A simple quasi-3-D model of suspended sediment transport in a nonequilibrium state

Coastal Engineering, 15 ( 1991 ) 459-474 Elsevier Science Publishers B.V., Amsterdam

459

A simple quasi-3-D model of suspended sediment transport in a nonequilibrium state Kazuo Nadaoka, Hiroshi Yagi and Hirohisa Kamata Dept. Civil Eng., Tokyo Inst. of Technology 2-12-1, O-okayama Meguro-ku, Tokyo 152, Japan (Received June 26, 1989; revised and accepted January 24, 1991 )

ABSTRACT Nadaoka, K., Yagi, H. and Kamata, H., 1991. A simple quasi-3-D model of suspended sediment transport in a nonequilihrium state. CoastalEng., 15: 459-474. A new simple model to evaluate quasi-3-D suspended sediment transport in a non-equilibrium state has been developed through a formulation based on a kind of weighted residual concept. The model includes only two unknown variables to be solved in a horizontal 2-D domain (x,y); i.e., the sediment concentration at the bottom Cb (x,y,t) and the shape factor A (x,y,t) in the assumed exponential curve for the vertical profile of the concentration, both of which are the most important factors governing the nonequilibrium sedimentary process. From the computational results for the 1-D deposition and erosion problems, the model has been confirmed to give satisfactory results in accuracy and in computational time. The present model has also been applied to the 3-D suspended sediment transport problem in the nearshore region including the surf zone. The comparison between the computational results by the present model and those by the conventional models assuming the local equilibrium concentration profile and the nonequilibrium but vertically uniform profile, have demonstrated the importance of taking the effect of the nonequilibrium 3-D condition into the consideration for such problems.

INTRODUCTION

In the studies of suspended sediment in coastal area, its transport rate has been usually estimated by multiplying the equilibrium vertical sediment concentration by the local horizontal advection velocity (Fleming and Hunt, 1976; Stive and Battjes, 1984; Dally and Dean, 1984; etc. ). However, except for the special cases such as longshore sediment transport on a uniform beach, the suspended sediment can be considered generally in a nonequilibrium state; i.e., its transport rate at a location concerned is governed not only by the local hydraulic conditions. This is because the sediment is transported mainly through a process that it is lifted up into suspension in the surf zone, then advected elsewhere by nearshore currents and during this process the hydraulic conditions may change to an appreciable degree. One of its typical examples is the case when the sediment suspended in the surf zone is trans0378-3839/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

460

K. NADAOKA ET AL.

ported offshore by rip currents and then deposited. In this case, the phenomenon entirely depends upon the degree how far is it from the equilibrium state. The nonequilibrium condition of the suspended sediments is directly related to the difference between the upward and downward sediment flux at the bottom (Fig. 1 ); the former is the pick-up rate P of the bottom sediments and the latter can be represented as Cb" Ws, where Cb and Ws are the sediment concentration at the bottom and the settling velocity of the sediment, respectively. Therefore the accurate representation of the nonequilibrium condition requires to obtain the information of Cb and thus the vertical distribution of the sediment concentration. In other words, the sediment transport with highly nonequilibrium condition should be treated as two- or three-dimensional problems including the determination of the vertical distribution of the concentration. For this purpose, a finite difference method may be used by discretizing a domain not only in horizontal coordinates x and/or y, but also in a vertical coordinate z (Miller, 1984; O'Connor and Nicholson, 1988; etc). This is one of possible and acceptable methods to solve a two-dimensional problem in a vertical plane (x,z), such as the siltation problem of a dredged trench (Delft Hydraulic laboratory, 1980). For more general three-dimensional problems in a nearshore region, however, the application of the finite difference method is very costly because it needs vast memory area and very long CPU time. One of the practical ways to avoid this difficulty is to assume uniform vertical distribution of the sediment concentration. However, this approach obviously leads to an underestimate of the downward flux Cb" W~. Another possible approach is to divide the vertical domain into several layers. For example, Irie and Kuriyama (1988) have adopted such a model with three layers. This layered model may yield appreciable error in the estimation of the vertical sediment flux, because such rough division of the vertical domain is not Z STz

/J I/CIZ) ~'ff \~I P=CbWs : equilibrium ;um bWs n0n-equillbr

I z //)'////I

~

CbWs Fig. 1. Equilibrium and nonequilibrium bottom conditions.

A SIMPLE QUASI-3-D MODEL OF SUSPENDED SEDIMENT TRANSPORT

461

enough to represent the vertical gradients of the sediment concentration, especially in the region close to the bottom. Hence, in the present study, an attempt has been made to derive a new model which can estimate the three-dimensional suspended sediment transport in a highly nonequilibrium state under the requirements of small storage area and short CPU time. In the derivation of the model, a kind of weighted residual formulation has been made to express quasi-3 dimensionality of the distribution of the sediment concentration. THEORY

Basic equation and boundary conditions Usually problems of suspended sediment transport can be treated by solving the diffusion equation 1 for the time-averaged concentration C of the sediments under the specified boundary conditions:

(1) where Kx, Ky, Kz and U, V, W are the diffusion coefficients and mean velocities in the x, y, z directions indicated in Fig. 2, and Ws is the settling velocity of the sediments. The boundary condition at the mean water surface rlis that there is no net sediment flux normal to the surface; i.e.:

KzOC ,. OCOrl OCOr1 ff~z-r,,xO--x--~x Ky o - ~ + WsC=O, at z=~

(2)

In what follows the overbars of C and ~will be dropped for simplicity of the notation.

~w/trJ

W,

V

>,X

............................... Z =- h Fig. 2. Definition of coordinates system.

462

K. NADAOKA ET AL.

At the bottom of the water depth h, on the other hand, the boundary condition for C has been expressed in several ways. For example, Galappatti and Vreugdenhil ( 1985 ) imposed the boundary condition by specifying the nearbed concentration as Cb = Ca. Its validity, however, is questionable because the near-bed concentration should be treated as a dependent variable to be determined as a result of the sediment transport within the domain. In other words, the difference between the instantaneous bottom concentration Cb and its equilibrium value governs the evolutional process from nonequilibrium to equilibrium state. A more reasonable way to prescribe the bottom boundary condition is to impose the sediment pick-up rate P, which is an outer variable to be determined by the local hydraulic conditions at the location concerned (e.g., O'Connor and Nicholson, 1988). Hence, in the present study, the following boundary condition is adopted, in which the re-entrainment coefficient is assumed to be zero:

P--K -

O C - K OCOh K OCOh ~ Oz

x OxOx

Y OyOy'

at z=

- h

(3)

In the equilibrium condition, the pick-up rate P is balanced with the deposition flux Cbe" Ws where Cbe is the bottom sediment concentration in the equilibrium state. Hence, the pick-up rate can be evaluated as P = Cbe" Ws by utilizing the existing methods to estimate the value of Cbe.

Procedure of formulation There have been several simplified methods for the modelling of nonequilibrium suspended sediment transport. Galappatti and Vreugdenhil (1985 ) derived an asymptotic solution for the 2-D problem by perturbing the equilibrium concentration profile as the 0th order solution. The applicability of this asymptotic solution, however, is restricted to the case that the deviation from the equilibrium state is not so appreciable (Wang and Ribberink, 1986 ). As a much simpler model, Kuroki et al. ( 1988 ) have proposed the following expression of the vertical distribution of the suspended sediment in a nonequilibrium state: C(~) = Cb'exp ( - - R ~ ) + R ( C b - Cbe)~'exp [ (-- ( R + 1 )~}]

(4)

where ~= (h + z)/h, R = Ws" h/K~ and Cb is the sediment concentration at the bottom. The above equation expresses quite simply the nonequilibrium condition by its second term. However, it has an essential difficulty that if the pick-up rate P is made suddenly zero from an equilibrium state, in which only the first term is effective because Cb= Cbe, the value of the second term and thus of C increase instantaneously through the entire depth, because no memory effect is included in Eq. 4. Namely, the influence of the nonequilibrium

A SIMPLE QUASI-3-DMODEL OF SUSPENDED SEDIMENT TRANSPORT

463

condition at the bottom expressed as Cb-- Cbe in Eq. 4 propagates throughout the depth without any time lag. Hence, in the present study, it has been attempted to derive a new model which can estimate such nonequilibrium suspended sediment transport quite simply but without loss of physical background. In what follows, the outline of the present model will be described. The principle of the present idea for the quasi-3-D modelling, i.e., for the way to incorporate simply the information of the vertical C-profile into the model, is in the use of a weighted residual formulation. In this formulation, the functional shape of the solution of C is usually assumed to be: nmax

C(x,y,z,t) = ~ an(x,y,t)'~n(z/h)

(5)

n=l

where ~n ( n = 1. . . . . ?/max) are trial functions and an ( n = 1. . . . . ?/max) are the unknown parameters to be determined so as to minimize the integrated residual of the governing equation (Eq. 1 ) substituted with Eq. 5 and multiplied by weighting functions wn. In the Galerkin method, e.g., the trial functions 0n are adopted as the weighting functions wn. Through such formulation, the original problem for the one unknown variable C in the 3-D (x,y,z) domain will be converted into an equivalent problem with nmax equations to determine the unknown variables an (n = 1..... ?/max) in the 2-D (x,y) domain, under the assumption that the residual becomes zero as ?/maxincreases infinitely. If the series expansion of Eq. 5 can be truncated into a small number of terms without any appreciable loss of the accuracy, the above formulation yields the significant advantages in practical computational applications because it needs only smaller storage area and shorter CPU time as compared with the ordinary 3-D finite difference methods. However the formulation based on Eq. 5 is not adequate for the application to the present problems of suspended sediment, because the time-dependent evolutional process of Cprofile under nonequilibrium condition cannot be well expressed with limited number of the expansion terms, in which any kind of complete sets of polynomials such as the Laguerre's polynomials or more simply the sinusoidal functions may be used for the trials functions. From this reason, in the present study, another type of formulation has been made, although it is also based on the weighted residual concept. As is well known, the sediment concentration in the equilibrium state Ce can be expressed with the following exponential type of function:

Ce=Cbe'eXp[-Ws/Kz(h+z) ]

(6)

In the nonequilibrium state, on the other hand, the shape of the C-profile differs from such simple exponential distribution. However, the empirical knowledge shows that the difference of its overall shape from the exponential

464

K. NADAOKA ET AL.

distribution is not appreciable, although the vertical decaying rate of the exponential distribution is considerably different from Ws/Kz for the equilibrium condition (e.g., Dobbins, 1944). Based on this empirical knowledge, in the present model, the vertical dependence of the solution ofEq. 1 is assumed to be:

C(x,y,z,t) --Cb(X,y,t).exp[-A(x,y,t) (h+z)/(h+rl) ]

(7)

in which Cb and A are the sediment concentration at the bottom and the shape parameter, respectively, to be determined under the requirement that the Eq. 7 satisfies Eq. 1 through the entire depth region - h < z_< ~/as well as possible. This assumption replaces the original 3-D problem for C(x,y,zd) by the quasi3-D problem for only two unknown variables, Cb(x,y,t) and A (x,y,t). If the local distribution of C is observed in more detail, of course, the actual distribution of C differs from such simple exponential shape; e.g., for the case that the pick-up rate P is zero, the C-profile must be uniform in the region close to the bottom, because OC/Oz=O at z = - h . For most practical purposes, however, the detailed evaluation of the local distribution is not required. It is only enough to include the information of the vertical distribution of C into the model to the degree that the bottom concentration Cb is estimated well enough to express the nonequilibrium condition Cb" Ws-P. Further, it should be noted that in the present formulation the gradient of C is not locally prescribed at the bottom by the condition of Eq. 3 because the weighted residual concept imposes only the weaker requirement that the Cprofile must satisfy the basic equation not locally, but to minimize its residual integrated over the whole depth region. Hence, for the present model, the assumption of the exponential shape for the vertical distribution of C is adopted as the simplest and the practically acceptable form. (When the more complicated vertical distribution is assumed, almost the same formulation would be possible by including additional shape functions, though the number of the unknown parameters will increase. ) The assumption of Eq. 7 includes two unknown variables Cb and A. In order to derive the two equations for these two variables from the original basic equation (Eq. 1 ), the weighted residual concept has been used, in which these two variables are to be so determined that the assumed function of Eq. 7 satisfies the basic equation 1 close enough over the whole depth region - h -
465

A SIMPLE QUASI-3-D MODEL OF SUSPENDED SEDIMENT TRANSPORT

0

OC

0

OC

1

-"x0-~x ^yo-~7*~z~ ~-Ws z=+ I

OC K O C O h OCOh-] - Kz a T - ~ o-~x K~ oTgs ]:= ~ - W~c~

1 0

2

0

2

0

(8)

2 2

0

OC 2

0

OC

Kx OC

dz-

1

dz-2Ws(C.+Cg)+ c . I _ ^ x... OC&I o_x_g~x Ky _ 0~7+Kz-OTz Ocoq _. Oc + wscJz=

+

F OCOh _ OCOh K z O C ] coL-Kxo-77x Ky0705 OZjz=h

(9)

where C, is the sediment concentration at the water surface. The terms in the first square bracket of the right hand side of both Eqs. 8 and 9 is equal in all to zero from the free surface boundary condition 2 and the terms in the second brackets can be replaced with the pick-up rate P by the bottom boundary condition 3. Substituting these boundary conditions and then Eq. 7 into Eqs. 8 and 9, the following final results of a set of equations for Co and A can be obtained: S, "Co,-Q~ .A, + SI " U'Cox - Q I " U'Ax + SI " V'Coy - Q I " V.Ay - (D. U)x'R1 - (D. V)y.Rl = =Kx" TX1 +Ky. TY~ + GX.hx + GY.hy +FX'~Ix +FY.qy + P - Co" Ws+ [ (Cox - Co " A ' h x / D ) ' m l + (Cb "A'Dx/D 2 - Cb "Ax/D)"M21 "Kx~+ [ ( C b y - C b ' A ' h y / D ) ' M , + (Cb'A'Dy/DZ-Cb'Ay/D)'M2]'Ky~

(10)

S2 "Cbt-Q2 "At +S2" U'Cox -Q2" U'Ax + S2" V'Coy -- Qz" V.Ay - (D. U)x'R2 - (D. V)y.R2 = =Kx" TX2 + Ky" TY2 + Co" GX'hx + Co" GY'hy + Cb "FX'qx + Co .FY. ~v+

466

K.NADAOKAETAL. p o C b - C ~ " Ws" (E2 + 1 ) / 2 + C~-A-K~" (E 2

-

-

1)/(2D)+

[( Cbx--Cb "A'hx/D)"N l + ( C b "A'Dx/D2-Cb "Ax/D)"N2]-Kx, + [( Cby -- Cb "A "hy/D) "Nl + ( Cb "A "Dy/D 2- Cb "A~/D) "N2] "K~

(ll)

where: El = e x p ( - A ) , g 2 = exp ( - 2A ), Sl ---- - D . ( E I - I ) / A , --Cb'D" (E2-- 1 ) / ( 2 A ) , Ql = -Cb'D" [E,/A + ( E l - 1 )/A2], Q2 -- - C 2 . D . [ E 2 / ( 2 A ) + ( E 2 - 1 ) / ( 4 A 2) ], R l = Cb" ( E l - 1 )/A, R2 -- C~" ( E 2 - 1 ) / ( 4 A ) , GX= ( Cbx-Cb'A'hx/D) GY= (Cb), -- C b ' A " hy/D) .Ky, FX = (Cb~-- Cb'Ax- Cb'A'hx/D+ Cb'A'Dx/D )'EI'K~, FY = (Cb,,-- Cb'Ay- Cb'A'hy/D+ Cb'A'Dy/D )"El "Ky, T X l __-- [ Cbxx-- 2 C b ' A ~ ' h x / D - 2A'h~" Cb~/D- Cb'A'hxx/D+ =

Cb'A 2" ( h:~)2/D2+ 2Cb'A'hx'D~/D 2] "M1 + [2Cb'Ax'Dx/D2 + 2Cbx'A'Dx/D 2 - 2Cb'A 2.h .Dx/D 3+ 2Cb'A'Ax'hJD 2 2Cbx'Ax/D+ Cb'A "Dx:c/D2- Cb'Axx/D- 2 (D~) 2" Cb.A/D 3] "M2 + [ -2Cb'A'A:~'Dx/D3+ Cb'A 2" (Dx)2/O4+ Cb'Ax2/D 2] "M3 and: D = h+rl, m l = 31, M2

NI N3

= =

-D

2. [ E , / A +

(El-- 1 )/A2],

- D 3 / A • [El + 2E1/A + 2 ( E I - 1 )/A2],

= s2, = - C b ' D 2" [E2/(2A) + ( E 2 - 1 ) / ( 4 A 2) ], = - Cb'D3/A • [E2/2 +E2/2A + ( E 2 - 1 ) / (4A 2 ) ].

