STUDY OF IRREGULAR WAVE-CURRENT-MUD INTERACTION

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1 STUDY OF IRREGULAR WAVE-CURRENT-MUD INTERACTION Mohsen Soltanpou, Fazin Samsami, Tomoya Shibayama 3 and Sho Yamao The dissipation of egula and iegula waves on a muddy bed with the existence of following and opposing cuents ae investigated though a seies of wave flume laboatoy expeiments. The commecial kaolinite is used as fluid mud laye. The laboatoy esults show the incease of both egula and iegula wave heights due to opposing cuents. On the othe hand, the following cuents esult to the decease of the wave heights. In the numeical teatment, the defomation of wave due to cuent was fist calculated based on the consevation equation of wave action and then the attenuation of this defomed wave due to the muddy bed is simulated by a multi-layeed wave-mud inteaction model. Acceptable ageements wee obseved between the numeical esults and laboatoy data. Keywods: Wave-cuent inteaction; iegula wave; fluid mud; wave dissipation; wave flume expeiments INTRODUCTION Waves and cuent inteaction is a complex phenomenon in coastal and estuaine aeas. Longuet- Higgins and Stewat (96) used the adiation stess concept to find out solutions fo wave-cuent inteaction and examined the enegy tansfe between waves and cuents. They poposed an equation to detemine the wave height vaiation fo steady cuents and no enegy loss condition. The othe ealy eseach was developed by Betheton and Gaett (968). Instead of employing enegy equation, the wave action equation (i.e. wave enegy density/wave fequency elative to the cuent) was deived in thei theoy to simplify the wave-cuent inteaction. The late theoy has given a good pediction on the wave height vaiation and it is a simple solution to be adopted in the coastal pocesses. The effect of wave-cuent field on a muddy bed is of consideable pactical inteests too. Howeve, liteatue shows few studies on wave-cuent-mud inteaction. An and shibayama (99) pefomed a seies of wave flume expeiments of egula wave-cuent inteaction on fluid mud laye. Thei esults indicated highe values of wave dissipation ates and mud mass tanspot velocities due to existence of opposing cuents. In the othe study by De Wit and Kanenbug (996), expeimental studies of wave-fluid mud inteaction with the pesence of a net cuent wee examined in a wavecuent flume using two atificial mud. Modifying Gade s model (958), they found a well ageement in compaison between the measued and calculated wave attenuation ates and wave-induced velocities. Zhao et al. (6) pefomed numeical and expeimental studies on wave-cuent-mud inteaction. They poposed a numeical model based on an eddy viscosity model fo wave and cuent inteaction and then coupled the motion equation with mud laye. Soltanpou et al. (8) poposed a numeical wave-mud-cuent inteaction model to simulate wave tansfomation and mud mass tanspot with pesence of a unifom cuent using the consevation equation of wave action by Thomas (98). They also compaed the esults of thei numeical model with the laboatoy data of An and Shibayama (99). Kaihatu and Tahvildai () poposed a fequency-domain model to investigate the combination effects of wave nonlineaity, mud-induced dissipation and wave-cuent inteaction. The calculated dissipation ates wee compaed with the laboatoy data of An and Shibayama (995) and Zhao et al. (6). The popagation of cnoidal waves on muddy bed unde the opposing and following cuents was also investigated in thei modeling study. A seies of wave flume expeiments was conducted in this study to investigate the attenuation of waves along soft mud layes with the pesence of cuents. Both egula waves and iegula waves with two diffeent wave specta, i.e., JONSWAP and Betschneide, wee employed. The consevation equation of wave action was employed to model the defomation of incident wave due to cuent. The wave attenuation is then computed using the numeical multi-layeed wave-mud inteaction model of Soltanpou et al. (7). Civil Engineeing Depatment, K. N. Toosi Univesity of Technology, Vali-As St., Tehan, Ian Fomely, Civil Engineeing Depatment, K. N. Toosi Univesity of Technology, Vali-As St., Tehan, Ian 3 Civil & Envionmental Engineeing Depatment, Waseda Univesity, Tokyo, Japan Civil & Envionmental Engineeing Depatment, Waseda Univesity, Tokyo, Japan

