A circular cylinder in water waves

A circular cylinder in water waves EVEN MEHLUM Sentralinstitutt For Industriell Forskning, Forskningsv. 1, P.B. 350 Blindern, Oslo 3, Norway

(Received 28 November 1979) This paper treats the problem in linearized water wave theory that arises when a wave travels across a submerged cylinder. This problem is classical and was first treated by Dean. The solution presented is novel and explicit. This is achieved through the use of certain recursive relations. The basic results are of course the same as in the older literature: (1) the coefficient of reflection is zero; (2) the only effect of the cylinder, is a phase shift relative to the undisturbed wave. However, the practical computation of the velocity potential and thephase shift is reduced almost to hand calculations. This is in contrast to the older literature which requires, in principle, the inversion of an infinite matrix. The work is motivated by the Institue's efforts to focus water waves. Cylinders are possible bodies to create the necessary phase shifts to achieve such focusing.

THE PROBLEM We shall study the following boundary value problem in linearized water wave theory. A circular cylinder of infinite length is completely submerged at a fixed position in the fluid. There is a simple harmonic wave travelling from the left and is modified upon passing the cylinder. The wave front is parallel to the axis of the cylinder (see Fig. 1). We seek stationary wave solutions. The problem is to obtain an expression for the velocity potential. It is clear that under the stated conditions the velocity potential will everywhere by harmonic in time so that the complex potential can be written: ~(Xo, yo)e -'~'t

(1)

where tI) = an analytic funtion of z o = x o + iy o everywhere in the fluid, e~=angular frquency, t--time, x o, Y0 = coordinates in the liquid with the origin located at the centre of the cylinder so that the surface is located by Yo = h. With a being the radius of the cylinder we have h > a. With % being the wave number and g the acceleration of gravity we have s 0 =0)2/,(].

Our problem is now reduced to the followingS: (I) satisfies the Laplace equation everywhere in the liquid. At the free surface we shall require 8~ --=~(~ 8y

(2)

On the cylinder we must have: 8~ --=0

(3)

8n

where n = surface normal. Under the assumptions given above, the principle of superposition is valid for the velocity potential. If, therefore, the incoming wave contains a frequency spectrum, we can find the complete solution by adding together simple harmonic solutions as obtained in this paper. We shall work with the complex coordinate (4)

z = x + iy

We shall also need the quantity Ro:

i = J - - ~ = e i~/ 2

0~
(5)

We can now scale the coordinates so that the dimensionless cylinder radius is unity. X--

XO

'Yo

a surface

y-

Y0 a

d

h>l

Yo ~ h

I-X 0

a

Figure 1

0141-I 187/80/040171~37$2.00 © 1980CML Publications

Applied Ocean Research, 1980, Vol. 2, No. 4

171

A circular cylinder in water waves." E. Mehlum g

At p = R o equations (10) and (11) with equation (7) gives: /oi, =o~ Oy

v2~ : o

2R0Z(1 - sin0)

G

1

Ro 1

J

Ro.:_2 C..,,(R;-

Ro")ei"°

d~

(]2)

The identity sign in equation (12) is used to signify that it shall be fulfilled for all 0. We must therefore match coefficients in front of all terms e i"°. Remembering that ei°-

sinO .~

Figure 2 THE

-iO

e

2i -

(13)

this match leads to the following system of recursive relations:

SOLUTION

We introduce the new coordinates: w = u + i v = p e ~° by means of a conformal, bilinear mapping: w -

at p = R o

(7)

at p = l

(8)

In the annular region we have: V2~ = 0

(9)

Finally, our solution must be bounded at: z = or, w = iR o. With w* being the complex conjugate of w, every function w" + w*-" =(p" + p - " ) e ~"° clearly satisfies equa-

tion (8). n = integer because the solution must have period 27t in 0. This is so because the solution must be continuous everywhere inside the annular region (Fig. 3). This argument also applies along the line C-(E, G, H). The point E, G, H is, however, exceptional since all points at infinity are mapped into that point. Inside the annular region we can now write for ¢: =

