Effect of bow shape on free surface shear flow

Effect of Bow Shape on Free- Surface Shear Flow

March 1985

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Effect of Bow Shape on Free-Surface Shear Flow

1. Introduction

Ship wave resistance may roughly be defined as a

resist-ance component connected with the free surface disturbresist-ance

around the ship's hull apparent through the wave profile along the water line, a domain of wave breaking both around bow and stern, and a wave pattern trailing in the

rear. It was realized hardly more than two decades ago that

wave breaking in particular may contribute a major part

to the total resistance in case of full ship forms. According-ly, efforts have been made for a deeper understanding of

this phenomenon with the

aim of improving design

methods both for better prediction and reduction of this

resistance component.

In this paper, we have compiled experimental findings

related to this complex physical phenomenon which

according to our own observations seems to be intrinsically interrelated with the occurrence of some kind of shear flow and the generation of transverse vortices ahead of the ship's

bow, finally deforming to necklace vortices around the

bow. It has been suggested by Baba (1981) that there might be a partial analogy of this process with the generation of a horseshoe vortex (i.e. of "secondary vorticity") around a vertical strut piercing a flat plate through distortion of the

i) Containership (C', = 0.56 Fn 0.27)

(2) Cargo ship (Ch = 0.65 Fn = 0.24)

MTB 168 March 1985

Katsuyoshi Takekuma*

Assuming the existence of free-surface shear flow in front of blunt ship bow, investigations are made on the relationship between the major bow form parameters such as draft, entrance angle and protruding bulb, and the production of the secondai-v_vortices (necklace vortex) around the bow. The analysis showed that the bow form with fine entrance angle and the protruding bulb_are effective in reducing the secondary vortices around the bow. In parallel with these analytical studies, experimental_observations were made by means of flow visualization. As a result of image processing of observed data, a qualitative agreement with the ana-lytical studies is obtained.

boundary (shear) layer along this plate.

We have evaluated an approach by Hawthorne (1954)

for the generation of such secondary vorticity and its

distribution around the strut in order to obtain a qualitative

model for the relative strength of this disturbance as a

possible measure of wave breaking intensity of such struts

in the presence of a free surface under uniform flow. In particular, the influence of certain ship form parameters

such as draft, bow entrance angle, location of a protruding bulb and waterline shape was considered. Such numerical results were qualitatively compared with corresponding

experimental observations.

In the last part of this paper, the author reports on an

attempt

to measure the velocity components on and

beneath the free surface around the bow by some image

processing technique, i.e. by particle tracing. It should be mentioned that within our limited work we could not treat

the questions connected with modeling. The effect of

Froude and Reynolds number on our experiments (quite apart from surface tension, contamination and of instation-arity) can not be dealt with.

2. Review of investigations on free surface shear flow

During model tests as well as full scale trials, marked

Cargo ship (Ch = 0.77 Fn = 0.21)

Tanker in ballast cond.(Cb 0.84 Fn = 0.17)

(3) Cargoship (C), = 0.69 Fn = 0.23) _{(6) Tanker in full load cond.(Ch = 0.84 Fn = 0.15)}

Fig. I Wave patterns around bow of typical conventional ships

aNagasaki Technical Institute, Technical Headquarters _{This paper was presented to the 15th Symposium on Naval} Hydrodynamics held in Hamburg, 1984 in collaboration with Prof.

PP

Head loss due 10 turbulent motion at bow

/=0.260 020 n 0.05 0.10 * Estunated effective breadth 0 10 - d=0130 S.S 8.5 0.15 ¿=0.260 0.05 0.10 0.05 d=0.390 0.254mm 0.0S I. /=0.6490 o 00 300 SòOmm

M.1715A Water line in 2% aft trim condition

S.S.7 10mm 0.10 z/d0.130 010 u d0.260 a /0.519

surface disturbance can be observed around bows as shown

in Fig. 1. On the way of the study on separation of ship

resistance components. Baba (1969) could provide

quanti-tative evidence of a resistance component due to wave

breaking around the bow through measuring loss of head by

wake survey methods; he could even trace this loss on the way from the bow where it is generated to the wake area

(Fig. 2).

