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Hydraulic Performance of PK Weirs Based on Experimental Study and Kernel-based Modeling


A piano key weir (PK weir) is a non-linear, labyrinth-type weir that benefits of a high discharge capacity, and is well suited for low head dams. Determination of the discharge coefficient (Cd) is considered as one of the most important issues, which plays a substantial role in reducing structural and financial damages caused by floods. The main aim of the present study is to experimentally investigate the variations of PK weirs discharge coefficient (Cd) through altering the geometric parameters. The obtained results revealed that in modified PK weirs (by an 11.5% increase in weir height, changing the crest shape, and fillet installation), the Cd values were about 5–15% more than those of the standard PK weirs. The Cd values of the non-contracted weirs were increased by increasing the inlet/outlet width ratio by 1.4, while this relation was adverse for contracted weirs. In the modified PK weirs, the submergence would occur faster than the standard weirs, while the complete submergence would occur later. Moreover, robust kernel-based approaches (kernel extreme learning machine and support vector machine) were successfully employed to the extensive experimental dataset by taking into consideration the Cd as a function of dimensionless geometric variables of PK weirs. The obtained results showed that the ratio of the upstream hydraulic head (H0) to total weir height (P) plays a significant role in the modeling process.

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Data Availability

The data and materials that support the findings of this study are available on request from the corresponding author.


PK weir:

Piano Key weir

Cd :

Discharge coefficient

H0 :

Upstream hydraulic head (m)

Hd :

Downstream hydraulic head (m)

H* :

Total submerged-flow upstream head (m)


Total weir height (m)

Bi/Bo :

Length of inlet/outlet cantilever overhang

Wi/Wo :

Inlet to outlet width ratio

Lc :

Length of crest centerline (m)

Si :

Inlet slope

So :

Outlet slope


Total width of weir (m)


Weir length (m)

Ts :

Wall thickness


Cycles number

Pi :

Height of the inlet at entrance measured from the PK weir crest (m)

Po :

Height of the outlet at entrance measured from the PK weir crest (m)

Pb :

Height of the apron level at inlet key (m)

Pb :

Outlet key intersection (m)

Pb :

Height of parapet wall in modified PK weir (m)


Flow discharge passing over the PK weir (m3/sec)




Kinematic viscosity


Surface tension


Acceleration of gravity (m/s2)


Velocity (m/s)


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Author information




K.R. conceived the study and were in charge of overall direction and planning. M.M.A performed the experiments and contributed to the interpretation of the results. S.S. took the lead in writing the manuscript and carried out the kernel-based modeling. All authors provided critical feedback and helped shape the research, analysis and manuscript.

Corresponding author

Correspondence to Mahdi Majedi Asl.

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Roushangar, K., Majedi Asl, M. & Shahnazi, S. Hydraulic Performance of PK Weirs Based on Experimental Study and Kernel-based Modeling. Water Resour Manage 35, 3571–3592 (2021).

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  • PK weirs
  • Submergence
  • Geometric parameters
  • Hydraulic performance
  • Contracted weirs
  • Kernel extreme learning machine
  • Support vector machine