Effect of bed roughness prediction on morphodynamic modelling: Application to the Dee estuary (UK) and to the Gironde estuary (France)

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1 34 th IAHR World Congress - Balance and Uncertainty 26 June - 1 July 2011, Brisbane, Australia 33 rd Hydrology & Water Resources Symposium 10 th Hydraulics Conference Effect of bed roughness prediction on morphodynamic modelling: Application to the Dee estuary (UK) and to the Gironde estuary (France) C. Villaret 1,2 N. Huybrechts 2, A.G. Davies 3, O.Way 3 1 Laboratoire National d Hydraulique et Environnement 6 quai Watier Chatou FRANCE 2 Laboratoire d Hydraulique St Venant 6 quai Watier 78400Chatou FRANCE 3 School of Ocean Sciences, Bangor University, Menai Bridge, Anglesey LL59 5AB, U.K. Catherine.villaret@edf.fr Abstract : The bed roughness coefficient is identified as a highly sensitive parameter in morphodynamic models. Different methods can be used to prescribe the bottom roughness: the bed roughness can be either considered as a calibration coefficient or predicted as a function of the mean flow and sediment parameters. The effect of the friction coefficient is illustrated here, based on two applications: the large macro-tidal Gironde estuary, located in southwest France on the Bay of Biscay and the Dee estuary in Liverpool Bay on the Irish Sea. Both 2D hydrodynamic and morphodynamic models of the Telemac hydro-informatic system have been applied in a coupled way, to calculate the medium term bed evolution. Keywords : Morphodynamic modelling, estuary, bed roughness, friction. 1. INTRODUCTION For modelling purposes, it is necessary to parameterize the geometrical bed roughness by a single length scale, which represents the inhomogeneous geometrical bedforms whose dimensions are averaged over the mesh size which can be typically of the order of 100 m in large scale models. The bottom roughness has been identified as a highly sensitive parameter for sediment transport applications (van Rijn 2007). First of all, the presence of bedforms (medium to large scale) modifies the flow pattern, through the generation of vortices in the lee of the bedforms. This produces an overall enhancement in the spatially-averaged bed shear stress and causes a mean flow velocity modification. Secondly, sediment transport predictions are highly sensitive to the local skin friction on individual grains and this depends, in turn, on the mean flow which must, therefore, be modelled as accurately as possible. The effect of bed roughness on morphodynamics is illustrated here for two estuaries, the Dee estuary and the Gironde estuary. We compare model results obtained (1) by applying a constant bed roughness and (2) by predicting the bed roughness as a function of the flow and sediment parameters, assuming equilibrium conditions. The modelling framework is the open-source Telemac hydroinformatics system (cf. We apply the tidal model Telemac-2D, internally coupled to the two-dimensional morphodynamic model Sisyphe (release 6.0). The modelling approach and the method of feedback between both hydrodynamic and morphodynamic models are described in detail in Parts 2 and 3. Applications to the Gironde and to the Dee estuary are presented in Parts 4 and 5. We end with a discussion on the importance and feasibility of the use of bed roughness predictors in Part COUPLED HYDRODYNAMIC/MORPHODYNAMIC MODELLING Hydrodynamic model Telemac-2D solves St Venant s depth-integrated equation on an unstructured grid: h hu hv + + t x y = 0 (1) ISBN Engineers Australia

2 U U U Z τ 1 + U + V = g s + x + div( hν grad( U )) t x y x h h e (2) V V Z τ V y 1 + U + V = g s + + div( hν grad( V )) t x y y h h e where h is the water depth, and U, V the horizontal mean velocity components. The first term on the right hand side of the equations of motion is the pressure gradient (Z s is the free surface elevation), the second one the bottom friction, and the third one the horizontal diffusion. Different numerical methods are available, as described in Hervouet (2007). The method of characteristics, kinetic schemes and others can be applied to calculate the convective terms in the momentum equation. The wave equation as well as providing a method of smoothing free surface instabilities is particularly well suited for large scale applications. For the treatment of tidal flats, a new algorithm based on segments ensures positive water depths (Hervouet et al. 2010). Morphodynamic model Sediment transport rates calculated in the module Sisyphe are decomposed into bed-load and suspended load. A choice between 10 classical transport formulae is available to predict the bed load as a function of the bed shear stress, corrected for skin friction. The bed evolution due to bed-load is calculated by solving the Exner equation, including correction terms for sloping bed effects. For the suspension, an additional 2D transport equation for the depth-averaged concentration is solved including erosion/deposition fluxes calculated at a reference level and convection together with a velocity correction (Huybrechts et al., 2010a). Method of coupling Sisyphe can be internally coupled to Telemac-2D: all hydrodynamic variables, including the updated flow velocity and water depth, and friction term (total friction) are sent at each time step to the morphodynamic model which sends back the updated bed level to the hydrodynamic model. 3. BED ROUGHNESS Bedform dimensions In nature, the dimensions of observed bedforms vary spatially and with time, as a function of the timevarying flow field and sediment characteristics. Bedforms can be classified based on measurable length scales, with a partition between: microscale grain roughness, typically of the order of mm, mesoscale ripples and megaripples, typically of the order of 0.1 m and 1 m, respectively, macroscale dunes with dimensions scaling with the local water-depth. For modelling purposes, it is necessary to parameterize this wide spectrum of bedform dimensions, within each of the above categories, through a single length, the equivalent bed roughness, which represents the bedform dimensions averaged over the grid scale. Total bed friction/skin friction The mean bottom shear stress averaged over bedforms τ 0 (τ x, τ y ) enters the momentum equation (Eq. 2). In 2D models, it is related to the mean (depth-averaged) flow velocity U by a quadratic friction coefficient denoted C D : 1 2 τ 0 = ρc D U (3) 2 Assuming the logarithmic velocity profile to be valid up to the free surface, the friction coefficient can be related to the representative equivalent bed roughness, denoted k s : h CD = 2κ Ln( ) (4) eks where κ (=0.4) is the von Karman constant and e=exp(1). In the presence of bedforms, the drag component increases which increases the turbulence level with an overall enhancement of the vertical mixing of the sediment in suspension. However only the local 1150

3 skin friction component affects the near bed transport processes. The total bed shear stress needs therefore to be partitioned in order to obtain the skin friction, before calculating the bedload transport and equilibrium concentrations. Review on bed roughness predictors Different bed roughness predictors can be found in the literature: existing models are generally based on controlled laboratory experiments and equilibrium steady flow conditions. Their applicability in the presence of time varying hydrodynamic forcing is questionable and should probably involve the introduction of an adaptation time scale. The apparent bed roughness is not only a physical parameter based on bedform geometry but should also include a further modification due to wave-current interaction. The presence of high sediment concentration in the near bed region also affects the near bed hydrodynamics. We assume here the bed roughness to be under equilibrium conditions and we do not consider the effects of wave-current interaction and sediment transport on the roughness coefficient. Method implemented For currents only, van Rijn s (2007) predictor for the total equivalent bed roughness has been implemented as this formula is still the most efficient (Huybrechts et al. 2010b) for the lower alluvial regime (ripples and dunes) as typically met in the Gironde and Dee estuaries. Such predictive equations are generally calibrated on equilibrium data of non tidal rivers. For an unsteady configuration, the influence of time adaptation for the sediment transport and bed morphology should be ideally taken into account into the methodology. The total bed roughness can be decomposed into a grain roughness k s, a small-scale ripple roughness k r, a mega-ripple component k mr, and a dune roughness k d. These are combined by van Rijn (2007) as follows: ' k = k + k + k + k (5) s s r mr d For waves and combined waves and currents, ripple dimensions are calculated as a function of the wave parameters. Here such estimations have been implemented only in the case of the Dee estuary, again based on the methodology of van Rijn (2007). Method of feedback The effect of the total bed roughness, including skin friction, ripples, mega-ripples and dunes, is calculated as a function of the flow velocity, wave characteristics and sediment grain size by the morphodynamic model and sent to the 2D- hydrodynamic model. The total bed shear stress is partitioned in order to identify the effect of small scale ripples which is then used for the sand transport predictions. The feasibility of this bed roughness predictor and method of feed-back between both hydrodynamic and morphodynamic models is here demonstrated in two different estuarine applications: the Dee estuary (U.K.) and the Gironde estuary (France). The methodology has appeared to be robust in both applications, with no unstable feedbacks occurring between the interacting mean flow and the bed forms. 