A statistical analysis of performance test results

A STATISTICAL ANALYSIS OF PERFORMANCE TEST RESULTS by

Ir. J. Holtrop*)

L Introduction

A statistical evaluation of model and trial test re-sults, selected from the archives of the Netherlands Ship Model Basin, was carried out using multiple re-gression analysis methods. The objective of this study was to develop a numerical description of the ship's resistance, the propulsion properties and the scale effects between the models and the full size. The most important applications of the obtained results are the determination of the required propulsive power with-out doing specific model tests and further the refine-ment of the extrapolation method by which model test results are scaled up.

The evaluation was performed by applying mul-tiple regression analysis to the results of 1707 resistan-ce measurements, 1287 propulsion measurements carried out with 147 ship models and the results of 82 trial measurements made on board 46 new ships. This material has been used partially in a previous study [ 1 ] , while most of the mentioned full-scale measure-ments were involved in an extensive model-ship cor-relation study initiated and co-ordinated by the Per-formance Committee of the International Towing Tank Conference [ 2 ] . A survey of the parameter ranges and ship types is given in Table 1.

*) Netherlands Ship Model Basin, Wageningen, The Netherlands.

2. Resistance prediction

In order to make the resistance prediction valid for ships and models of different size, the resistance com-ponents have to be expressed as dimensionless quan-tities depending on their respective scaling parameter. This dependency varies from model to model owing to differences in hull form. Applied to components of viscous and wave making origin, disregarding inter-action, we can thus express each component non-dimensionally as a function of the scaling parameter and the hull form:

— = f i (R„, form) a n d ^ = (F^, form)

Here, R^^ is the Reynolds number and F^ the Froude number, while Ry/A and R^y/A are the specific resis-tances of viscous and wave-making character respect-ively. Normally, the viscous resistance is determined from a flat plate friction formula which is corrected for the effect of the ship form. This additional form resistance is in most cases expressed as a fraction of the resistance of the flat plate of equal length and wetted surface at the same speed as the actual ship. This adaptation of the original Froude method as pro-posed by Hughes is generally referred to as the form factor method. Another component, which can be con-sidered of a viscous origin in most cases, is the resist-ance of the appendages. It is known that the appen-Table 1

Parameter ranges for different ship types

Type of ship Cp L/B Numbe

single screw r of ships twin screw max. min. max. min. max. model full _{scale model} full

scale Tankers, bulkcarriers 0.24 0.73 0.85 5.1 7.1 48 13 3 2 General cargo 0.30 0.58 0.72 5.3 8.0 21 17 3 2 Fishing vessels, tugs 0.38 0.55 0.65 3.9 6.3 35

-

2 4

dage resistance is under-estimated in general when only the wetted surface of the appendages is added to the equivalent flat plate surface. This under-estimation can be explained by the fact that appendages pierce to some extent through the boundary layer of the hull, have a short length en hence a high specific re-sistance. In the present analysis the resistance of the appendages was calculated separately, using the flat plate friction formula with the Reynolds number based on a virtual appendage length. This virtual length equals the length of the streamline over the ap-pendages if the appendage is more or less outside the boundary layer of the hull. When this virtual length equals the ship length, the resistance contribution of the appendage is the same as if it were part of the hull surface. This approach is vaUd only in the case that the local boundary layer of the appendage is turbulent and no flow separation occurs. From series of model resistance tests, in which the appendage configuration was varied systematically, several values of the vir-tual appendage length were determined. Average values for this virtual length are given in Table 2. The total viscous resistance of a ship model with appendages and a form factor 1 -i- k can be written as:

Ry =y2pv2cp(i+k)s^^j

with p the water density, V the speed, Cp the coef-ficient of frictiohal resistance and S^^j the total wetted surface of both the huh and the appendages. The form factor 1 -t- k can be divided into the form factor of the single hull form 1 -I- k j and a coefficient describing the contribution of the appendage resistance:

l + k = l + k i + ( k 2 - k i ) S ^ p p / S , „ j

The appendage factor 1 -i- k^ is presented in Figure 1 as a function of the virtual appendage length L^pp. Throughout the analysis all plate friction coefficients Cp were calculated from the ITTC-1957 formula:

0.075 (log R„ - 2f

Table 2

Virtual appendage length

Appendage configuration L / L

app' rudder — single screw 0.1 - 0 . 5

rudders — twin screw 0.03

rudders -i- shaft brackets - twin screw 0.01 rudders + shaft bossings — twin screw 0.02

stabilizer fins 0.01

bilge keels 0.20

dome 0.01

The wetted surface, without appendages, was correlat-ed with the hull-foi-m parameters and the following formula, having a standard deviation of a = 2.1 per cent, was deduced:

S = L(2T+B)VCM \ 0.5303368-H0.6321359CB -0.360327(C^j-0.5)-0.0013553L/Tt

In this formula L is the length on the waterline, B is the moulded breadth, T is the moulded draught, Cg is the block coefficient based on the waterhne length and Cy^ is the midship section coefficient.