For full hull forms, this contributes a significant portion

of the total resistance, and attempts have been made to

decrease it by ship form improvement. Investigations into

hydrodynamic mechanism of the wave breaking around

bow have been made by many researchers. Those investiga-tions can be classified into the following three areas.

Pioneering early experimental studies by means of

wake survey and resistance tests on the effect of ship form parameters, by protruding bulb in particular, by

Taneda (1969), Eckart and Sharma (1971), and

Taniguchi et al. (1972).

Analytical studies on mathematical flow models for the flow around a bow and the associated free surface

deformation by Dagan and Tulin (1969), Baba and

Takekuma (1975), Inui (1981), Eggers (1981), Maruo and Fukazawa (1981), and Mori (1983).

Numerical studies in solving the full inviscid flow boundary value problem (i.e. without smallness assump-tions for flow components) either for potential flow by Gadd (1976), Rankine source method by Chan & Chan

(1980), or with models including rotational flow by

Miyata (1983).

After the above primary experimental investigation, mainly performed in his institution, the author (1972), aiming to provide material for constructing a rational mathematical model to describe this phenomenon as a useful tool for ship design, conducted a series of flow

measurements around the bow of a full ship model by use

¿=0.130

a Velocity component perpendicular to the control surface

Ho : Total head of uniform flow in mm water 0.05 H Total head In wake, Model speed t'=1.973m s, Fo=0.237 p

Undisturhed free sari ace 2 : Height of water surface O

L1np : Length of the ship model (7.000m) d Draft of the ship model (192.51mm) Distance from the side wall of model (500mm from

Fig. 2 Separation of ship's resistance components (Wake survey)

wnht mm S.S.3

a ¿=0.260

zd=0,130

lead loss due to turbulent motion at bow

HoJi at 1=0130

o.os

0,05

O

w0 u,t1 ii/U, o/t' OE5

Shailow rfltf o=t75mm

i =175mm, i'=O

Fig. 3 Velocity profile around bow

0.5Lppfrom AP z/d0.130 ¿=0.260 a ¿=0.519 a d0.779 v-0 l'O p loo 200 300

of five-hole Pitot tubes. Comparing the results with those calculated by potential theory, it was found that:

except the thin layer beneath the free surface, velo-city components obtained by the measurement coincide well with those obtained by numerical calculation based on the assumption of double model flow, and

in this thin layer, velocity components change

abrupt-ly in the vertical direction (Fig. 3).

These observations suggested the importance of detailed studies on this phenomenon inside the thin layer near the

free surface.

Head loss due to friction on the hull surface 3 2 1 AP S.S9 8 7 6 5 0.10 _..i=0,519 200 400 600mm 600mm 200 400 600 800 1000mm 200 400

More recently, the flow beneath the free surface around some ship models and semi-submerged circular cylinders was visualized by use of metal flakes and water color dye (Kayo et al. 1981, 1982, 1983). A shear layer beneath the free surface corresponding to the abrupt change of velocity

profile was visualized as illustrated in Fig. 4 and Fig. 5.

It was observed that the intensity of the shear layer

and vortices increased with advance speed. In particular, an artificial increase of the shear layer caused wave

break-ing already at a much lower speed than without such

manipulation as illustrated in Fig. 6. lt was considered that the phenomena had similar properties to the flow around a vertical cylinder on a flat plate with generation of

horse-shoe vortices, as shown in Fig. 7. However, neither the

Circular without vinyl sheet with vinyl sheet surf ace Shear flow U = 0.05 ni/s

Fig. 4 Shear flow and vortices observed by flow visualization (1)

U = 0700rn/s Fn = 0091

analytical investigations nor the purely numerical ones

resulted into models for the shear flow and vortex

genera-tion observed.

3. Effect of bow shape on the free surface shear flow

In the course of ship form design, it is imperative to

make efforts to decrease wave breaking resistance for

improvement of propulsive performance. Thus, an attempt was made for explanation of the effect of some typical ship

form parameters on the free surface phenomena around

bow by the secondary flow theory. lt has been recognized that some ship form parameters have remarkable effect on the wave breaking phenomena around bow and

according-ly wave breaking resistance.

