# Analytical Analysis of Linear Discretization Strategies in Unsteady Open Channel Flows

• Mohamed S. Ghidaoui
• Bryan W. Karney
• Duncan A. McInnis
Conference paper

## Abstract

The equations governing unsteady flow in channels are ([8], [7], [1], [2])
$$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial v}}{{\partial x}} + \frac{{\partial y}}{{\partial x}}+g\left({{S_1} - {S_0}} \right) + \frac{{vq}}{A} = 0$$
(1)
$$T\frac{{\partial y}}{{\partial t}} + vT\frac{{\partial y}}{{\partial x}} + A\frac{{\partial v}}{{\partial x}} - q = 0$$
(2)
in which t = time, x = distance along the channel, v(x, t) = the average velocity, y(x,t) = depth of the fluid, A(x,t) = the wetted cross sectional area of the channel, T(x,y) = the top width of A(x,t), S f = slope of the energy grade line, S 0 = slope of the bed of the channel, q = lateral inflow per unit length of the channel, and g = the acceleration of gravity. The first of these equations represents the momentum equation while the second is a statement of mass conservation or continuity. It is clear that these two equations are nonlinear hyperbolic partial differential equations (PDEs) which can seldom be solved analytically. As a result, numerical solutions are used to approximate the flow equations. Because of their hyperbolicity, these PDEs can be transformed into ordinary differential equations (ODEs). The result is
$$\frac{{dv}}{{dt}} \pm \frac{g}{c}\frac{{dy}}{{dt}} + g\left( {{S_1} - {S_0}} \right) + \frac{q}{A}\left( {v - c} \right) = 0$$
(3)
in which the positive (negative) sign refers to the so-called C + (C -) compatibility equation which is valid along the positive (negative) characteristic equation defined by $$\frac{{dx}} {{dt}} = v \pm c$$( with similar sign conventions).

## Keywords

Characteristic Line Geometric Error Interpolation Error Analytical Analysis Space Interpolation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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