Convergent-Divergent Nozzle

Technical Background
The usual configuration for a converging diverging (CD) nozzle is shown in the figure. Gas
flows through the nozzle from a region of high pressure (usually referred to as the chamber) to
one of low pressure (referred to as the ambient or tank). The chamber is usually big enough so
that any flow velocities here are negligible. The pressure here is denoted by the symbol pc. Gas
flows from the chamber into the converging portion of the nozzle, past the throat, through the
diverging portion and then exhausts into the ambient as a jet. The pressure of the ambient is
referred to as the 'back pressure' and given the symbol pb.

A simple example
To get a basic feel for the behavior of the nozzle imagine performing the simple experiment
shown in figure 2. Here we use a converging diverging nozzle to connect two air cylinders.
Cylinder A contains air at high pressure, and takes the place of the chamber. The CD nozzle
exhausts this air into cylinder B, which takes the place of the tank.

Imagine you are controlling the pressure in cylinder B, and measuring the resulting mass flow
rate through the nozzle. You may expect that the lower you make the pressure in B the more
mass flow you'll get through the nozzle. This is true, but only up to a point. If you lower the back
pressure enough you come to a place where the flow rate suddenly stops increasing all together
and it doesn't matter how much lower you make the back pressure (even if you make it a
vacuum) you can't get any more mass flow out of the nozzle. We say that the nozzle has become
'choked'. You could delay this behavior by making the nozzle throat bigger (e.g. grey line) but
eventually the same thing would happen. The nozzle will become choked even if you eliminated
the throat altogether and just had a converging nozzle.

The reason for this behavior has to do with the way the flows behave at Mach 1, i.e. when the
flow speed reaches the speed of sound. In a steady internal flow (like a nozzle) the Mach number
can only reach 1 at a minimum in the cross-sectional area. When the nozzle isn't choked, the
flow through it is entirely subsonic and, if you lower the back pressure a little, the flow goes
faster and the flow rate increases. As you lower the back pressure further the flow speed at the
throat eventually reaches the speed of sound (Mach 1). Any further lowering of the back pressure
can't accelerate the flow through the nozzle any more, because that would entail moving the
point where M=1 away from the throat where the area is a minimum, and so the flow gets stuck.
The flow pattern downstream of the nozzle (in the diverging section and jet) can still change if
you lower the back pressure further, but the mass flow rate is now fixed because the flow in the
throat (and for that matter in the entire converging section) is now fixed too.

The changes in the flow pattern after the nozzle has become choked are not very important in our
thought experiment because they don't change the mass flow rate. They are, however, very
important however if you were using this nozzle to accelerate the flow out of a jet engine or
rocket and create propulsion, or if you just want to understand how high-speed flows work.

The flow pattern

Figure 3a shows the flow through the
nozzle when it is completely subsonic
(i.e. the nozzle isn't choked). The flow
accelerates out of the chamber through
the converging section, reaching its
maximum (subsonic) speed at the throat.
The flow then decelerates through the
diverging section and exhausts into the
ambient as a subsonic jet. Lowering the
back pressure in this state increases the
flow speed everywhere in the nozzle.

Lower it far enough and we eventually

get to the situation shown in figure 3b.
The flow pattern is exactly the same as
in subsonic flow, except that the flow
speed at the throat has just reached Mach 1.
Flow through the nozzle is now choked
since further reductions in the back pressure
can't move the point of M=1 away from the
throat. However, the flow pattern in the
diverging section does change as you lower
the back pressure further.

As pb is lowered below that

needed to just choke the flow a region of
supersonic flow forms just downstream of the
throat. Unlike a subsonic flow, the
supersonic flow accelerates as the area gets
bigger. This region of supersonic acceleration is terminated by a normal shock wave.

The shock wave produces a near-instantaneous deceleration of the flow to subsonic speed. This
subsonic flow then decelerates through the remainder of the diverging section and exhausts as a
subsonic jet. In this regime if you lower or raise the back pressure you increase or decrease the
length of supersonic flow in the diverging section before the shock wave.

If you lower pb enough you can extend the supersonic region all the way down the nozzle until
the shock is sitting at the nozzle exit (figure 3d). Because you have a very long region of
acceleration (the entire nozzle length) in this case the flow speed just before the shock will be
very large in this case. However, after the shock the flow in the jet will still be subsonic.

Lowering the back pressure further causes the shock to bend out into the jet (figure 3e), and a
complex pattern of shocks and reflections is set up in the jet which will now involve a mixture of
subsonic and supersonic flow, or (if the back pressure is low enough) just supersonic flow.
Because the shock is no longer perpendicular to the flow near the nozzle walls, it deflects it
inward as it leaves the exit producing an initially contracting jet. We refer to this as
overexpanded flow because in this case the pressure at the nozzle exit is lower than that in the
ambient (the back pressure)- i.e. the flow has been expanded by the nozzle to much.

A further lowering of the back pressure changes and weakens the wave pattern in the jet.
Eventually we will have lowered the back pressure enough so that it is now equal to the pressure
at the nozzle exit. In this case, the waves in the jet disappear altogether (figure 3f), and the jet
will be uniformly supersonic. This situation, since it is often desirable, is referred to as the
'design condition'.

Finally, if we lower the back pressure even further we will create a new imbalance between the
exit and back pressures (exit pressure greater than back pressure), figure 3g. In this situation
(called 'underexpanded') what we call expansion waves (that produce gradual turning and
acceleration in the jet) form at the nozzle exit, initially turning the flow at the jet edges outward
in a plume and setting up a different type of complex wave pattern.

The pressure distribution in the nozzle

A plot of the pressure distribution along the nozzle (figure 4) provides a good way of
summarizing its behavior.
To understand how the pressure behaves you have to remember only a few basic rules

 When the flow accelerates (sub or supersonically) the pressure drops

 The pressure rises instantaneously across a shock
 The pressure throughout the jet is always the same as the ambient (i.e. the back pressure)
unless the jet is supersonic and there are shocks or expansion waves in the jet to produce
pressure differences.
 The pressure falls across an expansion wave.

The labels on figure 4 indicate the back pressure and pressure distribution for each of the flow
regimes illustrated in figure 3. Notice how, once the flow is choked, the pressure distribution in
the converging section doesn't change with the back pressure at all.