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Higher-order waves propagating on constant vorticity currents in deep water
Coastal Engineering, 2 (1979) 237--259
237
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
H I G H E R - O R D E R WAVES PROPAGATING ON CONSTANT VORTICITY C U R R E N T S IN DEEP WATER
IVER BREVIK
Luftkrigsskolen, Trondheim-Mil, N- 7000 Trondheim (Norway) and Division of Port and Ocean Engineering, The University of Trondheim, N- 7034 Trondheim-NTH (Norway) (Received June 1, 1978; accepted November 14, 1978)
ABSTRACT Brevik, I., 1979. Higher-order waves propagating on constant vorticity currents in deep water. Coastal Eng., 2: 237--259. Using a Stokes expansion procedure, the third-order wave theory is developed for periodic waves travelling on a surface current whose undisturbed current profile is varying linearly down to some fixed depth h and zero otherwise. Deep water is assumed beneath the current. The main result is the expression for the first non-vanishing correction t o the phase velocity c; it is necessary to go to the third order to calculate this expression.
INTRODUCTION
Within a Stokes expansion approach it is known that, to the second order in wave steepness, the following expression holds for the phase velocity c of periodic waves propagating on quiescent, deep water (Wehausen and Laitone, 1960): c = x/g/k
(1 + 1/~k2a2)
(1)
in which k = 2 ~ / L is the wave number, L is the wave length, a is the wave amplitude, and g is the gravitational acceleration. Surface tension and viscosity have been neglected. (In fact, eq. 1 holds up to the third order.) If, in addition, there is a uniform opposing current present, whose undisturbed velocity U is constant everywhere (U taken positive for the opposing current), the absolute phase velocity can be found simply by subtracting U from the expression of eq. 1. This is an immediate consequence of the fact that the phase velocity transforms like a particle velocity under Galilean transformations. The main objective of the present paper is to derive the analytical expression for the phase velocity, correct to second order in the wave steepness, for periodic waves propagating on a shear current whose undisturbed velocity profile is varying linearly .from the free surface down to some depth h. The
238
l_ i3 LI y V ×
Fig. 1. Velocity profile in undisturbed current.
v e l o c i t y profile o f the h o r i z o n t a l u n d i s t u r b e d c u r r e n t is s h o w n in Fig. 1; with the c o o r d i n a t e axes o r i e n t a t e d as shown, the h o r i z o n t a l v e l o c i t y reads: u = -(1 +y/h)U
y >1 - h
u = 0
y < -h
(2)
Thus U d e n o t e s t h e c u r r e n t v e l o c i t y at the free surface. T h e m o t i o n is restricted t o be t w o - d i m e n s i o n a l , and d e e p w a t e r is assumed b e n e a t h t h e current. Dissipation is ignored t h r o u g h o u t . G e n e r a l l y speaking, t h e r e are f r o m a practical v i e w p o i n t g o o d reasons f o r pursuing the d e v e l o p m e n t o f the t h e o r i e s o f the w a v e - - c u r r e n t i n t e r a c t i o n . In the first place, the p r e s e n c e o f c u r r e n t s m a y alter the b e h a v i o u r o f sea waves drastically. As an e x t r e m e e x a m p l e , let us m e n t i o n t h e a b n o r m a l high waves t h a t can a p p e a r in the Agnlhas C u r r e n t n e a r S o u t h Africa ( S c h u m a n n , 1974, 1975). Here t h e surface v e l o c i t y was r e p o r t e d t o be a b o u t 2.5 m/s; if fully d e v e l o p e d sea waves with wave heights o f 6 m m e e t this c u r r e n t as an o p p o s i n g c u r r e n t , t h e result m a y be giant waves with wave heights o f 18 m or m o r e . Assuming t h a t t h e wave p e r i o d is a b o u t T = 12 s one finds, f o r instance b y use o f t h e curves given b y T a y l o r (1955), t h a t t h e absolute phase v e l o c i t y o f t h e waves (relative t o t h e g r o u n d ) is a b o u t 15 m/s. E v i d e n t l y such sea c o n d i t i o n s are dangerous even f o r the largest ships. In the s e c o n d place, it is desirable t o k n o w t h e wave---current i n t e r a c t i o n a c c u r a t e l y in o r d e r t o be able t o give reliable p r e d i c t i o n s f o r the forces on s u b m e r g e d structures, at least w h e n detailed m e a s u r e m e n t s o f t h e c o m b i n e d flow field are lacking. S i m p l y neglecting t h e p r e s e n c e o f c u r r e n t s m a y be un-
239 realistic. A third place where the wave- current interaction plays a major role is in the boundary layer near the b o t t o m . At present, rather little is known about shear stresses, b o t t o m stability and sediment transport in a combined wave--current system. The present work, however, is concerned only with the wave- c u r r e n t interaction in the upper regions of the fluid and thus has no bearing on the flow conditions near the b o t t o m . The assumed linearity of the undisturbed current velocity, as given in eq. 2, is of course an idealization of the real physical situation. This idealization is motivated by its mathematical simplicity, as well as by the observed fact that the linear profile in many situations of practical interest gives a fairly good approximation to the real profile. For instance, as shown experimentally b y Bulson (1961) and others, the two horizontal current branches set up by a two-dimensional air curtain will reasonably well be described by linear velocity profiles in the surface layer. In a pneumatic breakwater, one of the branches is directed against the oncoming waves in order to damp them partially, or even stop them. Adopting the linear form of the velocity profile, the linear wave theory for this kind of breakwater was given by Taylor (1955) for the case of deep water, and was elaborated by Brevik (1976) to include the case of finite water depths. Until recently, the case of rotational currents was n o t much studied. Thompson (1949) and Biesel (1950) derived the dispersion equation using linear wave theory and assuming a linear velocity profile. Sun' Tsao (1959) gave expressions for surface displacement and particle velocities assuming the same t y p e of current. The essential difference of the situation considered by Sun' Tsao from the situation considered herein is that he assumed a single layer model, corresponding to a current velocity profile varying linearly from the free surface downwards to the bottom. Recent developments on the theory of shear currents are covered in review papers by Fenton (1973) and Peregrine (1976). The reader is referred also to the report b y Brink-Kjaer (1976) and to the paper by Jonsson et al. (1978), wherein low-amplitude waves were assumed. As regards higher-order wave theories, the numerical stream function theories should especially be noted. This kind of theory was developed by Dean (1974, and further references therein) for the case of irrotational currents, and was generalized by Dalrymple (1973, 1974) to the cases of linear and bilinear currents. The m e t h o d consists in expanding the stream function to arbitrary orders in the steepness, and then determining the expansion coefficients by minimizing the boundary condition errors on a computer. The advantage of this kind of approach is its analytical accuracy, whereas a drawback is the necessity of presenting the results in tabular or graphical form. The next section presents the basic equations of the problem and the general approximation scheme. Thereafter the calculations are carried out in full to the third order. As expected, the first non-vanishing correction to the phase velocity, itself being of second order, is determined from the thirdorder boundary equations at the t w o interfaces of the current. Explicit expres-
240 sions are given for all third-order expansion coefficients appearing in potentials and surface elevation functions. Also, the simplifying case of deep currents is discussed. BASIC EQUATIONS AND EXPANSION SCHEME
In order to have some idea about the practical applicability of the theory developed in this paper, consider, for instance, the typical situation encountered in a pneumatic breakwater: periodic waves of initial wavelength Lin, initial amplitude ain and angular frequency w propagate on deep water from minus infinity into the opposing surface current in the breakwater and gradually become deformed as the current strength gradually increases. In practice, it often gives a reasonable approximation to assume that the undisturbed current satisfies the two requirements. (1) Its horizontal velocity profile is given by eqs. 2 for any value of the horizontal coordinate x within the established region of the current. (Actually, in a real breakwater there is also a reverse current, i.e. a current propagating in the positive x-direction, beneath y = - h . However, this reverse current will have only a moderate effect on the behaviour of the waves as it is usually weak and moreover directed parallel to the wave velocity, so it may to a good approximation be neglected.) (2) The variation of the current in the x-direction is "slow". That is, the variation of U over a distance equal to the wave length L is small: L d U / d x << U
(3)
This means that the waves may be considered to travel in locally homogeneous water. And such a physical situation may be adequately described by the idealized model assumed in this paper, viz. that the surface velocity U and the current depth h are constants, independent of x. In other words, our model is the one of an infinite train of finite-amplitude periodic waves propagating on a uniform current of constant vorticity U/h. Now it is possible to introduce a velocity potential in the following way. The linearity of the current profile permits separating off the rotational part of the velocity as -(1 + y / h ) Uex, where ex is the unit vector in the x-direction. As the remaining part of the velocity is irrotational, it may be written as the gradient of some potential function (P. (This property was discovered by Sun' Tsao (1959), and was emphasized also by Brink-Kjaer (1976).) As Bernoulli's equation will be used in the expansion procedure below it is necessary, because of the rotational properties of the motion, to leave the inertial frame to which the above considerations pertain (the laboratory frame, also called the absolute frame), and transform to the moving inertial frame where the profile of the free surface does n o t change with time. If c = w / k denotes the phase velocity in the laboratory frame, it is clear that the velocity of the moving frame with respect to the laboratory frame is equal to c. In the moving
241
frame the horizontal and vertical components of the total velocity in the rotational region can thus be written: u(x,y) = - ( 1 + y [ h ) U +
Cx(X,Y) (4)
v(x,y) = ~y(x,y)
in which indices mean partial derivatives. Here q~x includes also the term - c (see eq. 9a below). In the irrotational region beneath the current the motion may be described simply by a velocity potential ~ (x,y). We have in this region: u(x,y) = ~(x,y),
v(x,y) = ~y(x,y)
(5)
We let ~ and ~ denote the elevations of the free surface and the lower interface of the current, respectively. The fundamental independent variable is the wave number k (or wavelength L). Mathematically speaking, the whole flow region is divided by the lower interface of the current (y = - h + ~') into t w o simply connected regions in which the velocity potentials are each single-valued. The potentials • and are discontinuous across this interface due to the appearance of the first term to the right in the first of eqs. 4. For the determination of the velocity potentials, surface elevations and dispersion relation one has at one's disposal the incompressibility conditions: V 2 q~ = 0
(6a)
V2~ = 0
(6b)
the kinematic conditions at the upper and lower interfaces of the current: u(x,~)~x(X) - v(x,~) = 0
(7a)
u(x,-h
(7b)
+ ¢ ) ¢ x ( x ) - v ( x , - h + ~) = 0
and the Bernoulli equation: PIp + IA(u2 + v 2) + g y = C
(8)
In the rotational region, the Bernoulli " c o n s t a n t " C in general varies from streamline to streamline. In the irrotational region, on the other hand, it is constant throughout the fluid. As the lower interface of the current is a streamline, it follows from Bernoulli's equation that (u 2 + v 2 ) = continuous across this interface. Having written down the basic equations and boundary conditions, we n o w consider the expansion scheme. We expand the two potentials, the elevation of the two interfaces, and the phase velocity (in the laboratory frame) in a
242 small parameter e representing the wave steepness of the free surface: q~(x,y) = - c x + ecp, ( x , y ) + e 2 ~ 2 ( x , y ) + e3cP3(x,y)
(9a)
¢ ( x , y ) = - c x + edpl (X,y) + e2i~2(x,y) + e3¢3(x,y)
(9b)
~(x) = e~l (x) + e ~ 2 ( x ) + e3~3(x)
(9c)
~(x) = e~l (x) + e2~2(x) + e3~3(x)
(9d)
c = Co + ecl + e2c2 + e3c3
(9e)