They are fore draft, entrance angle, underwater protrud-ing bulb and waterline curvature around the shoulder part.

Color dye

MTB 168 March 1985

U= 1.00 rn/s U0.05 rn/s

Fig. 5 Shear flow and vortices observed by flow visualization (2)

U =1.089rn/s Fn = 0.142

Fig. 6 Effect of an artificial increase of shear layer

U1.284m/s Fn0.168

To start with, the effect of these parameters on the

behav-jour of a

free surface shear layer and on the vorticity

distribution was examined. Both of flow visualization

tech-niques and calculations of the secondary flow theory as explained in more detail in Appendix A were applied to

help to understand the mechanism. In this section, results of the examinations are described together with the

know-ledge obtained in the previous investigation.

3.1 Effect of draft

It is well known that the wave breaking phenomena

forward of the bow of a full ship are stronger in ballast

condition (shallow draft) than in full load condition (deep draft), while those around shoulder parts are more signi-ficant in full load condition (deep draft) than in ballast

condition (shallow draft). An attempt was made to examine

the effect of draft on breaking bow waves by using a

vertical circular cylinder as a simple model of full ships

(Fig. 8).

Calculation of the relative secondary vorticity=

and = were performed along certain streamlines

around a vertical circular cylinder taking the double body potential flow as the basic one as shown in Fig. 9. U is the

(a) Retarded stagnation flow with separatton

lb) Horseshoe vortex

Fig. 7 Examples of similar flow patterns

o

far upstream velocity in horizontal direction; .. meant to be its variation in downward direction due to shear.

Here stands for the secondary vorticity component along

the streamline of the basic flow whereas the vorticity

component of intensity is oriented horizontally in normal

direction to such line. The results are as follows.

Around the vertical circular cylinder, and are

higher in shallow draft than in deep draft.

Almost the same tendency as above is found in the relative secondary vorticity around the bow of full ships. No substantial difference in relative secondary vorti-city can be observed around shoulder parts as resulting from the effect of change in draft.

A visualization of the flow beneath the free surface

for-ward of a circular cylinder shows that intensity of the

vortical motion is higher in shallow draft than in deep draft (Fig. 10). Thus, it can be said that tendency of wave break-ing and shear flow, experimentally observed forward of the bow, coincides with that of magnitude of

andcalculated

by the secondary flow theory. However, on the phenomena around the shoulder parts, the secondary flow theory pro-vides little information, in spite of the significant feature of the phenomena observed there.

3.2 Effect of underwater protruding bulb

Around 1969, it was realized that resistance of full ship

Fig. 9 Effect of draft on anda for a vertical circular cylinder

d = lI8D LT _{10 m/} d1f4.D U=l.Ont/s d= 1/2D U 1.0 rn/c

Fig. 8 Effect of draft on wave breaking around a circular cylinder

0 10 20 30 40 50 60 70 80

in ballast condition can be decreased by fitting bulbous

bow. The decrease of resistance was shown to be caused by decrease of wave breaking phenomena around bow by fitting bulb(7)(22) as illustrated in Fig. 11. In the present study, investigation was extended to effect of underwater

protruding bulb on the shear flow and vortical motion

beneath the free surface by flow visualization and

calcula-d l-2D L O6Orii/s

d = l/8.D L' = 0.60 rn/s

Fig. 10 Effect of draft on behaviour of free surface shear layer and vortical motion in it

Without protruding bulb U (t

With protrudingbulb U = 0.8 rn/s

Fig. 12 Flow beneath the free surface around a circular cylinder with and without underwater protruding bulb (1)

x10 4

tion of the relative secondary vorticity.

