Analytical determination of nearshore wave height variation due to refraction shoaling and friction

Coastal Engineering, 7 (1983) 233--251

233

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

ANALYTICAL DETERMINATION OF NEARSHORE WAVE HEIGHT VARIATION DUE TO REFRACTION SHOALING AND FRICTION

PETER NIELSEN 1

Coastal Studies Unit, Department of Geography, University of Sydney, 2006 N.S. W. (Australia) (Received July 7, 1982; revised and accepted January 24, 1983)

ABSTRACT Nielsen, P., 1983. Analytical determination of nearshore wave height variation due to refraction shoaling and friction. Coastal Eng., 7: 233--251. Explicit wave formulae derived from the dispersion relation for linear waves are used to find an analytical solution to the problem of wave height variation on a simple topography; i,e. topographies with incrementally constant slope and straight parallel contours. The solution accounts for shoaling, refraction and frictional dissipation and will be sufficiently accurate for practical purposes considering the simplifying assumptions that are necessary for treatment of this problem by any method. The solution is simple enough to be handled on a personal calculator and has the advantage over numerical solutions that it can be solved for other parameters, for example to give friction factors from observed wave height data. The last chapter contains updated formulae for wave friction factors over movable beds.

INTRODUCTION

Waves that propagate over a sloping bed will change their height and direction. If there is no loss of energy, and the bed contours are straight and parallel, the governing equation is: d ds (El cos 5)

0

(1)

where Ef is the energy flux per unit length of wave crest, a is the angle between wave crests and bed contours, and s is the distance travelled along a wave orthogonal. In most cases there will be considerable loss of energy due to either opposing winds, bed friction, percolation and internal (viscous) friction. The dominating mode of dissipation (at least in the absence of I Present address: Coastal and Oceanographic Engineering, College of Engineering, University of Florida, Gainesville,F L 32611 (U.S.A.)

0378-3839/83/$03.00

© 1983 Elsevier Science Publishers B.V.

234

strong opposing winds) will normally be the bed friction. See e.g. Svendsen and Jonsson (1976). If we include bed friction, eq. 1 becomes: d (2)

ds (Ef cos e) = -TUb

where r is the instantaneous bed shear stress and Ub the instantaneous nearbed velocity. We define the energy dissipation factor fe by:

ub

2 =

Ofe

(3)

Ub,rnax

where p is the density of water. If r ( t ) varies as UbIUbl and ub(t) is a simple harmonic, this definition is equivalent to Jonsson's definition of the wave friction factor: rmax = lhP fw u2b , m a x

(4)

with fe = fw; in reality fw and fe are not quite equal as discussed in the section "Estimation o f friction factors". Introducing eq. 3 and the linear wave theory expressions for Ef and Ub,max into eq. 2, we get: d

d--ss (~ pgH2cg cos ~) =

_ 2 ( 7 [ H ) 37[ Pfe T sinh kh

3 (5)

where H is the wave height, Cg is the group velocity, and T is the wave period. After differentiation and some algebra, eq. 5 takes the form

ds

2Cg ds

2cos a

ds

-JH

3gT 3

CgCOS~ sinh3kh

Here we introduce the non-dimensional independent variable: 47r 2

koh = ~ h gT 2

(7)

after which eq. 6 can be written:

(_~)'- (in,vf~gCOSa),l=

kofe

dh

37[ -

Co

(8)

cg sinh 3 kh cos 2 a

where the primes denote differentiation with respect to ko h. Equation 8 is linear with respect to 1/H and can readily be integrated between the depths hi and h2 (see e.g. Kamke, 1959). For convenience we introduce: =

kof

3~ dh dx

(9)

235 and get:

t-~g cosal

,

Cg cos~

egsinh 3 k h

cos2~

and assuming constant/3, that is constant beach slope: H2=H'

k0~ I =

k .h 1

lt /Ccgg,' cosa['/cosa2 / [1 + fill,

V cg'co___sO,'CoI].]

C°l's (COS a) -2"s d(k0h) Cg~.s sinh 3 k h

(11)

(12)

The integral I cannot be evaluated analytically unless we write the integrand as an explicit function of the independent variable koh. We will use the technique introduced by Nielsen (1982) to do this and obtain an approximate analytical expression for eq. 12. EXPLICIT FORMULAE FOR LINEAR WAVEPROPERTIES The main difficulty in deriving explicit formulae for kh, sinh kh and other linear theory functions in terms of koh or h/Lo is that they are not analytical functions at h = 0. Therefore, a straight Taylor expansion approach is not feasible. It was shown by Nielsen (1982} how this difficulty can be evaded by extracting the singular part, which is the shallow-water limit (found for koh --* 0) and deriving a Taylor expansion for the rest. This is done for k h by inserting the assumed form: k h = x / ' - ~ (1 + oe koh +/3(k0h) 2 + . . . )

(13)

into the dispersion relation: koh = k h tanh k h

(14)

with: tanh x = x - ~ 1 X3 + ~ Xs - . . .

(15)

and requiring identity for all powers of koh. The result is: kh = v ~ h ' ( 1

1 11 (koh)2 + ... ) + ~koh + ~o

(16)

By inserting this into the expressions in terms of k h (see e.g. Svendsen and Jonsson, 1976), we can find similar expansions for other properties like: tanh k h = x / k - ~ (1 - ~koh - ~-~ (koh) 2 + . . . )

(17)

s i n h k h = x / ~ ( l + ~ k o h1

(18)

+ ~~1 ( k o h ) 2 +

..)

236

cg/Co

=

k~v'~oh(1 - ~ koh + ~ (koh) 2 + . . . )

Ks

= (4k0h) - ' / ' (1 +L4 koh + ~

(koh) 2 + "'" )

(19) (20)

These formulae are all within one percent of absolute error for h/Lo < 0.2 (hob < 1.26) and the first-order versions, neglecting terms of the order (koh) 2, are within one percent for h/Lo < 0.05 (k0h < 0.31), see Fig. 1. All the formulae are exact for h -+ 0. I 0 "2 .

6

,

l

ERROR

I0-3

o

i

l

1

i

//

/

~

i j

"

kh(I)

;

,

kh(Z)

,

Ks(i)/'i

0.01

1

' s

0"02

0-04 0"06

0"1

0"2

0'4

0"6

Fig. 1. A b s o l u t e e r r o r f o r t h e f o r m u l a e 1 6 - - 2 0 . T h e n u m b e r s in b r a c k e t s d e n o t e o r d e r o f

the highest-order term. W h e n the bottom contours are straight and parallel w e can use Snell's law: sin ~2

c2

- -sin (~, c,

(21)

to determine the local angles between wave crests and contours and the refraction coefficient for depths h, and h= becomes:

(22)

KR, = ~ Shell's law (eq. 21) is by trigonometrical identities equivalent to: cos 2 a2 = 1

tan2 ~, l+tan2al

(c212 ~"~,/

(23)

237

by using C/Co = tanh k h and the first-order version of eq. 17 we get: cos2a2

=

1-

tan2at

koh 2 ( 1 - ~ koh2) 2

l+tan2al

k o h l ( 1 - ~ i kohl)2~

(24)

This formula is valid when both koh2 and kohl are within the domain of eq. 17, koh <~ 1.26. When the starting depth, hi is "deep water" (kohl > lr) the relevant formula is: tan 2 a o l+tan2ao koh2(1-~koh2) 2

cos2a2 = 1

(25)

ANALYTICAL EVALUATION OF THE INTEGRAL eq. 12

We can n o w proceed towards an analytical form of:

k•hi

I =

c~.S (12)

l.s sinh3kh (cos a) -2"s d(koh)

kohl Cg

Using eqs. 18 and 19 we get: c~:s

CgLs

sinh a

kh

= ( k o h ) -2"2s

[1

i

- ~ koh - ~

(koh) ~]

(26)

and combining eq. 24 with

(1- x) -l'2s ~ 1 + 0.7x + 4x 2

(27)

we get: (cosa2)-2"s

=

koh+p(koh) 2

I+~

(28)

where 0.7 tan2 a~

=

(1 + tan2 a l ) k o h l (1

-

(29)

~kohi)

and P

4 tan 4 a 1 (1 +tan2al)2(kohl) 2 - I ~

(30)

We insert eqs. 28 and 26 into eq. 12 and get: I=~

a

(kohi) -i'2s [ 1 - ( h ~

+ (~p .

3~o

~)

hi

f . 2 s j + ( 1 - 4 5 ) ( k o h i ) -0"2s [(-~2)

(kohl) °'Ts

rL~h 2 / , - 1 ]

o.2s

-

1~

J

(31)

238

(

~.s

2"0

L+O 7x ~

1.5

/

t-xl~ /

TAYLOREXPANSION

t.O 0I.,

[

z

01,2 0"3 0"4 0.5 0"6

x

--~

Fig. 2. The least-squares fit (eq. 27 ) is a better "all round" approximation than the Taylor expansion (eq. 32).

The formula 27 is a least-squares fit for the interval 0 < x < 0.5 corresponding to angles less than 45 ° (see eq. 24). We use it instead of a Taylor expansion because the latter:

(l-x)

-1"2s ~ l + ~ xs +

4s ~ .2

(32)

is rather inaccurate for x > 0.3. See Fig. 2. EXAMPLES AND ERROR ANALYSIS

In the case of no refraction (a = 0) we can compare the approximate analytical solution to the numerical results of Bretschneider and Reid {1954) and of Putnam and Johnson (1949). For ~ = 0 eq. 31 becomes:

+o

+' -

and Fig. 3 shows the solution corresponding to eqs. 11 and 34 together with discrete values obtained with the two different numerical models. We see that the analytical approximation is as close to the numerical models as they are to each other. For perpendicular wave incidence (~ = 0) it is often unnecessary to include all the terms in eqs. 34, 19 and 20. In Fig. 4 we have replotted the solution from Fig. 3 together with the two-term version and the shallowwater version:

H2=

4re 1 + 15~

H+

(dh/dx}ht

~" hi .1.2s L(-~2) -1]

(35)

239

160

WAVE ~HT o Putnam and Johnson

155 150

o

o~~~x~

145

Equations

If and 34

1.40

0!05

01.10

o'.15

hlL o

0'20m"-

Fig. 3. Comparison o f the n o - r e f r a c t i o n - s o l u t i o n (eqs. 11 + 3 4 ) t o t h e numerical results o f Bretschneider and Reid ( 1 9 5 4 ) and o f P u t n a m and J o h n s o n ( 1 9 4 9 ) . T = 12s, h ~ / L o = 0.2, H x = 1 , 3 9 7 m , fe = 0.02, d h / d x ffi - 1 / 3 0 0 .

H

[m]

MS

o&

o~2 o~3 o.~A o.~ o.I~ o-~7 oh8 o.o9 o:,o"=

Fig. 4. T h e n o - r e f r a c t i o n - s o l u t i o n (eqs. 1 2 + 3 4 ) and the t w o l o w e r order a p p r o x i m a t i o n s . Because the f o r m u l a e are identical at h ~ 0 w e have started at h / L o = 0 . 0 1 and w o r k e d outwards.

240

We see that two terms is enough for h/Lo < 0.05 and the shallow-water solution is adequate for h/Lo < 0.015. To illustrate the significance of the refraction we have used the same example as in Fig. 3 except that the angle of wave incidence is 45 degrees at h / L o = 0.2 instead of zero. The result is plotted in Fig. 5. The dashed line shows the wave height variation due to shoaling and refraction (Snell's law). By comparison with Fig. 3 we see that the frictional dissipation from one depth to another is enhanced by the obliqueness. That is simply because the distance between two given depths, measured along wave, orthogonals is longer when the waves come in at an angle.

145H(m) 14

\

~\j./HLKSlKRI

1.3

/

hI L0 o'-o2 0:04 o.'o~ o.'os o'.,o o~q2 o'.14 o'.,6 o.'Ls o-'2o---

Fig. 5. Wave h e i g h t variation due t o shoaling r e f r a c t i o n a n d friction. T = 12s, h ~ / L o = 0.2, H~ ffi 1 . 3 9 7 m , fe = 0.02, d h / d x = - 1 / 3 0 0 , a t ffi 45 °.

When refraction is involved, it is n o t generally possible to omit the higher order terms in eq. 31 because the coefficients will increase for decreasing hi, see eqs. 29 and 30, Also the error corresponding to the use of the full solution (eqs. 11 + 31) will increase for decreasing hi. This effect is illustrated in Fig. 6 where the relative error of the approximated integrand: l

61

( k o h ) -2"2s + (~ - ¼)(k0h) -1"2s + (p - "~5 - ~ )

"~ (cos

(koh) -°'2s ~

co

,~)-2.~

l.s sinh 3 k h Cg

(36)

is plotted for different starting depths and different starting angles a l. For h/Lo < 0.15 and al < 45 degrees the error of the integrand is less than 6% and so the cumulative error on the integral (eq. 31) will be only a couple of percent whenever the friction is important. The resulting relative error on the wave height will normally be much less than this because the second term within the brackets of eq. 11 is usually much smaller than unity.