4. APPLICATION TO THE GIRONDE ESTUARY Extending from the Bay of Biscay to 170 km inland, the Gironde estuary is the largest estuary in France and Western Europe. In terms of tides, the Gironde estuary can be classified as macrotidal (amplitude higher than 4 m), hyper-synchronous (the amplitude of the tide grows upstream) and with an asymmetric tide (4h for the flood, 8h25 for the ebb). The unstructured triangular mesh comprises nodes extending from 50 m in the refined central part and extending to more than 2km in the maritime boundary. In the central part, recent bathymetric data, collected in 2005 by Grand Port Maritime de Bordeaux, are interpolated on the grid. At the upstream boundaries in the tributaries (Garonne and Dordogne rivers), measurements of the flow rates are imposed. At the location of the downstream tidal boundary, no water level measurement is available. The tidal height must thus be predicted through a sum of harmonics (Schureman 1958). 1151

4 The mean amplitude and phase lag of the different harmonics (46 waves) are provided from a largescale tidal model (TUGO model, Letellier 2004). The hydrodynamics are calibrated based on a trial and error procedure. The calibrated values of the constant friction coefficients were obtained in order to reproduce in the best way, water levels and velocity measurements made in August 2006 (neap and spring events). For the Strickler coefficient, a value of 37.5 m 1/3 /s is selected at the mouth and 67.5 m 1/3 /s in the central part of the estuary. In order to predict the bed roughness k s, information on the bed material composition (such as d 50 ) needs to be provided. At the mouth and in the coastal area, a median diameter equal to 0.31 mm is selected. In the central part, a value 0.03 mm is imposed (Huybrechts et al 2010b). At each node of the grid, a time and space variable roughness can be predicted according to the method of van Rijn (see Fig 1). The predicted bed roughness has been converted into a Strickler coefficient for comparison with calibrated results and both spatial distribution and time variation are shown on Figure 1. Variation of the Strickler coefficient as predicted by the VR method Fig. 1 (on the left): Spatial distribution of the mean (time-averaged) coefficient Fig. 2 (on the right): Time variation at Pauillac and Verdon stations during a tidal cycle At Verdon (mouth of the estuary) and Pauillac (centre part) stations, time evolutions of the predicted friction coefficient are plotted (Fig 2). During a tidal cycle, the Strickler coefficient at Verdon varies from 28.8 to 41 m 1/3 /s with a time-averaged value of 32.6 m 1/3 /s whereas it varies from 63.6 to 66 m 1/3 /s with a time-averaged value of 64.8 m 1/3 /s at Pauillac. The maximum Strickler coefficient corresponds to low or high tide (Fig. 2). At the mouth, the Strickler coefficient exhibits larger differences between the minimum and maximum values since bed forms such as dunes can develop. In the central part, the median diameter corresponds to fine cohesive material and the model predicts a quasi-constant value of the roughness which is in good agreement with the calibrated value. More significant differences are observed at the mouth where the van Rijn equation seems to predict too much friction, which is possibly due to the neglected effect of waves and wave-current interactions as discussed in Davies and Thorne (2008). The hydrodynamic model results obtained with the predicted time-varying bed roughness and the calibrated constant roughness are compared in Figure 3 with the measurements and the results previously obtained with the calibration. The top figures show the tidal signal at two points (Verdon and Pauillac stations) and the bottom figures, the velocities. As expected, the tidal signal obtained with the van Rijn equation has slightly smaller amplitude than the measurement at the Verdon station, which is consistent with a possible overestimate of the friction in the maritime part. This may be due to the neglected effect of waves. 1152

