Artificial Neural Networks in Estimating Marine Propeller Cavitation

Constantinos C. Neocleous Department of Mechanical and Marine Engineering, Higher Technical Institute

P. O. Box 20423, Aglantzia, Cyprus. Email: costas@ucy.ac.cy

Christos N. Schizas Department of Computer Science, University of Cyprus

75 Kallipoleos, Nicosia, Cyprus. Email: schizas@ucy.ac.cy

Abstract - Cavitation in marine propellers can be a serious

problem that may result in severe deterioration in performance. This is particularly important in heavily loaded propellers, commonly encountered in small craft. Efforts have been made to generate polynomials that fit experimental data on propeller performance and hence to facilitate the propeller selection pro-cedures (Blount and Hubble, 1981). These polynomial fits are not accurate in capturing the performance of propellers, and also do not account for cavitating conditions. In the present work, neu-ral networks have been developed that predict the performance of marine propellers in all tested conditions, including cavitation. The USN-series of experimental data (Denny et al, 1989) were applied on different neural network architectures and learning parameters, aiming at establishing a near optimum setup. The results of the networks are superior to those of the polynomial fit, and give an acceptable accuracy even in the cavitating condi-tions, thus enabling a naval architect/engineer to improve on the propeller selection process.

I. NOMENCLATURE AE = Expanded area of the propeller blades

AO = Disk area of the propeller = πD2

4

D = Propeller diameter

EAR = Expanded Area Ratio = AEAO

h = Total static head at propeller centerline

J = Advance coefficient = VaND

KQ = Torque coefficient = Q

ρN2D5

KT = Thrust coefficient = T

ρN2D4

N = Propeller angular speed [revs/sec]

P = Pitch

PD = Pitch to diameter ratio

Q = Torque

T = Thrust

Va = Speed of advance = VS(1 – wt)

VS = Speed of the vessel

wt = Taylor wake factor

z = Number of propeller blades

η = Propeller efficiency = KTJ

2πKQ

ρ = Water density

σ = Cavitation number = 2ghVa

2

II. INTRODUCTION The estimation of the performance of marine propellers is a

laborious and intensive work often requiring extensive indi-vidual propeller design and performance evaluation through model testing in suitable experimental tanks. This procedure is highly expensive and many of the smaller companies can-not bare the high cost or afford the time. An alternative ap-proach, used by naval architects/engineers, is to rely on ex-perimental data obtained from an extensive parametric and systematic series of propellers, done over many years by au-thorities in well-equipped laboratories, using exemplar meth-ods. When cavitating conditions exist, the process of propeller performance evaluation becomes even more difficult. For small naval architectural offices (especially in small coun-tries) the use of published marine propeller performance data from systematic series, is a common design task. The classical procedure employed is to use test results of parametric series of propellers that have been arranged in some suitable form, usually graphical. Any accurate electronic form of data mod-eling will help in making the task easier. Furthermore, if the system is a learning system such as the neural networks, any future developments may be learned easier, and experiences from use may subsequently be added, thus resulting in an in-telligent system for propeller selection.

Propeller cavitation is a phenomenon that is very difficult to

model in basic theoretical terms, as it depends on various pa-rameters that are hard to estimate and vary considerably, often with stochasticity. The development of fundamental analytic-mathematical functions, using principles of fluid mechanics, that fit the performance of marine propellers under cavitating

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conditions, is also difficult and not very accurate. Neural networks have been shown to be a powerful ap-

proach for achieving complex non-linear mappings and ex-hibit learning capabilities ([2], [6], [8], [10], [12]), and hence the motivation for using them in the present work for captur-ing the propeller performance characteristics even under cavi-tating conditions.

The main parameters, which are used for describing the

performance of marine propellers, are the thrust coefficient (KT), torque coefficient (KQ) and the efficiency (η) for differ-ent values of the advance coefficient (J). These are given for particular propeller geometry (D, P, z, EAR) and particular characteristic cavitating conditions (σ). Such characteristic curves for the USN series propeller [3] having z = 4, EAR = 0.72 and P/D = 1.058 are shown as Figures 1, 2, and 3. Ex-perimental results for this specific propeller are presented because they are characteristic for all the other propellers. The results for the neural network modeling (Figures 5, 6, and 7) are shown also for this specific propeller for comparison.

Figure 1. Thrust coefficient KT versus advance coefficient J, for different cavitation numbers σ, for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058.

Figure 2. Torque coefficient KQ versus advance coefficient J, for different cavitation numbers σ, for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058.

Figure 3. Efficiency η, versus advance coefficient J, for different cavitation numbers σ, for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058.

