Numerical modelling of the hydraulic behaviour of lateral storm weirs in a transcritical regime
In recent years, French and European legislation has introduced regulations about wastewater discharge into natural environments and particularly about combined sewer overflows. As a consequence, it has become essential to control the hydraulic behaviour of these structures and to estimate the pollution loads released at this level. The side weir is the regulation structure that permits the hydraulic regulation of the waste water carried by the sewer system. When the upstream flow intensity exceeds a value referred to as reference flow, the side weir directly rejects part of the waste water to the natural environment. The hydraulic behaviour of the sewer side weir was shown to involve a discontinuous evolution of the water depth, characterized by a hydraulic jump (transition from supercritical to subcritical flow) and also by a rapidly varying transcritical evolution (subcritical to supercritical). Initially, the side weir flow was determined with the use of empirical relations. Using formulae of Engels, Coleman and Smith, Balmaceda and Gonzales or Dominguez, it was possible to calculate the outflow according to the water level at the upstream and/or the downstream region of the weir. These relations were applicable only for certain flow regimes and in certain cases in which the geometry of the side weir was specified. Subsequently, a more physical approach, initiated by Ackers, was based on the assumption of constant energy along the side weir. This approach made it possible to focus not only on an assessment of the side channel flow, but also on the water profile at the crest. Unfortunately, as the study of El Kashab shows, this method falls short in certain cases because the equations are inappropriate. For example, in the case of hydraulics the constant energy approach was not applicable. Finally, the method that is currently used is based on a momentum equation, which makes it possible to establish equations for shallow water. This approach seemed the most appropriate in the case of the side weir. The numerical solution of these equations was always based on an algorithm that describes all the possible cases according to the flow regime and the hydraulic conditions in the side weir. One must know the flow regime a priori. These models don’t properly simulate the transitory behaviour of these kinds of works. In this article we propose hydraulic modelling of a sewer side weir that integrates the geometrical characteristics of the flow (height and length of the crest, variation of width along the crest), and avoids the need for a priori knowledge of flow conditions in the side weir. The model also takes into account hydraulic discontinuities (hydraulic jump, transitions from free surface to pressurised flow) and the transitory character of the flow. The numerical results were compared with measured values obtained from a test bench. For the 1D approach, the solution was found using the 1D shallow water equations written in a conservative form for a transitory situation. The conservative characters of the equations permit us to consider gradually and rapidly varied flows in a single system of equations. In order to account for the lateral overflow, we used the Hager relation, which involves the intensity and direction of the lateral velocity vector and also the influence of width variation of the side weir. The shallow water equations system couldn’t be analytically solved. As a consequence, numerous numerical methods have been developed such as characteristic methods or finite difference methods. Unfortunately, these methods are inadequate when discontinuities such as hydraulic jumps or flow regime transitions (froude number close to 1) appear. To solve these problems, numerical (shock capturing) schemes were developed, based on a finite volume formulation. Godunov was the initiator of this type of finite volume numerical scheme. Eleuterio improved the precision and the ability of these numerical schemes to converge. The result was a combination of total variation diminishing (TVD) interpolation with an appropriate Riemann solver. We used a second order TVD Upwind shock capturing numerical method associated with the Roe Riemann solver. In order to validate the numerical model, we have built a sewer side weir physical test bench at the Obernai site. The variable parameters were the downstream pipe diameter in comparison with the upstream pipe diameter, the side weir length, slopes of the upstream and downstream pipes and the height of the crest. The tested cases permitted us to sweep a slope ranging from 0.5 ‰ to 1 % for the upstream and downstream pipes. Globally, 114 configurations have been tested with the 1D numerical model. The operating curve represents a criterion for characterising the operating of the storm overflow. As long as the upstream flow does not reach the reference value, there is no overflow. As soon as the upstream flow exceeds the value, the downstream flow remains close to the reference value. Because comparisons were made in relation with the upstream flow, the criterion used for judging the modelling performance was the absolute error value in relation to the upstream flow. It is important to note that the results need to be weighted as experimental measurements have a margin of error of approximately 5%. Comparisons between numerical and experimental results permit the following conclusions: 1. The distribution of the number of errors was very close to the normal Gaussian distribution curve, with a slight shift towards the positive side. This indicates that the errors were random and therefore very close to experimental values. 2. In general, the maximum errors varied from -10 to 13%, with the majority of cases occurring between –5 and +5%. This shows that the performance of the tool was very useful in cases that are as complex as side weirs. However, the model has a slight tendency to overestimate the overflow rate as compared to experimental measurements.