~ pl =

(14)

2fl = ~(R o ' - Ro)

v

C,(w"+w*-")

C

,~

~

C , ( p " + p - " ) C "°

(10)

n = --1

Assuming convergence we can write: ap - p , Y ~

C"'n(p"-p-")ei"°

(11) Figure 3

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Applied Ocean Research, 1980, Vol. 2, No. 4

(15)

Once two C,'s, C 1 and C ~ say, are given values equation (14) makes possible an explicit evaluation of all coefficients C, in the expansion (10) for the velocity potential. We are thus formally in the position to evaluate the potential at any point x, y in the fluid without solving a system of linear equations. The series (10) will, however, show bad convergence for points x, y far away from the cylinder. This is clear from the following consideration: when z--*oo, w ~ i R o. (Point (E, G, H) in Fig. 3.) At Ixl = ~(E, G) we must expect a sinusoidal solution oscillating between limits depending on the depth. At x =0, y = - i ~ ( p o i n t H) we must expect the potential to vanish. Since all this is going to take place at the point w = iRo, the series (10) cannot converge to a definite value at that point. The series (10) is thus not adequate for calculations of asymptotic behaviour. We will therefore work out an equivalent solution which gives direct information about the asymptotic behaviour as Ixl-,m. At the same time this 'practical solution' shows far better convergence than the series (10). Other questions on convergence are postponed until this 'practical solution" is achieved.

B

=

all n

We have defined the constant fl by:

The condition at the cylinder surface is: =0

1}Cn+ 1 q'- 2i[n( R"o - R o") + fl( R~o +

(6)

The fluid is now contained in the annular region, shown in Fig. 3. The cylinder surface is the circle Iwl = 1, while the surface of the fluid is the circle Iwl = R0. From the mapping function (6) it is easy to show that the free surface condition (2) transforms into:

8p

1)(R~ +1 - Ro n -

R o ' ) ] C . - ( n - 1 ) ( R " o 1 - Ron+ I)C n 1 = 0 ,

i - Roz R o + iz

2Rg(1 - sin0) dqb Roz - 1 ~?p = ~0

(t/+

A circular cylinder in water waves: E. Mehlum However, before doing so, we will make one observation and thereby reduce the forthcoming complications somewhat without loss of generality. From the recursion relation (14) we deduce the following relations:

Co_C1 - C - 1

THE PRACTICAL S O L U T I O N We proceed in a somewhat mystifying manner by defining two functions of the variable

F,(0= ~ C.R"o("

(16)

(22)

n=l

i~z

C_,=(- 1)"+1~C,

Introducing these relations and restoring the coordinates x, y in the expression for the potential (10) we get:

~=C1-C-1 i~

F2(0= ~. C.Ro"~"

(17)

n~>l

F ~. C,(w"+w*-")-

Our first goal is to find a relationship between El(() and F2(0. To this end we multiply equation (21) by (" and form the sum from 1 to infinity. We get:

~ (n+l)C.+lR~o+l( "- ~ (n+l)C.+lRo~"+1)("+

.= 1

n=l

C_ CI

1

L

(-

1)"C,(w*"

n=l

+ w-")

n=l

2i ~ nC,R~o(" - 2i ~ n=l

-

C1 - C - 1

&

(23)

n=l

nC.Ro"~"+

.=1

F/i-Rox-Roiy~"

