Mud scour on a slope under breaking waves

63

Coastal and Estuarine Fine Sediment Processes W.H. McAnally and A.J. Mehta (Editors) 9 2001 Elsevier Science B. V. All rights reserved

Mud scour

on a slope under

breaking

waves

H. Yamanishi a, O. Higashi b, T. Kusuda a and R. Watanabe c Department of U r b a n and Environmental Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812-8581, J a p a n a

bCity and Environment Planning, Research, EX Corporation, 2-17-22 Takada, Toshima-ku, Tokyo, 171-0033, J a p a n CDepartment of Civil Engineering, F u k u o k a University, 8-19-1 N a n a k u m a , Jonan-ku, Fukuoka, 814-0180, J a p a n

Scour of mud by breaking waves was investigated in an experimental flume with a sloping bed and in a tidal river. It was found that: (1) the impact of breaking wave on the slope can be formulated in terms of the conservation equation of m o m e n t u m considering reflection of waves; (2) the impact of breaking wave decreased with increasing wave steepness under the same wave height; (3) cohesive sediment was scoured as pieces and accumulated as clasts at the foot of the slope; and (4) the mass of sediment scoured by breaking waves, W, per N number of waves and per unit area can be formulated as (Wg/As)/T~s/N = m[(p ,n/T.s)-(p m/X)c], where pm and ~ are the m a x i m u m impact of breaking wave and the bottom shear strength, respectively. From the experiments, m and (p m/Xs)c were found to be 0.14 and 0.37, respectively.

1. I N T R O D U C T I O N Waves and currents cause sediment transport, and turbidity maxima are formed and travel u p s t r e a m and downstream in estuaries. F u t a w a t a r i and Kusuda (1993) performed long term field observations in the Rokkaku River in Japan. This river is located in Ariake Bay, on the western side of Kyushu Island. The results showed t h a t the minimum and m a x i m u m concentrations of suspended solids in a fortnightly cycle took place over two to three days after a neap tide and a spring tide, respectively. Suspended solids concentration in the upper water layer tended to be lower during a neap tide t h a n usual, which means that suspended solids settled and accumulated on the bottom and banks of the river. Kusuda (1994) reported t h a t an annual sedimentation rate at the banks was over 20 cm in height. Thus, it is difficult to maintain the cross-section of

Yamanishi et al.

64

rivers with a high turbidity w a t e r body. Not only mud accumulated on banks prevents m a i n t e n a n c e of the cross-section, but also its dredging can be costly. Therefore, it becomes essential to remove m u d efficiently. A paucity of studies on n a t u r a l removal of m u d prompted us to investigate this problem. Breaking waves impose a high pressure on a slope, and their successive attacks m a y cause scour. The purpose of this paper is to examine the effects of breaking waves as a method to scour cohesive sediment accumulated on the banks of rivers.

2. I M P A C T O F B R E A K I N G WAVE A C T I O N

Figure 1 defines the notation for the estimation of the impact of breaking wave action on a slope. The x-axis is t a k e n along the slope, and the z-axis normal to the slope. Here, 0 is the incident angle, ~ is the slope angle, ~, is the reflected angle, H b is the b r e a k i n g wave height, h b is the breaking water depth, C b is the phase velocity, and v is the free fall velocity. For estimation of the impact of breaking waves, conservation of m o m e n t u m is stated for the direction normal to the slope as:

~" Fzdt = ~" pQ2V2zdt - ~" pQ1Vlzdt =~:'pA2V2V2zdt-~:'pA1V1VI~dt

(1)

Here, F is the vertical breaking wave force onto the slope, t ~ is the impulse duration, p is the w a t e r density, Q1 is the incident flow rate, Q2 is the splash flow rate, and V~ and V~ are the vertical components of V1 and V2, respectively. As seen, this equation considers splash in the breaker zone.

v

89

V _-=-

j

0+~

Figure 1. Notation for the formulation of the wave b r e a k i n g and scour problem.

