Test case I: Idealized dam break problem in a rectangular channel with dry

A 1200m long rectangular frictionless channel with unit width is considered with a dam located 500m from the upstream end of the channel. Initially the upstream water depth is 10m and the downstream channel is dry. The dry bed is treated by specifying the depth of the water as a very small number h_dry 1010m to avoid numerical problems associated with division by zero. In the computation, a spatial grid sizex2m and a temporal grid size t0.01s are used. Assuming the dam breaks at time t0, then the numerical results are generated 30 seconds after the dam failure. Figures 3.3 - 3.5 show comparisons of the numerically computed water depth, flow rate and velocity variations with the exact solution.

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Figure 3. 3: Water level variation along a channel of the idealized dry bed dam break problem after 30 seconds.

The depth of the water varies parabolically from the undisturbed upstream edge and the dry bed. Three regions can be clearly seen in the flow; the undisturbed stationary region; the moving water front region and the dry bed region. The positive wave travels downstream on the dry bed, while a negative wave travels upstream into the undisturbed region. The depth of the flow towards the upstream direction reduces with time as a result of the flow mass flowing downstream.

Figure 3. 4:Flow rate variation along the channel of the idealized dry bed dam break problem after 30 seconds.

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Figure 3. 5: Velocity variation along the channel of the idealized dry bed dam break problem after 30 seconds.

As can be seen in the comparisons above, the numerical results are in good agreement with the exact solution. In Figure 3.5, it can be seen that the velocity near the leading edge of the flow is under predicted. Many numerical models have difficulties in predicting the exact values of the velocities near the wave front edge, as the water depth in the area is very small [56] but as the focus of the present work is on the inverse problem, the quality of the numerical solution is deemed satisfactory for our purposes.

3.3.2. Test case II: Idealized dam break problem in a rectangular channel with wet bed

In this test case the initial downstream water depth is set to 2 m while the domain and the numerical conditions are set to be similar as that of test case I. This test case is used to investigate the performance of the numerical scheme for a flow with discontinuities in the domain. The numerical results are generated 30 seconds after the dam failure. Water depth, flow rate, and velocity variations are compared with the exact solution in Figures 3.6 - 3.8.

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Figure 3.6: Water depth versus distance for the idealized wet bed dam break problem after 30 seconds.

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Figure 3.8: Velocity of the flow versus distance for the idealized wet bed dam break problem after 30 seconds.

As can be seen from the above Figures, there is good agreement between the computed results and exact solutions. Due to the presence of stationary water downstream of the dam, there exists a discontinuity at the interface of the moving front and the undisturbed downstream stationary water. The numerical scheme has accurately handled the discontinuity, at the moving wave front of the flow. The upstream depth of the water reduces with time in a similar fashion as in the case of the dry bed.

The above two test cases are presented for the sole purpose of evaluating the capability of the numerical scheme to solve unsteady shallow water flows. The test cases used to generate the free surface data for the inverse problem are described in the following section.

3.3.3. Test case III: Steady flow in a frictionless rectangular channel with a bump

A 1m wide, 25m long channel is considered to test transcritical flow over a bump. The frictionless channel bed is assumed to have a rectangular cross section and a bump. The bed topography is defined by

52           12 8 10 05 0 2 0 12 8 0 2 x ) x ( . . x and x z (3. 7)

The depth at the downstream boundary is set as h0.33m and the water inflow condition is specified asQ0.18m3/s. This test is used to study the convergence of the numerical solution towards steady-state flow conditions and confirm the conservation of discharge along the channel. A spatial grid size



x0.1m and a temporal grid size ∆t = 0.01 s are used. Steady-state results have been generated and the flow rate over the domain was conserved after 400 seconds in the numerical computation. Figure 3.9 shows the comparison of computed water level with the analytical solution.

Figure 3. 9:Water level variations along a channel with transcritical flow

As an additional test case for the flow over the same bed topography, subcritical flow conditions of Q4.42m3/s at the upstream boundary and h2m at the downstream boundary are imposed. This test case is performed to evaluate the performance of the scheme in predicting the steady-state water level for subcritical flow. The computed steady-state free surface profile is compared with its analytical solution in Figure 3.10.

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Figure 3. 10: Water level variations along the channel with subcritical flow

It can be seen that the numerical results are in good agreement with the analytical solution which can be obtained by applying the Bernoulli equation for the points outside the hydraulic jump in the domain. The jump region in the transcritical flow case has been modelled with the help of conservation of momentum. These results are also in a good agreement with the results presented in [56, 74, 75].

From the forward problem analysis, it is now established that the numerical scheme is capable of modelling steady and unsteady; subcritical and transcritical shallow water flows. Wetting and drying fronts numerical implementation is omitted due to the ranges of applicability of the whole solution approach presented in this thesis.

In document Algorithms for Bed Topography Reconstruction in Geophysical Flows (Page 67-73)