Discussion of "transcritical flow due to channel contraction" by O. Castro-Orgaz, J. V. Giraldez, and J. L. Ayuso

The design of channel contraction is not obvious when transcritical or near-critical flows take place. The concept of critical flow conditions was first developed by Bélanger (1828) and later expanded by Bakhmeteff (1912, 1932). Bélanger and Bakhmeteff both defined the concept of critical flow in relation to the singularity of the backwater equation. Herein further applications of transcritical flow in channel contraction are discussed and a solution of the critical flow singularity is presented

Minimum Specific Energy and Critical Flow Conditions in Open Channels

In open channels, the relationship between the specific energy and the flow depth exhibits a minimum, and the corresponding flow conditions are called critical flow conditions. Herein they are re-analysed on the basis of the depth-averaged Bernoulli equation. At critical flow, there is only one possible flow depth, and a new analytical expression of that characteristic depth is developed for ideal-fluid flow situations with non-hydrostatic pressure distribution and non-uniform velocity distribution. The results are applied to relevant critical flow conditions : e.g., at the crest of a spillway. The finding may be applied to predict more accurately the discharge on weir and spillway crests

Bernoulli theorem, minimum specific energy and water wave celerity in open channel flow

One basic principle of fluid mechanics used to resolve practical problems in hydraulic engineering is the Bernoulli theorem along a streamline, deduced from the work-energy form of the Euler equation along a streamline. Some confusion exists about the applicability of the Bernoulli theorem and its generalization to open-channel hydraulics. In the present work, a detailed analysis of the Bernoulli theorem and its extension to flow in open channels are developed. The generalized depth-averaged Bernoulli theorem is proposed and it has been proved that the depth-averaged specific energy reaches a minimum in converging accelerating free surface flow over weirs and flumes. Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics

Free-surface profiles, velocity and pressure distributions on a broad-crested weir: a physical study

Basic experiments were conducted on a large-size broad-crested weir with a rounded corner. Detailed free-surface, velocity, and pressure measurements were performed for a range of flow conditions. The results showed the rapid flow distribution at the upstream end of the weir and next to the weir brink at large flow rates. The flow properties above the crest were analyzed taking into account the nonuniform velocity and nonhydrostatic pressure distributions. Introducing some velocity and pressure correction coefficients, it is shown that critical flow conditions were achieved above the weir crest for 0.1 < x/L-crest < 1. The velocity measurements highlighted a developing boundary layer. The data differed from the smooth turbulent boundary layer theory, although the present results were consistent with earlier studies. On average, the boundary stress was approximately tau(o)/(rho x g x H-1) 0.0015-0.0025. DOI: 10.1061/(ASCE)IR.1943-4774.0000515. (C) 2012 American Society of Civil Engineers

Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828)

A hydraulic jump is the sudden transition from a high-velocity to a low-velocity open channel flow. The application of the momentum principle to the hydraulic jump is commonly called the Bélanger equation, but few know that Bélanger's (1828) treatise was focused on the study of gradually varied open channel flows. Further, although Bélanger understood the rapidly-varied nature of the jump flow, he applied incorrectly the Bernoulli principle in 1828, and corrected his approach 10 years later. In 1828, his true originality lay in the successful development of the backwater equation for steady, one-dimensional gradually-varied flows in an open channel, together with the introduction of the step method, distance calculated from depth, and the concept of critical flow conditions