Opti­miza­tion and Sur­ro­gate Modeling of Tip Rake Propellers

In the maritime industry, there is a growing demand for energy effi­ciency and emission reduc­tions to meet envi­ron­men­tal reg­u­la­tions. Tra­di­tional pro­peller designs are no longer suf­fi­cient, prompt­ing the need for uncon­ven­tional designs, such as tip-rake pro­pellers, that can improve fuel effi­ciency and minimize envi­ron­men­tal impact. Com­pu­ta­tional Fluid Dynamics (CFD) has rev­o­lu­tion­ized pro­peller design by allowing for in-depth explo­ration of the design space and opti­miza­tion of pro­peller geometry for enhanced effi­ciency and per­for­mance. Fur­ther­more, sur­ro­gate modeling has evolved sig­nif­i­cantly in recent years, offering accel­er­ated solu­tions for complex design tasks by lever­ag­ing sim­u­la­tion results to provide instant insights into pro­peller per­for­mance. This stream­lined approach not only improves effi­ciency, but also facil­i­tates the explo­ration of a broader range of design alter­na­tives, result­ing in inno­v­a­tive and opti­mized pro­peller solutions.

Lit­er­a­ture Data

The lit­er­a­ture data incor­po­rated in this study focuses on Wagenin­gen B‑series pro­pellers, renowned for their exten­sive usage in marine propul­sion. One notable advan­tage of the B‑series is the ability to swiftly cal­cu­late open water char­ac­ter­is­tics due to the imple­men­ta­tion of poly­no­mial regres­sion based on exper­i­men­tal results [1]. Addi­tion­ally, the Burrill Diagram, a widely rec­og­nized cri­te­rion for fixed-pitch con­ven­tional pro­pellers in the pre­lim­i­nary design stage [2], had been dig­i­tized within CAESES® for this study.

Para­met­ric Model for Uncon­ven­tional Propellers

The Para­met­ric Model for Uncon­ven­tional Pro­pellers (PMUP) was set up and param­e­trized in CAESES®. It allows for the mod­i­fi­ca­tion of existing pro­peller designs to add features like tip rake and high skew angles. By adjust­ing key radial dis­tri­b­u­tion func­tions such as pitch, chord, skew, and rake using specific design vari­ables, the PMUP trans­forms con­ven­tional pro­peller designs into uncon­ven­tional ones. Espe­cially para­me­ters like modRake and modSkew are essen­tial for the gen­er­a­tion of uncon­ven­tional pro­pellers, allowing detailed control over the tip region.

Numer­i­cal Sim­u­la­tion Setup

The blade geometry modeled in CAESES®, rep­re­sented as a NURBS surface, was trans­formed into a panel mesh as input for the sim­u­la­tions. The mesh con­fig­u­ra­tion included 30 chord-wise and 20 span-wise panels, deter­mined through sen­si­tiv­ity analysis. Denser spacing was allo­cated at the leading edge and tip of the blade. A sharp trailing edge was imple­mented to satisfy the Kutta con­di­tion for accurate sim­u­la­tions. The hub geometry was not included in the modeling process, as illus­trated in the Figure below.

A Blade Element Momentum (BEM) code (panMARE from TUHH; based on poten­tial theory) was used to analyze the fluid flow around the pro­peller blades [3]. Open water tests were con­ducted to evaluate the pro­peller per­for­mance. A heuris­tic model accounted for flow friction effects. Cavity area and volume were cal­cu­lated to assess cav­i­ta­tion, pro­vid­ing insights for the pro­peller design optimization.

Opti­miza­tion Framework

Opti­miza­tion Algorithms

Firstly, a Design of Exper­i­ments (DoE) applying the Sobol method uses a quasi-random sequence to homo­ge­neously dis­trib­ute designs and scan the design space.

Secondly, the tangent search method (T‑Search) focuses on single-objec­tive opti­miza­tion, employ­ing tangent moves within hyper­space, although it requires a care­fully chosen initial point and may converge to local minima. Usually, the best Sobol design is a promis­ing can­di­date as a starting point for the T‑Search optimization.

