An interesting article. Strange though that Blair mentions Fluent many times, but does not name the 1D code anywhere. I wonder if it was his own or someone else's?
To sum up the article: The aim of a bell mouth is to minimise the flow separation at the duct entry. Flow separation leads to a reduction in the available flow area, the ratio of available flow area to actual area is the Cd. The amount of separation depends on the shape (primarily radius) of the corner and the speed the air is going around it. Hence it depends on geometry and pressure ratio. CFD can be used to make predictions of Cd vs pressure ratio and geometry.
The effective runner length issue is equally interesting but not touched upon by the article.
If you take an open pipe and calculate a natural frequency based on its length and the speed of sound, you'll get a value slightly higher than what's measured. The pipe behaves as if it is slightly longer than its geometric length (caused by complex interactions between the sound propagation in the duct and the sound radiation at the exit). This has obvious implications for runner length tuning - a simple (unflanged) pipe behaves as if it were about 0.3d longer than its geometric length (d is the pipe diameter). Stick a flange on the exit and the additional length goes up to 0.4d. This additional length is commonly called the "acoustic end correction".
As expected, the end correction is highly dependent on the form of the duct end. For a bell mouth, it's common to define the geometric length as the length of the straight part of the duct and then look up an end correction based on the duct and bell mouth radii. The sum of these is the effective length of the duct. For example, if the bell mouth radius is 2x the duct radius, the acoustic end correction would be about 1.0d. So the effective length of the duct would be its length without the bell mouth, plus its diameter. If the radii were equal, the end correction would be about 0.6d.
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The above values are taken from an excellent paper on the subject:
"Wave reflections from duct terminations", A.Selamet & Z.L.Ji (Ohio State University), R.A.Kach (Ford Motor Company). J. Acoust. Soc. Am. 109 (4), April 2001.
The paper shows end corrections (and reflection coefficients) for many different duct terminations.
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So.. the answer to the original question is:
Yes, but then so do all forms of termination.
How much depends on the radius, but the data is available.
- Steve