5 At Pauillac, the accuracy of both model results is equivalent in comparison with the data. However, the amplitudes of the velocity signal are significantly under-predicted using the van Rijn equation. The observed differences especially for the water level at Verdon and the velocity are still too significant to integrate the van Rijn equation into an operational model for the Gironde estuary. Further efforts are necessary to determine if a finer description of the bed material at the estuary mouth would allow an improvement in the accuracy of the results. As a first approach, the van Rijn bed roughness predictor can be quite useful to determine a first set of friction values. Indeed, the predicted friction values are relatively close to the values obtained by calibration. A possible improvement would be to include also phase lag effects in the predictions. Measurement at Verdon calibrated van Rijn measurement at Pauillac calibrated van Rijn Z (m) Time (days) measurement at P1 calibrated van Rijn Z (m) Time (days) measurement at P3 calibrated van Rijn V (m/s) V (m/s) Time (days) Time (days) Fig. 3. Comparison between the model results In dotted line are the calibrated results in red the results obtained with the van Rijn predictor. The black points are the data. Time=0 corresponds to August 1st at 0h UT. 5. APPLICATION TO THE DEE ESTUARY, U.K. The present day Dee Estuary has a total length of 30km, with a 12km canalized tidal channel at its inner end. The mouth of the estuary has a maximum width of 8.5km and the tidal limit of the river at Chester is 35km from the estuary mouth. The main channel on the western side of the estuary bifurcates 12km seaward from the canalised river at the head of the estuary, creating two deep channels which flow out into Liverpool Bay (Moore et al., 2009). These two main deep water channels in the mouth of the estuary, the Welsh Channel and the Hilbre Channel, are separated by West Hoyle Bank and both have a maximum depth of approximately 25m. The inner estuary landforms are dominated by extensive areas of intertidal mud, sand flats and salt marsh. The Dee Estuary is macrotidal, with semi-diurnal tides where the mean spring and neap tidal ranges at Hilbre Island are 7.6 and 4.1 m respectively. Waves in Liverpool Bay are mainly locally generated, as swell does not easily penetrate into this part of the Irish Sea from the North Atlantic. The mouth of the estuary experiences direct exposure to wave conditions with an average fetch of approximately 175 km. Waves are monitored by a wave buoy operated by National Oceanography Centre in the Hilbre 1153

6 Channel adjacent to Hilbre Island at a depth of m. Banks located at the mouth of the estuary provide a high degree of protection against waves generated in Liverpool Bay. However locally generated moderate to high waves from the north-west do have a significant effect. The model domain for the Dee Estuary was created using bathymetry obtained from LIDAR data collected in 2006 for the inner estuary, while POLCOMS Liverpool Bay 180 m grid data was used for the outer estuary and open sea. The domain consists of 25,880 points over a finite element grid. The inner estuary has a resolution of 100 m, while a coarser resolution of 400 m is applied offshore of the mouth of the estuary. The POLCOMS Liverpool Bay model for 2008 was used to obtain the tidal constituents, M2, S2 and M4, to generate a generic spring-neap cycle. The amplitudes and phases for each of the three tidal constituents are interpolated from the POLCOMS Liverpool Bay model. Elevation and velocity components are imposed along the open sea boundary of the domain. A mean river discharge of 31 m 3 s -1 was applied at the river boundary. Significant wave heights are imposed as H s = 0.80 m with a 5 sec period at a direction of 140, which represents incoming waves from the north-west. Along the western and northern open sea boundaries the conditions are set to prescribed values, whereas free boundary conditions are set along the eastern open sea boundary. This allows waves to enter from the north west of the domain and leave through the eastern open sea boundary. A uniform grain size of 0.22 mm is applied throughout the domain, based on particle sieve analysis of sediment samples collected from West Kirby Sands (McCann, 2007). The bathymetry of the Dee estuary is shown in Figure 4 where the two main entry channels exist in the mouth of the estuary before merging into one channel on the Welsh side of the