Effect of Wave-Current Interaction on the Wave Parameter

(1)

26K.17

EFFECT

OF

WAVE-CURRENT

INTERACTION

ON

THE

WAVE

PARAMETER

BV

VU-CHENG LI AND

.J OH N B. HER BIC H

OCEAN ENGINEERING PROGRAM

TEXAS ENGINEERING EXPERIMENT STATION TEXAS AaM UNIVERSITY

COLLEGE STATION TEXAS 77843

DEPARTMENT OF CIVIL ENGINEERING .TEXAS ENGINEERING EXPERIMENT STATION

CO E REPORT NO. 257 FEBRUARV 19S2

(2)

EFFECT OF WAVE-CURRENT INTERACTION ON

THE WAVE PARAMETER

by

Yu-Cheng Li and

John B. Herbi eh Oeean Engineering Program

Department of Civi1 Engineering Texas Engineering Experiment Station

COE Report No. 257

(3)

PREFACE

The study described in this report was conducted as part of a continuing research effort within the Ocean Engineering Program.

The Texas Engineering Experiment Station provided funds to defray

part of the computer costs and assisted in preparation of the drawings. The manuscript was typed for publication by Joyce Hyden.

(4)

TABLE 0 F CONTENTS

Page

PREFACE . . . ii

TABLE OF CONTENTS iii

LIST OF TABLES iv

LIST OF FIGURES v

ABSTRACT . . 1

INTRODUCTION 2

THEORETICAL CONSIDERATIONS 3

A. Employing the wave energy flux equation 5

B. Employing the wave action flux equation 12

DATA ANALYSIS AND RESULTS . . . 13 A. The resul ts of modifi ca tion of wave 1ength by the

1inear wave theory due to

wave-current interaction ..

13

B. Modification of wave length by non-linear wave

theory and error estimates due to 1inear theory

.

17

C. Modification of wave height

...

. . . .

17

D. Comparison of the resu1ts of the wave height change

by numerical calcu1ation with model test data

36 CONCLUSIONS

36

(5)

LIST OF TABLES

Table Page

The ratio of wave 1enqth LN/L . . . 18 2 A comparison of wave length of Fig. 4 with

test data and non-linear wave theories results 19 3 A comparison of the wave length by Fig. 4 with

the model test data of Hales and Herbich 20 25

4 5

Wave height change H/Hs .

Comparison of calculated values with Hales' and

Herbichls (H&H). . . 37

(6)

Fi gure 1 2

LIST OF FIGURES

Wave Current Interaction Process . . . . .

Wave Length Change Computed by Linear Wave Theory (L/Ls versus U/C) •••...•••••••••.

3 The Ratio of Relative ~/ave Celerity Due to Wave-Current Interaction to the Wave Celerity in Still

4 5 6 7 8 9 10 11 12 13 Wate r • . . • • . . • . . • • • .

The Wave Length Change Computed by Linear Wave Theory (:L/Ls versus U/Cs) .•. . ..••.•

The Wave Height Change Computed by Linear Wave Theory Using the Equation of Conservation of Wave Energy Flux .••..••.••..•••.••.

The Wave Height Change Computed by Linear Wave Theory Using the Equation of Conservation of Wave Action Fl ux . . . . The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equation of Conservation of Wave Energy Flux for H/Ls

=

0.01 . The Wave Height Change by Old Stokes Third Order Wave Theory Usi ng the Equation of Conserva ti on of Wave Energy Flux for Hs/Ls

=

0.03 . The Wave Hei ght Change by Ol d Stokes Thi rd Order Wave Theory Using the Equation of Conservation of Wave Energy Flux for Hs/Ls

=

0.05 .

The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equation of Conservation of Wave Energy Flux for Hs/Ls

=

0.06 •••••..•

The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equation of Conservation of Wave Energy Flux for Hs/Ls

=

0.07 ••....••

The Wave Height Change by New Stokes Third Order Wave Theory Using the Equation of Conservation of Wave Energy Fl ux for H/Ls = 0.01 •..•..••

The Wave Height Change by New Stokes Third Order Wave Theory Using the Equation of Conservation of Wave Energy Flux for Hs/Ls

=

0.03 .

Page 6 14 15 16 22 23 26 27 28 29 30 31 32

(7)

Figure

14

15

16

LIST OF FIGURES

The Wave Height Change by New Stokes Third Order

Wave Theory Using the Equation of Conservation of

Wave Energy Flux for Hs/Ls

=

0.05 .

The Wave Height Change by New Stokes Third Order

Wave Theory Using the Equation of Conservation of

Wave Energy Flux for H

_ss

IL

_·

=

0.06 .

The Wave Height Change by New Stokes Third Order

Wave Theory Using the Equation of Conservation of

Wave Energy Flux for Hs/Ls

=

0.07 .

vi

Page

33

34

(8)

EFFECT OF WAVE-CURRENT INTERACTION ON THE WAVE PARAMETER

by

Yu-Cheng Li* and John B. Herbich**

ABSTRACT

The interaction of a gravity wave with a steady uniform current is discussed in this paper. Analysis indicates there is no dominant dif-ference in the resu1ts obtained when emp10ying either the equation of conservation of wave energy flux or the equation of conservation of wave action flux. Numerical ca1cu1ations of the wave 1ength change by different non-1inear wave theories show that errors in the resu1ts computed by the 1inear wave theory are less than 10 percent within

H

the range of 0.15 S ~ ~ 0.40, 0.01 ~ ~ ~ 0.07 and -0.15 ~ ~ ~ 0.30,

s s s

where

d

=

water depth,

Hs

=

wave height in still water,

Ls

=

wave length in still water,

Cs

=

wave celerity in still water, and

U

=

surface current velocity.

Numerical calculations of wave height change employing different wave

theories show that errors in the results obtained by the linear wave

*V;siting Scholar, Ocean Engineering Program, Texas A&M University;

Department of Hydraulic Engineering, Da1ian Institute of Technology,

Dalian, People's Republic of China, where the author is an Associate

Professor.

**Professor and Head, Ocean Engineering Program, Texas A&M University,

College Station, Texas 77843, U.S.A.

(9)

theory in cOOlparison with the non-linear theories are greater when the opposing relative current and wave steepness become larger. However. within range of the following currents such errors will not be

significant. These results were verified by model tests. Nomograms for the modification of wave length and height by the linear wave theory and Stokes' third order theory are presented for a wide range of d/Ls' Hs/Ls and U/Co

These nomograms prövide the design engineer

with a practical guide for estimating wave lengths and heights affected

by currents.

INTRODUCTI ON

With increasing human activities in both the coastal and immediate

offshore reg ion, the problem of wave-current interaction has been

evaluated by a number of researchers.

From an engineering practice

point of view, the effect of wave current interaction on the wave

para-meter must be known.

Previous research involved different assumptions

and considerations.

Several researchers employed the linear wave theory

combined with the idea of conservation of energy flux4,12,13, or

com-bined with the idea of conservation of wave action flux8,9; others

em-ployed the non-l inear wave theory6, 15.

The prob1em concerni ng the

interaction of waves and currents in an inlet has also been studied2,7.

Additional research5,11 describes the change of velocity distribution

due to the interaction of waves with currents.

One purpose of this study was to evaluate the difference between

the change in wave parameter when employing either the equation of

conservation of wave energy flux or the equation of conservation

:

of

(10)

wave action flux. The results were evaluated with reference to practical engineering applications. Another purpose of this study was to determine the error in the calculated wave parameter by employ-ing both the linear wave theory and non-linear wave theor1es (Stokes' third-order wave theory was mainly considered). A two-dimensional case was considered and both the following and the opposing currents were analyzed.

THEORETICAL CONSIDERATIONS

Evaluation of change in the wave parameter may be obtained in two steps: a) evaluation of the change in wave length and, b) evaluation of the change in wave height. In both cases, the energy dissipation in wave propagation was neglected.

Change in Wave Length

The linear wave theory was considered first. The influence of non-linearity of waves is evaluated in the latter part of this section.

Wave celerity in still water (Cs) may be determined by the following linear wave equation

(1)

where ks

=

the wave number in still water

,

ks

=

21T ,

and

Ls

Th

=

hyperbolic tangent

The wave celerity is affected by the current and may be calculated as

(2 )

follows:

C

=

U

+

C

(11)

where

Ca

=

actua1 wave ce1erity as seen by an observer, and

Cr

=

relative ce1erity to the current, which may be obtained by

the fo11owing equation

(4 )

where

2'IT 2'IT

kr

=

L

=

T ' and

r

Lr

=

L (signifying the wave length propagated in a current).

Equation 3 can be rewritten as

(5)

wh ere

T

=

T (signifying the actual period of a wave which is not

a

affected by the interaction of waves and currents).

From Equations 1,4

and 5, the fo11owing equation can be obtained

(6)

and L

=

"

La'

C = C

a

In engineering applications, the value of C

=

Ca is unknown, therefore,

the ratio U/C is a1so unknown.

Thus the fo11owing formu1a may be used

(l)

(12)

Combining Equations 6 and 7, the re1ationship between ~ with ~

s

and ksd can be obtained from

LIL

_s

=

CIC

_sC'

=

f(JL ~)

_L

s s

(8)

and

Change of Wave Height

The process of the interaction of progressive waves with a steady

uniform current (the fo11owing or the opposing current on1y) may be

eva1uated as fo11ows:

(a) in the first step on1y currents and waves

propagate separate1y,

(b) in the second step the combined current

and waves interact and finally a steady motion of wave-current

combination propagates as shown in Figure 1.

If we neg1ect the change

in the current profile because of the wave-current interaction, the current

/

energy flux wi11 remain the same before and after the interaction

pro-cess.

A. Emp10yina the wave energy flux equation

Under steady flow conditions, the fo11owing equation (Equation 9)

(first derived by Longuet-Higgins and Stewart13), can be used

:x [E(U

+

Cg)

+

Sxx

~~J

=

0 (9)

where

(13)

1

Steady Uniform Current (only) 0\

2

Progressive Waves (only)

Fig. 1.

Wave-Current Interaction Process

3

Steady Wave-current

(14)

Cg

=

group velocity of the wave,

Sxx

=

radiation stress, and

x

=

horizontal axis.

Comparing cases

1

and

2

with case

3

(Figure

1)

(the current being

steady in both cases), it is noted that the radiation stress term in

Equation 9 equals zero, i.e.

Sxx

l!:!

=

0

ax

(10)

Therefore Equation

9

can be rewritten as

(11 )

Thus the wave energy flux before and after the interaction of

wave-current should be the same.

If the value in the case of waves only

(i.e. still water) is denoted with subscript s, Equation

11

can be

re-placed by

(12)

or

(13)

Since

1 2kd

Cg

= ~

Cr(l

+

Sh2kd)

where Sh

=

hyperbolic sine

(14 )

The first part of the right hand of Equation

13

may be written as

1

=

(l+2U. 1 . _A1)-1 (15)

1.JL

Cs

_f_ _

JL

(15)

where

2kd

A = 1 +

Sh2kd

( 16)

Assuming

( 17)

where

ks

=

wave number in still water,

k

=

2'1r and

s

L_s '

Ls

=

wave length in still water.

Then Equation 13 can be rewritten as

(18)

Considering the linear wave theory, the wave height change can be

cal-culated as follows

H _

2U

1 1 -1/2

Cs/C

1/2

As 1/2

(1 + - U A) ( U) (A) Hs - C 1 1

- ë

(19 )

Employing Stokes' third order wave theory, and assuming that Equation 14

may be utilized, it

is

assumed that Equation 18

is

correct.

The results

by Skjelbreia14, Tsuchiya and Yasuda17 were employed in calculating the

wave energy.

Employing Skjelbreia's Method

Wave energy,E,may be calculated by the following equation

_ ~ 2 H)2 3 2 H)2

E - 8 {l + 'Ir (r [B +

32x256

'Ir (r F]} (20)

(16)

where Ch41Td

B - 1 (

1

)2(1 +

3)2+

f:i

L

6 - 2

Th2~d

1+2(Sh2~d)Z

(Sh2~d)

Ch

=

hyperbolic eosine

(21)

and

F

=

(22)

Accordingly, wave energy in still wate~ Es'can be ca1cu1ated

similarly as

(23)

where Bs and Fs may be obtained from a similar formula as Band

F in

Equations 21 and 22 (chanqing L to Ls only).

The wave height change

can then be calculated as follows

H _ 2U

1 -1/2

Cs/C

1/2

As 1/2

(1 + - U

A)

(

U) (A) Hs - C 1 -

C

1 -

C

1/2

F]}

(24)

This equation must be solved by iteration.

Employing Tsuchiya's Method

(17)

(25)

where C is wave ce1erity, A is defined by Equation 16, and G is

G

=

2Sh4kd

(26)

Since

1 (

"2kd)

W

=

2 ECr

1

+

1

+

Sh2kd

(27)

Thus wave energy

m~

be determined as follows

(28)

and wave energy in still water Es is

'!TH 2 G

E

= ~

yH 2 {1 + 2 ( s) s}

sos

-ç

7Ç

(29)

where Gs can be determined simi1ar1y as G from Equation 26, and

wave number k is changed to ks in the case of still water.

As

aresult,

the wave height change can be ca1cu1ated as fo11ows

" "1THs 2 Gs

1/2

1 -1/2

C /C

1/2

A

1/2

{1 +

2(-L-)

r}

H = (1 + 2U 1 ) (s U) (+) s s Hs C 1 U Ä '" H 2 G

1/2

- C 1 - C {1 + 2('!TL) A} (30) 10

(18)

Equation 30 must also be solved iteratively. When the non-linear wave theory is used, wave ce1erity is determined not on1y by the parameter kd but a1so by wave steepness ~. Oue to the interaction of wave and current, the wave steepness ~ can be changed, since

the wave length determined by the non-1inear wave theory LN is not

the same as that determined by the 1inear theory L, i.e. LN ~ L.

The va1ue of LN/L can be determined as follows

Emp10ying Skje1breia's Method

Wave celerity C

=

(~Thkd)1/2

{1 +

(nH)

2 14+4Ch

2 2kd}

(31)

N

k

L

16Sh

4

kd

and

L /L

=

1 + __

1 (n~)2

14+4Ch

2 2kd

=

N

(32)

N

16 L

Sh

4

kd

1 where wave steepness ~ is the value under wave-current interaction.

LN

L

Thus, L /L

= . =

N

__

(33)

N

s

L

Ls

1 Ls

Employing Tsuchiya's Method

Wave celerity CN

=

(~Thkd)1/2{1

+

J:

(n!!)2

1

(8Ch

4 kd

k 16 L

Sh

4

kd

(34)

and

(35)

Thus

(36)

(19)

Coefficients

Nl and N2 show the influence of non-linearity

of waves.

B. Employing

the wave action flux equation

Similarly

to Case

A,

the following equation derived by Jonsson

8

may be used

(37)

where

w

r

=

21T/Tr represents the relative angular frequency;

Cga is the

apparent wave group velocity.

Equation 37 indicates that wave action flux before and after

interaction must be the same, i.e.

(38)

where

W

=

w -

kU

=

w -

kU

ras

(39) (40)

C

=

1 C A

gs

"2

s s

(41)

Cga

=

U

+

Cgr

= U + }

CrA

(42)

T

hus

E

quation

38

can be rewr

i

tten as follows

_E __ U Ls As U

2-A

-1

Es

.

(1 -

C)

T

A (1 +

ë

T) (43)

(20)

According1y, the wave height change can be determined by the f~llowing equations

Linear Wave Theory

1/2 L 1/2 A 1/2 - 1/2

H:

=

(1 - ~) (-r-) (i) (1 + ~ 2 ÄA)

=

R ( 44 )

/

Skje1breia's Stokes' Third-Order Wave Theory (Old)

H

1Ç=

(45)

Tsuchiya's Stokes' Third-Order Wave Theory (New)

H _

Hs

-lTH 2 G 1/2 {1+2(_s) 2}

Ls

As

R 2 1/2 {l+2(lTH) §_}

L

A

(46)

If the non-linear wave theory is used, the wave 1ength given by the

1inear wave theory must be modified.

