The Tainter Gate (also known as Radial Gate) is modelled based on a simplified calculation as used in the HEC-RAS simulation software. More specifically the following documentation is taken from the HEC-RAS Hydraulic Reference Manual on Radial Gates.
The calculation of the flow through the gate is approximated for three different regions:
$$Q = C_d W T^{T_E} B^{B_E} H^{H_E} \sqrt{2g} \tag{1} $$ (valid for gate opening higher than the downstream water level)
$$Q = C_d W T^{T_E} B^{B_E} (3H)^{H_E} \sqrt{2g} \tag{2} $$
(valid for the region where the ratio of downstream water level to upstream water level (H_ratio = h_o/h_i) is between 0.67 and 0.8)
$$Q = C_{dx} A \sqrt{2gH} \tag{3}$$
(otherwise with Cdx being a discharge coefficient tuned for a smooth transition between partially and fully submerged)
Note:
The use of Cdx is different to the implementaion as done in HEC-RAS. This was done in order to have a smoother transition from the partially to fully submerged region.