Holtrop, J.

A statistical re-analysis of resistance andpropulsion data

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Holtrop, J., A statistical re-analysis of resistance and propulsion data. In: International ShipbuildingProgress, Vol. 28, (n363_Nov), pp.272-276. Delft University Press, 1984. © Delft University Press.

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272

A STATISTICAL RE�ANALYSIS OF RESISTANCE AND PROPULSION DATA

by

J. Holtrop*

1. Introduction

In a recent publication [1] a power predictionmethod was presented which was based on a regressionanalysis of random model and full�scale test data.For several combinations of main dimensions and formcoefficients the method had been adjusted to testresults obtained in some specific cases. In spite of theseadaptations the accuracy of the method was found tobe insufficient for some classes of ships. Especiallyfor high speed craft at Froude numbers above 0.5 thepower predictions were often wrong. With the ob�jective to improve the method the data sample wasextended covering wider ranges of the parameters ofinterest. In this extension of the data sample thepublished results of the Series 64 hull forms [2] havebeen included. The regression analyses were now basedon the results of tests on 334 models. Beside theseanalyses of resistance and propulsion properties amethod was devised by which the influence of thepropeller cavitation could be taken into account. Inaddition some formulae are given by which the effectof a partial propeller submergence can tentatively beestimated. These formulae have been derived in a studycarried out in a MARIN Co�operative Research pro�gramme. Permission to publish these results is grate�fully acknowledged.

2. Re�analysis of resistance test results

The results were analysed using the same sub�divis�ion into components as used in [1]:

where:

RF = frictional resistance according to theITTC�1957 formula

1 +K1 = form factor of the hullRAPP = appendage resistanceRw = wave resistanceRB = additional pressure resistance of bulbous

bow near the water surfaceRTR = additional pressure resistance due to

transom immersionRA = model�ship correlation resistance.

A regression analysis provided a new formula forthe form factor of the hull:

*) Maritime Research Institute Netherlands, Wageningen, The Nether�lands.

In this formula B and T are the moulded breadth anddraught, respectively. L is the length on the waterlineand is the moulded displacement volume. C P is theprismatic coefficient based on the waterline length.LR is defined as:

where Icb is the longitudinal position of the centre ofbuoyancy forward of 0.5 L as a percentage of L.The coefficient c14 accounts for the stern shape. Itdepends on the stern shape coefficient Cstern for whichthe following tentative figures can be given:

Afterbody form C'stern

Pram with gondola � 25V�shaped sections �10Normal section shape 0U�shaped sections

with Hogner stern 10

C14= 1+0.011Cstern

As regards the appendage resistance no new analysiswas made. For prediction of the resistance of the ap�pendages reference is made to [1].

A re�analysis was made of the wave resistance. Anew general formula was derived from the data sampleof 334 models but calculations showed that this newprediction formula was not better in the speed rangeup to Froude numbers of about Fn = 0.5. The resultsof these calculations indicated that probably a betterprediction formula for the wave resistance in the highspeed range could be devised when the low speed datawere left aside from the regression analysis.By doing so, the following wave resistance formulawas derived for the speed range Fn > 0.55.

where:

The coefficients c2, c5, d and λ have the same definit�ion as in [1]:

The midship section coefficient! "# and the trans-verse immersed transom area at rest! $% and the trans-verse area of the bulbous bow! $&% have the samemeaning as in [1]. The vertical position of the centreof! $&% above the keel plane is!'&( The value of! '&

should not exceed the upper limit of 0.6! %)(Because attempts to derive prediction formulae for

the wave resistance at low and moderate speeds wereonly partially successful it is suggested to use for theestimation of the wave resistance up to a Froude num-ber of 0.4 a formula which closely resembles the orig-inal formula of [1]. The only modification consistsof an adaptation of the coefficient that causes thehumps and hollows on the resistance curves. Thisformula, which is slightly more accurate than theoriginal one reads:

For the speed range 0.40 <!)*! + 0.55 it is suggestedto use the more or less arbitrary interpolation formula:

Here! ,-.$/(0 is the wave resistance prediction for

)*!1 0.40 and! ,-! .! &/(22 is the wave resistance for!)*!1

0.55 according to the respective formulae.No attempts were made to derive new formulations

for the transom pressure resistance and the additionalwave resistance due to a bulb near the free surface.The available material to develop such formulae israther scarce. As regards the height of the centre ofthe transverse bulb area! '& it is recommended to obeythe upper limit of 0.6!%) in the calculation of the ad-ditional wave resistance due to the bulb.

!"#$%&'(')*+,+#-.#/0-/1)+,-(#2'3'

The model propulsion factors and the model-shipcorrelation allowance were statistically re-analysedusing the extended data sample. This data sample in-cluded 168 data points of full-scale trials on new builtships. In the analysis the same structure of the wakeprediction formulae in [1] was maintained. By theregression analyses new constants were determinedwhich give a slightly more accurate prediction.A point which has been improved in the wake predict-ion formula is the effect of the midship section coef-ficient!"# for full hull forms with a single screw.The improved formula for single screw ships with aconventional stern reads:

The coefficient! 34 depends on the coefficient c8

defined as:

273

274

The coefficient CV is the viscous resistance coef�ficient with

As regards the thrust deduction of single screwships a new formula was devised of comparable ac�curacy :

For the relative�rotative efficiency an alternativeprediction formula was derived but because its ac�curacy is not better than that of the original one it issuggested to use the prediction formula of [1]:

For multiple�screw ships and open�stern single�screwships with open shafts the formulae of [1] were main�tained.

The model�ship correlation allowance was statis�tically analysed. It appeared that for new ships underideal trial conditions a CA �value would be applicablewhich is on the average 91 per cent of the CA �valueaccording to the statistical formula of [1]. Apparent�ly, the incorporation of more recent trial data hasreduced the average level of CA somewhat. It is sug�gested, however, that for practical purposes the origin�al formula is used.

4. The influence of propeller cavitation and partialpropeller submergence

Especially on high speed craft propeller cavitationcan effect the propulsive performance.

Tests on B�series propellers in uniform axial flowunder cavitating conditions were reported in [3], butthe representation of the results was confined to agraphical form only.

The KT—KQ—J relationship of the 16 B�series

propellers tested under cavitating conditions werefed into the computer for a statistical analysis. Thedata used consisted of the changes of KT and KQ dueto cavitation at certain J�values. The unaffected KT

and KQ values of the propellers were supposed to bedetermined accurately by the polynomials given in[4] and [5]. From preliminary analyses it appeared that

for each propeller the conditions where influence ofthe suction�side cavitation begins can be represented

well by a certain value of the speed�independent coef�ficient:

This coefficient is indicated as (KT / (J2σ0))BI.

Here KT is the thrust coefficient, J is the advancecoefficient and σo is the cavitation number defined as

where pv is the vapour pressure, po + pgh is the staticpressure in the undisturbed flow at the level of theshaft centre line, p is the density of the water and V isthe advance speed of the propeller.

From the data of the B�series (K T /(J 2 σ o )) B I wasdetermined for each propeller and by means of mul�tiple regression analysis these ( K T / ( J 2 σ o ) ) B I valueswere correlated to the main propeller parameters.This resulted into the following formula:

Here AE /AO is the expanded blade area ratio and Z isthe number of blades.The pitch ratio appeared to have no significant in�fluence on the K T / ( J 2 σ o ) value where cavitation beginsto affect the propulsive performance. Of course, thiswill not be true for the effect of the pitch setting of acontrollable�pitch propeller because then the radialload distribution is changed.

If KT / ( J 2 σ o ) exceeds the value given by the predic�tion equation cavitation influence is present andshould be accounted for. This influence was represent�ed in relation to the characteristics of the non�cavitat�ing propeller because these are well defined by the

polynomial representation in [4] and [5]. This wasdone by analysing the ratios

and

Coefficient FN is the factor by which the rotation raten should be increased, whereas FP is the factor bywhich the propulsive power is increased due to cavitat�ion. The factors FN and FP were considered as a func�tion of KT/J2 for each cavitation number becauseK T / J 2 can be regarded the same for non�cavitatingconditions and for conditions in which the propulsiveproperties are affected.

