On the structure of oscillatory boundary layers

Coastal Engineering, 9 (1985) 261- 276 Elsevier Science Publishers B.V., Amsterdam - - P r i n t e d in The Netherlands

261

ON THE STRUCTURE OF OSCILLATORY BOUNDARY LAYERS

PETER NIELSEN*

Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL 32611, U.S.A. (Received February 13, 1984; revised and accepted March 25, 1985)

ABSTRACT Nielsen, P., 1985. On the structure of oscillatory boundary layers. Coastal Eng., 9: 261-276. An empirical analysis is performed on the most detailed, recent measurements of turbulent oscillatory boundary layer flow. The measurements show that throughout elevations where the flow can be considered horizontally uniform, one deficit model is sufficient for describing the fundamental mode. Some general properties of the non dimensional velocity deficit DI (z) appear with striking consistency. First of all the identity In ID 1 I --- Arg D1, which is a theoretical result for smooth laminar flow, seems to hold with great accuracy for a large range of turbulent flow conditions as well. This is of principal theoretical interest because all previous analytical eddy viscosity models (Kajiura, 1968; Grant and Madsen, 1979) as well as numerical mixing length models (van Doorn, 1981) predict a consistent and fairly large difference between Arg DI and In ID~ I. If the identity between In ID 11 and Arg D1 extends all the way to the bed, it means that the bed shear stress leads the free stream velocity by 45 degrees. It is also found that the structure of smooth turbulent oscillatory flows as measured by Kalkanis (1964) corresponds to a sharp maximum in the normalized energy dissipation rate.

INTRODUCTION

The following is an empirical analysis of the structure of oscillatory boundary layers based mainly on the data of Kalkanis {1964), Jonsson and Carlsen (1976), van Doorn (1981, 1982, 1983), and 81eath (1982). The aim is to point out very clear regularities shown by the data, but not predicted by any existing models, in order to hopefully instigate new ideas and lead to a better understanding of the structure of oscillatory boundary layer flow. We shall use the complex formulation by which a simple harmonic free stream velocity with amplitude ac0 is written:

*Present address: Coastal Engineering Branch, N.S.W. Dept. of Public Works, 140 Philip Street, Sydney, N.S.W. 2000, Australia.

0378-3839/85/$03.30

© 1985 Elsevier Science Publishers B.V.

262 u~(t)

=

a¢o

exp(icot)

(1)

and it is understood that only the real part of the complex exponential is given physical meaning. We restrict ourselves to consider only simple harmonic time dependence or the basic harmonic c o m p o n e n t of more complicated flows. The physical and mathematical structure of the problem makes it convenient to work with the non dimensional velocity deficit D1 (z) defined by: ul(z,t)

=

[1-D~(z)]u~(t)

(2)

Please note t h a t this definition of velocity deficit differs from the one used by Jonsson (1980) by a minus sign. The meaning of D, (z) is illustrated in Fig. 1.

Arg Di

Fig. 1. T h e c o m p l e x v e l o c i t y deficit D 1 (z) gives t h e local v e t o e i t y a d i f f e r e n t p h a s e as well as t h e d i f f e r e n t a m p l i t u d e f r o m t h e free flow u~(t).

We shall first give a brief treatment of the smooth, laminar case in order to familiarize ourselves with the terminology and then proceed to an analysis of smooth and rough, turbulent data using the formalism which comes naturally from treating the linearized, smooth, laminar problem. LAMINAR FLOW OVER SMOOTH WALL

When the flow is laminar we have: au r

=

pv

(3)

~

az

where r is the shear stress, p is the fluid density and v is the kinematic viscosity. Then the linearized equation of motion reads: au

1 ap -

at

pax

a2u

+ v -

az 2

(4)

263

The flow is essentially horizontal, so the pressure distribution is hydrostatic and we have: 1 0p p ax

Ou® at

(5)

everywhere. Thus we can replace the pressure term in eq. (4) by au~/at. Then by assuming the following form for u (z, t): u(z,t) = [1-D~(z)]u~(t)

= [1

(2) (6)

D1(z)]awexp(icot)

-

we get the differential equation for D, (z): d2Dl

iw =

dz 2

- -

p

(7)

Dl

This, together with the b o u n d a r y conditions D1 (0) = 1 and D1 -~ 0 for z ~oo leads to: z

T:~rn{D11

]Irn (a)

0'5{

'5

-o.svi

io~I~

/

T 0"5~

-o.s~

(c)

(b} 1

1

5

-~Z !{d)

C a~

Fig. 2. V e l o c i t y v a r i a t i o n s w i t h e l e v a t i o n in o s c i l l a t o r y l a m i n a r flow over a s m o o t h wall. (a) T h e deficit f u n c t i o n D 1(z) m o v e s a l o n g a l o g a r i t h m i c spiral s t a r t i n g at 1 a n d a p p r o a c h ing 0 as z increases. N u m b e r s o n t h e curves r e f e r t o t h e n o n - d i m e n s i o n a l e l e v a t i o n zx/'wi2v. (b) C o r r e s p o n d i n g v a r i a t i o n o f l - D 1 (z) w h i c h is t h e r a t i o b e t w e e n u(z,t) a n d u~(t), see eq. (2). (c) In t h e s i m p l e case o f l a m i n a r flow over a s m o o t h wall w h e r e u(z, t) is simple h a r m o n i c , we c a n c o n s t r u c t u(z, t) g e o m e t r i c a l l y b y using t h e spiral o f Fig. 2b. (d) T h e v a r i a t i o n o f t h e v e l o c i t y a m p l i t u d e U(z) = awl 1-D~ (z)J w i t h elevation.

264

The variation of D, with the dimensionless elevation z/x/2v/w is shown in Fig. 2. The locus of DI in the complex plane is a logarithmic spiral with the constant angle ~/4 between tangent and radius vector. That is a general character obtained whenever D~ has the form: D~ = e x p [ - ( l + i ) F ( z ) ]

(9)

where F(z) is a real valued function, i.e. if: lnlDll-

(10)

ArgD~

Figure 2b shows the corresponding variation of 1-D1 which, according to eq. (2) is the complex ratio between the local velocity u(z,t) and u~(t). The argument of 1-D~, which is predominantly positive is the phase angle b y which the velocities near the wall are ahead of the free stream. When the free stream veloCity varies as acoexp(ia~ t) we can derive the total velocity field from Fig. 2c as suggested by Jonsson (1980). Since the modulus of l - D 1 can exceed unity, there are elevations where the velocity amplitude exceeds that of the free stream. The maximum is approximately 1.069 a ~ and occurs at z = 2 . 2 8 4 x / 2 v / ~ . THE LAYER STRUCTURE

At large values of the Reynolds' N u m b e r a2w/v, we would expect the flow to become turbulent far from the wall, while it may remain laminar throughout a thin laminar sublayer with thickness of the order lOv/u,, as discussed by Kajiura (1968). For a flow over a smooth wall, we would expect the phase averaged velocities to be horizontally uniform at all elevations. With a rough wall, the situation is different. There we find a thin layer with large horizontal velocity gradients and with vertical velocities comparable in magnitude to the horizontal ones, see Fig. 3. The roughness geometry shown is the one used by van Doorn (1981, 1982) and the right hand part of the figure shows corresponding measurements of lnIDl]. We see that the CREST values and the

rr" LLJ m,

-3 n~

z (mm) 10

• x: TROUGH

o:

o

CREST

8

x

o x

t'

o

x

o

f

JJJ//J//JJ

2

-~n IDiI

x I

I

I

IlL

Fig. 3. L a y e r s t r u c t u r e c o r r e s p o n d i n g t o van D o o r n ' s e x p e r i m e n t s . T h e r o u g h n e s s elem e n t s are 2 m m high. T h e m e a s u r e d values o f In ID~ I are identical over r o u g h n e s s crests ( × ) and ow~,t r o u g h n e s s t r o u g h s (©) for e l e v a t i o n s larger t h a n 2.5 r a m , i n d i c a t i n g t h a t t h e flow is h o r i z o n t a l l y u n i f o r m a b o v e t h i s level.

265

T R O U G H values become identical for z/> 2.5 mm indicating that the flow is horizontally uniform upwards from that level. We shall from n o w on use the term outer layer for all elevations where the flow can be considered horizontally uniform. Note that this terminology is completely different from the one used by Kajiura (1968) who suggested a three-layer eddy viscosity model based on steady flow theory. In the following we shall see that the velocity field can be described adequately throughout the horizontally uniform layer by one fairly simple parametric description. A SIMPLE PARAMETRIC DESCRIPTION

Based on the measurements by van D o o m (1981, 1982), Kalkanis (1964), Jonsson and Carlsen (1976), and Sleath (1982) I shall n o w develop and discuss a simple parametric model for turbulent oscillatory boundary layers. The model is empirical, b u t apart from being well suited for practical use due to its simplicity, it is meant to draw attention to some basic qualities of turbulent oscillatory boundary layers which call for theoretical attention. These qualities appear with striking consistency in the presently available data b u t cannot be accounted for by simple adaptation of steady flow models. The model attempts to describe the fundamental m o d e of the motion only, and the form is inspired by the s m o o t h laminar solution [eqs. (2) and (8)l. For the smooth laminar solution we found that the argument and the logarithm of the modulus of D~ have identical values: ArgD1 - In tOC

I

IDll I

I

(10) I

I

II

II

I

I

1

1

I01/ I

VAN DOORN(1982) oOJ=106 m/s r~ :Q062

+ ~o

4c

/

w •

_~ -3

Z

-~ ic

.85

~)

-

X

Z o_

~

o ,+

21 I

t

o +/ o +

o +

TOP OF ROUGHNESS ELEMENT

'~ l

I

02

I

I

0.4

I

I

Illl

l

1,0

2.0

I

+

-~O n IDII

o

-Arg I

4.0

I

D t I

I II

I0.0

Fig. 4. M e a s u r e m e n t s f r o m van D o o r n ( 1 9 8 2 ) . T h e i d e n t i t y b e t w e e n In n o t e d even inside the r o u g h n e s s trough.

IO,l

and A r g O 1 is

266 and this appears to hold with remarkable accuracy for rough turbulent flows as well over the range of relative roughnesses that have primary practical importance for natural sand beds. See Figs. 4 and 5. Even for smooth turbulent flow, Kalkanis' data in Fig. 6 show that Arg D1 and In IDll are very closely related to each other. I

I

I

I

I

I 111

I

I

z(rnm) 200--

I00

I

I

I I I

JONSSON AND CARLSEN (197G) TEST ]I / ~ °

--

+

. ÷ ....

60-

~

40-

,/+ /

20-

~

,,;,

\ ' , Di

=exp ~(,+i}(3-~) 0:~

÷

ID,I

• -~n

IC - -

+ - Arg Dr

~:"

6

--

%" --

-~-

Z

4--

I--

I

I

2

t

4

III

6

Fig. 5. M e a s u r e m e n t s 0.035. r

l

IO from

i

i

i

I

2.0

I

40

I Illll 6.0 IO0

Jonsson and Carlsen (1976).

i

i

T = 7 . 9 s, aoa = 1 . 5 3 m / s , r l a =

f

KALKANIS TEST 105 z

2O xo

x

I0

o

~

8

x

6

x x

4

o o

o

o

x o

2

x -gn IDfl o -Arg DI I 2

I

! 4

I

I 6

I

I 8

I0

Fig. 6. T y p i c a l v a r i a t i o n o f In D , f r o m o n e o f K a l k a n i s ' s m o o t h t u r b u l e n t t e s t s .

