resistance wave making and wave decay of thin ships with emphasis on the effect of viscosity

8/13/2019 Resistance Wave Making and Wave Decay of Thin Ships With Emphasis on the Effect of Viscosity

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Resistance, Wave-Making andWave-Decay of Thin Ships, with

Emphasis on the Effects of Viscosity

Leo Victor Lazauskas

Thesis submitted for the degree of Doctor of Philosophy

in Applied Mathematics

at The University of Adelaide

Discipline of Applied Mathematics

School of Mathematical Sciences

April 20, 2009


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Appendix A

2D-TBL Defect Integrals

Chapter SummaryIn this Appendix the boundary layer defect integrals C 1 and C 2

used in Chapter 4 are derived for two cases, namely, (i) the standardlog-law plus the Winter-Gaudet wake function as used by Grigson [45],and (ii) for the log-law plus the present wake model.

A.1 Log-law + Winter-Gaudet Wake

For the purposes of this section, the BL velocity prole is given by

u+ = 1

log(y+ ) + B0 + (). (A.1)

Winter and Gaudets wake function is

() = 0; 0 s=

{1 + sin[ w( w)]}; s (A.2)

withw =

1m s

and w = s + m

2 . (A.3)

Let A0 = w w and A1 = w(1 w). Then1 =

[1 + sin A1 ] . (A.4)

Dene

U + U

u =

1

log( + ) + B0 + 1 . (A.5)

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Therefore,

U + u+ =

1

log( + ) + 1 1

log(y+ ) ()= 1 ()

1

log() (A.6)

The rst defect integral is

C 1 = 1

0U + u

+ d

= 1 + 1 I 1 (A.7)

where

I 1 = 1

s() d

=

1

s

{1 + sin[ w(

w)]

} d

=

(1 s )

w

cos(A1 ). (A.8)

Therefore,

C 1 = 1 + 1

(1 s ) + w

cos(A1 )

= 1

+

s + sin( A1 ) + 1 w

cos(A1 ) (A.9)

The second defect integral is

C 2 1

0 U +

u+ 2

d=

1

0

21

2 1

log() + 12

log2 () d

+ 1

s

2 () 2 1 () + 2

()log() d

= 21 + 2 1

+ 22 2 1 I 1 + I 2 +

2

I 3 (A.10)

where

I 2 = 1

s

2 () d

= 2

2 1

s {1 + sin[ w( w)]}2 d

= 32

22 (1 s ) +

2

4 w2 [sin(2A1 ) 8cos(A1 )] (A.11)

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and

I 3 = 1

s() log() d

=

1

s {1 + sin[ w(

w)]

}log() d

=

1

slog() d +

1

ssin[ w( w)] log() d

+ w {cos(A0 )[Ci( w) Ci( ws )] + sin( A0 )[Si( w) Si( ws )]}.

where Ci and Si are the Sine and Cosine integrals [1] given by

Si = z

0

sin tt

dt (A.12)

andCi = + log z + z0 cos(t 1)t dt (|arg z | < ). (A.13)

We can also write1 =

c0 (A.14)

C 1 = 1

(1 + c11 ), (A.15)

andC 2 =

12

(2 + c21 + c22 2 ) (A.16)

where c0 = 1 + sin( A 1 ) (A.17)

c11 = s + sin( A1 ) + 1 w

cos(A1 ) (A.18)

c21 = 2 1 [s + 1 w

cos(A1 )]

+2[s 1 s log(s )]+

2 cos( w w) w

[Ci( w) Ci( ws )]+ 2 sin( w w)

w[Si( w) Si( ws )] (A.19)

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and

c22 = 21 2 1 [1s 1 w

cos(A1 )] (A.20)

+3

2 1

w[2 cos(A1 ) +

1

4 sin(2A1 )]. (A.21)

Grigson [45] uses s = 0.1 and m = 0.776 for which c0 = 1.5054, c11 =0.4197, c21 = 2.1101, and c22 = 0.7657.

A.2 Log-law + Modied Winter-Gaudet Wake

The mean velocity prole is given by

u+ = 1

log(y+ D) + B0 + () (A.22)

where D = +v 1/ and where +v is the distance from the wall at which theow is fully turbulent.Dene

U + U

u =

1

log( + D) + B0 + 1 , (A.23)which gives

U + u+ =

1

log( + D) + 1 1

log(y+ D) ()

= 1

() 1

log

y+

D

+ D . (A.24)For + large,

logy+ D + D

logy+

+ = log(). (A.25)

The (piece-wise continuous) wake function is given by

() = 0; 0 s=

{1 + sin[ w( w)]}; s m

= 2

1 + [m + m ( m )]( m )2 ; m 1

= 1 1

log(); 1 (A.26)A-4


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where

w = 1m s

, w = s + m

2 ,

m = 6 + (1

m) + 3 1

2(1 m )2 and m =

4

(1

m)

2 1

2(1 m )3 .

