Bernoulli Equations

The Equations

Some of the equations we have discussed are posed in terms of state variables that do not include pressures. In these cases (e.g. the potential flow equations) the differential equations and boundary conditions allow one to compute the local velocities, but not the pressures.

Once the velocities are known, however, the momentum equation can be used to find the local pressure. Such equations are known as Bernoulli equations and they come in various forms, depending on the assumptions that can be made about the flow.

The conservation of momentum principle is the source of the relation between pressure and velocity. It can be used very simply to derive the Bernoulli equation.

To illustrate the basic physics behind the Bernoulli equations, we can derive a simple form: that for steady, incompressible flow.

In this case we show that along a streamline:

When the flow is not steady, the Euler equations can be integrated to obtain a more general form of this result: Kelvin's equation, the Bernoulli equation for irrotational flow.

Where f is a body force per unit mass (such as gravity) and F is an arbitrary function of time.

If we do not assume that the flow is irrotational, we cannot introduce the potential and the expression is not so nicely integrable. If, however, we assume that the flow is steady with no "body forces", but not necessarily irrotational we can write the following expression that holds along a streamline:

While the above equations hold for steady flows along a streamline, for irrotational flows they hold throughout the fluid.

We can derive a more useful form of the Bernoulli equation by starting with the expression for steady flow without body forces shown just above.

If the flow is assumed to be isentropic flow (no entropy change or heat addition): p = constant * ργ

Substitution yields the compressible Bernoulli equation:

This actually works for adiabatic (no heat transfer) flows as well as isentropic flows.

In summary, we often deal with one of two simple forms of the Bernoulli equation shown below.

The Pressures

In both the incompressible and compressible forms of Bernoulli's equation shown above there are 3 terms. The quantity pT is the total or stagnation pressure. It is the pressure that would be measured at points in the flow where V = 0. The other p in the above expressions is the static pressure.

Note that in incompressible flow, the speed is directly related to the difference in total and static pressure. This can be measured directly with a pitot-static probe shown below.

The dynamic pressure is defined as:

The static pressure coefficient is defined as:
where p is the freestream static pressure.

In incompressible flow, the expression for Cp is especially simple:

If the local velocity is expressed as a small perturbation in the freestream:
Then the incompressible Cp relation can be written:
Be careful with this expression! It is often not a good approximation and the correct expression is not very difficult.

The expression for Cp in compressible isentropic flow (sometimes called the isentropic pressure rule) is derived from the compressible Bernoulli equation along with the expression for the speed of sound in a perfect gas. In terms of the local Mach number the expression is:

In air with gamma = 1.4

Some interesting results follow from this expression...

We can tell if the flow is supersonic, just by looking at the value of Cp. The critical value of Cp, denoted Cp* is found by setting M = 1 in the above expression:

Also, we see that there is a minimum value of Cp, corresponding to a complete vacuum. Setting the local Mach number to infinity yields:

Cp cannot be any more negative than this. Experiments show that airfoils can get to about 70% of vacuum Cp. This can limit the maximum lift of supersonic wings.

Freestream Mach:
Local Mach: