The focus of this lesson will be on analyzing compressible gas flow through converging-diverging
nozzles. Also known as De-Laval Nozzles, these are devices that are used to accelerate fluids
to very high speeds. CD nozzles are commonly used in propulsion systems of supersonic aircraft
and spacecraft, and in supersonic wind tunnels. A CD nozzle is shaped such that the cross-sectional
area varies with axial distance along the length of the nozzle, as shown in the image
here. The shape of a CD nozzle resembles that of an hourglass. There is a minimum area section
called the throat, which is preceded by a converging section, and succeeded by a diverging
section. Using continuity and isentropic relations, we can obtain the following relation for the
variation of Mach number with nozzle cross-sectional area. This equation is called the area-mach
number relation. Notice the presence of a variable called A* which is the sonic throat
area. From this equation, it is clear that the Mach number at any location in the nozzle
is a function of the ratio of the local duct area to the sonic throat area. Here is a plot
of area-mach number relation. If you notice, there are two values of Mach number that satisfy
the equation for any given area ratio, a subsonic value, and a supersonic value.
Let us now understand a generic 1-D flow through the CD nozzle. Consider a CD nozzle with a
very high area ratio at the inlet. The inlet station feeds from a large reservoir operating
at stagnation conditions of p0, rho0, and T0. When the ambient or back pressure at the
exit of the nozzle is small enough, a supersonic flow will be established. In the converging
section, the flow is always subsonic and the exact value depends on the local area ratio.
At the throat, the area ratio is 1 i.e. At=A* and therefore the flow attains a sonic velocity.
This sonic flow then enters the diverging section where it is further accelerated. The
variation of pressure, temperature, and Mach number through the nozzle is shown here. Notice
that pressure and temperature decrease whereas the Mach number increases, monotonically.
The variables with subscript e refer to the conditions at the nozzle exit. Except for
two special conditions which we will discuss later, the exit pressure will always be equal
to the back pressure. For our discussion, we will be using the two terms interchangeably
and make the distinction when necessary. Let us now analyze the flow behavior at different
values of the pressure ratio. Here, the pressure ratio is referred to as the ratio of the exit
pressure to the inlet total pressure. For our analysis, we will consider the setup shown
here where the CD nozzle is connected to a tank supplying the inlet section with air
at a fixed total pressure and temperature. A valve attached to the system beyond the
nozzle exit is adjusted to vary the back pressure. Initially, let us assume that the back pressure
is equal to the total pressure at the inlet. At this condition, no gas flow is possible.
Let us now reduce the back pressure slightly such that pb=pe=peA. At this condition, a
low speed flow is established inside the nozzle. As the back pressure is further decreased,
the flow is accelerated more, leading to a higher Mach Number throughout the nozzle,
with the highest being at the throat. In the diverging section beyond the throat, the flow
is decelerated. The acceleration of the flow continues up to a back pressure, say pb=pe=peC,
where the flow becomes sonic at the throat i.e. M=1 and At=A*. At this value of exit
pressure, the mass flow rate through the nozzle attains its maximum value. Even with any further
reduction of the exit pressure, the Mach number at the throat will always be 1.0 and the mass
flow rate will remain constant. This situation is referred to as a choked flow. For a calorically
perfect gas at gamma=1.4, the sonic flow at the throat corresponds to a throat pressure
ratio i.e. p*/p0 of 0.528. Now, if the back pressure is decreased below
the value of peC, there will not be any change in the flow characteristics in the converging
section. For values of back pressure less than peC but greater than design condition,
which we will refer to as pb=pe=peD, a normal shock is formed in the diverging section of
the CD nozzle. The flow in front of the shock is supersonic, while behind the shock it is
subsonic. The plots here show the representative pressure and mach number distribution for
the condition where a normal shock exists in the diverging section. Before the shock,
the mach number of the flow increases through the diverging section, and subsequently, the
pressure reduces. Beyond the shock, the flow becomes subsonic and starts decelerating as
it flows through the rest of the nozzle, while the pressure dramatically increases across
the shock and keeps increasing till the nozzle exit, where it becomes equal to the prescribed
back pressure, peD. If the value of peD is closer to peC, the normal shock is located
closer to the throat of the nozzle. As peD is decreased, the shock wave moves downstream
and closer to the nozzle exit. At a certain value of back pressure, the shock sits exactly
at the nozzle exit. At this condition, the nozzle exit pressure and the back pressure
will become distinct. The back pressure, say peE, is the static pressure behind a normal
shock at the design Mach number of the nozzle. In this figure, peF represents the proper
isentropic value for the design condition and it exists just upstream of the normal
shock. From this point on, the nozzle exit pressure remains unchanged at peF, irrespective
of the back pressure. Since we know that peF is the design condition exit pressure, let
us look at how the flow behaves for back pressure values between peE and peF. As soon as the
back pressure is reduced below peE, the shock wave moves out of the nozzle, and the flow
inside the nozzle is fully supersonic and isentropic. However, since the exit pressure
(peF) is still less than the back pressure, oblique shock waves are formed at the nozzle
lip through which the gas pressure is increased. In such a case, the nozzle is said to be overexpanded.