A.2.1 The 1st Defect Integral C 1

The rst defect integral is

C 1 = 1

0U + u

+ d

= 1 1

s() d

1

1

0log() d

= 1 + 1 I 1 (A.27)

whereI 1 =

1

s() d. (A.28)

LetI 1 = I 1S + I 1 M (A.29)

where

I 1 S =

m

s {1 + sin[ w( w)]} d

=

(m s ) (A.30)and

I 1 M = 2

1

m1 + ( m )

2 m + ( m )3 m d

=

(1 m ) +

14

(1 m ) + 112

(1 m )2 . (A.31)

Therefore,

I 1 =

(1 s ) +

14

(1 m ) + (1 m )2

12 (A.32)

and so

C 1 = 1

(1 m )2

12

(1 s ) + 1 1

4 (1 m ). (A.33)

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A.2.2 The 2nd Defect Integral C 2

The second defect integral is

C 2

1

0U + u

+ 2 d

= 21 2 1 1

0log() + 1

2 log2 () d

+ 1

s

2 () 2 1 () + 2

()log() d

= 21 + 2 1

+ 22

+ I 2 2 1 I 1 + I 3 (A.34)where

I 2 = 1

s

2 () d (A.35)

and

I 3 = 2 1

s() log() d. (A.36)

LetI 2 = I 2S + I 2 M (A.37)

where

I 2 S = m

s

2S () d

= 2

2 m

s {1 + sin[ w( w)]}2 d

= 32

22 (m s ). (A.38)We also require

I 2 M = 1

m

2M () d

= 42

2 1

m1 + m ( m )

2 + m ( m )3 2 d

= (1 m )

10521562 + 54 1 + (13 + 11 1 )(1 m )

+ 39 2 2

1 + (1 m )2

. (A.39)

LetI 3 = I 3 S 1 + I 3 S 2 + I 3 M 1 + I 3 M 2 + I 3 M 3 (A.40)

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where

I 3 S 1 = 2

2 m

slog() d

= 2

2 [(s

m ) + m log(m )

s log(s )] . (A.41)

Let A() = w( w)] and let AS 0 = w w , and note that A(s ) =/ 2 and A(m ) = / 2. Then

I 3 S 2 = s

msin[A()] log() d

= 22 w

cos(AS 0 )[Ci( wm ) Ci( ws )] (A.42)

+ sin( AS 0 )[Si( wm ) Si( ws )] . (A.43)

We also require

I 3 M 1 = 42

1

mlog() d =

42

[m 1 m log(m )] , (A.44)

I 3 M 2 = 4m

2 1

m( m )

2 log() d

= 4m

2116 log(m )

3m3

2m +

m2

19

, (A.45)

and

I 3 M 3 = 4 m

2 1

m( m )

3 log() d

= 4 m

24m4

log(m ) 116

+ m

3 32m

4 + 3m

254m48

.(A.46)

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Appendix B

TBL Velocity Proles

Chapter SummaryIn this Appendix we describe the experimental BL velocity

prole data used in the tting procedure. Results of the ttingprocedure are shown for three pairs of log-law constants andB0 .

B.1 Experimental Data

Several different experimental data sets are used to estimate certain BL quan-tities required in the ship resistance model described in Chapters 5 and 7.Other, less complete, data sets are used in comparisons with the predictionsof the model.

In 1958, Smith and Walker [119] conducted experiments on a at plate fora wide range of Rn . The BL was tripped at the leading edge (i.e. x = 0) usinga special air-jet arrangement. Their tabulated data set comprises 61 usefulvelocity proles and 385 values of the skin-friction measured independentlyusing oating balances that were claimed to be accurate to 2%.These experiments have been criticised by several authors. Osaka et al[104] suggest that the two-dimensionality of the ow is in question becauseof the particular BL trip. Grigson [45] on the other hand maintains that anear-perfect trip was achieved.

Smith and Walkers [119] analysis of their own (uncorrected) experimentalvelocity proles led them to adopt = 0.461 and B0 = 7.15 for the log-law

constants, the largest values shown in Table 3.1. After applying correctionsto the velocity proles, lower values for both constants were obtained by bothPatel and Landweber (see Table 3.1) as discussed by Grigson [45].

The ERCOFTAC [33] database contains tabulated data in electronic form

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of Spalarts [120] direct numerical studies, Roach and Brierleys [113] T3experiments, and Osterlunds [105] at plate experiments.

Only two small sets of data from Roach and Brierleys [113] low-Rn ex-periments are used in the present thesis. The free-stream turbulence, T u, forthe seven T3A proles is between 1.1% and 1.5%. For the nine proles inthe T3B set, Tu is between 2.4% and 4.3%. These values of T u should beconsidered as high for wind tunnel tests.

Osterlunds [105],[106] experiments were conducted on a 7m long atplate at KTH in Sweden. The BL was tripped with embossing tape near theleading edge. The free-stream turbulence was 0.02% which is considered tobe very low. There are 70 velocity proles in the complete set. Skin-frictioncoefficients were measured independently using oil-lm interferometry tech-niques. Twenty values of cf and R were taken from a much-enlarged copyof Fig. 4 in Osterlund et al [106].

Barenblatt et al [2] concede that Osterlunds [105] data is reliable, however

they disagree with the processing and interpretation of that data. They arealso critical of the low values of the log-law constants ( 0.38 and B0 4.1)used by Osterlund because the constants are substantially less than thosepresented in the literature as standard [2]. This seems to be a somewhatpetty criticism: rstly, Table 3.1 shows that low values of and B0 havepreviously been deduced by several other researchers; secondly, the values of the constants considered as standard (e.g. = 0.41 and B 0 = 5.0) are, inreality, rarely accepted uncritically - most workers in this eld appreciate thatColes arrived at those values after a great deal of massaging and modicationof the data available up to 1968 [45].

Nagibs [100] experiments at IIT in the USA were conducted on a 9mlong axisymmetric body. A short fetch of sandpaper roughness was used totrip the BL. There are 15 velocity proles in this set.

Thus, in total, there are 161 boundary layer velocity proles that will beused to estimate quantities such as BL thicknesses and skin-friction coeffi-cients.

The three parts of Figure B.1 show the experimental velocity values forNagibs velocity proles for three pairs of log-law constants and B0 . Alsoshown are continuous curves of the log-law, Squires prole and Muskersprole considered in Chapter 3. The actual velocities from the experiments(before processing) are shown as a cloud of pink dots in the plots, and it isclear that they lie above the approximations for most of the range of y+ .