Here the subscripts refer to the order of a p p r o x i m a t i o n (not to be confused with derivatives.) As m e n t i o n e d above, we are adopting a model in which periodic waves (of infinite extension) are propagating on a u n i f o r m rotational current. The periodicity enables us t o make the assumption t hat the potentials q~ and ~ are periodic functions in x, i.e. t he y are assumed n o t to contain any term proportional to x or t. Physically, this means t ha t the mean Eulerian horizontal fluid velocity below the troughs o f the interface is equal to zero in the l aborat ory inertial frame. (Recall t hat in the analogous situation in ordinary wave t h e o r y (w i t h o u t currents) the r e q u i r e m e n t of a periodic velocity potential implies that the mass tr an s por t in the l a bor at or y frame is different from zero.) Note, however, th at the periodicity r e q u i r e m e n t u p o n q~ and ¢ does n o t imply t hat the expansion functions q~n and Cn are necessarily simple harmonic functions, like sin ( n k x ) , say. Actually it will turn out t h at q~3 has to contain terms with sin k x in addition to terms with sin 3 k x (see eq. 41). A n o t h e r assumption made in the calculation below is t hat the level y = 0 is taken to coincide with the m e a n level o f the free surface. Thus the mean value of ~ vanishes, ~ = 0, so that the various expansion functions Vn in eq. 9c are periodic in x with zero mean. Physically, this means t hat within a horizontal region of one wavelength there is just the same a m o u n t of fluid above the level y = 0 as below it. As the fluid is regarded as incompressible, the mean level of the l o w e r interface of the current must t her e f ore be lying at the position y = - h . This implies in turn that ~"= 0, so that the functions ~n in eq. 9d are similar to ~?n periodic in x with zero mean. If we fix the origin of coordinates vertically below a wave crest, it follows that the expansion functions Vn and ~n contain only terms symmetric in x. It will turn out later that the third order functions 73 and ~3 are containing terms with cos k x in addition to terms with cos 3 k x . The Bernoulli constant in eq. 8 at the free surface may be written as: C = p o / p + 1/2(U+co) 2 +eel +e2C2 +e3C3
(10)
(free surface) where P0 is the atmospheric pressure. The values of the various expansion coefficients Cn in eq. 10 will be written out below, at each perturbative step, although these coefficients are of no importance for the calculation of poten-
243 tials and elevations. In each case Cn may be calculated by taking the mean of the Bernoulli equation over a wavelength. Note that the value of C is calculated as a consequence of the periodicity assumptions stated above; C is n o t an input parameter here. Before embarking upon the calculation, it is convenient for later reference to give here the explicit expressions for the velocity components u and v at the two interfaces of the current, when all terms up to the third order are included. At the free surface we have:
u(x,,7)=-(U+co)+~ + ~IdPlxy
-C~-h~+*~x
+E~ - e ~ - h ~ + * ~ x +
--C3 -- h 773 + dp3x + 771(1)2xy + ~2dPlxy + I~7721(I~IXyy (lla) v(x,~) = e,t,~y +e2(,t,2y +~,:Ihyy) + + e3
+ e3(dP3y + ~ldP2yy + ~72dPlyy + 1~WI2 dPlyyy)
(lib)
where the terms to the right are to be evaluated at y = 0. At the lower interface w e have to distinguish between the rotational and the irrotational regions. If we let the R o m a n numeral I refer to the rotational region, then:
u(x,-h+~)lI=-C o +e
-c, - - ~ 1
+*,x
+e 2
-c2--~2
+ e3 --C3 -- -~ f3 + dP3x + fldP2xy + f2gPlxy + 1/2f21~Plxyy
v(x,-h + ~) II = eCbly + e~(Cb2y + ~lCblyy) + e3(&3y + ~lCb2yy + + ~2dPlyy + 1/2~dPlyyy)
+*2x +[1*ixy (12a)
(12b)
where now the terms on the right refer to y = -h. Similarly, if the numeral II refers co the irrotational region, we have: u ( x , - h +~)ll.i = - C o + e ( - c 1 +(/)ix) +E2(-c2 +~)2x + ~'l{~lXy) +
+ 63(-C3 + Cax + ~'l~b2xy + ~'2¢lxy + l&~'~¢lxyy)
(13a)
o(x,-h + ~) 1~I = e~)ly + E2(~2y + ~l¢lyy) + e3(~)3y + ~1~)2yy + + f2¢lyy + 1hf2~blyyy )
(13b)
again referring to y = - h on the right hand side. Finally, the following point regarding the boundary conditions at the lower interface of the current should be noted. Firstly, the kinematic condition,
244 eq. 7b, implies that the fluid at the lower interface is moving in a direction parallel to the profile tangent. Further, as the Bernoulli eq. 8 requires that (u 2 + v 2) be continuous across the interface, it follows that the continuity property must hold for the velocity components u and v separately. In other words, the equations:
)pii
114)
v ( x , - h + ~')l I = v ( x , - h + f ) l i i
(15)
u(x,-h +
)1I = u ( x , - h +
hold at the lower interface of the current. FIRST-ORDER APPROXIMATION
Insertion of the Stokes expansions in the equations of motion yields to the first order in e: (U+co)V~x
+q)ly=O
(16a)
(y=O)
U
[ g + h ( U + e°)]7~l
(U+e°)t~lX
=
Codex + ~Piy = 0
(y = - h )
U -- --
~1
+ (I)lx
(Y = - h )
= ~lX
h qhly = ely
(y = - h )
(y = O)
(16b) (16c) (16d) (16e)
To the first order, eq. 16a is the kinematic condition 7a at the free surface. Eq. 16b is the Bernoulli eq. 8 at the same surface. Note that the rotational part of the current, as expressed by the term -(1 + y / h ) U in eq. 4, contributes to eq. 16b. The first order Bernoulli coefficient C1 follows as: C~ = ( U + Co)C,
Equation 16c is the first-order version of the kinematic condition 7b at the lower interface of the current, whereas eqs. 16d,e express the velocity conditions 14, 15. For the potentials we make the following assumptions: U + Co c~eky + fie - k y qh ( x , y ) = ~ sinkx
k
¢~1( x , y ) -
a-~
U + c o 7 e k(y+h)
k
a-~
sinkx
(17a)
(17b)
These expansions satisfy eqs. 6. We choose the expansion parameter as e = k (a - ~). The linear dispersion equation can now be derived by inserting
245 eqs. 17 in eqs. 16. It is convenient to give the dispersion equation in dimensionless form. As in an earlier work {Brevik, 1976) we use the dimensionless variables Y = U/co,
(18)
Z = hg/U 2
(actually Z -- F -2 , where F is the Froude number}. The dispersion equation can then be written as:
[
][
-(1+Y}2 +~{l+y+zy) =
-
-
1 + y)2 +
kh
2+~
(1 + Y + Z Y ) k-£
=
]
(19)
e -2kh
which agrees with the equation given by Taylor (1955) in his study on pneumatic and hydraulic breakwaters. Now assume that the current strength U, current depth h, and wave number k are the independent variables. Thus the independent dimensionless variables are Z and k h , and the remaining dependent variable Y can be calculated from eq. 19, which is actually a cubic equation for Y. Fig. 2, which has been plotted from fig. 6 in Taylor's paper (1955), shows Y as a function of kh with Z as a parameter. Once Y has been determined from the diagram, the lowest order phase velocity follows immediately as co = U / Y . 6
[
I
--
I
I
I
1
5
4
/
0
[ 0
I 1
I 2
I 3
I 4
I0~
I 5
6
kh
Fig. 2. Solutions of eq. 19. "I~e figures give the constant value of Z for each curve.
246 With the above c h o i c e f o r the p a r a m e t e r e, o n e finds for the f u n c t i o n V~: 1 k coskx
V,
(20)
T h e first o r d e r elevation o f the free surface t h e n b e c o m e s : n=en,=
acoskx
(21)
where: a=O/--~
(22a)
is t h e a m p l i t u d e and: e -2kh c~
(22b)
1 + 2kh/Y
T h e e x p a n s i o n p a r a m e t e r is thus simply e = k a . At t h e l o w e r i n t e r f a c e the elevation f u n c t i o n ~ b e c o m e s : Zl ~1 = - -
e-kh
coskx
(23a)
k where I+Y z, =
Y
1 +
(23b) [1 - e x p ( - 2 k h ) ]
2kh
T h e p o t e n t i a l f u n c t i o n (P~ in the r o t a t i o n a l region is n o w d e t e r m i h e d b y eqs. 17a and 22b. F o r later use it is c o n v e n i e n t t o write it as follows: C0 ~P, = ~ (A ~e ky + B i e - ky ) sin k x
(24a)
where: Al =
( Y) 1+
Zl
(245)
2kh B, -
Y
z, e -2kh
(24c)
2kh
are dimensionless quantities. In t h e i r r o t a t i o n a l region b e n e a t h the c u r r e n t we find t h e f o l l o w i n g p o t e n tial f u n c t i o n : Co ~1 = --~ z , e k y s i n k x
(25)
247 SECOND-ORDER
APPROXIMATION
The first-orderexpressions calculated above are n o w used as input values in the second-order equations. W e firstwrite d o w n the latter equations in general form:
(U÷co)~12x ÷dP2y= g +~
( U + eo)
Co~2x ÷ ~2y -~
U -
-
(
( -el-h~71 "
÷dPl "rhx-'fh~lyy
(y=O)
r~ - ( U + eo)~2x =C2 -1/~e~ - ( U + eo)e2 + e l ~ x
)
--Cl -- h ~1 ÷ (~ix ~ix - ~l(~lyy
~: + ~2x - ¢~x = 0
(Y = -h)
(y = -h)
(26a)
-
(26c) (26d)
h ~b2y -- ~2y =
--(dPlyy--~)lyy)~i
(26e)
(Y = -h )
The expansions used here are given in eqs. 9, 10, 11, 12 and 13. Eq. 26a is the kinematic condition at the free surface, eq. 26b the Bernoulli equation at the same surface, eq. 26c the kinematic condition at the lower interface, and eqs. 26d, e are the two conditions on the velocity given by eqs. 14, 15 at the lower interface. The equations have been written such that the periodic second-order functions appear on the left-hand sides. Substituting for the first-order expressions on the right-hand sides of eqs. 26, we obtain for eq. 26a: (1 + Y)r~2x + - - q~2y = - - sin k x Co
A, + B~ -
sin 2 k x
(27a)
Co
(y = 0)
T h e BernouUi eq. 26b b e c o m e s : Y(I+Y+ZY)r~2
1 (I+Y)~
el
h
Co
e0
÷--4
3 ( 1 ÷ y)2 _ A~ +B~ - ~ -
(
A~ +B~ -
cos2kx
5)
coskx+
(y -- 0)
(27b)
In writing this e q u a t i o n we have m a d e use o f t h e r e q u i r e m e n t t h a t t h e lefth a n d side be a p e r i o d i c f u n c t i o n in x with zero mean. This d e t e r m i n e s the
248 s e c o n d - o r d e r Bernoulli c o e f f i c i e n t C2 at t h e free surface:
C2 =1//2C21 "t'(Ul-c°)c2 --4 (U'{'cO
4 CO2 A1 +Bl - ~-h
Finally, the t h r e e e q u a t i o n s 26c, d, e at t h e lower i n t e r f a c e take t h e f o r m :
c,
~2x + - - ~P:y=-- zl C0
sinkx -
(i
+
sin2kx
(27c)
C0
(y = - h ) Y 1 - -- ~2 + - - (¢P2x - ~ ) h c0
= 0
1 - - (¢P2y - ¢2y ) - -
z~e -2kh sin 2kx
Co
Y
2kh
(y = - h )
(27d)
(y = - h )
(27e)
F r o m t h e t w o e q u a t i o n s at t h e level y = 0 (eqs. 27a and 27b), we m a y eliminate 72 a n d t h u s o b t a i n a s e c o n d - o r d e r differential e q u a t i o n f o r ¢P2- With the following a s s u m p t i o n : C0
¢P2 = --~ (A2 e2ky + B2 e - 2 k y )sin 2kx
(28)
which is analogous t o eq. 24a, it follows i m m e d i a t e l y that: cl = 0
(29)
T h e r e f o r e , t h e first non-vanishing c o r r e c t i o n t o t h e phase v e l o c i t y is at least o f the s e c o n d o r d e r in e. This b e h a v i o u r is just analogous t o t h a t e n c o u n t e r e d in the case o f waves p r o p a g a t i n g o n still w a t e r (Wehausen and Laitone, 1960). Similarly i n t r o d u c i n g t h e f o l l o w i n g a s s u m p t i o n s f o r t h e t w o s e c o n d - o r d e r elevation f u n c t i o n s : e2
72 = -
cos 2 k x
(30)
k
Z2
~2= -
e - 2 k h cos 2 k x
(31)
k we find t h e following expressions f o r the s e c o n d - o r d e r coefficients: A2 = 4
1+
-3(1+Y)
2 +3(A1 +B~)
+k2h2
A~ +B~ +
kh - (A1 + e l )
1 +--e
2J~h
-2kh sinh2~
Y , e-2kh cosh 2kh ) 2kh (32)
249
Y(z2 + l + ~,,2) 4kh
B2 = - ~ e2
1 + Y
z~ = ~,~z 2, +
(33)
e-4kh
1,4 A t + B t -
(34)
+A2 -B2
A2 1 + Y/4kh
(35)
Finally, for the potential function ~2 we obtain the following expression: Co ¢2 =--~ (Z2 -
lhZ~)e2ky sin 2kx
(36)
These expressions enable us to calculate velocities and surface profiles to the second order. However, the degree of approximation is still too low to permit determination of the first non-vanishing correction c2 to the phase velocity. A main purpose of the next section is to calculate c2 from the third-order equations. THIRD O R D E R APPROXIMATION; THE PHASE VELOCITY
The starting point is the set of third-order equations at the two boundaries of the current: ( U + Co)173x + ~3y ffi "c2'01x + ~Plx172x + ¢~2x~lx - r/2~lyy - 171~ 2 y y -
U -h
(y = 0)
(Y~l~2X + 1~21~1x) -I-l~ll,~lx{i)txy - 1 / ~ 2 ~ l y y y
g+-:"(U+co
73-(U+coI¢3x=C3-(U+co)c3
+ ( U + Co)(~2,I, lxy + ~ 2 ~ l x y y + ~ 2 = y ) - ~2x -- 171~txy)
Co~3x + '~3y = (-'c2
-- ~ l y ( ~ 2 y
+ 171~lyy)
+ ~t - ¢ 1 x
o c2 +~,72 -
(Y = 0)
(37b)
U ~2 + ~.2x + ~tCtxy)~,x - ( U -~
-- ~1 ¢I~2yy -- ~2CI}lyy -- t/'Jfl~I~lyyy
(37a)
-
(y ffi - h )
(37c)
U - ~ ~3 + &3x - ~b3x = l ~ ( ~ t x y y - @txyy )~2t + ( ~ l y y - ~ l y y ) ~ l ~ l x
(37d)
(y = -h) ~3y -- ~b3y = --(dP2yy -- {.b2yy)-~'t -- (¢I~lyy - ~blyy )~'2
(y = - h )
(37e)
250
To the third order, eq. 37a is the kinematic condition at the free surface, eq. 37b is the Bernoulli equation at the same surface, eq. 37c is the kinematic condition at the lower interface, and eqs. 37d, e are the third order versions of eqs. 14, 15. In writing the equations we have made use of the first order eqs. 16 to obtain some simplifications, and we have taken into account that cl = 0 . The known first- and second-order expressions are now inserted on the righthand sides of eqs. 37. Eq. 37a takes the form: ( l + Y ) ? 3 x + - - ~P3y= C0 --3
[:
(I+Y)-A2
--8
- B 2 -1A A1 +B~
(
(I+Y)+A2
+B2 +IA A1 +B1 -~-~ e2
sin3kx
)e2] sinkx (y = O) (38a)
and similarly for the Bernoulli eq. 37b: -
-
_
h
_
=
Co +
A1 + B 1 -
-~ Y)] -
-8(1+ ---A2-B2 00
+IA 3 ( 1 + Y ) 2 + ~
e2 I c o s k x + t - 8 l(l1 + Y )
A~ +B~ -~-£
(A2 +B2)+~A
+ -kh - A~ +B1 - ~-~
e2
Y) A1 +Bl -
( At + BI - 1 - 8~ Y h)
+ Al + B l -
-
7(1+Y) 2 +
cos 3 k x
(y = 0)
(38b)
Here we have put: C3 = (U + co)C3
in order to make the expression on the right-hand side periodic in x with zero mean. At the lower interface of the current it is advantageous to make use of the first and second-order boundary conditions at the same interface so that z2 appears as the only second-order coefficient on the right-hand sides of the equations. For the kinematic condition 37c we obtain:
[c +lz
~3x + - - dP3y . . Co - Co
. 8
.