Flow beneath the free surface around a vertical circular

cylinder with and without underwater protruding bulb was visualized as illustrated in Fig. 12 and Fig. 13. It is found that the vortical motion beneath the free surface

has been eliminated and flow pattern has been remarkably

improved by fitting an underwater protruding bulb. The

relative secondary vorticity and around the vertical cylinder with and without bulb was calculated as illustrated

C1 Total resistance measured by dynamometer Wave-breaking resistance of M.t7t5 0.

..&_---.

8 o

Pure viscous resistance due to friction ori ship surface (L.- minus wave-breaking resistance)

TIC 197 l'ne

O W + M.17)5-D )wi h protruding bulb) at ballast condition

. G 4 M17l5-A(withoat protruding bulb) 2% aft trim

0.10 0,t4 0.18 Fraude number Fe=V gLn,

Fig. li Effect of protruding bulb on decrease of wave breaking resistance C,. : Viscous resistance by wake sarsey Wave-breaking resistance of M.17t5-D Wave-breaking resistance f M. 1715-A

Without protrudingbulb U = 1 .0 rn/s

MIB 168 March 1985

022

by direct measurement

With protrudingbulb U 1.0 rn/s

Fig. 13 Flow beneath the free surface around a circular cylinder with and without underwater protruding bulb (2)

s

Crr Wave-making resistance by wave analysis

in Fig. 14. As a result, magnitude of and was found to

decrease largely by the protrusion to the cylinder.

Further, almost the same tendency as above is also

shown in magnitude of Tand calculated around bow of

the full ship as shown in Fig. 15. Accordingly, it is said

that flow beneath the free surface around bow of full ships

is much improved by fitting an underwater protruding

bulb and the secondary flow

theory provides helpful

information in the design of bulbous bow.

3.3 Effect of entrance angle of the bow

The bow entrance angle also has a significant effect on the flow characteristics. Observation and measurement of

wave pattern made on a series of mathematical models

with systematically varied entrance angle of bow show that

d=D r: d= i f) 4 du 3 70 60 50 EIdU 40 dz 30 20 lo - Without bulb ---With bulb A With bulb B dU 'dz 3 z 1)

Fig. 14 Effect of underwater protruding bulb on relative secondary vorticity around a vertical cylinder calculated by the secondary flow theory

70 60 50 z 40 30 20 10 o Eni,ince angle lO - Without bulb With bulb rD

the entrance angle strongly influences the flow geometry as illustrated in Fig. 16. For a closer study, the flow around a vertical circular cylinder and a cylinder with wedge front part was examined by flow visualization and the relative

secondary vorticity components were calculated along

certain streamlines. The results are shown in Fig. 17 and

10 10 8 dU o 12 _{- M.1715D} M1715A 02 0.3 Entrance angle 17.5" dU

1/

0.5 (FP) 0.6 a L,rp

Fig. 15 Effect of protruding bulb on the relative secondary vorticity around bow of a full ship

o

without wedge front

D

with wedge front

Fig. 17 Free surface shear flow and vortical motion forward of a vertical cylinder (d = D/4)

Fig. 16

Wave pattern around a series of mathematical models with systematical varied entrance angle of bow

aD aID

loo 80 60 40 20 A ---B

-- C

D - Circular cylinder - - - Biconvex sect, on

Fig. 18 Relative secondary vorticity around a vertical cylinder and a biconvex strut, calculated by secondary flow theory

r)

Fig. 19

Relative secondary vorticity for a series of struts with different entrance angles

Fig. _{18. The intensity of the observed shear flow and}

vortical motion beneath the free surface increases with

entrance angle of the bow; this tendency is confirmed by the calculations of the secondary vorticity intensity.

Further, the calculation of the relative secondary

vortici-ty around bows with varied entrance angle was made as

illustrated in Fig. 19. Result shows a tendency correspond-ing to the behaviour of wave pattern observed around the bows (Fig. 16). Wave breaking phenomena around bow with relatively small entrance angle such as that of cargo ships are found to be characterized by the wave breaking

lines diverging afterward of bow with increase advance speed (Fig. 20). But such change in flow geometry with advance speed cannot be taken care of by our numerical approach. Further studies are needed to make clear the

marked effect of the advance speed on the wave breaking

around bow.