Figure B.2 show the velocities from Osterlunds [105] experiments. Thedata is much less scattered than those from Nagibs experiments shown inFigure B.1. The approximations t this set of data quite well. For low values

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of y+ , Muskers prole ts the data better than Squires prole for the threepairs of log-law constants.

Smith and Walkers [119] experimental velocities are shown in Figure B.3.The scatter is much larger than that of Osterlunds data and does not extendinto the laminar sub-layer. The approximations given by the log-law and thetwo other proles are quite good, but as we will now show, the agreementcan be improved by adjusting the Base velocities for experimental errors.

B.1.1 Results of the Fitting Procedure

The residual function 2 is dened by Kendall and Koochesfahani [73] as

2 = 1N

N

i=0|u

+i (data ) u

+i (model)|

u+i (data ) (B.1)

where u+i (data ) are the measured velocities in wall units, and u+

i (model) are

the velocites predicted using Muskers prole described in Chapter 3. Theindex i runs from 0 to N because the point ( y+ = 0 , u+ = 0) is included soas to satisfy the no-slip condition at the wall [73]. Kendall et al [73] found(and we have conrmed) that equation(B.1) gives slightly better results thanthe more usual residual function, i.e. the rms of the differences betweenexperimental points and the model equation.

Figure B.5 shows the residual function 2 for three pairs of log-law con-stants. Figures B.6 and B.7 show the adjustments made to u / and y0 ,respectively, for the same pairs of log-law constants. The numbers along thex-axis in these plots are the indices of the velocity proles in the set of data:

e.g. 1 . . . 16 are Nagibs proles; 154. . . 161 are the T3B proles.It can be seen from Figure B.5 that residuals are reduced signicantly

by choosing the best values of u and y0 . For example, in the top plot of the gure, the residual is around 1% for proles 1 to 145. The differencesbetween the original (i.e. unadjusted) data and the model is much largerthan the residuals of the tted data. For Nagibs experiments (proles 1to 15) the differences are between 5% and 9%: after adjustment they areabout 1% or better. Of course, whether we are free to adjust u and y0 bythe amounts shown in Figures B.6 and B.7 to achieve the smaller residualsis open to question.

Comparing the residuals in the three plots of Figure B.5 shows that thebest residuals (both original and tted) occur when = 0.418 and B0 = 5.45for most proles except Osterlunds where = 0.384 and B 0 = 4.08 seem togive better results.

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The adjustments shown in Figures B.6 and B.7 are quite small for mostproles except those of Nagib (proles 1 to 15). For Osterlunds data, theadjustment of u / is less than about 2%; the adjustment of y0 is very small,less than about 5 microns. It is heartening that these amounts are withinthe accuracy claimed by Osterlund for his experiments. It is also clear fromFigures B.6 and B.7 that the adjustments for the other experiments aregenerally much larger. The adjustments required when using = 0.384 andB0 = 4.08 are quite large for all experiments except those of Osterlund.When = 0.418 and B0 = 5.45 are used as the log-law constants, therequired adjustments are much smaller and far less scattered.

Figure B.4 shows the velocity prole across the entire BL (as well as out-side the edge) for one of Osterlunds experimental boundary layers for threepairs of the log-law constants. The graphs shows the base (or unadjusted)data points, the data points after adjusting u / and y0 using the tting pro-cedure described above, as well as the model curve, which is Muskers prole

plus the wake function, (). It can be seen that the curves are in reasonableagreement with the (adjusted) data points. Although it is difficult to judgeby eye, the pair of log-law constants = 0.384, B 0 = 4.08 is the best for thisparticular prole.

The effect of the tting procedure is shown as the clouds of blue dots inFigures B.1 to B.3. It can be seen that the tting procedure considerablyreduces the scatter in both Nagibs (Figure B.1) and in Osterlunds (FigureB.2) the data. The method is, however, a little less successful at low y+ forthe Smith and Walker data shown in Figure B.3.

It is possible to devise more sensitive techniques to assess the extent of the log-law region and the appropriate values of the log-law constants. Forexample, the diagnostic function dened by

= y+du+

dy+ 1

(B.2)

should be constant and equal to in the log-law region [106]. Similarly, thediagnostic function dened by

= u+ 1

log(y+ ) (B.3)

should be equal to B 0 in the same region.

Figure B.8 is a plot of calculated using (unadjusted) experimental data.The portion of the velocity proles for which > 0.15 were omitted fromthe plot: the log-law region should have ended before that value. The samelimit was used by Osterlund [105].

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The scatter in Figure B.8 is clearly very large and no doubt motivatedOsterlund et al [106] to use interpolated values of the velocities to reduce thescatter in their own data.

For Osterlunds set there seems to be a attening out of the data for300 < y + < 500 which corresponds to

0.38. This is consistent with

the ndings of Osterlund et al [106] who further claim that this behaviouris evidence that the log-law region begins at a much higher value than thetraditionally accepted value of y+ 50. Although the scatter for Smith andWalkers data is very large, the plot suggests a value of around 0.43. Thescatter is far too great to discern much from either Nagibs data or the twoT3 data sets.

Figure B.9 shows calculated using (unadjusted) experimental data forthree values of the log-law constant . As before, only the part of the velocityproles for which < 0.15 are used.

