DAM-BREAK FLOW IN A CHANNEL WITH A SUDDEN ENLARGEMENT

Transcription

1 THEME C: Dam Break 221 DAM-BREAK FLOW IN A CHANNEL WITH A SUDDEN ENLARGEMENT Soares Frazão S. 1,2, Lories D. 1, Taminiau S. 1 and Zech Y. 1 1 Université catholique de Louvain, Civil Engineering Department 2 Fonds National de la Recherche Scientifique Civ. Eng. Dept., Université catholique de Louvain Place du Levant 1, B Louvain-la-Neuve, Belgium Tel : / fa : / soares@gce.ucl.ac.be Abstract: Dam-break flows are usually simulated by solving the shallow-water equations, neglecting the turbulent stresses. However, in some situations like a flow in a channel with a sudden enlargement, features such as re-circulating flows may appear, that are generally not accurately modelled. The aim of the work presented in this paper is to try to answer the question whether turbulent stresses should be included in the shallow-water equations to improve the numerical results in such situations. Based on the CADAM eperience, new laboratory eperiments on both steady and dam-break flow in a channel with a sudden enlargement were performed and carefully observed by means of several measurement devices. Then, a Roe-type finite-volume scheme is developed to solve the adapted shallow-water equations including the turbulent stresses. Both first- and second-order spatial accuracy are considered. Finally, the numerical simulations are compared to the measurements in both the steady flow situation and in the dam-break flow. Interesting results are obtained, but at this stage no clear answer can be given to the question whether turbulent stresses are needed to accurately model dam-break flows. Keywords: dam break, sudden enlargement, finite volumes, turbulence 1. INTRODUCTION Dam-break flows in channels with a sudden enlargement were studied within the frame of the CADAM concerted action (Soares Frazão et al., 2000). The purpose was to focus on twodimensional behaviour occurring when a water front arrives in large flood plain. Onedimensional modelling appeared to be unable to reproduce the circular spreading of the wave (figure 1a), resulting in erroneous prediction of both water levels and wave propagation speed. Besides, a re-circulation zone was identified in the corner at the beginning of the enlarged cross section (figure 1b). Figure 1 : Circular spreading of the wave and re-circulation observed in the CADAM eperiments at LRH-Châtelet (Belgium)

2 222 XXX IAHR Congress AUTh, Thessaloniki, Greece In order to investigate more deeply those two-dimensional features and especially the recirculating flow, a new eperimental campaign has been launched in the laboratory of the Civil Engineering Department of the Université catholique de Louvain, at a smaller scale than the CADAM eperiments. Use of various measurements devices allowed to obtain an accurate description of the flow, both under steady and unsteady flow condition (dam-break flow). Those measurements are then used to validate a series of numerical models, with the aim of answering the question whether a turbulence model should be considered to accurately reproduce the re-circulating flow. 2. NUMERICAL SCHEME AND TURBULENCE MODELLING The two-dimensional shallow-water equations can be written in vector form as: U F( U) G( U) + + = S( U) t y h with = q, qy 2 2 U F ( U) = q h + g h 2, ( U) q q q y h 0 S ) ( U) = gh ( S0, S f, ( ) gh S0, y S f, y (1a) qy G = qqy h (1b,c,d) 2 2 qy h + g h 2 In order to take momentum echange through turbulence into account, adapted shallow-water equations are used, where depth-averaged terms related to turbulent stresses are included in the source terms. This consists in writing the source term (1e) in the following way ( U) = gh ( S0, S f, ) + ( h τ ρ) + ( h τy ρ) y ( ) ( ) ( ) gh S S + h τ ρ + h τ ρ (1e) 0 S (2) 0, y f, y y y Neglecting the turbulent kinetic energy, the turbulent stresses τ, yy epressed as (Rodi, 1993) yy τ, τ and τ are τ u τ y τ y u v τ y y v = 2 νt, = = νt +, = 2 νt (3) ρ ρ ρ y ρ y where the depth-averaged turbulent viscosity ν t is given by the relation ν t = λu* h where u * is the friction velocity (Rodi, 1993). The homogeneous part of equations (1) is solved by a finite-volume numerical scheme with a Roe solver for the flu calculation (Soares Frazão and Zech, 2002). Both first- and secondorder spatial accuracy, obtained by a MUSCL approach (Hirsch, 1997), are considered. For the turbulent stresses, the source terms given by (2) are calculated by a finite-difference scheme (Yulistiyanto, 1998).The complete eplicit numerical scheme is subjected to a CFL restriction on the time step t, taking the turbulent stresses into account. Finally, four different numerical schemes are used in this paper, as summarised in table 1 : y y

