Joukowskyligningen

Joukowskyligningen, commonly known in English as the Joukowsky equation, is a foundational relation in fluid dynamics that describes the instantaneous pressure change in a liquid caused by a sudden change in flow velocity, most notably in pipelines experiencing water hammer. The equation is typically written as p2 − p1 = ρ c (V1 − V2), or equivalently Δp = ρ c ΔV, where p1 and p2 are the fluid pressures before and after the change, ρ is the fluid density, V1 and V2 are the velocities, and c is the speed of sound in the liquid inside the pipe. The signs indicate that a rapid decrease in velocity produces a pressure rise.

Derivation and assumptions: The Joukowsky equation emerges from one-dimensional water hammer theory. It assumes a long, straight pipe filled with a liquid of finite compressibility, with a sudden boundary condition change (such as valve closure or pump trip). Friction, large nonlinearity, and complex three-dimensional effects are neglected in the basic form. The resulting disturbance travels as a pressure wave at speed c and converts a velocity change into a surge pressure.

Applications: It is widely used in hydraulic engineering to estimate surge pressures during valve closures or pump trips, informing the design of surge tanks, air chambers, and pressure-relief devices. It provides a quick, first-order estimate of maximum surge pressures in water supply networks, wastewater systems, and other piping systems.

History and naming: The relationship is attributed to Nikolai A. Joukowsky (Nikolai Andreevich Joukowsky), a Russian engineer and physicist, who developed the concept in the late 19th to early 20th century as part of early studies on transient flow in pipes.

Limitations: Real systems involve friction, nonlinearity, variable flow conditions, and complex valve dynamics. In practice, the basic equation is augmented with corrections for friction losses and multi-dimensional effects to improve accuracy.