For = 0.384 both Osterlunds and Nagibs data atten out for a region

roughly in the range 150 < y+

< 400, after which they both curve upwards,indicating that we are then beyond the log-law region. Nagibs data suggeststhat B0 5.0 in this region. Osterlunds data indicates that B0 4.0 whichis consistent with his estimate of B0 = 4.08 using the same value of . Noconstant region is discernible in Smith and Walkers data; the cloud of pointsdrops from B 0 4.5 at y+ 50 to B0 3.8 at y+ 3000. For the two T3data sets, B0 4.5 for 30 < y + < 100.For = 0.410, Nagibs data suggests that B0 5.7 for 35 < y + < 150.This constant region is at signicantly lower y+ -values than exhibited in thetop plot. Osterlunds data shows that B0 4.8 for 40 < y + < 150 whichwould not support his contention that the log-law begins at y+

200 if the

value of was also correct. Smith and Walkers data does not seem to havea constant region for = 0.410, however the (downward) slope of the data isnot as great as in the top plot where = 0.384. If there is a constant regionin the T3 data it is not very extensive.

For = 0.418, Nagibs data suggests that B0 5.9 for 35 < y + < 100.Osterlunds data suggests B0 5.0 for 50 < y + < 150 . Smith and Walkersdata indicates that for 75 < y + < 1000, B0 5.5, which is consistent withthe values found by Patel and by Landweber and shown in Table 3.1.

B.1.2 Estimating BL Wake Parameters

The BL wake function (), is quite sensitive to the value of the log-lawconstants. Figure B.10 shows wake functions tted to three representativeproles and for three pairs of log-law constants and B0 .

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The top plot shows wake functions based on three pairs of log-law con-stants tted to Nagibs prole bl125. The curves t the data reasonably wellfor > 0.3 It can be seen that the green curve ( = 0.384, B0 = 4.08) iszero for < 0.15 and that the wake region begins at smaller values of forthe other two curves. The values for the start of the wake region s and theposition m where it attains its maximum value for all BL velocity prolesare shown in Figures B.15 to B.17 to be described later.

The middle plot of Figure B.10 shows wake functions for Osterlundsprole SW981113F. For this BL, R 27300 which is the highest value of this data set. The jaggedness of the curve around = 0.6 for this prole forthe last two pairs of log-law constants is a graphical artefact caused by thesparseness of experimental data points in that region. With so few points it isdifficult to get reliable estimates of the maximum value of , and the -valuewhere that maximum occurs while simultaneously satisfying the correct slopeat = 1. On the other hand, for = 0.348 and B 0 = 4.08, values which are

believed to be correct, the t is quite satisfactory.The bottom plot of Figure B.10 shows the wake functions for Smith andWalkers prole number 70. Comparing this prole to the middle plot wecan see that the maximum value of and m can be located quite accuratelyfor all pairs of log-law constants. For < 0.3 the wake function oscillatesaround = 0, however the tting procedure seems to do a reasonable jobat estimating the location of the start of the wake region. Note that Coleswake function would start at = 0 and would not t the data as well as themodied wake law used here.

Figures B.11 to B.14 show the relative differences for all proles for eachset of experiments for three pairs of log-law constants and B 0 .

Comparing the separate plots in Figure B.11, it seems that = 0.418,B0 = 5.45 leads to smaller errors than = 0.348, B0 = 4.08 for Nagibsexperiments. The same is true of all the other experiments except Osterlundswhere = 0.384, B 0 = 4.08 is a better choice.

Fairly large errors for the Smith and Walker experiments are apparentaround y+ = 100 in Figure B.13. According to Lindgren et al [84], thereare at least three possible explanations for this overshoot in experimentaldata. The rst is that it is due to an inaccuracy in the probe positionrelative to the wall. (If that is indeed a difficulty, then the present ttingprocedure should have been able to correct it.) The second possibility isthat it is coupled to the B0 constant in the log-law. However this seems tobe a problem associated with hot-wire anemometry which is not relevant toSmith and Walkers experiments. Another is that the streamwise velocity isover-predicted by Pitot tubes. Perry et al, [109] add a fourth possibility -

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that the free-stream turbulence is also partly responsible for the kick-upin the blending region. They describe a technique to correct for free-streamturbulence as well as for shear effects on Pitot-tube readings. The correctionis only signicant for y+ < 200.

The kick-up at the start of the log-layer and the kick-up at the beginningof the wake give the velocity prole a slightly concave appearance. It isinteresting that the power-law often ts the data well for a wider range of y+

than does the log-law. This apparent better performance could, therefore,be accidental.

Referring to the plots at the bottom of Figures B.11 to B.14 it can beseen that an accuracy better than 2% is achieved for most of the BL, andregardless of which of the pairs of constants and B0 are chosen. If thebest pair of log-law constants is chosen for each set of experiments, errorsare typically much less than 1% over the majority of the BL.Figures B.15 to B.17 show the variation with R of s , m and m s ,respectively, for three pairs of log-law constants. Also shown are simpleapproximating functions, here single constants, given in Table 3.2. Theseconstants were calculated from ts to Osterlunds data set for each pair of log-law constants, except for the pair = 0.418, B 0 = 5.45 where the constantswere derived from Smith and Walkers data. There seem to some discernibletrends with Rn , however the large scatter and the different trends for eachset of experiments makes it difficult to devise approximating functions thatare better than the simple constants used in the present model. Further workon this area seems warranted but is beyond the scope of the present thesis.