3.4 Effect of waterline shape

It is known that the effect of waterline shape on the

- dt Ç) ₆ d 2 0.1 0.2 Model A Model B 0.3 04 Fp MTB 168 March 1985

Fig. 20 Wave breaking lines around a ship bow with small entrance angle

05 (FP) x Lpp

Fig. 21 Wave patterns and the relative secondary vorticity around a bow with different water line shape

wave breaking around bow is characterized by wave

break-ing around _{the shoulder parts. The relative secondary}

vorticity and calculated seem to indicate some

informa-tion on this phenomenon (Fig. 21). However, it should be noted that difference of and calculated on both water line shapes is not so significant as that of intensity of the

free surface phenomena observed around the shoulder

parts.

It may be assumed that the influence of shoulder

curva-ture on wave breaking is rather associated with the differ-ence in viscous boundary layer configuration, since Baba (198 1) could show that an artificial roughening of the ship

7

o

Port side is roughened for ballast condition

surface may drastically change the free surface phenomena around the shoulder part as shown in Fig. 22.

The results of investigations above are summarized as

follows.

The free surface shear flow and vortices in it are

closely related to the wave breaking around the bow.

Major parameters of ship form, namely, draft,

under-water protruding bulb, entrance angle

of bow and

waterline shape influence largely on the free surface

shear flow and vortices in it.

The secondary flow theory is considered to provide some guidance in the course of ship form design on the basis of understanding the hydrodynamic mechanism

of the free surface shear flow and vortices in it.

4. Quantitative measurement of flow around a

semi-sub-marged circular cylinder

As described in the previous sections, the character of the free surface phenomena around bow of full ships was made clear to some extent. The next step to be taken will

be quantitative measurement of flow beneath the free

surface. Thus, an attempt was made to measure the flow around a semi-submerged vertical circular cylinder with and without an underwater protruding bulb. On the basis of the results obtained, property of flow beneath the free surface, namely, velocity components, perturbation vortici-ty etc. was quantitatively examined.

4.1 Flow measurement by image processing of the

visu-alized flow

Since the free surface phenomena are sensitive to

in-stationary effect beneath the free surface, flow measure-ment should be made without causing any disturbance in

flow such as by Laser Doppler Velocimeter. In the

pre-Port side is roughened for full load

Ballast condition Fn = 0.183 Full load Fn = 0.183

Fig. 22 Effect of roughing surface on wave breaking around bow

20 10 o 10 20 30 40 ,,,,,, lt 20 30 40 50 St 70 80 90n,n, 10 20 30 40 30" 1=070,,, s w,th bulb

Fig. 23 Flow patterns obtained by tracing particle (Interval 1/25 sec.)

liminary study, an alternative approach was made by

application of an image processing technique to video

recorded or photographed flow pattern obtained by flow visualization. They were quantified by tracing particle and

plotting at a certain interval as illustrated in Fig. 23 (a). The velocity components along stream line close to the

center line of the flow field were obtained by measuring

the distance of particles which moved in that interval as

50 60 70 80 90m,,, 30 20 lt O lt 20

-u

rnr 10 20 30 20 o 10 20 30 ,,,n IO 20 30 20 10 10 20 30 40 Bulb

(b) I =O.70'/s wOO bulb

Fig. 24 Velocity vector obtained by image processing

shown inFig. 24.

4.2 Effect of underwater protruding bulb

The chief interest was concentrated on the

determina-tion of the free surface configuradetermina-tion. the flow

compo-nents and the vorticity to be calculated therefrom by

differentiation.

The flow field forward of the vertical cylinder with protruding bulb was analyzed by the same procedure as

above and illustrated in Fig. 25. It is found that the longi-tudinal velocity component increases largely beneath the

free surface. Magnitude of the circulation (f) indicating intensity of vortical motion was calculated as shown in

the following table.

r-348coIs

--)a) 1=0.70w/s w050ut bulb

30 40 50 60 70 80 90n,n,

r= 8,8crn/5

30 40 50

-This shows the significant decrease of intensity of vortical motion by fitting an underwater protruding bulb.

