Weir Hydraulics for Flow Measurement and Flow Splitting: Sharp-Crested, V-Notch, Broad-Crested Weirs and Flumes

Why the simple weir formula works, and why it is wrong

Textbooks derive the rectangular sharp-crested weir equation by assuming horizontal flow over the crest, hydrostatic pressure above it, no drawdown of the surface and an irrotational approach, integrating the velocity across two thirds of the head and arriving at q = (2/3)C_d√(2g)H^3/2 with C_d around 0.6. Every one of those assumptions is false: the flow turns upward before the crest, the pressure is far from hydrostatic, and the nappe surface sits about 0.85H above the crest at the point where the pressure returns to atmospheric. Fenton (2015) shows that dimensional analysis alone gives the same form, q/√(gH³) = constant, without any of the physics, and that the coefficient of discharge simply absorbs the errors. The practical consequence is that C_d is an empirical quantity that depends on the ratio of head to crest height H/P, on the ratio of weir width to channel width b/B, and weakly on viscosity and surface tension through the head itself. Any design that uses a fixed 0.6 or 0.62 is accepting an error of up to 5 to 10 percent at high H/P.

Rectangular sharp-crested weirs: Kindsvater-Carter and Fenton

ISO 1438, BS 3680 Part 4A and the Bureau of Reclamation Water Measurement Manual all adopt the Kindsvater and Carter (1957) formulation, which handles fully contracted, partially contracted and full-width (suppressed) weirs in one expression by replacing width and head with effective values:

Fenton fitted the Kindsvater and Carter curves with a single closed expression, C_d = 0.589 − 0.008(H/P) + (b/B)²[0.013 + 0.083(H/P)], which reduces to 0.602 + 0.075 H/P for a full-width weir and matches Rehbock within rounding. Either route gives a coefficient error below 1 percent for a weir built with reasonable care, provided the limits hold: h at least 0.03 m, h/p no more than 2.0 (critical depth forms upstream beyond about 5), p at least 0.10 m, b at least 0.15 m, tailwater at least 0.05 m below the crest and the nappe fully aerated. Nappe aeration is the failure most often seen at works: a clinging or partly drowned nappe on a launder or overflow weir reads several percent high and the error varies with tailwater.

V-notch weirs

For low flows, and for clarifier launders, the triangular notch gives a head that is large relative to its uncertainty. ISO 1438 and the USBR manual use the Kindsvater-Shen form:

A 90 degree notch at 0.20 m head passes 1.365 × (0.20085)^2.5 = 24.7 L/s. Sensitivity is dQ/Q = 2.5 dh/h, so 1 mm of head error at 0.2 m is 1.25 percent of flow, and the same 1 mm at 0.05 m is 5 percent, which is why ISO sets a 50 mm minimum head. On a clarifier launder with notches at 300 mm centres the head per notch is small; a 30 m diameter tank with 314 notches passing 300 L/s runs at 0.96 L/s per notch, which is 54 mm of head, and a 5 mm out-of-level along the weir plate puts 10 percent more flow through the low end. Levelling tolerance on launder plates should therefore be ±2 mm, and adjustable plates are worth specifying.

Broad-crested weirs, Crump weirs and flumes

Sharp-crested plates are for measurement in clean water and for splitting; they are not robust against rag and grit, and they need a free fall. Structures with a long control section are preferred for raw sewage, settled sewage and final effluent, both because they tolerate submergence and because parallel streamlines at the control make the coefficient nearly constant.

• Rectangular broad-crested weir (crest length L in the direction of flow, square upstream edge): Q = C_d C_v (2/3)(2g/3)^1/2 b h^3/2 = 1.705 C_d C_v b h^3/2. C_d is about 0.85 for 0.08 ≤ h/L ≤ 0.33 and rises towards 1.0 as h/L approaches 1.0 and streamline curvature grows; h/L should not exceed 0.5 (0.7 with spare head). Modular limit (the submergence ratio H₂/H₁ at which discharge drops 1 percent) is about 0.66 for h/L up to 0.33 and falls to about 0.38 at h/L = 1.5, with linear interpolation between. A vertical back face limits it to about 0.80 at best; a 1 in 4 downstream slope improves it.
• Triangular profile (Crump) weir, 1 in 2 upstream and 1 in 5 downstream (BS ISO 4360): C_d about 0.63 in the same form of equation, with the benefit that it keeps measuring to a submergence ratio of about 0.75 and can be rated in the drowned range with a crest tapping, which is why it is the UK river gauging standard.
• Long-throated flumes (ISO 4359), with a gradual contraction, a parallel throat and an expanding outlet, achieve modular limits of 0.65 to 0.95 depending on the exit energy recovery, and can be rated by computation rather than calibration. Flow is Q = C_d C_v (2/3)^3/2 √g b_c h^3/2 for a rectangular throat.
• Parshall flumes are empirical and must be built to the published dimensions; their free-flow limit is a submergence of 50 percent for the 25 to 75 mm sizes, 60 percent for 150 and 225 mm, 70 percent for 0.3 to 2.4 m and 80 percent for larger throats. Beyond those a second head is needed and accuracy falls badly.

