Pressure variation at cylinders of cross section similar to ship water lines

By Professor Sv. Aa. Harvald, D. Sc.

Calculation of pressure distribution in the potential field at cylinders of various cross section has been performed. The sections were of the shape and fullness similar to ship water lines. The calculations have been made for cylinders in in finite medium as well as for cylinders between two parallel planes. The influence of the distance between the planes has been examined. The sources and sinks method has been employed in the calculations ignoring the influence of friction.

Introduction.

At

navigation along a shore and at sailing on riverslaunchings where the ship slides along a quay, at

and canals a heavy suction may occur between ship and quay. In such cases the water depth will be greatly limited

and the flow will therefore be nearly two-dimensional.

By calculating the pressure distribution at cylinders with cross sections as water lines, it should be possible to form

an idea of the existing forces. Calculation of pressure

distribution in two-dimensional flow has previously been made, for instance by D. W. Taylor in "On Ship-Shaped Stream Forms" published in 1894 [1].

Fig. 1. Source-sink distribution.

629. 12.07 :632.58

Mathematical Basis.

The method used was first developed by Rankine [2)

and simplified later by D. W. Taylor [1).

If a source-sink fiéld is combined with a parallel field of velocity y0, the stream function will get the value

r=k1Sv0y(1)

S is determined by

S =

q 9 (2)

where O is the angle between the X-axis and the line

connecting the source (or sink) with the point where the

Fig. 2. S-y-diagrain, sources placed in i - 70, 80 and 90, sinks

placed in i 70, 80 and 90. S

I

Á1 70 100 -100 -70 0 A

J

fOjIji\'!./

/1KV

I::4VI;.

"!«'

41

j

/1/4

.'

./

/

, D

1,' / "

g

40 Y .1./NE6 L/NE4 o LIAZ LINEf SOURCES f00 SINKS -40 .LINE3 0 140 60 80 fOL .LINE5 s s s

Strtryk af Ingeniuiren. Volume 6. No. 3. 1962

Lab. y.

che::psbouwkunde

ARCHIEF

Pressure Variation at Cylinders

Technische HogschooI

F

\\ \\ \\\ \\\\'\\\\\\\\ \\\\\\ \\\\\\\\\\\\\\\\\\

B

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\-Fig. 3. Stream line pattern, distance netween wall aoci symmetric line equal to 30.

stream function is desired to be

found. q0 is a factor of the

rela-tive strength of the sources and sinks.

-As = O corresponds to the

stream line dividing the flow into

an inner and an outer flow, the

limit curve is determined by

O=k1S-v0y

(3) If the distance from the X-axis to the limit curve is chosen equal to B/2 the following is obtained

B/2

(4)

M

where SM is the value of S at the

point x = O, y = B/2.

For an arbitrary source-sink

di-stribution, S is determined in a series of points, and a

S-y-dia-gram is constructed. By means of lines parallel to the line (equation (3)) a stream line pattern can be made -òut.- The velociti parallel to the X-axis is determined by

d (y1 dS

_\

1)(5)

The velocity in a tube of flow is

vx

V = -

_{CO5 a} (6)

where a is the angle between the X axis and the be cf

flów.

By. means of Bernoulli's equation

pp0

vv2

₍₇ w 2g is obtained

/

/-4-i\2

W 21 IS1 dy

PTPo=iiVo\l\ cosa

The quantity p0 _{- p} is the. difference between the normal pressure and that where the velocity is y, w specific'weight

of the liquid, g acceleration due to gravity

(8)

The quantity P - Po is shown in the diagramLas the

function of x. p is the mass density and equal to

When sources and sinks .are.placed syrnmetrically abOut

the origin bn the X-axis, the flow round qlincers of

different sections can be determined. I

Flow around cylinders situated between two ptral1el

planes can be obtained by placing sources and sinks imilar i

pV O

Fig. 4. .S-y-diagram, sources placed in x = 70, 80 and 90, sinks placed in x 70, 80 and 90L

,-

-

__

o .20

ti

__

_-

__.,

80 Y L

1,7

//,,

41111

I

. D

\\

\

to those on the X-axis on an infinite number of equidistant lines parallel to the X-axis. The source-sink distribution must be the same on all lines. The -distance between the lines will be equal tO the distance of the planes between which the cylinders are placed.

Calculation Practice.

For cylinders in flows of infinite extension, S-y-diagrams from previous calculations t3J have been used. Streamline patterns have been constructed by means of S-y-diagrams

and have been used for the graphical determination of

cos a. Furthermore has been determined graphically on S-y-diagrams.

