Downstream boundary condition of the model

Entering the downstream boundary condition(s) of the network can be done in different ways, by:

  • Menu Tools > Downstream node
  • Double-clicking on the downstream node on the graph or on the tree view

A hydraulic model necessarily has boundary conditions upstream and downstream, which indicate how the network that is modeled is connected to the rest of the world at its borders. In the absence of such explicit precisions (ie if we do not enter "offtake" at these nodes), SIC will assume that the boundary condition(s) at the upstream nodes (as well as at the nodes possible intermediates) is $Q(t)=0$, and that the downstream boundary condition(s) is the critical depth $Y_c(Q)$, which is a function of the rate $Q$. Why the critical depth $Y_c (Q)$ and not the normal depth $Y_n(Q)$? Especially since the critical depth is more difficult to manage in a hydraulic model, especially under transient conditions. In fact, by definition, the critical depth corresponds to a Froude number $F_r=1$, with very strong backwater slopes, which SIC can process in a very good way in steady state and with sometimes more difficulties in unsteady state, or in any case with simplifications (ex: partial or total suppression of the terms of inertia, option Preissmann-Sart, etc.). The reason is simple, it is that the uniform regime, which would be another good candidate, does not always exist, contrary to the critical regime (in the event of a null slope or of slope). In spite of these choices of upstream and downstream limit conditions by default, it is always better to define them explicitly, to better control what the model does.

If the network is meshed, there may be more than one node of this type (downstream node). In addition, it is also possible to enter one or more "offtake" at each node, whether upstream, intermediate or downstream, and each of these offtake must contain a downstream condition. The details and information given here are therefore applicable to other cases than to the conditions at the downstream node stricto-sensu.

The calculation of the backwater curve in steady state requires fixing or being able to calculate the water elevation at the downstream nodes of the model. This elevation is set by the user (fixed elevation $Z(t)$), or is calculated by a relationship between discharge and elevation ($Z(Q)$ or $Q=Z^{\alpha}$).

For unsteady calculation, all boundary conditions are possible ($Z(t)$, $Q(t)$, $Q(Z)$).

If you want to get a $Q(Z)$ (or rather $Z(Q)$) downstream law of type uniform flow or critically flow, you can use the calculator available by pressing the button "Calculator".

This calculator allows to recover the local cross section (upstream of the node) or any other section by moving upstream or downstream along the system, using the arrows on the upper part of the windows, and to generate the corresponding law.

If several reaches reach this node, a question will be asked to choose the desired upstream reach, to retrieve the section and its geometric description. The left and right arrows in the upper part of the window make it possible to change this reach if necessary.

The calculator makes it possible to calculate a law $Q(Z)$ corresponding to the uniform regime ($Y_n$), as well as to the critical regime ($Y_c$).

For the uniform regime the equation used is the classical Manning-Strickler equation:

$Q_n=K_{s} SR^{2/3}\sqrt{I}$, where $K_{s}$ is the Strickler coefficient, and $I$ is the bottom slope, $S$ the wet surface, $R$ the hydraulic radius ($R=\frac{S}{P}$, where $P$ is the wetted perimeter).

For the critical regime, the following equation is used:

$Q_c=\sqrt{gS^{3}/L}$, where $L$ is the top width, $g=9.81 ms^{-2}$.

By clicking on the button "Normal $Q(Y_n)$ Calculation" or "Critical $Q(Y_c)$ Calculation", EdiSIC will calculate the corresponding $Q$ values ​​for different elevations:

  • firstly for the elevations corresponding to the description of the section in width - elevation format. These are the values ​​that appear on the right pane in the window. Intermediate or auxiliary variables P, S, R and V are calculated.
  • in order to have a finer definition of these laws, one can specify the number of calculated values, as well as the min (bottom elevation) and max (bank elevation). These min and max dimensions are pre-filled (at the bed and bank elevations of the studied section) but they can be modified. The number of default values ​​entered is 20 but can also be changed. The automatically generated elevations, for which the flows will be calculated, are a priori equidistributed. But in order to preserve the points of the width-elevation description, these particular points replace the points closest to the initial equi-distributed generation.

This calculator also handles sections with a medium bed, if any. In this case, the overflow elevation between the minor bed and the medium bed must be entered (it is pre-filled, with the correct value, but can possibly be modified). As well as 2 coefficients of friction, one for the minor bed, the other for the medium bed. The exchange variables between the minor bed and the medium bed are also calculated and graphed.

If the selected section has no medium bed, it is still able to do calculations in this configuration by increasing the maximum elevation. In this case, the old maximum elevation becomes the minor-medium bed elevation, and a new Strickler coefficient can be entered for that medium bed.