A circular cylinder in water waves

A circular cylinder in water waves

EVEN MEHLUM

Sentralinstiturt For Industriell Forskning, Forskiiingsv. 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 the

phase 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

poten-tial will everywhere by harmonic in time so that the

complex potential can be written:

(1)(x0,Yo)e (I)

where (1> =an analytic funtion of zo=x0 +iyo everywhere in

the fluid,

co =angular frquency, t = time, xo, yo =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 xo being the wave number and g the acceleration of gravity we have ao =co'lg.

= 1 = ehr 2

We can now scale the coordinates so that the

dimen-sionless cylinder radius is unity.

I' =.-1.0

d=

Ii> 1 a

Our problem is now reduced to the following':

(I) satisfies the Laplace equation everywhere in the

liquid. At the free surface we shall require

On the cylinder we must have:

EC,

- =

CO

where 11= surface normal.

Under the assumptions given above, the principle of

superposition is valid for the velocity potential. If, there-fore, 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

z=x+iy

(4)

We shall also need the quantity Ro:

0...5,R0=d,1/412 <1 ₍₅₎ TECHNISCHE UNIVERSITET Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD Delft Tel.: 015 - 786873 - Fax: 015- 781836 (3) zumegampoixixamingpar,:cr =2olt _{Eigtre 1} 0141 1187 SO 040171 07S2.00

0 I 90) (-NIL Publications Applied Ocean Research. I 980. Vol. 1 N,. 4 171

xo

=

(1 E(1) C.17

=

(2)

A &ire-ir1-1r ,,cylinder in water wares:' E. Meliloin

_

E

THE SOLUTION

We introduce the new coordinates:

w =ii + = pew by means of a conformal,

bilinear mapping: L.R0:

w-Ro+i:

(6) 00:1)

ap=0

at p= 1

In the annular region we have:

v2;:p=0

Finally, out; solution must be- bounded at: z= sc,

= iRo. With w* being the complex conjugate of w, every function w"+ w (p"-F p -")e'" -clearly satisfies equa-tion (8). n= integer because the soluequa-tion must have period 27r 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 (1):

,n= -I

=

C(p"+ p 7 le"

Assuming- cdnyergence we al.i,w.r-itt!..

(11) ; .

--

(,

- n)e- ha?

p

at p=Ro

(7)

172 Applied Ocean Research, 1980. rot. 2, .Vo. 4.

(10)

At P = Ra equations (10) and (11) with "equation (7) Ltives: 2R0(l - sini)) '

R8 C n(R"07, R")(:!""

[

+. , C,;(R7)---F R, ")einu .

]

n=

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`nu.

Remembering that

i0

-Figure 2

sin -

(13)

(12)

this match leads. to the following system of recursive relaticins:

1)(R;-1- -n- 1

_R)+13(R;+

Re)]C-(n-1)(Rr

RcT"' ')C;,_ =0: all n (14)

We have defined the constant 13 by:

213=2(Rc71 -R0) (15)

Once two C.-s, C, and C_ , say, are given values equation

(14) makes possible,. an explicit evaluation of all coef-ficients in the expansion (10) kir the velocity potential.

We are thus formally in the position to evaluate the potential at any point 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 oc, w-oiRo. (Point (E, a H) in Fig. 3.)

At :c(E, G) we must expect a sinusoidal solution

oscillating between limits depending on the depth. At x

=0, y= -ix(point H) we must expect the potential to

variish. 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 'xi-. x..At the same tinie this 'practical

solution shows far better convergence than the series (10).

Other questions on convergence 'are postponed until this 'practical solution' is achieved.

The fluid is now contained in the annular region, shown in Fig. 3. The cylinder surface_ is the circle Itvl =1, while the surface of the fluid is the circle 1w1 =Ro.

From the mapping function (6) it is easy to show that the free surface condition (2) transforms into:

2R(1 sin0) -e(l)

Rg- 1 = 2

However, before doing so, we will make one

obser-vation and thereby reduce the forthcoming complications somewhat without loss of generality.

From the recursion relation (14) we deduce the follow-ing relations:

Coci

11

c

_

C_,,=(

_{on 1---1C}

Introducing these relations and restoring the coordinates

x, y in the expression for the potential (10) _{we get:}

c

_1 to:.