Figure B.18 shows the variation of the wake parameter with R for threepairs of log-law constants. Also shown on each plot are the approximatingfunctions used in the algorithm to be described later. These functions are of the form

(R) = 0; r x1 ,= y1 sin3

(r x0 )2(x1 x0 )

; x1 r x2 ,

= y2 + ( y1 y2 )cos3 (r x1 )

2(x2 x1 ); r x2 ,

where r = log10 (R). The constants in the equation for various pairs of log-law constants are given in Table 3.2. These constants were calculated from

ts to Osterlunds data set for each pair of log-law constants, except for thepair = 0.418, B 0 = 5.45 where the constants were derived from Smith andWalkers data and the additional experimental data of Wieghardt and Karls-son shown in the bottom plot of Figure B.18. Also shown in the plot is the

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wake function used by Grigson [45]. The difference between Grigsons wakefunction and the present model are primarily due to the tting procedure.Grigson varied u but not y0 . Another minor difference is that Grigson usescubic splines whereas we use piecewise continuous trigonometric functions.

There is a tremendous amount of scatter in the results for the T3B setwhich is probably due to the high freestream turbulence. Results for the T3Aset, which were conducted with a lower freestream turbulence, are far moreconsistent. The scatter in the data for Nagibs and Osterlunds experimentsis a little larger than for those of Smith and Walkers experiments.

The results for Smith and Walkers experiments in the bottom plot of Figure B.18 are consistent with the values found by Grigson [45] and alsoreported by Lazauskas [77]. Although the values of 0 .6 < < 0.8 seem highfor Osterlunds set, they are consistent with the analysis of Guo et al [46]who found (using = 0.4) that = 0 .75770.07577.Although the scatter in these plots is quite high, Grigson [45] reminds usthat this should be expected because is less than about 5% of u

+

. Evenlarger inaccuracies should be expected at low Rn because the BL is verythin and it is difficult to measure the velocity prole accurately and hencedetermination of is uncertain.

Whether asymptotes to a constant value as R is still an openquestion in TBL theory. For the Smith and Walker data, the value of seems to level out at the largest values of R in Figure B.18 after reaching amaximum at R 103 .8 . Nagib et al [100] believe that this type of behaviouris possibly due to experimental problems where high Reynolds are achievedwith inadequate fetch, i.e. by using large uid speeds or by some othermeans. These BL are described as anaemic [100].

The behaviour of for the Osterlund set is less clear. For the largestvalues of R , seems to be decreasing but the scatter is too large to becertain.

For low values of R , the T3A data suggests that decreases as Rdecreases and that it would fall to zero at R 300. This is consistent withthe nding of Grigson [45] and many other researchers, but these low Rn ,high T u, data should be treated with great caution.

Figure B.19 shows the variation of 1 with log10 R for three pairs of log-law constants and the approximating functions used in the algorithm to bedescribed later. These functions are of the form

1 (R) = c1 (R). (B.4)

Values of c1 in this equation for various pairs of log-law constants are given inTable 3.2. As with the other wake parameters, the constants were calculated

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from ts to Osterlunds data set for each pair of log-law constants, except forthe pair = 0.418, B 0 = 5.45 where the constants were derived from Smithand Walkers data.

At very low R , 1 is negative due to the inner and outer layers not beingsufficiently distinct. As with the wake parameter it is a controversial issueas to whether the value of attains a maximum and then falls back to aconstant value at high R, or whether it attains a maximum constant value.

Figures B.20 and to B.21 show the variation of the defect velocity with and y / , respectively, for three pairs of log-law constants. Also shown inthe latter, is the approximation of Saric et al [116] given by

ud = 6.74 4.70 log(y/ ) . (B.5)The scatter is clearly much greater for the = y/ scaling than for the

y/ scaling which is an issue deserving of attention, but beyond the scopeof the present work.

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0 2 4 6 8

10 12 14 16 18

20 22 24 26

1 10 100 1000

u +

y+

Nagib Profiles =0.384, B 0=4.08

Base Adj

Log-Law Squire Musker

0 2 4 6 8

10 12 14 16 18 20 22

24 26

1 10 100 1000

u +

y+

Nagib Profiles =0.410, B 0=5.00

Base Adj Log-Law Squire Musker

0 2 4 6 8

10 12 14 16 18 20 22 24 26

1 10 100 1000

u +

y+

Nagib Profiles =0.418, B 0=5.45 Base Adj Log-Law Squire Musker

Figure B.1: Effect of log-law constants on ts to Nagib proles.

B-10


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0 2 4 6 8

10 12 14 16 18

20 22 24 26

1 10 100 1000

u +

y+

Osterlund Profiles =0.384, B 0=4.08

Base Adj


0 2 4 6 8

10 12 14 16 18 20 22

24 26

1 10 100 1000

u +

y+

Osterlund Profiles =0.410, B 0=5.00


0 2 4 6 8

10 12 14 16 18 20 22 24 26

1 10 100 1000

u +

y+

Osterlund Profiles =0.418, B 0=5.45 Base Adj Log-Law Squire Musker

Figure B.2: Effect of log-law constants on ts to Osterlund proles.

B-11


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0 2 4 6 8

10 12 14 16 18

20 22 24 26

1 10 100 1000

u +

y+

Smith and Walker Profiles =0.384, B 0=4.08

Base Adj


0 2 4 6 8

10 12 14 16 18 20 22

24 26

1 10 100 1000

u +

y+

Smith and Walker Profiles =0.410, B 0=5.00


0 2 4 6 8

10 12 14 16 18 20 22 24 26

1 10 100 1000

u +

y+

Smith and Walker Profiles =0.418, B 0=5.45 Base Adj Log-Law Squire Musker

Figure B.3: Effect of log-law constants on ts to Smith and Walker proles.

