Computational Mechanics ’95 pp 888-893 | Cite as

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

Conference paper

## Abstract

The equations governing unsteady flow in channels are ([8], [7], [1], [2]) in which in which the positive (negative) sign refers to the so-called

$$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial v}}{{\partial x}} + \frac{{\partial y}}{{\partial x}}+g\left({{S_1} - {S_0}} \right) + \frac{{vq}}{A} = 0$$

(1)

$$T\frac{{\partial y}}{{\partial t}} + vT\frac{{\partial y}}{{\partial x}} + A\frac{{\partial v}}{{\partial x}} - q = 0$$

(2)

*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)

*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

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