On the second approximation to mass transport in the bottom boundary layer

Coastal Engineering, 6 (1982) 93--120 Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

93

ON THE SECOND APPROXIMATION TO MASS T R A N S P O R T IN THE BOTTOM B O U N D A R Y L A Y E R

B.D. DORE

Department of Mathematics, University of Reading, Whiteknights, Reading RG6 2AX (Great Britain) (Received January 27, 1981; revised and accepted October 6, 1981)

ABSTRACT Dore, B.D., 1982. On the second approximation to mass transport in the bottom boundary layer. Coastal Eng., 6: 93--120. The second approximation is obtained for the mass transport velocity within the oscillatory bottom boundary layer beneath sinusoidal progressive and standing waves of finite amplitude. This approximation includes a simple new term, which essentially ensures continuity of the vertical gradient of mass transport at the edge of the layer and is of thirdorder in the perturbation (or wave-slope) parameter. For long progressive waves in conditions of zero net mass flow, the term represents a moderate reduction in mass transport at the edge of the layer, compared with the first approximation of Longuet-Higgins. For standing waves of arbitrary length, the mass transport is reduced (increased) far from (near) the bottom, except near nodal locations where an increase (a reduction) is predicted. The proposed correction to the first approximation yields clearly improved results when compared with appropriate experimental evidence. Deficiencies in the higher-order theories of Sleath and Isaacson for propagating waves are disclosed.

1. INTRODUCTION

Mass transport velocity, a secondary quantity in the c o n t e x t of water-wave theory, is associated with a time-averaged displacement of a fluid element. It was calculated as a "first approximation" (i.e. to second order in the perturbation, or wave-slope, parameter) by Longuet-Higgins (1953, 1958), who used concepts of boundary-layer theory as applied to sinusoidal waves. Due to its substantial effects on movement of sediment, the velocity, near both smooth and rough, horizontal or near-horizontal beds, has been the subject of several experimental investigations -- see, for example, Bagnold (1947), Russell and Osorio (1958), Allen and Gibson (1959), Brebner and Collins (1961), Brebner et al. (1966), Mei et al. {1972), Isaacson (1978) for sinusoidal progressive waves, and Noda (1968) for standing waves.

0378-3839/82/0000---0000/$02.75 © 1982 Elsevier Scientific Publishing Company

94 Progressive waves. Although there is considerable scatter in the data for the

oscillatory b o t t o m boundary layer, it is clear from the results of Allen and Gibson (1959, figs. 4 and 7), Brebner and Collins {1961, figs. 2, 3 and 5), Brebner et al. (1966, figs. 1 and 3) and Isaacson (1978, figs. 3 and 4), that, overall, the measurements are noticeably less than the corresponding theoretical predictions of Longuet-Higgins (1953), at least for waves of sufficiently large amplitude. An attempt to improve the accuracy of the theoretical expression for the mass transport velocity in conditions of laminar flow was made by Sleath (1972), who sought a "second approximation" by working to f o u r t h order in wave-slope. However, his theory is basically incorrect, as explained in § 3.2. Isaacson (1976a) calculated the first approximation in the oscillatory layer for cnoidal waves. Subsequently, Isaacson (1976b) attempted to calculate "the second approximation" via a third-order theory, and obtained some improvement in comparison of his results with measurements of Brebner and Collins {1961). In fact, it appears from § 3.2 that this theory is incomplete, in the sense that an additional correction, formally comparable with that of Isaacson (1976b), is required for consistency. S t a n d i n g waves. The data obtained by Noda (1968), within the oscillatory

layer and between a node and an adjacent anti-node, also have considerable scatter. But it is clear, especially from his figs. 5, 8 and 9, that the measured mass transport is generally much smaller than that predicted by LonguetHiggins (1953), the difference seemingly increasing as the wave amplitude becomes larger. However, no theory is known which adequately explains this reduction in the case of standing waves. In the present work, we calculate the dominant part of the correction to the expression of Longuet-Higgins (1953) for the mass transport in the oscillatory b o t t o m boundary layer beneath sinusoidal waves. By employing standard matching procedures in the method of matched asymptotic expansions, together with recent double boundary-layer theories of Dore (1976, 1977), we readily obtain an extension of the existing expression, by inclusion of a third-order term in the wave-slope parameter. This extension constitutes "the second approximation" to the mass transport velocity within the oscillatory layer. Subject to certain limitations on the comparisons, it is found that measurements of this velocity near the bed appear to correspond more closely, both for progressive and, in the main, for standing waves, to the predictions of our "second approximation" than to those of the first approximation of Longuet-Higgins (1953). 2. THE OSCILLATORY BOUNDARY LAYER Let two-dimensional waves take place in water of uniform equilibrium depth h, and define a stream function qJ such that the velocity q = (u, w) = (2 q / / ~ z , - 2 ~ / ~ x ) , z being measured vertically upwards from the b o t t o m and x horizontally. If the wave period and wavelength are T = 2~/a and X = 27r/k,

95

respectively, and Vw (= constant) is kinematic viscosity, variables are nondimensionalized as follows: f = kr,

[ = at,

~ = k2~p/a,

~ = (vwk2/a) ~

Henceforth, we omit carets, so that the vorticity equation is: aco/at + a ( c o , $ ) D ( x , z )

(1)

= e2v2co

where co = V2 ~ is the vorticity. The displacement of the air--water interface is: s exp[i(t - x ) ] + O(a 2)

zi - h =

I

exp(it)cos x + O(~ 2)

(progressive waves) (standing waves)

where ~ ~ min(l#~) is the m a x i m u m wave-slope and equals a k in terms of the (dimensional) wave amplitude a. The interfacial boundary conditions on z = z i ( x , t) do not concern us directly, but, at the bottom, we impose the condition: q = O,

on z = 0

Further, the boundary-layer approximation: e ~ m i n ( 1 , h) is made, since, for laminar flow, 10 -2 < e < 10 -6 throughout the entire gravity. wave range. Where possible, weak viscous damping is ignored; but explicit consideration of effects of spatial decay is made in § 4.1. Adjacent to the bottom, there is the oscillatory boundary layer of thickness O(e), and, within it, we write: Z = zl2~e

~b = ~ 1 (x,Z, t ; e , ~ ) + ~2~2(x,Z, t ; e , a , ~ ? ) + ...

(2)

where ~ = (viscosity of air)/(viscosity of water) = 1.35 X 10 -2, together with similar expansions for q, u, w and co. The function ~bl of linear theory is independent of a , but ~b2 may not be. Moreover, if a bar denotes a time average over a wave period, the d o m i n a n t parts of ~1 and ~2 axe shown by Longuet-Higgins (1953) to be O(e), whence Ul, u~ are O(1); wl, w2 are O(e); co ~, w2 are O(e -1 ). The vorticity co i -} 0 as Z -~ ~ since, just beyond the oscillatory layer, the linear velocity field ql is h'rotational. However, only the d o m i n a n t part, O (e -~ ), of ~ 2 decays to zero, since mean vorticity, of a higher order, diffuses into the interior, whence: ~2

= 2-~e-lau~w2/aZ

= o(e q) as Z-+ ~

(3)

96 [Equation 3 is valid for the problem under discussion, b u t is not necessarily true for waves in a liquid-liquid system, Dote (1970 , pp. 125--126).] It is readily shown that: ~,

(4)

= { e ~ , z ( x , Z ) + e~q;,: + . . . } e it

where 3 2 4 , , / 3 Z 2, 32~,2/3Z 2,½32~,3/~Z: + 32qs,,/ax 2,.. -* 0 as Z-* ~o, since c0, -* 0 as Z -* oo. Then, from eq. 2, we have: u,

= 2qe-'3C~,/3Z

wl =-3~1/3x

= ( U , o ( x , Z ) + e u , , + . . . } e it = (wl,(x,Z)+ew,2+...}e

it

where u,o = 2q~qs , , / ~ Z , w , , = -3~p , , / 3 x , etc. Similarly, at O(a2), it is seen from the w o r k of Longuet-Higgins (1953), Dote (1978), etc. that: (a) $ ~ only contains terms in e x p ( 2 i m t ) (m = 0, 1); (b) the d o m i n a n t term, O(e), in ~ 2 is independent of e, ~ and ~; (c) ~2 has the form:

~

= {eG,(x,Z)+,.,+

c~

+...+ ~

+...}

(5)

wherein the possibility must, in general, be allowed that there exists at least one term whose order of magnitude is much smaller than O(e) and much larger than O(e2), etc. - - s e e § 3, 4; (d) the dominant term in the mean Eulerian velocity: -u2 = 2 - ~ e - ' 3 ~ 2 / O Z = ( U ~ o ( x , Z ) + . . . +eu21 + . - . +e2u22 + . . . }

has the forms: _ U20

~ ~fp(Z) cosech 2 h, ~fs(Z) sin 2x cosech2h,

fp(OO) = 3

[progressive waves]

(6)

fs(°o) = - 3

[standing waves]

(7)

where fp(0) = 0 = fs(0) to satisfy the no-slip condition at the b o t t o m . The expressions for u20 as Z -* = yield boundary conditions on the mean, Eulerian velocity q2 at the outer edge of the oscillatory layer. 8. THE MASS T R A N S P O R T VELOCITY

This Lagrangian velocity, being a time-averaged velocity for a particular fluid element, was given by Longuet-Higgins (1953) as:

(8) and, when higher-order terms are neglected, is i n d e p e n d e n t o f to, the initial time at which the element is considered. Within the oscillatory layer, the second term (representing the Stokes drift velocity) on the right-hand side yields terms O(a2e r) (r = 0, 1, 2 . . . . ), on account of eq. 4. However, as regards

97

the Eulerian mean velocity, a2~2 ~it may only be said, without further consideration of ~ 2 beyond the oscillatory layer, that the dominant part is O(a 2) and that some, at least, of the higher-order corrections are O(a2e s) (s = 1, 2, 3,... ). The "first approximation" to the mass transport velocity within the layer consists of the dominant part of eq. 8, as given by LonguetHiggins (1953). Thus, we write: =

U1L

=

q,dt'Vu,

u20 + l i m e+0

(9)

Z

Wl[1]

=

W1L

= -2½e

f

(~UIL/ax)dZ

0

3.1. The first correction

When ~2~ ~ e2 (progressive waves) or when ~2~ ~ e3 (standing waves), the work of Dote (1978) shows that mean vorticity diffuses throughout the entire fluid, whence both q2 and ~ 2 are O(~/e) beyond the oscillatory layers at the b o t t o m and air--water interface. Equations 6 and 7 yield boundary conditions for this rotational, interior motion and, when determined, the corresponding mean vorticity matches with that at the outer edge of the b o t t o m layer and is equivalent to a mean tangential stress, O(~2,7e), acting on the water within the layer. In consequence, eq. 3 becomes: ~:

= 2 - { e - l a ~ 2 / a Z = O(?/e)

as Z - + oo

(10)

so that, near the edge of the oscillatory layer, u2 must contain a linear term in Z which is O(,7). Within the layer, the first correction to UIL is therefore O(a2,7), being due partly to u2 in eq. 8 if ,7 = O(e) (}, ~ 1 m), but solely to ~: if ~ >> e (h >> 1 m). [Mathematically, the dominant part of u2 is made continuous at the edge of the layer by use of eq. 6 as a boundary condition; eq. 10 is required to make the vertical gradient continuous.] Generally, ~ : is mainly confined, because of greater wave amplitudes and correspondingly enhanced convection, to only p a r t of the interior* of the fluid. Near the bottom, this takes the form of a second (or double) boundary layer adjacent to the oscillatory layer, as indicated by Dote (1976, 1977). Progressive waves. In the model suggested by Dore {1977), the double layer •

,

1

has non-umform thmkness O(~x~), where: e~6

= e/a
and x is measured from an idealized origin of wave generation. Since the mean *See pp. 104 a n d 1 1 2 f o r progressive and standing waves, respectively.

98 velocity scale is a 2 , mean vorticity is O(a2/6x ~) within the double layer. However, the model requires U1 >I 0 at the outer edge of the layer, whence, with the exception of very long waves (h < 1), the net mass flow must be positive. Calculations based on the model for h = O(1) cannot therefore relate to some published results of wave-tank experiments, b u t should be relevant in the open sea or to the forward region of a long tank before return currents O(~ 2) occur. Mass transport profiles corresponding to positive net mass flows have b e e n obtained by l]nliiata and Mei (1970), although significant effects of air drag were not considered.

Standing waves. Mean vorticity is shown by Dore (1976) to be O{a2/5) within the double layer of essentially uniform thickness O(5). Since the dominant term, O(a2), of U! -~ 0 at the outer edge of this layer, the theory has no restriction analogous to that stated above for progressive waves. When a >> e and, for progressive waves, 5x-~ < 1, mean motion in the double layer near the b o t t o m is, predominantly, quite independent of the mean flow in the air--water interfacial layers; ~ is O(a 2 ) for progressive and standing waves. Within the oscillatory layer, eq. 2 requires:

t=O(1/~x~)l ~o2 = 2-~e-ta-u2/aZ

O(1/6)

as Z ~ o o ,

[progressive waves] [standing waves]

(11) (12)

In consequence, u2 contains a term O(a), and so the first correction to glL within the layer is, formally, O(a 3) for sinusoidal waves, and is calculated in § 4. In relation to standing waves, the presence of this correction has been noted, in a different physical example, by Riley (1965, p. 169).

3.2. Other writers' work To the author's knowledge, only two writers have sought to determine U1, in the oscillatory b o t t o m layer, to a higher degree of accuracy than that, O(a2), of U I L "

Work of Sleath (1972). Sleath ignored higher-order terms generated in eq. 8 by the expansions of ql and q: • Instead, in his consideration of sinusoidal

progressive waves, he obtained "a second approximation" to U! by working to O ( a 4) - - since q3 only involves terms in exp[i(2n -- 1)t] (n = 1,2), and q4 has terms in exp(2int)(n -~ 0,1,2). However, by ~ 3.1, mean vorticity >~O(~ 2 ) is necessarily present b e y o n d the oscillatory layer, so that the first harmonic part of q~ is rotational. Therefore, the calculation by Sleath (pp. 298--300) at O(a 3 ) and O(~ 4) is erroneous, since it employs a value for this harmonic as derived by potential theory. [A related mistake was also made by Sleath (1973, eq. 4).] Beyond terms O(a2), the net drift of any fluid element over a wave period not only depends on its initial position but also on the initial instant to within the cycle. Consequently, Sleath defined the average

99 mass transport via the average of each such drift motion for to ~< t ~< to + 2~. After considerable algebra, he calculated:

#2~

1 to QIS - 2~ Jr0

Ql(t) d t

(13)

to O(~4), and re-produced VlL at O(~2). Work o f Isaacson (1976b). Isaacson (1976a) obtained a first approximation to the mass transport velocity in the oscillatory b o t t o m layer due to cnoidal waves. Subsequently, Isaacson (1976b) sought "the second approximation", employing an expansion analogous to eq. 2, but with a parameter ~ = (wave height)/(depth at trough). Because the dominant term, ~/, say, involves an infinite number of harmonics e x p [ i n ( t - x ) ] ( - ~ < n < ~), a correction to the first approximation for UI arises at O(~ 3 ) for cnoidal waves, compared with Sleath's correction O(~ 4) for sinusoidal waves. Thus, Isaacson (1976b), after much algebra, also used the expression (eq. 13) of averaged mass transport velocity. However, like Sleath's, Isaacson's calculations take no account of the manner in which the rotational, mean motion O(~2), near the edge of the oscillatory layer, merges, through a double boundary layer, with the corresponding irrotational motion in the interior [cf. Isaacson (1976a); eqs. 2.36, 3.2 and 3.5]. In essence, Isaacson (1976b, eq. 2.9] has incorrectly applied the condition, stated b y Batchelor (1967, p. 360), on mean vorticity at the adge of the oscillatory boundary layer. It seems highly probable that the double boundary-layer theory suggested b y Dore (1977, § 3) can be readily extended to cnoidal waves, so that a new term, O (~3) in u and arising from a relationship like eq. 11, must also be included within the oscillatory b o t t o m layer. Such a term is formally comparable (except, possibly, where x~ >> 1) with Isaacson's "first" correction. The dominant part, O(~ 2), of the mass transport velocity is zero b e y o n d the double boundary layer, under the condition of zero net mass flow -- Isaacson (1976a, eq. 3.5) -- so that the theory of Dore (1977), adapted for cnoidai waves, would apply for this condition. Standing waves. No previous calculations are known of the first correction to UIL in the oscillatory boundary layer. In § 4.2, we therefore obtain the new term arising from eq. 12. The topic has, however, been mentioned by Hsu (1979) in his theoretical and experimental investigation of mass transport in the b o t t o m boundary layer of short-crested progressive waves. 4. THE FIRST CORRECTION FOR SINUSOIDAL WAVES The mean motion just b e y o n d the oscillatory b o t t o m layer is governed by the equation: (QI-V)~2 = 52~/2~2,

