Bottom Shear Stress Formulations to Compute Sediment Fluxes in Accelerated Skewed Waves

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1 Journal of Coastal Researh SI ICS2009 (Proeedings) Portugal ISSN Bottom Shear Stress Formulations to Compute Sediment Fluxes in Aelerated Skewed Waves T. Abreu, F. Sanho and P. Silva Civil Engineering Department LNEC-DHA, National Civil Polytehni Institute of Viseu, ESTV Engineering Laboratory Campus de Repeses, Viseu Av. do Brasil, 1, Lisboa , Portugal , Portugal tabreu@estv.ipv.pt fsanho@lne.pt University of Aveiro & CESAM, Physis Department Campus de Santiago, Aveiro , Portugal psilva@ua.pt ABSTRACT ABREU, T., SANCHO, F. and SILVA, P., Bottom shear stress formulations to ompute sediment fluxes in aelerated skewed waves. Journal of Coastal Researh, SI 5 (Proeedings of the th International Coastal Symposium), Lisbon, Portugal, ISSN As waves travel and shoal towards a beah, their surfae elevation beomes peaky (sharp rests) and asymmetri relative to the vertial. Within a wave period, the orresponding orbital veloity near the bottom shows a similar (time) variation and it is reognized that these nonlinearities are responsible for sediment transport. The resulting wave-indued bottom shear stresses are required for most sediment transport model omputations. In this work, simple bottom shear stresses parameterisations for rough seabeds, under nonlinear waves, are analysed. The effets of the wave shape on the bottom shear stresses estimates is investigated by analysing a new set of experimental data, under aelerated skewed waves (SILVA et al., 2008). Results indiate a good agreement between all the formulations presented. ADITIONAL INDEX WORDS: Seabed shear stress, sediment transport, skewed waves, wave asymmetry, flow tunnel experiments. INTRODUCTION As waves propagate into shallower water, several nonlinear effets our and play a fundamental role in the motion of partiles and transport of sediments. The wave s orbital veloity beomes more peaked at the rest and flatter at the trough and, in the surf zone, a rapid hange during the steep wave front gives rise to large fluid aelerations. At the sloping rear fae of the wave, the orresponding aelerations are muh smaller. Due to the great omplexity of the wave hydrodynamis and non-linear interations, it is very diffiult to aurately predit the sediment fluxes. Under high wave onditions large near-bed orbital veloities and shear stresses are reated near and within the surf zone. If shear stresses are strong enough to move a thin layer of bed-load partiles of the order mm, the so-alled sheet-flow regime is attained. In this layer, extremely large sediment onentrations develop and the ombination of these and strong veloities an ause large sediment transport rates. Bottom shear stress estimations are a ruial step to most sediment transport models and, when the bedload is dominant, sediment transport an be aurately parameterized in terms of the seabed shear stress (HSU AND HANES, 2004). Here we onentrate on the bottom shear stresses aused by non-linear, skewed, surfae waves in shallow water For the present analysis, it suffies to onsider only the effet that the bottom boundary layer has on the flow above, namely, to generate a bottom shear stress. Some of the formulations used herewith (see Setion 3) do not resolve the details of the boundary flow. They have a strong empirial harater and rely on physial insights in ombination with data olleted in laboratory and field ampaigns. In general, the seleted formulations may inorporate both veloity- and aeleration-related terms all at one, or may inlude a time-varying frition fator and phase-lag (e.g., GONZALEZ-RODRIGUEZ AND MADSEN, 2007, SUNTOYO et al., 2008). The overall objetive of this study is to, under aelerated skewed waves, evaluate several bottom shear stress formulations that an be ombined with existing pratial sediment transport formulae. The formulae are applied to a reent laboratory data set (SILVA et al., 2008) and substantial insight is gained by onsidering the influene of several parameters. Suh analysis an provide further knowledge in the orret predition of the morphologial bottom evolution. TRANSKEW EXPERIMENT In the Large Osillating Water Tunnel (LOWT) of WL Delft Hydraulis a new series of experiments in sheet flow onditions, under aelerated skewed waves, was performed by SILVA et al. (2008). The tunnel as the shape of a U-tube, onsisting of a long (14m) retangular horizontal setion onneted to a vertial ylinder at eah end (Figure 1). The tunnel was designed for fullsale simulation of the near-bed horizontal osillating water motion, whih an be ombined with a steady urrent. The desired motion in the test setion is reated by the movement of the piston in one of the ylinders. A detailed desription of the faility an be found in RIBBERINK and AL-SALEM (1994). Journal of Coastal Researh, Speial Issue 5,

