KUNUL. TIKNISKA HOGSKOLANs
HAND LINGAR
TRANSACTIONS OF THE ROYAL INSTITUTE OF TECHNOLOOY
STOCKHOLM, SWEDEN
Nr24 1948
MODEL TESTS ON SINGLE-STEP
PLANING SURFACES
PUBLICATION NO 1/1947
OF THE SHIP TESTING LABORATORY BY
CURT FALKEMO AND JOHAN ADLERCREUTZ
GOTEBORG 1948
Introduction
The forces acting on flying boat hulls and seaplane floats can be studied under the most idealised conditions on a flat plate, which may be regarded as the simplest form of planing surface. The drag and the lift of a planing surface of this shape can be calculated by
purely theoretical methods on certain simplified assumptions by the
aid of PRANDTL'S wing theories (2), (5). H. WAGNER (2) has made use of these theories in some papers, and has evolved methods of calculation which can be employed for this purpose An extensive series of tests carried out at the Hamburgische Schiffbauversuchs-anstalt has contributed to the experimental study of this problem. These tests have been dealt with by Dr. -Ing. W. SOTTORF (3) in several papers. He describes the various stages of the investigation, and gives, in conclusion, a set of diagrams for calculating the drag and the centre of pressure of a flat plate moving over the surface of the water at a given load and a given angle of incidence. Further-more, the experimental results obtained by Sottorf throw light on the effects produced by the wedge form, the longitudinal curvature or camber, and the transverse curvature of the bottom surface.
From an engineering point of view, the problem of the simple planing
surface can therefore be considered to be sufficiently elucidated.
However, in many structures the supporting bottomsurface is divided
into several planes by transverse steps. Consequently, the aft planes
move in the water disturbed by the fore plane. It is obvious that the theoretical treatment of this problem presents great difficulties.
A co'mplete experimental investigation would be very intricate since the problem involves a large number of variables. Furthermore, this
investigation would require a testing tank which is equipped with a
high speed carriage, and which permits towage over a long distance.
For these reasons, the investigation dealt with in this paper had to be confined to tests made at the same speed and the same load
4
angles. The results obtained from the investigation made by Sottorf
have been taken into account in these tests which are primarily to be regarded as a supplement to his investigation.
Flat Planing Surface
Fig. 1 shows the
reso-lution into components of the total force R acting on
a flat plate which travels
over the surface
of the
water at an angle of inci-dence a. The normal force N gives rise to a profile diag or resistance W7 Atg. a, where A is the load, and the frictional force T pro-duces a skin friction drag or frictional resistanceWR=-T/cos a.
The total drag W = Atg a + T/cos
Sottorf uses the following notations:
A = Lift = Load, kg.
W = Drag or resistance, kg = Normal force, kg.
T = Tangential force or frictional force, kg.
= Speed, m per sec.
* =
- ViTg . (A/1i)113
- Reynolds number. w
--- =
Drag to lift ratio.A
- Lift
coefficient.albq
Fig 1. Forces on a flat planing surface.
a = WF + W5.
Froude number with a length dimension referred to the load.
5 A
a°0v2F
10 WE=
Fq
- Coeffièient of friction. V2 q=
2 - Stagnation pressure or dynamic pressure, kg per rn2.=' Mass density, kg sec2 per m4.
= Specific gravity, kg per m3. v = Kinematic viscosity, m2 per sec2.
= Mean length of wetted surface, m.
1 = Distance from centre of pressure to trailing edge of planing
surface, m.
F = Area of wetted surface,
b = Width of planing surface, m.
a = Angle of incidence.
For a flat planing surface of the width b subjected toany arbitrary load, we shall calculate the value of * and the lever arm of the external forces 4,. The relation between 4,, 1, and b is graphically
represented, in Fig. 2. From this diagram, we obtain by trial and
error method .the side ratio i/b corresponding to the load, and hence
the area of the wetted surface F
= 1 . b and R = 1 . v/v. Thediagram. shown in Fig. 3 gives the value of Ca, and we can therefore
calculate the angle of incidence. From a technical point of view, theproblem is now solved since we can determine the drag
compo-ev2
nents W = Atg
a and WE = . F. The coefficient of fric-tion C; = f(R) is taken from some experimental curve (ScmIcHTncG,SCHöNHERR).
When &' is large and the side ratio is small, 4,/i is independent'
of i/b. In that case, we calculate the drag for a number of appropriate
values of i/b and plot a curve representing the drag as a function of the side ratio. This curve has a minimum which determines the optimum side ratio.
The boundary line for the condition of pure planing is indicated in Fig. 3. This line, which is given by Ca = 0.94 (l/b) 0.375 (when
1/b> 0.75), determines the point at which the lateral surfaces of the planing plate begin to be wetted. When the Reynolds number is
i
aa cc to ii i. ' '., '"
Fig. 2. Relation betweon the width, the position. of the centre of pressure, and the mean length of the wetted surface of plane plates for various values of the Froude number.
6
r
V 09 Oa 0 0mIera meoiy / .2 /0 09
0;.
07 ,n, inje, 'c&'r' V
#FCtOfl9flS7fp/atA
No b'"
Ar
tmkvr
________________________ / 0,5'
009 11 374 . nflfl
-I
0 0 .0- 0I
b8
less than 106, the surface tension also indicates a tendency to wetting A
of the lateral surfaces if the mean pressure Pm = is
at the
same time less than 20 to 50 kg/rn2.Consequently, the two conditions for pure planing of flat plates are
/j
\_O.375Ca<0.94)
pm> 20-50 kg/rn2
It is to be noted that those model tests in which the area of the wetted surface cannot be accurately measured, and the total drag cannot be exactly resolved into the profile drag and frictional
resistance components, e. g. tests made on actual boat models, must
give doubtful results in the ranges limited by the above conditions.
Tests on Stepped Planing Surfaces
At a given angle of incidence and a given load, the drag can be reduced by providing the supporting surface with a transverse step. The reduction in drag has been attributed to the fact that the step increases the mean pressure on the bottom surface (the stagnation
pressure occurs at two points). A smaller friction surface is therefore
required for taking the lift correspOnding to. the load. At the same
time, the longitudinal stability of the hull is increased since the load
is carried by two supporting surfaces.
Nb information on tests made on stepped planing surfaces has been
found in the litterature. The shape of the wetted surface of the aft plane, the appropriate position of the centre of pressure, aid the
effects of the height of the step and the angle between the two planes
have only been studied in a purely speculative manner. At high speeds (or small loads), the wave motion is slight, and the aft plane
can approximately be considered to be separte, as if it moved through undisturbed water. At low velocities, on the other hand,
interesting interference phenomena must take place.
In order to elucidate these phenomena in detail, a systematic investigation was made at the testing tank of the Royal Institute
-fOo Z)
1065
300 0O
Fig. 4. Data on mode shapes obtained by varying the height of step and the angle between the planing surfaces.
(0
6000
Fig. 5. Set-up for model tests.
224o 2290
(3286-2436
9 Nàde/o4°
5,,,,,, 81 0 10 0 20 8 0 .30 Cr -c -z ao C f 20 C +1 20 C Z 20 C 2010
To segregate the effect of the step from the other factors, and to facilitate comparison with previous tests, the planing surfaces were made of practically smooth plane glass plates ruled with lines so as to form a network of squares for determining the shape of the
wetted surface. Nine shapes of the supporting surface were obtained
by varying the height of the step and the angle between the planing
surfaces, see Fig 4. Each model shape was tested at a load of 1.0
kg and a speed of 3.5 m per sec. The position of the centre of pressure was successively varied within the limits determined by the instants at which the fore and aft planing surfaces lifted from the water surface. The change in the position of the centre of pressure was brought about by varying the weights attached to the ropes running over the fore and aft sheaves used for suspending the plate, see the test set-up shown in Fig. 5. The angle of incidence corresponding to a given position of the centre of gravity (centre of pressure) was calculated from the angle through which the sheaves moved when the position at the model was changed. The tests were made with
great accuracy. For instance, in order to obtain results which exactly
crrespond to the velocity of 3.5 m per sec, the plates were towed in the tank from 4 to 6 times at speeds ranging. from 3.0 to 3.9 m per sec, and complete readings were taken throughout this range.
After that, all measured 'quantities were plotted against the velocity,
and the ordinates corresponding to 3.5 m per sec were taken from
the diagrams.
Test Results
A.
Check Tests on Single Planing Surface
The analytical methods for the determination of the planing of
a flat plate afford a possibility of checking the test set-up. To begin
with, we availed ' ourselves of this possibifity by testing a flat, i. e. non-stepped plate at a velocity of 3 m per sec and a load of 0.75 kg. In these tests, the conditions in respect of pure planing were
more critical than in the main tests made on the stepped plates.
The results of the check tests are reproduced in Fig. 6. In this case,
our plots lie above the theoretical curves.') At 'a given angle of
9 The term ,)theoretical' refers in what foows to the chart plotted by Sorro
(pW
Cmk9 s/75A0.TSo/r9 yjom_1
- - - - Sptfp,7tj tbQd!w'1co/ VO/UiS 7 -VQIURS\
\\
8 45
a
Fig. 6. Comparison of'nlnes observed in model tests with theoretical values obtained by Sottorf.
Single Plate tests.
0.00B -coo7 O ooS O,00 0,002 . S S O,00/ A è
,ô'
S Q 1,.' 0.4.55bowday /aye!r
ouna'ary layer (çFig. 7. Comparison of the coefficients of friction obtained from the model tests made at the Stockholm Tank and
at the Hamburg Tank with theoretical curves. Single plate tests.
brn 0./so 4 kg
O.o
7 per Sec .oc,o 7e ,nadeaf#te 5kkth!?EPk_ 0.o7S 0.80 3oo 6-"-
0.100 0.o .3.460Conditions for pure planing
Pm> 20-50 kg/rn2
1 - 0.375
Ca
<0.94(i)
13 incidence, the wetted length is greater and the drag is higher In
connection with this comparison, it is to be noted that the coefficients of friction used in calculating the theoretical values of the drag were
taken from the curve for turbulent boundary layers given by
SCHLICHTING, see Fig.. 7. Actually, the test results show that thefrictional resistance does not agree with this curve when the Reynolds number is less than .106. The same observation has also been made at the Hamburg Tank. The mean curves of the coefficient of friction
obtained in the Hamburg .tests are marked by a and b in Fig.. 7. When the plates are practically smooth, and the Reynolds numbers are as low as in the case under consideration, the boundary layer ought to be of stable laminar character, see the curve for laminar
flow (Blausius) in Fig. 7. SOTTOBF (3) says on this subject: oDass diese c-Werte im Bereich der Kurve fir turbulente Grenzschicht liegen, ist physikalisch nicht begrundet und als Ztifallsergebnis zu bewerten.o (8The 'fact that these values of c lie in the range of the curve for the turbulent boundary layer cannot be attributed to
any physical causes, and is to be regarded as an accidental result.)>)
This judgement seems, somewhat hasty considering that ti.e curve given by SCHLICHTING and other similar curves apply to plates towed at a zero angle' of incidence. In the case of a planing surface, the pressure gradient is high afore the stagnation point. However, it is
extremely difficult to estimate the influence of this fact on the point of transition from laminar to turbulent flow, without making pressure measurements. Table I.' Ca 0.04 (1/b) 0.375 19.2 0.705 0.764 22.0 0.643 0.806 25.6 0.664 0.84 6 36.1 ' ' ' 0.812 0.07 2 42.2 0.868 1.036 56.8 0.073 1.143
14
Table 1 shows that the conditions fOr pure planing are only just satisfied. By vIsual examination of the model during the tests, it
was found that the lateral surfaces of the plate were not wetted.
From this test it was inferred that the general trend of the curves
is correct, and that reliable conclusions can therefore be drawn from
a comparison of the various bottom shapes.
B.
Effect of Height of Step
The heights of steps on the test mo4els BI, BIT, and Bill were 1, 2, and 3 cm respectively. Fig. 8 represents the total drag W as a
function of the angle of incidence a. Furthermore, this diagram
gives the profile drag of a flat plate W = Atg a and the total drag
W of a plate determined in the tests made by SOTTORF, and contains
schematic sketches showing the shape of the wetted surface cor-responding to some points of the drag curve. The shape of the drag curves is very peculiar, particularly in the case of the test models BIT and BITT. At first, the angle of incidence increases as the aft trim moment becomes greater, whereas the drag decreases. This trend continues to a definite critical point (corresponding to 6.10 the case of the test model BIT). If the aft trim moment is further increased, the angle of incidence decreases, while th drag becomes greater. However, the drag does not vary according to the same
curve as before, and forms a curve branch located abovethe previous. curve. At a definite point (>5.55° in the case of the test model BIT), this curve branch changes into the drag curve of the single planing surface. The same tendency can be observed in the curve obtained in the tests oh the model BI, but the height of the step is too low
in this case, and no upper curve branch can therefore be formed. If
a flying boat provided with a stepped bottom moves at a speed corresponding to 3.5 m per see, three different drags and effects wifi
be produced at a given angle of incidence owing to the external
moment which counterbalances the water pressure.
The absolute reduction in drag obtained by the use of the stepped
bottom surface as compared with a flat planing surface seems to be
very considerable. (This Comparison refers to the curve marked
S0TT0RE's test series No. 5, see also Section C Effect of Angle between Planing Surfaces.) In our test serIes No. ,28 made on the model BIll, the total drag is sliglitly higher than the profile drag W
0,2 0,i Fore
a a
-8LP/oni,2,Jh
5o/#o
t.it
8e/796 Ne 5Br P/n,,,9 on aft .5wr/aGe ot4i
az_
ôir /6 -J.S6. A/oo/y V=J3mper 56G. ,-jg
a-
2+4-
5 6 7 8 9 ioFig. 8. Total drag of pitrallel I)lates provided with steps of varying height.
Br Ne,qhf of step 1cm
-"- 2cm
-"--- icm
V.J.5mpd!r
AR1cnk5
h cm 0
8!
M' f a#p 1cm 0 + anr -20 10 +10 +20 M kg.cm18
.f6
A
iooy
Fig. 10 A. The total drag of parallel planing surfaces as a function of the positiQn of the centre of pressure for various heights of. step.
19
surface by a frictionless fluid. The shape of the drag curves is ma-terially affected by the height of the step. If the height of the step is too small, the model cannot be moved at those favourable angles of incidence which result in an. optimum drag. Fig. 9 shows the variation of the distance from the centre of pressure to the step with the angle of incidence. This diagram illustrates our previous
remarks regarding the dependence of the angle of incidence on the trim moment. As the aft trim morent increases, the portion of the load carried by the aft planmg surface becomes greater Then the moment of the water pressure will tend to reduce the angle of
incidence of the planing surface. In order to bring about a variation
in the shape of the wetted surface at a given angle of incidence, a
change must be produced in the vertical elevation pf the step above
the stifiwater surface. This is shown in Fig. 10. The curve of the test model BI is continuous, whereas the other curves have a point of discontinuity just before the centre of pressure passes the step. This relationship is unambiguous in so far as a given trim moment correspmds to a single value of the vertical elevation of the step.
Fig. .10 A shows the variation in drag with the position of the centre
of pressure. .
C. Effect
f Angle betwêCn Planing Surfaces
The test models CI to CVI were provided with steps 2 cm in height. The angle between the planing surfaces was varied, see Fig. 4. Fig. 11 shows the total drag as a function of the angle of incidence of the fore plane. The test programme did'not comprise any complete
investigation of the models planing on a single surface at the velocity,
and the load used in these tests. Nevertheless, a few separate test ;:
values were obtained at relatively large angles of incidence. Further-more, the investigations made by SorroRF include a test series (No. 5)
with the data (b = 0.1 rn) A = 0.66 kg, and v = 3.4Gm per sec It is obvious that this eries falls within the range of the Reynolds
numbers met with in our tests. The results of this series reduced
to A 1.0 kg (W = A) are reproduced in Figs. 8 and 11. Our
test values are in fairly close agreement with S0TT0RF's series No. 5. By comparing our results with those of the series No. 5, it is found that the drag of the stepped bottom surface is lower than that of
the flat plate throughout the range of the angle of incidenceemployed
k1 Oo 020 a/a C-
- - - -
O) sio9ie_w#frce Va/ir,eo made athe.S/xkha/r" 7'k P/a,, on of/ ouqace on/f.BIE CN C CFE
-R
46
5
67
- -
54-Fig. 11. The total drag of planing surfaces provided with stops of equnl height, butvarying in angle between the surfaces.
MQ/_ dtep rn#va1w9hL 20 -2 20 c
'
20eN
f
£0 c +2 20 CYt +.3 20 81r 0 20?'- 356 Ao/.00 kg v3.s mper sec
59 6/
21
a.
0, we obtain a curve of the ambiguous shape referred to in the above. The fact that the relation expressed by the curves fora -
fi< 0 is unambiguous can, of course, be attributed to the shape of the models used in the tests since the angle of incidence of the fore plane cannot reach the critical point unless the step is clear of the water surface. In fact, the reduction of drag observed in the case of the test. models C is remarkably great. For instance, thedrag curve of the test model CVI is tangent to the profile drag curve
for a single planing surface at a = 8° to 9°. Fig. 12. shows the
variation in the position of the centre of pressure with the angle of
incidense. The corresponding position of the centre of pressure of a
non-stepped. bottom surface is represented by the dash-line curve in
the same diagram The longitudinal stability of the test models is
expressed by the, slope of the curves dM/da -
dl/da. When the
centre of pressure is situated close to the fore, the slope of thecurves is of the same. order of magnitude as in the case of a flat planing surface, but this slope increases as the angle of incidence becomes greater. At the critical point, dl/da = oc, and the initial stability is very great. At the critical point, the centre of pressure is located
right abaft the step The vertical elevation of the step above the
still water level is shown in Fig. 13. The order of magnitude of this elevation is the same for all test models C. The diagram gives a. mean curve for all these models, but it is evident that this curve
indicates a general trend only Fig 13 A represents the variation in
drag with the position of the centre of pressure.
D.
Tests Made on Non-Stepped Bottom to Check the
LiftCoefficient of Planing Surface
In order to determine the respective0 portions of the load carried by the two planing surfaces in a given case, it is necessary to know the relation expressing the lift coefficient Of the fore. plane
as a
function of the velocity, the angle of incidence, and the side ratio i/b.The lift comprises a static component and a dynamic component.
For this reason, the. relation referred to in the above cannot be
expected to be unambiguous when the Froude number is small, i. e.
when the effect of gravitation is. relatively strong.. However, it
follows from SorroEF's chart shown.., in Fig. 3 that the influence of the Froude number is slight when the side ratio i/b is less than unity.
+10 -/0 -20 50 -7et4e,i _NQ_ 32 / Cl CZ
ARokg Vntpa
PioAy
I
JOc.,, 3(,, C 'I 39 48. C2f C2 C 9 /0:
-62 6/ 59 5 4 5 6 7Fig. 1 2. The distance from the centre of pressure of planing surfaces from the stop as a function of the angle of incidence for various
angles between the planing surfaces
66
?rP7-6'
0 I!/
20, _'_f
+/ 20c.'
20 +3 20+ Zo 0 + + S
CI
OGlE &CzrICZ
:LC= -/0 .5 0 +5 1/gcin 10Fig. 13. .The vertical elevation of the step above the still water level, as a function of the trim moment of planing'surfaces provided
-/0
-5
0.5
(pcmFig. 13 A. The total drag as a function of the position of the centre of pressure for various angles between the planing surfaces.
/5ó
/0.0
Z80-550 o(
8oundorq I/ne IQr ØUP
\
/pylcQ,a'i'/o
AO,,o/
A :0:25 oso 0:75
: 'N -0 - --S S + 0:2 04 06 fo
Fig. 4 The ligt coefficient of a non'steppocl 1)oltom surface as a function of the side ratio for several values of the load and
26
Consequently, the relation to be found can be computed from
SOTTORF'S tests within certain limits. On the other hand, the
abso-lute values observed in The tests described in Section A proved to be dependent on the scale to such a degree that it was deemed
necessary to make independent tests in order to obtain a curve which can be used for further calculations.
The results of these tests are shown in Fig. 14. The mean curve
obtained by SOTTOBF in the range of &5 from 2.75 to 8.50 i.s slightly
above our values in the load range A = 0 75 to 1.00 kg. The scatter from the mean curve is of the same order of magnitude as in the tests made by SOTTORF.
At loads below 0.75 kg, the value of c rapidly decreases as the
load becomes smaller. For A = 0..1 to 0.25 kg, the lift coefficient
is therefore only half as high as it might otherwise be expected to be
considering the value of the side ratio i/b.
It is evident that the
calculation of the load acting on the fore plane in a given case is rendered very difficult by this circumstance. However, the cal-culated partial loads can be checke4 with the help of the position of the centre of pressure determined by measurements, see Figs. 9 and 12.SOTTORF'S condition for pure planing Ca < 0.94 (i/b)°3'5 or c <
8.22 (i/b)°375 is represented in Fig. 14. This condition is satisfied
by all measurements, even if the margin is narrow at comparatively
high 1oads
General Considerations Regarding Double Planing
Surfaces
A hollow or trough in the water surface is produced abaft the moving fore plane. The length of this aft trough is primarily a function of the velocity, while the depth of the aft trough is chiefly determined by the load applied to the fore planing surface. Water
tends to fifi up the aft trough on all sides, and therefore flows towards
the centre line of the trough. As the water collapses at the tail edge of the aft trough, its kinetic energy is converted to pressure energy, with the result that an aft wave, which is termed *the fountairn>, is formed at this edge. F. WEncTu') compares the fountain abaft a
27
planing surface with the collapse of the water behind a cavitation
area. A phenomenon described by WINIG as sWassersprungs takes place abaft a planing surface of infinite width. In a frictionless fluid, the distance from the sWassersprungs to the leading edge of the planing surface is determined by the formula
v2
2.g
where v is the speed of advance of the plate. If the width of the
planing surface is finite, the lateral motion of the water must also be
taken into account, and the above-mentioned distance can be represented by the function
V
Ii' .sina
b
where 1' is the wetted length of the fore planing surface, and b is its width. This function has been represented empirically by SOTTORF in a diagram for v = 6.0 m persec, b = 0.15 to 0.60 m, and a load
of 18 kg.
The shape of the wetted surface of the aft plane is determined by the plane sections formed by the intersection of the planing surface with the aft trough and the associated wave system. These sections are subsequently deformed by the effect of the aft plane on the motion of the water.
The wetted surface of the' aft plane can have two fundamentally
different shapes according as the fountain is formed below or abaft the plate. In the former case, see Fig. 15 a, the wetted surface comprises two lateral areas and a vortex area in the middle. In the latter case, see Fig. 15 d, the wetted surface consists of two lateral
areas only. Figs. 15 b to c represent the transitional shapes of the wetted surface occurring between the two principal shapes. Fig. 15 e shows the shape of the wetted surface in the case where the load on the fore plane is sthall and the wave motion produced by this plane is slight.
The pure planing condition is usually defined by the fact that the lateral surfaces of the plate are not washed or wetted by water, and that the water is expelled along planes which are tangent to the
28
a
b
C
e
Fig. 15. Variation in the shape of the wetted surface with the position of the aft
plane in the wave system produced by the fore plane.
increased by any additional momentum due to the sprays. When use is made of two or several planing surfaces, the rear.planes wifi be submitted to the transverse motion of the water at the aft trough. When this water strikes the lateral surfaces, it produces a stagnation
pressure, and gives rise to a phenomenon having the character of lateral wetting. In this case, however, this cascade cannot bring about a momentum acting on the bottom surfaces; its energy has already been taken from the fore planing surface which forms the aft trough.
Consequently, in examining the condition of pure planing df .a test model in connection with the phenomenon of lateral wetting, this phenomenon should be referred to the fore planing surface.
29
Relation between Drag and Shape of Wetted Surface
Figs. 8 and 11 show the change produced ir e shape of the
wetted surface when the drag curve passes .tbrOugh a minimum. The wetted surfaces of the test model Bli corresponding to the four characteristic points of the thag curve (test series Nos. 16, 19, 23 and 24) are represented in Fig. 8. It is seen from Fig. 15 that the
drag decreases, while the shape of the wetted surfaces changes from
the type c to the type d, see Fig. 15. The drag increases when the shape of the wetted surfaces corresponds to the types b and a. The test series 24 is on the verge of planing on a single plate, ancf the shape of the wetted surface is of the type e.
A similar tendency in the variation of the drg with the shape of the wetted' surfaces can be observed in all test series, but this tendency is not marked as strongly as, in the case considered in the above since the step was clear of the water surface before the drag reached a minimum. In all those cases where the drag did reach a mmimum, the shape of the wetted surface corresponded to the type d. Tt follows from the variation in drag that the fountain must be caused to pass the aft planing surface. A continuous wetted surface on the aft plane should preferably be avoided. This can be explained by the fact that the direction of the sprays swept forward
by the planing surface (Spritzer, see WAGNIER) is more favourable
in the case of the wetted surface of the type d..
According to the Jahrbueh der Schiffbautechnischen Gesellschaft, 1933, p. 222, the order of magnitude of this spray resistance>>
(*Spritzerwiderstand*) can be calculated as follows.
For a plate of infinite width,
"vs
where ó = thickness of sprays, 1 = length of wetted surface,
= *effective angle of incidence W8 = spray resistance, V = velocity of planing. Assume 1 = 0.1 m, = 3.00, b = 0.15 m, v = 3.5 m per sec.
30
On these assumptions 'we obtain the, thièkness of spray '= 0,22 mm and the spray resistance W8 = 0.08 kg. The order of' magnitude
of the spray resistance is such that it may quite well account for the 'importance of a continuous wetted surface.
Mean Pressure and Frictional 'Resistance
It has been pointed out in the above that the lower drag of the stepped bottom has usually for the most part been attributed to thehigher pressure, which implies a smaller surface, of friction at a given' angle of incidence.
Figs. 11 and 8 show that the total drag of all test mOdels is lower
than that of a single planing 'surface throughout the' greater part
'of the respective ranges of, the. angle of incidence. The mean pressure
on the total wetted surface is represented in Figs. 16 and 17 as
a function of the angle of incidence. Moreover, these diagrams show
'the variation in the mean pressure on a non-stepped bottom with
the angle of inôidence, other, conditions being equal. As has already
been mentioned, the mean pressure obtained by SOTTORF closely' agrees with that observed in our test D at the load in' question, i. e. A = Lo kg.
Fig. 17 shows that the mean pressure on the test models CI to CIII was considerably higher than that on the non-stepped bottom surface. In the case of the test model CIII, this statement holds good for a> 4.2°. The mean pressure on the other test models CIV to CVI and BI to BITT, see Fig. 16, is, as a rule, close to the curve relating to the flat plate, hut 'is 'in some cases fa below this curve. Consequently, the assumption that the lower drag of the stepped" bottom surface is to be ascribed to the higher, mean pressure can: only be applied to the test models CI to CIII. We shall now find out whether the reduction in drag observed in the tests on' these models as compared with 'So'rToRF's test series No. 5, see Fig. 11, is of 'an order of magnitude that is determined by the decrease in the surface of friction. The coefficient of friction for each planing surface of the stepped bottom model can be calculated from SCIILICH-TING'S curve for turbulent boundary layers, see Fig; 7, if the
Rey-nolds numl?er is computed 'from the speed of advance of, 3.5 m per
see and the' mean lengths of the respective wetted surfaces. The sum of the' resistances calculated by means of these coefficients of
'1
A_ "
2ZF "
200 50 - 54,yk AI7tace,3Q/7'of5i>q/e planing swjac test 0
--00
0 £ 3.66 A /.00kg V 31 2 C, 4 5 678
9Fig. 16. Comparison of the mean pressures on the wetted surface' of parallel planing
surface with the mean pressure on a non-stepped bottom surface.
friction is approximately equal to the skin friction drag of the model.
From the curve representing the mean pressure on .a non-stepped bottom surface, see Figs. 16 and 17, we can determine that surface
/50
32 A ,Tm zoo /50
.cz.
C.ff0 c.m
clv-E CV. x I - I I 5 6 7 8 9Fig. 17. Mean pressures on the wetted surface of planing surfaces provided with steps of equal height, but differing in the angle between the planing surfaces, compared
with the mean pressure on a non-stepped bottom surface.
of friction, and hence that frictional resistane, which would be required in order that the non-stepped bottom surface should be able to carry the load under consideration, i. e. A = 1.0 kg, at the
33
The results of this cothparison are shown n Table 2. It is seen from this table that the ca]culated reduction of the drag in the case
of the test models CI and CII is slightly greater than that observed in the tests, with the exception of the test series No. 38. In view
of the effects produced by many other factors, such as wave motion,
etc., the decrease in friction may be regarded as a satisfactory explanation of the low drag of these test models. In the case of the test model CIII, on the other hand, the calculated reduction. in drag is less than that obtained from the tests. This difference must natur-ally be still more pronounced in the case of the other models CIV
to CVI and BI to Bill.
It follows from the above that the decrease in surface of friction
due to the increase in mean pressure on the wetted surface Of stepped
bottoms can provide an explanation for the reduction of drag in the case of two test models only.
I Table 2.
The last column gives the reduction in drag observed in the tests on the models CI to
C III and computed in Fig. Ii by comparison with the test series No. 5 made. by
SOTTO Cr.
Causes of Reduction in Drag
The following two assumptions are discussed at some length in
this section.
1. Part of the energy contained in the wave system produced
by the fore planing surface is recovered by the aft planing surface.
3 Test Model Test Se. a Stepped Ste ed Reduction in Drag Calculated Reduction m Drag Observed ries 'R WR Wscries No. 5 Pm Fore Aft Z TVR p,, 1V11 lVfl-Z V11 32 i.G5 19.9 0.101 0.063 0.184 10.0 0.250 0.086 0.oss C I 30 2.30 41.0 0.030 0.000 0.090 14.5 0.181 0.091 0.064 38 2.62 27.0 0.o8 0.048 0.134 17.0 0.isi 0.017 0.042 C II 42 3.38 47.5 0.020 0.058 0.078 23.0 0.123 0.045 0.035 44 3.i8 2&2 0.103 0.038 0.141 .21.5 0.131 -0.010 0oo2
34
2. The boundary layer is changed so that the coefficients of
friction conform to the curve given by .BIAsrns, which corresponds to laminar flow.
The aft planing surface moves on the slope of the aft trough of the fore plane. Under favourable conditions, the aft plane may be assumed to be in such a position that its drag approaches zero and that this plane carries a definite part of the load. The aft planing surface can be imagined to be surf-ridings on the front edge of the fountain. If we assume, to begin with, that the drag of the aft
planing surface is zero, it ought to be found from the tests that the total drag of the test model can become identical with the total drag of the fore planing surface, and this drag can he calculated.
A comparison will be made with the test models Bill and
CIV to CVI. In the first place, we shall compute the load on the
fore planing surface in the various test series. The test D has shown
that the determination of the load from the value of c must be
unreliable. The hydrodynamic lift calculated from the mean curve
for loads A <0.5 kg can sometimes involve an error of 100 per cent.
However, the position of the centre of pressure computed from the
known external forces can be used for checking the loads calculated
from c on condition that the total load on the bottom surfaces
does not exceed 1.0 kg. (We can imagine that a region of low
press-ure is produced below the aft planing surface, and that capifiary forces and momenta due to the sprays come into play, with the result that the load acting on the bottom surfaces is greater than that which is determined by the excess weight of the test model.) The position of the centre of pressure on the fore planing surface is obtained from Fig. 2. The position of the centre of pressure on the aft planing surface is determined by direct measurements made on the diagrams representing the shape of the wetted surface which were obtained in the tests. When the load A,, in kg, aôting on the fore planing surface is knowii, the profile drag is computed from the formula W, = A, tg a.
The variation in frictional resistance with the Reynolds number is extremely irregular. En consideration of the tests made by SOTTORF, we can state that c is above the curve given by SCHLICHTING when R.< 4 1 o. In the range from B = 4. 1 o to 4. 06, the values of c vary between PRANDTL'S transitional curve andSCHLICBTING'S curve.
In this range, the values of the coefficient of friction are liable to abrupt changes, and their scatter is dependent on the width of plate
a
7 8 CL° 9 6 7 8 6 7 8 O(°9
6 7 8 x.o Fig. 18.Comiarison between the total drag of four typical models observed in the tests and the total drag of the fore planing
surface calculated from SCHLICHTING'S (I) and BLASIUS' (IT) formulae.
0.20 H 0.20 1
V
aa
.CY ZV
420 A -T CP
V 010 .a / ?*= ao - GQicefqldyoIu,.0 Aioek(fJ)
V5'nperec
36
Table 3.
Column 1. The load on the fore planing surface calcuiated from the value of the
lift coefficient c obtaihed from the test D.
Column 2. The load on the fore planing surface calculated from the distance between
the centre of pressure and the trailing edge of the planing surface
(used for the other calculations).
Column 3. Angle of incidence.
Column 4. The profile drag of the fore planing surface calculated from Columns
2and3.
Column 5. The frictional resistance of the fore planing surface calculated from
SCIILICHTING'S formula.
Column 6. The frictional resistance of the fore planing surface calculated from
Bi..&sius' formula.
Column 7 The total drag of the fore planing. surface calculated from SCIILICHTINC's formula.
Column 8 The total drag of the fore planing surface calculated from BxAsIus'
formula.
Column 9. The total drag of the model observed in the tests.
Test erie A1 A1 a WF, WRI WR/ WI W1 W
1 2 3 4 5
67
8 9 26 0.893 0.006 6.85 0.108 0.047 0.023 0.155 0.131 0.151 27 0.967 0.876 7.43 0.133 0.04O 0.020 0.153 0.133 0.144 B III 28 0.944 0.848 7.09 0.118 0.033 0.016 0.151 0.134 0.153 29 0.916 0.781 8.25 0.112 0.030 0.015 0.142 0.127 0.157 30 0.814 0.703 8.21 0.100 0.027 0.014 0.127 0.114 0.167 31 0.680 0.497 7.85 0.068 0.025 0.013 0o93 0.081 0.186 49 0.858 0.945 6.43 0.106 0.052 0.026 0.158 0.132 0.136 50 0.874 0.928 6.95 0.112 0.044 0.022 0.156 0.134 0.146 C IV 51 0.756 0.888 7:15 0.110 0.035 0.018 0.145 0.128 0.153 52 0.765 0.806 7.28 0.102 0.035 0.018 0.137 0.120 0.150 53 0.608 0.725 7.19 0.091 0.028 0.014 0.119 0.105 0.157 54 . 0.553 0.625 6.08 0.075 0.028 0.014 0.103 0.0800.175 55 0.780 0.975 5.86 0.100 0.050 0.030 0.159 0.130 0.137 .56 0.897 0.864 .6.61 0.100 0.051 0.026 0.151 0.126 0.137 57 0.942 0.851 7.28 0.108 0.042 0.021 0.150 0.129 0.142 C V 58 1.045 0.926 7.95 0.128 0.037 0.019 0.165 0.147 0.153 59 . 1.035 0.865 '8.22 0.124 0.034 0.017 0.188 0.141 0.157 60 0.022 0.824 8.40 0.122 0.028 0.014 0.150 0.136 0.171 61 p.661 0.634 8.21 0.091 0.022 0.011 0.113 0.102 0.187 62 0.808 0.874 5.90 0.00 0.060 0.030 0.150 0.120 0.147 63 . 0.845 0.847 6.48 0.00e 0.051 0.026 0.147 0.122 0.141 C VI 64 0.986 0.827 8.03 0.116. 0.035 0.017 0.151 0.183 0.148 65 1.ó78 0.741 9.20 0.118 .0.028 0.014 0.146 0.132 0.164 66 0.918 0.689 9.os 0.112 0.022 0.011 0.134 0.123 0.10137
and on theThload. For R> 4.106, the curves become continuous, and
are close to the curve given by SCHLICHTnG. No attempts have been made to systematise the variation in frictional resistance with the pressure gradient, initial turbulence, and surface roughness. For this reason, it is quite impossible to estimate the frictional resistance of the fore plane in the various tests made on stepped
bottom models. It may nevertheless be presumed that the coefficients of friction can never be below the curve given by BrAsIus for laminar boundary layers. To the profile drag calculated in the above,we have therefore added the skin friction drag computed according to BLASIIJS and according, to Scm.acHTING. Table 3 gives the results of the
calculations based on the train of reasoning expounded in this section.
The load on the fore planing surface computed from the value of c coniderably differs from the results of the calculations based on a fixed position of the centre of pressure.
Table 3 shows, moreover, the variation in the total drag of the fore planing surface with the character of the boundary layer.
Fig. 18 represents the total drag of the test model compared with the calculated drag of the fore planing surface. The minima of the
drag curves of all four models are below the curve showing the drag
of the fore planing surface if the frictional contribution to this drag
is calculated from SCHLICHTING'S formula. (I) On the other hand,
if the friction is assumed to agree with the curve given by Blasius, (II) the drag of the fore planing surface is lower than the total drag observed in the tests.
The shape of the drag curves can be attributed to the interplay of the phenomena characterised by the recovery of wave energy and the variation in the coeffièients of friction with the angle of incidence. In order to elucidate this relation in detail, it is necessary
to acquire accurate knowledge of the circumstances determining the
character of the boundary layer. Tests on double planing surfaces
made in a range that is less critical can possibly give further
inform-ation on this subject.
Model Scale
The purpose of' Table 4 is to,compare the test load of 1.0 kg acting on the planing surfaces with the loads carried by the bottom surfaces of actual boats.. This table shows that the load of 1.0 kg corresponds to an unusually great weight. of the boat, and the speeds
correspond-38
ing to the velocity of 3.5 m per sec are below the speed range f
these boats. It is to be noted, however, that the test velocity was chosen for technical reasons in order to ensure the planing condition
and wave interference between the planing surfaces. Table 4.
Acknowledgements
The Authors wish to express their gratitude to Professor E.
HOG-NER, who has subjected the test results to a detailed examination, and has given valuable advice regarding their interpretation. Thanks
also are due to Mr. C. BORGENSTAM, who has prompted these tests,
and to Mr. A. 0. WARHOLM, who has designed the test set-up, and
has taken an active part in the investigation.
Summary
1. At a given angle of incidence in still water, the drag of planing
surfaces moving at loads and speeds corresponding to a value of *
of the order of magnitude of 3.50 can always be reduced, and their
longitudinal stability can be improved 1f the planing surface is divided, by means of a transverse step of an adequate size.
Name of Boat Width at Step Model Width Scale 0.15 m Scale Load Speed Corres. ponding to 3.5mpersec m g Knots Baatz Record-Boat . . 1.12 7.45 428 18.6 R IV, R VII, 1.32 8.80 310 20.2 Topken 1.48 9.88 271 21.4
Percy Step Boat 1.48 9.88 561 21.4
Thornycroft I 1.60 l065 578 22.4
Thornycroft 2 1.50 10.00 900 21.5
Thornycroft 3 1.64 10.02 824 22.5
Thornycroft 4 1.80 12.00 750 23.5
Miss Britain III , 2.40 16.00 390 2.7.2
Blue Bird 2.50 1667 612 27.s
1iss England II 3.15 21.00 646 31.2
Thornycroft 5 3.28 21.80 1056 31.7
39
The drag and the longitudinal stability are considerably affected by the position of the centre of prssure on the bottom surface, which
is determined by the external forces.
In the case of double planing surfaces, there is an optimum
drag range determined by the angle of incidence for any velocity at any given load. An appropriate height of step and a suitable angle between the two planing surfaces are required in order that the planing surfaces should move within this range.
The reduction in drag can be attributed to two different causes
according to the shape of the bottom surfaces and the speed of
advance of the planing surfaces. In some cases, the reduction in drag
is due to an increase in the mean pressure on the bottom surface. In other cases, the reduction in drag is caused by the recovery of
the energy of the waves at the aft planing surface.
The investigation dealt with in this paper gives an idea of the
most appropriate shape of the bottom surfaces and the most suitable position of the centre of pressure, particularly in the case of planing surfaces having = 3.50. In order to arrive at general conclusiOns
in this respect, further tests made on models of other shapes.at
different loads and speeds will be required.
References
BORGENSTAM-GA WELL. Modellforsok med glidande stegbát. Examensarbete
nr 44 vid Kungi. Tekniska Hogskolan, institutionen f Or teknisk hydro.
mekanik, 1942. (Model Tests on Planing stepped Hull. Final Examination
Paper No. 44, The Royal Institute of Technology, Stockholm, Institution of Applied Hydromechanics, 1942.
H. WAGNER. Uber das Gleiten von Wasserfahrzeugen. Jahrbuch der Sehifi.
bautechnischen Gesellsehaft. 1933. W. SOTTOR'. Versuche mit Gleitfläehen
Werft-Reederei-Hafen 1929 pp. 425 to 432 1932 pp. 286 to 290
1933 pp. 43 to 47, 61 to 66 1938 pp. 51 to 56, 65 to 70.
W. G. A. PRRUcG, W. JOHNSTON. Hydrodynamic Forces and Elements on a Simple Planing Surface and on a Flying Boat Hull. Aeronauticalresearch comm R & M, No. 1646; 1935.
A. SAMBREUS. Gleitflächenversuche bei grossen Froudeschen Zahlen und Trag. flügelvergleich. Luftfahrtforsehung,. Vol. 13, No. 8, Aug. 1936.
Frictional Resistance. Discussion by EISNER, KARMAN, PBANDTL, KEMI'F, and
SCHONRERR. Werft-Reederei.Hafen, 1932, pp. 207 to 214.
Table of Contents
Page
Introduction, 3
Flat Planing Surface 4
Tests on Stepped Planing Surfaces V
8
Test Results 10
Check Tests on Single Planing Surface 10
Effect of Height of Step V 14
Effect of Angle between Planing Surfaces 19
Tests Made on Non-Stepped Bottom to Check the LiftCoefficient of
Planing Surface 21
General Con sideratiohs Regarding Double Planing Surfaces 26
Relation between Drag and Shape of Wetted Surface 29
Mean Pressure and Frictional Resistance V 30
Causes of Reduction in Drag V 33
Model Scale V 37
Acknowledgements 38
Summary 38
References
