full-range equation of friction coefficient and phase difference in a wave-current boundary layer

ELSEVIER Coastal Engineering 22 (1994) 237-254

COASTAL ENGINEERING

Full-range equation of friction coefficient and phase difference in a wave-current boundary layer

Hitoshi Tanaka, Aung Thu Division of Water Resources Engineering, Asian Institute of Technology, G.P.O. Box 2754, Bangkok 10501,

Thailand

(Received 14 January 1993; accepted after revision 6 July 1993 )

Abstract

Approximate formulae of wave-current friction coefficient and phase difference, spanning all flow regimes, are obtained. Both of them are based upon the theoretical results of Tanaka and Shuto ( 1981). No restriction is made regarding the intersection angle between waves and current. Since all the formulae are given in an explicit form, they can reduce computing time when they are applied to practical problems. Comparisons are made not only with the exact solutions but also with experimental data. The accuracy of these formulae is found to be satisfactory.

I. Introduction

Investigations into the friction coefficient under a wave-current combined motion have been made by many researchers based on various kinds of turbulence model. The studies presented up to now can be broadly classified into three types: (i) the time-independent eddy viscosity model proposed by Smith (1977), Grant and Madsen (1979), Tanaka and Shuto (1981 ), Tanaka et al. (1983), Asano and Iwagaki (1984), and Christoffersen and Jonsson (1985) ; (ii) a more sophisticated model based, for example, on the k-E turbulence model or the mixing length model presented by Bakker (1974), Bakker and Van Doom (1978), Tanaka (1986), Aydin (1987), and Davies et al. (1988); and (iii) other approaches such as those of Freds0e (1984) and of Myrhaug and Slaattelid (1990). Using the second category of the model, not only the friction coefficient, but also detailed infor- mation of the turbulent structure, such as the turbulence energy, the energy dissipation rate and the eddy viscosity can be deduced. From a practical viewpoint, however, this kind of turbulence model is not convenient, since an analytical solution is impossible and a numerical calculation should be made to solve the highly non-linear governing equations. In most

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238 1-1. Tanaka, A. Thu / Coastal Engineering 22 (1994) 237-254

practical situations, therefore, the friction coefficient based upon the first model has fre- quently been applied to estimate the bottom shear stress.

As far as sandy bottom is concerned, a friction coefficient formula for rough turbulent flow is commonly applied to calculate a force acting on the bottom, since it is plausible to assume that the sand surface is hydrodynamically rough. Recently, mud transport due to waves and current motion has been studied extensively, in which the combined motion might not always be rough turbulent; it may be sometimes laminar, smooth turbulent, or even transitional. Up to the present, however, the flow regime criteria under waves and current have not been clarified satisfactory.

Tanaka and Shuto (1984) proposed a flow regime diagram for wave-current motion which is based on an experimental study in a steady current. However, their criteria are not easy to apply, since an iteration procedure is needed to determine the flow regime. Another disadvantage of their study is that they did not develop a method which provides the friction coefficient in the transitional region. A method proposed by Christoffersen and Jonsson (1985) enables us to calculate the friction coefficient in the transitional zone between smooth turbulent and rough turbulent regime. However, according to the Kamphuis' experiment (1975) in a wave boundary layer, the transition takes place not only from smooth turbulent to rough turbulent but also from laminar to smooth turbulent or from laminar to rough turbulent regime. Therefore, their interpolation method is not sufficient in a coexistent flow with a predominant wave component.

The aim of this study is to establish a full range equation for a wave-current friction coefficient, as well as for phase difference. At first, approximate formulae of the friction coefficient and the phase difference for each flow regime are derived in an explicit form based upon the theoretical result by Tanaka and Shuto (1981). Secondly, the criteria of flow regimes are reexamined using the experimental data of previous studies and are expressed in an explicit form. Finally, approximate formulae of the friction coefficient and the phase difference for each flow regime are combined to yield full-range equations, by considering the flow regime criteria. Comparisons are made with experimental data to verify the present theory.