3 THEME C: Dam Break 223 Table 1 : Summary of numerical schemes used Spatial accuracy Turbulence modelling Roe2D-1o First order No Roe2D-1oT First order Yes Roe2D-2o Second order (MUSCL) No Roe2D-2oT Second order (MUSCL) Yes 3. PRELIMINARY EXPERIMENTS UNDER STEADY-FLOW CONDITIONS 3.1. EXPERIMENTAL SET-UP AND MEASUREMENTS The eperimental set-up is located in the laboratory of the Civil Engineering Department of the Université catholique de Louvain. The system is sketched in figure 2. Figure 2 : Eperimental set-up and location of the gauging points and measured water profiles Measurements were made using several techniques : the water level at some specific points, S1 S5 indicated in figure 2, was recorded using water-level gauges, water profiles along lines indicated in figure 2 were measured using a WAVO (Water Level Follower), water profiles along the channel walls were measured using digital imaging by filming the flow through the glass walls of the channel, and finally, digital imaging (Capart et al., 2002) was used to obtain the surface-velocity field by filming the flow from above the channel. The steady discharge was of m³/s, and the Manning friction coefficient was found to be s m -1/3. When entering the enlarged part of the channel, the flow separates and a reflection occurs against the wall located on the side opposite to the narrow channel. This results in the formation of a steady oblique hydraulic jump, and a second reflection further downstream, as shown in figure 3 showing the trajectories obtained by the digital imaging measurements. A re-circulation zone at the beginning of the enlarged part of the channel can also be clearly identified. Figure 4 shows the water surface reconstructed from the measured water profiles. Figure 3 : Flow trajectories on the free surface reconstructed from the digital imaging measurement

4 224 XXX IAHR Congress AUTh, Thessaloniki, Greece Figure 4 : Water surface reconstructed from the measured water profiles 3.2. COMPARISON WITH NUMERICAL RESULTS All computations were run on a m square grid, with a 0.9 CFL number. When turbulent stresses are considered, a free-slip condition is set on the walls of the channel. This appeared to be the most adapted model for the very smooth glass walls. Figure 5 shows the eperimental water surface and results from numerical computations taking the turbulent stresses into account. Both first order and second order numerical schemes reproduce the hydraulic jump, but its shape is different, and the second-order results are closer to the eperimental free surface. However, the height of the jump is overestimated by the numerical models. (c) Figure 5 : Water surface reconstructed from the measured profiles, first order and (c) second order numerical results with turbulence model

5 THEME C: Dam Break 225 Figure 6 compares eperimental and numerical water profiles along the two lines L1 and L5 indicated in figure 2. It appears that second-order accuracy does improve more the numerical results than the introduction of turbulent stresses. These profile confirm the overestimation of the height of the hydraulic jump by the numerical models. Those all produce a smooth water surface while the actual jump is completely irregular, as can be seen in the reconstruction of the free surface of figure 4. (c) Figure 6 : Comparison between eperimental and numerical water profiles (first and second order) along line L1 without turbulence model, L1 with turbulence model, (c) L5 without turbulence model, (d) L5 with turbulence model When comparing the velocity vectors and the velocity magnitude in the re-circulation zone, the conclusions are similar. The hydraulic jump is more accurately represented by the second order schemes. However, it appears that the velocity magnitude is slightly underestimated by the numerical schemes (figure 7). (d) Figure 7 : Measured and computed velocity magnitude (m/s) 4. DAM-BREAK FLOW After the preliminary study of a steady flow, a dam-break flow was simulated, with an initial water depth in the reservoir of 0.2 m and an initially dry bed in the flood plain.

