2013/11/09 by S. Ziada, Samir Ziada, P. Lafon +1
Engineering · Mathematics · #Acoustic Wave Phenomena Research #Acoustic radiation #Acoustic resonance #Acoustics #Aerodynamics and Acoustics in Jet Flows #Amplitude #Atomic physics #Excited state #Flow (mathematics) #Fluid Dynamics and Vibration Analysis #Mathematics #Mechanics #Optics #Physics #Piping #Radiation #Resonance (particle physics) #Scaling #Thermodynamics
paper · doi:10.1115/1.4025958
openalex publication_date 2013/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26
Dr. Moody raises two issues related to the prediction of flow-excited acoustic resonances; namely, the scaling approach of resonance prediction and the influence of flow, including vortex formation and convection in the shear layer, on the resonance frequency and mode shape. These are indeed important issues and warrant additional comment and clarification. We first elaborate on the usage of the proposed source terms, such as those shown in Figs. 21 and 24, in predicting acoustic resonances for piping systems of different sizes, geometries, and flow conditions, and then discuss the main difficulties involved in amplitude scaling when small-scale model tests are performed to assess the liability of a full-size installation to flow-excited acoustic resonances. This will be followed by a discussion of the effect of the pipe flow on the acoustic resonance frequencies and mode shapes.The scaling of pressure amplitudes measured within the resonance lock-in range of a subscale model, for example, to a larger size similar system, operating with different flow conditions, is by no means a straightforward process. This is because flow-excited acoustic resonances are excited by a highly nonlinear feedback mechanism and the steady-state amplitude at resonance is determined from the balance between the acoustic energy generated by the source and that dissipated by viscothermal losses and radiation damping. Note that the energy generated and absorbed is amplitude-dependent and that radiation damping includes both radiation losses from the subsystem into the main piping system and radiation from the terminations of the main piping system. For example, Figs. 16 and 22 show the effect of viscothermal losses on the resonance amplitude and Fig. 27 illustrates the effect of radiation losses from the side-branches into the main pipe. These losses can never be replicated in a small-scale model, and it is therefore very difficult to compensate for these differences because the acoustic resonance mechanism is highly nonlinear. The pipe size, medium properties, and static pressure affect substantially the acoustic attenuation coefficient. Furthermore, it is difficult to simulate the impedance of the main pipe terminations in a small-scale model.Because of these difficulties, we propose the usage of the normalized aeroacoustic source term, which is independent of system properties, such as attenuation losses and the arrangement of the attached piping system. The source term can be combined with a linear acoustic model of any piping system containing one or more side-branches and/or cavities. Any changes in the acoustic characteristics of the piping system, such as the acoustic-mode shapes and frequencies, are adequately accounted for in the acoustic model, but the normalized source characteristics, such as those shown in Fig. 21, remain the same. This feature is illustrated in Fig. 22, which shows the predicted resonance amplitude, by using the source term of Fig. 21 for three branch-pipe arrangements with different levels of viscothermal losses. Another example is given in Fig. 26, which shows excellent prediction of the resonance amplitude in a pipeline housing a cavity, although the source term used in the prediction was determined from a different test arrangement. The effect of radiation from the branches into the main pipe, which is illustrated in Fig. 27(a), is also well predicted by using the same source term of Fig. 21, as reported in Fig. 13 of Ref. [41].Given the difficulties involved in scaling the pressure amplitude from small-scale model tests, one may ask: what are scale model tests good for? They are very useful in indicating the critical flow velocity for the onset of resonance and for determining the source strength. Scaling of acoustic pressure at resonance from scale-model measurements to the full-size conditions is likely to underpredict the resonance amplitude in the plant. As an example, for the Quad Cities problem discussed in the article, if scale model tests are to be performed with 10-cm pipes and at a static pressure of 10 bar, the acoustic attenuation coefficient in the model would be more than 30 times higher than that of the plant. As mentioned earlier, this large difference cannot be accounted for in the amplitude-scaling process because the flow-sound interaction mechanism is highly nonlinear, especially at large pulsation amplitudes.Since the normalized source terms given in Figs. 21 and 24 are complex variables, they include the effect of the shear layer on the resonance frequency and mode shape. As discussed in Sec. 4, the real (or active) part of the complex source term is responsible for the generation of acoustic power, whereas the imaginary (or reactive) part represents an added mass (or stiffness) effect at the source, which is located at the mouth of the side-branch or cavity. As the flow velocity is increased, corresponding to a decrease in the Strouhal number from 0.5 to 0.3 in Fig. 21, the negative imaginary part first decreases to zero, becomes positive, and then increases with flow velocity, which indicates that the source, or the shear layer, acts first as an added mass and thereby decreases the resonance frequency. As the flow velocity increases further, the added mass effect continuously decreases until it vanishes at zero imaginary part of the source, and then the source starts to act as an added stiffness, which increases the resonance frequency. Thus, the imaginary part of the source has the effect of continuously changing the resonance frequency as the velocity is increased within the lock-in range. This frequency change is very small but discernible and is generally observed in experimental investigations. This discussion clarifies how the source term accounts for the effect of the shear layer on the resonance frequency.The effect of mean flow on the resonance mode can also be important, especially at relatively high Mach number [68]. Regarding the effect of the mean flow, this is also discussed in Sec. 2 and in more detail in Ref. [69]. As can be seen in Fig. 7, high-speed flow can affect the acoustic mode shape and its associated particle velocity distribution and must be taken into consideration when assessing the excitation mechanism of this case. The reader may notice that no source terms were provided for the excitation of diametral cross modes of ducted shallow cavities, which is because the aeroacoustic source in this case is not compact but rather distributed over a domain size comparable with the wavelength of the resonant acoustic mode. For this reason, the source cannot be modeled as a lumped element, which is different from the other case of longitudinal resonance modes, for which the acoustic wavelength of the resonant mode is much longer than the characteristic length of the shear layer and aeroacoustic source.