MATHEMATICAL MODELLING OF TRANSIENT FLOWS IN ALLUVIAL STREAMS

Abstract

Mathematical modelling has proved to be an efficient tool for the study of morphological processes in alluvial streams. Generally, these processes occur over long time periods. However, some extreme events like a flood due to a dam break may cause extensive changes in river morphology over a short period. Therefore, a study involving pronounced unsteadiness of flow in alluvial rivers is important. A onedimensional coupled model is proposed in the present study for simulation of sharp unsteady flows in alluvial rivers. Also the mixed supercriticalsubcritical flows involving a hydraulic jump in an alluvial river are simulated in the present model. In addition, the physical processes involving bed material grain sorting, transport of wash load and nonequilibrium transport of bed material load are modelled. A number of mathematical models exist for the study of alluvial river processes. In these mainly the governing equations for one- dimensional flow have been solved using the finite difference methods (Garde and Ranga Raju, 1985). Most of these models are uncoupled in nature. Lyn (1987) pointed out important limitations of uncoupled models. Applying perturbation analysis he showed that the uncoupled approach is only valid in cases where temporal changes in boundary conditions are negligible. For studying cases when discharge and sediment inputs are variable, a coupled model is therefore necessary. In recent years, several coupled models have been developed. Lyn (1987) presented a coupled model in which the governing equations were (i) solved simultaneously using generalized Preissmann scheme. His model was applied for the study of sediment deposition in a reservoir with constant and variable sediment input and constant flow discharge. Rahuel et al. (1989) proposed a coupled model named CARICHAR which deals with bed load dominant transport processes. The CARICHAR model also simulated the bed material grain sorting. SEDICOUP (Holly and Rahuel, 1990) is another coupled model which simulates both bed load and suspended load transport. Both in CARICHAR and SEDICOUP, the non- equilibrium transport of bedload was also studied. Correia et al. (1992) proposed another coupled mobile boundary model namely the FCM. In FCM the governing equations used were explicitly coupled. FCM offers for use a wide range of alluvial resistance equations and sediment transport equations. Correia et al. (1992) also proposed a solution scheme for linear system of discretized governing equations. A close study of the coupled models indicated that these models could be applied successfully for the simulation of gradual bed and flow transients. Also none of the existing models simulates the changes occurring in the state of flow with distance e.g. transformation of flow from supercritical state to subcritical state. Processes such as nonequilibrium transport of bed material and transport of wash load also need to be incorporated in the model based on the latest information available on these aspects. The present study has been taken up with an objective to fill these gaps existing in the available models. The model proposed in the present study is based on the following set of governing partial differential equations for one- dimensional flow. (ii) i) Flow Continuity Equation § +P|?+B§ =0 ...(D ii) Flow Momentum Equation aQ 2Q aQ p Q2 ah . ah . dz „AC n ,0. ^f + -A-ax-BA^5x + gAax + gA^ + gASf = ° '"[2) iii) Sediment Continuity Equation aQb aQs az ah 3Cau i£ + id + p"-»i + BC-w + A^f = ° -i3i Here A = cross-sectional area, B= flow width, P = flow perimeter, Q = discharge, h = depth of flow, z = bed elevation, g = acceleration due to gravity, S = slope of energy grade line, Q = volumetric bed load discharge, Q = volumetric suspended load discharge, C = average concentration of s av suspended load, A= porosity of bed material, x = distance along flow direction and t = time. Equations (1) to (3) can be numerically solved in conjunction with the auxiliary equations for determination of S, Q , Q f b s and C in terms of local flow parameters. In the present study five different resistance equations and seven different sediment transport equations have been provided for use. The discretization of these equations has been done using the generalized Preissmann scheme. For a total of NX computational nodes, 3(NX-1) discretized equations are obtained. Three boundary conditions are then supplied to close the system of 3NX equations with as many number of unknowns. However, the discretized algebraic equations adopted herein form a set of non- linear simultaneous equations. This set of non-linear equations has been solved using the Newton-Raphson (iii) iterative method. In this method each of the unknowns (viz. h, Q and z) is first assigned a trial value and the same is updated through iterations so as to achieve convergence. During these operations the set of the discretized form of non-linear governing equations is reduced to another set of linear equations involving the Jacobians and residues of the original equations. During each iteration of the Newton-Raphson method this set of 3NX linear equations needs to be solved. In the above set of equations, at each computational node the unknown values of the variables are linked to their values at the adjacent nodes. In order to exploit this special character in the linear equations an efficient and compact solution procedure has been proposed, which is based on an extension of the two-stage successive substitution method of Fread (1971). This solution procedure has been explained for both subcritical and supercritical flows along with appropriate boundary conditions. The proposed model is tested for its completeness using several proof-of-concept tests. Also it was applied to study of the Quail Creek Dike failure in Washington County, Utah, USA (Trieste, 1992). A total alluvial reach of 8 km downstream of the location of dike was modelled. The inflow discharge varied from 283 m3/s to 2350 m3/s over a very short period of about 100 minutes. Large magnitudes of scour and deposition which occurred in the 2.5 km reach immediately downstream of the dike location are ably reproduced by the model results. However, the FCM model of Correia et al. (1992) failed to simulate this particular event. The proposed mathematical model also simulates the mixed supercritical-subcritical flows. The case of supercritical flow followed by subcritical flow, which involves the formation of a hydraulic (iv) > jump has been studied due to its inherent importance. It is well known that the change in state of flow from supercritical to subcritical flow is accompanied by change in the directions of the characteristics involved. This aspect has been simulated by introduction of the compatibility equations involving a fictitious computational node at the location where the jump occurs. This method of computation has been extended for simulation of the moving hydraulic jump. Several proof-ofconcept tests have been performed to validate the proposed formulation for the simulation of mixed supercritical-subcritical flows. In addition to the above, the proposed method is also extended for simulation of the different sediment transport rates of the different fractions in non-uniform river bed materials. The changes occurring in the composition of the active bed layer are accounted for by the solution of the grain sorting equation. The composition of the substrate is also documented by considering the substrate to consist of different horizontal layers. The transport of wash load is treated as per limiting concentration criterion. The non-equilibrium transport of bed load and suspended load is also considered in the model for application to cases in which the incoming load is substantially different from the capacity. Several observations from laboratory experiments involving nonequilibrium sediment transport and development of static armour coat are simulated using the present model. The existing coupled unsteady-flow mobile-boundary models do not simulate the sharp unsteadiness due to flow and bed level variations. The proposed model duly accounts for the non-linear terms appearing in the equations governing unsteady flow in alluvial rivers and is thus capable (v) of producing physically realistic solution for the flow cases involving pronounced unsteadiness. The model is also extended to deal with situations involving mixed supercritical-subcritical flows. In addition the phenomena of grain sorting, transport of wash load and non-equilibrium transport of bed material are also accounted for. The model has been validated with the help of a limited amount of field data and a number of laboratory studies available from earlier investigations. (vi)