The St Venant hydraulic routing equations (equations 14.25 and 14.30 or 14.32 and 14.33) are non-linear in v and y. The use of the St Venant equations in flood routing by their integration down the length of a channel depends on the use of approximate solutions. Three sorts of approximate solutions have been used in the past. Approximate analytical solutions depend on linearising the equations around some reference flow state and relying on changes around that state being small. This results in a form of linear transfer function. More accurate solutions can be obtained by the method of characteristics, which uses a mathematical property of partial differential equations in that the solution can be separated into ordinary differential equations (the characteristic equations) that represent how changes in the upstream and boundary conditions propagate in the space-time domain (e.g. Amein, 1966). It is simpler to integrate these ordinary differential equations. Where the characteristic lines cross, solutions for v and y can be obtained, but to obtain values at specific points in space and time may then require interpolation. The method of characteristics is interesting because the characteristic equations are the expression of how the wave velocity or celerity propagates any change in the flow through space and time. For sub-critical flow, the celerity, c, is given, in one dimension, by:

The two celerities given by (14.34) represent propagation in the upstream and downstream directions (think of the waves resulting from the perturbation caused by throwing in a stone into a deep river). Both are different from the mean flow velocity. This explains both why a flood wave propagates downstream at a rate faster than the flow, and why changes downstream can have a backwater effect in the upstream direction (such as upstream of a confining bridge during a flood event). For super-critical flow, v > (gy)0'5, and disturbances can have an effect only in the downstream direction (try throwing a stone into the fast, shallow, super-critical flow downstream of a weir crest). The kinematic approximation to the full St Venant equations, referred to in the previous section, has only a downstream celerity and therefore cannot represent the backwater effects that will affect flood wave propagation in sub-critical flows.

Approximate numerical solutions are more frequently used now to solve the St Venant equations, particularly finite difference solutions for the one-dimensional (1D) case (Abbott and Minns, 1998). Finite element and finite volume solutions are also used in solving the two-dimensional (2D) equations, with the advantage that the element size can be more easily made smaller where more detail in the solution is needed. There are many different computer packages now available, both commercially and freeware, that solve the 1D and 2D St Venant equations (or simpler versions of the equations that neglect one or more terms in the energy equation).

Examples of 1D models that are widely used are MIKE 11 from the Danish Hydraulics Institute (DHI),4 SOBEK from Deltares in the Netherlands,5 ISIS from Halcrow in the UK6 and HEC-RAS from the US Army Corps of Engineers.7 Examples of 2D models are MIKE FLOOD from DHI,8 TUFLOW from BMT WBM Pty. Ltd,9 RMA-2 from the US Army Corps of Engineers,10 TELEMAC from EdF/SOGREAH in France,11 and DIVAST-TVD from the University of Cardiff (Liang et al., 2007). Examples of simplified 2D models are LISFLOOD FP12 (e.g. Hunter et al., 2006;

Bates et al., 2006) and JFLOW (Bradbrook, 2006). Both of these last models use the diffusion approximation to the full St Venant equations for routing flow across the flood plain, either coupled to a 1D kinematic solution for the channel or, for large-scale mapping purposes under steady flow assumptions, with an allowance made for the capacitance of the channel in each reach (see Chapter 16).

As an example of the use of this type of model, the 2D LISFLOOD-FP has been used to simulate the inundation in Carlisle as a result of the January 2005 flood. LISFLOOD is a grid-based 2D model, with simplified dynamics. For this application, the simulation made use of a LIDAR survey of the flood plain topography, and a grid resolution of 25 m. After the Carlisle event, the maximum extent of the flooded area was surveyed by differential GPS positioning of wrack marks and water marks on buildings. Fig. 14.17 shows a comparison of the depths and maximum extent of inundation predicted by the model to the flooded area estimated from the survey.

The JFLOW model has been used by JBA Consulting to produce national maps of areas at risk of flooding, also making use of remotely sensed topographic survey data. Chapter 16 describes a regional application of this technology for flood risk modelling to support catchment flood management planning in north-east England.

14.4.1 Stability and uncertainty in hydraulic models

The full St Venant equations are hyperbolic partial different equations. The celerity puts some constraints on the time step that can be used in an approximate numerical

Fig. 14.17 Application of LISFLOOD FP to predict flooding at Carlisle during the 2005 flood event.

Colours show predicted maximum depths during the event, wrack marks and water marks show maximum depths from water marks and wrack marks surveyed after the event (after Neal et al., 2009).

Fig. 14.17 Application of LISFLOOD FP to predict flooding at Carlisle during the 2005 flood event.

Colours show predicted maximum depths during the event, wrack marks and water marks show maximum depths from water marks and wrack marks surveyed after the event (after Neal et al., 2009).

solution since (on a gridded model) the approximate solution will deteriorate or go unstable if the local celerity is sufficient to propagate the peak (or other disturbance) more than one grid length in one time step. Most of the numerical codes described above have automatic time step controls to avoid this, but this can add significantly to computer run times because the celerity has to be calculated everywhere in the flow domain to check the maximum time step allowable, especially if in a finite element or finite volume solution the element size is very different in different parts of the flow domain. It is also the case that particular solutions can generate instabilities at time steps less than the theoretical limit imposed by the local celerity (this may be the case at the edge of the inundated area when part of the domain may oscillate between being wet and dry; or where there is inadequate description of the channel transition between a tributary and the main stream). Thus other conditions on the time step are often imposed and the actual time steps used in the solution are often very short.

A further consideration in the application of flood routing models based on the St Venant equations is uncertainty in the upstream and downstream boundary conditions and parameters that need to be specified to run the model. The St Venant equations involve two unknown variables, mean velocity and depth, at every boundary point. It is normal to have measurements of stage at both upstream and downstream boundaries in a forecasting problem, or an estimate of a design discharge in a flood risk mapping problem. In both cases, therefore, one condition on depth and velocity can be specified but, because there are two unknowns, another is needed. Very often this is imposed in a model by assuming that, at the boundary, the friction slope can be specified (generally by assuming it is equal to the bed slope so that flow at the boundary is uniform). One of the uniform flow equations, Manning or Chezy, can then be used to provide the second condition relating velocity and depth. This is, however, an approximation that is somewhat inconsistent with using the fully dynamic St Venant equations to route the flood wave (which will predict water surface slopes that are different on the rising and falling limbs of the hydrograph such that there is a hysteresis in the stage-discharge relationship).

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