Water Hammer in a Pipeline

What is modelled

Water flows from a reservoir through a pipe to a valve. When the valve closes the moving water has to stop, the fluid compresses and the pipe walls stretch, and a pressure wave travels up the pipe at the wave speed a (about 1000 m/s in steel, much less in plastic). It reflects at the reservoir and comes back, so the pressure at the valve rings with period 4L/a.

Equations

• Joukowsky: ΔH = a Δv / g, i.e. Δp = ρ a v for a closure faster than 2L/a.
• Method of characteristics for the 1-D equations ∂H/∂t + (a²/g) ∂v/∂x = 0 and ∂v/∂t + g ∂H/∂x + f v|v|/(2D) = 0, solved along C± lines with time step dt = dx/a.
• Valve: orifice law v = k(τ) √(2 g H) with a linear opening schedule; reservoir end: fixed head.
• Slow closure estimate (Michaud): ΔH ≈ 2 L v / (g tc).

Checks run in the page

• Near-instant closure against Joukowsky (5 %).
• The duration of the pressure plateau against the round trip 2L/a.
• A slow closure against the Michaud order of magnitude.

Each check builds a fresh, almost frictionless copy of the solver and compares it with the closed formula.

Honest simplifications

One-dimensional, quasi-steady friction, constant wave speed, no column separation or cavitation model (the page only flags negative head), no surge tank or air vessel, and a rigid linear valve schedule. Real design uses unsteady friction and cavitation models.

• Press Sudden slam and watch the red pressure front run to the reservoir and back.
• Raise the closure time above 2L/a (about 1.1 s here) and watch the peak on the right-hand graph fall.
• Choose Soft plastic pipe: the wave is slower, the surge smaller, the ringing slower.