At a sewage works the final effluent flume is the regulatory meter. Under the Environment Agency MCERTS self-monitoring standard the total daily volume must be known to within ±8 percent, which a correctly installed ISO flume or Crump weir with a certified level sensor meets comfortably, but a flume set too low, with the downstream channel backing up at high flow, will not. Checking the modular limit at the flood tailwater case is part of the hydraulic profile, not an afterthought.

Approach conditions, head measurement and uncertainty

Three installation rules recover most of the accuracy that the formulae promise. First, the approach channel should be straight for about 10 times its width, with a cross-sectional area at the gauging point at least 10 times the flow area over the crest, so that velocity is uniform and low. Second, the head is measured 3 to 4 times the maximum head upstream of a sharp crest (USBR: at least four heads), or at the ISO-specified tapping for a flume, in a stilling well connected below the water surface. Third, the crest or throat level must be surveyed to within a millimetre or two relative to the gauge zero; a datum error enters the flow directly through the 1.5 or 2.5 power.

Where the approach velocity is not negligible, the energy head H = h + v²/2g replaces h, which is handled either by iterating on Q or by the approach velocity coefficient C_v = (H/h)^3/2 read from the area ratio C_dA*/A₁ (Bos, Figure 1.12). Values rise from 1.00 at an area ratio of 0.1 to about 1.05 at 0.5, so a weir with a shallow approach channel reads 5 percent low if the correction is skipped.

Combining errors: for a rectangular thin-plate weir the coefficient contributes under 1 percent, the effective width and head corrections about ±0.3 mm each, and the head reading whatever the sensor achieves (±1 mm for a good ultrasonic or pressure sensor in a stilling well, ±3 mm for a staff gauge read by eye). At a working head of 150 mm the root-sum-square uncertainty is about 1.5 to 3 percent; at 50 mm it exceeds 5 percent. USBR advises that even a standard, well-maintained field device should not be expected to beat ±2 percent, and that submergence corrections to a drowned sharp-crested weir are only good to ±15 percent and should be treated as emergency estimates. Cipolletti and other trapezoidal weirs carry a ±5 percent coefficient uncertainty and offer no advantage over a Kindsvater-rated rectangular plate.

Weirs for flow splitting

A splitter chamber divides one flow among several tanks by passing it over identical weirs fed from a common pool. Equal flow requires equal head over equal crests with free discharge, and the arithmetic is unforgiving: with 150 mm of head on each weir, a 3 mm difference in crest level between two outlets produces a 3 percent difference in flow, and a 10 mm difference produces 10 percent. Common faults are a chamber too small to still the inlet jet, so the pool surface slopes; outlets that drown at peak flow because the downstream channel backs up; and weirs that are too wide, so that the head is small and tolerances dominate. Good practice is to keep the head at design flow above 100 mm, to use a drop of at least 50 mm below the lowest crest at the highest tailwater, to make crests adjustable in the field, and to size the chamber so the approach velocity is below 0.3 m/s. Where one tank must take a fixed fraction, a weir of proportionate width does it; where the fraction must change with flow, an adjustable penstock downstream of the weir is the usual control.

Physical and computational modelling earn their cost when the geometry is cramped. For the Wigan inlet works a 1 in 10 scale model of the interceptor chamber, wet well, screens channels and clustered detritor split showed the flow division within 5 percent of the calculated values and needed only minor benching changes; CFD on a Prague distribution structure found a 15 percent imbalance in the original layout that a baffle removed. For most works a hand calculation of weir head, a check of the tailwater at peak flow and a survey of the crests after construction will do; for a new works design with six or more tanks off one chamber, model it. Flow measurement for permit compliance is discussed further in our effluent compliance guidance.