The discharge coefficient of SMBF flumes under free and submerged conditions

Abstract

In this study, the discharge coefficients (Cd) of SMBF flumes under free and submerged flow conditions were analytically investigated. The dimensionless parameters involved in the discharge coefficient, derived from the dimensional analysis, are the contraction ratio [rcrn = ratio of flume width (w) to channel width (B)], relative head (hw: the ratio of the upstream head (h) to the w\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w$$\\end{document}) and, in the case of submerged flow, also the submergence ratio [Sr=ht/hu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{{\ ext{r}}} = h_{{\ ext{t}}} /h_{{\ ext{u}}}$$\\end{document}: downstream flow depth (ht\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{{\ ext{t}}}$$\\end{document}) to upstream flow depth (hu)]. Cd decreases logarithmically from 1.2 to 0.75 in the range of hw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{w}$$\\end{document} between 0.4 and 1.8. The submerged condition does not reduce the Cd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{\ ext{d}}}$$\\end{document}, but it reduces the discharge capacity (up to 50%), so that in some cases, to pass a given flow discharge, hu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{{\ ext{u}}}$$\\end{document} should increase by about 100% compared to the free condition.