When a supersonic flow encounters a concave corner, we know that this causes the flow
to turn into itself.
This leads to the formation of an Oblique shock.
But what if the flow encountered a convex corner?
In this case, the supersonic flow would effectively be turning away from itself.
This interaction leads to the formation of Expansion Waves, and in this lesson, we will
try to understand the physics behind it. So let's get started.
Expansion waves are formed when a supersonic flow turns in such a way that it expands in
the direction of the flow.
This is exactly opposite of the situation in which the flow turns into itself leading
to the formation of oblique shocks.
As can be seen in this Mach number contour plot, when a Mach 2.5 flow encounters this
cone the leading edge essentially acts as a convex corner.
We can see that this encounter results in the formation of an oblique shock.
Across the shock, the Mach number decreases, i.e. the flow decelerates, and the static
flow properties increase.
Now, as the flow moves further downstream it encounters a convex corner.
This creates a system of expansion waves.
As the flow expands, the Mach number of the flow increases, i.e. the flow accelerates,
and the static flow properties decrease.
So if you think about it, expansion waves are basically the opposite of oblique shocks.
Let’s look at the contour plot again – Do you see something strikingly different between
the oblique shock and the expansion wave? Did you notice that unlike a shock, expansion
region is continuous, and the properties vary relatively smoothly and continuously, but without
any abrupt changes.
For an abrupt corner as shown here, a group of expansion Mach waves is formed such that
the flow turns and becomes parallel to the downstream wall.
Expansion waves are often called expansion fans.
Each line in the expansion fan represents a line of constant properties.
In fact, there are an infinite number of waves originating from the corner, and each wave
turns the flow by an infinitesimal amount, and accelerates the flow by an infinitesimal
amount as well, this leads to a differential model.
Now that we have some physical understanding of expansion waves, let us try to formulate it
by deriving the governing equations for a planar expansion wave.
Consider a single Mach wave for a flow being turned through a differential angle d-theta.
From geometric and trigonometric considerations, we can get the following equation.
Now, for a small d-theta this can be simplified to the following expression.
If we now use Taylor series expansion up to the first order term and use the relationship
between the mach angle and the mach number of the given Mach wave, we can get the ‘Prandtl-Meyer Equation’.
This is a differential equation relating the change in the flow angle, dθ, to the corresponding
change in the flow velocity dV as the flow turns around this corner.
Keep in mind that this equation was derived strictly on the basis of geometric reasoning
and the definition of the mach wave.
And therefore, it is a general equation which can be applied to ideal gases, real gases
or even chemically reacting gases!
A solution for the expansion wave can be obtained by integrating this equation.
We can simplify this to obtain an explicit expression for an ideal gas by using these relations.
The resulting equation is called the ‘Prandtl-Meyer Function’ and is represented using the Greek
symbol ν (‘nu’).
Using this function, we can express the deflection angle θ2 as the difference of Prandtl-Meyer
functions corresponding to M2 and M1.
We can now calculate the flow properties across a Prandtl-Meyer expansion wave by following
these steps.
First, we determine ν(M1) from the Prandtl-Meyer function using the known upstream Mach value of M1
Since we know the turning angle θ2, we can then calculate ν(M2) from this equation.
Once we have ν(M2), we can calculate M2 that is the Mach number downstream from the expansion fan using the Prandtl-Meyer function.
One thing to keep in mind here is that the calculation of Mach number for nu would require
inversion of this non-linear equation.
Therefore, this calculation is usually carried out using a numerical technique.
The Mach number can also be obtained from tabulated or plotted representation of ν vs M.
Since the expansion wave is assumed to be isentropic, the total pressure and total
temperature are conserved across the expansion and thus we can calculate their static values
downstream of the expansion using these relations.
As we had discussed earlier, the Prandtl-Meyer function was derived using the definition
of a Mach wave and geometric considerations.
We never actually enforced any condition that explicitly stated that the flow has to be expanding.
Therefore, the Prandtl-Meyer function holds in all situations where the flow is turning
isentropically through an envelope of mach waves – irrespective of whether the flow
is expanding or compressing.
When we discussed oblique shocks, we did consider a situation of supersonic flow compression
through an envelope of Mach Waves over a curved corner, but we did not offer any theory at
that time.
Well, it turns out that the near-wall compression flow across a Mach wave fan can be analyzed
using the same Prandtl-Meyer function approach for expansion waves, but with the opposite
sign as shown here.
Keep in mind that this analysis is limited to the near-wall region where the envelope
Mach waves exist.
Further away from the wall, these Mach waves coalesce into an oblique shock and the oblique
shock theory needs to be applied.
Before we wrap up this lesson, let us look at a comparison of expansion and compression
wave analysis.
Let us consider two cases where supersonic flow at Mach 2 encounters a 10 degree expansion
and a 10 degree compression respectively.
Based on the incoming mach number we get a ν(M1) of 26.38 for both the cases.
Now depending on whether the flow encounters an expansion or a compression of θ2 equal
to 10 degrees, we get ν(M2) of 36.38 degrees for expansion and 16.38 degrees for compression.
Using this we can see that the resultant Mach number M2 becomes 2.386 after the 10-degree
expansion and 1.652 after the 10 degree compression.
With that let's wrap up this lesson