[9 2 9( Y) + 8zl -~ 1 + 3kh
1 + 2
z2
z2e - 2 k h
]
z l e -kh sinkx +
3kh
]Zle_3kh
sin3kx
(y = - h )
(38c)
251 and for the velocity conditions 37d,e we have: Y
- h ~s + -
1
Co
(¢ax - ~ b s x )
1 -- (q)3y - ~3y ) Co
Y 8kh
z ~ e - 3 k h ( c o s k x + 3 cos 3 k x )
Y - - Z l Z 2 e - 3 k h ( s i n k x + 3 sin 3 k x ) 2kh
(y
-h)
(38d)
(y = - h )
(38e)
The task is now to solve the five eqs. 38. It is convenient first to eliminate the elevation functions 77 and ~. Differentiating eq. 38b with respect to x and combining with eq. 38a, we have: 1 1 (I + Y)&3~ + - - (A, + B,)¢3y = kCo Co
(
+2
A, +B,-~'~
-2
, +B,
-
-A:-B:
-
(I + Y)
- 2
e2
B1
-
+
) [
+1~ 3 ( 1 + y ) 2 _
(A2 + B 2 ) + I A
A, + B~ - ~'~
+
1
7(1+Y)2
_
(y ; 0)
sin 3 k x
(39)
Correspondingly, differentiating eq. 38d and combining with eq. 38c, we have: 1
Co
((P3~ - ~ 3 ~ ) + - -
(I)3y
hco
9Y 2h
h
-2
ZlZ2e - a k h s i n 3 k x
1 +
1 +
\)
z2e -2kh
(y
-h)
3k
z 1
-
(40)
3kh
There are thus three equations at our disposal for the determination of the potentials I,, 4~ and the velocity correction c2, viz. eq. 39 at y = 0 and eqs. 40, 38e at y = -h. Each of these equations contains actually t w o equations, as the equality must hold when comparing terms with sin k x and sin 3 kx separately. As sin k x appears in the equations at both the two interfaces of the current, it is n o t possible to satisfy the equations unless some terms with sin k x are present in the third-order potentials. We shall solve the equations on the basis of the following assumptions: CO
q~3 = ~ - [(A31e ky + B 3 t e - h Y ) s i n k x + (A33e 3ky + B 3 3 e - 3 k Y ) s i n 3 k x ]
(41)
CO
~3 = -~- f33e 3ky sin 3 k x
(42)
252
These are the simplest expressions one can construct, requiring that the Laplace equation is satisfied and also that there are enough coefficients to permit a consistent solution of the boundary equations. We may thus assume that @3 varies with x as sin 3 k x , whereas ¢P3 must contain terms with sin k x also. The equations arising when comparing terms with sin k x are the most interesting ones, as the expression for the velocity correction may be derived from them. Inserting eqs. 41 and 42 in eqs. 39, 40 and 38e, we obtain for the velocity correction:
c~t=
Co
4 (1 + Y )
(
A1 + B I -3kh2Y
) ( +
~-)
A~ +B1 - 2 k h
(A2 + B 2 ) -
A1 + B1 - 2 k ~
and the second-order phase velocity e in the laboratory frame follows as: (43b)
c = Co + k2a2c2
where a is the first-order wave amplitude. Note that a is an external parameter in the theory; it has no relation to the calculated first-order coefficients A~ and B1. The expression 43a for the velocity correction is the main result of this paper. The comparison of terms with sin k x also gives the following expressions for the coefficients A3,, B3~: A31
=-2kh
B31
=--Y 2kh
(cj0 -
+z2e -2kh
)
(44)
z1
( - - -c2 + 2 z 2 e - 2 k h ) z~ e_2k h
(45)
Co
whereas the comparison of terms with sin 3 k x determines the remaining coefficients in eqs. 41, 42: A33 =
t
-2(1+Y)
+IA
t
(
A1 +B1 -
7(1+Y) 2-
1
+B~ -
A1 + B ~ - 3 ( I + Y ) - ~ - ~ [ A ,
"
-2
(
Al +B1
e2 + ~ ' £
[A1 +B1 + 3 ( l + Y ) l z ~ z 2
+B~ + 3 ( 1 + Y ) 1
(
2+
e-6kh
e-~
1
(46)
253 f33 =
(
B33
~-~ ~zlz2
1 +
Y
A33
(47)
1
e -6~h
(48)
The forms of the t w o surface profiles are determined b y going back to the original boundary equations 38. With the following assumptions: 1 n3 = -~ (e31 cos kx + e33 cos 3kx)
(49)
1 ~3 = ~ (z31 e -kh cos kx + z33 e -3kh cos 3kx)
(50)
(cf. eqs. 30, 31), we obtain from the kinematic eq. 38a at the free surface:
e3~ =-8 + ~ I + Y e33 = - - 8 + ~ l + y
[°:: -
(
+A2 +B2 +1,~ A~ +B~ -
A2 +B2 +1,~ A~ +B~ -
5)
e2 +A3~ -B31
]
e: + A 3 3 - B 3 3
(51) (52)
and similarly from one of the velociW conditions, eq. 38d, at the lower interface.: Z31 ----
=
----+2,z2-~-~ e0 -
z,
+
Zl
(53)
(54)
In eqs. 53, 54 we have used the boundary equations at the lower interface to simplify the expressions. The expansion scheme to the third order is n o w completed, and it may be convenient to summarize here the use of the formulas. The strength and depth of the current, and the wavelength, are parameters assumed given initially. This determines the nondimensional quantities Z and hh. From Fig. 2 we may determine Y graphically, and thereafter calculate the phase velocity according to first-order wave theory as Co = U/Y. The first order coefficients z~, A 1, B~ follow from eqs. 23, 24. The second-order coefficients A2, B2, e2, z2 are thereafter calculated from eqs. 32--35. The first non-vanishing correction to the phase velocity is calculated in third-order theory and is given by eqs. 43. The other third-order coefficients are given by eqs. 43--54. One final remark is in order, as regards the initial assumptions 41, 42 for the potentials. As mentioned earlier, these are the simplest expressions that one may construct, compatible with the Laplace equation and also containing the necessary number of coefficients to be determined by the boundary equations.
