Abstract. 1 Introduction

Transcription

1 One-dimensional unsteady flow computation in channels with floodplains D. Bousmar, R. Scherer & Y. Zech Civil Engineering Dept., Universite Catholique de Louvain, Place du Levant, 1, B-1348 Louvain-la-Neuve, Belgium Abstract A new one-dimensional unsteady flow model is presented for overbank flow computation in natural rivers. This model takes into account floodplains acting not only as a storage volume but also as a river element carrying a fraction of the total discharge and interacting with the main channel. The model is incorporated in a hydraulic software and simulations are compared with an experimental case found in the literature. 1 Introduction During flood events, river floodplains generally act as a storage volume contributing to peak flow mitigation. Nevertheless, if not too much obstructed by vegetation or constructions, they also can concur to the channel conveyance. In both cases, there will be an interaction between the water flowing fast in the main channel and the low-velocity water in the floodplains. Two different effects are observed. The first is the momentum transfer due to the shear layer at the interface between main channel and floodplains. This momentum transfer has been widely studied for steady

2 206 Hydraulic Engineering Software 3) (D Figure 1: Division of a compound channel in 3 subsections flow in prismatic channels (see e.g. Selling Knight & Shiono^). A second effect is the momentum transfer associated to mass exchanges through the interface. These exchanges may occur in steady-flow as a consequence of flow redistribution due to non-prismaticity, but also in unsteady flow, filling or emptying the floodplains. Depending on the predominating floodplain effect, two kinds of computational models have been developed. Thefirstone considers only a storage effect: the discharge only flows in the main channel and lateral discharge to the storage ponds is included in the continuity equation (see e.g. Fender^). The second model also includes floodplain conveyance : the discharge is estimated by the well known Divided Channel Method (DCM) as the sum of the discharges corresponding to each subsections (main channel and floodplains, see Figure 1). A correction factor can take into account the shear layer momentum transfer (see e.g. Stephenson & Kolovopoulos^, Abida & Townsend*, Lyness et al.^). Based on the authors' steady flow Exchange Discharge Model (EDM) (Bousmar & Zech^), a new model is presented where the lateral discharge between main channel and floodplains due to both nonprismaticity and filling or emptying is taken into account not only as a mass exchange but also as a momentum transfer. 2 Exchange Discharge Model for steady flow In this model, the effect of the momentum transfer between main channel andfloodplainsis taken into account as an additional loss & to be added to the friction slope ^estimated with the DCM, in order to get the total head loss Se: S,=S+S, (1)

3 Hydraulic Engineering Software 207 The value of the additional loss is given by the Saint-Venant equations written with a lateral inflow #,- and a lateral outflow #<%,, per unit length for each subsection (Bousmar and Zech ) (2) and,, A + ga ^ (3) where x and t are respectively the abscissa along the channel and the time, g is the gravity constant, z and U are the water level and the mean velocity in the section and %, is the lateral inflow velocity component in the direction of the main flow. The inflow and outflow of each channel subsection / are on one hand an oscillating turbulent exchange discharge c{ due to the shear layer and the resulting vortices at the interface between the floodplain and the mainchannel (Figure 2). On the other hand, a geometrical transfer discharge (f* is due to the redistribution of flow which occurs for steady flows in nonprismatic channel (for instance, widening of thefloodplains) : ds (4) Turbulent exchange Geometrical transfer Figure 2: Exchange discharges in a compound channel

4 208 Hydraulic Engineering Software Getting the particular additional losses in each subsection, it is possible to evaluate a global additional loss for the whole channel, leading to accurate steady-flow discharge and water-profile prediction. 3 Momentum transfer in unsteady flow In order to extend the Exchange Discharge Model to unsteady flow, two assumptions are necessary. The first is the classical assumption that the friction slope in Saint-Venant equation can be estimated by a uniform-flow equation like the Manning's one, in the same way as for steady-flow modelling. The additional loss is thus added to the friction term of the Saint-Venant equation. The second assumption is to take into account an unsteady flow geometrical transfer discharge (f" resulting from thefillingor emptying of the floodplains. This unsteady transfer discharge is to be added to the turbulent exchange discharge q* and to the steady-flow geometrical one <f for the estimation of the momentum transfer between main channel and floodplains. The value of the unsteady geometrical transfer discharge can be found from the conservation of volume written for a floodplain reach (subsection /) between two cross-sections (Figure 3) (5) where FJ is the water volume and #, the mean width of the floodplain, ds is the length of the reach, H is the water level in the channel and Q, is the floodplain discharge. Assuming that (f* and (f are constant during the time interval Af and that the discharge g, vary linearly, the integration of eq (5) leads after simplifications to