2. Friction coefficient for each flow regime

The wave-current friction coefficient, few, considered here is defined in terms of the amplitude of wave-induced velocity as follows (Tanaka and Shuto, 1981 ):

- - a * 2 few Uw TOmax--P cw ~___p_~ ^2 (1)

where ~'Omax is the maximum bottom shear stress under waves and current, p the density of fluid, u *cW* the maximum shear velocity under waves and current, and Uw the maximum of the wave-induced velocity just outside the boundary layer. Hereafter, the subscripts cw, c and w denote quantities under wave--current combined motion, steady current, and wave motion, respectively. In addition, subscripts (L), (R), ( S ) refer to laminar, rough turbulent and smooth turbulent flow.

H. Tanaka, A. Thu / Coastal Engineering 22 (1994) 237-254 239

2.1. Laminar flow

The friction coefficient for laminar flow has already been derived by Tanaka and Shuto (1984) theoretically. Their solution can be simplified as follows, by eliminating the con- tribution of the water depth in the wave component as done by Tanaka (1989) for turbulent flow.

few(L) 2 I/2 = {fctL) +2f~(L,fwtL)COS~b' +fw(L,2 }

where

6 (~wt2

(2)

(3)

2 fw(L)---- '~a (4)

Uc Zh Re = (5) /.I

Uwam R a = (6)

p

where Kc is the depth-averaged mean velocity of the steady current component, Zh the water depth, am = Uw/tY, oz. the angular frequency of the wave motion (tz= 27r/T, T: wave period), v the kinematic viscosity of the fluid, and

~b' = c o s - l( IcosOS[ ) (0 ~< ~b' .<< 7r/2, 0 ~< &~< 27r) (7)

where ~b is the angle between the wave orthogonal and the current vector. Note that o- and T are not relative values, but absolute ones. Eq. (2) is an explicit expression with respect tOfcw~ L) and is a function of four dimensionless parameters: Re, Ra, u~//-Tw and oh.

2.2. Rough turbulent flow

Tanaka and Shuto ( 1981 ) derived an analytical solution of the friction coefficient for turbulent wave-current motion with an arbitrary angle of intersection. Since their exact solution is expressed in terms of the complex-valued Bessel and Neuman functions, they also derived an approximate solution which is expressed by means of simple functions. From a practical viewpoint, however, it still has a disadvantage; it is given in an implicit form, which means an iteration procedure is needed to find a final solution. Recently, Tanaka (1992) developed an explicit expression of friction coefficient for rough turbulent flow based on Tanaka and Shuto (1981). A coefficient, •(R), which arises from a non-linear interaction of waves and current, was introduced and the following approximate expression was proposed.

f~w~R, =f~R) + 2v/f~{R)fl~R,fw{R, COS$' + fl~R)fw~R~ (8)

where

240 H. Tanaka, A. Thu / Coastal Engineering 22 (1994) 237-254

2K2 (Uc) 2 f~(R) - {ln(z./zo) - 112 ~ (9)

- ^ .--0.100 l

1 {1 + 0.863otexp( - 1 . 4 3 a ) ( - ~ f } (11) 13~R) = 1 +0.769oe °83°

1 ~ a=ln(Zh/Zo) -- 1 Uw (12)

where K is the yon Karman constant (=0 .4 ) , zo the roughness length ( zo=kJ30, ks: Nikuradse's equivalent roughness), and the angle 4~ in Eq. ( 11 ) is in radians. In summary, Lw(R~ is a function of four dimensionless parameters: 0w/(O'Zo), z,/zo, ~c/0w and 4~ or three factors since the second and the third parameters can be combined as seen in Eq. (12).

2.3. Smooth turbulent flow

Provided that an expression of the friction coefficient for rough turbulent flow is obtained as Eq. (8), it is possible to extend it to smooth turbulent flow by considering a relationship between the roughness height, Zo, and the thickness of the viscous sublayer, 6L, expressed as Eq. (13) (Jonsson, 1966; Tanaka and Shuto, 1981; Myrhaug and Slaattelid, 1990).

