Applied Mathematics and Mechanics (English Edition, Vol.8, No.I, Jan. 1987)

Published by SUT, Shanghai, China

HIGHER-ORDER THEORY OF I N T E R N A L SOLITARY W A V E S

W I T H A F R E E S U R F A C E IN T W O - L A Y E R F L U I D S Y S T E M OF F I N I T E D E P T H Zhou Chin-fu ( ~ / ~ ] ) (Department of Mathematics and Mechanics, Zhongshan University, Guangzhou)

(Received Feb. 10, 1985 Communicated by Jiang Fu-ru) Abstract A higher-order approximation theory of internal solitary waves with a free surface is presented. Using the method of strained co-ordinates, the third-order approximqtion evolution equation of interface has beenfound. An analytic expression of the wave velocity is given. The evolution equation has been solved numerically. It isfound that the effects of free surface on the shape and wave velocity of solitary wave are O( e!) , and the thirdorder numerical solutions are closer to experimental data than the first-and second-order solutions.

Following the KDV and Benjamin-One theories, Kubota et allq (1977) derived an evolution equatibn governing the propagation of long internal waves in stratified fluids of finite depth. In this model, the thermocline is much smaller than the total fluid depth. The wavelengths are much longer than the thickness of the thermocline, but may be comparable to the total fluid depth. In order to examine the validity of the theory, Keep and Bulter[21(1981) and Scgur and Hammackl31 (1982) performed experimental investigations respectively. They used a two-layer fluid system instead of continuous density distribution model. The second-order evolution equation, with a rigid lid instead of free surface, was also derived. By comparing measured data with theoretical solution shapes they found, that the first-order solution is good only for smaller wave amplitude. With the value of amplitude increased, the predictiveamplitudes of the first-order solitary waves are no longer in agreement with measured data, but the second-order theoretical values are in close agreement with experiment. When we continue increasing the values of amplitude the noticeable difference between prediction by the second-order theory and measured value occurs again. It is dear that in theory of solitary waves in stratified fluids of finite depth the study of higher-order approximations is important. As will be mentioned below, derivative of potential ~l on free surface in upper layer is the order of magnitude as follows

-~-=O(e,d)where ~bl' is velocity potential in upper layer, z/~- (P~--Pl)/Pt , e is the ratio of depth of lower layer to depth of upper layer. That means that the influence of the free surface on the internal solitary waves must be felt in the second-order solutions. ! In this paper, a higher-order approximation theory for internal solitary waves with a free 7i

72

Zhou Chin-fu

surface is presented. The method of strained co-ordinates is used to avoid the secular terms. We deal with this problem to fourth-order to obtain approximate equations for rh , rh. and r/3.. The contribution of free surface to the shape of solitary wave and wave velocity are given. In order to obtain the solition shape, ~t is necessary to numerically solve a linear integral-differential equation. Using the fast-Fourier transform technique and least squares method, several numerical experiments are performed. This numerical method can approximate the solutions to any desired degree of accuracy. I. B a s i c E q u a t i o n s We consider a two-layer fluid system with the disturbed interface y * = r / * ( t * , x*) . The scales and co-ordinate system are shown in Fig. 1. The theory of internal solitary waves with finite depth is based on three assumptions: (1) Thickeness h 2 of the lower layer is much less than thickness h, of upper layer

(i.I)

e=h2/hl <
(2) Interface wavelength is much longer than depth h v but is comparable to depth hi; (3) The ratio of amplitude of interface wave to h2 is the same order as e According to the above three assumptions, we introduce dimensionless variables:

x=x*/hl,

y=y*lhz,

t=ct*/h,,

r/*=eh:r/

1 }

(1.2)

9 ,,rj*frt

J

s

'v. ~t*, Pt

for upper layer, where c denotes the wave velocity, and

x=x*/h~,

y, =y./h,

t=~*/h,,

~*=e.'h2r } J

,)1"_ ehze~l

tx* ~*-----0

"l 9

(t.3)

9

9

.