estuary in its interior. The canalised river at the upper end of the estuary is omitted from this figure. At mid-tide with current speeds reaching 1.5 m/s in the deep channels, combined with typical waves of period 5 s having height 0.8 m in the outer domain before breaking on the banks, the roughness variation predicted by van Rijn s formulation is shown in Figure 5. Dunes are predicted in the outer part of the domain (red and yellow) while smaller features are predicted in the inner areas. In Figure 6 the predicted sediment transport rates are shown for the cases of a spatially constant bed roughness of k s =0.01 m (6a, left) and the spatially varying roughness from Figure 5 (6b, right). Significant transport is evident in wider areas in the latter case due to the imposed roughness variation in Telemac-2D, with potentially significant implications for seabed morphodynamic change. Fig.4 (on the left): Dee estuary bathymetry within the model domain. The upper rectangular boundaries represent open sea. 1154

7 Fig. 5 (on the right): Seabed roughness (ks) variation fed back into Telemac2D at mid-tide, SOLID_DISCH (M2/S) Fig 6: Transport rates at mid tide Fig 6.a on the left is for constant ks=0.01m, while in Fig 6.b on the right, the method of feedback has been used to predict ks. Bijker s (1992) sediment transport formulation has been used here EVOLUTION (M) EVOLDIFFks () Fig 7 : Seabed evolution at mid tide (m) Fig 7.a on the left is for constant ks=0.01m, while Fig 7.b on the right shows difference in evolution (in m) arising from the use of the method of feedback. The predictions in Fig. 7a are augmented 1155

8 significantly by roughness feedback. Positive/negative evolution corresponds to deposition/erosion respectively. 6. CONCLUSIONS The van Rijn method to predict the bed roughness has been programmed in Sisyphe, the 2D morphodynamic model of the Telemac system. The bed roughness predictor and method of feedback of the bed roughness in coupled hydrodynamic and morphodynamic models, has been illustrated here in two estuaries, the Dee estuary in the UK and the Gironde estuary. In the Gironde estuary, the VR predictor approach could be validated and leads to reasonable estimates of the friction coefficients. Model results compare favourably well with both tidal amplitudes and mean flow velocity measurements. In the Dee estuary, the use of a constant bed roughness leads to an overall reduction in the transport rates and bed evolution results, in comparison to the bed roughness predictor approach. The effect of waves on the bed roughness appears to be important. Here we have considered waves only in the context of the Dee estuary. The validity of the bed roughness predictor needs to be further improved by including an adaption time scale, to account for non-stationary effects. 7. REFERENCES Bijker, E.W., (1971). - Longshore transport computations. Journal of Waterways, Harbours and Coastal Engineering Division, American Society of Civil Engineers, 97, WW4, Davies A.G. and Thorne P.D., Advances in the Study of Moving Sediments and Evolving Seabeds. Surveys in Geophysics, DOI /s x, 36pp. Hervouet, J.M. (2007). Hydrodynamics of free surface flow modelling with the finite element method. Wiley. ISBN , 341p Hervouet J.M., Razafindrakoto E., Villaret C., 2010: Dealing with dry zones in the free surface flow, a new class of advection schemes. (paper submitted to AIRH). Huybrechts, N. Villaret, C. and Hervouet, J.M. (2010) Comparison between 2D and 3D modelling: application to the dune evolution, Riverflow 2010, Braunschweig, 7p Huybrechts, N., Luong, G. V., Zhang, Y. F.,Villaret, C. and Verbanck, M. A. (2010) Dynamic routing of flow resistance and alluvial bed form changes from the lower to the upper regime. (Journal of Hydraulic Engineering in press) Letellier, T.(2004) Etude des ondes de mare sur les plateaux continentaux, PhD thesis, Université Toulouse III- Paul Sabatier, 279p McCann D.L. (2007). Dune tracking in the Dee with X-band radar: a new remote sensing technique for quantifying bedload transport. Unpublished thesis, University of Bangor. Moore R.D., Wolf J., Souza A.J., Flint S.S. (2009). Morphological evolution of the Dee Estuary, Eastern Irish Sea, UK: A tidal asymmetry approach. Geomorphology, 103, van Rijn, L.C. (2007). Unified view of sediment transport by currents and waves. 1. Initiation of motion, bed roughness, and bed-load transport. Journal of Hydraulic Engineering, 133, 6, Schureman, P. (1958) Manual of harmonic analysis and prediction of tides, US Department of Commerce, Coast and Geodetic Survey, Washington, US.p