~L Zo -= - - (13)

105

where

6L=11"6 cw = 1 1 ' 6 - ~ U w _ (14)

Since ~ is expressed in terms off~w(s), the smooth turbulent expression thus obtained is not in an explicit form with respect to few(s), and, accordingly, it is not convenient for practical use. Tanaka and Thu (1993) recently developed a convenient formula, Eq. (15), which approximates Tanaka and Shuto's exact solution ( 1981 ) for smooth turbulent flow.

fcw(s~ =f~(s) + 2~/fc~s)/3(s~fw~s)cos~b' +/3(s)fw(s) (15)

where - - 2

fc(s) = e x p { - 7.60 + 5.98Rc °'°977 }(U~ww) (16)

fw(s~ = exp{ - 7.94 + 7.35R~ -°°748 } (17)

-2 F 2 " 5

/3(s) - 1 . o.,,-,,,c Jc(s) . .L R t ~ A D ' - - 0 ' 0 3 0 3 r r 0 " 3 7 9 ( 1 8 )

This approximate formula is effective in the range of 105~<Ra~< 107, 103~<Rc~< 106, 0~<~c/~w~<5.0 and no restriction should be made to the angle ~b. The dimensionless parameters involved in the equation are the same as those in the laminar expression. It is seen that the term fl(s) given by Eq. (18) is slightly complicated as compared with/3(R) for rough turbulent flow, Eq. ( 11 ). This is due to the fact that no parameters out of the four can be combined as was done in the case of rough turbulent flow. The possible error of Eq. (15) is examined in the range described above and found to be less than 3% (Thu, 1992), sufficiently small for a practical purpose.

3. Phase difference for each flow regime

3.1. Turbulent flow

40

According to Tanaka's theory (1989), the phase difference in a turbulent wave-current boundary layer can be obtained from the argument of the following complex function.

H~l)(~o) -,n/4 (19) f(~:o) Ho(I) (~o ~ e

t9 (deg)

e x a c t solution ~f

30

20

I0


0 [ J I 1 I I I I I I I I I l l l l l 1 J I I I I I I I I I I I I I I I I I I I I I I I

10-5 10-4 10-3 C 10-2 10-1 10 0

Fig. 1. Relationship between 0 and C.


0 (deg) 14

12

I0

35

exact solution ] ~ 0 ~ A i-] approx, solution

~ ' ~ , ~ Rc.IO 6

D ~ R c - 10 6

_ . , , . _ . + . 1 o ~ ~ - - - ~ 5 ~ = ~ . . . uj 0~-1.o Rc'10 ~ ~ ~. 0 °

(a) I J [ I I I I I I I I I I I I L I

0 ~ 106 107 Ra

0 (deg)

exact solution

30 , ~ - ~ 0 ~ /~ [-~ approx, solution

20

10 ~)" 0 o

(b)

~ 1 o ' ' " + ' ~ " , , , , , , . . . . . . , , , , , , , , , , , , , , t ' ' ' ' " " ' , 0 6 ' ' ' ' " ' 1 102 103 A 104 105 10 Uw/CTZ o

Fig. 2. (a) Phase difference under waves and current ( smooth turbulent, u~/Ow = 1.0, ~b = 0°). (b) Phase difference under waves and current (rough turbulent, ~b = 0°).


where H~ l) is the Hankel function of the first kind of the nth order, ~o = 2e- "/4 c~zo, and c = ~r/( Kff*w ). It is easily seen that Eq. (19) is a function of ~o or C = CZo alone. Hence, the relationship between 0 and C = CZo is shown in Fig. 1 as a solid line, while the open circles denote an approximate formula, Eq. (20), which provides a reasonably good prediction.

1 + 0.00279C-°357 0= 42'4C°153 1 +0.127C °'563 (deg) (20)

By definition, C in the above equation can be expressed in terms of the friction coefficient as given by either Eqs. (21) or (22), depending on whether the boundary layer is smooth turbulent or rough turbulent.

0.111 smooth: C (21)

fcw(S) K - - ~ Ra

l rough: C= /~cw,R,~ Uw (22)

Here, Eqs. ( 1 ), (13) and (14) are used to correlate C with few. The friction coefficients in Eqs. (21) and (22) can be evaluated using the explicit expressions given by Eqs. (8) and (15), respectively. Accordingly, Eq. (20) is in a fully explicit form. Comparison between the approximate formula and the exact solution is made for both rough turbulent and smooth turbulent flow in Fig. 2. More extensive comparison has been made elsewhere for various combination of the parameters (Thu, 1992; Tanaka and Thu, 1993), and it was found that the accuracy of the present formula is satisfactory in the range of practical use.

3.2. Laminar f low

As far as the flow is laminar, the phase difference is constant and, as already known, is given by Eq. (23).

0~L~ =45 (deg) (23)

4. Flow regime criteria

4.1. Laminar turbulent transition

At first, flow regime under steady flow is discussed in this section. According to the experimental data reported by Chow (1973) and Patel and Head (1968), the lower and the upper limit of the transitional range between laminar and smooth turbulent occur at

lower limit: Rc = 500 (24)

upper limit: Rc = 1200 (25)

For pure wave motion, on the other hand, Kamphuis (1975) stated on the basis of comprehensive experiments that the upper limit of laminar range is located at Ra --- 1.0 × 10 4. A closer inspection of his measurements shows, though, that the upper limit of laminar range can be extended up to

244 1t. Tanaka, A. Thu / Coastal Engineering 22 (1994) 237-254

R, = 2.5 X l0 s (26)

A slightly different Reynolds number for the upper limit of the transition range has been proposed by Jonsson (1980), R a = 3 . 0 X 10 5.

According to the direct measurement of bed shear stress by Kamphuis (1975), the lower limit of the smooth turbulent regime seems to be given as

R a =6.0X 105 (27)

Jensen et al. (1988) reported that the transition to fully developed turbulent flow occurs at much higher Reynolds number, R, = 3.0 X 10 6. In the present study, however, the results based on Kamphuis' laboratory experiment will be used consistently. Later, a comparison will be made between the present study and the measurements in which experimental data by Jensen et al. ( 1988, 1989) will also be included (see Figs. 5b and 8 below).

In order to satisfy the above flow regime criteria for steady flow and for wave motion simultaneously, the Reynolds number for wave-current combined motion should be defined as

R = 500Re + R a (28)

Referring to Eqs. (24) through (28), the transitional region from laminar to smooth turbulent flow under wave-current motion is given as follows in terms of the newly defined Reynolds number:

lower limit: R = 2.5 X 105 (29)

upper limit: R = 6.0 X 105 (30)

When Ra tends to zero, the above equations approach Eqs. (24) and (25), respectively, whereas Eqs. (29) and (30) reduce to Eqs. (26) and (27) in the case of Re=0. Thus, it is concluded that Eqs. (29) and (30) are the generalized conditions which do not contradict the previous studies in the two limiting cases.

4.2. Smooth rough transition

Tanaka and Shuto (1984) and Myrhaug and Slaattelid (1990) commonly applied the following expressions to find the transitional regime from smooth turbulent to rough turbulent under wave-current motion:

aSwk~ =5 (31)

ff*k~ =70 (32) V

Jonsson (1966) employed similar expressions with a slightly different constant on the right hand side of the above equations.

From a practical viewpoint, Eqs. (31) and (32) are not convenient, since they are not straightforward expressions in terms of the known quantities. Here, the flow regime under


8 10

~,, 10 7

o

o tq 106

,.c: t4 o to 5 co 10

o t'4 4 Io 10

10:3

c , , ) . . . . 1

2~ 1 0 3

10 4 6 7 8 g

10 10 5 10 10 10 10

Ra, 500R c or R

Fig. 3. Flow regime diagram for wave motion, steady current and wave-current combination.

pure wave motion and under pure steady flow will be reexamined and will be expressed as a function of given parameters. After that, these will be generalized to wave-current coexistent motion.

For wave motion, a simple replacement of Ucw"* by Uw* in Eqs. (31) and (32) and a substitution of Eqs. (10) and (17) lead to a relationship between Ra and O,J (OZo). The open circles and rhombuses in Fig. 3 show a flow regime criterion under pure wave motion thus obtained.

The flow regime under steady flow can also be obtained by similar replacement and substitution, using Eqs. (31) and (32) together with the current friction coefficient, Eqs. (9) and (16). The flow regime criterion can be expressed as a relationship between Rc and Zh/Zo as shown in Fig. 3 using open squares and asterisks. It is found that the flow regime diagram for a steady current and that for a pure wave motion overlap considerably well, provided that Rc and Zn/Zo are multiplied by 500 and 350, respectively. Here, the former constant is based on Eq. (28).

In order to generalize the results shown in Fig. 3 to wave-current co-existent motion, it is necessary to introduce the following variable:

~'= 3503/z~+ ( I - ,y)/~/w (33) Zo O"Zo


Gzo a 1 0 8

1 0 5

1 0 4.

i0 s

1 0 2

1 0 3 1 0 4. 1 0 5 1 0 6 1 0 7

Gzo b

~ uc/'Uw=2"O, q ~=0" smooth ~(D /~f~

0 5 zhlam=l. 0 -I / ~D

~ ~ .,S in(z=/z )=8

10 2 ~ 1 0 ~ 1 0 ~ 1 0 ~ 1 0 6 0 7

R a

R a

Fig. 4. (a) Comparison of flow regime (uc//~w = 1.0, ~b=O°). (b) Comparison of flow regime (uJOw=2.0, 4~= 0°).

14. Tanaka, A. Thu / Coastal Engineering 22 (1994) 237-254 247

where

o/Ow "Y= 1 +Kc//-]w (34)

Then we get the following relationships which asymptotically reduce to zh/zo vs. R~ relationship and f/w/(O7o) vs. R, relationship in the limits of 3' ~ 0 (or Kc/0,~--* w) and y ~ 0 (or uc/Ow --* 0), respectively.

lower limit: Rl =0.501( H5 (35)

upper limit: Rz = 7.00( Z 15 (36)

Eqs. (35) and (36) as well as the laminar-rough criteria, Eqs. (29) and (30), are shown in Fig. 3.

Since there have been very few studies of flow regime under wave-current combined motion either experimentally or theoretically, it is impossible to make a thorough verification of the above equations. Here, a comparison with the theoretical result obtained by Tanaka and Shuto (1984) is made in Fig. 4. As mentioned above, they used Eqs. (31) and (32) directly. It is observed that the result of the present study is very close to that of the previous study. Similar agreement can be seen for other combinations of Kc//)w and 4), although they are not shown here.

5. Full-range equations

5.1. Friction coefficient

The wave-current friction coefficient formula spanning all flow regimes is proposed as follows, by combining the explicit equations for each flow regime.

f~w =f2 {flfcw(L) + ( 1 --fl )fcw,S) } + ( 1 --f2)f~w(R> (37)

wherefl and f2 are the weight functions. The two terms in the braces on the right-hand side show an interpolation between laminar flow and smooth turbulent flow, while the second interpolation is made in the smooth-rough transitional regime using another weight function f2. The functional forms offj and f2 are obtanied as:

J .4.65

fl = exp { - 0.0513 ( ~ ) ) (38) R

f2 =exp ~-0 .0101 ( R ]2.()6 \~-j} ) (39)

Bothf~ and f2 are determined so that they approach 1.0 and 0.0 when the combined motion tends to a lower and an upper critical value in the transitional region, respectively. In a particular case, when it is confirmed that the flow is fully turbulent, we can substitutef~ = 0 to eliminate the first intepolation in the braces of Eq. (37).

As already shown, the approximate formulae off~w for each flow regime are in an explicit


fw

100

1 0 -I

10 -2

I 0 -3

f w

10 0

I 0 - 1

1 0 -~

1 0 -3

- - E q , ( 3 7 ) ( , U c / O w = O ) I . . . . . . . . I . . . . . . . . [ . . . . . . . . I ~ , , , , , I ~ 1 1

4- . . . . . . . . L . . . . . . . . I . . . . . . . . ] ', I ', I * , ' , ' , :

=2xlO ~ . 293r~ ~ 9 7 . 5 -- 5

x o

~ . . . ~ 4 0 ~ ~..¢.~, 615 ¢ ~ ) 4 5 0 2 • ~ = ~ 5

_ ix103 I080 (--~----m1275 ~ 2

~ 5 4 0 0 . ix104

90O 5

2 |~20250 /__ 43500 5 ixl0 s

''o 82500 ! 2

^ • 5

Numbers refer to U-~ a (~Z°

........ i ........ I ........ i I . . . . . ' " 1 ' ' ' ' . . . . [ ' ' ' . . . . ' l ', I t ~ I : : ', : I I I : ~ ,

02 103 104 105 1 0 e 07RG

- - E q . (37) (Uc/Uw=O)

O'Zo

=2x10 --- 140

~7 ~712 5 ?236 ixl0 z 209 2

~ 6 4 5 ~7 ~7261 852 /

f 150~ ~7 v ¢ 5 2187~7831 372(3 ix103

0 5 ixlO 4

4140~13100 ~ 2

~ 56400~'~ 5 ixT06

2 2 ~ 2

?I 33360 21600~ ~T~ 5

Numbers refer to ~ 813000 \i b ~z I o iii000

I L I I l l , I : " """"::1 : ~ ~zl::ll : ~ I IIIIII " " : :::t

02 1 0 ~ 1 04 1 05 1 O 6 1 0 7 R~

Fig. 5. (a) Wave friction coefficient (data from Kamphuis, 1975 ). (b) Wave friction coefficient. For explanation of symbols see Fig. 6.


form, as well as Eqs. (38) and (39). Hence, the full-range equation for fc,~, Eq. (37), is fully explicit.

Fig. 5a is drawn to show a friction coefficient under wave motion alone. The solid lines show Eq. (37) with uc//~/w = 0.0, and the circles denote the experimental data reported by Kamphuis (1975). The theory shows good agreement with the measurements. In particular, the transitional behavior of the data for Uw/(O7o) = 3000, 20250 and 43500 is well predicted by the present theory. In Fig. 5b, other sets of data cited in Fig. 6 are plotted together with the theory for pure wave motion. The theoretical results are in good agreement with the experimental data. A diagram similar to Fig. 5 has already been proposed by Jonsson (1966). Recently, Myrhaug (1989) also derived a wave friction coefficient on the basis of the analogy between wave boundary layer flow and planetary boundary layer flow. Myrhaug used the following equation originated by Christoffersen and Jonsson (1985) to generalize the rough turbulent solution to smooth turbulent regime:

1 - e x p - ~ ] ) 4 (40) 9ac*

According to Myrhaug's report, Sleath's data for/Jw/(aZo) = 22290 and 33360 showed disagreement with his theory and stated that the reason for this was unclear. In Fig. 5b, these particular data also show good coincidence with the present theory. As explained

author(s)

symbols

smooth rough

Kamphuis (1975) • O

Jonsson & Carlsen (1976) O

Hino et al . (1983) •

Sleath (1987) V

Sumer et a l . (1987) A

Jensen et al . (1989) X

Sawamoto ~ Sato (1990) I~I t>~

Fig. 6. Definition of symbols.

