Sound propagation in Poiseuille flow

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Martin Manscher - 03 Aug 2005 10:16 GMT

A well-known technique for measuring flow in a pipe is comparing the
time-of-flight or phase of a sound signal (propagating in the
longitudinal direction) from point A to point B with that from point B
to point A. The difference in phase or time-of-flight is a measure of
the flow rate.

In a straight pipe with circular cross-section, Poiseuille flow can be
established (depending on the geometry and other conditions). Now, the
velocity in the Poiseuille flow is close to zero near the walls, and
maximum in the centre. How does this affect the sound wave propagation?
Will there be a single phase/time-of-flight, or will the velocity
distribution create dispersion? I am especially interested in the
lowest-order mode, but also (out of curiosity) in a more general answer
for lower and higher frequencies.

Book or article references are also appreciated.

Martin Manscher

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David Wilkinson - 03 Aug 2005 12:28 GMT

> A well-known technique for measuring flow in a pipe is comparing the
> time-of-flight or phase of a sound signal (propagating in the
[quoted text clipped - 14 lines]
>
> Martin Manscher

Hi Martin

A method you could consider is Huygens principle (from any College
Physics book), used in optics but valid for sound waves as well. The
basis of this is that you draw a line at right angles to the direction
of propagation of the wave and each point on the line is considered to
be a point from which a new wave element propagates at the local wave
speed. If the wave speed varies along the line then the faster moving
wave elements will move further in a given time than the slow ones. This
means that the wave front will tilt towards the lower velocity side and
the sound as a whole will be refracted in that direction. This sort of
thing happens in the atmosphere when there is a temperature inversion,
higher as you go up, tending to confine sound near the ground and
enabling it to travel long distances without much dissipation. I think
exceptionally distant gunfire was heard like this in one of the world wars.

Just a couple of points on your Poiseuille flow. All pipe flow has zero
velocity at the wall and a maximum in the middle, whether it is laminar
or turbulent. Also laminar flow only occurs at very low Reynolds numbers
so both pipe diameter and velocity tend to be low. It would be much
easier, and probably much more accurate, to just measure the pressure
drop in a length of pipe and use the standard formula in any fluid
mechanics textbook to find the velocity

However, if you want to do it that way, how will the velocity profile
affect the sound propagation? There are two case, sound with the flow
and against it. If the sound is going with the flow then the waves will
be fastest in the middle of the pipe and slowest at the wall. This
means, from Huygens principle, that the wave will refract towards the
wall and the sound will mainly tend to travel there with very little
speed increase.

For the other case of sound going upstream the sound will be slowest in
the middle of the pipe and higher at the wall so the wave will tend to
refract towards the middle and tend to travel there with maximum
retardation. The difference in the down and upstream wave speeds will
therefore by about equal to the maximum velocity but the average will be
lower than the wave speed in a still fluid.

However this is all qualitative and the actual refraction effects may be
negligible in practice. If you know your fluid velocity and pipe
diameter and the wave speed in water of about 5,000 ft/sec then you can
work out the rate of refraction and what the development length will be
for the sound to refract towards the wall or middle. You may well find
with small diameter pipes and low laminar flow velocities that the
refraction effect is so small that the wave effectively travels as a
plane wave. Then the wave speeds will just be c+u and c-u where u is the
average fluid velocity and you can just do the timing sums.

This is all off the cuff so excuse any conceptual mistakes.

Hope this helps.

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s.macintosh - 04 Aug 2005 14:01 GMT

You can think of the velocity distribution as a layered system. As soon
as you have a layered system you will have dispersion. Do a search for
"Lamb Waves" or "Generalized Lamb Waves" for some info on this
phenomenon, if you need some background info. There is also a Russian
author, I beleive it is Victorov, which has written a good book on the
subject.

One feature of having a faster sound speed in the middle will be to
cause the turn-on of the higher order modes to shift to a higher
frequency.

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ari - 08 Aug 2005 16:03 GMT

Few ideas after reading the mails from both of you.

To my understanding the measurement technique is usually used to
determine flow rate (volume or ultimately mass flow rate). Is this case
the actual velocity profile is not so much of an interest, it is enough
to average the axial velocity on a cross section.

In the measurement set up, i'd expect you to get a some what "strechted"
pulse form (as a cross sectional average). I'm not sure what is the
correct English word, but this is something like average or gross or
lumped sound speed. The "actual" speed of sound is frequency dependent.
In your case, i think, you are looking for very low frequency speed of
sound (or ideally 0 Hz). The frequency dependency is strongly dependent
on the liquid and the actual "form" of it (like posssible gas bubbles,
concentration and so on) - not just the profile.
I have seen some rough sound velocity "approximations" for liquids like
water in fluid mechanics/dynamics books. (Wylie&Streeter, for example.)

I know, that there are sound velocity profile considerations for pipes,
but can't remember them by heart, sorry. There are also many articles
which are based on vortex sound, which could be usufull in understanding
how to "combine" flow and sound. This might be useful if have months to
spend on the subject (Authors, by heart, Powell, Howe, Doak and most
probably some other as well. Howe has written a book also).

Hope this helps,

ari

>> A well-known technique for measuring flow in a pipe is comparing the
>> time-of-flight or phase of a sound signal (propagating in the
[quoted text clipped - 67 lines]
>
> Hope this helps.

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Natural Science Forum / Physics / Acoustics / August 2005

Sound propagation in Poiseuille flow