6 226 XXX IAHR Congress AUTh, Thessaloniki, Greece 4.1. DESCRIPTION OF FLOW Figure 8 shows successive pictures of the flow. White tracers placed on the channel bed show the propagation of the dry front, while tracers thrown on the free surface allow to better identify the flow features and direction. The spreading of the front is as described in the CADAM eperiments, with a circular shape (figures 8a). Then, the front reflects on the channel wall, forming a hydraulic jump that moves from the right to the left side of the channel (figures 8b) as the incoming discharge decreases du to the emptying of the reservoir. When it reaches the left bank, this hydraulic jump reflects again (figure 8c). (c) Figure 8 : Dam-break flow in the UCL channel with a sudden enlargement 4.2. COMPARISON WITH NUMERICAL RESULTS Like in the steady flow case, the results obtained with the four numerical schemes show only minor differences among them as shown in figure 9. Figure 9 : Comparison between eperimental measurements and numerical results at gauge S2 and gauge S4

7 THEME C: Dam Break 227 In figure 9, measurements at the gauging points are compared with numerical results (first and second order, with and without turbulence model). Besides the secondary undulations measured on the head of the hydraulic jumps, which is a typical feature of such measurements, the water depth is well reproduced by the numerical schemes. A delay can be observed (figure 9b) in the arrival time of the hydraulic jump formed by the second reflection of the wave. Eperimental velocity fields obtained from digital-imaging measurements are then compared to the computations run by the second-order scheme with turbulence (Roe2D-2oT). The propagation of the wave, the reflections and the progressive slowing down of the flow can be followed in figure 10 at time t = 2.75 s and t = 7.00 s. Figure 11 compares the measured and computed velocity fields at time t = 9.25 s. The re-circulation zone is well reproduced, but like in figure 10, it appears that the numerical model seems to underestimate the velocity in the main flow, and to overestimate the velocity in the re-circulation zone. eperimental, t = 2.75 s Roe2D-2oT, t = 2.75 s (c) eperimental, t = 7.00 s (d) Roe2D-2oT, t = 7.00 s Figure 10 : and (c) eperimental and and (d) computed velocity magnitude in (m/s) eperimental Roe2D-2oT Figure 11 : eperimental and numerical (Roe2D-2oT) velocity field at t = 9.25 s

8 228 XXX IAHR Congress AUTh, Thessaloniki, Greece 5. CONCLUSION An important eperimental work has been achieved to get more insight into both steady and unsteady flow in a channel with a sudden enlargement. These data are then used to validate four different numerical models. The main features of the flow are well reproduced by the numerical models, with a better accuracy for the second-order schemes. However, there is still an intriguing question as it seems that including the turbulence stresses in the shallow-water equations and in the numerical scheme does not improve so much the results, contrarily to using second order spatial accuracy. It is clear that only a very simple turbulence model was adopted here, and other more sophisticated models should be investigated. Another possible eplanation is given by Abbott and Basco (1989), who state that, although the flow in a sudden epansion is an ideal eample for studying separating flows, re-circulation might appear by itself from the intrinsic diffusion of the numerical scheme, and resembles the physical features. For the dam-break flow, the origin of the recirculation might also be found in the first reflection of the wave, forming on the one hand a hydraulic jump in the downstream direction and a new front propagating in the upstream direction, into the initially empty area opposite to the narrow channel. This new front then reflects against the upstream end of the enlarged channel, and finally bumps into the first wave again, resulting in circular trajectories in that area. Finally, the question whether turbulence effects are significant in dam-break flows remains open and should be studied more in depth by means of new test cases, trying to avoid unphysical although realistic re-circulating flow in numerical modelling. REFERENCES Abbott M.B. and Basco D.R. (1989), Computational fluid dynamics, Longman, Singapore Capart H., Young D.L.; Zech Y. (2002), Voronoï imaging methods for the measurement of granular flows, Eperiments in Fluids, vol. 32, Hirsch C. (1997), Numerical Computation of Internal and Eternal Flows, Wiley, Great Britain Rodi W. (1993), Turbulence Models and Their Application in Hydraulics, Balkema, Rotterdam. Soares Frazão S., Morris M. and Zech Y. (2000), editors, Concerted Action on Dambreak Modelling : Objectives, Project Report, Test Cases, Meeting Proceedings" (CD-ROM), Université catholique de Louvain, Civ. Eng. Dept., Hydraulics Division, Louvain-la- Neuve Soares Frazão S. and Zech Y. (2002), Dam-break in channels with 90 bend, J. Hydraul. Eng., 128(11), Yulistiyanto B., Zech Y. and Graf H. (1998), Flow around a cylinder: shallow-water modelling with diffusion-dispersion, J. Hydr. Eng., 124(4),