Thus, it can be concluded that the underwater protrud-ing bulb influences largely the increase of velocity

compo-nents beneath the free surface and this effect provides

improvement of flow and decrease of wave breaking around the bow of full ships.

However, the results obtained so far both at IfS in

Hamburg and at MHI Nagasaki cannot be considered

conclusive yet, some discrepancies between results from different methods have to be resolved. Further

investiga-tions are necessary for the quantitative evaluation of the free surface disturbance around bows with the aim of

mm 30 N 20 lo o E o

20

30

u/U 0 0.1 0.2 0.3 0.4 0.5

Stil) w'ater level

MTB 168 March 1985

14/ U

0 0.1 0.2 0.3 0.4 0.5

Still water level

(a) U=0.70ms without bulb (b) (=0.70m s with bulb

Fig. 25 Variation of horizontal velocity u/U in the vertical direction

providing material for a more rational mathematical model. 5. Conclusion

For better understanding of the hydrodynamic mecha-nism of the free surface phenomena around bow of ships, investigations were made into behaviour of the shear flow

and vortices induced in the upstream of the bow. In the

study, calculation of vorticity on the basis of the secondary flow theory, flow visualization and flow measurement by

means of image processing technique on the video recorded

were conducted for evaluation of the effect of the major

ship form parameters, such as draft, entrance angle of bow,

underwater protruding bulb and waterline shape on the

shear flow and vortices in it. The results are summarized as

follows.

(I) _{The free surface shear flow and vortices in it are}

closely related to wave breaking around bow.

Of the major parameters of ship form, draft,

under-water protruding bulb. entrance angle

of bow and

waterline shape influence largely the free surface shear

flow and vortices in it.

Underwater protruding bulb gives acceleration of

velocity components beneath the free surface and this effect provides improvement of flow, suchas

remarka-ble decrease of vortical motion, and decrease of wave

breaking around bow of full ships.

The secondary flow theory is considered to be a useful guidance for explanation of the hydrodynamic

mechanism of the free surface shear flow and vortices in it and seems to provide some helpful suggestions on

bow shape in the course of design of full ship form.

Measurement of flow beneath the free surface around a vertical circular cylinder by image processing of video recorded gave materials for quantitative evaluation of

the free surface shear flow and vortical motion in it.

Free surface phenomena around bow of full ships are

characterized by the free surface shear flow, vortices in

it and wave breaking phenomena. The hydrodynamic

mechanism of these phenomena has been clarified tosome

9

o 10

E _Et

d

I C i LO without bulb 34.8 0.70 with bulb

8.8

0.70

Case Circulation r (cm2 /s) Speed(m/s)

SOrnn, 60 70 80 mm 30

2'

o o w 10 E o

lo

o

20

3°

extent by investigations made up to the present, but there still remain various aspects to be studied further.

Acknowledgement

The author expresses sincere gratitude to Prof. K. Eggers,

Dr. S.D. Sharma and Dr. J. Kux, Institut für Schiffbau der

Universität Hamburg for their cooperation.

Reference

Baba E., A New Component of Viscous Resistance of Ships, Journal of the Society of Naval Architects of Japan Vol. 125 (1969). 23-34

Baba E. and Takekuma K., A Study on Free Surface Flow around Bow of Slowly Moving Full Forms, Journal of the

SocietyofNaval Architects of Japan Vol. 137 (1975) Baba E.. Some Free Surface Phenomena around Ships to be challenged by Numerical Analysis, Third International Conference on Numerical Ship Hydrodynamics, Paris (1981) Batchelor G.K., Introduction to Fluid Dynamics, Cambridge Univ. Press (1970)

Chan R.K.-C. and Chan F.W.-K., Numerical Solution of Transient and Steady Free Surface Flows about a Ship of

General Hull Shape, Proc. 13th Symp. Naval Hydrodynamics, Tokyo (1980). 257-280

Dagan G. and Tulin M.P., Bow Waves before Blunt Ships, Hydronautics, Inc., Technical Report 117-14 (1969)