.....

9

y * f - . hz

Fig. 1

for lower layer. Where ~* denotes elevation of free surface in upper layer, ~P~ and lp$ are potentials of upper and lower layers respectively. Because r satisfies Laplace's equation and a rigid boundary condition on the bottom, we can show that the dimensionless potential ~z can be expressed as a Taylor series as follows:

(y+ 1) z. .-o

(2n) !

(

8z",p2 Ox z"

),._:

Then, we can see that velocity components u ' a n d o ' i n lower layer have the orders of m a g n i t u d e as"

v* = eaco Remainder conditions satisfied by ~z c o n t i n u i t y condition on interface. They are Or/ Or/

""0t + eu--~-- =v

cz{

8~b,

_

~-~.Te

1

s

--az/h2rl:O

(U~+V') -

(1.5)

are k i n e m a t i c c o n d i t i o n a n d pressure o n . y = --1 +r/

(1.6)

8~2 1 , z s z Ot - - ~ - t e . + e v ) )

ony=--l+r/

(I.7)

,Higher-Order Theory of Internal Solitary Waves

73

where U, and V are dimensionless velocity components in upper layer, 9 is gravity acceleration. Now we describe the equations and boundary conditions in upper fluid layer.Laplace equation must be satisfied by ~0t VZCs=0 *Pt

and

r/

(1.8)

satisfy eq. (1.7) also. Kinematic conditions is given by Ou

e( eU--u) ~ = V - - v

(1.9)

on V=r/

combining the kinematic and pressure continuity conditions on free surface in upper layer gives a governing equation for tbt as

a , ~.~1 ~ 2 . , , ~+ 1 ( 0 . ~ ' ~ a + ~1 ( 0 , ) z } cZ ,eOz~1 I &h +e3 c~.__~.~e ghz Ot 2 OYz gnz k oroYt "~k Ox 1. 2 OYt + ~3

"1,"~'~

Ox

O---x-

OxOva

Ox j

on V, = 1 + es~

0 (1.10)

Summarily, velocity component, u, v in upper layer is prescripted by (1.4), which satisfy equation (1.6), (I.7), (1.9) together with r/ ; In upper layer tPi satistied eqs. (I.7), (1.8), (1.9) and (1.10). So equations (1.6), (!.7), (1.9) are connective conditions between upper and lower physical quantities.

II. The Perturbation S o l u t i o n s We introduce the following wave frame

r=(x*--ct*)/ht=x--t

(2.1)

In order to avoid secular terms in third-order, the strained co-ordinate transformation

r--~-e0(D

(2.2)

where the unknown variable 0(~) will be determined in successive approximations, is necessary. We now expand these variables as a power series in e c=cz + ec2 + e~cs+ . . .

O(D=o, (D +eo~(~)+...

[

u=ul (~) +eu~(~) +e~us(D +... '7='h (~) + e,h(e) + e:,h(D +'-. r (~) + e~(~) + e~L(D +... ,~1= r + ~ ( t , ~ , y ) +...

(2.3)

The transformation between derivatives is

O (1_e01~ +sz (0,r br-- =

+... ) a a~

(A) First-order evolution equation Substituting expression (2.3) into eqs. (1.6) - (1.10) leads to the first-, second- and third-order equations. From eqs. (1.6), (1.7) and (1.9), the first-order relations on y = 0 are

74

Zhou Chin-fu rhr162

I/'t+u~r

c~u~--gAh=rh=O

Then,

cI =gAb2,

77,=ut,

0

F ! = a-~-L OYl = --u=r

(2.4)

The first-order wave velocity has been determined from the first equation of (2.4), and boundary condition of ffl on Yl = 0 comes from the third equation of (2.4). Boundary value problem determining r is given by eq. (2.4) and the leading terms of(l.8) and (l.10):