5 Hydraulic Engineering Software 209 Q,+ 8Q,/5scte Figure 3: Volume conservation for a floodplain reach dt ds (6) Given the definition of #** by eq (4), for the time f, we get finally the value of the unsteady geometrical transfer discharge dq, r-* + -, '' ' dt 2 55 f+af ds (7) showing that the water entering the floodplain originates either from the main channel, or from the upstream reach of the floodplain. The distribution between both supplying sources will depend on the relative flood wave celerity. For floodplains acting only as storage volume, the floodplain discharge g, is equal to zero and, according to eq (7), the water level increase on the floodplain results only from a filling by the main channel. If the floodplain conveyance is not negligible, the floodplain discharge will also lightly contribute to the storage feeding. For example, it counts for 5 to 10 % of the feeding in the test case presented in the next part.

6 210 Hydraulic Engineering Software 4 Flood wave simulations This extended EDM model was incorporated in a one-dimensional unsteady flow computational model solving the Saint-Venant equations by an explicit predictor-corrector McCormack scheme (Garcia-Navarro & Saviron*). Also an implicit Preissmann 4-points scheme has been tested, allowing longer time steps. Nevertheless, in the McCormack scheme, the unsteady geometrical transfer discharge is easier to estimate : while a first value of <f* for the predictor step is given as a result of the previous time step, a second one is then evaluated using the predictor-step water levels before the corrector step. Numerical simulation were performed for comparison with the experimental data of Tominaga et al.*. The test flume is 11.5 m long, the 0.20 m width main channel isflankedby two 0.20 m width symmetrical floodplains, of which the bed is 59 mm higher than the main-channel bottom. A test section is located 7.5 m downstream from the entrance. The slope was fixed at & = A controlled flood wave is imposed upstream, with discharge varying from 3 to 20 1/s and a peak time Tp of either 60s or 120 s. The downstream end of the channel is a control section. As no roughness values were available, they were arbitrary estimated as n^ = nf= 0.010, corresponding to the perspex walls of the flume. These values seem to be confirmed by the correct prediction of a steady flow water level at the begin of the flooding. Figure 4 presents the water level evolution in the test section, for a hydrograph with peak time at 60s. Three simulations where performed : one with the classical DCM, one with the steady-flow EDM (including turbulent and steady geometrical exchange), and the last one with the unsteady-flow EDM (also including unsteady geometrical exchange). The rising stage is well estimated by the three models. For the falling stage, the unsteady EDM offers a better prediction than the other methods, while not perfect. Figure 5 presents velocity-water level loop curves for both main channel and floodplain. All the three methods present curves of similar aspect when compared to experimental data but fail to model it accurately. One explanation is to be found in Figure 6 which presents the velocity as a function of time : the EDM is only modelling mean velocities of each subsection and could still be improved by accurate modelling of velocity distribution.

7 Hydraulic Engineering Software f Measured values DCM EDM (steady) EDM (unsteady) Time (s) Figure 4: Water level evolution with time (Tp = 60s) 0.80 EDM (steady) EDM (unsteady) Water depth (m) Figure 5: Velocity-stage loop curve (Tp = 120s) Measured values EDM (unsteady) Time (s) Figure 6: Subsection velocities evolution with time (Tp = 120s)

8 212 Hydraulic Engineering Software y I Time (s) Figure 7: Friction slope and additional loss evolution with time (Tp = 60s) o.ooio - turbulent geom. steady -geom. unsteady Time (s) Figure 8: Exchange discharges evolution with time (Tp = 60s) The evolution according to the time of thefrictionslope S/and of the additional head loss & is presented on Figure 7. During the stagerise,the relative discharge increase is greater than the rise of the water level and of the corresponding conveyance. As a result, the friction slope is higher than the bottom slope & = of the channel. It is only later, during the recession, that the discharge decrease leads to lower friction. This may explain the loop shape of the velocity-stage curves of Figure 5. The friction peak at 170 s corresponds to the end of floodplain emptying : as the momentum transfer disappears, the water level decreases suddenly while the friction arises.