+._-z.,ql

) +

2ifl ~ C.R"o("+ 2ifl ~ C.Ro"f"-

C_,C,.~,C"L[ ~ F/'i+R°x-:-R°iy'X"~)+_(

-i+R°x+R°iy~-"~~ y

n=l

n=l

'- i-- Rox + R oiy Ro_ix_y )-"]-

• (n-1)C._lR~o-l("+ ~, (n-1)C._lRo("-l)("=O ]

n=2

n=2

(24) (18) The second sum in equation (18) is seen to be equal to the first except for the sign of the variable x. Bearing in mind that equation (18) is to be multiplied with e-/~" it is therefore clear that the two sums represent two equal disturbances travelling in opposite directions. The relative amplitude and phase of the two contributions results from a specification of the complex constant - C_ 1/C~. The remaining freedom to specify C_ ~ and C~ is a result of the fact that we have not given any specification of the incident wave, except that it shall be harmonic in time. The difference in phase and amplitude between the two contributions must not be mixed up with the phase shift and possible reflection within each contribution. We are now in the position that we can give values to C~ and C_ ~ without loss of generality. We chose C1 = ict and C_ ~=0. The equations we will have to work from in the rest of the develpment will therefore be;

O=1+ ~C.(w"+w*-")=l+ ZC.(p"+p-")e i"° n=l

(19)

.=1

(1 + i 0 2 d ~ 0 =(1 + i ( ) 2 ~

2iflF2( Q-ifl

(25)

+ 2iflF,(Q + ifl

This is the desired relationship between Fx(0 and F2(~). Regarding FI(0 (22) as a known function (25) is a differential equation for F2( 0. This differential equation is solved with comparative ease:

F2(()=e(2i#om +io_ 1 + ~, C.R~o(("+ J.(O) (26) n=l

with

jn=4ifle-2#/(l +iOt ~ 2 # / ( l JU + u~ 0

C I = i~

+it)dt

(27)

(20)

(n+ 1){R~ +1 -Ro"-I)C.+, +2i[n(R~-Ro")+ fl(R~ + R o " ) ] C " -

(n-IXR~-I-Ro"+I)C,_I=O

Adjusting indices and remembering equation (20) we are led directly to:

n>l

(21)

This choice of C_ 1 and C~ implies an indirect specification of the incoming wave.

The integration path is not to encircle the point t = i. It is observed that F2(0)= 0 from equation (26) as it should according to equation (23). Through partial integration the integrals J, are shown to be connected recurrently starting from the two integrals Jo and J1. We have:

J. +~= - 2in~fl2{" + J.) + 2iJ. + J._ 1

n > 1 (28)

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173

A circular cylinder in water waves: E. Mehlum J0 =2( e<2i#~}/H+i~}__1)

{29)

J 1: 4flJ + iJo

The asymptotic behaviour of Laguerre polynomials as n __.307 gives strong reason to believe that P, asymptotically will behave like:

with:

P,=,,-3/4{A cos~/8/Jn + B sin x/8~n}

17---4 30

(36)

J = e - Za/{l+i;}{Trsign( I m ~

) - i[ Ei(2fl) + Ex (-12~fl~ ) ]} {3o)

Here all difficulties are collected in the expression for J(30). We have used the standard definitions of the exponential integrals Ei and E~ as given in ref. 6. The function E~ as defined in that reference has a branch cut along the negative real axis. In our application of exponential integrals we must have: J = 0 for ~ = 0. As will become clear later we must also require that J behaves analytically when the argument of E 1 crosses the negative real axis. The apparent discontinuity introduced by the signum function in equation (30) is just what is needed to meet these two requirements. For the practical computation of exponential integrals there exist tables and rapidly converging Taylor and asymptotic expansions 6. The rest of the computation of F 2 is straightforward algebra if the series in equation (26) can be shown to have good convergence. The result (26) will now be applied to kill the bad convergence in equation (19). Defining w*

(31)

we get from equation (19):

In equation (36) A and B are constants of finite magnitude. The correctness of this guess is easily verified by direct substitution in equation (34) while maintaining only the most significant terms in a Taylor expansion in n-1/4 Thus we have:

IC.I =0(n 3/4R~}

n=l = e(21#¢}/{1 + i¢} -t- ~ C n ( w n "4- R"o~" + n=l

R"oJ.(~))

(32)

Equation (32) is the 'practical solution'.