2 Bottom shear stress formulations Figure 1. General outline of the wave tunnel. The purpose of those experiments was to obtain measurements of the sediment transport rates in ertain hydrauli onditions. These onsisted on a repetition of regular osillatory flows with equal root-mean-square veloities ( U rms 09. m s ), different wave periods ( T =7 and s), and different veloity and aeleration asymmetries, R and, respetively, defined as RUmax Umax Umin, amax amax amin, where U is the flow veloity and a is the flow aeleration. A seond wave skewness parameter is defined, 2T p T, where T p is the time interval between the zero-up ross point and the wave rest in the veloity reords. The Reynolds number magnitude Re U, where is the fluid kinemati visosity and 2 max is the angular frequeny, loates the experiments in the rough turbulent regime. Table 1 synthesizes the harateristis of the experiments analysed in the present work: - Series A (A1 to A4) onsisted of regular osillatory flows with different degrees of aeleration skewness ; - Series C (C1 to C3) onsidered both veloity- and aeleration-skewed osillatory flows. The bottom of the tunnel was overed with a 30 m layer of well-sorted sand with a median diameter, d mm. Flow veloity measurements above the wave boundary layer, representing the free stream veloity, U t, were performed using an Eletromagneti Flow Meter (EMF). The sampling frequeny was 0Hz and the signals were olleted 30m above the bed. As an example, in Figure 2, the ensemble-averaged veloity time series for Tests A1 and C1 are shown. The orresponding loal aeleration time series, au t, were obtained by numerial differentiation of the (high-frequeny filtered) veloity time series (Figure 2). Figure 2. Time series of the measured (ensemble averaged) veloities, and the orresponding aelerations for the experimental onditions A1 and C1. BOTTOM SHEAR STRESS PARAMETERISATIONS In this setion, bottom shear stress parameterisations found in the literature for non-linear waves are presented. Most of them an be used as an input to sediment transport models beause, when the bedload is dominated, sediment transport an be aurately parameterized in terms of the seabed shear stress (HSU and HANES, 2004). Method 1 (Quasi-steady approah) The quasi-steady approah assumes that the instantaneous bottom shear stress 0 t an be estimated diretly from U t under the hypothesis that the bottom stress is in phase with the free-stream veloity, 1 0 t fw U t U t, (1) 2 where f w is the wave frition fator, assumed onstant, and the water density. This method is the basi oneptual model that several authors refined to aount for nonlinear effets. Sometimes, it leads to onsiderable errors (e.g., DRAKE and CALANTONI, 2001; NIELSEN and CALLAGHAN, 2003). Method 2 (TANAKA AND SAMAD, 200) To aount for the asymmetry of the bottom shear stresses in non-linear wave boundary layers, TANAKA and SAMAD (200) proposed the introdution of a phase differene between the free stream veloity and the bottom shear stress,. This originates the Table 1. Hydrauli harateristis obtained for the test onditions. Condition U rms (m/s) R T (s) Re A A A A C C C Journal of Coastal Researh, Speial Issue 5,

3 Abreu et al. seond method given by the following equation: 1 0 t fwut Ut 2. (2) Method 3 (NIELSEN, 1992, 2002) In ontrast with the previous approahes, NIELSEN (1992, 2002) inluded the effet of the fluid aeleration ating on the sediment partiles on the bottom shear stress. The parameterization is formed in terms of the instantaneous wave frition veloity, t, and is based on harmoni boundary layer theory: f sin U w t U(t) * os Ut, (3) 2 t t U t U t. (4) 0 * * Method 4 (GONZALEZ-RODRIGUEZ and MADSEN, 2007) Reently, GONZALEZ-RODRIGUEZ and MADSEN (2007) presented a simple oneptual model to ompute bottom shear stresses under asymmetri and skewed waves. The new formulation uses a timevarying frition fator, fw t, and a time-varying phase lead, t, that aounts for the variability in the wave shape. They justify the use of a variable frition fator over the wave period by examining the physis of the boundary layer. The values are alulated at the wave rest and at the wave trough, and are then assumed to be linearly interpolated for the rest of the wave phases: 1 t t 0 t fw t Ut Ut. (5) 2 Method 5 (SUNTOYO et al., 2008) Reently, SUNTOYO et al. (2008) proposed slight hanges to NIELSEN (1992, 2002) formulation, developing a new aeleration oeffiient under sawtooth-shape waves. This parameterization is formed in terms of the instantaneous wave frition veloity, t, inluding a new aeleration oeffiient, a, determined empirially. The instantaneous wave frition veloity is thus expressed as: f w a U t t Ut, () 2 t and the bottom shear stress is given by Equation (4). RESULTS To establish the omparison of the different methodologies presented above, some remarks are pointed out. The impliit wave frition fator proposed by TANAKA and THU (1994) was used in all the methods. Therefore, the magnitudes of bed shear stress are ompromised by the solutions of this formula. The phase shift between the bed shear stress and the free stream veloity,, adopted for this omparison was obtained assuming the methodology presented in SUNTOYO et al. (2008) for the rough turbulent regime. Their approah is based in the formula proposed by TANAKA and THU (1994) for the sinusoidal ase, s, and, to aount for the wave nonlinearities, a orretion fator depending on is added. For the majority of the TRANSKEW experiments, varies between 8.7º and 13.2º. It is remarked that the alulation of the aeleration oeffiient, a, proposed by SUNTOYO et al. (2008) onerns only sawtooth waves where the veloity skewness R 0. 