267

The fact t h a t all the data fall on essentially straight lines in the logarithmic diagrams suggests the following form of D l:

Dl(z) = exp

[

(1+i)

(11)

and this applies t h r o u g h o u t the outer layer (Fig. 3). There is thus no evidence for the need of multi layer models as far as the fundamental mode of the flow is concerned. The vertical length scale z0 is of the order of magnitude of the roughness height but it depends equally strongly on the semi excursion a. For predictive purposes, we can recommend the formula: z0 = 0 . 0 9 V ~

(12)

where r is the Nikuradse roughness. See Fig. 7. As mentioned above, Kalkanis' smooth turbulent data show a consistent difference between Arg D~ and In IDII: Arg D1

= 1.214 +-0.056

(13)

In 1911

.06 05 D4

Zo/O

Zo~o~s /

0/

D3 .02 .01 I

I

0.2

I

I

I

I.

0.4

Fig. 7. Values of the vertical length scale z0 from the experiments of Jonsson and Carlsen (1976): I, II, and from van Doom (1981) and (1982): M,S,D, corresponding to the experiments MOO, S00 and V00 respectively.

Kalkanis measured the motions induced into an otherwise still fluid by an oscillating plate, and whether the 21% difference would be present in oscillatory flow over a smooth fixed wall is so far unresolved. But if we apply the description:

o z,: exp[

14,

268 to the smooth turbulent data, we find: Z0

- 0.50 +0.16

(15)

without any consistent dependence on the Reynolds Number. The power p varies s m o o t h l y from being marginally larger than unity for very large relative roughnesses to the value of: p = 0.326 -+0.056

(16)

found from Kalkanis' smooth turbulent measurements. See Fig. 8. i

20

i

I

i

i i ill

i

]

P

i

I

i

i I i i

i

i

I IT: JONSSON AND CARLSENO,S,M~VAN DOORN

I.O

---D-S~.M.~.I

o59~p

T

,-(~) -I / ~ ~BI /

59

0.2

I+

-7-~---

_m~__K~js/

(~-r) ] /

SMOOTH / G/r

2

I

I

4

IlllJI

2J

I0

0

,

Z~L

,ll

Ii~

2

~

1

Fig. 8. Variation of the power p with the relative smoothness a/r.

This variation of p for rough turbulent flows is reasonably well described by:

p = 0.59 exp

.59

l( r)

(17)

It is interesting to note here that the value of p f o u n d by curve fitting to Kalkanis' data coincides with a very sharp m a x i m u m of the time-averaged, normalized energy dissipation DE(/)). See Fig. 9. This may be a clue to understanding how the turbulence structure in a smooth turbulent boundary layer adjusts itself. The energy dissipation is calculated from: (18)

DE = u (t) r(O,t)

where r(O,t) in turn is determined by the equation of m o t i o n on the form: T(z,t) = p

f z

a

~

(u~ - u ) d z

(19)

269

R~Ic,} z~/e 16 ~

Y-J~LKANIS'DATA

L~

1.2-

io-

0.8-

o.6

04

Q2 P i

I

I

I

i

i

i

~

O( b 0.2 04 0.6 0.8 1.0 1.2 Fig. 9. V a r i a t i o n o f t h e n o r m a l i z e d e n e r g y d i s s i p a t i o n w i t h t h e p o w e r p . O 1 is given b y D E = 0.5 p(a¢o) 3 R e ( e l ).

which reduces to: r(0,t) =

[i; -

a

D~(z)dz

]

p(aw)2exp(iwt)

(20)

0

with the final result [for u . ( t ) = a w exp ( i ~ t)] : DE

=

-

~ p(aco ) 3 Im

Dt(z)dz 0

1

(21)

Please note that the formula (20) will underestimate the total bed shear stress in a rough flow considerably because the expression for D1 is n o t valid in the inner, non uniform layer, and the form drag and vortex shedding which takes place in this layer contributes most of the drag force over for example sharp crested ripples. See Longuet-Higgins (1981). E D D Y V I S C O S I T Y IN O S C I L L A T O R Y

FLOW

The eddy viscosity PT provides the simplest possible connection between velocity gradients and shear stresses in a turbulent flow: T = PET

~u --

Oz

(22)

270

and this is very useful because it leads to a linear version of the equation of motion: au at

lap -

3 +

pax

-

(

az

au) v T

- -

az

(23)

