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Finally, the total flux corresponding to the continuity equation for all cells is reduced to the flux through the external boundary OF.

Figure 8 Finally, the total flux corresponding to the continuity equation for all cells is reduced to the flux through the external boundary OF.

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Free surface flows in rivers and estuaries are generally modeled by the shallow-water equations. Considering an incompressible fluid flow in a domain Q and neglecting diffusion effects, Coriolis terms and wind forces, the two-dimensional shallow-water equations are written as follows: his the water depth, with h=1)—z where 17 is the free surface level, z the bottom level, (u, v) are the averaged velocity components and g the gravitational acceleration. n is the Manning roughness coefficient, S; the friction vector and T; the time duration.
Free surface flows in rivers and estuaries are generally modeled by the shallow-water equations. Considering an incompressible fluid flow in a domain Q and neglecting diffusion effects, Coriolis terms and wind forces, the two-dimensional shallow-water equations are written as follows: his the water depth, with h=1)—z where 17 is the free surface level, z the bottom level, (u, v) are the averaged velocity components and g the gravitational acceleration. n is the Manning roughness coefficient, S; the friction vector and T; the time duration.
Fig. 1. Beffa and Connell's [12] local hydrostatic reconstruction.
Fig. 1. Beffa and Connell's [12] local hydrostatic reconstruction.
Fig. 2. Brufau et al.'s [14] bed slope modification in steady wetting/drying fronts over adverse steep slopes in real and discrete representations Some of the first W/D front treatments over fixed grids were in the works of Leendertse and Gritton [2], Sielecki and Wurtele [41] in the context of the finite difference method and Holz and Nitsche [42] in the framework of the finite element method. In [2,43] any dry or partially dry cell is removed. In [44] the model used a thin layer of water everywhere in the domain, but this does not respect mass conservation and may become instable when dry zones appear within the computational domain. In the finite elements framework, some authors tolerate negative water depths when treating adverse slopes to control the mass conservation [11,45]. Positive depth corresponds to wet areas and negative depth to dry ones. Besides the lack of meaning of negative depths, conservation is not completely satisfied in general situations. A water depth limitation technique (WDL) was rather used with a default minimal value ¢ for the entire computational domain. The value of ¢ is assigned arbitrarily or could be set by a calibration based on the bed roughness, as proposed in [46]. 2.2.2.2. Global mass conservation. The criteria required to identify a wet or dry cell, as observed by Balzano [21], is a crucial feature in W/D modeling. A model built on positive water depth values still proves sensitive to the value of ¢. Indeed, the smaller ¢ is the more accurate is the model, but this could generate spurious velocities at wet-dry fronts if the velocity components are obtained by dividing the discharge by the water depth. A reasonable value has to be chosen to satisfy both accuracy and stability. A threshold of e=10~°m is generally considered. However, it has been observed that for a supercritical flow and under a certain limit of bed slope an unphysical negative water depth could still be generated [5]. Updating this negative water depth to the threshold as often done supposes adding an amount of water to the global system and this violates a strict global mass
Fig. 2. Brufau et al.'s [14] bed slope modification in steady wetting/drying fronts over adverse steep slopes in real and discrete representations Some of the first W/D front treatments over fixed grids were in the works of Leendertse and Gritton [2], Sielecki and Wurtele [41] in the context of the finite difference method and Holz and Nitsche [42] in the framework of the finite element method. In [2,43] any dry or partially dry cell is removed. In [44] the model used a thin layer of water everywhere in the domain, but this does not respect mass conservation and may become instable when dry zones appear within the computational domain. In the finite elements framework, some authors tolerate negative water depths when treating adverse slopes to control the mass conservation [11,45]. Positive depth corresponds to wet areas and negative depth to dry ones. Besides the lack of meaning of negative depths, conservation is not completely satisfied in general situations. A water depth limitation technique (WDL) was rather used with a default minimal value ¢ for the entire computational domain. The value of ¢ is assigned arbitrarily or could be set by a calibration based on the bed roughness, as proposed in [46]. 2.2.2.2. Global mass conservation. The criteria required to identify a wet or dry cell, as observed by Balzano [21], is a crucial feature in W/D modeling. A model built on positive water depth values still proves sensitive to the value of ¢. Indeed, the smaller ¢ is the more accurate is the model, but this could generate spurious velocities at wet-dry fronts if the velocity components are obtained by dividing the discharge by the water depth. A reasonable value has to be chosen to satisfy both accuracy and stability. A threshold of e=10~°m is generally considered. However, it has been observed that for a supercritical flow and under a certain limit of bed slope an unphysical negative water depth could still be generated [5]. Updating this negative water depth to the threshold as often done supposes adding an amount of water to the global system and this violates a strict global mass
Fig. 3. Neighbouring cells to the calculated cell K; for a triangular mesh.
Fig. 3. Neighbouring cells to the calculated cell K; for a triangular mesh.
This second model leads to a better resolution of discontinuities (compared to model-1), as it will be shown by numerical tests, since the nonlinear term 1/2 is kept in the convection flux so that a better satisfaction of the Rankine-Hugoniot condition is obtained (Toro [52]). where Z is an averaged value of the bed level, to be defined more precisely in accordance with conservation properties. The vectors G and H in Eq. (3) are then modified into:
This second model leads to a better resolution of discontinuities (compared to model-1), as it will be shown by numerical tests, since the nonlinear term 1/2 is kept in the convection flux so that a better satisfaction of the Rankine-Hugoniot condition is obtained (Toro [52]). where Z is an averaged value of the bed level, to be defined more precisely in accordance with conservation properties. The vectors G and H in Eq. (3) are then modified into:
Fig. 4. Riemann problem associated with interface Ij. Many choices for Dy exist in the literature, for instance Loukili and Soulaimani [54] used a Lax-Friedrichs scheme with: Dy = | | [UinVin| " | + ./gh;. Davis [55] took into account the neighbouring cell: Dpwis = max{ Uin—V ghi Ujin—/shy, Uin + J gh; 5 luin of sh}. A more robust and simpler scheme was proposed by Toro [53]: Dee = max_{ |i. + Vghi,|ujn| + stu}. ’
Fig. 4. Riemann problem associated with interface Ij. Many choices for Dy exist in the literature, for instance Loukili and Soulaimani [54] used a Lax-Friedrichs scheme with: Dy = | | [UinVin| " | + ./gh;. Davis [55] took into account the neighbouring cell: Dpwis = max{ Uin—V ghi Ujin—/shy, Uin + J gh; 5 luin of sh}. A more robust and simpler scheme was proposed by Toro [53]: Dee = max_{ |i. + Vghi,|ujn| + stu}. ’
It is therefore easy to verify the conservation properties. Fig. 5. Dry cell flooded due to inertia and gravity effects. However, the flux should address the global mass conservatior property. In this paper, we evaluate Dj in Eqs.(24) and (25) as:
It is therefore easy to verify the conservation properties. Fig. 5. Dry cell flooded due to inertia and gravity effects. However, the flux should address the global mass conservatior property. In this paper, we evaluate Dj in Eqs.(24) and (25) as:
Fig. 6. Free surface correction conditions for a wet cell with a dry neighbour. must be avoided when h is zero or very small. Furthermore, the numerical flux computation at these interfaces (or internal boundary) should allow flooding whenever it is physically possible. For instance, at an interface Ij; between a wet cell (where h;>¢) and a dry cell (where hj<é), when z;<1; gravity and inertia forces make it possible for water to flow from the upper wet to the dry cell (Fig. 5). In this case, we use Eq. (24) or (25) with V;=[z, 0 oy’. The flux expression at I; for each model is detailed as follows: NS tJ For a 1 water front advancing over an adverse steep slope z;>1); a free surface correction (FSC) is enforced. The underlying idea is similar to that invoked in [12,14,16]. The aim is to avoid the creation of a free surface gradient (V7 4 0)which is responsible for an unphysical flux across I; leading to numerical spurious oscillations. The FSC technique proposed here is stated as follows:
Fig. 6. Free surface correction conditions for a wet cell with a dry neighbour. must be avoided when h is zero or very small. Furthermore, the numerical flux computation at these interfaces (or internal boundary) should allow flooding whenever it is physically possible. For instance, at an interface Ij; between a wet cell (where h;>¢) and a dry cell (where hj<é), when z;<1; gravity and inertia forces make it possible for water to flow from the upper wet to the dry cell (Fig. 5). In this case, we use Eq. (24) or (25) with V;=[z, 0 oy’. The flux expression at I; for each model is detailed as follows: NS tJ For a 1 water front advancing over an adverse steep slope z;>1); a free surface correction (FSC) is enforced. The underlying idea is similar to that invoked in [12,14,16]. The aim is to avoid the creation of a free surface gradient (V7 4 0)which is responsible for an unphysical flux across I; leading to numerical spurious oscillations. The FSC technique proposed here is stated as follows:
Fig. 7. Free surface correction conditions for a dry cell with a wet neighbour.
Fig. 7. Free surface correction conditions for a dry cell with a wet neighbour.
Fig. 8. Dam break flow on wet bottom: initial conditions. Relations (39) and (40) differ by the terms 2ghjn; and g(1n? — 2z;n;) which, in principle, represent the effect of the free surface gradient. The flux expression (Eq. (40)) could be problematic when treating a dry cell as it will be described in the following.
Fig. 8. Dam break flow on wet bottom: initial conditions. Relations (39) and (40) differ by the terms 2ghjn; and g(1n? — 2z;n;) which, in principle, represent the effect of the free surface gradient. The flux expression (Eq. (40)) could be problematic when treating a dry cell as it will be described in the following.
Fig. 9. Dam break flow on wet bottom: predicted free surface and velocity at t=4s. Remark 1. While updating the solution, the water depth in the former dry cell is evaluated from the computed free surface level 1, as hj=1j—Zjoriginal Where Zjoriginaa denotes the unmodified bed
Fig. 9. Dam break flow on wet bottom: predicted free surface and velocity at t=4s. Remark 1. While updating the solution, the water depth in the former dry cell is evaluated from the computed free surface level 1, as hj=1j—Zjoriginal Where Zjoriginaa denotes the unmodified bed
It is clearly shown in Eq. (42) that the dry cell will be inundated due to the inertia effects of the adjacent wet cell. In fact, the discretized gradient of the free surface flow is zero. However, for model-2, the discretized free surface gradient leads to the term: g (nf — 2znj). Considering that z;= 1, then g (nf — 2zinj) = — gnj. This term implies the existence of an unphysical pressure force. Actually, and as said before, the flux expression (25) is valid only for h=n-—z>0, thus it should not be used for the case of wet/dry interfaces. However, expressions (39) and (42) do not introduce any spurious force. Therefore, in the case of a dry cell which may be flooded, the flux at the wet/dry interface is computed using Eqs. (39) or (42).
It is clearly shown in Eq. (42) that the dry cell will be inundated due to the inertia effects of the adjacent wet cell. In fact, the discretized gradient of the free surface flow is zero. However, for model-2, the discretized free surface gradient leads to the term: g (nf — 2znj). Considering that z;= 1, then g (nf — 2zinj) = — gnj. This term implies the existence of an unphysical pressure force. Actually, and as said before, the flux expression (25) is valid only for h=n-—z>0, thus it should not be used for the case of wet/dry interfaces. However, expressions (39) and (42) do not introduce any spurious force. Therefore, in the case of a dry cell which may be flooded, the flux at the wet/dry interface is computed using Eqs. (39) or (42).
Dam break flow on a wet bottom: relative errors in the L2-norm. Table 1
Dam break flow on a wet bottom: relative errors in the L2-norm. Table 1
Fig. 10. Dam break flow on wet bottom: grid convergence study at t=4s.
Fig. 10. Dam break flow on wet bottom: grid convergence study at t=4s.
Fig. 11. Dam break flow on a dry bottom: initial conditions. This ensures that there is sufficient energy in the wet cell so that the flow reaches the dry cell center. Relations (38) and (41) are applied only if conditions (44) are satisfied, otherwise a wall condition is enforced for the wet cell. Therefore, bottom modification and easy water rush into the dry upper cell are avoided when too- 3.4, Flux approximation at the external boundaries
Fig. 11. Dam break flow on a dry bottom: initial conditions. This ensures that there is sufficient energy in the wet cell so that the flow reaches the dry cell center. Relations (38) and (41) are applied only if conditions (44) are satisfied, otherwise a wall condition is enforced for the wet cell. Therefore, bottom modification and easy water rush into the dry upper cell are avoided when too- 3.4, Flux approximation at the external boundaries
Fig. 12. Dam break flow on a dry bottom: predicted water free surface at t= 2.5 s compared with analytical solutions for a dam break flow on a dry bed. External boundaries concern the edges at the fixed boundary of the computational domain. The corresponding boundary condition is either: an inflow, outflow or a wall condition. The flux is calculated
Fig. 12. Dam break flow on a dry bottom: predicted water free surface at t= 2.5 s compared with analytical solutions for a dam break flow on a dry bed. External boundaries concern the edges at the fixed boundary of the computational domain. The corresponding boundary condition is either: an inflow, outflow or a wall condition. The flux is calculated
Fig. 13. Dam break on a dry bed: sensitivity analysis of the velocity prediction to water depth threshold.