2 COASTAL ENGINEERING MODELING OF WAVE-CURRENT-MUD INTERACTION Regula Wave-Cuent Inteaction The fist attempt to deal with the wave popagation ove non-unifom cuent was poposed by Longuet-Higgins and Stewat (96). They intoduced the tem of Radiation Stess based on steady and no enegy loss condition as follows.[ E( C U )] () g S ij ij whee E is the enegy density, C g is goup velocity, U is cuent velocity, S ij is adiation stess tenso and γ ij is the ate of stain tenso associated with the cuent. In anothe simple teatment, Betheton and Gaett (968) used the wave action, i.e. wave enegy density divided by wave fequency elative to the cuent, to study the wave-cuent inteaction. The wave action equation can be witten as E E ( ).[ ( Cg U )] () t whee σ is the elative angula fequency defined by ku, ω is angula fequency, U is the mean depth-aveaged cuent velocity and k is wave numbe. The adiation stess and wave action equations ae identical fo steady state. Many eseaches followed these two theoies to investigate diffeent aspects of egula wave-cuent inteaction (e.g., Whitham 96; Dalymple 97; Peegine 976; Bevik and Aas 98; Thomas 98). Following Thomas (98), the defomed wave height and wavelength can be pedicted unde the steady condition and the wave action equation as: o E ( C g U ) Const. (3) H ( C g U ) Const. Consideing the iotational dispesion equation () gk tanh kd (5) ( ku ) gk tanh kd (6) whee d is wate depth. In ode to define the constant value, it is necessay to specify a efeence level as no cuent condition. Newton-Raphson method is used in numeical calculation to find out the wave height vaiations unde cuent condition. Spectal Wave-Cuent Inteaction In a simple teatment, the moe complicated changes in wave specta caused by cuent can be obtained following the equation poposed by Huang et al. (97) as E( C g U ) [ ] (7) x whee ω is angula wave fequency in fame of efeence moving with cuent (intinsic angula wave fequency). As wave popagates fom no cuent condition to a cuent one, Eq. (7) can be ewitten as E Cg E( Cg a U ) (8) whee the subscipt efes to no cuent condition and ω a is angula wave fequency in stationay fame of efeence (appaent angula wave fequency). Consideing the linea wave theoy, the goup velocities ae defined as

3 C COASTAL ENGINEERING 3 k d a ( (9) k g ) sinh kd kd Cg ( ) () sinh kd k The spectal density in this tansition condition can be witten as follows S ( a, U ) S ( a, U ) da S ( a ) S ( a ) da E E k d sinh kd () kd ku sinh kd k Assuming that the waves ae not efacted by cuent and wate depth is sufficiently deep, the defomed wave spectum will be S ( a, U ) S ( a ) a U U g g U g / U a U a g g Simulation of wave-cuent inteaction on fluid mud In spite of consideable pactical applications, thee has been few studies on wave-cuent-mud inteaction (e.g., Zhao and Li 99; An and Shibayama 99; Zhao et al. 6; Soltanpou et al. 8). It is assumed hee that the wave attenuation pocess will take place afte egula/iegula waves ae defomed due to the cuents. The theoy of Thomas (98) is followed to calculate the change of egula wave. Fo the case of a spectal wave, the defomed wave spectum due to cuent action is fist calculated based on Huang et al. (97). Employing Fouie theoy, the spectal wave is then pesented as supeposition of many simple, egula hamonic wave components having thei own amplitudes and peiods. The numeical model of Soltanpou et al. (7) is used fo the calculation of attenuation of both egula waves and hamonic components of spectal waves. EXPERIMENTAL INVESTIGATIONS Wave flume expeiments wee pefomed in Coastal Engineeing Laboatoy of Depatment of Civil and Envionmental Engineeing at Waseda Univesity, Japan. The glass-sided flume has a coss section of.5 cm wide by 6 cm deep in coss-section and the total length of 3.8 m. Iegula waves wee geneated by a flap-type wavemake. Following and opposing cuents can be geneated by a pump system though an inlet o outlet located at the bottom of flume. Thee capacitance wave gauges wee employed to measue wave heights while the cuent velocity was ecoded by an electomagnetic cuent mete in a fixed location about cm above the muddy bed. The data of gauge pesents the measued defomed iegula wave due to cuent and gauge 3 shows the wave unde both effects of cuent and fluid mud. Two false beds wee placed in the flume to confine the m long mud section with a thickness of cm. Fig. shows the sketch of expeimental setup. A well mixtue of commecial kaolinite with tap wate was used as muddy bed. The chemical chaacteistics of the used kaolinite wee SiO =5.9%, Al O 3 =37.8% and Fe O 3 =.6%; and the paticle size distibution follows D and D 86 of m and m, espectively. The heological behavio of kaolinite was tested with an Anton Paa Physica MCR3 instument at the Rheomety Laboatoy of Institute fo Coloants, Paint, and Coatings (ICPC). This instument allows any type o combination of heological tests, both in otational and oscillatoy modes. Viscoelastic chaacteistics of the fluid mud wee measued using the oscillatoy fequency sweep test. Applying the least squae method to the data, Fig. epesents the Kelvin-Voigt viscoelastic paametes μ and G fo the used kaolinite whee W is the wate content atio of the mud sample (%). It is obseved that viscoelastic paametes depend not only on the wate content atio of the mud but also stongly on the fequency. Table lists the fitted paametes of G (Pa) and μ (Pa.s) to the heological tests. / ()

4 COASTAL ENGINEERING Fig.. Sketch of expeimental setup, units in metes W=.3 % W=3.3 % W=36.85 % W=57.76 % W=.3 % W=3.3 % W=36.85 % W=57.76 % G (Pa) μ (Pa.s) 6 8 T (s) Fig.. Elastic modulus and dynamic viscosity of used kaolinite 6 8 T (s) Table. Relationship of viscoelastic paametes of kaolinite vesus peiod and wate content atio W=.3 % G = e.3t μ = 85.98T W=3.3 % G = 5898 e -.3T μ =.7T -.75 W=36.85 % G = 7 e -.8T μ = 79.8T W=57.76 % G = 35.8 e.5t μ = 99.5T Regula monochomatic waves and iegula waves of JONSWAP and Betschneide specta with following, opposing, and no cuent conditions wee examined. Table pesents the measued wave chaacteistics, cuent velocities and calculated wave height attenuation ates assuming an exponential wave height decay. H m, H mean, and T m stand fo significant wave height by spectal calculation ( H m m ), mean wave height ( H mean m ) and spectal mean wave peiod, espectively. The plus and minus signs of velocities indicate following and opposing cuents, espectively. RESULTS AND DISCUSSIONS As an example, Fig. 3 shows the measued Betschneide wave specta at gauge No. (beginning of mud section) with following, opposing, and no cuent conditions. Because of shot duation of wave action of about 6-9 seconds, the spectal shape has not been completely developed. The discepancy is mainly attibuted to the flap-type wavemake that is not capable to exactly geneate the same input

5 COASTAL ENGINEERING 5 Table. Measued wave chaacteistics and cuent velocities of test uns Wave spectum Betschneide JONSWAP Regula Wave peiod Rep. Wave height (cm) k Test No. T wave m (s) i (/m) Gauge Gauge 3 Gauge Gauge 3 BTN Hm Hmean BTN Hm Hmean BTN3 Hm Hmean BTN Hm Hmean BTN5 Hm Hmean BTN6 Hm Hmean JTN7 Hm Hmean JTN8 Hm Hmean JTN9 Hm Hmean JTN Hm Hmean JTN Hm Hmean JTN Hm Hmean RTN3 Hm Hmean RTN Hm Hmean RTN5 Hm Hmean RTN6 Hm Hmean RTN7 Hm Hmean RTN8 Hm Hmean U (cm/s) W (%) incident waves fo all thee conditions (i.e. opposing, following and no cuent). When simila incident waves ae applied, the wave enegy inceases due to an opposing cuent and deceases with the pesence of a following cuent. Two un tests of measued incident wave specta at gauge No. and attenuated wave specta at gauge No. 3 ae illustated in Fig.. It is noted that because of the shot length of mud section, the diffeence of the wave specta at two wave gauges is small. The wave spectal defomation due to cuent effect is calculated based on the theoy of Huang et al. (97). In ode to simulate the enegy dissipation on mud laye, the defomed wave specta due to cuent at the beginning of mud section is discetized to a numbe of hamonic egula waves (Zhang and Zhao, 999). The attenuated wave specta is then calculated by the numeical model of Soltanpou et al. (7). Table 3 shows the epesentative wave paametes of the measued and simulated wave specta due to cuent action at gauge No.. It is assumed that the geneated input wave specta fo the conditions of opposing and following cuents ae both simila to no cuent condition.

6 6 COASTAL ENGINEERING.5 BTN-Opposing cuent BTN-No cuent BTN3-Following cuent 6 5 BTN-Opposing cuent BTN5-No cuent BTN6-Following cuent Spectal density (cm s).5 Spectal density (cm s) Fequency (Hz) Fequency (Hz) Fig. 3. The measued wave specta in gauge No. with opposing, following, and no cuent conditions Spectal density (cm s) Test case: BTN Gauge No. Gauge No. 3 Spectal density (cm s) Test case: BTN6 Gauge No. Gauge No Fequency (Hz) Fequency (Hz) Fig.. Sample cases of measued wave specta befoe and on the muddy bed Table 3. Compaisons of measued and calculated waves at gauge Test No. U (cm/s) Measued wave (cm) Tansfomed wave (cm) Hm Hmean Hm Hmean BTN BTN BTN BTN Howeve, as it was mentioned befoe, it was not possible to geneate the same incident wave conditions in thee compaable uns, i.e. opposing, following and no cuent conditions, because of the flap-type wavemake. In ode to investigate the attenuation of the defomed wave specta due to muddy bed, the dissipation of iegula wave on mud section was modeled. Fig. 5 shows the compaison of the measued and simulated wave specta at gauge 3. The coesponding epesentative wave paametes obtained fom the attenuated wave specta ae also listed in Table. The gaphical and tabulated compaisons indicate that the iegula attenuated waves can be pedicted by numeical model.