B-12


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4 6 8

10 12 14 16 18 20 22 24 26 28 30 32

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

u +

log 10(y+)

Osterlund Profile SW981113F =0.384, B 0=4.08

Base Adj

Fit

4 6 8

10 12 14 16 18 20 22 24 26 28 30 32

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

u +

log 10(y+)

Osterlund Profile SW981113F =0.410, B 0=5.00

Base Adj Fit

4 6 8

10 12 14 16 18 20 22 24 26 28 30

32

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

u +

log 10(y+)

Osterlund Profile SW981113F =0.418, B 0=5.10 Base Adj Fit

Figure B.4: Effect of log-law constants on ts to Osterlund prole SW981113F.

B-13


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

2

Profile

=0.384, B 0=4.08 Nagib (1999) Osterlund (1999) S&W (1958) T3A (1992) T3B (1992)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

2

Profile


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08 0.09

0.1

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

2

Profile

=0.418, B 0=5.45

Nagib (1999) Osterlund (1999) S&W (1958) T3A (1992) T3B (1992)

Figure B.5: Residual function. Open symbols - t to original data; solid symbols- t to adjusted data.

B-14


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-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

A d j u s

t m e n

t o f u

/

Profile

=0.384, B 0=4.08 Nagib (1999) Osterlund (1999) S&W (1958)

T3A (1992) T3B (1992)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

A d j u s

t m e n

t o f u /

Profile

=0.410, B 0=5.00 Nagib (1999)

Osterlund (1999) S&W (1958) T3A (1992) T3B (1992)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

A d j u s

t m e n

t o f u /

Profile

=0.418, B0=5.45

Nagib (1999) Osterlund (1999) S&W (1958) T3A (1992) T3B (1992)

Figure B.6: Adjustment of u / .

B-15


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-120

-100

-80

-60

-40

-20

0 20

40

60

80

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

y 0 ( m i c r o n s

)

Profile


-120

-100

-80

-60

-40

-20

0

20 40

60

80

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

y 0 ( m i c r o n s

)

Profile


-120

-100

-80

-60

-40

-20

0

20

40

60

80

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161

y 0 ( m i c r o n s

)

Profile


Figure B.7: Adjustment of y0 .

B-16


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 10 100 1000

y+

Nagib Osterlund SW T3A T3B

Figure B.8: Diagnostic function . Only the part of the velocity proles for which < 0.15 are used.

B-17


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0

1

2

3

4

5

6

7

8

1 10 100 1000

y+

=0.384 Nagib Osterlund SW T3A

T3B

0

1

2

3

4

5

6

7

8

1 10 100 1000

y+

=0.410 Nagib Osterlund

SW T3A T3B

0

1

2

3

4

5

6

7

8

1 10 100 1000

y+

=0.418

Nagib Osterlund SW T3A T3B

Figure B.9: Diagnostic function for three values of the log-law constant . Onlythe part of the velocity proles for which < 0.15 are used.

B-18


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-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

( )

Nagib Profile bl125 =0.384, B 0=4.08 =0.410, B 0=5.00 =0.418, B 0=5.45

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

( )

Osterlund Profile SW981113F =0.384, B 0=4.08 =0.410, B 0=5.00 =0.418, B 0=5.45

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

( )

Smith and Walker Profile 70 =0.384, B 0=4.08 =0.410, B 0=5.00 =0.418, B 0=5.45

Figure B.10: Wake functions tted to three velocity proles.

B-19


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29/63


30/63

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)

Smith and Walker: =0.384, B 0=4.08

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)


-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)


Figure B.13: Relative error of t to Smith and Walker proles for three differentpairs of the log-law constants and B 0 .

B-22


31/63

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)

T3A: =0.384, B 0=4.08 T3B: =0.384, B 0=4.08

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)

T3A: =0.410, B 0=5.00 T3B: =0.410, B 0=5.00

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5

( d a t a -

f i t ) / f i t

log10(y+)

T3A: =0.418, B 0=5.45

T3B: =0.418, B 0=5.45

Figure B.14: Relative error of t to T3A and T3B proles for three different pairsof the log-law constants and B 0 .

B-23


32/63

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4 4.5 5 5.5 6 6.5 7 7.5 8

s

log10(Rn)

=0.384, B 0=4.08 Approx. Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4 4.5 5 5.5 6 6.5 7 7.5 8

s

log10(Rn)


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

4 4.5 5 5.5 6 6.5 7 7.5 8

s

log10(Rn)

=0.418, B 0=5.45

Approx. Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

Figure B.15: Behaviour of s with Rn and the approximating functions used inthe present model for three pairs of log-law constants = and B 0 .

B-24


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 4.5 5 5.5 6 6.5 7 7.5 8

m

log10(Rn)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 4.5 5 5.5 6 6.5 7 7.5 8

m

log10(Rn)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9

1

4 4.5 5 5.5 6 6.5 7 7.5 8

m

log10(Rn)


Figure B.16: Behaviour of m with Rn and the approximating functions used inthe present model for three pairs of log-law constants = and B 0 .

B-25


34/63

0

0.1

0.2

0.3

0.4

0.5 0.6

0.7

0.8

0.9

4 4.5 5 5.5 6 6.5 7 7.5 8

m

- s

log10(Rn)


0

0.1

0.2

0.3

0.4

0.5

0.6 0.7

0.8

0.9

4 4.5 5 5.5 6 6.5 7 7.5 8

m

- s

log10(Rn)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4 4.5 5 5.5 6 6.5 7 7.5 8

m

- s

log10(Rn)


Figure B.17: Behaviour of m s with Rn and the approximating functions usedin the present model for three pairs of log-law constants = and B0 .

B-26


35/63

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 2.5 3 3.5 4 4.5 5

log 10(R)

=0.384, B 0=4.08 Present Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 2.5 3 3.5 4 4.5 5

log 10(R)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 2.5 3 3.5 4 4.5 5

log 10(R)

=0.418, B 0=5.45

Present Grigson Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P. Wieghardt (1951) Karlsson (1980)

Figure B.18: Behaviour of with R for three pairs of log-law constants andB0 and the approximating functions used in the present model.