Eckert E. and Sharma S.D.. Bugwülste für Langsame, Völlige Schiffe, Jahrb. Schiffbautech. Ges. 64 (1970). 129-171 Eggers K., Non-Kelvin Dispersive Waves around Non-Slender Ships, Schiffstechnik Vol. 28 (1981), 223-250

Gadd G.E.. A Method of Computing the Flow and Surface

Wave Pattern around Full Form, Trans. Roy. Inst. Naval

Arch. 188 (1976). 207-219

Hawthorne W.R.. Secondary Flow about Struts and Airfoils, J. Aeronautical Soc. Vol. 21(1954). 588-608

(il)

Inui T., From Bulbous Bow to Free Surface Shock Wave-Trends of 20 Years' Research on Ship Waves at the Tokyo University Tank. J. Ship Res. 25 (1981), 147-180

Kayo Y. and Takekuma K., On the Free Surface Shear Flow Related Bow Wave-Breaking of Full Ship Models. Journal of the Society of Naval Architects of Japan Vol. 149 (1981).

11-20

Kayo Y.. Takekuma K.. Eggers K. and Sharma S.D.,

Observa-tion of Free Surface Shear Flow and its RelaObserva-tion to Bow

Wave-Breaking on Full Forms. Inst. Schiffbau, Univ.

Hamburg Rept. 420 (1982)

Kayo Y. and Takekuma K.. Shear Layer and Secondary Vortical Flow Beneath Free Surface around Bow of

Full-Form Ship Model, Trans. West Japan Soc. Nay. Arch. No. 65 (1983). 17-25

Maruo H. and Fukazawa M.. On the Free Surface Flow

around a Two-Dimensional Body Fixed in a Uniform Stream, Proceedings of the 29th Japan National Congress for Applied Mechanics. 1979, University Tokyo Press (1981)

Maruo H.. On the Breaking of Waves at the Bow, Symposium

on New Developments of Naval Architecture and Ocean

Engineering, Shanghai (1983)

Miyata H. et al.. Numerical and Experimental Analysis of

Nonlinear . Bow and Stern Waves of a Two-Dimensional

Body. J. Soc. Nay. Arch. Japan Vol. 154 (1983)

Mori K.. A. Calculation of Wave Resistance and Sinkage by Rankine-Source Method. B. Prediction of 2-D Near Wake Flow by Making Use of Time Dependent Vorticity Trans-port Equation. C. Free Surface Boundary Layer and Necklace Vortex Formation, IIHR Report No. 262 (1983)

Prosperetti A. et al.. Small-Amplitude Waves Produced by a Submerged Vorticity Distribution on the Free Surface of a Viscous Liquid, Phys. of Fluids 25 (12), (1982), 2188-2192 Takekuma K., Study on the Non-linear Free Surface Problem around Bow. J. Soc. Naval Arch. Japan Vol. 132 (1972). 1-9

Taneda S. and Amamoto H.. On the Necklace Vortex,

Bulletin of Res. Inst. AppI. Mech., Kyushu Univ. No. 31,

Japanese (1969)

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Appendix A Summary of Hawthorne's theory

Hawthorne studied the flow around a strut placed in

the approach velocity varying in the depthwise direction.

As described in the previous sections, the flow around a

blunt bow with forward free surface shear is considered to belong to this type of flow.

The disturbance induced by the strut or bow of ship may be understood as due to the variation in stagnation

pressure, at the leading edge, which causes an upward velo-city and thus horseshoe vortex. For analytical expression of such a flow field, Hawthorne supposed that the flow can be expressed as a sum of the two-dimensional flow with

varying approaching velocity and the three-dimensional disturbance, namely, according to the notation shown in

Fig. 26, the velocity field is approximated by

U (z) ¡(x, y) + v(x, y, z) dU (dû (x, y, z) = 2q

jj

1 _q2 (A.3) ' (x, y, z) = O

where, , 77. are the perturbation vorticity components in

the direction of the streamline V(x,y), normal to it in the horizontal plane and upwards; q is I VI as mentioned above and û is the angle of this streamline against the x-axis.