254 In principle t h e r e is, however, n o t h i n g which prevents one f r o m adding a n o t h e r t e r m , o f the f o r m (co/k)f31 e x p ( k y ) sin k x , t o the p o t e n t i a l 03 in the i r r o t a t i o n a l region. Does such an e x t r a t e r m lead t o a n y change in o u r main result, which is the e x p r e s s i o n f o r c2/Co? The answer is no: w h e n calculating c2/Co the e x t r a t e r m with f31 disappears f r o m the formalism, and we end up with the same e x p r e s s i o n as t h a t given in eq. 43a, c o r r e s p o n d i n g t o f3~ = 0. On the o t h e r hand, it turns o u t t h a t the t e r m with f31 induces changes in the expressions f o r A3~, B3~, e3~, z3~, as t h e b o u n d a r y e q u a t i o n s emerging f r o m the c o m p a r i s o n o f t e r m s with sin k x and cos k x b e c o m e changed. We shall n o t give these m o r e general expressions here. By c o n t r a s t , the c o e f f i c i e n t s A33, B33, f~s, e33, z33 will n o t b e c o m e changed, as these c o e f f i c i e n t s are d e t e r m i n e d f r o m t h e e q u a t i o n s emerging f r o m t h e c o m p a r i s o n o f terms with sin 3 k x and cos 3 k x . These e q u a t i o n s are u n a f f e c t e d b y fsl- D i f f e r e n t values o f f31 corres p o n d to slightly d i f f e r e n t fluid velocities and surface elevations. EXAMPLES In o r d e r t o illustrate t h e m a g n i t u d e o f t h e various t e r m s in the expansions, we shall give t w o exampleS. Example 1
We shall first c o n s i d e r a relatively small scale situation. Let U be c h o s e n equal t o 1.50 m/s; this is a c t u a l l y a measure o f the m a x i m u m surface v e l o c i t y o b t a i n e d in Bulson's p n e u m a t i c b r e a k w a t e r e x p e r i m e n t ( 1 9 6 1 , 1 9 6 3 , 1968). F o r the c u r r e n t p a r a m e t e r we c h o o s e Z = 4, which c o r r e s p o n d s t o a c u r r e n t d e p t h h = Z U 2/g = 0 . 9 2 m. T h e wavelength L is d e t e r m i n e d b y the i n d e p e n d e n t p a r a m e t e r kh; we c h o o s e k h = 1.5, c o r r e s p o n d i n g t o L = 27r/k = 3.85 m. F r o m Fig. 2 we m a y read o f f Y = 1.03 (alternatively, eq. 19 m a y be solved t o o b t a i n this a c c u r a t e value), so t h a t Co = U / Y = 1.46 m/s. We obtain: Z 1 1.531 A2 = - 0 . 4 0 4 z 2 = 0.827 A3, = - 0 . 3 4 2 B33 = 0 . 0 0 0 0 5 5 e33 ~- 0 . 0 6 2 4 ----
A, = 2 . 0 5 6 B2 = 0 . 0 0 0 8 5 e:/co = 0 . 6 9 2 Bs, = - 0 . 0 1 6 0 f33 = 0 . 1 0 4 z31 = - 0 . 9 8 7
B, = 0 . 0 2 6 2 e2 = 0 . 2 2 9 A33 = 0 . 1 1 6 e3, = - 0 . 2 4 6 z33 = 0 . 6 5 8
We m a y write o u t t h e expressions f o r t h e t w o surface elevations, and the phase v e l o c i t y in t h e l a b o r a t o r y frame: = a (1 - 0 . 6 5 9 a 2 )cos k x + 0 . 3 7 4 a 2 c o s 2 k x + 0 . 1 6 7 a 3 c o s 3 k x = a ( 0 . 3 4 2 - 0 . 5 8 9 a 2 )cos k x + 0 . 0 6 7 a 2 c o s 2 k x + 0 . 0 2 0 a 3 cos 3 k x C/Co = 1 + 1 . 8 5 1 a 2
(55)
255
w h e r e t h e a m p l i t u d e a is t o b e i n s e r t e d in m e t e r s . T h e s e e x p r e s s i o n s m a y be c o m p a r e d w i t h t h e c o r r e s p o n d i n g e x p r e s s i o n s f o r waves p r o p a g a t i n g o n still, d e e p w a t e r (Wehausen a n d L a i t o n e , 1 9 6 0 ) : ~ ( U = 0) = a ( 1 + 0 . 3 3 4 a 2 )cos k x + 0 . 8 1 8 a 2 cos 2 k x + 1 . 0 0 2 a 3 cos 3 k x (56) c -
( U = 0) = 1 + 1 . 3 3 7 a 2
-
C0
Example 2
C o n s i d e r n o w a large scale s i t u a t i o n w i t h s t r o n g c u r r e n t : U = 2.5 m / s , L = 80 m. T a k i n g h = 6.37 m we o b t a i n Z = 10, k h = 1/2. F r o m Fig. 2, o r m o r e acc u r a t e l y b y solving t h e c u b i c dispersion e q u a t i o n 19 w i t h r e s p e c t t o Y, we d e t e r m i n e Y = 0.243. T h e l o w e s t o r d e r p h a s e v e l o c i t y is Co = U / Y = 1 0 . 2 9 m/s. F u r t h e r c a l c u l a t i o n yields f o r t h e coefficients: zl = 1 . 0 7 7 A: = -0.137 z2 = 0 . 4 5 8 A3, = - 0 . 0 8 1 4 B33 = 0 . 0 0 5 9 6 e33 = 0 . 1 5 7
AI = 1 . 3 3 9 B2 = 0 . 0 1 7 1 c2/Co = 0 . 4 7 9 B31 = - 0 . 0 1 3 7 f33 = - 0 . 0 0 2 9 3 z3, = - 0 . 3 0 2
B1 = 0 . 0 9 6 3 e2 = 0 . 3 5 6 A33 = - 0 . 0 0 3 1 7 e31 = - 0 . 0 2 5 9 z33 = 0 . 2 6 8
so t h a t t h e surface e l e v a t i o n s a n d t h e p h a s e v e l o c i t y m a y be w r i t t e n as: = a (1 - 0 . 0 0 0 2 a 2 )cos kx + 0 . 0 2 7 9 a 2 cos 2 k x + 0 . 0 0 1 0 a 3 cos 3 k x 3 k x = a ( 0 . 6 5 4 - 0 . 0 1 4 4 a 2 ) c o s k x + 0 . 0 1 3 2 a 2 cos 2 k x + 0 . 0 0 0 4 a 3 c o s
(57)
C/Co = 1 + 0 . 0 0 2 9 6 a 2
T h e f o r m u l a s c o r r e s p o n d i n g t o t h e case o f still, d e e p w a t e r are given b y eqs. 56, as b e f o r e . THE CASE OF DEEP CURRENTS
In p r a c t i c e it o f t e n t u r n s o u t t h a t t h e c u r r e n t is s u f f i c i e n t l y d e e p in c o m p a r i s o n to t h e w a v e l e n g t h t o p e r m i t certain s i m p l i f i c a t i o n s in t h e f o r m u l a s . F r o m t h e linear dispersion eq. 19 it is clear t h a t t h e r i g h t - h a n d side b e c o m e s negligible if t h e c o n d i t i o n : kh~
2.5
or:
(58)
h~O.4L
is satisfied. F o r instance, it t u r n s o u t t h a t this c o n d i t i o n is satisfied w i t h i n t h e region o f t h e r a t h e r s t r o n g c u r r e n t s p r o d u c e d in Bulson's b r e a k w a t e r experi-
256 ment (Bulson, 1963, 1968). For a typical long wave in that experiment, with incident wavelength on still water being about 9 m, it turns out that the parameter k h reaches the value 2.5 already when the supplied power is about one half of the power required to stop the waves completely. Approximative formulas will now be given for the case that the condition 58 is satisfied. First of all, the left-hand side of eq. 19 can be put equal to zero, giving a quadratic equation that can be solved explicitly to give: Y-
-
k h -1/2 +
co
(59)
khZ +
Z + 1 - kh
It is thus easy to calculate Y, and hence Co, when the wavelength and the current data are known. When writing out the expansions, it is further convenient to transform to the laboratory inertial frame in which the terms ( - c x ) in the potentials, cf. eqs. 9, are absent. We shall give the approximative expressions for potentials and surface elevations up to the second order in e, and also the expression for the phase velocity: ¢ = ac0(1 + Y)e ky sin(kx - cot) +
Y(
+ a : c o ~-~
1)
- 3 +
e 2ky sin 2 ( k x - cot)
(60)
khl+Y
I+Y = aco i + Y / 2 k h
eky sin(kx - w t )
ka [
= a c o s ( k x - cot) + ~ -
1
(61)
2Y
y2
1 -
-
.4.
kh I + Y
-
1
] cos 2 ( k x - co t)
-
2k2h 2 ( l + Y ) 2
(62) I+Y
e - k h cos(kx - cot)
=a
(63)
1 + Y/2kh c
co
l+k2a --
y3
2
(1 +Y)2 2(l+Y-Y/2kh) 1
k3h 3 (l+y)3
y4 +- -
r1 L
--
1
8k4h 4 (l+y)4
2Y
1
kh
I + Y
1 j
1 4k2h 2 (l+Y)29Y 2
"b
-
-
-
-
(64)
Here co is the angular frequency in the laboratory frame. We have made use of the fact that the wave phase is a Galilean invariant. Terms with exp ( - 2 k h ) have been neglected in comparison with unity. In particular, in the limiting case where h -~ o% we obtain: = ac0(1 + Y)e ky sin(kx - cot)
(65)
257 = a cos(kx - cot) + 1/2k2a2 cos 2 ( k x - cot)
(66)
Co = V/~K - U
(67)
c = c0[1 + 1~(1 + Y ) k 2 a 2 ] = g v ~ ( 1 + l h k 2 a 2) - U
(68)
The expression 66 is identical to that obtained if U = 0 (Wehausen and Laitone, 1960). This is just what one should expect: the limit h -* oo corresponds to the case where the waves are propagating on a current uniform over depth, and this situation is differing only by a Galilean transformation from the simple situation in which the waves propagate on still water. Obviously, a Galilean transformation has no influence upon the surface elevation. Similar features are shared by the other equations above. For instance, the expression for c in eq. 68 is equal to the expression obtained by subtracting U from the expression in eq. 1, as one would expect. O n t h e w a v e p e r i o d u s e d as i n d e p e n d e n t p a r a m e t e r
The theory developed above uses k h as independent wave parameter. The wavelength L = 27r/k in the current is thus assumed to be a known quantity. In practice it is, however, the absolute period Ta in the laboratory frame, rather than the wavelength, which is the convenient external parameter. Under steady conditions, the value of T a is constant when waves propagate from still water into a current, whereas the wavelength is changing. Usually, it is necessary to observe the phenomenon in practice in order to determine L. For this reason it is desirable to give a short description of how to handle the problem if Ta, rather than k h , is the independent wave parameter. For concreteness, one may again imagine the breakwater situation alluded to in the beginning of this paper. Let us consider the general case with arbitrary current depth first. The basic condition is that the period Ta, or angular frequency co = 21r/Ta, is a conserved quantity. Thus: co = k c = kc0(1 + k2a2c2/Co)
(69)
The procedure for determining k is now rather complicated: firstly, one may obtain an analytic expression for Co = U / Y by solving the cubic dispersion eq. 19 with respect to Y; secondly, use is to be made of the expression 43a for c2/Co. By solving eq. 69 numerically, one may thus determine the value of k, if the wave amplitude a in the current is known. Once k is determined, the formalism developed earlier is applicable directly. The phase velocity follows most simply as c = co/k. The special case of deep currents simplifies the situation, as eqs. 59 and 64 become applicable for Y and c / c o , respectively. The case of an infinitely deep current, h -* 0% plays a particular role, as it is now possible to predict the value of the wave amplitude a in the current when the initial amplitude ain for the incoming waves on still water is known.
258 Actually, assuming an entirely t w o - d i m e n s i o n a l m o t i o n in w h i c h the transverse c o m p o n e n t o f the c u r r e n t was equal to zero, Longuet-Higgins and S t e w a r t ( 1 9 6 1 ) derived t h e f o l l o w i n g expression f o r the a m p l i t u d e ratio a/ain {our n o t a t i o n ) :
a/ain = [
gv~ co2(g v ~ ~ 2 U ) ]
1/2 (70)
Using this expression, and the e x p r e s s i o n 6 8 for c, it is n o w relatively easy t o d e t e r m i n e k n u m e r i c a l l y w h e n co a n d ain are k n o w n . The f o r m u l a 70 leads t o an infinite a m p l i t u d e ratio w h e n U a p p r o a c h e s ½x/g/k, and it is t h e r e f o r e applicable o n l y t o cases well b e l o w this limit. ACKNOWLEDGEMENTS I wish t o express m y g r a t i t u d e for valuable r e m a r k s f r o m Ivar G. J o n s s o n at the I n s t i t u t e o f H y d r o d y n a m i c s a n d H y d r a u l i c Engineering (ISVA), Technical University, C o p e n h a g e n , D e n m a r k and also f r o m Per B r u u n , NTH, T r o n d h e i m , Norway. This w o r k was s u p p o r t e d in p a r t b y Norges T e k n i s k - N a t u r v i t e n s k a p e l i g e Forskningsr~id.
REFERENCES Biesel, F., 1950. Etude th6orique de la houle en eau courante. La Houille Blanche, 5, Spec. A, pp. 279--285. Brevik, I., 1976. The stopping of linear gravity waves in currents of uniform vorticity. Phys. Norvegica, 8: 157--162. Brink-Kjaer. O., 1976. Gravity waves on a current: The influence of vorticity, a sloping bed, and dissipation. Ser. Pap., 12, Inst. Hydrodyn. Hydraul. Eng., Tech. Univ., Denmark. Bulson, P.S., 1961. Currents produced by an air curtain in deep water. Dock Harbour Auth., 42(487): 15--22. Bulson, P.S., 1963. Large scale bubble breakwater experiments. Dock Harbour Auth., 44(516): 191--197. Bulson, P.S., 1968. The theory and design of bubble breakwaters. Proc. Conf. Coastal Eng., l l t h , 2: 995--1015. Dalrymple, R.A., 1973. Water wave models and wave forces with shear currents. Tech. Rep., 20, Coastal Oceanogr. Eng. Lab., Univ. Florida, Gainesville, Fla. Dalrymple, R.A., 1974. Water waves on a bilinear shear current. Proc. Conf. Coastal Eng., 14th, 1: 626--641. Dean, R.G., 1974. Evaluation and development of water wave theories for engineering application. Spec. Rep. 1, Vols. I and II, U.S. Army, Corps Eng., Coastal Eng. Res. Center, Fort Belvoir, Va. Fenton, J.D., 1973. Some results for surface gravity waves on shear flows. J. Inst. Math. Appl., 12: 1--20. Jonsson, I.G., Brink-Kjaer, O. and Thomas, G.P., 1978. Wave action and set-down for waves on a shear current. J. Fluid Mech., 87: 401--416.
