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 reanalyzed 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 nonhydrostatic pressure distribution and nonuniform 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. DOI: 10.1061/ ASCE 0733-9437 2006 132:5 498 CE Database subject headings: Energy; Critical flow; Open channel flow; Weirs; Spillways; Coefficients. Introduction Considering an open channel flow, the free surface is always at atmospheric pressure, the driving force of the fluid motion is gravity, and the fluid is incompressible and Newtonian. Newton’s law of motion leads to the Navier–Stokes equations. The integration of the Navier–Stokes equations along a streamline, assuming that the fluid is frictionless, the volume force potential i.e., gravity is independent of the time, for a steady flow i.e., V / t=0 and an incompressible flow i.e., =constant , yields P + g z + v 2 = constant 1 where =fluid density; g=gravity acceleration; z=elevation aligned along the vertical direction and positive upward ; P=pressure; and V=velocity Henderson 1966; Liggett 1993; Chanson 1999, 2004 . Eq. 1 is the local form of the Bernoulli equation. In this study, the singularity of the depth-averaged Bernoulli principle for open channel flow is detailed, i.e., the critical flow conditions. Detailed expressions of the critical flow properties are derived for the general case of nonhydrostatic and nonuniform velocity distributions. The results are then applied to the rating curve of weir crest acting as discharge meter. Application to Open Channel Flows In open channels, it is common to use the depth-averaged Bernoulli equation within the frame of relevant assumptions e.g., Liggett 1993 Reader, Environmental Fluid Mechanics, Dept. of Civil Engineering, The Univ. of Queensland, Brisbane QLD 4072, Australia. E-mail: [email protected] Note. Discussion open until March 1, 2007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on June 2, 2005; approved on December 9, 2005. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 132, No. 5, October 1, 2006. ©ASCE, ISSN 0733-9437/2006/5-498–502/ $25.00. 498 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE J. Irrig. Drain Eng. 200 H = 1 d 0 d v y 2 2 g + z y + P y g dy = V 2 g + d + z0 = constant 2 where H=depth-averaged total head; z0=bottom elevation; d=flow depth; =momentum correction coefficient or Boussinesq coefficient ; and V=depth-averaged velocity