7 COASTAL ENGINEERING 7.5 Measued Test case BTN. Measued Test case BTN Spectal density (cm s).5 Spectal density (cm s) Fequency (Hz) Measued Test case BTN Fequency (Hz) Measued Test case BTN Spectal density (cm s).8.6. Spectal density (cm s) Fequency (Hz).5 Measued Fequency (Hz) Measued Test case BTN5.6 Test case BTN6 Spectal density (cm s).5 Spectal density (cm s) Fequency (Hz) Fequency (Hz) Fig. 5. Compaison of measued and calculated wave specta on muddy bed SUMMARY AND CONCLUSIONS The iegula wave inteaction with opposing and following cuents and the spectal changes of waves passing ove a muddy bed ae studied thoughout a seies of wave flume expeiments. It is evealed that the exponential decay is a good estimation fo egula and iegula wave heights attenuation ove muddy beds. Both appoaches of epesentative and spectal wave analyses wee used in numeical modeling showing acceptable ageements with laboatoy data. In geneal, wave enegy inceases due to an opposing cuent and it deceases in the pesence of a following cuent. Futhemoe, the dissipation of both egula and iegula waves along the mud can be appoximated by the exponential decay.

8 8 COASTAL ENGINEERING Table. Compaison of measued and modeled epesentative wave heights at gauge 3 Test No. wave Measued wave Hm (cm) Hmean (cm) Hm (cm) Hmean (cm) BTN BTN BTN BTN REFERENCES An, N.N., and T. Shibayama. 99. Wave-cuent inteaction with mud bed, Poc. of th Coastal Engineeing Confeence, ASCE, Betheton, F.P., and C.J.R. Gaett Wavetains in inhomogeneous moving media, Poc. of the Royal Society, 3(7), Bevik, I., and B. Aas. 98. Flume expeiment on waves and cuents I. Rippled bed, Coastal Engineeing, 3, Dalymple, R.A. 97. A finite amplitude wave on a linea shea cuent, J. Geophysical Reseach, 79, De Wit P.J. and C. Kanenbug On the effects of a liquefied mud bed on wave and flow chaacteistics, Jounal of Hydaulic Reseach, 3:, 3-8. Gade, H.G Effects of non igid, impemeable bottom on plane suface wave in shallow wate, J Maine Reseach, 6, 6-8. Huang, N.E., D.T. Chen, and C.C. Tung. 97. Inteactions between steady non-unifom cuents and gavity waves with applications fo cuent measuements, J. Physical Oceanogaphy,, -3. Kaihatu, J.M., and N. Tahvildai.. The combined effect of wave-cuent inteaction and mudinduced damping on nonlinea wave evolution, Ocean Modelling,, 3. Longuet-Higgins, M.S., and R. W. Stewat. 96. The changes in amplitude of shot gavity waves on steady non-unifom cuents, J. Fluid Mech.,, Peegine, D.H Inteaction of wate waves and cuents, Advances in Applied Mechanics, 6, Academic Pess, New Yok, 7 pp. Soltanpou M., S.A. Haghshenas, and T. Shibayama. 8. An integated wave-mud-cuent inteaction model, 3 st Coastal Eng. Conf., ASCE, Hambug, Gemany, Soltanpou, M., T. Shibayama, and Y. Masuya. 7. Iegula wave attenuation and mud mass tanspot, Coastal Eng. Jounal, 9, 7-8. Thomas, G.P. 98. Wave-cuent inteactions: an expeimental and numeical study, Pat linea waves. J. Fluid Mech., Whitham, G.B. 96. Mass, momentum and enegy flux in wate waves, J. Fluid Mech.,, Zhang Q.H. and Z.D. Zhao Wave-mud inteaction: wave attenuation and mud mass tanspot, Poc. of Coastal Sediments, 99, Zhao, Z.D. and H.Q. Li. 99. Viscous damping of iegula wave popagation ove mud seabeds, Poc. of Intenational Confeence in Hydo-Technical Engineeing fo Pot and Habo Constuction, 9-3. Zhao, Z.D., J.J. Lian, and J.Z. Shi. 6. Inteactions among waves, cuent, and mud: numeical and laboatoy studies, Advances in Wate Resouces, 9, 73 7.