5, i.e., the veloity under the rest is equal to the veloity under the trough. Consequently, the proposed expression is based only on the aeleration skewness. However, in the ase of Series C (C1 to C3), in addition to different degrees of aeleration skewness, several degrees of veloity skewness an be found. In the present study the same expression was extended to Series C, but it might be advisable, in further studies, to find a more generi expression that aounts for the ombination of both asymmetries in the oeffiient a. Figures 3 and 4 synthesise the time-variation of the bottom shear stresses, omputed with the methods desribed above, for Series A and C, orrespondingly. All methods (exept Method 1) yield similar results and it is possible to observe an overlapping between them. Clearly, this superposition does not apply to Method 1 beause the formulation does not enter with any phase lead between free-stream veloity and bottom shear stress. Fousing on Series A (Figure 3), it is notied, for the same wave period (onditions A1 and A3 orrespond to T 7s, while A2 and A4 orrespond to T s ), that all the methods predit an inrease in the asymmetry of bottom shear stress with the inrease of the aeleration skewness. The method whih aentuates suh differene is Method 5, where the asymmetry is muh more pronouned, originating the time-varying shear stress to detah from the others preditions. Analysing the influene of the wave period for the same aeleration skewness, i.e, omparing onditions A1 with A2 and A3 with A4, all methods present a slight redution in the magnitudes of the shear stresses with an inrease in the wave period. That pattern is straightforwardly understandable sine inreasing values of the wave period ause dereasing values of the wave frition fator (TANAKA and THU, 1994). Looking to Method 4 results, it is possible to further distinguish a slight lower magnitude of the time-variation of the shear stress, whih is more pereptible for the peak values, partiularly at the trough of the wave. Those differenes are a diret outome of the method beause it uses a time-varying frition fator, t. fw t, and a time-varying phase lead, Though it seams that Method 3 overlaps Method 2 for this range of values, it is pointed out that the inrease of the aeleration skewness parameter aentuates more the shear stresses at the rest with Method 3 than with Method 2. This is due to the seond term of the instantaneous wave frition veloity, t, whih is linked to the aeleration and assumes a larger relative magnitude when ompared to the first veloity-related term. Fousing on Series C, it is seen that the peak magnitudes resulting from Methods 4 and 5 are somewhat larger than those from the other methods. However, it should be kept in mind that the aeleration oeffiient, a, applied in Method 5 was developed for sawtooth waves. It is possible to detet, trough the omparison of Series A with Series C, that the veloity skewness has a muh greater interferene on the time-mean shear stress than the aeleration skewness. Comparing ondition C1 with A1 we see that the veloity skewness indued a onsiderable redution of the peak values of the bottom shear stress under the trough, when ompared with the peak values under the rest. In general, all methodologies provided similar trends, even inluding Method 1 that does not Journal of Coastal Researh, Speial Issue 5,

4 Bottom shear stress formulations a) b) ) d) Figure 3. Comparison between the results of different bottom shear stress methodologies for Series A. Panels a), b), ) and d) represent onditions A1, A2, A3 and A4, respetively. aount for any phase lead with the free-stream veloity. Nevertheless, for sediment transport purposes, that assumption an sometimes introdue onsiderable errors and the other methods are reommended. CONCLUSIONS In this work, five simple bottom shear stresses parameterisations found in the literature are investigated and applied to the reent data obtained in the Large Osillating Water Tunnel (LOWT) of WL Delft Hydraulis. The experiments performed in the tunnel ontemplate osillatory flows with different degrees of veloity and aeleration skewnesses, under sheet flow onditions. The auray of the parameterisations to aount for nonlinearwave effets on the bed shear stress formulations is ruial and an be used in some pratial sediment transport formulae like MEYER-PETER-MÜLLER (1948). Conerning the formulations used, the quasi-steady approah