Automated Marine Propeller Optimal Design Combining Hydrodynamics

Models and Neural Networks

Danilo Calcagni, CNR-INSEAN, Rome/Italy, d.calcagni@insean.it

Giovanni Bernardini, University Roma Tre, Rome/Italy, g.bernardini@uniroma3.it

Francesco Salvatore, CNR-INSEAN, Rome/Italy, f.salvatore@insean.it

Abstract

In the present paper, a computationally efficient methodology to develop fast and reliable propeller

selection procedures based on a fully automated optimization technique is described. To this aim, a

comprehensive propeller hydrodynamics model is combined with performance prediction acceleration

techniques based on Neural Networks. Under given operating conditions, screw characteristics and

blade shape details are optimized around a baseline configuration via general-purpose numerical op-

timization software based on genetic algorithms and via a parametric model. Numerical applications

concern the propulsion retrofitting of marine vessels. A off-design performance verification study is

presented to evaluate the robustness of the identified optimal configurations.

1. Introduction

The present work addresses the development of theoretical and computational models for the analysis

and the preliminary design of marine propellers. From a general standpoint, the improvement of

propulsor efficiency is a challenging problem in ship design and has received considerable attention in

the last decade; the reduction of the operating costs accounting for environmental issues (fuel

consumption and NOx emissions, for instance) represents one of the most urgent needs for designers.

Practical interest is given to the design of new propellers to be installed on existing aged vessels. The

replacement of propulsion units represents one of the most viable solutions to achieve remarkable

trade-off between gains in hydrodynamic efficiency and budget necessary. Nowadays, such a design

update strategy (retrofitting) is very popular among ship-owners. The availability of fast and accurate

numerical prediction tools during the pre-design stage is a crucial point to achieve effective

propulsion systems optimization strategies. In fact, guidelines derived at early stage may result into

faster and less expensive iterative procedures within the classical design spiral steps. The standard

technique based on the use of experimental-based systematic propeller series and semi-empirical,

knowledge-based performance prediction models represents the oldest, but still largely used design

strategy, see e.g. Oosterveld and Van Oossanen (1975).

During the last decades, several attempts to enhance design procedures by a full exploitation of state-

of-art computational hydrodynamics models have been proposed. While Computational Fluid

Dynamics (CFD) models for the numerical solution of the Navier-Stokes equations still have a very

limited appeal as preliminary design tools because of computational effort and complexity of

applications to real-life problems, extensive applications of inviscid-flow methods like lifting-line,

lifting surface and panel methods are the subject of a vast literature. Limiting to consider applications

to marine propellers, Coney (1989) develops an inverse design method based on a cascade of a lifting

line/lifting surface models for the determination of the geometrical shape for given performances.

Improvements of this technique are presented by Stubblefield (2008) to design multi-component

propulsors. Druckenbrod et al. (2010) present an improved methodology by coupling a RANSE

solver and a first order panel method for the hydrodynamics analysis of a propulsors in multi-body

configuration. A direct design method is proposed through the application of a genetic algorithm-

based optimizer with design variables controlling blade pitch and chord distributions. Mishima (1996)

and Geisbert and Schroeder (2008) improve the computational efficiency of a direct design method

by applying a regression model to synthesize the response surface of the system. Deng (2005) applies

this technique to cavitating propellers operating in non-uniform onset flow in multi-body

configurations. Constraints on the power delivered by the engine, on the thrust generated at given ship

speed and on acceptable cavitation risk are taken into account.

Following the philosophy of design procedures reviewed above, the present work is aimed at

developing an efficient, fully automated procedure to determine the best propulsive configuration

fitting given operational requirements and constraints. For the optimal solution seeking procedure, a

set of design variables to characterize the screw geometry is firstly defined. Different geometric

configurations may be obtained by changing values of these variables within a suitable bounded

domain: hence, a virtual systematic propeller series is defined and the optimal solution is searched in

its context. The strategy used to find the optimal solution consists of a two step procedure, where two

different sets of design variables are allowed to vary, respectively. In detail, global and local design

variables are introduced and two optimizations (global and local) are performed, where the starting

point for the final optimization (local) is the optimal solution of the first one (global). The

hydrodynamic efficiency is chosen as objective function and contributes to the definition of the

response surface of the system.

