The discharge coefficient of SMBF flumes under free and submerged conditions

Abstract

Abstract In this study, the discharge coefficients ( C d ) 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 [ r crn = ratio of flume width ( w ) to channel width ( B )], relative head ( h w : the ratio of the upstream head ( h ) to the $$w$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>w</mml:mi> </mml:math> ) and, in the case of submerged flow, also the submergence ratio [ $$S_{{\text{r}}} = h_{{\text{t}}} /h_{{\text{u}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mtext>r</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mtext>t</mml:mtext> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mtext>u</mml:mtext> </mml:msub> </mml:mrow> </mml:math> : downstream flow depth ( $$h_{{\text{t}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mtext>t</mml:mtext> </mml:msub> </mml:math> ) to upstream flow depth ( h u )]. C d decreases logarithmically from 1.2 to 0.75 in the range of $$h_{w}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:math> between 0.4 and 1.8. The submerged condition does not reduce the $$C_{{\text{d}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mtext>d</mml:mtext> </mml:msub> </mml:math> , but it reduces the discharge capacity (up to 50%), so that in some cases, to pass a given flow discharge, $$h_{{\text{u}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mtext>u</mml:mtext> </mml:msub> </mml:math> should increase by about 100% compared to the free condition.

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