TYI in Eqs. 10 and 11 is expressed by replacing x-derivative in TXI with y-derivative. And TX2 and TY2 are obtained by replacing Ml, M2 andM3 with N~, N2 and N3 in TXI and TYI, respectively. The subscripts t, x and y excluding for the diffusion coefficients represent the partial derivatives with respect to t, x and y. Equations 10 and 11 constitute a solvable system for the u n k n o w n variables Cb and A. Hence, C(x,y,z,t) is obtained from Eq. 7 by numerically solving Eqs. 10 and 11 for Cb and A in a horizontal 2-D d o m a i n with a usual finite difference scheme. In the derivation of Eqs. 10 and 1 1, the m e a n convection velocity U and V and the diffusion coefficients Kx, Ky and Kz are assumed to be vertically uni-

467

A SIMPLE QUASI-3-D MODEL OF SUSPENDED SEDIMENT TRANSPORT

form only for the reason of simplicity. It should be noted that no other assumptions are introduced in the derivation of Eqs. 10 and 11. In other words, if specific conditions are prescribed for the bottom slope and others, Eqs. 10 and 11 may become simpler by neglect of the terms being small in their order of magnitudes. VERIFICATION OF THE THEORY IN 1-D PROBLEM

To verify the present theory, it has been applied to a 1-D problem of nonequilibrium sediment suspension and the results of the computation have been compared with those of the analytical solution and the experiments by Dob-

1.O (=+h)/h .8 .6

1,o _]

Ws=0.264Cm/s Kz=4.14cmz/s h =45.2(ra

, ~ =360s

,8

--

,6

--

,4

--

"~,

~ \f:~

Ws=0.264crn/s h =41.6cm

=i20s

~ t

.4

,

,2

.~ \ \ . ~ .

,2

~

--

T

(a) case 1

C/the

1.0

(b) case 2

C/cbe

Ws=0.264cm/s

t =O

Kz=10.

/

(z+h)/![6824

\

.2 .4 .6 .8 LO

.2 .4 .6 .8 l,O

Oca'2/s n

h =50.0an t:3[Ls

t=120s

0

.4 (c) case 3

.6

.8

1,0

O/Obe

Fig. 3. Time evolution of the vertical concentration profile for deposition processes (case 1 and 2 ) and for an erosion process (case 3 ). The full lines represent the computational results by the present model, and the broken lines and the open circles indicate the analytical solution and the experimental value by Dobbins (1944), respectively.

468

K. NADAOKA

I

I

I

I

4

2

I

O

(a)cose

ETAL.

I

8(n~In)

1

C/eb,

8(rain) (b) case

1, O

2

~

C/Cbo

0 Y

I

I

2

I (c) case

I

I

4

I

6

8(min)

B

Fig. 4. Time variation of the bottom and depth-averagedconcentration, Cb and C m, The full and broken lines represent the computationalresults by the present method and the analytical solutions by Dobbins (1944), respectively. bins (1944). For this case, Eqs. 10 and 11 become quite simple as follows:

S, "Cb,-Qj "A,=P--Cb" Ws

(12)

32 "Cbt--Q2 "A,=P'Cb-C 2" Ws" (E2 + 1 ) / 2 + C2"A'K~ • ( E 2 - 1 ) / ( 2 D ) (13) Figure 3a and b shows the results for the deposition process, referred to here as case 1 and case 2, in which the pick-up rate P suddenly decreases from the value of the initial equilibrium state ( P = Cbe" Ws) to 0.3" Cbe" Ws and to zero, respectively. On the other hand, Fig. 3c represents the results for an erosion process, referred to as case 3, where the pick-up rate P of a non-zero value is suddenly supplied to an initially no-suspended sediment condition. The computations have been carried out with the time step At= 0.1 s under the conditions of h, Ws and K~ indicated in each figure. The horizontal and vertical coordinates in these figures are normalized by Cbe and h, respectively. In Fig. 3c, the experimental values are not indicated, because they are not included in the original paper of Dobbins (1944). Comparison of these results by the present method with those by the analytical solution and the ex-