One should observe that O will increase along the

stream-line along which it has to be integrated from zero far ahead to a maximum aside of the bow and then decrease to zero again. Hence the integrands for the integrals as displayed

in Fig. 9 are integrated and is related to the area

enclos-(Al)

(A.2)

(A.4)

Fig. 26 Coordinate system and shear layer en countering surface-piercing body

ed by these curves. Whereas far ahead of the bow we have

q 1. 0 = O (i.e. parallel flow), q must vanish at the

stagna-tion point, leading to infinite values of 17 there and a

divergent integral for along the bow waterline.

Hawthorne excluded this domain when formulating his

theory. By choosing stream lines passing the stagnation point

closely enough, we may obtain secondary vorticity of

whatever magnitude. Thus the absolute value of the second-ary vorticity is not representative for the magnitude of the global disturbance created. Hence for evaluating the merits of different bow geometries, the comparison was based on relative secondary vorticity components ₌ and = 17/d _{along two stream lines with the same origin far ahead}

of the different bows.

Appendix B Relation between surface curvature and shear

flow at a free surface

Hawthorne's theory was derived for the case of an uni-form linear vertical shear profile, with the shear layer along

a horizontal plane wall of unlimited extent ahead of the

bow. In such case, there is no need to impose a dynamical

boundary condition along this wall.

A shear layer generated by a ship bow can have only

finite extent; moreover, the dynamic boundary condition on the free surface can not be disregarded. Hence there is

the need for an improved analysis.

Consider a point po at a stationary free surface Sf and a

local Cartesian x-y-z coordinate system adjusted to this point such that the z-axis is directed normal to Sf in the coordinate origin po and the x-axis coincides with the

direction of the flow in Po which then must he tangential to Sf. Let Sf be described by z = Z(x, y) in the vicinity of P0 and let u, t, w stand for the velocity components in x-, y-, and z-direction (thus v= w = O in P0). Then the

require-ment that there should be no viscous force component

acting across Sf in Po can only be satisfied under the three

conditions.

uz+wx=O.

¿'z+wyO. wzO

(BI)

If we even had = O at P0 (i.e. flow were essentially two-dimensional), the continuity equation would require = O, that means the flow variation with x, y and z would

MTB 168 March 1985

be purely rotational like that of a fluid mass under angular velocity w (u7 Wx)/ 2 = = Wx = /2, where î is

the vorticity component along the y-axis which is the axis of rotation. Hence under dynamic equilibrium at Sf there

can not be shear in the z-direction only.

The kinematic boundary condition on Sf may be

ex-pressed as

G(x,y,z)0

(B.2)

with

a a a

G(x,y,z)u a+l'+w

[zZ(x,y)] (B.3)

Observing that through our conventions we have Z = = Zy = O and i w = O in P0, through differentiation along a line y = const on Sf we obtain

= + GfZx = W - U'Zxx at po (B.4)

i.e. /2= -Wx = _UZ = -uk, wherek isthe curvature of

Sf at Po in a cut with the xz plane (as Z, is zero at Po,

Zxx equals ka!). Batchelor (1970) proved this relation using tools of differential geometry, i.e. with a general curvilinear

coordinate system adjusted to Sf and the flow direction

within this surface. Note that the only assumption needed is stationarity of the flow and the free surface in an appro-priate system of reference. Note that there is no influence of any viscosity coefficient to this relation, though positive viscosity (however small) is assumed. One may observe

that the sense of vorticity corresponds to that of orbital motion in case of an infinitesimal irrotational wave

pro-gressing with speed u in the crest or in the trough.

According to above analysis, no stationary free surface shear flow can exist where the free surface does not display curvature. Expressed otherwise: On a plane free surface the vertical derivatives of all flow components must be zero. Thus Hawthorne's secondary flow model is not appropriate

for quantitative predictions, apart from being unable to take account of the influence of advance speed on flow geometry. The above more refined conditions are taken care of in a recent study by Prosperetti (1 982) together with a genuine influence of gravity, though for time

de-pendent flow only so far.