9 Hydraulic Engineering Software 213 The evolution of additional losses is easier to explain with Figure 8 that presents the discharge exchanged through the interface between main channel and floodplains. As the floodplains in the actual case are not so wide, the turbulent exchange discharge is significantly higher than the geometrical ones. It should be noted that its value higher than 1 1/s/m is not negligible compared to the peak value of the total channel discharge of 201/s. The steady-flow geometrical transfer discharge is negative during the flooding, indicating that the floodplain conveyance decreases downward. Indeed, since the channel discharge increases according to the time, the water profile is rather steep at the downstream end of the channel, leading to a downward decreasing water level in the floodplains. During the recession, the profile will be more parallel to the channel bottom and the steady-flow geometrical transfer will reduce to zero. The geometrical transfer discharge due to unsteadiness is positive toward the floodplains during the rising stage as the water level is increasing. For this particular example, it approximately counterbalances the steady-flow geometrical transfer, resulting in a sum near zero. For this reason, the additional loss from the unsteady EDM computation is lower than from the steady-flow model, giving lower water profiles due to lower total head losses (Figure 4). During the stage recession, the floodplains are emptying (with a maximum discharge just when flow is leaving floodplains, at 165 s), the geometrical transfer due to unsteadiness is negative and the associated momentum transfer slows down the main channel, corresponding to higher additional loss (Figure 7), with a peak when water is leaving the floodplains. 5 Conclusions The new model developed for one-dimensional unsteady flow in compound channels takes into account an additional head loss. This loss corresponds to a momentum transfer given by the mass exchange between main channel and floodplains when the floodplains water level is varying. This model seems to reproduce appropriately existing phenomena : during flooding, the momentum transfer increases the floodplains discharge and does not interfere with main channel. During recession, it enlarges the main-channel losses in such a way thatfloodplainsemptying may be delayed.

10 214 Hydraulic Engineering Software References [1] Abida, H. & Townsend, R.D., A model for routing unsteady flows in compound channels, Journal of Hydraulic Research, 1AHR, 32 (1), pp , "2] Bousmar, D. & Zech, Y., Water profile computation in compound channels, Hydrosoft98, Como, Italy, [3] Bousmar, D. & Zech, Y., A model of momentum transfer for practical flow computation in compound channels, submitted for publication in Journal ofhyd. Engrg., ASCE, [4] Garcia-Navarro, P. & Saviron, J.M. McCormack's method for the numerical simulation of one-dimensional discontinuous unsteady open channel flow, Journal of Hydraulic Research, IAHR, 30 (1), pp , 1992 [5] Knight, D.W. and Shiono, K., Turbulence measurements in a shear layer region of a compound channel, Journal of Hydraulic Research, IAHR, 28 (2), pp , 1990 [6] Lyness, J.F., Myers, W.R.C. & Wark, J.B., The useof different conveyance calculations for modelling flows in a compact compound channel, Journal ofthelnst of Water and Envir. Mgmt, 11 (5), pp , [7] Fender, G, Maintaining numerical stability of flood plain calculations by time increment splitting, Proc. ICE, Wat., Marit. and En. Journal, 96, pp , 1992 [8] Sellin, R.H.J, A laboratory investigation into the interaction between the flow in the channel of a river and that over its flood plain, La Houille Blanche, 7, pp , [9] Stephenson, D. & Kolovopoulos, P., Effects of momentum transfer in compound channels, Journal of Hyd. Engrg, ASCE, 116 (12),pp , 1990 [10] Tominaga, A., Liu,I, Nagao, M. and Nezu, I., Hydraulic characteristics of unsteady flow in open channels with flood plains, Proc. 26th Congress of JAHR, Hydra 2000, London, 1, pp , 1995