Analysis of unsteady flow in open channel using Fourier series

Introduction: The shallow-water equations in unidirectional form namely as Saint Venant equations (SVE) are a set of quasi-linear hyperbolic partial differential equations, having a wide range of applications in open channel and river flow analysis. Because of intrinsic non-linearity, there are no analytical solutions for these equations in most practical applications except for simplified versions. On the other hand, numerical solutions by finite difference or finite element methods are time-marching and for forecasting and timely management of floods are relatively lengthy and time-consuming. Recently, new solutions of SVE in frequency domain, using Laplace Transform (LT) or Fourier Series (FS) have been proposed to overcome these difficulties. In the LT method, input wave is converted into a unit hydrograph, a unit step, or a unit pulse. Despite of unconditional stability, the accuracy of this method depends on time step of decomposition of input information. In this research, however, the FT method is proposed to reduce the execution of real-time flood forecasting. Unlike finite difference models, this is not a marching method and the results may be generated at a given time, directly. Moreover, there is not any restriction in the decomposition of input data due to their independence from time.

Methodology: The complete form of SVE, namely as full dynamic equations are used in the present work. Initial conditions are non-uniform and the up-and downstream boundary conditions are inflow hydrograph and stage-discharge rating curve. SVE are linearized around a steady-state situation using the Taylor expansion. Assuming that the changes in water depth and discharge follow a sine pattern, the linear equations of continuity and momentum are transferred from time domain to frequency domain using the FS and sine functions. The input wave to the model, not necessarily harmonic and periodic, is converted to a set of periodic waves using Fast Fourier Transform (FFT). Considering the initial condition of non-uniform flow in the model, the channel is divided into some intervals that may have equal or non-equal lengths with uniform flow at each part. All channel characteristics such as mean flow depth are computed at each interval separately. Then, transition matrices are constructed to interconnect the channel intervals at the boundaries. Finally, the frequency response of flow discharge and water level are obtained at each part of the channel.

Results and discussion: This method could be used for all kinds of prismatic and non-prismatic channels, natural rivers with various types of flow (critical, sub-critical, and super-critical), different boundary conditions at the up- or downstream ends, and point or distributed lateral inflow. Rashid and Chaudhry (1995) performed their experiments in a rectangular flume. The flow was unsteady and non-uniform. FFT was used to decompose the input hydrograph into a complex sum of periodic waves. In this research, 256 waves with a frequency of 0.002 to 0.5 were used for accurate matching between the input hydrograph of the laboratory model and the hydrograph of the total waves analyzed by the fast Fourier transform. The result of the proposed method was compared with laboratory results of Rashid and Chaudhry, analytical model of Cimorelli, and numerical method of Preissman in time domain. The Nash–Sutcliffe efficiency coefficient (NSE) in the present study is more accurate than other models and in stations (2) and (5) are equal to 0.9893 and 0.9872, respectively. The peak of hydrograph in our model is more than the Cimorelli analytical model. The lag time of mean peak of hydrograph in the model is equal to the experimental results of Rashid and Chaudhry (1995). Execution time of the model is 11.84 seconds in comparison with Preissmann implicit method that is 54.48 seconds with the same computer. This run time is important in forecasting and warning models of floods. Visual comparison of theoretical and experimental hydrograph curves are satisfactory.

Conclusions: The proposed method is unconditionally stable. Full dynamic unsteady flow equations of Saint Venant is solved using FFT and Transition Matrix. The upstream boundary condition is stage-hydrograph and the downstream boundary condition is a stage-discharge relationship. The effects of lateral inflows and non-uniform initial conditions are considered in the model. To evaluate the accuracy of the model, the results compared with experimental data of Rashid and Chaudhry, analytical model of Cimorelli and numerical model of priessmann in time domain, were satisfactory both quantitatively and qualitatively. Regarding the unconditional stability and the appropriate run time of computer, the code is suitable for flood forecasting, warning and optimization models. This method can be used to analyze the flow in natural rivers and irrigation canals with any type of flow regime