241

RELATIVE ERROR

010

0"08 h%./Lo, 0-03 ~ . = 30 °

0'06

,,,, , ~ - -

,,,,

h,,./Lo = 0.2 c < 1 = 45 =

0'04 ~

--" -~-"'~-.. j

..~

-"~'<

'%

~

h t / L o = 0.2

~

~,~ =3o"

0.02

0

0.006

o.o,

.o

~ , ~ . . . . h t / L O = 0-03

-0"02

. o'2o

•

'

~

I

-0'04

-0"06

-008

Fig. 6. Relative error o f the integrand (eq. 36) for different starting depths h I and starting angles az. For h/Lo < 0.15 and al < 45 degrees the relative error is less than 6% which means that the errors on predicted wave heights will be insignificant for practical purposes.

F o r large angles of incidence ( > 4 5 degrees) one should not use eq. 31 uncritically, b u t preferably incorporate a better approximation than eq. 27 for the relevant interval [ 0; a i ]. REAL WAVES AND THE MODEL The major limitation of the analytical model presented above is n o t the approximations that have been used in order to get an explicit solution; but the fact that it is based on linear wave theory, because it is well known (see e.g. Svendsen and Buhr Hansen, 1976) that, for example, the shoaling coefficient obtained from linear wave theory is not very accurate. Furthermore, the presence of winds and currents can have damping effects of the same order of magnitude as friction. The assumption of straight, parallel bed contours is n o t very restrictive since most natural topographies can be divided into sections that will fullfil it reasonably well. The time-averaged dissipation rate rUb is proportional to the meanabsolute-cube of the water velocity as stated by eq. 3:

• ub

=

2

3

-~-p fe u b , ~

(3)

242

which assumes that the velocity varies as a simple harmonic so that u~ = 3 x. Particle velocities near the bed under natural IA Umax~ and lu-u-u-u~l= (4/3U)Uma nearshore waves are n o t simple harmonics, and in fact the ratio: q

''5

(37)

is often significantly different from the sine wave value of 8 x / ~ 3 u ~ 1.20. The typical range seems to be: 1.3 ~< q < 2.7

(38)

for nearshore waves (h/Lo < 0.1) with L0 determined from the spectral peak period. This has important implications for the frictional dissipation since the dissipation is proportional to lu-'~-J while the energy flux is proportional to u~--", and it can quite easily be taken into account. If we replace the "sinus value" 8 x / ~ 3 , in eqs. 11 and 9 with q we get: Cg~ cos ~1 H: = H~ ~ g ~ cos a2

g~ COS ~1

1+ dh

(39)

Co

or:

fe -

g~cos~i

ko q H I I

~

cg~ cos~2

that is, re-values derived from measured wave heights are inversely proportional to q. For irregular waves one should use the root-mean-square height: HRM S = / 8 ~ "~

(41)

in connection with eq. 40; ~ is the instantaneous water surface elevation above the mean water level. ESTIMATION

OF FRICTION

FACTORS

In order to use eq. 11 for predictive purposes, we must be able to estimate the friction factor fw or re)- In the following we use Jonsson's {1966) definition for the wave friction factor fw: _- 1 pfwu

7ma x

,m

(4)

where r is the bed shear stress, and we define the energy dissipation factor fe by:

TUb

=

2

3

"~"~PfeUb,ma x

(31

243

where r u(~ is the time-averaged energy dissipation. Those two definitions lead to identical values for fe and fw if the near-bed velocity Ub is simple harmonic, e.g.: U b ( t ) = Ub,ma x sin cot

(42)

and r (t) varies as:

Ti t)

= t"max Isin cotl sin cot

(43)

Lofquist {1980) has measured the variation of r(t) over natural wave ripples and found that it can be quite a lot different from fsin cotJ sin cot, see Fig. 7. Lofquist also derived values for fw and fe in accordance with definitions eqs. 3 and 4. Figure 8 shows that fe and fw are generally nearly equal. The two extreme deviations correspond to tests where r (t) has a narrow peak. The friction factor is a function of the Reynolds n u m b e r aUb,max/P and of the roughness to semi excursion ratio r]a (Jonsson, 1966). When the flow is rough turbulent, as is normally the case under field conditions, fw (and fe) depend only on a/r.

