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.