Pressure, speed, and height relationship

Summary

Bernoulli’s principle describes how pressure, speed, and height (elevation) are related in fluid flow. For a steady, horizontal, incompressible flow with negligible viscous effects, an increase in fluid speed corresponds to a decrease in static pressure, and vice versa. In general, within a flow region where the total mechanical energy per unit mass is constant, the sum of kinetic energy (from speed), potential energy (from height), and pressure energy remains unchanged along a streamline. In its common incompressible form, Bernoulli’s equation is written as v^2/2 + gz + p/ρ = constant, where v is speed, g is gravitational acceleration, z is elevation, p is static pressure, and ρ is density. This shows the trade-off: if elevation and density effects are small or constant, then higher speed must come from lower pressure; if speed decreases without a height change, it must be due to an increase in static pressure. The principle is also often expressed using “heads” (velocity head, elevation head, and pressure head), which makes the pressure–speed–height relationship especially clear for engineering applications.

Key Points

• 1

  In steady, incompressible, low-viscosity flow, higher speed occurs where static pressure is lower, and lower speed occurs where static pressure is higher.

• 2

  Bernoulli’s equation links speed, elevation, and pressure through conservation of energy: v^2/2 + gz + p/ρ = constant along a streamline (under applicable assumptions).

• 3

  The relationship can be expressed as a constant sum of “heads” (velocity head, elevation head, and pressure head), showing how changes in one term require compensating changes in the others.