259 Longuet-Higgins, M.S. and Stewart, R.W., 1961. The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech., 10: 529--549. Peregrine, D.H., 1976. Interaction of water waves and currents. Adv. Appl. Mech., 16: 9--117. Schumann, E.H., 1974. South Africa National Research Institute for Oceanology, Internal General Report IG 74/14. Schumann, E.H., 1975. A.S. Shipping News and Fishing Review, 30(3). Sun' Tsao, 1959. Behaviour of surface waves on linearly varying flow. Moskow. Fiz.-Tekh. Inst. Issted. Mekh. Prikl. Mat., 3: 66--84. Taylor, G.I., 1955. The action of a surface current used as a breakwater. Proc. R. Soc. Lond., Set. A, 231: 466--478. Thompson, P.D., 1949. The propagation of small surface disturbances through rotational flow. Ann. N.Y. Acad. Sci., 51: 463--474. Wehausen, J.V. and Laitone, E.V., 1960. Surface waves. Encycl. Phys., 9: 466--778.
237
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H I G H E R - O R D E R WAVES PROPAGATING ON CONSTANT VORTICITY C U R R E N T S IN DEEP WATER
IVER BREVIK
Luftkrigsskolen, Trondheim-Mil, N- 7000 Trondheim (Norway) and Division of Port and Ocean Engineering, The University of Trondheim, N- 7034 Trondheim-NTH (Norway) (Received June 1, 1978; accepted November 14, 1978)
ABSTRACT Brevik, I., 1979. Higher-order waves propagating on constant vorticity currents in deep water. Coastal Eng., 2: 237--259. Using a Stokes expansion procedure, the third-order wave theory is developed for periodic waves travelling on a surface current whose undisturbed current profile is varying linearly down to some fixed depth h and zero otherwise. Deep water is assumed beneath the current. The main result is the expression for the first non-vanishing correction t o the phase velocity c; it is necessary to go to the third order to calculate this expression.
INTRODUCTION
Within a Stokes expansion approach it is known that, to the second order in wave steepness, the following expression holds for the phase velocity c of periodic waves propagating on quiescent, deep water (Wehausen and Laitone, 1960): c = x/g/k
(1 + 1/~k2a2)
(1)
in which k = 2 ~ / L is the wave number, L is the wave length, a is the wave amplitude, and g is the gravitational acceleration. Surface tension and viscosity have been neglected. (In fact, eq. 1 holds up to the third order.) If, in addition, there is a uniform opposing current present, whose undisturbed velocity U is constant everywhere (U taken positive for the opposing current), the absolute phase velocity can be found simply by subtracting U from the expression of eq. 1. This is an immediate consequence of the fact that the phase velocity transforms like a particle velocity under Galilean transformations. The main objective of the present paper is to derive the analytical expression for the phase velocity, correct to second order in the wave steepness, for periodic waves propagating on a shear current whose undisturbed velocity profile is varying linearly .from the free surface down to some depth h. The
238
l_ i3 LI y V ×
Fig. 1. Velocity profile in undisturbed current.
v e l o c i t y profile o f the h o r i z o n t a l u n d i s t u r b e d c u r r e n t is s h o w n in Fig. 1; with the c o o r d i n a t e axes o r i e n t a t e d as shown, the h o r i z o n t a l v e l o c i t y reads: u = -(1 +y/h)U
y >1 - h
u = 0
y < -h
(2)
Thus U d e n o t e s t h e c u r r e n t v e l o c i t y at the free surface. T h e m o t i o n is restricted t o be t w o - d i m e n s i o n a l , and d e e p w a t e r is assumed b e n e a t h t h e current. Dissipation is ignored t h r o u g h o u t . G e n e r a l l y speaking, t h e r e are f r o m a practical v i e w p o i n t g o o d reasons f o r pursuing the d e v e l o p m e n t o f the t h e o r i e s o f the w a v e - - c u r r e n t i n t e r a c t i o n . In the first place, the p r e s e n c e o f c u r r e n t s m a y alter the b e h a v i o u r o f sea waves drastically. As an e x t r e m e e x a m p l e , let us m e n t i o n t h e a b n o r m a l high waves t h a t can a p p e a r in the Agnlhas C u r r e n t n e a r S o u t h Africa ( S c h u m a n n , 1974, 1975). Here t h e surface v e l o c i t y was r e p o r t e d t o be a b o u t 2.5 m/s; if fully d e v e l o p e d sea waves with wave heights o f 6 m m e e t this c u r r e n t as an o p p o s i n g c u r r e n t , t h e result m a y be giant waves with wave heights o f 18 m or m o r e . Assuming t h a t t h e wave p e r i o d is a b o u t T = 12 s one finds, f o r instance b y use o f t h e curves given b y T a y l o r (1955), t h a t t h e absolute phase v e l o c i t y o f t h e waves (relative t o t h e g r o u n d ) is a b o u t 15 m/s. E v i d e n t l y such sea c o n d i t i o n s are dangerous even f o r the largest ships. In the s e c o n d place, it is desirable t o k n o w t h e wave---current i n t e r a c t i o n a c c u r a t e l y in o r d e r t o be able t o give reliable p r e d i c t i o n s f o r the forces on s u b m e r g e d structures, at least w h e n detailed m e a s u r e m e n t s o f t h e c o m b i n e d flow field are lacking. S i m p l y neglecting t h e p r e s e n c e o f c u r r e n t s m a y be un-
239 realistic. A third place where the wave- current interaction plays a major role is in the boundary layer near the b o t t o m . At present, rather little is known about shear stresses, b o t t o m stability and sediment transport in a combined wave--current system. The present work, however, is concerned only with the wave- c u r r e n t interaction in the upper regions of the fluid and thus has no bearing on the flow conditions near the b o t t o m . The assumed linearity of the undisturbed current velocity, as given in eq. 2, is of course an idealization of the real physical situation. This idealization is motivated by its mathematical simplicity, as well as by the observed fact that the linear profile in many situations of practical interest gives a fairly good approximation to the real profile. For instance, as shown experimentally b y Bulson (1961) and others, the two horizontal current branches set up by a two-dimensional air curtain will reasonably well be described by linear velocity profiles in the surface layer. In a pneumatic breakwater, one of the branches is directed against the oncoming waves in order to damp them partially, or even stop them. Adopting the linear form of the velocity profile, the linear wave theory for this kind of breakwater was given by Taylor (1955) for the case of deep water, and was elaborated by Brevik (1976) to include the case of finite water depths. Until recently, the case of rotational currents was n o t much studied. Thompson (1949) and Biesel (1950) derived the dispersion equation using linear wave theory and assuming a linear velocity profile. Sun' Tsao (1959) gave expressions for surface displacement and particle velocities assuming the same t y p e of current. The essential difference of the situation considered by Sun' Tsao from the situation considered herein is that he assumed a single layer model, corresponding to a current velocity profile varying linearly from the free surface downwards to the bottom. Recent developments on the theory of shear currents are covered in review papers by Fenton (1973) and Peregrine (1976). The reader is referred also to the report b y Brink-Kjaer (1976) and to the paper by Jonsson et al. (1978), wherein low-amplitude waves were assumed. As regards higher-order wave theories, the numerical stream function theories should especially be noted. This kind of theory was developed by Dean (1974, and further references therein) for the case of irrotational currents, and was generalized by Dalrymple (1973, 1974) to the cases of linear and bilinear currents. The m e t h o d consists in expanding the stream function to arbitrary orders in the steepness, and then determining the expansion coefficients by minimizing the boundary condition errors on a computer. The advantage of this kind of approach is its analytical accuracy, whereas a drawback is the necessity of presenting the results in tabular or graphical form. The next section presents the basic equations of the problem and the general approximation scheme. Thereafter the calculations are carried out in full to the third order. As expected, the first non-vanishing correction to the phase velocity, itself being of second order, is determined from the thirdorder boundary equations at the t w o interfaces of the current. Explicit expres-
240 sions are given for all third-order expansion coefficients appearing in potentials and surface elevation functions. Also, the simplifying case of deep currents is discussed. BASIC EQUATIONS AND EXPANSION SCHEME
In order to have some idea about the practical applicability of the theory developed in this paper, consider, for instance, the typical situation encountered in a pneumatic breakwater: periodic waves of initial wavelength Lin, initial amplitude ain and angular frequency w propagate on deep water from minus infinity into the opposing surface current in the breakwater and gradually become deformed as the current strength gradually increases. In practice, it often gives a reasonable approximation to assume that the undisturbed current satisfies the two requirements. (1) Its horizontal velocity profile is given by eqs. 2 for any value of the horizontal coordinate x within the established region of the current. (Actually, in a real breakwater there is also a reverse current, i.e. a current propagating in the positive x-direction, beneath y = - h . However, this reverse current will have only a moderate effect on the behaviour of the waves as it is usually weak and moreover directed parallel to the wave velocity, so it may to a good approximation be neglected.) (2) The variation of the current in the x-direction is "slow". That is, the variation of U over a distance equal to the wave length L is small: L d U / d x << U
(3)
This means that the waves may be considered to travel in locally homogeneous water. And such a physical situation may be adequately described by the idealized model assumed in this paper, viz. that the surface velocity U and the current depth h are constants, independent of x. In other words, our model is the one of an infinite train of finite-amplitude periodic waves propagating on a uniform current of constant vorticity U/h. Now it is possible to introduce a velocity potential in the following way. The linearity of the current profile permits separating off the rotational part of the velocity as -(1 + y / h ) Uex, where ex is the unit vector in the x-direction. As the remaining part of the velocity is irrotational, it may be written as the gradient of some potential function (P. (This property was discovered by Sun' Tsao (1959), and was emphasized also by Brink-Kjaer (1976).) As Bernoulli's equation will be used in the expansion procedure below it is necessary, because of the rotational properties of the motion, to leave the inertial frame to which the above considerations pertain (the laboratory frame, also called the absolute frame), and transform to the moving inertial frame where the profile of the free surface does n o t change with time. If c = w / k denotes the phase velocity in the laboratory frame, it is clear that the velocity of the moving frame with respect to the laboratory frame is equal to c. In the moving
241
frame the horizontal and vertical components of the total velocity in the rotational region can thus be written: u(x,y) = - ( 1 + y [ h ) U +
Cx(X,Y) (4)
v(x,y) = ~y(x,y)
in which indices mean partial derivatives. Here q~x includes also the term - c (see eq. 9a below). In the irrotational region beneath the current the motion may be described simply by a velocity potential ~ (x,y). We have in this region: u(x,y) = ~(x,y),
v(x,y) = ~y(x,y)
(5)
We let ~ and ~ denote the elevations of the free surface and the lower interface of the current, respectively. The fundamental independent variable is the wave number k (or wavelength L). Mathematically speaking, the whole flow region is divided by the lower interface of the current (y = - h + ~') into t w o simply connected regions in which the velocity potentials are each single-valued. The potentials • and are discontinuous across this interface due to the appearance of the first term to the right in the first of eqs. 4. For the determination of the velocity potentials, surface elevations and dispersion relation one has at one's disposal the incompressibility conditions: V 2 q~ = 0
(6a)
V2~ = 0
(6b)
the kinematic conditions at the upper and lower interfaces of the current: u(x,~)~x(X) - v(x,~) = 0
(7a)
u(x,-h
(7b)
+ ¢ ) ¢ x ( x ) - v ( x , - h + ~) = 0
and the Bernoulli equation: PIp + IA(u2 + v 2) + g y = C
(8)
In the rotational region, the Bernoulli " c o n s t a n t " C in general varies from streamline to streamline. In the irrotational region, on the other hand, it is constant throughout the fluid. As the lower interface of the current is a streamline, it follows from Bernoulli's equation that (u 2 + v 2 ) = continuous across this interface. Having written down the basic equations and boundary conditions, we n o w consider the expansion scheme. We expand the two potentials, the elevation of the two interfaces, and the phase velocity (in the laboratory frame) in a
242 small parameter e representing the wave steepness of the free surface: q~(x,y) = - c x + ecp, ( x , y ) + e 2 ~ 2 ( x , y ) + e3cP3(x,y)
(9a)
¢ ( x , y ) = - c x + edpl (X,y) + e2i~2(x,y) + e3¢3(x,y)
(9b)
~(x) = e~l (x) + e ~ 2 ( x ) + e3~3(x)
(9c)
~(x) = e~l (x) + e2~2(x) + e3~3(x)
(9d)
c = Co + ecl + e2c2 + e3c3
(9e)
Here the subscripts refer to the order of a p p r o x i m a t i o n (not to be confused with derivatives.) As m e n t i o n e d above, we are adopting a model in which periodic waves (of infinite extension) are propagating on a u n i f o r m rotational current. The periodicity enables us t o make the assumption t hat the potentials q~ and ~ are periodic functions in x, i.e. t he y are assumed n o t to contain any term proportional to x or t. Physically, this means t ha t the mean Eulerian horizontal fluid velocity below the troughs o f the interface is equal to zero in the l aborat ory inertial frame. (Recall t hat in the analogous situation in ordinary wave t h e o r y (w i t h o u t currents) the r e q u i r e m e n t of a periodic velocity potential implies that the mass tr an s por t in the l a bor at or y frame is different from zero.) Note, however, th at the periodicity r e q u i r e m e n t u p o n q~ and ¢ does n o t imply t hat the expansion functions q~n and Cn are necessarily simple harmonic functions, like sin ( n k x ) , say. Actually it will turn out t h at q~3 has to contain terms with sin k x in addition to terms with sin 3 k x (see eq. 41). A n o t h e r assumption made in the calculation below is t hat the level y = 0 is taken to coincide with the m e a n level o f the free surface. Thus the mean value of ~ vanishes, ~ = 0, so that the various expansion functions Vn in eq. 9c are periodic in x with zero mean. Physically, this means t hat within a horizontal region of one wavelength there is just the same a m o u n t of fluid above the level y = 0 as below it. As the fluid is regarded as incompressible, the mean level of the l o w e r interface of the current must t her e f ore be lying at the position y = - h . This implies in turn that ~"= 0, so that the functions ~n in eq. 9d are similar to ~?n periodic in x with zero mean. If we fix the origin of coordinates vertically below a wave crest, it follows that the expansion functions Vn and ~n contain only terms symmetric in x. It will turn out later that the third order functions 73 and ~3 are containing terms with cos k x in addition to terms with cos 3 k x . The Bernoulli constant in eq. 8 at the free surface may be written as: C = p o / p + 1/2(U+co) 2 +eel +e2C2 +e3C3
(10)
(free surface) where P0 is the atmospheric pressure. The values of the various expansion coefficients Cn in eq. 10 will be written out below, at each perturbative step, although these coefficients are of no importance for the calculation of poten-
243 tials and elevations. In each case Cn may be calculated by taking the mean of the Bernoulli equation over a wavelength. Note that the value of C is calculated as a consequence of the periodicity assumptions stated above; C is n o t an input parameter here. Before embarking upon the calculation, it is convenient for later reference to give here the explicit expressions for the velocity components u and v at the two interfaces of the current, when all terms up to the third order are included. At the free surface we have:
u(x,,7)=-(U+co)+~ + ~IdPlxy
-C~-h~+*~x
+E~ - e ~ - h ~ + * ~ x +
--C3 -- h 773 + dp3x + 771(1)2xy + ~2dPlxy + I~7721(I~IXyy (lla) v(x,~) = e,t,~y +e2(,t,2y +~,:Ihyy) + + e3
+ e3(dP3y + ~ldP2yy + ~72dPlyy + 1~WI2 dPlyyy)
(lib)
where the terms to the right are to be evaluated at y = 0. At the lower interface w e have to distinguish between the rotational and the irrotational regions. If we let the R o m a n numeral I refer to the rotational region, then:
u(x,-h+~)lI=-C o +e
-c, - - ~ 1
+*,x
+e 2
-c2--~2
+ e3 --C3 -- -~ f3 + dP3x + fldP2xy + f2gPlxy + 1/2f21~Plxyy
v(x,-h + ~) II = eCbly + e~(Cb2y + ~lCblyy) + e3(&3y + ~lCb2yy + + ~2dPlyy + 1/2~dPlyyy)
+*2x +[1*ixy (12a)
(12b)
where now the terms on the right refer to y = -h. Similarly, if the numeral II refers co the irrotational region, we have: u ( x , - h +~)ll.i = - C o + e ( - c 1 +(/)ix) +E2(-c2 +~)2x + ~'l{~lXy) +
+ 63(-C3 + Cax + ~'l~b2xy + ~'2¢lxy + l&~'~¢lxyy)
(13a)
o(x,-h + ~) 1~I = e~)ly + E2(~2y + ~l¢lyy) + e3(~)3y + ~1~)2yy + + f2¢lyy + 1hf2~blyyy )
(13b)
again referring to y = - h on the right hand side. Finally, the following point regarding the boundary conditions at the lower interface of the current should be noted. Firstly, the kinematic condition,
244 eq. 7b, implies that the fluid at the lower interface is moving in a direction parallel to the profile tangent. Further, as the Bernoulli eq. 8 requires that (u 2 + v 2) be continuous across the interface, it follows that the continuity property must hold for the velocity components u and v separately. In other words, the equations:
)pii
114)
v ( x , - h + ~')l I = v ( x , - h + f ) l i i
(15)
u(x,-h +
)1I = u ( x , - h +
hold at the lower interface of the current. FIRST-ORDER APPROXIMATION
Insertion of the Stokes expansions in the equations of motion yields to the first order in e: (U+co)V~x
+q)ly=O
(16a)
(y=O)
U
[ g + h ( U + e°)]7~l
(U+e°)t~lX
=
Codex + ~Piy = 0
(y = - h )
U -- --
~1
+ (I)lx
(Y = - h )
= ~lX
h qhly = ely
(y = - h )
(y = O)
(16b) (16c) (16d) (16e)
To the first order, eq. 16a is the kinematic condition 7a at the free surface. Eq. 16b is the Bernoulli eq. 8 at the same surface. Note that the rotational part of the current, as expressed by the term -(1 + y / h ) U in eq. 4, contributes to eq. 16b. The first order Bernoulli coefficient C1 follows as: C~ = ( U + Co)C,
Equation 16c is the first-order version of the kinematic condition 7b at the lower interface of the current, whereas eqs. 16d,e express the velocity conditions 14, 15. For the potentials we make the following assumptions: U + Co c~eky + fie - k y qh ( x , y ) = ~ sinkx
k
¢~1( x , y ) -
a-~
U + c o 7 e k(y+h)
k
a-~
sinkx
(17a)
(17b)
These expansions satisfy eqs. 6. We choose the expansion parameter as e = k (a - ~). The linear dispersion equation can now be derived by inserting
245 eqs. 17 in eqs. 16. It is convenient to give the dispersion equation in dimensionless form. As in an earlier work {Brevik, 1976) we use the dimensionless variables Y = U/co,
(18)
Z = hg/U 2
(actually Z -- F -2 , where F is the Froude number}. The dispersion equation can then be written as:
[
][
-(1+Y}2 +~{l+y+zy) =
-
-
1 + y)2 +
kh
2+~
(1 + Y + Z Y ) k-£
=
]
(19)
e -2kh
which agrees with the equation given by Taylor (1955) in his study on pneumatic and hydraulic breakwaters. Now assume that the current strength U, current depth h, and wave number k are the independent variables. Thus the independent dimensionless variables are Z and k h , and the remaining dependent variable Y can be calculated from eq. 19, which is actually a cubic equation for Y. Fig. 2, which has been plotted from fig. 6 in Taylor's paper (1955), shows Y as a function of kh with Z as a parameter. Once Y has been determined from the diagram, the lowest order phase velocity follows immediately as co = U / Y . 6
[
I
--
I
I
I
1
5
4
/
0
[ 0
I 1
I 2
I 3
I 4
I0~
I 5
6
kh
Fig. 2. Solutions of eq. 19. "I~e figures give the constant value of Z for each curve.
246 With the above c h o i c e f o r the p a r a m e t e r e, o n e finds for the f u n c t i o n V~: 1 k coskx
V,
(20)
T h e first o r d e r elevation o f the free surface t h e n b e c o m e s : n=en,=
acoskx
(21)
where: a=O/--~
(22a)
is t h e a m p l i t u d e and: e -2kh c~
(22b)
1 + 2kh/Y
T h e e x p a n s i o n p a r a m e t e r is thus simply e = k a . At t h e l o w e r i n t e r f a c e the elevation f u n c t i o n ~ b e c o m e s : Zl ~1 = - -
e-kh
coskx
(23a)
k where I+Y z, =
Y
1 +
(23b) [1 - e x p ( - 2 k h ) ]
2kh
T h e p o t e n t i a l f u n c t i o n (P~ in the r o t a t i o n a l region is n o w d e t e r m i h e d b y eqs. 17a and 22b. F o r later use it is c o n v e n i e n t t o write it as follows: C0 ~P, = ~ (A ~e ky + B i e - ky ) sin k x
(24a)
where: Al =
( Y) 1+
Zl
(245)
2kh B, -
Y
z, e -2kh
(24c)
2kh