O5 fw = 2 ~mox

Ub,rnox

x

Q,

x/

04 03

~, : 31.8cm, a * 2 4 c m

T

0.5 o

-

~

'

~

s-~7,

~ 2T(t) ~ u2 b.moz

~

~ ~

__I (gt :

fr -2

~

~

-

'

4"55s

02

3"71 s

~

3'13s

o

X

04 /

1T ~-

~

3fr

T

00

~

"Z-Ub ,e - 2_.&£,,3

" I 0 I

J 02

311" ~b,mox I -_. 03

Fig. 7. Variation o f r ( t ) for different wave periods (T), f r o m L o f q u i s t (1980). The corres p o n d i n g free f l o w velocity varies as sin w t. Fig. 8. Comparison o f measured values o f fw and fe f r o m Lofquist (1980). We see t h a t fw and fe are generally nearly equal. The few e x t r e m e deviations s t e m f r o m experiments where r(t) has a p r o n o u n c e d narrow peak.

244

Several authors (Jonsson, 1966; Kajiura, 1968; Kamphuis, 1975; Grant and Madsen, 1982) have suggested different forms of the relation between fw (fe) and a/r derived in analogy with steady boundary layer flow. According to Grant and Madsen (1982) this analogy seems reasonably well satisfied when air is large (a/r >1 28) as in Jonsson and Carlsen's experiments; however, the variation of r (t) as measured by Lofquist shows that the structure of the boundary layer over large natural ripples may be very different from that of a steady boundary layer. Because of these uncertainties it seems n o t worth while to compare the theoretical merits of the different formulae in great detail. Instead, we will use the most handy one which turns o u t to be Jonsson's in the explicit form given by Swart (1974): fe = e x p [ 5 . 2 1 3 ( r / a ) °''94 - 5.977]

for r/a < 0.63

(44)

fe = 0.30

for r/a >~ 0.63

(45)

Figure 9 shows the variation of fe with r/a. 0 63 f, ,f,

/

0'5

r

/ f =exp(5213(g)- 5 9 7 7 ) ~ .

.

.

.

.

.

.

.

.

.

.

~

..........

024

---

L

i

, , , ,,,~L

L

o l J i itlll

0.02

o.o~

o~

0.5

o.L

030 ~

" - - ~ , o . 7 ~

02 O.iO

002

.

o.194

~o

Fig. 9. Variation of fe with r/a following eqs. 44 + 45 or eq. 52. The upper limit of 0.30 is rather arbitrary (Grant and Madsen argue that it should be only 0.23}, and it is traditionally based on Bagnold's observation: that fe did n o t increase above 0.24 over his artificial ripples in sediment-free water. However, the more recent measurements by Carstens et al. (1969) and Lofquist (1980} using natural sand beds indicate that fe can be 0.30 and even larger, and it is n o t unlikely that the presence of loose sand may lead to friction factors larger than the maximum values for fixed beds. This is supported by the observation of Nakat.o et al. (1977) that the presence of loose sand enhanced the turbulence intensity by a factor of 2 over ripples, all other things being equal. The influence of moving sand on the roughness is shown in Fig. 10 where we have plotted measured fw-Values from Kamphuis (1975) and re-values from Carstens et al. (1969). Kamphuis measured the shear force on a fiat bed

245 1

I

i

T I I I I

I

0'2

I

i

I

I

I I 11

I

I

I

I

I

I I

Rough turbulent flow •

F

OI

02

.01 F Corstens ef al Loose sand

,o

~o ',;o'

I

J

I

',;o

0

2;o

I

J

400

1

I

J

',b'oc

2000 '

I

i

i

.¢ooo

I

I

I

10O00

Fig. 10. Friction factors measured over flat beds o f fixed and loose sand. We see that the friction is nearly an order o f magnitude stronger over loose a n d than over fixed sand for the same values of a/d.

of fixed sand grains and Carstens et al. measured the energy dissipation over a flat bed of loose sand. We see that the friction is nearly one order of magnitude stronger over the loose sand for the same values of d/a. Kamphuis' fixed-bed-measurements are well predicted by eq. 44 with: r = 2.5 d