V =~1 = 0,

Oyt ~y,=l

=0,

Og~ lIy , = 0 = - - r h r

(2.5)

and the solution is

1

q~t = " ~ " ~ - 0 o c~ k(1-Y=)fflCksinhk e x p ( i k ~ ) d k

(2.6)

where ~la denotes the Fourier transformation of r/1r . The first-order relations are not sufficient to determine evolution equation of rh . To do that; the second-order equation coming from eqs. (1.6), (1.7) and (1.9) must be used. In the same way, evolution equation of rh should be determined from the second-order equations of (1.8) and (1.10) together with the third-order equations of (1.6), (1.7) and (1.9); Evoluation equation of r/3 should be determined from the third-order equation of (1.8), (1.10) together with the fourth-order equation of (1.6), (1.7) and (1.9). First, eliminating rh and U=from the second-order equation of (1.6), (1.7) and (1.9), then substituting ~i into it, and using the following relation 1 I~_ ~ 2~r

0 f ~176 (k coth k)j ~ exp(ik~) :/k---- - 1~ ~-~ - ~ / ( y ) c o t h -~r -~-(~--y)dy

the first-order evolution equation for

rh

(2.7)

is obtained:

3ctr/tr/l~-- 2c2r/tr -- 8~= 2 J _ ~ r/t (y) coth~- ( t - - Y ) dy The solitary wave solution for this kind of equations is given by Joseph ~4] 3 1sin1 2 r/t-- cos~-I-cosh2(~-t-~0)

(2.8)

(2.9)

1 c~=- ~ct2

where

-~o is a parameter,

I

cot 2 (2.10) is a parameter too, and its value is limited as 0 < I ~

(B) The second-order evolution equation Eliminating rh equation as follows:

and u3 from the third-order equations of (1.6), (1.7) and (1.9) gives an

1 2r11r r ~ = O. --6clc2rlirh~ + ( c: + 2c, cs)rhc + 6c: rl: rll~---~c~

(2.11)

Higher-Order Theory of Internal Solitary Waves where 01r

and ~c~

75

is to be determined. We priori take a simple form for 01

0x(~) =aS

(2.12)

where a is a constant and is to be determined on the basis of eliminating the secular terms. According to the second-order equations of (1.8), (1.9) and (1.10), the boundary value problem determining Cz may be otained VZ~b2----2ar162

l

Cz,, lie-1 = --dCar r

(2.13)

I

(2.14)

la-0 = --r/2r q- ar/lr

The solution of the above boundary value problem can be expressed as

2u

i~lEkylsinhk(l'yt)

r162

+ e o s h k ( 1 - - y ) ] exp -

- o o

' sinh

-1-r ~' and

(ik~)db k

-'"

osh~y 1 q~2u J-~o f ~~ i k ~ l ,csinhZk exp(i~k)dk

(2.15)

o~

-- 2~

-co

i(#z--a#l) e~

sinhk

oo

1 I L/~l~(,a) 2~ -oo -t

cosh ky I exp(i~k)dk hsinhk

(2 16)

where ( -: ) denotes Fourier transformation of quantity in the parentheses. It can be shown that ~l~(~l=l)=--

ikZ

(2.17)

sinhk

Then, the second term in the right-hand side of r

~71 can be rewritten as

1 f~176 coshky 1 '-2~r J_oo (/k'd~l) sinh zk

(2.18)

exp(i~k)dk

We can prove that existence of (2.18) must lead to a secular term a* foo --~'rj_ (~-y)rlz

u

eoth-~(~-y)dy

in the second-order evolution equation. To avoid this situation we take a=--z:] then,

(2.19)

Cz become i

oo

Cz = - - ~ " - I _ ~ I-~zeoshk(1--Yl) --~k71ylsinhk(1--Yl) Substituting

~l

and

~z

] ex.p (:~k,)dk $1nn/~

(2.20)

into (2.1 l), using the relations [31

" ~ =,t~r/l -.I-9czr/r~--~r/~

(2.21)

76

Zhou Chin-fu

we can rewrite equation (2.11) as 1 J-oo f ~ r/zcoth~2 (~--y)dy 2c~rh~--3cl(rhrlz)~ +cr-~ -~