Mud scour under waves

65

The following relations are derived from geometry: A 1 = A sin(0 + ~)\ A 2 = A sin ~,

(2)

f

V~z = -V~ sin(0 + ~)~ V2z = V2 sin 7 J

(3)

Then, s u b s t i t u t i n g (2) and (3) into (1), the following equation is obtained:

~:*Fzdt = ~:*pAV12 sin 2 (0 + ~)dt + ~:*pAV22 sin 2 Tdt

(4)

A s s u m i n g t h a t p, A, V , V2 and 0 are constant w h e n a wave b r e a k s on the slope, (4) is t r a n s f o r m e d as follows:

~:*pdt = pt*V~2 sin 2 (0 + ~) + pt*V22 sin2 ~/

(5)

where, the i m p a c t of b r e a k i n g wave pressure p is F~/A. F u r t h e r , the impulse t e r m on the left-hand side of (5) is a p p r o x i m a t e d by (6):

~* pdt ~ kPmt*

(6)

where, k is a coefficient of the impulse term, the product Of Pm and t*, in which Pm is the m a x i m u m b r e a k i n g wave pressure. S u b s t i t u t i n g (6) into (5), (7) is obtained: Pm=

k[V~ sin~ (0 + ~) +V~ sin~ ~1

(7)

Here, we a s s u m e t h a t a portion of Q1 becomes Q2, i.e.,

Q2 =klQ1

(8)

Then, (9) is obtained as:

V 2 = K1V 1

(9)

Finally, (10) is obtained by s u b s t i t u t i n g (8) and (9) into (7):

P--~m:aV12[sin2(O+B)+K12sin2Tl pg

2g

(10)

66

Yamanishi et al.

Thus, the i m p a c t of b r e a k i n g wave on a slope results in (10), where, g is the acceleration of gravity, and ~ and K 1 are the coefficients based on experimental results. Some p a r a m e t e r s in (10) m u s t be d e t e r m i n e d to calculate the impact of b r e a k i n g wave. The incident velocity V 1 and the incident angle 0 are described by (11) and (12):

V 1 = ~v 2 +Cb 2

(11)

0 = t a n -~ ( v / C b )

(12)

The free falling velocity v and the phase velocity C b are defined by (13) and (14), respectively:

v = ~/2gH b

(13)

Cb = ~/g (hb + rib)

(14)

where, H b is the b r e a k i n g wave height, ~lb is the wave crest height (the height from m e a n w a t e r level to the top of b r e a k i n g wave height, see Figure 1), and h b is the b r e a k i n g w a t e r depth. Here, we define the phase velocity C b as the wave velocity of a solitary wave, because the small a m p l i t u d e wave theory is not applicable to the b r e a k i n g wave phenomenon. Reflected angle ~, and coefficients a and K 1 m u s t be e s t i m a t e d from e x p e r i m e n t a l results.

3. B R E A K I N G W A V E E X P E R I M E N T S

The b a n k s of the R o k k a k u River have steep slopes (1/4-1/6). Because m a n y e x p e r i m e n t s on s e d i m e n t t r a n s p o r t have in the past been conducted on gentle slopes, we performed some e x p e r i m e n t s on breaking waves on a rigid/mud bed with a slope of I in 5. The wave t a n k used for the e x p e r i m e n t s was 0.5 m wide, 0.3 m high and 14 m long, with a sloping bed. The e x p e r i m e n t a l a p p a r a t u s is shown in Figure 2. Regular waves were generated u n d e r various wave conditions, and all waves were plunging breakers over the slope. 3.1. B r e a k i n g w a v e h e i g h t H b Le M~haut6 and Koh (1967) suggested the following empirical equation to describe the relation between the d e e p w a t e r wave height and the breaking wave height,

Ub = 0.76(tan~j)1/7 (Uo1-1/4 [,L-o0)

H~

(15)

Mud scour under waves

67

WAVE Pr_~ADDL~

"

-=

"

,0.3

~

\/ V

rWAVE

~BSORBER

FALSE FLOOR -~

2.5

7.0

~-

14.0

-

- - 1. D~ -

(WIDTH:0 . 5 m , U N I T : m) Figure 2. Elevation view of the experimental apparatus.

