Applied Mathematics and Mechanics (English Edition, Vol.9, No.7, July 1988)

Published by SUT, Shanghai, China

AN ANALYTICAL SOLUION OF WAVE FORCES ON

LARGE S Q U A R E CYLINDER

Huang He-ning (~-~"l'J:)

(hl.s'titute o f Mar#w Environmental Protection State Oceanic Administration, Dalian)

(Received Nov. 10. 1986. Communicated by Chien Wei-zang)

A b s t r a c t

Wave ./orees on large square c:vlimler are determhwd hy u.vhlg the eonformal

tran.r method. B is found that only.for square cylinder, the governing equation is

still the Hehnholt- equation after the con/ormol trans.formation. An analytical solution o/"

it 'a ve.fi~rc'es on square c l'linder is presented h.l" ushtg the sohttion of wa re forces on a circular

cylhuler.

I. I n t r o d u c t i o n

An exact solution of wave forces on large rectangular cylinders can not be obtained because of

the angles of the rectangular sectionl'L The numerical methods such as finite element method~-'l or

source distribution method TM are usually used for determining wave forces on large rectangular

cylinders. Zhao and Lulq transformed the outer domain of the rectangle to that of the unit circle by

using the conformal mapping, then the wave forces on large rectangular cylinders were determined

by use of the exact solution of wave forces on circular cylinders. But Zhao and Lu did not show if

their transformation was valid. They only presented the pressures on several points on the cylinder

surface in their numerical example.

In this paper, the conformal transformation method for determining wave forces on

rectangulz2r cylinders are examined. It is found that only for square cylinders, the govering equation

is still the Helmholtz equation after the conformal transformation. In this case, the solution for the

circular cylinder problem can be utilized. F o r generally rectangular cylinder problem, the Heimholtz equation.is transformed into a noni'near equation which is more complex and difficult to

solve and the transformation is not useful.

H. F o r m u l a t i o n s o f the P r o b l e m and Confor mal M a p p i n g

Under the assumptions that the flow is inviscid, imcompressible, and irrotational and that linear wave theory is applicable, the complex velocity potential of the scattered wave induced by a cylinder can be.written as:

ch~(z-~ t- d)~bs (x, y) exp[ -- icot] (2 .1 ) ~ a ( x , y , z , t ) = ~

where k= wave ntjmber, co = angular frequency, d= water depth, and ~b s is the scattered wave

potential for horizontal two-dimensional problem, which satisfies the governing equation:

701

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

702 Huang He-ning

and the boundary conditions:

V:4,s + h~,;bS=O (2.2)

OcpS a~bl (cylinder surfaces) (2 3) an = ---O-n-n

ack~ -i/~(~=o (r->~) (2.4) Or

where dfl:-i2-1a~-lgHexp[ikx], is the incident wave potential for horizontal two-dimensional

problem, with the incident direction parallel to the x axis.

The outer domain of the rectangular section with the length A and width B is transformed into

that of the unit circle section (See Fig. 1). The mapping function is given bym

a+n (a--~) ~ (a~--a') (a--a) ] z : R ~-F--~--F 24~a + 80C~ ~--.. (2.5)

where z = x + j t j , ~ = ~ + j r l , R is a constant which characterizes the size of the rectangle,

a = e x p l 2 l a j ] ,a =exp[--2 I azj],j--._~,/--,-l_ 1, /"is a constant which varies with the thickness ratio

A/B of the rectangle. The determination for the R and I can be found in reference [1].

I!) / rl

i n A ! z plan ~ phm

Fig. 1 Conformai mapping of the .7- plan onto the ~ plan

Based on the theory of the conformal transformation14L in the C plan, equation (2.2) is

transformed into

1 k2~b~ = 0 (2.6) v:'k~ + la~/azl~

If the first and second term in equation (2.5) are retained, it is easy to have

OC 1 ~2 . . . . (2.7) az R ~--c

Where c = (a+ a ) / 2 , From equation (2.7) it is shown tht

1 z - - R z ( 1 2c.cos2q3 F 'c (2.8) laC/azl p= . ,

Substituting equation (2.8) into equation (2.6)

VZ~b s-F (kR)" (1 2c'cos2q~ pZ F @ ) ~ " = O (2,9)

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

An Analytical Solution of Wave Forces on Large Square Cylinder 703

It is clear that from the mapping function (2.5), the Helmholtz equation (2.2) in the : plan is transformed into the nonlinear equation (2.9) in the ~ plan. which is more complex and difficult to solve. In a general way, therefore, the transformation is not useful for rectangular cylinder

problem.

