The most common applications of compressible internal flow problems are often represented by
geometries with varying cross-sectional areas. Propulsion systems or advanced wind tunnels
used in research facilities, take advantage of the variations of their cross-sectional area
in order to accelerate subsonic flows to supersonic speed or decelerate
supersonic flows to a subsonic state. In this lesson we will analyze the underlying relations
that describe and govern these kinds of flows. Let’s explore quasi 1D flows.
Quasi 1-D flows are very similar to the 1D flows we analyzed before.
We assume the flow is compressible and steady. The gas behaves as an ideal gas and its property
variations are assumed to be isentropic, except across shockwaves. Viscous effects are neglected,
the solid boundaries are adiabatic and there is no work done to or by the fluid.
The main difference between 1D and Quasi 1D flows is the cross-sectional area of the passage.
For quasi 1D flows, it is no longer constant, but is a function of the x location.
The properties of the flow are assumed to be uniform across the passage and only vary in the x direction.
The general form of the governing equations for quasi 1D flows does not differ from the one we
have for 1D flows. We have mass, momentum and energy conservation, as well as the relations for
ideal gas and isentropic flow. The only thing to remember is that when applying the mass equation
between two locations, we cannot simplify the area term anymore.
We can also recall the isentropic relations of the stagnation quantities, for our convenience.
The continuity equation and the isentropic relations can be combined to obtain an expression
for the mass flow rate. As we can see, the mass flow rate can be increased if we increase the flow
total pressure and the section of the passage, or if we decrease the total temperature of the fluid.
Often, when discussing quasi 1D flows, it is assumed that the incoming fluid is stored in
a pressurized tank at stagnation conditions. Then, the fluid accelerates isentropically
to the inlet of the duct, so that the stagnation conditions are still valid at the inlet.
Now, let's explore one of the most important relations of quasi 1D
flows - the area velocity relation. In the differential form of the continuity equation
we have three terms: the density one, the velocity one and the one related to the area.
Now, using the momentum equation in its differential form and the definition
of speed of sound based on isentropic assumptions, we can find an expression for the density term.
Substituting the expressions into the continuity equation we obtain the final
area-velocity relation. We can see that area changes are proportional to velocity changes and,
depending on the Mach number, the proportionality can be direct or inverse. Let's analyze this.
If we consider a subsonic flow, we can see that the proportionality coefficient will be negative.
This means that for a subsonic flow moving through a converging geometry,
the velocity will increase, while a subsonic flow in a diverging geometry would decelerate. This
is the reason why when we blow on something, we contract our mouth to accelerate the air,
instead of just blowing with open mouth. We can also test the opposite trend simply blowing air
in an open cone, the air exiting the cone would be slower than just blowing it with your mouth.
If the Mach number is larger than one, we have a supersonic flow. In this case
the coefficient is always positive, meaning that when we increase the area of a duct
the flow will accelerate and if we decrease the area of the duct the flow will decelerate instead.
This may not be immediate to grasp, since we do not commonly deal with supersonic flows in our
everyday lives. Here is a simple analogy to remember this. Think about a highway with
road works. As the lines available get reduced the cars slow down, like the supersonic flow
would in a converging geometry. Past the roadworks the cars accelerate as the road opens up and the
numbers of lines increases back, exactly like a supersonic flow would in a diverging duct.
The area velocity relation and its implications are really what is “behind the scenes” of
supersonic wind tunnels and propulsive systems. If we take those pieces
and combine them together we can achieve something that those pieces alone
would not. If we use these configurations, we can accelerate a subsonic flow
into a supersonic flow and decelerate a supersonic flow into a subsonic one.
Let’s now look at another aspect: the critical state. Let's assume a subsonic flow going through
a converging nozzle. As the flow accelerates, it will eventually reach Mach equal to 1. In this case,
we have reached the maximum mass flow rate that can go through the passage
and the flow reached a critical state. Since the flow has reached sonic conditions
the state quantities and the area of the passage can be denoted by using the star superscript.
The energy equation can be used to obtain an expression for the velocity,
by combining the definition of enthalpy and the isentropic relations.