(~2 = ~/2~2)

(14)

100 Longuet-Higgins {1953), Dore (1976). When 6 ~ 1, the mean flow has a boundary-layer character, and ~ 2 decays to zero within a double layer whose thickness, we assume, satisfies 5 (x) ~ min (1, h).

4.1. Progressive waves In the theory of Dore (1977, § 3), the boundary-layer version of eq. 14 is integrated with respect to the stretched variable z/5 from z/d to 0% and the waves are regarded as being generated from a localized origin at x = 0. The similarity solution: ~2/8 = ( 2 x ) ~ [ G ( ~ ) - ~ ] c o s e c h h = (z cosech h)/5 (2x)}

(15)

then arises, wherein G ([) satisfies a boundary-value problem of Blasius type:

V'"+GV" = 0

(0~<~
(16)

V(0) = 0, V ' ( 0 ) = 5/4,

(17)

V'(oo) = A + J 2

(18)

'

where ' denotes differentiation with respect to ~. If Co is the horizontal component, O(1), of 712 in the irrotational region just beyond the outer edge of the double layer: Co = A cosech 2 h The boundary conditions on G' (~') arise from asymptotic matching at the edges of the double layer (cf. eq. 6), and, in fact: Vl ~ 52 G'(~),

(& = a cosech h)

The boundary-value problem has a solution in (0,oo) as follows: 0

A > 3/4 j !

G"(~)

- 0,

G'(~)=5/4,

A = 3/4 1

(19)

I

< 0

-5 < A < 3/4

'

1

If A < --~, eq. 14 has no solution of the form of eq. 15, and backflow exists near the b o t t o m (but beyond the oscillatory layer). In terms of A and h, the net mass flow per unit depth is given by:

Q/h = 5 2 [A + (sinh 2h)/4h] _

1

and is > 0 when A > -3, but = 0 for A - -5 in the long-wave limit h -~ 0. By eq. 11, we are particularly interested in: lim (~2) ~ 5-' (2x) --~ G " ( 0 ) cosech 3 h z 16 +0

(20)

101

Within the oscillatory layer, eq. 5 shows that the dominant term in ~ 2 is ~e-lO2~2(x)/OZ2 , but this ~ 0 as Z -*~ because there is no mean vorticity O{a2e -1 ) b e y o n d this layer; further, the next term, shown explicitly in eq. 5, yields a contribution to ~2 which is O(1), that is, small compared with the order of magnitude o f the right-hand side of eq. 20. Hence, matching conditions at the outer edge o f the oscillatory layer require the solution, within this layer, of the time-averaged vorticity eq. 1 at O{a28-1 ). Since qx and its space derivatives involve only powers of e, we have: 0 =

(21)

04~)20)/0Z '*

where = o =

(z = o)

(22) ~ 2 ~ 2 ( 1 ) / a Z 2 ~ (2x)-½G"(0) cosech a h

(Z-~ ~)

and the term e2~ -1 ~2o)(x, Z) must be introduced between those O(e) and O(e 2) in eq. 5. Thus, we obtain a contribution: (23)

a2-52o) = ~3 x-½ G " (O)Z

to the mean horizontal velocity ~, in agreement with § 3.1. The quantity G " ( 0 ) is plotted against A in Fig.1. Of course, it can n o w be seen that a2~2(1) could, alternatively, be included in the term aau3. However, since the convection terms O(a a ) within the oscillatory layer have zero mean, we actually have: a~u2(l) = aa~3

(24)

for sinusoidal waves (cf. p. 98). Equation 23 represents a uniform shear, which decays as x-½, and, for realistic wave amplitudes (see below), yields the first correction to U1L. Thus, the "second approximation" to the mass transport I

I

l

t

I

r

t

l

I '7o) 0

-I

0

A

7

2

Fig. i . The f u n c t i o n G " ( 0 ) o f eq. 20 p l o t t e d against A .

102 velocity within the oscillatory layer is given by: e l [2] = UI[11 + {X3U2(1) + O(a 4, 0t2~,

a2e) ~ 62Fp [2] (x,Z; 6)

where Fp[2]

=

[~ (5 - 8e -Z cos Z + 3e -2Z) + 6P(x)Z]

(25)

and P(x) = x-~ G "(0). Although U1[1] = UIL > 0 for Z > 0, u2o~ ~ 0 for Z > 0 according as G " ( 0 ) ~ 0. In fact, Ultfl/62 = Fp hI has a global m a x i m u m value of 1.391, within the layer, at Z hI = 2.303, together with local minima and maxima at Z = (r - ¼)n (r = 2, 3, ... ). (The corresponding values, Fp [1] = 1.376 and Z [~1 = 2.306, given by Longuet-Higgins (1953) and quoted by many subsequent writers, are slightly inaccurate.) The effect of ~2¢1~ is as follows. (1) If 6P(x) > 0.090, no stationary values of U1[21 occur across a section x = constant of the oscillatory layer, and a Ult2]/OZ > 0 for Z >~ 0. On taking Z~ = 2n to correspond to the outer edge of the layer, we therefore have: max U1[21 = UI[2 ]= 62[ s + 2n6P(x)] Other values of Z~ may be assumed; for example, Z -~ 4.3 represents the position b e y o n d which Fp [~1 lies within 1% of its asymptotic limit 5/4, whereas, for Z/> 5.2, the magnitude of the shear d Ulttl/dZ is less than 1% of its value at the b o t t o m . (2) If -~ < 6 P(x) < - 0 . 0 0 3 7 , there is a single stationary value (maximum) at Z t2] -~ 2.303 + 3.74 & P(x), which is closer to the b o t t o m than LonguetHiggins' position Z ill. We find: max U1[21 = U1[21 z[:] = 62[ 5.5--65 + 2.303 6P(x) ] k 4 U1[2] rz[2] / U I = [2] = 1.113 - 3.752 6P(x) (3) If - 0 . 0 0 3 7 < 6P(x) < 0.09, two stationary values of U1[2] occur for 0 < Z < 2~, of which that corresponding to the smaller value, Z [2] , of Z is a maximum. The relative effect of u2o) is given by E = 6P(x)Z/Fp [11 (Z). As Z -* 0, E -~ 2KP(x); at Z = Z tl] , E = 1.6596P(x); and at Z = Z=, E = 5.0276P(x). Thus, fi~tl) has most influence on the mass transport velocity within the outermost part of the b o u n d a r y layer. The following comments are made about the derivation of eq. 25. (a) The values of both Ul~ l and UI~] are independent of the constant kinematic viscosity vw. Although Longuet-Higgins (1958) showed that Ul~ ] is actually independent of any vertical variation of viscosity in the layer, the counterpart of eq. 21 demonstrates that there is no corresponding result for 1

103

(b) For constant Vw, the present approach is equivalent to forming the c o m p o s i t e asymptotic expansion:

Ul ~ (Ul[~l - lim U1tll ) + G ' ( [ ) Z÷oo

(26)

uniformly valid, to O(1), within both the oscillatory and double boundary layers. The structure of the expression (eq. 26), comprising the double-layer function G ' ( [ ) and the oscillatory-layer quantity in braces (decaying to z e r o at the edge of this layer), is similar to that which is frequently used to represent the primary velocity O(a) within the whole fluid. The important difference lies in the manner of the construction; for the O(a) velocity field, the outer flow is determined first, whereas the oscillatory boundary-layer function U1[~] is obtained initially in eq. 26. Within the oscillatory layer, where ~ should be small, expansion of G '(~) gives: U1 ~ Ulel] + f G " ( 0 ) + f2G'"(0)/2! + f3G'"'(0)/3! + . . . = U1[11 + G " ( O ) ( ~ Z / x ½ ) - 5 G " ( 0 ) ( ; , Z / x ½ ) 3 / 2 4 + . . .