A SIMPLE QUASI-3-D MODEL OF SUSPENDED SEDIMENT TRANSPORT

469

periments clearly demonstrates that the present theory gives satisfactory resuits in accuracy, though it is a quite simple m e t h o d with only two unknown variables. Figure 4a-c shows the time evolution of the bottom and depth-averaged concentration, Cb and Cm, for each case, representing remarkably good agreement between the computational results and the analytical solution of these quantities. It should be particularly emphasized that the high accuracy in estimation of Cb means that the present model may work as a reliable tool also in the prediction of seabed deformation because the deformation rate Oh/Ot can be calculated with the relation Oh/Ot=P- Cb" Ws. The agreement in Cm indicates that the condition for the total mass conservation of the sediment in the entire water column of - h < z < 0 can be satisfied also with good accuracy by the present model. The CPU time required for the computation by the present method is obviously shorter than that by the usual 3-D finite difference scheme. In this 1D problem, e.g., the C P U time was about 1/100 times shorter than that by a usual implicit finite difference scheme with 25 mesh points in the vertical coordinate. Further it has also been confirmed that the accuracy of the present method is not so affected by the variation in the time step At as the usual finite difference scheme is. APPLICATION TO A 3-D PROBLEM IN COASTAL ZONE

For the application of the present method to the 3-D problem, the suspended sediment transport in the coastal zone including the surf zone has been analyzed. As the pre-computation for the analysis, the nearshore current field associated with the supposed linear wave field was calculated with the basic equation for the wave-induced currents based on the radiation stress concept. The computation of the suspended sediment and the resultant seabed deformation was carried out under a typical condition of the pick-up rate, which was most appreciable inside the surf zone.

Computational conditions The conditions of computation, both for the nearshore currents and the suspended sediment transport, are summarized in Tables I and 2, respectively. The values of the diffusion coefficients and the pick-up rate in these tables are not sufficiently reliable in magnitude, because the way to estimate these values itself is not well established at present and still remains as one of important problems to be solved. Further, in the present computation procedure, the feed-back effect of the seabed deformation onto the wave and current field is not incorporated.

470

K. NADAOKAET AL.

TABLE 1 Computational conditions for the nearshore currents Computational area

336 m in the on-offshore (x) direction and 126 m in the longshore (y) direction

Mesh size Sea bottom slope Incident waves

3x=3y=7 m

Horizontal diffusion coefficients Boundary conditions Finite difference scheme

constant slope of 1/50 wave height o f H ( m ) = 1 - 0 . 3 c o s ( y n / 1 2 6 ) and normal incident angle at the offshore end ~x=xo) and wave period T = 8 s K~=Ky=O.Olxx/gh (Longuet-Higgins, 1970) inside the surf zone, and = 0 . 0 l Y b x ~ b outside the surf zone, where the subscript b denotes the values at the wave breaking point U = 0 at the onshore end ( x = 0 ) , Oq/Ox=O at X=Xo, V = 0 at the longshore boundaries ADI method

TABLE 2 Computational conditions for suspended sediment Horizontal diffusion coefficients Vertical diffusion coefficients

Pick-up rate

Settling velocity Boundary conditions Finite difference scheme

same as those in Table 1 for inside the surf zone, K~= 0.1877/3L1/3h2/3 (gh) I/2S 1/3 with y = H/h = 0.8, wave length L and bottom slope s (Nadaoka and Hirose, 1986), which includes the effect of wave breaking, and for outside the surf zone, Kz= O.16h~/g( U2+ V 2)/Ch, where Ch is Chezy coefficient ( Bijker, 1980 ) P = Cbe" Ws, where Cb~ was estimated from the formula by Lane-Kalinske ( 1939 ) by adding the effect of orbital wave motion on the friction velocity u. according to Willis (1978), i.e., u .z = ( U2+ W2)/C~ +fwu~/4, wherefw is Jonsson's wave friction factor and u0 is maximum orbital velocity at the bottom ~=0.01 m/s C b = 0 at x = 0 , OCb/Ox=O and OA/Ox=O at x=xo, OCb/Oy=O and OA/ Oy= 0 at the longshore boundaries explicit scheme - - forward difference in the time marching, up-wind difference for the convection terms, central difference for the diffusion terms

Computational results and discussion Figures 5 and 6 show the computational result of the nearshore currents and the pick-up rate, respectively. In Fig. 5, the development of the circulation pattern can be found with the rip current along the lower boundary. For the representation and comparison of the results of the calculation of suspended sediment, the resultant deformation rate of the seabed - 0h/Ot has been computed by the following several methods including the present method:

A SIMPLEQUASI-3-DMODELOF SUSPENDEDSEDIMENTTRANSPORT

471

I m/s :breaker llne

I ~ -~~ ? I E ~ I ~ , ~ ~ _ ' _

o

"~_ _" ~_ " " . . . . . . . . . . . . . . . . . . . . . .

