the energy recovery. Simpliﬁed but reliable models are then required to reduce the computational burden of the hydrodynamic predictions, allowing the use of design by optimization strategies as an aﬀordable design approach. To this aim, a self-propulsion prediction method relying on radially varying actuator disk has been validated and used to account for the inﬂuence of the propeller in the design process. Three design runs, widening the design space in terms of additional shape modiﬁcations of the nozzle geometry, were analysed. One optimal geometry for each design case, i.e. the one providing the highest reduction of the ship resistance in self-propulsion, was selected for additional analyses. All the three selected WEDs showed model scale improved performance conﬁrmed by high-ﬁdelity RANSE calculations of the self-propulsion functioning with the fully resolved propeller. A base reduction of the total ship resistance larger than 2% was achieved

Francesco Furcas: Methodology, Investigation, Validation. Giuliano Vernengo: Writing - original draft, Visualization. Diego Villa: Software, Visualization. Stefano Gaggero: Conceptualization, Methodology, Writing - original draft, Writing - review & editing, Supervision. Declaration of Competing Interest The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. References [1] H.-J. Shin, J.-S. Lee, K.-H. Lee, M.-R. Han, E.-B. Hur, S.-C. Shin, Numerical and experimental investigation of conventional and un-conventional preswirl duct for VLCC, Inte. J. Naval Archit. Ocean Eng. 5 (3) (2013) 414–430. [2] T. van Terwisga, On the working principles of energy saving devices, Proceedings of the Third International Symposium on Marine Propulsors. Launceston, Tasmania, Austrailia, (2013).

16

Applied Ocean Research 97 (2020) 102087

F. Furcas, et al.

[26] A. Coppedé, S. Gaggero, G. Vernengo, D. Villa, Hydrodynamic shape optimization by high ﬁdelity CFD solver and Gaussian process based response surface method, Appl. Ocean Res. 90 (2019) 101841. [27] G. Vernengo, T. Gaggero, E. Rizzuto, Simulation based design of a ﬂeet of ships under power and capacity variations, Appl. Ocean Res. 61 (2016) 1–15. [28] E.-J. Foeth, F. Lafeber, Systematic propeller optimization using an unsteady Boundary Element Method, Fourth International Symposium on Marine Propulsors (SMP’15). Austin, Texas, USA, (2015). [29] F. Vesting, R. Gustafsson, R.E. Bensow, Development and application of optimisation algorithms for propeller design, Ship Technol. Res. 63 (1) (2016) 50–69. [30] J. Huisman, E.-J. Foeth, Automated multi-objective optimization of ship propellers, Proceedings of the Fifth International Symposium on Marine Propulsors (SMP’17), Espoo, Finland, (2017). [31] K. Mizzi, Y.K. Demirel, C. Banks, O. Turan, P. Kaklis, M. Atlar, Design optimisation of propeller boss cap ﬁns for enhanced propeller performance, Appl. Ocean Res. 62 (2017) 210–222. [32] D. Peri, E.F. Campana, Multidisciplinary design optimization of a naval surface combatant, J. Ship Res. 47 (1) (2003) 1–12. [33] L. Bonﬁglio, P. Perdikaris, G. Vernengo, J.S. de Medeiros, G. Karniadakis, Improving swath seakeeping performance using multi-ﬁdelity Gaussian process and Bayesian optimization, J. Ship Res. 62 (4) (2018) 223–240. [34] L. Bonﬁglio, P. Perdikaris, S. Brizzolara, G. Karniadakis, Multi-ﬁdelity optimization of super-cavitating hydrofoils, Comput. Method. Appl. Mech. Eng. 332 (2018) 63–85. [35] SiemensPLM, StarCCM+ Users’ manual, v. 12.04, SiemensPLM, 2017. [36] Esteco, modeFRONTIER Users’ manual, Esteco, 2017. [37] L. Larsson, F. Stern, M. Visonneau, Numerical Ship Hydrodynamics: An Assessment of the gothenburg 2010 Workshop, Springer, 2013. [38] L. Larsson, F. Stern, M. Visonneau, T. Hino, N. Hirata, J. Kim, Tokyo 2015: A workshop on CFD in ship hydrodynamics, Workshop Proceedings, Tokyo, Dec, (2018), pp. 2–4. [39] NMRI, Japan Bulk Carrier geometry, Technical Report, National Maritime Research Institute, 2015. https://t2015.nmri.go.jp/ [40] S. Gaggero, D. Villa, M. Viviani, An extensive analysis of numerical ship self-propulsion prediction via a coupled BEM/RANS approach, Appl. Ocean Res. 66 (2017) 55–78. [41] S. Gaggero, T. Gaggero, G. Tani, G. Vernengo, M. Viviani, D. Villa, Ship self-propulsion performance prediction by using OpenFOAM and diﬀerent simpliﬁed propeller models, Progress in Maritime Technology and Engineering: Proceedings of the 4th International Conference on Maritime Technology and Engineering (MARTECH 2018), May 7–9, 2018, Lisbon, Portugal, CRC Press, 2018, p. 195. [42] D. Villa, M. Viviani, G. Tani, S. Gaggero, D. Bruzzone, C.B. Podenzana, Numerical evaluation of rudder performance behind a propeller in bollard pull condition, J. Mar. Sci. Appl. 17 (2) (2018) 153–164. [43] S. Gaggero, D. Villa, Steady cavitating propeller performance by using OpenFOAM, StarCCM+ and a boundary element method, Proc. Inst. Mech. Eng. Part M 231 (2) (2017) 411–440. [44] L. Eça, M. Hoekstra, A procedure for the estimation of the numerical uncertainty of CFD calculations based on grid reﬁnement studies, J. Comput. Phys. 262 (2014) 104–130. [45] A. Coppedè, S. Gaggero, G. Vernengo, D. Villa, Hydrodynamic shape optimization by high ﬁdelity CFD solver and gaussian process based response surface method, Appl. Ocean Res. 90 (2019). [46] D. Villa, S. Gaggero, T. Gaggero, G. Tani, G. Vernengo, M. Viviani, An eﬃcient and robust approach to predict self-propulsion coeﬃcients, Appl. Ocean Res. (2019). [47] M. Hoekstra, A RANS-based analysis tool for ducted propeller systems in open water condition, Int. Shipbuild. Prog. 53 (3) (2006) 205–227.

