A )'
r"
Siudjecentrum T. N. O.
voor Scheeps0 _{en}
Aid. Scheepsbouw
SKIP SM ODE L L T A N K E N
NORGES TEKNISKE HÖGSKOLE
ON THE EFFECTS OF DIFFERENT TTJRBULENCEEXCITERS ON
B.S.R.A. 0.75 BLOCK MODELS MADE TO VARIOUS SCALES
by
I
E. SUND
(WITH A NOTE BY J.K. LUNDE)
SKIPSMODELLTANKENS MEDDELELSE NR. 11 AUGUST 1951
NTH  Trykk
Trondheim 195e
Lab. y. Scheepsbouwkunde
Technische Hogeschool
DeLftON THE EFFECTS OF DIFFERENT TURBULENCEEXCITERS ON
B.SR.A. 0.75
BLOCK MODELS MADE TO VARIOUS SCALESby
E. SUND
(WITH A NOTE BY J.K. LUNDE)
1. GENERAL DESCRIPTION OF THE TESTS. In order to study the
effect of different turbulenceexciters, a series of experimental tests on a full type ship were carried out, last year, at the
Skipsmodelltanken in Trondheim. The
0.75
block hull form whichhas been the subject of these experiments originated at B.S.R.A. It has also been studied at other towing tanks in connection with the nature of frictional resistance.
This liné form has been selected for investigation because of the relatively great change in towing resistance which occurs for models of this form with, and without,
turbulenceexciters. Similar experiments carried out at the
National Physical Laboratory, Teddington, are described by J.F. Allan and J.F. Conn in Transactions of I.N.A., vol. 92,
1950, p.
107:
"Effect of Laminar Flow on Ship Models." OnFig. 5 of this paper are shown the lines of the model.
The tests here described were planned to bring out the effectiveness of different turbulenceexciters with
various model sizes and values of Froudes number. Stated
another way, an attempt has been made to ascertain the limits of velocity and model sizes within which reliable test results may be obtained with the various turbulenceexciters.
To obtain results which could bring this out, the

zi) The models tested had tobe geometrically similar,
but to different scales.
The type of turbulenceexciter and its location
had to be similar for the different models. It had also to
conform to types customarily used in model tests.
The speed range of the tests had to extend down to low velocities in order to include the case where the
turbulenceexciter in question completely or partially ceased
to cause turbulence.
Different draughts had to be tested to determine whether this effected the relationship between turbulence and
laminar flow.
Customary model building techniques had to be
used.
From the same prototype four models were made to scales of 1:15, 1:22.5, 1:30, and 1:45, using ths stock wax
of the Tank. Line draughtings were prepared on thin steel
plates to eliminate paper shrinkage and the accompanying
inaccuracy between different models. After the lines were
cut, the hulls were finished with the same degree of precision
as commercial models.
Dimensions and coefficients of the prototype and
the scale models are shown in the tables i and 2.
Scales have been so selected that any model used In
customary testing should lie between these limits.
The tests have included some of the devices which
are regarded as suitable for producing turbulence. It is
known that sharp edges along the stem in itself can disturb
an otherwise stable, laminar flow. This phenomenon has also
been investigated.
To conform with the shape of this ship, the water
line should terminate in a rounded bow. For the specific
purpose of producing sharp edges along the stem, the
Scale
LLWL LBP BdLE
' LWL d1 WL 1(MLD)dL
II
VL 11(MLD) TABLE1.
Ì1AlN DIMENSiONS .Prototype
M i
M 2 M 3M 4
1:1
1:15
1:22.5
l:3o
1:45
41o'o"
124.87 in
8.325 in
5,55 in
4.162 ni
2.773 'a
400'o"
121.92 in
8.128 in
5,42 in
4.o64 ni
2.710 ni
55'_"
16,76 n'
1.118 in
o.745 in
o.559 ni
0.373 ni
26'o" =
7.92 ni
o.528 ni
o.352 ni
o.264 in
o.176 in
ll9oo tons (MLI))
l2loo ni3
3.585 ni3
l.o61
ni3o.448 ni3
o.133o
21'o"
6.4o in
o.427 ni
o.285 ni
o.213 ni
o.142 in
936o tons (MLD)
9515
nì32.819 In3
o,835 ni3
o.352
in3o.1o45 ni3
16'o"
4.88 ni
o.325 ni
o.217 in
o.163 in
o.1o85 ni
692o tons (MLD),
7o33
ni32,o84 ni3
o.617
ni3o.261 ni3
o.o772 ni3
*Mti
Units
4
a transverse surface at the bow. This transverse surface
tapers off below, after which the bow becomes a knifeedge.