We can find v T from velocity measurements by using the definition (eq. 22) together with the equation of motion (eq. 19):

f

7"

PT -

~-~ ( u~ - u ) dz

z

au P ~-

=

au az

(24)

This seems simple enough, However, when one looks carefully at the r and values of Jonsson and Carlsen it becomes clear that the phase averaged values of these two quantities are generally not zero at the same phase. See Fig. 10. a u/az

~z

% Fig. 10. Local velocity gradients and shear stresses f r o m Jonsson and Carlsen (1976), T E S T I. Elevation above Ripple crest: 4.5 cm.

This leads to a somewhat radical variation of vT(t ) as defined by eq. (24), encompassing negative and infinite values. If we look at the fundamental m o d e only and use real-valued functions:

°

az

ul(z,t)

z~(z,t)

=

=

-az l I°u

cos cot

Irllcos(cot+O~)

(25) (26)

we find: vl (z,t)

(cos~l - sin¢l tan cot)

=

(27)

P 1 ~ul so if we insist on the real formalism we must accept a strongly time dependent eddy viscosity which goes to plus and minus infinity twice every period.

271 A more smooth interpretation is obtained if we use the complex formalism: -

-

az

ul(z,t)

rl(z,t)

=

-

exp(iwt)

-

az

(28)

Ir~l exp[i(cot+@~)]

=

(29)

which leads to: ITll

vl(z)

=

exp(i@l)

~

°

az

Ivllexp(iArg 121)

=

(30)

I

We see that in this case v l is n o t time dependent -- but it is a complex number of which the argument is equal to the local phase lead of r,(z,t) relative to

~

ul(z,t).

The existence of a complex eddy viscosity, or more basically, the existence of finite shear stresses simultaneously with zero-velocity gradients (for phase-averaged velocities) is so far theoretically unexplained. But if we want a description which matches the measured, phase-averaged velocities, and particularly the fundamental mode of these, then we will have to accept it and use it formally. Figure 11 shows the variation of Arg v~ for the data of Jonsson and Carlsen (1976). The values are obtained by harmonic analysis and the use of eq. (30). zi~, 40o

3.0-

X

o X

20-

o x o

JONSSON 8, CARLSEN x TEST I

x

o

I0 ~

T E S T IT

°°

Arg /]1

o i° -20 °

x'

I

I

I

I

2C) °

'40 °

6()"

80 °

I

I-

IC)O °

Fig. 11. Variation of Argv~ for the experiments of Jonsson and Carlsen (1976). The vertical length scale ~ is Re{i

f o

D~ (z)dz}.

272

We see t h a t the trend is very consistent and similar for the t w o tests, and in general we find Arg v l > 0 indicating t hat the local shear stress is ahead of the local velocity gradient. So in general we must e x p e c t the e d d y viscosity for rough t u r b u l e n t flows to be complex. Under very rough conditions however, as in van Doorn's test VOO, ( r / a = 0.25) it looks like ,1 becomes a real valued constant t h r o u g h o u t the layer of practical importance. Figure 12 shows the variation of l n D , ( z ) for van Doorn's VOO-data and we see that it is well represented by: z

ArgD~ = lnlD~l = - ( l + i )

(31)

4 mm

which is co mp letel y analogous to the laminar solution. T herefore we assume that vl is a constant in this case, and we can find t he magnitude of ~1 by comparing eq. (30) with t he laminar solution: z

Arg D1 = l n l D l l = - (1 +i) ~/~_/w

(32)

by which we get: = 4 mm

(33)

and P 1

--

1

2~

2

T

IOO 60

e

4: ~ 25 m m : s-'

i ) i i i [i I MEASUREMENTS 8Y VAN

z(mm)

RIPPLE

40

TROUGH

(34)

i l i I ~ ]) DCX:~N (198[)

(VOORA)

z(mm) 51

20

~n D' :-(' ÷i)z~ +

+

IO 6

j

4 TOPOF

~

ROUGHNE~ E,-EMENTS I

02

= I)

+-Arg

Ol

( I I

I

I ] I ill

o4 06

)

Lo

2o

I

I

I i )+

40 60

Ioo

Fig. 12. Variation of Arg D~ and In ID~t for van Doorn's TEST VOORA. T = 2.0 s, a = 8.48 em, Nikuradse roughness 2.1 cm.