Fig. 13. Dam break on a dry bed: sensitivity analysis of the velocity prediction to water depth threshold.
Dam break flow on a dry bottom: relative errors the L2-norm for different values of the threshold.
Dam break flow on a dry bottom: relative errors the L2-norm for different values of the threshold.
Fig. 14. Dam break flow on a dry bottom: grid convergence study at t=2.5s.
Fig. 14. Dam break flow on a dry bottom: grid convergence study at t=2.5s.
Fig. 16. C-property verification in the CADAM test 2. The second test is more severe in the sense that the water propagates on a dry bed. Geometric configurations are identical to those in the previous case apart from the dry zone downstream of the dam where hg=O (Fig. 11). The dam is assumed to fail instanta- neously. Again, the water depth and velocity predicted are compared with the analytical solutions at time t= 2.5 s in Fig. 12. For the dry bed case, there is no shock but an expansion wave and a smooth transition The dam is initially located at x = L/2 where L= 100 mis the length of the horizontal channel. Initially, the water is stagnant. The water depths are h; = 10 m upstream of the dam and hp = 1 m downstream (Fig. 8). The dam break flow is simulated with an instantaneous breach of the dam. Numerical predictions of the water depth and the velocity obtained through the present model are compared to the analytical solutions in Fig. 9. The numerical results at t=4s were
Fig. 16. C-property verification in the CADAM test 2. The second test is more severe in the sense that the water propagates on a dry bed. Geometric configurations are identical to those in the previous case apart from the dry zone downstream of the dam where hg=O (Fig. 11). The dam is assumed to fail instanta- neously. Again, the water depth and velocity predicted are compared with the analytical solutions at time t= 2.5 s in Fig. 12. For the dry bed case, there is no shock but an expansion wave and a smooth transition The dam is initially located at x = L/2 where L= 100 mis the length of the horizontal channel. Initially, the water is stagnant. The water depths are h; = 10 m upstream of the dam and hp = 1 m downstream (Fig. 8). The dam break flow is simulated with an instantaneous breach of the dam. Numerical predictions of the water depth and the velocity obtained through the present model are compared to the analytical solutions in Fig. 9. The numerical results at t=4s were
Fig. 15. CADAM test 2: reservoir and channel with a triangular bump. obtained with 1000 grid points and are very close to the analytical solutions. The excellent match reveals that the mass conservation property is satisfied. A quantitative comparison is achieved through the relative error in the L?-norm calculation: For a prescribed inflow Q, the numerical flux is computec ccording to the flow rate per unit length Q = Q/I:
Fig. 15. CADAM test 2: reservoir and channel with a triangular bump. obtained with 1000 grid points and are very close to the analytical solutions. The excellent match reveals that the mass conservation property is satisfied. A quantitative comparison is achieved through the relative error in the L?-norm calculation: For a prescribed inflow Q, the numerical flux is computec ccording to the flow rate per unit length Q = Q/I:
Fig. 17. CADAM test-2: water level predictions compared with experimental data and the predictions of Castro et al. (2006) [65].
Fig. 17. CADAM test-2: water level predictions compared with experimental data and the predictions of Castro et al. (2006) [65].