As expected, the lower the pressure, the higher will be the velocity.
We can also introduce a new variable that defines the mass flux to the passage
by simply dividing the mass flow rate by the area. Now, assume that the minimum area
is at the end of this converging nozzle and let's call this location as the throat of the duct.
By the definition of mass flux,
for a given mass flow rate the maximum mass flux will occur at the smallest section of the duct.
Let's denote the flow properties of the throat with the subscript T.
Then doing the logarithmic differentiation of the mass flux expression and dividing by dx,
we can achieve an interesting conclusion. The derivative of the mass flux is zero at the throat,
also combining this equation with the momentum equation and the assumption of isentropic flows,
we find that the velocity at the throat is equal to the speed of sound.
This means that, if the converging section of the duct is small enough so that the flow
reaches the speed of sound, then the throat becomes the point of maximum
mass flux. This physical effect is called choking and the mass flow rate
at this condition location is the maximum that can be achieved in the geometry for a given stagnation
condition. We cannot exceed the mass flow rate that we achieve in choking conditions.
With the throat at sonic conditions we can easily estimate the fluid temperature,
pressure and density using the sonic relations.
Also, the expression we found before for the mass flow rate can be simplified using Mach equal to 1.
The mass flow rate for a chocked flow is function of the total conditions
and the choking area. This equation can be useful to estimate the mass flow rate
in a duct using choked converging passage nozzles. The mass flow rate can be found if we measure the
upstream total pressure and total temperature using a static tube and a thermocouple, assuming
that the nozzle throat area is known. This approach is valid as long as the flow is chocked.
We can use the continuity equation to find that the ratio of the duct
area to the critical throat area is a function of the Mach number. As the Mach
number goes to 1 the area ratio goes to 1 as well. Now why is this relation so important?
Using this equation we can estimate the local Mach number at each section of known geometry.
Assume we know the passage geometry, hence A(x), the inlet conditions or stagnation properties
and the Mach number. We can then calculate the value of A*. Using the ratio of A/A*, for a
specific location in the duct, we can extrapolate the Mach number using tables or numerical methods.
Then we can use the isentropic relations to estimate all the flow properties based
on the inlet total conditions, since the flow is isentropic.
The critical area is based on the inlet conditions, but the actual passage can
present a minimum area that is larger than A*. In that case the flow will not choke at the throat.
On the other hand, if the minimum area is smaller than A*, then the Mach number
prescribed will not be possible. The throat will reach sonic conditions and the inlet Mach number
will change accordingly. Let’s analyze this in detail.
From the continuity equation we can obtain this relation between Mach and area variations.
Let's start with a converging passage. If we have a subsonic inflow, we would
have an area variation less than 0 and a Mach variation larger than 0,
hence the exit Mach number will be larger than the inlet Mach Number. For a supersonic inflow
the Mach variation is negative hence the Mach decreases along the channel.
This means that a converging passage is a nozzle for a subsonic inflow and a diffuser
for a supersonic inflow. Let's assume now that we have a sonic inflow and a supersonic outflow.
In this case the area variation is still less than zero and the Mach variation would be negative too,
meaning that the Mach at the end of the throat will be less than the inflow,
hence this situation is impossible. Now, if we assume to have a sonic inflow and
a subsonic outflow, we would have a Mach variation larger than zero
so the flow at the exit would be faster than the inlet , and this again is impossible.
The conclusion is that a Sonic flow cannot possibly enter a converging passage.
Now we can analyze a diverging passage, hence the area variation would be larger than zero.
If we have a subsonic inflow the Mach variation would be less than zero,
meaning that the flow decelerates along duct. On the other hand, with a supersonic inflow the Mach
variation is larger than 0 meaning that the Mach number increases through the duct. The conclusion
is that the diverging passage is a diffuser for a subsonic inflow and a nozzle for a supersonic
inflow. Now let's see what would happen if we have a sonic inflow. In this case both supersonic and
subsonic outflows would be possible, as verified by the Mach variation for each scenario. Hence a
sonic flow can enter a diverging passage and the outflow can be either subsonic or supersonic.
This concludes our lesson on the analysis of quasi 1D flows.