[3] W. Van Lammeren, Enkele constructies ter verbetering van het rendement van de voorstuwing, Ship en Werf van 1 (1949). [4] H. Schneekluth, The wake equalising duct, International Maritime and Shipping Conference (IMAS’89), 4th, (1989). [5] N. Sasaki, T. Aono, Development of energy saving device ’SILD’, Sumitomo. Tech. Rev. 135 (1997) 47–50. [6] Q. Wald, et al., Performance of a propeller in a wake and the interaction of propeller and hull, J. Ship. Res. 9 (02) (1965) 1–36. [7] J.-T. Lee, M.-C. Kim, J.-C. Suh, S.-H. Kim, J.-K. Choi, Development of a preswirl stator-propeller system for improvement of propulsion eﬃciency : a symmetric stator propulsion system, J. SocNaval. Archit. Korea 29 (4) (1992) 132–145. [8] F. Mewis, H. Peters, Power savings through a novel ﬁn system, SMSSH Conference Proceedings, (1986). [9] K. Ouchi, T. Kawasaki, M. Tamashima, Propeller eﬃciency enhanced by PBCF, 4th International symposium of Marine Engineering, Kobe, Japan, (1990), pp. 15–19. [10] S. Gaggero, Design of PBCF energy saving devices using optimization strategies: a step towards a complete viscous design approach, Ocean Eng. 159 (2018) 517–538. [11] P. Andersen, S.V. Andersen, L. Bodger, J. Friesch, J.J. Kappel, Cavitation considerations in the design of Kappel propellers, Nct’50. International Conference on Propeller Cavitation, (2000). [12] S. Gaggero, J. Gonzalez-Adalid, M.P. Sobrino, Design of contracted and tip loaded propellers by using boundary element methods and optimization algorithms, Appl. Ocean Res. 55 (2016) 102–129. [13] F. Mewis, A novel power-saving device for full-form vessels, First International Symposium on Marine Propulsors SMP, 9 (2009), pp. 443–448. [14] F. Mewis, T. Guiard, Mewis duct–new developments, solutions and conclusions, Second International Symposium on Marine Propulsors, Hamburg, Germany, (2011). [15] T. Guiard, S. Leonard, The Becker Mewis duct–challenges in full-scale design and new developments for fast ships, 3rd international symposium on marine propulsors, Launceston, (2013). [16] J.-H. Kim, J.-E. Choi, B.-J. Choi, S.-H. Chung, H.-W. Seo, Development of energysaving devices for a full slow-speed ship through improving propulsion performance, Int. J. Naval Archit. Ocean Eng. 7 (2) (2015) 390–398. [17] H.W. Lerbs, Moderately loaded propellers with a ﬁnite number of blades and an arbitrary distribution of circulation, Trans. Soc. Naval Archit. Marine Eng. 60 (1952) 73–117. [18] D. Bertetta, S. Brizzolara, S. Gaggero, M. Viviani, L. Savio, CPP Propeller cavitation and noise optimization at diﬀerent pitches with panel code and validation by cavitation tunnel measurements, Ocean Eng. 53 (2012) 177–195. [19] S. Gaggero, D. Villa, G. Tani, M. Viviani, D. Bertetta, Design of ducted propeller nozzles through a RANSE-based optimization approach, Ocean Eng. 145 (2017) 444–463. [20] S. Gaggero, G. Tani, D. Villa, M. Viviani, P. Ausonio, P. Travi, G. Bizzarri, F. Serra, Eﬃcient and multi-objective cavitating propeller optimization: an application to a high-speed craft, Appl. Ocean Res. 64 (2017) 31–57. [21] S. Gaggero, G. Vernengo, D. Villa, L. Bonﬁglio, A reduced order approach for optimal design of eﬃcient marine propellers, Ships Oﬀshor. Struct. 0 (0) (2019) 1–15. [22] S. Gaggero, Numerical design of a RIM-driven thruster using a RANS-based optimization approach, Appl. Ocean Res. 94 (2020) 101941. [23] G. Vernengo, L. Bonﬁglio, S. Gaggero, S. Brizzolara, Physics-based design by optimization of unconventional supercavitating hydrofoils, J.Ship Res. 60 (4) (2016) 187–202. [24] J. Royset, L. Bonﬁglio, G. Vernengo, S. Brizzolara, Risk-adaptive set-based design and applications to shaping a hydrofoil, J. Mech. Des. 139 (10) (2017). [25] G. Vernengo, L. Bonﬁglio, A computational framework to design optimally loaded supercavitating hydrofoils by diﬀerential evolution algorithm and a new viscous lifting line method, Adv. Eng. Softw. 133 (2019) 28–38.

17

Design of Wake Equalizing Ducts using RANSE-based SBDO

Design of Wake Equalizing Ducts using RANSE-based SBDO

Recommend Documents