Properties of still water
Water at rest is called still water. It pushes on every wall and floor it touches. This push per area is pressure.
Deeper water has more water above it, so the pressure is bigger: p = ρ g h. Here ρ (rho) is density (1000 kg/m³ for water), g is 9.81 m/s², h is depth in metres. The answer is in pascals; 1 kPa = 1000 Pa. For water, ρg = 9.81 kN/m³, so p (kPa) = 9.81 × h.
On a vertical gate the pressure makes a triangle: zero at the top, biggest at the bottom. The total force on a gate of width b is the area of the triangle: F = ½ ρ g h² b. It acts at h/3 above the bottom, where the triangle balances. Water also lifts floating things: buoyancy equals the weight of water pushed aside.
Properties and measurement of water flow
The flow rate (discharge) Q is the volume that passes each second. Unit: m³/s. Q = A × v, where A is the cross-section area of the flow and v is the average speed.
If water fills a pipe and the pipe gets narrower, A falls, so v must rise. This is the continuity rule: A₁v₁ = A₂v₂. Where speed is high, pressure is lower (Bernoulli).
How do we measure flow? Use a float and a stopwatch for speed, a weir (a small wall water spills over, depth tells Q), or a Venturi meter (a pipe with a narrow throat, pressure drop tells speed).
Channel calculations
An open channel (canal, river, drain) has a free water surface. Three shape numbers are needed: flow area A, wetted perimeter P (the length of wall and bed that touches water), and hydraulic radius R = A / P.
Manning's equation gives the speed: v = (1/n) × R^(2/3) × S^(1/2). Here n is the roughness number (smooth concrete about 0.013, earth canal about 0.025) and S is the bed slope (drop per length). Then Q = A v.
Steps: find A, find P, find R, put R, S, n into Manning, then multiply by A. Deeper water gives a bigger R, so faster flow.
Forces of currents and waves
A moving current hits a pier like a hand pushing on it. The drag force is about F = ½ C ρ A v², where C is a shape number (about 1 for a flat face, less for a rounded one) and A is the face area. Double the speed and the force becomes four times.
A wave moves energy, not water. In shallow water the wave speed is c = √(g h). When a wave strikes a sea wall it adds a push on top of the still-water push. Engineers design for the tallest wave expected in a storm, plus a safety margin.
Try it
At home: poke three holes at different heights in a plastic bottle and fill it with water. Which jet shoots farthest? The lowest one. That shows pressure grows with depth.
In the 3D: first guess, then check. If you double the depth, will the force on the gate double or become four times? Move the depth slider and read F.
Key formulas and definitions
- p = ρ g h (water: p in kPa = 9.81 × h in m)
- F = ½ ρ g h² b, acting at h/3 above the bottom
- Q = A × v; A₁v₁ = A₂v₂
- R = A / P (hydraulic radius)
- v = (1/n) R^(2/3) S^(1/2) (Manning)
- c = √(g h) (shallow-water wave speed)
- Drag F = ½ C ρ A v²
Worked examples
1. Find the water pressure at the bottom of a tank 10 m deep.
p = 9.81 × 10 = 98.1 kPa (gauge, i.e. without the air above).
2. A gate is 3 m wide and holds 2 m of water. Find the force on it and where it acts.
F = ½ × 9.81 × 2² × 3 = 58.9 kN. It acts at h/3 = 0.67 m above the bottom.
3. A canal has flow area 4 m² and water speed 2.5 m/s. Find Q.
Q = A v = 4 × 2.5 = 10 m³/s.
4. A pipe of area 0.2 m² narrows to 0.05 m². Water enters at 1.5 m/s. How fast is it in the narrow part?
A₁v₁ = A₂v₂, so v₂ = 0.2 × 1.5 / 0.05 = 6 m/s.
5. A concrete channel is 3 m wide, water depth 2 m, slope 0.002, n = 0.013. Find v and Q.
A = 3 × 2 = 6 m². P = 3 + 2 × 2 = 7 m. R = 6/7 = 0.857 m. v = (1/0.013) × 0.857^(2/3) × √0.002 = 76.9 × 0.902 × 0.0447 = 3.10 m/s. Q = 6 × 3.10 = 18.6 m³/s.
6. How fast does a shallow-water wave travel where the water is 4 m deep?
c = √(g h) = √(9.81 × 4) = √39.2 = 6.26 m/s (about 22 km/h).
Common mistakes
- Using F = ρgh × area for a vertical gate. The pressure is not the same everywhere; use the average, which gives ½ρgh²b.
- Using total depth as hydraulic radius. R = A / P, not h.
- Forgetting to change units. Use metres, seconds and kN with ρg = 9.81 kN/m³.
- Thinking a wider tank gives more pressure. Pressure depends on depth only, not on width or shape.