Mass transport in water waves propagated over a permeable bed

Coastal Engineering, 1(1977) 79--96 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

MASS T R A N S P O R T IN WATER WAVES P R O P A G A T E D O V E R A P E R M E A B L E BED

PHILIP L-F. LIU School of Civil and Environmental Engineering, CorneU University, Ithaca, N. Y. 14853 (U.S.A.)

(Received August 16, 1976; revision accepted November 24, 1976)

ABSTRACT Liu, Ph.L-F., 1977. Mass transport in water waves propagated over a permeable bed. Coastal Eng., 1: 79--96. Mass transport in water waves propagated over a permeable bottom is investigated. Boundary-layer approximations are incorporated into the Lagrangian equations of motion, and the mass transport velocity is obtained for both monochromatic and random waves. It is found that inside the permeable bed, although the Eulerian steady streaming vanishes, the mass transport always exists in the direction of wave propagation. The effects of porosity and permeability of the porous bed on the mass transport are discussed.

INTRODUCTION Due to increasing off-shore activities, knowledge of wave-induced flow p h e n o m e n a in a permeable b o t t o m and their effects on the wave systems is in great demand. In particular, in evaluating the erosion capability o f waves and in determining the stability o f a permeable bed, i nform at i on concerning the wave-induced pressure gradient and mass transport is essential (Henkel, 1970; Carter et al., 1973). In the last tw o decades, several a t t e m pt s have been made t o study the wave-induced pressure in a p o r o u s bed (e.g. Reid and Kajiura, 1957; Hunt, 1959; Murray, 1965; Sleath, 1970; Liu, 1973; Moshagen and T~rum, 1975; Massel, 1976 etc.). T h e results have been used to analyze t he stability o f u n d e r w a t e r soft (Henkel, 1970; Madsen, 1974). On the o t h e r hand, the basic t h e o r y o f mass t r ans por t in water waves over a rigid b o t t o m is well known (Longuet-Higgins, 1953) and has been e m p l o y e d t o explain certain rhythmical sedimentary features in the off-shore area (Bowen and Inman, 1971; Carter et al., 1973). Though there is experimental evidence t h a t the mass t ransport velocity is r ed u ced due t o the appearance o f a permeable b o t t o m (Lhermitte, 1961), an analytical t h e o r y f or the relation am ong wave characteristics, mass transport, and the por os i t y and permeability of a permeable bed appears t o be wanting.

80 In the present paper, based on a laminar flow theory, the mass transport in both permeable beds and free-fluid regions is examined. The first-order (of wave slope) velocity field which has been obtained by Liu (1973) is first used to calculate the second-order mean currents in the permeable bed. It is found that although the wave-induced steady-streaming Eulerian description vanishes in the porous medium, the mass transport (Lagrangian description) is in general non-zero. In the free-fluid region, boundary-layer arguments are incorporated into the Lagrangian equations of motion and the mass transport velocity is derived for monochromatic waves. The effects of the porosity and the permeability of the permeable bed on the mass transport are discussed. SUMMARY OF FIRST-ORDER WAVE FIELD We consider small-amplitude waves in a fluid of mean depth h above a permeable medium of infinite depth. The fluid is of kinematic viscosity v and the porous medium has a permeability K and a porosity n. If Cartesian coordinates (x,y) are used and fixed on the mean free surface, y=o, where y is positive upwards, we assume the free surface of the form:

(1)

r/= a exp (i0), 0 = k x - o t

where a is the wave amplitude, h the wave number and o the wave frequency. The corresponding velocity components in both the free fluid region and the permeable medium have been obtained by Liu (1973). The detailed derivations of the solutions are referred Liu (1973); we state briefly the m e t h o d of solution as follows: (a) the physical problem is first linearized due to the smallness of wave motions, i.e., k a < < l ; (b) in the free-fluid region, the fluid motion is assumed to be essentially irrotational except near the interfacial boundary, y= - h , where a viscous boundary layer of thickness of the order O ( x / ~ ) is formed; (c) the unsteady Darcy's equations are employed to model the flow in a permeable bed, which leads to a potential equation for the pressure variations in the permeable medium; and (d) the solutions for the pressure field and velocity field are found by requiring the continuity of the pressure and velocity components along the interface up to the order of O(vrff). Referring to Liu (1973), the irrotational velocity components in the free fluid region and the velocity components in the permeable medium can be summarized as follows: (1) = iA [sinh k (y~+h) -i(Q/o)cosh k(y+h)] exp(i0) u1

