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Definition of Weir : A type of small overflow dam that can be used for flow measurement. The Linear Flow Orifice Meter is a mimic of this weir.
Definition of Sutro Weir : The discharge (flow) through the weir is proportional to the head (water depth above a reference plane located at one third of the depths of the crest of the base weir).
Development : The linear-proportional weir was developed by Stout in 1897 and was theoretically based, the design stipulated the width at the base as infinite. In 1908 Sutro modified the design to create a practical linear-proportional weir. The weir has a rectangular base and the flow through the weir is proportional to the height of the water through the curved portion of the weir plus

{latex} $$ 2\over 3 $$ {latex}

of the height of the rectangular base ie.

{latex} $$ Q = c [h + {2 \over 3} s] $$ {latex}

Equations Page Here

Source 1: Prof. B.S. Thandaveswara from the Indian Institute of Technology Madras website

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W = base of rectangular weir
s = height of rectangular weir
h = weir height above rectangular weir
c = constant of proportionality

{latex} $$ C_d $$ {latex}

= coefficient of discharge, ranges from 0.0597 to 0.619

{latex} $$ q_w $$ {latex}

= Flow through rectangular weir

{latex} $$ q_u $$ {latex}

= Flow through upper portion of weir,Important Parameter
Q = Total Discharge

{latex} $$ C_0 $$ {latex}

= vena contracta area ratio, average value is 0.62
g = acceleration due to gravity

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