(46)

For the loose-sand data the corresponding value is r -~ 120d. When the bed is covered by ripples or other bed forms, these will add to the friction. Following Grant and Madsen (1982) we will assume that the combined roughness from bed forms and moving sand can be described by:

r = (s + CM)F(O')d + K ~ 7"1

(47)

where ~ is the ripple height, X the ripple length, 0' is the skin friction Shields parameter, s is the relative density of the sand, and CM is the added mass coefficient for sand grains (-~ 0.5}. Grant and Madsen derive the following formula for F(O '):

f(O') = 160 [ v r ~ - 0 . 7 v ~ c l 2

(48)

in analogy with Owen (1964) who studied sand grain saltation in air. The weak point in this analogy is that the vertical length scale of the sediment motion is derived from the idea that a sand grain which hits the bed with horizontal velocity vi m a y bounce off vertically and reach a height of the order of magnitude v~/2g. This is probably true in air where the relative density (sand to air) is of the order of 2000 and inertia and gravity may dominate over the drag force, but in water, where the relative density is a thousand times smaller, a sand particle will only move a distance comparable

246 to its own diameter before an initial upward velocity is annihilated by the drag force. Furthermore, sand grains do not bounce off in water because of the water's cushioning effect. However, the general form of eq. 47 is physically sound, so in the following we will derive an empirical formula based on eqs. 44 and 47. The skin friction Shields parameter e' is calculated from: 1 e'

U2

"~ f w b,max (s-1)gd

-

(49)

where fw is derived from eq. 44 with r = 2.5d. First we look for the value of the constant K, which can be determined from the data by the approximation r = K~72/X. This approximation to eq. 47 is valid for fixed bed and for small values of 8 '. ]

500

i

i

I

i ] [ I

i

,

i

,

,

, T

ro.s

200

I00

50 B

B

20

~GNOLO

c

B

c B

I0

IONSSON ~---c

~

I c

J_02s_s0. ~[ L

C

~,

~C C ¢~ CBCB CC B c c BB

B

~

L

S

~L

B

H

L L

5

i e'

'

'o:o;"o

~,

L

i

02

,

i

05

i

,

J~|

,.o

Fig. 11. Observed values of r~./n = as function of the skin friction Shields parameter. The broken lines labelled "BAGNOLD", "JONSSON I" and "JONSSON II" correspond to experiments with solid, artificial ripples. Legend as Fig. 12. Figure 11 shows observed values of r/(~?2/X) versus 0 '. The line labelled "Bagnold" is derived from Bagnold's result (Bagnold, 1946): fB = 0.072 (X/a) °:Ts

(50) 1

By definition fB is equal to i r e and the ripple steepness 7/X was 0.149 in Bagnold's experiments, so eq. 50 can be transformed into: fe = 3.75 072/Xa) °'75

(51)

If we force eq. 51 to coincide with: fe = 0.392 ( r / a ) °'Ts

(52)

247

which is a good approximation to eq. 44 for 0.1 < r/a < 0.5, we find that Bagnold's result corresponds to: r

=

20.3~

~

(53)

The lines labelled "Jonsson I" and "Jonsson II" refer to the two experiments from Jonsson and Carlsen (1976) where rXh? 2 was found to be 10.9 and 7.4. As 0' approaches unity, rX/~?2 tends to infinity because ~/X goes to zero (see e.g. Nielsen, 1981). The optimal value for K turns out to be approximately 8, and using this, we may find the moving-sand-contribution as the residual rd = rOBS - 8~ 2/k. The relation between rd and 0' is sought by plotting rd/d versus 8 '; see Fig. 12. Again the scatter is very considerable but there seem to be no systematic deviations between different grain sizes. The flat-bed data from Carstens et al. (1969) conform well with the rest although they correspond to an unstable situation, from which ripples would have developed sooner or later. IO00

soo

. . . .

~'"l

robs-8tla/'~ d

~ - 7

200 IO0

'-

.

20 c

I0 5

2

| / c I /

CARSTENS ET AL A RIPPLED d=Ol9Omm

~ / |

S R IPPLED c RIPPLED

LL

| I I 1 J ,.,£ 0"02 O"0 5

d =0.297rnm d = 01585 mrll

F FLATBED d=O1297mrn LL OI

LOFOUIST L RIPPLED d = O 5 5 m m I-) I I I I ' ' ' ' l ' t I .... 012 0" 5 1'0 2 5

e'

I0

Fig. 12. For the presently available data the roughness contribution from the moving sand can be approximated by r d = 1 9 0 ~ d .