H~

: K ~ K d KwH o

(16)

where, tan~ is the slope, H 0is the equivalent deepwater wave height, H 0 is the deepwater wave height, L 0 is the deepwater wave length (=gT2/2~), K is the refraction coefficient, K~ is the diffraction coefficient, and K r is the friction coefficient. Here, for the sake of simplicity, we have assumed t h a t all coefficients in (16) are unity. Therefore, we use Ho/L o as the characteristic wave steepness. Figure 3 shows the relation between Ho/L o and Hb/H o. In the plot, (15) does not agree with the experimental data, but the trend can be regarded as similar. Equation (15) is applicable only to a gentle slope (~1/10), and its applicability to a steep slope has not been verified. However, it is convenient to use the form of (15), in order to simulate measured data. On the basis of (15), two empirical equations were obtained for the initial and steady states of various experimental conditions. Here, the state of a few waves except for the first wave is defined as the initial state [see Figure ll(a)], and after t h a t it is considered to be steady state. Each experimental result shows the mean value. Figure 3 relates to the initial state, and plots (17) with a dotted line:

Hb : 0.63(tan[3)l/7 (Ho ~-1/4 H--o ~,-~o )

(17)

Figure 4 is for the steady state, and plots (18) with a dotted line:

Hb H0

= 0.55(tan ~)1/7(Ho 1-1/4 <-~-o)

(18)

Comparison of Figures 3 and 4 shows that the trend gradually changes due to the effect of r e t u r n flow. Equations (17) and (18) are empirical, but are applicable in this experimental region, because of the observed agreement with experimental data. In this way, it is possible to predict measured values of H b.

Yamanishi et al.

68

al 9

< Steady

State>

EXP.data

o

EXP.data

(is)

(is) 2

k

2

I,i~~ ............(,17)i l"

~

...............Q.......~ ......,.--,~

.........I' ............Q'.............. 0

0

0

0.02

0.04

0.06

0.08

(18)

...................

0.1

i

0

Ho/Lo

0.02

I

0.04

I

0.06

I

0.08

0.1

Ho /Lo

Figure 3. Relation between wave steepness Ho/Lo and Hb/H o (in the initial state).

Figure 4. Relation between wave steepness Ho/L o and Hb/H o (at steady state).

3.2. B r e a k i n g w a t e r d e p t h h b

S u n a m u r a (1983) derived an empirical equation describing a critical breaking wave on the basis of experimental data:

Hb : 1.09(tan~)O 17(hb / -~

(19)

Figures 5 and 6 show the relation between h b/L o and H b/h b. In order to estimate the breaking wave water depth hb, the following empirical equations were obtained from these results"

Hb =0.97(tan~)~ hb

~

-~

H~ : 0.89(tan~)O 17(hb ~-~

t<)

(20)

(21)

Equation (20) is in agreement with the initial state, and (21) with the steady state. Both are applicable in the experimental domain. 3.3. E s t i m a t i o n o f b r e a k i n g w a v e p r e s s u r e

Figure 7 shows the relation between Ho/L o and the incident angle 0, and comparison of calculated results with experimental data. Each angle 0 was m e a s u r e d by images from a digital video camera. Some lines in Figure 7 have

69

M u d scour u n d e r waves

2.5

2.5

< Initial State>

2.0

< Steady State > 9

EXP.data

2.0 i

(19)

.a 1.5 1.0

$

Q

.~

1.5

EXP.data

...................

1.0

....................................................

.... .............

0.5

0.5 t 0

0

(19)

................... (20)

I

0

0.02

I

0.04

I

0.08

'

I 0

I

0.06

0.1

0.02

hb/Lo

9

Ho = 5 c m

55

................... H o = 6 c m

.,-,

c-

0.1

EXP.data

i CAL.line from (12) ................... Ho=8

(D C)3 c-

I 0.08

F i g u r e 6. R e l a t i o n b e t w e e n h b / L o a n d H b / h b (at s t e a d y s t a t e ) .

60

v

I 0.06

hb/Lo

F i g u r e 5. R e l a t i o n b e t w e e n h b / L o a n d H b/h b (in t h e i n i t i a l s t a t e ) .

C)') "O

I 0.04

cm

......

Ho = 1 0 c m

.......

Ho=12 cm

5.2cm 50

..... = .................l . o . . ~ o r ~ ...... ~ - - ~ . - . . - ' 1 - . o ' : ~ ' ~

.................= = " " ' " "

.....w..........................9 ................ - 9 -"" ..........~u -................................ ---.~.... , ~ u............................. m

'13 r,.) c-

s.~:

............

~.~;

-~_ .........

"T2:3~n-

o _ _

9

...,,,.

45

! 0.02

| 0.04

0.06

0.08

0.1

Ho /Lo F i g u r e 7. C o m p a r i s o n of c a l c u l a t e d r e s u l t s w i t h e x p e r i m e n t a l d a t a on t h e r e l a t i o n s h i p b e t w e e n 0 a n d H o / L o.

b e e n o b t a i n e d f~om (12). I n o r d e r to c a l c u l a t e t h e i n c i d e n t a n g l e 0 f r o m (12), it is n e c e s s a r y to s e t v a l u e s of t h e u n k n o w n p a r a m e t e r s i n c l u d e d in (13) a n d (14). T h e r e f o r e , w e u s e (17) or (18) to o b t a i n t h e b r e a k i n g w a v e h e i g h t Hb, (20) or (21) to o b t a i n t h e b r e a k i n g w a t e r d e p t h h b a n d (22) to e s t i m a t e t h e w a v e c r e s t h e i g h t qb (see F i g u r e 8):

70

Y a m a n i s h i et al.

0.10

2.0 EXP.data

0.08

o

/

EXP.data

I/2=0.691/1 o /

1.5

O

E

U)

0

O oO

E,E1.0

v

:r

~" 0.04

0.5

0.02 #"

I

I

0.05

0.10

0.15

H~(m) Figure 8. Relation between H b and rib

rib : 0 " 9 1 ( H b - 0 . 0 3 6 )

0

0.5

1.0

1.5

2.0

2.5

V1(m/s) Figure 9. Relation between incident velocity V~ and reflected velocity V~.

(22)

Extrapolated values of K 1 and y in (10) are required to determine the impact of breaking wave from (10). Figure 9 shows the relation between the incident velocity V1 and the reflected velocity V2 from experimental results. If V2 is assumed to be linear with V~, then K 1 becomes 0.69. Figure 10 shows the relation between the deepwater wave steepness H o / L o and the reflected angle y. The angle ~, was estimated from water motion monitored by the digital video camera. Equation (23) as the best-fit relation with the experimental results are shown in Figure 10. \-0.07

(23)

Figures 11(a) and (b) indicate typical records of the impact of breaking wave on the rigid slope. The impact pressure was measured with semiconductor type transducers. The sampling speed to record the impact was 200 Hz. As shown in Figure 11, the impact pressure shows spikes during a short time period. However, none of the measured maxima of the pressure Pm indicated the same value, but were scattered. We defined each maximum of pro under the initial and the steady state as a mean value over several waves. The point where plunging water penetrated the water surface at first is referred to as the first plunging point (P.P.1). Further, the point where splashed w a t e r e n t e r s t h r o u g h w a t e r surface again is called the second p l u n g i n g point

71

Mud scour under waves

so 9o

411

~

9

3o

~ t'-"

~ -o

20

__r

lO

(1.) nr"

0

EXP.data

(1)

(23)

0

i

i

i

l

0.02

0.04

0.06

0.08

0.1

HolLo

Figure 10. Relation between Ho/L o and y.

.-.

30

P.P.1

.

.

.

.

(a)

i

3O P.P.1

(b)

~o

.

I'

T=1.5sec,Hi=12.1cm (on the rigid slope) -30

. . . . 0

i 5

T=1.5sec, H i =12.1cm (on the rigid slope)

. . . .

-30 10

15

Time(sec)

0

5

10

15

Time(sec)

Figure 11. T e m p o r a l changes of b r e a k i n g wave p r e s s u r e for: (a) initial state and (b) steady state.

(P.P.2). Figure 12 shows the relation b e t w e e n the m a x i m u m values of the b r e a k i n g wave pressure (Pl,m) at the first plunging point and values of pg(V~2/2g)[ sin2(0+~)+K12sin~7 ] based on (10). The best-fit line from Figure 12 yields a = 0.79 (k = 2.5) in the initial s t a t e and 0.72 (k = 2.7) at s t e a d y state. For these results, Figure 13 can be used to e s t i m a t e the i m p a c t of b r e a k i n g wave. Figure 13 also shows v a r i a t i o n s of b r e a k i n g wave impact at the first p l u n g i n g point PVn with wave steepness H i l L i. As the wave steepness increases, the b r e a k i n g wave pressure decreases and the value of PI,~ at steady state is seen to be smaller t h a n in the initial state. Using this figure, b r e a k i n g wave i m p a c t for a given wave steepness can be e s t i m a t e d .

