WaterSubject

A short look to any handbook of hydraulic engineering or to classical textbooks on free surface flows is sufficient to guess the importance of the concepts of specific energy and total force in the crucial aspects of open channel flows. For example, the complex and rich physics of fluid motion around obstacles, at abrupt contractions or expansions of the channel width, near localized steps or sudden changes in bed elevation, in flowing out from different hydraulic structures, at the occurrence of a sudden change of the flow status (supercritical–subcritical), giving rise to stationary and moving hydraulic jump, requires the mastery of the concepts of specific energy and total force.

Often, to reproduce the main features of a rapidly varied flow, it is sufficient to understand if a problem is dominated by energy conservation or by total force conservation. In the former case, the meticulous evaluation of the small sources of dissipation changes the details of the problem, but not the substantial aspects, whilst, in the latter case, the accurate estimate of the forces supported by the boundaries gives final refinements, leaving more or less unchanged the main aspects of the flow field.

Two immediate examples of that are the flow under a sluice gate and the stationary hydraulic jump: in the former case, the specific energy is conserved and a portion of the total force of the flow is supported by the gate; in the latter case, neglecting the small force supported by the wetted boundary, the total force of the stream is conserved and the specific energy is dissipated in the inviscid shock. In the former case, abrupt variations of the flow field are determined by the geometry of the obstacle; in the latter case, from the internal nonlinearity of the flow field, which is governed by hyperbolic equations, admitting discontinuous solutions.

The same concepts can be useful in the different context of shallow flow computational dynamics. While the numerical treatment of the homogeneous form of shallow water equations can be considered a solved problem, at least from the engineering point of view, the source terms treatment in the momentum balance equation is still an open question. This is particularly true when the balancing of the scheme is put into evidence for practical purpose calculations. The development of novel models in this field often requires accurate procedures, involving the energy or the total force conservation.

In this context, the present work is conceived and carried out. A simple, practical, correct, conservative (not propagating errors) framework to manipulate, in an “exact” form, the specific energy vs depth relationship and the total force vs depth relationship is found. Such a framework allows to pass easily from one of these physical quantities to the others, using analytical expressions.

A rectangular cross-section of an open wide channel, neglecting bank effects, is considered. Expressions of specific energy (depth plus kinetic energy) and of total force (hydrostatic plus inertial force) as functions of water depth are the starting point of the present work.

The specific energy and the total force are made nondimensional, writing them as functions of the nondimensional water depth. Inversion of such functions involves algebraic equations of third degree.

A new, analytical, solution of the problem is found here. A certain value of the flow discharge per unit width is considered as known, which gives an unique value for the critical depth, critical specific energy and critical total force. For each value of the specific energy (greater than the critical value), a subcritical and a supercritical value of the corresponding depth are found analytically. Similarly, for each value of the total force (greater than the critical value), a subcritical and a supercritical value of the corresponding depth are found analytically. The further, real, root is shown to be negative both for energy and force, so that it can be rejected, having no physical meaning. Similarly, for physical reasons, analysis of inversion for energy values (force values) smaller than critical value is not performed.

The proposed solution is simple, both in the case of specific energy and in the case of total force. Such solutions have the important advantage that the nature of the three algebraic roots is known a priori: for the proper range of parameters, the first solution is always negative (so it must be discarded), the second solution is always subcritical and the third solution is always supercritical.

Some simple examples from classical open channel hydraulics show how such analytical results can be used in engineering practice.

A remarkable added value of the herein presented method is to be recognized in its utilization inside numerical applications, concerning shallow water equations integration. In such applications, particularly at singularities which require the control over the specific energy and the total force near the critical stage, the inversion of the depth–specific energy relationship or of the depth–total force relationship can be faced a huge number of times, like it happens in nonstationary simulations which can require millions or ten of millions time steps. In such cases, analytical expressions allows: (a) to obtain precise estimates up to error machine, knowing that a specific part of the procedure does not propagate errors in computations; (b) to completely avoid iterative methods, which represent potential sources of computational inefficiency; (c) to avoid ambiguities or departure from convergence when working in a narrow range around the critical condition.

All these aspects stress the potentials of the found analytical solutions as an efficient tool in open channel hydraulics.