(27)

since G '"(0) = 0 and G'"'(0) = -5G"(0)/4 by eqs. 16 and 17. The second term on the right-hand side corresponds to eq. 23, whilst the third term is insignificant in practice. W e emphamze, however, that this method, ~ia a compo• find the first correction . ~__ site expansion, c a n n o t be used to ~o urT~[ 1 1] i f t h e viscosity varies vertically within the oscillatory layer (see (a), above). (c) Whereas U1[1] is independent of net mass flow Q and horizontal distance x, UI[2] depends on both these quantities. However, with a view to experimental comparisons, the x-dependence is weak. Thus, for example, if ~ = 0.2, h = 0.5 and taking A = -~, for which G "(0) = -0.877, we obtain max Ul[2] = 1.264, 1.310 and 1.327 at x = 10, 25 and 40, respectively. (d) In contrast to calculations of Sleath (1972) and Isaacson (1976b), the first correction O(a 2) in the phase velocity is not required for Ul[21 . Moreover, Ul[2] does not depend on the averaged definition (eq. 13) used by these writers. (e) Since Ul[l] is independent of the mean flow outside the oscillatory layer, the corresponding time-scale which must elapse, following initiation of waves, is O(1). For Ul[23 , however, the double boundary layer must be established, and this requires a longer time-scaie O ( x / a 2). (f) The mean flow, just b e y o n d the double boundary layer, is necessarily irrotational in the calculation of Ul[2] . In addition, it is required that: 5x] ~ rain(l, h)

(28)

(g) The spatial damping length, over which wave amplitudes decay by a factor e-1, is given by: x d = (2h + sinh 2h)]2½e + O (w/e)

104

so that, by eq. 28, a necessary condition that the present solution (A I> -~), modified by the influence of the damping factor e -x/xd, is valid for x = Xd is that a satisfies the stronger condition: 1

1

>~ e~ max(h~, h-~)

(29)

The theory of Dore (1977, 1978) shows that, when eq. 29 is strongly satisfied, the double layers at the b o t t o m and near the air-water interface do n o t intersect. If, however, a satisfies eq. 28, b u t n o t eq. 29, the present theory becomes invalid for x/> O(8-2) min(1, h2), where the mean flow is rotational t h r o u g h o u t the depth. (h) For laminar flow, Xd might be as small as 500 in a wave-tank experiment, b u t as large as 104 or 10 s in oceanic conditions. In the latter case, turbulence might increase the effective viscosity, and, thereby, reduce such values of Xd by a factor o f a b o u t 10. The effect of the new term eq. 23 on Ul is shown in Fig.2, where profiles of Fp[2] for various values of ~P(x) are given. For long waves (h <1) and zero net mass flow, we m a y take A = -~, which corresponds to negative values of 52(x) and to a reduction in mass transport compared with Ul[1] , throughout the oscillatory layer. This feature was also found in the incomplete theory of Isaacson (1976b) for cnoidal waves. In Fig. 3, the ratio U~]/UI~ ] is given as

6O

1i

I

O~

&P(x)=O OO2 ~ \ ~

u/'~'

°o; 01

1

01~5

"

- ~•I 0

5

06 _ _ _ L _ _ 02 04

FS

_ i 06 h

_! . . . . OB

1 1

Fig. 2. Mass transport velocity profLes across the oscillatory layer for progressive waves, second approximation Fp[2] = UI[~ ]/& 2 of eq. 25. First approximation of Longuet-Higgins (1953), &P(x) = 0; dashed line, curve through m a x i m u m values. Fig. 3. Ratio of the second to the first approximation for the mass transport velocity at the .~dge of the oscillatory layer;x/h = 25,103(G"(O)/G"(O){A=_4)(~/x) = 0, 4 and 8 corre:pond to curves (a), (b) and (c), respectively: q

~

105

a function of h for x/h = 25 and fixed negative values of aG"(O)/x; if we take A = -~, then the two curves correspond to a/x = 4 X 10 -3 (upper) and 8 X 10 -3 (lower). It is seen that the second approximation is noticeably less than the first-- as in the theory of Isaacson (1976b) -- but that, in contrast to curves (a) and (b) of Isaacson's fig.5, the reduction diminishes as h increases. For h = 0.4, the percentage reduction is about 13.6 and 27.2 for a/h = 0.1 and 0.2, respectively*. Reduced values of the present theory, as compared with those of Longuet-Higgins (1953), are also seen in Fig. 4, where m a x UI[2l/(a/h)2 is plotted as a function of h for three values of a/h. A n interesting feature is the predicted decrease as h + 0 (c~/h fixed, non-zero). This considerably clarifiesthe statement, Isaacson (1976a, p. 411), that such a trend "... is not accounted for by sinusoidal wave theory", although his fig.2 clearly suggests a marked reduction as h -~ 0, according to both his cnoidal-wave theory and to his comparison with experimental data of Allen and Gibson (1959). [In relation to his fig.2, we note that Isaacson has, apparently, compared his theoretical values at the edge of the oscillatory layer with Allen and Gibson's data for max UI. ] 1"4;---.--

I

--

==~ u f ~ 1.3

1.7

0'I

013

b

OL5

0.7

Fig. 4. Maximum values of mass transport velocity within the oscillatory layer; dashed line, asymptote for curve having a/h = 0 ( f ~ t approximation).

Comparison with experiment. In order to attempt any comparison of results of the present theory with existing experimental data, the latter should, strictly, relate to circumstances when mean vorticity can be conceived as having been confined to a double boundary layer. Thus, measurements should have been taken when sufficient time r(x) has elapsed, after initiation of wave motion, for this layer to have become established in the observation region, but not so much later that mean vorticity has been diffused and convected, from boundaries of the water/wave-tank system, throughout the whole fluid. If tc(x) is the time for a hypothetical fluid element, supposedly always having the (local) maximum mass transport velocity, to be convected from the neigh*If we take Zooffi 4.3 instead of 2% the corresponding reductions are 9.3 and 18.6, respectively.

106

b o u r h o o d of the origin to station x, then r(x) > tc(X); whilst, if tc(X) is the corresponding time for any real fluid element, tc(X) > tc(X). Also, the present 1 t h e o r y only corresponds to zero net mass flow when A = -5 and h -* 0. With these considerations in mind, a nominal comparison of the present results for Ul~ ] has been made in Fig. 5 with a typical set of data of Brebner and Collins (1961). The experimental points correspond to a water depth of 38 cm, with h -~ 0.57 and x = 12, for which the appropriate theoretical curve has A = -3, and P(x) -~ -0.26. From computations of G(~), a practical representation of the thickness of the double boundary layer gives: 5(x) = u g(2x)~,

(5 = e/&)

(30)

so t h a t the data have 6 (x)/h <. 0.26 for &/> 0.1. Thus, for smaller &, the theory might n o t be sufficiently accurate, partly because eq. 15 neglects the variation in Stokes drift across the double layer. Moreover, the measurements were made several minutes after wave motion was started (A. Brebner, pers. c o m m u n . , 1980), so t h a t the double layer would probably n o t have been fully established in the observation region, where tc -~ 6 min. Nevertheless, despite disparities in conditions of theory and experiment, the c o m m o n trend, of an increasing difference f r o m U1[1] = U1L as a increases, is apparent. Other data obtained by Brebner and Collins for h < 1 correspond to values of P(x) between - 0 . 2 and -0.38, so that Fig. 5 also includes a curve for P(x) = -0.4. In addition, the dashed line represents, roughly, the mean of the experimental results, for &/> 0.1, where "significant departure" from UI[~=] is obtained. Brebner and Collins (1961) and Brebner et al. (1966) attributed this

01 P(x) = 0 ~_ O.Z6 ~

/ /

i

.~

04~ U[2]

loo

0-0;

0-001

0025

I 01

06

&

Fig. 5. Mass transport velocity at the edge of the oscillatory layer; e, experimental data of Brebner and Collins (1961) for P(x) = 0.26; dashed line, mean of experimental results for &> 0.1.

107 departure to turbulence in the oscillatory layer, which occurred when the Reynolds n u m b e r R = ~-~ t> 160/2~ (~ = 0.117 in Fig. 5). This conclusion was queried by Isaacson (1976a), and the effect of turbulence appears from Fig. 5 to be somewhat less significant than originally supposed by Brebner et al. It is possible, perhaps, that this w e a k e r effect on Ul.is related to item (a), above. Far field: 52 ~ 77. As noted after eq. 29, although the double boundary-layer

theory is formally valid in 0 < x < 0(8-2), for h = O(1), say, the mean flow becomes rotational everywhere in the far field, 0(8 -2 ) < x <~ O(e -t ), where the conduction solution of Longuet-Higgins (1953), significantly modified by Dore (1978) to allow for air drag, should come into evidence. The mean vorticity at the edge of the oscillatory bottom boundary layer varies in some continuous manner, from the algebraically decaying expression of § 4.1: ~o.. = ~3 e - l ( 2 x ) - ~ G " ( O ) ,

to the expression: ~ ® = 3 Q ( ~ / h ) 2 _ ~2 [2-s/2(cosh h + sinh h)2~ea I +

+ ~{2sinh 2h + ~3h-l(h -1 sinh 2h + 3))] e -2x/xd

(31)

of Dore (1978, § 4.1) in the far field. Although eq. 31 generates a contribution of the form a 2 ( e / e a ) Z 0(7 coth 2 h) to the mean velocity ~ in the oscillatory layer, analogous to eq. 23, its significance, relative to ~2ill , is negligible in practice. [The quantity ea2 represents a wave Reynolds number for air.] 4.2 S t a n d i n g waves

For 8 ~ 1, write D

42 = 8 cosech h [ 4 + o(1)],

z = gN

in the formulation of Dore (1976). Then 4 ( x , N ) satisfies the boundary value problem: 04

024

aN axaN

4=0,

04 024

034

ax aN 2

aN 3

a 41~N

=

a4/aN

~ 0

- ~

a41aN = 0

sin 2x

(0
(N = 0) (N-," oo)

(32)

(33)

1

(x = (r + ~),r, r = O, +1, + 2 . . . )

within the double boundary layer in any cell of width ~;~ bounded by verticals through a node and an adjacent anti-node. The solution has been computed, in a different context, by Davidson and Riley (1972), whilst a colleague, Dr. D.J. Crampin, has determined 4 ( x , N ) as a special case of a

108

study of mass transport in short-crested waves, and her results are in excellent agreement with those of Davidson and Riley. For present purposes, we require: a2~2

N=0 ~

a26-1 cosechah(a2¢/bN2)N=O

= _&2~-1 DM/ax

(34)

where M ( x ) = [ (~¢/aN) 2 d N 0

is representative of the m o m e n t u m flux in the double layer. On writing: g ( X ) = (~2¢}/~N2)N= O,

X = x-(r+

1)~

(35)

so that X = 0 at a node, we obtain: ~20) = g ( X ) Z2 cosech3 h a2u2(1) = g3 2 ] g ( X ) Z

(36)

within the oscillatory layer, in order t h a t the mean vorticity match, as Z -~ o% with t h a t given by eq. 34 [cf. § 4.1, eqs. 21 to 24]. Correspondingly: ~2~2(, ) = -e& 3 g ' ( X ) Z 2

(37)

and eqs. 36 and 37 represent the first correction to 01L. The function g ( X ) ~ -33~2X/8 as X-* 0 and is shown in Fig.6, from which it is seen that g(X) = O(1) and vanishes at X = 0 and at X = X0 = 0.714 n/2; also, g~(X) has zeros at X = Xz = 0.347n/2 and at X = X2 = 0.93n/2. The stream function for the mass transport in the oscillatory layer is given by:

O/l[2]

=

q'l [1] + a2e 2d-1~2(,) = a2eF[s 2] (X, Z)

where 2-~ sin 2x F s [2]

=

{ 3 Z - 4 1 1 - e - z ( s i n Z + c o s Z ) ] --32(1- e-2Z)) + 5g(X)Z 2 (38)

8

Thus, the second approximation to QI is: Ul[2] = Ul[,] + a2u2(, ) = &2[~ sin 2X(3 - 8e-Zsin Z - 3e -2Z) + &S(X)Z] (39) Wl[21 = _& 2 ea F[s~l/~ X

(40)

where S(X) = 2 ~ g(X). The function g(X)/sin 2X is shown in Fig.6, and is seen to be m o n o t o n i c increasing from the value -33/2/16 as X ~ 0. According to the first approximation ogf,] of Longuet-Higgins (1953), the streamlines in 0 ~< X ~< ] u are closed for 0 <~ Z ~< 1.473, symmetrical about

109 01375

0-125

//

0 125

0

g IX)

-

0"25

0

012~.

g (X) stn2X

- 0 125

-0'25

-

0.25

0.2

O.Z,

0.6

0.8

- 0375

1.0

2 Xlrf

Fig.6. Solid line, the f u n c t i o n g(X) of eq. 35 for standing waves; dashed line, g ( X ) / s i n 2X.

X = ¼~, and the mean motion has a stagnation point at Xs = ~ ~, Zs = 0.928. Profiles of Ull~] are independent of X, U1[~] <> 0 according as Z <>Zs, and 8Ul[~]/& 2 sin 2X -~ 3 as Z-* oo ;within the oscillatory layer (Z ~< 2u, say), there is a single minimum value - 0 . 4 3 9 at Z = 0.435, together with a single maximum value 3.110 at Z = 3.937. In specific numerical calculations, eq. 39 is not always found to be completely adequate throughout the oscillatory layer for 0 < X < ~1 ~. The reasons are: (1) g ( X ) vanishes for X = X0 ; (2) calculations required for ~ as large as about 0.2; (3) investigation of max Ul for Z as large as about 4; (4) in the middle and outer parts of the layer, Z t> 2 say, the two terms in square brackets in eq. 39 are of opposite sign in 0 < X < X0 and, depending on &, may be numerically comparable in magnitude. Consequently, we have also made use of an improved second approximation to U1 in the middle and outer parts of the oscillatory layer, viz.: U112] = Ul[2] + (3/8)2a 4 Z 2 sin 4X

(41)

This expression*, an extension of § 3.1 and obtained from the asymptotic matching process, makes the curvature of U1 continuous, to lowest order, at Z~. It can be seen that eq. 41 is equivalent, to O(& 4 ), to: U1 = lUlls] - l i m U1D] ] + Z-.oo

(~dP/aN)g=2~&Z

(42)

where the oscillatory-layer function in square brackets decays to zero at the edge of the layer, cf. eqs. 26 and 27.

*Generally, max Ul[21 -~ max UI[~1 . But the level, Z = ZM, of m a x U! is quite sensitive to higher-order terms.

110

We make a f u r ther i m p o r t a n t c o m m e n t regarding the expression (41) and numerical applications. F o r & = 0.1, and Z = 0.5, the ratio of the magnitudes o f the second and first terms on the right-hand side is 0.67 >< 10 -2 at X = u/8, and 0.98 × 10 -2 at X = 3u/8; thus, for all practical purposes, U1[2] is, indeed, sufficiently accurate in this inner part of the oscillatory layer. [ F o r Z = 3, the ratios are 5.39 X 10 -2 at X = u/8, and 4.65 X 10 -2 at X = 3u/8.) However, it must be recognised t hat there exist terms such as a 4 ~4 which, in reality, c o n t r i b u t e to U1 within this layer, but which are finite as Z -+ ~ . In addition, when ~/e = O(1) say, the mean m o t i o n b e y o n d the double b o u n d a r y layer has velocity field O(a2d), which induces a horizontal velocity O(~25), and vorticity O(a 2), within the double l a y e r - cf. Riley (1975). In consequence, a term of the f o r m &2eZ is required in ~2 within the oscillatory layer. Finally, on taking a c c o u n t of eq. 42, all higher-order corrections to U1['] are included in: U 1 = ~1t2] + O ( & 4 , ~ 2 e z , ~ s z 3 )

.

