Numerical study of coastal changes

Advances in Engineering Software 19 (1994) 85-89 © 1994 Elsevier Science Limited Printed in Great Britain. All fights reserved 0965-9978/94/$07.00

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Numerical study of coastal changes Kazumasa Mizumura Department of Civil Engineering, Kanazawa Institute of Technology, 7-1, Ogigaoka, Nonoichimachi, Ishikawa Pref. 921, Japan (Received 13 September 1993; revised version received 6 May 1994; accepted 17 May 1994) Two-dimensional changes of beach profile are computed by the continuity equation of sediment. Wave transformation used in the continuity equation is obtained by solving the shallow water equations numerically.The application of this method for different wave height, period, and sand particle diameters shows that this approach is reasonable. Applying this method and the result of field observations to Nakisuna beach, the coastal changes there are found to be stable without any protection.

Key words: coastal changes, shallow water equations, sediment transport, open boundary condition, beach profile, water depth. 1 INTRODUCTION

height was 0.2m-0.5m. The foreshore slopes were 1/100-1/10 and the mean diameter of bed materials was 0.1 mm-0.4 mm.

In Japan, most of the beautiful sand beaches which represent typical Japanese scenary were lost because of the failure of river sediment control. Therefore, keeping the remaining sand beaches is the most important work for coastal engineers. Coastal changes are usually predicted by hydraulic model tests or numerical computation. Recently the principle tool for the prediction of coastal changes became the numerical computation method because of its cost and simplicity. As the first step of this research, the coastal changes are computed when on-offshore sand transport is dominated. In this case longshore movement of sand transport is neglected. This computation consists of two parts. One is the computation of wave transformation by using shallow wave equations. This computation gives the correct wave run up and back wash process. The other is the computation of water depth change due to sand movement. This study investigates the method of computation of the beach profile change due to on-offshore drift of sand. This method is applicable to Nakisuna beach and the beach condition is determined (or simulated).

CHARACTERISTICS OF NAKISUNA BEACH Nakisuna means that sand sings when we walk on it. This kind of beach is found in Japan. Nakisuna beach is located on the west side of Noto peninsula as shown in Fig. 1. Since most of sand consists of quartz, its color is white. The mean diameter of the sand is 0"l-0.2mm. Wave climates form typical patterns of the sea of Japan such that waves are very high during winter and calm during summer. The wave type which frequently occurs forms beaches of the typical bar type (see Fig. 2). In the range from the shoreline to the offshore 200-300m hard rock sea bottom is found. This is natural submerged offshore breakwater and resists erosion due to wave attack. The comparison of aerophotos indicates a very stable shoreline and does not find an influence of coastal structures. After a sea flood the beach is eroded and stones under sand are exposed. Beautiful sand moves offshore. After several weeks, most sand comes back and covers the stones.

FIELD OBSERVATIONS To study the characteristics of Nakisuna beach we measured foreshore slopes, alongshore currents, and bed materials in August 1992. During the field observations wave direction was N - N W and wave

GOVERNING EQUATIONS To compute the beach profile change, the wave transformation must be known. This is described by 85

K. Mizumura

86

Noto Pen/nsula

/

x=0

( Honshu Island

Fig. 3. Coordinate system. The computational process of the run-up and back-wash are done according to the paper of Hibberd and Peregrine. 3 The continuity equation of sediment (see Fig. 4) is given by

NaJdsuna Beach Fig. 1. Locationof Nakisuna Beach. the following shallow water equations:

Oy Ou Oh Ot ~-u-o--xx+ g - ~ = g( So - Sf ) Oh

(1)

Ohu

Ot + ~

= 0

(2)

in which u = depth averaged velocity; h = water depth; g = the gravitational acceleration; t = time; x = coordinate system (see Fig. 3); So = sea bottom slope; and ST = friction slope. To solve eqns (I) and (2) numerically, Lax and Wendroff's numerical scheme 2 is used. Equations (1) and (2) are transformed into the following equation: 3

vj,.+, = uj,. - ~{I [Fj+,,° - 6-,,n] + AxGj,.} /~2

+ "-~ {gj,n -- gj-l,n -- Axgj,n}

(3)

in which

[:]

U=

,

m = uh,

A = Ax,

Ax + 5 - Cj+l,. + ~j,.)], g h - m]2 /0h 2

1

m

The empirical expression of sediment transport is given by 4

Bwwov'-/ v~,- vL qx - (1 - n o ) s v / ~

g

(5)