An unsteady, three-dimensional hydrodynamics solver based on a Boundary Element Method (BEM)

for the analysis of the flow around thrusting ducted propellers is chosen to provide predictions of

propulsor performance. The response surface is synthesized through a suitable regression model,

based on Neural Networks (NN). The neural network is then interfaced with the optimization tool in

order to explore the surrogate response surface, to define the optimal solution. Two optimization

models are considered here: a Genetic Algorithm (GA) and a simple Parametric Modelling (PM)

approach based on a systematic variation of the design variables. Hull boundary-layer induced effect

is herein modeled through global parameters known from the characterization of the performance of

the existing vessel. Suitable constraints are derived by accounting for hull-propulsor coupling and

from requirements on advance speed, allowable cavity extension and maximum overall dimension.

Results of the application of the proposed methodology to the retrofitting problem of a ducted

propeller of a fishing boat are presented and discussed in details.

2. Problem overview and definition of an optimal design procedure

The development of a fully automated optimal design procedure for marine propellers is carried on

here by integrating the following aspects:

1. definition of a baseline configuration (the reference propeller and its operating conditions)

and parameterization through a limited number of geometry/operational parameters

2. generation of a virtual propeller series by systematic variations of parameters characterizing

the geometry of the baseline configuration

3. construction of a fast computational model to predict the hydrodynamic performance of any

individual from the virtual propeller series

4. selection of the best propulsive configuration by exploring the hyper-surface representing the

hydrodynamic performance of the virtual propeller series.

The first two steps of the overall procedure allow achieving an effective parameterization of the

propeller geometry. Shape manipulations are then obtained by controlling a limited set of parameters

and arbitrary geometry variants to be analysed during the optimal configuration search are easily

obtained. Quantities to describe the propeller geometry fall into two classes: global parameters and

local distributions. A conventional screw propeller can be defined in terms of global parameters as

diameter D, hub/blade diameter ratio H/D, number of blades Z, expanded area ratio EAR, nominal

pitch to diameter ratio P/D. A reference propeller is then uniquely characterised by specifying local

shape details as radial distributions of blade chord C = C(r), pitch P = P(r), skew S = S(r), rake Rk =

Rk(r), maximum sectional thickness TMAX= TMAX (r), camber fMAX= fMAX(r), and blade section offset

distributions along the chord. In the present work a virtual propeller series is built by choosing a

baseline configuration and varying its global parameters D, Z, EAR, P/D within given ranges. All

other global parameters and local distributions are kept the same as for the baseline propeller blade.

In order to make possible extensive investigations at preliminary design stage, a crucial issue is to

develop fast hydrodynamic models to simulate the response of a large number of guess configurations

during the optimal layout search (step 3 above). A distinguishing feature of the proposed procedure is

that hydrodynamic predictions at negligible computational time are achieved by developing a

surrogate model of a standard propeller hydrodynamics model based on a Boundary Element Method

(BEM). The surrogate model is derived by means of a Neural Network model. Finally, the selection of

the best propeller (step 4 above) is accomplished by an optimal design procedure in which the

(surrogate) propeller hydrodynamics model is combined with a fast numerical optimization model

based on Genetic Algorithms.

The optimal design problem involves the definition of design parameters, design objectives and

constraints. Two types of design parameters are considered: geometry parameters to uniquely identify

a given candidate propeller through the smallest number of variables, and operational parameters to

specify ship and propulsion system operating conditions (design conditions). It is worth noting that

the larger the number of geometry parameters, the larger the variety of candidate propellers that can

be explored with inherent advantages in terms of quality of results of the optimization study. A natural

limit to the number of geometry parameters is enforced by the computational effort required by the

overall design procedure. Objective of numerical applications proposed in this paper is to demonstrate

that a design exercise with potential relevance for real-life problems can be addressed by using a very

limited number of design parameters. Specifically, only four global geometry parameters are

considered: propeller diameter D, number of blades Z, expanded area ratio EAR, nominal pitch to

diameter ratio P/D. Such a parameterization is the same used to define the virtual propeller series as

mentioned above. Next, as operational design parameter, the propeller advance coefficient is

considered. In order to cope with hull/propeller interaction effects, the following expression of the

advance coefficient is used: J = Va / (nD), in which Va = (1-w) VShip is the averaged speed at propeller

plane in behind conditions, w is the wake fraction and VShip is the ship advance speed.

Design and optimization goals are defined through one objective function, to maximise propeller

hydrodynamic efficiency, and design constraints to ensure blade strength and to avoid risk of

cavitation-induced vibrations and erosion as clarified later. Global geometry parameters D, Z, EAR,

P/D are varied to determine the best propeller among the propeller models populating the virtual

series defined at the beginning of the procedure. Using this approach, configurations outside the

boundaries of the virtual series cannot be investigated. This introduces some stiffness in the overall

procedure and limits potential efficiency gains achieved through the optimal design analysis. In order

to introduce more flexibility in propeller shape variations addressed during the optimization

procedure, a shape refinement process is then performed once the result of optimal design based on

global parameters is obtained. Specifically, radial distributions of parameters describing blade

planform are varied. To this aim, radial base functions are introduced to define modified distributions

of blade chord, pitch, skew, rake, thickness and camber representative of the virtual series considered

here. Radial base functions are parameterized in order to recast any particular combination of radial

distributions through a limited set of scalar parameters, akin to the manipulation of global geometry

variables. Fig.1 sketches the coupling of different tools integrated into the proposed optimal design

procedure.

Fig.1: Design flow-chart example for the determination of an optimal propeller configuration

3. Analysis and evaluation models

The optimal design procedure outlined in the previous section is general in that different models to

analyse the response of the system to optimise (here, a marine propeller) and to evaluate a number of

guess configurations to find the optimal configuration (the optimization model) can be plugged into

the structure sketched in Fig.1. In the present work, the optimization model is based on Genetic

Algorithms and the hydrodynamics model is based on a Boundary Element Method and its surrogate

model developed through Neural Networks. Basic features of these models are discussed in the

following.

3.1 Numerical Optimization: Genetic Algorithms (GA) and Parametric Model (PM)

In this work two different optimization models are used to seek the optimal solution. Optimization

methods based on genetic algorithms (GA) mimic the biological evolution to find the optimal solution

of a given problem, Holland (1975). Candidate solutions are called individuals and the set of

individuals’ population. Since binary-based (GA) are used here, a chromosome/gene bit-string

notation is used to encode each individual, Raymer(2002). The optimization process starts from a

completely random-generated population (each one related by different chromosome informations)

and evaluates objective functions and constraints. Individuals are compared and a selection of the best

ones is done in order to generate a new offspring. The constraints are included in the optimization

process through a penalty function approach, Hatfka and Gurdal (2008), which enhances the breeding

possibility of individuals potentially able to generate good offsprings. In this sense, constraints are

taken into account indirectly, turning the constrained optimization into a sequence of unconstrained

minimization (maximization) problems. A uniform crossover technique is applied over parents’

chromosomes for child chromosome definition. The individuals selected as parents are the result of a

tournament selection, for which, starting from a random selection of four parents, they are compared

one-vs-one in two pair and the couple of 'winners' are selected to be parents of two children with two

independent crossover operations, in order to keep constant the amplitude of the population. Once the

crossover is performed, a mutation operation is applied, to avoid premature convergence to local

optima. In order to prevent possible negative aspects of the evolution process, at each step of the

optimization process the best individuals are selected to become part of an elite group. They are

represented in the next generation unchanged; this technique avoids the possibility to obtain worse

generation during the process itself. A purely random generation is chosen at the beginning of the

process, in order to cover the wider region of the design variables as possible, therefore, maximizing

the probability to find the global optimum. A non random generation is selected during final phases of

the process, in order to obtain the best solution accordingly to the evolutionary strategy, for which the

genetic patrimony changes not only in view of random factors governed by phate, but even for

characteristics of the ambient in which the population growths. Therefore, initially, bits have a

random distribution, whereas, at final stages, all individuals converge toward the same gene pool

(identical chromosomes). The convergence of the process is evaluated on the basis of the individuals

bit-string affinity, Raymer(2002).

A parametric model is here used in the design process parallel to the genetic algorithm-based

optimizer. The entire hyper-space of configurations is covered using a systematic variation of design

variables. Uniform rates of changing are defined to values representing a good trade-off between

computational efforts and resulting resolution. The optimal solution is estimated within all

configurations that satisfy all the constraints, trying to maximize the objective function.

3.2 Synthesis of complex systems using Neural Networks (NN)

Among all models able to synthesize systems characterized by strong non-linear relationships

between input and output variables, neural networks represent a typical example and are used in this

work. Neural networks were initially developed to simulate learning and synthesis mechanisms in

human brains. Information is elaborated and stored by a complex system of neurons, strictly

interconnected by so called “synapses”. A neural network is able to learn implicitly relationships

between input and output variables of a given system once connections are tailored during the training

phase. Generally speaking, neural networks denote a particular subclass of non linear regression

models applied to synthesize functional relationships between input and output variables in a complex

system. Examples of linear regression models can be found in Myers and Montgomery (1995). Such

models, based on polynomial expansion of the input variables, provide fast predictions of expansion

coefficients, but are strictly dependent on the polynomial order used to express relationships.