(2a) -h
v(1) = A [cosh k(y+h) -i(Q/o)sinh k(y+h)]exp(iO)

(2b)

Ul(2) = iA exp[k(y+h)] exp(i0) ~

(2c)

v(2) 1 = A exp[k(y+h)] exp(i0)

f

y<-h

(2d)

81 with

A = kag/QD Q = ia/n - r/K D = cosh kh + (io/Q) sinh kh

(3a) (3b) (3c) The rotational velocity c o m p o n e n t in the direction of wave propagation inside the boundary layer adjacent to the interface, y = - h , is:

U 1 = A(i-Q/o) e x p [ - ( 1 - i ) a ~ - ~ (y+h)] exp(i0)

(4)

We remark that all the velocity components cited above are in the order of o(ka) and the rotational velocity decays exponentially far away from the porous boundary. Moreover, it is noted that the velocity in the permeable medium diminishes exponentially as one moves away from the interface and the influence depth is about one-half of the wavelength, (2~/k). In the following section, the second-order (in wave slope) mean currents in the permeable m e d i u m are sought. Both Eulerian and Lagrangian descriptions will be presented. WAVE-INDUCED MEAN C U R R E N T S IN A PERMEABLE BED

Steady streaming (Eulerian description) To determine the second-order wave-induced steady streaming the nonlinear convective terms in the governing equations of flow motions in the permeable medium (i.e., eqs.5 in Liu, 1973) can no longer be neglected. Assuming that u~) + u~ ) , v~ ) + v~ ), and/~1 ) + P~2) satisfy the complete nonlinear equations and o(u~ ), v~ ), P~)= o(ka)o(u~ ), v~ ), P~)), the time-averaged governing equations for u2a~, v~ ) and P~)become: ap(2)

au (2)

au (2)

1

K

2

pax

n

a~(2) v ( 2 ) 1 --V 4 - - -2 K 2 p ay u (2) a 2 ax

+

av (2) 2 ay

-0

ax

av (2) 1[u(12) - - +1 n ax

1 Oy av (2) v(2) 1 ] 1 ay

(5)

(6)

(7)

in which the over bar " ~ " d e n o t e s the time average over one wave period and e~) represents the pressure field in the permeable medium. (u~) , v~ )) is the wave-induced steady streaming in the porous bed.

82 Substituting eqs. 2c, d into eqs. 5 and 6 and cross-differentiating these m o m e n t u m equations, it follows that:

a2~) o.2=(2) 1-"2 ax 2

2pk 2 ]A

ay 2

12 exp [2k(y +h ) ]

n

(8)

The general solution for the preceding equation can be readily written as follows:

l

P~2) =

PlAI2 2n + cl

sin

2kx

+ c2 cos

2kx 1 exp[2k(y÷h)]

where cl and c2 are u n k n o w n constants. It is, however, evident that the second-order mean pressure in the free-fluid region ( - - h < y < 0 ) , which may be calculated from Bernoulli's equation, is independent of x since the pressure is approximately uniform inside the boundary layer. The u n k n o w n constants Cl and c2 must be zero, so as to match the pressure along the interface, thus: p(2) 2

PlAI 2

=

--

2n

exp[2k(y+h)], y<-h

(9)

Consequently, from eqs. 5 and 6 it becomes obvious that the steady streaming vanishes throughout the permeable medium, i.e.: u(2)

-(2)= O,

up to

o[(ka)2].

y<-h

2 = V2

(10)

Lagrangian mass transport To o [ (ka)2], the Lagrangian mass transport can be readily calculated b y the following relations (Phillips, 1966): au (2) u (2) = u (2) +

m

2

ax

DU(2)

1 f u ( 2 ) dt + - - 1 1 ay

~v (2)

f

2

ax

(lla)

dt

(11b)

~v (2)

1 /u(2 ) dt+ -- 1/(2) V(2) =V (2) + m

v (12)dt

1

ay

Vl

Upon using the information of the first-order Eulerian velocity field, i.e., eqs. 2c, d, the mass transport velocity reads:

83

u(2)

klAI 2

- - m

(12a)

exp[2k(y+h)]

o

y'(-h v (2) = 0

(12b)

m

This indicates t h a t the mass transport current exists only in the direction of wave propagation. The maximum velocity occurs at the interface and the magnitude of the velocity decays exponentially as (y+h) decreases. It should also be noted that the influence depth of the mass transport velocity is onehalf t h a t of the first-order Eulerian velocity. WAVE-INDUCED MEAN CURRENTS IN THE FREE FLUID REGION Here it is convenient to utilize the Lagrangian description at the onset rather than the Eulerian. Once the Lagrangian mass transport is known it is always possible to obtain the Eulerian induced streaming by the relation given in eqs. 11. Let Xo= (b,c) be the mltlal position of a fired partmle, x" = [~(xO , t), y(xO, t)] its coordinates at time t. For convenience, the b-axis is designated to be tangential to the local mean free surface and c is positive upwards. Therefore, we can simply specify the free-surfaced b o u n d a r y condition on c = o. Since the first-order (in wave slope) governing equations of fluid motions in the Lagrangian description are identical to the linearized Navier-Stokes equations in Eulerian form (Pierson; 1962), we may rewrite the velocity solutions (i.e., eqs. 2 and 4) in Lagrangian form as follows: --}

.

.

.

.

.

.

.

x (p) = u (1) = iA [sinh k (c+h)-i (Q/o)cosh k (c+h)] exp (i0) it 1

-h~c(O

yit

(

P

)

= v (1) = A[cosh k (c+h)-i (Q/o) sinh k (c+h)] exp (i0) 1

(13) with:

0 = kb-at

(14)

where A and Q are given in eqs. 3. The superscript (p) denotes the potential velocity field. Inside the boundary layer near the interface, c = - h , a rotational part of velocity must be added, from eq. 4 it reads:

84

x (r) = U 1 = A(i-Q/a) exp [ - ( 1 - i ) x / ~ f ~ ( c + h ) ] exp (i0)

(15)

It

For later use, we summarize the total solution in the interracial boundary layer as follows:

Xlt

= x (p) + x (r) = (AQ/a) It it

l

E

1 + (ia/Q - 1) exp - ( 1 - i ) V ~ ] ~ ( c + h )

11

exp (i0)

(16a) y(p)

.

(r) =

Y l t = It t Y l t

Aexp(i0)

(16b)

up to O(ka). It should be noted that the vertical rotational velocity component, y~), which may be calculated using the continuity equation, is in the order of o~k2 ~ x ~ ) and is neglected here. To complete the first-order Lagrangian velocity field, we note that a rotational velocity field should be added within the free-surface b o u n d a r y layer of thickness ~/-2~o so as to satisfy the boundary condition of no tangential stress on the free surface. In the boundary layer the boundarylayer behavior is only significant in the vertical direction, i.e., a/ab < < a/ac. The rotational velocity c o m p o n e n t in the b-direction satisfies the following equation approximately Unliiata and Mei, 1970): x (r) = ~x (r) l tt l tcc

(17)

The solution of this equation is subjected to the zero tangential stress condition on the free surface. Upon using eq. 13, we have: x (r) = " (P) + y(P)) = - 2 i k A [ c o s h kh - i (Q/o) sinh kh] exp(i0) ltc -(Xltc ltac= 0

(18)

The solution can be readily obtained as follows:

x(r)= l (1-i)k exp(i0)

2x/~ i A[c°sh k h -t i (Q/°)sinh kh ] exp [ ( 1 - i ) x / ~ c (19)

Similar to the interfacial boundary layer, the vertical c o m p o n e n t of rotational velocity is of the order o(k~/2~]a xl~)t) and may be obtained b y integrating the

85 continuity equation. We remark that although the rotational velocity in the free-surface boundary layer is small, o ( k 2 ~ ) , its contribution to the vorticity field x lte ¢~ is, however, of o(1). T o conclude this section, we summarize the total solution ha the freesurface boundary layer as follows:

x lt = iA {sinh kh-i( Q/o)cosh kh - (l+i)kv/2v]o [cosh kh-i( Q/o)sinh kh ] exp [ (1-i o)x/-~c]} exp(i0) x

(20a)