.

in which no = void ratio; qx : rate of sediment transport; ~ = constant parameter to include the effect of beach profile change; s = P s / P - 1; p = water density; Ps = sand density; d = mean diameter of sand particle; f = g ~ 2 / h l / 3 ; h = M a n n i n g ' s roughness coefficient; B w = 7.0 (constant); U, = v/-fu; U, c = ~/~c(P, - p)gd; ~c = the critical Shields number; and w0 = the falling velocity of sand particle. The value of qx is zero if U, is less than U**. This formula for qx is to describe the equilibrium motion of sediment transport. Therefore, to express the equilibrium and non-equilibrium conditions together, the following equation is suggested: 5

/ d \ 1/3

L=H 2.4[,~)

sj° =/xx o6j,. '

(4)

in which A = mean step length of sediment particles. A is selected to be 100d. When the incident waves come from offshore, sand particles in the deeper region than hi do not move. But if the water depth becomes shallower than hi, sand particles move as the wave passes. The limiting water depth hi is expressed by 6

'

F~

Oh}

IqBl~

A dq8 -~x + qB = qx

gj,n : 1 [A(Uj+I,n) + A(Uj,n)][Fj+I, n _ Fj,n

A - o u - O F [2ml/h

Oh OqB O{ Ot - O ~ + e - - ~ x

27rhi

sinh L

Ot

=Ax

[ g,o

N +g

--~- - ~ + --0-fluO-t

I

II

T

/

.

I

A j,u

q*-T

I

• o

Fig. 2. Beach type.

Fig. 4. Derivation of sand continuity equation.

ar

87

Numerical study of coastal changes

in which H = wave height and L = wave length. The continuity equation of sand is applied in the shallower water-depth region than hi.

Type I

BOUNDARY AND INITIAL CONDITIONS The boundary conditions with respect to wave motion are located at the up-wave side (offshore boundary) and the down-wave side (onshore boundary). At the up-wave side the form of the incident wave is given by H . 27r ( = ~ sm-~-

he - hM uMhM t - = 0 At Ax'

(7)

The vaue of At is the time increment for integration. The water depth on the offshore boundary is given by hp - h0 + ¢ + h0 2

(8)

in which h0 -- the stationary water depth on the offshore boundary and Ax' = At gv~0 (see Fig. 5). The value of 2 of the denominator on the fight hand side of eqn (8) means that the reflected wave height by a vertical wall is twice that on the open boundary. The water depth is stationary and velocity is zero at the initial time. The initial condition for sediment transport is given by the initial water depth. The boundary conditions for sediment transport at the up-wave and down-wave sides are given by qB = 0

(9)

~

[ o

Type 11I

Fig. 6. Pattern of beach profile changes. BEACH PROFILE CHANGES

The two-dimensional changes of beach profile which start from the uniform slope and are caused by on-offshore sediment transport are classified into the following three types: 6 Type I is characterized by sediment transport from onshore to offshore. The direction of the sediment transport of Type III is opposite to Type I. For Type II sediment moves to the onshore and offshore. Their sketches are given in Fig. 6. Considering the direction of sediment transport, Types I and III correspond to the bar and step type of beach profile. The empirical equation to classify these three types is given by H/L

(d/L)°.67/sO.27

= C

(10)

The constant value C is a parameter to discriminate between these types of beach profile. For the hydraulic model beach, the value is suggested as follows: Type I TypelI Type III

(11)

For the prototype beach, the value is recommended by

A~ Aa?

~

C> 8 8>C>4 C< 4

P M

Type II

(6)

in which T = wave period. The reflected waves from the beach are transmitted through this offshore boundary. Due to Hino's open boundary condition7 assuming the velocity is zero,

he=

j

C > 18 18>C>9 C< 9

AX

X

Fig. 5. Coordinate system at offshore boundary.

Type I TypelI Type III

(12)

Considering the momentum equation on a sand particle, the theoretical derivation of this kind of equation is given by Hayasaka and Mizumura. s Herein, we apply

88

K. Mizumura

• Prototype Type I ® Prototype Type II o Model Type I • Model Type HI © Model Type H

the numerical result to eqn (10) and check whether the numerical result is appropriate or not.