Therefore, their use is limited to cases where input/output relationships show a regular and smooth

behavior. Moreover, although the order of polynomial expression is directly related to the accuracy of

the model in training points, such trend isn't assured everywhere; often, a deviation from the real

curve is observed. On the other side, a neural network with two layers (one hidden) has been

demonstrated to play the role of universal approximator for continuous function in Rn, using a wide

class of activation functions, Grippo and Sciandrone (2003), whereas three-layers based neural

networks are characterized by an unbounded lower limit for error, Maiorov and Pinkus (1999).

Here, a feed-forward architecture is used: the signal flow runs from input to output neurons, through

neurons. In Fig. 2, a typical scheme reproducing a feed-forward neural network is shown, as well as

the one of the generic neuron. Formally, the relationship between output and input signals of neuron

“i” of the hidden layer “l” is described in the Fehler! Verweisquelle konnte nicht gefunden

werden.).

(1)

where the sum is extended to all neurons “j” of the hidden layer “l-1”. The relationship between the

output signals of the network and the input variables is therefore determined for given values of

weight connections. The hyperbolic tangent function is here used as “F”.

In the training phase, “examples” of Input/Output combinations are presented to the neural network

that “learn” relationships through an optimization problem expressed in terms of the error Ep, related

to the difference between the output prediction of the network and the value used to train. A

Levenberg-Marquardt technique is applied through a back-propagation scheme to determine the

gradient of the error with respect to connection weights, Grippo and Sciandrone (2003); thus, the

unknown weights are evaluated through a recursive procedure.

Details of the NN architecture proposed in the present work are given in Calcagni et al. (2010) and

Calcagni (2011), where results of the application of the NN to reproduce hydrodynamic performance

of the Wageningen B-series propellers, Kuiper (1992), are discussed. The neural network presented

above is used to synthesize the response surface, built on the dataset of performances predicted by a

potential flow solver applied to propellers of a systematic series in different configurations.

Fig. 2: Typical scheme of a neural network (NN or ANN), based on a multilayer "feed-forward"

architecture. Left: general sketch of the NN. Right: particular view of a generic neuron.

The dataset of performances exerted by propulsors in different configurations represents the complex

system considered in this work. It is used as a black box in the design and optimization procedures:

performances are determined in a fast way with an acceptable accuracy. The use of a surrogate model

is justified when the number of points on which the surface is built is lower then the number of

configurations examined during the optimization phase. In this case consistent computational time

savings can be achieved.

3.3 Propeller hydrodynamics model

The propeller hydrodynamics model used for the training of the NN is based on a Boundary Element

Model (BEM) for potential flows. Comparing computational models used for marine propulsion

studies, BEM are widely recognised as a good trade-off between accuracy of predictions and

computational burden. Here, an approach for open screw propellers developed and validated over the

last decade, Pereira et al. (2004) and Salvatore et al. (2006), has been extended to analyse ducted

propulsors, Salvatore et al. (2006b), where details of the computational methodology are given. Basic

features are here briefly reviewed for the sake of completeness.

Assuming the onset flow is inviscid and incompressible, propulsor-induced perturbation velocity v

may be expressed in terms of a scalar potential as, v = ∇∇∇∇φ where φ is governed by the Laplace

equation ∇2φ = 0. Here, the Laplace equation is solved through a boundary integral formulation,

Morino (1993), that gives φ as a superposition of singularities distributed over the fluid domain

boundary represented by the propulsor solid surface and a surface describing the vortical wake

generated by propeller blades. Impermeability on solid boundaries and vorticity convection along

wakes are imposed to define suitable boundary conditions.

A key feature of the methodology is that unknowns are distributed only over these boundary surfaces

and hence the additional burden of unknowns distributed throughout the fluid region is prevented.

Once equations are written in discrete form, the problem is recast as the solution of a linear system of

algebraic equations. Fig. 3 shows a BEM model computational grid of a ducted propeller with vortical

wakes emanated by blades and by the duct trailing edge. Once the perturbation potential φ is

determined, propeller-induced velocity is determined and pressure follows through Bernoulli's

theorem. Propeller thrust TP, torque QP and duct thrust TD are calculated by integrating the pressure

field over propeller and duct surfaces. A simple equivalent flate-plate approach is used to include

viscous losses on predicted potential forces. In view of applications to optimal design studies, the

capability of BEM to describe open as well as ducted propeller performance is briefly discussed,

whereas more details can be found in Calcagni (2011). As a case study representative of ducted screw

configurations, the CNR-INSEAN E1622 model is considered. Main geometry details are given in

Table I, whereas propeller open-water performance results are shown in Fig. 5, where numerical

predictions by BEM are compared to model tests performed at CNR-INSEAN.

Fig. 3: Three-dimensional model of ducted propeller flow analysis by BEM: computational grids on

propeller, duct and wake surfaces. For clarity, only wake portions shed by one blade and the

corresponding duct sector are shown.

Table I: E1622 full scale main characteristics

DP Z Ae/A0 P/D.7R

2.6 m 4 .69 0.97

Fig. 4: Fishing vessel: fully appended model used for propulsion tests (left) and set-up for ducted

propeller open-water measurements (right). Model tests performed at CNR-INSEAN

Propeller thrust, torque, duct thrust coefficients and hydrodynamic efficiency are defined as in

Fehler! Verweisquelle konnte nicht gefunden werden.); the total amount of thrust is defined as

.

; ; ;

(2)

All quantities are related to the advance ratio J=V/(nD), where V is the advance speed and n, D

denote, respectively, propeller rotational speed and diameter. Reliable predictions of unducted open

water propeller thrust and torque are observed from left Fig. 5. A trend to underestimate propeller

thrust at advance ratio higher than 0.6 is observed. This corresponds to under-predictions of

hydrodynamic efficiency. A similar level of accuracy results from ducted propeller flow studies, right

Fig. 5. In fact, discrepancies between calculations and model test data are present at low advance ratio

and close to zero thrust condition. Lack of accuracy can be partly explained recalling that the inviscid-

flow model by BEM provides only a rough estimate of viscous-flow contributions to propeller loads.

Dealing with ducted propellers in off-design conditions, the effects of boundary layer flow separation

on the outer surface of the nozzle are then largely missed, Caja et al. (2008). Nevertheless, Fig. 5

highlights that propeller loadings in both open and ducted configurations are predicted with

reasonable accuracy over a range of operating conditions close to design condition.

Fig. 5: CNR-INSEAN E1622 model propeller. Open-water performance (thrust, torque, efficiency).

Comparison between numerical predictions by BEM and experimental data. Left: open propeller;

right: ducted propeller.

4. Numerical applications: a propeller design exercise

The proposed computational approach is applied to a propeller design problem taken as representative

of studies commissioned by ship-owners interested to update the propulsion system of existing vessels

(propulsion retrofitting study). The test case is a 1440 DWT deep sea fishing vessel whose main

characteristics are summarized in Table II.

Table II: Fishing vessel: main characteristics (full scale)

LPP GT PMCR nMCR Prop.type

67 m 1440 Gt 2930 Hp 375 rpm FPP in nozzle

This ship has a single-screw hull equipped with a four bladed ducted propeller with similar global

geometrical parameters as the E1622 model addressed in Table I. A λ=12 scaled model of the fully

appended ship was manufactured by CNR-INSEAN to perform a comprehensive set of measurements

in the framework of a research project, see Salvatore et al. (2006) and Calcagni (2011) for details).

Photographs of the ship model and of the ducted propeller during towing tank tests at CNR-INSEAN

are given in Fig. 4. Design conditions considered here reflect the ship operating at low-medium

speed, under high loading conditions occurring during trawling. The optimal configuration is searched

here within a defined virtual systematic propeller series built by varying global shape parameters Z,

EAR (Ae/A0), P/D in the following ranges: 3 ≤ Z ≤ 5 ; 0.5 ≤ Ae/A0 ≤ 0.8 ; 0.85 ≤ P/D ≤ 1.25. Examples

of three configurations obtained by different combinations of these parameters are shown in Fig.6.

Fig.6: Virtual systematic propeller series: examples configurations

4.1. Global design

The developed tool for the optimal design of propellers, has been applied to the test-case described in

the previous section. A suitable optimization technique is based on the recursive query of a surrogate

model, represented by a neural network, trained on performance dataset of the examined propulsion

system. By varying design variables, different propeller configurations are compared with respect to

the defined objective function; the one with the highest objective function while all constraints are

satisfied is the optimal solution. The optimization process consists of two sub-problems, solved

sequentially. Here the first design problem is considered. This case is referred to as “global” design

problem due to the characteristics of the design variables.

The solution is here sought within the (hyper-)space defined by the following geometrical quantities:

diameter (DP), number of blades (Z), expanded area ratio (Ae/A0), pitch to diameter ratio (P/D); all

local geometrical distributions (pitch, chord, skew, camber, rake) are fixed to those of the baseline

propeller. An additional variable represents the operative condition: the speed of the vessel VShip. The

objective is to maximize the objective function (i.e. the hydrodynamics efficiency η0). The working

point represents the matching condition for the system “hull-propeller-engine”. Here, the engine

working point is fixed. Thus the first requirement is represented by a developed power at a given rate

of revolution of the shaft axis. The 85% of the Maximum Continous Rate (MCR) is considered, at a

rate of revolution fixed to n = 250 rpm. Assuming to perform the solution for the vessel in free

running condition, a lower bounded speed has been chosen: VShip min = 7.31 m/s.

Geometrical issues limits the maximum allowable diameter: DP max = 2.7 m. Additional constraints on

torque and on thrust are:

• torque Q absorbed by the propeller must match the torque Qd delivered by the engine, given

the rate of revolution of the shaft axis n. A 2.5% allowance is given to have |(Q - Qd)/Qd| ≤ 2.5

%.

• modified thrust T, exerted by the propeller, must balance the resistance R of the hull, given

the advance speed VShip: |(Th-R)/R| ≤ 2.5 %, where Th=T/(1-t).

The thrust deduction factor t is deduced by self-propulsion tests, with respect to the advance ship

velocity VShip, as well as the wake fraction w.

In order to prevent the risk of cavitation inception on blade surface a simple model, based on the

Keller's formulas, Kuiper (1992), was adopted. The lowest value of the expanded area ratio avoiding

cavitation is determined: it depends on thrust, number of blades, propeller diameter and on the

typology of vessel. Specifically, the constraint is Ae/A0 ≥ (Ae/A0)min, where (Ae/A0)min = (1.3+0.3Z) T

DP2/(p0-pv)+k; here, k = 0.2 for single shaft; p0 represents the static pressure at shaft axis, equal to the

atmospheric pressure added up to the hydrostatic pressure ρ g h; pv represents vapour pressure of the

fluid.

Results of the global optimization exercise are shown in Fig. 7 to 9, where solutions are compared to

the four-bladed reference propeller. Specifically, optimal values of the hydrodynamic efficiency on

the best performing 3-bladed, 4-bladed and 5-bladed screws are shown in left Fig. 7. As an example

of optimal geometry parameters, pitch to diameter ratio for the three different propellers are shown in

right Fig. 7, whereas other quantities (EAR, D, VS) present similar trends and are not shown here,

Calcagni (2011). The optimization model based on genetic algorithms identifies configurations very

close to those obtained by varying systematically design variables (parametric model). Discrepancies

are judged to depend on the step resolution by which the expanded area ratio EAR is modified using

the parametric model and genetic algorithms. In particular, the rate of variation of EAR is higher when

the former approach is used. Here, global parameters take account for the hull-induced wake (thrust

deduction factor t and wake fraction w). Recalling uniform flow operating conditions are addressed

here, the number of blades influences the hydrodynamic efficiency less than what is expected when a

non-uniform onset flow is applied to propeller blades, causing peaks in blade loading distributions.

Fig. 7: Global optimization problem of a fishing vessel in free running condition. Left: hydro-

dynamic efficiency over the number of blades. Right: pitch to diameter ratio over the number of

blades.

In Table III main data of different optimal solutions by means of the number of blades are presented

and compared to those of the reference propeller, whereas corresponding performances are listed in

Table IV. It should be noted that the three optimal propellers achieve the highest efficiency values

compliant with existing constraints at different operating conditions, as reflected by different values

of ship velocity in Table III. Matching ship velocity and hull resistance as well as hull-propeller

interaction parameters t and w from available self propulsion-test results, the new operating

conditions in Table III follow. Recalling optimization is based on a neural network-based surrogated

model of propeller hydrodynamics by BEM, it is worth to compare efficiency gain results outlined

above with results of optimised propeller flow studies by full BEM calculations.

