4-Point FDF of Muskingum method based on the complete St Venant equations

https://doi.org/10.1016/j.jhydrol.2005.10.010Get rights and content

Abstract

Using the Froude number, a nonlinear convection–diffusion equation was derived from the St Venant equations of continuity and momentum. It was made applicable to discrete space using a mixing-cell method, resulting in a first-order nonlinear ordinary differential equation. A 4-point finite difference scheme was used to solve the first-order nonlinear ordinary differential equation to achieve a nonlinear algebraic form that is very similar in structure to the traditional Muskingum form thus its naming as the ‘4-point FDF of Muskingum Method (4-point FDF M-M)’. In this method, the space interval Δx is a characteristic channel length and a variable that is dependent on flow discharge, water depth, and flow velocity. An iterative search method was applied to simultaneously obtain the flow discharge and the optimal space interval Δx. The method developed was tested with both synthetic numerical examples and observed events and the results were compared with those of the Lambda scheme and the method of characteristics. The outflow hydrographs produced by this new 4-point FDF M-M were of comparable accuracy. The parameters used in the new method are based on the physical attributes of the channel and thus do not need calibration as required for the Muskingum method.

Introduction

Flow routing is a mathematical procedure for predicting the changing magnitude, speed, and shape of a flood wave in a channel as a function of time. Many ways have been sought to predict the characteristic features of a flood wave in order to estimate the degree and scope of flooding effects and to plan for control measures. In addition, flow routing has the potential to be used economically to improve the transport of water through natural or man-made watercourses. The traditional lumped Muskingum channel routing method introduced by McCarthy (1938) is simple and easy to use in practice. The model parameters, however, must be calibrated using inflow and outflow hydrograph data. The simplified distributed models based on the Muskingum-Cunge method (e.g. Ponce and Yevjevich, 1978; Ponce and Chaganti, 1994; Ponce and Lugo, 2001) utilize the 4-point finite difference method to produce a discrete form of the kinematic wave equation on an xt plane, with weighting factors X and Y. The resultant equations are similar to the traditional Muskingum method and may be either linear or nonlinear, with the latter requiring an iterative solution procedure. In these methods, the parameters do not have to be calibrated but determined from channel characteristic features instead. However, the space interval Δx is a constant, and the relationship between the storage in the channel reach and the discharge at the downstream cross-section is not a single-valued curve. The numerical solutions of the full St Venant equations include characteristic (Cunge, et al., 1980) methods as well as explicit and implicit finite difference methods; all of which are more complicated than the hydrologic routing and simplified distributed models. Recently, Wang et al., 2003a, Wang et al., 2003b proposed a semi-analytical solution of the complete St Venant equations that is superior to the numerical solutions. However, that semi-analytical solution is limited to the case of a rectangular channel.

The objective of this paper is to extend the work of Wang et al., 2003a, Wang et al., 2003b using the Froude number to derive a nonlinear convection diffusion equation from the complete St Venant equations for various channel cross-sections. The resultant nonlinear convection diffusion equation will be made discrete in space using the mixing-cell method of Singh et al. (1997), and by transforming it into a first-order nonlinear ordinary differential equation for which the optimal space interval Δx is the same as the characteristic reach length. Lastly, a varying-weights four-point finite difference (Cunge, et al., 1980) method will be used to convert the first-order nonlinear ordinary differential equation into a form that is similar in application to that of the traditional Muskingum method.

Section snippets

Lumped hydrologic flow routing methods

Since its development around 1934 and report by McCarthy (1938), the Muskingum method for routing flood waves through channels has been the subject of many investigations (Singh, 1988). The continuity equation and the storage equation in channel routing is given asdWdt=IQandW=k[XI+(1X)Q]

Combining the continuity Eq. (1a) and the storage Eq. (1b) and expressing the combination in the following finite difference form (FDF) results in the approximate solution, which forms the basis of the

A nonlinear convection diffusion equation

For a trapezoidal channel reach, the momentum equation becomes (Wang et al., 2003a, Wang et al., 2003b)(1Fr24B0+αhB0+2αh)hx=S0Sfwhere B0, channel bottom width, and α, lateral section slope, Sf, a friction slope, and for Chezy's equation, Sf=Q2/[c2h3(B0+αh)2].

