Understanding Expansion Fans and Reflected Waves in Propulsion Systems

[TimeRip496]

"For simplicity, let us assume the flow turns around a corner downstream of the throat, where
the Mach number is M₀ > 1, to an angle θ₁, and let us indicate just three of an infinite number of waves making up the "expansion fan". Since the centerline is an axis of symmetry, the
waves must reflect from it while canceling the flow deviation, so that the flow downstream of
the reflection is axial. When the flow passes through these reflected waves it is deflected
toward the axis. If the wall turns to match this deflection where the reflected wave hits it, the
wave is just cancelled. The net result is that the flow arrives at the exit with an axial
direction and a uniform Mach number."

Can anyone explain what this passage means? Isn't the centerline just something we draw and thus how can it reflect the waves? And how does the reflected waves cancel out the flow deviation?

Source: https://ocw.mit.edu/courses/aeronau...pring-2012/lecture-notes/MIT16_50S12_lec8.pdf

 

[boneh3ad]

The centerline reflecting the waves is something of a mathematical convenience. Really, the waves pass right through it, but since the whole flowfield is symmetric, the math and the picture are identical to the waves reflecting off of it. In fact (for an inviscid flow), you can chop off half of the domain and replace the centerline with a flat wall and the solution would be identical. In that case, the waves really would be reflecting.

Perhaps don't think about the reflected waves canceling out flow deviation, per se. Instead, view them as causing flow deviation that results in the flow after them being exactly along the centerline. The flow coming out of the throat is angled away from the centerline (except for right on the centerline) since the flow is expanding. However, along the centerline, the flow must be exactly in that direction or else voids would form and the continuum approximation breaks down. So, every wave (and its reflection) hitting the centerline must result in the flow remaining parallel to it.

 

[NatSusan]

Hey there,

So basically, this passage is talking about the flow of air or gas through a curved duct. The flow starts out with a high Mach number (speed) at the throat (narrowest point of the duct) and then turns around a corner downstream. As the flow turns, it creates an "expansion fan" of waves, which are essentially disturbances in the flow.

Now, because the centerline of the duct is an axis of symmetry, the waves will reflect off of it and cancel out the flow deviation. This means that the flow downstream of the reflection will be in an axial (straight) direction and will have a uniform Mach number.

To understand this better, picture a wave in water hitting a wall and reflecting back. When the reflected wave meets the original wave, they cancel each other out and the water becomes still again. This is the same concept with the flow in the duct - the reflected waves cancel out the original waves, resulting in a uniform flow.

As for your question about the centerline, it is not just something we draw - it is an actual physical line that extends through the center of the duct. So when the waves reflect off of it, they are essentially bouncing off of the physical centerline.

I hope this helps clarify things for you. Let me know if you have any other questions!