Lastly, Response Surface Method­ol­ogy (RSM) con­structs input-output sur­ro­gates using the Kriging method, enabling finding optimal solu­tions without sim­u­la­tions, with a sample size require­ment of at least five times the input vector size (design vari­ables used in the opti­miza­tion process).

Input Para­me­ters

The main goal was to develop effi­cient pro­peller designs suitable for com­mer­cial vessels. Key para­me­ters such as pro­peller diameter and inflow velocity, chosen within a range tailored for com­mer­cial applic­a­bil­ity, sig­nif­i­cantly impact per­for­mance char­ac­ter­is­tics. Uti­liz­ing non-dimen­sional coef­fi­cients enhances design flex­i­bil­ity across diverse oper­a­tional con­di­tions. Inte­gra­tion of 5‑bladed pro­pellers meets modern design stan­dards. The Sobol method effi­ciently explores the design space, ana­lyz­ing input para­me­ter sen­si­tiv­ity and the influ­ence on pro­peller performance.

Opti­miza­tion Process

The opti­miza­tion was per­formed in CAESES and adopted a dual-phase approach aimed at max­i­miz­ing pro­peller effi­ciency, min­i­miz­ing cav­i­ta­tion and meeting required thrust. T‑Search opti­miza­tion was employed in both phases. In the initial phase, the opti­miza­tion relies on the afore­men­tioned input para­me­ters, tailored to a B‑series setup. Uti­liz­ing B‑series poly­no­mi­als to assess the open water char­ac­ter­is­tics and the Burrill diagram to estimate the cav­i­ta­tion, elim­i­nates the need for sim­u­la­tions. In the sub­se­quent phase, panMARE sim­u­la­tions were con­ducted using the para­met­ric CAESES model to optimize both con­ven­tional and uncon­ven­tional designs. This approach ensures man­age­able design space dimen­sion­al­ity, shape diver­sity, and suitable con­straints. The initial values in this phase were derived from the optimal B‑series designs iden­ti­fied in the first phase.

	1st Phase	2nd Phase
Name	B‑series setup	Uncon­ven­tional Pro­peller setup
Algo­rithm	T‑search	T‑search
Initial Values		Optimal B‑series design from the 1st phase
Open Water Tests	B‑series poly­no­mi­als	panMARE
Cav­i­ta­tion	Burrill Diagram	panMARE
Find	P/D , AE / A0	Design vari­ables of Para­met­ric Model for Uncon­ven­tional Propeller
Objec­tive Function	maximize pro­peller effi­ciency η0	maximize pro­peller effi­ciency η0
Con­straints	c=Burrill(σ0.7R,τc ) < 30% Treq ≤ Tpan ≤ 1.3Treq	Acav < 0.1 AE Treq ≤ Tpan ≤ 1.3Treq

Opti­miza­tion with CAESES and panMARE

As an exem­plary con­fig­u­ra­tion, to demon­strate this opti­miza­tion phase, a diameter of 7 m, an inflow velocity of 20 knots, an advance coef­fi­cient of 0.85, a required thrust coef­fi­cient of 0.125, and 5 blades were used. The con­ver­gence history of the second opti­miza­tion is depicted in the figure below. The initial T‑Search eval­u­a­tion yields a B5-73 pro­peller, rep­re­sent­ing an optimal B‑series pro­peller. Sub­se­quently, the opti­miza­tion aims at finding a tip rake pro­peller with targeted thrust, limited cav­i­ta­tion, and enhanced effi­ciency. Red tri­an­gles indicate design instances vio­lat­ing con­straints, while green circles denote feasible designs. The blue circle high­lights the optimal tip rake variant with the highest effi­ciency. Instances with over­es­ti­mated effi­ciency have higher thrust than desired, whereas those with under­es­ti­mated effi­ciency exhibit lower thrust values.