4. R H E O L O G I C A L C H A R A C T E R I S T I C S S h e a r s t r e n g t h is one of the typical rheological characteristics of m u d a c c u m u l a t e d on riverbanks. It depends on the state of a c c u m u l a t e d mud. K u s u d a et al. (1989) s h o w e d t h a t t h e critical s h e a r s t r e s s for erosion a n d t h e m a s s of

72

Yamanishi et al.

3.0

o 9 -

EXP.data(initial state) _ EXP.data(steady state/~ Initial state /..... .......... ................... Steady s t a t e ~ .............. a. ~ ....... '" 2 O0 ..''"""

\

Initial State

/-/i(cm) ............ Steady State A(g 2 . 0

a.

/-/i (cm) 12 10

1.0

r

8 6 4

..........................~

0

I

I

!

1

2

3

pg(V1212g)[sin

2(e+

~)+K12

4

sin 2 7 ]

0.0

9

I

I

.

0.02

0.04

I

0.06

,

I

0.08

.

0.1

Hi/Li

(kPa)

Figure 12. Relation between P and pg(V~2/2g) [sin2( 0+~ )+K12sin27]. 1,m

Figure 13. Variations of Pl,m with wave steepness H~/L i.

erodible m u d depend on the disturbance of mud. The yield stress was m e a s u r e d in the p r e s e n t study by a vane shear meter. After inserting a vane to a fixed depth, the m a x i m u m of working torque was m e a s u r e d by r o t a t i n g the m e t e r with a constant speed of 0.5 deg/s. In addition, in order to e x a m i n e the relation between the w a t e r content and the shear strength, n a t u r a l m u d s were collected in acrylic columns and the vertical profiles of w a t e r content were measured. Mud used was obtained in the u n d i s t u r b e d state from the Rokkaku River, where the m a x i m u m tidal range is about 5 m. The m e a n particle diameter and density were 6.0 m m and 2,540 kg/m 3, respectively. The vertical profiles of the w a t e r content W are shown in Figure 14. The w a t e r content decreases toward the bottom due to consolidation. Figure 15 indicates changes in shear s t r e n g t h x with W. The shear s t r e n g t h of both u n d i s t u r b e d and disturbed m u d decreases with increasing w a t e r content. According to these results, the relation between 9 and W is expressed as" U n d i s t u r b e d muds: %,1 = 1.42x 101~ -447

(24)

Disturbed muds: ~s,2 = 4.49 • 109W -447

(25)

where, xs,1 and ~s,2 are shear s t r e n g t h of u n d i s t u r b e d and disturbed muds, respectively. Figure l5 implies a decrease in shear s t r e n g t h due to remolding of mud. Here, a p a r a m e t e r r for gaging the remolding effect is selected to be

Mud

scour under

73

waves

w(%) 100 0

150

200

250

I

I

I

\

O

\ \ ~

\

4

E

\

\

.

"a_~ ' 3

0 ,.C

O

Column A 9 Column B A Column C 9 Column D

15-

. 9

9

": ....

~2 2

10-

", '",..

D. a

9 Non-disturbed data --(24) O Disturbed data

9

O""~......o"

1 -'~5 ............. .....O." ' " " ~ " - - o ~

0 100

i

I

120

i

I

i

140

I

160

i

I

180

i

200

w(%)

Figure 14. W a t e r content profiles of bottom m u d in the R o k k a k u River, Ariake Bay.

r = ~ x s -, 1~s,2

Figure 15. C h a n g e s in s h e a r s t r e n g t h % with w a t e r content W.

(26)

"~S ,2

S u b s t i t u t i n g (24) and (25) into (26), r is found to be equal to 2.16. Therefore, the shear s t r e n g t h of u n d i s t u r b e d m u d is about twice t h a t of d i s t u r b e d mud.

