Water Hammer 2D — Pressure Surge from Valve Closure

Pipe

Pipe length L 500 m

Wave speed c 1200 m/s

Flow

Initial velocity v₀ 2.0 m/s

Valve closure time T꜀ 1.00 s

Live Readouts

flow steady — valve open

Critical time 2L/c

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Closure regime

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Joukowsky ΔP (full)

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Expected peak ΔP

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Live valve pressure

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Recorded peak pressure

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How it works

Water hammer: closing a valve suddenly stops the fluid at the valve, but the fluid further up the pipe keeps moving until a pressure wave — traveling at the pipe's wave (celerity) speed c — arrives and stops it too. The wave bounces between the closed valve and the open reservoir every 2L/c seconds.

ΔP = ρ·c·Δv                (Joukowsky surge, instantaneous closure)
T_cr = 2L/c                 (critical time — one round trip)
ΔP_slow ≈ ΔP · (T_cr / T_c)  (closure slower than T_cr reduces the peak)

If the valve closes faster than the critical time T_cr = 2L/c ("rapid closure"), the full Joukowsky surge develops before the relief wave from the reservoir can return — the pipe sees the worst-case pressure. Closing slower than T_cr ("slow closure") lets the relief wave partially cancel the surge before the valve finishes closing, so the peak pressure is reduced roughly in proportion to T_cr/T_c. This simulation solves the pipe with the method of characteristics (ρ=1000 kg/m³ water), so the surge, the 2L/c reflection period and its gradual decay from friction all emerge from the same equations rather than being drawn in by hand.

Overpressure (surge) Underpressure