What is a weir? #

A weir is a raised crest across a channel or pond outlet. When the upstream water rises above the crest, water spills into the downstream channel. At a detention pond, an emergency weir provides an overflow path when the pond reaches its crest. Its rating relates the upstream water level to the discharge; the opening geometry affects that relation.

To trace the energy equation back to its source, read From Newton’s Second Law to Bernoulli’s Equation. That article derives Bernoulli’s equation from Newton’s second law; here we combine Bernoulli with continuity to derive discharge ratings for common weir shapes.

We begin with free overflow over a sharp crest: a thin plate, the nappe (the sheet of water flowing over the crest) ventilated from below, and tailwater below the crest. The broad-crested case has a different control condition and follows later.

Let z_c be the crest elevation and z_{WS,u} the upstream water-surface elevation, both measured from the same datum. The upstream head above the crest is

\displaystyle H=z_{WS,u}-z_c.

Free-overflow discharge as a function of head #

Let Q be discharge, y height above the crest with 0\le y\le H , and w(y) the opening width at that height. Let C_d be the dimensionless discharge coefficient calibrated for the crest and operating conditions. For the upstream head H defined above, the free-flow rating is

\displaystyle Q=C_d\sqrt{2g}\int_0^H w(y)\sqrt{H-y}\,dy.

We derive this rating by applying Bernoulli to the speed at each height, then using continuity to sum the slice discharges. See more below.

Represent the opening as horizontal slices #

To see the relationship, note that the slices run from the crest, y=0 , to the upstream water surface, y=H . A slice at height y is called a strip in the integral. Its thickness is dy and its area is dA=w(y)\,dy .

Find slice velocity with Bernoulli #

We then apply Bernoulli between point 1 at the upstream surface and point 2 in the free jet at height y . Their elevations are z_1=z_c+H and z_2=z_c+y , as shown in the first diagram. Let p_1 and p_2 be the pressures at those points, V_1 the mean approach speed, and v(y) the local jet speed. With losses neglected, the energy equation is

\displaystyle \frac{p_1}{\gamma}+z_1+\frac{V_1^2}{2g}=\frac{p_2}{\gamma}+z_2+\frac{v(y)^2}{2g},

where \gamma is water’s specific weight and g is gravitational acceleration. The pressure heads cancel because both points are at atmospheric pressure. The upstream area is large compared with the overflow area, so we neglect V_1 . Substituting the elevations gives

\displaystyle z_c+H=z_c+y+\frac{v(y)^2}{2g},\qquad v(y)=\sqrt{2g(H-y)},

The calculated strip velocity v(y) is greater nearer the crest, where the driving head H-y is larger.

Add the strip discharges #

The discharge through each strip is its water velocity multiplied by its area: dQ=v(y)\,dA . Adding the strip discharges from the crest to the upstream surface gives

\displaystyle Q_{ideal}=\sqrt{2g}\int_0^H w(y)\sqrt{H-y}\,dy.

The coefficient C_d accounts for contraction and other departures from the ideal strip model. It is empirical, not a Bernoulli result; use a value calibrated for the crest geometry and operating conditions.

Discharge equations for common crest shapes #

These cases use the same free, sharp-crested assumptions. Only the opening width w(y) changes.

Rectangular crest #

For a horizontal rectangular crest of width b , w(y)=b . Thus

$latex \displaystyle Q=C_db\sqrt{2g}\int_0^H(H-y)^{1/2}\,dy =\frac{2}{3}C_db\sqrt{2g}\,H^{3/2}. $

The head contributes one factor of H through the overflow height and \sqrt{H} through the strip speeds. Integration accounts for the factor 2/3 . The result is the rectangular sharp-crested weir equation.

V-notch #

For a symmetrical V-notch with included angle \theta , the width grows with height: w(y)=2y\tan(\theta/2) . Substitution gives

$latex \displaystyle Q=2C_d\tan(\theta/2)\sqrt{2g}\int_0^H y(H-y)^{1/2}\,dy =\frac{8}{15}C_d\tan(\theta/2)\sqrt{2g}\,H^{5/2}. $

The extra power comes from the width, which is proportional to y . A V-notch therefore gives a stronger change in discharge for a change in head than a rectangular crest.

Trapezoidal opening #

For a symmetric trapezoidal opening with bottom width b and side slope m horizontal to one vertical, w(y)=b+2my . Split the integral into its rectangular and triangular parts:

\displaystyle Q=C_d\sqrt{2g}\left[\frac{2}{3}bH^{3/2}+\frac{8}{15}mH^{5/2}\right].

This is the ideal integral multiplied by one coefficient. A Cipolletti weir uses side slopes selected to compensate approximately for the side contractions of a rectangular weir; its tabulated coefficient is a calibration for that geometry, not a new Bernoulli result.

Broad crest: critical flow sets the rating #

At a pond overflow, we raise the crest to set the water level at which spilling begins. The overflow may be built as a concrete sill with a flat top extending along the flow direction. That top gives the structure a broad crest: water flows across a raised bed before descending downstream. Its length is along the flow; its width b is across the channel.

This geometry changes how we calculate discharge. Water remains supported by the crest, so we need to find both its depth and its velocity there. As water approaches the raised bed, it becomes shallower and accelerates. For a sufficiently long crest with low enough tailwater, a critical-flow control forms on the crest. That condition supplies the depth–velocity relationship needed to calculate discharge from the upstream energy. We will derive that relationship below.

The blue shading is water; the grey shading is the pond bed and raised weir structure. The flat crest sits above the pond bed but below the upstream water surface, so water can flow over it. The blue surface drops as the water accelerates towards the crest.

The grey dashed line extends the crest elevation into the upstream pond. We measure H from this datum to the upstream water surface, not from the pond bed. On the flat crest, d_c measures the water depth from the crest to the surface at the critical control section. These are two different vertical distances at two different locations.

At the upstream measuring section, the approach flow moves at mean speed V_a . The orange dashed line marks the upstream energy level, which lies above the upstream water surface by the approach velocity head. The orange arrow E_c measures this energy level above the crest datum. Neglecting losses between the measuring section and the crest control gives

\displaystyle E_c=H+\alpha_a\frac{V_a^2}{2g},

where \alpha_a accounts for the approach-velocity distribution. This calculates E_c from upstream measurements and the approach speed. We then use the critical-flow condition to find d_c ; d_c is an output, not an input to the energy calculation. When approach speed is negligible, E_c\approx H .

For a rectangular control section of width b , let q=Q/b be discharge per unit width. At depth d , area per unit width is d , so continuity gives q=Vd and V=q/d . The specific energy at depth d is therefore

\displaystyle E(d)=d+\frac{V^2}{2g}=d+\frac{q^2}{2gd^2}.

Differentiate with respect to depth, treating q and g as constants:

\displaystyle \frac{dE}{dd}=1-\frac{q^2}{gd^3}.

For a fixed discharge q , the critical depth is where specific energy reaches its minimum: it is the least energy that can carry that flow. Turned around, a given available energy E_c sets the maximum discharge at critical flow. We find that depth by setting the derivative to zero; the positive second derivative 3q^2/(gd^4) confirms the minimum:

\displaystyle 1-\frac{q^2}{gd_c^3}=0,\qquad q^2=gd_c^3.

Here d_c is the critical depth.

Since V_c=q/d_c , this critical relation gives V_c^2=q^2/d_c^2=gd_c . At the crest, the specific energy is E_c . Substituting the critical velocity into E_c=d_c+V_c^2/(2g) gives

$latex \displaystyle E_c=d_c+\frac{V_c^2}{2g} =d_c+\frac{gd_c}{2g}=\frac{3}{2}d_c, \qquad d_c=\frac{2E_c}{3}. $

Physical interpretation #

If d>d_c , flow is subcritical: water-surface changes can travel upstream, so tailwater may alter the depth and discharge at the crest. We must include the downstream level. If d<d_c , flow is supercritical; downstream changes cannot travel upstream, but the critical rating below still does not apply. That rating assumes d=d_c .

Continuity gives Q=bq=bd_cV_c . Using V_c=\sqrt{gd_c} and d_c=2E_c/3 gives the ideal rating above. For a real crest, a calibrated coefficient may be applied:

\displaystyle Q=C_{d,b}\,b\sqrt{g}\left(\frac{2E_c}{3}\right)^{3/2}.

Here C_{d,b} is the coefficient for the specified broad-crest geometry and calibration.

Check the conditions before using a rating #

The rating assumes the head represents the undisturbed approach flow, the overflow width matches the crest, and the nappe remains free. The sketches show how each condition can fail.

Head and approach flow #

Our sharp-crested equation assumes a slowly moving upstream pool, with head H measured above the crest. Near the crest, water accelerates and its surface drops (drawdown). Using that lower water level in the equation underestimates discharge. The remedy is to measure farther upstream, outside the drawdown region shown in the left-hand sketch.

For the thin-plate measuring weirs covered by the USBR Water Measurement Manual, Chapter 7 §5, the measuring point must be at least four times the maximum measuring head upstream of the crest. For example, a maximum head of 0.30\,\mathrm{m} requires a measuring point at least 1.20\,\mathrm{m} upstream.

If the water is moving appreciably even at the upstream measuring section, neglecting its velocity omits part of the energy driving the overflow. Let V_a be an approximately uniform approach velocity and h_a=V_a^2/(2g) its velocity head. Retaining this term in our ideal strip calculation gives

\displaystyle v(y)=\sqrt{2g(H+h_a-y)}.

For a rectangular opening, integrating over the same physical opening, 0\le y\le H , gives

\displaystyle Q_{ideal}=b\sqrt{2g}\int_0^H\sqrt{H+h_a-y}\,dy=\frac{2}{3}b\sqrt{2g}\left[(H+h_a)^{3/2}-h_a^{3/2}\right].

Setting h_a=0 recovers our original result. This shows how approach velocity changes the ideal calculation; for actual discharge, the coefficient must match this treatment of approach flow, as discussed in the USBR thin-plate weir derivation.

Crest width and side contraction #

When the crest ends are set in from the channel walls, water turns inward around them and narrows the overflowing sheet. This side contraction reduces discharge, so we account for it with a smaller effective width in place of b , or with a coefficient C_d that already includes the contraction.

If the crest spans the channel and meets both side walls, these end contractions do not form, so no side-contraction correction is needed. This arrangement is called a suppressed weir.

Ventilation and tailwater #

Before using the sharp-crested discharge equation, check that air can reach beneath the falling sheet of water and that downstream water is low enough to leave it free. These are the conditions that allowed us to take atmospheric pressure on both sides of Bernoulli’s equation. If downstream water rises enough to support the sheet, the pressure beneath it changes and discharge also depends on the downstream level. For that condition, we use a submerged-flow equation based on both water levels, with coefficients obtained from measurements for the relevant weir shape and submergence conditions.

Conclusion #

Using Bernoulli’s equation and continuity, we derived free-overflow ratings for rectangular, V-notch and trapezoidal openings. The trapezoidal result also covers a Cipolletti crest when its calibrated coefficient is used. For a rectangular broad crest, we derived the rating from the minimum-specific-energy condition and continuity.