NUMERICAL RESULT By solving eqns (3) and (4) numerically, time-varying beach profiles are computed. The result is plotted in Fig. 7. The unit of length in Fig. 7 is meters. The horizontal direction in the figure shows the coordinate along the sea bed from the offshore to the onshore. The origin corresponds to the point where sediment motion does not exist. The incident wave height, period, bottom slope, and Manning's roughness coefficient are 0.077 m, 1 see, 0.10, and 0.02, respectively. The spatial increment Ax is selected to be 1 cm and the resultant time increment A t = 0 . 0 2 ( A x / ~ ) . hmax means the maximum water depth in the computational region. This satisfies the following Courant condition: At 1 < Ax lu ±

(13)

This is the necessary condition which makes the computation stable. The pattern of this beach profile change corresponds to Type I. That is, sand in the onshore moves to the offshore. Figure 8 compares the numerical result with eqns (11) and (12). This result shows that this numerical computation estimates the experimental result very well. Circles in Fig. 8 show the computed result based on the condition of waves and beaches. Four solid lines correspond to eqns (11) and (12). The parameter C is given by eqn (10). Therefore, the regions C > 8, 4 < C < 8, and C < 4 show that the model beach becomes Type I, II, and III, respectively. The computational result of model beaches (white circles, left-white circles, and white-line circles) is coincident with the empirical relationship of eqn (11). The region C > 18, 18 > C > 9, and C < 9 describe that the prototype beach becomes Type I, II, and III, respectively. The computational result of prototype beaches (black circles and left-black circles) is also

t

10-a

t=Os

shoreline 10 m

10-3t

~ = 200s ......

,

_

0 t 10-3

t" . . . . . . . . .

t "- 4 0 0 s ". . .

0

"''''"

~.. 10-3

0

t

.

: "'"

= 600s

.-'.-" .... -...

% • . "..

Fig. 7. Numerical beach profile changes.

C = 18

0.01

0.001

0.001

0.01

(a/L )O.eT Fig. 8, Classification of beach profile changes. coincident with the empirical relationship of eqn (12). Therefore, solid circles in Fig. 8 represent the coastal changes under the wave condition (wave height and period) which most frequently occurs in this study area. Therefore, the predominant wave condition forms the bar-shape of beach as shown in Fig. 2. The stability condition of numerical computation is given by the Courant condition [eqn (13)]. To exclude the wave breaking in the offshore region, the following condition is employed to select incident waves for numerical computation. H < 0.142 tanh 27rh (14) L L The waves to satisfy eqn (14) are stable in the wave form.

CONCLUDING REMARKS The wave transformation is expressed by the shallow water equations and wave run-up and back-wash are also strictly investigated. The beach profile change is computed by using the continuity equation of sediment. The sediment transport formula is described by the experimental equation. As a result, two-dimensional coastal change is numerically modelled. The determination of several parameters is not fixed, but this model predicts beach profile changes very well. As a result, the most frequent wave condition gives the beach type to be bar-shaped. The beach type is always bar-shaped and stable during a year. Therefore, Nakisuna beach is protected by the bars in the offshore, if we human beings do not touch the beach (i.e. large coastal structures are not constructed).

Numerical study of coastal changes

REFERENCES 1. Construction Institute of Fishery Ports. Report on Wave Climate in the sea of Japan, Mar., 1988 (in Japanese). 2. Lax, P. & Wendroff, B. Systems of conservation laws. Comm. Pure Appl, Math., 1960, 13, 217-237. 3. Hibberd, S. & Peregrine, D.H. Surf and runup on a beach: a uniform bore. Journal of Fluid Mechanics, 1979, 95(2), 323 -345. 4. Ifuku, M., Kanazawa, T. & Yumiyama, Y. Computation of two-dimensional wave transformation and coastal changes by method of characteristics. Japanese Conf. on Coastal Eng., 1991, 38, 376-380 (in Japanese).

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5. Nakagawa, H. & Tsujimoto, T. Loose Boundary Hydranlics, Gihoudou Shuppan Co. Ltd., Tokyo 1986 (in Japanese). 6. Mizumura, K. Coastal and Ocean Engineering, Kyouritsu Shuppan Co., Ltd., Tokyo, 1992 (in Japanese). 7. Hino, M. A very simple scheme of non-reflection and complete transmission condition of waves for open boundaries, Technical Report, Dept. of Civil Engng., Tokyo Inst. of Tech., 1987, No. 38, 31-38 (in Japanese). 8. Hayasaka, M. & Mizumura, K. Theoretical considerations on two-dimensional coastal changes. Proc. 34th Annual Conf. of JSCE, 1979, 2, 669-670 (in Japanese).