B-27


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-2-1.5

-1-0.5

0 0.5

1 1.5

2 2.5

3 3.5

4

2 2.5 3 3.5 4 4.5 5

1

log 10(R)


-2-1.5

-1-0.5

0 0.5

1 1.5

2 2.5

3 3.5

4

2 2.5 3 3.5 4 4.5 5

1

log 10(R)


-2-1.5

-1-0.5

0 0.5

1 1.5

2 2.5

3 3.5

4

2 2.5 3 3.5 4 4.5 5

1

log 10(R)

=0.418, B 0=5.45

Present Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

Figure B.19: Behaviour of 1 with R for three pairs of log-law constants andB0 and the approximating functions used in the present model.

B-28


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0

5

10

15

20

25

0.001 0.01 0.1 1

u d

=0.384, B 0=4.08 Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

0

5

10

15

20

25

0.001 0.01 0.1 1

u d

=0.410, B 0=5.00 Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

0

5

10

15

20

25

0.001 0.01 0.1 1

u d

=0.418, B 0=5.45

Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

Figure B.20: Variation of defect velocity with outer scaling = y/ for threepairs of log-law constants and B 0 .

B-29


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0

5

10

15

20

25

0.001 0.01 0.1 1

u d

y/

=0.384, B 0=4.08 Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P. Saric (1996)

0

5

10

15

20

25

0.001 0.01 0.1 1

u d

y/

=0.410, B 0=5.00 Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P. Saric (1996)

0

5

10

15

20

25

0.001 0.01 0.1 1

u d

y/

=0.418, B 0=5.45

Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P. Saric (1996)

Figure B.21: Variation of defect velocity with outer scaling y/ for three pairsof log-law constants and B 0 .

B-30


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40/63


41/63

1.2

1.3

1.4

1.5

1.6

1.7

1.8

5 5.5 6 6.5 7 7.5 8

H 1 2 =

* /

log10(Rn)

=0.384, B 0=4.08 Present 2% Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

1.2

1.3

1.4

1.5

1.6

1.7

1.8

5 5.5 6 6.5 7 7.5 8

H 1 2 =

* /

log10(Rn)

=0.410, B 0=5.00 Present 2% Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

1.2

1.3

1.4

1.5

1.6

1.7

1.8

5 5.5 6 6.5 7 7.5 8

H 1 2 =

* /

log10(Rn)

=0.418, B 0=5.45

Present 2% Nagib V.P. Osterlund V.P. S&W V.P. T3A V.P. T3B V.P.

Figure C.2: Predicted variation of H 12 with Rn compared to values extractedfrom velocity proles for three pairs of log-law constants and B0 .

C-3


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C.2 The Skin-friction Coefficient c f Figures C.3 and C.4 show the predicted variation with low and high R ,respectively, of the skin-friction coefficient cf for three pairs of log-law con-stants compared to values calculated using experimental velocity proles.

The points are very scattered in Figure C.3 however Osterlunds two datapoints lie within the 2% bands in the top and middle plots. At the higher Rshown in Figure C.4 it can be seen that Osterlunds data falls within the 2%bands in the top plot. The agreement is not quite as good in the middle plot.In the top and bottom plots, Smith and Walkers data lies above predictions;Nagibs data is well above.

Except for one data point, Smith and Walkers data falls within the bandsin the bottom plot of the gure. Osterlunds data lies signicantly belowmodel predictions using this pair of log-law constants (and wake parametersfrom Smith and Walkers proles); Nagibs data are larger than the predic-tions.

Figures C.5, C.6 and C.7 show the predicted variation with low, moderateand high Rn , respectively.

Figure C.5 shows that the agreement with experiments for Rn < 106 isquite poor, but as expected.

For the moderate Rn -range shown in Figure C.6, results are more coher-ent. As with the R , cf plots we see that Osterlunds data is reproduced wellby the rst two pairs of log-law constants, and that Smith and Walkers datais better predicted by the model based on the = 0.418, B 0 = 5.45 pair. Itis interesting that on a Rn scale, Nagibs data is more similar to Smith andWalkers data than to Osterlunds data.

At high Rn the three plots in Figure C.7 show the same type of agreementas for moderate Rn .

C-4


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1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

1 0 0 0 c f

log 10(R)

=0.384, B 0=4.08 Present 2% Nagib V.P. Osterlund V.P. S&W V.P.

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

1 0 0 0 c f

log 10(R)


1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

1 0 0 0 c f

log 10(R)

=0.418, B 0=5.45

Present 2% Nagib V.P. Osterlund V.P. S&W V.P.

Figure C.4: Predicted variation of cf with high R compared to estimates fromvelocity proles for three pairs of log-law constants and B 0 .

C-6


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46/63

2.3 2.4 2.5 2.6 2.7 2.8 2.9

3 3.1 3.2

3.3 3.4 3.5 3.6 3.7

1 2 3 4 5 6 7 8 9 10

1 0 0 0 c f

Rn (millions)


2.3 2.4 2.5 2.6 2.7 2.8 2.9

3 3.1 3.2 3.3 3.4

3.5 3.6 3.7

1 2 3 4 5 6 7 8 9 10

1 0 0 0 c f

Rn (millions)


2.3 2.4 2.5 2.6 2.7 2.8 2.9

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

1 2 3 4 5 6 7 8 9 10

1 0 0 0 c f

Rn (millions)

=0.418, B 0=5.45

Present 2% Nagib V.P. Osterlund V.P. S&W V.P.

Figure C.6: Predicted variation of cf with moderate Rn compared to estimatesfrom velocity proles for three pairs of log-law constants and B0 .

C-8


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1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

10 15 20 25 30 35 40 45 50

1 0 0 0 c f

Rn (millions)

=0.384, B 0=4.08 Present 2% Osterlund V.P. S&W V.P.

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

10 15 20 25 30 35 40 45 50

1 0 0 0 c f

Rn (millions)

=0.410, B 0=5.00 Present 2% Osterlund V.P. S&W V.P.

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

10 15 20 25 30 35 40 45 50

1 0 0 0 c f

Rn (millions)

=0.418, B 0=5.45

Present 2% Osterlund V.P. S&W V.P.

Figure C.7: Predicted variation of cf with high Rn compared to estimates fromvelocity proles for three pairs of log-law constants and B 0 .

C-9


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Appendix D

Wave-making Equations

D.1 Free-Surface Boundary Condition

Consider the generalised Kelvin free-surface boundary condition given byequation (5.18). Scale all lengths by the fundamental wave number k0 = g/U 2

so that ( X,Y,Z ) = ( k0 x, k0 y, k0 z ). Then equation (5.18) becomes

Z + XX + 2 XZ Z = 0 on z = 0 (D.1)

where 2 is the non-dimensional viscosity parameter given by equation (5.19).

D.2 Velocity Potential

The velocity potential = G(X,Y,Z ; X 0 , Y 0 , Z 0 ) of a unit Havelock sourcelocated at ( X,Y,Z ) = ( X 0 , Y 0 , Z 0 < 0) is

G = GS + GW (X X 0 , Y Y 0 , Z + Z 0 )where GS = (GS + GS + ) is due to a simple Rankine source and its image inthe undisturbed free surface, and where GW is analytic for all Z 0, [132].Tuck [132] obtained a solution for G for a surface layer of viscous uidusing Fourier transformations in the X and Y directions.

The potential of the Rankine source terms are

GS

= 1

4 (X X 0 )2 + ( Y Y 0 )2 + ( Z Z 0 )2

= 18 2

d

d

ei k|Z Z 0 |

k

D-1


49/63

where k = 2 + 2 and = X + Y .The other component of the velocity potential is, [132],

GW (X,Y,Z ) = 142

d

d

2 eiX + iY + kZ

k(2 k i 2 k 2 ) (D.2)

The ow is laterally symmetric, i.e. around Y = 0, so that

GW (X,Y,Z ) = 142

d

0d

2 eiX + kZ cos(Y )k(2 k i 2 k 2 )

.

In polar coordinates we have

GW (X,Y,Z ) = 142

/ 2

/ 2d

0dk

keikX cos + kZ cos(kY sin )k sec i 2 k2 sec

. (D.3)

The X -derivative of the odd part in X of GW is

GW OX X = 122 / 2

/ 2 d

0 dk 2

k4

ekZ

cos[k(X cos + Y sin )](k sec2 )2 + ( 2 k2 sec )2Substituting k = q sec2 and assuming Y = 0 gives

GW OX X = 122

/ 2

/ 2d sec3

0dq q 2 eqZ sec

2 cos(qX sec) V (D.4)

where V is given by equation (5.24). Applying the limit given earlier inequation (5.25) yields

GW OX X 12

/ 2

/ 2d sec3 eqZ sec

2 cos(qX sec). (D.5)

which corresponds to the single integral part of the potential of a Havelocksource, [132].

D.3 Wave Resistance of a Thin Ship

For a thin ship with offsets y = f (x, z ) the disturbance velocity potentialis U and the linearised hull boundary condition isy = f x (x, z ) on y = 0 . (D.6)

Dene a distribution of Havelock sources over the (vertical) region R :xb xe, y = 0, z b 0 with strength 2 Uf (, ) at ( , ), namely(x,y,z ) = 2 k0 R ddf (, )G(k0 x, k0 y, k0 z ; k0 , 0, k0 ) (D.7)

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where G is the potential for a unit Havelock source.The hydrodynamic pressure at a point ( x ,y,z ) in the ow eld is

p(x ,y,z ) = U 2 x (x,y,z ). (D.8)

The wave resistance is obtained by integration of the x-component of force over both sides of the ship, [132],

RW = 2 R dxdzf x (x, z ) p(x, 0, z ). (D.9)Substituting the expression for the pressure gives

RW = 4U 2 k20 R dxdzf x (x, z ) R ddf (, )GW X (k0 x, 0, k0 z ; k0 , 0, k0 ). (D.10)

The remainder of this integral is unchanged if ( , ) and (x, z ) are inter-changed, so only that part of GW X which is even in X contributes [132].Substituting equation (D.4) for GW OX X gives

RW = 4U 2 k20 2

2 / 2

0d

0dk

k4 |(k0 k cos, k0 k)|2(k sec2 )2 + 22 k4 sec2

(D.11)

where(, k ) = R dxdzf x (x, z )eix + kz . (D.12)

Repeating the substitution k = q sec2 , we have

RW = 4U 2 k20 2

2 / 2

0 d sec6

I V (q ; , 2 ) (D.13)where

I V =

0dq

q 4 |(qk1 , qk2 )|2(q 1)2 + 22 q 4 sec6

= k21

0dq

q 6 (P 2 + Q2 )(q 1)2 + ( 2 q 2 sec3 )2

). (D.14)

Another form for RW that is convenient is

RW =

2g4

2 U 6 / 2

/ 2 d sec5

0 q 4

[P 2

+ Q2

] V dq.Using the limit given in (5.25) on this expression recovers Michells waveresistance integral (5.26).

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resistance wave making and wave decay of thin ships with emphasis on the effect of viscosity

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