Penstock Water Hammer Simulator

Penstock Setup

Reservoir head H₀ 120 m

Penstock length L 220 m

Pressure-wave speed a 1200 m/s

Steady flow velocity V₀ 3.5 m/s

Valve closing time T_c 2.0 s

Surge tank enabled

Live Readout

Valve opening

100%

Head at valve

120.0 m

Peak surge above H₀

+0.0 m (0%)

Wave travel time 2L/a

0.37 s

Surge tank level z

0.0 m

Closure regime

—

How it works

Water flowing down a penstock carries momentum. When the turbine's guide vanes close, that momentum can't vanish instantly — the flow deceleration launches a pressure wave that travels up and down the pipe at the elastic wave speed a (set by the fluid's compressibility and the pipe wall's stiffness, typically 900–1400 m/s for steel).

Joukowsky equation (instant closure):
  ΔH = a·ΔV / g

Rapid closure  if  Tc < 2L/a  → full Joukowsky surge
Slow closure   if  Tc > 2L/a  → surge reduced roughly by 2L/(a·Tc)

The pipe is solved with the classic method of characteristics (Wylie & Streeter): discretise the penstock into reaches of length Δx = L/N and step time by Δt = Δx/a. Head H and flow Q at each interior node come from two characteristic equations:

C+:  H = C_P − B·Q   (from the upstream node)
C−:  H = C_M + B·Q   (from the downstream node)
B = a / (g·A)   — pipe's hydraulic impedance

At the closing valve the two equations are combined with the orifice law Q ∝ τ(t)·√H to solve for the new head and flow every step, which is what produces the pressure spike you see colour the pipe red.

A surge tank — an open vertical shaft between a long low-pressure tunnel and the short high-pressure penstock — protects the penstock by absorbing the rejected flow as a slow, undamped mass oscillation (rigid water-column theory) instead of letting the full elastic shock reach the valve:

d z/dt  =  (q_tunnel − q_penstock) / A_surge
d q_tunnel/dt = (g·A_tunnel / L_tunnel) · (−z − friction)

Toggle the surge tank off and close the valve fast to see the unmitigated Joukowsky spike; turn it back on to see the same closure produce a much smaller surge at the turbine, at the cost of a slow oscillation in the tank itself — exactly the trade-off real hydro plants are engineered around.

Below static head Static head H₀ Surge (above H₀)

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