Considering the square cylinder problem, from reference [11. in that case I = 1 / 4 ,

a=exp[2-1:rj],~=exp[--2-J:rj], then c = 0 and equation (2.9) becomes

VZ~ s + (kR) zqbs = 0 (2.10)

and corresponding boundary conditions are

.ar = _ ar (p=1) (z . i l ) an :an

a@s _ikR@8=o (p-->oo) (2.12) ap

Equations (2.10) - (2.12) define a scattered problem in the ~ plan withlhcwave~a(hnberkR

in the dimensionless form and incident wave potential ~ t = _ _ i 2 - z o g - ~ H e x p [ i k R ~ ] .

For square cylinder problem, therefore the mapping fu,~ction (2.5} i.~ succc~fully used tt,

transform the problem in the outer domain of the rectangle il~ the- plar into thai, ,! the .:.rclc i~ the

plan, and both of the problems are governed by the Heln~l~,~ltz , :q t la t iun. [~t_lu;.ttion~ (2. Ill) {o (2.12) can be solved easily.

III. Analytical Solution for Square Cylinder

In the ~ plan, the solution for equations (2.10) to (2.12) is shown as

.gH ~ J " (kR) H~,~ (kRp)c.osncp ( 3 . 1 ) r ( - f l , ) , He," ( kR)

where ,80=1, ,8,,----2i", n ~ l .

The dynamic pressure p i~ related to the velocity potential by

, 0 ~ . t chk(z+d) ( r 1 6 2 ] (3 .2 ) p = - - p ~ = t o 2 p �9 ehkd

where pt =the density of the fluid.

The horizontal wave force in the x direction on the square cylinder is written as _..s

2

,3.3, 2 2 2 '1

Substituting q~ and ff~ into(3.2), i t is noticed that p =1 is corresponding to the

boundary of the square..Then equation (3.2) is substituted into equation (3.3)~ After being integrated, the horizontal wave force on large square cylinders in the case with the incident wave direction parallel to the x axis is given by

F 1 , . . . . . thkd f 2 is in--~--+2 2v J',(kR) n - 1 , 3 , 5 ' ' " Jff(l)t(kR)

H~".( kR)cosnqa}exp[ --icot ] (3.4)

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

704 Huang He-ning

in which, the cosn~0 (n= 1,3,5,,..N) are the furiction of the eosq~ by using

cos3q~= 4cosSq , - 3coscp

cos5~p--= 16c0.sSq~ - 20eosSq0+ 5cosq0 j

cos 7q,= 6,1COS7q J - - 112C OSr'qg+ 56 C OS3q; -- 7C OSty l' ( 3 ; 5 )

cosgq0= 256cosgqo-- 576cosTq0+ 432cos~p-- 120cos3q~+ 9cose; . . . . , , . . . . . .

Fron', equation (2.5). thc x and y are determined by

O 1 " 1 2 3 x=R(pcoscp+eo.s,Frp- cos~p----~sin 21nO- eos3q0+ ...) (3.6~

u = R ( l 1 2 s i ,os.ing--cos2I~p- s i n g + - ~ s i n 2Drp- s n3q~+ .. .) (3 .7)

In equation (3.6) ,def ining x = A / 2 and p =1, then the cosnq~can bedetermmed from equations (3.5) and (3.6).

The equivalent.inertia coefficient is defined as

F = , , (3.8) C*. ----- 2 =lp' g H A B t h k d

If thc incident ~a\,c direction is at an ange of "Y to the x axis. the incident wave potential hOCOlllC~,

~b~= --i g--~ exp[. ik(xcosy+ysiny) ] (3.9).

It is easy to obtain the solution of the scattered potential in the' ~ plan

, . g H ~ J I ( kR) = --s=---2_ ~. (--ft.). , H~)(kRp)cosn(q~--?)