= ikA[cosh k h - i (Q/o) shah kh] {1-2 exp [(1

ltc

s~qrs~c] } exp(i0) (20b)

(20c)

Y l t = A[cosh k h - i (Q/o) sinh kh ] exp(i0) This first-order motion given in eqs. 13, 16 and ' for the second-order motion of o [(ka) 2 ] for which averaged equations is recorded as follows (Pierson,

provides forcing terms te complete set of time~62):

P2b/p - v~ 2 Fc2t = 01

(21)

P2c/p

(22)

-

v~2:Y2t = 0 2

with: G1

=

-XlttXlb-YlttYlb + v {

(++ )+ ltbb

3xltcc Xlb

2XltbbYlc

+ Xltb(Ylb c- Xlcc) - 2 Xltbc(Ylb + Xlc ) - Xlt c (Ylbb - Xlbc) (23a)

+ Ylb (Yltbb + Yltcc ) } G 2 = - XlttXlc - y l u Y l c + v {(Yltcc + 3Yltbb) Ylc + 2YltccXlb

+

Yltc(Xlbc

+X

and:

lc (Xltce

-

Ylbb ) -2Yltbc (Xlc + Ylb ) -Yltb (Xlcc +X

ltbb )}

-

Ylbc ) (23b)

86 +

+

X2b Y2c XlbYlc-XlcYlb

--

0

(24)

where P2 represents the second-order mean Lagrangian pressure and all time averages over a wave period axe denoted by overbars. It should be noted that in presenting eqs. 21 and 22 we have already assumed that a second-order mean vorticity field has been established in the entire free-fluid region; hence (-X2tt' "Y2tt) = O. The analysis presented herein follows closely that of 0nliiata and Mei (1970); the mass transport will be calculated first in the free-surface boundary layer and in the interfacial boundary layer, the corresponding results evaluated at the outer edge of the boundary layers will then be used as boundary conditions for the calculation of the second-order mean flow in the interior region.

The free-surface boundary layer Substituting the first-order solutions, eqs. 20, into eqs. 21--23 it follows after some algebra that: a2U (1)

P2b/p - v - ac

m

- 2(1-i) (2v/a)l/2k2LA[ 2 [(1 + IQ/al 2) sinh kh cosh kh

2

+ i(Q*/a) cosh 2 kh - i(Q/a) sinh 2 kh] exp [(1-i) ~/a/2v c] +O(2~,/a) P2c/p = klAI 2

(25a)

{[sinh k (c+h) cosh k (c+h) -i(Q/o) cosh 2 (c+h) +i(Q*/a) *

sinh 2 k (c+h)] - [sinh kh cosh kh -i(Q/a) cosh 2 kh + i(Q /o) sinh kh] exp [ ( 1 - i ~ c ] }

+O(~/2vla)

(25b)

where the asterisk denotes the complex conjugate. The boundary layer approximations, i.e., a/ab << a/ac and O(x2t = Uam)) = O( ox/~2-v)O(Y2t), have been employed. Eq. 25b shows that the vertical gradient of the second-order Lagrangian mean pressure is of the order O(1). If we stipulate that the horizontal pressure gradient,/T2b. , outside the boundary layer is of the order O(2~/o), P2o,- is also in the orde/Prof O(2v/a) throughout the boundary layer. Therefore', P2blp in eq 25a may be neglected up to O(~/~-a). To solve eq. 25a, it is necessal~, to impose a second-order free-surface boundary condition as follows (Unliiata and Mei, 1970):

87

au (D m

ac

= X l X l t b - 2 Xl t X l b

c=O

Upon substitution of eqs. 20 into the right-hand side of the preceding equation, we obtain: au (1) m

ac

=0(2~)

c=O

(26)

Employing this condition, we integrate eq. 25a to give: au (1) m ac

4k 2 IA 12 * 2 - [(1+ IQ/al2)sinh kh cosh kh + i(Q /o)cosh kh - i o (Q/o)sinh 2 kh] { 1 - e x p [ ( 1 - i a ) x / - ~ c ] }

(27)

At the outer edge of the free-surface boundary layer, (ox/~2~c')-* - ~ , we have: au (1) m

ac

4 k 2 IA 12

[(1 + IQ/ol 2) sinh kh cosh kh + i(Q*/o) cosh 2 kh

o -i(Q/a) sinh 2 kh]