Table III: Results of global optimization of E1622 model propeller: main geometrical data and

operative conditions

D(m) Z EAR P/D VS(m/s) 1-w 1-t RS(t) J

E1622 2.60 4 .67 0.98 7.59 .769 1.034 16.7 .513

OPTG3 2.69 3 .60 0.89 7.58 .768 1.034 16.6 .518

OPTG4 2.47 4 .77 1.06 7.63 .770 1.035 17.1 .571

OPTG5 2.45 5 .85 1.07 7.65 .771 1.036 17.2 .578

Table IV: Results of global optimization of E1622 model propeller: performances. Comparison by

means of different working points, each satisfying thrust and torque constraints. Symbol (*) denotes

data not available because referred to a propeller not taken from the propeller series.

J KTT 10KQ ηBEM ∆ηBEM(%) ηNN ∆ηNN(%) E1622 .513 .200 .317 .516 - - (*) -

OPTG3 .518 .170 .264 .532 +3.06 .536 +3.78

OPTG4 .571 .248 .425 .530 +2.63 .534 +3.47

OPTG5 .578 .260 .450 .531 +2.93 .537 +4.07

Fig. 8: Global optimization problem of a fishing vessel in free running condition. Pressure coefficient

distribution over the blade surface. Left: E1622 reference model propeller (J=0.514). Center: 4-

bladed solution OPTG4 (J=0.578). Right: 5-bladed solution OPTG5.

Fig. 9: Global optimization problem of a fishing vessel in free running condition. Efficiency over the

advance coefficient. Comparison between reference propeller and optimized solutions with 4 and 5

blades.

As example, predictions by full BEM show that efficiency improvements in the order of 2.7% are

achieved for the 4-bladed solution OPTG4 compared to the baseline propeller, whereas ∆η predictions

by NN is 3.5%, as reported in the last column in Table IV. This result can be taken as a global

indicator of the confidence of the surrogate model considered here. Blade pressure distributions on the

baseline propeller and on 4-bladed and 5-bladed configurations are shown in Fig. 8.

Finally, robustness of the optimal solutions found at design conditions with advance ration J close to

0.55 (Tables III and IV above) has been checked over a wide range of operational conditions between

J = 0.1 and J = 1.1, including off-design conditions.

Numerical calculations of propeller efficiency are plotted in Fig. 9. Compared to the baseline

propeller, optimized solutions are characterized by increased efficiency at high values of the advance

coefficient, whereas limited loss of performance are observed at low J. These results highlight the

importance to develop the present methodology in order to include multiple design points over a wide

range of operational conditions, as to ensure a robust optimization.

A general conclusion from the global optimization exercise discussed so far is that similar amounts of

hydrodynamic efficiency gains have been achieved with propellers having 3, 4 and 5 blades. Results

in left Fig. 7 and in Table IV point out some advantage of the 5-bladed configuration, where the

smallest blade loading density is expected and a smoother pressure distribution is evaluated, Fig. 8.

For these reasons, the 5 bladed solution (labelled as OPTG5) is chosen to be further optimized through

the local refinement approach.

4.2. Local refinement

Once the “global” design problem is addressed, a new propeller family is built around the optimal

configuration; thus, a “local” refinement is possible. Global variables are now frozen and the

geometrical radial distributions are free to vary within limited boundaries. All local quantities g0 are

modified by suitable functions ∆g, expressed by polynomial functions of the radius, such that

, with -1 ≤ k ≤ 1. All individuals of the new propeller series are

characterized by the same dataset Z, D, Ae/A0, P/D, J, whereas differences characterize the following

radial dimensionless geometrical distributions: pitch P(r)/D; chord C(r)/D; skew Sk(r)/D; maximum

camber fmax(r)/C; rake Rk(r)/D. Here, the same procedure of the global design stage is used, and the

results obtained by different optimization models are shown. Specifically, pitch and chord

distributions are presented in Fig. 10, where optimal distributions are shown as well as bounding

curves (indicated by purple lines); skew and rake are characterized by similar trends and are not

shown here. Right Fig. 10 shows that the optimal configuration is characterized by a chord

distribution that overlaps one of the two bounding curves. This is a rather frequent result due to the

introduction of constraints in the optimization procedure.

Fig. 10: Local optimization problem of a fishing vessel in free running condition. Left: radial

distribution of pitch. Right: radial distribution of chord.

