On numerical modeling of overland flow
Introduction
Overland flow is the first flow that develops after precipitation over the land surface and infiltration to the subsurface. It can be characterized as very thin sheet flow, as illustrated in Fig. 1. The depth and flow rate of this thin sheet of water depends, among others, on the rate of rainfall, ground surface characteristics (e.g., roughness, slope) and antecedent and immediate subsurface conditions (e.g., infiltration, moisture). Developing numerical models that can predict accurately surface runoff is of interest to a wide variety of users, including city planners and irrigation practitioners. These models can also be easily extended to predict the fate and transport of contaminants along the surface. It is widely accepted that a significant amount of this contamination originates from nonpoint sources and is carried across the landscape by overland flow.
Flood wave propagation in overland flow may be described by the complete equations of motion for unsteady nonuniform flow, known as the dynamic wave equations, first proposed by St. Venant in 1871. These equations are highly nonlinear and therefore do not have analytical solutions. Under a different set of simplifying assumptions, more practical kinematic-wave and diffusion-wave models can be derived from the dynamic wave equations: these approximations are presented in hydrology textbooks, e.g. [1]. These models constitute a relatively accurate physical representation of the flow [2]: both approaches allow distributed overland flow and channel flow routing. By neglecting the acceleration and pressure terms in the momentum equation for dynamic waves, the kinematic wave model [3] substitutes a steady uniform flow (stage-discharge) relationship for the momentum equation. However, unsteady flow is preserved through the continuity equation. Henderson [4] noted that kinematic waves behave closely to observed natural flood waves in steep rivers (slopes > 0.002). A kinematic wave does not subside or disperse, but changes its shape because its velocity depends upon depth (introducing nonlinearity). Numerical solutions introduce small amounts of diffusion, and thus some wave attenuation. By introducing physical diffusion into the kinematic wave continuity equation (resulting mathematically in a second-order term), the diffusion wave is obtained. Cunge [5] also obtained the diffusion wave equation by linearizing the dynamic wave equation around a perturbation and ignoring inertial terms. Diffusion occurs in most natural unsteady open channel flows and in overland flow [1], [3], [6]. Diffusion wave theory applies to the milder slopes (0.001–0.0001), for which kinematic wave theory is insufficient.
There have been numerous studies published which attempt to solve both kinematic and diffusion waves with different numerical methods, e.g. [7], [8], [9], [10], [11], [12]. These studies have focused on obtaining more accurate and computationally effective results. A thorough review of the literature has shown that the MacCormack explicit finite difference method, first introduced to solve nonlinear fluid dynamics aeronautical problems [13], has not been previously applied to overland flow. This paper presents such an application for both kinematic and diffusion waves, and its performance is compared to available analytical solutions and, in particular, to a 4-point implicit finite difference method.
Section snippets
Governing equations
The dynamic wave equations for overland flow consist of continuity and momentum equations, as follows, respectively [14]:where y is depth of water (L), Sf is friction slope (L/L), S0 is bed slope (L/L), V is water velocity (L/T), i is rainfall intensity (L/T), f is infiltration rate (L/T), g is acceleration of gravity (L/T2), t is time (T), and x is distance (L). These equations are for moderately wide overland flow, small bottom slope (θ ≈
Numerical methods
Finite difference techniques are employed commonly in solving partial differential equations numerically. The solutions obtained from explicit finite difference schemes are conditionally stable and the stability condition is given by the Courant–Friedrichs–Lewy (CFL) restriction [15]. Although unconditional stability is an advantage of the implicit finite difference techniques, they are computationally intensive and difficult to apply to nonlinear problems [16]. Splitting methods are introduced
Application to synthetic examples
Two synthetic examples are employed in this section in order to demonstrate the practical application of the theory presented in the previous sections. The numerical methods discussed above are applied to these overland flow problems and their results are compared with each other, as well as to their analytical solutions.
Conclusions
In this study, the theory of kinematic and diffusion waves is reviewed and summarized for overland flow. Numerical approximations are derived, and the results are compared by solving two synthetic overland flow problems with both kinematic and diffusion wave formulations, along with their analytical solutions. Although the 4-point implicit method provides the highest accuracy for kinematic wave solutions when compared to a specific analytical solution, it is much slower than the MacCormack
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2017, Advances in Water ResourcesCitation Excerpt :Here, we examine the basic case of overland flow occurring from uniform rainfall over an impervious plane (see West, 2015 for an application of the DG-SAKE model to the moving storm test case of Singh and details on the analytic solutions). This is a commonly used verification test case for overland flow models; see, for example, Kazezyilmaz-Alhan et al. (2005), Kim et al. (2012) and Yu and Duan (2014) Figs. 8 and 9 can also be compared to Fig. 4 of Kim et al. (2012), which compares the analytic solution of this test case to numerical results from two different models using a normalized mesh spacing (Δx/L) of 0.0025.
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An improved solution for diffusion waves to overland flow
2012, Applied Mathematical ModellingCitation Excerpt :The stability condition is given by the Courant–Friedrichs–Lewy (CFL) number [17] and the method is second order accurate. Kazezyılmaz-Alhan et al. [18] have shown the accuracy and efficiency of MacCormack method as compared to the classical explicit and implicit finite difference methods particularly in solving diffusion wave equations for overland flow problems. The formulations of the MacCormack method for the diffusion wave equation for overland flow problem are given as follows: