Joukowsky Equation (Water Hammer)

When flow is suddenly stopped, kinetic energy converts to pressure. The Joukowsky equation calculates the maximum pressure rise:

dP = rho * c * dV

• dP = Pressure spike (Pa)
• rho = Fluid density (kg/m3)
• c = Speed of sound in fluid (m/s)
• dV = Velocity change (m/s)

Speed of Sound in Hydraulic Fluid

c = sqrt(B / rho)

For mineral oil: c is approximately 1200-1400 m/s

With entrained air, effective speed of sound decreases significantly.

Closure Time Effect

If valve closure time is longer than the critical time (2L/c), the spike is reduced:

dP_actual = dP_max * (tc / t_closure) for t > tc

Where tc = 2L/c (critical closure time)

Inertia Effects

Moving masses create pressure spikes when decelerated:

dP = (M * a) / A = (M * dV) / (A * dt)

Prevention Strategies

• Use proportional or soft-shift valves
• Install accumulators near quick-acting valves
• Add flow controls or deceleration ramps
• Use cylinder cushions for end-of-stroke