2c,c3 +1c~22+ 2Llc,c2 )r/, +(3c2 - ~3 zlc, )rl:t ----~-cff/, 7 ~ ~ (2 22)

=-~(c~(3c]--

The coefficient of Th in the right-hand side of(2.22) must be zero in order to avoid a secular term like ~rh pl. Finally, we get the second-order equation and correction of wave velocity as follows:

02

1

oo

~r

2c~=~- 3c~( ~ ) r + c~-~-r -YI_ o~~ coth~-(~--V) dv

=~_.~((3cz_ T3 J e t )rh--~-ctr/, \, 7 , ~

(2.23)

~___~_~2cot=2.t_ -~-~. 1 2 --~-2cot~. L/

(2.24)

Effects of frce surface on the wave shape and wave velocity are found by parameter

(C) The third-order evolution equation The procedure for obtaining the third-order evolution equation is the same as that in section (B), we only explain points briefly. Function 0z(~) is taken as Oz(~) =.B~

(2.25)

where fl to be determined. The following equation comes from the fourth-order equations of (1.6), (1.7-) and (1.9)

{ -- c~ ~3r162 -t" c~ 01r

-- 2ctcd~=r162 },l.o--3c=~(tilth)

1 =

.q- 2c, czrh, q- ..ffc, (U ] -{-F'~ ) , - F ~ ( [ - -

2cz( ct (,Jz--fl) + ( ct q- 2c3) -F

2

+ c~ + 2c~c~+ 2c,c=~d t--~c~ . 1 ,-~. X -- 3c~c~-- -~c, 3 ,'!.j%'-I-[IOcIcz-F3C:LI]r/:

--4c~rlt

+[c~ +2ctcs]rl~--6c,c=r/,rh-- .-~c 3 ''~11=

5 =r/~r - - -.1~ c ,i , ~ + 6c~,7~,7~- --ffcl

(2.26)

Boundary value problem satisfied by ~ is given by the third-order equations of (1.8), (1.9) and (1.10). After it is solved, the value of ~ on Y~=0 can be written as i

oo

1

~

z

'

3

""

=

1

OO

1 cothk "k d 2~ f _ o o ( U t r / t ' ) - - ' T " - e x p ( ' ~ ) k + - ~ - I _ * * ( - - - ~ ' t - ~ '- ~2--~ Jz

Higher-Order Theory of Internal Solitary Waves

77

}rl~exp(t~/~),lk

_ f l + 2cz /_/+2~12 cl

(z.z7)

let final term of the right-hand side be zero (to eliminate the secular term), thcn value determined

2cz _4

fl=l~z"/z+ 15

fl

is

(2.28)

c1

It is easy to show that

N

)

U~,7,r )=ih c~ ~ _ 1 7 ~ If one substituted

r Cz

and

Ca

if

into (2.26), a term

:.,o

- e l OC" 2 J_or

(o~

,

1,7,, '~

r/,--T

~

~

'

)coth~--(~--y)dy

(2.29)

Would occur in (2.26). Now we mtroduce the transform r/3=r/~ ,

+c~_rhC= =_ 21_.,/~

(2.30)

To changc variable r/~ to r/s(1~ , so that term (2.29) does not occur in the evolution equation, in fact, by using the following relations c~ O__~

-5-

0fl

2 z

. 9 ~ ,'~ ~-3c"7'~.

( u ' ~ + z ~ ) = a-~l.~ ~''7'~+z~;'7'-3c'~0~'

3 z (,2'")

=

- - ( Z. eot

the third-order evolution equation of

r/st ' )

3 - (1Ltcl + ~

),'. --, :

is expressed as

-- ~=c,--

6z/c=

c~ 3

]r/: cl

~

1

2

+ 3ctL/r/ffh -- ~-ctr/, -- -~ Cdhr162 + 6clq, qz ~

(2.3'1)

Summarily, we list our results as follows:

(2.33)

78

Zhou Chin-fu

where rh, rh and wave velocity is

are governed by (2.5), (2.23) and (2.31) respectively. The expression of

rh ~

c=cl

+ ec2 + e=cs+..-

cI c~=gh2zl, cz=---~ cs=c,

\8(32'e~

cot 2

+ --~1 ~__~_ttJ. cot

2~,

(z.34)

III. Numerical Method for Solving Equation Equations (2.23) and (2.31) can be written more generally as

C<.

