🌊 Discharge by the Velocity-Area Method
A virtual re-creation of stream gauging in the glass flume. Traverse a pitot-static tube up three mid-verticals of the cross-section, read the twin 30°-inclined manometer tubes, build the velocity profiles, and integrate them into a discharge — checked against the supply venturimeter. Then test the two classic stream-gauging shortcuts: the 0.6-depth rule and the 0.2/0.8-depth average.
Objectives
(1) To measure the discharge passing a given cross-section of a rectangular open channel; (2) to verify the adequacy of two common assumptions regarding the velocity distribution in a vertical, used in stream gauging.
Theory
Velocity is zero at the boundaries and greatest near the surface, so a single reading cannot give the discharge. The section is divided into n vertical strips; point velocities along each mid-vertical are combined into a weighted mean, and:
Q = Σ v̄ᵢ·Aᵢ • v̄ᵢ = Σ(uᵢⱼ·aᵢⱼ)/Aᵢ • u = √(2g·Δh), Δh = (R_total − R_static)·sin 30°
Field tests show the vertical velocity distribution is roughly parabolic/logarithmic, giving two practical rules: v̄ ≈ u at 0.6·depth below the surface, and v̄ ≈ ½(u at 0.2·depth + u at 0.8·depth). The inclined tubes stretch each head reading by 1/sin30° = 2, halving the reading error.
Apparatus
Glass-walled flume with stilling tank and entry screens; traversing point gauge; Prandtl pitot-static tube on a slidable plank, connected to two open glass tubes inclined at 30°; venturimeter on the supply line.
Channel width B = 0.30 m Depth at station O = 15 cm n = 3 verticals · m = 11 points each Lowest point 0.3 cm above the bed
Procedure — perform it here
Observations
| No. | Vertical | y (cm) | Rt (mm) | Rs (mm) | Δh (mm) | u (m/s) |
|---|---|---|---|---|---|---|
| No points yet — position the pitot tube and press “Record point”, or Auto-traverse a vertical. | ||||||
Results — discharge & the gauging rules
| Vertical | points | v̄ integrated (m/s) | u@0.6-depth (m/s) | err % | ½(u₀.₂+u₀.₈) (m/s) | err % |
|---|---|---|---|---|---|---|
| Traverse the verticals to evaluate the two stream-gauging rules. | ||||||
Velocity profiles
Recorded points on the three mid-verticals. Dashed levels mark 0.2, 0.6 and 0.8 of the depth below the surface — the levels the gauging rules use.
Discussion & Precautions
Why is the maximum velocity slightly below the surface?
Ideally it would be at the surface, but secondary currents driven by the side walls and the air resistance at the free surface push the velocity maximum a little below it — the "velocity dip", most pronounced in narrow channels where the width-to-depth ratio is small.
Why do the 0.6-depth and 0.2/0.8-depth rules work?
For a logarithmic (or near-parabolic) vertical velocity distribution, the depth-averaged velocity mathematically occurs at almost exactly 0.6 of the depth below the surface, and equals the average of the velocities at 0.2 and 0.8 depth. Stream gaugers exploit this to replace a full traverse with one or two current-meter readings per vertical.
Why are the manometer tubes inclined at 30°?
The head difference at these velocities is only a few millimetres of water. Inclining the tubes stretches each reading by 1/sin30° = 2, doubling the scale length and halving the relative reading error — the same trick as an inclined-tube manometer.
Sources of error
Misalignment of the pitot tube with the flow direction, air bubbles in the connecting tubing, the finite size of the probe near the bed, unsteadiness of the supply, and reading errors on the inclined tubes — largest for the slow near-bed points where Δh is a millimetre or two.
Precautions: purge all air from the pitot tubing before readings; keep the horizontal limb submerged and aligned with the flow; wait for the inclined-tube menisci to steady at each point; take the lowest point with the limb just touching the bed (measurement centre 0.3 cm up).