An inverse problem for modeling open channel flow with movable bed

Unsteady flow numerical simulation over mobile beds is theoretically challenging and practically important in the field of river engineering. So far, a large number of numerical river modeling methodologies of water-sediment-morphology fluvial systems have been developed (Van & Chua 2020). These methodologies have been applied as the primary methods in river engineering studies (Liao et al. 2014). The one-dimensional (1D) modeling of the free surface flow and bed evolution in alluvial channels are most often performed via applying the St. Venant equations (De St. Venant 1871) with the Exner equation (Graf 1971). In recent years, a great variety of numerical techniques have been applied in river engineering issues considering two basic aspects (Chen & Duan 2008). The first is associated with the numerical solution procedure of the governing equations system. Alluvial flows on erodible beds are different from streams on a fixed bed, because the stream may remove sediments from the bed or, on the contrary, sediments carried by the stream may deposit on the bed, which usually leads to river bed degradation or aggradation (Liang & Marche 2009; Daneshfaraz et al. 2021). It is supposed that the rate of bed morphological evolution has a lower order of magnitude compared to flow variations with low sediment concentrations (Estakhr et al. 2022). Therefore, the equation of water-sediment continuity is assumed to be identical for a fixed-bed clear water flow, regardless of the mobility of the channel bed, i.e. or the third term in Equation (1) is relinquished (Needham & Hey 1991; Sieben 1999; Sieben et al. 2009). As it will be described in the mathematical expressions in the next section, the water-sediment-morphology fluvial system is strongly coupled with the auxiliary closure relations. In available numerical approaches, these equations are often solved by asynchronous methods. In particular, at a given time step, the equations of mixture continuity and momentum are solved by supposing a small bed variation rate. In the next step, via the newly obtained flow variables, the sediment continuity equation is solved. Methods that contain asynchronous solutions are commonly referred to as uncoupled. Models involving the asynchronous solution are usually termed uncoupled. There are some-semi coupled methods that use iterative solutions of the bed and flow equations at a given time step (Palumbo et al. 2008). On the other hand, there is not a universal consensus about the best approach for morphological river modeling. The second aspect is the identification problem of alluvial channel friction factors. Mathematical modeling of open-channel flow systems involves wave propagation over channel reaches which is governed by the St. Venant equations, which describe the conservation of mass and momentum. The roughness coefficient used in the momentum equation is often considered as a lumped empirical coefficient that indicates the overall flow resistance of specific river channels. The literature on identifying and estimating parameters for unsteady open channel flow is sparse. Roux and Dartus investigated uncertainty in open-channel inverse problems using inundation observations (Roux & Dartus 2008). Sellier (2008) and Gessese et al. (2011) extended the inverse problem of reconstruction of river bed topography.

In the present research, the equations of 1D unsteady flow (i.e. St. Venant equations) and sediment continuity equation were solved numerically using the second-order accurate, explicit finite difference two steps scheme (Richtmyer & Morton 1967). The governing equations were solved in both coupled/uncoupled solution procedures. By evaluating the effect of bed load transport formulas and simplification of the continuity equation on simulation of alluvial flow, the model was validated with new laboratory experimental data. Then, the inverse problem of evaluating the open channel flow roughness was investigated using a genetic algorithm optimization model. Flow depths and discharges experimental datasets at several locations and time steps were used as inputs of the optimization model. In order to dynamically identify the roughness parameter the genetic algorithm and the unsteady alluvial flow model were coupled by time interval. Calibration was repeated with the selection of different objective functions.

Step 1:

In which is the Courant number. For scheme's stability, Equation (20) must satisfy the Equation (21) at every grid point.

After establishing an equilibrium state, the stream flow which was capable to move the sediment was observed in the channel. Then, sediment was fed at upstream boundary, with a much higher rate than the flow transport capacity. The water supply rate was kept constant at 68 lit/sec, and the sediment supply at the flume entrance was 0.008 kg/s. The oversupply of sediment resulted in the sediment deposition and the aggradation of the stream bed near the flume entrance. The initial bed slope was S_x = 0.0005. The bed and water level profiles were recorded by traversing the bed and water level sensors along the flume length at various time intervals. The initial conditions were determined based on the water and bed level profiles measured at t = 0. The constant flow rate, bed level measured at the upstream, and water depth measured at downstream were considered as boundary conditions. The flume bed was covered with dunes during the experiments. The aggradation process occurred mainly in the first two mod channel length (with respect to the flume entrance). After passing this region, the movement of the dunes occurred, however, without significant change in the mean bed level. A better perspective of these features could be seen in Figure 2(b). The simulations of the proposed model were carried out to predict the average bed level and water surface profile, and good agreement was observed between the measurements and simulations.