For cylinders between two planes, rather simple source-sink distributions have been used to limit the calculation

work. On each line, three sources and three sinks have been placed. The distance between the lines as well as the distance between the sources and sinks have been varied. On the other hand th same source-sink strength distribution has been used in all calculations. As the

distri-bution has been as shown in Fig. 1, the calculations can be considered as carried out either for a continual triangular strength distribution, where the summation is performed

according to Simpson's rule or for a distribution with

three sources with the strength 1, 2 and 1. För

determin-ing S as a function of x and y, a summation must be

performed. Components of S, derived from the sources

and sinks on the lines i and 2, 3 and 4,, 5 and 6, 7 and8,

9andlO

,

l3andl4

,

l9and2O and 29and30, have

been calculated. All the pairs of linee are placed

sym-metrically about the mid-line between lines 1 and 2. The

S components are calculated within the field between the Y-axis (x 0) and x = 120 and the X-axis (line1) and

the above-mentioned mid-line which will represent the bounding planes. The S-components for the remaining

line-pairs (11and 12 , 15 and 16 , 17and 18 , 21 and 22 23 and 24 , 25 and 26 , -27 and 28) have been determined graphically. Then a summation of the components for the

lines i and 2 to 29 and 30 has been performed adding

1/3X20 X the value of S for the uñes 29 and 30 as a cor-rection for the contribution outside the line 29 and 30. S-y diagrams have been constructed (example shown Fig. 2) and the pressure distribution has been determined as for the unlimited flow.

Owing to the symmetry round the X-axis, the S-values

for points on this axis (y

= 0) can be found without summation, as the contribution from the lines 2 and 3, 4 and 5, 6 and 7 etc. cancel each other out. By- turning

the S-y curves a little, they, have been made to pass through

these calculated points. The disagreement is of course

due to the fact that summation to ± infinity has riot

been perfectly correct. A stream-line pattern has been

shown in Fig. 3.

Calculati9ns have also been carried out for cylinders at

a certain distance from a single plane. Identical sources

and sinks have been placed on only two parallel lines, in other respects the process has been similar to that above.

An example of the S-y-diagram has been given (Fig. 4)

and a corresponding flow picture (Fig. 5). It will be

observed that the cylinders are no longer symmetric. By

altering the placing of the source-sink it is possible to

produce flows round a symmetrical body, but check

cal-culations showed that it is rather difficult and-extremely

laborious to find the correct distribution. -

-Pressure Distributions.

For the -cross sections -shown in Figs. 6 and 7, the pressure distribution along the surfaces of the cylinders is shown in Figs. 8 and 9. The length/breadth ratios for

the cylinders are:

A: L/B = 200/20 B: LIB = 200/30 C: L/B 200/40 D: L/B = 200/50

\\\\\\\\\'

B A

\\ \'

C

20 20 20 20 40 40 40 60 60 60 20 40 60

Fig. 6. \Vater lines for cylinders, Series I.

40 60

Fig. 7. Water lines for cylinders, Series H.

80 80 80 80 80 --100 100 100 100 .100

Fig. 8. Presseure distribution along cylinders (Series I) In a flow unlimited In extent.

Fig. 9. Pressure distributiòn along cylinders (Series Il) in a flow unlimited in extent.

P-Po

-IA

o 20 40 --05

--1.0 p-pa

ZB

0.5

-

_lr s Q 41 -1.0 p

-IC

4k

;

o2 o o 20 40

71W

-

-_

ilA

lpv2 0.5 o 20 40 100 -1.0 p,po

ilB

fpvj 0.5 0 20 40

-4

No.5

r'

-0. -1.0

BC

0.5 0 -1.0 20

_LW

40 100

P

ilD

0.5 o 20 40 60 700 -0.5

V,

--1.0

Fig. 10. Pressure distribution along cylinders between parallel walls. (fullforms).

The diagrams (Figs. 8 and 9) show the variatiob of the pressure when the liquid field is unlimited. The cdrves

have been drawn without regard to the coefficient dçpited

having the value i at the point of stagnation, as. the rea where the high pressure is prevalent is quite diminuive for these forms. The curves are drawn through the baI-Series

- . culätéd points without any cross-fairing, and the

irrgu-No

_{larities òf the curves are partly due to the metho}

of

summation partly due to errors resulting from the graphical differentiation prócedure used.

. .

On the diagrams (Fig. lo) the pressure distribution

is given fór cylinders with water line coefficients of

próx. 0.80 put between parallel planes, The distatces from the symmetric lines of the cylinders to the plaies

have been shown on the figure. Corresponding curves

forms with water line coefficients of approx. 0.66 hive been given in Fig.

11. For a = 0.55 calculations hve

only bèen made for the distance 30 between the wall .hd

lines of the cylinders (Fig. 12).

With the asymmetric flow, the pressure curves for tie

two sides are different. The curves calcalted are given in Fig. 13, crresponding to the stream line pattern in Fig.5,

Table 1. Water line coefficients for the cylinders, around which the flo is calculated. p-p0

B

1 -

!.

