Define water hammer; continuity and momentum equations

Water hammer

Water hammer is the rise and fall of pressure in a pipe caused by a sudden change of flow velocity, such as the quick closing or opening of a valve or turbine gate, or a pump failure. The kinetic energy of the moving water changes into elastic (strain) energy of the water and the pipe wall. A pressure wave travels along the pipe at celerity $c$ and is reflected at the ends. It may burst or collapse the pipe and causes a hammering noise.

Equation of motion (momentum / Euler's equation)

Newton's second law for a water element in a pipe of diameter $D$, with $x$ along the pipe, velocity $V$ and piezometric head $H$:

$$\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial H}{\partial x}+\frac{fV|V|}{2D}=0$$

For water hammer, the convective term $V\,\partial V/\partial x$ is small compared with $\partial V/\partial t$ and is neglected:

$$\frac{\partial V}{\partial t}+g\frac{\partial H}{\partial x}+\frac{fV|V|}{2D}=0$$

Without friction, and for a horizontal pipe, this becomes the Euler equation $\dfrac{1}{\rho}\dfrac{\partial p}{\partial x}+\dfrac{\partial V}{\partial t}=0$.

Continuity equation

Mass conservation, including the compressibility of water (bulk modulus $K$) and the elasticity of the pipe:

$$\frac{1}{\rho}\frac{dp}{dt}+c^2\frac{\partial V}{\partial x}=0\quad\text{or}\quad \frac{\partial H}{\partial t}+\frac{c^2}{g}\frac{\partial V}{\partial x}=0$$

where $c=\sqrt{\dfrac{K/\rho}{1+KD/(Et)}}$ is the wave celerity.

These two equations (momentum and continuity) are solved together, e.g. by the method of characteristics, to find $H$ and $V$ along the pipe at any time.