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 * ρ^{γ}

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.

In both the incompressible and compressible forms of Bernoulli's equation
shown above there are 3 terms. The quantity p_{T} 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 C_{p} is especially simple:

If the local velocity is expressed as a small perturbation in the freestream:

Then the incompressible C_{p} 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 C_{p} 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 C_{p}.
The critical value of C_{p}, denoted C_{p}* is found by setting M = 1 in the above
expression:

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

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

Freestream Mach: | |

Local Mach: | |

Cp: |