+ L C(w" +w*-")

_n=1 C

L

IrC(w*" +w-^)

n=1

iRoxRoiy

2_, C ix

+

,

"ft

Ro+ ix y

( i Rox+ Roiy)-"]_

R, ix

y C71r.

E

'

c

n

[(i+RoxRoiyy (i+Rox+Roiy1

R0 ix

y

Ro+ixy

)

(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 -^i" 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/C1. 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 C, =ix and}

C_ =0. The equations we will have_{to work from in the}

rest of the develpment will therefore be:

0=1+

_{c(wn+ w*)= 1+ E C(pn+p-")ein9}

₍₁₉₎

n=1 n= I

This choice of C_, and C, implies

_{in indirect} specifi-cation of the incoming wave.

r*rofecifdr,faigikeififiaifitlY3f tZ; n 1

A circular cylinder in water wat-es: E. _Mehlum

THE PRACTICAL SOLUTION

We proceed in a somewhat mystifying manner by defining two functions of the variable

Our first goal is to find a relationship between FI(ç) and F2(). To this end we multiply equation_{(21) by i;" and form} the sum from 1 to infinity. We get:

7 . (n + 1)C C"

Eold-oc,,RV",)r,n+

n= 1 _n=1 F1(;)= E CRW:" ₍₂₂₎ n=

F2()= E C.R(;nl,"

n = I

2i E nCR;C"-2i

ItCRo-"C"+ n=I _n=1 7. 2i13 E CRN"+2if3 E CRcT"Cn-n=1 n=1 E(n-1)C_,R72,-ii;"+ E (n-1)C_ IR0-1"-1}C"= 0 n=2 n= (24)

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

. ,dF2(C)

(1 +

di: 20F2(C)

, _2dF,(C)

= (1 + ic) dc +2if3F1(C)+0 (25)

This is the desired relationship between F1(ç) and F2(0.

Regarding F1(C) (22) as a known function (25) is a

differential equation for F2().

This differential equation is solved with comparative ease: F 2(C)= e12lig)'"

_{1 + E CR(C" + J))}

₍₂₆₎ n=1 with ;

f

J=4if3e- 2" + ic) tn ..2111(1 -I- iodt

(1 +i()2' (27)

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 are shown to be connected recurrently starting from the two integrals Jo and J,.-We have:

Applied Ocean Research, 1980, Vol. 2, No. 4 173 (23) C1= ix (20) (n+ 1)(R

_{R6-"-')C+2i[n(R'01 R")+}

13(R+ RcT")]C

(n 1)(R'' R)C_

, =0

17 1 (21) =

_{(17-+J1+2iJ+J_,}

_{n1 (28)}

A circularcylinder in water waves: E. Mehhim

j 0 = 1(e(2i11;1. + i;)_ )

J1=40 +if,

with:

2fl

.j=e 211"+

iizsign(1

_{i[Ei(213)+E1(}

2/3

1+L. ±

(30)

Here all difficulties are collected in the expression for J(30).

We have used the standard definitions of the exponen-tial 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 C=0. As will become clear later we must also require that J behaves analytically when the argu-ment of E, crosses the negative real axis. The apparent

discontinuity introduced by the signum function in

equa-tion (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'. The rest of the computation of

F2 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

Ro w*

we get from equation (19):

D= l+ E cne+F2(c)

n= 1

=e(2inogi +iC)±

E

_{n(tt.n+Kgn+ ((:))}

= 1

Equation (32) is the 'practical solution'.

BOOK-KEEPING

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

-=( ) P

R

n

Since 0 Ro< I. the recursion relation (21) tends to:

One solution of this difference equation in n

is the Laet4rre polynomial:

174 ,Applied Ocean Research, 1980. I 2. No. 4

(29) The asymptotic behaviour of Laeuerre polynomials as 11

7_ gives strong reason to believe that

asymptoti-cally will behave like:

This is the same recursion relation as the one describing the asymptotic behaviour of the Pn's (34).

Q will therefore behave the same way as P,, asymptoti-cally. Therefore

1./.1=0(n1/4) 7: (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 Ro. For practical values of Ro this is quite good. The 'worst' case arises when ht.' =1 when we have to sum the series

E

C n=1

(33)

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 (1.: can give trouble only for:

R0+i:=0

(44)

Equations (44) and (45) both define pointsoutsidethe fluid

Secondly. we have the term: er'ircw" +14roccurring in -equation (32) and in the definition -of Jo- (29).

- From thedefinition ofL.- and w(31, 6) %ye...have c120:1 1 =e (4() 'SAM. 1R01< 1 R0 1 RO 1

With the definition

J=r-the recursion relation (28) tends to:

(n +1)0

(2n 2/3)Q +(n 1)Q.

0 (38) (39) (40) (41) (42) (n +

+, (2n

2/3)P,; + (n 1 )P_ , =0 11> 7D and i+ Ro.:* =0 (45)

P=1,' 'IA cos, STin +B sin Ain)

11> 7:

(36)

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

Thus we have:

=0(11 3 4 RP) 11-0 7: ₍₃₇₎

Equation (46) is' evidently allright. The remaining

contri-bution to equation(32) that needs checking is J (30).

We have

J =e-'' 14°) '

i3X[

asi en x -i(Ei(2fl)+E

le

K2+71' ÷

iXX))]

(47). Here it is clear from the definition of Ro that the argument

of E, is always in the left half plane. For E, the following relationship holds6:

- Ei( a > 0

limE,( -a+ iE)= a)+ in (48)

c-o E> 0

- _

It follows directly that J is continuous across x = Q. From the definition of El it is also clear that all the derivatives enJ/ex" are continuous.

The signum function. in equation (47) is necessary to

counteract the discontinuity in the standard definition of E1. 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:

and

0(1)

x(Ds= 0 for.y=d, (;=e'8,

=Roe'

ery.

7i13 r

-(1

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 0:

1 -Ri

(l ±i02

(i+Ro..7*)=

1 kg _(R0+;,02

(R0+i=)2 1-R

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

(10 ,e;c: di CC)01;)

E

+Knn

--ey Fr cy .., dr, oy

j

c

RN^-+:R'cli(C)) (55)

in equation (19).

(49) _{By definition C = Ro/w* so that we have to prove the}

following identity:

1.111111111111111r9VMPirWrreP741Piajazie

'--A circular cylinder in water wares: E. Mehlum

Introducing equations (50)-(54) above we find that equa-tion (55) vanish term by term for all points at the surface,

where we have It.= Roe' and i:=e. 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

GRv..01)"

The necessary convergence is therefore assured whenever

110> Ro. Within the liquid and at the cylinder surface this is trivial since there we have Ro <11%14 1.

As for the equivalence part of the proof it is sufficient to demonstrate that F2(C) (26) from the practical solution is identical to

E

n = 1

E C RV

(r,))1-

0

11=1

Performing the necessary algebra inside the brackets followed by the differentiation and readjustment of

in-dices 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 Cn (equations 20 and 21).

This completes the proof. We have at the same time

shown that the two solutions are equivalent whenever Ro

< lwl I.

CONCLUSION Asymptotic. behaviour

For convenience the_results are summarized below. In

- terms of cartesian coordinates sailed to cylinder radius

= 1 we have (Fig. 2)

(57)

Applied Ocean Research, 1980, Vol. 2., No. 4 175

,!,(4/. ...ioniewwdaowwr'

The first term in equation (32) fulfils equation (49) trivially.

Secondly, we have _{e(2100/11+10_ 1+ E cniv(c.+J(0)a. E CnRP"c" (56)}

n= 1 n=

This identity can be controlled from the very construction of F,(C). 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 C=0 and since emil" +4) does not vanish, we have to show:

d _{p2ptil +10 el2ip,;)/(1+ic)_ 1 + E}

C(fq-Ro-)n

A circular cylinder in water .warts: E.. MeOwl

=wave nuinber' -= x +iy

Our -eapires-sion for the space ,deperident, 'part of the

potential (32) is: 4)=e- 'R.

+

R" +RZ,J ()) -

(61) n=1 with e-ziRn+ rrsi Ro:

Ro+i:

Re ic* r, Rol:*

d =depth to centre of Cylinder > 1

Ro=d-=2(RET

.=;(e-

Sit. + i2X +3t

x_

.+E1(--L+'ay+ia.x))]'

Ro

(65)-J 1= 40 +i(65)-Jo

(66)

2C" + J.)+2iJ+J.- 1

/IT? 1 (67)

= a

(68) (n +4)(RV. = RcT" 1)C4.1 =

RcTn)-1-13(R;+R)]C,;+(n--1)(R;-1

n 1:. (69),

Net let us look into the asyrnptcitic behaviour of the

.

,

potential:, Let 1)cH co, C--i,

First, we obserie that for large 'modulus nf the argument, the function E, vanishes.

'The reclirSie relation ,(67) beccimes:

= (2r +J)+21.1+1

(70)

where an and are real qiiaritities not depending on x. Equation (79) is verified by direct substitution in the defining eqUatioris,for.the sc-quence S..(72 and 73). Taking

. (71) real and imaginary partc; this leads lb ,realif-siVe relations

176 _{AOlied.Ocean Research. 1.980.. Vol. 2. No. 4}

Equation (70)

becOrrieS:-..-With

Si =4/3e ---;P(ztsi = iti2fi)) +.2i (73)

From equation (61) we now get the following asymptotic

expression- as _

which reduces readily to .

. .

_ = eleg) -141

=-- purely real.

It is also clear that

e- xRe+12.! _{:E.. tz(iRo_)4_,373in.t.}

.EqUation (75) now becomes:

----

-eev-R.),

(72)

-1Z1,;(p12-v+.R?S2in)] (74)

(64) The factor in paranthesis deserves a-new symbol.

(75)

It is- clear- that tfr is a complex constant apart from _an abrupt change as Changes from negative to positive values. This juthp. 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

damping with dePth. We observe that 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:jump in the asymptotic e-xpreSSion (77) is filled in by the evanescent wave so that -the complete scilution (61) is continuous across a = O.

As regards phase Shift and reflection We must look_more closely into the quantities entering into the Constant

From equations (68) and (69) it is immediately clear that is purely imaginary when n is odd and purely real When n is even. i.e.:

(78)

Figure 4

for the sequences and as well as starting values to

trigger off the recursion.

For we now get:

z

0

=(i

+ >(i)

n-,,,)

_&I

an +i.."sisr..n

x))

R)

n= 1

which reduces to:

0=1 +

_{(Rof7,,o-+i sign x}

_{(R0)"7. (80)}

n= 1 n = 1

It is seen that the only change in kir as x changes from oo

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 done to the wave, is a phase shift.

These results are not new, and were first given by

Dean'.

The wave disturbed by the submerged cylinder will lag behind acorresponding undisturbed wave by an amount given by the phaseshift angle AO:

D=2 arcryIn*

ROI

We have achieved an explicit solution to our problem,

(81) 400 300 200 100 Figure 5_

.4 circular cylinder in water wares: E Mehl-Lim

0, 035 030 025 060045 050

rlpplied Ocean Research, 1980. Vol."), No. 4 177

07 070 065 060 055 080085

making it possible to compute 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

appli-cations. 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

performedusing equations (80) and (81).

REFERENCES

1 Dean, W. R. On the reflexionof_{surface waves by a submerged}

circular cylinder, Proc. Camb. Phil. Soc. 1948, p. 483

2 Ursell. F. Surface waves on deep water in the presence of a submerged circular cylinder, Proc. Camb. Phil. Soc. 1950. p. 141

3 _{Ogilvie. T. F. First- and second-order forces on a cylinder}

submerged under a free surface. J. Fluid Mech. 1963, p. 451

4 Longuet-I-ligzins, 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

5 Whitman. G. B. linear and nonlinear waves, John Wiley, New York, 1974

6 Abramowitz. M. and Stetzun, A. Handbook of Mathematical Functions National BureauofStandards. Washington DC, 1970

7 Gradshteyn. I. S. and Ryzhik. I. M. Tableofintegrals, series and products Academic Press. New York, 1965

EmEneammsgstffrf1,7,!,5,0x,.,1,1,:,,,,,;;;775r7ff-yr,57771.777717755,:,:i,:t7,777777r7rrr!f,:,,,:ipnr.7,-.-7,77,7777, 5".onnrrro7177,7

500 ? ss

02 04 0