Chapter 3 Model Validation
3.2 Validation of Bed Evolution Model
The bed evolution model is validated with five types of bedforms, including two of forced bars (side bars and central bars), one of free migrating alternate bars and two of coexistence of free and forced bars. These results suggest that the CDG scheme may be applied to the sediment continuity equation without incorporating any artificial smoothing [Vasquez et al., 2007].
3.2.1 Forced bars – side bars
The bed topography measured in run C1-11 of Bittner (1994) is used to validate the bedform of side bars. The geometry of the channel has been described in 3.1.1, but the computational domain is changed as shown in Figure 3-6a, which consists of a total of 3,192 nodes and 5,460 elements. The zoomed-in element mesh in a cycle of width variation is demonstrated in Figure 3-6b. There are totally 19 cycles included in this domain, the upstream and downstream reaches are extended, where the bed elevations remain fixed. This ensures the conservation of sediment in the channel and that bed evolutions at the upstream and downstream ends would not be affected by unstable flows [Defina, 2003; Zech et al., 2006].
The VA model is used for the computation of flow dynamics. The boundary condition imposed at the upstream is the specified unit discharge q0 0.0073 m2/s.
No-penetration condition is used at the sidewalls. Initially, flow depths are all set as 0.022 m, the bed is flat with a slope of 0.004. The helical flow coefficient a3. The sediment influx is calculated with the flow conditions at the extended upstream reach.
The upwind coefficient () used in flow dynamics model and bed evolution model are 0.75 and 0.5, respectively.
The experimental results compared with the linear solution [Wu and Yeh, 2005]
and the computed bed topography of side bars are shown in Figure 3-7, the measured
bed deformations have a pile of deposition along the near-wall regions at the widest channel width section of a cycle and V-shape scour at the narrowest section of a cycle, where satisfactory agreement is demonstrated. To further compare the bed deformation patterns at different locations, the cycle-averaged measurements and the computed bed topography at four specified sections of a width-variation cycle are shown in Figure 3-8, where the linear solution and numerical results agree well with the measurements. Both linear solution and numerical computation slightly under predict the depth of scour at 1/4π and 2/4π of the width-variation cycle and slightly over predict the depth of scour at 0/4π and 3/4π of the width-variation cycle. However, the numerical computations are more close to the measured than linear solution.
Figure 3-6 (a) Computation domain of Run C1-11 [Bittner, 1994] used in the validation of the bed evolution model; (b) zoomed-in element mesh in a cycle of width variation, the wavelength of cycle (π) is 1.6m
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Figure 3-7 Experimental results of Run C1-11 [Bittner, 1994] are compared with the linear solution and computed numerical bed topography of side bars
Figure 3-8 Comparison of measured and computed Run C1-11’s [Bittner, 1994]
results of bed topography at four specified sections of a width-variation cycle
3.2.2 Forced bars – central bars
A series of experiments were conducted in a variable-width channel by Wu and Yeh (2005). One of the experiments, S6, is used here to validate the simulated result of central bars. The sinusoidal variation of the channel width is described by
0 the wavelength of the channel whose value was 3.351 m; x is the coordinate in the longitudinal direction. The computational domain is shown in Figure 3-9a, which consists of a total of 2,457 nodes and 4,200 elements. The zoomed-in element mesh in a cycle of width variation is demonstrated in Figure 3-9b. There are totally 6 cycles included in this domain
The VA model is used for flow computations and compared with the result from linear solution. The boundary condition imposed at the upstream is the inflow unit discharge q0 0.0197 m2/s. No-penetration of water flow is imposed at the sidewalls. The roughness height K is 0.004; the helical flow coefficient s a is 5.
The initial flow depth is given by Dw 0.0049 m, the unit discharge Q is set equal x to q . The initial flat bed has a slope of 0.003. In both the upstream and downstream 0 extended reaches the fixed-bed condition is imposed, the sediment transport rate in the upstream extended reach is used as the sediment influx to the variable-width channel. The upwind coefficient ( ) used in flow dynamics model and bed evolution model are 0.75 and 0.5, respectively.
The computed bed topography of central bars and the experimental result are shown in Figure 3-10, where satisfactory agreement between the experimental and
computed results is demonstrated. The measured central bars have bullet-shape fronts, which are captured more successfully by the linear solution and the numerical models.
To further demonstrate this, the lateral bed profiles at the widest and narrowest sections are shown in Figure 3-11. At the widest section of a cycle the linear solution and numerical computation successfully predict the bed elevation in the central region of variation-width channel, but under predict the scour in the near-wall region. At the widest section of a cycle both the linear solution and numerical computation under predict the depth of scour in the central region of variation-width channel, but over predict the scour in the near-wall region. However, the numerical computation model slightly outperforms the linear solution.