273

Let us n o w briefly examine the results of using one of the real valued eddy viscosities that has been suggested namely: vw

=

(35)

KU,Z

which has been used by Grant and Madsen (1979), u, is defined It turns o u t that such a model fails to predict the identity: In IDll -- ArgD1

as

%/Vmax/p .

(10)

which is so c o m m o n l y shown by measurements. The solution: k e r 2 V ~ + i kei 2 ~/[ ker2~/~-0 + i kei 2 ~/~0

D1

(36)

which corresponds to eq. (35) with ~ = z / ( • u , / ¢ o ) and u(~0) = 0, is compared with the data from Sleath (1982), TEST 6 in Fig. 13. This data set which has the largest relative smoothness ( a i d = 140) should, if any, be well described by the PT = Ku , z - m o d e l b u t we notice a large, consistent difference between the predicted values of ln IDll and Arg D1. i

20

,

,

,

,

i

I

z [mm]

,

i

i

i

i

SLEATH (1982)TEST 6

Io

xo~o

8 6 4

/

2

/ / J

10-4

I 0"6

/

ox/ / i

I i I 0"8 I'0

x - t,nlDtl o - ArgDt I 2-0

I

i 4.0

I

i 60

I 8-0

Fig. 13. C o m p a r i s o n of t h e Kelvin f u n c t i o n s o l u t i o n ( 3 6 ) w i t h t h e m e a s u r e m e n t s o f S l e a t h ( 1 9 8 2 ) , T E S T 6.

PRANDTL'S MIXING LENGTH AND OSCILLATORY FLOWS

We saw above h o w one of the a priori very reasonable eddy viscosity models fails to predict one of the main features of many oscillatory boundary

274

layers namely the identity of ln lDll and ArgD1. The same shortcoming is exhibited by numerical solutions based on Prandtl's mixing length model: r =

pl2

0_zUl_0u DZ

(37)

with = ~z

l

(38)

See Fig. 14. The failure of the mixing length model may be due to the fact that most of the turbulence in rough turbulent, oscillatory boundary layers is convected upwards from the bed rather than being generated locally, and since it is not locally generated, it is not necessarily related to (Kz)21Ou/OzI. The convective m o m e n t u m transfer mechanism has been modelled by Sleath (1982). His model is based on the observation of jets originating at the boundary and injecting "low m o m e n t u m fluid" into the main stream. I

I

i

i

i

] i i i

[

I00 -VAN DOORN 1982, MIORAL - T(z,t)= p(K'z) ~

-

I

i

/ /

i

i

IXF

//o: -.~n

IDq I

x

~- 20

_z z O t7-

Z

o"~

Y

1,

15ram

+1

or ~z~ . . . . . . . .

,,~-,-~

,_,o,°,, o-Arg DI I

0.1

0.2

I

I

0.4

I

I

II

I

I

I0

20

~

I

4D

I

1 I1

IO0

Fig. 14. N u m e r i c a l results o b t a i n e d b y van D o o r n ( 1 9 8 2 ) c o m p a r e d t o d a t a f r o m test M 1 0 R A L . N o t e t h a t in this test t h e r e was a s t e a d y c u r r e n t s u p e r i m p o s e d o n t h e wave m o t i o n . This d o e s n o t d i s t u r b t h e i d e n t i t y b e t w e e n Arg D 1 and In [D~ [ h o w e v e r .

DISCUSSION

The previous pages are intended to draw attention to some major features of turbulent oscillatory boundary layers, which do not follow from steady flow laws via quasi steady speculations.

