The Bernoulli equation

The Bernoulli equation is the first line integral of the equation of motion. The equation can take various forms, depending on the particular kinematical or dynamical assumptions relating to the motion. The equation

The physical meaning of this result is particularly important; it is the well-known Bernoulli theorem. The total mechanical energy content of a unit mass of fluid is constant along any streamline or vortex line of a steady barotropic flow,

arbitrary horizontal plane at which the potential energy of the fluid is zero, then the potential energy of the fluid varies directly with the distance above the arbitrary plane. If h is the distance above the chosen plane, the potential energy of a unit mass of fluid is

These equations all relate to an arbitrary point in the flowing gas. To evaluate the constant on the right-hand side of the equation, a reference point or a reference state must be chosen. This reference point may be that where the gas is at rest. Thus any gas flow may be considered to be a flow coming from an infinite reservoir. The conditions in this reservoir are denoted by the subscript 0. The flow variables of this fictitious reservoir are identical to those at any point of the system at which the velocity is zero. Thus these variables are called reservoir or stagnation variables. For example the constant of Eq. (4.42) is the stagnation enthalpy. The stagnation pressure, density, temperature, etc. are defined in a similar manner. Such a stagnation point occurs at the nose of an immersed body, where the flowing fluid is brought to rest and the kinetic energy of the fluid is thus converted into enthalpy. This stagnation enthalpy is equal to the enthalpy in the reservoir irrespective of the location of the stagnation point. (Note, that this statement is valid only for isentropic flow.) The stagnation pressure can be measured by a Pitot tube. This measuring device consists of three main parts as shown in Fig. 4.4. The head is placed in the flow with its nose directed in the opposite direction to the velocity. At the nose the fluid is

brought to rest, and a stagnation point occurs. This point is connected to a pressure-measuring device; commonly an U-tube manometer. Thus the manometer measures the stagnation pressure.

The development of the stagnation point is an isentropic process. The stagnation enthalpy is equal to the reservoir enthalpy. Thus the temperature read by a stationary thermometer is equal to the temperature read in the reservoir. For a thermometer to measure the temperature of the flowing gas it would be necessary for it to move at the same velocity as the gas. This would be an impractical measuring technique, thus the temperature of the flowing gas is determined indirectly, applying a kinetic energy correction

For supersonic flows the Pitot tube is unsuitable for determining stagnation parameters.