O~ e' 062 2 I_~orheoth~(6--y)dv=G(6)

3cl(lhrh)r162

(3.1)

Generally speaking, the analytic solutions o f (3.1) do not exist. Now we present a new numerical method which avoids the calculation of the principal value integral. By taking the Fourier transform of both sides of equation (3.1), we get

3~hrh=~z

keothk+

cx

Following reference [3], we choose the Hermite's Polynomials as a set of basis functions, and expand rh into a series on it. Then, using a procedure of the least squares method leads to a set of linear algebraic equations. In order to examine the efficiency of this method, let G ( ~ ) be (r/~)c In this case, a precise solution of (3.1) is .rh=2rh/3cl We find that the relative error between numerical solution f-/2 and precise solution rh is less than 2 • 10.2 for N = ! 2, t E (~r/10, ~r/4) . Numerical results for N = 14 and 16 are also listed in Fig. 2, and we can sec that the numerical solutions are stable with N increased. The values o f the second-order numerical solution r{ z) =r/t + e r h are shown in figure 3. The dashed line denotes averaged exper.i,-acntal values from Koop and Butler. G o o d agreement between theoretical values and experimental data can be found. Comparison of the third-order solution to the experimental results given by Segur and Hansmack is shown in Fig. 4, where , d = 0 . 0 4 8 , ~=107~r/180 and r/( s~ -----r/1-J- erh 4- etrh In this case, the radio of wave amplitude to h, is 0.476. The second-order solution departs from the curve of experiment data. But a good improvement is presented by the third-order solution.

Higher-Order Theory of Internal Solitary Waves q2.Ui:(

J 0.9i

79

0 )

~.

0.8[

- - Exact solution Numerical solution x for N =12 o N:-I4

\

0.7]I

k

\

A N~16

0.6

,7,t,h(0)

1.0

0.5

t0.~

OA

,1=./6

h~./h. -----~~ 0 . 1 9 7

0~6

0.3

y. Averaged experimential values from Koop and

tO.4

O~

I

0.0 L----1.46 . . . . , _~ , 2.93 4.39 5.86 7.32 8.76

*~"~

x/h,

. 11.6

,0.'

Fig.2. Comparison between the second-order numerical solution and exact solution for G(~)=( ~2 )~ . L0, 0,,9

\

O,P ~ 0.~ ;O.e

'

i~

'

,'~

:,/):(o..~)

Fig.3

~ = I07."r/LSo

\%

,a:o.o4s ~=I/9 \\\

q:.~>

~

0.~ 0..4

~

02,

0.2] ,0..1

Fig.4.

,crh~ 0.176 , 0.52g 9 O. , ~ ._=, " , ...... =~N'0~ 0.352 O.70..1: 1.06L~ 1.,lI0 L.760 l..qIO q(*>, rfia:, r/(a> are the first-, second- and third-order numerical solutions respectively; [] denotes the experimental results given by Segur and Hansmack.

References [ I ] Kubota, T., D.R.S. Ko, a n d L . Dobbs, J. Hydronaut, AIAA 12 (1978), 157.. [2 ] Koop, C.G. and G. Buiter, J. l@uid M e r h . , l l ) (1981), 225-251. [ 3 ] Segur, H. and J.L. Hammack, J. Fluid Mech.,ll8 (1982), 285-304. [4] Joseph, R.I. and R. Egri, J. Phys. A. Math. and Gne.,ll (1978). [ 5 ] Whitham, G.B., Linear and Nonlinear Waves (1974).