275

First of all, the fundamental m o d e of the flow can be modelled by a single deficit model: ul(z,t) D,(z)

=

[1-D~(z)]u~(t)

=[exp-(1+i)

(2)

(z~) p ]

(11)

through all elevations where it makes sense to ignore dependence on the horizontal co-ordinates. The complex deficit function D1 (z) satisfies the identity: lnlDll - ArgD1

(10)

with great accuracy for all data obtained over fixed beds while some discrepancies occur for measurements by fixed instruments over oscillating trays. Fulfilment of the identity (10) is of great theoretical interest for two reasons. Firstly because it is a uniting feature between laminar and turbulent flows. Secondly because it is not predicted by quasi steady models, based on well established strain stress relations from steady flow, such as u T x u.z~ or the Prandtl von Karman stress model (eq. 37). Analysis of Kalkanis' (1964) data in the framework of the model (eq. 11) shows z0 to be a constant for smooth turbulent flows (eq. 15) and that p takes the value corresponding to m a x i m u m energy dissipation (for fixed z0). Finally it is worth noticing that the behaviour of DI (z) is very little sensitive to the superposition of steady currents, see Fig. 15. This means that =

1

l

I

l

l

I

I

I

I

I

I

l

I

l

I

t

T

MEASUREMENTS BY VAN DOORN (1981), RIPPLE TROUGH z(mm) X Xe

30 X X

2C

eX om oe

IC

•

o

o X %.

o~

~e

4 ~

o N O CURRENT

o,~~"

x aja,.=.s,

TOP OF

ROL~HNESS ELEMENTS

O'

0 '2

--en I D~ '

' ' 0.6 ' '0.8 ' ''1.0 0.4

2 '.0

'

' 4'0 ' 6 '0' 80100

Fig. 15. M e a s u r e d values o f In ID, I for waves s u p e r i m p o s e d o n c u r r e n t s o f d i f f e r e n t s t r e n g t h , f r o m v a n D o o r n ( 1 9 8 1 ) . fi-, a n d u', d e n o t e t h e f r i c t i o n v e l o c i t y t o t h e s t e a d y c u r r e n t a n d t h e p u r e wave m o t i o n , respectively.

276

the formulae {12) and (17) derived from the purely oscillatory flow data may be used even in the presence of a fairly strong steady current. ACKNOWLEDGEMENT

The present study was supported by the U.S. Office of Naval Research, Coastal Sciences Program, Task No. 388-189, Contract No. N00014-83-K0198 through the Virginia Institute of Marine Science and University of Florida.

REFERENCES Grant, W.D. and Madsen, O.S., 1979. Combined wave and current interaction with a rough bottom. J. Geophys. Res., 84(C4): 1797--1808. Jonsson, I.G., 1980. A new approach to oscillatory, rough turbulent boundary layers. Ocean Eng., 7 : 1 0 9 - - 1 5 2 + 567--570. 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 b o t t o m boundary layer in water waves. Bull. Earthquake Res. Inst. Univ. Tokyo, 46, Chapter 5, pp. 75--123. Kalkanis, G., 1957. Turbulent flow near an oscillating wall. Beach Erosion Board, T.M. 97. Kalkanis, G., 1964. Transport of bed material due to wave action. C.E.R.C., Tech. Memo No. 2. Longuet-Higgins, M.S., 1981. Oscillating flow over steep sand ripples. J. Fluid Mech., 107: 1--35. Sleath, J.F.A., 1982. The effect of jet formation on the velocity distribution in oscillatory flow over flat beds of sand or gravel. Coastal Eng., 6: 151--177. van Doorn, Th., 1981. Experimental investigation of near-bottom velocities in water waves without and with a current. Report Investigation, Delft Hydraulics Laboratory, Rep. M1423, Part 1. van Doorn, Th., 1982. Experimenteel onderzoek naar her snelheidsveld in de turbulente bodemgrenslaag in een oscillerende stroming in een golftunnel. Delft Hydraul. Lab., Rep. No. M1562-1b (in Dutch). van Doorn, Th., 1983. Computations and comparison with measurements of the turbulent bottom boundary layer in an oscillatory flow. Delft Hydraul. Lab., Rep. No. M1562, Part 2.