We see that the formula: (s+CM)F((~') = 190x/O' - 0.05

(54)

gives a fair representation of the trend while: (S + C M ) F ( 0 ') = 504[V/~ -- 0.16] ~

(55)

which is Grant and Madsen's formula with CM = 0.5, s = 2.65 and 0c - 0.05 seems t o o steep.

248 We will therefore recommend the following formula for the combined roughness: r = 8n2/X + 1 9 0 x / 8 ' - 0.05"d

(56)

for quartz sand with s ~ 2.65 and CM ~ 0.5. Figure 13 shows a comparison between r/a derived from measured fe values by eq. 44, and values predicted by eq, 56. The agreement between measured and predicted values is acceptable and the agreement between the d [~ corresponding [e values will be closer because ~1e d(r/a)

is considerably

smaller than unity over most o f the practical range. i

i

o.o

t

I

L

i

i

I-l

i

i

°

i

i

i

i

J

/

OBS

CC C L

060

q:c ,5%_ LL B C C C ~ , f LL

L

-Cj C L

0.40

,~/A B

L

BB

0'20

0'10

B

F~

OOB i

I

2 B

~ i

I

0-o8 0-Io

F} I

020

, j

L

0.40

J

I

l

0.60 0.8o

Fig. 13. Comparison between predicted and measured roughness to semi-excursion ratios,

The only systematic trend seems to be that Carstens et al.'s measurements over non-equilibrium flat beds give slightly larger re-values than measurements over stabilized rippled beds at the same O' values. This may mean that non-equilibrium beds are rougher because they are more active at the same values of O'. Savage (1953) observed similarly that the friction factors were larger when the ripple pattern was not in equilibrium with the wave conditions. Under irregular field waves, the bedforms can never be in equilibrium with the instantaneous wave conditions, so we may expect slightly larger dissipation factors in the field than in laboratory experiments with regular waves. Figure 14 shows the roughness derived from Carstens et al.'s "fiat-bed data" versus the skin friction Shields parameter. We see that the best fit for these data alone is: (57) which is approximately t w e n t y percent larger than the overall best fit given by eq. 54.

(s + C M ) F ( O ' )

= 280x/0' - 0.05

249 i

r d

200

1

,

,

,

23o./6'--0--5~

,

, , ]

~ F

F I00 F

60

F

FF F F~F

40 F 20

f

I0 0.02

O'(r = 2.5d)

'

0"04 0'06 0"10

0"20

' ..... 0.40 0.60

J 1.00

Fig. 14. Roughness values derived by eq. 44 from Carstens et al.'s "flat-bed data". Note that only four highest 0'-values correspond to fully developed rough turbulent flow according to the criteria given by Jonsson (1980).

In conclusion we recommend determination of fe (fw) for rough turbulent flow over natural sand beds by the following formulae: fe = e x p [ 5 . 2 1 3

(r/a)

0"194 -

5.977]

fe = 0 . 3 0

for r/a < 0.63

(44)

for r/a ~ 0.63

(45)

with: r = 8~2/~ + 1 9 0 x / 0 ' - 0.05"d

(56)

For prediction of the ripple geometry under field conditions, one may use:

~/X = 0.342 - 0 . 3 4 ~

(58)

)-1.85

U2

~/a = 21

for 0 ' < 1

b,max (s-1)gd

(59)

as suggested by Nielsen (1981). For 0 '>1 the ripple height may be taken as zero. LIST OF SYMBOLS a c cg Ef fe

fw

m m/s m/s W/m --

-

Water semi excursion Wave celerity Group velocity Energy f l u x Energy dissipation factor Friction factor

250 g

mls 2

H h k

m m m -I

KR(h ) KR1

--

Ks(h )

gs~ L

-

-

m

q r

m

T

s

Ub

m/s

m -~ 5 kt N/m z m

~"

m

m Subscript " 0 " Subscript "1" Subscript " 2 "

