ARCHIEF
Lab. V.Scheepshouw!wde
Technische Hogsschool
Reprinted from Reported of the Research Institute for Fluid Engineering Vol. 6. No.D#11)
A Contribution to the Theory of the Frictional
Resistance of Ships
By
RESISTANCE OF SHIPS" By
Jun-ichi OKABE
From \oI. 4. No. 2.
OUTLINE
There have been published a number of investigations, theoretical
and experimental, on the resistance of ships and it has been takenfor
granted that our formula is satisfactory enough for practical purpose. It
is true that the estimation of the resistance established by William Froude
predicts quite reasonable values of resistance for those types of ships which have constituted the greater part of the objects of our research works, but
it does not follow necessarily that his theory remains valid always for any form of ships. Actually the fact that the effective horse-power of smaller ships with full water-lines has been experimentally revealed to be
consider-ably greater than were calculated from this theory suggests us some
necessity of the further examination of the fundamentals underlying our
prevalent method. The following is one tentative for the theory of the
frictional resistance.
When the frictional resistance of a ship is put to the estimation it is
assumed without exception that the friction experienced by a ship is equal
to that experienced by a flat plate with the equal wetted area when they
are moving with the same speed. This simplification is readily expected
to be legitimate when LIB of the ship (L being the length ; B, the breadth)
is very large, but on the contrary if the ship has a full water-line many factors accompanying the variation of the form from bow to stern can be
no longer negligible thus tending to make the discrepancy quite noticeable:
acceleration of the velocity on the surface, gradient of static pressure,
development of the velocity profile in the boundary layer formed over the
flank, complicated distribution of the traction on the surface and finally 1) Reports of the Research Institute for Fluid Engineering, vol. IV, No. 2, December, 1947.
Frictional Resistance of Ships 53
separation of the boundary layer, etc. All of these become conspicuous all
the more as LIB of the ship becomes smaller;, the object of the present paper is to examine as closely as practically possible the effects of the fullness upon the frictional resistance, being separated for the time being
from other ambiguities arising from miscellaneous causes (among them we
must mention above all the interactions between the boundary layer and the free surface, between the frictional resistance and the wave resistance
in other words).
For this purpose we choose a series of lenticular ship forms of
infinite draft whose LIB varies among, 4, 5, 6, 7 and 8 respectively (Fig. 1) and study two dimensional flows round these models. It is hardly possible to find the exact correspondence between these two types of flows : round
an actual ship (three dimensional) on one hand and past a cylinder of
infinite draft (two dimensional) on the other, but nevertheless wc can
expect with certainty that the close investigation of the latter will serve to
make it clear whether or not the simplification by Froude is applicable to
those ships with full water-lines. (About two dimensional models similar
to the curves of areas of ships reference is made to Baker ; ship Design,
vol. 1, p. 8, 1933.)
Before we set about the calculation of boundary layers we have to know the distribution of velocity of the inviscid flow round the same
models ; it is enabled by the method developed by Moriya (Journal of the
Aeronautical Science of Nippon, vol. 5, No. 33, 1938) and Fig. 2 embodies the result.
Assumptions :
The surface of the ship is smooth.
The boundary layer of the ship is turbulent right from its beginn-ing at the bow.
The effect of the free surface on the boundary layer is utterly neglected.
In the numerical calculation of turbulent boundary layers a method
proposed by Howarth (Goldstein, Modern Developments in Fluid Dynamics,
§ 2.
1938, vol. II, 194, pp. 436-439) which is an alternative to that of Buni
(Goldstein, ibid. 194.) has been used. But the curves of (31/19,--T) and of
(cr) (Goldstein, ibid. 166. P. 375) which play an essential part in the
calculation are not prepared for the accelerated region. Therefore the use
of this method from bow to stern throughout is not possible.
Another method, simpler than the preceding, found by Millikan (A. S.
lg. E. Trans., vol. 54, No. 2, 1932) which is applicable to a body with
slender form is to assume an invariable velocity profile within the boundary
layer neglecting the effect of the adverse pressure gradient. Namely by
assuming power laws for the distribution of velocity and for the surface traction respectively just as we do when we discuss the flow past a flat
plate (Goldstein loc. cit. 163, pp. 361-362), we can readily integrate the
momentum equation. If we denote by U the velocity in the boundary
layer, by U the velocity on its outer edge, J the thickness of the layer,
11 the normal distance from tne surface, ro the surface traction, S the
arc-length from the bow, by n, m and the numerical constants depending on
the Reynold:, number and m = 2n/(1+n), and further if we assume
(UIUi) = (YIJ)n (3.6)
and
rolPW = x(U1-110'n. (3.8)
then our final result is
)1-n, (1 + 2n)(1.+ 3n) U1-(3"")(' ""
LT)
. 0
An example of calculation :
L (the length of the ship) = 4 m.,
U, (the speed of the ship) = 1.3036 m./sec., Uo/VL = 0.7 kt./1/fl.
and v (kinematic viscosity of water at 15°C) = 0.01141 cm.2/sec., accordingly
the Reynolds number RL = UoLlu is found to be 4.57 x 106. For this value of the Reynolds number the following values of the constants are
appro-priate: 1 1 n = , m x = 0.0232. 4 ' UI 3+6" dS vc, L (3.9) ,
Frictional Resistance of Ships 55
(This value of x is the revised one 0.0225 in order to arrive at the formula
Cf = 0.074 RI:* instead of 0.072 for a flat plate, cf. Goldstein, ibid. 163.)
Making use of the formulae (3.9) and (3.8), since U1 is a known function of S reckoned in the preceding section, we can tabulate J and r0 as the
functions of S from bow to midship.
Starting from the initial value given by Millikan's method at the
midship, the differential equation of Howarth has been solved numerically.
In order, however, to avoid the slight discontinuity of C at the junction of these two methods, a minor correction had to be made (rather arbitrarily)
to the curve of c-r above-mentianed. Fig. 3 embodies the result : the
growth of the momentum thickness (±L) of the model ship from bow to
stern for five values of LIB. The surface traction r0plUO2 is shown in
Fig. 4.
Finally, with respect to the discussion of the separation of turbulent
boundary layers the parameter F used by Bun i (Goldstein, ibid. 194, p. 437)
gives criterion : at the point of separation 1 was observed experimentally
to be about --0.06 (according to Prandtl, however, the critical value lies in
rather broad range 0.05---0.09. In Fig. 5 the behaviours of F in the
retarded region has been reproduced and the points corresponding to F= 0.06 have been marked. In all these figures from Fig. 3 to 5, curves in full lines in the retarded region (the afterpart of the ship) show the results
obtained by Howarth's more reliable but at the same time very troublesome
method, while on the other hand the dotted lines correspond to that of Millikan, which may become rather crude when the adverse pressure
gradient is no longer negligible but may be integrated by the quadrature, used continuously from the forebody into the region of the retarded flow
in spite of this expected danger.
In conclusion, let us consider the coefficient of the frictional
resistance of the ship. There is still a considerable and unsurmountable ambiguity left about it. For naturally arises a question how we can
estimate the friction experienced by that region of the surface from the separation point downstream to the stern, where owing to large eddies
generated even the friction with adverse direction may be expected. But
let us assume here without further deliberate discussions that in this region of the wake the traction on the surface is completely negligible because of the very small relative velocity between the surface and the fluid.
Cf is defined by
Cf f41° o, cos 50 dsSe
2
where S and So denote the arc-lengths measured from the bow to the
point of separation and to the stern respectively, SO is an angle shown in
Fig. 2; the wetted area of the ship is represented by So Cf calculated for
various values of LIB are shown in Fig. 6
Table VI
and in the following table. For ready cf c ficf(prate) comparison Cf for a flat plate with the
4 0.00418 1.214 same Reynolds number is added. If
re-5 406 1.179 ference is made to both of the table and 6 398 1.154 the figure, we shall arrive at a conclusion
7 392 1.137 which may be summarized as follows :
8 387 1.124 In ordinary ship forms effects of the 345 1.000 separation of the turbulent boundary layer
upon the :resultant frictional resistance are not sensible and the coefficient of friction C.?, defined above becomes fairly
greater than the value of the flat plate with the equal wetted area (as LIB
becomes smaller) owing to the velocity U1 exceeding Llo (the general flow)
over the broad range on the surface of the ship, cf. Fig. 2.
Appendix. Relation between the length of the ship and the roughness.
If the absolute roughness (by this word we mean for example the parameter
k used in the experimental works of Nikuradse), determined exclusively by the engineering work, is invariable in spite of the variety of scales of ships, then the relative roughness defined by kIL becomes greater neces-sarily as L becomes smaller. Therefore a plausible explanation may be made that the greater value of the relative roughness may tend to increase
the frictional resistance of smaller ships considerably, thus bringing about
the discrepancy in their effective horse-power as was mentioned in § 1
between the value predicted by the theory and that measured actually on
=
F tr
Frictional Resistance of ships .57
the sea. In the appendix the examination of this supposition is described briefly.
By Means of the method of Prandtl and Schlichting (TV .R.H., 15, 1934)
it is concluded after some calculations that the increase in Cf due to
roughness of k = 0.0559 cm. (const.) makes its appearance by almost the
same quantity independent of the length of the ship and that accordingly
we cannot ,ascribe the disagreement in the resistance between the theory
and the experiment, conspicuous especially in smaller ships, to larger values of the relative roughness. Fig. 7 shows two series of Cf for various values of L (Apr, 18, 1950) 5 :4= 125 0
TV:
0.003
0.002
CI 0.0045 0.0040 0.0035 0.00.30 40 JO 20 /0 4
s
7 LIB FFrictional Resistance ofShips 59
RL.=4.5 7x/04
LA, .ca (pat pitlfe) cf . 000345 /m/s\ V L 75/3k 'uric. to,3) e .1001h , "''teS --L"S5 4914-5,71 0 50 100 450 200 Lon)
ERRATA
Approximate Calculation of Laminar Jets (Vol. 5, No. 1, 1948)
By Jun-ichi OKABE
Page 9, Table I. line 7 (from the bottom), for "0.32" read "0.30." Page 21, Table II, line 3 (heading), for "ii" read "i"
Page 21, Table III, line 6 (from the bottom), jro "4.6465" and "9.4221"