It appeared that the influence of the cavitation num�ber could be expressed well by using

as an independent variable.By means of selective regression analysis the proportionality was correlated with the main propeller particulars, and the following prediction equations werederived:

and

It should be noted, however, that the scatter in thedata was fairly large. It is suggested that the parameters AE/AO and Z are not used outside the ranges of

The formula for FN is valid for

whereas the formula for Fp is valid only for

In all other cases FN and Fp are 1.0.

In the optimization of the performance of shipsin ballast conditions the behaviour of not fully immersed propellers can be of importance.

For practical use the following equations werederived from model experiments on the assumptionthat by introducing a fictitious increase G of the entrance velocity the influence of the partial emergencecan be accounted for over the range of propellerloadings of interest:

VE is the resultant entrance velocity of the propeller.This increase factor G was related to coefficientsdescribing the emergence of the propeller and thepropeller loading.As a parameter indicating the emergence the variableU is used with:

Where D is the diameter, ho is the vertical distancefrom the keel plane to the blade tip in its lowestposition, TA is the draught aft and wh is a measure forthe wave height at the location of the propeller, ap

proximated by:

275

where:

From experiments it appeared that the speed increase factor G could be expressed as a linear functionof the emergence coefficient U and the propellerloading KT/J2 = T/(pD 2 (1 w) 2 V 2 ) . Hence, for positive values of U the factor G can be determinedfrom:

where the coefficient 3 is an empirical constant.When the propeller emergence is not excessive the

thrust deduction and the relative rotative efficiencycan be regarded to be unaffected.

5. Numerical example

For the following hypothetical twin screw shipthe still water powering performance is calculatedover the speed range from 25 to 35 knots.

Main particulars

LB

TF

TA

S appC stern

50.0012.003.103.30

900500

mmmmm3

m2

ABT

iE

CMlcbAT

1+K2

CWP

025 degrees

0.784.5% L aft of ½ L.

10 m2

30.80

Related coefficients

C P

LR1+K1

C17

m3

C2

0.6009614.1728 m

1.297

1.41332.02981.0

CB

S hull

CA

C5

C 15

0.46875584.9 m2

0.00064

0.73290.74401.69385

Results resistance calculation

Speed m4cos( /F2

n) m3Fd

n Rw RAPP RTR R(knots) (kN) (kN) (kN) (kN)

252729313335

0.32790.18200.0409

0.08340.18760.2730

3.31003.08832.89622.72742.57802.4453

475512539564590618

212428313539

25162000

662715756807864925

Results propeller design and calculation of propulsion factors

t = 0.054 D = 3.231 m 0 = 0.705 (30 knots)w = 0.039 P/D = 1.136

R = 0.980 AE/AO = 0.763

!"

!"

276

,567896!:5;<=;>?*35! 3?8378?9@=*

Speed total! AB! CDB! )A! )C! ABB! C6BBthrust

(knots) (kN) (RPM) (kW) (RPM) (kW)

25 699 259.3 12670 1,000 1.000 259.3 1279827 756 275.7 14707 1.000 1.000 275.7 1485629 799 291.1 16617 1.000 1.000 291.1 1678531 853 307.1 18915 1.008 1.000 309.6 1910633 913 326.2 21508 1.019 1.011 329.8 2196435 978 340.2 24406 1.033 1.027 351.4 25318

* without effect of propeller eavitation.** including effect propeller cavitation.

$%.%0%(4%+

l.Holtrop. J. and Mermen, G.G.J., 'An approximate powerprediction method', International Shipbuilding Progress,Vol. 29, July 1982.

2. Yeh, H.Y.H., 'Series 64 resistance experiments on high-speed displacement forms', Marine Technology, July 1965.

3. Lammeren, WP.A. van, Manen, J.D. van, and Oosterveld,M.W.C., 'The Wageningen B-screw series', SNAME, November1969.

4. Oosterveld, M.W.C. and Oossanen, P. van, 'Further computeranalysed data of the Wageningen B-screw series', Internation-al Shipbuilding Progress, July 1975.

5. Oosterveld, M.W.C. and Oossanen, P. van, 'Representationof propeller characteristics suitable for preliminary shipdesign studies', International Conference on ComputerApplications in Shipbuilding, Tokyo, 1973.