AULOS software models interconnected open and closed channels

An informed discussion of Current Issues in Hydraulic modelling

Dr Barnett answers questions, basesd on his extensive modelling experience with many computer packages

Implicit or Explicit Solutions: Which Are Best?

The comments to date on implicit solutions relate only to the simplest systems of equations. In more mathematically challenging hyperbolic problems such as fluid dynamics, implicit solutions are no longer unconditionally stable.

Stability fundamentally relates to the propagation of waves of information through the solution. Dynamic waves result from a balance between gravity and inertia, while kinematic waves result from a balance between gravity and resistance.

1. Where kinematic waves travel faster than dynamic waves, the problems will be inherently unstable at microscopic level (hydrodynamic instability). Under these conditions both implicit and explicit solutions have been shown to be unstable.

2. Otherwise numerical stability depends on the relative scales of the control element chosen for analyis. In the most general case this is a 4-dimensional finite hypervolume or CELL, following the locus of a finite volume over a finite lifetime. The distance travelled by a wave during the control element lifetime then relates to a spatial dimension of the CELL by a ratio known as the Courant Number.

3. Boundary conditions can be viewed relativistically as a timewise extension of initial conditions. This means that the spacewise direction in which the equations are solved becomes important.

4. Explicit solutions for each CELL use information only from within that CELL. This means they are stable only for Courant Numbers <1.

5. Implicit solutions solve simultaneously for groups of CELLS, so they are generally still stable for Courant Numbers >1. However they may be unstable at low Courant Numbers if they are solved in the wrong direction from the boundaries.

6. Stability is no guarantee of accuracy, which depends on a balance of influence between boundary and initial conditions, reached at Courant Number = 1. Here both explicit and implicit schemes are generally most accurate.

7. However there are as many Courant Numbers as there are wave types and spatial dimensions, so solution schemes which perform well over a range of Courant Numbers are needed. Here implicit solutions are much to be preferred, especially as they can also be coded to run far faster than explicit solutions.



Calibration and Validation: Is there a methodology to deal with errors in calibration and validation at a gauging station?

Often modellers are asked to make an estimate of calibration and verification accuracy based on errors in matching the hydrographs measures at gauging sites. I suggest this starts at the wrong end in analysing these errors, because the modeller has to deal with a mixture of level errors (at the gauge) and flow errors (everywhere else).

If instead the gauge levels are specified as boundary conditions, the model will then be forced to estimate the flows at the gauging point, and these can be compared with any flow measurements at the same point. Then the model will exactly match the gauged levels and all errors will be flow errors. This method of calibration is called Residual Flow Analysis, because it examines the residual left after all measured and estimated flows have been taken into account.

For example, the computed model should indicate that at the same gauge level more flow is passing through on a rising flood than during a flood recession, and one significant error may be ignoring this loop rating effect. Another error which can be explored by this method is the choice of the wrong rain gauge for the rainfall-runoff model, as a sudden burst of rainfall should create a delayed response in the computed flow hydrograph.

Such elimination of gauge level errors effectively uses hydraulic modelling to produce a lateral inflow hydrograph for the modelled river reach, and this should match with the local catchment outflow hydrograph produced by hydrological rainfall/runoff modelling. If this match is poor, both reach inflow and catchment outflow must be reconsidered and adjusted to improve the match.

This approach provides a systematic methodology for measuring and improving the accuracy of fit between the lateral flow hydrographs produced by hydraulic and hydrological modelling.

More details can be read via a download of the paper Barnett A.G., C. Hellberg “Storm Lateral Runoff Calibration by Direct Hydraulic Balances” Stormwater 2013, Water New Zealand, Auckland, May 2013 - click here (This is an updated version of Barnett A.G., C. Hellberg “Residual Flow Modelling: An Advance in Channel Resistance Calibration” Paper 3370, Proc. 34th IAHR World Congress, Brisbane, Australia June-July 2011.)

When do we need a supercritical flow solution?

The textbook analysis of the direction of flow control in steady flow makes life  unnecessarily complicated for modellers, as it assumes the solutions along the forward and backward dynamic wave characteristic are of equal importance. In practice the forward (downstream) characteristic is always dominant, as this takes the same direction as the kinematic wave characteristic, which determines the flow response to resistance effects.

The change from supercritical to subcritical flow merely changes the direction of the weak backward characteristic from downstream to upstream, which makes no difference in the usual case where flow is governed by the bed slope and resistance.

In model applications, the problem should not be seen as the selection of a supercritical or subcritical flow regime, but rather the selection of optimum locations for the upstream and downstream boundary of the study reach. Both should be sited where a good record of levels is available (or even better, a good level recorder should already have been sited at reach boundaries where the stream bed is naturally stable or fixed by an artificial structure).

Once the upstream and downstream boundaries are established, both should be specified as levels if that is what has been measured, and a correctly written solution code should solve the flow problem regardless of flow regime. If, rather than being estimated from a rating curve, the flow is continuously being measured directly (for example, through a turbine) then flow may also be used as an upstream boundary condition.

However a downstream flow boundary should be avoided, as this works against the dominant forward and kinematic wave characteristics which already govern the flow from upstream in both subcritical and supercritical flow, causing incompatibility which will usually crash the solution. Obviously this applies especially if an upstream flow boundary is given, as even if the boundary flows are identical the water mass within the reach will be fixed by some (arbitrary) initial calculation level rather than by the desired steady flow profile.

Testing on a model reach of alternating subcritical and supercritical flow is instructive in exploring the response of a package such as HEC-RAS in practice, and a verified test model is presented in Exercise QA6 "Mixed sub- and super-critical profile analysis" in the manual provided with the AULOS download.