(37)

From the definitions of the various quantities it follows: IRol < 1

(38)

Ro ~
(39)

Ro ~< I~'1~< 1

(40)

J.=i"'n.Q,

(41)

With the definition

the recursion relation (28} tends to:

(n+ l}Q,+a-(2n-2fl)Qn+(n-1}Q,_a=o

n~30 (42)

This is the same recursion relation as the one describing the asymptotic behaviour of the P.'s (34). Q, will therefore behave the same way as P, asymptotically. Therefore IJ,I = 0(nl/4)

= 1 + ~ C.w"-4-F2(~ )

n ---~.oo

n --* 30

(43)

We can now state that for all points x, y in the liquid including the boundaries the series expansion in the practical solution (32) will converge better than a geometrical series with factor R o. For practical values of R o this is quite good. The 'worst' case arises when Iwl = 1 when we have to sum the series

BOOK-KEEPING

~C n.

n=l

We now check the convergence of the series in our result (32) and check the end result against the original problem. We start with investigating what happens to C, as n tends to infinity. We define

Since we can now differentiate term by term, let us check the 'practical solution' against the original problem. First, let us verify that equation (32) defines a function with harmonic real and imaginary parts everywhere in the fluid including the boundaries. It is clear that w and ~ can give trouble only for:

Since 0~
rt---~ 30

(34) One solution of this difference equation in n is the Laguerre polynomial:

P. = L / ' (2/~)

174

Applied Ocean Research, 1980, Vol. 2, No. 4

(35}

and

R o + iz = 0

(44)

i + Roz* = 0

(45)

Equations (44) and (45) both define points outside the fluid Secondly, we have the term: e Cz~;~/~x+~;} occurring in equation (32) and in the definition of Jo (29). From the definition of ( and w (31, 6) we have e(2 i//~}/{1 +i~)~c-~Ro+i~r+Ty

(46}

A circular cylinder in water waves: E. Mehlum Equation (46) is evidently aliright. The remaining contribution to equation (32) that needs checking is J (30). We have

J=e-t'/Ro'+"~+~Y[nsign x-i(Ei(2fl)+E~(-L+cty+ictx))] (47) Here it is clear from the definition ofR 0 that the argument of E~ is always in the left half plane. For E~ the following relationship holds6: limE~( - a + ie) = - Ei(a) + in ~o

--

--

a>0 8>0

(48)

Iw*l/

It follows directly that J is continuous across x = Q. From the definition of E 1 it is also clear that all the derivatives 8"J/Ox" are continuous. The signum function in equation (47) is necessary to counteract the discontinuity in the standard definition of E~. In the practical use of equations (47)(and 32) one must of course be consistent in the choice of the sign for x = 0. It is now clear that equation (32) defines a function with harmonic real and imaginary parts everywhere in the fluid. Next let us check the free surface condition, i.e: --8y

~ = 0 for y =d, ( = e i°, w = Ro el°

Introducing equations (50)-(54) above we find that equation (55) vanish term by term for all points at the surface, where we have w = Ro ei° and ~ = e i°. Thus the free surface condition is verified for the 'practical solution'. Finally, we must verify that the boundary condition at the surface of the cylinder is fulfilled. To this end it is sufficient to demonstrate the equivalence between equation(19) and (32) close to the cylinder, and to demonstrate that equation (19) has the necessary convergence to make the very comparison valid. Let us look into the convergence question first. In virtue of equation (37) it is evident that equation (19) converges better than a geometrical series with factor

(49)

The first term in equation (32) fulfils equation (49) trivially. Secondly, we have

The necessary convergence is therefore assured whenever Iwl>R0. Within the liquid and at the cylinder surface this is trivial since there we have R o < Iwl ~<1. As for the equivalence part of the proof it is sufficient to demonstrate that F2(() (26) from the practical solution is identical to

LCn W•-n

n=l

in equation (19). By definition ( = Ro/w* so that we have to prove the following identity: e(2,ao/(, +,o_ 1 + ~ C . R ~ ( f " + J . ( ( ) ) - L C.Ro"(" (56) n=l

0w Oy

i(1 - R 2) (R o d-iz) 2

~3( iRo(1-R~) c3y (i+Roz*) 2

(50)

(51)

and dJ. 2ifl -- (1 + i() 2(Jn + 2(n) d~-

(52)

Equations (50)-(52) follow directly from the definition of the quantities involved. By trivial algebraic calculations we also have (from the definitions of w and (): 1 -

R 2

(i+Roz*) -

(1 + i ( ) 2

1-R 2

(53)

1- R 2 (R 0 + iw) 2 (R o+iz) 2 - 1 - R 2

(54)

n=l

This identity can be controlled from the very construction of F2((). However, since decent book-keeping should be doublechecking, let us do something else. Since the left and right side of equation (56) both vanish at ( = 0 and since e 2#/(1 +io does not vanish, we have to show:

d(Ie2fl/(l + i~)(e(21fl°/(l + i° - l -~-n = l~Cn(R~ - R°n)(n ~-

~ I C , R~J,

-0

(57)

n=

Performing the necessary algebra inside the brackets followed by the differentiation and readjustment of indices we are led directly to the vanishing of the left hand side of equation (57) when we bring in the recurrent relations defining the sequence C, (equations 20 and 21). This completes the proof. We have at the same time shown that the two solutions are equivalent whenever R o <[wl~l.

From equations (32) and (46) we have: By

,=,

\

~ f + nR ° ¢ - ~y + K ° d ~ - ~ ) -

L C.(w"+ R~"+ R~J.(~)) n=l

(55)

CONCLUSION Asymptotic behaviour For convenience the results are summarized below. In terms of cartesian coordinates scaled to cylinder radius = 1 we have (Fig. 2)

Applied Ocean Research, 1980, Vol. 2, No. 4

175

A circular cylinder in water waves: E. Mehlum Equation (70) becomes:

-- wave n u m b e r

z = x + iy

(58)

i - Roz R o + iz

w .....

(59)

R R ° - iz*

= _

S.+ 1 = 2 i ( 1 - f l n ) S . + S . _ ,

S0=2

(60)

S 1 = 4fie zP(rtsign x - iEi(2fl)) + 2i

d = depth to centre of cylinder > 1 O u r expression for the space dependent part of the potential (32) is:

(61)

C.(w"+R"o("+R"oJ.(C))

(73)

F r o m equation (61) we now get the following asymptotic expression as X----~

?1=1

(72)

with

oi + R o z ,

~=e-=ro+'=~+=-'+~

n ~> 1

OO."

(I) = e - ~Ro+ i'x + ~Y+ ~'~ C.[(iRo)"+R"oi"+ n=l

with

0~, ~

R o = d - x/d ~ - 1

(62)

2fl = ~(R o i _ Ro )

(63)

Jo = 2( e-~R°+~'~+'~'- 1)

(64)

"

~n

(74)

which reduces readily to: ¢ = ei~e~(Y-g°(1 +~=1C.S.R"o)

(75)

The factor in paranthesis deserves a new symbol.

qJ = 1 + ~ C,S.R"o J=e-'/ro<~+'Ylrtsign x - i ( E i ( 2 f l ) + E ~ ( - R ~ o +Cty+i~tx)) ]

(76)

n=l

Equation (75) now becomes: (65)" = qJ" ei'Xe'(Y- g°/ (66)

J1 = 4flJ + iJ o

J n + l ~-

-2in-~2(n + J,)+ 2iJ. + J,-1

n~> 1

C l = is

(67)

(68)

(n + 1)(R~ + 1 - R o " - ~)C. + ~ = - 2i[n(R"o - R o") +

fl(R"o+Ro")]C,+(n-lXR"o-l-Ro"+~)C,_x

n>~l (69),

Next let us look into the asymptotic behaviour of the potential: Let I x [ ~ ,

Jn+l

= -

(2i"+J.)+2iJ.+J._~

(70)

Defining J , = ei'x +'y-'RoS. - 2i"

176

It is clear that qJ is a complex constant apart from an abrupt change as x changes from negative to positive values. This j u m p is caused by the signum function in equation (73). It is therefore clear from equation (77) that we have obtained the expected sinusoidal wave with exponential d a m p i n g with depth. We observe ttiat the indirect specification we did of the incoming wave at the end of the solution section has led to a wave travelling to the right. The j u m p in the asymptotic expression (77) is filled in by the evanescent wave so that the complete solution (61) is continuous across x = 0. As regards phase shift and reflection we must look m o r e closely into the quantities entering into the constant ~. F r o m equations (68) and (69) it is immediately clear that C, is purely imaginary when n is odd and purely real when n is even. i.e.:

~ ~i, w ~ i R o

First, we observe that for large modulus of the argument, the function E~ vanishes. The recursive relation (67) becomes:

Applied Ocean Research, 1980, Vol. 2, No. 4

(71)

(77)

C, = i"?,

(78)

S, = i"(a. + i~. sign x)

(79)

7, = purely real. It is also clear that

where a. and ~. are real quantities not depending on x. Equation (79) is verified by direct substitution in the defining equations for the sequence S. (72 and 73). Taking real and imaginary parts, this leads to recursive relations

A circular cylinder in water waves: E. Mehlum

2O3

160

f

~ :ogs

~ O

O -a

,/

~,

/

oorol j / 0 85 0 90

0.05

0"06

400

'

'

o

0-07 045O.50 0 " ~

O.lf 0.12

80

0.13

200

8:IS'oo, 40

"~-90"18 IO0

0,

0.2

04

a/~,

O~

0-8

1.0

,

0

02

Figure 4

04

a/~

i 06

08

i

Fiyure 5 for the sequences a, and ¢,, as well as starting values to trigger off the recursion. F o r ~O we now get: ~b=(1 +~=l(i"7,).(i'(tr,+i~,sign x))" R"o) which reduces to: ~O= 1 + ~ ( - Ro)"7.tr" + i sign x n=l

( - Ro)"7,¢ . (80) n=l

It is seen that the only change in ~ as x changes from to + oo is a change in sign of the imaginary part. Therefore, under the assumptions given earlier, (A) the cylinder does not reflect energy; (B) asymptotically, the only change d o n e to the wave, is a phase shift. These results are not new, and were first given by D e a n 1. The wave disturbed by the submerged cylinder will lag behind a corresponding undisturbed wave by an a m o u n t given by the phaseshift angle A~:

line

A ~ = 2 arctg~z~_, r~etg

(81)

We have achieved an explicit solution to o u r problem,

making it possible to c o m p u t e the velocity potential, and thereby the particle velocities anywhere in the fluid. The considerations earlier showed that the series in equation (61) converges better than a geometrical series. It is therefore safe to truncate the series in practical applications. It turns out that less than 10 terms are needed to achieve pr. mille accuracy in practice. Figures 4 and 5 show the phase shift as a function of the radius and depth of the cylinder. The calculations were performed using equations (80) and (81).

REFERENCES Dean, W. R. On the reflexion of surface waves by a submerged circular cylinder, Proc. Camb. Phil. Soc. 1948, p. 483 Ursell, F. Surface waves on deep water in the presence of a submerged circular cylinder, Proc. Camb. Phil. Soc. 1950, p. 141 Ogilvie, T. F. First- and second-order forces on a cylinder submerged under a free surface, J. Fluid Mech. 1963, p. 451 Longuet-Higgins, M. S. The mean forces exerted by waves on floating or submerged bodies with application to sand bars and wave power machines, Proc. R. Soc. (A) 1977, 352, 463 Whitman, G. B. linear and nonlinear waves, John Wiley, New York, 1974 Abramowitz, M. and Stegun, A. Handbook of Mathematical Functions National Bureau of Standards, Washington DC, 1970 Gradshteyn, I. S. and Ryzhik, I. M. Table of integrals, series and products Academic Press, New York, 1965

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