and the methods proposed by NIELSEN (1992, 2002), TANAKA and SAMAD (200), GONZALEZ-RODRIGUEZ and MADSEN (2007) and SUNTOYO et al. (2008) are employed. An overall good agreement between eah other was found, despite some small disrepanies introdued by the inorporation of both veloity and aeleration terms, or the inlusion of time-varying frition fators and timevarying phase differenes. For the aelerated skewed waves (Series A), all the methods predit an inrease in the asymmetry of bottom shear stress with the aeleration skewness parameter. Furthermore, a slight redution in the magnitudes of the shear stresses were notied when inreasing the wave period. For the osillatory flows with both veloity- and aelerationskewness (Series C), the methods show that the veloity skewness parameter has a muh greater interferene on the time-mean shear stress than the aeleration skewness. In addition, the veloity skewness parameter indues a onsiderable redution of the peak values under the trough in detriment with the peak values under the rest. This kind of asymmetries should also happen for the sawtooth waves (Series A) ausing net sediment transport rates. However, the small values of introdued in the methodologies presented are responsible for that behaviour. As suggested by NIELSEN 1992 (p. 128), it may be that the bed-load transport lags slightly behind the bed shear stress. Thus, for sediment transport purposes, it ould be reasonable to introdue a larger value of. In the future, it will be interesting to ombine the instantaneous bed shear stress formulations with the sediment transport formulae, like MEYER-PETER and MÜLLER (1948), and see if the apparent agreement between the methods remains when averaged over the entire wave period. The validation with new experiments will, moreover, yield greater insight in the seletion of the wave frition fator, attending the variability of the formulae for this range of values. We also intend to extend the formulations to the ase of wave plus urrents flows, and ombine the methods with sediment transport formulae and validate them with the net sediment transport rates measured in the TRANSKEW experiment. Journal of Coastal Researh, Speial Issue 5,

5 Abreu et al. a) b) ) Figure 4. Comparison between the results of different bottom shear stress methodologies for Series C. Panels a), b) and ) represent onditions C1, C2 and C3, respetively. LITERATURE CITED DRAKE, T.G. and CALANTONI, J., Disrete partile model for sheet flow sediment transport in the nearshore. Journal of Geophysial Researh, (C9), GONZALEZ-RODRIGUEZ, D. and MADSEN, O. S., Seabed shear stress and bedload transport due to asymmetri and skewed waves. Coastal Engineering, 54(12), HSU, T.J. and HANES, D.M., Effets of wave shape on sheet flow sediment transport. Journal of Geophysial Researh, 9 (C05025). doi:.29/2003jc MEYER-PETER, E. and MÜLLER, R., Formulas for bed-load transport. In: Report from the 2nd Meeting of the International Assoiation for Hydrauli Strutures Researh (Stokholm, Sweden, IAHR), pp NIELSEN, P., Coastal bottom boundary layers and sediment transport, Volume 4, Advaned Series on Oean Engineering. Singapore: World Sientifi Publiation. NIELSEN, P., Shear stress and sediment transport alulations for swash zone modeling. Coastal Engineering, 45, NIELSEN, P. and CALLAGHAN, D.P., Shear stress and sediment transport alulations for sheet flow under waves. Coastal Engineering, 47, RIBBERINK, J.S. and AL-SALEM, A.A., Sediment transport in osillatory boundary layers in ase of rippled beds and sheet flow. Journal of Geophysial Researh, 99 (C), SILVA, P.; Abreu, T.; Freire, P.; Kikkert, G.; Mihallet, H.; O Donoghue, T.; Pleha, S.; Ribberink, J.; Ruessink, G.; Sanho, F.; Steenhauer, K.; Temperville, A.; Van der A., D. and Van der Werf, J., Sand transport indued by aeleration-skewed waves and urrents The TRANSKEW projet. In: PECS08 Physis of Estuaries and Coastal Seas (Liverpool, UK). SUNTOYO; TANAKA, H. and SANA, A., Charateristis of turbulent boundary layers over a rough bed under saw-tooth waves and its appliation to sediment transport. Coastal Engineering, 55(12), TANAKA, H., and THU, A., Full-range equation of frition oeffiient and phase differene in a wave-urrent boundary layer. Coastal Engineering, 22, TANAKA, H., and SAMAD, M.A., 200. Predition of instantaneous bottom shear stress for turbulent plane bed ondition under irregular wave. Journal of Hydrauli Researh, 44 (1), 94. ACKNOWLEDGEMENTS The experimental work was supported by the European Community's Sixth Framework Programme through the Integrated Infrastruture Initiative HYDRALAB III, Contrat no (RII3). This work has also been done within the framework of the researh projet BRISA - BReaking waves Interation with SAnd transport, PTDC/ECM/7411/200 supported by Portuguese National Sine Foundation ( Fundação para a Ciênia e a Tenologia, FCT). The first author of this work has been supported by FCT through a PhD grant (SFRH/BD/41827/2007). Journal of Coastal Researh, Speial Issue 5,