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Experimental investigation of the drag induced by bilge vortices


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y. Scheepsbouwkunde


Technische Hogeschool




'al drag experienced by a ship in steady, longitudinal has bern commonly divided into two components, the

drag and the wave-making drag. However, whenever tirit have been made to measure these two drags separately

tìdependentiv frulli each other, the wave resistance has been found to be smaller than the difference, called ry drag. between the total drag and the viscous resist-bas btu-u suggested that, besides the errors in the of measurements, there may exist a third component

'nta drai. namely the resistance induced by the


along the bilges of tite ship. Wieghardt [I])


iii aurements in tite flow field around simple ship nd f.mnd that in the case of a rectangular

cross-troii longitudinal vortices were present. These

ces were successfully visualized by Ediavez 121 derived an expression for their induced drag from Lagn!lv theorem, and computed that the vortical drag

could be as much as 20 per cent of the surface drag

Wh('fl the l)ilge Curvature is infinite. in the present work an attempt is made to determine more accurately the importance af the vortice1 drag and to study the influence of various

fac-ors Itv direct measurements of the velocity field in the stern cgion of an ogive representing an idealized double ship


This Research was performed under sponsorship of the Bureau

! ShipS Fundamental Ifydromcchanics Research Program,

ad-ninistered bt,i the Naval Ship Research and Devetopment Center.

'Ofltract Nonr 1611 (05).

Xtesearch Associate, Ïnstitttte of Hydraulic Research, The

Jniversity of Iowa, Iowa Citmj, Iowa.

Numbers In brackets designate References at the end of



lieft 87, Mai 1970 (17. Band)

Experimental Investigation of the Drag Indtced

by Bilge Vortices

Jean-Claude Tatinclaux1) 2)

Wind-tunnel experiments were conducted, using a five-hole directional probe and a hot-wire

anemometer, to investigate the vortices generated at the bilge of an ogive, representingan

idealized double ship-model. Tite geometric characteristics of the vortices and the circulation

were determined experimentally. These data were used to evaluate the vortical drag by

appivnng an analytical expression derived herein for the case of n bilge vortices. The influence

of the bilge curvature was investigated and it was found that the vortical drag decreases rapidly from 14 to 2 per cent of the surface drag when the ratio of the radius ofcurvature to the maxinmunn width of the ogive is increased from O to 0.12. Subsequent experiments

4iowed that the presence of bilge keels induces an increase in the vortical drag from 2 to

11 per cent of the surface drag when the ratio of the bilge height to the maximuni width

of tile ogive increases from O to 0.08.

Kinetic Energy per Unit Length of a System

of n Bilge Vortices

It is first necessary to derive an analytical expression for the drag induced by vortices generated at the bilge of a ship as a function of physical characteristics, such as circulation, core radius and distance between vortices, which can then be

determined experimentally. The quantity of energy transmitted

to the fluid by generation of line vortices when the body has traveled a distance s, namely Ex\s, where E is the energy per unit length of the vortices is equal to that of the work done by the body acting on the fluid, namely DxAs, where D

is the drag induced by the bilge vortices. Therefore the in-duced drag is equal to the energy per unit length of tile


The total kinetic energy E is the sum of the energy E asso-ciated with the rotational vortex cores and of the energy E0

of the exterior irrotational region.

In the case of vortices generated at the bilges of a ship, assuming that tile free surface acts like a rigid wall, or in the case of ti double model studied in a wind-tunnel, sudi as the one investigated in this report, tite energy E0 is equal to half

the kinetic energy of the system of 4n vortices formed by the

n pairs generated at the bilges and of their image system with respect to the free surface, or in the case of tile double model, formed by n vortices generated at each of the four bilges. Such

a system' of 4n vortices is shown in Fig. 1, where n has been

taken equal to 2 for the sake of clarity.

The expression for the nergy E0 will now be derived using potential theory. If it is assumed that the complex potential W (P) of the irrotational region is identical to that due to n




PathofIntegration in the Case of TwoSysemsofFour Vortices

syste'ns of 4 point-vortices located at C31 with i = 1,4 and

j i, n, then


W(F ik


(P) + iW(P)

wh.'r, k3 ¡S tile strength of a vortex of the jth group, Z is the

complex coordinate of the point P, Z. is the complex

coordi-nate of the point iJ (P) is the potential function at point P,

and ' ' P) S tile stream function at point P. following notation is introduced:

Z Zj3 = r33 e 33

Z Zj4 = r34 e 34

where rJk (k = 1, 2, 3, 4) is the distance between the point P

and the point CJk and jk (k = 1,2,3,4) is the angle between

an horizontal axis and tile radius vector CJkP (See Fig. 2). The )otcnial and stream functions can then be expressed as


[(D31-39) + ()J =

(i1 +



k3 [log

: +lo


The potential functions

= k- (* -

) and 14ii =

k3 log ri/r are associated with the first pair (Oi , O) of

the jth group of four vortices, and the functions


k3 (j and 4'


k3 log r33/r34 are associated with

the second pair 034) Wilere O represents the closed

streamline which encloses the core of the ith vortex of the jth

Fig. 2. Bennitton Sketch

group, i. e. it is the streamline which forms the boundary be-tween the rotational region of the core and the outer

irr'tatio-nal zone. Therefore the stream function 14! is constant along cadi 03. In the case of a simple system of two vortices these

streamlines have a circular shape, but in the present case,

be-cause of the interaction between the various vortices of dif-ferent strength and sizes, they have an unknown oval shape.

From Green's first identity it is well known that the kinetic

energy 2E0 of the region exterior to the vortex cores can be

expressed as




with the restriction that the region enclosed by the path of

integration c be a simple connected domain in which the

func-tions and 14i are regular and single valued. To meet these requirements the contour of integration has been chosen as

shown in Fig. 1, where the ovals U31 have been represented by circles as an approximation to the unknown shapes of the

vor-tex cores.

Because of symmetry the total strength of the system of vortices is equal to zero, and therefore the induced velocity

vanishes, when R goes to infinity, fast enough for the integral

of 1 d W around the circle of radius R to go to zero when R

approaches infìinity. Thus

um d , = O.

Furthermore, since the closed curves 013 are streamlin, i. e.

i4i is constant and dW = O along them, then




j = 1,n

The expression for the kinetic energy 2 E11 of the irrotational region is then reduced to the simple form

A-. &' A3 A'34

j = I A A'3., A34

and because of the symmetry of the system of 4n vortices the drag induced by the formation of n vortices at each bilge of a

ship is given by

A'31 A31

E0 = p/2 LS1>d14Y + j(1d14J. (2)

j = i A'3 A3,,

Expanding this expression in terms of the various components of the potential and stream functions, we get

n /A

E0 = p/2


(1>qt + 'q2) d [ ("Vta i ± "l'm&] +

j=1 \A31 .=1 zn=i

A'31 n ri

(qi + iF3) d [ (Wart + W12)I

A'3q=1 n-1 or n n n A3., E0 = pl2 [j' (1>i + q2)d (14i + 14'rn2) + 3=1 q=lrn=1 A1 A31 S (4> + F02) d


'I') I


Init k Log

r11 and


k Log

Frn3 SchIfl'stechnikBd. 17 1970Heft87 38

-Z-Zj = r3t e'9i1

Z - Zj.

r. e

and with = k0 (*(tt - *q2)

= k


For any value of q the potential function takes the same

value at any point along A1A3 as at the same point along

since the quantity (- f4) is not affected by the

rotation around and therefore

(1q:j - Ait (Oq:ì - 0q4)Ajl

Futhermore. since the paths Aj A2 and A'J2 A'31 are described in opposite senses, and the functions Wml and take iden-tical values along the two paths of integration, we have


d (-t4'i + Wm2) =


J q2d (''m1 + Wm2) Ai1 and n n n A32

E =

[Soid(Wmi +W,2) +

j=1 q=1 ni1 À31 -V1 S d (lmi +14,2)J

lt 'Wan be shown that for similar reasons, when q j we also



S d ('l'mi +

l',2) =

.1 i

d (('mi + 4')


F'inally the only terms which contribute to the expression of the kinetic energy E0 are those for which q j. The

expres-pion to be evaluated is then:

E(, Ql2 [S d (Wmi + 14m2) +

3=1 r,i=i Ai1 vjl

j 'I

d + '4''4]

Along the path A31 A0 the value of the potential function is

= - k

t and along the path A'30A'1,

= + k


'Therefore, since the W's are single valued,

n ta "ji


k j d ('l'mi + 'Prn2)

j=1 t&i=1 -"32

E0 = k3 (A31) j t III I

'l'mi (A3.1) + WflI' (A31) - Wn12 (A32)]

Furthermore since the system of vortices is symmetric with

respect to the vertical Z-axis then:


(A31) = ip



lm2 (A31) = - 'l'ni2 (A30)



k [W1 (A31) +1(I (A31)] - (3)

j= i Ial

Now let us assume that the streamlines bounding the vor-tìc's are circles, with "point" vortices at inverse points of =!''iC pair' i and 2, and 3 and 4, as shown in Fig. 2. We then


r,,,1 (A1) k,,2

- A3.'


= -


r,,-, (A31) 2

(d3 - d,,,)2 -f- [(b1 - e212) (b3 - a3) ]2 (d3 - d,,,)2 + [(b,,, - c,) + (b3 - a3)]2


-r,5 (A1) km

(A31) = - k,,, Log

- Log

r,,,4 (A31) - 2

(d + d,)2 + I(b,1, - e,,,) + (b3 - a)]2 (d3 -I- d1,,)2 -1- [(b,,, - en,) (b3 - a3)]2

with c,, = bm - Vb,,,2 - a,,,2


E0 = 31

j =1



(d - d

j m/2 +[Vbm2- am2 - (b3 a)]2


(d3 - dm)2 + [1/bm2 - am2 + (b3 - a3)]2

with k1 = (F11231), where F1 is the circulation around the ith


We have to add to this expression the kinetic energy of the vortex cores to obtain the total kinetic energy of the entire system. Assuming that the oval shaped cores of the vortices can be approximated by circles, and that the vorticity w3 is constant within the cores for each type of vortex, then each

element of a core has a constant angular velocity w3/2, Hence

each core is rotating as a solid body with angular velocity

wl2, and has a kinetic energy per unit length

E3 = '/1 (wj/2)2

where I is the moment of inertia



a4 (0.2

E,.3 = 31)


If r3 = Jta2 w3 is the circulation of each vortex then

E.. =



and the total energy of the system is then

E = E0 + 2 E3 or

r r

3=1 j=1 1,1=1 (d3 - d,,)2 + [1/bm2 - amt - (b3 - a3)]t Lo


(d3_d,,,)2 + [Vbm2-am2 + (b3a3)]'

(d3 + d,)2

+ (b3a3)]'

-+ Log

V . (4) (d3 + ln)2 + [3/b,,,2 - a,,,2 - (b3 - a)]' I It is to be noted that, in the present derivation of the vorti-cal drag, two assumptions were made: (1) The potential

out-side the vortex cores was identical to the potential of a system

of point vortices, (2) the cores of tile vortices could be repre-sented by circles containing a constant vorticity distribution. These two assumptions are obviously incompatible and the final expression for the vortical drag is, therefore, only an


Schifl'stethnik Bd. 17 1970 - Heft 87

(d3 + d,,,)2 + [3/bm2 - am2 + (b3 - a3)]2





Experimental Equipment and Procedure

The experimental work was conducted in an open throat

wind tunnel, on an ogival prism, the dimensions of which are presented in Fig. 3. This simplified double ship model was sus-pended in the test section of the wind tunnel by means of eight thin wires so located as to interfere as little as possible

with the vortices.

A five-hole directional probe was used to measure the local

total head and the velocity components u, y, w at numerous

points forming a grid of mesh size 0.025' X 0.025'. The probe

was held horizontally by a support which permitted rotation

about a vertical axis passing through the tip of the probe. Thus

it was possible to orient the probe in the vertical plane of the

velocity vector, such a position being readied when the diffe-rential pressure between the two horizontal side holes was zero. The angle u between the x-axis and the vertical plane through the velocity vector could lie measured directly on a protractor attached to the probe (see Fig. 4). According to Rouse [3] the angle (3 between the velocity vector and the horizontal plane, when it is not greater than 20 degrees, is directly proportional to the ratio of the differential pressure between top and bottom

holes to the stagnation pressure registered by the center hole.

This technique permits us to determine at any point the direction of the velocity vector as well as its magni-tude. All the presents measurements were performed with

alcohol manometers, precise to a thousandth of an inch

of alcohol. In order to check the measurements, the

velo-city components were also ¡iieasured in the transverse plane at the stern using a crossed-wire probe connected

lo a Hubbard hot-wire anemometer, Old Gold Model

type 4-2 [4], and an electronic counter, Hewlett-Packard type

5214L. The apparatus is shown in Fig. 5. It was found that the results of the two methods of measurement were in very

good agrvemetIt.

FIg. 4. VelocIty Vector Diagram


Table i

Variation of <be Circulation with Distance from the Ste-rn

Using the value of the circulation at the stern, the drag induced by the bilge vortices with n equal to I was computed from Eq. 4 to be 14 P cent of the surface drag of tise ogive

obtained from the ITTC formula for the sanie value of tise

Reynoldsnunsber, i. e., 1.05 X 10e.


Development of the Vortices

In the first phase of tile study, measurements of the total head were performed in a number of transverse planes at

dif-ferent locations along the x-axis in order to show the growths and development of the vortices. Tise results of these

measure-ments are presented in Fig. 6 as lines of constant ratio of the local total head H to the total head H0 of the nican flow. The

two zones of low pressure exhibited by these diagrams indicate

the presence of two vortices at cadi bilge, in apparent dis-agreement with the results presented by Wieghardt [i]. It is possible that Wieghardt's measurements did not extend suf. ficiently far away from the hull of his models to detect the outer low-pressure zone. On the other hand the results of tise

nseasurements of the velocity consponents y and w in the

trans-verse plane at the stern, which are presented in Fig. 8a, seem

to indicate the presence of a single vortex. This apparent con-tradiction between tile two sets of nseasurements is explained by the fact that the outer vortex, though strong enough to induce a noticeable drop in total head, was too weak to create

a secondary flow which could be measured directly either by the five-hole directional probe or by tise hot wire. When this secondary vortex becomes stronger, as was found ip sub. sequent experiments (see section, 'influence of Bilge. Keels

on Vortical Drag"), it can be visualized by measurement of the velocity component in the transverse plane. In the present series

of experiments, the influence of the secondary vortex was

neglected in tile consputation of tise vortical drag.

The circulation F around rectangular paths enclosing the.

bilge vortex was determined. lt first increases as the flow

pro-ceeds downstream up to its maximum value F1115 at tile stern, then decreases when tite flow Isas passed the body. Beyond the

stern, no additional vorticity is generated and, due to energy

loss through viscous action, the vortex weakens and tise

circu-lation decreases. Table i shows the calculated circucircu-lation as

tise ratio r/r,11 for the various planes investigated.

z - X0


- 0.2'


0.0' 0.2' 0.32 0.70 5.85 1.00 0.83 3 0901' 0.41611 o i o

Fig. 3. Ogive Dimensions

Schlffstechnlk Bd. 17 1970 - Heft 87 40


Fig. Ga. xx1 =-0.7'

Fig. Gd. x - x0 0.0'

Inf!uence of the Radius of Curvature at the Bilge

The previous results were obtained for the extreme case of

an ozive with infinite bilge curvature, while actual barges and

ships have more rounded hulls. The measurements of secon

dar., íow made by Wieghardt on a well-rounded afterbody did not indicate the presence of a bilge vortex such as for hi "rectangular model". Furthermore, the attempts made by Echavez 2J to visulaize by the smoke technique the presence of vortices along the bilges of a double ship model of bilge

'urvature equal to 12.5 ft» were unsuccessful.

T(. radius f curvature at the bilge of the ogive was then :v'reased from O to 1/4 inch in increments of 1/4 indi, and for

.'iluc th

usual measurements of the total-head and

velo-'.r\ ¶:1flponents in the transverse plane at the stern were per. c'rni'il. T}'e results of these measurements are presented in 7 Çor the .otal head und Fig. 8 for the flow pattern. The

CreUIfllOr ' and the vorillal drag D0. were determined in cadi (55C a 'd ar' presented in Table 2, together with the geometric paramcter characterizing th vortex, its radius "a", and the approkimat" coordinates "b" and "d" of the center with respect to the two planes of symmetry of the double model.

Table 2. Variations of the Circulation and Vortex Drag as a Functloji ut the Radius of Curvature

oI. 41 -r Fig. lb RcIh 0.08 'o 15.

Fig. 6a-6e Lines of Constant Total Head ¡n Transverse plane

Fig. la Re/h 0.04 o 's 5 'o Fig. 7e RJh 0.12 o 75 5 i. FIg. 6e. X - X11 0.2' .11 CO IO

( /

f 7O The variations of F/UL and DV/DO are also plotted versus

Re/h in Fig. 9, where h is the maximum width of the ogive. From these data it appears that even a slight curvature of the bilge strongly reduces the strength of the bilge vortex, as

shown by the drop in the circulation, and therefore in the vortical drag. This tends to a value equal to about 2 per cent

of the surface drag of the model (as the curvature



Fig. 7a-7c

Lines of Constant Total head in the Transverse Plane at the Stern

Schifistechnik Bd. 17 1970 Heft 87 in ft. b in ft. d In ft. UL 0.1) (1.050 0.100 0.173 0.0273 0.140 (fl4 0.050 0.075 0.180 0.0170 0.042 LOs 0.025 0.050 0.155 0.0 120 0.028 0.12 0.0125 0.0375 0.158 0.0092 0.021 'O es

Fig. 6b. xx0 0.2'

FIg.6c. xx0 8.1'


- - - ---:

, /


--o t t o',

- / /

/ /


-'///r////i 1\\


- ,

-.----..--- 1


__-//// N'

,//I\ \

- - -',

f//I i

T i Ftg. Ob RIh 0.04 Fig. 8e = 0.08 Fig. 8d Re/h 0.12 Fig. 8a-8d

Secondary Flow Pattern in the Transverse Plane at the Stern

is further diminished). It is to be noted also that the size

of the vortex decreases, as shown by the decrease in its

radius "a", and that, while it stays at an approximately constant distance from tise horizontal plane of symmetry

sir'sulating the undisturbed water surface, it moves closer to the

vcrtia centerplane, as indicated by the variations in "b". An interpretation of tisis phenonsenon is that, as tise radius of curvature is increased, the flow lines along the hull depart from the body farther downstream before developing into a vortex. In order to confirm this conjecture, more detailed studies of tise flow pattern along the ship model extending from bow to stern are necessary and are suggested for future

work. $chlffstechnlkBd. 17 1910Heft87 - 42 -0 -03 002 r UL 00 o o 004 008 012

Fig. 9. Circulation and Vortical Drag versus Radius of Curvature

Influence of the Bilge Keels on the Vortical Drag

Many ships are equipped with bilge keels to reduce their amplitude of rolling. In practice, the location of the keels is chosen as a mean flow line along the bilges, determimed by model tests in a towing tank. In the present case, the only possible location of the keels is along the edges of the ogive, where a thin plate of brass, sinsulating the bilge keels, was

glued in groove cut at 45 degrees to the horizontal plane. The model has a radius of curvature at tise bilge R1./h = 0.12 (see

Fig. 10). Two keel heights were investigated, bk/h = 0.04

and 0.08. The results of the usual measurements of total-head

and flow pattern in the stern plane are presented in Figs. 11

and 12, respectively. It is seen that the presence of bilge keels has the inverse effect to the rounding of the bilges. The vortex

increases greatly in size and strength and is located farther

away from the hull.

A most interesting result is obtained in the case hk/h = 0.08,

for which the presence of the small secondary vortex could

finally be shown by measurements of the velocity component

jis the transverse plane. As was stated earlier, the presence of this second vortex had been suspected because of the

cor-responding zone of low stagnation pressure, but was yet to be confirmed. The vortical drag was then determined from Eq. 4

with n = 2. The circulation was measured around each of the two vortices as well as along a path enclosing both vortices. The fact that the total circulation was almost identical to the sum (l'i + l') of the circulations around individual

vor-tices provides an excellent check on the nieasurensents, viz.

"total = 3.14 ft.2/sec.

Fj + I = 4.12 + (-0.94) = 3.18 ft.3!sec.

From the computations of the vortical drag, which are given in Table 3 together with the values of tise circulation and the


Fig. iO. Sketch of the Double Model Equipped with Bilge Keels 010 005 Q Fig. 8a Re/h 0.0 1 I



O O.'




Fig. lia and b

lines of Const.ant Total Head in the Transverse at the Stern

geometric characteristics of the vortices, it can be seen that the effect of the bilge keels is to offset almost completely the reduction in vortical drag obtained by the rounding of the


Table 3. Variation of the Vortical Drag as a Function of the Bilge-Keel Height

Discussion and Conclusion

For a well-rounded ship from the drag induced by the

gene-at ion of vortices gene-at the bilges can be expected to be of the )rder o only a few per cent of the surface drag. However, he presence of bilge keels may increase appreciably the trengih of the vortices and therefore their induced drag. In encra' the value of the vortical drag is still insufficient to xplain completely the discrepancy between the measured esiduary drag and the computed wave-making resistance

vhidi usually observed, and improvements in the methods

f determination of both the viscous drag and the

wave-iaking drag are still needed.

It was also found that in some cases the presence of two ortices at cadi bilge could be detected. It confirms the ex-' erimental results obtained by Tagori [51 who visualized the

ortices by means of a grid of tufts. He claims that one vortex,

errespending to the weak, outer one described here, is

enerated at the bow, while the second and stronger vortex generated at approximately midship. The experimental

esults presented by Wieglìurdt show the presence of only this

itter vortex, but Wieghurdt's studies were conducted on two afterbodies" attached to a long cylinder, and thus the effect f the bow was not present. In fact, there are indications that ic preenee of u bow does affect tile generation and

develop-sent of both vortices, and therefore the shape of the bow must

e an important factor iii bilge-vortex formation. Takahei [6]

Fig. lia hk/h = 0.04 Fig. lib hk/h 0.08

-- -- / /

/I \\\

--"t t

t O." O.? - -

/ /






// \\___

= = =-/1 ¡

1 \

----f I!


-Fig. lia and b

Flow Pattern in the Transverse Piane at the Stern

reports that the bilge vortices were practically eliminated when a bulbous bow was substituted for a more conventional bow on the ship forni he was investigating. This result has been suggested as the explanation that, at low Froude numbers, when the wave-making resistance is small, the total drag is appreciably smaller for a ship form equipped with a bulbous

bow than the same ship with a standard bow shape.

It would therefore be of great interest to undertake a systematic survey of the flow field around a ship form, with and

without bulb, by determining the flow pattern in a series of transverse planes along the body from bow to stern, in order

to obtain a detailed picture of the generation and development

of the bilge vortices and to determine more precisely tise

in-fluence of the bow shape.


This study was conducted at the Institute of Hydraulic Re-search of The University of Iowa, under tile sponsorship of the Bureau of Ships Fundamental Hydromethanics Research

Program, Project Nonr 1611(05), technically adnsinistered by the Naval Ship Research and Development Center. The writer

would like to express his gratitude to Dr. L. Landweber who suggested the study and provided useful advice und criticism throughout. Many thanks are due to Dr. J. R. Glover for his consultation 00 tile proper use of the hot-wire anemometer,

and to the Institute workshop stair for their help in tise design and construction of the experimental ctuipnlent.


[i] K. Wieghardt, Messungen 1m Strömungsfeld an zwei

Hinter-schlffsmodellen', Schlfistechnik Band 4, 1957, Heft 20.

Also available as "Measurements in the Field of Flow on Two Afterbody Models", David Taylor Model Basin Translation 277,


[2) G. Echavez, "Induced Drag due to Bilge Vortices", M. S. Thesis, the University of Iowa, Iowa city, February, 1966.

[3] H. Rouse, "Measurements of Velocity and Pressure Fluctuations In the Turbulent Flow of Air and Water", Extraits des Mémoires sur la Mécanique des Fluides offerts a M. D. Riabouchinsky à l'occasion de son Jubilé Scientifique. Pubtications Scientifiques el Techniques du Ministère de l'Air, Paris, 1957.

Fig. lia = 0.04 Fig. l2b hkfh = 0.0 - 43 - Schlffstechnik Bd. 17 1970 lieft 87 b1 ir. ....n ft. In ft.d1 a., in ft. b., in ft. d., in ft. FIUL F,JUL DtD 0.0 0.0125 j 0.0375 0.158




0.0113 -. 0.021 0.94 0.' 25 0.1125 0.143

- -




0.060 'Jot 0.Q"25 0.1625 0.155 0.0125 0.338 0.205 0.0242 - 0.0055 0.114 IO



-)4J J. R. Glover, "Old Gold Model, Type 4-2H Hot-Wire Anemo-meter and Type 2 Mean-Product Computer", uHR Report N. 105, Institute of Hydraulic Research, The University of Iowa, July. 1967.

[5) T. Tagori, "Investigations ori Vortices Generated at the Bilge", Proceedings of the 11th International Towing Tank Conference,

Tokyo, 1967.

t6) T. Takahei, "Investigations on the Flow around the Entrances of Full Hull Forms", Proceedings of the 11th International

Towing Tank Conference, Tokyo. 1967.


a radius of vortex core

b half the distance between two parallel vortices C distance from one of the inverse points with respect to

the vortex cores to the center of the vortex core

cl half the distance between two vortex pairs

vortical drag

surface drag of the model

E energy per unit length of a system of two vortices E0 energy per unit length of the region exterior

to the vortex cores

energy per unit length of a vortex core h maximum width of the ogive

keel height

SchIllstechnlk Ed. 17 1970 Heft 87 44

-k vortex strength

t half the distance between the two inverse points with

respect to the vortex cores

L length of the ogive

R radius of the circle enclosing the system of vortices

iR Reynolds number = ULJv

U mean-flow velocity V instantaneous velocity u V velocity components w W complex potential

X0 abscissa of the stern of the ogive z complex coordinate

r, * polar coordinates

'1 coaxial coordinates

(i angle between the X-Z plane and the vertical plane

through the velocity vector

tI angle between the XY plane and the velocity vector

r circulation

dynamic viscosity of the fluid u kinematic viscosity of the fluid

potential function

il, stream function

Q mass density of fluid


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