In the coupled system of governing equations, selecting the appropriate numerical solution approach is so important. The water sediment-morphology fluvial system was strongly coupled, as mentioned in Equations (2)–(4) along with Equations (5) and (6). In existing numerical approaches, the mentioned equations are solved using an asynchronous procedure. In particular, at a given time step, at first, the mixture continuity and momentum equations are solved, supposing inconsiderable bed variation (or fixed bed). The sediment continuity equation is then solved through the newly achieved flow variables. Methods that include the asynchronous solution are commonly referred to as uncoupled (Cao & Carling 2002). The basis of the asynchronous solution strategy is the ‘fixed bed’ and ‘quasi-steady’ flow assumptions and often the flow is supposed to be steady when the channel bed evolution is investigated. On the other hand, the river bed is implicitly supposed to be fixed within a time step, whereas the flow over a mobile bed is of primary interest.

Bed load discharge is the main parameter in specifying the morphologic development of alluvial channel reaches. An essential task with the application of 1D numerical model is the use of an appropriate bed load discharge equation. Since most of the existing relationships depend on limited data and untested assumptions, the application of most relationships is limited to their development conditions. In this study, considering the validity of bed load transport formulas, Meyer-Peter & Müller (1948), Einstein (1950), and Ackers & White (1973) were applied for steady flow. The formulas mentioned above along with Song & Graf (1997) were used for unsteady flow.

There is an ongoing debate about using coupled or uncoupled approaches in alluvial flows. For most of the practical cases, the application of uncoupled methods may be justified, mainly due to different time scales in the motion of water and sediment, as well as the inclusion of the empirical relations with different degrees of approximations, which largely affect the accuracy of the outcomes. The coupled and uncoupled approaches were tested for steady and unsteady flow conditions. For unsteady flow, the uncoupled approach at t = 30 min and t = 90 min gave a close solution to the coupled method, however, at t = 120 min some oscillations were observed.

Strong oscillations of false bed and water level took place around t = 240 min. In the steady flow, at t = 120 min, it was found that the asynchronous solution procedure entailed significant inaccuracy for both aggradation and degradation processes. In the uncoupled procedure, the rapid changes of the bed level related to the sediment component were not introduced simultaneously to the flow component. Since the flow and sediment computations were done separately, the flow component received the new bed levels after Δt. The obtained flow conditions (new water depths and velocities) related to the sudden variations of bed level significantly differed from the conditions extracted from the coupled method, which included a simultaneous solution of both water and sediment components at the previous time step. This deviation of the outputs was magnified through uncoupling of the components at the subsequent time steps, resulting in larger artificial oscillation. Conversely, in the coupled method, slight fluctuations of the bed level were gradually reduced due to simultaneous interference of the flow conditions.

In automatic optimization, a systematic iterative process is used to search a set of parameter values for which a chosen function assumes its extreme valves. The model implemented is an explicit finite difference scheme of Richtmyer and the process is automated using a genetic algorithm for estimating Manning's roughness coefficients using flow measurements data in a laboratory flume. The comparison between the water and bed level profiles (Figure 5(b)–5(d) and Table 2) monitored at various instants before/after optimization shows that the optimized model reduces computational errors, improves the simulation results, and performs satisfactorily in all cases (the evaluation herein is limited to subcritical flow conditions).

In an unsteady flow, by optimization of the roughness coefficient, the form of the objective function can be elated. Both the optimization procedure and optimization criterion type have an effect on the duration and accuracy of computations. Also, if the selection process of these parameters has not been done properly, they may lead to incorrect values. In fact, the values, type, and quality of datasets play an important role in outcomes and often specify the selection of objective functions. The results of the performance of the objective functions, defined by f₁–f₅ on error percentages (see Figure 7), show that the objective function significantly affects the quality of the calculations. The best outcomes are determined via functions based on relative errors (f₄ and f₅). The impacts of the traditional f₁ and f₂ are relatively poor. The difference between f₁ and f₄ indicates that the objective function is particularly significant when both water stage and discharge are measured.

In this research, an explicit finite difference two steps scheme developed by Richtmyer is presented for the simulation of 1D steady/unsteady flow and bed morphology in alluvial channels. Some new laboratory experiments were performed in a mobile-bed flume for both steady and unsteady flow to validate the proposed model. For improving the simulation, first, the role of the sediment transport formula, coupled/uncoupled approaches, and simplification in the mass continuity equation is investigated. Comparisons between experimental data and model performance highlight the advantage of the coupled method over the uncoupled method, the domain role of the sediment transport model in the bed morphological simulation, and the inaccuracy of aggradation processes due to simplifying the mixture continuity equation. Second, the importance of changing alluvial roughness is established by testing the model with calibrated and optimized friction factors. The inverse problem of estimating alluvial flow roughness is solved using a GA optimization model coupled with an alluvial flow model. The study demonstrates that the use of GA in the process of searching for optimal values of roughness coefficients can significantly reduce computational errors and improve the computed water stage hydrographs. The results of the performance of the objective functions show that the form of the objective function influences the quality of calculations and the best results are obtained by functions based on relative errors.

The authors did not receive support from any organization for the submitted work.

Kiyoumars Roushangar: Conceptualization, supervision, methodology, review & editing; Roghayeh Ghasempour: Project administration, investigation, data curation, methodology, writing; Hazi Mohammad Azamathulla: Conceptualization, methodology, review & editing.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.