Thus, in the middle and out e r parts of the oscillatory layer, it is necessary that: 5Z >> ~ = e/a if eq. 41 is to be a consistent a p p r o x i m a t i o n to U1. Additionally, whilst it might be t h o u g h t t ha t ~Z ,~1 is necessary, it is found in practice (by comparison of eqs. 41 and 42) that: Z ~< min(2~, 1 / 2 ~ )

(43)

yields sufficiently accurate values for Ul. This is due, in part, to the factor 3/8 occurring in eq. 33. Thus, for 6 ~< 0.113, the present results are satisf a c t o r y f o r Z ~< 2~; if 6 > 0.113, t h e y are only reasonable over the complete X-range if Z ~< 0.707/&. The m a x i m u m value of U1 near the b o t t o m is of particular theoretical and practical interest. According to lowest-order theory, O(a2), t h r o u g h o u t b o t h the oscillatory and double b o u n d a r y layer, c o m p u t e d results within the double b o u n d a r y layer show that, for 0 < X ~< X0 = 0.714~/2, the only maximum value of Ul is 62(3.110/8)sin 2X, which occurs at Z = ZM = 3.937 Ul/(~ 2, seemingly, has two within the oscillatory layer. But, for X0 < X < ~Tr, ' m a x i m u m values, viz 0.389 sin 2X at Z -- 3.937, t oget her with a second value within the double layer (Davidson and Riley, 1972; fig. 3). The latter value decreases m o n o t o n i c a l l y with X, from (3/8)sin 2X = 0.293 at X = X0 + to 1 0.150 at X = ~n, and occurs at a level N = NM(X) which increases m o n o t o n i c a l l y with X, f r o m N = 0 (effectively, the edge of the oscillatory layer, within the c o n t e x t o f lowest-order t h e o r y ) to N = 0.91 at X = ' n. Schematic illustration of the m a x i m a o f U1, according t o lowest-order t h e o r y , is seen in Fig. 7. When terms o f higher order than 62 are included in U1, the occurrence of

111 maxima changes significantly. For Ul is given (approximately) b y eq. 40 or eq. 42 in the oscillatory layer, and the transport in the d o u b l e layer must n o w be obtained by using a ¢ / a N of eq. 32 n o t for N ~ 0 but for N > / 2 3 / 2 ~ (on taking Z® = 2n, as before). Since -g(X) cosec 4 X increases monotonically to oo in 0 ~< X ~< 1A~,it is readily shown that Ul[21 has a single maximum which, for 0 < X < X0, occurs at a variable level ZM (X; &), < 3.937, provided that Z < 0.208. Moreover, for Xo < X < lh~, it is found numerically that there is only o n e maximum of ~ 2 l throughout b o t h b o t t o m boundary layers. For a certain interval X0 < X < ~(&), the maximum occurs within the oscillatory layer where ZM(X; &) < 3.937; there is n o maximum in the double layer, where we have 23/2~ & > NM(~). For X = ~(d), Z M = 3.937, and, as X increases further, Z M increases b e y o n d the value 3.937. If 2 3 / 2 ~ > 0.91, that is: > 0.102

(44) 1

max Ul occurs at ZM < 2~ (within the oscillatory layer) f o r X o < X < ~ . But if & < 0.102, ZM = 2~ for X = ,~ in (X0, ~ ) such that NM(X) = 23/2~& < 0.91. Then, for X0 < X < )~, the maximum occurs in the oscillatory layer, whilst, f o r X < X < ~1 , it occurs in the d o u b l e layer. The position of the single m a x imum of U 1 according to the present higher-order theory is shown schematically in Fig. 8. Example. If & = 0.05, 2~/2~& = 0.444 = NM(.~), and computations within the double

boundary layer give .~ = 0 . 9 0 8 ~ / 2 , where m a x UI[2] = 0 . 1 8 8 , Z M = 2n and max Ul[l] = 0.111, Z M = 3 . 9 3 7 .

J

0

0

0 u~

0

ut

Fig. 7. (Schematic.) Illustration o f maxima o f U! according to the first approximation; the dotted line represents the edge of the oscillatory layer. Fig. 8. ( S c h e m a t i c ) Illustration o f m a x i m a o f U! according t o the s e c o n d a p p r o x i m a t i o n , & < 0 . 1 0 2 : (a) X < X 0 < X; (b) X > X. The d o t t e d line represents the edge o f the.oscillatory layer.

112

In Fig.9, profiles for various values o f 2X/~ are s h o w n for t h e case & = 0.2, and it is seen t h a t t h e y are significantly d e p e n d e n t on X; t h e c o n d i t i o n (43) limits the t h e o r e t i c a l profiles o f eq. 41 t o Z < 3.536. H o w e v e r , b y eq. 44, m a x U1[2] necessarily o c c u r s w i t h i n t h e oscillatory layer. Near the b o t t o m , t h e h o r i z o n t a l v e l o c i t y t o w a r d s the n o d e :X = 0 is increased f o r 0 < X < Xo and r e d u c e d f o r X0 < X < ~ ~. !

]

--

:

T "-

i ": h'

"

- 1|

410 L

2)(/17:01

3.0

]

"~ }~08: !

Z I i

/

2o~ 10 ~I

O: -01

_

[

01

0

,

02

__

U

Fig. 9. Mass transport velocity profiles across the oscillatory layer for standing waves; second approximation, & = 0.2.

T h e t h e o r y o f this section breaks d o w n within a h o r i z o n t a l distance O(5) o f anti-nodal l o c a t i o n s X = ~1 , where eq. 39 shows t h a t t h e p r e d i c t e d correct i o n to UI[I] is n o n - z e r o . T h e reason is, o f course, t h a t a similar b r e a k d o w n occurs in the d o u b l e b o u n d a r y - l a y e r t h e o r y o f Davidson and Riley ( 1 9 7 2 ) and D o r e ( 1 9 7 6 ) a r o u n d these locations, w h e r e the m o m e n t u m flux each side o f t h e a n t i - n o d e is n o n - z e r o , leading t o a collision o f t h e m e a n flows and a resulting 'jet-like' m o t i o n d i r e c t e d vertically upwards. Since we have m a d e use of t h e f u n c t i o n ~ f r o m the d o u b l e b o u n d a r y - l a y e r t h e o r y , it is inevitable t h a t such use w i t h i n the oscillatory layer is invalid near anti-nodal locations. For, whereas the m o m e n t u m flux associated with U1L is zero as X -* ~, so t h a t the leading t e r m in the oscillatory layer does not c o n t r i b u t e to the f o r m a t i o n o f the " j e t " , the m o m e n t u m flux associated with Ul [21 o f eq. 39 c o n t r i b u t e s a n o n - z e r o t e r m o:S2(1A~) at anti-nodal locations. F u r t h e r , since eq. 39 gives: gives: Ul[2] ~ ~ a ~ ^ 2 Z [ & S ( X ) _ 2 s i n 2 X ]

as

Z -* 0

113

there exists the possibility of a flow reversal, near the b o t t o m and close to anti-nodal locations, where:

g(X)/sin 2 X > 2~/~ However, by what has been written above, this reversal is only predicted on theoretical grounds if the stronger condition: ~ >> e

( o r ~ ~ ~)

holds.

