Analytical Analysis of Linear Discretization Strategies in Unsteady Open Channel Flows

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


The equations governing unsteady flow in channels are ([8], [7], [1], [2])
$$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial v}}{{\partial x}} + \frac{{\partial y}}{{\partial x}}+g\left({{S_1} - {S_0}} \right) + \frac{{vq}}{A} = 0$$
$$T\frac{{\partial y}}{{\partial t}} + vT\frac{{\partial y}}{{\partial x}} + A\frac{{\partial v}}{{\partial x}} - q = 0$$
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
$$\frac{{dv}}{{dt}} \pm \frac{g}{c}\frac{{dy}}{{dt}} + g\left( {{S_1} - {S_0}} \right) + \frac{q}{A}\left( {v - c} \right) = 0$$
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).


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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  1. [1]
    H.M. Chaudhry, Applied Hydraulic Transients, Van Nostrand Reinhold, (1987).Google Scholar
  2. [2]
    H.M. Chaudhry, Open Channel Flow, Prentice Hall, NJ, (1993).MATHGoogle Scholar
  3. [3]
    M.S. Ghidaoui and B.W. Karney, Equivalent differential equations in fixed-grid characteristics method, J. Hydr. Engng., ASCE, 120(10), 1159–1175.Google Scholar
  4. [4]
    M.S. Ghidaoui and B.W. Karney, Equivalent Hyperbolic Differential Equations in Advective Problems With Discontinuous Conditions, Sec. Annu. Conf. CFI) Soci. Canada, Toronto, 1994, pp.465–471.Google Scholar
  5. [5]
    D.E. Goldberg and E.B. Wylie, Characteristics method using time-line interpolations. J. Hydr. Engng., ASCE, 109(5) (1983), pp.670–683.CrossRefGoogle Scholar
  6. [6]
    C. Lai, Comprehensive Method of Characteristics Models for Flow Simulation, Hydr. Engrg., ASCE, 114(9) (1989), pp.1074–1095.CrossRefGoogle Scholar
  7. [7]
    E.B. Wylie, Inaccuracies in the characteristics method, Proc, Spec. Conf. on Comp, and Physical modelling in Hydr. Engrg., ASCE, Chicago, Illinois, (1980), pp.165–176.Google Scholar
  8. [8]
    E.B. Wylie and V. Streeter, Fluid Transients in Systems, Prentice Hall, Englewood Cliffs, New Jersey, (1993).Google Scholar
  9. [9]
    D.C. Wiggert and M.J. Sundquist, Fixed-grid characteristics for pipeline transients. J. Hydr. Engng., ASCE, 103(12) (1977), pp.1403–1415.Google Scholar
  10. [10]
    J.C. Yang and E.L. Hsu, Time-line interpolation for solution of the dispersion equation, J. Hydr. Res., IAHR, 28(4) (1990), pp.503–523.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Mohamed S. Ghidaoui
    • 1
  • Bryan W. Karney
    • 2
  • Duncan A. McInnis
    • 1
  1. 1.Department of Civil & Structural EngineeringThe Hong Kong University of Science & TechnologyKowloonHong Kong, China
  2. 2.Department of Civil EngineeringUniversity of TorontoTorontoCanada

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