# 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

Expense Liner Lution

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