Abstract
In the previous chapter, the resonance frequency , ω o, of a Helmholtz resonator was calculated. When driven at that frequency, the predicted pressure amplitude inside the resonator’s volume (compliance) became infinite. This was because the theory used to model that inertance and compliance network in Figs. 8.11 and 8.15, and in Eq. (8.50), did not include any dissipation. By introducing DeltaEC, we were able to calculate the amount of power dissipated in the neck (inertance) and volume (compliance) of a 500-mL boiling flask. In this chapter, those losses will be calculated from hydrodynamic “first principles.”
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Notes
- 1.
William Hewlett, an electrical engineering student at Stanford University, used a current-dependent resistance (a flashlight bulb) to stabilize the amplitude of an audio oscillator circuit as his master’s thesis. That oscillator subsequently became the HP-200. It was the first product developed commercially by the original Silicon Valley startup: Hewlett-Packard. The first five of their “production” models were purchased by Walt Disney to produce sound effects in the movie Fantasia.
- 2.
Before the French dominated Le Système International d’Unités was adopted, the unit of conductance was the mho (ohm spelled backward). They renamed the unit of electrical conductance the siemens [S]. These are the same folks, who in their collective wisdom, renamed the unit of frequency from the obscure cycles per second (cps) to the hertz [Hz].
- 3.
This material parameter is also sometimes called the thermometric conductivity and abbreviated as χ. For instance, see L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Butterworth-Heinemann, 1987); ISBN 0-7506-2767-0. See §50.
- 4.
To take the square root of –j = j −1, it is sometimes useful to draw a diagram on the complex plane by expressing j −1 as –j, drawn vertically downward at −90° from the positive real axis. The square root operation divides that angle by two, resulting in a unit vector rotated only −45° away from the real axis having a projection of (2)–½ along the real axis and –(2)–½ along the imaginary axis demonstrating that √–j = (1 – j)/√2.
- 5.
The term “evanescent” is also applied to sound fields that decay rapidly without an oscillating part (i.e., no imaginary component of the wavenumber), such as the response of a waveguide that is excited at a frequency which is below the cutoff frequency (see Sect. 13.5).
- 6.
Evolution has found the evanescent waves useful for the control of light reaching the optic nerve of insects with compound eyes. Did you ever notice that it is hard to swat a fly under a wide range of lighting conditions? How can you control the light levels when your eye has a thousand lenses? One iris for each lens is clearly out of the question, since each lens has a diameter of only about 30 μm. The mechanism employed by insects to control light levels makes each lens the entrance to an optical waveguide (called an ommatidium by the entomologists), much like an optical fiber used in telecommunications, that channels light from the lens down to the optic nerve (see Fig. 11.6). In bright light, a chemical change in the fluid surrounding the waveguides causes a precipitate to form that scatters the evanescent wave decaying (exponentially) beyond the waveguide into the fluid. Scattering of the evanescent wave reduces the light that makes it down the waveguide to the optic nerve. For further information on the physics of insect vision and some relevant graphics, see R. P. Feynman, Lectures on Physics, Vol. I (Addison-Wesley 1963), §36–4.
- 7.
It is difficult for most people to visualize the spatial and temporal dependence of the temperature based only on plots of the real and imaginary parts of the solution as a function of position, such as those provided in Figs. 9.7 and 9.8. An animation of the temperature variation for an ideal gas near a solid (isothermal) wall is available at the Los Alamos National Laboratory Thermoacoustics Home Page: http://www.lanl.gov/thermoacoustics/Book/index.html. In the second paragraph on the page at that site, you have the option to download a zipped animation file. You can “unzip” the file, and run the DOS-executable animation THERMAL.EXE. The animation starts with the pressure and velocity in a standing wave and then zooms into the solid boundary to animate the temperature in the fluid as a function of space and time. The animation goes further to calculate the work done by an imaginary piston moving with the fluid to demonstrate that power is dissipated during the transition from adiabatic compressions far from the wall to isothermal compressions at the fluid–solid boundary.
- 8.
In liquids, ω crit is even higher. In water, f crit = 1210 GHz, although this is less significant because the difference between the adiabatic and isothermal sound speeds is so small due to the smaller thermal expansion coefficient. At 4 °C, the density of water reaches its maximum value, so the thermal expansion coefficient vanishes and the isothermal and adiabatic sound speeds are equal. The existence of life on this planet probably owes much to the fact that ice is less dense than water.
- 9.
This transition from hydrodynamic to ballistic propagation is known as the “Knudsen limit.” In that regime, the wave becomes evanescent. Based on M. Greenspan, “Propagation of Sound in Five Monatomic Gases,” J. Acoust. Soc. Am. 28(4), 644–648 (1956), the hydrodynamic results should not be used for frequencies above ω ≅ c 2/10α. An excellent review article by Greenspan appears in Physical Acoustics, Vol. II A, edited by W. P. Mason and R. N. Thurston (Academic Press, 1964), pp. 1–45.
- 10.
The entire field of Rheology is dedicated to the study of non-Newtonian fluids and plastic flows. Nondrip paints are thixotropic fluids; their viscosities decrease with increasing strain rate. You want a paint that has a low viscosity when it is being applied to reduce drag on the paint brush (and wrist of the painter), but a high viscosity once it is applied to a surface to keep it from dripping. Viscoelastic fluids, like Silly Putty™ are also non-Newtonian fluids. A detailed discussion of viscosity and of non-Newtonian flow is provided by R. B. Bird, W. E. Steward, and E. N. Lightfoot, Transport Phenomena (J. Wiley & Sons, 1960); ISBN 0-471-07392-X, in Chapter 1.
- 11.
- 12.
- 13.
See Ref. [5], for solutions in tubes of other cross-sections, in §17, pp. 53–54 for (Problem 1) annular, (Problem 2) elliptical, and (Problem 3) triangular ducts.
- 14.
When flow enters a smaller tube from a larger reservoir, it must travel some distance before the flow becomes “organized” into the parabolic profile shown in Fig. 9.12. This distance is known as the “entrance length ,” L, and is usually expressed in terms of the tube diameter, D, and the Reynolds number, Re (see the next footnote): L/D ≅ 0.06 Re.
- 15.
The criterion for the transition to turbulence in smooth-walled round pipes is usually given in terms of the nondimensional Reynolds number, Re = ρ<v>D/μ, where <v> is the flow velocity averaged over the pipe’s area, πD 2/4. The transition from laminar (Fig. 9.12) to turbulent flow is usually taken to occur at a Reynolds number greater than 2200 ± 100 in circular pipes with a “smooth” surface finish. Further details regarding the transition to turbulence are given in most fluid dynamics textbook, such as Refs. [2], [3], and [5].
- 16.
- 17.
For an animated visualization of the fluid in the oscillatory viscous boundary layer, we can again turn to the Los Alamos Thermoacoustics Home Page and run OSCWALL.EXE to see exactly the situation (rotated by 90°) that is diagrammed in Fig. 9.13. The reverse case of a stationary wall and fluid moving uniformly far from the wall is animated in VISCOUS.EXE. In that animation, a vibrating piston sets up the fundamental λ/2 = L standing wave in a tube and then focuses in on a portion of the resonator’s wall at the velocity antinode near the center of the tube.
- 18.
A derivation of Eq. (9.38) is provided in G. W. Swift, Thermoacoustics: A unifying perspective for some engines and refrigerators (Acoust. Soc. Am., 2002); ISBN 0-7354-0065-2, Chapter 5.1, pp.106–109.
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Garrett, S.L. (2017). Dissipative Hydrodynamics. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_9
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