Civil engineering mechanics & structural foundations of Weir Flow & Discharge.
In open channel hydraulic engineering, sharp-crested weirs are fundamental obstruction structures utilized for precise discharge measurement, flood routing, irrigation water allotment, and storm runoff control. When water spills over an aerated sharp-crested plate, it detaches cleanly to form a free-falling jet known as a nappe. For a 90° triangular V-notch weir, the integration of elementary horizontal strips across the triangular geometry yields the theoretical discharge relation Q_th = (8/15) · tan(θ/2) · √(2g) · H^(5/2), where θ is the notch angle (90°), g is gravitational acceleration (9.81 m/s²), and H is upstream head above the vertex. Viscous boundary drag, surface tension, and contraction of streamlines reduce actual discharge relative to theoretical flow, necessitating a calibrated discharge coefficient C_d: Q = C_d · (8/15) · √(2g) · H^(5/2). For standard 90° brass notches, C_d typically ranges between 0.58 and 0.62, giving an empirical coefficient multiplier of approximately 1.38 m^(0.5)/s. Maintaining full aeration under the falling nappe is mandatory; insufficient venting creates an under-nappe partial vacuum that depresses pressure, sucks the jet downward, and induces catastrophic flow measurement errors up to 25%. In this laboratory flume experiment, water head H is accurately monitored with a vernier hook-and-point gauge at systematic discharge rates to establish the head-discharge rating curve and calibrate the weir discharge coefficient.
Hydraulic Formulation of Sharp-Crested Weirs
Sharp-crested weirs are open-channel control structures where flow separates from a sharp upstream edge. Consider an infinitesimal horizontal strip of height $dh$ at depth $h$ below the free surface over a triangular notch with included angle $\theta = 90^\circ$. The strip width is $b(h) = 2(H - h)\tan(\theta/2)$. Applying Torricelli's theorem for frictionless velocity $v(h) = \sqrt{2gh}$, theoretical elemental discharge is:
\[ dQ_{th} = b(h) \cdot dh \cdot \sqrt{2gh} = 2 (H - h) \tan\left(\frac{\theta}{2}\right) \sqrt{2gh} \, dh \]Integrating from $h = 0$ at the free surface to $h = H$ at the vertex:
\[ Q_{th} = 2 \tan\left(\frac{\theta}{2}\right) \sqrt{2g} \int_0^H (H h^{1/2} - h^{3/2}) \, dh = \frac{8}{15} \tan\left(\frac{\theta}{2}\right) \sqrt{2g} \, H^{5/2} \]For a standard $90^\circ$ notch ($\theta = 90^\circ$, $\tan 45^\circ = 1$):
\[ Q_{th} = \frac{8}{15} \sqrt{2 \times 9.81} \, H^{5/2} = 2.361 \, H^{5/2} \]Real fluids possess viscosity, surface tension, and streamline convergence, contracting the jet into a vena contracta. The actual volumetric discharge is modified by the discharge coefficient $C_d$:
\[ Q = C_d \frac{8}{15} \sqrt{2g} \, H^{5/2} = k_{weir} \cdot H^{5/2} \]where $k_{weir} = C_d \times 2.361$. For a calibrated weir with $C_d = 0.5845$, $k_{weir} = 1.3800\text{ m}^{0.5}\text{/s}$.
Rectangular Weirs and End Contractions
For a rectangular weir of crest length $L$, the theoretical flow equation is $Q_{th} = \frac{2}{3} L \sqrt{2g} H^{3/2}$. When the weir crest is narrower than the flume width ($L < B$), side wall contractions pinch the stream inward. The Francis equation incorporates effective crest length: $L_{eff} = L - 0.1 n H$, where $n$ is the number of end contractions (usually $n = 2$).
Viva voce questions & analytical theory
Why is a triangular V-notch weir preferred over a rectangular weir for measuring low discharges?
At low flow rates, a rectangular weir produces an extremely shallow head H across a wide crest that is difficult to measure accurately; in contrast, a triangular notch concentrates flow toward its vertex, generating a substantial, easily measurable head even at minimal discharges.
What physical phenomenon occurs if the under-nappe air cavity is not ventilated?
Air entrainment by falling water evacuates the pocket beneath the nappe, generating a sub-atmospheric partial vacuum that draws the jet downward (clinging nappe), increasing flow rate by up to 25% and invalidating standard rating equations.
Why must head H be measured at least 4H upstream of the weir crest?
As water approaches the weir, streamlines accelerate and curvature depresses the free surface downward in a drawdown curve; measuring at least 4H upstream ensures head is recorded in the undisturbed hydrostatic flow zone.
What is the theoretical head-discharge power exponent for a 90° V-notch weir versus a rectangular weir?
For a triangular V-notch weir, discharge scales with H^(5/2) (exponent 2.5); for a rectangular sharp-crested weir, discharge scales with H^(3/2) (exponent 1.5).
How do surface tension and fluid viscosity affect the discharge coefficient C_d at very small heads (H < 20 mm)?
At very low heads, water clings to the crest face due to surface tension and viscous boundary layer drag, causing C_d to increase sharply and deviating from standard empirical calibration constants.
What is the Cipolletti weir and what hydraulic design advantage does it offer?
A Cipolletti weir is a trapezoidal sharp-crested weir with side slopes of 1 horizontal to 4 vertical (14°); this specific slope exactly compensates for end contractions, making discharge directly proportional to crest length L.
Experimental precautions and laboratory safety
- Position the hook gauge point at least 4 times the maximum expected head upstream from the weir plate to avoid surface drawdown curvature.
- Ensure side aeration vent pipes remain completely unobstructed to prevent under-nappe vacuum formation and clinging nappe behavior.
- Wait until steady-state uniform conditions are established in the flume before logging head readings and volumetric tank collection intervals.
- Keep the brass weir knife edge clean and free from oil films, algal accumulation, or mechanical nicks that distort jet contraction.
Theoretical assumptions & boundary condition limits
- The civil engineering physical models apply within the specified calibrated operational window (0.05 to 0.25 m^(2.5)).
- Assumes homogeneous, isotropic continuum material behavior without unmodeled micro-fracture anomalies.
- Continuous sensor readings reflect ordinary least-squares regression across successive observations.
- Extreme inelastic yielding or cyclic degradation requires non-linear constitutive continuum damage mechanics.