Acceleration of gravity Wave height Water depth Wave number 2 n I L Refraction coefficient, x/cos~0/cos~ Refraction coefficient, x/cosa~/cos~ Shoaling coefficient K s (h)IK s (h,) Wave length See eq. 37 Hydraulic roughness Wave period Water particle velocity Angle between wave crests and bottom contours. See eq. 9 See eq. 29 See eq. 30 Bed shear stress Ripple height Instantaneous water surface elevation Ripple length Skin friction Shields parameter, eq. 49 means deep water value refers to starting depth h~ refers to final depth h:

ACKNOWLEDGEMENT

This study has been supported by the U.S. Office of Naval Research, Coastal Science Program, Task NR 388-157, Grant N00014-80-G-0001. REFERENCES Bagnold, R.A., 1946. Motion of waves in shallow water: interaction between waves and sand bottoms. Proc. R. Soc. London, Ser. A, 187: 1--15. Bretschneider, C.L. and Reid, R.O., 1954. Modification of wave height due to bottom friction, percolation, andrefraction. Beach Erosion Board, Tech. M e m o No. 45. Carstens, M.R., Neilson, F.M. and Altinbilek, H.D., 1969. Bed forms generated in the laboratory under oscillatory flow. Coastal Eng. Res. Centre, Tech. M e m o 28, Washington D.C. Grant, W.D. and Madsen, O.S., 1982. Movable bed roughness in unsteady oscillatory flow. J. Geophys. Res., 87 : 469--481. Jonsson, I.G., 1966. Wave boundary layers and friction factors. Proc. Int. Conf. Coastal Eng., Tokyo, Chapter 10. Jonsson, I.G., 1980. A n e w approach to oscillatory, rough turbulent boundary layers. Ocean Eng., 7 : 109--152. Jonsson, I.G. and Carlsen, N.A., 1976. Experimental and theoretical investigations in an "oscillatory rough turbulent Boundary Layer. J. Hydraul. Res., 14(1): 45--60. Kajiura, K., 1968. A model of the bottom boundary layer in water waves. Bull. Earthquake Res. Inst. Univ. Tokyo, 46(5): 75--123. Kamke, E., 1959. Differentialgleichungen L6sungs-methoden und LiSsungen. Chelsea Publishing Company, N e w York, 3rd ed.

251 Kamphuis, J.W., 1975. Friction factors under oscillatory waves. Proc. ASCE, 102(WW2): 135--144. Lofquist, K.E.B., 1980. Measurements of oscillatory drag on sand ripples. Proc. 17th Coastal Eng. Conf. Sydney, pp. 3087--3106. I~onguet-Higgins, M.S., 1981. Oscillating flow over steep sand ripples. J. Fluid Mech., 107: 1--35. Nakato, T., Locher, F.A., Glover, J.R. and Kennedy, J.F., 1977. Wave entrainment of sediment from rippled beds. Proc. ASCE, 103(WWl): 83--100. Nielsen, P., 1980. Sand bed friction factors for oscillatory flows (Discussion with Philip Vitale). Proc. ASCE, 106(WW4): 498--499. Nielsen, P., 1981. Dynamics and geometry of wave generated ripples. J. Geophys. Res., 86(C7): 6467--6472. Nielsen, P., 1982. Explicit formulae for practical wave calculations. Coastal Eng., Vol. 6: 389--398. Owen, P.R., 1964. Saltation of Uniform Grains in Air. J. Fluid Mech., 20(2): 225. Putnam, J.A. and Johnson, J.W., 1949. The dissipation of wave energy by bottom friction. Trans. Am. Geophys. Union, 30: 349--356. Savage, P.R., 1953. Laboratory study of wave energy loss by bottom friction and percolation. Beach Erosion Board, Tech. Memo 31. Svendsen, I.A. and Jonsson, I.G., 1976. Hydrodynamics of Coastal Regions. Den Private Ingenioerfond, Lyngby, Denmark. Svendsen, I.A. and Buhr Hansen, J., 1976. Deformation up to breaking of periodic waves on a beach. Proc. 15th Int. Conf. Coastal Eng., Hawaii, pp. 477--495. Swart, D.H., 1974. Offshore sediment transport and equilibrium beach profiles. Delft Hydr. Lab., Publication No. 131. Tunsdall, E.B. and Inman, D.L., 1975. Vortex generation by oscillatory flow over rippled surfaces. J. Geophys. Res., 80: 3475--3485. Vitale, P., 1979. Sand bed friction factors for oscillatory flows. Proc. ASCE 106(WW3): 229--245.