Muskingum-Cunge Method
  • 07 Nov 2022
  • 2 Minutes to read
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Muskingum-Cunge Method

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In addition to the standard Muskingum method, Flood Modeller also includes the three variants of the more accurate Variable Parameter Muskingum-Cunge (VPMC) method to route the flow:

  • the VPMC Routing unit requires you to supply a table of wavespeed and attenuation parameters against flow
  • the VPMC Cross Section needs a user supplied cross section from which Flood Modeller will generate the wavespeed-attenuation table. Whilst this variant allows more flexibility than the VPMC Routing, the relationship between wavespeed and attenuation may become slightly inaccurate in the region of bankfull flows.
  • the inaccuracy at bankfull in variant two is largely overcome by the VPMC Routing Section. This variant generates the wavespeed-attenuation table from an idealised user supplied cross section, giving a smoother relationship.

The VPMC Routing is based on the diffusion equation, and uses a numerical scheme which is similar to the Muskingum equation with variable parameters k and x. The algorithm is described in Price R.K. (1977) and further discussed in Reeves A. (1995). A very similar algorithm was applied in the earlier flow routing software RIBAMAN.

A minimum of two VPMC Routings, VPMC Cross Sections or VPMC Routing Sections are required for each river or channel reach. The downstream section need not have any wavespeed/attenuation or section data, and such data are not used even if they are supplied. In general, it is only necessary to supply two nodes for any one reach, even if the distance between them is considerable.

Flood Modeller automatically selects a distance step based on the Courant condition at each timestep, but within constraints set by the user. The distance increment also cannot exceed the distance between adjacent sections. Intermediate nodes are generated internally, and so it is only necessary to provide a routing node at each end of a reach. Flows at these internal nodes cannot be examined directly, but the user can control how many such nodes are generated.

The equation used in the VPMC Routing is a form of the convective diffusion equation derived from the St Venant equations, written here in conservative form:

 

(1)

where:

Q = discharge [m3/s]

c = c(Q) = wavespeed [m/s]

m = m(Q) = attenuation [m2/s]

x = chainage along reach [m]

t = time [s]

q = lateral inflow per unit length [m2/s].

In linear form, the finite difference scheme for this equation is the same as for the Muskingum-Cunge equation with variable parameters.

The user supplies a table of discharges Q, wavespeeds c and attenuation parameters a (m = aQ). Guidance on derivation of these parameters is given in standard textbooks and in the Flood Studies Report (1975) Volume 3. Alternatively, a VPMC Cross Section or a VPMC Routing Section, which generate wavespeed/attenuation curves from section details, can be used.


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