Comparison with experiment. The same precautionary comments, as made in the initial paragraph of the analogous discussion in § 4.1, are taken for granted. However, since the double boundary layer for standing waves is essentially of uniform thickness, the time r for its establishment is expected to be shorter in practice than its counterpart for progressive waves. The most extensive set of experimental data for present purposes are due to Noda (1968), whose measurements were started about 10 minutes after the initiation of the standing waves (H. Noda, pets. commun., 1980). Although he obtained some reasonable agreement with VlL for 0.05 < ~ < 0.1 (but not for ~ < 0.05), the data for max U1 deviate more and more from max U1L, as increases. In fact, for ~ > 0.125, such data are only a fraction -- between about a quarter and a half -- of Longuet-Higgins' first approximation (cf. Noda's fig. 8). Noda, following Collins (1963) and Brebner et al. (1966), attributed this departure to the onset of turbulence at a "critical" Reynolds number: R (X) = Isin 2XI/~ = 160/2-~ = 113 if vw is taken as 10 -2 cm2sec -1 . In the present laminar theory, significantly reduced m a x i m u m values of mass transport velocity are obtained for X < X*(~), where 0.75~/2 ~< X* ~< 0.8~/2 approximately, but beyond this position the theoretical values of TABLE I Values of the ratio (max U1l~] )/(max U1[l] ) for standing waves 2X/~&

0.072

0.127

0.192

-*0 0.1 0.25 0.5 0.75 0.85

0.740 0.744 0.761 0.830 1.003 1.165

0.610 0.615 0.628 0.709 0.958 1.210

0.524 0.526 0.520 0.577 0.871 1.199

0.2 0.519 0.520 0.509 0.562 0.858 (1.194)

114

the second a p p r o x i m a t i o n exceed t h o s e o f Longuet-Higgins ( 1 9 5 3 ) , since changes sign at X = X0. These f e a t u r e s are e v i d e n t in Table I, in which the b r a c k e t e d value occurs at a level w h i c h does n o t satisfy eq. 43. T h e results o f t h e s e c o n d a p p r o x i m a t i o n f o r m a x i m a o f U1m and U1t2] are comp a r e d in Fig.10, b o t h with m a x U1D] and with N o d a ' s data, in w a t e r o f d e p t h 40 cm, f o r & = 0 . 0 7 2 , g = 1.13 × 10 -2 (a = 0.038, h = 0.503), a = 3 cm, m a x R (x) = 88.5. It is seen t h a t m a x Ull2I and m a x U1[21 are v e r y close, the s e c o n d a p p r o x i m a t i o n , in c o m p a r i s o n with N o d a ' s results, is m u c h s u p e r i o r t o

g(X)

20r

T

l

1

T

I i !

1.5 max u~

0.389

(a)

10

i (c)

05

0

05

075

ex/~

1.0

Fig. 10. Maximum values of mass transport velocity within the oscillatory layer, & = 0.072: (a) max UI[~I/0.389 = sin 2X, first approximation of Longuet-Higgins (1953); (b) max UI[2I/ 0.389, (c) max U112]/0.389. The hatched regions represent experimental data of Noda (1968). 6oL

-7

,

T

,

~

'°I

1

i

+

i

~0 Z 30

1

.f''"

0

i

05

-025

0

i 1

~_ .

025

05

.

.

.

.

.

0-75

i

10

U~Imax U:

Fig. 11. Mass transport velocity profiles across the oscillatory layer at X = 77r,1& = 0.072: solid line, second approximation UI[2]/max Ul[2] ; dashed line, first approximation of Longuet-Higgins ( 1 9 5 3 ) UI[I]/max U1DI ; +, experimental data of Noda (1968).

115

the first for X = ~1 , where max Ul[l ] is an overestimate, and both first and second approximations differ somewhat from the data near X = 0.6~/2. Profiles of UI[ll and Ul[2] at X = ~1 are compared in F i g . l l for this same value of &, and it is clear that the present approximation U1[2] is mostly closer to _1 data for X - ~ in Noda's fig. 7 than is the first approximation. In particular, the present results exhibit the reduction in Ul, as compared with UI[~] , found by Noda for Z > 3. Figure 12 refers to the case & = 0.127, ~ = 0.947 × 10 -2 (~ = 0.151, h = 1.005), a = 6 cm, max R (x) = 105.5. Again, the theoretical predictions of the second approximation, max U1[2] , are in very satisfactory agreement with Noda's data near X = ~ , and are generally superior in comparison with max Ul[~] for 2X/n between about 0.7 and 0.8. The above-mentioned measurements of Noda (1968) were made by observing dye streaks emanating from grains of potassium permanganate. As increases, with a and kh fixed, Noda states that the flow in the oscillatory layer becomes turbulent. Consequently, the dye-streak m e t h o d was succeeded by observation of nylon particles. Figure 13 corresponds to the case & = 0.192, = 4.24 × 10 -3 (a = 0.101, h = 0.503), a = 8 cm, max R (x) = 235.9. Although it can be seen t h a t the results of the present theory are generally far better, in comparison with Noda's measurements with nylon particles, than those of Longuet-Higgins (1953), such results now account (at this much larger value of &) for only part of the very substantial reduction in max Ul shown by Noda's comparison with max Ul[~]. We note, also, that max R (x) easily exceeds the critical value of Noda, so that turbulence may, indeed, be responsible for the remaining part of the difference. This seems to be a reasonable deduction on qualitative grounds, since it is indicated by calculations of Johns (1970), that standing waves are more sensitive to the influence of a z-dependent eddy viscosity (within the oscillatory layer) than progressive waves.

I

[

I-

~

70

fa/

0" 0

I

Z I

02

I

I

04

0'6

I

0"8

I0

2X/~

Fig. 1 2. M a x i m u m values o f mass transport velocity within the oscillatory layer, & = 0.127: (a) m a x UI[I]/0.389 ffi sin 2X, first approximation; (b) m a x U112l/0.389; + , e x p e r i m e n t a l data o f N o d a (1968).

116

maxU/0

a

)

0389

0

02

06

04

08

~0

2X/?i

Fig. 13. Maximum values of mass transport velocity within the oscillatory layer, & = 0.192: (a) max u1D]/0.389 = sin 2X, first approximation; (b) max U112]/0.389; +, experimental data of Noda (1968).

5. REVIEW AND CRITIQUE The t h e o r y presented here has been compared, somewhat tentatively, with experimental results of Brebner and Collins (1961) for progressive waves and of Noda (1968) for standing waves. In the former case, it is important to notice that some observers, e.g. Bagnold (1947), Allen and Gibson (1959), measured Ul~, whilst others, notably Brebner and Collins (1961), and Russell and Osorio (1958) measured max Ul. Moreover, data corresponding to these velocities are generally associated with very considerably different percentage reductions from the predictions UI~ l and max UIIll of Longuet-Higgins (1953) Thus, from fig. 4a, b of Isaacson (1978), measurements of Brebner and Collins (1961) give: ,rr[ll

~l~

_

.56 Vl)/gl~] ~ t 00.39

(h = 0.38) (h = 0.60)

(45)

for a / h ~ 0.185. The relative difference in max [/1, as measured by Bagnold (1947) for a / h = 3/16 and h < 1 (see, also, table 5 of Allen and Gibson, 1959), is given in Table II and is seen to be m u c h s m a l l e r than the values in eq. 45 corresponding to Ul~. This feature is broadly confirmed on inspection of fig. 7 of Allen and Gibson, where the data for 1.0 ~> h >~ 0.5, albeit with various and x, suggest values of the relative reduction averaging about 17% at h = 1.0, increasing to about 20% at h = 0.5. Unfortunately, Bagnold did not state the location of his observation station (s) (in a channel of length 10 m and mean depth 16 cm), so t h a t accurate comparison of Table II with the present theory is n o t possible. If we assume a single station at x / h = 3.5 m/0.16 m -~21.9, our second approximation yields percentage reductions of 7.4, 8.1, 9.4 and 10.6 for h = 0.916, 0.810, 0.647 and 0.531, respectively; the corresponding values of 8 ( x ) / h are 0.30, 0.29, 0.27 and 0.26, which are satisfactory for the double

117

TABLE II Percentage reductions in the values of max U1 measured by Bagnold (1947) from the (corrected) value, max UI['] -- 1.391& 2 , of Longuet-Higgins (1953)