Building the Sur­ro­gate Model

The workflow of building the Sur­ro­gate Model for Uncon­ven­tional Pro­pellers (SMUP) employs the afore­men­tioned dual-stage opti­miza­tion process orig­i­nat­ing from a vector of four primary inputs and result­ing in an uncon­ven­tional pro­peller design. This yields two results: a vector detail­ing the geom­e­try’s design vari­ables values and another cap­tur­ing the open water char­ac­ter­is­tics, aug­mented with cav­i­ta­tion esti­mates. Sub­se­quently, this opti­mized data set is fed into the sur­ro­gate model for training purposes, with the final target of being able to use the sur­ro­gate model to quickly and accu­rately predict the opti­miza­tion outcomes and the optimal pro­peller geometry from just the four primary inputs, bypass­ing the exten­sive sim­u­la­tion phase.

Opti­miza­tion Results

An Sobol sequence effi­ciently gen­er­ated a set of 24 input vectors, which serve as the initial training data set for the sur­ro­gate model. The figure below (top left) shows how the design space was effec­tively explored. B‑series based opti­miza­tion iden­ti­fies the most effi­cient pro­peller, focusing on thrust and cav­i­ta­tion lim­i­ta­tions, with pro­peller geometry deter­mined by expanded area ratio and pitch ratio (bottom left).

Fol­low­ing the sim­u­la­tion-based opti­miza­tion phase, enhanced per­for­mance of tip rake designs is evident (top right), with an average increase of approx­i­mately 3.3% in effi­ciency. Design vari­ables (bottom right) indicate a tendency towards backward and tip rake pro­pellers, like Kappel pro­pellers, with low chord values proving effec­tive. However, mod­i­fy­ing the skew curve seems less prefer­able, as B‑series pro­pellers already possess a favor­able skew curve.

Con­clu­sion and Outlook

A suc­cess­ful para­me­ter­i­za­tion of tip rake pro­pellers and a dual-stage opti­miza­tion approach led to uncon­ven­tional designs with enhanced effi­ciency. This stream­lined workflow seam­lessly inte­grates insights from lit­er­a­ture and sim­u­la­tion data inside CAESES. The sur­ro­gate model gen­er­ated on the basis of a training data set from previous opti­miza­tion runs performs effec­tively, facil­i­tat­ing the iden­ti­fi­ca­tion of emerging patterns in the optimal outcomes of uncon­ven­tional designs, par­tic­u­larly with back rake and tip rake propellers.

However, it should be men­tioned that chal­lenges in the sur­ro­gate model devel­op­ment occurred, notably in extrap­o­la­tion. It is essen­tial to minimize data noise to resolve dis­crep­an­cies between pre­dicted sim­u­la­tions and opti­miza­tion outcomes. This requires refining the para­met­ric model through an iter­a­tive trial-and-error process, which may take some time due to its com­plex­ity. Moving forward, further plans encom­pass incor­po­rat­ing not only B‑series profiles, but also NACA66 profiles, inte­grat­ing ship hull wake field data using panMARE, and accom­mo­dat­ing 3‑bladed and 4‑bladed pro­pellers. The ultimate objec­tive is to estab­lish a robust tool for early-stage marine pro­peller design.

Acknowl­edg­ment

This research was con­ducted as part of the research & devel­op­ment project Deff­Pro­Form, funded by the Federal Republic of Germany, Federal Ministry for Economic Affairs and Energy (now Federal Ministry for Economic Affairs and Climate Action) on the orders of the German Bun­destag (Förderkennze­ichen 03SX516C).

Ref­er­ences

[1] Bar­nit­sas, Michael M., D. Ray, and P. Kinley. KT, KQ and effi­ciency curves for the Wagenin­gen B‑series pro­pellers. Uni­ver­sity of Michigan, 1981.

[2] Burrill, L. C., and A. Emerson. ​“Pro­peller cav­i­ta­tion: Further tests on 16in. pro­peller models in the King’s College cav­i­ta­tion tunnel.” Inter­na­tional Ship­build­ing Progress 10.104 (1963): 119 – 131.

[3] panMARE code. Avail­i­able via: https://www.tuhh.de/panmare/home