(28)

For the special case of an impermeable bottom, n, K ~ 0 and QA kag/cosh kh, eq. 28 becomes: a u (1)

m.= 4ok 2a2coth kh ac

(29)

which is the result of Longuet~Higgins (1953). The interracial boundary layer Upon substitution of eqs. 16 into eqs. 21--23, we obtain a set of approximate equations for mass transport velocity in the interfacial boundary layer:

88 a2u (1) ik P 2 b / p - v C2 m

-

2 2 I'IA--Q-Q o

{4(Q-1-'- )exp [-(1-i o)x/-~ (c+h)+ 3i

ti.o _ 112 exp [-2Vro-]~(c+h)l } +O(~/2v/o) Q -

P2clp = 1 2~AQ[2 ]o] ~

(30a)

io

(1-i) (~--1) exp[-(1-i o)M~]~(c+h)]+ 0(1)

(30b)

Cross-differentiating eqs. 30a and 30b, we have: a3u(D m

io

- iko~-]-~ IAQ/ol 2 { 2(1-i) ( u - l ) exp[-(1-i)~/o]2~(c+h)

ac 3 + 3 i ~ - 1 2 exp [-2 aVr~f~(c+h)] } + O(1)

(31)

Similarly to the situation arising in the free-surface boundary layer, we anticipate that at the outer edge of the interfacial boundary layer P2blp = O(2v/o). Equivalently, we imposed the following boundary conditions: a2u(D v ac 2

m

~ O(2r/a) as V ~ v (c+h )-.=

ou )

(32)

ac Furthermore, along the interface, c=-h, the boundary condition reads: u (1)

u (2) =

m

k[AI 2 -

m

c=-h

(33)

o

The right-hand side of the equation is the mass transport in the permeable bed at the interface, eq. 12a. Integrating eq. 31 three times and applying conditions 31--33, we have:

89 u (1) = -k IQA - 12 { 8(io/Q - 1 ) [exp [ - (1-i a}Vra-f~(c+h)] - 1 ] +3 lia/Q -112 m 403 [exp[-2 oV~(c+h)]-l]

+402/IQ 12 }

(34)

At the outer edge of the boundary layer, mass transport becomes: u (1) _ kIQAI 2 m

{8(1-ia/Q)-3 lio/Q-112 + 4o2/IQI 2 }

(35)

4o 3

This again checks with t h a t obtained by Longuet-Higgins for an impermeable b o t t o m ; n, K-*0, and QA -~ kag/cosh kh: u (1) ~ 5a2ok

(36)

4sinh 2 kh

m

The interior region Ignoring the weak spatial damping, it has been shown that for twodimensional progressive waves in a constant water d e p t h there is no second boundary layer outside the Stokes b o u n d a r y layer (Mei and Liu, 1972). Therefore, the conduction solution similar to that of Longuet~Higgins (1953) can be readily obtained in the interior region. Substituting the potential velocity c o m p o n e n t s (eqs. 13) into eqs. 21--24, we have: a2u(D

a2u (1)

v ( -----m--m + ~ m Ob2

) = P2b /p +

ac 2

O

(1Q 1Q . ) a2v 0 ) a2v 0 ) _ m v - - m + ac 2 )= e2c/p ab au 0 )

1

(37a)

(37b)

av (1)

In

--+ ab

sinh 2k(c+h)

m

ac

'=0

(37c)

90 where v ma) is the mass transport velocity c o m p o n e n t in the vertical direction. These equations are subject to the following boundary conditions: aUm (1)

2k 2 IA 12 o

ac

[(1 + IQ/ol2)sinh 2kh - i(Q/o)cosh 2kh]

v(1) = 0

c=O

(38a)

c=O

(38b)

c=-h

(38c)

c=-h

(38d)

m

u (1) = klQ AI2

{ 8(1-io/Q)-31io/Q-112 + 4o2/IQI 2 }

403

m

v (D = 0 in

It is easy to observe that the vertical c o m p o n e n t of mass transport must vanish in the interior region and the mass transport is independent of the b-axis. Hence, in view of eq. 37c we introduce a stream function such that: u( D = a_~@ m ac

v(D _ a$ _ 0 m ab

(39)

Cross-differentiating eqs. 37a and 37b and employing eq. 39, we have: a4@

4k 4 IQA 12

ac 4

03

{ (Io/QI2+l)sinh 2k(c+h)+io(1/Q - 1/Q*)cosh 2k(c+h)} (40)

subject to the conditions: a2~

2k 2 IA 12 - - [(1 + IQ/ol2)sinh 2 k h - i(Q/a)cosh 2kh]

c=0

(41a)

@=0

c=O

(41b)

4--0

c=-h

(41c)

c=-h

(41d)

ac 2

a

a~

klQAI 2 -

ac

-

{ 8(1-io/Q)-31io/Q-1L 2 +4o2/IQl 2 }

403

It should be noted that conditions 41 b and c imply zero flux across any

91

fixed b plane. The mass transport can be readily obtained by integrations, thus: u(1)m -kIQA]2{~3 C l [ ( h ) 2_31]+4__3khc2[()2hC +34(h)+31] g

+ c 3 [cosh 2k(c+h)

4L\h/

kh

_ ]

(42)

with: C1 =

-2(ia/Q -1) - ~ (l-i) lio ~ - 1]2 _ 12

(I_Io/Q]2)

c 2 = [ la/QI2 +l]sinh 2kh + 2(ia/Q)cosh 2kh (43) 1

c 3 = ~ (1+ Jo/Q 12) C4 = i o / Q

We iterate that only the real part of the solution has physical importance. The horizontal Lagrangian mean pressure gradient may be obtained by substituting eq. 42 into eq. 37a: I Cl +kh 2~ c2 + P2b/p = 3vkIQA]2 --

2kh

sinh 2kh + -c4 cosh 2 k h - 1 } 2kh

(44)

Eqs. 42 and 44 may be reduced to Longuet~Higgins' solutions for an impermeable bottom by setting n, K -+ 0. E F F E C T S O F P O R O S I T Y A N D P E R M E A B I L I T Y ON T H E MASS T R A N S P O R T

It has been reported (Lhermitte, 1961) that for some waves over a smooth solid bed the mass transport velocity along the bottom (at the outer edge of Stokes boundary layer) is greater than that observed over the silt bed. Physi-

92 cally, this is expected since the wave must transmit energy into the permeable bed so as to generate the mean current. To demonstrate the effects of porosity and permeability of the permeable medium on the mass transport, two regimes of the present results, O(~ K) <>><1, are examined in this section. Case 1: o/n >> v/K In this case, the constants Q and A can be approximated as follows:

(45)

(46a) (465)

Q "-" i o / n

A " i k a g n / [ o ( c o s h kh + n sinh kh)]

The mass transport velocity in different regions is recorded below: u(2) = k 3a2n2g2exp [2k(y+h ) ] y<-h m

U(1) = m

(47)

o3[cosh kh + n sinh kh] 2

_

3 (n2+2n_3)

3

a (cosh kh + sinh kh) 2 [(n2+l)sinh2kh

cosh 2k(y+h ) +

1

~

-

+ 2nco .

+

-3- k h 4

+4(h )

(I l )1

(n2+ 1)

+ n[sinh 2k(y+h) +

3(cosh 2 k h - 1 )

-h
4kh

(48)

and in the interfacial boundary layer the mass transport velocity becomes: k 3a 2g2 u (1) = { 3 ( n - l ) 2 (exp[-2Vro-]~(y+h)] m 402 [cosh kh + n sinh kh] 2 - 1 ) + 8 ( n - l ) [cos(ox/~-]~ (y+h)) exp[-2x/~-]~ ( y + h ) ] - l ] + 4n 2 }

(49)

It should be pointed out that in the preceding equations the Lagrangian vertical axis, c, has been replaced by the Eulerian vertical coordinate y. The accuracy of the results remains the same, since OOy-cl ) = O(ka).

93

The mass transport velocity profile in three different regions is presented in Figs. 1 and 2. Longuet~Higgins' (1953) results for a solid bed are also computed as a comparison. It is noticed that the mass transport velocity in the free-fluid region is influenced significantly by the appearance of a permeable bed. In particular, the mass transport at the outer edge of the interfacial boundary layer is always smaller than that of Longuet.Higgins (1953) with a solid bed. On the other hand, the mass transport velocity on the free-surface is always greater than Longuet-Higgins' result in the direction of wave propagation.