"/.co ..o H~. t) ( kR)

= _ i g H '~ .l~(kR) (t> - ~ - ] ~ ' ( - - f l , ) , H , (hRp)[eosnq~.cosny+sinnq~.sianl~] (3.10)

,-o H~, l> (kR)

From equations (3.9). (3.10) and (3.3). the horizontal forces in the x direction is written as

1 p ' g H A B t h kd {- 4i A B sin ( - f f - -hsin y) 17x= 2 ABkZsin~ ' sin ( - - - ~ kcos v )

B '

+ .A--'k2 ,,-1,3,6,..~-' ( --2i") H~, '>'g;(kR)(kR): H~t)(kR) .cosn-/cosnq0 }exp[.--ioot] (3.11)

The horizontal wave force in the y direction is given by -..A.x ,A.x

Fv==~_ dx -- dx dz exp[---icot] (3.12) z Y--T _.4: y = _ as

after rearrangemem it becomes

. ABkZcos~,

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

An Analytical Solution of Wave Forces on Large Square. Cylinder 705 /r

+ Bk2 ,-~,s,6..~' ( - -2 i " ) H r HC,~'(kR)sinny.sinnqo}exp[-icot (3 .13) J " ( kR)

�9 _ . , ( k R )

in Which the sinnqo (n= 1,3,5,.-.At) are the function of the sintp by using

sin3~p~ 3s in~-- 4sinS~o "]

sinS~p= 5s in~-- 20sinS~v+ 16sin~p

sin 7~p= 7sin~-- 56sinSr 112s in~ - 8dsinT~ (3. l,t )

sin9cp= 9sin~p-- 120sinS~p+ 432sinS~-- 578sin7cp+ 25t-;si, ~q~ H . H ~ - . . ~

In equation (3.7), dcfining.~=B/2 and P = I, then the sind can be determined from equations (3.14) and (3.7).

If y = s t / 2 , the first term in the parenthesis of equation (3.13) is replaced by 2i . k B

~ s m 2 - IV. Results

The equivalent inertia coefficients for large square cylinders are determined by using the

present method. The results are compared with those obtained by Isaacsonl~L using source

distribution method and by Mogridge and Jamieson!51 in their experiments. Fig.2 shows that good agreement has been achieved, specially for the case with y = 4 5 " and k A ~ l . 0 .

It is found in the calculating that the series in equations (3.4). (3.1 I) and (3.13) converge rapidly. The maximum relative error of the results for N= 5 and N = 7 is less than 0.002.

~c~ Analytical solution

~----Wave source method [3]

u = 0 ~ 3 ~ IJ~45 "'t/Experiment results [5 j

2 - - ~ v--=d5 1 v=O"

, , kA 2 2 3 ,1

Fig. 2. Eqmvalent inertia coefficients Ibr square cylinders R e f e r e n c e s

[ I ] Zha.o Zi-dan and Lu Jung-liang, Action of the small amplitude wave oil large rectangular cylinders, Acta Oceanologica Sinicu.-2. 3 I 1980). 137 ~ 152. (in Chinese)

[ 2 ] Shankar, N.J., T. Balendra, and C.E. Soon, Wave lbrces On large vertical cylinders of square and rectangular sections, The Institution o/l-ngineers, Australia (1982). 36 43.

[3 ] Isaacson, M.de St.Q., Wave forces on large square cylinders. Mechanic.s" O/ Wuve-lnduced Forces on Cylinder.v. ed.T.L. Shaw, Pitman, London (1979), 609--622.

[ 4 ] Liang Kun-miao, Mathematical and Physical Methods. Bei.jing (1979). 441 - 446. (in Chinese)

[ 5 ] Mogridge, G.R.'and W,W. Jamieson, Wave forces on square caissons, Proc. 15th Coastal Eng. Conf., Honolulu, 8 (1976), 2271- 2289.

Abstract

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter pur- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

1. Introduct ion

Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest.

Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):

ap +u_~_xp + au --ff =o,

au au 1 y =0,

aS as a--T =o,

p =p(p, s),

(i.0

293

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products