I ~ ].'/,.~'~~t~'-_'-_'b_'>.".~." 11 ~ , ' ~ _ " . " . " . " I

"4'~k~i.d[~l[Vlll'flP'A"~lg.l..~.

I ~ .$.% ~ , ~

" ®

I ~ ~

I

-.

.':"." _" • . . . . . . . . . . . . . . . . . . . . . . ,"'::': ~. ": .......................

_ • % % % I "~ '~ t ~ t ! s , t ,

,

• ~ q-.'"_".'."::.:" ' . . . . . . . . . . . . . . . . . . . . . . .

~%~,~.~/%,~.--'~A--"~_.I.":'."."."_'~ . . . . . . . . . . . . . . . . . . . . . . .

- r -

-

"----r

0

H~

.....

100

r . . . . . .

/'

I

200

/

300m

offshore

Fig. 5. Calculated nearshore currents associated with the supposed linear wave field.

Fig. 6. Spatial distribution of the pick-up rate P estimated with the method indicate in Table 2.

(a) Present method:

~t=P-Cb.W~

(14)

(b) Conventional method based on the local equilibrium profile of suspended sediment concentration C:

Oh Oq~ Oq~ Ot-Ox + Oy

(15)

where:

q~= Cb~(Kz/W~). [ 1 - e x p ( qy= Cb~(KJW~)" [1 --exp(

--

WJK=.h) ] U, WJK~.h ) ] V.

and

(c) Nonequilibrium calculation based on the assumption of vertically uniform distribution of C:

Oh-p-Cm'Ws Ot-

(16)

472

K. NADAOKA ET AL.

where Cm is the depth-averaged sediment concentration. Among the above methods, the method (b) can easily yield the spatial distribution of -Oh/Ot, because it needs computation only for an equilibrium state. On the contrary, the methods (a) and (c) need time marching of the computation, which was continued, in the present cases, until nearly stationary states were achieved with the time step of At=0.1 s. The initial condition for these time-dependent problems is C = 0 at t = 0. Figure 7a-c shows the contours of the values of - Oh/Ot by the methods (a) to (c), respectively, where for (a) and (c) the values for almost stationary state are represented. Figure 7a for the present method (a) shows that the convection of suspended sediment by the rip current results in the accretional area extending far offshore beyond the breaker line. On the contrary, Fig. 7b for the local equilibrium method (b) shows that the accretion occurs in a quite limited region around the breaker line, and that the most eroded and deposited points are located near the breaker line and the absolute values at these points are considerably larger than those by the method (a). This is owing to the fact that, in case (b), the spatial variation of the suspended sediment concentrala)

:eroslon

'"i

xlO-3mm/s

(b)

i

<:i:2

1,

• :o0,mul=ti0n

+,r000

(c)

Fig. 7. Seabed deformation rate -Oh/Ot calculated with (a) the present method, (b) the method based on a local equilibrium assumption, and (c) nonequilibrium estimation based on vertically uniform distribution of the concentration.

A SIMPLE QUASI-3-DMODEL OF SUSPENDED SEDIMENT TRANSPORT

473

tion and thus of the sediment transport rate becomes most noticeable around the breaker line, corresponding to the distribution of the pick-up rate shown in Fig. 6. The occurrence of such unrealistic deformation of seabed is inevitable for the methods based on the local equilibrium assumption, because the magnitude of hydraulic agitating factors for sediment suspension such as orbital velocity, unidirectional current and turbulence suddenly varies across the breaker line. In other words, this result emphasizes the importance of the incorporation of the nonequilibrium condition into the model especially for the case of sediment transport across the breaker line. In Fig. 7c for case (c), the accretion area appears more extensively toward offshore as compared with that for case (a), though its overall pattern is similar to that of case (a). Furthermore, the deposition rate in the region close to the shore line attains relatively large value, suggesting the augmentation of the sediment convection toward the shore. These are attributable to the fact that the assumption of the vertically uniform distribution of C in case (c) leads to an underestimate of Cb and hence of the deposition flux Cb" Ws, and results in an overestimate of the horizontal sediment flux. CONCLUSIONS