5. S C O U R E X P E R I M E N T S

E x p e r i m e n t s to e s t i m a t e the a m o u n t of s e d i m e n t scoured by b r e a k i n g waves were performed by using the described e x p e r i m e n t a l flume w i t h a slope. Along the slope, a t r e n c h was installed as the test section, in which u n d i s t u r b e d m u d was filled to a thickness of 0.1 m. In all experiments, the w a t e r d e p t h w a s 0.3 m. The e x p e r i m e n t a l conditions are s u m m a r i z e d in Table 1. The b r e a k i n g wave pressure at the first plunging point (Pl,m) was obtained from Figure 13 for steady state, because of s u s t a i n e d wave action on the slope. Figure 16(a) shows a profile of the m a x i m u m b r e a k i n g wave pressure on the slope (for an incident wave period T = l . 7 sec and height H i = l l . 5 cm). The results indicate t h a t the b r e a k i n g wave pressure Pm was high a r o u n d the plunging point, especially at the first plunging point (P.P.1), and was stronger t h a n at the second point (P.P.2). Figure 16(b) shows t e m p o r a l changes of scour u n d e r the same conditions as Figure 16(a). There is a correlation b e t w e e n the i m p a c t pressure and the scour of the inclined m u d bed. Mud scoured e n m a s s e and a c c u m u l a t e d as clasts at the foot of the slope. This phenomenon, which is also found in the field experiments u n d e r ship waves, depends on the m u d characteristics.

74

Y a m a n i s h i et al.

Table 1 Experimental conditions T(sec)

H/(cm)

9 (kPa)

1.0 1.5 2.0 1.5 1.5 1.5 1.5 1.3 1.5 1.7

9.2 9.6 6.6 12.3 11.3 9.9 8.4 11.2 11.0 11.5

2.30 1.39 2.30 1.39 1.39 2.46 2.30 4.20 1.96 2.54

Pl,m(kPa) 1.00 1.24 1.05 1.51 1.42 1.28 1.13 1.33 1.40 1.51

2

(a)

0 -50 10

E

-25 0 P.P.2

o ....,,

o

r .o

.10

m

-20 -30, -50

25 50 P..1 BP.

b

~ ~

0nin ~

'-",._, '....... "('"..,~ "",, .-'.':.~:'...~'.:~

"", .'~:--J='-'~% "'.

, 0

5rnn lOmin .......... 30 rain .......... .......

" ....

,"'"" 50

100

""

bottom

150

2()0

Distance from the shoreline (cm)

Figure 16. (a) Profiles of maximum impact pressure and (b) temporal changes of mud surface elevation.

Figure 17 shows relations between N, the number of breaking waves, and the amount of scoured mud per unit width W / B . Here, W is the mass of mud scoured by breaking waves, and B is the width of the experimental flume. W was evaluated by assuming t h a t mud with a constant water content within the scoured layer was eroded uniformly along the transverse direction of the experimental flume. As seen in Figure 17, W / B increases with N. Furthermore, it is believed t h a t mud was scoured step by step as the shear strength of mud decreased by successive attacks of breaking waves.

Mud scour under waves

75

150

O T=1.3s Hi = 11.2cm 9 T=l.5s Hi = 1 1 . 0 c m A T=1.7s Hi =11.5cm E 100

~

so

AO

AO 0

0

I

I

500

1000

1500

Accumulated number of acting waves N

Figure 17. Relation between n u m b e r N of b r e a k i n g waves acting on a m u d slope and mass of scoured m u d per unit width (WJB).

When the b r e a k i n g wave pressure exceeds the sum of the weight and internal shear s t r e n g t h of mud, scour occurs. Accordingly, W per wave per area A s is formulated as

mElZ)( Zl

(27)

where P m and ~ are the m a x i m u m b r e a k i n g wave pressure and m u d shear strength, respectively. Figure 18 shows the relation b e t w e e n (pm/Xs) and (Wg/As)/xs/N, w h e r e Pm is at steady state at the first plunging point (from Figure 16), W is the value for 500 waves (N=500), and A s is the area scoured. In addition, ~s was calculated from vane shear test results. Figure 18 suggests an increase in ( W g / A s ) / x s / N with (Pm/Xs), and the existence of a critical shear strength (p m/Xs)c. When (27) is applied to d a t a obtained in the laboratory tests, the coefficient m and (p m/Xs)c are found to be 1.4 and 0.37, respectively. Then, (26) is t r a n s f o r m e d as: Xs,1 = 0.32~s,1

"Cs'2- l + r

(28)

Comparison of "~s,2 obtained from (28) and the critical b r e a k i n g wave pressure, pmc(=O.37Zs:,), obtained from Figure 18 shows t h a t both are almost equal. Therefore, the m a x i m u m of b r e a k i n g wave pressure Pm to SCOUr an u n d i s t u r b e d m u d is seen not to necessarily exceed the shear s t r e n g t h of m u d xs,1, b u t xs.2. Now, considering

76

Yamanishi et al.