250

f C W

10 °

1 0 -1

1 0 -~

H. T a n a k a , A . T h u / C o a s t a l E n g i n e e r i n g 22 ( 1 9 9 4 ) 2 3 7 - 2 5 4

+ * * E q . ( 3 7 ) . . . . . . . I o m i n u r ( E q . ( 2 ) )

smooth , u r b u l en~ . (Eq :8 ( ;~ ) ) rough t u r b u l e n t ( E q . (

I , , , , , , , , , , ~ , , , , , I , , ~ , , , , , I , , , , , , . . . . . . . . . . I

-~tx=O. 5 ~u---~=l. 0 ,q>=O" -~a=103 i am Uw t Zo +

am ~. GZ °

q~ '+'.~. ,~ _}~ W~ ~A~''~AA'~'~;'A'~ ;'~'AA ;'~ "[ =ix102

" ~ . i ~ _ ; ~ . . . * , * - : . . . . . . . . . . . . . . . . . . . . . . . . w 5 X i 0 2

\~'>-.'*.2-~.~* + + + + + : ' , : : : : : : : ~ - ~ = i o ~ , ~=o.s 2~i0~ ~,~ "~--4..... z . 2

. . . . . . . ,I . . . . . . . . I . . . . . . , , I , , , ~ " " r - ~ , , , ,

........ I ........ I ........ I ........ I ........

02 1 03 1 04 1 0 ~ 1 06 07 Ra

fcw

10 0

1 0 -I

1 0 -2

+ * * E q . ( 5 7 ) . . . . . . . . . l a m i n a r ( E q . ( 2 ) ) - - - s m o o t h t u r b u l e n t ( E q . ( 1 5 ) ) - - - rough t u r b u l e n t ( E q . [ 8 ) )

o-z o

. " ~ . . " , * * * ~ = = = : = = ~ ~ : : ~ ~ ~ ~ - - , .++++++:.,,,::,.::: sx10 2. o. -" . 4- Z h

. . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . . I , , . . . . . .

. . . . . . . . I . . . . . . . . , . . . . . . . . I . . . . . . . . I . . . . . . ; ' 0 7 R Q 02 1 03 1 04 1 0 ~ 1 06

Fig. 7. Wave-current friction coefficient, u ~ / U w = ( a ) 1.0, (b ) 2.0. qS=O °, z h l z ( ~ = 10 3, z h / a m = 0 . 5 ( + ); 2 ( * ) 10 (~#).

previously, Eq. (40) is valid as long as the flow is in turbulent regime. As seen in Fig. 5b, two specific experimental data from Sleath are located in the lower regime of the transitional region between laminar and rough turbulent. This is probably the reason why Myrhaug's theory deviated from these experimental results.


a(d g) 50

5 0 -

4 0

3 0

2 0

10

0

A

- - E q . ( 4 - 1 ) ( 5 c / U w = 0 )

. . . . . . . . l a m i no r

. . . . . . . . s m o o t h t u r b u l e n t . . . . . . . . I . . . . . " H . . . . . . . . I . . . . . . . . I

, , , ,,,,,[ I I I I I I I I

" - ffZo 852 O "

"--. ~ \ ~ I <>~ l x i 0 ~ 4- l x 1 0 3

Numbers refer to w xZ ~ 1~ o~

,,,,,, t,J,,J,, l ,~ i:::::~[ ::::::::, : :::::::, : :::::::,

0 2 1 0 ~ 1 0 4 1 0 5 1 0 ~ 1 0 7 Ra

Fig. 8. Phase difference under pure wave motion. For explanation of symbols see Fig. 6.

Fig. 7 shows an example of a friction coefficient diagram under waves and current including all flow regimes. With the increase of the Reynolds number, the transition from laminar to smooth turbulent takes place and, finally, proceeds to the rough turbulent region.

5.2. Phase di f ference

Similar to the full-range equation for few, the generalized equation for the phase difference can be derived using the same weight functionsf~ and f2.

