Classifying fluid flows with examples

Fluid flow is classified by how density, the motion of particles and the flow properties behave in space and time.

Compressible versus incompressible flow

• Compressible flow: density changes from point to point, i.e. $\rho \neq$ constant. Examples: flow of gases at high speed, air in a compressor or nozzle, flow of steam in a turbine.
• Incompressible flow: density is (practically) constant, $\rho =$ constant. Liquids are almost always treated as incompressible. Examples: water flow in a penstock, river flow, oil in a pipe; air at low speed (Mach number below about 0.3).

Laminar versus turbulent flow

• Laminar flow: particles move in smooth, parallel layers (laminae) that slide over each other without mixing. Viscous forces dominate. In pipes $Re < 2000$. Examples: flow of thick oil in a thin tube, blood flow in capillaries, groundwater flow.
• Turbulent flow: particles move in a random, zig-zag way with eddies and strong mixing. Inertia forces dominate. In pipes $Re > 4000$. Examples: flow in rivers and canals, water in a penstock, smoke rising far from a chimney.
• Reynolds number: $Re = \dfrac{\rho V D}{\mu} = \dfrac{VD}{\nu}$; between 2000 and 4000 the flow is transitional.

Steady versus unsteady flow

• Steady flow: flow properties (velocity, pressure, density) at a point do not change with time:

$$\frac{\partial V}{\partial t} = 0,\quad \frac{\partial p}{\partial t} = 0,\quad \frac{\partial \rho}{\partial t} = 0$$

Examples: flow through a pipe from a large reservoir at constant level; constant discharge in a canal.

• Unsteady flow: properties at a point change with time, $\dfrac{\partial V}{\partial t} \neq 0$. Examples: flood flow in a river, flow during valve closure (water hammer) in a penstock, emptying of a tank.