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Wave-current interaction in complex nearshore flows NJ. MacDonald, J. Nicholson, B.A. O'Connor University ofLiverpool, Department of Civil Engineering, P.O. Box 147, Liverpool, L69 3BX, UK Abstract A computer model was applied to two coastal situations in order to investigate the effect of wave-current interaction on the local hydrodynamics. It was concluded that wave-current interaction was important in situations subjected to large incident wave conditions, and was more significantif regular waves, rather than irregular waves, were employed. However, when small waves were incident, irregular wave cases were found to be more affected. Introduction It is well known that the presence of strong currents can greatly affect the wave climate. Away from the surf zone, where currents are likely to be meteorologically or astronomically forced and, therefore, only weakly coupled to the surface waves, it is relatively easy to account for wave-current interaction (WCI) in a wave climate computer model by implementing an independent shallow water flow model. However, where wave-induced flows dominate, such as in the surf zone, this is less easily done because the waves and currents are fully coupled. In these situations, the incorporation of full WCI can lead to a substantial increase in the amount of computing time required for design studies, and may prove to be uneconomical for smaller coastal engineering consulting firms. It is important, therefore, to identify common situations where the phenomenon of wave-current interaction should be considered, and those where its effects are less important and can be ignored. To investigate the influence of WCI in two common coastal engineering situations, the results from a computer model, run with and without WCI, are presented and analysed. Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

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  • Wave-current interaction in complex nearshore

    flows

    NJ. MacDonald, J. Nicholson, B.A. O'Connor

    University of Liverpool, Department of Civil Engineering, P.O. Box

    147, Liverpool, L69 3BX, UK

    Abstract

    A computer model was applied to two coastal situations in order to investigatethe effect of wave-current interaction on the local hydrodynamics. It wasconcluded that wave-current interaction was important in situations subjectedto large incident wave conditions, and was more significant if regular waves,rather than irregular waves, were employed. However, when small waveswere incident, irregular wave cases were found to be more affected.

    Introduction

    It is well known that the presence of strong currents can greatly affect thewave climate. Away from the surf zone, where currents are likely to bemeteorologically or astronomically forced and, therefore, only weakly coupledto the surface waves, it is relatively easy to account for wave-currentinteraction (WCI) in a wave climate computer model by implementing anindependent shallow water flow model. However, where wave-induced flowsdominate, such as in the surf zone, this is less easily done because the wavesand currents are fully coupled. In these situations, the incorporation of fullWCI can lead to a substantial increase in the amount of computing timerequired for design studies, and may prove to be uneconomical for smallercoastal engineering consulting firms. It is important, therefore, to identifycommon situations where the phenomenon of wave-current interaction shouldbe considered, and those where its effects are less important and can beignored.

    To investigate the influence of WCI in two common coastalengineering situations, the results from a computer model, run with andwithout WCI, are presented and analysed.

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • 334 Environmental Problems in Coastal Regions

    Description of the computer model

    The computer model selected for the present work was the WC2D nearshorecirculation model, which is based on the work of Yoo and O'Connor* butsubstantially improved since its initial development. The model is composedof two fully coupled modules, one to compute the wave-period-averaged wavefield, and other to compute the depth-averaged nearshore flow field.

    Wave kinematics are obtained through solution of the kinematicconservation equation, modified to account for combined refraction-diffractionand the presence of slowly-varying depth-averaged currents:

    a*,

    dt

    KC-L *U.5.

    dx

    aG dh

    2h dx.

    2kA *. dx

    duK.—L

    (1)

    = 0

    where K. is the wave number vector, /= x,y, U. is the depth-averagedvelocity vector, k is the wave separation factor, h is the depth, g isgravitational acceleration, A is the wave amplitude, C is the group velocityand G = 2kh/smh 2kh . *

    The wave amplitude is obtained from the wave energy conservationequation for linear surface waves:

    dA_

    dt

    I d

    2/1 a*- 0 (2)

    where S. is the radiation stress tensor and p is the density of water. Thefinal term in Eq. 2 has been introduced to include the effects of frictionaldissipation. The calculation of C,, using an improved Bijker approach, isdescribed in O'Connor and Yoo\

    Irregular waves are simulated using the truncated-Rayleigh distributiontechnique proposed by Battjes and Jannsen*.