In the present study, a new simple model to evaluate quasi-3-D suspended sediment transport in a nonequilibrium state has been developed through a formulation based on a kind of weighted residual concept. The present model includes only two unknown variables to be solved in a horizontal 2-D domain (x,y); i.e., the sediment concentration at the bottom Cb (x,Y, t ) and the shape factor A(x,y,t) in the assumed exponential curve for the vertical C-profile, both of which are the most important factors governing the nonequilibrium sedimentary process. Several examples of the computation has confirmed the validity and the computational efficiency of the present model and demonstrated the importance of the nonequilibrium 3-D condition for the suspended sediment transport problem in the coastal region, especially in the case of sediment transport across the breaker line. Although, in the present model, the horizontal mean current and the diffusion coefficients are dealt with as vertically uniform only for the reason of simplicity. However, the vertical variation of these quantities may be incorporated into the model without any essential change of the formulation procedure, though this remains as the subject for future studies. ACKNOWLEDGEMENT

The present study has been supported by a grant in aid for scientific research, No 62550370, by the Ministry of Education, Science and Culture.

474

K. NADAOKAETAL.

REFERENCES Bijker, E.W., 1980. Sedimentation in channels and trenches, Proc. 17th Int. Conf. on Coastal Eng., ASCE, pp. 1708-1718. Dally, W.R. and Dean, R.G., 1984. Suspended sediment transport and beach profile evolution, J. WW Div., ASCE, 110 (1): 15-33. Delft Hydraulic Laboratory, 1980. Storm Surge Barrier Oosterschelde, Computation of siltation in dredged trenches. Rep. 1267-V/M 1570. Dobbins, W.E., 1944. Effect of turbulence on sedimentation, Trans. ASCE, 109: 629-656. Fleming, C.A. and Hunt, J.N., 1976. Application of a sediment transport model, Proc. 15th Int. Conf. on Coastal Eng., ASCE, pp. 1184-1202. Galappatti, G. and Vreugdenhil, C.B., 1985. A depth-integrated model for suspended sediment transport, J. Hydraul. Res., 23 (4): 359-377. Irie, I. and Kuriyama, Y., 1988. Reproduction of sedimentation in harbors by combined physical/ mathematical modeling. Proc. of IAHR Symp. on Mathematical Modelling of Sediment Transport in the Coastal Zone, IAHR, pp. 79-88. Kuroki, M., Shi, Y. and Kishi, T., 1988. Distribution of suspended sediment concentration in non-equilibrium state, Proc. 32nd Japanese Conf. on Hydraulics, JSCE, pp. 407-412 (in Japanese). Lane, E.W. and Kalinske, A.A., 1939. The relation of suspended to bed material in rivers, Trans. AGU: 637-643. Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves, 1,2, J. Geophys. Res., 75: 6778-6801. Miller, H.P., 1984. Three-dimensional free-surface suspended particles transport in the South Biscayne Bay, Florida, Int. J. Numer. Methods Fluid, 4:901-914. Nadaoka, K. and Hirose, F., 1986. Modelling of diffusion coefficient in the surf zone based on the physical process of wave breaking, 33rd Japanese Conf. on Coastal Eng., JSCE, pp. 2630 (in Japanese). O'Connor, B.A. and Nicholson, J., 1988. A three-dimensional model of suspended particulate sediment transport, Coastal Eng., 12:157-174. Stive, M.J.F. and Battjes, J.A., 1984. A model for offshore sediment transport, Proc. 19th Int. Conf. on Coastal Eng., ASCE, pp. 1420-1436. Wang, Z.B. and Ribberink, J.S., 1986. The validity of a depth-integrated model for suspended sediment transport, J. Hydraul. Res., 24 ( 1 ): 53-66. Willis, D.H., 1978. Sediment load under waves and currents, Proc. 16th Int. Conf. on Coastal Eng., ASCE, pp. 1626-1637.