On further reduction of exit pressure such that pb=peF, we arrive at the design condition
of the nozzle. At this condition, there is a perfect supersonic expansion and no shock
waves are present either inside or outside the nozzle. For this condition, the back pressure
and exit pressure will match identically. Now, if the back pressure is further reduced
below peF, the nozzle is said to be underexpanded and in order for the exit pressure (peF) to
be made equal to the back pressure, expansion waves are generated at the nozzle lip
Till now, we have analyzed the qualitative flow behavior in a CD nozzle at different
pressure ratios. It is now time to understand the overall process quantitatively. Let us
begin by assuming that the flow through the nozzle is entirely isentropic. As a result,
the total conditions can be considered constant throughout the nozzle. For a known value of
exit pressure p2, the following expression can be written using the isentropic relations,
from which exit Mach number M2 can be calculated. Using A2 and M2, we can calculate A* from
the area ratio equation. This calculated value A* can be compared with the actual cross-sectional
area of the throat, At. If At is greater than A*, it indicates that the throat is too large
for a sonic flow and thus the entire nozzle flow is subsonic. If At is less than A*, it
indicates that the throat is too small for subsonic flow and thus the nozzle is choked.
Let us now look at each of these conditions in more detail. When the flow is fully subsonic
in the nozzle, using A* as the reference quantity, we calculate the area ratio at any location
in the nozzle. From Mach number – area ratio relation, we obtain the corresponding flow
Mach number at this location. Knowing the total properties and the Mach number, it is
straight forward to calculate all the flow properties and the mass flow rate. For the
condition where the throat is choked, i.e. At=A*, we know that the flow in the converging
section is subsonic and isentropic. Hence, we can use the same procedure as we just discussed
to obtain the flow properties at any location in the converging section. For the diverging
section however, things are a little different. There are two possibilities based on the prescribed
back pressure. 1. The flow at the nozzle exit is subsonic and 2. The flow at the nozzle
exit is supersonic. In the case of a subsonic exit flow, as discussed earlier, there must
be a normal shock wave in the diverging section. For a supersonic flow at the exit, we first
compute the design condition exit Mach number using the exit area ratio A2/A*. For this
Mach number, we can compute the exit pressure, let us say p2’, using isentropic relation
for pressure ratio. If we compare the value of p2’ to our prescribed back pressure,
we encounter two scenarios. If p2 is less than p2’, the flow needs further expansion,
in order for the exit and back pressures to match, which happens outside the nozzle. This
is the underexpanded condition that we discussed previously. If p2 is greater than p2’, there
must be a normal shock in the diverging section and the flow must pass through this shock
and further diffuse subsonically to the prescribed back pressure. In this case, there is one
problem though. How do we determine the location of the normal shock? Well, we have to use
a trial and error strategy. Let us look at this using the following flowchart.
We first assume a location for the normal shock in the diverging section of the nozzle.
Let us consider the location just in front of the normal shock as A and just behind the
normal shock as B. Since the flow from the nozzle inlet to point 𝐴 is isentropic,
we know the total properties at 𝐴. From the area-ratio equation we can calculate 𝑀𝐴
using the supersonic solution. Using 𝑀𝐴 and the total properties at 𝐴, we then
calculate the static properties at 𝐴 (𝑝𝐴, 𝑇𝐴, 𝜌𝐴). We can now use the normal
shock wave relations to obtain the static properties at location B from the known values
of MA and static properties at A. We can obtain the total quantities at B which will prevail
through the remainder of the nozzle (where the flow is isentropic). Using MB and the
area-mach number relationship, we can calculate a new reference critical area Ab*. This critical
area will also prevail from the downstream side of the shock to the exit. Using the exit
area 𝐴2, we calculate 𝐴2/𝐴b* and then obtain a provisional subsonic 𝑀2,
again using the area-ratio equation. Finally, using the isentropic relation for the total
to static pressure ratio, we solve for a provisional exit pressure 𝑝2′. If 𝑝2′ is greater
than the prescribed exit pressure 𝑝2, the shock must be closer to the throat and the
assumed axial position 𝑥 is reduced and rerun the analysis. Otherwise, we increase
𝑥 and rerun the analysis. A small increment Δ𝑥 is chosen for these adjustments.
That brings us to the end of this lesson