For a trapezoidal channel reach, the continuity equation of St Venant equations can be written asht=qB0+2αh1B0+2αhQx

Differentiating Eq. (16) with respect to t, obtainst(hx)=2Qpc2h3(B0+αh)2Qt+(2αSfp(B0+αh)+3Sfph)htFr2B0α(S0

4-Point FDF M-M

Eqs. (20), (20a), (20b) are nonlinear convection diffusion equations of the parabolic type having two boundaries in a reach channel. The inflow hydrograph serves as the upstream boundary condition, and ∂Q(L,t)/∂x=0 serves as the lower boundary condition.

The structure and boundary conditions of Eq. (20) suggest that the use of the mixing-cell method is appropriate. Applying Taylor series expansion backward in space to the first-order term of Eq. (20), and neglecting the terms of the third and

Procedure for 4-point FDF M-M

The 4-point FDF M-M is a nonlinear channel routing form, where the iterative calculation is used to obtain the discharge. The boundary condition is the inflow hydrograph, that is, Q11,Q12,Q13,,Q1n+1; and initial values at different locations are given, that is, Q11,Q21,Q31,,Qm1. For time index k=1 and spatial index i=1, one can execute the following steps:

  • (1)

    The first guess value Qg of iteration for Q22

  • (2)

    Calculate Q1 using Eq. (29), that is, Q1=(Q11+Q21+Q12+Qg)/4

  • (3)

    Calculate h from Q1 using Eq. (25)

  • (4)

Numerical simulation and applications of the 4-point FDF M-M

A program of numerical examples was designed for testing the 4-point FDF M-M developed in this paper. A wide rectangular channel was assumed in order to carry out computations in terms of unit-width discharge, flow depth, and velocity. For calculation purposes, the bottom slope was fixed at S=0.0005, the channel length L=30 km was chosen, and the channel friction coefficient c=25 was selected. The inflow hydrograph is assumed asQ(t)=A0.5cos2πtTwhere Q(t), inflow at time; A=1.5; and T, flood

Parameter's analysis of 4-Point FDF M-M

For the 4-Point FDF M-M, the estimated hydrograph depends upon channel characteristics, such as channel slope, surface roughness, shape of cross section and weight factors X and Y. As mentioned earlier, in the 4-Point FDF M-M, X is a weighting factor of time difference, Y is a weighting factor of space difference, and the values for both Y and X are between 0 and 0.5. For Y=0.5 the hydrographs calculated using X=0, 0.3, and 0.5 are shown in Fig. 6 and clearly show that the outflow peak for X

Discussion and conclusion

The 4-point FDF M-M is derived from a nonlinear convection diffusion Eq. that is based on the complete St Venant equations. When the mixing-cell approach is used to make the space discrete and keep time in continuity, it leads to a nonlinear ordinary differential equation and Δx equates to a characteristic channel reach. A 4-point finite difference method is used to make the equation spatially discrete with weighting factors X and Y to obtain the 4-point FDF M-M.

In the traditional Muskingum

References (17)

  • K.M. O'Connor

    Derivation of discretely coincident forms of continuous linear time-invariant models using the transfer function approach

    J. Hydrol.

    (1982)
  • J.A. Cunge

    On the subject of a flood propagation computational method

    J. Hydraul. Res.

    (1969)
  • J.A. Cunge et al.

    Practical Aspects of Computational River Hydraulics

    (1980)
  • J.C. Dooge

    Linear theory of hydrologic systems

    US Dept. Agric. Tech. Bull.

    (1973)
  • G.P. Kalinin et al.

    O raschete reustanovivshegosya dvizheniya vady v atkrytykh ruslak”(on the computation of unsteady flow in open channels)

    Meteorogy I Gidrologiya, Leningrad

    (1957)
  • A.D. Koussis

    Comparison of Muskingum Method Difference Schemes

    J. Hydraul. Div. ASCE

    (1980)
  • McCarthy, G-T., 1938. The Unit Hydrograph and Flood Routing. Conference North Atlantic Division, US Army Corporation of...
  • V.M. Ponce et al.

    The Muskingum-Cunge method with variable parameters

    ASCE J. Hydraul. Div.

    (1978)
There are more references available in the full text version of this article.

Cited by (0)

View full text