The form factors could be obtained in the case of 91 resistance tests with a reasonable reliability from a numerical or graphical evaluation of:

l + k = lim (R/Rp) V

This procedure is based on the assumptions that the boundary layer is turbulent in all measured points and that the Froude-dependent resistance components vanish at low Froude numbers. The 91 form factors

RESISTANCE O F APPENDAGES

\

\

\

\

Figure 1. Appendage resistance as a function of the virtual ap-pendage length.

sistance when necessary and were correlated with the form parameters. By means of regression analysis the following formula was derived having a standard de-viation of a = 4.6 per cent

1 + k = 0.93 + ( T / L ) ° - 2 2 2 8 4 ( g / L ^ ) 0 . 9 2 4 9 7

(0.95-Cp)-0 " i 4 4 8 . (i_Cp+0.0225 lcb)0-6906

In this formula Cp is the prismatic coefficient based on the waterline length. Lj^ is the length of the run and is approximated by:

= L I l-Cp+0.06Cp.lcb/(4Cp-l) !

In the formulae Icb is the position of the centre of buoyancy forward of 0.5L given as a percentage of the waterline length L.

With regard to the resistance components that de-pend on the Froude number, practically the same procedure as fohowed in the previous analysis [ 1 ] , was applied. The equations are based on a simple wave-making theory, originally derived by Havelock [ 3 ] :

R^y/A = C j C

- m . F ^ •^19 - m . F

+e ' " ICj+CjCosCXF-^) In this equation C j , C j , C3, X and m are coefficients which depend on the hull form. This expression des-cribes the wave-making resistance of two pressure dis-turbances of infinite width with the first term as a correction to account for the induction of the diverg-ing waves. The distance between the centres of the disturbances X.L can be regarded as the wave-making length. The interaction between the transverse waves, accounted for by the cosine term, results into the typical humps and hollows on the resistance curves. By an analysis of 31 experimental resistance curves, the wave-making length could be related to the pris-matic coefficient Cp and the length-breadth ratio by:

X= 1.446.Cp -0.03L/B

For practical use as regression and prediction formula the wave-making equation was simpHfied to:

From the regression analysis the following expressions were derived for the coefficients c, d, m^ and m j :

C = 569 • (B/L) 2-984. . c l ; 2 6 5 5

d = -0.9

mj = -4.8507B/L-8.1768Cp+14.034C2-7.0682C3

In these expressions the waterplane coefficient C^y ^ > the prismatic coefficient Cp and the Froude number are based on the length on the waterline. Application of this prediction formula for the wave-making resist-ance in combination with the given formula for the form factor to the basic material resulted in a standard deviation of 6.9 per cent of total model resistance values.

3. Prediction of the delivered power

When the resistance is known the power delivered to the propeller P^ can be predicted from an estimated' propulsive efficiency 7 ? ^. The propulsive efficiency is generally sub-divided into the propulsion factors w, t,

and 7 ? ^ :

1-t 1-w

Here w is the effective wake fraction, t is the thrust deduction fraction, defined as t = 1-R/T, r;^ is the open-water propeller efficiency and I?R is the relative-rotative efficiency. Throughout this study the effective wake w is based on thrust identity which means that the advance coefficient J, defined as V(l-w)/nD, is determined from K.^ = K.p^(J). In these definitions K.J. and K.j, ^ are the thrust coefficients for the behind and the open-water condition respectively with:

T pn^D^

Further, the torque coefficient KQ is defined as: 75.P,

Qo

pn^D^ 27rpn^D^

Similarly, the open-water torque coefficient K exists and rj^ and are defined as:

JK.j.„

Q o

From the results of model propulsion tests, together with the measured characteristics of 77 propeller models, the values of w, t and T^J^ were computed. It is generally known that the effective wake is composed of both a viscous fraction w^ and a potential wake fraction Wp. For this reason the effective wake was correlated With both a characteristic boundary layer variable as well as geometrical parameters. As boun-dary layer parameter the quantity with:

2 6

was used. In this definition D is the propeller diameter, L is the waterline length and Cj and C^y are the coef-ficients of total and wave-making resistance. The para-meter was obtained by considering the propeller to be placed at the trailing edge of a flat plate with length L and having equal viscous resistance as the ac-tual hull form. From an integration of the velocity over the screw disk it followed that the nominal wake is directly depending on D^, provided the velocity in the boundary layer follows a n-th power law and the propeller radius is less than the boundary layer thick-ness. Although the latter condition is not always ful-filled, especially not at the full scale, proved to be significant in the regression analysis. For single-screw configurations the fohowing prediction formula for the effective wake was obtained:

w = 0.177714B2/(L-L.Cp)2-0.577076B/L +0.404422Cp+7.65122/D2

This formula is based on 1176 model values and 68 values derived from trial test measurements with a standard deviation of CT = 0.048. In a similar way a prediction formula for twin-screw ships with a stan-dard deviation of a = 0.041 was determined:

w = 0.4141383C2-0.2125848Cp+5.768516/D2 The thrust deduction t and the relative-rotative ef-ficiency were also correlated with the hull-form and propeller parameters but as no Reynolds scale ef-fects are assumed to be present on both these two pro-pulsion factors in modern analysis methods, the boun-dary layer parameter was not considered in the sta-tistical analysis. For single-screw configurations the formula

t = 0.088775+0.2992778Cp-0.2355184C^

+0.04302C|+0.0355997B2/(L-L.Cp)^

with a standard deviation a = 0.037 and for tvwin-screw ships the prediction formula:

t = 0.0994+0.125B/Lj^

with a = 0.06 were deduced. With respect to the latter formula the remark is made that for twin-screw ships with open-stern arrangement the thrust deduction can be up to 0.07 lower than indicated by the formula. Also smaller wake fractions can be expected for this class of ships. For the relative-rotative efficiency with a 3 per cent standard deviation were found:

T?j^ = 0.9922-0.05908Ag/A^+0.07424Cp^

(single screw) and

7 ? ^ = 0.9737+0.11136Cp^-0.06325P/D (twin screw)

In these formulae A^/A^ is the propeller blade-area ratio, P/D is the pitch-diameter ratio of the propeller and Cp ^ is the prismatic coefficient of the afterbody, approximated by:

Cp A =Cp-0.0225 Icb

It is noted here that the influence of the propeller par-ticulars is not considered to reflect a physical phenom-enon. The last propulsion factor that has to be predict-ed is the open-water propeller efficiency j j ^ . However, for practical purposes the open-water characteristics of most propellers can be approximated fairly well by those of a B-series screw. Polynomials for the thrust and torque characteristics of this extensive propeller series are given in [ 4 ] . Using the prediction formulae a standard error of 8.6 per cent for single screw and 7.8 per cent for twin-screw configurations has to be considered for the propulsive efficiency T J ^ .

4. Scale effects between model and ship

The numerical model for the powering characteris-tics as described in the previous sections is based main-ly on the results of model tests and direct application to full-scale ships would lead to unrealistic results. In present day extrapolation and correlation methods scale effects are considered to be present at the resis-tance, the effective wake and the propeller characteris-tics, while the thmst deduction and the relative-rotative efficiency are considered equal for model and ship and independent of the propeller load. The cor-relation analysis from which the scale effects have been derived was carried out along the following lines: - the trial test results were corrected for deviating

draught, water depth, wind force above number 2 Beaufort, deviating propeller pitch and sea water temperature,

- a friction loss of 1 per cent in the stern tube was as-sumed,

- the open-water torque and thrust coefficients were determined using ri^ and the model open-water characteristics corrected for the proper Reynolds number and average full-scale blade roughness, - the full-scale effective wake, based on thrust

iden-tity and the resistance were determined,

- comparison with the model values for the wake and the resistance then yielded the scale effects.

From this correlation analysis it was found that the resistance components as described so far, cover only

built ships under trial condition. The additional re-sistance that accounts for this discrepancy is generally referred to as the incremental resistance or correlation allowance with:

and is partially due to the roughness of the hull plating and the air resistance. Besides, systematic effects, aris-ing from the model testaris-ing technique and inaccuracies in the appUed extrapolation method are considered to contribute to the correlation allowance. Based on 82 measurements made on board 4 6 new built ships the statistical -formula

= ( 1 . 8 + 2 6 0 / L ) 0 . 0 0 0 1

with a standard deviation a = 0.00025 was derived. There appeared no obvious difference between the

