Bernoulli’s equation

An aircraft flying through the air causes local changes in both velocity relative to the aircraft and pressure. These changes are linked by Bernoulli’s equation. This equation can be written in many forms, but originally it was given by

t) V2

t – + — + gz = a constant

P 2

where z is the height, p is the pressure, p is the density, У is the flow speed, and g is the gravity constant.

The equation is sometimes called Bernoulli’s integral, because it is obtained by integrating the Euler momentum equation for the case of a fluid with con­stant density. Since the equation involves a constant density, it should really only be applied to incompressible fluids. Completely incompressible fluids do not actually exist, although liquids are very difficult to compress. Air is defi­nitely compressible, but nevertheless, airflow calculations using Bernoulli’s equation give good answers unless the speed of the flow starts going above about half the speed of sound. Bernoulli’s equation will also not apply in regions where viscosity is important.

The terms in this equation all have the units of energy per unit mass, and the equation looks temptingly similar to the steady flow energy equation that you will meet if you ever study thermodynamics. The second and third terms do in fact represent kinetic energy per unit mass and potential energy per unit mass respectively. The true energy equation is, however, significantly different, and contains an important extra term, internal energy, which cannot be neg­lected in compressible airflows. However, whatever Bernoulli’s equation is or is not, it remains a useful and simple means for getting approximately correct answers for low-speed flows. Aerodynamicists usually prefer it in the form below:

p + IpV2 + pgz = a constant

This is obtained by multiplying the original equation by the density, which makes all of the terms come out in the units of pressure (N/m2). The last term is usually ignored because changes in height are small in most of the calcula­tions that we perform for airflows around an aircraft, so we write

p + IpV2 = a constant

The first term represents the local pressure of the air, and is called the static pressure. The second term jpV2 is associated with the flow speed and is called dynamic pressure. For convenience it is sometimes represented by the letter q, but in this book we will use the full expression.

If we ignore the third term, as above, Bernoulli’s equation says that adding the first two terms, the static and the dynamic pressure, produces a constant result. Therefore, if the flow is slowed down so that the dynamic pressure decreases, then to keep the equation in balance the static pressure must increase. If we bring the flow to rest at some point, then the pressure must reach its highest possible value, because the dynamic pressure becomes zero. This maximum value is called the stagnation pressure because it occurs at a point where the air has stopped or become stagnant. Using the version of Bernoulli’s equation above, we can write

p + IpV2 = a constant = p (stagnation) + 0 This gives the important result that

Static pressure + Dynamic pressure = Stagnation pressure