Siphons

It is possible to model siphons in the SIC software through the so-called "Preissmann slot " [1] [2] [3]. This method is applicable both for steady state calculations (Fluvia) or transient calculations (Sirene). The program for calculating the geometry (Talweg) automatically detects the sections which are closed at their upper end. This is possible for sections of width-elevation, abscissa-elevation, culvert (always closed), circular (if bank level high enough zbank >= zbed + 2*r) and trapezoidal (with negative slope and adhoc data) type. Anyway, internally, all these sections are converted into width-elevation format, and then the algorithms are the same for all these sections. The technique known as the Preissmann slot is to create an artificial slot at the top of the section from the ceiling to transform a closed pipe into an open free surface section. But in this artificial slot friction is not taken into account. To be more specific, the wetted perimeter P, section S, velocity V, hydraulic radius Rh=S/P used in the calculation of the pressure drop J=\frac{Q^2}{(K S R_{h}^{2/3})^2} is limited to the actual section of the siphon. But elevation Z is calculated in the full section including the artificial slot.

In the calculation of the head H, the velocity V is taken in the actual section, and Z in the slot. The width-elevation tables are calculated by Talweg once for all and written in the xml file. When there is a Preissmann slot we can verify that P does not change in the slot. One can also see that S does not change either in the slot in the tables.

The advantage of this is that all calculations are the same (except for the trick on P, S, V, Rh and H) with or without slot in permanent or transitory flow conditions. The disadvantages are that the volume is stored in this slot, but it must be limited if the slot is thin.

We can show that the equations of Saint Venant pose problem with a width L=0, which one would have without the Preissmann slot from where precisely the idea of ​​this trick.

On the other hand beware: one can have the impression that this is not the case, because when one looks at the wet section for example in the results, one does not find exactly what one would expect. For example, if we have a 9 m wide by 4 m high culvert we would expect to have a surface of 36 m2. In fact we will have 36.0045 m2 (36 + 9/2*0.001). This is due to the fact that the ceiling of the section is slightly modified during the treatment by Talweg, putting a section starting the Preissmann slot to 1mm more than the initial ceiling. Instead of having a flat horizontal ceiling we have a light roof hanging 1mm high, so with a small additional volume. This is to improve the numerical stability of the schema and to have a better defined \frac{dL}{dZ} derivative. To better visualize this one has increased the writing format of S, P and L on the log file of F8.2 to F9.3 (cf Talweg.ans, messages 53, 54, 55 and 56). The wet perimeter P is calculated without this small modification because it only intervenes in the friction term.

The slot width must be chosen small (L=0.01 m in SIC by default, but it can be changed for each section in the section description or using the automatic modification tool) in order not to store too large volumes of water. In sewage systems, where there are manholes, some software uses a trick to reduce the section of those manholes to reduce the volume artificially stored in the slot to get a better volume conservation of the total volume of water into the network.

One can also choose this slot width L (using the automatic modification tool) having a wave speed c=\sqrt{\frac{gS}{L}} close to that of sound waves in water which is about 1500 ms-1.

This modification using the wave speed can be done also using the automatic modification tool. This can lead for example to a slot width in the order of 10-4 m to 10-6 m. With explicit numerical schemes this was a problem because of the very small time step required for the calculation in order to have a Courant number Cr< 1 with C_r=\frac{(V+c)DT}{DX} <1, where V is the velocity of the water, c the celerity of the waves, DT the computation time step and DX the space step between 2 calculation sections. But the implicit Preissmann scheme is not subject to the same constraints. It is unconditionally stable regardless of the Courant number. But a large numerical Cr (or also a very small one, which is not our case) introduces numerical bias (diffusion and phase shift, cf thesis Cunge). By default in SIC we choose a slot width of 0.01 m. For example in a culvert of 1 m2, for a flow of 1 m3s-1, we get a celerity c of the order of 30 ms-1 with a slot of 0.01 m. For c of the order of 1500 ms-1 we should take a slot width of approximately 0.000005 m. The advantage of a small slot is that the stored volume in the slot is limited. The drawback is that it could be less stable numerically. This is a compromise to find in case of difficulties.

Bibliographical References:

[1Cunge J.A., Wegner M., 1964. "Intégration numérique des équations d’écoulement de Barré de Saint Venant par un schéma implicite de différences finies. Application au cas d’une galerie tantôt en charge, tantôt à surface libre". La Houille Blanche, n°1-1964, pp 33-39

[2Vasconcelos J.G., Wright S.J., 2004. "Numerical modeling of the transition between free surface and pressurized flow in storm sewers". Innovative Modeling of Urban Water Systems, Monograph 12, Chap 10, W. James, Ed., pp 189-214

[3Ukon T., Shigeta N., Watanabe M., Shiraishi H., Uotani M., 2008. "Correction methods for dropping of simulated water level utilising Preissmann and MOUSE slot models". 11th International Conference on Urban Drainage, Edinburgh, Scotland, UK, pp 1-9