In itself a bow of this form may cause turbulence. Only to
an insignificant degree, however, was this shown to be true. A significant effect was obtained only with the largest model at
maximum draught. Further tests with this device were
consequently only carried out with the models loaded to maximum
draught. Upon conclusion of these tests the bow was rounded
off to its proper shape.
Turbulence was also produced by an incrustration
of sand on the surface of the model. Granular sand was sieved
to yield a uniform size with a mean diameter of 0.7 mm. This
relatively homogeneous sand was applied with a density of
approximately 300 grains per cm2.
TABLE 2. COEFFICIENTSP
LWL
WLI
WLII
CB 0.746 O.72 0.705
C
O.96
O.9l
0.9760.757 0.742 0.723
O.42
O.l0
O.77
LCB
 LBP 1.49 % F 1.71 % F 1.72 % F
LBP/B _{7.27} _{7.27} _{7.27}
B/d _{2.12} 2.62 3.44
*
The coefficients are calculated from the displacements
including cruiser stern and using the different waterline
5Two methods of placing the sand were used, and each
subject to complete tests. In the first method a one cm. wide
sand strip was laid along, and one cm. from, the bow, running from the waterliné to section 9 1/2
In the second alternative the original sand strip was retained, but in addition a similar strip was placed just under the waterline, running from the bow and aft to section
8 on both sides.
A trip wire was the device in another set of tests.
For this the trip wire equipment _{common in most towing tanks}
was used. _{The i rnm.diameter, circular wire was placed around}
the model at a distance from the FP 1/20 of the total length
of the load waterline. The trip wire was kept at this station
independent of the particular draught run on.
Two criteria determined the velocity range for the
tests with the abovementioned turbuience.exciters. _{One, that}
the tests should serve to shed light on model tests in general.
The velocity range must therefore cover the general test range
for this type of hull. Two, that it must be, if possible,
clearly established where the various turbulenc'e.exciters cease
to function satisfactorily. The velocity range could therefore
not be completely determined aforehand, but after a few
preliminary tests it was selected between the limits 0.04 to
0.22 of Froudes number. _{This corresponds to a range of Reynolds}
number from about 0.6 x io6 _{to 2.8 x} 106 _{for the smallest model,}
and 3.1 x 106 to 1.44 x lO for the largest. In the prototype
the range is 3.35 to 14.75 knots.
Earlier, in other towing tanks, experiments with this
hull type have been carried out with three different _{draughts}
of the prototype, to wit, 26'O", 21'O", and l6'O". For ease
in comparing earlier results, the same draughts have been
selected here. These values are suitable in other respects,
In that they include the generally occuring draughts.
The models were formed in wax and treated in the same manner as commercial models tested for clients.
6
Results of all towing tests are converted to the
dimensionless form CR=R/(l/2f SV2), where R is the towing
resistance in kg., the density of the tank water
(102 kg.rn sec ), V the model velocity in rn/sec., and S the
wetted surface of the model in m2. A test series has included
from 15 to 22 individual towings. The computed C values from
the tests are plotted on the basis of Froudes number F, and a
mean curve drawn through the points. The degree of accuracy in
this procedure has been the same as for ordinary model tests,
and as objective. Thereafter the test results are plotted on
the basis of Reynolds number, and values of resistance are
taken from the mean curve of the first plotting. Reynolds
number, Re, is here designated as Re
VL/V
where V is themodel velocity in rn/sec., L the length of the model in m and
' the kinematic viscosity in m2/sec. As previously
mentioned, the models were first made with sharp edges along
the stem, and tested in this condition. Thereafter the bows
were rounded to the appropriate shape, and all subsequent
tests made in this condition.
All models with rounded bows were first subjected to
complete towing tests with a smooth hull, i.e., without
turbulenceexciters.
Tests were similarly carried out using the different
turbulenceexciters. Results are transferred to diagrams where
the spesific resistance, CR , is plotted on the basis of
Reynolds number, Re (Fig. 111). For comparison purposes the
results with a smooth hull and rounded bow are also presented.
There is further shown Schoenherr's curve for frictional
resistance with turbulent flow, and H. Lackenby's curve from
his publication: "Reanalysis of William Froude's Experiments
on Surface Friction and their Extension in Light of Recent
Developments" (Transactions of I.N.A., vol. 79, 1937, p. 120).
The following considerations are taken into account
inì the analyses of the results: For the models the specific
residual resistance shall be the same at the same Froudes
7models, the increase or decrease in specific residual resistance
should he the same. On the other hand, if the flow is turbulent,
the magnitude of the specific frictional resistance will be a
function of Reynolds number. With the different models the
same change in Froudes number will result in the same difference between specific total resistance and frictional resistance, and
will represent the specific residual resistance. If, for any
model, the flow changes from turbulent to laminar, the frictional
resistance will be effected. The resulting change in total
resistance will then not conform to turbulent conditions.
In the diagrams the specific resistance of the different models is plotted to a base of Reynolds number, and on the curves are indicated values of Froudes number from 0.05 to 0.22 in equal
increments of 0.01. One has attempted to ascertain the extent
to which test results wi.th the individual models will yield
equally valid values of specific residual resistance. This should
be the case if the frictional resistance of all models is of the
same turbulent nature at the corresponding velocity. On the
diagrams, therefore, curves are drrawn through points of specific
resistance with the same Froudes number, i.e., isoF curves. If
the frictional resistance is turbulent for all models, these
curves will be uniformly spaced. If turbulent flow could be
maintained down to zero velocity, the curve through F O for
all models would represent the exterpolator to be applied in
computing the frictional resistance for models within the range
of Reynolds number here tested. Turbulence, however, will cease
with lower velocities, i.e., with smaller Reynolds numbers. It
is therefore impossible to determine the pure frictional
resistance on the basis of such tests. _{The superimposed isoF}
curves are only drawn as far as they mutually agree, i.e., _{are}
equally spaced. _{The hypothetical curve for zero velocity,}
assuming turbulent flow, should have been equidistant from _{any}
one of the isoF curves. _{It is because of the interest in}
8and Schoenherr are also drawn on the diagrams.
It will be noted that some points o not lie on the
corresponding isoF curve. Such irregularities may be
attributed to instrumental errors and permissible inaccuracies
in readings. In drawing the isoF curves regard is given to
the probable magnitude of these influences.
2. DESCRIPTION OF THE INDIVIDUAL RESULTS. Consider first the
results with smooth hulls. Here it must be assumed that the
flow, particularly with low velocity, tends to be laminar. This tendency is confirmed in the diagrams, in that the curves lie considerably below those where artificial turbulence has
been produced. As will he discussed later, the set of points
for the same Froudes number, are quite scattered for the different
models.
Test results with sbarr edges at the, stern but otherwise
smooth hull are shown on Fig. 1, for LWL, where the resistance
is plotted on the basis of Reynolds number. The two largest
models indicate that turbulence develops at relatively low
velocities. With a Froude number 0.130.14, the isoF curves
of these two models are approximately equidistant from both
Lackenby's and Schoenherrts curves. Below F = 0.13 the
resistance of the smaller model drops rapidly and a iarked
laminar influence develops. The
larger model alDparentiv retains
its turbulence to a lower Froude number, but in the lower portion,
i.e., under F 0.0e, the sharp drop in the curve indicates that
laminar conditions are present to a marked degree. Both models
rapidly lose turbulence for Reynolds numbers less than about
4.5 x io6. For the third model, M 3, the resistance curve lies
far under the corresponding curves for the first two. Model M ¿4.
was not tested with sharp edges at the stem since its Reynolds
9
With the turbulenceexciter _{here used, i.e. sharp}
edges at the stem, the results _{obtained indicate that towing}
tests with M i produce turbulence _{down to F} _{o.og, with M 2}
to F = 0.13, and with M 3 probably _{to F} _{0.20.}
Tests with sharp edges at the stem and attached trip
wire for LWL were made _{immediately afterwards and the}
results
shown on Fig. 2. _{Data was here obtained for comparison later}
with the trip wire and rounded _{bow.} _{Since sharp edges at the}
stem and the trip wire would both _{produce turbulence separately,}
one was interested in the determining _{whether the turbulence}
could be retained down _{to a lower limit.} _{As later tests showed,}
the wire alone gave virtuelly _{the same results as the devices}
used, i.e., sharp edges at _{the stem + trip wire.} _{Turbulence}
weakened at a Reynolds number of _{about 4 x} io6 _{for M 1, or at}
F _{0.06, for M 2 at about F}
0.09, and for M 3 at about F _{0.13}
Since the results otherwise are in so close agreement
with those achieved with trip wire alone, it is sufficient to
refer to the description of _{these.}
Tests with a sand strip along _{the how were carried out}
for all draughts and for _{the entire velocity range between Froudes}
number 0.05 and 0.22.
For LWL the resistance (Fig. _{3) for the two largest}
models can be cornrsred, down to a Froudes number _{0.12, or}
rerhaps 0.11. _{Models M 3 and M 4} _{can perhaps also be compare.d}
to the first two for Froudes _{number ranging from 0.22 to 0.19.}
At this latter limit the _{turbulence decreases sharply for}
the
two smaller models.
Turbulence ceases irregularly with the different model
sizes, i.e., the transition _{does not accur with the}
same Reynolds
number. _{It thus appears that M i}
rapidly loses its turbulence
with a Reynolds number of 6 x 106 _{and less.} _{With M 2 the break}
comes at Reynolds number 4 _{x} io6.
In M 3 turbulence occurs only
at the upper velocity limit, _{but even then its complete}
presence
lo
satisfactory turbulence down to Reynolds number 2.4. x
io6.
For WL I (Fig. 4.) models M i and M 2 can be compared
down to F 0.12, while neither M 3 nor M /+ can at all be
considered. With this draught M 2 shows an unexpected high
resistance with low Froudes numbers, immediately before turbulence
ceases. The cause is difficult to determine, and cannot be
attributed to instrumental or experimental errors.
For WL II (Fig. 5) the conditions are the same as for
the previous draught, but the two largest models show a uniform
residual resistance down to F = 0.10.
Sand strips of the type here used are not as effective
a turbulenceexciter as a trip wire. With the latter, turbulence
commences with a definite Reynolds number, whilst with sand strips
turbulence commences irregularly with the different model sizes.
In the test series with two sand strips, the sand strip
on the bow was retained from the previously described test whilst
a similar strip was placed on the model on each side just under
the waterline in still water. This sand strip was laid from
the bow aft to section .
This arrangement produces a slightly better turbulence
than the sand along the bow alone, but laminar tendencies are
noticeable even at relatively high velocities.
For LWL (Fig. 6) M 1 produces a decided turbulent flow
at velocities down to F 0.10, whilst model M 3 already deviates
from the others at F 0.16 or nerhaps 0.15. M 4 does not agree
with the others when Froudes number is less than 0.l.
For WL I (Fig. 7) it can be assumed that M 2 has good
turbulence down to F = 0.11, and M i even lower. At F 0.19
or 0.l, M 3 and M 4 suffer from laznínar conditions. The spread
in the test results is somewhat greater than for the tests on
LWL. This is particularly true for M 2 and M 1.
For WL II (Fig. ) it can be said that
M i and M 2 run
uniformly down to F 0.11, whilst M 3 can be followed somewhat
uncertainly to F = 0.15. The results should indicate that
although the points are somewhat spread. M 4 agrees down to
F = 0.17, but the further tendency of this curve and its scale
values must be considered rather unreliable.
These tests also show that turbulence occurs with
different Reynolds number for the different model sizes. The
tendency towards turbulence is apparent at Reynolds number
about 6 x io6 _{for the largest model, whilst the}
same is true for
M 2, M 3 and M 4 at Reynolds number about 4 x 106,
34
x io6and 2.5 x io6 _{respectively.}
The results with a trip wire (Fig. 911) show that
this turbulenceexciter gives the most satisfactory _{results of}
all the exciters analysed. _{As previously mentioned,} _{a 1 }
mm.diameter wire was used on all models. This is placed around
the model at a distance from the FP 1/20 of the _{total load}
waterline length and kept at this station independent _{of the}
particular' draught run on.
For LWL (Fig. 9) the results agree well for all
models down to F = 0.13. _{Here M 3 and M 4 drop out, whilst the}
two larger models may be compared down to F _{0.09.} _{The isoF}
curve for the largest model indicates that turbulence is present
in the entire velocity range test, i.e., down _{to Reynolds number}
of about 3 x 106. It will be noted that at lower velocities the
curve is parallel to and immediately4 under Lackenby's _{curve.}
This tendency of parallelism is reasonable, _{when one assumes}
that turbulence is intact and that changes in velocity will
result in only smaller changes in residual _{resistance.} _{At these}
low velocities, however, it is difficult _{to obtain absolute test}
values. _{One is, therefore, on the safe side in} _{judging only the}
relative tendency of the _{curve and not its scale values.}
For WL I (Fig. 10) all models may be compared down to
F _{0.14, after which the resistance for M 4. decreases.} _{The}
same is true for M 3 at F = 0.12. _{The two larger models may be}
compared down to F 0.10.
12
down io F = 0.09, but ceases for M 3 and M 4. at F 0.12.
From the results with a trip wire it can be concluded that the reliability of the tests, i.e., the comparison between models, is considerably better than with other
turbulenceexciters employed. The transition from a partly laminar to a
turbulent range evidently occurs here at a lower Reynolds number. The deviations in values of resistance for the different models, with regard to isoF curves, are considerably less in the tests with a trip wire than with sand strips.
It will be seen from the diagrams that even at high
values of Reynolds number the curves for smooth and for turbulence.
stimulated models are not comparable. The variation is greatest
in the smaller models, tapering off, and even ceasing entirely,
in the largest model. This difference in resistance is due tc
two important causes. One cause is the resistance inherent in
the turbulence device itself. In the case of the trip wire, the
magnitude of this can be determined from approximate formulae
which have been developed. Lb is, however, a question whether
these formulae give values of sufficient accuracy. As a
consequence it is of problematic worth to separate the
turbulenceexciter in this difference between resistance curves. In two
of the tests with a trip wire, for LWL and WL II, the curves for the largest model are comparable at high Froudes number,
whilst for WL I there is still a small difference. This should
indicate that the trip wire in itself does not result in a
significantly large resistance. With decreasing velocities its
inherent resistance will also decrease. The other cause is the
extend of the area of laminar flow for the smooth models. This
area is comparatively larger for the smaller models leading to a greater difference in the smooth and stimulated curves for these
models.
In conjunction with the test carried out with the trip wire, the trimming of the different models has also been
recorded. To determine if he trimming was similar it was
13
satisfactory turbulenceexciter. From fig. 12 it will appear
that the values obtained show no significant difference between
the models.
3. SUMMARY OF THE TEST RESULTS. The results of the aifferent
tests are Dresented in purely graphical form. _{Figures 13,} 14.
and 15 show, in terms of Froudes number, that part of the total
specific resistance which lies above the Schoenherr line. Where
turbulenceexciters have been used, only that part of the curve
is shown, where it could be assumed that turbulence was present,
i.e., that rart of the curve for which there was an agreement
between the different models. The curves for smooth hulls of
the different models are quite different over the entire range.
These are drawn down to the selected datum, i.e., to the point
of intersection with the Schoenherr line. These curves for
smooth hulls strikingly illustrate how the laminar effect is
peculiar for each individual model. It is significant that M 2
lies above M 1, M 3 and M 4 for LWL and WL I.
As previously mentioned, isoF curves have been
superimposed on the resistance curves, insofàr as they mutually
agreed. _{Since various values of resistance lie} _{partly over and}
partl.y under the curves, due to experimental errors, the curves
on figures 13, 14 and 15 will represent the _{mean for all models.}
The deviation, however, is small and of _{no importance when}
comparing with results for smooth hulls.
Figures 16, 17 and 1 _{illustrate graphically the}
limitations of the effective _{range of turbulenceproducing devices.}
The diagrams show the lower limit _{of Rroids number, for which}
turbulence exists depending _{on the length of the different models.}
The limits must not be strictly interpreted, _{in that they are set}
up from diagrams where specific resistance has been _{plotted on}
the basis of Reynolds number. _{In general it can be said that the}
transition from turbulent to laminar flow _{is slightly nebulous}
14
model re most difficult to determine, in that the lower part
of the resistance curve cannot be compared with any other model. The determination of transition from turbulence is then made by considering the resistance tendency compared with the tendency
o.f the curves for frictional resistance.
In general these tests have shown that the trip wire must he designated as the superior turbulenceexciter among
those tested. The results using this exciter have been far more
uniform, with considerably smaller spread, than with the other
excit ers.
The tests further show that turbulence arises about
the same lower limit for the various draughts. This is rather
conclusively shown in the data collected from an extended range
of draughts.
These tests have been so planned, and the results so presented, that they can serve as a guide for the ordinary model tests which frequently are performed with similar full type models.
On the basis of themany sided problems which arises in connection with the phenomenon of flow, the field which has
here been investigated may unfortunately be somewhat limited. It
would therefore be of great importance to undertake further tests to investigate, in a pure form, how turbulence arises in models.
Thus, it could be of interest to undertake experiments with a
view to clarifying the effect that different degrees of
smoothness on the surface of the model has on turbulence. A desired
continuation of the exneriments here performed would be an
extensive series of tests for this purpose.
4. ACKNOWLEDGEMENTS. The research described in the foregoing
was sponsored by the Royal Norwegian Council for Scientific and
Industrial Research.
The Author wishes to record his thanks to those of
the staff at the Skipsmodelltanken. who have assisted with both
15
A NOTE BY J.K. LUNDE. It has been suggested in the tast that the
isoF curves should all be parallel to each other in order that
Froude's law of comparison shall be satisfied. It will be noted,
however, that this suggestion of parallelism requires the isoF
curves to be straight lines, and it therefore appears to be more
correct to suggest that the vertical distance between any two
of the isoF curves for a family of geosims, should be constant
for varying values of Reynolds number. This, as indeed Froude's
law, requires the conception of a perfect fluid. On the other
hand, if we use the residual resistance, including as it does,
items which do not follow this law, the suggestion of constant
vertical distance between any two isoF cu'ves may not be
strictly true.
It is also of interest to note that the wave resistance
itself in the case of a real fluid, is influenced by the scale
effect. The vertical distance between any one of the isoF
curves and the minimum turbulent friction line can, therefore,
never be constant for varying Reynolds number. To show this we
takethe origin O amidships in the free fluid surface. Ox in the
direction of the motion, Oz vertically upwards and Oy transversely.
