Developing Semi-Analytical solutions for Saint-Venant Equations in the Uniform Flow Region

Unsteady flow in irrigation systems is the result of operations in response to changes in water demand that affect the hydraulic performance networks. The increased hydraulic performance needed to recognize unsteady flow and quantify the factors affecting it. Unsteady flow in open channels is governed by the fully dynamic Saint Venant equation, which express the principles of conservation of mass and momentum. Unsteady flow in open channels can be classified into two types: routing and operation-type problems. In the routing problems, The Saint Venant equations are solved to get the discharge and water level in the time series. Also, they are used in the operation problem to compute the inflow at the upstream section of the channel according to the prescribed downstream flow hydrographs. The Saint Venant equation has no analytical solution and in the majority cases of such methods use numerical integration of continuity and momentum equations, and are characterized by complicated numerical procedures that are not always convenient for carrying out practical engineering calculations. Therefore, approximate methods deserve attention since they would allow the solution of dynamic problems in analytical form with enough exactness. There are effective methods for automatic controller synthesis in control theory that provide the required performance optimization. It is therefore important to get simplified models of irrigation canals for control design. It would be even more interesting to have linear models that explicitly depend on physical parameters. Such models would allow one to, handle the dynamics of the system with fewer parameters, understand the impact of physical parameters on the dynamics, and facilitate the development a systematic design method. Many analytical models have been proposed in the literature, Most of them have been obtained in the frequency domain by applying Laplace transform to linearized Saint-Venant equations. The got transcendental function can then be simplified using various methods to get a model expressed as a rational function of s (the Laplace variable), possibly including a time delay. It is therefore important to develop simple analytical models able to accurately reproduce the dynamic behavior of the system in realistic conditions.
Changes in water demand can create transient flow in irrigation networks. The Saint Venant equations are the equations governing open channel flow when unsteady flow propagates. In this research, the finite volume method using the time splitting scheme was employed to develop a computer code for solving the one dimensional unsteady flow equations. Considering stationary regime and small variations around it, the Saint-Venant equations around initial condition was linearized.
The Laplace transform is applied to the linearized saint venant equations, leading to an ordinary differential equation in the space variable x and parameterized by the Laplace variable s. The integration of this equation lead to a transfer matrix, and gives the discharge Q*(x, s) at any location with respect for the upstream discharge. This matrix is coupled with the downstream boundary condition and developed an equation that solved using Simpson integration algorithm. It should be noted numerical solution of developed equation is easier than solving fully dynamic saint venant and is less sensitive to the spatial step and the researcher simply writing code.
Froud Number (F), variation of inflow discharge (ΔQ/Q), and dimensionless parameter of KF2 in which K is the kinematic flow number, are effective factors on accuracy of developed equation. In order to determine the effect of the factors on accuracy of presenting formula, several simulations were performed using numerical model. The presented formula and numerical model were compared for 10, 20 and 30 percent discharge variation and error calculated, the maximum error increases with increasing ΔQ/Q.
To assess the importance of Froud Number and KF2, also several simulations were carried out, the results showed that the maximum error in the development equation for various Froud Number and KF2>1, is less than 3.8 percent.
Using Laplace transform to the saint venant equations and with respect to upstream and downstream boundary a formula for routing discharge presented. Investigation of the applicability range of presenting formula and cognitive effective factors on accuracy is necessary. So, the finite volume method using the time splitting scheme was employed to develop a computer code for solving the one dimensional unsteady flow equation. Then some tests of unsteady flow were simulated and verified the equations. The results showed that the maximum error increases with decreasing KF2 and increasing the rate of sudden changes of discharge. The maximum error in the presented formula for all tests with KF2>1, less than 3.8 percent.