Fig. 18. CADAM test-2: projected water levels compared to the experimental data registered in gauges G4, G10, G11, G13 and G20 3rd, 1998 in Wallingford UK [64]. The test focuses on the bottom slope effect and was carried out experimentally by Prof. J.M. Hiver and his group at the Laboratoire de Recherches Hydrauliques in Chatelet (Université Libre de Bruxelles). As shown in Fig. 15, the experimental setup is composed of a reservoir initially filled to 0.75 m water depth, and a dry rectangular channel containing a symmetrical triangular obstacle of slope 7.59°. Gauges are located strategically to measure the water elevation at 0.15 time intervals for a period of 40s. A gate located 10 m upstream of the bump, is suddenly removed to simulate an instantaneous dam break. The numerical tests were conducted with a Manning roughness coefficient n=0.0125 m~ 7s, a struc- tured mesh of 12,000 triangular elements and(a=1.0, CLF=0.9). Two preliminary tests were carried out to verify the C-property by imposing a null velocity and water depths of 0.3 m and 0.5 m over the whole domain (Fig. 16). The fixed boundaries were considered as solid walls. No velocity was generated and the computed water levels in the reservoir and the channel remained constant. The dam break test was then performed with a single outflow condition on the free outlet at the end of the channel. Fig. 17 shows the predicted water depths compared with the results of Castro et al. [65] and the experimental data. The results satisfy the aim of this benchmark test. The model indeed propagates the dam break wave along the dry bed, overtops the bump and reproduces further shock and rarefaction wave interactions. At the gauging points G4 and G10 located upstream from the bump, the predictions indicate a good estimation of the arrival time of the wave when G13 gives a satisfactory simulation of the flooding and drying (Fig. 18). Gauging points G13 and G20 show an underestimation and an overestimation of the water level
Fig. 18. CADAM test-2: projected water levels compared to the experimental data registered in gauges G4, G10, G11, G13 and G20 3rd, 1998 in Wallingford UK [64]. The test focuses on the bottom slope effect and was carried out experimentally by Prof. J.M. Hiver and his group at the Laboratoire de Recherches Hydrauliques in Chatelet (Université Libre de Bruxelles). As shown in Fig. 15, the experimental setup is composed of a reservoir initially filled to 0.75 m water depth, and a dry rectangular channel containing a symmetrical triangular obstacle of slope 7.59°. Gauges are located strategically to measure the water elevation at 0.15 time intervals for a period of 40s. A gate located 10 m upstream of the bump, is suddenly removed to simulate an instantaneous dam break. The numerical tests were conducted with a Manning roughness coefficient n=0.0125 m~ 7s, a struc- tured mesh of 12,000 triangular elements and(a=1.0, CLF=0.9). Two preliminary tests were carried out to verify the C-property by imposing a null velocity and water depths of 0.3 m and 0.5 m over the whole domain (Fig. 16). The fixed boundaries were considered as solid walls. No velocity was generated and the computed water levels in the reservoir and the channel remained constant. The dam break test was then performed with a single outflow condition on the free outlet at the end of the channel. Fig. 17 shows the predicted water depths compared with the results of Castro et al. [65] and the experimental data. The results satisfy the aim of this benchmark test. The model indeed propagates the dam break wave along the dry bed, overtops the bump and reproduces further shock and rarefaction wave interactions. At the gauging points G4 and G10 located upstream from the bump, the predictions indicate a good estimation of the arrival time of the wave when G13 gives a satisfactory simulation of the flooding and drying (Fig. 18). Gauging points G13 and G20 show an underestimation and an overestimation of the water level
Fig. 20. Double expansion: water levels. The water volume inside the domain results from the initial volume Voli3't and the inflow and/or outflow contributions: Fig. 19. CADAM test-2: time evolution of mass error.
Fig. 20. Double expansion: water levels. The water volume inside the domain results from the initial volume Voli3't and the inflow and/or outflow contributions: Fig. 19. CADAM test-2: time evolution of mass error.
compared to experimental measurements respectively. Relative errors between the predictions and the experimental data do not exceed 4% and it should be noted that Prof. Hiver's data (Laboratoire de Recherches Hydrauliques in Chatelet) were obtained with 5% of accuracy. The comparison between the present predictions and those in [14,65] and the experimental results is satisfactory. A check on the global mass conservation is carried out over all the simulations. The predicted global water volume is calculated as
compared to experimental measurements respectively. Relative errors between the predictions and the experimental data do not exceed 4% and it should be noted that Prof. Hiver's data (Laboratoire de Recherches Hydrauliques in Chatelet) were obtained with 5% of accuracy. The comparison between the present predictions and those in [14,65] and the experimental results is satisfactory. A check on the global mass conservation is carried out over all the simulations. The predicted global water volume is calculated as
Fig. 21. Double expansion: predicted water levels at times 0 s, 0.05 s, 0.25 s, 0.45 s and 0.65 s.
Fig. 21. Double expansion: predicted water levels at times 0 s, 0.05 s, 0.25 s, 0.45 s and 0.65 s.
Fig. 22. Double expansion: discharge (a) and (b) velocity and velocity pattern (c) in the channel at different times.
Fig. 22. Double expansion: discharge (a) and (b) velocity and velocity pattern (c) in the channel at different times.
The fixed boundaries are solid walls while positions x=0O and x=25m correspond to the outlets. The results were obtained for a mesh of 3900 triangles. Fig. 20 shows the predicted water depths compared with those of Castro et al. [17]. Some discrepancies are observed but the results are broadly similar. ae a
The fixed boundaries are solid walls while positions x=0O and x=25m correspond to the outlets. The results were obtained for a mesh of 3900 triangles. Fig. 20 shows the predicted water depths compared with those of Castro et al. [17]. Some discrepancies are observed but the results are broadly similar. ae a
Fig. 23. Double expansion: time evolution of the wetting/drying front caused by a tidal wave in a basin with a variable slope, water level (a), discharge (b), velocity (c). The results were compared with those in [14] and [11] who used the finite volumes technique and the finite elements technique respectively. The domain was discretized with a constant space step Ax = Ay=5 m and the Manning roughness coefficient was assumed to be equal to n=0.03. In Fig. 23a, at time t= 36min, water recedes from the upstream part of the channel due to the tidal wave at the inlet. Flooding then occurs until the channel is completely inundated at t= 54 min. The numerical models show relatively close predictions, however tiny differences on the water front prediction are observed, namely at times t=36min and t=54min as the wetting front advances over the second and the third slope. This highlights the
Fig. 23. Double expansion: time evolution of the wetting/drying front caused by a tidal wave in a basin with a variable slope, water level (a), discharge (b), velocity (c). The results were compared with those in [14] and [11] who used the finite volumes technique and the finite elements technique respectively. The domain was discretized with a constant space step Ax = Ay=5 m and the Manning roughness coefficient was assumed to be equal to n=0.03. In Fig. 23a, at time t= 36min, water recedes from the upstream part of the channel due to the tidal wave at the inlet. Flooding then occurs until the channel is completely inundated at t= 54 min. The numerical models show relatively close predictions, however tiny differences on the water front prediction are observed, namely at times t=36min and t=54min as the wetting front advances over the second and the third slope. This highlights the
Fig. 24. Sketch of the Bordeaux stockade bathymetry. The steady state was reached with a 16.34 m water averaged free surface. Then, from this steady flow considered as the new initial condition, we imposed an inflow flux varying with the tidal cycle:
Fig. 24. Sketch of the Bordeaux stockade bathymetry. The steady state was reached with a 16.34 m water averaged free surface. Then, from this steady flow considered as the new initial condition, we imposed an inflow flux varying with the tidal cycle:
Fig. 25. Bordeaux stockade: 2-D and 3-D Bordeaux stockade mesh (25,976 elements, 13,217 nodes).
Fig. 25. Bordeaux stockade: 2-D and 3-D Bordeaux stockade mesh (25,976 elements, 13,217 nodes).
Fig. 26. Bordeaux stockade: mass errors time evolution for the dam break test.
Fig. 26. Bordeaux stockade: mass errors time evolution for the dam break test.
Fig. 27. Bordeaux stockade: velocity field in the river during the extreme wetting anc drying phases after 1200 s and 2700s. Fig. 28. Bordeaux stockade: zoom on the velocity field in the south-east part of the domain at times 1200 s and 2700 s.
Fig. 27. Bordeaux stockade: velocity field in the river during the extreme wetting anc drying phases after 1200 s and 2700s. Fig. 28. Bordeaux stockade: zoom on the velocity field in the south-east part of the domain at times 1200 s and 2700 s.
Fig. 29. Bordeaux stockade: streamlines in the river during wetting and drying phase: at times 1200 s and 2700s.
Fig. 29. Bordeaux stockade: streamlines in the river during wetting and drying phase: at times 1200 s and 2700s.
Fig. 30. Bordeaux stockade: contour lines of water depths during wetting and drying phases at times 1200 s and 2700 s.
Fig. 30. Bordeaux stockade: contour lines of water depths during wetting and drying phases at times 1200 s and 2700 s.
Fig. 31. Bordeaux stockade: 3D view of the contour lines of water levels during wetting and drying phases at times 1200s and 2700 s.
Fig. 31. Bordeaux stockade: 3D view of the contour lines of water levels during wetting and drying phases at times 1200s and 2700 s.
Fig. 32. Bordeaux stockade: 3D view of wetting/drying after 1200 s and 2700s.
Fig. 32. Bordeaux stockade: 3D view of wetting/drying after 1200 s and 2700s.
Related topics:
EngineeringFinite Volume MethodsMesh generationModeling and SimulationConservation LawsMass ConservationFinite Volume MethodNumerical StabilityShallow Water Equation (SWE)Free Surface Flow
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