Figure 3-9 (a) Computation domain of Run S6 [Wu and Yeh, 2005] used in the bed evolution model; (b) Zoomed-in element mesh in a cycle of width variation, the wavelength of a cycle is 3.351m
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Figure 3-10 Experimental result of Run S6 [Wu and Yeh, 2005] are compared with the linear solution and computed numerical bed topography of central bars
Figure 3-11 Experimental result and computed lateral bed profiles of Run S6 [Wu and Yeh, 2005] at the wide and narrow sections
3.2.3 Free migrating alternate bars
Migrating alternate bars belong to the advection-dominated bed evolution. The traditional Gelerkin method is numerically unstable when used to simulate such cases.
Defina (2003) performed a series of numerical experiments on the alternate bars migration using the streamline upwind scheme. In Defina’s flow dynamics model, some empirical parameters were used and the sediment transport rate was assumed constant in order to avoid the numerical instability. In our study the CDG scheme is for the first time applied to solve the Exner equation. It is thus of our interest to see if the advection-dominated bed evolution can be simulated reasonably well without degrading the accuracy of sediment transport dynamics.
The numerical simulation performed here follows the numerical experiment of Defina (2003) that is based on the flume experiment of alternate bars conducted by Lanzoni (2000). In the numerical experiment of Defina (2003), an initial disturbance with a single bump was created at the upstream end inducing a train of alternate bars.
It should be noted here that even with the same hydraulic condition, different initial disturbances may induce different characteristics of alternate bars in terms of the wavelength, bar height, and celerity. Thus our simulated result of migrating alternate bars is only compared qualitatively with the result of Defina (2003) to validate the bed evolution model.
The parameters used in the numerical simulation are based on the hydraulic condition of the experimental run P1505 [Lanzoni, 2000]. The flume is 1.5 m wide by 55 m long; a length of 120 m is used in the numerical simulation to offer a sufficient space for the alternate bar development. The fixed-bed reaches are extended in both the upstream and downstream of the flume. The computational domain is discretized with a total of 9 481 nodes and 7,680 elements, as shown in Figure 3-12. The initial disturbance is given by the following expression [Defina, 2003] :
,
cos
/ 2 0
sin 2 /
respectively. Equation (3-3) was used by Defina (2003) to create a single bump that only covered half the channel width, but an initial disturbance covering the whole width (shown in Figure 3-13) is used in this study to accelerate the bar growth.The VA model is used here for the flow computation. The boundary condition at the upstream is the inflow unit discharge q0 0.002 m2/s. The inflow sediment transport rate is calculated with Equation (2-16) using the hydraulic condition in the extended upstream fixed-bed reach, with the roughness height Ks 0.0048 m. The calculated sediment transport rate is 2.9110-5 m2 /s, which is slightly greater than 2.610-5 m2 /s measured in the flume. The no-penetration condition is imposed at the sidewalls. The initial bed is flat with a slope of 0.00452. The initial flow depth is assumed equal to 0.044 m; the unit discharge in the x-direction is 0.002 m2/s; the unit discharge in the y-direction is zero. The upwind coefficient ( ) used in flow dynamics model and bed evolution model are 0.75 and 0.75, respectively.
The simulated evolution of the alternate bars is shown in Figure 3-14. In the first two hours, the initial disturbance migrates downstream and triggers the formation of alternate bars. The upstream alternate bars continue to grow, migrate downstream, and trigger the alternate bars further downstream. After three hours, the diagonal fronts of the alternate bars become more obvious. The bed returns to flat after the train of alternate bars passes by. The bar height, defined as the difference between the extreme elevations in the left and right halves of a cross section, grows with time, as shown in Figure 3-15. The growth rates are steep in the first two hours, and then become mild.
The height of the first generated bar, denoted as bar No.1, almost reaches a steady
value after 7 hours. A comparison between the calculated and observed longitudinal profiles of bar height is shown in Figure 3-16. The simulation results are in agreement with the observed bar height and wavelength. The simulated wavelength of the free bars is 12 m, which is slightly greater than 10 m that was observed in the experiment of Lanzoni (2000). The quasi-steady bar height is approximately 6 cm, close to the bar height of 7 cm observed by Lanzoni (2000). The average migration speed is 4 m/hr, greater than 2 and 2.8 m/hr obtained by Defina (2003) and Lanzoni (2000).
The morphodynamic model developed in this study is validated with the forced and free bars. The results indicate that both the linear solution and numerical models would simulate the flowfield reasonably well. However, the numerical model generally outperforms the linear solution. The numerical simulation of free migrating alternate bars reveals that the CDG scheme is applicable to solving the Exner equation.
Figure 3-12 (a) Computation domain of Run P1505 [Lanzoni, 2000] used in bed evolution model; (b) zoomed-in element mesh
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Figure 3-13 Initial disturbance used to trigger the formation of alternate bars in the simulation of Run P1505 [Lanzoni, 2000]
Figure 3-14 Development and migration of alternate bar trains in the simulation of Run P1505 [Lanzoni, 2000]
Figure 3-15 Growth of bar height with time in the simulation of Run P1505 [Lanzoni, 2000]. The height of bar No.1 almost reaches a steady value.
Figure 3-16 Comparison between the calculated and measured Run P1505’s [Lanzoni, 2000] longitudinal profiles of bar height. (Bar height is defined as the difference between the extreme elevations in the left and right halves of a cross section.)
3.2.4 Coexistence of free and forced bars
The validated case of coexistence of free and forced bars is chosen from two of a series of experiments conducted in a variable-width channel by Wu and Yeh (2005), labeled as F2 and F7. The experimental bed topography of F2 is central bar paved by an alternative pattern in the channel, as shown in Figure 3-17, thus deduced that the distorted central bar is result of coexistence of free and forced bars. In the experiment of F7, the order of free bars are larger than the force bars, as showed in Figure 3-19, as the results the experimental and simulated topography display a free bars dominated bed forms. The dominated bed deformation of forced bar or free bars in the case of coexistence is attributed to the aspect ratio β [Lanzoni, 2000], which are aspect ratio of F2 and F7 are 5.1 and 13.5, respectively. Higher aspect ratio results in free bars dominated bed forms.
The sinusoidal variation of the channel width is described by Eq. 3-2 with A equal to 0.156 and c equal to 3.35m. The geometry of the channel of F2 and F7 is the same as prescribed in S6 case, and the computation domain is the same as shown in Figure 3-9 which consists of a total of 2,457 nodes and 4,200 elements.
The VA model is used for flow dynamic model. Required computation condition include the roughness height K is 0.004, the helical flow coefficient s a is 5 and a slope of the initial flat bed is 0.005. The boundary condition imposed at the upstream is the inflow unit discharge (q ) equal to 0.0137 and 0.005 m0 2/s in F2 and F7, respectively. No-penetration of water flow is imposed at the sidewalls. The initial flow depth (D ) is given by 0.00313 m, the unit discharge w Q is set equal to x q . In 0 both the upstream and downstream extended reaches the fixed-bed condition is imposed, the sediment transport rate in the upstream extended reach is used as the sediment influx to the variable-width channel. The upwind coefficient ( ) used in flow dynamics model and bed evolution model are 1.25 and 0.75, respectively.
The computed bed topography and the experimental result of F2 and F7 are shown in Figure 3-17 and 3-19, respectively, where satisfactory agreement between the experimental and computed results is demonstrated. In Figure 3-17 the measured bed topography have distorted bullet-shape fronts, which are similar to central bars superposed by an alternative pattern and are captured more successfully by the numerical models. To further demonstrate this, the comparison of experimental measurement and numerical computation result are made with the lateral bed profiles at four specified sections of a width-variation cycle, as shown in Figure 3-18. The elevation of bed forms are predicted by numerical model exactly. The numerical results agree well with the measurements, except that at 2/4π of the width-variation cycle numerical results under predict the scour depth of the right side of the channel.
In Figure 3-19 the measured bed topography have an alternative scour and deposition pattern, which are similar to free bars are the dominate bed forms in the channels with variable-width and are captured more successfully by the numerical models. The comparison of experimental measurement and numerical computation result are made with the lateral bed profiles at four specified sections of a width-variation cycle, as shown in Figure 3-20, where the computed results agree well with the measurements, except that at 1/4π and 2/4π of the width-variation cycle the model over predicts the deposition of bed elevation.
Figure 3-17 Experimental result and computed bed topography of coexistence of forced bar dominated case in Run F2 [Wu and Yeh, 2005].
Figure 3-18 Comparison of measured and computed results of bed topography at four specified sections of a width-variation cycle in the case of coexistence of forced bars dominated case in Run F2 [Wu and Yeh, 2005].
Figure 3-19 Experimental result and computed bed topography of coexistence of free bar dominated case in Run F7 [Wu and Yeh, 2005].
Figure 3-20 Comparison of measured and computed results of bed topography at four specified sections of a width-variation cycle in the case of coexistence of free bars dominated case in Run F7 [Wu and Yeh, 2005].