-values for single and twin-screw ships. In Figure 2 the -values are given on a base of the ship's length. In this diagram the height of the columns corresponds with the maximum deviation found between the

re-MODEL - SHIP CORRELATION ALLOWANCE I T T C - 5 7 W I T H F O R M F A C T O R

F R I C T I O N L O S S : 1 P E R C E N T 0 0 0 1 5 . ^ " T ; A C C . T O L I N D G R E N

sister ships. With respect to the scale effect on the ef-fective wake use can be made of the numerical sub-division of the wake into a potential and a viscous part. The Reynolds scale effect can then be calculated from the prediction formulae for the viscous wake fractions w^ with

w^ = 7.65122/D2 (single screw) and hence:

Aw= 7.65122 ( C T „ - C , ^ ) ( C ^ ^ + C , - 2C,,)L^/D^ In a similar way for twin-screw ships the wake scale effect can be calculated from:

Aw = 5.769 ( C ^ ^ - C , , ) ( C , „ - 2 ^ , /D^ In Figure 3 the viscous component of the effective wake fraction w^ is presented as a function of the boundary layer parameter D^. In these diagrams both the results of trial measurements and corresponding model tests are given. The diagrams were set up in such a way that the representation of the potential part of the wake by the prediction formulae was assumed to be perfect. The effective wakes determined from the full-scale measurements have been based on

open-WAKE SCALE EFFECT F R I C T I O N L O S S : 1 P E R C E N T A K ^ , A K Q A C C . T O L I N D G R E N SINGLE SCREW N -O 0 . 1 0 1 5 0 2 0 0 2 5 0 SHIP'S LENGTH (m) TWIN SCREW

-Figure 2. Correlation allowance C. as function of length.

L ( C , - C „ )

2 8

water characteristics that have been corrected for aver-age blade roughness and Reynolds effects according to the method proposed by Lindgren [ 5 ] . The scale ef-fect on the propeller characteristics was investigated by applying three different methods for propeller scale effects:

- AK.J, = AKQ = 0,

- Reynolds effect according to B-series polynomials, - Lindgren's method.

For each of these methods the effective wake on the full-scale was calculated and compared with the sta-tistical curve in Figure 3 as determined from model ' experiments. From these calculations it appeared that if no scale effect is considered to be present at all, the effective wake is somewhat overestimated when the prediction formulae are used and the full-scale calculated wake fractions fall well below the curves in Figure 3. Contrarily, when the Reynolds effect on the propeller characteristics is considered by using the B-series polynomials [ 4 ] the full-scale effective wake fractions were too high when compared with the statis-tical lines.

A probable explanation is that blade roughness, which is not considered in the B-series polynomials, plays an important role in the propeller performance. It turned out that Lindgren's method, accounting for both Rey-nolds and roughness effects, yielded wake fractions that well complied with the statistical lines as can be seen from Figure 3. It is noted that the scale effects should be employed only in their specific combination, because of the mutual dependency and relation to the method of determining the resistance components.

In order to check the accuracy of the presented for-mulae for resistance, propulsion factors and scale ef-fects combined as a power prediction method the powering characteristics of 4 9 ships were estimated. The computations resulted in a standard deviation be-tween calculated and measured power values of a = 8.83 per cent and a standard deviation between pre-dicted and observed propeller rotative speed of a =

3.08 per cent. Compared with the power prediction from model tests, where standard deviations of the same magnitude are obtained [ 2 ] , the accuracy ofthe presented statistical power calculation looks surpris-ingly good. It is likely, however, that as most of the full scale test data have been used also in the deriva-tion of the presented formulae, less promising values for the accuracy will be found when the method is ap-plied to more general cases.

5. Conclusions

The numerical description of the resistance and pro-pulsion properties can be considered to be useful for the estimation of the required propulsive power of a ship in early stage design. However, a more critical assessment of the accuracy of the presented method for power prediction is desired. With the presented method scale effects can be determined that have to be appHed when the performance properties are predicted from model experiments. Especially in view of the evaluation of model test results the derived system of correlation factors will be gradually im-proved and implemented in the facilities of the Neth-erlands Ship Model Basin.

References

1. Holtrop, J., "Evaluation of performance model tests and the power prediction from model test statistics", IV Inter-national Symposium on Ship Automation, Geneva, Novem-ber 1974.

2. Report of the Performance Committee, 14th International Towing Tank Conference, Ottawa 1975.

3. Havelock, T.H., "Ship resistance, the wave making properties of certain travelling pressure disturbances", Proc. of the Royal Society, A , Vol. 89.

4. Oosterveld, M.W.C. and Oossanen, P. van, "Representation of propeller characteristics suitable for preliminary ship design studies", International Conference on Computer Ap-plications in Shipbuilding, Tokyo, 1973.

5. Lindgren, H., "Ship model correlation method based on theoretical considerations", 13th International Towing Tank Conference, Berlin and Hamburg, 1972.