Mesoscopic Transport Phenomena
Eindhoven University of Technology
Chapter in CISM Lecture Series: Noise Sources in Turbulent Shear Flows
18-22 April 2011 Udine, Italy
Abstract After a review of basic equations of fluid dynamics, the Aeroacoustic analogy of Lighthill is derived. This analogy describes the sound field generated by a complex flow from the point of view of a listener immerged in a uniform stagnant fluid. The concept of monopole, dipole and quadrupole are introduced. The scaling of the sound power generated by a subsonic free jet is explained, providing an example of the use of the integral formulation of the analogy. The influence of the Doppler Effect on the radiation of sound by a moving source is explained. By considering the noise generated by a free jet in a bubbly liquid, we illustrate the importance of the choice of the aeroacoustic variable in an aeroacoustic analogy. This provides some insight into the usefulness of alternative formulations, such as the Vortex Sound Theory. The energy corrolary of Howe based on the Vortex Sound Theory appears to be the most suitable theory to understand various aspects of self-sustained oscillation due to the coupling of vortex shedding with acoustic standing waves in a resonator. This approach is used to analyse the convective energy losses at an open pipe termination, human whistling, flow instabilities in diffusers, pulsations in pipe systems with deep closed side branches and the whistling of corrugated pipes.
Due to the essential non-linearity of the governing equations it is difficult to predict accurately fluid flows under conditions at which they do produce sound. This is typical for high speeds with non-linear inertial terms in the equation of motion much larger than the viscous terms (high Reynolds numbers). Direct simulation of such flows is very difficult. When the flow velocity remains low compared to the speed of sound waves (low Mach numbers) the sound production is a minute fraction of the energy in the flow,
R. Camussi (Ed.), Noise Sources in Turbulent Shear Flows: Fundamentals and Applications, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1458-2_1,
© CISM, Udine 2013
making numerical simulation even more difficult. It is not even obvious how one should define the acoustic field in the presence of flows. Aeroacoustics does provide such definitions. The acoustic field is defined as an extrapolation of an ideal reference flow. The difference between the actual flow and this reference flow is identified as source of sound. Using Lighthill’s terminology, we call this an “analogy” [Lighthill (1952-54)].
In free field conditions the sound intensity produced by flows is usually so small that we can neglect the effect of acoustics on the flow. Furthermore, the listener is usually immerged in a uniform stagnant fluid. In such cases the convenient reference flow is the linear inviscid perturbation of this stagnant, uniform fluid. It is convenient to use an integral formulation of the aero-acoustical analogy. This integral equation is a convolution of the sound source by the Green function: the response of the reference state to a localized impulsive source. The advantage of the integral formulation is that random errors in the source term average out. One therefore often uses such an integral formulation to extract acoustic information from direct numerical simulations of the flow which are too rough to directly predict the acoustic field. Such an approach is used so as to obtain scaling laws for sound production by turbulent flows when only global information is available about the flow. When flow dimensions are small compared to the acoustical wave length (compact flow) we can locally neglect the effect of wave propagation within the source region. Here the analogy of Lighthill provides again a procedure which guarantees that we keep the leading order term where brute force would predict no sound production at all or would dramatically overestimate it [Crighton et al. (1992)]. In compact flows at low Mach numbers the flow is most efficiently described in terms of vortex dynamics, allowing a more detailed study of the sound production by nonlinear convective effects.
Walls have a dramatic effect on the production of sound because it becomes much easier compressing the fluid than in free space. In internal flows acoustic energy can accumulate into standing waves, which correspond to resonances. Even at low Mach numbers acoustical particle velocities of the order of magnitude of the main flow velocity can be reached when hydrodynamic flow instabilities couple with the acoustic standing waves. This relatively high amplitude facilitates numerical simulations considerably. Such self-sustained oscillations are best described qualitatively in terms of vortex dynamics.
In a pipe the main flow does not necessarily vanish when travelling
away from the source region. For these reasons another analogy should be used, called the Vortex-Sound Theory. Whilst Powell (1964) initially developed this theory for free space, Howe generalised it for internal flows [Howe (1975), Howe (1984), Howe (1998), Howe (2002)]. In Howe’s approach the acoustic field is defined as the unsteady irrotational component of the flow, which again stresses the fact that vortices are the main sources of sound in isentropic flows. An integral formulation can also be used in this case.
When considering self-sustained oscillations, one is interested in conditions at which they appear and the amplitude they reach. While a linear theory provides information on the conditions under which self-sustained oscillation appears, the amplitude is determined by essentially non-linear saturation mechanisms. We will show that when ever the relevant non-linear mechanism is identified, the order of magnitude of steady self-sustained pulsation amplitude can be easily obtained. A balance between the acoustic power produced by the source and the dissipated power will be used.
A summary of the equations of fluid dynamics is given in (section 2). In Section 3 we introduce the acoustic field by means of Lighthill’s analogy, followed by basic concepts of the acoustics of a stagnant uniform fluid, such as elementary solutions of the wave equation, acoustic energy, the Green function, multipole expansion, Doppler effect and convective effects due to a uniform main flow (section 4). We use the analogy of Lighthill to derive the scaling law for sound production by a subsonic isothermal free jet. The influence of the difference in speed of sound between the source region and the listener is discussed by using the example of bubbly liquids (section 5). We then introduce the acoustics of pipes, derive the low frequency limit of acoustic properties of a pipe discontinuity and of an open pipe termination (with and without main flow). In Section 6 we introduce the concepts of resonators and discuss closed-side branch and Helmholtz resonators. In section 7 we introduce vortex sound theory and apply it to the analysis of whistling, from human whistling to whistling of corrugated pipes. Some aspects introduced here are discussed in depth in the following chapters.
Our discussion is inspired by the book of Dowling and Ffowcs Williams (1983), which is an excellent introductory course. Basic acoustics is discussed in the books of Morse and Ingard (1968), Pierce (1990), Kinsler et al. (1982), Temkin (2001), Blackstock (2000) and Bruneau (2006). Aeroacoustics is treated in the books of Goldstein (1976), Blake (1986), Crighton et al. (1992), Howe (1998)and Howe (2002). In this introduction
we ignore the effect of wall vibration [Junger and Feit (1986), Cremer and Heckl (1988) and Norton (1989)]. Acoustics of musical instruments is discussed by Fletcher and Rossing (1998) and Chaigne and Kergomard (2008). In an earlier course Hirschberg et al. (1995) and a review paper [Fabre et al. (2012)] we discussed the aeroacoustics of woodwinds. In the Lecture notes of Rienstra and Hirschberg (1999) provide more details on the mathematical aspects.