Using the nondimensional coordinates ,', , defined by
7=Y/6,
z/d
where
21
,26
and ci' are the length, maximum beam and draughtrespectively pf the form, we have that the wave resistance in
the case of perfect fluid may be written
(1)
_{R =}
¿2dy2''2 Q,de)5e658
de
where
cos(fk.b'sec
9)
16
and here the model is given by
It has been known for sorne years now, that it is not only the form of' the model hull itself which is the cause of the
wave resistance in a real fluid, but rather, that it is the shape
of the hull modified by the boundary layer. As the displacement
thickness
6
gives a measure of the outward deflection of thestreamlines, "the wave making form" of the model may be taken to
be given by
The wave resistance may now be written
R
pydI/(Ç2Q,22se39
da
a#jï(p2
_{c)secade}
q Qa)3û
d1/
o owhere
P
_{,} _{Q} are given by 2) and where , , Q2 are also givenby (2) hut with replaced by
If the model is very narrow we may exnress the displacement thickness as
(6)
Olo46es6,,)v
17
the nondimensional f.rm,
Rer'aV'/i
is
_{the Reynolds number.}This expression of the displacement thickness will of cource not
alter the first term in
_{(5)}
_{where} ' , Q, are given by (2),and which is the part corresponding _{to the wave resistance}
experienced by the model moving in _{a perfect fluid. Neither will}
the next two terms in (5) be altered, _{but ,'} _{now becomes}
(7)ìO37e'/h/
oJ %4,) '/s
E
(
_{sec il}
ep'ds4. SS/ a'1
From (2), _{(5) and (7) we note that the wave resistance}
experienced by a narrow model, moving _{at constant speed of advance}
in a real fluid, is the sum of three terms, namely:
The wave resistance experienced _{by the model when moving in}
a perfect fluid.
The wave resistance experienced by _{a form corresponding to}
the boundary layer represented _{by its displacement thickness}
and moving in a perfect fluid.
The interference between the wave resistance due to (a) arid
(b).
It will be seen from (7) that both (b) and (c) depend
upon Reynolds number, and that its influence _{is greater the}
smaller the model is made. _{This in itself advocates the use of}
the largest possible models in order _{to obtain reliable resistance}
results. _{As no calculations have been carried}
out, it is perhaps,
unsafe to prophesise beforehand the extent of the influence of
(b) and (c) on the general _{run of the isoF curves, although} _{some}
calculations by Havelock (Calculations _{Illustrating the Effect}
of the Boundary Layer _{on Wave Resistance, Transactions of I.N.A.,}
Vol.. 90, l94, _{p. 259) on a plank indicates that the influence}
18
These terms, however, cause the wave resistance to be
a function of Reynolds number and accordingly influenced by the
scale effect. Thus the specific wave resistance for a real fluid
is not strictly constant along the isoF curves as it would be for a perfect fluid in accordance with Froudets law, nor will the vertical distance between any one of the isoF curves and the minimum turbulent friction line be constant for varying Reynolds
number. The use of (6) does, however, make the vertical distance
between any two of the isoF curves constant for varying values
of Reynolds number.
A different expression of the displacement thickness has, no doubt, to be used when considering more ship shaped models for which account also will have to be taken of the
possibilities of both transition and separation. For such models
the rapidly increasing thickness and appreciable boundary layer at
the stern, caused by the curvature of the form in this region,
plays a greater r1e on the wave resistance result than does the
more slowly increasing boundary layer thickness along the sides
FIG. I. SHARP EDGES AT BOTH SIDES OF STEM. LWL.
SCALE OF REYNOLDS NUMßER, eF
FIG. 2. SIIARP EDGES AT BOTH SIDES or SEEM AND TRIP WIRE..
L
SCALE OF REYNOLDS NUMBER,
00. ROUt'IØE STEM (MOOTH AT STEM H LL) EDGES
.
ooo4t.
0.003 1x106
2 3______EÇ
4 5 6 Ô LAC N8y 7 9 I*I0  I I 1.2 1.3 1.4 ROLNDEP STEM (.MC.TH 6TEr1 NU D TkIP WI L) E. EDE5 AF
40.0O4UáAl
6pii
____
1.3 003 1x106 2 3 5 7 i)L 11H 8 9 I.I0Ii
1,2 ER 4SCALE OF REYNOLDS NUMBER,
R.9NG BOTH SIDES OF STEM. WL I
SCALE OF REYNOLDS NUMBES, Re4
FIG. 5. SAND STRIPS ALONG BOTH SIDES OF STEM. WLL
SCALE OF REYNOLDS NUMBER, R&'»
.00 J5JD 5TR%5 Tr1 (SM ALON6 0m HULL 0.003
o___
__
NO6 a .4 5 6 7 8 9 L2 1.3 sj07 1.1 0.00 . ROiJNPt) 5AND ST STEM (S IPS ALON 00TH 5TE.I1 NUL J . 0004 0003*..
4 5 6 7 8 9 1x107 I!!t:
I 1.2__
1.3 4l. bJ06 2 1.4 'q0"
p'__
0003J
h
6 8 9 IxI0.??:?T
I la NO6 2 'I 4 7 I 1.3 1.4FIG. 3. SAND STRIPS ALONG BOTH SIDES OF STEM
>
L) u L)
HG. 6. SAND STRIPS ALONG BOTII SIDES OF STEM AND WATERLINES, LWL.
SC.AL
OF REYNOLDS NUMBER,
7. SAND STRIPS ALONG BOTH SIDES OF STEM AND WATERLINE5.
VIL I.
SCALE OF REYNOLDS NUMBER Re
F STElI AND WATERLINES.
VIL
.
SCALE OF REYNOLDS NUMBER, Re
00. 5 NDWAT LINE fr106
3__
5 6Ei_.5
._______

_{}
4 8 1.3 7 2 4 9 fr107 I i 1.2 Q . sSTEM TRIPS ALO .Öc,TH,,tLJ 6 STEM NC) WATE . INES ________ RcUNOD . 5ANO
u

1
'LTr'
 .. 6 7 ==r.8 9 lxlO SC i 1.2'i
0.003 I s NIIRR 1.4 1s106 ¿ 3 4 5 1.300'.
i,1.1I.0 C) ,flI1 ALO ).MQ(PI1 I f.rLM NOATLLir4E' .5414t) rRlps_=__I
0004 O 00.3h.
. . r L.Ai.
IçENB,' fric6 Z .3 4 5 6 7 8 9 fr101 1.1'L4
1.2 1.3 1.4L) U] t.) z Li L) U) ûti) L. o LI J
SCALE OF REYNOLDS NUMBER, Re
WL
SCALE OF REYNOLDS NUMBER, Re
\JL. 11. SCALE O REYNOLDS NUMBER, R CZD &TE (SilOOT HULL) L 5 6 7 8 9
1.1 LACN,' SCHØENER z 3 1.2O'.
________ . IUNOD 5TE.1 IP '..jIP.t. (SMOOT HULL)T 000
7 _____________ 8 9 lI,o' I i l.a 6 LACKNBy HERR. 1406 2 5 1.3 1.4 s s s
__mi4
r itji L) 2 _. . s00 M I IA 106:OR
1.4'J
FIG.
IS. SINKAGE
FIG
14.
PART 0F SPEC. RESISTA'LCE EXCEETNIS SOENHRR FRICTION LiNE. WL.
iao : : : ° o. 55 cl,, 300 4(10 ..& G ,, , + LWL loo soc . 000. SG G ' ' 300 400 e. LS. WL I 300 400 MODEL Na , 0I 016 0.14 I 5 S 4 0.17 AP, PP
_{o.,..}
L 0.10 WLt[ 0.11 0,12 OIS 018 019 QaO oaI 022UL
U
.1IIUPÁ1
!!!!!!! .I!!!!4
000)4 ME N 5 00TH IP WIR t. . 0001 THE ESTS IPS AT 510M111111
r
VA
:
__
..!I!,
14 16 IB 20 25 24 26 ) MCSJ4f'I.
_
111111
I..,.II
u..
'02 '04 24 26SCALE OP FROUDES NUIIBER, F.
SCALE OP PROUDES NUMBER, F.
FIG.
IS.
PART OF SPEC. RESISTANCE EXC.EDIN6 SCNOEJ'IIILRR FRICTON LINE LWL.
F1615.
PART OF SPEC RESISTANCE EXCEEDING SCNOENIIERR FRICTION LINE WLJI.
SCALE 0F FROUDE5 NUMBER, F
24f16. 6 LOWER IJMIT CF Tb.B&JL E, LWL.
SCALE OF RVNOLO& NUMBtR
F16. 7. LOWER U?9T CF Tt&ILENCZ. '1 I
5C.ALE OF REVLD5 NUM8R
b/
5CUX O RYt6LC8 NIJM8R
e 9 4 5 6 e s