1.5 []

EXP.data in lab.

~

1.0

~

0.5

EXP.d ata in s i t u

[] /

6 ....% t-,,

2

v

0

0

I

I

0.5

1.0

,/q

0

1.5

P m / "r,s

Figure 18. Relation between (p J % ) and (W~g/A,)/%/N.

1

i

I

I

2

3

4

5

p ,,, I ,c,

Figure 19. Relation between (p J % ) and (W~g/AJ/%/N.

Pmc = ncXc

(29)

and substitution of (24) and (25) into (29) yields Pmc,1 = 51W n

(30)

Pmc,2 = 52W n

(31)

where, n c is determined from experimental data, 51=5.11x109, 5~=1.62x109 and n=4.47. From (30) and (31), it is possible to predict the value ofpm c for a given state of mud and w a t e r content. For an estimation of the mass of sediment scoured in the field, in situ experiments were carried out at the riverbank 6.8 km upstream from the river mouth, where the m a x i m u m tidal range was 5 m. Ship waves were produced by a fishing boat, which moved about 30 times per set of experiments. During these experiments, wave height was measured and the surf zone was recorded by a video camera. In the same m a n n e r as above, results of the field experiments were derived. Figure 19 is a comparison of (32) with results of the field experiments. Equation (32) was obtained from the laboratory experiments as follows:

( W sxsgN/ A s ) = l ' 4 [ ( P-~s m ) - 0"371

(32)

The field data are not sufficient; however, they are in accordance with the results from (32). Consequently, the results obtained both in the laboratory and in the

Mud scour under waves

77

field indicate the predictability of sediment mass scoured by breaking waves, especially by ship waves.

6. C O N C L U S I O N S We investigated the effects of the impact of breaking waves on mud scour. The conclusions of this study are as follows: (1) Breaking wave pressure acting on a slope can be formulated by the equation of conservation of momentum considering reflection. From pressure measurements, a relation between Pl.m and wave steepness Hi/L o was derived. (2) As wave steepness increases, the breaking wave pressure decreases, and Pl,m at steady state is smaller t h a n in the initial state. (3) The shear strength of mud, measured by a van~ shear meter, is a function of the water content. Due to the remolding effect, the shear strength of undisturbed m u d was about twice the shear strength for disturbed mud. (4) The mass of sediment scoured by breaking waves, W, per wave and per unit area is given by (Wg/As)/xs/N=m[(p/xs)-(p/Xs)c]. From the experiments, m=l.4 and (p ~/~,)c =0.37 were obtained.

7. A C K N O W L E D G M E N T

This study was initiated as a joint work by the Takeo office of the Ministry of Construction, and was supported in part by a G r a n t in Aid from the F u n d a m e n t a l Scientific Research and Scholarship of the Wesco Foundation.

REFERENCES

Futawatari, T., and Kusuda, T., 1993. Modeling of suspended sediment transport in a tidal river. In: Nearshore and Estuarine Cohesive Sediment Transport, A. J. Mehta ed., American Geophysical Union, "/r DC, 504-519. Kusuda, T., 1994. Application of Self-Purification's Functions, Gihodo Publications, Tokyo, J a p a n , 166p. (in Japanese). Kusuda, T., Yamanishi, H., Yoshimi, H., and F u t a w a t a r i , T., 1989. Experimental study on erosion of disturbed and non-disturbed mud. Proceeding, Coastal Engineering Conference, J a p a n e s e Society of Civil Engineers, Tokyo, Japan, 36, 314-318 (in Japanese). Le M~haut~, B., and Koh, R.C.Y., 1967. On the breaking waves arriving at an angle to the shore. Journal of Hydraulic Research, 5(1), 67-88. Sunamura, T., 1983. Determination of breaker height and depth in the field. Annual Report No. 8, Institute of Geoscience, University of Tsukuba, Tsukuba, Japan, 53-54.