DATA ANALYSIS AND RESULTS

A. The resu1ts of modification of wave 1ength by the 1inear wave

theory due to wave-current interaction

Using Equations 6, 7 and 8 different combinations of dimension1ess

parameters ~ (or

iL),

and

[L

may be calculated.

The resu1ts are shown

s s

in Figures 2, 3 and 4.

As shown in Figure 2, it is clear that the

wave length change ~

is greater in deep water with the relative

(21)

2.40

LINEAR THEORY

2.00-; /d/L=0.40 0.35 0.30 0.25 0.20 0.15 1.60-1

//////

₀_.₁₀ 0.80{~ .~

en

...J 1.20 <, ...J 0.40

O

.

OO+---r----...,...--_...,...

-.--

...

-0.20 -0.10 0.00 0.10

U/Cr

Fi g. 2. Wave Length Change Computed by Linear Wave Theory (L/Ls versus u/tr)

(22)

o

<,

:::>

0.05 <..n 0.70

Cr/Cs

Figure 3.

The Ratio of Relative Wave Celerity Due to Wave-Current Interaction to the

(23)

=

0.50

o.

o

o.

L/Ls

or

C/C

s

Fig. 4. The Wave Length Change Canputed by Linear Wave Theory (U/Cs versus L/Ls)

(24)

C

current ~ than in shallow water. The variation of ratio

tf

with the ratio

g

is affected in a similar manner.

B. Modification of wave length by non-linear wave theory and error estimates due to linear theory

Numerical calculations of the wave length were made employing

Stokes' third order wave theory14,17. The ratio of wave lenqth computed by the non-linear wave theory to that computed by the linear wave theory is given in Table 1. It can be seen that the wave length calculated by

the non-linear wave theory is usually greater than that calculated by H

the linear theory. Within the range of 0.15 s 0.40, 0.01 ~ ~ ~ 0.07

U s

and -0. 15 ~

c- ~

0.30, the wave lengths computed by the linear wave s

theory are approximately 10 percent less than those computed from Stokes' third-order wave theory. Table 2 shows a comparison of the results obtained by the 1inear wave theory with data obtained by the

non-1inear wave theories and with model test data. It appears that the r~-su1ts obtained employing different methods are quite similar. A1so a

comparison of the wave 1ength (obtained from the 1inear wave theory) with Hales' and Herbich's model tests data2 are given in Table 3; the

agreement is considered quite good. C. Modification of wave height

The results obtained with the linear wave theory using both the principle of conservation of wave energy flux (Equation 19) and

wave action flux (Equation 44) are shown in Figures 5 and 6, respectively. The results given by Skjelbreia's Stokes' third-order wave theory for different wave steepnesses (i.e. Hs/Ls

=

0.01 - 0.07) using the equation of conservation of wave energy flux (Equation 24) are shown in

(25)

Tab1e 1. The ratio of wave 1ength LN/L***

Hs U/C = -0.15 -0. 10 0.00 0.10 0.20 0.30

d s

ç

_ç

0.20 0.01 1.002 1.002 1.001 1.002 1.001 1.002 1.001 1.001 1.001 1. 001 1.001 1.002 0.03 1.010 1.019 1.008 1.016 1.008 1.014 1.007 1.013 1.008 1.013 1.010 1.017 0.05 1.025 1.048 1.024 1.042 1.021 1.037 1.021 1.034 1.023 1.038 1.029 1.050 0.06 1.034 1.064 1.032 1.057 1.031 1.054 '.031 1.049 1.035 1.056 1.044 1.077 0.07 1.043 1.082 1.042 1.074 1.041 1.072 1.043 1.067 1.049 1.078 1.065 1. 117 0.30 0.01 1.002 1.002 1.002 1.002 1.001 1.001 1.001 1.001 1.001 1.001 1.000 1.001 0.03 1. 007 1.018 1.006 1.014 1.005 1.010 1.004 1. 007 1.004 1.006 1.004 1.006 ... 0.05 1.013 1.043 1.013 1.035 1.014 1.026 1.013 1.020 1.013 1.017 1.013 1.018 cc 0.06 1.017 1.058 1.018 1.047 1.020 1.037 1.020 1.028 1.020 1.025 1.021 1.026 0.07 1.022 1.072 1.023 1.060 1.027 1.049 1.029 1.037 1.029 1.034 1.032 1.035 0.40 0.01 1.001 1.002 1. 001 1.002 1. 001 1.001 1.001 1.001 1.001 1.001 1.000 1.000 0.03 1.004 1.018 1.004 ,1.014 1.005 1.009 1.004 1.006 1.004 1.005 1.004 1.004 0.05 1.008 1.043 1.009 1.034 1.013 1.024 1.015 1.017 1.015 1.013 1.015 1.012 0.06 1.Ol0 1.057 1.013 1.045 1.018 1.034 1.023 1.024 1.018 1.019 1.025 1.017 0.07 1.013 1.071 1.016 1.058 1.024 1.045 1.033 1.032 1.039 1.026 1.041 1.023 .*Skjelbreia IS resu1t -·*i~uchiya's resu1t

(26)

d(m) wave height H(m) T (sec)

u

(m/s) Lexp. (m) d/Ls U/Cs (L/Ls)cal Lca1(m) error in L l-L

wave

(%)=

ca eXPxl00 Note

length Lexp

Table 2. A comparison of wave length of Fig. 4 with test data and non-linear wave theories results.

0.587 0.021 1.25 -0.12 2.05 0.26 -0.067 0.88 1.98 -3.4 exp.by G. P. Thomas16 _. 1.0 0.570 0.024 1 .25 -0.20 1.82 0.26 -0.11 0.81 1.81 -0.5

_

"_-30.5 15.2 10.0 +0.90 152.5 0.22 +0.07 1.10 151 -1.0 cal. by Dalrymplel cal. by T.S. Hedges* 10 2.5 12.0 +1.0 130 0.09 +0. 11 1.09 126 -3.0

(27)

Tab1e 3~ A comparison of the wave 1ength by Fig. 4 with the model test data of Ha1es and Herbich

.. th L 1-L

error m ~%)

=

ca mea x 100 d(m) _Lmea(m) U(m/s) d/Ls U/Cs (L/Ls)cal Lca1 wave 1ength L .

(from Fig. 4} mea 0.098 0.79 +0.277(flood 0.17 0.34 1.41 0.80 +1.2 0.94 current)0.14 0.32 1.37 0.96 +2.0 1.16 0.11 0.30 1.32 1.20 +3.4. 1.39 0.09 0.30 1.32 1.44 +3.6 1.58 0.08 0.29 1.31 1.63 +3.2 2.00 0.06 0.28 1.30 2.06 +3.0 2.25 0.055 0.28 1.28 2.29 +1.8 0.098 0.69 +0.143(f1ood 0.17 0.17 1.21 0.69 O. N 0.83 current)O.14 0.16 1.20 0.84 +L2 0 _1.04 _0.11 _0.16 _1.18 _1.07 _+2.9 1.26 0.09 0.16 1.18 1.29 +2.3 1.44 0.08 0.15 1.16 1.45 +0.7 1.79 0.06 0.15 1.15 1.83 +2.2 2.02 0.055 0.15 1.15 2.06 +2.0 0.098 0.50 -0.076(ebb 0.17 -0.09 0.88 0.50 ~ 0 0.62 current)O.14 -0.09 0.88 0.62 0 0.82 0.11 -0.08 0.91 0.82 -1.0 1.00 0.09 -0.08 0.91 0.99 0 1.14 0.08 .,.0.08 0.91 1.14 0 1.49 0.06 -0.08 0.92 1.46 -2.0 1.65 0.055 -0.08 0.92 1.65 0 0.098 0.42 -0~165(ebb 0.17 -0.20 0.70 0.40 -4.7 0.53 current)0.14 -0.19 0.74 0.52 1.9 0.71 0.11 -0.18 0.78 0.71 0 0.88 0.09 -0.18 0.79 0.86 -2.3 1.03 0.08 -0.17 0.80 0.99 -3.9 1.32 0.06 -0.17 0.82 1.30 -1.4 1.50 0.055 -0.17 0.82 1.47 -2.0

(28)

Tab1e 3. (continued)

L -L

error in th~%) = cal mea x 100

d(m) _Lmea(m) U(m/s) d/Ls U/Cs (L/Ls)ca1 Lca1 wave 1ength Lmea

(from Fig. 4) 0.445 0.65 +0.265(f1ood 1.07 0.33 1.22 11 _{current) 0.47} _0.22 _1.39 1.31 +7.4 1.72 11 _0.32 _0.19 _1.29 _1.77 +2.9. 2.41 11 _0.22 _0.16 _1.20 _2.46 +2.1 3.04 11 _0.17 _0.15 _1.18 _3.06 +0.7 4.07 11 _0.12 _0.14 _1.16 _4.15 +1.9 0.445 0.54 +0.134(f1ood 1.07 0.17 1.10 11 _{current) 0.47}

o.

₁₁ _1.20 _1.13 +2.7 1.57 11 _0.32 _0.09 _1.14 _1.56 _-0.6 N 2.23 11 _0.22 _0.08

_1.11

_2.26 +1.3 __. 2.84 11 _0.17 _0.08 _1.09 _2.83 _-0.4 3.81 11 _0.12 _0.07 _1.08 _3.86 +1.3 0.445 0.37 -0.055(ebb 1.07 -0.07 0.92 11 _{current) 0.47} _-0.05 _0.92 _0.87 -5.4 1.34 11 _0.32 _-0.04 _0.94 _1.29 _-3.7 1.95 11 _0.22 _-0.03 _0.95 _1.94 _-0.5 2.51 11 _0.17 _-0.03 _0.97 _2.51 ₀ 3.48 11 _0.12 _-0.03 _0.97 _3.47 _-0.3 _.0.445 0.29 -0.168(ebb 1.07 -0.21 0.76 11 _{current) 0.47} _-0.14 _0.73 _0.68 -10.5 1.18 11 _0.32 _-0.12 _0.80 _1.09 -7.6 1.80 11 _0.22 _-0.10 _0.84 _1.72 _-4.4 2.29 11 _O.17 _-0.09 _9.88 _2.29 0 3.20 11 _0.12 _-0.09 _0.89 _3.18 _-0.6

(29)

H/HS LINEAR THEORY WAVE ENERGY FLUX

1.50 dil =0.10 0.15 0.20 0.25 0.30 0.35 0.40 1.30 N N 1.10

en

I

<,

I

0.90 0.70 0.50, I I I I I • 0.20 _-0.n -en_IV ... 0.00 0.10

U/C

0.20 ₀_.₃₀ _0.40

Fig. 5.

_{The Wave Height Change Computed}

_by

_{Linear Wave Theory Using the Equation}

_of

(30)

en

:r:

<, 0.80

:r:

0.60 0.40 0.20

H/HS, LINEAR THEORY WAVE ACTION FLUX

1.20 1.00 d/L=0.10 0.20 0.30 0.40 O.OOI-+---r---r---r-- __ ---. ----..~--___. -0.20 -0.10 0.00 0.10

U/C

0.20 0.30 .0.40

(31)

Figures 7-11. The resu1ts given by the .equation of conservation of wave action flux are shown in Tab1e 4.

The resu1ts shown by Tsuchiya's modification of Stokes' third-order wave theory are presented in Figures 12-16 for wave steepness Hs/Ls

=

0.01 - 0.07, using the equation of conservation of wave energy

f1ux.

A comparison of the resu1ts emp10ying the 1inear theory (Figures 5

and 6) wit~ that of Skje1breia's method (Figures 7 through 11) is

shown in Tab1e 4.

The fo11owing observations were made:

(1)

In a iteady current,the modification of wave height due to

wave-current interaction ca1cu1ated by the equation of

con-servation of wave energy flux and by that of wave action flux

is quite simi1ar

.

(2)

In the following current, errors in wave height change by

the 1inear wave theory are minimal in comparison with the

resu1ts obtained by the non-1inear wave theory.

(3)

When there is an opposing re1ative current and the wave

steepness is sufficient1y large, errors in wave height

em-p10ying the 1inear theory wi11 be greater than errors

ob-.

tained emp10ying the non-1inear theory.

(4)

When the re1ative current velocity U/C increases, the change

of wave height by Tsuchiya's theory differs from the(above

resu

1t

s

.

U

s

i

n

g

his method, unde

r

deep water conditions the

wave height change H/H

s

increases with the increase of U/C

and reaches a peak va1ue, after which the va1ue of H/Hs

de-creases with the increase of U/Co

(32)

Tab1e 4. Wave height change H/Hs

d Hs U/C = -0.15 -0.10 0.00 0.10 0.20 0.30

ç ç

Old 01d 01d Old 01d 01d

Airy Stokes 3 Airy Stokes 3 Airy Stokes 3 Airy Stokes 3 Airy Stokes 3 Airy Stokes 3

0.20 0.01 1.217* 1.213 1.137 1.135 1.000 1.000 0.886 0.886 0.792 0.792 0.709 0.709 (1.305~*(1.300) (1.193) (1.190) (1.000) (1.000) (0.841) (0.841) (0.708) (0.708) (0.593) (0.593) 0.03 1.184 1.115 1,000 0.883 0.792 0.710 (1.264) (1.167) (1.000) (0.838) (0.709) (0.595) 0.05 1.139 1.082 1.000 0.877 0.791 0.712 (1.210) (1.129) (1.000) (0.834) (0.709) (0.597) 0.06 1.115 1.063 1.000 0.873 0.790 0.713 (1.182) (1.108) (1.000) (0.830) (0.710) (0.599) 0.07 1.091 1.044 1.000 0.869 0.790 0.715 (1.154) (1.086) (1.000) (0.827) (0.710) (0.601 ) 0.30 0.01 1.267* 1.262 1.171 1.167 1.000 1.000 0.859 0.859 0.742 0.742 0.646 0.646 N (1.359)**(1.353) (1.228) (1.224) (1.000) (1.000) (0.815) (0.815) (0.664) (0.664) (0.541) (0.541) U1 _0.03 _1.227 _1.144 _1.000 _0.856 0.743 0.648 (1.310) (1.197) (1.000) (0.813) (0.665) (0.542) 0.05 1.173 1.106 1.000 0.852 0.743 0.651 (1.245) (1.154) (1.000) (0.810) (0.666) (0.545) 0.06 1.146 1.086 1.000 0.849 0.744 0.652 (1.213) (1.130) (1.000) (0.807) (0.667) (0.547) 0.07 1.118 1.064 1.000 0.845 0.744 0.655 (1.182) (1.106) (1.000) (0.805) (0.669) (0.550) 0.40 0.01 1.266* 1.260 1.172 1.168 1.000 1.000 0.850 0.849 0.722 0.722 0.616 0.616

0 .

.

357)1r*(1.351) (1.229) (1.225) (1.000) (1.000) (0.806) (0.806) (0.646) (0.646) (0.515) (0.515) 0.03 1.224 1.144 1.000 0.847 0.723 0.617 (1.306) (1.197) (1.000) (0.804) (0.647) (0.517) 0.05 1.169 1.106 1.000 0.843 0.724 0.621 (1.240) (1.153) (1.000) (0.802) (0.649 ) (0.520) 0.06 1.140 1.084 1.000 0.841 0.725 0.623 (1.207) (1.129) (1.000) (0.800) (0.650) (0.522) 0.07 1.113 1.063 1.000 0.838 0.726 0.625 (1.174) (1.104) (1.000) (0.798) (0.652) (0.524)

(33)

H/L

==

0.01, STOKES 3 THEORY WAVE ENERGY FL'

1.3 1.00

en

:c

<, 0.9

J:

0.80 0.70 0.60 d/L=0.15 0.20 0.30 0.35 0.40 0.45 0.50~--r----...---'r----__"'---r----...., -0.10 0.00 0.10 0.20

_0.30

I 0.40

u/c

Fig. 7.

The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equat1

of Conservation of Wave Energy Flux for Hs/Ls

=

0.01

(34)

H/L

=

0.03, STOKE5 3 THEORY WAVE ENERGY FLUX

1.30 1.20 1.10 1.00

en

I

<, 0.90

I

0.80 0.70 0.60 0.20 0.25 0.30 0.35 0.40 0.45 0.50--'---.---.---,---,r---,----ï -0.10 0.00 0.10 0.20 0.30 0.40

u/c

Fig. 8

.

The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equation

of Conservation of Wave Energy Flux for Hs/Ls

=

0.03

(35)

H/L

=

0.05, STOKES 3 THEORY WAVE ENERGY FLUX

Cf) ~ 0.90

:r:

1.20 1.10 1.00 0.80 -0.10 0.00 0.10 0.20

..

U

/

_

C

0.30 0.40

Fig

.

9. The Wave Height Change by Old Stokes Third Order Wave Theory Using the Equation

of Conservation of Wave Energy Flux for Hs/Ls

=

0.05 )

(36)

H/L

=

0.06, STOKES 3 THEORY WAVE ENERGY FLUX

1.30 0.70 1.20 1.10 1.00

en

~ 0.90 J: - 0.80 0.50...J..._--r---r----r---,.---r---, -0.10 0.00 0.10 0.20 0.30 0.40

u/c

Fi g. 10. The Wave Hei ght Change by Old Stokes Thi rd Order Wave Theory Usi ng the Equati on of Conservation of Wave Energy Flux for H /L

=

0.06

(37)

H/L

=

0.07,

STOKES

3 THEORY WAVE ENERG

,

Y

FLUX

1.30 1.00

en

J:

_..._0.₉₀

J:

0.8 0.7 0.6 dIL

=

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.501~-"",,---__'....---r---""""----"""'---' -0.10 0.00 0.10 0.20

U/C

0.30 0.40

Fig. 11. The Wave Hei~ht Change by 01d Stokes Third Ü;der W-~;~-Th-~-or.yU-;i~gthe- Equ~ti~-~---

--of conservatten of ~Iave Energy Flux for Hs'Ls

=

0.07

0.03 .

0.30

(40)

H/L

=

0.05, NEW STOKES 3 THEORY ENERGY FLUX

1.24 0.84 d/L= 0.45 1.16 1.08 1.00 Cf)

I

<, 0.92

I

0.76 0.68 0.60-L---r---,r----~---,----~---. -0.10 0.00 0.10 0.20

U/C

0.30 0.40

Fig. 14.

The Wave Height Change by New Stokes Third Order Wave Theory Using the Equation

of Conservation of Wave Energy Flux for Hs/Ls

=

0.05

(41)

H/L

=

0.06, NEW STOKES 3 THEORY ENERGY FLUX

1.40 1.30 0.90 dil::: 0.45

en

J:

<,

J:

0.80 0.70 0.601...&....--r- r----__ --y- _ -0.10 ₀_.₀₀ 0.10 0.20

U/C

0.30 ₀_.₄₀

Fig. 15.

The Wave Height Change by New Stokes Third Order Wave Theory Using the Equatie

of Conservation of Wave Energy Flux for Hs/ls

=

0.06

(42)

H/L

=

0.07, NEW STOKES 3 THEORY ENERGY FLUX

1.40 1.10 1.30 1.20 Cf)

I

<,

I

0.90 0.80 0.70 0.60-'----r----.----...---r----....,...---. -0.10 0.00 0.10 0.20

U/C

0.30 0.40

(43)

D. Comparison of the resu1ts of the wave height change by numerical calculation with model test data

Tab1e 5 shows a comparison of the resu1ts of the wave height change by numerical ca1cu1ation (Equation 24) w1th the results of Hales' and Herbich's empirical formu1a2. Their formulas are as fol1ows:

Wave he1ght change due to the following current

H 0.90760 _ 0.98801 CU +

0.21123 (CU

)2 +

0.00164 ~

+

0.00006 (LdS)2

Hs

=

s

S IJ H H 2 U

Ls

U .H +

0.88017 ~

+

1.05971

(2.) -

0.00312

t-"<a)+

0.88371

~_(2)

s

Ls

s

Cs Ls

H

S

L

+

0.24931

l

(-f)

s

(47)

Since Ha1es' and Herbich's model test data were discrete during the

opposing current, a

· comparison of numerical ca1cu1ation with their

em-pir

i

ca1 relation was not made.

From Tab1e 5 it can be seen that the

wave height change H/Hs decreases with the increase of re1ative depth

d/Ls both by the theoreti

c

al method and by Hales' and Herbich's formula.

It appears that the wave steepness variab1e has a greater inf1uence in

Hales' and Herbichls empirica1 formu1a than in the theoretica1 equation.

On an average, the difference

i

nthe

resu1ts given by these two methods

is within a range of 1 to 9 percent.

CONC

L

USIONS

1. There

i

s no dominant d

i

fference in the resu1ts

obtetned

for the

change of wave length due to wave-current interaction using either

the 1inear or the non-linear wave theor

y

.

Within the range of

p

.15 ~ ~

. 's

(44)

Table 5. Comparison of calculated values with Hales' and Herbich's (H&H)

-- --- .--- .--- - -- -- ._- - -_-.

d Hs U/Cs =

o.

la 0.20 0.30

[s

ç

H&H Cal. H&H Cal. H&H Cal.

0.20 0.01 0.841 0.900 0.748 0.830 0.659 0.770 0.03 0.886 0.897 0.795 0.826 0.689 0.780 0.05 0.932 0.889 0.843 0.823 0.726 0.780 0.07 0.979 0.884 0.891 0.820 0.764 0.776 0.30 0.01 0.834 0.879 0.742 0.795 0.653 0.740 0.03 0.871 0.875 0.780 0.790 0.687 0.738 0.05 0.929 0.865 0.820 0.790 0.724 0.735 0.07 0.976 0.861 0.860 0.791 0.762 0.735 0.40 0.01 0.832 0.869 0.738 0.783 0.650 0.725 0.03 0.865 0.865 0.773 0.783 0.686 0.720 0.05 0.898 0.863 0.808 0.782. 0.723 0.720 0.07 0.933 0.858 0.844 0.781 0.761 0.725

(45)

H

s

~ 0.40, 0.01 ~

r-

$

0.07 s

wave theory are less than 10

and -0.15 s ~

S 0.30, errors by the linear

s

percent as compared with the results

ob-tained employing the non-linear wave theory.

2. There is no great difference in the results of wave height

change by employing

ef

ther the equation of conservat+on of wave epergy

flux or the equation of wave action flux.

3. Numerical calculations of wave hèight change by different wave

theories indicate that the errors resu1ting fr~

the use of the 1inear

wave theory in comparison with errors resu1ting from the non-1inear

theory are greater when the opposing re1ative current and wave steepness

both become larger .

.

However, within the range of the fo11owing currents

such errors wil1 be minimal.

4. The important factors inf1uencing the change of wave parameter

are relative current velocity ~ (or ~),

re1ative water depth ~

and

H

s

wave steepness~.

The re1ative current velocity

g

is the most

s

_

.

_

important parameter.

In consideration of the wave-current interaction,

not on1y the wave parameter is changed but a1so the velocity distribution

of steady flow with depth is changed.

In this paper the change of

steady surface velocity due to wave-current interaction is negl.ected.

This shou1d be considered in further research.

5.

For engineering purposes, the nomogram provided in this paper

may be used to estimate the change of wave parameter due to wave-current

interaction.

Figure 4 and Figures 7 through 11 are recommended for the

ca1culation of wave 1ength and wave height changes, respective1y.

(46)

REFERENCES

1. Dalrymple, R.A., IlMode1s for Non1inear Water Waves in Shear turrents ," 6th Offshore Techno10gy Conference, 1974, No. 2114.

2. Hales, L.Z. and Herbich, J.B., "Effects of a Steady Nonuniform Current on the Characteristics of Surface Gravity Waves," Misce1-laneous Paper H-74-11, Hydraulic laboratory, U.S. Army Engineer Waterways Experiment Station, Dec. 1974.

3. Hedges, T. S., "Measurement and Ana1ys i s of Waves on Currents ,11 from 'Mechanics of Wave-induced Forces on Cy1inders,' Editor: Shaw, r.l., Pitman Advanced Publishing Program, 1979, pp. 249-259.

4. Horikawa, K., Mizuguchi, M., Kitazawa, 0., Nakai, M., "Hydrodynamic Forces on a Circular Cylinder," Annual Report of the Engineering Research Institute, Faculty of Engineering, University of Tokyo, Vol. 36, 1977, pp. 37-48.

5. Van Hoften , J. D.A. and Karak i, S., 11Interacti on of Waves and a Tur-bulent Current," Proceedings, Fifteenth Conference on Coastal

Engineering, Vol. 1, 1976, pp. 402-422.

6. Hunt, N.J., "Gravity Waves in F10wing Water," Proceedings, Royal Society, london, A23l, 1955, pp. 496-504.

7. Ismai l , Awadalla N., "Wave-current Interaction," Ph.D. Dissertation, Hydraulic Engineering laboratory, College of Engineering, University of Ca1ifornia, Berke1ey, October, 198Q.

8. Jonsson, l.G., "Combinations of Waves and Currents, 11 Institute of Hydrodynamics and Hydrau1ic Engineering, Technica1 University of Denmark, 1979.

9. Jonsson, l.G., Skougaard, C., Wang, J.D., "Interaction Between Waves and Currents," Proceedings, Twelfth Conference on Coastal Engineering, 1970.

10. li, Y.C., "A Nomogramfor the Determination of Wave length and Wave

Celerity during the Interaction of Waves and Currents," Technical Report No. COE-256, Ocean Engineering Program, Texas A&M-'University, 1981 11. Liu, Y.B., "The Interaction of Waves and currents rA study and

experi-ment," Graduate Thesis, Tienjin University. 1965 (in Chinese). 12. longuet-Higgins, M.S. and Stewart, R.W., "Changes in the Form of

Short Gravity Waves on long Waves and Tida 1 Currents, 11 Journa 1,

Fluid Mechanics, Vol. 8, Part 4, 1960, p. 565.

13. longuet-Higgins, M.S. and Stewart, R.W., "Radiati on Stress in Water Waves; A Physical Discussion, with App1ications," Deep Sea Research, Vol. 11, 1964, p. 529.