    Depth-averaged currents are computed by coupling the wave model toa shallow water flow model. The surface elevation is computed using thecontinuity equation:

    = 0 (3)

    where r| is the free surface displacement.The depth-averaged currents are computed from the horizontal

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • Environmental Problems in Coastal Regions 335

    momentum equations:

    U.dt ' a* p (h + r\) &

    (4)

    where v is the eddy viscosity and C^ is a friction coefficient. The formeris determined using the Battjes* method, and the latter using an improvedBijker approach (O'Connor and Yoô ).

    The numerical scheme uses a simple explicit finite differenceformulation. The solution is obtained on a staggered grid in an iterativeapproach until a steady-state solution is achieved. Boundary conditions at theoffshore boundary are supplied as a wave amplitude, period and direction.

    Although the wave and current modules of the WC2D model are fullycoupled, the effects of WCI can be ignored by neglecting all terms containing U.in Eqs. 1 and 2.

    Wave-current interaction tests

    The first test situation studied was a prototype-scale offshore breakwater ona plane beach (see Fig. 1). The WC2D model, as well as six other models,was validated for this situation using laboratory-scale data (Pechon et al/).

    The second situation studied was a complex nearshore bathymetryinvolving two inclined elliptical shoals on a plane beach (see Fig. 2).Preliminary validation of the WC2D model for this situation, as well as a fulldescription of the hydraulic model tests, is given in O'Connor et al A

    Effects of WCI in the offshore breakwater layout

    The following test conditions were examined for the offshore breakwater:

    TEST

    1

    2

    3

    4

    5

    6

    WAVE TYPE

    Irregular

    Irregular

    Irregular

    Irregular

    Regular

    Regular

    WAVE DIRECTION

    0*

    0*

    30*

    30"

    0*

    30"

    WAVE HEIGHT

    1.0m

    2.0m

    1.0m

    2.0m

    2.0m

    2.0m

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • 336 Environmental Problems in Coastal Regions

    All other parameters were kept constant, the wave period being maintained ata value of 8.0s. Note that in tests where irregular waves are employed, waveheights are expressed as //^, while H is used for regular wave test results.

    Results from the most 'extreme' conditions, Test 1 (small, normally-incident, irregular waves) and Test 6 (large, obliquely-incident, regular waves)are contained in Figures 3 to 8. Figures 3a and 3b show the wave field forTest 1 with WCI included and omitted, respectively. Similar results are givenin Figures 4a and 4b for the current field. In addition, a more detailedindication of the effects of WCI was obtained by comparing wave height andcross-shore velocity distributions along transect XX in Figures 5a and 5b; thelocation of transect XX is plotted in Figure 1. This whole exercise was thenrepeated in Figures 6 to 8 for the results of Test 6, Figure 8 involvinglongshore velocity distributions and transect Y-Y (see Figure 1).

    The contents of Figures 3 to 8 indicate that, for the offshorebreakwater layout, WCI effects are insignificant in situations involving small,normally-incident waves (Test 1, see Figures 3 to 5), but become appreciablein situations involving large obliquely-incident waves (Test 6, see Figures 6to 8). In addition, WCI effects are enhanced by the presence of regular,rather than irregular, waves.

    Effects of WCI in the double-elliptical-shoal layout

    The following test conditions were used for the double-elliptical-shoal tests:

    TEST

    7

    8

    9

    10

    11

    12

    WAVE TYPE

    Regular

    Irregular

    Regular

    Irregular

    Regular

    Irregular

    WAVE PERIOD

    1.5s

    1.5s

    1.0s

    1.0s

    1.0s

    1.0s

    WAVE HEIGHT

    10 cm

    10 cm

    10 cm

    10 cm

    5 cm

    5 cm

    All tests were performed with normally incident waves. These test conditionswere selected to ensure that significant wave-induced currents were generatedby the wave transformation over the shoals. The combination of refraction,diffraction and breaking resulted in the generation of a marked rip currentbetween the two shoals which had a strong influence on the wave field.

    Figure 9 shows a detail of the wave height field in the lee of the right-hand shoal for Test 7, with and without WCI. While the surf zone near theshore is similar in each case, the wave height patterns in the immediatevicinity of the shoal show significant differences. In particular, the areas ofhigh and low wave height in the WCI case, located along the centreline and

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • Environmental Problems in Coastal Regions 337

    behind the shoal, respectively, are reversed compared with the no-WCI case.Figure 10 shows a detail of wave height contours for Test 8 in which

    irregular waves were used. As with the previous regular wave case, the waveheight near the centreline is larger with WCI, because of the influence of therip current returning between the shoals. The general trend for the largerwaves is towards a lessened importance of WCI with irregular waves.

    In contrast, however, consider the results of two less extreme incidentwave fields shown in Figure 11. This figure shows the WCI and no-WCIwave heights along a shore-parallel line at 15 m for Tests 11 and 12. Here,because of the low wave-induced currents, only a very small differencebetween the two approaches can be observed. Note that while neither no-WCIrun predicts the increased wave height at the centreline, there is, arguably, agreater difference between the two simulations of Test 12, the irregular wavetest, than Test 11.

    As expected, the current fields produced by the model for the WCI andno-WCI cases differ since the two wave fields differ. Figure 12 shows acomparison of the WCI and no-WCI results for Test 9. Here the wave-induced current field for the no-WCI case is significantly more vigorous thanthe WCI case and there are two, rather than four, distinct gyres in the lee ofthe shoal. As in the previous case, there is less variation in the results of thecorresponding irregular wave test, Test 10, shown in Figure 13. In thiscomparison, only the magnitude of the velocities are substantially different;the general flow pattern is fairly similar in both cases.

    The differences between the current fields for the two smallest incidentwaves, Tests 11 and 12, are illustrated in Figure 14, which shows the WCIand no-WCI current speeds along a shore-parallel line at 15 m. It is evidentfrom this figure that the influence of WCI is more pronounced in the irregularwave results, Test 12, than in the regular wave results, Test 11.

    Discussion and Conclusions

    For the breakwater layout, WCI effects were found to increase as the waveheight increased and the obliquity of the wave incidence increased, the latterbeing synonymous with larger currents. The wave type was also found to bea factor, regular waves (swell) being associated with more pronounced WCIeffects than irregular waves (sea). This was caused by the fact that both thelongshore current and the breaker zone become more 'concentrated' as thewaves become more regular. Quantitatively, it was found that WCI producedan order of magnitude increase in the wave height (see Figure 8a) and at leasta doubling of the current speed (see Figure 8b) in the lee of the breakwater.

    For the case of complex nearshore bathymetry, here represented by aninclined double-elliptical-shoal layout, the effects of WCI were found to begreatest with the larger incident waves. This is to be expected because thelarger waves have a greater current-generating potential which, in the shallowwater of the surf zone, is fully realized. The WCI effects were also found to

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • 338 Environmental Problems in Coastal Regions

    be more pronounced with regular waves when the incident waves were large,and with irregular waves when the incident waves were small. The cause ofthe former finding is reasoned to be the result of the lower current-generatingpotential of the irregular waves in shallow water. This is due to the greaterloss of energy offshore, where the larger waves in the spectrum break. Thecause of the latter finding is, at this point, still uncertain.

    Wave-current interaction affects both waves and currents in thenearshore zone. Apart from limiting the accuracy of nearshore hydrodynamicpredictions, failure to properly account for WCI is likely to have an evenstronger influence on the associated morphodynamics.

    Acknowledgements

    This work has been carried as part of the G8 Coastal Morphodynamicsresearch programme and was funded by the Commission of the EuropeanCommunities, Directorate-General for Science, Research and Developmentunder MAST Contract No. MAS2-CT92-0027. Funding was also providedby the UK Engineering and Physical Sciences Research Council under GrantsGR/J25017 and GR/J76361.

    References

    1. Yoo, D. and O'Connor, B.A. (1986): Mathematical modelling ofwave-induced nearshore circulation. In: B.L. Edge (Editor), Proc.Twentieth Coastal Engrg. Conf., ASCE, 3: 1667-1682.

    2. O'Connor, B.A., Sayers, P.B. and MacDonald, N.J. (1995):Combined refraction-diffraction-wave-current interaction over acomplex nearshore bathymetry. Coastal Dynamics'95, Gdansk, Poland.

    3. O'Connor, B.A. and Yoo, D. (1988): Mean bed friction of combinedwave-current flow. Coastal Engineering, 12(1), 1-21.

    4. Battjes, J.A. and Jannsen, J.P.F.M (1978): Energy loss and set-updue to breaking of random waves. In: B.L. Edge (Editor), Proc.Sixteenth Coastal Engrg. Conf., ASCE, 1: 569-587.

    5. Pechon, P., Rivero, F., Johnson, H., Chesher, T., O'Connor, B.A.,Tanguy, J.-M., Karambas, T., Mory, M. and Hamm, L. (submitted),Intercomparison of Wave-Driven Current Models, Coastal Engineering.

    6. Battjes, J.A. (1975): Turbulence in the surf zone. In: Symp. onModeling Techniques, ASCE, 1050-1061.

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • Environmental Problems in Coastal Regions 339

    0 100 200 300 400 500 600 700 BOODISTANCE (m)

    Figure 1.Prototype-scale offshorebreakwater layout (contours - mto still water level)

    10 20DISTANCE (m)

    Figure 2.Double-elliptical-shoal layout(contours - cm to still waterlevel)

    BOO 700 BOO

    (a) With WCI (b) Without WCI

    Figure 3. Test 1: Wave field

    400-

    300-

    100-

    : : : : : : ; { \'~. , .Ŷ ) ( /t'.~-~'̂} } } I '. '•'• '•''•'\ \\ » • - M i l / ' ' '////'NSVX / I I U ,%/%/

    400-

    300-

    200-

    100-

    \ \\XV- -v^ - ' / ) { ( » - "SS/t / -

    100 200 300 400 500 600 700 BOO 0 100 200 300 400 500 600 700 800DISTANCE (m) DISTANCE (m)

    (a) With WCI (b) Without WCI

    Figure 4. Test 1: Flow field

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • 340 Environmental Problems in Coastal Regions

    100 150 200 250 300 350 400 450 500 550 600DISTANCE (m)

    0 50 100 150 200 250 300 350 400 450 500 550 600DISTANCE (m)

    (a) Waves (b) CurrentsFigure 5. Test 1: WCI effects

    0 100 200 300 400 500 600 700 BOO 0 100 200 400 500 600 700 800

    (a) With WCI (b) Without WCIFigure 6. Test 6: Wave field

    0 100 200 300 400 500 600 700 800DISTANCE (m)

    (a) With WCI

    0 100 200 300 400 500DISTANCE (m)

    (b) Without WCIFigure 7. Test 6: Flow field

    50 100 150 200 250 300 350DISTANCE (m)

    50 100 150 200 250 300 35(DISTANCE (m)

    (a) Waves (b) CurrentsFigure 8. Test 6: WCI effects

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • Environmental Problems in Coastal Regions 341

    17.5

    20.0 22.5 25.0DISTANCE (m)

    27.5 20.0 22.5 25.0DISTANCE (m)

    27.5

    (a) With WCI (b) Without WCIFigure 9. Test 7: Wave field

    17.5

    20.0 22.5 25.0DISTANCE (m)

    27.5 20.0 22.5 25.0DISTANCE (m)

    27.5

    (a) With WCI (b) Without WCIFigure 10. Test 8: Wave field

    10-

    10 20DISTANCE (

    (a) Test 11Figure 11. Wave height at 15 m (-

    10 20 30DISTANCE (m)

    (b) Test 12WCI, no WCI)

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

  • 342 Environmental Problems in Coastal Regions

    17.5 T

    ft»>/.i-

    17.5

    "̂20.0 22.5 25.0 27.5

    DISTANCE (m) 20 cm/s20.0 22.5 25.0

    DISTANCE (m)27.5

    (a) With WCI (b) Without WCIFigure 12. Test 9: Current field

    20.0 22.5 25.0DISTANCE (m)

    27.5 20 cm/s 20.0 22.5 25.0DISTANCE (m)27.5

    (a) With WCI (b) Without WCIFigure 13. Test 10: Current field

    20-

    ~T0 10 20

    DISTANCE (m)

    (a) Test 11Figure 14. Current speed at 15 m (-

    , . - . . i . . I10 20 30

    DISTANCE (m)

    (b) Test 12WCI, no WCI)

    Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541