0----f2 {fl 0(L) -~ ( 1 --fl ) 0(S) } + ( 1 --f2) O(R) (41)

A diagram of phase difference under pure wave motion is drawn in Fig. 8 to compare with the experimental data listed in Fig. 6. Agreement with the experimental data over a smooth bottom (6'w/(a%) = oo) is fairly good, while the theory tends to give a smaller phase lag in the rough turbulent regime. The phase difference under combined motion is also drawn in Fig. 9 for-ffc/Uw = 1.0 and 2.0.

6. C o n c l u s i o n s

The principal results of this study are: (1) Explicit formulae for the wave-current friction coefficient, f~w, are obtained for

laminar, smooth turbulent and rough turbulent flow on the basis of the theoretical study by Tanaka and Shuto (1981). Approximate formulae for the phase difference of the bottom shear stress are also derived in an explicit form for all the flow regimes.


+ * * Eq . ( 4 1 ) . . . . . . . . l ami n a r - - s m o o t h t u r b u l e n t

~ ( d e g ) r o u g h t u r b u l e n t 5 0 I . . . . . . . . = . . . . . . . . i . . . . . . . . i . . . . . . . . , . . . . . . t } . . . . . . . . , . . . . . . . . , , , , _ , , , , , . . . . . . . . , . . . . . . .

f : *~- ' -+ '"+ .; ~=1.o, ,~=o' ,Z":1o ~, :F

* l~ ~ ~ I( ~ ............. ?-I- 5xlO

++++++ : : : : : : : : : : 2xlO s

1 0 '

I 0 2 I 0 "~ I 0 4 I 0 ~ 1 0 8

+ * , Eq.(41) . . . . . . . I om~ n o r - - s m o o t h t u r b u l e n t

4 ( d e , )

5O

40

3 0

20

10

0

0 7 R o

I r o u g h t u r b l u l e n t I i I I """"""1 " " """"'dl ~ ~ ~ " ~ 1 " : : I : H ~ I " I I ~ H ~ L

~ + + + - . i - . . . . . . u-~=2 0 ~,=0" z h = 1 0 3 4- * - - - - u ~ z " = 2 O,, " '~" 'Zo j - -- am " 4-

[_ + ~ 0"Zo - .k .a, ~ ,A, .g.,~ ;, ~ ~., ~ ~ ,~ ~ ~, ~ ~ ~ . . . . . . . =ixlO a

, * * * , , , . . . . . . . . .'. [ ?. ? [? . : t

.-~--to -~--2 ~-~--o.s ~ T

. a,,,, an, a~ b , , , , , , , , I , , . . . . . . I . . . . . . . . i . . . . . . . . I . . . . . . . . l . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . . I

02 10 -~ 104 1 0 ~ 108 107 Ro

Fig. 9. Phase difference under waves and current, u¢lUw = (a) 1.0, (b) 2 .0 .4 ,= 0 °, Zh/Zo = 103, Zh/a,n = 0.5 + );

2 ( * ) 10 (* ) .

(2) The flow regime criteria for steady current and for wave motion are examined separately using the data of the previous studies by Chow (1973), Patel and Head (1968) and Kamphuis (1975). Moreover, these can be generalized to cover wave-current coexistent motion, which is found to be very close to the theory proposed by Tanaka and Shuto (1984).

(3) The equations for the friction coefficient and the phase difference spanning all flow


regimes are derived by combining the expressions for each flow regime in collaboration with the flow regime criteria. Since all the approximate formulae as well as the flow regime criteria are given in a simple explicit form, the full-range equation for few and 0 not only gives the solution without a trial-and-error procedure, but also avoids interpolation and provides unique value in the transitional region.

References

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Aydin, I., 1987. Computation and Analysis of Turbulent Flow on Flat Bottom and over Rigid Ripples. Ph.D. Dissertation, Tohoku University, 193 pp.

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full-range equation of friction coefficient and phase difference in a wave-current boundary layer

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