0.164 0.169 0.175 0.179

h

[1 - (max Ul)/(max UI['] )] × 102

0.916 0.810 0.647 0.531

13.88 16.91 18.90 13.76

b o u n d a r y 4 a y e r theory. The second approximation U1[21, then, would appear to give reasonably accurate information, confirms the observed feature of smaller reductions for max U1 than for U l , (cf. footnote, p. 109), but, in general, remains a comparatively-slight overestimate. The latter may, of course, be due to several reasons, viz experimental scatter, methods of measuring Ul, some effects of turbulence (see § 4.1), condition of zero net mass flow {satisfied theoretically only for h = 0+), only partial establishment, in observations, of the double boundary layer, influence of higher-order terms (in particular, due to air drag), etc. Isaacson did not compare his theoretical results for max Ul with observed values, b u t it appears from his figs. 1 (1976a, b) and fig. 2 (1976b) that the {incomplete) cnoidal-wave theory does not yield the above-mentioned feature concerning max U1, at least for 1.0 ~> h ~> 0.5. Moreover, it appears from Isaacson (1976b, fig. 6) and Isaacson {1978, fig. 4b, c,d) that the cnoidal theory and/or the (erroneous) fourth-order, sinusoidal theory of Sleath {1972) generally yield significant underestimates of the observations of Brebner and Collins (1961) for UI~ when a/h > 0.1 and 0.45 ~< h ~< 0.75. In Isaacson's theory, at least one possible reason for this discrepancy suggests itself. If UI~ = ~2 C2(K ) + ~3C3 (K), where a = 2a/(h - ~), fig, 3 of Isaacson (1976b) shows that C3/C2 varies between about - 2 . 4 and - 1 . 6 as the modulus K of the Jacobian elliptic function cn increases from 0.8 to 1.0. There is clearly a strong possibility that fourth-order terms, at least, are highly significant in the cnoidal-wave theory if a/h > 0.1, i.e. ~ > 0.22. In the propagating-wave case, further experiments would seem to be desirable which adhere, as closely as possible, to the restrictions of the present theory. Given that 6 (x)/h is, indeed, reasonably small at the observation stations, careful measurements might, for example, detect an x-dependence in U1. In the stationary case, inspection of the mean flow near the nodes, say from X = 0.8n/2 to ~/2 {where Noda's results are scanty), would be of particular interest in relation to the nodal "jet", the increase in mass transport (Z >> 1) predicted by the second approximation, and the possible reversal in direction of U1 near the b o t t o m , provided that 62 >> e.

118 LIST OF SYMBOLS a

g h k q, ql . . . . r

t U~ U l , W,

...

W 1 , ...

X,Z

Xd zi A Fp, F s G N P Q

wave amplitude boundary-layer function for standing waves uniform equilibrium depth wave number (= 2n/h) velocity vectors position vector time horizontal Eulerian components of velocity vertical Eulerian components of velocity Cartesian co-ordinates spatial damping length position of air--water interface

X0 Z

constant in double boundary-layer solution for progressive waves double boundary-layer functions for progressive, standing waves double boundary-layer function for progressive waves double boundary-layer variable for standing waves (= z/6) mass transport function for progressive waves net mass flow mass transport velocity vector Reynolds number (= 1/6) mass transport function for standing waves wave period (= 2n/o) horizontal, vertical components of mass transport velocity zero of g (X) boundary-layer variable (= z/2~ e)

5 6

maximum wave-slope (= ak ) a/sinh kh measure of double boundary-layer thickness (= e/a)

Ql R S T U1, W1

1

Vw o 7

qJ, qJ, . . . . ¢...0 ~ ( . D 1 ~ . . .

,p

1

measure of thickness of oscillatory boundary layer (= V~vk/O~) double boundary-layer similarity variable (viscosity of air)/(viscosity of water) wavelength kinematic viscosity of water wave frequency (= 2n/T) time-scale for establishment of double boundary layer Eulerian stream functions components of vorticity (= V2 $, V2 qJ 1, -..) double boundary-layer function for standing waves stream function for mass transport velocity

119

(-)

( )[,l ( )12l (), ( )2 ( )l ( )lL

( )p,( ()~

s

time average over a wave period first approximation, according to Longuet-Higgins (1953) second approximation, according to present theory function calculated from first-order, or O(a), theory function calculated from second-order, or O(a 2), theory Lagrangian mean, or mass transport, function mass transport function of Longuet-Higgins (1953) progressive, standing-wave functions function evaluated at the edge of the oscillatory boundary layer

REFERENCES Allen, J. and Gibson, D.H., 1959. Experiments on the displacement of water by waves of various heights and frequencies. Proc. Inst. Civ. Eng., 13: 363--386. Bagnold, R.A., 1947. Sand movement by waves: some small-scale experiments with sand : of very low density. J. Inst. Civ. Eng., 27 : 447--469. Batchelor, G.K., 1967. A n Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 615 pp. Brebner, A. and Collins, J.I., 1961. The effect on mass-transport of the onset of turbulence at the bed under periodic gravity waves. Trans. Eng. Inst. Can., 5: 55--62. Brebner, A., Askew, J.A. and Law, S.W., 1966. The effect of roughness on the mass-transport of progressive gravity waves. Proc. 10th Int. Conf. Coastal Eng., N e w York, Ch. 12, pp. 175--184. Collins, J.I., 1963. Inception of turbulence at the bed under periodic gravity waves. J. Geophys. Res., 68: 6007---6014. Davidson, B.J. and Riley, N., 1972. Jets induced by oscillatory motion. J. Fluid Mech., 53: 287--303. Dore, B.D., 1970, Mass transport in layered fluid systems. J. Fluid Mech., 40: 113--126. Dore, B.D., 1976. Double boundary layers in standing surface waves. Pure Appl. Geophys., 1 1 4 : 6 2 9 ~,37. Dore, B.D., 1977. On mass transport velocity due to progressive waves. Q.J. Mech. Appl. Math., 30: 157--173. Dore, B.D., 1978. Some effects of the air-water interface on gravity waves. Geophys. Astrophys. Fluid Dyn., 10: 215--230. Hsu, J.R.C., 1979. Short-crested Water Waves. Ph.D. Thesis, Dept. Civ. Eng., University of Western Australia, 229 pp. Isaacson, M. de St. Q., 1976a. Mass transport in the bottom boundary layer of cnoidal waves. J. Fluid Mech., 74: 401--413. Isaacson, M. de St.Q., 1976b. The second approximation to mass transport in cnoidal waves. J. Fluid Mech., 78: 445--457. Isaacson, M. de St.Q., 1978. Mass transport in shallow water waves. ASCE J. Waterway, Port, Coastal Ocean Div., 104: 215--225. Johns, B., 1970. On the mass transport induced by oscillatory flow in a turbulent boundary layer. J. Fluid Mech., 43: 177--185. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser. A, 245: 535--581. Longuet-Higgins, M.S., 1958. The mechanics of the boundarydayer near the bottom in a progressive wave. (Appendix to a paper by R.C.H. Russell and J.D.C. Osorio) Proc. 6th Int. Conf. Coastal Eng., Miami, pp. 184--193. Mei, C.C., Liu, P.L-F. and Carter, T.G., 1972. Mass transport in water waves. Ralph M. Parsons Laboratory for Water Resources and Hydrodynamics, Massachusetts Institute of Technology, Report No. 146, 287 pp.

120 Noda, H., 1968. A study on mass transport in boundary layers in standing waves. Proc. 11th Int. Conf. Coastal Eng., London, Ch. 15, pp. 227--247. Riley, N., 1965. Oscillating viscous flows. Mathematika, 12 : 161--175. Riley, N., 1975. The steady streaming induced by a vibrating cylinder. J. Fluid Mech., 68: 801--812. Russell, R.C.H. and Osorio, J.D.C., 1958. An experimental investigation of drift profiles in a closed channel. Proc. 6th Int. Conf. Coastal Eng., Miami, Ch. 10, pp. 171--183. Sleath, J.F.A., 1972. A second approximation to mass transport by water waves. J. Mar. Res., 30: 295--304. Sleath, J.F.A., 1973. Mass-transport in water waves of very small amplitude. J. Hydraul. Res., 11: 369--383. lJnliiata, U. and Mei, C.C., 1970. Mass transport in water waves. J. Geophys. Res., 75: 7611--7618.