III. DATA AND PREVIOUS WORK Test data are available for different series of propellers,

such as the Gawn-Burrill series [4], the Troost series (Wagen-ingen-B) [11] and the Gawn Open Water series [5]. These series are either tested in suitable water tunnels or in open-water towing tank facilities.

The USN propeller series data, as reported by Denny et al

[3], has been used for the training of the neural networks that were attempted in the work presented in this paper. This is a set of 3 and 4-bladed, commercially available marine propel-lers, having pitch to diameter ratios ranging from 1.004 to 1.504, expanded area ratios between 0.55 to 0.72 and tested under cavitation numbers ranging from 17.2 to 1.2. The data were experimentally obtained from tests done on a suitably modified USN 36-ft boat, under actual sea conditions. The test craft had a realistic shaft angle arrangement and was pro-vided with two powerful outboard engines to enable the pro-pellers to be tested under various loading conditions.

In the work reported by Blount and Hubble [1], similar pro-

pellers have been modelled through polynomial expressions that could give the propeller thrust and torque coefficients as functions of the advance coefficient, number of blades, pitch to diameter ratio and the expanded area ratio. The form of the expressions is that shown in equations 1 and 2.

KT = ∑i=1

39 Cti ( )J

si

P

Dti( )EAR

ui( )z

vi

(1)

KQ = ∑i=1

47 Cqi ( )J

si

P

Dti( )EAR

ui( )z

vi

(2) Specific values of Cti, Cqi, si, ti, ui, and vi have been ob-

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.100.150.200.250.300.35

0.400.450.500.550.60

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

10K

Q

17.2 11.75.5 3.42.2 1.81.2

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

KT

17.2 11.75.5 3.42.2 1.81.2

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.300.350.400.450.500.550.600.650.700.750.80

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

Effic

ienc

y

17.2 11.7

5.5 3.4

2.2 1.8

1.2

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tained and presented in the referred paper. It is noted that these equations do not account for cavitation phenomena, even though they have been used extensively in various pro-peller design programs. Furthermore, these polynomials do not estimate the propeller performance in a satisfactory degree of accuracy as it is evidenced from Figures 5, 6 and 7.

In previous work [9] it was shown that artificial neural net-

works may be used to successfully capture and display the performance characteristics of marine propellers. That work is further extended now with particular focusing on the problem of modeling the cavitation phenomena as they were experi-mentally specified.

IV. METHOD Different neural network architectures and learning strate-

gies were implemented, aiming at finding a setup that meets a desired accuracy and it is confined to within acceptable com-plexity constraints. The architectures that were attempted were of feedforward, multilayer and multislab structure. The input vectors to the networks were composed of five ele-ments, namely the geometric parameters of z, EAR, P/D and the operating conditions J and σ. The network output was a three element vector corresponding to values of the thrust coefficient, the torque coefficient and the propeller efficiency.

Figure 4. The neural network architecture used. Based on systematic network explorations and on previous

experience in similar applications, the architecture which was ultimately used was the multi-slab feedforward network hav-ing different activations in each slab as depicted in Figure 4.

This architecture has proved to be very efficient in numerous applied function approximation problems.

The learning procedure that was used was the backpropaga-

tion in which the learning rate was set to 0.1 and the momen-tum factor to 0.1. The weights for all the links were initialized to a value of 0.3.

V. RESULTS The available patterned data for the neural network imple-

mentations were 301 tests. These data were split into 265 for a training file and the remaining 36 for an unknown test file. The various networks attempted have been trained until 1000 epochs exceeded the set minimum average error of 0.001 without any further improvement on the test set file.

The selected network suggests the desired performance lev-

els (KT, KQ, η) with high accuracy levels. The correlation co-efficients for the series of mappings done were of the order of 0.99 for all cases. The maximum absolute error was 0.064 for KT, 0.081 for KQ and 0.117 for the efficiency.

These results are shown in detail in Figures 5, 6 and 7, for a

particular propeller (z = 4, EAR = 0.72 and P/D = 1.058), so that the reader may compare them with the actual data shown in Figures 1, 2 and 3. The light-solid line in Figures 5 and 6 show the results of the artificial neural network fit. Similar results were obtained for all other propellers, but are not shown here in order to keep the size of the paper within speci-fied limits.

The thick-dotted lines in Figures 5 and 6 show the results of

the polynomial fittings for the Wageningen series-B propel-lers [11] and for the segmental-type propellers [1]. It is noted that the predictive accuracy of this modeling is inadequate, as the discrepancies between experimental data and mathemati-cal fits are high. In the contrary, the neural network model captures the propeller performance with adequate accuracy.

VI. CONCLUDING COMMENTS A neural network system has been developed that can help

a naval architectural designer to estimate the performance of a USN-series propeller and thus make better selection of a suit-able marine propeller that satisfies desired propulsion re-quirements even in cavitating conditions. The network is shown to be superior to existing polynomial fitting techniques and thus may be used effectively in modeling the performance of a series of marine propellers and in directing a naval archi-tect in areas of search where efficient propeller designs may be found. The thrust, torque and efficiency under propeller cavitating conditions is well-modeled and the network can be used for effective propeller design.

(hidden)

SLAB 4

(10 neurons)

Tanh Activation

(hidden)

SLAB 2

(10

neurons)

Gaussian Activation

(hidden)

SLAB 3

(10

neurons)

Gaussian Comple-

ment Activation

OUTPUT

SLAB 5

(3 neurons)

Logistic Activation

INPUT

SLAB 1

(5 neurons)

Linear Activation

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VII. REFERENCES

[1] D. L. Blount and E. N. Hubble, “Sizing Segmental Section Com-mercially Available Propellers for Small Craft,” Propellers 81 Sympo-sium, SNAME, May 1981.

[2] G. Cybenko, “Approximation by superpositions of a sigmoidal func-tion,” Mathematics of Control, Signals, and Systems. 2:303-314. 1989.

[3] B. Denny, T. Puckette, E. Hubble, K. Smith, and F. Najarian, “A New Usable Marine Propeller Series,” Marine Technology, Vol. 26. 3:173-191, 1989.

[4] R. W. L, Gawn. and L. C. Burrill, “Effect of Cavitation on the Per-formance of a Series of 16-Inch Model Propellers,” Trans. Institution of Naval Architects, Vol. 99, 1957.

[5] R. W. L. Gawn, “Effect of Pitch and Blade Width on Propeller Per-formance,” Trans. Institution of Naval Architects, Vol. 95, 1953.

[6] Y. Ito. “Approximation of Functions on a Compact Set by Finite Sums of a Sigmoid Function with and without Scaling”. Neural Networks. Vol. 4. 817-826. 1992.

[7] V. Lewis, “Principles of Naval Architecture,” Society of Naval Archi-tects and Marine Engineers, 1988.

[8] M. Nabhan, Y. Zomaya. “Toward Generating Neural Network Struc-tures for Function Approximation”. Neural Networks. Vol 7. 1:89-99.1994.

[9] C. Neocleous and Chr. Schizas, “Artificial Neural Networks in Marine Propeller Design,” Proceedings of the IEEE International Conference on Neural Networks ICNN95, pp. 1098-1102, 1995.

[10] T. Pogio and F. Girosi, “Networks for Approximation and Learning,” Proceedings of the IEEE, 78:1481-1497, 1990.

[11] Ir. L. Troost, “Open-Water Test Series with Modern Propeller Forms,” Trans. North East Coast Institution of Engineers and Shipbuilders, Vol. 67, 1951.

[12] H. White, “Connectionist Nonparametric Regression: Multilayer Feed-forward Networks Can Learn Arbitrary Mappings”, Neural Networks, 3:535-550, 1990.

Figure 5. Thrust coefficient versus advance coefficient for different cavitation numbers for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058 as evaluated by the artificial neural network and the polynomial functions.

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

KT

Test, σ=17.2

Test, σ=11.7

Test, σ=5.5

Test, σ=3.4

Test, σ=2.2

Test, σ=1.8

Test, σ=1.2

B-Series model

Segmental model

ΑΝΝ, σ=17.2

ANN, σ=11.7

ANN, σ=5.5

ANN, σ=3.4

ANN, σ=2.2

ANN, σ=1.8

ANN, σ=1.2

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Figure 6. Torque coefficient versus advance coefficient for different cavitation numbers for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058 as evaluated by the artificial neural network and the polynomial functions. Figure 7. Efficiency versus advance coefficient for different cavitation numbers for the four-bladed USN propeller having EAR = 0.72 and P/D = 1.058 as evaluated by the artificial neural network.

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

10K

Q

Test, σ=17.2

Test, σ=11.7

Test, σ=5.5

Test, σ=3.4

Test, σ=2.2

Test, σ=1.8

Test, σ=1.2

B-Series model

Segmental model

ANN, σ=17.2

ANN, σ=11.7

ANN, σ=5.5

ANN, σ=3.4

ANN, σ=2.2

ANN, σ=1.8

ANN, σ=1.2

USN Propeller z = 4 EAR = 0.72 P/D = 1.058

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10J

Effi

cien

cy

Test, σ=17.2

Test, σ=11.7

Test, σ=5.5

Test, σ=3.4

Test, σ=2.2

Test, σ=1.8

Test, σ=1.2

ANN, σ=17.2

ANN, σ=11.7

ANN, σ=5.5

ANN, σ=3.4

ANN, σ=2.2

ANN, σ=1.8

ANN, σ=1.2

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