are dimensionless quantities. In t h e i r r o t a t i o n a l region b e n e a t h the c u r r e n t we find t h e f o l l o w i n g p o t e n tial f u n c t i o n : Co ~1 = --~ z , e k y s i n k x
(25)
247 SECOND-ORDER
APPROXIMATION
The first-orderexpressions calculated above are n o w used as input values in the second-order equations. W e firstwrite d o w n the latter equations in general form:
(U÷co)~12x ÷dP2y= g +~
( U + eo)
Co~2x ÷ ~2y -~
U -
-
(
( -el-h~71 "
÷dPl "rhx-'fh~lyy
(y=O)
r~ - ( U + eo)~2x =C2 -1/~e~ - ( U + eo)e2 + e l ~ x
)
--Cl -- h ~1 ÷ (~ix ~ix - ~l(~lyy
~: + ~2x - ¢~x = 0
(Y = -h)
(y = -h)
(26a)
-
(26c) (26d)
h ~b2y -- ~2y =
--(dPlyy--~)lyy)~i
(26e)
(Y = -h )
The expansions used here are given in eqs. 9, 10, 11, 12 and 13. Eq. 26a is the kinematic condition at the free surface, eq. 26b the Bernoulli equation at the same surface, eq. 26c the kinematic condition at the lower interface, and eqs. 26d, e are the two conditions on the velocity given by eqs. 14, 15 at the lower interface. The equations have been written such that the periodic second-order functions appear on the left-hand sides. Substituting for the first-order expressions on the right-hand sides of eqs. 26, we obtain for eq. 26a: (1 + Y)r~2x + - - q~2y = - - sin k x Co
A, + B~ -
sin 2 k x
(27a)
Co
(y = 0)
T h e BernouUi eq. 26b b e c o m e s : Y(I+Y+ZY)r~2
1 (I+Y)~
el
h
Co
e0
÷--4
3 ( 1 ÷ y)2 _ A~ +B~ - ~ -
(
A~ +B~ -
cos2kx
5)
coskx+
(y -- 0)
(27b)
In writing this e q u a t i o n we have m a d e use o f t h e r e q u i r e m e n t t h a t t h e lefth a n d side be a p e r i o d i c f u n c t i o n in x with zero mean. This d e t e r m i n e s the
248 s e c o n d - o r d e r Bernoulli c o e f f i c i e n t C2 at t h e free surface:
C2 =1//2C21 "t'(Ul-c°)c2 --4 (U'{'cO
4 CO2 A1 +Bl - ~-h
Finally, the t h r e e e q u a t i o n s 26c, d, e at t h e lower i n t e r f a c e take t h e f o r m :
c,
~2x + - - ~P:y=-- zl C0
sinkx -
(i
+
sin2kx
(27c)
C0
(y = - h ) Y 1 - -- ~2 + - - (¢P2x - ~ ) h c0
= 0
1 - - (¢P2y - ¢2y ) - -
z~e -2kh sin 2kx
Co
Y
2kh
(y = - h )
(27d)
(y = - h )
(27e)
F r o m t h e t w o e q u a t i o n s at t h e level y = 0 (eqs. 27a and 27b), we m a y eliminate 72 a n d t h u s o b t a i n a s e c o n d - o r d e r differential e q u a t i o n f o r ¢P2- With the following a s s u m p t i o n : C0
¢P2 = --~ (A2 e2ky + B2 e - 2 k y )sin 2kx
(28)
which is analogous t o eq. 24a, it follows i m m e d i a t e l y that: cl = 0
(29)
T h e r e f o r e , t h e first non-vanishing c o r r e c t i o n t o t h e phase v e l o c i t y is at least o f the s e c o n d o r d e r in e. This b e h a v i o u r is just analogous t o t h a t e n c o u n t e r e d in the case o f waves p r o p a g a t i n g o n still w a t e r (Wehausen and Laitone, 1960). Similarly i n t r o d u c i n g t h e f o l l o w i n g a s s u m p t i o n s f o r t h e t w o s e c o n d - o r d e r elevation f u n c t i o n s : e2
72 = -
cos 2 k x
(30)
k
Z2
~2= -
e - 2 k h cos 2 k x
(31)
k we find t h e following expressions f o r the s e c o n d - o r d e r coefficients: A2 = 4
1+
-3(1+Y)
2 +3(A1 +B~)
+k2h2
A~ +B~ +
kh - (A1 + e l )
1 +--e
2J~h
-2kh sinh2~
Y , e-2kh cosh 2kh ) 2kh (32)
249
Y(z2 + l + ~,,2) 4kh
B2 = - ~ e2
1 + Y
z~ = ~,~z 2, +
(33)
e-4kh
1,4 A t + B t -
(34)
+A2 -B2
A2 1 + Y/4kh
(35)
Finally, for the potential function ~2 we obtain the following expression: Co ¢2 =--~ (Z2 -
lhZ~)e2ky sin 2kx
(36)
These expressions enable us to calculate velocities and surface profiles to the second order. However, the degree of approximation is still too low to permit determination of the first non-vanishing correction c2 to the phase velocity. A main purpose of the next section is to calculate c2 from the third-order equations. THIRD O R D E R APPROXIMATION; THE PHASE VELOCITY
The starting point is the set of third-order equations at the two boundaries of the current: ( U + Co)173x + ~3y ffi "c2'01x + ~Plx172x + ¢~2x~lx - r/2~lyy - 171~ 2 y y -
U -h
(y = 0)
(Y~l~2X + 1~21~1x) -I-l~ll,~lx{i)txy - 1 / ~ 2 ~ l y y y
g+-:"(U+co
73-(U+coI¢3x=C3-(U+co)c3
+ ( U + Co)(~2,I, lxy + ~ 2 ~ l x y y + ~ 2 = y ) - ~2x -- 171~txy)
Co~3x + '~3y = (-'c2
-- ~ l y ( ~ 2 y
+ 171~lyy)
+ ~t - ¢ 1 x
o c2 +~,72 -
(Y = 0)
(37b)
U ~2 + ~.2x + ~tCtxy)~,x - ( U -~
-- ~1 ¢I~2yy -- ~2CI}lyy -- t/'Jfl~I~lyyy
(37a)
-
(y ffi - h )
(37c)
U - ~ ~3 + &3x - ~b3x = l ~ ( ~ t x y y - @txyy )~2t + ( ~ l y y - ~ l y y ) ~ l ~ l x
(37d)
(y = -h) ~3y -- ~b3y = --(dP2yy -- {.b2yy)-~'t -- (¢I~lyy - ~blyy )~'2
(y = - h )
(37e)
250
To the third order, eq. 37a is the kinematic condition at the free surface, eq. 37b is the Bernoulli equation at the same surface, eq. 37c is the kinematic condition at the lower interface, and eqs. 37d, e are the third order versions of eqs. 14, 15. In writing the equations we have made use of the first order eqs. 16 to obtain some simplifications, and we have taken into account that cl = 0 . The known first- and second-order expressions are now inserted on the righthand sides of eqs. 37. Eq. 37a takes the form: ( l + Y ) ? 3 x + - - ~P3y= C0 --3
[:
(I+Y)-A2
--8
- B 2 -1A A1 +B~
(
(I+Y)+A2
+B2 +IA A1 +B1 -~-~ e2
sin3kx
)e2] sinkx (y = O) (38a)
and similarly for the Bernoulli eq. 37b: -
-
_
h
_
=
Co +
A1 + B 1 -
-~ Y)] -
-8(1+ ---A2-B2 00
+IA 3 ( 1 + Y ) 2 + ~
e2 I c o s k x + t - 8 l(l1 + Y )
A~ +B~ -~-£
(A2 +B2)+~A
+ -kh - A~ +B1 - ~-~
e2
Y) A1 +Bl -
( At + BI - 1 - 8~ Y h)
+ Al + B l -
-
7(1+Y) 2 +
cos 3 k x
(y = 0)
(38b)
Here we have put: C3 = (U + co)C3
in order to make the expression on the right-hand side periodic in x with zero mean. At the lower interface of the current it is advantageous to make use of the first and second-order boundary conditions at the same interface so that z2 appears as the only second-order coefficient on the right-hand sides of the equations. For the kinematic condition 37c we obtain:
[c +lz
~3x + - - dP3y . . Co - Co
. 8
.
[9 2 9( Y) + 8zl -~ 1 + 3kh
1 + 2
z2
z2e - 2 k h
]
z l e -kh sinkx +
3kh
]Zle_3kh
sin3kx
(y = - h )
(38c)
251 and for the velocity conditions 37d,e we have: Y
- h ~s + -
1
Co
(¢ax - ~ b s x )
1 -- (q)3y - ~3y ) Co
Y 8kh
z ~ e - 3 k h ( c o s k x + 3 cos 3 k x )
Y - - Z l Z 2 e - 3 k h ( s i n k x + 3 sin 3 k x ) 2kh
(y
-h)
(38d)
(y = - h )
(38e)
The task is now to solve the five eqs. 38. It is convenient first to eliminate the elevation functions 77 and ~. Differentiating eq. 38b with respect to x and combining with eq. 38a, we have: 1 1 (I + Y)&3~ + - - (A, + B,)¢3y = kCo Co
(
+2
A, +B,-~'~
-2
, +B,
-
-A:-B:
-
(I + Y)
- 2
e2
B1
-
+
) [
+1~ 3 ( 1 + y ) 2 _
(A2 + B 2 ) + I A
A, + B~ - ~'~
+
1
7(1+Y)2
_
(y ; 0)
sin 3 k x
(39)
Correspondingly, differentiating eq. 38d and combining with eq. 38c, we have: 1
Co
((P3~ - ~ 3 ~ ) + - -
(I)3y
hco
9Y 2h
h
-2
ZlZ2e - a k h s i n 3 k x
1 +
1 +
\)
z2e -2kh
(y
-h)
3k
z 1
-
(40)
3kh
There are thus three equations at our disposal for the determination of the potentials I,, 4~ and the velocity correction c2, viz. eq. 39 at y = 0 and eqs. 40, 38e at y = -h. Each of these equations contains actually t w o equations, as the equality must hold when comparing terms with sin k x and sin 3 kx separately. As sin k x appears in the equations at both the two interfaces of the current, it is n o t possible to satisfy the equations unless some terms with sin k x are present in the third-order potentials. We shall solve the equations on the basis of the following assumptions: CO
q~3 = ~ - [(A31e ky + B 3 t e - h Y ) s i n k x + (A33e 3ky + B 3 3 e - 3 k Y ) s i n 3 k x ]
(41)
CO
~3 = -~- f33e 3ky sin 3 k x
(42)
252
These are the simplest expressions one can construct, requiring that the Laplace equation is satisfied and also that there are enough coefficients to permit a consistent solution of the boundary equations. We may thus assume that @3 varies with x as sin 3 k x , whereas ¢P3 must contain terms with sin k x also. The equations arising when comparing terms with sin k x are the most interesting ones, as the expression for the velocity correction may be derived from them. Inserting eqs. 41 and 42 in eqs. 39, 40 and 38e, we obtain for the velocity correction:
c~t=
Co
4 (1 + Y )
(
A1 + B I -3kh2Y
) ( +
~-)
A~ +B1 - 2 k h
(A2 + B 2 ) -
A1 + B1 - 2 k ~
and the second-order phase velocity e in the laboratory frame follows as: (43b)
c = Co + k2a2c2
where a is the first-order wave amplitude. Note that a is an external parameter in the theory; it has no relation to the calculated first-order coefficients A~ and B1. The expression 43a for the velocity correction is the main result of this paper. The comparison of terms with sin k x also gives the following expressions for the coefficients A3,, B3~: A31
=-2kh
B31
=--Y 2kh
(cj0 -
+z2e -2kh
)
(44)
z1
( - - -c2 + 2 z 2 e - 2 k h ) z~ e_2k h
(45)
Co
whereas the comparison of terms with sin 3 k x determines the remaining coefficients in eqs. 41, 42: A33 =
t
-2(1+Y)
+IA
t
(
A1 +B1 -
7(1+Y) 2-
1
+B~ -
A1 + B ~ - 3 ( I + Y ) - ~ - ~ [ A ,
"
-2
(
Al +B1
e2 + ~ ' £
[A1 +B1 + 3 ( l + Y ) l z ~ z 2
+B~ + 3 ( 1 + Y ) 1
(
2+
e-6kh
e-~
1
(46)
253 f33 =
(
B33
~-~ ~zlz2
1 +
Y
A33
(47)
1
e -6~h
(48)
The forms of the t w o surface profiles are determined b y going back to the original boundary equations 38. With the following assumptions: 1 n3 = -~ (e31 cos kx + e33 cos 3kx)
(49)
1 ~3 = ~ (z31 e -kh cos kx + z33 e -3kh cos 3kx)
(50)
(cf. eqs. 30, 31), we obtain from the kinematic eq. 38a at the free surface:
e3~ =-8 + ~ I + Y e33 = - - 8 + ~ l + y
[°:: -
(
+A2 +B2 +1,~ A~ +B~ -
A2 +B2 +1,~ A~ +B~ -
5)
e2 +A3~ -B31
]
e: + A 3 3 - B 3 3
(51) (52)
and similarly from one of the velociW conditions, eq. 38d, at the lower interface.: Z31 ----
=
----+2,z2-~-~ e0 -
z,
+
Zl
(53)
(54)
In eqs. 53, 54 we have used the boundary equations at the lower interface to simplify the expressions. The expansion scheme to the third order is n o w completed, and it may be convenient to summarize here the use of the formulas. The strength and depth of the current, and the wavelength, are parameters assumed given initially. This determines the nondimensional quantities Z and hh. From Fig. 2 we may determine Y graphically, and thereafter calculate the phase velocity according to first-order wave theory as Co = U/Y. The first order coefficients z~, A 1, B~ follow from eqs. 23, 24. The second-order coefficients A2, B2, e2, z2 are thereafter calculated from eqs. 32--35. The first non-vanishing correction to the phase velocity is calculated in third-order theory and is given by eqs. 43. The other third-order coefficients are given by eqs. 43--54. One final remark is in order, as regards the initial assumptions 41, 42 for the potentials. As mentioned earlier, these are the simplest expressions that one may construct, compatible with the Laplace equation and also containing the necessary number of coefficients to be determined by the boundary equations.
254 In principle t h e r e is, however, n o t h i n g which prevents one f r o m adding a n o t h e r t e r m , o f the f o r m (co/k)f31 e x p ( k y ) sin k x , t o the p o t e n t i a l 03 in the i r r o t a t i o n a l region. Does such an e x t r a t e r m lead t o a n y change in o u r main result, which is the e x p r e s s i o n f o r c2/Co? The answer is no: w h e n calculating c2/Co the e x t r a t e r m with f31 disappears f r o m the formalism, and we end up with the same e x p r e s s i o n as t h a t given in eq. 43a, c o r r e s p o n d i n g t o f3~ = 0. On the o t h e r hand, it turns o u t t h a t the t e r m with f31 induces changes in the expressions f o r A3~, B3~, e3~, z3~, as t h e b o u n d a r y e q u a t i o n s emerging f r o m the c o m p a r i s o n o f t e r m s with sin k x and cos k x b e c o m e changed. We shall n o t give these m o r e general expressions here. By c o n t r a s t , the c o e f f i c i e n t s A33, B33, f~s, e33, z33 will n o t b e c o m e changed, as these c o e f f i c i e n t s are d e t e r m i n e d f r o m t h e e q u a t i o n s emerging f r o m t h e c o m p a r i s o n o f terms with sin 3 k x and cos 3 k x . These e q u a t i o n s are u n a f f e c t e d b y fsl- D i f f e r e n t values o f f31 corres p o n d to slightly d i f f e r e n t fluid velocities and surface elevations. EXAMPLES In o r d e r t o illustrate t h e m a g n i t u d e o f t h e various t e r m s in the expansions, we shall give t w o exampleS. Example 1
We shall first c o n s i d e r a relatively small scale situation. Let U be c h o s e n equal t o 1.50 m/s; this is a c t u a l l y a measure o f the m a x i m u m surface v e l o c i t y o b t a i n e d in Bulson's p n e u m a t i c b r e a k w a t e r e x p e r i m e n t ( 1 9 6 1 , 1 9 6 3 , 1968). F o r the c u r r e n t p a r a m e t e r we c h o o s e Z = 4, which c o r r e s p o n d s t o a c u r r e n t d e p t h h = Z U 2/g = 0 . 9 2 m. T h e wavelength L is d e t e r m i n e d b y the i n d e p e n d e n t p a r a m e t e r kh; we c h o o s e k h = 1.5, c o r r e s p o n d i n g t o L = 27r/k = 3.85 m. F r o m Fig. 2 we m a y read o f f Y = 1.03 (alternatively, eq. 19 m a y be solved t o o b t a i n this a c c u r a t e value), so t h a t Co = U / Y = 1.46 m/s. We obtain: Z 1 1.531 A2 = - 0 . 4 0 4 z 2 = 0.827 A3, = - 0 . 3 4 2 B33 = 0 . 0 0 0 0 5 5 e33 ~- 0 . 0 6 2 4 ----
A, = 2 . 0 5 6 B2 = 0 . 0 0 0 8 5 e:/co = 0 . 6 9 2 Bs, = - 0 . 0 1 6 0 f33 = 0 . 1 0 4 z31 = - 0 . 9 8 7
B, = 0 . 0 2 6 2 e2 = 0 . 2 2 9 A33 = 0 . 1 1 6 e3, = - 0 . 2 4 6 z33 = 0 . 6 5 8
We m a y write o u t t h e expressions f o r t h e t w o surface elevations, and the phase v e l o c i t y in t h e l a b o r a t o r y frame: = a (1 - 0 . 6 5 9 a 2 )cos k x + 0 . 3 7 4 a 2 c o s 2 k x + 0 . 1 6 7 a 3 c o s 3 k x = a ( 0 . 3 4 2 - 0 . 5 8 9 a 2 )cos k x + 0 . 0 6 7 a 2 c o s 2 k x + 0 . 0 2 0 a 3 cos 3 k x C/Co = 1 + 1 . 8 5 1 a 2
(55)
255
w h e r e t h e a m p l i t u d e a is t o b e i n s e r t e d in m e t e r s . T h e s e e x p r e s s i o n s m a y be c o m p a r e d w i t h t h e c o r r e s p o n d i n g e x p r e s s i o n s f o r waves p r o p a g a t i n g o n still, d e e p w a t e r (Wehausen a n d L a i t o n e , 1 9 6 0 ) : ~ ( U = 0) = a ( 1 + 0 . 3 3 4 a 2 )cos k x + 0 . 8 1 8 a 2 cos 2 k x + 1 . 0 0 2 a 3 cos 3 k x (56) c -
( U = 0) = 1 + 1 . 3 3 7 a 2
-
C0
Example 2
C o n s i d e r n o w a large scale s i t u a t i o n w i t h s t r o n g c u r r e n t : U = 2.5 m / s , L = 80 m. T a k i n g h = 6.37 m we o b t a i n Z = 10, k h = 1/2. F r o m Fig. 2, o r m o r e acc u r a t e l y b y solving t h e c u b i c dispersion e q u a t i o n 19 w i t h r e s p e c t t o Y, we d e t e r m i n e Y = 0.243. T h e l o w e s t o r d e r p h a s e v e l o c i t y is Co = U / Y = 1 0 . 2 9 m/s. F u r t h e r c a l c u l a t i o n yields f o r t h e coefficients: zl = 1 . 0 7 7 A: = -0.137 z2 = 0 . 4 5 8 A3, = - 0 . 0 8 1 4 B33 = 0 . 0 0 5 9 6 e33 = 0 . 1 5 7
AI = 1 . 3 3 9 B2 = 0 . 0 1 7 1 c2/Co = 0 . 4 7 9 B31 = - 0 . 0 1 3 7 f33 = - 0 . 0 0 2 9 3 z3, = - 0 . 3 0 2
B1 = 0 . 0 9 6 3 e2 = 0 . 3 5 6 A33 = - 0 . 0 0 3 1 7 e31 = - 0 . 0 2 5 9 z33 = 0 . 2 6 8
so t h a t t h e surface e l e v a t i o n s a n d t h e p h a s e v e l o c i t y m a y be w r i t t e n as: = a (1 - 0 . 0 0 0 2 a 2 )cos kx + 0 . 0 2 7 9 a 2 cos 2 k x + 0 . 0 0 1 0 a 3 cos 3 k x 3 k x = a ( 0 . 6 5 4 - 0 . 0 1 4 4 a 2 ) c o s k x + 0 . 0 1 3 2 a 2 cos 2 k x + 0 . 0 0 0 4 a 3 c o s
(57)
C/Co = 1 + 0 . 0 0 2 9 6 a 2
T h e f o r m u l a s c o r r e s p o n d i n g t o t h e case o f still, d e e p w a t e r are given b y eqs. 56, as b e f o r e . THE CASE OF DEEP CURRENTS
In p r a c t i c e it o f t e n t u r n s o u t t h a t t h e c u r r e n t is s u f f i c i e n t l y d e e p in c o m p a r i s o n to t h e w a v e l e n g t h t o p e r m i t certain s i m p l i f i c a t i o n s in t h e f o r m u l a s . F r o m t h e linear dispersion eq. 19 it is clear t h a t t h e r i g h t - h a n d side b e c o m e s negligible if t h e c o n d i t i o n : kh~
2.5
or:
(58)
h~O.4L
is satisfied. F o r instance, it t u r n s o u t t h a t this c o n d i t i o n is satisfied w i t h i n t h e region o f t h e r a t h e r s t r o n g c u r r e n t s p r o d u c e d in Bulson's b r e a k w a t e r experi-
256 ment (Bulson, 1963, 1968). For a typical long wave in that experiment, with incident wavelength on still water being about 9 m, it turns out that the parameter k h reaches the value 2.5 already when the supplied power is about one half of the power required to stop the waves completely. Approximative formulas will now be given for the case that the condition 58 is satisfied. First of all, the left-hand side of eq. 19 can be put equal to zero, giving a quadratic equation that can be solved explicitly to give: Y-
-
k h -1/2 +
co
(59)
khZ +
Z + 1 - kh
It is thus easy to calculate Y, and hence Co, when the wavelength and the current data are known. When writing out the expansions, it is further convenient to transform to the laboratory inertial frame in which the terms ( - c x ) in the potentials, cf. eqs. 9, are absent. We shall give the approximative expressions for potentials and surface elevations up to the second order in e, and also the expression for the phase velocity: ¢ = ac0(1 + Y)e ky sin(kx - cot) +
Y(
+ a : c o ~-~
1)
- 3 +
e 2ky sin 2 ( k x - cot)
(60)
khl+Y
I+Y = aco i + Y / 2 k h
eky sin(kx - w t )
ka [
= a c o s ( k x - cot) + ~ -
1
(61)
2Y
y2
1 -
-
.4.
kh I + Y
-
1
] cos 2 ( k x - co t)
-
2k2h 2 ( l + Y ) 2
(62) I+Y
e - k h cos(kx - cot)
=a
(63)
1 + Y/2kh c
co
l+k2a --
y3
2
(1 +Y)2 2(l+Y-Y/2kh) 1
k3h 3 (l+y)3
y4 +- -
r1 L
--
1
8k4h 4 (l+y)4
2Y
1
kh
I + Y
1 j
1 4k2h 2 (l+Y)29Y 2
"b
-
-
-
-
(64)
Here co is the angular frequency in the laboratory frame. We have made use of the fact that the wave phase is a Galilean invariant. Terms with exp ( - 2 k h ) have been neglected in comparison with unity. In particular, in the limiting case where h -~ o% we obtain: = ac0(1 + Y)e ky sin(kx - cot)
(65)
257 = a cos(kx - cot) + 1/2k2a2 cos 2 ( k x - cot)
(66)
Co = V/~K - U
(67)
c = c0[1 + 1~(1 + Y ) k 2 a 2 ] = g v ~ ( 1 + l h k 2 a 2) - U
(68)
The expression 66 is identical to that obtained if U = 0 (Wehausen and Laitone, 1960). This is just what one should expect: the limit h -* oo corresponds to the case where the waves are propagating on a current uniform over depth, and this situation is differing only by a Galilean transformation from the simple situation in which the waves propagate on still water. Obviously, a Galilean transformation has no influence upon the surface elevation. Similar features are shared by the other equations above. For instance, the expression for c in eq. 68 is equal to the expression obtained by subtracting U from the expression in eq. 1, as one would expect. O n t h e w a v e p e r i o d u s e d as i n d e p e n d e n t p a r a m e t e r
The theory developed above uses k h as independent wave parameter. The wavelength L = 27r/k in the current is thus assumed to be a known quantity. In practice it is, however, the absolute period Ta in the laboratory frame, rather than the wavelength, which is the convenient external parameter. Under steady conditions, the value of T a is constant when waves propagate from still water into a current, whereas the wavelength is changing. Usually, it is necessary to observe the phenomenon in practice in order to determine L. For this reason it is desirable to give a short description of how to handle the problem if Ta, rather than k h , is the independent wave parameter. For concreteness, one may again imagine the breakwater situation alluded to in the beginning of this paper. Let us consider the general case with arbitrary current depth first. The basic condition is that the period Ta, or angular frequency co = 21r/Ta, is a conserved quantity. Thus: co = k c = kc0(1 + k2a2c2/Co)
(69)
The procedure for determining k is now rather complicated: firstly, one may obtain an analytic expression for Co = U / Y by solving the cubic dispersion eq. 19 with respect to Y; secondly, use is to be made of the expression 43a for c2/Co. By solving eq. 69 numerically, one may thus determine the value of k, if the wave amplitude a in the current is known. Once k is determined, the formalism developed earlier is applicable directly. The phase velocity follows most simply as c = co/k. The special case of deep currents simplifies the situation, as eqs. 59 and 64 become applicable for Y and c / c o , respectively. The case of an infinitely deep current, h -* 0% plays a particular role, as it is now possible to predict the value of the wave amplitude a in the current when the initial amplitude ain for the incoming waves on still water is known.
258 Actually, assuming an entirely t w o - d i m e n s i o n a l m o t i o n in w h i c h the transverse c o m p o n e n t o f the c u r r e n t was equal to zero, Longuet-Higgins and S t e w a r t ( 1 9 6 1 ) derived t h e f o l l o w i n g expression f o r the a m p l i t u d e ratio a/ain {our n o t a t i o n ) :
a/ain = [
gv~ co2(g v ~ ~ 2 U ) ]
1/2 (70)
Using this expression, and the e x p r e s s i o n 6 8 for c, it is n o w relatively easy t o d e t e r m i n e k n u m e r i c a l l y w h e n co a n d ain are k n o w n . The f o r m u l a 70 leads t o an infinite a m p l i t u d e ratio w h e n U a p p r o a c h e s ½x/g/k, and it is t h e r e f o r e applicable o n l y t o cases well b e l o w this limit. ACKNOWLEDGEMENTS I wish t o express m y g r a t i t u d e for valuable r e m a r k s f r o m Ivar G. J o n s s o n at the I n s t i t u t e o f H y d r o d y n a m i c s a n d H y d r a u l i c Engineering (ISVA), Technical University, C o p e n h a g e n , D e n m a r k and also f r o m Per B r u u n , NTH, T r o n d h e i m , Norway. This w o r k was s u p p o r t e d in p a r t b y Norges T e k n i s k - N a t u r v i t e n s k a p e l i g e Forskningsr~id.
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259 Longuet-Higgins, M.S. and Stewart, R.W., 1961. The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech., 10: 529--549. Peregrine, D.H., 1976. Interaction of water waves and currents. Adv. Appl. Mech., 16: 9--117. Schumann, E.H., 1974. South Africa National Research Institute for Oceanology, Internal General Report IG 74/14. Schumann, E.H., 1975. A.S. Shipping News and Fishing Review, 30(3). Sun' Tsao, 1959. Behaviour of surface waves on linearly varying flow. Moskow. Fiz.-Tekh. Inst. Issted. Mekh. Prikl. Mat., 3: 66--84. Taylor, G.I., 1955. The action of a surface current used as a breakwater. Proc. R. Soc. Lond., Set. A, 231: 466--478. Thompson, P.D., 1949. The propagation of small surface disturbances through rotational flow. Ann. N.Y. Acad. Sci., 51: 463--474. Wehausen, J.V. and Laitone, E.V., 1960. Surface waves. Encycl. Phys., 9: 466--778.