Optimal configurations (subscript G for global design, L for local refinement) are compared. In

particular, Fig. 11 presents the pitch and chord distributions. Pitch is increased with respect to the

baseline configuration, but minor differences occur after the local refinement. On the other side, the

chord is increased during the global optimization, whereas during the local refinement its radial

distribution is adjusted to increase blade width at low radius and decrease towards the tip. Next,

performance prediction results are shown in Fig. 12. Thrust and torque are increased in both global

and local cases as a consequence of the overall increase of blade pitch, as clearly observed from left

Fig. 11. The hydrodynamic efficiency shows improvements for high values of the advance velocity,

whereas a loss is found at low velocities. As expected, local refinement yields a small improvement

with respect to the global optimal configuration.

In Table V main data of different optimal solutions (global and local) are summarized and compared

to those of the reference propeller. Akin to global optimization, results of local refinement are based

on performance predictions by neural networks. Findings are then checked by running directly the

BEM model at design points and for a wide range of the advance coefficient, in order to verify the

accuracy of models involved in the optimization. As a result, BEM simulations certify efficiency

improvements of about 3%, instead of a slightly more optimistic 4% predicted by the NN.

Fig. 11: Optimization problem of a fishing vessel in free running condition. Comparison between

reference and optimized propellers. Radial pitch (left) and chord (right) distributions.

Finally, pressure distributions over propeller and duct surfaces is shown in Fig. 13. A detailed insight

shows that small differences are observed over propeller blades from global and local optimizations.

Nevertheless, local chord and pitch changes after local refinement determine a smoother blade-tip

interaction with the duct inner surface.

Fig. 12: Optimization problem of a fishing vessel in free running condition. Comparison between

reference and optimized propellers. Thrust (left), torque (center) coefficients and efficiency (right)

over the advance coefficient.

Table V: Geometrical parameters and performances of propeller resulting in the optimization

problem of ducted propellers. Comparison by means of different working points, each satisfying

thrust and torque constraints. Symbol (*) denotes data not available because referred to a propeller

not taken from the propeller series.

J KTT 10KQ ηBEM ∆ηBEM(%) ηNN ∆ηNN(%) E1622 .513 .200 .317 .516 - - (*) -

OPTG5 .578 .260 .450 .531 +2.93 .537 +4.07

OPTL5 .578 .247 .423 .537 +4.01 .538 +4.12

OPTG4 .578 .245 .422 .533 +2.63 .534 +3.47

Fig. 13: Local refinement optimization problem of a fishing vessel in free running condition.

Pressure coefficient distribution over the blade surface: local refined propeller: OPTL5 (J=0.578).

5. Conclusion

The present work addresses the development of theoretical and computational models for the analysis

and the preliminary design of marine propellers. The present work has been aimed at developing an

efficient, fully automated procedure to determine the best propulsive configuration fitting given

operational requirements and constraints. A virtual systematic propeller series has been defined and

the optimal solution has been searched in its context. The strategy used to find the optimal solution

consists of a two step procedure, where two different sets of design variables were allowed to vary,

respectively. In detail, global and local design variables were introduced and two optimizations

(global and local) were performed, where the starting point for the final optimization (local) was

represented by the optimal solution of the first one (global). The hydrodynamic efficiency has been

chosen as objective function. An unsteady, three-dimensional hydrodynamics solver based on a

Boundary Element Method (BEM) for the analysis of the potential flow around thrusting ducted

propellers has been chosen to provide predictions of propulsor performance. The response surface was

synthesized through a suitable regression model, based on Neural Networks (NN). The neural network

has been interfaced with the optimization tool in order to explore the surrogate response surface, to

define the optimal solution.

Two optimization models were considered here: a Genetic Algorithm (GA) and a simple Parametric

Modelling (PM) approach based on a systematic variation of the design variables. Hull boundary-

layer induced effect has been herein modeled through global parameters known from the

characterization of the performance of the existing vessel. Suitable constraints were derived by

accounting for hull-propulsor coupling and from requirements on advance speed, allowable cavity

extension and maximum overall dimension. Results of the application of the proposed methodology to

the retrofitting problem of a ducted propeller of a fishing boat in free running condition were

presented and discussed in details. Results are judged to be satisfactory because: a) different

optimization models provide similar solutions; b) optimal solutions show gain both in hydrodynamics

efficiency and speed limit; c) the surrogate model, queried by the optimization model, approximates

accurately the response surface of the system. The analysis undertaken for the verification of the

robustness of the optimization procedure suggests that a definition of multiple design points may be

an optimal approach to improve performances of the design procedure. Hence, further improvements

can be achieved by extending the present methodology to robust optimization procedures and by

improving the accuracy of the present hydrodynamics flow model considering a suitable model for the

boundary layer estimation.

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