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Acoustic Attenuation in Thin Tubes2

Acoustic Attenuation in Thin Tubes

(OP)
I am studying the acoustic wave attenuation in thin tubes. One of the classic papers that discuss the subject is “On the propagation of sound waves in cylindrical tubes” by H. Tijdeman. It contains all the formulas and tables needed to calculate the attenuation in thin tubes filled with air. Since this is an advanced text, it does not dwell on the in-depth description of the physical mechanism besides noting, that the acoustic power is getting dissipated in the viscous boundary layers, and by heat conduction.

My problem is in understanding the difference between power dissipation of an acoustic wave vs. power dissipation of a simple fluid flow (laminar or turbulent) in the same tube & medium. When I calculate the attenuation (power loss) of a wave, it is much higher than the power dissipated by a simple fluid flow within the same tube having the same velocity as the fluid molecules in the wave. This is especially noticeable when the medium is not gas but liquid, and the molecule velocities are very small.

If both acoustic and simple flows encounter the same viscous and heat conduction losses in the boundary layers, then why is there a big difference between the power losses?

There is an analogy between fluid dynamics and electricity. In electrical engineering the thin tube is analogous to a resistor. The resistor dissipates the same power as long as the applied effective voltage is the same, be it direct current (like simple steady fluid flow), alternating current (alternating fluid flow), or conducted EM waves (analogous to acoustic waves). If the analogy stands in this regard, shouldn’t the thin tube behave in a similar way?

Any input on this subject is appreciated.

Joe

RE: Acoustic Attenuation in Thin Tubes

Presumably the paper identifies the mechanism, directly or indirectly. Without seeing it I'd guess there's a lot of power lost at the boundary in shear as you set up the alternating bulk flow velocity.

Cheers

Greg Locock

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RE: Acoustic Attenuation in Thin Tubes

A copy of the paper is here: https://www.researchgate.net/profile/H_Tijdeman/pu...

One thing to note is that sound propagation is not a fluid flow, it's a pressure disturbance flow, i.e., the air molecules in sound propagation are like the water molecules in ocean waves, their motion is sinusoidal or circular, and there is no net movement.

TTFN (ta ta for now)
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RE: Acoustic Attenuation in Thin Tubes

(OP)
Thanks for the comments Greg and IRstuff.

Yes, the mechanism of dissipation is not a mystery to me, as mentioned in the first post. Kinetic energy is getting converted into heat due to viscous friction of the molecules; to some degree in the bulk fluid, but especially in the narrow boundary layers near the walls. This supposed to work the same way in both acoustic wave propagation, as well as in the simple bulk flow of fluid mechanics (as far as I understand).

In both cases the fluid molecules move with a specific average speed, but there is a velocity gradient across the tube’s cross section due to viscous drag. Different layers of the fluid move at slightly different speeds next to each other, and these layers experience viscous friction, converting kinetic energy into heat. Due to the loss of kinetic energy an external pressure gradient is required as the source of input power to maintain the bulk fluid flow; or in the case of acoustic waves, both the acoustic pressure amplitude and molecule velocity will get attenuated.

The other mechanism of power dissipation in the acoustic waves is the heat conduction. In the regions of the acoustic wave where the pressure is above the ambient pressure, the density of the medium and also the temperature are slightly higher than the average. This disturbance carries power, to which the increased local instantaneous temperature increase also contributes. If the heat conductivity of the fluid is significant, and the walls of the duct also conduct heat away from the wave, then heat energy will leave the pressurized regions of the wave diminishing the pressure amplitude, and also the velocity amplitude. The same mechanism works in the opposite direction at the regions of the wave, where the local instantaneous pressure and temperature is lower than the average. By conducting external heat into these regions the pressure disturbance will get diminished, causing the loss of acoustic power.

Now after writing these thought I realize that this second principle of power loss via heat conduction might be responsible for the increased dissipation of acoustic waves in tubes compared to simple bulk flow.

The heat loss through the walls of the tube in a bulk flow does not result in kinetic energy loss of the fluid (if I understand it right). In fact it supposed to increase it, because as the fluid cools down along the tube length its density will increase, and the fluid volume will shrink. This will cause a suction effect which will increase its kinetic power. This mechanism is partially responsible for providing the kinetic power of the hurricanes.

IRstuff:
“One thing to note is that sound propagation is not a fluid flow, it's a pressure disturbance flow, i.e., the air molecules in sound propagation are like the water molecules in ocean waves, their motion is sinusoidal or circular, and there is no net movement.”

Indeed, there is no bulk flow in a linear acoustic wave (though there can be in high amplitude nonlinear acoustic waves). The molecules are oscillating around a point of rest forth and back. But they still move, and have a specific velocity when they do so. This molecule movement supposed to experience the same laws of energy loss due to viscous friction, just like they do in a common bulk flow.

This is what confused me, because I had the impression that this must be the major loss mechanism in the acoustic waves, and the heat loss through isothermal walls would not contribute that much. If this would be true, then I would have expected that the power dissipation in fluids having the same effective velocity would be the same for both simple bulk flow (unidirectional, or alternating), and also for acoustic waves. But as mentioned earlier, I realize now that this power loss of acoustic waves via heat conduction through isothermal walls might be the major (or only) cause for the different attenuation magnitudes of acoustic vs. bulk flow in the same tubes and same medium.

Is there any other reason for the difference? It would be great to get some confirmation or detailed explanation from a professor (if they read this forum) or by an expert in this field of science who might happen to be a member engineer.

If I have got something wrong in these explanations, please let me know.

Joe

RE: Acoustic Attenuation in Thin Tubes

You seem to be missing a point about the boundary layer. In continuous and laminar flow, the boundary layer is a nice "layer" and the frictional losses are minimized, since the velocity of the air next to the wall is close to zero. When you have oscillation, the boundary layer is either constantly flipping direction or not there at all, either of which should significantly increase the frictional losses, since the velocity of the air at the wall is unlikely to be zero.

TTFN (ta ta for now)
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RE: Acoustic Attenuation in Thin Tubes

while it may be outside your scope of studies, the book by A.H. Bhatia dealing with accoustic losses will be helpful,

RE: Acoustic Attenuation in Thin Tubes

(OP)
IRstuff, in theory there can be such thing as a slip boundary condition, which is often used in first approach simulations. This corresponds to an ideal fluid which has got zero viscosity. But in practice I am not aware of any common fluid that would not have at least a little viscosity, and consequently a boundary layer. In non ideal fluids with a finite viscosity a no-slip condition exists at the walls, which means that the fluid velocity must be zero at the wall. There will be a velocity gradient from the wall towards the interior of the volume forming the boundary layer. The thickness of this viscous boundary layer in acoustics can be calculated as dv=sqrt(2*mu/(om*rho)). You can see that it depends on the viscosity, density, and frequency as well. I wouldn’t say that “the frictional losses are minimized” by this boundary layer. The correct statement is that the major part of the viscous losses are dissipated within this boundary layer, and only a minor part of these losses take place in the rest of the fluid, where the velocity gradient is small. In acoustics the boundary layer constantly oscillates and changes direction.

Hacksaw, what is the exact title of Bhatia’s book that you are recommending?

Joe

RE: Acoustic Attenuation in Thin Tubes

The boundary layer isn't going to be fully developed for the purposes of acoustic wave propagation. If you assume a fully developed boundary layer for your friction corelation, you'll not get enough acoustic attenuation. Similarly, if you decrease the boundary layer thickness to match the acoustic attenuation, you'll over-predict steady-state pressure drop.

Steve

RE: Acoustic Attenuation in Thin Tubes

(OP)
After some more pondering of this subject of the “boundary layer not being fully developed”, I have got the point, and now understand its consequences on the viscous power dissipation of acoustic waves. IRstuff was hinting on it, but SomptingGuy expressed it more clearly. Let me summarize what we have understood so far.

There are (at least) two major differences between the power dissipation of a unidirectional bulk flow, and an acoustic wave in thin tubes.

1) The thickness of the viscous boundary layer in acoustics can be calculated as dv=sqrt(2*mu/(om*rho)), which is inversely proportional to the frequency. With other words, the higher the frequency, the thinner is the viscous boundary layer, because there is not enough time between each period for the boundary layer to expand (thicken) away from the wall. However, the velocity still must be zero at the no-slip boundary of the wall, and the velocity still must be a specific value determined by the wave amplitude in the middle (or inner volume) of the tube. The thinner the viscous boundary layer the greater is the velocity gradient within this layer. The greater the velocity gradient, the greater the viscous friction within this layer, and the greater is the viscous power dissipation. As the frequency is diminished, the boundary layer becomes thicker, and at f=0 it approaches the steady state bulk flow case.
2) In acoustics the heat conduction within the fluid and between the wall and the fluid adds an extra power loss component that does not exists in simple steady bulk flow. In the regions of the acoustic wave where the pressure is above the ambient pressure, the density of the medium and also the temperature are slightly higher than the average. This disturbance carries power, to which the increased local instantaneous temperature increase also contributes. If the heat conductivity of the fluid is significant, and the walls of the duct also conduct heat away from the wave, then heat energy will leave the pressurized regions of the wave diminishing the pressure amplitude, and also the velocity amplitude. The same mechanism works in the opposite direction at the regions of the wave, where the local instantaneous pressure and temperature is lower than the average. By conducting external heat into these regions the pressure disturbance will get diminished, causing the loss of acoustic power. If the heat conductivity of the fluid is high, then a significant amount of heat energy may leave the high pressure regions and move into low pressure regions even if the walls of the duct don’t conduct heat (adiabatic).

Thanks to all contributors for the comments and helping me understand the process.
Is there any other phenomenon that significantly contributes to the difference between power dissipation of acoustic waves and steady bulk flow within thin ducts? Did I explain something wrong?

Joe

RE: Acoustic Attenuation in Thin Tubes

joedunai,

Ultrasonic Absorption (1967) reprint 1985 as paper back. Covers quite a range of factors, not a casual read. Of course the losses in accoustic propagation in tubes, is vastly more complex than boundary layer effects. Hardly a 1D problem....

RE: Acoustic Attenuation in Thin Tubes

(OP)
Thanks for the useful tip Hacksaw, I will take a look at it.

Joe

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