Case 2: o/n < < v/K (50) When the permeable bed consists of sands and clay, the magnitude of (v/K) is usually much greater than that of (a/n); for example for regular sand, K ~ 10 -6 cm 2 and v/K ~ 104 sec. -1 The constants Q, D and A may be approxim a t e d as follows: Q = -v/K

(51a)

D = cosh kh-io(K/u)sinh kh

(51b)

=

/

/3=.4

,,..

in?r'f ( ~ ~ . ~ . . . ~ , "~ ~ .......

, Um

0 A

.

-1A)

B

1

2

3

Um

4

Fig.lA. Mass transport velocity profile in the interior region and the permeable bed; o/n > > v/K, kh =1.0 and the mass transport velocity is normalized by k 3a292/(o3cosh ~ kh ). B. Mass transport velocity profile inside the interfacial boundary layer; o/n > > v/K,

k h = l . 0 , Z= (y+h),~-/2v and the mass transport velocity is normalized by k Sa2g~/ (4 o3cosh 2kh ).

94

\\\\~. i~4-.6 n=©.

interface

L~

~"~'-..

-o.'5\ o'-....... 0.5 ~.o -1~

Um 115 '

1

um"

0

permeable bed

1

2

3

4

5

B

-1.4

Fig.2A. Mass transport velocity profile in the interior region and the permeable bed; a]n >> v/K, kh=0.5 and the mass transport velocity is normalized by kSa2g2](a3 cosh 2 kh). B. Mass transport velocity profile inside the interfacial boundary layer; o/n > > v/K, kh=0.5, Z= (y+h)vra-/2v and the mass transport velocity is normalized by k2a292/ (4aScosh 2 kh ).

kag A = -(K/v) - cosh kh

(51c)

The mass t r a n s p o r t velocity in the permeable m e d i u m and in the free fluid region m a y then be rewritten as:

k

3a 2g2 u (2) (K/v) 2 e x p [ s k ( y + h ) ] , m ocosh 2 kh u 0) + k2a2g2 m = UL.H. o c o s h 2 k h ( K / v ) 2

sinhkh2kh ] 1

t

y<-h

)2 _83_ [ ( h

(52a)

3

_1]

+ - khsinh kh 4

-h
(52b)

95 where u_ H denotes Longuet-Higgins' solution (1953) for a solid b o t t o m and readLs': " k3a2g2 -

t

3 ~ sinh 2kh 3 + 2 cosh k ( y + h ) +

UL'H" 4o3cosh2 kh

+3) [(Y)2-1]

+khsinh2kh

[1+4

(h)+3(h)2J

I

(53)

In view of the order of magnitude of (K/v), the intensity of the mass transport velocity in the permeable bed and the modification of the mass transport velocity in the free fluid region due to the appearance of a permeable bed are quite small. To obtain more accurate solutions for this situation, higher-order (in wave slope) problems must be studied. Nevertheless, this transport phenomenon can not be overlooked, especially over a long period of time. CONCLUDING REMARKS A theory of mass transport velocity in water waves propagated over a permeable bed is given in this paper. Laminar flow has been assumed throughout the analysis. The mass transport velocity differs from Longuet-Higgins' (1953) solution significantly as o/n > > v/K. On the other hand, little influence of a permeable b o t t o m has been found when °/n > > v/K. Controlled laboratory experiments and further improvement on the theory to include turbulent effects are certainly desirable. The present results can clearly be used as a basis for the calculation of the mass transport velocity in random waves. If the free-surface profile can be considered as a stationary random function of both position and time, it can be represented as a Fourier--Stieltjes integral (Phillips, 1966): i0 P = J cL4(a)e According to 0nliiata and Mei (1970), the corresponding mass transport velocity may be represented as follows: < x2t> = 4f ok s ~ 7 0

(o) U

m

(°,y)d°

where < ~ denotes the ensemble average, S is the spectral density of the wave height and u m is given by eqs. 27 and°'~4 for free-surface and interfacial boundary layers and is given by eq. 42 for the interior region.

96

ACKNOWLEDGMENT

The author wishes to acknowledge the financial support of the NSF under the contract (ENG 76-09421) with Come]] University.

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