-20D D/ST-60 D/ST30 D/ST20 a..79b

a.826

a..874 p_pa

w

0 . 20 40 60

4

-.

Ø

TT:f

:''

-3.0

-

D/ST - a . 778

----D/ST-60

D/sT-30 DIST-20

a.785

a.811

a.832

PPo 20 40

D

60 80

L

0. 40.0 L -201.1 3Qß

----

= L781 -«w DIST = 60 D/ST3O a 812 aJ.9O1 20 40

C

60 ptçi . o -2.5

.-.--

/

/

/'

.. -5.0 /1/11 -7.5

----DIST

a=..777

--70.0

---.--

_---.-

_{DIST - 60, a .807} - D15T30 c,.=.859 0.837 0.831 0832 O31 3 .778 .774 .777 .781 4 .728 .719 .723 .729 5 .664 .664 .670 .677 6 .607 .612 .621 .634 7 .549 .557 .573 .585 8 .497 .510 .533 .551 9 .443 .471 .499 . .524 Io .427 .458 .489 . .515 -erzes No 0.777 0i76 0777 0784 3 .697 .697 .698 .706 the centre 4 .609 .615 .625 .637 5 .536 . .551 .568 .585

- Figures written on the curves are serial numbers. The

water line coefficients A

a =

L.D® for the cylinders are given in Table 1.

Conclusions.

Pressure distributions round cylinders in a flow of un-limited extension vary considerably with the lengthJbreadth

ratio and with the fullness of the waterline section. At the full forms over the greater part of the surface there will, be almost a constant under-pressure and for a distance of only about io % from the ends there will

be an over-pressure.

At the slender foniis an under-pressure will exist for about half the ..length 'amidships", the other half at -the

ends 'will have over-pressure. The under-pressure will be considerably higher than the over-pressure, numerically. With cylinders placed between planes the pressure

di-stribution will be altered essentially and the smaller the

distance is between these planes the greater the alteration will be.

Fig. 11. Pressure distribution along cylinders between parallel walls (medium forms).

Fig. 12. Pressure distribution along cylinders between parallel walls

(slender forms). P-P0 20 40

A

60 o

-.--.---.-.---1.0

-/

-20 3Q

DfST ¿x-664

-0

-

i2(ST-60D/ST.30 D/5T- 20 a-.610 a=.635 a - . 659

B

--inc DIST ¿x =.664 20.0 D/ST=60 D/ST-30 DISt- 20 cc'.623 a..660 a = .724 20 40 60

C

_80;L1-..__

f 0 -2.5

-.,-..--,.-..--.,-.,-,,-..-,.-..--.--

-,/'

-.--

--/

---5.0 -7.5 .

'1"

--

D/ST m a

-tao - . D/s7=60 a=.639 D/ST-iO a.- .702 P - Po

D

0 20 40 60

_0.--,04_

-10.0.. -20.0

/

/

-30.0 ...

/'

--

P1ST a .677 D/ST=60 cc=.655 - D/$T- 30 a - .777 -40.0 -f yPV0 2 0 20 40 60

--ç--A

D -30.0 _A

---'A--.475 D/ST-30

- .521 570 -400 a0..626

If the cylinders ar p1acd near a single plane there will, be a great difference between the distribution' curves for

the side turning to the surface and the side to the free. The differences and thus the forces on the hqdy wUl bc

so great tha± it is impossible by sinple precutiorÏs'to alter this condition. At a launching where the ship passes closely

by a quay the suction between ship and quay cannqt for

this reason be balanced only by adjusting the rudder. To prevent the ship from leaving its track, essential alterations to the slip and qi.zay are neçcs.ry, This can be done either by creating the highest grade of symmetry or by providing

P, PIANSENS ßoGrPYKKEPI

ALONG 5/DE FACING WALL

ALONG S/be AWAY FROM WALL

DISTANtE 3Q

Fig. 13. Pressurè distribution along asymmetrical cylinders near a single wall.

tÌie.. ship and the quay with steering surfaces allowing the

ship to slide against the quay, in this way absorbing the

suction pressur.

References.

L

Taykr, D. W., »On Ship-Shaped Stream Forms«, TINA

1894, P. 385..

Rankine W J M »On Plane Water Lines in Two Di

rnensions«, Philosophical Transactions 1864, P. 369: Harvald, Svend Aàge, »Wake of Merchant Ships«,n'Co-penhagen 1950, p. 24. - P - Po

B

0 20 40 60 80 /1<- 100 -20 .829 1JL767

-A

o 20 40 60

/

100 -20 . a,-.814 .754

C

o 20 40 60 / 80 _/ 100 -Lo -2.0 - -cc.868 -781'